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              The English Works of Thomas Hobbes: Volume 1

[Illustration:

  THOMAS HOBBES.
  _W Humphrys sc._
  _London. Published March 1st 1839, by John Bohn. 17, Henrietta Street,
    Covent Garden._
]

                                  THE

                             ENGLISH WORKS

                                   OF

                             THOMAS HOBBES

                             OF MALMESBURY;

                     NOW FIRST COLLECTED AND EDITED

                                   BY

                     SIR WILLIAM MOLESWORTH, BART.

                                  ---

                                VOL. I.

                                  ---




                                LONDON:
                               JOHN BOHN,
                    HENRIETTA STREET, COVENT GARDEN.

                                  ---

                              MDCCCXXXIX.




                                LONDON:
                C. RICHARDS, PRINTER, ST. MARTIN'S LANE.




                                   TO

                           GEORGE GROTE, ESQ.

                      M.P. FOR THE CITY OF LONDON.

                             --------------

DEAR GROTE,

          I dedicate to you this edition of the Works of Hobbes; first,
because I know you will be well pleased to see a complete collection of
all the writings of an Author for whom you have so high an admiration.
Secondly, because I am indebted to you for my first acquaintance with
the speculations of one of the greatest and most original thinkers in
the English language, whose works, I have often heard you regret, were
so scarce, and so much less read and studied than they deserved to be.
It now, therefore, gives me great satisfaction to be able to gratify a
wish, you have frequently expressed, that some person, who had time and
due reverence for that illustrious man, would undertake to edite his
works, and bring his views again before his countrymen, who have so long
and so unjustly neglected him. And likewise, I am desirous, in some way,
to express the sincere regard and respect that I feel for you, and the
gratitude that I owe you for the valuable instruction, that I have
obtained from your society, and from the friendship with which you have
honoured me, during the many years we have been companions in political
life.

                                                Yours, truly,
                                                     WILLIAM MOLESWORTH.

   February 25th, 1839.
79, Eaton Square, London.

------------------------------------------------------------------------

                        ELEMENTS OF PHILOSOPHY.

                           THE FIRST SECTION,

                            CONCERNING BODY,

                            WRITTEN IN LATIN

                                   BY

                      THOMAS HOBBES OF MALMESBURY

                                  AND

                        TRANSLATED INTO ENGLISH.




                                  THE

                       TRANSLATOR TO THE READER.

If, when I had finished my translation of this first section of the
Elements of Philosophy, I had presently committed the same to the press,
it might have come to your hands sooner than now it doth. But as I
undertook it with much diffidence of my own ability to perform it well;
so I thought fit, before I published it, to pray Mr. Hobbes to view,
correct, and order it according to his own mind and pleasure. Wherefore,
though you find some places enlarged, others altered, and two chapters,
XVIII and XX, almost wholly changed, you may nevertheless remain
assured, that as now I present it to you, it doth not at all vary from
the author's own sense and meaning. As for the Six Lessons to the
Savilian Professors at Oxford, they are not of my translation, but were
written, as here you have them in English, by Mr. Hobbes himself; and
are joined to this book, because they are chiefly in defence of the
same.[A]

Footnote A:

  They will be published in a separate volume, with other works of a
  similar description. W. M.


                    THE AUTHOR'S EPISTLE DEDICATORY,

                                 TO THE

                 RIGHT HONORABLE, MY MOST HONORED LORD,

                      WILLIAM, EARL OF DEVONSHIRE.

                                -------

This first section of the Elements of Philosophy, the monument of my
service and your Lordship's bounty, though, after the Third Section
published, long deferred, yet at last finished, I now present, my most
excellent Lord, and dedicate to your Lordship. A little book, but full;
and great enough, if men count well for great; and to an attentive
reader versed in the demonstrations of mathematicians, that is, to your
Lordship, clear and easy to understand, and almost new throughout,
without any offensive novelty. I know that that part of philosophy,
wherein are considered lines and figures, has been delivered to us
notably improved by the ancients; and withal a most perfect pattern of
the logic by which they were enabled to find out and demonstrate such
excellent theorems as they have done. I know also that the hypothesis of
the earth's diurnal motion was the invention of the ancients; but that
both it, and astronomy, that is, celestial physics, springing up
together with it, were by succeeding philosophers strangled with the
snares of words. And therefore the beginning of astronomy, except
observations, I think is not to be derived from farther time than from
Nicolaus Copernicus; who in the age next preceding the present revived
the opinion of Pythagoras, Aristarchus, and Philolaus. After him, the
doctrine of the motion of the earth being now received, and a difficult
question thereupon arising concerning the descent of heavy bodies,
Galileus in our time, striving with that difficulty, was the first that
opened to us the gate of natural philosophy universal, which is the
knowledge of the nature of _motion_. So that neither can the age of
natural philosophy be reckoned higher than to him. Lastly, the science
of _man's body_, the most profitable part of natural science, was first
discovered with admirable sagacity by our countryman Doctor Harvey,
principal Physician to King James and King Charles, in his books of the
_Motion of the Blood_, and of the _Generation of Living Creatures_; who
is the only man I know, that conquering envy, hath established a new
doctrine in his life-time. Before these, there was nothing certain in
natural philosophy but every man's experiments to himself, and the
natural histories, if they may be called certain, that are no certainer
than civil histories. But since these, astronomy and natural philosophy
in general have, for so little time, been extraordinarily advanced by
Joannes Keplerus, Petrus Gassendus, and Marinus Mersennus; and the
science of human bodies in special by the wit and industry of
physicians, the only true natural philosophers, especially of our most
learned men of the College of Physicians in London. Natural Philosophy
is therefore but young; but Civil Philosophy yet much younger, as being
no older (I say it provoked, and that my detractors may know how little
they have wrought upon me) than my own book _De Cive_. But what? were
there no philosophers natural nor civil among the ancient Greeks? There
were men so called; witness Lucian, by whom they are derided; witness
divers cities, from which they have been often by public edicts
banished. But it follows not that there was _philosophy_. There walked
in old Greece a certain phantasm, for superficial gravity, though full
within of fraud and filth, a little like philosophy; which unwary men,
thinking to be it, adhered to the professors of it, some to one, some to
another, though they disagreed among themselves, and with great salary
put their children to them to be taught, instead of wisdom, nothing but
to dispute, and, neglecting the laws, to determine every question
according to their own fancies. The first doctors of the Church, next
the Apostles, born in those times, whilst they endeavoured to defend the
Christian faith against the Gentiles by natural reason, began also to
make use of philosophy, and with the decrees of Holy Scripture to mingle
the sentences of heathen philosophers; and first some harmless ones of
Plato, but afterwards also many foolish and false ones out of the
physics and metaphysics of Aristotle; and bringing in the enemies,
betrayed unto them the citadel of Christianity. From that time, instead
of the worship of God, there entered a thing called _school divinity_,
walking on one foot firmly, which is the Holy Scripture, but halted on
the other rotten foot, which the Apostle Paul called _vain_, and might
have called _pernicious philosophy_; for it hath raised an infinite
number of controversies in the Christian world concerning religion, and
from those controversies, wars. It is like that _Empusa_ in the Athenian
comic poet, which was taken in Athens for a ghost that changed shapes,
having one brazen leg, but the other was the leg of an ass, and was
sent, as was believed, by Hecate, as a sign of some approaching evil
fortune. Against this _Empusa_ I think there cannot be invented a better
exorcism, than to distinguish between the rules of religion, that is,
the rules of honouring God, which we have from the laws, and the rules
of philosophy, that is, the opinions of private men; and to yield what
is due to religion to the Holy Scripture, and what is due to philosophy
to natural reason. And this I shall do, if I but handle the Elements of
Philosophy truly and clearly, as I endeavour to do. Therefore having in
the Third Section, which I have published and dedicated to your
Lordship, long since reduced all power ecclesiastical and civil by
strong arguments of reason, without repugnance to God's word, to one and
the same sovereign authority; I intend now, by putting into a clear
method the true foundations of natural philosophy, to fright and drive
away this metaphysical _Empusa_; not by skirmish, but by letting in the
light upon her. For I am confident, if any confidence of a writing can
proceed from the writer's fear, circumspection, and diffidence, that in
the three former parts of this book all that I have said is sufficiently
demonstrated from definitions; and all in the fourth part from
suppositions not absurd. But if there appear to your Lordship anything
less fully demonstrated than to satisfy every reader, the cause was
this, that I professed to write not all to all, but some things to
geometricians only. But that your Lordship will be satisfied, I cannot
doubt.

There remains the second section, which is concerning _Man_. That part
thereof, where I handle the _Optics_, containing six chapters, together
with the tables of the figures belonging to them, I have already written
and engraven lying by me above these six years. The rest shall, as soon
as I can, be added to it; though by the contumelies and petty injuries
of some unskilful men, I know already, by experience, how much greater
thanks will be due than paid me, for telling men the truth of what men
are. But the burthen I have taken on me I mean to carry through; not
striving to appease, but rather to revenge myself of envy, by encreasing
it. For it contents me that I have your Lordship's favour, which, being
all you require, I acknowledge; and for which, with my prayers to
Almighty God for your Lordship's safety, I shall, to my power, be always
thankful.

                           Your Lordship's most humble servant,

                                                          THOMAS HOBBES.

   London,
April 23, 1655.

                                  THE
                    AUTHOR'S EPISTLE TO THE READER.

Think not, Courteous Reader, that the philosophy, the elements whereof I
am going to set in order, is that which makes philosophers' stones, nor
that which is found in the metaphysic codes; but that it is the natural
reason of man, busily flying up and down among the creatures, and
bringing back a true report of their order, causes and effects.
Philosophy, therefore, the child of the world and your own mind, is
within yourself; perhaps not fashioned yet, but like the world its
father, as it was in the beginning, a thing confused. Do, therefore, as
the statuaries do, who, by hewing off that which is superfluous, do not
make but find the image. Or imitate the creation: if you will be a
philosopher in good earnest, let your reason move upon the deep of your
own cogitations and experience; those things that lie in confusion must
be set asunder, distinguished, and every one stamped with its own name
set in order; that is to say, your method must resemble that of the
creation. The order of the creation was, _light_, _distinction of day
and night_, the _firmament_, the _luminaries_, _sensible creatures_,
_man_; and, after the creation, the _commandment_. Therefore the order
of contemplation will be, _reason_, _definition_, _space_, the _stars_,
_sensible quality_, _man_; and after man is grown up, _subjection to
command_. In the first part of this section, which is entitled Logic, I
set up the light of reason. In the second, which hath for title the
Grounds of Philosophy, I distinguish the most common notions by accurate
definition, for the avoiding of confusion and obscurity. The third part
concerns the expansion of space, that is Geometry. The fourth contains
the Motion of the Stars, together with the doctrine of sensible
qualities.

In the second section, if it please God, shall be handled _Man_. In the
third section, the doctrine of _Subjection_ is handled already. This is
the method I followed; and if it like you, you may use the same; for I
do but propound, not commend to you anything of mine. But whatsoever
shall be the method you will like, I would very fain commend philosophy
to you, that is to say, the study of wisdom, for want of which we have
all suffered much damage lately. For even they, that study wealth, do it
out of love to wisdom; for their treasures serve them but for a
looking-glass, wherein to behold and contemplate their own wisdom. Nor
do they, that love to be employed in public business, aim at anything
but place wherein to show their wisdom. Neither do voluptuous men
neglect philosophy, but only because they know not how great a pleasure
it is to the mind of man to be ravished in the vigorous and perpetual
embraces of the most beauteous world. Lastly, though for nothing else,
yet because the mind of man is no less impatient of empty time than
nature is of empty place, to the end you be not forced for want of what
to do, to be troublesome to men that have business, or take hurt by
falling into idle company, but have somewhat of your own wherewith to
fill up your time, I recommend unto you to study philosophy. Farewell.

                                                               T. H.

                        TITLES OF THE CHAPTERS.


                              PART FIRST,

                               OR LOGIC.


    CHAP.                                                     PAGE.

       1. Of Philosophy                                           1

       2. Of Names                                               13

       3. Of Proposition                                         29

       4. Of Syllogism                                           44

       5. Of Erring, Falsity, and Captions                       55

       6. Of Method                                              65


                             PART SECOND,

                  OR THE FIRST GROUNDS OF PHILOSOPHY.

       7. Of Place and Time                                      91

       8. Of Body and Accident                                  101

       9. Of Cause and Effect                                   120

      10. Of Power and Act                                      127

      11. Of Identity and Difference                            132

      12. Of Quantity                                           138

      13. Of Analogism, or the Same Proportion                  144

      14. Of Strait and Crooked, Angle and Figure               176


                              PART THIRD,

             OF THE PROPORTIONS OF MOTIONS AND MAGNITUDES.

      15. Of the Nature, Properties, and divers                 203
          Considerations of Motion and Endeavour

      16. Of Motion Accelerated and Uniform, and of Motion      218
          by Concourse

      17. Of Figures Deficient                                  246

      18. Of the Equation of Strait Lines with the Crooked      268
          Lines of Parabolas, and other Figures made in
          imitation of Parabolas

      19. Of Angles of Incidence and Reflection, equal by       273
          Supposition

      20. Of the Dimension of a Circle, and the Division of     287
          Angles or Arches

      21. Of Circular Motion                                    317

      22. Of other Variety of Motions                           333

      23. Of the Centre of Equiponderation of Bodies            350
          pressing downwards in Strait Parallel Lines

      24. Of Refraction and Reflection                          374


                             PART FOURTH,

                OF PHYSICS, OR THE PHENOMENA OF NATURE.

      25. Of Sense and Animal Motion                            387

      26. Of the World and of the Stars                         410

      27. Of Light, Heat, and of Colours                        445

      28. Of Cold, Wind, Hard, Ice, Restitution of Bodies       466
          bent, Diaphanous, Lightning and Thunder, and of
          the Heads of Rivers

      29. Of Sound, Odour, Savour, and Touch                    485

      30. Of Gravity                                            508

------------------------------------------------------------------------

                         COMPUTATION OR LOGIC.


                                -------


                               CHAPTER I.

                             OF PHILOSOPHY.

1. The Introduction.—2. The Definition of Philosophy explained.—3.
    Ratiocination of the Mind.—4. Properties, what they are.—5. How
    Properties are known by Generation, and contrarily.—6. The Scope of
    Philosophy.—7. The Utility of it.—8. The Subject.—9. The Parts of
    it.—10. The Epilogue.


[Sidenote: Introduction.]

Philosophy seems to me to be amongst men now, in the same manner as corn
and wine are said to have been in the world in ancient time. For from
the beginning there were vines and ears of corn growing here and there
in the fields; but no care was taken for the planting and sowing of
them. Men lived therefore upon acorns; or if any were so bold as to
venture upon the eating of those unknown and doubtful fruits, they did
it with danger of their health. In like manner, every man brought
Philosophy, that is, Natural Reason, into the world with him; for all
men can reason to some degree, and concerning some things: but where
there is need of a long series of reasons, there most men wander out of
the way, and fall into error for want of method, as it were for want of
sowing and planting, that is, of improving their reason. And from hence
it comes to pass, that they who content themselves with daily
experience, which may be likened to feeding upon acorns, and either
reject, or not much regard philosophy, are commonly esteemed, and are,
indeed, men of sounder judgment than those who, from opinions, though
not vulgar, yet full of uncertainty, and carelessly received, do nothing
but dispute and wrangle, like men that are not well in their wits. I
confess, indeed, that that part of philosophy by which magnitudes and
figures are computed, is highly improved. But because I have not
observed the like advancement in the other parts of it, my purpose is,
as far forth as I am able, to lay open the few and first Elements of
Philosophy in general, as so many seeds from which pure and true
Philosophy may hereafter spring up by little and little.

I am not ignorant how hard a thing it is to weed out of men's minds such
inveterate opinions as have taken root there, and been confirmed in them
by the authority of most eloquent writers; especially seeing true (that
is, accurate) Philosophy professedly rejects not only the paint and
false colours of language, but even the very ornaments and graces of the
same; and the first grounds of all science are not only not beautiful,
but poor, arid, and, in appearance, deformed. Nevertheless, there being
certainly some men, though but few, who are delighted with truth and
strength of reason in all things, I thought I might do well to take this
pains for the sake even of those few. I proceed therefore to the matter,
and take my beginning from the very definition of philosophy, which is
this.

[Sidenote: Definition of Philosophy explained.]

2. PHILOSOPHY _is such knowledge of effects or appearances, as we
acquire by true ratiocination from the knowledge we have first of their
causes or generation; And again, of such causes or generations as may be
from knowing first their effects_.

For the better understanding of which definition, we must consider,
first, that although Sense and Memory of things, which are common to man
and all living creatures, be knowledge, yet because they are given us
immediately by nature, and not gotten by ratiocination, they are not
philosophy.

Secondly, seeing Experience is nothing but memory; and Prudence, or
prospect into the future time, nothing but expectation of such things as
we have already had experience of, Prudence also is not to be esteemed
philosophy.

By RATIOCINATION, I mean _computation_. Now to compute, is either to
collect the sum of many things that are added together, or to know what
remains when one thing is taken out of another. _Ratiocination_,
therefore, is the same with _addition_ and _substraction_; and if any
man add _multiplication_ and _division_, I will not be against it,
seeing multiplication is nothing but addition of equals one to another,
and division nothing but a substraction of equals one from another, as
often as is possible. So that all ratiocination is comprehended in these
two operations of the mind, addition and substraction.

[Sidenote: Ratiocination of the Mind.]

3. But how by the _ratiocination_ of our mind, we add and substract in
our silent thoughts, without the use of words, it will be necessary for
me to make intelligible by an example or two. If therefore a man see
something afar off and obscurely, although no appellation had yet been
given to anything, he will, notwithstanding, have the same idea of that
thing for which now, by imposing a name on it, we call it _body_. Again,
when, by coming nearer, he sees the same thing thus and thus, now in one
place and now in another, he will have a new idea thereof, namely, that
for which we now call such a thing _animated_. Thirdly, when standing
nearer, he perceives the figure, hears the voice, and sees other things
which are signs of a rational mind, he has a third idea, though it have
yet no appellation, namely, that for which we now call anything
_rational_. Lastly, when, by looking fully and distinctly upon it, he
conceives all that he has seen as one thing, the idea he has now is
compounded of his former ideas, which are put together in the mind in
the same order in which these three single names, _body_, _animated_,
_rational_, are in speech compounded into this one name,
_body-animated-rational_, or _man_. In like manner, of the several
conceptions of _four sides, equality of sides, and right angles_, is
compounded the conception of a _square_. For the mind may conceive a
figure of four sides without any conception of their equality, and of
that equality without conceiving a right angle; and may join together
all these single conceptions into one conception or one idea of a
square. And thus we see how the conceptions of the mind are compounded.
Again, whosoever sees a man standing near him, conceives the whole idea
of that man; and if, as he goes away, he follow him with his eyes only,
he will lose the idea of those things which were signs of his being
rational, whilst, nevertheless, the idea of a body-animated remains
still before his eyes, so that the idea of rational is substracted from
the whole idea of man, that is to say, of body-animated-rational, and
there remains that of body-animated; and a while after, at a greater
distance, the idea of animated will be lost, and that of body only will
remain; so that at last, when nothing at all can be seen, the whole idea
will vanish out of sight. By which examples, I think, it is manifest
enough what is the internal ratiocination of the mind without words.

We must not therefore think that computation, that is, ratiocination,
has place only in numbers, as if man were distinguished from other
living creatures (which is said to have been the opinion of
_Pythagoras_) by nothing but the faculty of numbering; for _magnitude,
body, motion, time, degrees of quality, action, conception, proportion,
speech and names_ (in which all the kinds of philosophy consist) are
capable of addition and substraction. Now such things as we add or
substract, that is, which we put into an account, we are said to
_consider_, in Greek λογίζεσθαι, in which language also συλλογίζεσθαι
signifies to _compute_, _reason_, or _reckon_.

[Sidenote: Properties, what they are.]

4. But _effects_ and the _appearances_ of things to sense, are faculties
or powers of bodies, which make us distinguish them from one another;
that is to say, conceive one body to be equal or unequal, like or unlike
to another body; as in the example above, when by coming near enough to
any body, we perceive the motion and going of the same, we distinguish
it thereby from a tree, a column, and other fixed bodies; and so that
motion or going is the _property_ thereof, as being proper to living
creatures, and a faculty by which they make us distinguish them from
other bodies.

[Sidenote: How Properties are known by Generation, and contrarily.]

5. How the knowledge of any effect may be gotten from the knowledge of
the generation thereof, may easily be understood by the example of a
circle: for if there be set before us a plain figure, having, as near as
may be, the figure of a circle, we cannot possibly perceive by sense
whether it be a true circle or no; than which, nevertheless, nothing is
more easy to be known to him that knows first the generation of the
propounded figure. For let it be known that the figure was made by the
circumduction of a body whereof one end remained unmoved, and we may
reason thus; a body carried about, retaining always the same length,
applies itself first to one _radius_, then to another, to a third, a
fourth, and successively to all; and, therefore, the same length, from
the same point, toucheth the circumference in every part thereof, which
is as much as to say, as all the _radii_ are equal. We know, therefore,
that from such generation proceeds a figure, from whose one middle point
all the extreme points are reached unto by equal _radii_. And in like
manner, by knowing first what figure is set before us, we may come by
ratiocination to some generation of the same, though perhaps not that by
which it was made, yet that by which it might have been made; for he
that knows that a circle has the property above declared, will easily
know whether a body carried about, as is said, will generate a circle or
no.

[Sidenote: Scope of Philosophy.]

6. The _end_ or _scope_ of philosophy is, that we may make use to our
benefit of effects formerly seen; or that, by application of bodies to
one another, we may produce the like effects of those we conceive in our
mind, as far forth as matter, strength, and industry, will permit, for
the commodity of human life. For the inward glory and triumph of mind
that a man may have for the mastering of some difficult and doubtful
matter, or for the discovery of some hidden truth, is not worth so much
pains as the study of Philosophy requires; nor need any man care much to
teach another what he knows himself, if he think that will be the only
benefit of his labour. The end of knowledge is power; and the use of
theorems (which, among geometricians, serve for the finding out of
properties) is for the construction of problems; and, lastly, the scope
of all speculation is the performing of some action, or thing to be
done.

[Sidenote: Utility of Philosophy.]

7. But what the _utility_ of philosophy is, especially of natural
philosophy and geometry, will be best understood by reckoning up the
chief commodities of which mankind is capable, and by comparing the
manner of life of such as enjoy them, with that of others which want the
same. Now, the greatest commodities of mankind are the arts; namely, of
measuring matter and motion; of moving ponderous bodies; of
architecture; of navigation; of making instruments for all uses; of
calculating the celestial motions, the aspects of the stars, and the
parts of time; of geography, &c. By which sciences, how great benefits
men receive is more easily understood than expressed. These benefits are
enjoyed by almost all the people of Europe, by most of those of Asia,
and by some of Africa: but the Americans, and they that live near the
Poles, do totally want them. But why? Have they sharper wits than these?
Have not all men one kind of soul, and the same faculties of mind? What,
then, makes this difference, except philosophy? Philosophy, therefore,
is the cause of all these benefits. But the utility of moral and civil
philosophy is to be estimated, not so much by the commodities we have by
knowing these sciences, as by the calamities we receive from not knowing
them. Now, all such calamities as may be avoided by human industry,
arise from war, but chiefly from civil war; for from this proceed
slaughter, solitude, and the want of all things. But the cause of war is
not that men are willing to have it; for the will has nothing for object
but good, at least that which seemeth good. Nor is it from this, that
men know not that the effects of war are evil; for who is there that
thinks not poverty and loss of life to be great evils? The cause,
therefore, of civil war is, that men know not the causes neither of war
nor peace, there being but few in the world that have learned those
duties which unite and keep men in peace, that is to say, that have
learned the rules of civil life sufficiently. Now, the knowledge of
these rules is moral philosophy. But why have they not learned them,
unless for this reason, that none hitherto have taught them in a clear
and exact method? For what shall we say? Could the ancient masters of
Greece, Egypt, Rome, and others, persuade the unskilful multitude to
their innumerable opinions concerning the nature of their gods, which
they themselves knew not whether they were true or false, and which were
indeed manifestly false and absurd; and could they not persuade the same
multitude to civil duty, if they themselves had understood it? Or shall
those few writings of geometricians which are extant, be thought
sufficient for the taking away of all controversy in the matters they
treat of, and shall those innumerable and huge volumes of _ethics_ be
thought unsufficient, if what they teach had been certain and well
demonstrated? What, then, can be imagined to be the cause that the
writings of those men have increased science, and the writings of these
have increased nothing but words, saving that the former were written by
men that knew, and the latter by such as knew not, the doctrine they
taught only for ostentation of their wit and eloquence? Nevertheless, I
deny not but the reading of some such books is very delightful; for they
are most eloquently written, and contain many clear, wholesome and
choice sentences, which yet are not universally true, though by them
universally pronounced. From whence it comes to pass, that the
circumstances of times, places, and persons being changed, they are no
less frequently made use of to confirm wicked men in their purposes,
than to make them understand the precepts of civil duties. Now that
which is chiefly wanting in them, is a true and certain rule of our
actions, by which we might know whether that we undertake be just or
unjust. For it is to no purpose to be bidden in every thing to do right,
before there be a certain rule and measure of right established, which
no man hitherto hath established. Seeing, therefore, from the not
knowing of civil duties, that is, from the want of moral science,
proceed civil wars, and the greatest calamities of mankind, we may very
well attribute to such science the production of the contrary
commodities. And thus much is sufficient, to say nothing of the praises
and other contentment proceeding from philosophy, to let you see the
utility of the same in every kind thereof.

[Sidenote: Subject of Philosophy.]

8. The _subject_ of Philosophy, or the matter it treats of, is every
body of which we can conceive any generation, and which we may, by any
consideration thereof, compare with other bodies, or which is capable of
composition and resolution; that is to say, every body of whose
generation or properties we can have any knowledge. And this may be
deduced from the definition of philosophy, whose profession it is to
search out the properties of bodies from their generation, or their
generation from their properties; and, therefore, where there is no
generation or property, there is no philosophy. Therefore it excludes
_Theology_, I mean the doctrine of God, eternal, ingenerable,
incomprehensible, and in whom there is nothing neither to divide nor
compound, nor any generation to be conceived.

It excludes the doctrine of _angels_, and all such things as are thought
to be neither bodies nor properties of bodies; there being in them no
place neither for composition nor division, nor any capacity of more and
less, that is to say, no place for ratiocination.

It excludes _history_, as well _natural_ as _political_, though most
useful (nay necessary) to philosophy; because such knowledge is but
experience, or authority, and not ratiocination.

It excludes all such knowledge as is acquired by Divine inspiration, or
revelation, as not derived to us by reason, but by Divine grace in an
instant, and, as it were, by some sense supernatural.

It excludes not only all doctrines which are false, but such also as are
not well-grounded; for whatsoever we know by right ratiocination, can
neither be false nor doubtful; and, therefore, _astrology_, as it is now
held forth, and all such divinations rather than sciences, are excluded.

Lastly, the doctrine of _God's worship_ is excluded from philosophy, as
being not to be known by natural reason, but by the authority of the
Church; and as being the object of faith, and not of knowledge.

[Sidenote: Parts of Philosophy.]

9. The principal parts of philosophy are two. For two chief kinds of
bodies, and very different from one another, offer themselves to such as
search after their generation and properties; one whereof being the work
of nature, is called a _natural body_, the other is called a
_commonwealth_, and is made by the wills and agreement of men. And from
these spring the two parts of philosophy, called _natural_ and _civil_.
But seeing that, for the knowledge of the properties of a commonwealth,
it is necessary first to know the dispositions, affections, and manners
of men, civil philosophy is again commonly divided into two parts,
whereof one, which treats of men's dispositions and manners, is called
_ethics_; and the other, which takes cognizance of their civil duties,
is called _politics_, or simply _civil philosophy_. In the first place,
therefore (after I have set down such premises as appertain to the
nature of philosophy in general), I will discourse of _bodies natural_;
in the second, of the _dispositions and manners of men_; and in the
third, of the _civil duties of subjects_.

[Sidenote: Epilogue.]

10. To conclude; seeing there may be many who will not like this my
definition of philosophy, and will say, that, from the liberty which a
man may take of so defining as seems best to himself, he may conclude
any thing from any thing (though I think it no hard matter to
demonstrate that this definition of mine agrees with the sense of all
men); yet, lest in this point there should be any cause of dispute
betwixt me and them, I here undertake no more than to deliver the
elements of that science by which the effects of anything may be found
out from the known generation of the same, or contrarily, the generation
from the effects; to the end that they who search after other
philosophy, may be admonished to seek it from other principles.


                                -------


                              CHAPTER II.

                               OF NAMES.

1. The necessity of sensible Moniments or Marks for the help of Memory:
    a Mark defined.—2. The necessity of Marks for the signification of
    the conceptions of the Mind.—3. Names supply both those
    necessities.—4. The Definition of a Name.—5. Names are Signs not of
    Things, but of our Cogitations.—6. What it is we give Names to.—7.
    Names Positive and Negative.—8. Contradictory Names.—9. A Common
    Name.—10. Names of the First and Second Intention.—11. Universal,
    Particular, Individual, and Indefinite Names.—12. Names Univocal and
    Equivocal.—13. Absolute and Relative Names.—14. Simple and
    Compounded Names.—15. A Predicament described.—16. Some things to be
    noted concerning Predicaments.


[Sidenote: Necessity of sensible Moniments or Marks for the help of
           Memory.]

1. How unconstant and fading men's thoughts are, and how much the
recovery of them depends upon chance, there is none but knows by
infallible experience in himself. For no man is able to remember
quantities without sensible and present measures, nor colours without
sensible and present patterns, nor number without the names of numbers
disposed in order and learned by heart. So that whatsoever a man has put
together in his mind by ratiocination without such helps, will presently
slip from him, and not be revocable but by beginning his ratiocination
anew. From which it follows, that, for the acquiring of philosophy, some
sensible moniments are necessary, by which our past thoughts may be not
only reduced, but also registered every one in its own order.[Sidenote:
A Mark defined] These moniments I call MARKS, namely, sensible things
taken at pleasure, that, by the sense of them, such thoughts may be
recalled to our mind as are like those thoughts for which we took them.

[Sidenote: Necessity of Marks for the signification of the conceptions
           of the Mind.]

2. Again, though some one man, of how excellent a wit soever, should
spend all his time partly in reasoning, and partly in inventing marks
for the help of his memory, and advancing himself in learning; who sees
not that the benefit he reaps to himself will not be much, and to others
none at all? For unless he communicate his notes with others, his
science will perish with him. But if the same notes be made common to
many, and so one man's inventions be taught to others, sciences will
thereby be increased to the general good of mankind. It is therefore
necessary, for the acquiring of philosophy, that there be certain signs,
by which what one man finds out may be manifested and made known to
others. Now, those things we call SIGNS are the _antecedents of their
consequents, and the consequents of their antecedents, as often as we
observe them to go before or follow after in the same manner_. For
example, a thick cloud is a sign of rain to follow, and rain a sign that
a cloud has gone before, for this reason only, that we seldom see clouds
without the consequence of rain, nor rain at any time but when a cloud
has gone before. And of signs, some are _natural_, whereof I have
already given an example, others are _arbitrary_, namely, those we make
choice of at our own pleasure, as a bush hung up, signifies that wine is
to be sold there; a stone set in the ground signifies the bound of a
field; and words so and so connected, signify the cogitations and
motions of our mind. The difference, therefore, betwixt marks and signs
is this, that we make those for our own use, but these for the use of
others.

[Sidenote: Names supply both those necessities.]

3. Words so connected as that they become signs of our thoughts, are
called SPEECH, of which every part is a _name_. But seeing (as is said)
both marks and signs are necessary for the acquiring of philosophy,
(marks by which we may remember our own thoughts, and signs by which we
may make our thoughts known to others), names do both these offices; but
they serve for marks before they be used as signs. For though a man were
alone in the world, they would be useful to him in helping him to
remember; but to teach others, (unless there were some others to be
taught) of no use at all. Again, names, though standing singly by
themselves, are marks, because they serve to recal our own thoughts to
mind; but they cannot be signs, otherwise than by being disposed and
ordered in speech as parts of the same. For example, a man may begin
with a word, whereby the hearer may frame an idea of something in his
mind, which, nevertheless, he cannot conceive to be the idea which was
in the mind of him that spake, but that he would say something which
began with that word, though perhaps not as by itself, but as part of
another word. So that the nature of a name consists principally in this,
that it is a mark taken for memory's sake; but it serves also by
accident to signify and make known to others what we remember ourselves,
and, therefore, I will define it thus:

[Sidenote: Definition of a Name.]

4. _A_ NAME _is a word taken at pleasure to serve for a mark, which may
raise in our mind a thought like to some thought we had before, and
which being pronounced to others, may be to them a sign of what thought
the speaker had, or had not before in his mind_. And it is for brevity's
sake that I suppose the original of names to be arbitrary, judging it a
thing that may be assumed as unquestionable. For considering that new
names are daily made, and old ones laid aside; that diverse nations use
different names, and how impossible it is either to observe similitude,
or make any comparison betwixt a name and a thing, how can any man
imagine that the names of things were imposed from their natures? For
though some names of living creatures and other things, which our first
parents used, were taught by God himself; yet they were by him
arbitrarily imposed, and afterwards, both at the Tower of Babel, and
since, in process of time, growing everywhere out of use, are quite
forgotten, and in their room have succeeded others, invented and
received by men at pleasure. Moreover, whatsoever the common use of
words be, yet philosophers, who were to teach their knowledge to others,
had always the liberty, and sometimes they both had and will have a
necessity, of taking to themselves such names as they please for the
signifying of their meaning, if they would have it understood. Nor had
mathematicians need to ask leave of any but themselves to name the
figures they invented, _parabolas_, _hyperboles_, _cissoeides_,
_quadratices_, &c. or to call one magnitude A, another B.

[Sidenote: Names are signs not of things, but of our cogitations.]

5. But seeing names ordered in speech (as is defined) are signs of our
conceptions, it is manifest they are not signs of the things themselves;
for that the sound of this word _stone_ should be the sign of a stone,
cannot be understood in any sense but this, that he that hears it
collects that he that pronounces it thinks of a stone. And, therefore,
that disputation, whether names signify the matter or form, or something
compounded of both, and other like subtleties of the _metaphysics_, is
kept up by erring men, and such as understand not the words they dispute
about.

[Sidenote: What it is we give names to.]

6. Nor, indeed, is it at all necessary that every name should be the
name of something. For as these, a _man_, a _tree_, a _stone_, are the
names of the things themselves, so the images of a man, of a tree, and
of a stone, which are represented to men sleeping, have their names
also, though they be not things, but only fictions and phantasms of
things. For we can remember these; and, therefore, it is no less
necessary that they have names to mark and signify them, than the things
themselves. Also this word _future_ is a name, but no future thing has
yet any being, nor do we know whether that which we call future, shall
ever have a being or no. Nevertheless, seeing we use in our mind to knit
together things past with those that are present, the name _future_
serves to signify such knitting together. Moreover, that which neither
is, nor has been, nor ever shall, or ever can be, has a name, namely,
_that which neither is nor has been_, &c.; or more briefly this,
_impossible_. To conclude; this word _nothing_ is a name, which yet
cannot be the name of any thing: for when, for example, we substract 2
and 3 from 5, and so nothing remaining, we would call that substraction
to mind, this speech _nothing remains_, and in it the word _nothing_ is
not unuseful. And for the same reason we say truly, _less than nothing_
remains, when we substract more from less; for the mind feigns such
remains as these for doctrine's sake, and desires, as often as is
necessary, to call the same to memory. But seeing every name has some
relation to that which is named, though that which we name be not always
a thing that has a being in nature, yet it is lawful for doctrine's sake
to apply the word _thing_ to whatsoever we name; as if it were all one
whether that thing be truly existent, or be only feigned.

[Sidenote: Names Positive and Negative.]

7. The first distinction of names is, that some are _positive_, or
_affirmative_, others _negative_, which are also called _privative_ and
_indefinite_. _Positive_ are such as we impose for the likeness,
equality, or identity of the things we consider; _negative_, for the
diversity, unlikeness, or inequality of the same. Examples of the former
are, _a man_, _a philosopher_; for a _man_ denotes any one of a
multitude of men, and _a philosopher_, any one of many philosophers, by
reason of their similitude; also, _Socrates_ is a positive name, because
it signifies always one and the same man. Examples of _negatives_ are
such positives as have the negative particle _not_ added to them, as
_not-man_, _not-philosopher_. But positives were before negatives; for
otherwise there could have been no use at all of these. For when the
name of _white_ was imposed upon certain things, and afterwards upon
other things, the names of _black_, _blue_, _transparent_, _&c._ the
infinite dissimilitudes of these with _white_ could not be comprehended
in any one name, save that which had in it the negation of white, that
is to say, the name _not-white_, or some other equivalent to it, in
which the word _white_ is repeated, such as _unlike to white_, _&c._ And
by these _negative names_, we take notice ourselves, and signify to
others what we have not thought of.

[Sidenote: Contradictory names.]

8. Positive and negative names are _contradictory_ to one another, so
that they cannot both be the name of the same thing. Besides, of
contradictory names, one is the name of anything whatsoever; for
whatsoever is, is either man, or not-man, white or not-white, and so of
the rest. And this is so manifest, that it needs no farther proof or
explication; for they that say _the same thing cannot both be, and not
be_, speak obscurely; but they that say, _whatsoever is, either is, or
is not_, speak also absurdly and ridiculously. The certainty of this
axiom, viz. _of two contradictory names, one is the name of anything
whatsoever, the other not_, is the original and foundation of all
ratiocination, that is, of all philosophy; and therefore it ought to be
so exactly propounded, that it may be of itself clear and perspicuous to
all men; as indeed it is, saving to such, as reading the long discourses
made upon this subject by the writers of _metaphysics_ (which they
believe to be some egregious learning) think they understand not, when
they do.

[Sidenote: A common name.]

9. Secondly, of names, some are _common_ to many things, as a _man_, _a
tree_; others _proper_ to one thing, as _he that writ the Iliad_,
_Homer_, _this man_, _that man_. And a common name, being the name of
many things severally taken, but not collectively of all together (as
man is not the name of all mankind, but of every one, as of Peter, John,
and the rest severally) is therefore called an _universal name_; and
therefore this word _universal_ is never the name of any thing existent
in nature, nor of any idea or phantasm formed in the mind, but always
the name of some word or name; so that when _a living creature_, _a
stone_, _a spirit_, or any other thing, is said to be _universal_, it is
not to be understood, that any man, stone, &c. ever was or can be
universal, but only that these words, _living creature_, _stone_, _&c._
are _universal names_, that is, names common to many things; and the
conceptions answering them in our mind, are the images and phantasms of
several living creatures, or other things. And therefore, for the
understanding of the extent of an universal name, we need no other
faculty but that of our imagination, by which we remember that such
names bring sometimes one thing, sometimes another, into our mind. Also
of common names, some are more, some less common. _More common_, is that
which is the name of more things; _less common_, the name of fewer
things; as _living creature_ is more common than _man_, or _horse_, or
_lion_, because it comprehends them all: and therefore a more common
name, in respect of a less common, is called the _genus_, or a _general
name_; and this in respect of that, the _species_, or a _special name_.

[Sidenote: Names of the first and second intention.]

10. And from hence proceeds the third distinction of names, which is,
that some are called names of the _first_, others of the _second
intention_. Of the _first intention_ are the names of things, a _man_,
_stone_, _&c._: of the _second_ are the names of names and speeches, as
_universal_, _particular_, _genus_, _species_, _syllogism_, and the
like. But it is hard to say why those are called names of the _first_,
and these of the _second intention_, unless perhaps it was first
intended by us to give names to those things which are of daily use in
this life, and afterwards to such things as appertain to science, that
is, that our second intention was to give names to names. But whatsoever
the cause hereof may be, yet this is manifest, that _genus_, _species_,
_definition_, _&c._ are names of words and names only; and therefore to
put _genus_ and _species_ for things, and _definition_ for the nature of
any thing, as the writers of _metaphysics_ have done, is not right,
seeing they be only significations of what we think of the nature of
things.

[Sidenote: Universal, particular, individual, and indefinite names.]

11. Fourthly, some names are of _certain_ and _determined_, others of
_uncertain_ and _undetermined_ signification. Of _determined_ and
_certain_ signification is, first, that name which is given to any one
thing by itself, and is called an _individual name_; as _Homer_, _this
tree_, _that living creature_, &c. Secondly that which has any of these
words, _all_, _every_, _both_, _either_, or the like added to it; and it
is therefore called an universal name, because it signifies every one of
those things to which it is common; and of _certain_ signification for
this reason, that he which hears, conceives in his mind the same thing
that he which speaks would have him conceive. Of _indefinite_
signification is, first, that name which has the word _some_, or the
like added to it, and is called a _particular name_; secondly, a common
name set by itself without any note either of universality or
particularity, as _man_, _stone_, and is called an _indefinite name_;
but both _particular_ and _indefinite_ names are of uncertain
signification, because the hearer knows not what thing it is the speaker
would have him conceive; and therefore in speech, particular and
indefinite names are to be esteemed equivalent to one another. But these
words, _all_, _every_, _some_, _&c._ which denote universality and
particularity, are not names, but parts only of names; so that _every
man_, and _that man which the hearer conceives in his mind_, are all
one; and _some man_, and _that man which the speaker thought of_,
signify the same. From whence it is evident, that the use of signs of
this kind, is not for a man's own sake, or for his getting of knowledge
by his own private meditation (for every man has his own thoughts
sufficiently determined without such helps as these) but for the sake of
others; that is, for the teaching and signifying of our conceptions to
others; nor were they invented only to make us remember, but to make us
able to discourse with others.

[Sidenote: Names univocal and equivocal.]

12. Fifthly, names are usually distinguished into _univocal_ and
_equivocal_. _Univocal_ are those which in the same train of discourse
signify always the same thing; but _equivocal_ those which mean
sometimes one thing and sometimes another. Thus, the name _triangle_ is
said to be _univocal_, because it is always taken in the same sense; and
_parabola_ to be _equivocal_, for the signification it has sometimes of
allegory or similitude, and sometimes of a certain geometrical figure.
Also every _metaphor_ is by profession _equivocal_. But this distinction
belongs not so much to names, as to those that use names, for some use
them properly and accurately for the finding out of truth; others draw
them from their proper sense, for ornament or deceit.

[Sidenote: Absolute and relative names.]

13. Sixthly, of names, some are _absolute_, others _relative_.
_Relative_ are such as are imposed for some comparison, as _father_,
_son_, _cause_, _effect_, _like_, _unlike_, _equal_, _unequal_,
_master_, _servant_, _&c._ And those that signify no comparison at all
are _absolute names_. But, as it was noted above, that universality is
to be attributed to words and names only, and not to things, so the same
is to be said of other distinctions of names; for no things are either
_univocal_ or _equivocal_, or _relative_ or _absolute_. There is also
another distinction of names into _concrete_ and _abstract_; but because
abstract names proceed from proposition, and can have no place where
there is no affirmation, I shall speak of them hereafter.

[Sidenote: Simple and compounded names.]

14. Lastly, there are _simple_ and _compounded names_. But here it is to
be noted, that a name is not taken in philosophy, as in grammar, for one
single word, but for any number of words put together to signify one
thing; for among philosophers _sentient animated body_ passes but for
one name, being the name of every living creature, which yet, among
grammarians, is accounted three names. Also a _simple name_ is not here
distinguished from a _compounded name_ by a preposition, as in grammar.
But I call a _simple name_, that which in every kind is the most common
or most universal; and that a _compounded name_, which by the joining of
another name to it, is made less universal, and signifies that more
conceptions than one were in the mind, for which that latter name was
added. For example, in the conception of _man_ (as is shown in the
former chapter.) First, he is conceived to be something that has
extension, which is marked by the word _body_. _Body_, therefore, is a
_simple name_, being put for that first single conception; afterwards,
upon the sight of such and such motion, another conception arises, for
which he is called an _animated body_; and this I here call a
_compounded name_, as I do also the name _animal_, which is equivalent
to an _animated body_. And, in the same manner, an _animated rational
body_, as also a _man_, which is equivalent to it, is a more compounded
name. And by this we see how the composition of conceptions in the mind
is answerable to the composition of names; for, as in the mind one idea
or phantasm succeeds to another, and to this a third; so to one name is
added another and another successively, and of them all is made one
_compounded_ name. Nevertheless we must not think bodies which are
without the mind, are compounded in the same manner, namely, that there
is in nature a body, or any other imaginable thing existent, which at
the first has no magnitude, and then, by the addition of magnitude,
comes to have quantity, and by more or less quantity to have density or
rarity; and again, by the addition of figure, to be figurate, and after
this, by the injection of light or colour, to become lucid or coloured;
though such has been the philosophy of many.

[Sidenote: A predicament described.]

15. The writers of _logic_ have endeavoured to digest the names of all
the kinds of things into certain _scales_ or degrees, by the continual
subordination of names less common, to names more common. In the scale
of _bodies_ they put in the first and highest place _body_ simply, and
in the next place under it less common names, by which it may be more
limited and determined, namely _animated_ and _inanimated_, and so on
till they come to _individuals_. In like manner, in the scale of
_quantities_, they assign the first place to _quantity_, and the next to
_line_, _superficies_, and _solid_, which are names of less latitude;
and these orders or scales of names they usually call _predicaments_ and
_categories_. And of this ordination not only positive, but negative
names also are capable; which may be exemplified by such forms of the
_predicaments_ as follow:

                  THE FORM OF THE PREDICAMENT OF BODY.

      Not-Body, or
        Accident.
             { Not animated.
      Body   {            { Not living Creature.
             { Animated   {
                          { Living     { Not Man.
                          {  Creature  {            { Not Peter.
                                       { Man        {
                                                    { Peter.

                        {                      { Quantity, or so much.
 Both Accident and Body { Absolutely, as       {
 are considered         {          or          { Quality, or such.
                        { Comparatively, which is called their Relation.

                THE FORM OF THE PREDICAMENT OF QUANTITY.

            { Not continual,
            { as Number.                        { Line.
 Quantity.  {                 { Of itself, as   { Superficies.
            {                 {                 { Solid.
            { Continual       {
                              {                 { Time, by Line.
                              {                 { Motion, by Line and
                              { By accident, as {   Time.
                                                { Force, by Motion and
                                                {   Solid.

Where, it is to be noted, that _line_, _superficies_, and _solid_, may
be said to be of such and such quantity, that is, to be originally and
of their own nature capable of equality and inequality; but we cannot
say there is either majority or minority, or equality, or indeed any
quantity at all, in _time_, without the help of _line_ and _motion_; nor
in _motion_, without _line_ and _time_; nor in _force_, otherwise than
by _motion_ and _solid_.

                THE FORM OF THE PREDICAMENT OF QUALITY.

                                         { Seeing.
                                         { Hearing.
                          { Primary      { Smelling.
                          {              { Tasting.
           { Perception   {              { Touching.
           {  by Sense    {
           {              {              { Imagination.
           {              { Secondary    { Affection    { pleasant.
  Quality  {                             {              { unpleasant.
           {              { By Seeing, as Light and Colour.
           {              { By Hearing, as Sound.
           { Sensible     { By Smelling, as Odours.
           {  Quality     { By Tasting, as Savours.
                          { By Touching, as Hardness, Heat,
                          {  Cold, &c.

                THE FORM OF THE PREDICAMENT OF RELATION.

              { Magnitudes, as Equality and Inequality.
              { Qualities, as Likeness and Unlikeness.
 Relation of  {            { Together     { In Place.
              {            {              { In Time.
              { Order      {
                           {              { In Place     { Former.
                           { Not together {              { Later
                                          { In Time      { Former.
                                                         { Later

[Sidenote: Some things to be noted concerning predicaments.]

16. Concerning which predicaments it is to be noted, in the first place,
that as the division is made in the first predicament into contradictory
names, so it might have been done in the rest. For, as there, _body_ is
divided into _animated_ and _not-animated_, so, in the second
predicament, _continual quantity_ may be divided into _line_ and
_not-line_, and again, _not-line_ into _superficies_ and
_not-superficies_, and so in the rest; but it was not necessary.

Secondly, it is to be observed, that of _positive_ names the former
comprehends the latter; but of _negatives_ the former is comprehended by
the latter. For example, _living-creature_ is the name of every man, and
therefore it comprehends the name _man_; but, on the contrary, _not-man_
is the name of everything which is not-living-creature, and therefore
the name _not-living-creature_, which is put first, is comprehended by
the latter name, _not-man_.

Thirdly, we must take heed that we do not think, that as names, so the
diversities of things themselves may be searched out and determined by
such distinctions as these; or that arguments may be taken from hence
(as some have done ridiculously) to prove that the kinds of things are
not infinite.

Fourthly, I would not have any man think I deliver the forms above for a
true and exact ordination of names; for this cannot be performed as long
as philosophy remains imperfect; nor that by placing (for example)
_light_ in the predicament of _qualities_, while another places the same
in the predicament of bodies, I pretend that either of us ought for this
to be drawn from his opinion; for this is to be done only by arguments
and ratiocination, and not by disposing of words into _classes_.

Lastly, I confess I have not yet seen any great use of the predicaments
in philosophy. I believe _Aristotle_ when he saw he could not digest the
things themselves into such orders, might nevertheless desire out of his
own authority to reduce words to such forms, as I have done; but I do it
only for this end, that it may be understood what this ordination of
words is, and not to have it received for true, till it be demonstrated
by good reason to be so.


                                -------


                              CHAPTER III.

                            OF PROPOSITION.

1. Divers kinds of speech.—2. Proposition defined.—3. Subject,
    predicate, and copula, what they are; and abstract and concrete
    what. 4. The use and abuse of names abstract.—5. Proposition,
    universal and particular.—6. Affirmative and negative.—7. True and
    false.—8. True and false belongs to speech, and not to things.—9.
    Proposition, primary, not primary, definition, axiom, petition.—10.
    Proposition, necessary and contingent.—11. Categorical and
    hypothetical.—12. The same proposition diversely pronounced.—13.
    Propositions that may be reduced to the same categorical
    proposition, are equipollent.—14. Universal propositions converted
    by contradictory names, are equipollent.—15. Negative propositions
    are the same, whether negation be before or after the copula.—16.
    Particular propositions simply converted, are equipollent.—17. What
    are subaltern, contrary, subcontrary, and contradictory
    propositions.—18. Consequence, what it is.—19. Falsity cannot follow
    from truth.—20. How one proposition is the cause of another.


[Sidenote: Divers kinds of speech.]

1. From the connexion or contexture of names arise divers kinds of
speech, whereof some signify the desires and affections of men; such
are, first, _interrogations_, which denote the desire of knowing: as,
_Who is a good man?_ In which speech there is one name expressed, and
another desired and expected from him of whom we ask the same. Then
_prayers_, which signify the desire of having something; _promises_,
_threats_, _wishes_, _commands_, _complaints_, and other significations
of other affections. Speech may also be absurd and insignificant; as
when there is a succession of words, to which there can be no succession
of thoughts in mind to answer them; and this happens often to such, as,
understanding nothing in some subtle matter, do, nevertheless, to make
others believe they understand, speak of the same incoherently; for the
connection of incoherent words, though it want the end of speech (which
is signification) yet it is speech; and is used by writers of
_metaphysics_ almost as frequently as speech significative. In
philosophy, there is but one kind of speech useful, which some call in
Latin _dictum_, others _enuntiatum et pronunciatum_; but most men call
it _proposition_, and is the speech of those that affirm or deny, and
expresseth truth or falsity.

[Sidenote: Proposition defined.]

2. _A_ PROPOSITION _is a speech consisting of two names copulated, by
which he that speaketh signifies he conceives the latter name to be the
name of the same thing whereof the former is the name; or_ (which is all
one) _that the former name is comprehended by the latter_. For example,
this speech, _man is a living creature_, in which two names are
copulated by the verb _is_, is a _proposition_, for this reason, that he
that speaks it conceives both _living creature_ and _man_ to be names of
the same thing, or that the former name, _man_, is comprehended by the
latter name, _living creature_. Now the former name is commonly called
the _subject_, or _antecedent_, or the _contained name_, and the latter
_predicate_, _consequent_, or _containing name_. The sign of connection
amongst most nations is either some word, as the word _is_ in the
proposition _man is a living creature_, or some case or termination of a
word, as in this proposition, _man walketh_ (which is equivalent to
this, _man is walking_); the termination by which it is said he
_walketh_, rather than he _is walking_, signifieth that those two are
understood to be copulated, or to be names of the same thing.

But there are, or certainly may be, some nations that have no word which
answers to our verb _is_, who nevertheless form propositions by the
position only of one name after another, as if instead of _man is a
living creature_, it should be said _man a living creature_; for the
very order of the names may sufficiently show their connection; and they
are as apt and useful in philosophy, as if they were copulated by the
verb _is_.

[Sidenote: Subject, predicate, and copula, what they are, and abstract
           and concrete what.]

3. Wherefore, in every proposition three things are to be considered,
_viz._ the two names, which are the _subject_, and the _predicate_, and
their _copulation_; both which names raise in our mind the thought of
one and the same thing; but the copulation makes us think of the cause
for which those names were imposed on that thing. As, for example, when
we say _a body is moveable_, though we conceive the same thing to be
designed by both those names, yet our mind rests not there, but searches
farther what it is _to be a body_, or _to be moveable_, that is, wherein
consists the difference betwixt these and other things, for which these
are so called, others are not so called. They, therefore, that seek what
it is _to be_ any thing, as _to be moveable_, _to be hot_, _&c._ seek in
things the causes of their names.

And from hence arises that distinction of names (touched in the last
chapter) into _concrete_ and _abstract_. For _concrete_ is the name of
any thing which we suppose to have a being, and is therefore called the
_subject_, in Latin _suppositum_, and in Greek ὑποκέιμενον; as _body_,
_moveable_, _moved_, _figurate_, _a cubit high_, _hot_, _cold_, _like_,
_equal_, _Appius_, _Lentulus_, and the like; and, _abstract_ is that
which in any subject denotes the cause of the concrete name, as _to be a
body_, _to be moveable_, _to be moved_, _to be figurate_, _to be of such
quantity_, _to be hot_, _to be cold_, _to be like_, _to be equal_, _to
be Appius_, _to be Lentulus_, _&c._ Or names equivalent to these, which
are most commonly called _abstract_ names, as _corporiety_, _mobility_,
_motion_, _figure_, _quantity_, _heat_, _cold_, _likeness_, _equality_,
and (as Cicero has it) _Appiety_ and _Lentulity_. Of the same kind also
are infinitives; for _to live_ and _to move_ are the same with _life_
and _motion_, or _to be living_ and _to be moved_. But _abstract names_
denote only the causes of _concrete names_, and not the things
themselves. For example, when we see any thing, or conceive in our mind
any visible thing, that thing appears to us, or is conceived by us, not
in one point, but as having parts distant from one another, that is, as
being extended and filling some space. Seeing therefore we call the
thing so conceived _body_, the cause of that name is, that that thing is
_extended_, or the _extension_ or _corporiety_ of it. So when we see a
thing appear sometimes here, sometimes there, and call it _moved_ or
_removed_, the cause of that name is that it _is moved_ or the _motion_
of the same.

And these causes of names are the same with the causes of our
conceptions, namely, some power of action, or affection of the thing
conceived, which some call the manner by which any thing works upon our
senses, but by most men they are called _accidents_; I say accidents,
not in that sense in which accident is opposed to necessary; but so, as
being neither the things themselves, nor parts thereof, do nevertheless
accompany the things in such manner, that (saving extension) they may
all perish, and be destroyed, but can never be abstracted.

[Sidenote: The use and abuse of names abstract.]

4. There is also this difference betwixt _concrete_ and _abstract_
names, that those were invented before propositions, but these after;
for these could have no being till there were propositions, from whose
_copula_ they proceed. Now in all matters that concern this life, but
chiefly in philosophy, there is both great use and great abuse of
_abstract names_; and the use consists in this, that without them we
cannot, for the most part, either reason, or compute the properties of
bodies; for when we would multiply, divide, add, or substract heat,
light, or motion, if we should double or add them together by concrete
names, saying (for example) hot is double to hot, light double to light,
or moved double to moved, we should not double the properties, but the
bodies themselves that are hot, light, moved, &c. which we would not do.
But the abuse proceeds from this, that some men seeing they can
consider, that is (as I said before) bring into account the increasings
and decreasings of quantity, heat and other accidents, without
considering their bodies or subjects (which they call _abstracting_, or
making to exist apart by themselves) they speak of accidents, as if they
might be separated from all bodies. And from hence proceed the gross
errors of writers of metaphysics; for, because they can consider thought
without the consideration of body, they infer there is no need of a
thinking-body; and because quantity may be considered without
considering body, they think also that quantity may be without body, and
body without quantity; and that a body has quantity by the addition of
quantity to it. From the same fountain spring those insignificant words,
_abstract substance_, _separated essence_, and the like; as also that
confusion of words derived from the Latin verb _est_, as _essence_,
_essentiality_, _entity_, _entitative_; besides _reality_,
_aliquiddity_, _quiddity_, _&c._ which could never have been heard of
among such nations as do not copulate their names by the verb _is_, but
by adjective verbs, as runneth, readeth, &c. or by the mere placing of
one name after another; and yet seeing such nations compute and reason,
it is evident that philosophy has no need of those words _essence_,
_entity_, and other the like barbarous terms.

[Sidenote: Proposition, universal and particular.]

5. There are many distinctions of propositions, whereof the first is,
that some are _universal_, others _particular_, others _indefinite_, and
others _singular_; and this is commonly called the distinction of
_quantity_. An _universal_ proposition is that whose subject is affected
with the sign of an universal name, as _every man is a living creature_.
_Particular_, that whose subject is affected with the sign of a
particular name, as _some man is learned_. An _indefinite_ proposition
has for its subject a common name, and put without any sign, as _man is
a living creature_, _man is learned_. And a _singular_ proposition is
that whose subject is a singular name, as _Socrates is a philosopher_,
_this man is black_.

[Sidenote: Affirmative and negative.]

6. The second distinction is into _affirmative_ and _negative_, and is
called the distinction of _quality_. An _affirmative_ proposition is
that whose predicate is a positive name, as _man is a living creature_.
_Negative_, that whose predicate is a negative name, as _man is not a
stone_.

[Sidenote: True & false.]

7. The third distinction is, that one is _true_, another _false_. A
_true_ proposition is that, whose predicate contains, or comprehends its
subject, or whose predicate is the name of every thing, of which the
subject is the name; as _man is a living creature_ is therefore a true
proposition, because whatsoever is called _man_, the same is also called
_living creature_; and _some man is sick_, is true, because _sick_ is
the name of _some man_. That which is not true, or that whose predicate
does not contain its subject, is called a _false_ proposition, as _man
is a stone_.

Now these words _true_, _truth_, and _true proposition_, are equivalent
to one another; for truth consists in speech, and not in the things
spoken of; and though _true_ be sometimes opposed to _apparent_ or
_feigned_, yet it is always to be referred to the truth of proposition;
for the image of a man in a glass, or a ghost, is therefore denied to be
a very man, because this proposition, _a ghost is a man_, is not true;
for it cannot be denied but that a ghost is a very ghost. And therefore
truth or verity is not any affection of the thing, but of the
proposition concerning it. As for that which the writers of metaphysics
say, that _a thing_, _one thing_, and _a very thing_, are equivalent to
one another, it is but trifling and childish; for who does not know,
that _a man_, _one man_, and _a very man_, signify the same.

[Sidenote: True & false belongs to speech, and not to things.]

8. And from hence it is evident, that truth and falsity have no place
but amongst such living creatures as use speech. For though some brute
creatures, looking upon the image of a man in a glass, may be affected
with it, as if it were the man himself, and for this reason fear it or
fawn upon it in vain; yet they do not apprehend it as true or false, but
only as like; and in this they are not deceived. Wherefore, as men owe
all their true ratiocination to the right understanding of speech; so
also they owe their errors to the misunderstanding of the same; and as
all the ornaments of philosophy proceed only from man, so from man also
is derived the ugly absurdity of false opinions. For speech has
something in it like to a spider's web, (as it was said of old of
_Solon's_ laws) for by contexture of words tender and delicate wits are
ensnared and stopped; but strong wits break easily through them.

From hence also this may be deduced, that the first truths were
arbitrarily made by those that first of all imposed names upon things,
or received them from the imposition of others. For it is true (for
example) that _man is a living creature_, but it is for this reason,
that it pleased men to impose both those names on the same thing.

[Sidenote: Proposition, primary, not primary, definition, axiom,
           petition.]

9. Fourthly, propositions are distinguished into _primary_ and _not
primary_. _Primary_ is that wherein the subject is explicated by a
predicate of many names, as _man is a body_, _animated_, _rational_; for
that which is comprehended in the name _man_, is more largely expressed
in the names _body_, _animated_, and _rational_, joined together; and it
is called _primary_, because it is first in ratiocination; for nothing
can be proved, without understanding first the name of the thing in
question. Now _primary_ propositions are nothing but definitions, or
parts of definitions, and these only are the principles of
demonstration, being truths constituted arbitrarily by the inventors of
speech, and therefore not to be demonstrated. To these propositions,
some have added others, which they call _primary_ and _principles_,
namely, _axioms_, and _common notions_; which, (though they be so
evident that they need no proof) yet, because they may be proved, are
not truly principles; and the less to be received for such, in regard
propositions not intelligible, and sometimes manifestly false, are
thrust on us under the name of _principles_ by the clamour of men, who
obtrude for evident to others, all that they themselves think true. Also
certain petitions are commonly received into the number of principles;
as, for example, _that a straight line may be drawn between two points_,
and other petitions of the writers of geometry; and these are indeed the
principles of art or construction, but not of science and demonstration.

[Sidenote: Proposition necessary & contingent.]

10. Fifthly, propositions are distinguished into _necessary_, that is,
necessarily true; and true, but not necessarily, which they call
_contingent_. A _necessary_ proposition is when nothing can at any time
be conceived or feigned, whereof the subject is the name, but the
predicate also is the name of the same thing; as _man is a living
creature_ is a necessary proposition, because at what time soever we
suppose the name _man_ agrees with any thing, at that time the name
_living-creature_ also agrees with the same. But a _contingent_
proposition is that, which at one time may be true, at another time
false; as _every crow is black_; which may perhaps be true now, but
false hereafter. Again, in every _necessary_ proposition, the predicate
is either equivalent to the subject, as in this, _man is a rational
living creature_; or part of an equivalent name, as in this, _man is a
living creature_, for the name _rational-living-creature_, or _man_, is
compounded of these two, _rational_ and _living-creature_. But in a
_contingent_ proposition this cannot be; for though this were true,
_every man is a liar_, yet because the word _liar_ is no part of a
compounded name equivalent to the name _man_, that proposition is not to
be called _necessary_, but _contingent_, though it should happen to be
true always. And therefore those propositions only are _necessary_,
which are of sempiternal truth, that is, true at all times. From hence
also it is manifest, that truth adheres not to things, but to speech
only, for some truths are eternal; for it will be eternally true, _if
man, then living-creature_; but that any _man_, or _living-creature_,
should exist eternally, is not necessary.

[Sidenote: Categorical & hypothetical.]

11. A sixth distinction of propositions is into _categorical_ and
_hypothetical_. A _categorical_ proposition is that which is simply or
absolutely pronounced, as _every man is a living-creature_, _no man is a
tree_; and _hypothetical_ is that which is pronounced conditionally, as,
_if any thing be a man, the same is also a living-creature_, _if
anything be a man, the same is also not-a-stone_.

A _categorical_ proposition, and an _hypothetical_ answering it, do both
signify the same, if the propositions be necessary; but not if they be
contingent. For example, if this, _every man is a living-creature_, be
true, this also will be true, _if any thing be a man, the same is also a
living-creature_; but in contingent propositions, though this be true,
_every crow is black_, yet this, _if any thing be a crow, the same is
black_, is false. But an _hypothetical_ proposition is then rightly said
to be true, when the consequence is true, as _every man is a
living-creature_, is rightly said to be a true proposition, because of
whatsoever it is truly said _that is a man_, it cannot but be truly said
also, _the same is a living creature_. And therefore whensoever an
_hypothetical_ proposition is true, the _categorical_ answering it, is
not only true, but also necessary; which I thought worth the noting, as
an argument, that philosophers may in most things reason more solidly by
_hypothetical_ than _categorical_ propositions.

[Sidenote: The same proposition diversely pronounced.]

12. But seeing every proposition may be, and uses to be, pronounced and
written in many forms, and we are obliged to speak in the same manner as
most men speak, yet they that learn philosophy from masters, had need to
take heed they be not deceived by the variety of expressions. And
therefore, whensoever they meet with any obscure proposition, they ought
to reduce it to its most simple and categorical form; in which the
copulative word _is_ must be expressed by itself, and not mingled in any
manner either with the subject or predicate, both which must be
separated and clearly distinguished one from another. For example, if
this proposition, _man can not sin_, be compared with this, _man cannot
sin_, their difference will easily appear if they be reduced to these,
_man is able not to sin_, and, _man is not able to sin_, where the
predicates are manifestly different. But they ought to do this silently
by themselves, or betwixt them and their masters only; for it will be
thought both ridiculous and absurd, for a man to use such language
publicly. Being therefore to speak of _equipollent_ propositions, I put
in the first place all those for _equipollent_, that may be reduced
purely to one and the same categorical proposition.

[Sidenote: Propositions that may be reduced to the same categorical
           proposition, are equipollent.]

13. Secondly, that which is categorical and necessary, is _equipollent_
to its hypothetical proposition; as this categorical, _a right-lined
triangle has its three angles equal to two right angles_, to this
hypothetical, _if any figure be a right-lined triangle, the three angles
of it are equal to two right angles_.

[Sidenote: Universal propositions converted by contradictory names, are
           equipollent.]

14. Also, any two universal propositions, of which the terms of the one
(that is, the subject and predicate) are to the terms of the other, and
their order inverted, as these, _every man is a living creature_, and
_every thing that is not a living-creature is not a man_, are
_equipollent_. For seeing _every man is a living creature_ is a true
proposition, the name _living creature_ contains the name _man_; but
they are both positive names, and therefore (by the last article of the
precedent chapter) the negative name _not man_, contains the negative
name _not living-creature_; wherefore _every thing that is not a
living-creature, is not a man_, is a true proposition. Likewise these,
_no man is a tree_, _no tree is a man_, are _equipollent_. For if it be
true that _tree_ is not the name of any _man_, then no one thing can be
signified by the two names _man_ and _tree_, wherefore _no tree is a
man_ is a true proposition. Also to this, _whatsoever is not a
living-creature is not a man_, where both the terms are negative, this
other proposition is equipollent, _only a living creature is a man_.

[Sidenote: Negative propositions are the same, whether the negation be
           before or after the copula.]

15. Fourthly, negative propositions, whether the particle of negation be
set after the copula as some nations do, or before it, as it is in Latin
and Greek, if the terms be the same, are equipollent: as, for example,
_man is not a tree_, and, _man is not-a-tree_, are equipollent, though
Aristotle deny it. Also these, _every man is not a tree_, and _no man is
a tree_, are equipollent, and that so manifestly, as it needs not be
demonstrated.

[Sidenote: Particular propositions simply converted, are equipollent.]

16. Lastly, all particular propositions that have their terms inverted,
as these, _some man is blind, some blind thing is a man_, are
equipollent; for either of the two names, is the name of some one and
the same man; and therefore in which soever of the two orders they be
connected, they signify the same truth.

[Sidenote: What are subaltern, contrary, subcontrary, and contradictory
           propositions.]

17. Of propositions that have the same terms, and are placed in the same
order, but varied either by quantity or quality, some are called
_subaltern_, others _contrary_, others _subcontrary_, and others
_contradictory_.

_Subaltern_, are universal and particular propositions of the same
quality; as, _every man is a living creature_, _some man is a living
creature_; or, _no man is wise_, _some man is not wise_. Of these, if
the universal be true, the particular will be true also.

_Contrary_, are universal propositions of different quality; as, _every
man is happy_, _no man is happy_. And of these, if one be true, the
other is false: also, they may both be false, as in the example given.

_Subcontrary_, are particular propositions of different quality; as,
_some man is learned_, _some man is not learned_; which cannot be both
false, but they may be both true.

_Contradictory_ are those that differ both in quantity and quality; as,
_every man is a living creature_, _some man is not a living-creature_;
which can neither be both true, nor both false.

[Sidenote: Consequence, what it is.]

18. A proposition is said to _follow_ from two other propositions, when
these being granted to be true, it cannot be denied but the other is
true also. For example, let these two propositions, _every man is a
living creature_, and, _every living creature is a body_, be supposed
true, that is, that _body_ is the name of _every living creature_, and
_living creature_ the name of _every man_. Seeing therefore, if these be
understood to be true, it cannot be understood that _body_ is not the
name of _every man_, that is, that _every man is a body_ is false, this
proposition will be said to _follow_ from those two, or to be
necessarily _inferred_ from them.

[Sidenote: Falsity cannot follow from truth.]

19. That a true proposition may follow from false propositions, may
happen sometimes; but false from true, never. For if these, _every man
is a stone_, and _every stone is a living creature_, (which are both
false) be granted to be true, it is granted also that _living creature_
is the name of _every stone_, and _stone_ of _every man_, that is, that
_living creature_ is the name of _every man_; that is to say, this
proposition _every man is a living creature_, is true, as it is indeed
true. Wherefore a true proposition may sometimes follow from false. But
if any two propositions be true, a false one can never follow from them.
For if true follow from false, for this reason only, that the false are
granted to be true, then truth from two truths granted will follow in
the same manner.

[Sidenote: How one proposition is the cause of another.]

20. Now, seeing none but a true proposition will follow from true, and
that the understanding of two propositions to be true, is the cause of
understanding that also to be true which is deduced from them; the two
antecedent propositions are commonly called the causes of the inferred
proposition, or conclusion. And from hence it is that logicians say, the
_premises_ are causes of the _conclusion_; which may pass, though it be
not properly spoken; for though understanding be the cause of
understanding, yet speech is not the cause of speech. But when they say,
the cause of the properties of any thing, is the thing itself, they
speak absurdly. For example, if a figure be propounded which is
triangular; seeing every triangle has all its angles together equal to
two right angles, from whence it follows that all the angles of that
figure are equal to two right angles, they say, for this reason, that
that figure is the cause of that equality. But seeing the figure does
not itself make its angles, and therefore cannot be said to be the
_efficient-cause_, they call it the _formal-cause_; whereas indeed it is
no cause at all; nor does the property of any figure follow the figure,
but has its being at the same time with it; only the knowledge of the
figure goes before the knowledge of the properties; and one knowledge is
truly the cause of another knowledge, namely the _efficient cause_.

And thus much concerning _proposition_; which in the progress of
philosophy is the first step, like the moving towards of one foot. By
the due addition of another step I shall proceed to _syllogism_, and
make a complete pace. Of which in the next chapter.


                                -------


                              CHAPTER IV.

                             OF SYLLOGISM.

1. The definition of syllogism.—2. In a syllogism there are but three
    terms.—3. Major, minor, and middle term; also major and minor
    proposition, what they are.—4. The middle term in every syllogism
    ought to be determined in both the propositions to one and the same
    thing.—5. From two particular propositions nothing can be
    concluded.—6. A syllogism is the collection of two propositions into
    one sum.—7. The figure of a syllogism, what it is.—8. What is in the
    mind answering to a syllogism.—9. The first indirect figure, how it
    is made.—10. The second indirect figure, how made.—11. How the third
    indirect figure is made.—12. There are many moods in every figure,
    but most of them useless in philosophy.—13. An hypothetical
    syllogism when equipollent to a categorical.


[Sidenote: Definition of syllogism.]

1. A SPEECH, consisting of three propositions, from two of which the
third follows, is called a SYLLOGISM; and that which follows is called
the _conclusion_; the other two _premises_. For example, this speech,
_every man is a living creature_, _every living creature is a body_,
therefore, _every man is a body_, is a _syllogism_, because the third
proposition follows from the two first; that is, if those be granted to
be true, this must also be granted to be true.

[Sidenote: In a syllogism there are but three terms.]

2. From two propositions which have not one term common, no conclusion
can follow; and therefore no _syllogism_ can be made of them. For let
any two premises, _a man is a living creature_, _a tree is a plant_, be
both of them true, yet because it cannot be collected from them that
_plant_ is the name of a _man_, or _man_ the name of a _plant_, it is
not necessary that this conclusion, _a man is a plant_, should be true.
Corollary: therefore, in the _premises_ of a _syllogism_ there can be
but three _terms_.

Besides, there can be no term in the _conclusion_, which was not in the
_premises_. For let any two premises be, _a man is a living creature_,
_a living creature is a body_, yet if any other term be put in the
conclusion, as _man is two-footed_; though it be true, it cannot follow
from the premises, because from them it cannot be collected, that the
name _two-footed_ belongs to a _man_; and therefore, again, in every
_syllogism_ there can be but three _terms_.

[Sidenote: Major, minor and middle term; also major and minor
           proposition, what they are.]

3. Of these terms, that which is the _predicate_ in the conclusion, is
commonly called the _major_; that which is the _subject_ in the
conclusion, the _minor_, and the other is the _middle term_; as in this
syllogism, _a man is a living creature_, _a living creature is a body_,
therefore, _a man is a body_, _body_ is the _major_, _man_ the _minor_,
and _living creature_ the _middle term_. Also of the premises, that in
which the _major term_ is found, is called the _major proposition_, and
that which has the _minor term_, the _minor proposition_.

[Sidenote: The middle term in every syllogism to be determined in both
           propositions to one and the same thing.]

4. If the middle term be not in both the premises determined to one and
the same singular thing, no conclusion will follow, nor syllogism be
made. For let the minor term be _man_, the middle term _living
creature_, and the major term _lion_; and let the premises be, _man is a
living creature_, _some living creature is a lion_, yet it will not
follow that _every_ or _any man is a lion_. By which it is manifest,
that in every syllogism, that proposition which has the middle term for
its _subject_, ought to be either _universal_ or _singular_, but not
_particular_ nor _indefinite_. For example, this syllogism, _every man
is a living creature_, _some living creature is four-footed_, therefore
_some man is four-footed_, is therefore faulty, because the middle term,
_living creature_, is in the first of the premises determined only to
_man_, for there the name of _living creature_ is given to man only, but
in the latter premise it may be understood of some other living creature
besides man. But if the latter premise had been _universal_, as here,
_every man is a living creature_, _every living creature is a body_,
therefore _every man is a body_, the syllogism had been true; for it
would have followed that _body_ had been the name of _every living
creature_, that is of _man_; that is to say, the conclusion _every man
is a body_ had been true. Likewise, when the middle term is a _singular_
name, a syllogism may be made, I say a true syllogism, though useless in
philosophy, as this, _some man is Socrates_, _Socrates is a
philosopher_, therefore, _some man is a philosopher_; for the premises
being granted, the conclusion cannot be denied.

[Sidenote: From two particular propositions nothing can be concluded.]

5. And therefore of two premises, in both which the middle term is
particular, a syllogism cannot be made; for whether the middle term be
the _subject_ in both the premises, or the _predicate_ in both, or the
_subject_ in one, and the _predicate_ in the other, it will not be
necessarily determined to the same thing. For let the premises be,

    _Some man is blind_,            In both which the middle
    _Some man is learned_,         } term is the _subject_,

it will not follow that _blind_ is the name of any _learned man_, or
_learned_ the name of any _blind man_, seeing the name _learned_ does
not contain the name _blind_, nor this that; and therefore it is not
necessary that both should be names of the same man. So from these
premises,

 _Every man is a living-creature_,   } In both which the middle
 _Every horse is a living-creature_, } term is the _predicate_,

nothing will follow. For seeing _living creature_ is in both of them
_indefinite_, which is equivalent to _particular_, and that _man_ may be
one kind of _living creature_, and _horse_ another kind, it is not
necessary that _man_ should be the name of _horse_, or _horse_ of _man_.
Or if the premises be,

  _Every man is a                  In one of which the
  living-creature_,                   } middle-term
  _Some living creature is
  four-footed_,

                                      } is the _subject_, and in the

                                      } other the _predicate_,

the conclusion will not follow, because the name _living creature_ being
not determined, it may in one of them be understood of _man_, in the
other of _not-man_.

[Sidenote: A syllogism is the collection of two propositions into one
           sum.]

6. Now it is manifest from what has been said, that a syllogism is
nothing but a collection of the sum of two propositions, joined together
by a common term, which is called the _middle term_. And as proposition
is the addition of two names, so syllogism is the adding together of
three.

[Sidenote: The figure of a syllogism what it is.]

7. Syllogisms are usually distinguished according to their diversity of
_figures_, that is, by the diverse position of the middle term. And
again in figure there is a distinction of certain _moods_, which consist
of the differences of propositions in _quantity_ and _quality_. The
first figure is that, in which the terms are placed one after another
according to their latitude of signification; in which order the _minor
term_ is first, the _middle term_ next, and the _major_ last; as, if the
minor term be _man_, the middle term, _living creature_, and the major
term, _body_, then, _man is a living-creature, is a body_, will be a
syllogism in the first figure: in which, _man is a living creature_ is
the minor proposition; the major, _living creature is a body_, and the
conclusion, or sum of both, _man is a body_. Now this figure is called
_direct_, because the terms stand in direct order; and it is varied by
_quantity_ and _quality_ into four _moods_: of which the first is that
wherein all the terms are _positive_, and the minor term _universal_, as
_every man is a living creature_, _every living creature is a body_: in
which all the propositions are affirmative, and universal. But if the
major term be a negative name, and the minor an universal name, the
_figure_ will be in the second _mood_, as, _every man is a living
creature_, _every living creature is not a tree_, in which the major
proposition and conclusion are both universal and negative. To these
two, are commonly added two more, by making the minor term particular.
Also it may happen that both the major and middle terms are negative
terms, and then there arises another _mood_, in which all the
propositions are negative, and yet the syllogism will be good; as, if
the minor term be _man_, the middle term _not a stone_, and the major
term _not a flint_, this syllogism, _no man is a stone_, _whatsoever is
not a stone is not a flint_, therefore, _no man is a flint_, is true,
though it consist of three negatives. But in philosophy, the profession
whereof is to establish universal rules concerning the properties of
things, seeing the difference betwixt negatives and affirmatives is only
this, that in the former the subject is affirmed by a negative name, and
by a positive in the latter, it is superfluous to consider any other
_mood_ in direct _figure_, besides that, in which all the propositions
are both universal and affirmative.

[Sidenote: What is in the mind answering to a syllogism.]

8. The thoughts in the mind answering to a direct syllogism, proceed in
this manner; first, there is conceived a phantasm of the thing named,
with that accident or quality thereof, for which it is in the minor
proposition called by that name which is the subject; next, the mind has
a phantasm of the same thing with that accident, or quality, for which
it hath the name, that in the same proposition is the predicate;
thirdly, the thought returns of the same thing as having that accident
in it, for which it is called by the name, that is the predicate of the
major proposition; and lastly, remembering that all those are the
accidents of one and the same thing, it concludes that those three names
are also names of one and the same thing; that is to say, the conclusion
is true. For example, when this syllogism is made, _man is a living
creature_, _a living creature is a body_, therefore, _man is a body_,
the mind conceives first an image of a man speaking or discoursing, and
remembers that that, which so appears, is called _man_; then it has the
image of the same man moving, and remembers that that, which appears so,
is called _living creature_; thirdly, it conceives an image of the same
man, as filling some place or space, and remembers that what appears so
is called _body_; and lastly, when it remembers that that thing, which
was extended, and moved and spake, was one and the same thing, it
concludes that the three names, _man_, _living creature_, and _body_,
are names of the same thing, and that therefore _man is a living
creature_ is a true proposition. From whence it is manifest, that living
creatures that have not the use of speech, have no conception or thought
in the mind, answering to a syllogism made of universal propositions;
seeing it is necessary to think not only of the thing, but also by turns
to remember the divers names, which for divers considerations thereof
are applied to the same.

[Sidenote: The first indirect figure how made.]

9. The rest of the figures arise either from the inflexion, or inversion
of the first or direct figure; which is done by changing the major, or
minor, or both the propositions, into converted propositions equipollent
to them.

From whence follow three other figures; of which, two are _inflected_,
and the third _inverted_. The first of these three is made by the
conversion of the major proposition. For let the minor, middle, and
major terms stand in direct order, thus, _man is a living creature_, _is
not a stone_, which is the first or direct figure; the inflection will
be by converting the major proposition in this manner, _man is a living
creature_, _a stone is not a living creature_; and this is the second
figure, or the first of the indirect figures; in which the conclusion
will be, _man is not a stone_. For (having shown in the last chapter,
art. 14, that universal propositions, converted by contradiction of the
terms, are equipollent) both those syllogisms conclude alike; so that if
the major be read (like Hebrew) backwards, thus, _a living creature is
not a stone_, it will be direct again, as it was before. In like manner
this direct syllogism, _man is not a tree_, _is not a pear-tree_, will
be made indirect by converting the major proposition (by contradiction
of the terms) into another equipollent to it, thus, _man is not a tree_,
_a pear-tree is a tree_; for the same conclusion will follow, _man is
not a pear-tree_.

But for the conversion of the direct figure into the first indirect
figure, the major term in the direct figure ought to be negative. For
though this direct, _man is a living creature_, _is a body_, be made
indirect, by converting the major proposition, thus,

                 _Man is a living creature_,
                 _Not a body is not a living creature_,
                 Therefore, _Every man is a body_;

Yet this conversion appears so obscure, that this mood is of no use at
all. By the conversion of the major proposition, it is manifest, that in
this figure, the middle term is always the predicate in both the
premises.

[Sidenote: Second indirect figure how made.]

10. The second indirect figure is made by converting the minor
proposition, so as that the middle term is the subject in both. But this
never concludes universally, and therefore is of no use in philosophy.
Nevertheless I will set down an example of it; by which this direct

                   _Every man is a living creature_,
                   _Every living creature is a body_,

by conversion of the minor proposition, will stand thus,

                   _Some living creature is a man_,
                   _Every living creature is a body_,
                   Therefore, _Some man is a body_.

For _every man is a living creature_ cannot be converted into this,
_every living creature is a man_: and therefore if this syllogism be
restored to its direct form, the minor proposition will be _some man is
a living creature_, and consequently the conclusion will be _some man is
a body_, seeing the minor term _man_, which is the subject in the
conclusion, is a particular name.

[Sidenote: How the third indirect figure is made.]

11. The third indirect or inverted figure, is made by the conversion of
both the premises. For example, this direct syllogism,

                _Every man is a living creature_,
                _Every living creature is not a stone_,
                Therefore, _Every man is not a stone_,

being inverted, will stand thus,

          _Every stone is not a living creature_,
          _Whatsoever is not a living creature, is not a man_,
          Therefore, _Every stone is not a man_;

which conclusion is the converse of the direct conclusion, and
equipollent to the same.

The figures, therefore, of syllogisms, if they be numbered by the
diverse situation of the middle term only, are but three; in the first
whereof, the middle term has the middle place; in the second, the last;
and in the third, the first place. But if they be numbered according to
the situation of the terms simply, they are four; for the first may be
distinguished again into two, namely, into direct and inverted. From
whence it is evident, that the controversy among logicians concerning
the fourth figure, is a mere λογόμαχια, or contention about the name
thereof; for, as for the thing itself, it is plain that the situation of
the terms (not considering the quantity or quality by which the moods
are distinguished) makes four differences of syllogisms, which may be
called figures, or have any other name at pleasure.

[Sidenote: There are many moods in every figure, but most of them
           useless in philosophy.]

12. In every one of these figures there are many moods, which are made
by varying the premises according to all the differences they are
capable of, by quantity and quality; as namely, in the direct figure
there are six moods; in the first indirect figure, four; in the second,
fourteen; and in the third, eighteen. But because from the direct figure
I rejected as superfluous all moods besides that which consists of
universal propositions, and whose minor proposition is affirmative, I
do, together with it, reject the moods of the rest of the figures which
are made by conversion of the premises in the direct figure.

[Sidenote: An hypothetical syllogism when equipollent to a categorical.]

13. As it was showed before, that in necessary propositions a
categorical and hypothetical proposition are equipollent; so likewise it
is manifest that a categorical and hypothetical syllogism are
equivalent. For every categorical syllogism, as this,

                   _Every man is a living creature_,
                   _Every living creature is a body_,
                   Therefore, _Every man is a body_,

is of equal force with this hypothetical syllogism:

      _If any thing be a man, the same is also a living creature_,
      _If any thing be a living creature, the same is a body_,
      Therefore, _If any thing be a man, the same is a body_.

In like manner, this categorical syllogism in an indirect figure,

                   _No stone is a living creature_,
                   _Every man is a living creature_,
                   Therefore, _No man is a stone_,
                   Or, _No stone is a man_,

is equivalent to this hypothetical syllogism:

     _If any thing be a man, the same is a living creature_,
     _If any thing be a stone, the same is not a living creature_,
     Therefore, _If any thing be a stone, the same is not a man_,
     Or, _If any thing be a man, the same is not a stone_.

And thus much seems sufficient for the nature of syllogisms; (for the
doctrine of moods and figures is clearly delivered by others that have
written largely and profitably of the same). Nor are precepts so
necessary as practice for the attaining of true ratiocination; and they
that study the demonstrations of mathematicians, will sooner learn true
logic, than they that spend time in reading the rules of syllogizing
which logicians have made; no otherwise than little children learn to
go, not by precepts, but by exercising their feet. This, therefore, may
serve for the first pace in the way to Philosophy.

In the next place I shall speak of the faults and errors into which men
that reason unwarily are apt to fall; and of their kinds and causes.


                                -------


                               CHAPTER V.

                   OF ERRING, FALSITY, AND CAPTIONS.

1. Erring and falsity how they differ. Error of the mind by itself
    without the use of words, how it happens.—2. A sevenfold incoherency
    of names, every one of which makes always a false proposition.—3.
    Examples of the first manner of incoherency.—4. Of the second.—5. Of
    the third.—6. Of the fourth.—7. Of the fifth.—8. Of the sixth.—9. Of
    the seventh.—10. Falsity of propositions detected by resolving the
    terms with definitions continued till they come to simple names, or
    names that are the most general of their kind.—11. Of the fault of a
    syllogism consisting in the implication of the terms with the
    copula.—12. Of the fault which consists in equivocation.—13.
    Sophistical captions are oftener faulty in the matter than in the
    form of syllogisms.


[Sidenote: Erring & falsity how they differ. Error of the mind by
           itself, without the use of words, how it happens.]

1. Men are subject to _err_ not only in affirming and denying, but also
in perception, and in silent cogitation. In affirming and denying, when
they call any thing by a name, which is not the name thereof; as if from
seeing the sun first by reflection in water, and afterwards again
directly in the firmament, we should to both those appearances give the
name of sun, and say there are two suns; which none but men can do, for
no other living creatures have the use of names. This kind of error only
deserves the name of _falsity_, as arising, not from sense, nor from the
things themselves, but from pronouncing rashly; for names have their
constitution, not from the species of things, but from the will and
consent of men. And hence it comes to pass, that men pronounce
_falsely_, by their own negligence, in departing from such appellations
of things as are agreed upon, and are not deceived neither by the
things, nor by the sense; for they do not perceive that the thing they
see is called sun, but they give it that name from their own will and
agreement. Tacit errors, or the errors of sense and cogitation, are
made, by passing from one imagination to the imagination of another
different thing; or by feigning that to be past, or future, which never
was, nor ever shall be; as when, by seeing the image of the sun in
water, we imagine the sun itself to be there; or by seeing swords, that
there has been or shall be fighting, because it uses to be so for the
most part; or when from promises we feign the mind of the promiser to be
such and such; or lastly, when from any sign we vainly imagine something
to be signified, which is not. And errors of this sort are common to all
things that have sense; and yet the deception proceeds neither from our
senses, nor from the things we perceive; but from ourselves while we
feign such things as are but mere images to be something more than
images. But neither things, nor imaginations of things, can be said to
be false, seeing they are truly what they are; nor do they, as signs,
promise any thing which they do not perform; for they indeed do not
promise at all, but we from them; nor do the clouds, but we, from seeing
the clouds, say it shall rain. The best way, therefore, to free
ourselves from such errors as arise from natural signs, is first of all,
before we begin to reason concerning such conjectural things, to suppose
ourselves ignorant, and then to make use of our ratiocination; for these
errors proceed from the want of ratiocination; whereas, errors which
consist in affirmation and negation, (that is, the falsity of
propositions) proceed only from reasoning amiss. Of these, therefore, as
repugnant to philosophy, I will speak principally.

[Sidenote: A sevenfold incoherency of names, all of which make always a
           false proposition.]

2. Errors which happen in reasoning, that is, in syllogizing, consist
either in the falsity of the premises, or of the inference. In the first
of these cases, a syllogism is said to be faulty in the _matter_ of it;
and in the second case, in the _form_. I will first consider the matter,
namely, how many ways a proposition may be false; and next the form, and
how it comes to pass, that when the premises are true, the inference is,
notwithstanding, false.

Seeing, therefore, that proposition only is true, (chap, III, art. 7) in
which are copulated two names of one and the same thing; and that always
false, in which names of different things are copulated, look how many
ways names of different things may be copulated, and so many ways a
false proposition may be made.

Now, all things to which we give names, may be reduced to these four
kinds, namely, _bodies_, _accidents_, _phantasms_, and _names_
themselves; and therefore, in every true proposition, it is necessary
that the names copulated, be both of them names of _bodies_, or both
names of _accidents_, or both names of _phantasms_, or both names of
_names_. For names otherwise copulated are incoherent, and constitute a
false proposition. It may happen, also, that the name of a _body_, of an
_accident_, or of a _phantasm_, may be copulated with the name of a
_speech_. So that copulated names may be incoherent seven manner of
ways.

 1. If the name of a Body       }            { the name of an Accident.
 2. If the name of a Body       }            { the name of a Phantasm.
 3. If the name of a Body       }     be     { the name of a Name.
 4. If the name of an Accident  } copulated  { the name of a Phantasm.
 5. If the name of an Accident  }    with    { the name of a Name.
 6. If the name of a Phantasm   }            { the name of a Name.
 7. If the name of a Body,      }            { the name of a Speech.
    Accident, or Phantasm       }            {

Of all which I will give some examples.

[Sidenote: Examples of the first manner of incoherency.]

3. After the first of these ways propositions are false, when abstract
names are copulated with concrete names; as (in Latin and Greek) _esse
est ens_, _essentia est ens_, τὸ τί ἦν ειναὶ (_i._); _quidditas est
ens_, and many the like, which are found in Aristotle's _Metaphysics_.
Also, the _understanding worketh_, the _understanding understandeth_,
the _sight seeth_; a _body_ is _magnitude_, a _body_ is _quantity_, a
_body_ is _extension_; to _be a man_ is a _man_, _whiteness_ is a _white
thing_, &c.; which is as if one should say, the _runner_ is the
_running_, or the _walk walketh_. Moreover, _essence_ is _separated_,
_substance_ is _abstracted_: and others like these, or derived from
these, (with which common philosophy abounds.) For seeing no _subject_
of an _accident_ (that is, no _body_) is an _accident_: no name of an
_accident_ ought to be given to a _body_, nor of a _body_ to an
_accident_.

[Sidenote: The second.]

4. False, in the second manner, are such propositions as these; a _ghost
is a body_, or a _spirit_, that is, a thin body; _sensible species fly
up and down in the air_, or _are moved hither and thither_, which is
proper to bodies; also, a _shadow is moved_, or _is a body_; _light is
moved_, or _is a body_; _colour is the object of sight_, _sound of
hearing_; _space_ or _place is extended_; and innumerable others of this
kind. For seeing ghosts, sensible species, a shadow, light, colour,
sound, space, &c. appear to us no less sleeping than waking, they cannot
be things without us, but only phantasms of the mind that imagines them;
and therefore the names of these, copulated with the names of bodies,
cannot constitute a true proposition.

[Sidenote: The third.]

5. False propositions of the third kind, are such as these; _genus est
ens_, _universale est ens_, _ens de ente prædicatur_. For _genus_, and
_universale_, and _predicare_, are names of names, and not of things.
Also, _number is infinite_, is a false proposition; for no number can be
infinite, but only the word _number_ is then called an indefinite name
when there is no determined number answering to it in the mind.

[Sidenote: The fourth.]

6. To the fourth kind belong such false propositions as these, _an
object is of such magnitude or figure as appears to the beholders_;
_colour, light, sound, are in the object_; and the like. For the same
object appears sometimes greater, sometimes lesser, sometimes square,
sometimes round, according to the diversity of the distance and medium;
but the true magnitude and figure of the thing seen is always one and
the same; so that the magnitude and figure which appears, is not the
true magnitude and figure of the object, nor anything but phantasm; and
therefore, in such propositions as these, the names of accidents are
copulated with the names of phantasms.

[Sidenote: The fifth.]

7. Propositions are false in the fifth manner, when it is said that _the
definition is the essence of a thing_; _whiteness_, or some other
accident, _is the genus, or universal_. For definition is not the
essence of any thing, but a speech signifying what we conceive of the
essence thereof; and so also not whiteness itself, but the word
whiteness, is a genus, or an universal name.

[Sidenote: The sixth.]

8. In the sixth manner they err, that say the _idea of anything is
universal_; as if there could be in the mind an image of a man, which
were not the image of some one man, but a man simply, which is
impossible; for every idea is one, and of one thing; but they are
deceived in this, that they put the _name_ of the thing for the _idea_
thereof.

[Sidenote: The seventh.]

9. They err in the seventh manner, that make this distinction between
things that have being, that some of them _exist by themselves_, others
by _accident_; namely, because _Socrates is a man_ is a necessary
proposition, and _Socrates is a musician_ a contingent proposition,
therefore they say some things exist necessarily or by themselves,
others contingently or by accident; whereby, seeing _necessary_,
_contingent_, _by itself_, _by accident_, are not names of things, but
of propositions, they that say _any thing that has being, exists by
accident_, copulate the name of a proposition with the name of a thing.
In the same manner also, they err, which place some ideas in the
understanding, others in the fancy; as if from the understanding of this
proposition, _man is a living creature_, we had one idea or image of a
man derived from sense to the memory, and another to the understanding;
wherein that which deceives them is this, that they think one idea
should be answerable to a name, another to a proposition, which is
false; for proposition signifies only the order of those things one
after another, which we observe in the same idea of man; so that this
proposition, _man is a living creature_ raises but one idea in us,
though in that idea we consider that first, for which he is called man,
and next that, for which he is called living creature. The falsities of
propositions in all these several manners, is to be discovered by the
definitions of the copulated names.

[Sidenote: Falsity of propositions detected by resolving the terms with
           definitions.]

10. But when names of bodies are copulated with names of bodies, names
of accidents with names of accidents, names of names with names of
names, and names of phantasms, with names of phantasms, if we,
nevertheless, remain still doubtful whether such propositions are true,
we ought then in the first place to find out the definition of both
those names, and again the definitions of such names as are in the
former definition, and so proceed by a continual resolution till we come
to a simple name, that is, to the most general or most universal name of
that kind; and if after all this, the truth or falsity thereof be not
evident, we must search it out by philosophy, and ratiocination,
beginning from definitions. For every proposition, universally true, is
either a definition, or part of a definition, or the evidence of it
depends upon definitions.

[Sidenote: Of the fault of a syllogism consisting in implication of the
           terms with the copula.]

11. That fault of a syllogism which lies hid in the form thereof, will
always be found either in the implication of the copula with one of the
terms, or in the equivocation of some word; and in either of these ways
there will be four terms, which (as I have shewn) cannot stand in a true
syllogism. Now the implication of the copula with either term, is easily
detected by reducing the propositions to plain and clear predication; as
(for example) if any man should argue thus,

               _The hand toucheth the pen_,
               _The pen toucheth the paper_,
               Therefore, _The hand toucheth the paper_;

the fallacy will easily appear by reducing it, thus:

             _The hand, is, touching the pen_,
             _The pen, is, touching the paper_,
             Therefore, _The hand, is, touching the paper_;

where there are manifestly these four terms, _the hand_, _touching the
pen_, _the pen_, and _touching the the paper_. But the danger of being
deceived by sophisms of this kind, does not seem to be so great, as that
I need insist longer upon them.

[Sidenote: Of the fault which consists in equivocation.]

12. And though there may be fallacy in equivocal terms, yet in those
that be manifestly such, there is none at all; nor in metaphors, for
they profess the transferring of names from one thing to another.
Nevertheless, sometimes equivocals (and those not very obscure) may
deceive; as in this argumentation:--_It belongs to metaphysics to treat
of principles; but the first principle of all, is, that the same thing
cannot both exist and not exist at the same time; and therefore it
belongs to metaphysics to treat whether the same thing may both exist
and not exist at the same time_; where the fallacy lies in the
equivocation of the word principle; for whereas Aristotle in the
beginning of his _Metaphysics_, says, that _the treating of principles
belongs to primary science_, he understands by principles, causes of
things, and certain existences which he calls primary; but where he says
_a primary proposition is a principle_, by principle, there, he means
the beginning and cause of knowledge, that is, the understanding of
words, which, if any man want, he is incapable of learning.

[Sidenote: Sophistical captions are oftener faulty in the matter than in
           the form of syllogisms.]

13. But the _captions_ of sophists and sceptics, by which they were
wont, of old, to deride and oppose truth, were faulty for the most part,
not in the form, but in the matter of syllogism; and they deceived not
others oftener than they were themselves deceived. For the force of that
famous argument of Zeno against motion, consisted in this proposition,
_whatsoever may be divided into parts, infinite in number, the same is
infinite_; which he, without doubt, thought to be true, yet nevertheless
is false. For to be divided into infinite parts, is nothing else but to
be divided into as many parts as any man will. But it is not necessary
that a line should have parts infinite in number, or be infinite,
because I can divide and subdivide it as often as I please; for how many
parts soever I make, yet their number is finite; but because he that
says parts, simply, without adding how many, does not limit any number,
but leaves it to the determination of the hearer, therefore we say
commonly, a line may be divided infinitely; which cannot be true in any
other sense.

[Sidenote: Conclusion.]

And thus much may suffice concerning syllogism, which is, as it were,
the first pace towards philosophy; in which I have said as much as is
necessary to teach any man from whence all true argumentation has its
force. And to enlarge this treatise with all that may be heaped
together, would be as superfluous, as if one should (as I said before)
give a young child precepts for the teaching of him to go; for the art
of reasoning is not so well learned by precepts as by practice, and by
the reading of those books in which the conclusions are all made by
severe demonstration. And so I pass on to the way of philosophy, that
is, to the method of study.


                                -------


                              CHAPTER VI.

                               OF METHOD.

1. Method and science defined.—2. It is more easily known concerning
    singular, than universal things, that they are; and contrarily, it
    is more easily known concerning universal, than singular things, why
    they are, or what are their causes.—3. What it is philosophers seek
    to know.—4. The first part, by which principles are found out, is
    purely analytical.—5. The highest causes, and most universal in
    every kind, are known by themselves.—6. Method from principles found
    out, tending to science simply, what it is.—7. That method of civil
    and natural science, which proceeds from sense to principles, is
    analytical; and again, that, which begins at principles, is
    synthetical.—8. The method of searching out, whether any thing
    propounded be matter or accident.—9. The method of seeking whether
    any accident be in this, or in that subject.—10. The method of
    searching after the cause of any effect propounded.—11. Words serve
    to invention, as marks; to demonstration, as signs.—12. The method
    of demonstration is synthetical.—13. Definitions only are primary
    and universal propositions.—14. The nature and definition of a
    definition.—15. The properties of a definition.—16. The nature of a
    demonstration.—17. The properties of a demonstration, and order of
    things to be demonstrated.—18. The faults of a demonstration.—19.
    Why the analytical method of geometricians cannot be treated of in
    this place.


[Sidenote: Method and science defined.]

1. For the understanding of _method_, it will be necessary for me to
repeat the definition of philosophy, delivered above (Chap. I, art. 2.)
in this manner, _Philosophy is the knowledge we acquire, by true
ratiocination, of appearances, or apparent effects, from the knowledge
we have of some possible production or generation of the same; and of
such production, as has been or may be, from the knowledge we have of
the effects_. METHOD, therefore, in the study of philosophy, _is the
shortest way of finding out effects by their known causes, or of causes
by their known effects_. But we are then said to know any effect, when
we know _that there be causes of the same_, and _in what subject those
causes are_, and _in what subject they produce that effect_, and _in
what manner they work the same_. And this is the science of causes, or,
as they call it, of the διότι. All other science, which is called the
ὄτι, is either perception by sense, or the imagination, or memory
remaining after such perception.

The first beginnings, therefore, of knowledge, are the phantasms of
sense and imagination; and that there be such phantasms we know well
enough by nature; but to know why they be, or from what causes they
proceed, is the work of ratiocination; which consists (as is said above,
in the 1st Chapter, Art. 2) in _composition_, and _division_ or
_resolution_. There is therefore no method, by which we find out the
causes of things, but is either _compositive_ or _resolutive_, or
_partly compositive_, and _partly resolutive_. And the resolutive is
commonly called _analytical_ method, as the compositive is called
_synthetical_.

[Sidenote: It is easier known concerning singular than universal things,
           that they are; and contrarily it is easier known concerning
           universal than singular things, why they are, or what are
           their causes.]

2. It is common to all sorts of method, to proceed from known things to
unknown; and this is manifest from the cited definition of philosophy.
But in knowledge by sense, the whole object is more known, than any part
thereof; as when we see a man, the conception or whole idea of that man
is first or more known, than the particular ideas of his being
_figurate_, _animate_, and _rational_; that is, we first see the whole
man, and take notice of his being, before we observe in him those other
particulars. And therefore in any knowledge of the ὅτι, or that any
thing _is_, the beginning of our search is from the whole idea; and
contrarily, in our knowledge of the διότι, or of the causes of any
thing, that is, in the sciences, we have more knowledge of the causes of
the parts than of the whole. For the cause of the whole is compounded of
the causes of the parts; but it is necessary that we know the things
that are to be compounded, before we can know the whole compound. Now,
by parts, I do not here mean parts of the thing itself, but parts of its
nature; as, by the parts of man, I do not understand his head, his
shoulders, his arms, &c. but his figure, quantity, motion, sense,
reason, and the like; which accidents being compounded or put together,
constitute the whole nature of man, but not the man himself. And this is
the meaning of that common saying, namely, that some things are more
known to us, others more known to nature; for I do not think that they,
which so distinguish, mean that something is known to nature, which is
known to no man; and therefore, by those things, that are more known to
us, we are to understand things we take notice of by our senses, and, by
more known to nature, those we acquire the knowledge of by reason; for
in this sense it is, that the _whole_, that is, those things that have
universal names, (which, for brevity's sake, I call _universal_) are
more known to us than the _parts_, that is, such things as have names
less universal, (which I therefore call _singular_); and the causes of
the parts are more known to nature than the cause of the whole; that is,
universals than singulars.

[Sidenote: What it is philosophers seek to know.]

3. In the study of philosophy, men search after science either simply or
indefinitely; that is, to know as much as they can, without propounding
to themselves any limited question; or they enquire into the cause of
some determined appearance, or endeavour to find out the certainty of
something in question, as what is the cause of _light_, of _heat_, of
_gravity_, of a _figure_ propounded, and the like; or in what _subject_
any propounded _accident_ is inherent; or what may conduce most to the
_generation_ of some propounded _effect_ from many _accidents_; or in
what manner particular causes ought to be compounded for the production
of some certain effect. Now, according to this variety of things in
question, sometimes the _analytical method_ is to be used, and sometimes
the _synthetical_.

[Sidenote: The first part, by which principles are discovered, is purely
           analytical.]

4. But to those that search after science indefinitely, which consists
in the knowledge of the causes of all things, as far forth as it may be
attained, (and the causes of singular things are compounded of the
causes of universal or simple things) it is necessary that they know the
causes of universal things, or of such accidents as are common to all
bodies, that is, to all matter, before they can know the causes of
singular things, that is, of those accidents by which one thing is
distinguished from another. And, again, they must know what those
universal things are, before they can know their causes. Moreover,
seeing universal things are contained in the nature of singular things,
the knowledge of them is to be acquired by reason, that is, by
resolution. For example, if there be propounded a conception or _idea_
of some singular thing, as of a _square_, this square is to be resolved
into a _plain, terminated with a certain number of equal and straight
lines and right angles_. For by this resolution we have these things
universal or agreeable to all matter, namely, _line_, _plain_, (which
contains _superficies_) _terminated_, _angle_, _straightness_,
_rectitude_, and _equality_; and if we can find out the causes of these,
we may compound them altogether into the cause of a square. Again, if
any man propound to himself the conception of _gold_, he may, by
resolving, come to the ideas of _solid_, _visible_, _heavy_, (that is,
tending to the centre of the earth, or downwards) and many other more
universal than gold itself; and these he may resolve again, till he come
to such things as are most universal. And in this manner, by resolving
continually, we may come to know what those things are, whose causes
being first known severally, and afterwards compounded, bring us to the
knowledge of singular things. I conclude, therefore, that the method of
attaining to the universal knowledge of things, is purely _analytical_.

[Sidenote: The highest causes, and most universal in every kind, are
           known by themselves.]

5. But the causes of universal things (of those, at least, that have any
cause) are manifest of themselves, or (as they say commonly) known to
nature; so that they need no method at all; for they have all but one
universal cause, which is motion. For the variety of all figures arises
out of the variety of those motions by which they are made; and motion
cannot be understood to have any other cause besides motion; nor has the
variety of those things we perceive by sense, as of _colours_, _sounds_,
_savours_, &c. any other cause than motion, residing partly in the
objects that work upon our senses, and partly in ourselves, in such
manner, as that it is manifestly some kind of motion, though we cannot,
without ratiocination, come to know what kind. For though many cannot
understand till it be in some sort demonstrated to them, that all
mutation consists in motion; yet this happens not from any obscurity in
the thing itself, (for it is not intelligible that anything can depart
either from rest, or from the motion it has, except by motion), but
either by having their natural discourse corrupted with former opinions
received from their masters, or else for this, that they do not at all
bend their mind to the enquiring out of truth.

[Sidenote: Method from principles found out, tending to science simply,
           what it is.]

6. By the knowledge therefore of universals, and of their causes (which
are the first principles by which we know the διότι of things) we have
in the first place their definitions, (which are nothing but the
explication of our simple conceptions.) For example, he that has a true
conception of _place_, cannot be ignorant of this definition, _place is
that space which is possessed or filled adequately by some body_; and
so, he that conceives _motion_ aright, cannot but know that _motion is
the privation of one place, and the acquisition of another_. In the next
place, we have their generations or descriptions; as (for example) that
_a line is made by the motion of a point, superficies by the motion of a
line_, and _one motion by another motion_, &c. It remains, that we
enquire what motion begets such and such effects; as, what motion makes
a straight line, and what a circular; what motion thrusts, what draws,
and by what way; what makes a thing which is seen or heard, to be seen
or heard sometimes in one manner, sometimes in another. Now the method
of this kind of enquiry, is _compositive_. For first we are to observe
what effect a body moved produceth, when we consider nothing in it
besides its motion; and we see presently that this makes a line, or
length; next, what the motion of a long body produces, which we find to
be superficies; and so forwards, till we see what the effects of simple
motion are; and then, in like manner, we are to observe what proceeds
from the addition, multiplication, subtraction, and division, of these
motions, and what effects, what figures, and what properties, they
produce; from which kind of contemplation sprung that part of philosophy
which is called _geometry_.

From this consideration of what is produced by simple motion, we are to
pass to the consideration of what effects one body moved worketh upon
another; and because there may be motion in all the several parts of a
body, yet so as that the whole body remain still in the same place, we
must enquire first, what motion causeth such and such motion in the
whole, that is, when one body invades another body which is either at
rest or in motion, what way, and with what swiftness, the invaded body
shall move; and, again, what motion this second body will generate in a
third, and so forwards. From which contemplation shall be drawn that
part of philosophy which treats of motion.

In the third place we must proceed to the enquiry of such effects as are
made by the motion of the parts of any body, as, how it comes to pass,
that things when they are the same, yet seem not to be the same, but
changed. And here the things we search after are sensible qualities,
such as _light_, _colour_, _transparency_, _opacity_, _sound_, _odour_,
_savour_, _heat_, _cold_, and the like; which because they cannot be
known till we know the causes of sense itself, therefore the
consideration of the causes of _seeing_, _hearing_, _smelling_,
_tasting_, and _touching_, belongs to this third place; and all those
qualities and changes, above mentioned, are to be referred to the fourth
place; which two considerations comprehend that part of philosophy which
is called _physics_. And in these four parts is contained whatsoever in
natural philosophy may be explicated by demonstration, properly so
called. For if a cause were to be rendered of natural appearances in
special, as, what are the motions and influences of the heavenly bodies,
and of their parts, the reason hereof must either be drawn from the
parts of the sciences above mentioned, or no reason at all will be
given, but all left to uncertain conjecture.

After _physics_ we must come to _moral philosophy_; in which we are to
consider the motions of the mind, namely, _appetite_, _aversion_,
_love_, _benevolence_, _hope_, _fear_, _anger_, _emulation_, _envy_,
_&c._; what causes they have, and of what they be causes. And the reason
why these are to be considered after _physics_ is, that they have their
causes in sense and imagination, which are the subject of _physical_
contemplation. Also the reason, why all these things are to be searched
after in the order above-said, is, that physics cannot be understood,
except we know first what motions are in the smallest parts of bodies;
nor such motion of parts, till we know what it is that makes another
body move; nor this, till we know what simple motion will effect. And
because all appearance of things to sense is determined, and made to be
of such and such quality and quantity by compounded motions, every one
of which has a certain degree of velocity, and a certain and determined
way; therefore, in the first place, we are to search out the ways of
motion simply (in which geometry consists); next the ways of such
generated motions as are manifest; and, lastly, the ways of internal and
invisible motions (which is the enquiry of natural philosophers). And,
therefore, they that study natural philosophy, study in vain, except
they begin at geometry; and such writers or disputers thereof, as are
ignorant of geometry, do but make their readers and hearers lose their
time.

[Sidenote: That method of civil and natural science, proceeding from
           sense to principles, is analytical; and again, that which
           begins at principles is synthetical.]

7. _Civil_ and _moral philosophy_ do not so adhere to one another, but
that they may be severed. For the causes of the motions of the mind are
known, not only by ratiocination, but also by the experience of every
man that takes the pains to observe those motions within himself. And,
therefore, not only they that have attained the knowledge of the
passions and perturbations of the mind, by the _synthetical method_, and
from the very first principles of philosophy, may by proceeding in the
same way, come to the causes and necessity of constituting
commonwealths, and to get the knowledge of what is natural right, and
what are civil duties; and, in every kind of government, what are the
rights of the commonwealth, and all other knowledge appertaining to
civil philosophy; for this reason, that the principles of the politics
consist in the knowledge of the motions of the mind, and the knowledge
of these motions from the knowledge of sense and imagination; but even
they also that have not learned the first part of philosophy, namely,
_geometry_ and _physics_, may, notwithstanding, attain the principles of
civil philosophy, by the _analytical method_. For if a question be
propounded, as, _whether such an action be just or unjust_; if that
_unjust_ be resolved into _fact against law_, and that notion _law_ into
the _command_ of him or them that have _coercive power_; and that
_power_ be derived from the _wills_ of men that constitute such power,
to the end they may live in peace, they may at last come to this, that
the appetites of men and the passions of their minds are such, that,
unless they be restrained by some power, they will always be making war
upon one another; which may be known to be so by any man's experience,
that will but examine his own mind. And, therefore, from hence he may
proceed, by compounding, to the determination of the justice or
injustice of any propounded action. So that it is manifest, by what has
been said, that the method of philosophy, to such as seek science
simply, without propounding to themselves the solution of any particular
question, is partly analytical, and partly synthetical; namely, that
which proceeds from sense to the invention of principles, analytical;
and the rest synthetical.

[Sidenote: The method of searching out, whether anything propounded be
           matter or accident.]

8. To those that seek the cause of some certain and propounded
appearance or effect, it happens, sometimes, that they know not whether
the thing, whose cause is sought after, be matter or body, or some
accident of a body. For though in geometry, when the cause is sought of
magnitude, or proportion, or figure, it be certainly known that these
things, namely magnitude, proportion, and figure, are accidents; yet in
natural philosophy, where all questions are concerning the causes of the
phantasms of sensible things, it is not so easy to discern between the
things themselves, from which those phantasms proceed, and the
appearances of those things to the sense; which have deceived many,
especially when the phantasms have been made by light. For example, a
man that looks upon the sun, has a certain shining idea of the magnitude
of about a foot over, and this he calls the sun, though he know the sun
to be truly a great deal bigger; and, in like manner, the phantasm of
the same thing appears sometimes round, by being seen afar off, and
sometimes square, by being nearer. Whereupon it may well be doubted,
whether that phantasm be matter, or some body natural, or only some
accident of a body; in the examination of which doubt we may use this
method. The properties of matter and accidents already found out by us,
by the synthetical method, from their definitions, are to be compared
with the idea we have before us; and if it agree with the properties of
matter or body, then it is a body; otherwise it is an accident. Seeing,
therefore, matter cannot by any endeavour of ours be either made or
destroyed, or increased, or diminished, or moved out of its place,
whereas that idea appears, vanishes, is increased and diminished, and
moved hither and thither at pleasure; we may certainly conclude that it
is not a body, but an accident only. And this method is _synthetical_.

[Sidenote: The method of seeking whether any accident be in this or in
           that subject.]

9. But if there be a doubt made concerning the subject of any known
accident (for this may be doubted sometimes, as in the precedent
example, doubt may be made in what subject that splendour and apparent
magnitude of the sun is), then our enquiry must proceed in this manner.
First, matter in general must be divided into parts, as, into object,
medium, and the sentient itself, or such other parts as seem most
conformable to the thing propounded. Next, these parts are severally to
be examined how they agree with the definition of the subject; and such
of them as are not capable of that accident are to be rejected. For
example, if by any true ratiocination the sun be found to be greater
than its apparent magnitude, then that magnitude is not in the sun; if
the sun be in one determined straight line, and one determined distance,
and the magnitude and splendour be seen in more lines and distances than
one, as it is in reflection or refraction, then neither that splendour
nor apparent magnitude are in the sun itself, and, therefore, the body
of the sun cannot be the subject of that splendour and magnitude. And
for the same reasons the air and other parts will be rejected, till at
last nothing remain which can be the subject of that splendour and
magnitude but the sentient itself. And this method, in regard the
subject is divided into parts, is analytical; and in regard the
properties, both of the subject and accident, are compared with the
accident concerning whose subject the enquiry is made, it is
synthetical.

[Sidenote: Method of searching for the cause of any effect, propounded.]

10. But when we seek after the cause of any propounded effect, we must
in the first place get into our mind an exact notion or idea of that
which we call cause, namely, that _a cause is the sum or aggregate of
all such accidents, both in the agents and the patient, as concur to the
producing of the effect propounded; all which existing together, it
cannot be understood but that the effect existeth with them; or that it
can possibly exist if any one of them be absent_. This being known, in
the next place we must examine singly every accident that accompanies or
precedes the effect, as far forth as it seems to conduce in any manner
to the production of the same, and see whether the propounded effect may
be conceived to exist, without the existence of any of those accidents;
and by this means separate such accidents, as do not concur, from such
as concur to produce the said effect; which being done, we are to put
together the concurring accidents, and consider whether we can possibly
conceive, that when these are all present, the effect propounded will
not follow; and if it be evident that the effect will follow, then that
aggregate of accidents is the entire cause, otherwise not; but we must
still search out and put together other accidents. For example, if the
cause of light be propounded to be sought out; first, we examine things
without us, and find that whensoever light appears, there is some
principal object, as it were the fountain of light, without which we
cannot have any perception of light; and, therefore, the concurrence of
that object is necessary to the generation of light. Next we consider
the medium, and find, that unless it be disposed in a certain manner,
namely, that it be transparent, though the object remain the same, yet
the effect will not follow; and, therefore, the concurrence of
transparency is also necessary to the generation of light. Thirdly, we
observe our own body, and find that by the indisposition of the eyes,
the brain, the nerves, and the heart, that is, by obstructions,
stupidity, and debility, we are deprived of light, so that a fitting
disposition of the organs to receive impressions from without is
likewise a necessary part of the cause of light. Again, of all the
accidents inherent in the object, there is none that can conduce to the
effecting of light, but only action (or a certain motion), which cannot
be conceived to be wanting, whensoever the effect is present; for, that
anything may shine, it is not requisite that it be of such or such
magnitude or figure, or that the whole body of it be moved out of the
place it is in (unless it may perhaps be said, that in the sun, or other
body, that which causes light is the light it hath in itself; which yet
is but a trifling exception, seeing nothing is meant thereby but the
cause of light; as if any man should say that the cause of light is that
in the sun which produceth it); it remains, therefore, that the action,
by which light is generated, is motion only in the parts of the object.
Which being understood, we may easily conceive what it is the medium
contributes, namely, the continuation of that motion to the eye; and,
lastly, what the eye and the rest of the organs of the sentient
contribute, namely, the continuation of the same motion to the last
organ of sense, the heart. And in this manner the cause of light may be
made up of motion continued from the original of the same motion, to the
original of vital motion, light being nothing but the alteration of
vital motion, made by the impression upon it of motion continued from
the object. But I give this only for an example, for I shall speak more
at large of light, and the generation of it, in its proper place. In the
mean time it is manifest, that in the searching out of causes, there is
need partly of the analytical, and partly of the synthetical method; of
the analytical, to conceive how circumstances conduce severally to the
production of effects; and of the synthetical, for the adding together
and compounding of what they can effect singly by themselves. And thus
much may serve for the method of invention. It remains that I speak of
the method of teaching, that is, of demonstration, and of the means by
which we demonstrate.

[Sidenote: Words serve to invention as marks; to demonstration as
           signs.]

11. In the method of invention, the use of words consists in this, that
they may serve for marks, by which, whatsoever we have found out may be
recalled to memory; for without this all our inventions perish, nor will
it be possible for us to go on from principles beyond a syllogism or
two, by reason of the weakness of memory. For example, if any man, by
considering a triangle set before him, should find that all its angles
together taken are equal to two right angles, and that by thinking of
the same tacitly, without any use of words either understood or
expressed; and it should happen afterwards that another triangle, unlike
the former, or the same in different situation, should be offered to his
consideration, he would not know readily whether the same property were
in this last or no, but would be forced, as often as a different
triangle were brought before him (and the difference of triangles is
infinite) to begin his contemplation anew; which he would have no need
to do if he had the use of names, for every universal name denotes the
conceptions we have of infinite singular things. Nevertheless, as I said
above, they serve as _marks_ for the help of our memory, whereby we
register to ourselves our own inventions; but not as _signs_ by which we
declare the same to others; so that a man may be a philosopher alone by
himself, without any master; Adam had this capacity. But to teach, that
is, to demonstrate, supposes two at the least, and syllogistical speech.

[Sidenote: The method of demonstration is synthetical.]

12. And seeing teaching is nothing but leading the mind of him we teach,
to the knowledge of our inventions, in that track by which we attained
the same with our own mind; therefore, the same method that served for
our invention, will serve also for demonstration to others, saving that
we omit the first part of method which proceeded from the sense of
things to universal principles, which, because they are principles,
cannot be demonstrated; and seeing they are known by nature, (as was
said above in the 5th article) they need no demonstration, though they
need explication. The whole method, therefore, of demonstration, is
_synthetical_, consisting of that order of speech which begins from
primary or most universal propositions, which are manifest of
themselves, and proceeds by a perpetual composition of propositions into
syllogisms, till at last the learner understand the truth of the
conclusion sought after.

[Sidenote: Definitions only are primary, & universal propositions.]

13. Now, such principles are nothing but definitions, whereof there are
two sorts; one of names, that signify such things as have some
conceivable cause, and another of such names as signify things of which
we can conceive no cause at all. Names of the former kind are, _body_,
or _matter_, _quantity_, or _extension_, _motion_, and whatsoever is
common to all matter. Of the second kind, are _such a body_, _such and
so great motion_, _so great magnitude_, _such figure_, and whatsoever we
can distinguish one body from another by. And names of the former kind
are well enough defined, when, by speech as short as may be, we raise in
the mind of the hearer perfect and clear ideas or conceptions of the
things named, as when we define motion to be _the leaving of one place,
and the acquiring of another continually_; for though no thing moved,
nor any cause of motion be in that definition, yet, at the hearing of
that speech, there will come into the mind of the hearer an _idea_ of
motion clear enough. But definitions of things, which may be understood
to have some cause, must consist of such names as express the cause or
manner of their generation, as when we define a circle to be a figure
made by the circumduction of a straight line in a plane, &c. Besides
definitions, there is no other proposition that ought to be called
primary, or (according to severe truth) be received into the number of
principles. For those _axioms of Euclid_, seeing they may be
demonstrated, are no principles of demonstration, though they have by
the consent of all men gotten the authority of principles, because they
need not be demonstrated. Also, those _petitions_, or _postulata_, (as
they call them) though they be principles, yet they are not principles
of demonstration, but of construction only; that is, not of science, but
of power; or (which is all one) not of _theorems_, which are
speculations, but of _problems_, which belong to practice, or the doing
of something. But as for those common received opinions, _Nature abhors
vacuity_, _Nature doth nothing in vain_, and the like, which are neither
evident in themselves, nor at all to be demonstrated, and which are
oftener false than true, they are much less to be acknowledged for
principles.

To return, therefore, to definitions; the reason why I say that the
cause and generation of such things, as have any cause or generation,
ought to enter into their definitions, is this. The end of science is
the demonstration of the causes and generations of things; which if they
be not in the definitions, they cannot be found in the conclusion of the
first syllogism, that is made from those definitions; and if they be not
in the first conclusion, they will not be found in any further
conclusion deduced from that; and, therefore, by proceeding in this
manner, we shall never come to science; which is against the scope and
intention of demonstration.

[Sidenote: The nature & definition of a definition.]

14. Now, seeing definitions (as I have said) are principles, or primary
propositions, they are therefore speeches; and seeing they are used for
the raising of an _idea_ of some thing in the mind of the learner,
whensoever that thing has a name, the definition of it can be nothing
but the explication of that name by speech; and if that name be given it
for some compounded conception, the definition is nothing but a
resolution of that name into its most universal parts. As when we define
man, saying _man is a body animated, sentient, rational_, those names,
_body animated_, _&c._ are parts of that whole name _man_; so that
definitions of this kind always consist of _genus_ and _difference_; the
former names being all, till the last, _general_; and the last of all,
_difference_. But if any name be the most universal in its kind, then
the definition of it cannot consist of _genus_ and _difference_, but is
to be made by such circumlocution, as best explicateth the force of that
name. Again, it is possible, and happens often, that the _genus_ and
_difference_ are put together, and yet make no definition; as these
words, _a straight line_, contain both the _genus_ and _difference_; but
are not a definition, unless we should think a straight line may be thus
defined, _a straight line is a straight line_: and yet if there were
added another name, consisting of different words, but signifying the
same thing which these signify, then these might be the definition of
that name. From what has been said, it may be understood how a
definition ought to be defined, namely, _that it is a proposition, whose
predicate resolves the subject, when it may; and when it may not, it
exemplifies the same_.

[Sidenote: Properties of a definition.]

15. The properties of a definition are:

First, that it takes away equivocation, as also all that multitude of
distinctions, which are used by such as think they may learn philosophy
by disputation. For the nature of a definition is to define, that is, to
determine the signification of the defined name, and to pare from it all
other signification besides what is contained in the definition itself;
and therefore one definition does as much, as all the distinctions (how
many soever) that can be used about the name defined.

Secondly, that it gives an universal notion of the thing defined,
representing a certain universal picture thereof, not to the eye, but to
the mind. For as when one paints a man, he paints the image of some man;
so he, that defines the name man, makes a representation of some man to
the mind.

Thirdly, that it is not necessary to dispute whether definitions are to
be admitted or no. For when a master is instructing his scholar, if the
scholar understand all the parts of the thing defined, which are
resolved in the definition, and yet will not admit of the definition,
there needs no further controversy betwixt them, it being all one as if
he refused to be taught. But if he understand nothing, then certainly
the definition is faulty; for the nature of a definition consists in
this, that it exhibit a clear idea of the thing defined; and principles
are either known by themselves, or else they are not principles.

Fourthly, that, in philosophy, definitions are before defined names. For
in teaching philosophy, the first beginning is from definitions; and all
progression in the same, till we come to the knowledge of the thing
compounded, is compositive. Seeing, therefore, definition is the
explication of a compounded name by resolution, and the progression is
from the parts to the compound, definitions must be understood before
compounded names; nay, when the names of the parts of any speech be
explicated, it is not necessary that the definition should be a name
compounded of them. For example, when these names, _equilateral_,
_quadrilateral_, _right-angled_, are sufficiently understood, it is not
necessary in geometry that there should be at all such a name as
_square_; for defined names are received in philosophy for brevity's
sake only.

Fifthly, that compounded names, which are defined one way in some one
part of philosophy, may in another part of the same be otherwise
defined; as a _parabola_ and an _hyperbole_ have one definition in
geometry, and another in rhetoric; for definitions are instituted and
serve for the understanding of the doctrine which is treated of. And,
therefore, as in one part of philosophy, a definition may have in it
some one fit name for the more brief explanation of some proposition in
geometry; so it may have the same liberty in other parts of philosophy;
for the use of names is particular (even where many agree to the
settling of them) and arbitrary.

Sixthly, that no name can be defined by any one word; because no one
word is sufficient for the resolving of one or more words.

Seventhly, that a defined name ought not to be repeated in the
definition. For a defined name is the whole compound, and a definition
is the resolution of that compound into parts; but no total can be part
of itself.

[Sidenote: Nature of a demonstration.]

16. Any two definitions, that may be compounded into a syllogism,
produce a conclusion; which, because it is derived from principles, that
is, from definitions, is said to be demonstrated; and the derivation or
composition itself is called a demonstration. In like manner, if a
syllogism be made of two propositions, whereof one is a definition, the
other a demonstrated conclusion, or neither of them is a definition, but
both formerly demonstrated, that syllogism is also called a
demonstration, and so successively. The definition therefore of a
demonstration is this, _a demonstration is a syllogism, or series of
syllogisms, derived and continued, from the definitions of names, to the
last conclusion_. And from hence it may be understood, that all true
ratiocination, which taketh its beginning from true principles,
produceth science, and is true demonstration. For as for the original of
the name, although that, which the Greeks called ἀποδέιξις, and the
Latins _demonstratio_, was understood by them for that sort only of
ratiocination, in which, by the describing of certain lines and figures,
they placed the thing they were to prove, as it were before men's eyes,
which is properly ἀποδεικνύειν, or to _shew_ by the figure; yet they
seem to have done it for this reason, that unless it were in geometry,
(in which only there is place for such figures) there was no
ratiocination certain, and ending in science, their doctrines concerning
all other things being nothing but controversy and clamour; which,
nevertheless, happened, not because the truth to which they pretended
could not be made evident without figures, but because they wanted true
principles, from which they might derive their ratiocination; and,
therefore, there is no reason but that if true definitions were premised
in all sorts of doctrines, the demonstrations also would be true.

[Sidenote: Properties of a demonstration, and order of things to be
           demonstrated.]

17. It. is proper to methodical demonstration,

First, that there be a true succession of one reason to another,
according to the rules of syllogizing delivered above.

Secondly, that the premises of all syllogisms be demonstrated from the
first definitions.

Thirdly, that after definitions, he that teaches or demonstrates any
thing, proceed in the same method by which he found it out; namely, that
in the first place those things be demonstrated, which immediately
succeed to universal definitions (in which is contained that part of
philosophy which is called _philosophia prima_). Next, those things
which may be demonstrated by simple motion (in which geometry consists).
After geometry, such things as may be taught or shewed by manifest
action, that is, by thrusting from, or pulling towards. And after these,
the motion or mutation of the invisible parts of things, and the
doctrine of sense and imaginations, and of the internal passions,
especially those of men, in which are comprehended the grounds of civil
duties, or civil philosophy; which takes up the last place. And that
this method ought to be kept in all sorts of philosophy, is evident from
hence, that such things as I have said are to be taught last, cannot be
demonstrated, till such as are propounded to be first treated of, be
fully understood. Of which method no other example can be given, but
that treatise of the elements of philosophy, which I shall begin in the
next chapter, and continue to the end of the work.

[Sidenote: Faults of a demonstration.]

18. Besides those _paralogisms_, whose fault lies either in the falsity
of the premises, or the want of true composition, of which I have spoken
in the precedent chapter, there are two more, which are frequent in
demonstration; one whereof is commonly called _petitio principii_; the
other is the supposing of a _false cause_; and these do not only deceive
unskilful learners, but sometimes masters themselves, by making them
take that for well demonstrated, which is not demonstrated at all.
_Petitio principii_ is, when the conclusion to be proved is disguised in
other words, and put for the definition or principle from whence it is
to be demonstrated; and thus, by putting for the cause of the thing
sought, either the thing itself or some effect of it, they make a circle
in their demonstration. As for example, he that would demonstrate that
the earth stands still in the centre of the world, and should suppose
the earth's gravity to be the cause thereof, and define gravity to be a
quality by which every heavy body tends towards the centre of the world,
would lose his labour; for the question is, what is the cause of that
quality in the earth? and, therefore, he that supposes gravity to be the
cause, puts the thing itself for its own cause.

Of a _false cause_ I find this example in a certain treatise where the
thing to be demonstrated is the motion of the earth. He begins,
therefore, with this, that seeing the earth and the sun are not always
in the same situation, it must needs be that one of them be locally
moved, which is true; next, he affirms that the vapours, which the sun
raises from the earth and sea, are, by reason of this motion,
necessarily moved, which also is true; from whence he infers the winds
are made, and this may pass for granted; and by these winds he says, the
waters of the sea are moved, and by their motion the bottom of the sea,
as if it were beaten forwards, moves round; and let this also be
granted; wherefore, he concludes, the earth is moved; which is,
nevertheless, a paralogism. For, if that wind were the cause why the
earth was, from the beginning, moved round, and the motion either of the
sun or the earth were the cause of that wind, then the motion of the sun
or the earth was before the wind itself; and if the earth were moved,
before the wind was made, then the wind could not be the cause of the
earth's revolution; but, if the sun were moved, and the earth stand
still, then it is manifest the earth might remain unmoved,
notwithstanding that wind; and therefore that motion was not made by the
cause which he allegeth. But paralogisms of this kind are very frequent
among the writers of _physics_, though none can be more elaborate than
this in the example given.

[Sidenote: Why the analytical method of geometricians cannot be treated
           of in this place.]

19. It may to some men seem pertinent to treat in this place of that art
of the geometricians, which they call _logistica_, that is, the art, by
which, from supposing the thing in question to be true, they proceed by
ratiocination, till either they come to something known, by which they
may demonstrate the truth of the thing sought for; or to something which
is impossible, from whence they collect that to be false, which they
supposed true. But this art cannot be explicated here, for this reason,
that the method of it can neither be practised, nor understood, unless
by such as are well versed in geometry; and among geometricians
themselves, they, that have most theorems in readiness, are the most
ready in the use of this _logistica_; so that, indeed, it is not a
distinct thing from geometry itself; for there are, in the method of it,
three parts; the first whereof consists in the finding out of equality
betwixt known and unknown things, which they call equation; and this
equation cannot be found out, but by such as know perfectly the nature,
properties, and transpositions of proportion, as also the addition,
subtraction, multiplication, and division of lines and superficies, and
the extraction of roots; which are the parts of no mean geometrician.
The second is, when an equation is found, to be able to judge whether
the truth or falsity of the question may be deduced from it, or no;
which yet requires greater knowledge. And the third is, when such an
equation is found, as is fit for the solution of the question, to know
how to resolve the same in such manner, that the truth or falsity may
thereby manifestly appear; which, in hard questions, cannot be done
without the knowledge of the nature of crooked-lined figures; but he
that understands readily the nature and properties of these, is a
complete geometrician. It happens besides, that for the finding out of
equations, there is no certain method, but he is best able to do it,
that has the best natural wit.




                                PART II.

                                  THE
                      FIRST GROUNDS OF PHILOSOPHY.


                                -------


                              CHAPTER VII.

                           OF PLACE AND TIME.

1. Things that have no existence, may nevertheless be understood and
    computed.—2. What is Space.—3. Time.—4. Part.—5. Division.—6.
    One.—7. Number.—8. Composition.—9. The whole.—10. Spaces and times
    contiguous, and continual.—11. Beginning, end, way, finite,
    infinite.—12. What is infinite in power. Nothing infinite can be
    truly said to be either whole, or one; nor infinite spaces or times,
    many.—13. Division proceeds not to the least.


[Sidenote: Things that have no existence, may nevertheless be understood
           and computed.]

1. In the teaching of natural philosophy, I cannot begin better (as I
have already shewn) than from _privation_; that is, from feigning the
world to be annihilated. But, if such annihilation of all things be
supposed, it may perhaps be asked, what would remain for any man (whom
only I except from this universal annihilation of things) to consider as
the subject of philosophy, or at all to reason upon; or what to give
names unto for ratiocination's sake.

I say, therefore, there would remain to that man ideas of the world, and
of all such bodies as he had, before their annihilation, seen with his
eyes, or perceived by any other sense; that is to say, the memory and
imagination of magnitudes, motions, sounds, colours, &c. as also of
their order and parts. All which things, though they be nothing but
ideas and phantasms, happening internally to him that imagineth; yet
they will appear as if they were external, and not at all depending upon
any power of the mind. And these are the things to which he would give
names, and subtract them from, and compound them with one another. For
seeing, that after the destruction of all other things, I suppose man
still remaining, and namely that he thinks, imagines, and remembers,
there can be nothing for him to think of but what is past; nay, if we do
but observe diligently what it is we do when we consider and reason, we
shall find, that though all things be still remaining in the world, yet
we compute nothing but our own phantasms. For when we calculate the
magnitude and motions of heaven or earth, we do not ascend into heaven
that we may divide it into parts, or measure the motions thereof, but we
do it sitting still in our closets or in the dark. Now things may be
considered, that is, be brought into account, either as internal
accidents of our mind, in which manner we consider them when the
question is about some faculty of the mind; or as species of external
things, not as really existing, but appearing only to exist, or to have
a being without us. And in this manner we are now to consider them.

[Sidenote: What is Space.]

2. If therefore we remember, or have a phantasm of any thing that was in
the world before the supposed annihilation of the same; and consider,
not that the thing was such or such, but only that it had a being
without the mind, we have presently a conception of that we call
_space_: an imaginary space indeed, because a mere phantasm, yet that
very thing which all men call so. For no man calls it space for being
already filled, but because it may be filled; nor does any man think
bodies carry their places away with them, but that the same space
contains sometimes one, sometimes another body; which could not be if
space should always accompany the body which is once in it. And this is
of itself so manifest, that I should not think it needed any explaining
at all, but that I find space to be falsely defined by certain
philosophers, who infer from thence, one, that the world is infinite
(for taking _space_ to be the extension of bodies, and thinking
extension may encrease continually, he infers that bodies may be
infinitely extended); and, another, from the same definition, concludes
rashly, that it is impossible even to God himself to create more worlds
than one; for, if another world were to be created, he says, that seeing
there is nothing without this world, and therefore (according to his
definition) no space, that new world must be placed in nothing; but in
nothing nothing can be placed; which he affirms only, without showing
any reason for the same; whereas the contrary is the truth: for more
cannot be put into a place already filled, so much is empty space fitter
than that, which is full, for the receiving of new bodies. Having
therefore spoken thus much for these men's sakes, and for theirs that
assent to them, I return to my purpose, and define _space_ thus: SPACE
_is the phantasm of a thing existing without the mind simply_; that is
to say, that phantasm, in which we consider no other accident, but only
that it appears without us.

[Sidenote: Time.]

3. As a body leaves a phantasm of its magnitude in the mind, so also a
moved body leaves a phantasm of its motion, namely, an idea of that body
passing out of one space into another by continual succession. And this
idea, or phantasm, is that, which (without receding much from the common
opinion, or from _Aristotle's_ definition) I call _Time_. For seeing all
men confess a year to be time, and yet do not think a year to be the
accident or affection of any body, they must needs confess it to be, not
in the things without us, but only in the thought of the mind. So when
they speak of the times of their predecessors, they do not think after
their predecessors are gone, that their times can be any where else than
in the memory of those that remember them. And as for those that say,
days, years, and months are the motions of the sun and moon, seeing it
is all one to say, motion _past_ and motion _destroyed_, and that
_future_ motion is the same with motion which _is not yet begun_, they
say that, which they do not mean, that there neither is, nor has been,
nor shall be any time: for of whatsoever it may be said, _it has been_
or _it shall be_, of the same also it might have been said heretofore,
or may be said hereafter, _it is_. What then can days, months, and
years, be, but the names of such computations made in our mind? _Time_
therefore is a phantasm, but a phantasm of motion, for if we would know
by what moments time passes away, we make use of some motion or other,
as of the sun, of a clock, of the sand in an hour-glass, or we mark some
line upon which we imagine something to be moved, there being no other
means by which we can take notice of any time at all. And yet, when I
say _time_ is a phantasm of motion, I do not say this is sufficient to
define it by; for this word _time_ comprehends the notion of _former_
and _latter_, or of _succession_ in the motion of a body, in as much as
it is first _here_ then _there_. Wherefore a complete definition of
_time_ is such as this, TIME _is the phantasm of before and after in
motion_; which agrees with this definition of _Aristotle_, _time is the
number of motion according to former and latter_; for that numbering is
an act of the mind; and therefore it is all one to say, _time is the
number of motion according to former and latter_; and _time is a
phantasm of motion numbered_. But that other definition, _time is the
measure of motion_, is not so exact, for we measure time by motion and
not motion by time.

[Sidenote: Part.]

4. One space is called _part_ of another space, and one time _part_ of
another time, when this contains that and something besides. From whence
it may be collected, that nothing can rightly be called a PART, but that
which is compared with something that contains it.

[Sidenote: Division.]

5. And therefore to _make parts_, or to _part_ or DIVIDE space or
_time_, is nothing else but to consider one and another within the same;
so that if any man _divide_ space or time, the diverse conceptions he
has are more, by one, than the parts he makes; for his first conception
is of that which is to be divided, then of some part of it, and again of
some other part of it, and so forwards as long as he goes on in
dividing.

But it is to be noted, that here, by _division_, I do not mean the
severing or pulling asunder of one space or time from another (for does
any man think that one hemisphere may be separated from the other
hemisphere, or the first hour from the second?) but diversity of
consideration; so that division is not made by the operation of the
hands but of the mind.

[Sidenote: One.]

6. When space or time is considered among other spaces or times, it is
said to be ONE, namely _one of them_; for except one space might be
added to another, and subtracted from another space, and so of time, it
would be sufficient to say space or time simply, and superfluous to say
one space or one time, if it could not be conceived that there were
another. The common definition of _one_, namely, that _one is that which
is undivided_, is obnoxious to an absurd consequence; for it may thence
be inferred, that whatsoever is divided is many things, that is, that
every divided thing, is divided things, which is insignificant.

[Sidenote: Number.]

7. NUMBER is _one_ and _one_, or _one one_ and _one_, and so forwards;
namely, _one_ and _one_ make the number _two_, and _one one_ and _one_
the number _three_; so are all other numbers made; which is all one as
if we should say, _number is unities_.

[Sidenote: Composition.]

8. To COMPOUND space of spaces, or time of times, is first to consider
them one after another, and then altogether as one; as if one should
reckon first the head, the feet, the arms, and the body, severally, and
then for the account of them all together put _man_. And that which is
so put for all the severals of which it consists, is called the WHOLE;
and those severals, when by the division of the whole they come again to
be considered singly, are parts thereof; and therefore the _whole_ and
_all the parts taken together_ are the same thing. And as I noted above,
that in _division_ it is not necessary to pull the parts asunder; so in
_composition_, it is to be understood, that for the making up of a whole
there is no need of putting the parts together, so as to make them touch
one another, but only of collecting them into one sum in the mind. For
thus all men, being considered together, make up the whole of mankind,
though never so much dispersed by time and place; and twelve hours,
though the hours of several days, may be compounded into one number of
twelve.

[Sidenote: The whole.]

9. This being well understood, it is manifest, that nothing can rightly
be called a whole, that is not conceived to be compounded of parts, and
that it may be divided into parts; so that if we deny that a thing has
parts, we deny the same to be a whole. For example, if we say the soul
can have no parts, we affirm that no soul can be a whole soul. Also it
is manifest, that nothing has parts till it be divided; and when a thing
is divided, the parts are only so many as the division makes them.
Again, that a part of a part is a part of the whole; and thus any part
of the number _four_, as _two_, is a part of the number _eight_; for
_four_ is made of _two_ and _two_; but _eight_ is compounded of _two_,
_two_, and _four_, and therefore _two_, which is a part of the part
_four_, is also a part of the whole _eight_.

[Sidenote: Spaces and times contiguous and continual.]

10. Two spaces are said to be CONTIGUOUS, when there is no other space
betwixt them. But two times, betwixt which there is no other time, are
called immediate, as A B, B C.[Sidenote: A    B    C] And any two
spaces, as well as times, are said to be CONTINUAL, when they have one
common part, as A C, B D, where the part B C is common; [Sidenote:
A   B   C   D] and more spaces and times are continual, when every two
which are next one another are continual.

                             A   B   C   D
                             -------------

                              A    B     C
                              ------------

[Sidenote: Beginning, end, way, finite, infinite.]

11. That part which is between two other parts, is called a MEAN; and
that which is not between two other parts, an EXTREME. And of extremes,
that which is first reckoned is the BEGINNING, and that which last, the
END; and all the means together taken are the WAY. Also, _extreme parts_
and _limits_ are the same thing. And from hence it is manifest, that
_beginning_ and _end_ depend upon the order in which we number them; and
that to _terminate_ or _limit_ space and time, is the same thing with
_imagining their beginning and end_; as also that every thing is FINITE
or INFINITE, according as we imagine or not imagine it _limited_ or
_terminated_ every way; and that the _limits_ of any number are
_unities_, and of these, that which is the first in our numbering is the
_beginning_, and that which we number last, is the _end_. When we say
number is _infinite_, we mean only that no number is expressed; for when
we speak of the numbers _two_, _three_, _a thousand_, &c. they are
always _finite_. But when no more is said but this, _number is
infinite_, it is to be understood as if it were said, this name _number_
is an _indefinite_ name.

[Sidenote: What is infinite in power. Nothing infinite can be truly said
           to be either whole or one; nor infinite spaces or times,
           many.]

12. Space or time is said to _finite in power_, or _terminable_, when
there may be assigned a number of finite spaces or times, as of paces or
hours, than which there can be no greater number of the same measure in
that space or time; and _infinite in power_ is that space or time, in
which a greater number of the said paces or hours may be assigned, than
any number that can be given. But we must note, that, although in that
space or time which is infinite in power, there may be numbered more
paces or hours than any number that can be assigned, yet their number
will always be finite; for every number is finite. And therefore his
ratiocination was not good, that undertaking to prove the world to be
finite, reasoned thus; _If the world be infinite, then there may be
taken in it some part which is distant from us an infinite number of
paces: but no such part can be taken; wherefore the world is not
infinite_; because that consequence of the major proposition is false;
for in an infinite space, whatsoever we take or design in our mind, the
distance of the same from us is a finite space; for in the very
designing of the place thereof, we put an end to that space, of which we
ourselves are the beginning; and whatsoever any man with his mind cuts
off both ways from infinite, he determines the same, that is, he makes
it finite.

Of infinite space or time, it cannot be said that it is a _whole_ or
_one_: not a _whole_, because not compounded of parts; for seeing parts,
how many soever they be, are severally finite, they will also, when they
are all put together, make a whole finite: nor _one_, because nothing
can be said to be one, except there be another to compare it with; but
it cannot be conceived that there are two spaces, or two times,
infinite. Lastly, when we make question whether the world be finite or
infinite, we have nothing in our mind answering to the name _world_; for
whatsoever we imagine, is therefore finite, though our computation reach
the fixed stars, or the ninth or tenth, nay, the thousandth sphere. The
meaning of the question is this only, whether God has actually made so
great an addition of body to body, as we are able to make of space to
space.

[Sidenote: Division proceeds not to the least.]

13. And, therefore, that which is commonly said, that space and time may
be divided infinitely, is not to be so understood, as if there might be
any infinite or eternal division; but rather to be taken in this sense,
_whatsoever is divided, is divided into such parts as may again be
divided_; or thus, _the least divisible thing is not to be given_; or,
as geometricians have it, _no quantity is so small, but a less may be
taken_; which may easily be demonstrated in this manner. Let any space
or time, that which was thought to be the least divisible, be divided
into two equal parts, A and B. I say either of them, as A, may be
divided again. For suppose the part A to be contiguous to the part B of
one side, and of the other side to some other space equal to B. This
whole space, therefore, being greater than the space given, is
divisible. Wherefore, if it be divided into two equal parts, the part in
the middle, which is A, will be also divided into two equal parts; and
therefore A was divisible.


                                -------


                             CHAPTER VIII.

                         OF BODY AND ACCIDENT.

1. Body defined.—2. Accident defined.—3. How an accident may be
    understood to be in its subject.—4. Magnitude, what it is.—5. Place,
    what it is, and that it is immovable.—6. What is full and empty.—7.
    Here, there, somewhere, what they signify.—8. Many bodies cannot be
    in one place, nor one body in many places.—9. Contiguous and
    continual, what they are.—10. The definition of motion. No motion
    intelligible but with time.—11. What it is to be at rest, to have
    been moved, and to be moved. No motion to be conceived, without the
    conception of past and future.—12. A point, a line, superficies and
    solid, what they are.—13. Equal, greater, and less in bodies and
    magnitudes, what they are.—14. One and the same body has always one
    and the same magnitude.—15. Velocity, what it is.—16. Equal,
    greater, and less in times, what they are.—17. Equal, greater, and
    less, in velocity, what.—18. Equal, greater, and less, in motion,
    what.—19. That which is at rest, will always be at rest, except it
    be moved by some external thing; and that which is moved, will
    always be moved, unless it be hindered by some external thing.—20.
    Accidents are generated and destroyed, but bodies not so.—21. An
    accident cannot depart from its subject.—22. Nor be moved.—23.
    Essence, form, and matter, what they are.—24. First matter,
    what.—25. That the whole is greater than any part thereof, why
    demonstrated.


[Sidenote: Body defined.]

1. Having understood what imaginary space is, in which we supposed
nothing remaining without us, but all those things to be destroyed,
that, by existing heretofore, left images of themselves in our minds;
let us now suppose some one of those things to be placed again in the
world, or created anew. It is necessary, therefore, that this
new-created or replaced thing do not only fill some part of the space
above mentioned, or be coincident and coextended with it, but also that
it have no dependance upon our thought. And this is that which, for the
extension of it, we commonly call _body_; and because it depends not
upon our thought, we say is _a thing subsisting of itself_; as also
_existing_, because without us; and, lastly, it is called the _subject_,
because it is so placed in and _subjected_ to imaginary space, that it
may be understood by reason, as well as perceived by sense. The
definition, therefore, of _body_ may be this, a _body is that, which
having no dependance upon our thought, is coincident or coextended with
some part of space_.

[Sidenote: Accident defined.]

2. But what an _accident_ is cannot so easily be explained by any
definition, as by examples. Let us imagine, therefore, that a body fills
any space, or is coextended with it; that coextension is not the
coextended body: and, in like manner, let us imagine that the same body
is removed out of its place; that removing is not the removed body: or
let us think the same not removed; that not removing or rest is not the
resting body. What, then, are these things? They are _accidents_ of that
body. But the thing in question is, _what is an accident?_ which is an
enquiry after that which we know already, and not that which we should
enquire after. For who does not always and in the same manner understand
him that says any thing is extended, or moved, or not moved? But most
men will have it be said that _an accident is something_, namely, some
part of a natural thing, when, indeed, it is no part of the same. To
satisfy these men, as well as may be, they answer best that define an
_accident_ to be _the manner by which any body is conceived_; which is
all one as if they should say, _an accident is that faculty of any body,
by which it works in us a conception of itself_. Which definition,
though it be not an answer to the question propounded, yet it is an
answer to that question which should have been propounded, namely,
_whence does it happen that one part of any body appears here, another
there?_ For this is well answered thus: _it happens from the extension
of that body_. Or, _how comes it to pass that the whole body, by
succession, is seen now here, now there?_ and the answer will be, _by
reason of its motion_. Or, lastly, _whence is it that any body
possesseth the same space for sometime?_ and the answer will be,
_because it is not moved_. For if concerning the name of a body, that
is, concerning a concrete name, it be asked, _what is it?_ the answer
must be made by definition; for the question is concerning the
signification of the name. But if it be asked concerning an abstract
name, _what is it?_ the cause is demanded why a thing appears so or so.
As if it be asked, _what is hard?_ The answer will be, hard is that,
whereof no part gives place, but when the whole gives place. But if it
be demanded, _what is hardness?_ a cause must be shewn why a part does
not give place, except the whole give place. Wherefore, I define an
_accident_ to be _the manner of our conception of body_.

[Sidenote: How an accident may be understood to be in its subject.]

3. When an _accident_ is said _to be in a body_, it is not so to be
understood, as if any thing were contained in that body; as if, for
example, redness were in blood, in the same manner, as blood is in a
bloody cloth, that is, as a part in the whole; for so, an accident would
be a body also. But, as magnitude, or rest, or motion, is in that which
is great, or which resteth, or which is moved, (which, how it is to be
understood, every man understands) so also, it is to be understood, that
every other accident _is in_ its subject. And this, also, is explicated
by _Aristotle_ no otherwise than negatively, namely, that _an accident
is in its subject, not as any part thereof, but so as that it may be
away, the subject still remaining_; which is right, saving that there
are certain accidents which can never perish except the body perish
also; for no body can be conceived to be without extension, or without
figure. All other accidents, which are not common to all bodies, but
peculiar to some only, as _to be at rest_, _to be moved_, _colour_,
_hardness_, and the like, do perish continually, and are succeeded by
others; yet so, as that the body never perisheth. And as for the opinion
that some may have, that all other accidents are not in their bodies in
the same manner that extension, motion, rest, or figure, are in the
same; for example, that colour, heat, odour, virtue, vice, and the like,
are otherwise in them, and, as they say, _inherent_; I desire they would
suspend their judgment for the present, and expect a little, till it be
found out by ratiocination, whether these very accidents are not also
certain motions either of the mind of the perceiver, or of the bodies
themselves which are perceived; for in the search of this, a great part
of natural philosophy consists.

[Sidenote: Magnitude, what it is.]

4. The _extension_ of a body, is the same thing with the _magnitude_ of
it, or that which some call _real space_. But this _magnitude_ does not
depend upon our cogitation, as imaginary space doth; for this is an
effect of our imagination, but _magnitude_ is the cause of it; this is
an accident of the mind, that of a body existing out of the mind.

[Sidenote: Place, what it is, and that it is immovable.]

5. That space, by which word I here understand imaginary space, which is
coincident with the magnitude of any body, is called the _place_ of that
body; and the body itself is that which we call the _thing placed_. Now
_place_, and the _magnitude_ of the thing placed, differ. First in this,
that a body keeps always the same _magnitude_, both when it is at rest,
and when it is moved; but when it is moved, it does not keep the same
_place_. Secondly in this, that _place_ is a phantasm of any body of
such and such quantity and figure; but _magnitude_ is the peculiar
accident of every body; for one body may at several times have several
places, but has always one and the same magnitude. Thirdly in this, that
_place_ is nothing out of the mind, nor _magnitude_ any thing within it.
And lastly, _place_ is feigned extension, but _magnitude_ true
extension; and a placed body is not extension, but a thing extended.
Besides, _place is immovable_; for, seeing that which is moved, is
understood to be carried from place to place, if place were moved, it
would also be carried from place to place, so that one place must have
another place, and that place another place, and so on infinitely, which
is ridiculous. And as for those, that, by making _place_ to be of the
same nature with _real space_, would from thence maintain it to be
immovable, they also make place, though they do not perceive they make
it so, to be a mere phantasm. For whilst one affirms that place is
therefore said to be immovable, because space in general is considered
there; if he had remembered that nothing is general or universal besides
names or signs, he would easily have seen that that space, which he says
is considered in general, is nothing but a phantasm, in the mind or the
memory, of a body of such magnitude and such figure. And whilst another
says: real space is made immovable by the understanding; as when, under
the superficies of running water, we imagine other and other water to
come by continual succession, that superficies fixed there by the
understanding, is the _immovable place_ of the river: what else does he
make it to be but a phantasm, though he do it obscurely and in perplexed
words? Lastly, the nature of _place_ does not consist in the
_superficies of the ambient_, but in _solid space_; for the whole placed
body is coextended with its whole place, and every part of it with every
answering part of the same place; but seeing every placed body is a
solid thing, it cannot be understood to be coextended with superficies.
Besides, how can any whole body be moved, unless all its parts be moved
together with it? Or how can the internal parts of it be moved, but by
leaving their place? But the internal parts of a body cannot leave the
superficies of an external part contiguous to it; and, therefore, it
follows, that if place be the superficies of the ambient, then the parts
of a body moved, that is, bodies moved, are not moved.

[Sidenote: What is full and empty.]

6. Space, or place, that is possessed by a body, is called _full_, and
that which is not so possessed, is called _empty_.

[Sidenote: Here, there, somewhere, what they signify.]

7. _Here_, _there_, _in the country_, _in the city_, and other the like
names, by which answer is made to the question _where is it?_ are not
properly names of place, nor do they of themselves bring into the mind
the place that is sought; for _here_ and _there_ signify nothing, unless
the thing be shewn at the same time with the finger or something else;
but when the eye of him that seeks, is, by pointing or some other sign,
directed to the thing sought, the place of it is not hereby defined by
him that answers, but found out by him that asks the question. Now such
shewings as are made by words only, as when we say, _in the country_, or
_in the city_, are some of greater latitude than others, as when we say,
_in the country_, _in the city_, _in such a street_, _in a house_, _in
the chamber_, _in bed_, &c. For these do, by little and little, direct
the seeker nearer to the proper place; and yet they do not determine the
same, but only restrain it to a lesser space, and signify no more, than
that the place of the thing is within a certain space designed by those
words, as a part is in the whole. And all such names, by which answer is
made to the question _where?_ have, for their highest _genus_, the name
_somewhere_. From whence it may be understood, that whatsoever is
somewhere, is in some place properly so called, which place is part of
that greater space that is signified by some of these names, _in the
country_, _in the city_, or the like.

[Sidenote: Many bodies cannot be in one place, nor one body in many
           places.]

8. A body, and the magnitude, and the place thereof, are divided by one
and the same act of the mind; for, to divide an extended body, and the
extension thereof, and the idea of that extension, which is place, is
the same with dividing any one of them; because they are coincident, and
it cannot be done but by the mind, that is by the division of space.
From whence it is manifest, that neither two bodies can be together in
the same place, nor one body be in two places at the same time. Not two
bodies in the same place; because when a body that fills its whole place
is divided into two, the place itself is divided into two also, so that
there will be two places. Not one body in two places; for the place that
a body fills being divided into two, the placed body will be also
divided into two; for, as I said, a place and the body that fills that
place, are divided both together; and so there will be two bodies.

[Sidenote: Contiguous and continual, what they are.]

9. Two bodies are said to be _contiguous_ to one another, and
_continual_, in the same manner as spaces are; namely, _those are
contiguous, between which there is no space_. Now, by space I
understand, here as formerly, an idea or phantasm of a body. Wherefore,
though between two bodies there be put no other body, and consequently
no magnitude, or, as they call it, real space, yet if another body may
be put between them, that is, if there intercede any imagined space
which may receive another body, then those bodies are not contiguous.
And this is so easy to be understood, that I should wonder at some men,
who being otherwise skilful enough in philosophy, are of a different
opinion, but that I find that most of those that affect metaphysical
subtleties wander from truth, as if they were led out of their way by an
_ignis fatuus_. For can any man that has his natural senses, think that
two bodies must therefore necessarily touch one another, because no
other body is between them? Or that there can be no _vacuum_, because
_vacuum_ is nothing, or as they call it, _non ens_? Which is as
childish, as if one should reason thus; no man can fast, because to fast
is to eat nothing; but nothing cannot be eaten. _Continual, are any two
bodies that have a common part; and more than two are continual, when
every two, that are next to one another, are continual._

[Sidenote: The definition of motion. No motion intelligible but with
           time.]

10. MOTION _is a continual relinquishing of one place, and acquiring of
another_; and that place which is relinquished is commonly called the
_terminus a quo_, as that which is acquired is called the _terminus ad
quem_; I say a continual relinquishing, because no body, how little
soever, can totally and at once go out of its former place into another,
so, but that some part of it will be in a part of a place which is
common to both, namely, to the relinquished and the acquired places.
[Sidenote: A G B I E +-----+------+ : : : : +-----+------+ C H D K F]
For example, let any body be in the place A C B D; the same body cannot
come into the place B D E F, but it must first be in G H I K, whose part
G H B D is common to both the places A C B D, and G H I K, and whose
part B D I K, is common to both the places G H I K, and B D E F. Now it
cannot be conceived that any thing can be moved without time; for time
is, by the definition of it, a phantasm, that is, a conception of
motion; and, therefore, to conceive that any thing may be moved without
time, were to conceive motion without motion, which is impossible.

[Sidenote: What it is to be at rest, to have been moved, and to be
           moved. No motion to be conceived without the conception of
           past and future.]

11. _That is said to be at rest, which, during any time, is in one
place; and that to be moved, or to have been moved, which, whether it be
now at rest or moved, was formerly in another place than that which it
is now in._ From which definitions it may be inferred, first, that
_whatsoever is moved, has been moved_; for if it be still in the same
place in which it was formerly, it is at rest, that is, it is not moved,
by the definition of _rest_; but if it be in another place, it has been
moved, by the definition of _moved_. Secondly, that _what is moved, will
yet be moved_; for that which is moved, leaveth the place where it is,
and therefore will be in another place, and consequently will be moved
still. Thirdly, that _whatsoever is moved, is not in one place during
any time, how little soever that time be_; for by the definition of
rest, that which is in one place during any time, is at rest.

There is a certain sophism against motion, which seems to spring from
the not understanding of this last proposition. For they say, that, _if
any body be moved, it is moved either in the place where it is, or in
the place where it is not; both which are false; and therefore nothing
is moved_. But the falsity lies in the major proposition; for that which
is moved, is neither moved in the place where it is, nor in the place
where is not; but from the place where it is, to the place where it is
not. Indeed it cannot be denied but that whatsoever is moved, is moved
somewhere, that is, within some space; but then the place of that body
is not that whole space, but a part of it, as is said above in the
seventh article. From what is above demonstrated, namely, that
whatsoever is moved, has also been moved, and will be moved, this also
may be collected, that there can be no conception of motion, without
conceiving past and future time.

[Sidenote: A point, a line, superficies, and solid, what they are.]

12. Though there be no body which has not some magnitude, yet if, when
any body is moved, the magnitude of it be not at all considered, the way
it makes is called a _line_, or one single dimension; and the space,
through which it passeth, is called _length_; and the body itself, a
_point_; in which sense the earth is called a _point_, and the way of
its yearly revolution, the _ecliptic line_. But if a body, which is
moved, be considered as _long_, and be supposed to be so moved, as that
all the several parts of it be understood to make several lines, then
the way of every part of that body is called _breadth_, and the space
which is made is called _superficies_, consisting of two dimensions, one
whereof to every several part of the other is applied whole. Again, if a
body be considered as having _superficies_, and be understood to be so
moved, that all the several parts of it describe several lines, then the
way of every part of that body is called _thickness_ or _depth_, and the
space which is made is called _solid_, consisting of three dimensions,
any two whereof are applied whole to every several part of the third.

But if a body be considered as _solid_, then it is not possible that all
the several parts of it should describe several lines; for what way
soever it be moved, the way of the following part will fall into the way
of the part before it, so that the same solid will still be made which
the foremost superficies would have made by itself. And therefore there
can be no other dimension in any body, as it is a body, than the three
which I have now described; though, as it shall be shewed hereafter,
_velocity_, which is motion according to _length_, may, by being applied
to all the parts of a _solid_, make a magnitude of motion, consisting of
four dimensions; as the goodness of gold, computed in all the parts of
it, makes the price and value thereof.

[Sidenote: Equal, great, greater, and less, in bodies and magnitudes,
           what they are.]

13. _Bodies_, how many soever they be, that can fill every one the place
of every one, are said to be _equal_ every one to every other. Now, one
body may fill the same place which another body filleth, though it be
not of the same figure with that other body, if so be that it may be
understood to be reducible to the same figure, either by flexion or
transposition of the parts. And _one body is greater than another body,
when a part of that is equal to all this; and less, when all that is
equal to a part of this_. Also, _magnitudes_ are _equal_, or _greater_,
or _lesser_, than one another, for the same consideration, namely, when
the bodies, of which they are the magnitudes, are either _equal_, or
_greater_, or _less_, &c.

[Sidenote: One and the same body has always one and the same magnitude.]

14. One and the same body is always of one and the same magnitude. For
seeing a body and the magnitude and place thereof cannot be comprehended
in the mind otherwise than as they are coincident, if any body be
understood to be at rest, that is, to remain in the same place during
some time, and the magnitude thereof be in one part of that time
greater, and in another part less, that body's place, which is one and
the same, will be coincident sometimes with greater, sometimes with less
magnitude, that is, the same place will be greater and less than itself,
which is impossible. But there would be no need at all of demonstrating
a thing that is in itself so manifest, if there were not some, whose
opinion concerning bodies and their magnitudes is, that a body may exist
separated from its magnitude, and have greater or less magnitude
bestowed upon it, making use of this principle for the explication of
the nature of _rarum_ and _densum_.

[Sidenote: Velocity, what it is.]

15. Motion, in as much as a certain length may in a certain time be
transmitted by it, is called VELOCITY or _swiftness_: &c. For though
_swift_ be very often understood with relation to _slower_ or _less
swift_, as great is in respect of less, yet nevertheless, as magnitude
is by philosophers taken absolutely for extension, so also _velocity_ or
_swiftness_ may be put absolutely for motion according to length.

[Sidenote: Equal, greater, and less, in times, what they are.]

16. Many motions are said to be made in equal times, when every one of
them begins and ends together with some other motion, or if it had begun
together, would also have ended together with the same. For time, which
is a phantasm of motion, cannot be reckoned but by some exposed motion;
as in dials by the motion of the sun or of the hand; and if two or more
motions begin and end with this motion, they are said to be made in
equal times; from whence also it is easy to understand what it is to be
moved in greater or longer time, and in less time or not so long;
namely, that that is longer moved, which beginning with another, ends
later; or ending together, began sooner.

[Sidenote: Equal, greater, and less, in velocity, what.]

17. Motions are said to be equally swift, when equal lengths are
transmitted in equal times; and greater swiftness is that, wherein
greater length is passed in equal time, or equal length in less time.
Also that swiftness by which equal lengths are passed in equal parts of
time, is called _uniform_ swiftness or motion; and of motions _not
uniform_, such as become swifter or slower by equal increasings or
decreasings in equal parts of time, are said to be _accelerated_ or
_retarded uniformly_.

[Sidenote: Equal, greater, and less, in motion, what.]

18. But motion is said to be greater, less, and equal, not only in
regard of the length which is transmitted in a certain time, that is, in
regard of swiftness only, but of swiftness applied to every smallest
particle of magnitude; for when any body is moved, every part of it is
also moved; and supposing the parts to be halves, the motions of those
halves have their swiftness equal to one another, and severally equal to
that of the whole; but the motion of the whole is equal to those two
motions, either of which is of equal swiftness with it; and therefore it
is one thing for two motions to be equal to one another, and another
thing for them to be equally swift. And this is manifest in two horses
that draw abreast, where the motion of both the horses together is of
equal swiftness with the motion of either of them singly; but the motion
of both is greater than the motion of one of them, namely, double.
Wherefore _motions are said to be simply equal to one another, when the
swiftness of one, computed in every part of its magnitude, is equal to
the swiftness of the other computed also in every part of its magnitude:
and greater than one another, when the swiftness of one computed as
above, is greater than the swiftness of the other so computed; and less,
when less_. Besides, the magnitude of motion computed in this manner is
that which is commonly called FORCE.

[Sidenote: That which is at rest will always be at rest, except it be
           moved by some external thing.]

19. _Whatsoever is at rest, will always be at rest, unless there be some
other body besides it, which, by endeavouring to get into its place by
motion, suffers it no longer to remain at rest._ For suppose that some
finite body exist and be at rest, and that all space besides be empty;
if now this body begin to be moved, it will certainly be moved some way;
seeing therefore there was nothing in that body which did not dispose it
to rest, the reason why it is moved this way is in something out of it;
and in like manner, if it had been moved any other way, the reason of
motion that way had also been in something out of it; but seeing it was
supposed that nothing is out of it, the reason of its motion one way
would be the same with the reason of its motion every other way,
wherefore it would be moved alike all ways at once; which is impossible.

[Sidenote: That which is moved will always be moved, unless it be
           hindered by some external thing.]

In like manner, _whatsoever is moved, will always be moved, except there
be some other body besides it, which causeth it to rest_. For if we
suppose nothing to be without it, there will be no reason why it should
rest now, rather than at another time; wherefore its motion would cease
in every particle of time alike; which is not intelligible.

[Sidenote: Accidents are generated and destroyed, but bodies not so.]

20. When we say a living creature, a tree, or any other specified body
is _generated_ or _destroyed_, it is not to be so understood as if there
were made a body of that which is not-body, or not a body of a body, but
of a living creature not a living creature, of a tree not a tree, &c.
that is, that those accidents for which we call one thing a living
creature, another thing a tree, and another by some other name, are
generated and destroyed; and that therefore the same names are not to be
given to them now, which were given them before. But that magnitude for
which we give to any thing the name of body is neither generated nor
destroyed. For though we may feign in our mind that a point may swell to
a huge bulk, and that this may again contract itself to a point; that
is, though we may imagine something to arise where before was nothing,
and nothing to be there where before was something, yet we cannot
comprehend in our mind how this may possibly be done in nature. And
therefore philosophers, who tie themselves to natural reason, suppose
that a body can neither be generated nor destroyed, but only that it may
appear otherwise than it did to us, that is, under different _species_,
and consequently be called by other and other names; so that that which
is now called man, may at another time have the name of not-man; but
that which is once called body, can never be called not-body. But it is
manifest, that all other accidents besides magnitude or extension may be
generated and destroyed; as when a white thing is made black, the
whiteness that was in it perisheth, and the blackness that was not in it
is now generated; and therefore bodies, and the accidents under which
they appear diversely, have this difference, that bodies are things, and
not generated; accidents are generated, and not things.

[Sidenote: An accident cannot depart from its subject]

21. And therefore, when any thing appears otherwise than it did by
reason of other and other accidents, it is not to be thought that an
accident goes out of one subject into another, (for they are not, as I
said above, in their subjects as a part in the whole, or as a contained
thing in that which contains it, or as a master of a family in his
house,) but that one accident perisheth, and another is generated. For
example, when the hand, being moved, moves the pen, motion does not go
out of the hand into the pen; for so the writing might be continued
though the hand stood still; but a new motion is generated in the pen,
and is the pen's motion.

[Sidenote: Nor be moved.]

22. And therefore also it is improper to say, an accident is moved; as
when, instead of saying, _figure is an accident of a body carried away_,
we say, _a body carries away its figure_.

[Sidenote: Essence, form, and matter, what they are.]

23. Now that accident for which we give a certain name to any body, or
the accident which denominates its subject, is commonly called the
ESSENCE thereof; as rationality is the essence of a man; whiteness, of
any white thing, and extension the essence of a body. And the same
essence, in as much as it is generated, is called the FORM. Again, a
body, in respect of any accident, is called the SUBJECT, and in respect
of the form it is called the MATTER.

Also, the production or perishing of any accident makes its subject be
said _to be changed_; only the production or perishing of form makes it
be said it is _generated_ or _destroyed_; but in all generation and
mutation, the name of _matter_ still remains. For a table made of wood
is not only wooden, but wood; and a statue of brass is brass as well as
brazen; though Aristotle, in his _Metaphysics_, says, that whatsoever is
made of any thing ought not to be called ἐκεινὸ, but ἐκέινινον; as that
which is made of wood, not ξύλον, but ξύλινον, that is, not wood, but
wooden.

[Sidenote: First matter, what.]

24. And as for that matter which is common to all things, and which
philosophers, following Aristotle, usually call _materia prima_, that
is, _first matter_, it is not any body distinct from all other bodies,
nor is it one of them. What then is it? A mere name; yet a name which is
not of vain use; for it signifies a conception of body without the
consideration of any form or other accident except only magnitude or
extension, and aptness to receive form and other accident. So that
whensoever we have use of the name _body in general_, if we use that of
_materia prima_, we do well. For as when a man not knowing which was
first, water or ice, would find out which of the two were the matter of
both, he would be fain to suppose some third matter which were neither
of these two; so he that would find out what is the matter of all
things, ought to suppose such as is not the matter of anything that
exists. Wherefore _materia prima_ is nothing; and therefore they do not
attribute to it either form or any other accident besides quantity;
whereas all singular things have their forms and accidents certain.

_Materia prima_, therefore, is body in general, that is, body considered
universally, not as having neither form nor any accident, but in which
no form nor any other accident but quantity are at all considered, that
is, they are not drawn into argumentation.

[Sidenote: That the whole is greater than any part thereof, why
           demonstrated.]

25. From what has been said, those axioms may be demonstrated, which are
assumed by Euclid in the beginning of his first element, about the
equality and inequality of magnitudes; of which, omitting the rest, I
will here demonstrate only this one, _the whole is greater than any part
thereof_; to the end that the reader may know that those axioms are not
indemonstrable, and therefore not principles of demonstration; and from
hence learn to be wary how he admits any thing for a principle, which is
not at least as evident as these are. _Greater_ is defined to be that,
whose part is equal to the whole of another. Now if we suppose any whole
to be A, and a part of it to be B; seeing the whole B is equal to
itself, and the same B is a part of A; therefore a part of A will be
equal to the whole B. Wherefore, by the definition above, A is greater
than B; which was to be proved.


                                -------


                              CHAPTER IX.

                          OF CAUSE AND EFFECT.

1. Action and passion, what they are.—2. Action and passion mediate and
    immediate.—3. Cause simply taken. Cause without which no effect
    follows, or cause necessary by supposition.—4. Cause efficient and
    material.—5. An entire cause is always sufficient to produce its
    effect. At the same instant that the cause is entire, the effect is
    produced. Every effect has a necessary cause.—6. The generation of
    effects is continual. What is the beginning in causation.—7. No
    cause of motion but in a body contiguous and moved.—8. The same
    agents and patients, if alike disposed, produce like effects though
    at different times.—9. All mutation is motion.—10. Contingent
    accidents, what they are.


[Sidenote: Action and passion, what they are.]

1. A body is said to work upon or _act_, that is to say, _do_ something
to another body, when it either generates or destroys some accident in
it: and the body in which an accident is generated or destroyed is said
to _suffer_, that is, to have something _done_ to it by another body; as
when one body by putting forwards another body generates motion in it,
it is called the AGENT; and the body in which motion is so generated, is
called the PATIENT; so fire that warms the hand is the _agent_, and the
hand, which is warmed, is the _patient_. That accident, which is
generated in the patient, is called the EFFECT.

[Sidenote: Action and passion, mediate, and immediate.]

2. When an agent and patient are contiguous to one another, their action
and passion are then said to be _immediate_, otherwise, _mediate_; and
when another body, lying betwixt the agent and patient, is contiguous to
them both, it is then itself both an agent and a patient; an agent in
respect of the body next after it, upon which it works, and a patient in
respect of the body next before it, from which it suffers. Also, if many
bodies be so ordered that every two which are next to one another be
contiguous, then all those that are betwixt the first and the last are
both agents and patients, and the first is an agent only, and the last a
patient only.

[Sidenote: Cause simply taken.]

3. An agent is understood to _produce_ its determined or certain effect
in the patient, according to some certain accident or accidents, with
which both it and the patient are affected; that is to say, the agent
hath its effect precisely such, not because it is a body, but because
such a body, or so moved. For otherwise all agents, seeing they are all
bodies alike, would produce like effects in all patients. And therefore
the fire, for example, does not warm, because it is a body, but because
it is hot; nor does one body put forward another body because it is a
body, but because it is moved into the place of that other body.
[Sidenote: Cause without which no effect follows, or cause necessary by
supposition.]The cause, therefore, of all effects consists in certain
accidents both in the agents and in the patients; which when they are
all present, the effect is produced; but if any one of them be wanting,
it is not produced; and that accident either of the agent or patient,
without which the effect cannot be produced, is called _causa sine qua
non_, or _cause necessary by supposition_, as also the _cause requisite
for the production of the effect_. But a CAUSE simply, or _an entire
cause, is the aggregate of all the accidents both of the agents how many
soever they be, and of the patient, put together; which when they are
all supposed to be present, it cannot be understood but that the effect
is produced at the same instant; and if any one of them be wanting, it
cannot be understood but that the effect is not produced_.

[Sidenote: Cause efficient, and material.]

4. The aggregate of accidents in the agent or agents, requisite for the
production of the effect, the effect being produced, is called the
_efficient cause_ thereof; and the aggregate of accidents in the
patient, the effect being produced, is usually called the _material
cause_; I say the effect being produced; for where there is no effect,
there can be no cause; for nothing can be called a cause, where there is
nothing that can be called an effect. But the efficient and material
causes are both but partial causes, or parts of that cause, which in the
next precedent article I called an entire cause. And from hence it is
manifest, that the effect we expect, though the agents be not defective
on their part, may nevertheless be frustrated by a defect in the
patient; and when the patient is sufficient, by a defect in the agents.

[Sidenote: An entire cause is always sufficient to produce its effect.]

5. An entire cause is always sufficient for the production of its
effect, if the effect be at all possible. For let any effect whatsoever
be propounded to be produced; if the same be produced, it is manifest
that the cause which produced it was a sufficient cause; but if it be
not produced, and yet be possible, it is evident that something was
wanting either in some agent, or in the patient, without which it could
not be produced; that is, that some accident was wanting which was
requisite for its production; and therefore, that cause was not
_entire_, which is contrary to what was supposed.

[Sidenote: At the same instant that the cause is entire, the effect is
           produced.]

It follows also from hence, that in whatsoever instant the cause is
entire, in the same instant the effect is produced. For if it be not
produced, something is still wanting, which is requisite for the
production of it; and therefore the cause was not entire, as was
supposed.

[Sidenote: Every effect has a necessary cause.]

And seeing a necessary cause is defined to be that, which being
supposed, the effect cannot but follow; this also may be collected, that
whatsoever effect is produced at any time, the same is produced by a
necessary cause. For whatsoever is produced, in as much as it is
produced, had an entire cause, that is, had all those things, which
being supposed, it cannot be understood but that the effect follows;
that is, it had a necessary cause. And in the same manner it may be
shewn, that whatsoever effects are hereafter to be produced, shall have
a necessary cause; so that all the effects that have been, or shall be
produced, have their necessity in things antecedent.

[Sidenote: The generation of effects is continual. What is the beginning
           in causation.]

6. And from this, that whensoever the cause is entire, the effect is
produced in the same instant, it is manifest that causation and the
production of effects consist in a certain continual progress; so that
as there is a continual mutation in the agent or agents, by the working
of other agents upon them, so also the patient, upon which they work, is
continually altered and changed. For example: as the heat of the fire
increases more and more, so also the effects thereof, namely, the heat
of such bodies as are next to it, and again, of such other bodies as are
next to them, increase more and more accordingly; which is already no
little argument that all mutation consists in motion only; the truth
whereof shall be further demonstrated in the ninth article. But in this
progress of causation, that is, of action and passion, if any man
comprehend in his imagination a part thereof, and divide the same into
parts, the first part or beginning of it cannot be considered otherwise
than as action or cause; for, if it should be considered as effect or
passion, then it would be necessary to consider something before it, for
its cause or action; which cannot be, for nothing can be before the
beginning. And in like manner, the last part is considered only as
effect; for it cannot be called cause, if nothing follow it; but after
the last, nothing follows. And from hence it is, that in all action the
beginning and cause are taken for the same thing. But every one of the
intermediate parts are both action and passion, and cause and effect,
according as they are compared with the antecedent or subsequent part.

[Sidenote: No cause of motion but in a body contiguous and moved.]

7. There can be no cause of motion, except in a body contiguous and
moved. For let there be any two bodies which are not contiguous, and
betwixt which the intermediate space is empty, or, if filled, filled
with another body which is at rest; and let one of the propounded bodies
be supposed to be at rest; I say it shall always be at rest. For if it
shall be moved, the cause of that motion, by the 8th chapter, article
19, will be some external body; and, therefore, if between it and that
external body there be nothing but empty space, then whatsoever the
disposition be of that external body or of the patient itself, yet if it
be supposed to be now at rest, we may conceive it will continue so till
it be touched by some other body. But seeing cause, by the definition,
is the aggregate of all such accidents, which being supposed to be
present, it cannot be conceived but that the effect will follow, those
accidents, which are either in external bodies, or in the patient
itself, cannot be the cause of future motion. And in like manner, seeing
we may conceive that whatsoever is at rest will still be at rest, though
it be touched by some other body, except that other body be moved;
therefore in a contiguous body, which is at rest, there can be no cause
of motion. Wherefore there is no cause of motion in any body, except it
be contiguous and moved.

The same reason may serve to prove that whatsoever is moved, will always
be moved on in the same way and with the same velocity, except it be
hindered by some other contiguous and moved body; and consequently that
no bodies, either when they are at rest, or when there is an
interposition of vacuum, can generate or extinguish or lessen motion in
other bodies. There is one that has written that things moved are more
resisted by things at rest, than by things contrarily moved; for this
reason, that he conceived motion not to be so contrary to motion as
rest. That which deceived him was, that the words _rest_ and _motion_
are but contradictory names; whereas motion, indeed, is not resisted by
rest, but by contrary motion.

[Sidenote: The same agents and patients, if alike disposed, produce like
           effects, though at different times.]

8. But if a body work upon another body at one time, and afterwards the
same body work upon the same body at another time, so that both the
agent and patient, and all their parts, be in all things as they were;
and there be no difference, except only in time, that is, that one
action be former, the other later in time; it is manifest of itself,
that the effects will be equal and like, as not differing in anything
besides time. And as effects themselves proceed from their causes, so
the diversity of them depends upon the diversity of their causes also.

[Sidenote: All mutation is motion.]

9. This being true, it is necessary that mutation can be nothing else
but motion of the parts of that body which is changed. For first, we do
not say anything is changed, but that which appears to our senses
otherwise than it appeared formerly. Secondly, both those appearances
are effects produced in the sentient; and, therefore, if they be
different, it is necessary, by the preceding article, that either some
part of the agent, which was formerly at rest, is now moved, and so the
mutation consists in this motion; or some part, which was formerly
moved, is now otherwise moved, and so also the mutation consists in this
new motion; or which, being formerly moved, is now at rest, which, as I
have shewn above, cannot come to pass without motion; and so again,
mutation is motion; or lastly, it happens in some of these manners to
the patient, or some of its parts; so that mutation, howsoever it be
made, will consist in the motion of the parts, either of the body which
is perceived, or of the sentient body, or of both. Mutation therefore is
motion, namely, of the parts either of the agent or of the patient;
which was to be demonstrated. And to this it is consequent, that rest
cannot be the cause of anything, nor can any action proceed from it;
seeing neither motion nor mutation can be caused by it.

[Sidenote: Contingent accidents.]

10. Accidents, in respect of other accidents which precede them, or are
before them in time, and upon which they do not depend as upon their
causes, are called _contingent_ accidents; I say, in respect of those
accidents by which they are not generated; for, in respect of their
causes, all things come to pass with equal necessity; for otherwise they
would have no causes at all; which, of things generated, is not
intelligible.


                                -------


                               CHAPTER X.

                           OF POWER AND ACT.

1. Power and cause are the same thing.—2. An act is produced at the same
    instant in which the power is plenary.—3. Active and passive power
    are parts only of plenary power.—4. An act, when said to be
    possible.—5. An act necessary and contingent, what.—6. Active power
    consists in motion.—7. Cause, formal and final, what they are.


[Sidenote: Power and cause are the same thing.]

1. Correspondent to _cause_ and _effect_, are POWER and ACT; nay, those
and these are the same things; though, for divers considerations, they
have divers names. For whensoever any agent has all those accidents
which are necessarily requisite for the production of some effect in the
patient, then we say that agent has _power_ to produce that effect, if
it be applied to a patient. But, as I have shewn in the precedent
chapter, those accidents constitute the efficient cause; and therefore
the same accidents, which constitute the efficient cause, constitute
also the _power_ of the agent. Wherefore the _power of the agent_ and
the _efficient cause_ are the same thing. But they are considered with
this difference, that _cause_ is so called in respect of the effect
already produced, and power in respect of the same effect to be produced
hereafter; so that _cause_ respects the past, _power_ the future time.
Also, the _power of the agent_ is that which is commonly called _active
power_.

In like manner, whensoever any patient has all those accidents which it
is requisite it should have, for the production of some effect in it, we
say it is in the _power_ of that patient to produce that effect, if it
be applied to a fitting agent. But those accidents, as is defined in the
precedent chapter, constitute the material cause; and therefore the
_power of the patient_, commonly called _passive power_, and _material
cause_, are the same thing; but with this different consideration, that
in cause the past time, and in power the future, is respected. Wherefore
the power of the agent and patient together, which may be called entire
or _plenary power_, is the same thing with _entire cause_; for they both
consist in the sum or aggregate of all the accidents, as well in the
agent as in the patient, which are requisite for the production of the
effect. Lastly, as the accident produced is, in respect of the cause,
called an effect, so in respect of the power, it is called an _act_.

[Sidenote: An act is produced at the same instant in which the power is
           plenary.]

2. As therefore the effect is produced in the same instant in which the
cause is entire, so also every act that may be produced, is produced in
the same instant in which the power is plenary. And as there can be no
effect but from a sufficient and necessary cause, so also no act can be
produced but by sufficient power, or that power by which it could not
but be produced.

[Sidenote: Active and passive power are parts only of plenary power.]

3. And as it is manifest, as I have shewn, that the efficient and
material causes are severally and by themselves parts only of an entire
cause, and cannot produce any effect but by being joined together, so
also power, active and passive, are parts only of plenary and entire
power; nor, except they be joined, can any act proceed from them; and
therefore these powers, as I said in the first article, are but
conditional, namely, _the agent has power, if it be applied to a
patient; and the patient has power, if it be applied to an agent_;
otherwise neither of them have power, nor can the accidents, which are
in them severally, be properly called powers; nor any action be said to
be possible for the power of the agent alone or of the patient alone.

[Sidenote: An act, when said to be possible.]

4. For that is an impossible act, for the production of which there is
no power plenary. For seeing plenary power is that in which all things
concur, which are requisite for the production of an act, if the power
shall never be plenary, there will always be wanting some of those
things, without which the act cannot be produced; wherefore that act
shall never be produced; that is, that act is IMPOSSIBLE: and every act,
which is not impossible, is POSSIBLE. Every act, therefore, which is
possible, shall at some time be produced; for if it shall never be
produced, then those things shall never concur which are requisite for
the production of it; wherefore that act is _impossible_, by the
definition; which is contrary to what was supposed.

[Sidenote: An act necessary and contingent, what.]

5. A _necessary act_ is that, the production whereof it is impossible to
hinder; and therefore every act, that shall be produced, shall
necessarily be produced; for, that it shall not be produced, is
impossible; because, as is already demonstrated, every possible act
shall at some time be produced; nay, this proposition, _what shall be,
shall be_, is as necessary a proposition as this, _a man is a man_.

But here, perhaps, some man may ask whether those future things, which
are commonly called _contingents_, are necessary. I say, therefore, that
generally all contingents have their necessary causes, as is shewn in
the preceding chapter; but are called contingents in respect of other
events, upon which they do not depend; as the rain, which shall be
tomorrow, shall be necessary, that is, from necessary causes; but we
think and say it happens by chance, because we do not yet perceive the
causes thereof, though they exist now; for men commonly call that
_casual_ or _contingent_, whereof they do not perceive the necessary
cause; and in the same manner they used to speak of things past, when
not knowing whether a thing be done or no, they say it is possible it
never was done.

Wherefore, all propositions concerning future things, contingent or not
contingent, as this, _it will rain tomorrow_, or this, _tomorrow the sun
will rise_, are either necessarily true, or necessarily false; but we
call them contingent, because we do not yet know whether they be true or
false; whereas their verity depends not upon our knowledge, but upon the
foregoing of their causes. But there are some, who though they confess
this whole proposition, _tomorrow it will either rain, or not rain_, to
be true, yet they will not acknowledge the parts of it, as, _tomorrow it
will rain_, or, _tomorrow it will not rain_, to be either of them true
by itself; because they say neither this nor that is true
_determinately_. But what is this _determinately true_, but true upon
our knowledge, or evidently true? And therefore they say no more but
that it is not yet known whether it be true or no; but they say it more
obscurely, and darken the evidence of the truth with the same words,
with which they endeavour to hide their own ignorance.

[Sidenote: Active power consists in motion.]

6. In the 9th article of the preceding chapter, I have shewn that the
efficient cause of all motion and mutation consists in the motion of the
agent, or agents; and in the first article of this chapter, that the
power of the agent is the same thing with the efficient cause. From
whence it may be understood, that all active power consists in motion
also; and that power is not a certain accident, which differs from all
acts, but is, indeed, an act, namely, motion, which is therefore called
power, because another act shall be produced by it afterwards. For
example, if of three bodies the first put forward the second, and this
the third, the motion of the second, in respect of the first which
produceth it, is the act of the second body; but, in respect of the
third, it is the active power of the same second body.

[Sidenote: Cause, formal and final, what they are.]

7. The writers of metaphysics reckon up two other causes besides the
_efficient_ and _material_, namely, the ESSENCE, which some call the
_formal cause_, and the END, or _final cause_; both which are
nevertheless efficient causes. For when it is said the essence of a
thing is the cause thereof, _as to be rational is the cause of man_, it
is not intelligible; for it is all one, as if it were said, _to be a man
is the cause of man_; which is not well said. And yet the knowledge of
the _essence_ of anything, is the cause of the knowledge of the thing
itself; for, if I first know that a thing is _rational_, I know from
thence, that the same is _man_; but this is no other than an efficient
cause. A _final cause_ has no place but in such things as have sense and
will; and this also I shall prove hereafter to be an efficient cause.


                                -------


                              CHAPTER XI.

                      OF IDENTITY AND DIFFERENCE.

1. What it is for one thing to differ from another.—2. To differ in
    number, magnitude, species, and genus, what.—3. What is relation,
    proportion, and relatives.—4. Proportionals, what.—5. The proportion
    of magnitudes to one another, wherein it consists.—6. Relation is no
    new accident, but one of those that were in the relative before the
    relation or comparison was made. Also the causes of accidents in the
    correlatives, are the cause of relation.—7. Of the beginning of
    individuation.


[Sidenote: What it is for one thing to differ from another.]

1. Hitherto I have spoken of body simply, and accidents common to all
bodies, as _magnitude_, _motion_, _rest_, _action_, _passion_, _power_,
_possible_, _&c._; and I should now descend to those accidents by which
one body is distinguished from another, but that it is first to be
declared what it is to be _distinct_ and _not distinct_, namely, what
are the SAME and DIFFERENT; for this also is common to all bodies, that
they may be distinguished and differenced from one another. Now, two
bodies are said to _differ_ from one another, when something may be said
of one of them, which cannot be said of the other at the same time.

[Sidenote: To differ in number, magnitude, species, and genus, what.]

2. And, first of all, it is manifest that no two bodies are the _same_;
for seeing they are two, they are in two places at the same time; as
that, which is the _same_, is at the same time in one and the same
place. All bodies therefore differ from one another in _number_, namely,
as one and another; so that the _same_ and _different in number_, are
names opposed to one another by contradiction.

In _magnitude_ bodies differ when one is greater than another, as _a
cubit long_, and _two cubits long_, of _two pound weight_, and of _three
pound weight_. And to these, _equals_ are opposed.

Bodies, which differ more than in magnitude, are called _unlike_; and
those, which differ only in magnitude, _like_. Also, of unlike bodies,
some are said to differ in the _species_, others in the _genus_; in the
_species_, when their difference is perceived by one and the same sense,
as _white_ and _black_; and in the _genus_, when their difference is not
perceived but by divers senses, as _white_ and _hot_.

[Sidenote: What is relation, proportion, and relatives.]

3. And the _likeness_, or _unlikeness_, _equality_, or _inequality_ of
one body to another, is called their RELATION; and the bodies themselves
_relatives_ or _correlatives_; _Aristotle_ calls them τὰ πρὸς τί; the
first whereof is usually named the _antecedent_, and the second the
_consequent_; and the _relation_ of the antecedent to the consequent,
according to magnitude, namely, the equality, the excess or defect
thereof, is called the PROPORTION of the antecedent to the consequent;
so that _proportion_ is nothing but the equality or inequality of the
magnitude of the antecedent compared to the magnitude of the consequent
by their difference only, or compared also with their difference. For
example, the _proportion_ of three to two consists only in this, that
three _exceeds_ two by unity; and the proportion of two to five in this,
that two, compared with five, is _deficient_ of it by three, either
simply, or compared with the numbers different; and therefore in the
proportion of unequals, the proportion of the less to the greater, is
called DEFECT; and that of the greater to the less, EXCESS.

[Sidenote: Proportionals, what.]

4. Besides, of unequals, some are more, some less, and some equally
unequal; so that there is _proportion of proportions_, as well as of
_magnitudes_; namely, where two unequals have relation to two other
unequals; as, when the inequality which is between 2 and 3, is compared
with the inequality which is between 4 and 5. In which comparison there
are always four magnitudes; or, which is all one, if there be but three,
the middlemost is twice numbered; and if the proportion of the first to
the second, be equal to the proportion of the third to the fourth, then
the four are said to be _proportionals_; otherwise they are not
proportionals.

[Sidenote: The proportion of magnitudes to one another, wherein it
           consists.]

5. The proportion of the antecedent to the consequent consists in their
difference, not only simply taken, but also as compared with one of the
relatives; that is, either in that part of the greater, by which it
exceeds the less, or in the remainder, after the less is taken out of
the greater; as the proportion of two to five consists in the three by
which five exceeds two, not in three simply only, but also as compared
with five or two. For though there be the same difference between two
and five, which is between nine and twelve, namely three, yet there is
not the same inequality; and therefore the proportion of two to five is
not in all relation the same with that of nine to twelve, but only in
that which is called arithmetical.

[Sidenote: Relation is no new accident, but one of those that were in
           the relative, before the relation or comparison was made.
           Also the causes of accidents in correlatives are the cause of
           relation.]

6. But we must not so think of relation, as if it were an accident
differing from all the other accidents of the relative; but one of them,
namely, that by which the comparison is made. For example, the likeness
of one _white_ to another _white_, or its unlikeness to _black_, is the
same accident with its _whiteness_; and _equality_ and _inequality_, the
same accident with the _magnitude_ of the thing compared, though under
another name: for that which is called _white_ or _great_, when it is
not compared with something else, the same when it is compared, is
called _like_ or _unlike_, _equal_ or _unequal_. And from this it
follows that the causes of the accidents, which are in relatives, are
the causes also of _likeness_, _unlikeness_, _equality_ and
_inequality_; namely, that he, that makes two unequal bodies, makes also
their inequality; and he, that makes a rule and an action, makes also,
if the action be congruous to the rule, their congruity; if incongruous,
their incongruity. And thus much concerning _comparison_ of one body
with another.

[Sidenote: Of the beginning of individuation.]

7. But the same body may at different times be compared with itself. And
from hence springs a great controversy among philosophers about the
_beginning of individuation_, namely, in what sense it may be conceived
that a body is at one time the same, at another time not the same it was
formerly. For example, whether a man grown old be the same man he was
whilst he was young, or another man; or whether a city be in different
ages the same, or another city. Some place _individuity_ in the unity of
_matter_; others, in the unity of _form_; and one says it consists in
the unity of the _aggregate of all the accidents together_. For
_matter_, it is pleaded that a lump of wax, whether it be spherical or
cubical, is the same wax, because the same matter. For _form_, that when
a man is grown from an infant to be an old man, though his matter be
changed, yet he is still the same numerical man; for that _identity_
which cannot be attributed to the matter, ought probably to be ascribed
to the form. For the _aggregate of accidents_, no instance can be made;
but because, when any new accident is generated, a new name is commonly
imposed on the thing, therefore he, that assigned this cause of
_individuity_, thought the thing itself also was become another thing.
According to the first opinion, he that sins, and he that is punished,
should not be the same man, by reason of the perpetual flux and change
of man's body; nor should the city, which makes laws in one age and
abrogates them in another, be the same city; which were to confound all
civil rights. According to the second opinion, two bodies existing both
at once, would be one and the same numerical body. For if, for example,
that ship of Theseus, concerning the difference whereof made by
continual reparation in taking out the old planks and putting in new,
the sophisters of Athens were wont to dispute, were, after all the
planks were changed, the same numerical ship it was at the beginning;
and if some man had kept the old planks as they were taken out, and by
putting them afterwards together in the same order, had again made a
ship of them, this, without doubt, had also been the same numerical ship
with that which was at the beginning; and so there would have been two
ships numerically the same, which is absurd. But, according to the third
opinion, nothing would be the same it was; so that a man standing would
not be the same he was sitting; nor the water, which is in the vessel,
the same with that which is poured out of it. Wherefore the beginning of
_individuation_ is not always to be taken either from matter alone, or
from form alone.

But we must consider by what name anything is called, when we inquire
concerning the _identity_ of it. For it is one thing to ask concerning
Socrates, whether he be the same man, and another to ask whether he be
the same body; for his body, when he is old, cannot be the same it was
when he was an infant, by reason of the difference of magnitude; for one
body has always one and the same magnitude; yet, nevertheless, he may be
the same man. And therefore, whensoever the name, by which it is asked
whether a thing be the same it was, is given it for the matter only,
then, if the matter be the same, the thing also is _individually_ the
same; as the water, which was in the sea, is the same which is
afterwards in the cloud; and any body is the same, whether the parts of
it be put together, or dispersed; or whether it be congealed, or
dissolved. Also, if the name be given for such form as is the beginning
of motion, then, as long as that motion remains, it will be the same
_individual_ thing; as that man will be always the same, whose actions
and thoughts proceed all from the same beginning of motion, namely, that
which was in his generation; and that will be the same river which flows
from one and the same fountain, whether the same water, or other water,
or something else than water, flow from thence; and that the same city,
whose acts proceed continually from the same institution, whether the
men be the same or no. Lastly, if the name be given for some accident,
then the _identity_ of the thing will depend upon the matter; for, by
the taking away and supplying of matter, the accidents that were, are
destroyed, and other new ones are generated, which cannot be the same
numerically; so that a ship, which signifies matter so figured, will be
the same as long as the matter remains the same; but if no part of the
matter be the same, then it is numerically another ship; and if part of
the matter remain and part be changed, then the ship will be partly the
same, and partly not the same.


                                -------


                              CHAPTER XII.

                              OF QUANTITY.

1. The definition of quantity.—2. The exposition of quantity, what it
    is.—3. How line, superficies, and solid, are exposed.—4. How time is
    exposed.—5. How number is exposed.—6. How velocity is exposed.—7.
    How weight is exposed.—8. How the proportion of magnitudes is
    exposed.—9. How the proportion of times and velocities is exposed.


[Sidenote: Definition of quantity.]

1. What and how manifold _dimension_ is, has been said in the 8th
chapter, namely, that there are three dimensions, line or length,
superficies, and solid; every one of which, if it be determined, that
is, if the limits of it be made known, is commonly called _quantity_;
for by _quantity_ all men understand that which is signified by that
word, by which answer is made to the question, _How much is it?_
Whensoever, therefore, it is asked, for example, _How long is the
journey?_ it is not answered indefinitely, _length_; nor, when it is
asked, _How big is the field?_ is it answered indefinitely,
_superficies_; nor, if a man ask, _How great is the bulk?_ indefinitely,
_solid_: but it is answered determinately, the journey is a hundred
miles; the field is a hundred acres; the bulk is a hundred cubical feet;
or at least in some such manner, that the magnitude of the thing
enquired after may by certain limits be comprehended in the mind.
QUANTITY, therefore, cannot otherwise be defined, than to be _a
dimension determined_, or _a dimension, whose limits are set out, either
by their place, or by some comparison_.

[Sidenote: The exposition of quantity, what it is.]

2. And _quantity_ is determined two ways; one, by the sense, when some
sensible object is set before it; as when a line, a superficies or
solid, of a foot or cubit, marked out in some matter, is objected to the
eyes; which way of determining, is called _exposition_, and the quantity
so known is called _exposed quantity_; the other by memory, that is, by
comparison with some exposed quantity. In the first manner, when it is
asked of what quantity a thing is, it is answered, of such quantity as
you see exposed. In the second manner, answer cannot be made but by
comparison with some exposed quantity; for if it be asked, how long is
the way? the answer is, so many thousand paces; that is, by comparing
the way with a pace, or some other measure, determined and known by
exposition; or the quantity of it is to some other quantity known by
exposition, as the diameter of a square is to the side of the same, or
by some other the like means. But it is to be understood, that the
quantity exposed must be some standing or permanent thing, such as is
marked out in consistent or durable matter; or at least something which
is revocable to sense; for otherwise no comparison can be made by it.
Seeing, therefore, by what has been said in the next preceding chapter,
comparison of one magnitude with another is the same thing with
proportion; it is manifest, that quantity determined in the second
manner is nothing else but the proportion of a dimension not exposed to
another which is exposed; that is, the comparison of the equality or
inequality thereof with an exposed quantity.

[Sidenote: How line, superficies, and solids, are exposed.]

3. _Lines_, _superficies_, and _solids_, are exposed, first, by
_motion_, in such manner as in the 8th chapter I have said they are
generated; but so as that the marks of such motion be permanent; as when
they are designed upon some matter, as a line upon paper; or graven in
some durable matter. Secondly, by _apposition_; as when one line or
length is applied to another line or length, one breadth to another
breadth, and one thickness to another thickness; which is as much as to
describe a line by points, a superficies by lines, and a solid by
superficies; saving that by points in this place are to be understood
very short lines; and, by superficies, very thin solids. Thirdly, lines
and superficies may be exposed by _section_, namely, a line may be made
by cutting an exposed superficies; and a superficies, by the cutting of
an exposed solid.

[Sidenote: How time is exposed.]

4. _Time_ is exposed, not only by the exposition of a line, but also of
some moveable thing, which is moved uniformly upon that line, or at
least is supposed so to be moved. For, seeing time is an idea of motion,
in which we consider former and latter, that is succession, it is not
sufficient for the exposition of time that a line be described; but we
must also have in our mind an imagination of some moveable thing passing
over that line; and the motion of it must be uniform, that time may be
divided and compounded as often as there shall be need. And, therefore,
when philosophers, in their demonstrations, draw a line, and say, _Let
that line be time_, it is to be understood as if they said, _Let the
conception of uniform motion upon that line, be time_. For though the
circles in dials be lines, yet they are not of themselves sufficient to
note time by, except also there be, or be supposed to be, a motion of
the shadow or the hand.

[Sidenote: How number is exposed.]

5. _Number_ is exposed, either by the exposition of points, or of the
names of number, _one_, _two_, _three_, _&c._; and those points must not
be contiguous, so as that they cannot be distinguished by notes, but
they must be so placed that they may be _discerned_ one from another;
for, from this it is, that number is called _discreet quantity_, whereas
all quantity, which is designed by motion, is called _continual
quantity_. But that number may be exposed by the names of number, it is
necessary that they be recited by heart and in order, as one, two,
three, &c.; for by saying one, one, one, and so forward, we know not
what number we are at beyond two or three; which also appear to us in
this manner, not as number, but as figure.

[Sidenote: How velocity is exposed.]

6. For the exposition of _velocity_, which, by the definition thereof,
is a motion which, in a certain time, passeth over a certain space, it
is requisite, not only that time be exposed, but that there be also
exposed that space which is transmitted by the body, whose velocity we
would determine; and that a body be understood to be moved in that space
also; so that there must be exposed two lines, upon one of which uniform
motion must be understood to be made, that the time may be determined;
and, upon the other, the velocity is to be computed. As if we would
expose the velocity of the we would expose the velocity of the body A,
we draw two lines A B and C D, and place a body in C also; which done,
we say the velocity of the body A is so great, that it passeth over the
line A B in the same time in which the body C passeth over the line C D
with uniform motion.

                                 A    B
                                 ------
                                 C    D
                                 ------

[Sidenote: How weight is exposed.]

7. _Weight_ is exposed by any heavy body, of what matter soever, so it
be always alike heavy.

[Sidenote: How the proportion of magnitudes is exposed.]

8. The _proportion_ of two magnitudes is then exposed, when the
magnitudes themselves are exposed, namely, the proportion of equality,
when the magnitudes are equal; and of inequality, when they are unequal.
For seeing, by the 5th article of the preceding chapter, the proportion
of two unequal magnitudes consists in their difference, compared with
either of them; and when two unequal magnitudes are exposed, their
difference is also exposed; it follows, that when magnitudes, which have
proportion to one another, are exposed, their proportion also is exposed
with them; and, in like manner, the proportion of equals, which consists
in this, that there is no difference of magnitude betwixt them, is
exposed at the same time when the equal magnitudes themselves are
exposed. For example, if the exposed lines A B and C D be equal, the
proportion of equality is exposed in them; and if the exposed lines, E F
and E G be unequal, the proportion which E F has to E G, and that which
E G has to E F are also exposed in them; for not only the lines
themselves, but also their difference, G F, is exposed. The proportion
of unequals is quantity; for the difference, G F, in which it consists,
is quantity. But the proportion of equality is not quantity; because,
between equals, there is no difference; nor is one equality greater than
another, as one inequality is greater than another inequality.

                                A     B
                                -------
                                C     D
                                -------
                                E  F  F
                                -------

[Sidenote: How the proportion of times and velocities is exposed.]

The proportion of two _times_, or of two _uniform velocities_, is then
exposed, when two lines are exposed by which two bodies are understood
to be moved uniformly; and therefore the same two lines serve to exhibit
both their own proportion, and that of the times and velocities,
according as they are considered to be exposed for the magnitudes
themselves, or for the times or velocities. For let the two lines A and
B be exposed; their proportion therefore (by the last foregoing article)
is exposed; and if they be considered as drawn with equal and uniform
velocity, then, seeing their times are greater, or equal, or less,
according as the same spaces are transmitted in greater, or equal, or
less time, the lines A and B will exhibit the equality or inequality,
that is, the proportion of the times. To conclude, if the same lines, A
and B, be considered as drawn in the same time, then, seeing their
velocities are greater, or equal, or less, according as they pass over
in the same time longer, or equal, or shorter lines, the same lines, A
and B, will exhibit the equality, or inequality, that is, the proportion
of their velocities.

                                   A
                                  ---
                                   B


                                -------


                             CHAPTER XIII.

                 OF ANALOGISM, OR THE SAME PROPORTION.

1, 2, 3, 4. The nature and definition of proportion, arithmetical and
    geometrical.—5. The definition, and some properties of the same
    arithmetical proportion.—6, 7. The definition and transmutations of
    analogism, or the same geometrical proportion.—8, 9.. The
    definitions of hyperlogism and hypologism, that is, of greater and
    less proportion, and their transmutations.—10, 11, 12. Comparison of
    analogical quantities, according to magnitude.—13, 14, 15.
    Composition of proportions.—16, 17, 18, 19, 20, 21, 22, 23, 24, 25.
    The definition and properties of continual proportion.—26, 27, 28,
    29. Comparison of arithmetical and geometrical proportions.

  [Note, that in this chapter the sign + signifies that the quantities
  betwixt which it is put, are added together; and this sign - the
  remainder after the latter quantity is taken out of the former. So
  that A + B is equal to both A and B together; and where you see A - B,
  there A is the whole, B the part taken out of it, and A - B the
  remainder. Also, two letters, set together without any sign, signify,
  unless they belong to a figure, that one of the quantities is
  multiplied by the other; as A B signifies the product of A multiplied
  by B.]


[Sidenote: The nature and definition of proportion, arithmetical &
           geometrical.]

1. Great and little are not intelligible, but by comparison. Now that,
to which they are compared, is something exposed; that is, some
magnitude either perceived by sense, or so defined by words, that it may
be comprehended by the mind. Also that, to which any magnitude is
compared, is either greater or less, or equal to it. And therefore
proportion (which, as I have shewn, is the estimation or comprehension
of magnitudes by comparison,) is threefold, namely, proportion of
_equality_, that is, of equal to equal; or of _excess_, which is of the
greater to the less; or of _defect_, which is the proportion of the less
to the greater.

Again, every one of these proportions is two-fold; for if it be asked
concerning any magnitude given, how great it is, the answer may be made
by comparing it two ways; first, by saying it is greater or less than
another magnitude, by so much; as seven is less than ten, by three
unities; and this is called _arithmetical proportion_. Secondly, by
saying it is greater or less than another magnitude, by such a part or
parts thereof; as seven is less than ten, by three tenth parts of the
same ten. And though this proportion be not always explicable by number,
yet it is a determinate proportion, and of a different kind from the
former, and called _geometrical proportion_, and most commonly
_proportion simply_.

2. Proportion, whether it be arithmetical or geometrical, cannot be
exposed but in two magnitudes, (of which the former is commonly called
the _antecedent_, and the latter the _consequent_ of the proportion) as
I have shewn in the 8th article of the preceding chapter. And,
therefore, if two proportions be to be compared, there must be four
magnitudes exposed, namely, two antecedents and two consequents; for
though it happen sometimes that the consequent of the former proportion
be the same with the antecedent of the latter, yet in that double
comparison it must of necessity be twice numbered; so that there will be
always four terms.

3. Of two proportions, whether they be arithmetical or geometrical, when
the magnitudes compared in both (which Euclid, in the fifth definition
of his sixth book, calls the _quantities of proportions_,) are equal,
then one of the proportions cannot be either greater or less than the
other; for one equality is neither greater nor less than another
equality. But of two proportions of inequality, whether they be
proportions of excess or of defect, one of them may be either greater or
less than the other, or they may both be equal; for though there be
propounded two magnitudes that are unequal to one another, yet there may
be other two more, unequal, and other two equally unequal, and other two
less unequal than the two which were propounded. And from hence it may
be understood, that the proportions of excess and defect are quantity,
being capable of more and less; but the proportion of equality is not
quantity, because not capable neither of _more_, nor of _less_. And
therefore proportions of inequality may be added together, or subtracted
from one another, or be multiplied or divided by one another, or by
number; but proportions of equality not so.

4. Two equal proportions are commonly called _the same proportion_; and,
it is said, that the proportion of the first antecedent to the first
consequent is the _same_ with that of the second antecedent to the
second consequent. And when four magnitudes are thus to one another in
geometrical proportion, they are called _proportionals_; and by some,
more briefly, _analogism_. And _greater proportion_ is the proportion of
a greater antecedent to the same consequent, or of the same antecedent
to a less consequent; and when the proportion of the first antecedent to
the first consequent is greater than that of the second antecedent to
the second consequent, the four magnitudes, which are so to one another,
may be called _hyperlogism_.

_Less proportion_ is the proportion of a less antecedent to the same
consequent, or of the same antecedent to a greater consequent; and when
the proportion of the first antecedent to the first consequent is less
than that of the second to the second, the four magnitudes may be called
_hypologism_.

[Sidenote: The definition and some properties of the same arithmetical
           proportion.]

5. One arithmetical proportion is the _same_ with another arithmetical
proportion, when one of the antecedents exceeds its consequent, or is
exceeded by it, as much as the other antecedent exceeds its consequent,
or is exceeded by it. And therefore, in four magnitudes, arithmetically
proportional, the sum of the extremes is equal to the sum of the means.
For if A. B :: C. D be arithmetically proportional, and the difference
on both sides be the same excess, or the same defect, E, then B + C (if
A be greater than B) will be equal to A - E + C; and A + D will be equal
to A + C - E; but A - E + C and A + C - E are equal. Or if A be less
than B, then B + C will be equal to A + E + C; and A + D will be equal
to A + C + E; but A + E + C and A + C + E are equal.

Also, if there be never so many magnitudes, arithmetically proportional,
the sum of them all will be equal to the product of half the number of
the terms multiplied by the sum of the extremes.

For if A. B :: C. D :: E. F be arithmetically proportional, the couples
A + F, B + E, C + D will be equal to one another; and their sum will be
equal to A + F, multiplied by the number of their combinations, that is,
by half the number of the terms.

If, of four unequal magnitudes, any two, together taken, be equal to the
other two together taken, then the greatest and the least of them will
be in the same combination. Let the unequal magnitudes be A, B, C, D;
and let A + B be equal to C + D; and let A be the greatest of them all;
I say B will be the least. For, if it may be, let any of the rest, as D,
be the least. Seeing therefore A is greater than C, and B than D, A+B
will be greater than C + D; which is contrary to what was supposed.

If there be any four magnitudes, the sum of the greatest and least, the
sum of the means, the difference of the two greatest, and the difference
of the two least, will be arithmetically proportional. For, let there be
four magnitudes, whereof A is the greatest, D the least, and B and C the
means; I say A + D. B + C :: A - B. C - D are arithmetically
proportional. For the difference between the first antecedent and its
consequent is this, A + D - B - C; and the difference between the second
antecedent and its consequent this, A - B - C + D; but these two
differences are equal; and therefore, by this 5th article, A + D. B + C
:: A - B. C - D are arithmetically proportional.

If, of four magnitudes, two be equal to the other two, they will be in
reciprocal arithmetical proportion. For let A + B be equal to C + D, I
say A. C :: D. B are arithmetically proportional. For if they be not,
let A. C :: D. E (supposing E to be greater or less than B) be
arithmetically proportional, and then A + E will be equal to C + D;
wherefore A + B and C + D are not equal; which is contrary to what was
supposed.

[Sidenote: The definition and transmutations of analogism, or the same
           geometrical proportion.]

6. One geometrical proportion is the same with another geometrical
proportion; when the same cause, producing equal effects in equal times,
determines both the proportions.

If a point uniformly moved describe two lines, either with the same, or
different velocity, all the parts of them which are contemporary, that
is, which are described in the same time, will be two to two, in
geometrical proportion, whether the antecedents be taken in the same
line, or not. For, from the point A (in the 10th figure at the end of
the 14th chapter) let the two lines, A D, A G, be described with uniform
motion; and let there be taken in them two parts A B, A E, and again,
two other parts, A C, A F; in such manner, that A B, A E, be
contemporary, and likewise A C, A F contemporary. I say first (taking
the antecedents A B, A C in the line A D, and the conquents A E, A F in
the line A G) that A B. A C :: A E. A F are proportionals. For seeing
(by the 8th chap, and the 15th art.) velocity is motion considered as
determined by a certain length or line, in a certain time transmitted by
it, the quantity of the line A B will be determined by the velocity and
time by which the same A B is described; and for the same reason, the
quantity of the line A C will be determined by the velocity and time, by
which the same A C is described; and therefore the proportion of A B to
A C, whether it be proportion of equality, or of excess or defect, is
determined by the velocities and times by which A B, A C are described;
but seeing the motion of the point A upon A B and A C is uniform, they
are both described with equal velocity; and therefore whether one of
them have to the other the proportion of majority or of minority, the
sole cause of that proportion is the difference of their times; and by
the same reason it is evident, that the proportion of A E to A F is
determined by the difference of their times only. Seeing therefore A B,
A E, as also A C, A F are contemporary, the difference of the times in
which A B and A C are described, is the same with that in which A E and
A F are described. Wherefore the proportion of A B to A C, and the
proportion of A E to A F are both determined by the same cause. But the
cause, which so determines the proportion of both, works equally in
equal times, for it is uniform motion; and therefore (by the last
precedent definition) the proportion of A B to A C is the same with that
of A E to A F; and consequently A B. A C :: A E. A F are proportionals;
which is the first.

Secondly, (taking the antecedents in different lines) I say, A B. A E ::
A C. A F are proportionals; for seeing A B, A E are described in the
same time, the difference of the velocities in which they are described
is the sole cause of the proportion they have to one another. And the
same may be said of the proportion of A C to A F. But seeing both the
lines A D and A G are passed over by uniform motion, the difference of
the velocities in which A B, A E are described, will be the same with
the difference of the velocities, in which A C, A F are described.
Wherefore the cause which determines the proportion of A B to A E, is
the same with that which determines the proportion of A C to A F; and
therefore A B. A E :: A C. A F, are proportionals; which remained to be
proved.

Coroll. I. If four magnitudes be in geometrical proportion, they will
also be proportionals by _permutation_, that is, by transposing the
middle terms. For I have shown, that not only A B. A C :: A E. A F, but
also that, by _permutation_, A B. A E :: A C. A F are proportionals.

Coroll. II. If there be four proportionals, they will also be
proportionals by _inversion_ or _conversion_, that is, by turning the
antecedents into consequents. For if in the last _analogism_, I had for
A B, A C, put by inversion A C, A B, and in like manner converted A E, A
F into A F, A E, yet the same demonstration had served. For as well A C,
A B, as A B, A C are of equal velocity; and A C, A F, as well as A F, A
C are contemporary.

Coroll. III. If proportionals be added to proportionals, or taken from
them, the aggregates, or remainders, will be proportionals. For
contemporaries, whether they be added to contemporaries, or taken from
them, make the aggregates or remainders contemporary, though the
addition or subtraction be of all the terms, or of the antecedents
alone, or of the consequents alone.

Coroll. IV. If both the antecedents of four proportionals, or both the
consequents, or all the terms, be multiplied or divided by the same
number or quantity, the products or quotients will be proportionals. For
the multiplication and division of proportionals, is the same with the
addition and subtraction of them.

Coroll. V. If there be four proportionals, they will also be
proportionals by _composition_, that is, by compounding an antecedent of
the antecedent and consequent put together, and by taking for consequent
either the consequent singly, or the antecedent singly. For this
composition is nothing but addition of proportionals, namely, of
consequents to their own antecedents, which by supposition are
proportionals.

Coroll. VI. In like manner, if the antecedent singly, or consequent
singly, be put for antecedent, and the consequent be made of both put
together, these also will be proportionals. For it is the _inversion of
proportion by composition_.

Coroll. VII. If there be four proportionals, they will also be
proportionals by division, that is, by taking the remainder after the
consequent is subtracted from the antecedent, or the difference between
the antecedent and consequent for antecedent, and either the whole or
the subtracted for consequent; as if A. B :: C. D be proportionals, they
will by division be A - B. B :: C - D. D, and A - B. A :: C - D. C; and
when the consequent is greater than the antecedent, B - A. A :: D - C.
C, and B - A. B :: D - C. D. For in all these divisions, proportionals
are, by the very supposition of the analogism A. B :: C. D, taken from A
and B, and from C and D.

Coroll. VIII. If there be four proportionals, they will also be
proportionals by the _conversion of proportion_, that is, by inverting
the divided proportion, or by taking the whole for antecedent, and the
difference or remainder for consequent.

As, if A. B :: C. D be proportionals, then A. A - B :: C. C - D, as also
B. A - B :: D. C - D will be proportionals. For seeing these inverted be
proportionals, they are also themselves proportionals.

Coroll. IX. If there be two analogisms which have their quantities
equal, the second to the second, and the fourth to the fourth, then
either the sum or difference of the first quantities will be to the
second, as the sum or difference of the third quantities is to the
fourth. Let A. B :: C. D and E. B :: F. D be analogisms; I say A + E. B
:: C + F. D are proportionals. For the said analogisms will by
permutation be A. C :: B. D, and E. F :: B. D; and therefore A. C :: E.
F will be proportionals, for they have both the proportion of B to D
common. Wherefore, if in the permutation of the first analogism, there
be added E and F to A and C, which E and F are proportional to A and C,
then (by the third coroll.) A + E. B :: C + F. D will be proportionals;
which was to be proved.

Also in the same manner it may be shown, that A - E. B :: C - F. D are
proportionals.

7. If there be two analogisms, where four antecedents make an analogism,
their consequents also shall make an analogism; as also the sums of
their antecedents will be proportional to the sums of their consequents.
For if A. B :: C. D and E. F :: G. H be two analogisms, and A. E :: C. G
be proportionals, then by permutation A. C :: E. G, and E. G :: F. H,
and A. C :: B. D will be proportionals; wherefore B. D :: E. G, that is,
B. D :: F. H, and by permutation B. F :: D. H are proportionals; which
is the first. Secondly, I say A + E. B + F :: C + G. D + H will be
proportionals. For seeing A. E :: C. G are proportionals, A + E. E :: C
+ G. G will also by composition be proportionals, and by permutation A +
E. C + G :: E. G will be proportionals; wherefore, also A + E. C + G ::
B + F. D + H will be proportionals. Again, seeing, as is shown above, B.
F :: D. H are proportionals, B + F. F :: D + H. H will also by
composition be proportionals; and by permutation B + F. D + H :: F. H
will also be proportionals; wherefore A + E. C + G :: B + F. D + H are
proportionals; which remained to be proved.

Coroll. By the same reason, if there be never so many analogisms, and
the antecedents be proportional to the antecedents, it may be
demonstrated also that the consequents will be proportional to the
consequents, as also the sum of the antecedents to the sum of the
consequents.

[Sidenote: The definition of hyperlogism and hypologism, that is, of
           greater and less proportion, and their transmutations.]

8. In an hyperlogism, that is, where the proportion of the first
antecedent to its consequent is greater than the proportion of the
second antecedent to its consequent, the permutation of the
proportionals, and the addition of proportionals to proportionals, and
substraction of them from one another, as also their composition and
division, and their multiplication and division by the same number,
produce always an hyperlogism. For suppose A. B :: C . D and A. C :: E.
F be analogisms, A + E. B :: C + F . D will also be an analogism; but A
+ E. B :: C. D will be an hyperlogism; wherefore by permutation, A + E.
C :: B. D is an hyperlogism, because A. B :: C. D is an analogism.
Secondly, if to the hyperlogism A + E. B :: C. D the proportionals G and
H be added, A + E + G. B :: C + H. D will be an hyperlogism, by reason A
+ E + G. B :: C + F + H. D is an analogism. Also, if G and H be taken
away, A + E - G. B :: C - H. D will be an hyperlogism; for A + E - G. B
:: C + F - H. D is an analogism. Thirdly, by composition A + E + B. B ::
C + D. D will be an hyperlogism, because A + E + B. B :: C + F + D. D is
ah analogism, and so it will be in all the varieties of composition.
Fourthly, by division, A + E - B. B :: C - D. D will by an hyperlogism,
by reason A E - B. B :: C + F - D. D is an analogism. Also A + E - B. A
+ E :: C - D. C is an hyperlogism; for A + E - B. A + E :: C + F - D. C
is an analogism. Fifthly, by multiplication 4 A + 4 E. B :: 4 C. D is an
hyperlogism, because 4 A. B :: 4 C. D is an analogism; and by division ¼
A + ¼ E. B :: ¼ C.D is an hyperlogism, because ¼ A. B :: ¼ C. D is an
analogism.

9. But if A + E. B :: C. D be an hyperlogism, then by inversion B. A + E
:: D. C will be an hypologism, because B. A :: D. C being an analogism,
the first consequent will be too great. Also, by conversion of
proportion, A + E. A + E - B :: C. C - D is an hypologism, because the
inversion of it, namely A + E - B. A + E :: C - D. C is an hyperlogism,
as I have shown but now. So also B. A + E - B :: D. C - D is an
hypologism, because, as I have newly shown, the inversion of it, namely
A + E - B. B :: C - D. D is an hyperlogism. Note that this hypologism A
+ E. A + E - B :: C. C - D is commonly thus expressed; if the proportion
of the whole, (A + E) to that which is taken out of it (B), be greater
than the proportion of the whole (C) to that which is taken out of it
(D), then the proportion of the whole (A + E) to the remainder (A + E -
B) will be less than the proportion of the whole (C) to the remainder (C
- D).

[Sidenote: Comparison of analogical quantities, according to magnitude.]

10. If there be four proportionals, the difference of the two first, to
the difference of the two last, will be as the first antecedent is to
the second antecedent, or as the first consequent to the second
consequent. For if A. B :: C. D be proportionals, then by division A -
B. B :: C - D. D will be proportionals; and by permutation A - B. C - D
:: B. D; that is, the differences are proportional to the consequents,
and therefore they are so also to the antecedents.

11. Of four proportionals, if the first be greater than the second, the
third also shall be greater than the fourth. For seeing the first is
greater than the second, the proportion of the first to the second is
the proportion of excess; but the proportion of the third to the fourth
is the same with that of the first to the second; and therefore also the
proportion of the third to the fourth is the proportion of excess;
wherefore the third is greater than the fourth. In the same manner it
may be proved, that whensoever the first is less than the second, the
third also is less than the fourth; and when those are equal, that these
also are equal.

12. If there be four proportionals whatsoever, A. B :: C.D, and the
first and third be multiplied by any one number, as by 2; and again the
second and fourth be multiplied by any one number, as by 3; and the
product of the first 2 A, be greater than the product of the second 3 B;
the product also of the third 2 C, will be greater than the product of
the fourth 3 D. But if the product of the first be less than the product
of the second, then the product of the third will be less than that of
the fourth. And lastly, if the products of the first and second be
equal, the products of the third and fourth shall also be equal. Now
this theorem is all one with Euclid's definition of _the same
proportion_; and it may be demonstrated thus. Seeing A. B :: C. D are
proportionals, by permutation also (art. 6, coroll. I.) A. C :: B . D
will be proportionals; wherefore (by coroll. IV. art. 6) 2 A. 2 C :: 3
B. 3 D will be proportionals; and again, by permutation, 2 A. 3 B :: 2
C. 3 D will be proportionals; and therefore, by the last article, if 2 A
be greater than 3 B, then 2 C will be greater than 3 D; if less, less;
and if equal, equal; which was to be demonstrated.

[Sidenote: Composition of proportions.]

If any three magnitudes be propounded, or three things whatsoever that
have any proportion one to another, as three numbers, three times, three
degrees, &c.; the proportions of the first to the second, and of the
second to the third, together taken, are equal to the proportion of the
first to the third. Let there be three lines, for any proportion may be
reduced to the proportion of lines, A B, A C, A D; and in the first
place, let the proportion as well of the first A B to the second A C, as
of the second A C to the third AD, be the proportion of defect, or of
less to greater; I say the proportions together taken of A B to A C, and
of A C to A D, are equal to the proportion of A B to A D. Suppose the
point A to be moved over the whole line A D with uniform motion; then
the proportions as well of A B to A C, as of A C to A D, are determined
by the difference of the times in which they are described; that is, A B
has to A C such proportion as is determined by the different times of
their description; and A C to AD such proportion as is determined by
their times. But the proportion of A B to A D is such as is determined
by the difference of the times in which A B and A D are described; and
the difference of the times in which A B and A C are described, together
with the difference of the times in which A C and A D are described, is
the same with the difference of the times in which A B and A D are
described. And therefore, the same cause which determines the two
proportions of A B to A C and of A C to A D, determines also the
proportion of A B to A D. Wherefore, by the definition of the same
proportion, delivered above in the 6th article, the proportion of A B to
A C together with the proportion of A C to A D, is the same with the
proportion of A B to A D.

                             A   B   C   D
                             -------------

In the second place, let A D be the first, A C the second, and A B the
third, and let their proportion be the proportion of excess, or the
greater to less; then, as before, the proportions of A D to A C, and of
A C to A B, and of A D to A B, will be determined by the difference of
their times; which in the description of A D and A C, and of A C and A B
together taken, is the same with the difference of the times in the
description of A D and A B. Wherefore the proportion of A D to A B is
equal to the two proportions of A D to A C and of A C to A B.

In the last place. If one of the proportions, namely of A D to A B, be
the proportion of excess, and another of them, as of A B to A C be the
proportion of defect, thus also the proportion of A D to A C will be
equal to the two proportions together taken of A D to A B, and of A B to
A C. For the difference of the times in which AD and AB are described,
is excess of time; for there goes more time to the description of A D
than of A B; and the difference of the times in which A B and A C are
described, is defect of time, for less time goes to the description of A
B than of A C; but this excess and defect being added together, make D B
- B C, which is equal to D C, by which the first A D exceeds the third A
C; and therefore the proportions of the first A D to the second A B, and
of the second A B to the third A C, are determined by the same cause
which determines the proportion of the first A D to the third A C.
Wherefore, if any three magnitudes, &c.

Coroll. I. If there be never so many magnitudes having proportion to one
another, the proportion of the first to the last is compounded of the
proportions of the first to the second, of the second to the third, and
so on till you come to the last; or, the proportion of the first to the
last is the same with the sum of all the intermediate proportions. For
any number of magnitudes having proportion to one another, as A, B, C,
D, E being propounded, the proportion of A to E, as is newly shown, is
compounded of the proportions of A to D and of D to E; and again, the
proportion of A to D, of the proportions of A to C, and of C to D; and
lastly, the proportion of A to C, of the proportions of A to B, and of B
to C.

Coroll. II. From hence it may be understood how any two proportions may
be compounded. For if the proportions of A to B, and of C to D, be
propounded to be added together, let B have to something else, as to E,
the same proportion which C has to D, and let them be set in this order,
A, B, E; for so the proportion of A to E will evidently be the sum of
the two proportions of A to B, and of B to E, that is, of C to D. Or let
it be as D to C, so A to something else, as to E, and let them be
ordered thus, E, A, B; for the proportion of E to B will be compounded
of the proportions of E to A, that is, of C to D, and of A to B. Also,
it may be understood how one proportion may be taken out of another. For
if the proportion of C to D be to be subtracted out of the proportion of
A to B, let it be as C to D, so A to something else, as E, and setting
them in this order, A, E, B, and taking away the proportion of A to E,
that is, of C to D, there will remain the proportion of E to B.

Coroll. III. If there be two orders of magnitudes which have proportion
to one another, and the several proportions of the first order be the
same and equal in number with the proportions of the second order; then,
whether the proportions in both orders be successively answerable to one
another, which is called _ordinate proportion_, or not successively
answerable, which is called _perturbed proportion_, the first and the
last in both will be proportionals. For the proportion of the first to
the last is equal to all the intermediate proportions; which being in
both orders the same, and equal in number, the aggregates of those
proportions will also be equal to one another; but to their aggregates,
the proportions of the first to the last are equal; and therefore the
proportion of the first to the last in one order, is the same with the
proportion of the first to the last in the other order. Wherefore the
first and the last in both are proportionals.

[Sidenote: Composition of proportions.]

14. If any two quantities be made of the mutual multiplication of many
quantities, which have proportion to one another, and the efficient
quantities on both sides be equal in number, the proportion of the
products will be compounded of the several proportions, which the
efficient quantities have to one another.

                                A B.  A.
                                C B.  C.
                                C D.  E.

First, let the two products be A B and C D, whereof one is made of the
multiplication of A into B, and the other of the multiplication of C
into D. I say the proportion of A B to C D is compounded of the
proportions of the efficient A to the efficient C, and of the efficient
B to the efficient D. For let A B, C B and C D be set in order; and as B
is to D, so let C be to another quantity as E; and let A, C, E be set
also in order. Then (by coroll. IV. of the 6th art.) it will be as A B
the first quantity to C B the second quantity in the first order, so A
to C in the second order; and again, as C B to C D in the first order,
so B to D, that is, by construction, so C to E in the second order; and
therefore (by the last corollary) A B. C D :: A. E will be
proportionals. But the proportion of A to E is compounded of the
proportions of A to C, and of B to D; wherefore also the proportion of A
B to C D is compounded of the same.

Secondly, let the two products be A B F, and C D G, each of them made of
three efficients, the first of A, B and F, and the second of C, D and G;
I say, the proportion of A B F to C D G is compounded of the proportions
of A to C, of B to D, and of F to G. For let them be set in order as
before; and as B is to D, so let C be to another quantity E; and again,
as F is to G, so let E be to another, H; and let the first order stand
thus, ABF, CBF, CDF and CDG; and the second order thus, A, C, E, H. Then
the proportion of A B F to C B F in the first order, will be as A to C
in the second; and the proportion of CBF to CDF in the first order, as B
to D, that is, as C to E (by construction) in the second order; and the
proportion of CDF to CDG in the first, as F to G, that is, as E to H (by
construction) in the second order; and therefore A B F. C D G:: A. H
will be proportionals. But the proportion of A to H is compounded of the
proportions of A to C, B to D, and F to G. Wherefore the proportion of
the product A B F to C D G is also compounded of the same. And this
operation serves, how many soever the efficients be that make the
quantities given.

                               ABF.   A.
                               CBF.   C.
                               CDF.   E.
                               CDG.   H.

From hence ariseth another way of compounding many proportions into one,
namely, that which is supposed in the 5th definition of the 6th book of
Euclid; which is, by multiplying all the antecedents of the proportions
into one another, and in like manner all the consequents into one
another. And from hence also it is evident, in the first place, that the
cause why parallelograms, which are made by the duction of two straight
lines into one another, and all solids which are equal to figures so
made, have their proportions compounded of the proportions of the
efficients; and in the second place, why the multiplication of two or
more fractions into one another is the same thing with the composition
of the proportions of their several numerators to their several
denominators. For example, if these fractions 1⁄2, 2⁄3, 3⁄4 be to be
multiplied into one another, the numerators 1, 2, 3, are first to be
multiplied into one another, which make 6; and next the denominators 2,
3, 4, which make 24; and these two products make the fraction 6⁄24. In
like manner, if the proportions of 1 to 2, of 2 to 3, and of 3 to 4, be
to be compounded, by working as I have shown above, the same proportion
of 6 to 24 will be produced.

15. If any proportion be compounded with itself inverted, the compound
will be the proportion of equality. For let any proportion be given, as
of A to B, and let the inverse of it be that of C to D; and as C to D,
so let B be to another quantity; for thus they will be compounded (by
the second coroll. of the 12th art.) Now seeing the proportion of C to D
is the inverse of the proportion of A to B, it will be as C to D, so B
to A; and therefore if they be placed in order, A, B, A, the proportion
compounded of the proportions of A to B, and of C to D, will be the
proportion of A to A, that is, the proportion of equality. And from
hence the cause is evident why two equal products have their efficients
reciprocally proportional. For, for the making of two products equal,
the proportions of their efficients must be such, as being compounded
may make the proportion of equality, which cannot be except one be the
inverse of the other; for if betwixt A and A any other quantity, as C,
be interposed, their order will be A, C, A, and the later proportion of
C to A will be the inverse of the former proportion of A to C.

[Sidenote: The definition and properties of continual proportion.]

16. A proportion is said to be multiplied by a number, when it is so
often taken as there be unities in that number; and if the proportion be
of the greater to the less, then shall also the quantity of the
proportion be increased by the multiplication; but when the proportion
is of the less to the greater, then as the number increaseth, the
quantity of the proportion diminisheth; as in these three numbers, 4, 2,
1, the proportion of 4 to 1 is not only the duplicate of 4 to 2, but
also twice as great; but inverting the order of those numbers thus, 1,
2, 4, the proportion of 1 to 2 is greater than that of 1 to 4; and
therefore though the proportion of 1 to 4 be the duplicate of 1 to 2,
yet it is not twice so great as that of 1 to 2, but contrarily the half
of it. In like manner, a proportion is said to be divided, when between
two quantities are interposed one or more means in continual proportion,
and then the proportion of the first to the second is said to be
subduplicate of that of the first to the third, and subtriplicate of
that of the first to the fourth, &c.

This mixture of proportions, where some are proportions of excess,
others of defect, as in a merchant's account of debtor and creditor, is
not so easily reckoned as some think; but maketh the composition of
proportions sometimes to be addition, sometimes substraction; which
soundeth absurdly to such as have always by composition understood
addition, and by diminution substraction. Therefore to make this account
a little clearer, we are to consider (that which is commonly assumed,
and truly) that if there be never so many quantities, the proportion of
the first to the last is compounded of the proportions of the first to
the second, and of the second to the third, and so on to the last,
without regarding their equality, excess, or defect; so that if two
proportions, one of inequality, the other of equality, be added
together, the proportion is not thereby made greater nor less; as for
example, if the proportions of A to B and of B to B be compounded, the
proportion of the first to the second is as much as the sum of both,
because proportion of equality, being not quantity, neither augmenteth
quantity nor lesseneth it. But if there be three quantities, A, B, C,
unequal, and the first be the greatest, the last least, then the
proportion of B to C is an addition to that of A to B, and makes it
greater; and on the contrary, if A be the least, and C the greatest
quantity, then doth the addition of the proportion of B to C make the
compounded proportion of A to C less than the proportion of A to B, that
is, the whole less than the part. The composition therefore of
proportions is not in this case the augmentation of them, but the
diminution; for the same quantity (Euclid V. 8) compared with two other
quantities, hath a greater proportion to the lesser of them than to the
greater. Likewise, when the proportions compounded are one of excess,
the other of defect, if the first be of excess, as in these numbers, 8,
6, 9, the proportion compounded, namely, of 8 to 9, is less than the
proportion of one of the parts of it, namely, of 8 to 6; but if the
proportion of the first to the second be of defect, and that of the
second to the third be of excess, as in these numbers, 6, 8, 4, then
shall the proportion of the first to the third be greater than that of
the first to the second, as 6 hath a greater proportion to 4 than to 8;
the reason whereof is manifestly this, that the less any quantity is
deficient of another, or the more one exceedeth another, the proportion
of it to that other is the greater. Suppose now three quantities in
continual proportion, A B 4, A C 6, A D 9. Because therefore A D is
greater than A C, but not greater than A D, the proportion of A D to AC
will be (by Euclid, V. 8) greater than that of A D to A D; and likewise,
because the proportions of A D to A C, and of A C to A B are the same,
the proportions of A D to A C and of A C to A B, being both proportions
of excess, make the whole proportion of A D to A B, or of 9 to 4, not
only the duplicate of A D to A C, that is, of 9 to 6, but also the
double, or twice so great. On the other side, because the proportion of
A D to A D, or 9 to 9, being proportion of equality, is no quantity, and
yet greater than that of A C to A D, or 6 to 9, it will be as 0 - 9 to 0
- 6, so A C to A D, and again, as 0 - 9 to 0 - 6, so 0 - 6 to 0 - 4; but
0 - 4, 0 - 6, 0 - 9 are in continual proportion; and because 0 - 4 is
greater than 0 - 6, the proportion of 0 - 4 to 0 - 6 will be double to
the proportion of 0 - 4 to 0 - 9, double I say, and yet not duplicate,
but subduplicate.

If any be unsatisfied with this ratiocination, let him first consider
that (by Euclid V. 8) the proportion of A B to A C is greater than that
of A B to A D, wheresoever D be placed in the line A C prolonged; and
the further off the point

D is from C, so much the greater is the proportion of A B to A C than
that of A B to A D. There is therefore some point (which suppose be E)
in such distance from C, as that the proportion of A B to A C will be
twice as great as that of A B to A E. That considered, let him determine
the length of the line A E, and demonstrate, if he can, that A E is
greater or less than A D.

                                Ḅ  C̣   Ḍ
                                A—————E

By the same method, if there be more quantities than three, as A, B, C,
D, in continual proportion, and A be the least, it may be made appear
that the proportion of A to B is triple magnitude, though subtriple in
multitude, to the proportion of A to D.

17. If there be never so many quantities, the number whereof is odd, and
their order such, that from the middlemost quantity both ways they
proceed in continual proportion, the proportion of the two which are
next on either side to the middlemost is subduplicate to the proportion
of the two which are next to these on both sides, and subtriplicate of
the proportion of the two which are yet one place more remote, &c. For
let the magnitudes be C, B, A, D, E, and let A, B, C, as also A, D, E be
in continual proportion; I say the proportion of D to B is subduplicate
of the proportion of E to C. For the proportion of D to B is compounded
of the proportions of D to A, and of A to B once taken; but the
proportion of E to C is compounded of the same twice taken; and
therefore the proportion of D to B is subduplicate of the proportion of
E to C. And in the same manner, if there were three terms on either
side, it might be demonstrated that the proportion of D to B would be
subtriplicate of that of the extremes, &c.

18. If there be never so many continual proportionals, as the first,
second, third, &c. their differences will be proportional to them. For
the second, third, &c. are severally consequents of the preceding, and
antecedents of the following proportion. But (by art. 10) the difference
of the first antecedent and consequent, to difference of the second
antecedent and consequent, is as the first antecedent to the second
antecedent, that is, as the first term to the second, or as the second
to the third, &c. in continual proportionals.

19. If there be three continual proportionals, the sum of the extremes,
together with the mean twice taken, the sum of the mean and either of
the extremes, and the same extreme, are continual proportionals. For let
A. B. C be continual proportionals. Seeing, therefore, A. B :: B. C are
proportionals, by composition also A + B. B :: B + C. C will be
proportionals; and by permutation A + B. B + C :: B. C will also be
proportionals; and again, by composition A + 2B + C. B + C :: B + C. C;
which was to be proved.

20. In four continual proportionals, the greatest and the least put
together is a greater quantity than the other two put together. Let A. B
:: C. D be continual proportionals; whereof let the greatest be A, and
the least be D; I say A + D is greater than B + C. For by art. 10, A -
B. C - D :: A. C are proportionals; and therefore A - B is, by art. 11,
greater than C - D. Add B on both sides, and A will be greater than C +
B - D. And again, add D on both sides, and A + D will be greater than B
+ C; which was to be proved.

[Sidenote: The definition and properties of continual proportion.]

21. If there be four proportionals, the extremes multiplied into one
another, and the means multiplied into one another, will make equal
products. Let A. B :: C. D be proportionals; I say A D is equal to B C.
For the proportion of A D to B C is compounded, by art. 13, of the
proportions of A to B, and D to C, that is, its inverse B to A; and
therefore, by art. 14, this compounded proportion is the proportion of
equality; and therefore also, the proportion of A D to B C is the
proportion of equality. Wherefore they are equal.

22. If there be four quantities, and the proportion of the first to the
second be duplicate of the proportion of the third to the fourth, the
product of the extremes to the product of the means, will be as the
third to the fourth. Let the four quantities be A, B, C and D; and let
the proportion of A to B be duplicate of the proportion of C to D, I say
A D, that is, the product of A into D is to B C, that is, to the product
of the means, as C to D. For seeing the proportion of A to B is
duplicate of the proportion of C to D, if it be as C to D, so D to
another, E, then A. B :: C. E will be proportionals; for the proportion
of A to B is by supposition duplicate of the proportion of C to D; and C
to E duplicate also of that of C to D by the definition, art. 15.
Wherefore, by the last article, A E or A into E is equal to B C or B
into C; but, by coroll. IV. art. 6, A D is to A E as D to E, that is, as
C to D; and therefore A D is to B C, which as I have shown is equal to A
E, as C to D; which was to be proved.

Moreover, if the proportion of the first A to the second B be triplicate
of the proportion of the third C to the fourth D, the product of the
extremes to the product of the means will be duplicate of the proportion
of the third to the fourth. For if it be as C to D so D to E, and again,
as D to E so E to another, F, then the proportion of C to F will be
triplicate of the proportion of C to D; and consequently, A. B :: C. F
will be proportionals, and A F equal to B C. But as A D to A F, so is D
to F; and therefore, also, as A D to B C, so D to F, that is, so C to E;
but the proportion of C to E is duplicate of the proportion of C to D;
wherefore, also, the proportion of A D to B C is duplicate of that of C
to D, as was propounded.

23. If there be four proportionals, and a mean be interposed betwixt the
first and second, and another betwixt the third and fourth, the first of
these means will be to the second, as the first of the proportionals is
to the third, or as the second of them is to the fourth. For let A. B ::
C. D be proportionals, and let E be a mean betwixt A and B, and F a mean
betwixt C and D; I say A. C :: E. F are proportionals. For the
proportion of A to E is subduplicate of the proportion of A to B, or of
C to D. Also, the proportion of C to F is subduplicate of that of C to
D; and therefore A. E :: C. F are proportionals; and by permutation A. C
:: E. F are also proportionals; which was to be proved.

24. Any thing is said to be divided into extreme and mean proportion,
when the whole and the parts are in continual proportion. As for
example, when A + B. A. B are continual proportionals; or when the
straight line A C is so divided in B, that A C. A B. B C are in
continual proportion. And if the same line A C be again divided in D, so
as that A C. C D. A D be continual proportionals; then also A C. A B. A
D will be continual proportionals; and in like manner, though in
contrary order, C A. C D. C B will be continual proportionals; which
cannot happen in any line otherwise divided.

                             A        B   C
                             ----|----|----
                                 D

25. If there be three continual proportionals, and again, three other
continual proportions, which have the same middle term, their extremes
will be in reciprocal proportion. For let A. B. C and D. B. E be
continual proportionals, I say A. D :: E. C shall be proportionals. For
the proportion of A to D is compounded of the proportions of A to B, and
of B to D; and the proportion of E to C is compounded of those of E to
B, that is, of B to D, and of B to C, that is, of A to B. Wherefore, by
equality, A. D :: E. C are proportionals.

[Sidenote: Comparison of arithmetical and geometrical proportion.]

If any two unequal quantities be made extremes, and there be interposed
betwixt them any number of means in geometrical proportion, and the same
number of means in arithmetical proportion, the several means in
geometrical proportion will be less than the several means in
arithmetical proportion. For betwixt A the lesser, and E the greater
extreme, let there be interposed three means, B, C, D, in geometrical
proportion, and as many more, F, G, H, in arithmetical proportion; I say
B will be less than F, C than G, and D than H. For first, the difference
between A and F is the same with that between F and G, and with that
between G and H, by the definition of arithmetical proportion; and
therefore, the difference of the proportionals which stand next to one
another, to the difference of the extremes, is, when there is but one
mean, half their difference; when two, a third part of it; when three, a
quarter, &c.; so that in this example it is a quarter. But the
difference between D and E, by art. 17, is more than a quarter of the
difference between the extremes, because the proportion is geometrical,
and therefore the difference between A and D is less than three quarters
of the same difference of the extremes. In like manner, if the
difference between A and D be understood to be divided into three equal
parts, it may be proved, that the difference between A and C is less
than two quarters of the difference of the extremes A and E. And lastly,
if the difference between A and C be divided into two equal parts, that
the difference between A and B is less than a quarter of the difference
of the extremes A and E.

                              A         A
                              -         -
                              B         F
                             ---       ---
                              C         G
                            -----     -----
                              D         H
                           -------   -------
                              E         E
                          --------- ---------

From the consideration hereof, it is manifest, that B, that is A
together with something else which is less than a fourth part of the
difference of the extremes A and E, is less than F, that is, than the
same A with something else which is equal to the said fourth part. Also,
that C, that is A with something else which is less than two fourth
parts of the said difference, is less than G, that is, than A together
with the said two-fourths. And lastly, that D, which exceeds A by less
than three-fourths of the said difference, is less than H, which exceeds
the same A by three entire fourths of the said difference. And in the
same manner it would be if there were four means, saving that instead of
fourths of the difference of the extremes we are to take fifth parts;
and so on.

27. LEMMA. If a quantity being given, first one quantity be both added
to it and subtracted from it, and then another greater or less, the
proportion of the remainder to the aggregate, is greater where the less
quantity is added and substracted, than where the greater quantity is
added and substracted. Let B be added to and substracted from the
quantity A; so that A - B be the remainder, and A + B the aggregate; and
again, let C, a greater quantity than B, be added to and substracted
from the same A, so that A - C be the remainder and A + C the aggregate;
I say A - B. A + B :: A - C. A + C will be an hyperlogism. For A - B. A
:: A - C. A is an hyperlogism of a greater antecedent to the same
consequent; and therefore A - B. A + B :: A - C. A + C is a much greater
hyperlogism, being made of a greater antecedent to a less consequent.

28. If unequal parts be taken from two equal quantities, and betwixt the
whole and the part of each there be interposed two means, one in
geometrical, the other in arithmetical proportion; the difference
betwixt the two means will be greatest, where the difference betwixt the
whole and its part is greatest. For let A B and A B be two equal
quantities, from which let two unequal parts be taken, namely, A E the
less, and A F the greater; and betwixt A B and A E let A G be a mean in
geometrical proportion, and A H a mean in arithmetical proportion. Also
betwixt A B and A F let A I be a mean in geometrical proportion, and A K
a mean in arithmetical proportion; I say H G is greater than K I.

                            A  E   G   H   B
                            ---+---+---+----
                            ----+---+---+---
                            A   F   I   K  B

 For in the first place we have this      A B. A G :: B G. G E, by
   analogism                                     article 18.


 Then by composition we have this        A B + A G. A B :: B G + G E
                                             that is, B E. B G.


 And by taking the halves of the      ½A B + ½A G. A B :: ½B G + ½G E,
   antecedents this third                    that is, B H. B G.


 And by conversion a fourth             A B. ½A B + ½A G :: B G. B H.


 And by division this fifth               ½A B - ½A G. ½A B + ½A G
                                                :: H G. B H.


 And by doubling the first
   antecedent and the first
   consequent                         A B - A G. A B + A G :: H G. B H.


 Also by the same method may be
   found out this analogism           A B - A I. A B + AI :: K I. B K.

Now seeing the proportion of A B to A E is greater than that of A B to A
F, the proportion of A B to A G, which is half the greater proportion,
is greater than the proportion of A B to A I the half of the less
proportion; and therefore A I is greater than A G. Wherefore the
proportion of A B - A G to A B + A G, by the precedent lemma, will be
greater than the proportion of A B - A I to A B + A I; and therefore
also the proportion of H G to B H will be greater than that of K I to B
K, and much greater than the proportion of K I to B H, which is greater
than B K; for B H is the half of B E, as B K is the half of B F, which,
by supposition, is less than B E. Wherefore H G is greater than K I;
which was to be proved.

Coroll. It is manifest from hence, that if any quantity be supposed to
be divided into equal parts infinite in number, the difference between
the arithmetical and geometrical means will be infinitely little, that
is, none at all. And upon this foundation, chiefly, the art of making
those numbers, which are called Logarithms, seems to have been built.

29. If any number of quantities be propounded, whether they be unequal,
or equal to one another; and there be another quantity, which multiplied
by the number of the propounded quantities, is equal to them all; that
other quantity is a mean in arithmetical proportion to all those
propounded quantities.


                                -------


                               CHAP. XIV.

                    OF STRAIT AND CROOKED, ANGLE AND
                                FIGURE.

1. The definition and properties of a strait line.—2. The definition and
    properties of a plane superficies.—3. Several sorts of crooked
    lines.—4. The definition and properties of a circular line.—5. The
    properties of a strait line taken in a plane.—6.. The definition of
    tangent lines.—7. The definition of an angle, and the kinds
    thereof.—8. In concentric circles, arches of the same angle are to
    one another, as the whole circumferences are.—9. The quantity of an
    angle, in what it consists. —10. The distinction of angles, simply
    so called.—11. Of strait lines from the centre of a circle to a
    tangent of the same. —12. The general definition of parallels, and
    the properties of strait parallels.—13. The circumferences of
    circles are to one another, as their diameters are.—14. In
    triangles, strait lines parallel to the bases are to one another, as
    the parts of the sides which they cut off from the vertex.—15. By
    what fraction of a strait line the circumference of a circle is
    made.—16. That an angle of contingence is quantity, but of a
    different kind from that of an angle simply so called; and that it
    can neither add nor take away any thing from the same.—17. That the
    inclination of planes is angle simply so called.—18. A solid angle
    what it is.—19. What is the nature of asymptotes.—20. Situation, by
    what it is determined.—21. What is like situation; what is figure;
    and what are like figures.


[Sidenote: The definition end properties of a strait line.]

1. Between two points given, the shortest line is that, whose extreme
points cannot be drawn farther asunder without altering the quantity,
that is, without altering the proportion of that line to any other line
given. For the magnitude of a line is computed by the greatest distance
which may be between its extreme points; so that any one line, whether
it be extended or bowed, has always one and the same length, because it
can have but one greatest distance between its extreme points.

And seeing the action, by which a strait line is made crooked, or
contrarily a crooked line is made strait, is nothing but the bringing of
its extreme points nearer to one another, or the setting of them further
asunder, a _crooked line_ may rightly be defined to be _that, whose
extreme points may be understood to be drawn further asunder_; and a
_strait line_ to be _that, whose extreme points cannot be drawn further
asunder; and comparatively, a more crooked, to be that line whose
extreme points are nearer to one another than those of the other,
supposing both the lines to be of equal length_. Now, howsoever a line
be bowed, it makes always a _sinus_ or cavity, sometimes on one side,
sometimes on another; so that the same crooked line may either have its
whole cavity on one side only, or it may have it part on one side and
part on the other side. Which being well understood, it will be easy to
understand the following comparisons of strait and crooked lines.

First, if a strait and a crooked line have their extreme points common,
the crooked line is longer than the strait line. For if the extreme
points of the crooked line be drawn out to their greatest distance, it
will be made a strait line, of which that, which was a strait line from
the beginning, will be but a part; and therefore the strait line was
shorter than the crooked line, which had the same extreme points. And
for the same reason, if two crooked lines have their extreme points
common, and both of them have all their cavity on one and the same side,
the outermost of the two will be the longest line.

Secondly, a strait line and a perpetually crooked line cannot be
coincident, no, not in the least part. For if they should, then not only
some strait line would have its extreme points common with some crooked
line, but also they would, by reason of their coincidence, be equal to
one another; which, as I have newly shown, cannot be.

Thirdly, between two points given, there can be understood but one
strait line; because there cannot be more than one least interval or
length between the same points. For if there may be two, they will
either be coincident, and so both of them will be one strait line; or if
they be not coincident, then the application of one to the other by
extension will make the extended line have its extreme points at greater
distance than the other; and consequently, it was crooked from the
beginning.

Fourthly, from this last it follows, that two strait lines cannot
include a superficies. For if they have both their extreme points
common, they are coincident; and if they have but one or neither of them
common, then at one or both ends the extreme points will be disjoined,
and include no superficies, but leave all open and undetermined.

Fifthly, every part of a strait line is a strait line. For seeing every
part of a strait line is the least that can be drawn between its own
extreme points, if all the parts should not constitute a strait line,
they would altogether be longer than the whole line.

[Sidenote: The definition and properties of a plane superficies.]

2. A _plane_ or _a plane superficies, is that which is described by a
strait line so moved, that all the several points thereof describe
several strait lines_. A strait line, therefore, is necessarily all of
it in the same plane which it describes. Also the strait lines, which
are made by the points that describe a plane, are all of them in the
same plane. Moreover, if any line whatsoever be moved in a plane, the
lines, which are described by it, are all of them in the same plane.

All other superficies, which are not plane, are crooked, that is, are
either concave or convex. And the same comparisons, which were made of
strait and crooked lines, may also be made of plane and crooked
superficies.

For, first, if a plane and crooked superficies be terminated with the
same lines, the crooked superficies is greater than the plane
superficies. For if the lines, of which the crooked superficies
consists, be extended, they will be found to be longer than those of
which the plane superficies consists, which cannot be extended, because
they are strait.

Secondly, two superficies, whereof the one is plane, and the other
continually crooked, cannot be coincident, no, not in the least part.
For if they were coincident, they would be equal; nay, the same
superficies would be both plane and crooked, which is impossible.

Thirdly, within the same terminating lines there can be no more than one
plane superficies; because there can be but one least superficies within
the same.

Fourthly, no number of plane superficies can include a solid, unless
more than two of them end in a common vertex. For if two planes have
both the same terminating lines, they are coincident, that is, they are
but one superficies; and if their terminating lines be not the same,
they leave one or more sides open.

Fifthly, every part of a plane superficies is a plane superficies. For
seeing the whole plane superficies is the least of all those, that have
the same terminating lines; and also every part of the same superficies
is the least of all those, that are terminated with the same lines; if
every part should not constitute a plane superficies, all the parts put
together would not be equal to the whole.

[Sidenote: Several sorts of crooked lines.]

3. Of straitness, whether it be in lines or in superficies, there is but
one kind; but of crookedness there are many kinds; for of crooked
magnitudes, some are congruous, that is, are coincident when they are
applied to one other; others are incongruous. Again, some are
ὁμοιομερεῖς or uniform, that is, have their parts, howsoever taken,
congruous to one another; others are ἀνομοιομερεῖς or of several forms.
Moreover, of such as are crooked, some are continually crooked, others
have parts which are not crooked.

[Sidenote: Definition and properties of a circular line.]

4. If a strait line be moved in a plane, in such manner, that while one
end of it stands still, the whole line be carried round about till it
come again into the same place from whence it was first moved, it will
describe a plane superficies, which will be terminated every way by that
crooked line, which is made by that end of the strait line which was
carried round. Now this superficies is called a CIRCLE; and of this
circle, the unmoved point is the _centre_; the crooked line which
terminates it, the _perimeter_; and every part of that crooked line, a
_circumference_ or _arch_; the strait line, which generated the
_circle_, is the _semidiameter_ or _radius_; and any strait line, which
passeth through the centre and is terminated on both sides in the
circumference, is called the _diameter_. Moreover, every point of the
radius, which describes the circle, describes in the same time its own
perimeter, terminating its own circle, which is said to be _concentric_
to all the other circles, because this and all those have one common
centre.

Wherefore in every circle, all strait lines from the centre to the
circumference are equal. For they are all coincident with the radius
which generates the circle.

Also the diameter divides both the perimeter and the circle itself into
two equal parts. For if those two parts be applied to one another, and
the semiperimeters be coincident, then, seeing they have one common
diameter, they will be equal; and the semicircles will be equal also;
for these also will be coincident. But if the semiperimeters be not
coincident, then some one strait line, which passes through the centre,
which centre is in the diameter, will be cut by them in two points.
Wherefore, seeing all the strait lines from the centre to the
circumference are equal, a part of the same strait line will be equal to
the whole; which is impossible.

For the same reason the perimeter of a circle will be uniform, that is,
any one part of it will be coincident with any other equal part of the
same.

[Sidenote: The properties of a strait line taken in a plane.]

5. From hence may be collected this property of a strait line, namely,
that it is all contained in that plane which contains both its extreme
points. For seeing both its extreme points are in the plane, that strait
line, which describes the plane, will pass through them both; and if one
of them be made a centre, and at the distance between both a
circumference be described, whose radius is the strait line which
describes the plane, that circumference will pass through the other
point. Wherefore between the two propounded points, there is one strait
line, by the definition of a circle, contained wholly in the propounded
plane; and therefore if another strait line might be drawn between the
same points, and yet not be contained in the same plane, it would
follow, that between two points two strait lines may be drawn; which has
been demonstrated to be impossible.

It may also be collected, that if two planes cut one another, their
common section will be a strait line. For the two extreme points of the
intersection are in both the intersecting planes; and between those
points a strait line may be drawn; but a strait line between any two
points is in the same plane, in which the points are; and seeing these
are in both the planes, the strait line which connects them will also be
in both the same planes, and therefore it is the common section of both.
And every other line, that can be drawn between those points, will be
either coincident with that line, that is, it will be the same line; or
it will not be coincident, and then it will be in neither, or but in one
of those planes.

As a strait line may be understood to be moved round about whilst one
end thereof remains fixed, as the centre; so in like manner it is easy
to understand, that a plane may be circumduced about a strait line,
whilst the strait line remains still in one and the same place, as the
_axis_ of that motion. Now from hence it is manifest, that any three
points are in some one plane. For as any two points, if they be
connected by a strait line, are understood to be in the same plane in
which the strait line is; so, if that plane be circumduced about the
same strait line, it will in its revolution take in any third point,
howsoever it be situate; and then the three points will be all in that
plane; and consequently the three strait lines which connect those
points, will also be in the same plane.

[Sidenote: Definition of tangent lines.]

6. Two lines are said to _touch_ one another, which being both drawn to
one and the same point, will not cut one another, though they be
produced, produced, I say, in the same manner in which they were
generated. And therefore if two strait lines touch one another in any
one point, they will be contiguous through their whole length. Also two
lines continually crooked will do the same, if they be congruous and be
applied to one another according to their congruity; otherwise, if they
be incongruously applied, they will, as all other crooked lines, touch
one another, where they touch, but in one point only. Which is manifest
from this, that there can be no congruity between a strait line and a
line that is continually crooked; for otherwise the same line might be
both strait and crooked. Besides, when a strait line touches a crooked
line, if the strait line be never so little moved about upon the point
of contact, it will cut the crooked line; for seeing it touches it but
in one point, if it incline any way, it will do more than touch it; that
is, it will either be congruous to it, or it will cut it; but it cannot
be congruous to it; and therefore it will cut it.

[Sidenote: The definition of an angle, and the kinds thereof.]

7. An angle, according to the most general acceptation of the word, may
be thus defined; _when two lines, or many superficies, concur in one
sole point, and diverge every where else, the quantity of that
divergence is an_ ANGLE. And an angle is of two sorts; for, first, it
may be made by the concurrence of lines, and then it is a _superficial
angle_; or by the concurrence of superficies, and then it is called a
_solid angle_.

Again, from the two ways by which two lines may diverge from one
another, superficial angles are divided into two kinds. For two strait
lines, which are applied to one another, and are contiguous in their
whole length, may be separated or pulled open in such manner, that their
concurrence in one point will still remain; and this separation or
opening may be either by circular motion, the centre whereof is their
point of concurrence, and the lines will still retain their straitness,
the quantity of which separation or divergence is an _angle_ simply so
called; or they may be separated by continual flexion or curvation in
every imaginable point; and the quantity of this separation is that,
which is called an _angle of contingence_.

Besides, of superficial angles simply so called, those, which are in a
plane superficies, are plane; and those, which are not plane, are
denominated from the superficies in which they are.

Lastly, those are _strait-lined angles_, which are made by strait lines;
as those which are made by crooked lines are _crooked-lined_; and those
which are made both of strait and crooked lines, are _mixed angles_.

[Sidenote: In concentric circles, arches of the same angle are to one
           another, as the whole circumferences are.]

8. Two arches intercepted between two radii of concentric circles, have
the same proportion to one another, which their whole perimeters have to
one another. For let the point A (in the first figure) be the centre of
the two circles B C D and E F G, in which the radii A E B and A F C
intercept the arches B C and E F; I say the proportion of the arch B C
to the arch E F is the same with that of the perimeter B C D to the
perimeter E F G. For if the radius A F C be understood to be moved about
the centre A with circular and uniform motion, that is, with equal
swiftness everywhere, the point C will in a certain time describe the
perimeter B C D, and in a part of that time the arch B C; and because
the velocities are equal by which both the arch and the whole perimeter
are described, the proportion of the magnitude of the perimeter B C D to
the magnitude of the arch BC is determined by nothing but the difference
of the times in which the perimeter and the arch are described. But both
the perimeters are described in one and the same time, and both the
arches in one and the same time; and therefore the proportions of the
perimeter B C D to the arch B C, and of the perimeter E F G to the arch
E F, are both determined by the same cause. Wherefore B C D. B C :: E F
G. E F are proportionals (by the 6th art. of the last chapter), and by
permutation B C D. E F G :: B C. E F will also be proportionals; which
was to be demonstrated.

[Sidenote: The quantity of an angle, in what it consists.]

9. Nothing is contributed towards the quantity of an angle, neither by
the length, nor by the equality, nor by the inequality of the lines
which comprehend it. For the lines A B and A C comprehend the same angle
which is comprehended by the lines A E and A F, or A B and A F. Nor is
an angle either increased or diminished by the absolute quantity of the
arch, which subtends the same; for both the greater arch B C and the
lesser arch E F are subtended to the same angle. But the quantity of an
angle is estimated by the quantity of the subtending arch compared with
the quantity of the whole perimeter. And therefore the quantity of an
angle simply so called may be thus defined: _the quantity of an angle is
an arch or circumference of a circle, determined by its proportion to
the whole perimeter_. So that when an arch is intercepted between two
strait lines drawn from the centre, look how great a portion that arch
is of the whole perimeter, so great is the angle. From whence it may be
understood, that when the lines which contain an angle are strait lines,
the quantity of that angle may be taken at any distance from the centre.
But if one or both of the containing lines be crooked, then the quantity
of the angle is to be taken in the least distance from the centre, or
from their concurrence; for the least distance is to be considered as a
strait line, seeing no crooked line can be imagined so little, but that
there may be a less strait line. And although the least strait line
cannot be given, because the least given line may still be divided, yet
we may come to a part so small, as is not at all considerable; which we
call a point. And this point may be understood to be in a strait line
which touches a crooked line; for an angle is generated by separating,
by circular motion, one strait line from another which touches it, as
has been said above in the 7th article. Wherefore an angle, which two
crooked lines make, is the same with that which is made by two strait
lines which touch them.

[Sidenote: The distinction of angles, simply so called.]

10. From hence it follows, that _vertical angles_, such as are A B C, D
B F in the second figure, are equal to one another. For if, from the two
semiperimeters D A C, F D A, which are equal to one another, the common
arch D A be taken away, the remaining arches A C, D F will be equal to
one another.

Another distinction of angles is into _right_ and _oblique_. _A right
angle is that, whose quantity is the fourth part of the perimeter._ And
the lines, which make a right angle, are said to be _perpendicular_ to
one another. Also, of oblique angles, that which is greater than a
right, is called an _obtuse angle_; and that which is less, an _acute
angle_. From whence it follows, that all the angles that can possibly be
made at one and the same point, together taken, are equal to four right
angles; because the quantities of them all put together make the whole
perimeter. Also, that all the angles, which are made on one side of a
strait line, from any one point taken in the same, are equal to two
right angles; for if that point be made the centre, that strait line
will be the diameter of a circle, by whose circumference the quantity of
an angle is determined; and that diameter will divide the perimeter into
two equal parts.

[Sidenote: Of strait lines from the centre of a circle to a tangent of
           the same.]

11. If a tangent be made the diameter of a circle, whose centre is the
point of contact, a strait line drawn from the centre of the former
circle to the centre of the latter circle, will make two angles with the
tangent, that is, with the diameter of the latter circle, equal to two
right angles, by the last article. And because, by the 6th article, the
tangent has on both sides equal inclination to the circle, each of them
will be a right angle; as also the semidiameter will be perpendicular to
the same tangent. Moreover, the semidiameter, inasmuch as it is the
semidiameter, is the least strait line which can be drawn from the
centre to the tangent; and every other strait line, that reaches the
tangent, will pass out of the circle, and will therefore be greater than
the semidiameter. In like manner, of all the strait lines, which may be
drawn from the centre to the tangent, that is the greatest which makes
the greatest angle with the perpendicular; which will be manifest, if
about the same centre another circle be described, whose semidiameter is
a strait line taken nearer to the perpendicular, and there be drawn a
perpendicular, that is, a tangent, to the same.

From whence it is also manifest, that if two strait lines, which make
equal angles on either side of the perpendicular, be produced to the
tangent, they will be equal.

[Sidenote: The general definition of parallels; the properties of strait
           parallels.]

12. There is in Euclid a definition of strait-lined parallels; but I do
not find that parallels in general are anywhere defined; and therefore
for an universal definition of them, I say that _any two lines
whatsoever, strait or crooked, as also any two superficies, are_
PARALLEL; _when two equal strait lines, wheresoever they fall upon them,
make always equal angles with each of them_.

From which definition it follows; first, that any two strait lines, not
inclined opposite ways, falling upon two other strait lines, which are
parallel, and intercepting equal parts in both of them, are themselves
also equal and parallel. As if A B and C D (in the third figure),
inclined both the same way, fall upon the parallels A C and B D, and A C
and B D be equal, A B and C D will also be equal and parallel. For the
perpendiculars B E and D F being drawn, the right angles E B D and F D H
will be equal. Wherefore, seeing E F and B D are parallel, the angles E
B A and F D C will be equal. Now if D C be not equal to B A, let any
other strait line equal to B A be drawn from the point D; which, seeing
it cannot fall upon the point C, let it fall upon G. Wherefore A G will
be either greater or less than B D; and therefore the angles E B A and F
D C are not equal, as was supposed. Wherefore A B and C D are equal;
which is the first.

Again, because they make equal angles with the perpendiculars B E and D
F; therefore the angle C D H will be equal to the angle A B D, and, by
the definition of parallels, A B and C D will be parallel; which is the
second.

_That plane, which is included both ways within parallel lines, is
called a_ PARALLELOGRAM.

Coroll. I. From this last it follows, that the angles A B D and C D H
are equal, that is, that a strait line, as B H, falling upon two
parallels, as A B and C D, makes the internal angle A B D equal to the
external and opposite angle C D H.

Coroll. II. And from hence again it follows, that a strait line falling
upon two parallels, makes the alternate angles equal, that is, the angle
A G F, in the fourth figure, equal to the angle G F D. For seeing G F D
is equal to the external opposite angle E G B, it will be also equal to
its vertical angle A G F, which is alternate to G F D.

Coroll. III. That the internal angles on the same side of the line F G
are equal to two right angles. For the angles at F, namely, G F C and G
F D, are equal to two right angles. But G F D is equal to its alternate
angle A G F. Wherefore both the angles G F C and A G F, which are
internal on the same side of the line F G, are equal to two right
angles.

Coroll. IV. That the three angles of a strait-lined plain triangle are
equal to two right angles; and any side being produced, the external
angle will be equal to the two opposite internal angles. For if there be
drawn by the vertex of the plain triangle A B C (fig. 5) a parallel to
any of the sides, as to A B, the angles A and B will be equal to their
alternate angles E and F, and the angle C is common. But, by the 10th
article, the three angles E, C and F, are equal to two right angles; and
therefore the three angles of the triangle are equal to the same; which
is the first. Again, the two angles B and D are equal to two right
angles, by the 10th article. Wherefore taking away B, there will remain
the angles A and C, equal to the angle D; which is the second.

Coroll. V. If the angles A and B be equal, the sides A C and C B will
also be equal, because A B and E F are parallel; and, on the contrary,
if the sides A C and C B be equal, the angles A and B will also be
equal. For if they be not equal, let the angles B and G be equal.
Wherefore, seeing G B and E F are parallels, and the angles G and B
equal, the sides G C and C B will also be equal; and because C B and A C
are equal by supposition, C G and C A will also be equal; which cannot
be, by the 11th article.

Coroll. VI. From hence it is manifest, that if two radii of a circle be
connected by a strait line, the angles they make with that connecting
line will be equal to one another; and if there be added that segment of
the circle, which is subtended by the same line which connects the
radii, then the angles, which those radii make with the circumference,
will also be equal to one another. For a strait line, which subtends any
arch, makes equal angles with the same; because, if the arch and the
subtense be divided in the middle, the two halves of the segment will be
congruous to one another, by reason of the uniformity both of the
circumference of the circle, and of the strait line.

[Sidenote: The circumferences of circles are to one another as their
           diameters are.]

13. Perimeters of circles are to one another, as their semidiameters
are. For let there be any two circles, as, in the first figure, B C D
the greater, and E F G the lesser, having their common centre at A; and
let their semidiameters be A C and A E. I say, A C has the same
proportion to A E, which the perimeter B C D has to the perimeter E F G.
For the magnitude of the semidiameters A C and A E is determined by the
distance of the points C and E from the centre A; and the same distances
are acquired by the uniform motion of a point from A to C, in such
manner, that in equal times the distances acquired be equal. But the
perimeters B C D and E F G are also determined by the same distances of
the points C and E from the centre A; and therefore the perimeters B C D
and E F G, as well as the semidiameters A C and A E, have their
magnitudes determined by the same cause, which cause makes, in equal
times, equal spaces. Wherefore, by the 13th chapter and 6th article, the
perimeters of circles and their semidiameters are proportionals; which
was to be proved.

[Sidenote: In triangles strait lines parallel to the bases are to one
           another, as the parts of the sides which they cut off from
           the vertex.]

14. If two strait lines, which constitute an angle, be cut by
strait-lined parallels, the intercepted parallels will be to one
another, as the parts which they cut off from the vertex. Let the strait
lines A B and A C, in the 6th figure, make an angle at A, and be cut by
the two strait-lined parallels B C and D E, so that the parts cut off
from the vertex in either of those lines, as in A B, may be A B and A D.
I say, the parallels B C and D E are to one another, as the parts A B
and A D. For let A B be divided into any number of equal parts, as into
A F, F D, D B; and by the points F and D, let F G and D E be drawn
parallel to the base B C, and cut A C in G and E; and again, by the
points G and E, let other strait lines be drawn parallel to A B, and cut
B C in H and I. If now the point A be understood to be moved uniformly
over A B, and in the same time B be moved to C, and all the points F, D,
and B be moved uniformly and with equal swiftness over F G, D E, and B
C; then shall B pass over B H, equal to F G, in the same time that A
passes over A F; and A F and F G will be to one another, as their
velocities are; and when A is in F, D will be in K; when A is in D, D
will be in E; and in what manner the point A passes by the points F, D,
and B, in the same manner the point B will pass by the points H, I, and
C; and the strait lines F G, D K, K E, B H, H I, and I C, are equal, by
reason of their parallelism; and therefore, as the velocity in A B is to
the velocity in B C, so is A D to D E; but as the velocity in A B is to
the velocity in B C, so is A B to B C; that is to say, all the parallels
will be severally to all the parts cut off from the vertex, as A F is to
F G. Wherefore, A F. G F :: A D. D E :: A B. B C are proportionals.

The subtenses of equal angles in different circles, as the strait lines
B C and F E (in fig. 1), are to one another as the arches which they
subtend. For (by art. 8) the arches of equal angles are to one another
as their perimeters are; and (by art. 13) the perimeters as their
semidiameters; but the subtenses B C and F E are parallel to one another
by reason of the equality of the angles which they make with the
semidiameters; and therefore the same subtenses, by the last precedent
article, will be proportional to the semidiameters, that is, to the
perimeters, that is, to the arches which they subtend.

[Sidenote: By what fraction of a strait line the circumference of a
           circle is made.]

15. If in a circle any number of equal subtenses be placed immediately
after one another, and strait lines be drawn from the extreme point of
the first subtense to the extreme points of all the rest, the first
subtense being produced will make with the second subtense an external
angle double to that, which is made by the same first subtense, and a
tangent to the circle touching it in the extreme points thereof; and if
a strait line which subtends two of those arches be produced, it will
make an external angle with the third subtense, triple to the angle
which is made by the tangent with the first subtense; and so
continually. For with the radius A B (in fig. 7) let a circle be
described, and in it let any number of equal subtenses, B C, C D, and D
E, be placed; also let B D and B E be drawn; and by producing B C, B D
and B E to any distance in G, H and I, let them make angles with the
subtenses which succeed one another, namely, the external angles G C D,
and H D E. Lastly, let the tangent K B be drawn, making with the first
subtense the angle K B C. I say the angle G C D is double to the angle K
B C, and the angle H D E triple to the same angle K B C. For if A C be
drawn cutting B D in M, and from the point C there be drawn L C
perpendicular to the same A C, then C L and M D will be parallel, by
reason of the right angles at C and M; and therefore the alterne angles
L C D and B D C will be equal: as also the angles B D C and C B D will
be equal, because of the equality of the strait lines B C and C D.
Wherefore the angle G C D is double to either of the angles C B D or C D
B; and therefore also the angle G C D is double to the angle L C D, that
is, to the angle K B C. Again, C D is parallel to B E, by reason of the
equality of the angles C B E and D E B, and of the strait lines C B and
D E; and therefore the angles G C D and G B E are equal; and
consequently G B E, as also D E B is double to the angle K B C. But the
external angle H D E is equal to the two internal D E B and D B E; and
therefore the angle H D E is triple to the angle K B C, &c.; which was
to be proved.

Coroll. I. From hence it is manifest, that the angles K B C and C B D,
as also, that all the angles that are comprehended by two strait lines
meeting in the circumference of a circle and insisting upon equal
arches, are equal to one another.

Coroll. II. If the tangent B K be moved in the circumference with
uniform motion about the centre B, it will in equal times cut off equal
arches; and will pass over the whole perimeter in the same time in which
itself describes a semiperimeter about the centre B.

Coroll. III. From hence also we may understand, what it is that
determines the bending or curvation of a strait line into the
circumference of a circle; namely, that it is fraction continually
increasing in the same manner, as numbers, from one upwards, increase by
the continual addition of unity. For the indefinite strait line K B
being broken in B according to any angle, as that of K B C, and again in
C according to a double angle, and in D according to an angle which is
triple, and in E according to an angle which is quadruple to the first
angle, and so continually, there will be described a figure which will
indeed be rectilineal, if the broken parts be considered as having
magnitude; but if they be understood to be the least that can be, that
is, as so many points, then the figure described will not be
rectilineal, but a circle, whose circumference will be the broken line.

Coroll. IV. From what has been said in this present article, it may also
be demonstrated, that an angle in the centre is double to an angle in
the circumference of the same circle, if the intercepted arches be
equal. For seeing that strait line, by whose motion an angle is
determined, passes over equal arches in equal times, as well from the
centre as from the circumference; and while that, which is from the
circumference, is passing over half its own perimeter, it passes in the
same time over the whole perimeter of that which is from the centre, the
arches, which it cuts off in the perimeter whose centre is A, will be
double to those, which it makes in its own semiperimeter, whose centre
is B. But in equal circles, as arches are to one another, so also are
angles.

It may also be demonstrated, that the external angle made by a subtense
produced and the next equal subtense is equal to an angle from the
centre insisting upon the same arch; as in the last diagram, the angle G
C D is equal to the angle C A D; for the external angle G C D is double
to the angle C B D; and the angle C A D insisting upon the same arch C D
is also double to the same angle C B D or K B C.

[Sidenote: That an angle of contingence is quantity, but of a different
           kind from that of an angle simply so called; and that it can
           neither add nor take away anything from the same.]

16. An angle of contingence, if it be compared with an angle simply so
called, how little soever, has such proportion to it as a point has to a
line; that is, no proportion at all, nor any quantity. For first, an
angle of contingence is made by continual flexion; so that in the
generation of it there is no circular motion at all, in which consists
the nature of an angle simply so called; and therefore it cannot be
compared with it according to quantity. Secondly, seeing the external
angle made by a subtense produced and the next subtense is equal to an
angle from the centre insisting upon the same arch, as in the last
figure the angle G C D is equal to the angle C A D, the angle of
contingence will be equal to that angle from the centre, which is made
by A B and the same A B; for no part of a tangent can subtend any arch;
but as the point of contact is to be taken for the subtense, so the
angle of contingence is to be accounted for the external angle, and
equal to that angle whose arch is the same point B.

Now, seeing an angle in general is defined to be the opening or
divergence of two lines, which concur in one sole point; and seeing one
opening is greater than another, it cannot be denied, but that by the
very generation of it, an angle of contingence is quantity; for
wheresoever there is greater and less, there is also quantity; but this
quantity consists in greater and less flexion; for how much the greater
a circle is, so much the nearer comes the circumference of it to the
nature of a strait line; for the circumference of a circle being made by
the curvation of a strait line, the less that strait line is, the
greater is the curvation; and therefore, when one strait line is a
tangent to many circles, the angle of contingence, which it makes with a
less circle, is greater than that which it makes with a greater circle.

Nothing therefore is added to or taken from an angle simply so called,
by the addition to it or taking from it of never so many angles of
contingence. And as an angle of one sort can never be equal to an angle
of the other sort, so they cannot be either greater or less than one
another.

From whence it follows, that an angle of a segment, that is, the angle,
which any strait line makes with any arch, is equal to the angle which
is made by the same strait line, and another which touches the circle in
the point of their concurrence; as in the last figure, the angle which
is made between G B and B K is equal to that which is made between G B
and the arch B C.

[Sidenote: That the inclination of planes is angle simply so called.]

17. An angle, which is made by two planes, is commonly called the
inclination of those planes; and because planes have equal inclination
in all their parts, instead of their inclination an angle is taken,
which is made by two strait lines, one of which is in one, the other in
the other of those planes, but both perpendicular to the common section.

[Sidenote: A solid angle what it is.]

18. A solid angle may be conceived two ways. First, for the aggregate of
all the angles, which are made by the motion of a strait line, while one
extreme point thereof remaining fixed, it is carried about any plain
figure, in which the fixed point of the strait line is not contained.
And in this sense, it seems to be understood by Euclid. Now it is
manifest, that the quantity of a solid angle so conceived is no other,
than the aggregate of all the angles in a superficies so described, that
is, in the superficies of a pyramidal solid. Secondly, when a pyramis or
cone has its vertex in the centre of a sphere, a solid angle may be
understood to be the proportion of a spherical superficies subtending
that vertex to the whole superficies of the sphere. In which sense,
solid angles are to one another as the spherical bases of solids, which
have their vertex in the centre of the same sphere.

[Sidenote: What is the nature of asymptotes.]

19. All the ways, by which two lines respect one another, or all the
variety of their position, may be comprehended under four kinds; for any
two lines whatsoever are either parallels, or being produced, if need
be, or moved one of them to the other parallelly to itself, they make an
angle; or else, by the like production and motion, they touch one
another; or lastly, they are _asymptotes_. The nature of parallels,
angles, and tangents, has been already declared. It remains that I speak
briefly of the nature of asymptotes.

Asymptosy depends upon this, that quantity is infinitely divisible. And
from hence it follows, that any line being given, and a body supposed to
be moved from one extreme thereof towards the other, it is possible, by
taking degrees of velocity always less and less, in such proportion as
the parts of the line are made less by continual division, that the same
body may be always moved forwards in that line, and yet never reach the
end of it. For it is manifest, that if any strait line, as A F, (in the
8th figure) be cut anywhere in B, and again B F be cut in C, and C F in
D, and D F in E, and so eternally, and there be drawn from the point F,
the strait line F F at any angle A F F; and lastly, if the strait lines
A F, B F, C F, D F, E F, &c., having the same proportion to one another
with the segments of the line A F, be set in order and parallel to the
same A F, the crooked line A B C D E, and the strait line F F, will be
_asymptotes_, that is, they will always come nearer and nearer together,
but never touch one another. Now, because any line may be cut eternally
according to the proportions which the segments have to one another,
therefore the divers kinds of asymptotes are infinite in number, and not
necessary to be further spoken of in this place. In the nature of
asymptotes in general there is no more, than that they come still nearer
and nearer, but never touch. But in special in the asymptosy of
hyperbolic lines, it is understood they should approach to a distance
less than any given quantity.

[Sidenote: Situation, by what it is determined.]

20. SITUATION _is the relation of one place to another_; and where there
are many places, their situation is determined by four things; by _their
distances from one another_; by _several distances from a place
assigned_; by _the order of strait lines drawn from a place assigned to
the places of them all_; and by _the angles which are made by the lines
so drawn_. For if their distances, order, and angles, be given, that is,
be certainly known, their several places will also be so certainly
known, as that they can be no other.

[Sidenote: What is like situation; what is figure; and what are like
           figures.]

21. Points, how many soever they be, have _like situation_ with an equal
number of other points, when all the strait lines, that are drawn from
some one point to all these, have severally the same proportion to
those, that are drawn in the same order and at equal angles from some
one point to all those. For let there be any number of points as A, B,
and C, (in the 9th figure) to which from some one point D let the strait
lines D A, D B, and D C be drawn; and let there be an equal number of
other points, as E, F, and G, and from some point H let the strait lines
H E, H F, and H G be drawn, so that the angles A D B and B D C be
severally and in the same order equal to the angles E H F and F H G, and
the strait lines D A, D B, and D C proportional to the strait lines H E,
H F, and H G; I say, the three points A, B, and C, have like situation
with the three points E, F, and G, or are placed alike. For if H E be
understood to be laid upon D A, so that the point H be in D, the point F
will be in the strait line D B, by reason of the equality of the angles
A D B and E H F; and the point G will be in the strait line D C, by
reason of the equality of the angles B D C and F H G; and the strait
lines A B and E F, as also B C and F G, will be parallel, because A D. E
H :: B D. F H :: C D. G H are proportionals by construction; and
therefore the distances between the points A and B, and the points B and
C, will be proportional to the distances between the points E and F, and
the points F and G. Wherefore, in the situation of the points A, B, and
C, and the situation of the points E, F and G, the angles in the same
order are equal; so that their situations differ in nothing but the
inequality of their distances from one another, and of their distances
from the points D and H. Now, in both the orders of points, those
inequalities are equal; for A B. B C :: E F. F G, which are their
distances from one another, as also D A. D B. D C :: H E. H F. H G,
which are their distances from the assumed points D and H, are
proportionals. Their difference, therefore, consists solely in the
magnitude of their distances. But, by the definition of _like_, (chapter
I. article 2) those things, which differ only in magnitude, are like.
Wherefore the points A, B, and C, have to one another like situation
with the points E, F, and G, or are placed alike; which was to be
proved.

FIGURE _is quantity, determined by the situation or placing of all its
extreme points_. Now I call those points extreme, which are contiguous
to the place which is without the figure. In lines therefore and
superficies, all points may be called extreme; but in solids only those
which are in the superficies that includes them.

_Like figures_ are those, whose extreme points in one of them are all
placed like all the extreme points in the other; for such figures differ
in nothing but magnitude.

And like figures are _alike placed_, when in both of them the homologal
strait lines, that is, the strait lines which connect the points which
answer one another, are parallel, and have their proportional sides
inclined the same way.

And seeing every strait line is like every other strait line, and every
plane like every other plane, when nothing but planeness is considered;
if the lines, which include planes, or the superficies, which include
solids, have their proportions known, it will not be hard to know
whether any figure be like or unlike to another propounded figure.

And thus much concerning the _first grounds of philosophy_. The next
place belongs to _geometry_; in which the quantities of figures are
sought out from the proportions of lines and angles. Wherefore it is
necessary for him, that would study geometry, to know first what is the
nature of quantity, proportion, angle and figure. Having therefore
explained these in the three last chapters, I thought fit to add them to
this part; and so pass to the next.

------------------------------------------------------------------------

[Illustration:

  _Vol. 1. Lat. & Eng._
  C. XIV.
  _Fig. 1-10_
]

------------------------------------------------------------------------




                                PART III

                         PROPORTIONS OF MOTIONS
                            AND MAGNITUDES.


                                -------


                              CHAPTER XV.

                 OF THE NATURE, PROPERTIES, AND DIVERS
                      CONSIDERATIONS OF MOTION AND
                               ENDEAVOUR.

1.. Repetition of some principles of the doctrine of motion formerly set
    down.—2. Other principles added to them.—3. Certain theorems
    concerning the nature of motion.—4. Divers considerations of
    motion.—5. The way by which the first endeavour of bodies moved
    tendeth.—6. In motion which is made by concourse, one of the movents
    ceasing, the endeavour is made by the way by which the rest tend.—7.
    All endeavour is propagated in infinitum.—8. How much greater the
    velocity or magnitude is of a movent, so much the greater is the
    efficacy thereof upon any other body in its way.


[Sidenote: Repetition of some principles of the doctrine of motion
           formerly set down.]

1. The next things in order to be treated of are MOTION and MAGNITUDE,
which are the most common accidents of all bodies. This place therefore
most properly belongs to the elements of geometry. But because this part
of philosophy, having been improved by the best wits of all ages, has
afforded greater plenty of matter than can well be thrust together
within the narrow limits of this discourse, I thought fit to admonish
the reader, that before he proceed further, he take into his hands the
works of Euclid, Archimedes, Apollonius, and other as well ancient as
modern writers. For to what end is it, to do over again that which is
already done? The little therefore that I shall say concerning geometry
in some of the following chapters, shall be such only as is new, and
conducing to natural philosophy.

I have already delivered some of the principles of this doctrine in the
eighth and ninth chapters; which I shall briefly put together here, that
the reader in going on may have their light nearer at hand.

First, therefore, in chap. VIII. art. 10, _motion_ is defined to be _the
continual privation of one place, and acquisition of another_.

Secondly, it is there shown, that _whatsoever is moved is moved in
time_.

Thirdly, in the same chapter, art. 11, I have defined _rest to be when a
body remains for some time in one place_.

Fourthly, it is there shown, that _whatsoever is moved is not in any
determined place_; as also that the same _has been moved, is still
moved, and will yet be moved_; so that in every part of that space, in
which motion is made, we may consider three times, namely, the _past_,
the _present_, and _future time_.

Fifthly, in art. 15 of the same chapter, I have defined _velocity_ or
_swiftness to be motion considered as power, namely, that power by which
a body moved may in a certain time transmit a certain length_; which
also may more briefly be enunciated thus, _velocity is the quantity of
motion determined by time and line_.

Sixthly, in the same chapter, art. 16, I have shown that _motion is the
measure of time_.

Seventhly, in the same chapter, art. 17, I have defined motions to be
equally swift, when in equal times equal lengths are transmitted by
them.

Eighthly, in art. 18 of the same chapter, _motions_ are defined to be
_equal, when the swiftness of one moved body, computed in every part of
its magnitude, is equal to the swiftness of another, computed also in
every part of its magnitude_. From whence it is to be noted, that
_motions equal to one another, and motions equally swift_, do not
signify the same thing; for when two horses draw abreast, the motion of
both is greater than the motion of either of them singly; but the
swiftness of both together is but equal to that of either.

Ninthly, in art. 19 of the same chapter, I have shown, that _whatsoever
is at rest will always be at rest, unless there be some other body
besides it, which by getting into its place suffers it no longer to
remain at rest_. And that _whatsoever is moved, will always be moved,
unless there be some other body besides it, which hinders its motion_.

Tenthly, in chap. IX. art. 7, I have demonstrated, that _when any body
is moved which was formerly at rest, the immediate efficient cause of
that motion is in some other moved and contiguous body_.

Eleventhly, I have shown in the same place, that _whatsoever is moved,
will always be moved in the same way, and with the same swiftness, if it
be not hindered by some other moved and contiguous body_.

[Sidenote: Other principles
           added to them.]

2. To which principles I shall here add those that follow. First, I
define ENDEAVOUR _to be motion made in less space and time than can be
given_; that is, _less than can be determined or assigned by exposition
or number_; that is, _motion made through the length of a point, and in
an instant or point of time_. For the explaining of which definition it
must be remembered, that by a point is not to be understood that which
has no quantity, or which cannot by any means be divided; for there is
no such thing in nature; but that, whose quantity is not at all
considered, that is, whereof neither quantity nor any part is computed
in demonstration; so that a point is not to be taken for an indivisible,
but for an undivided thing; as also an instant is to be taken for an
undivided, and not for an indivisible time.

In like manner, endeavour is to be conceived as motion; but so as that
neither the quantity of the time in which, nor of the line in which it
is made, may in demonstration be at all brought into comparison with the
quantity of that time, or of that line of which it is a part. And yet,
as a point may be compared with a point, so one endeavour may be
compared with another endeavour, and one may be found to be greater or
less than another. For if the vertical points of two angles be compared,
they will be equal or unequal in the same proportion which the angles
themselves have to one another. Or if a strait line cut many
circumferences of concentric circles, the inequality of the points of
intersection will be in the same proportion which the perimeters have to
one another. And in the same manner, if two motions begin and end both
together, their endeavours will be equal or unequal, according to the
proportion of their velocities; as we see a bullet of lead descend with
greater endeavour than a ball of wool.

Secondly, I define IMPETUS, _or quickness of motion, to be the swiftness
or velocity of the body moved, but considered in the several points of
that time in which it is moved. In which sense_ impetus _is nothing else
but the quantity or velocity of endeavour. But considered with the whole
time, it is the whole velocity of the body moved taken together
throughout all the time, and equal to the product of a line representing
the time, multiplied into a line representing the arithmetically mean_
impetus _or quickness_. Which arithmetical mean, what it is, is defined
in the 29th article of chapter XIII.

And because in equal times the ways that are passed are as the
velocities, and the _impetus_ is the velocity they go withal, reckoned
in all the several points of the times, it followeth that during any
time whatsoever, howsoever the _impetus_ be increased or decreased, the
length of the way passed over shall be increased or decreased in the
same proportion; and the same line shall represent both the way of the
body moved, and the several _impetus_ or degrees of swiftness wherewith
the way is passed over.

And if the body moved be not a point, but a strait line moved so as that
every point thereof make a several strait line, the plane described by
its motion, whether uniform, accelerated, or retarded, shall be greater
or less, the time being the same, in the same proportion with that of
the _impetus_ reckoned in one motion to the _impetus_ reckoned in the
other. For the reason is the same in parallelograms and their sides.

For the same cause also, if the body moved be a plane, the solid
described shall be still greater or less in the proportions of the
several _impetus_ or quicknesses reckoned through one line, to the
several _impetus_ reckoned through another.

This understood, let A B C D, (in figure 1, chap. XVII.) be a
parallelogram; in which suppose the side A B to be moved parallelly to
the opposite side C D, decreasing all the way till it vanish in the
point C, and so describing the figure A B E F C; the point B, as A B
decreaseth, will therefore describe the line B E F C; and suppose the
time of this motion designed by the line C D; and in the same time C D,
suppose the side A C to be moved parallel and uniformly to B D. From the
point O taken at adventure in the line C D, draw O R parallel to B D,
cutting the line B E F C in E, and the side A B in R. And again, from
the point Q taken also at adventure in the line C D, draw Q S parallel
to B D, cutting the line B E F C in F, and the side A B in S; and draw E
G and F H parallel to C D, cutting A C in G and H. Lastly, suppose the
same construction done in all the points possible of the line B E F C. I
say, that as the proportions of the swiftness wherewith Q F, O E, D B,
and all the rest supposed to be drawn parallel to D B and terminated in
the line B E F C, are to the proportions of their several times designed
by the several parallels H F, G E, A B, and all the rest supposed to be
drawn parallel to the line of time C D and terminated in the line B E F
C, the aggregate to the aggregate, so is the area or plane D B E F C to
the area or plane A C F E B. For as A B decreasing continually by the
line B E F C vanisheth in the time C D into the point C, so in the same
time the line D C continually decreasing vanisheth by the same line C F
E B into the point B; and the point D describeth in that decreasing
motion the line D B equal to the line A C described by the point A in
the decreasing motion of A B; and their swiftnesses are therefore equal.
Again, because in the time G E the point O describeth the line OE, and
in the same time the point S describeth the line S E, the line O E shall
be to the line SE, as the swiftness wherewith OE is described to the
swiftness wherewith SE is described. In like manner, because in the same
time H F the point Q describeth the line Q F, and the point R the line R
F, it shall be as the swiftness by which Q F is described to the
swiftness by which R F is described, so the line itself Q F to the line
itself R F; and so in all the lines that can possibly be drawn parallel
to B D in the points where they cut the line B E F C. But all the
parallels to B D, as S E, R F, A C, and the rest that can possibly be
drawn from the line A B to the line B E F C, make the area of the plane
A B E F C; and all the parallels to the same B D, as Q F, O E, D B and
the rest drawn to the points where they cut the same line B E F C, make
the area of the plane B E F C D. As therefore the aggregate of the
swiftnesses wherewith the plane B E F C D is described, is to the
aggregate of the swiftnesses wherewith the plane A C F E B is described,
so is the plane itself B E F C D to the plane itself A C F E B. But the
aggregate of the times represented by the parallels A B, G E, H F and
the rest, maketh also the area A C F E B. And therefore, as the
aggregate of all the lines Q F, O E, D B and all the rest of the lines
parallel to B D and terminated in the line B E F C, is to the aggregate
of all the lines H F, G E, A B and all the rest of the lines parallel to
C D and terminated in the same line B E F C; that is, as the aggregate
of the lines of swiftness to the aggregate of the lines of time, or as
the whole swiftness in the parallels to D B to the whole time in the
parallels to C D, so is the plane B E F C D to the plane A C F E B. And
the proportions of Q F to F H, and of O E to E G, and of D B to B A, and
so of all the rest taken together, are the proportions of the plane D B
E F C to the plane A B E F C. But the lines Q F, O E, D B and the rest
are the lines that design the swiftness; and the lines H F, G E, A B and
the rest are the lines that design the times of the motions; and
therefore the proportion of the plane D B E F C to the plane A B E F C
is the proportion of all the velocities taken together to all the times
taken together. Wherefore, as the proportions of the swiftnesses, &c.;
which was to be demonstrated.

The same holds also in the diminution of the circles, whereof the lines
of time are the semidiameters, as may easily be conceived by imagining
the whole plane A B C D turned round upon the axis B D; for the line B E
F C will be everywhere in the superficies so made, and the lines H F, G
E, A B, which are here parallelograms, will be there cylinders, the
diameters of whose bases are the lines H F, G E, A B, &c. and the
altitude a point, that is to say, a quantity less than any quantity that
can possibly be named; and the lines Q F, O E, D B, &c. small solids
whose lengths and breadths are less than any quantity that can be named.

But this is to be noted, that unless the proportion of the sum of the
swiftnesses to the proportion of the sum of the times be determined, the
proportion of the figure D B E F C to the figure A B E F C cannot be
determined.

Thirdly, I define RESISTANCE _to be the endeavour of one moved body
either wholly or in part contrary to the endeavour of another moved
body, which toucheth the same_. I say, wholly contrary, when the
endeavour of two bodies proceeds in the same strait line from the
opposite extremes, and contrary in part, when two bodies have their
endeavour in two lines, which, proceeding from the extreme points of a
strait line, meet without the same.

Fourthly, that I may define what it is to PRESS, I say, that _of two
moved bodies one presses the other, when with its endeavour it makes
either all or part of the other body to go out of its place_.

Fifthly, _a body, which is pressed and not wholly removed, is said to_
RESTORE _itself, when, the pressing body being taken away, the parts
which were moved do, by reason of the internal constitution of the
pressed body, return every one into its own place_. And this we may
observe in springs, in blown bladders, and in many other bodies, whose
parts yield more or less to the endeavour which the pressing body makes
at the first arrival; but afterwards, when the pressing body is removed,
they do, by some force within them, _restore_ themselves, and give their
whole body the same figure it had before.

Sixthly, I define FORCE _to be the_ impetus _or quickness of motion
multiplied either into itself, or into the magnitude of the movent, by
means whereof the said movent works more or less upon the body that
resists it_.

[Sidenote: Certain theorems concerning the nature of motion.]

3. Having premised thus much, I shall now demonstrate, first, that if a
point moved come to touch another point which is at rest, how little
soever the impetus or quickness of its motion be, it shall move that
other point. For if by that impetus it do not at all move it out of its
place, neither shall it move it with double the same impetus. For
nothing doubled is still nothing; and for the same reason it shall never
move it with that impetus, how many times soever it be multiplied,
because nothing, however it be multiplied, will for ever be nothing.
Wherefore, when a point is at rest, if it do not yield to the least
impetus, it will yield to none; and consequently it will be impossible
that that, which is at rest, should ever be moved.

Secondly, that when a point moved, how little soever the impetus thereof
be, falls upon a point of any body at rest, how hard soever that body
be, it will at the first touch make it yield a little. For if it do not
yield to the impetus which is in that point, neither will it yield to
the impetus of never so many points, which have all their impetus
severally equal to the impetus of that point. For seeing all those
points together work equally, if any one of them have no effect, the
aggregate of them all together shall have no effect as many times told
as there are points in the whole body, that is, still no effect at all;
and by consequent there would be some bodies so hard that it would be
impossible to break them; that is, a finite hardness, or a finite force,
would not yield to that which is infinite; which is absurd.

Coroll. It is therefore manifest, that rest does nothing at all, nor is
of any efficacy; and that nothing but motion gives motion to such things
as be at rest, and takes it from things moved.

Thirdly, that cessation in the movent does not cause cessation in that
which was moved by it. For (by number 11 of art. 1 of this chapter)
whatsoever is moved perseveres in the same way and with the same
swiftness, as long as it is not hindered by something that is moved
against it. Now it is manifest, that cessation is not contrary motion;
and therefore it follows that the standing still of the movent does not
make it necessary that the thing moved should also stand still.

Coroll. They are therefore deceived, that reckon the taking away of the
impediment or resistance for one of the causes of motion.

[Sidenote: Divers considerations of motions.]

4. Motion is brought into account for divers respects; first, as in a
body _undivided_, that is, considered as a point; or, as in a _divided_
body. In an undivided body, when we suppose the way, by which the motion
is made, to be a line; and in a divided body, when we compute the motion
of the several parts of that body, as of parts.

Secondly, from the diversity of the regulation of motion, it is in body,
considered as undivided. sometimes _uniform_ and sometimes _multiform_.
_Uniform_ is that by which equal lines are always transmitted in equal
times; and _multiform_, when in one time more, in another time less
space is transmitted. Again, of multiform motions, there are some in
which the degrees of acceleration and retardation proceed in the same
proportions, which the spaces transmitted have, whether duplicate, or
triplicate, or by whatsoever number multiplied; and others in which it
is otherwise.

Thirdly, from the number of the movents; that is, one motion is made by
one movent only, and another by the concourse of many movents.

Fourthly, from the position of that line in which a body is moved, in
respect of some other line; and from hence one motion is called
_perpendicular_, another _oblique_, another _parallel_.

Fifthly, from the position of the movent in respect of the moved body;
from whence one motion is _pulsion_ or driving, another _traction_ or
drawing. _Pulsion_, when the movent makes the moved body go before it;
and _traction_, when it makes it follow. Again, there are two sorts of
_pulsion_; one, when the motions of the movent and moved body begin both
together, which may be called _trusion_ or _thrusting_ and _vection_;
the other, when the movent is first moved, and afterwards the moved
body, which motion is called _percussion_ or _stroke_.

Sixthly, motion is considered sometimes from the effect only which the
movent works in the moved body, which is usually called _moment_. Now
_moment is the excess of motion which the movent has above the motion or
endeavour of the resisting body_.

Seventhly, it may be considered from the diversity of the _medium_; as
one motion may be made in _vacuity_ or _empty place_; another in a
_fluid_; another in a _consistent medium_, that is, a _medium_ whose
parts are by some power so _consistent_ and _cohering_, that no part of
the same will yield to the movent, unless the whole yield also.

Eighthly, when a moved body is considered as having parts, there arises
another distinction of motion into _simple_ and _compound_. _Simple_,
when all the several parts describe several equal lines; _compounded_,
when the lines described are unequal.

[Sidenote: The way by which the first endeavour of bodies moved
           tendeth.]

5. All endeavour tends towards that part, that is to say, in that way
which is determined by the motion of the movent, if the movent be but
one; or, if there be many movents, in that way which their concourse
determines. For example, if a moved body have direct motion, its first
endeavour will be in a strait line; if it have circular motion, its
first endeavour will be in the circumference of a circle.

[Sidenote: In motion, which it made by concourse, one of the movents
           ceasing, the endeavour is made by the way by which the rest
           tend.]

6. And whatsoever the line be, in which a body has its motion from the
concourse of two movents, as soon as in any point thereof the force of
one of the movents ceases, there immediately the former endeavour of
that body will be changed into an endeavour in the line of the other
movent.

Wherefore, when any body is carried on by the concourse of two winds,
one of those winds ceasing, the endeavour and motion of that body will
be in that line, in which it would have been carried by that wind alone
which blows still. And in the describing of a circle, where that which
is moved has its motion determined by a movent in a tangent, and by the
radius which keeps it in a certain distance from the centre, if the
retention of the radius cease, that endeavour, which was in the
circumference of the circle, will now be in the tangent, that is, in a
strait line. For, seeing endeavour is computed in a less part of the
circumference than can be given, that is, in a point, the way by which a
body is moved in the circumference is compounded of innumerable strait
lines, of which every one is less than can be given; which are therefore
called points. Wherefore when any body, which is moved in the
circumference of a circle, is freed from the retention of the radius, it
will proceed in one of those strait lines, that is, in a tangent.

[Sidenote: All endeavour is propagated in infinitum.]

7. All endeavour, whether strong or weak, is propagated to infinite
distance; for it is motion. If therefore the first endeavour of a body
be made in space which is empty, it will always proceed with the same
velocity; for it cannot be supposed that it can receive any resistance
at all from empty space; and therefore, (by art. 7, chap, IX) it will
always proceed in the same way and with the same swiftness. And if its
endeavour be in space which is filled, yet, seeing endeavour is motion,
that which stands next in its way shall be removed, and endeavour
further, and again remove that which stands next, and so infinitely.
Wherefore the propagation of endeavour, from one part of full space to
another, proceeds infinitely. Besides, it reaches in any instant to any
distance, how great soever. For in the same instant in which the first
part of the full _medium_ removes that which is next it, the second also
removes that part which is next to it; and therefore all endeavour,
whether it be in empty or in full space, proceeds not only to any
distance, how great soever, but also in any time, how little soever,
that is, in an instant. Nor makes it any matter, that endeavour, by
proceeding, grows weaker and weaker, till at last it can no longer be
perceived by sense; for motion may be insensible; and I do not here
examine things by sense and experience, but by reason.

[Sidenote: How much greater the velocity or magnitude is of a movent, so
           much the greater is the efficacy thereof upon any other body
           in its way.]

8. When two movents are of equal magnitude, the swifter of them works
with greater force than the slower, upon a body that resists their
motion. Also, if two movents have equal velocity, the greater of them
works with more force than the less. For where the magnitude is equal,
the movent of greater velocity makes the greater impression upon that
body upon which it falls; and where the velocity is equal, the movent of
greater magnitude falling upon the same point, or an equal part of
another body, loses less of its velocity, because the resisting body
works only upon that part of the movent which it touches, and therefore
abates the impetus of that part only; whereas in the mean time the
parts, which are not touched, proceed, and retain their whole force,
till they also come to be touched; and their force has some effect.
Wherefore, for example, in batteries a longer than a shorter piece of
timber of the same thickness and velocity, and a thicker than a
slenderer piece of the same length and velocity, work a greater effect
upon the wall.


                                -------


                              CHAPTER XVI.

                 OF MOTION ACCELERATED AND UNIFORM, AND
                        OF MOTION BY CONCOURSE.

1. The velocity of any body, in what time soever it be computed, is that
    which is made of the multiplication of the impetus, or quickness of
    its motion into the time.—2-5. In all motion, the lengths which are
    passed through are to one another, as the products made by the
    impetus multiplied into the time.—6. If two bodies be moved with
    uniform motion through two lengths, the proportion of those lengths
    to one another will be compounded of the proportions of time to
    time, and impetus to impetus, directly taken.—7. If two bodies pass
    through two lengths with uniform motion, the proportion of their
    times to one another will be compounded of the proportions of length
    to length, and impetus to impetus reciprocally taken; also the
    proportion of their impetus to one another will be compounded of the
    proportions of length to length, and time to time reciprocally
    taken.—8. If a body be carried on with uniform motion by two movents
    together, which meet in an angle, the line by which it passes will
    be a strait line, subtending the complement of that angle to two
    right angles.—9, &c. If a body be carried by two movents together,
    one of them being moved with uniform, the other with accelerated
    motion, and the proportion of their lengths to their times being
    explicable in numbers, how to find out what line that body
    describes.


[Sidenote: The velocity of any body, in what time soever it be computed,
           is that which is made of the multiplication of the impetus or
           quickness of its motion into the time.]

1. The velocity of any body, in whatsoever time it be moved, has its
quantity determined by the sum of all the several quicknesses or
impetus, which it hath in the several points of the time of the body's
motion. For seeing velocity, (by the definition of it, chap, VIII, art.
15) is that power by which a body can in a certain time pass through a
certain length; and quickness of motion or impetus, (by chap. XV, art.
2, num. 2) is velocity taken in one point of time only, all the impetus,
together taken in all the points of time, will be the same thing with
the mean impetus multiplied into the whole time, or which is all one,
will be the velocity of the whole motion.

Coroll. If the impetus be the same in every point, any strait line
representing it may be taken for the measure of time: and the
quicknesses or impetus applied ordinately to any strait line making an
angle with it, and representing the way of the body's motion, will
design a parallelogram which shall represent the velocity of the whole
motion. But if the impetus or quickness of motion begin from rest and
increase uniformly, that is, in the same proportion continually with the
times which are passed, the whole velocity of the motion shall be
represented by a triangle, one side whereof is the whole time, and the
other the greatest impetus acquired in that time; or else by a
parallelogram, one of whose sides is the whole time of motion, and the
other, half the greatest impetus; or lastly, by a parallelogram having
for one side a mean proportional between the whole time and the half of
that time, and for the other side the half of the greatest impetus. For
both these parallelograms are equal to one another, and severally equal
to the triangle which is made of the whole line of time, and of the
greatest acquired impetus; as is demonstrated in the elements of
geometry.

[Sidenote: In all motion, the lengths which are passed through are to
           one another, as the products made by the impetus multiplied
           into time.]

2. In all uniform motions the lengths which are transmitted are to one
another, as the product of the mean impetus multiplied into its time, to
the product of the mean impetus multiplied also into its time.

For let A B (in fig. 1) be the time, and A C the impetus by which any
body passes with uniform motion through the length D E; and in any part
of the time A B, as in the time A F, let another body be moved with
uniform motion, first, with the same impetus A C. This body, therefore,
in the time A F with the impetus A C will pass through the length A F.
Seeing, therefore, when bodies are moved in the same time, and with the
same velocity and impetus in every part of their motion, the proportion
of one length transmitted to another length transmitted, is the same
with that of time to time, it followeth, that the length transmitted in
the time A B with the impetus A C will be to the length transmitted in
the time A F with the same impetus A C, as A B itself is to A F, that
is, as the parallelogram A I is to the parallelogram A H, that is, as
the product of the time A B into the mean impetus A C is to the product
of the time A F into the same impetus A C. Again, let it be supposed
that a body be moved in the time A F, not with the same but with some
other uniform impetus, as A L. Seeing therefore, one of the bodies has
in all the parts of its motion the impetus A C, and the other in like
manner the impetus A L, the length transmitted by the body moved with
the impetus A C will be to the length transmitted by the body moved with
the impetus A L, as A C itself is to A L, that is, as the parallelogram
A H is to the parallelogram F L. Wherefore, by ordinate proportion it
will be, as the parallelogram A I to the parallelogram F L, that is, as
the product of the mean impetus into the time is to the product of the
mean impetus into the time, so the length transmitted in the time A B
with the impetus A C, to the length transmitted in the time A F with the
impetus A L; which was to be demonstrated.

Coroll. Seeing, therefore, in uniform motion, as has been shown, the
lengths transmitted are to one another as the parallelograms which are
made by the multiplication of the mean impetus into the times, that is,
by reason of the equality of the impetus all the way, as the times
themselves, it will also be, by permutation, as time to length, so time
to length; and in general, to this place are applicable all the
properties and transmutations of analogisms, which I have set down and
demonstrated in chapter XIII.

3. In motion begun from rest and uniformly accelerated, that is, where
the impetus increaseth continually according to the proportion of the
times, it will also be, as one product made by the mean impetus
multiplied into the time, to another product made likewise by the mean
impetus multiplied into the time, so the length transmitted in the one
time to the length transmitted in the other time.

For let A B (in fig. 1) represent a time; in the beginning of which time
A, let the impetus be as the point A; but as the time goes on, so let
the impetus increase uniformly, till in the last point of that time A B,
namely in B, the impetus acquired be B I. Again, let A F represent
another time, in whose beginning A, let the impetus be as the point
itself A; but as the time proceeds, so let the impetus increase
uniformly, till in the last point F of the time A F the impetus acquired
be F K; and let D E be the length passed through in the time A B with
impetus uniformly increased. I say, the length D E is to the length
transmitted in the time A F, as the time A B multiplied into the mean of
the impetus increasing through the time A B, is to the time A F
multiplied into the mean of the impetus increasing through the time A F.

For seeing the triangle A B I is the whole velocity of the body moved in
the time A B, till the impetus acquired be B I; and the triangle A F K
the whole velocity of the body moved in the time A F with impetus
increasing till there be acquired the impetus F K; the length D E to the
length acquired in the time A F with impetus increasing from rest in A
till there be acquired the impetus F K, will be as the triangle A B I to
the triangle A F K, that is, if the triangles A B I and A F K be like,
in duplicate proportion of the time A B to the time A F; but if unlike,
in the proportion compounded of the proportions of A B to A F and of B I
to F K. Wherefore, as A B I is to A F K, so let D E be to D P; for so,
the length transmitted in the time A B with impetus increasing to B I,
will be to the length transmitted in the time A F with impetus
increasing to F K, as the triangle A B I is to the triangle A F K; but
the triangle A B I is made by the multiplication of the time A B into
the mean of the impetus increasing to B I; and the triangle A F K is
made by the multiplication of the time A F into the mean of the
_impetus_ increasing to F K; and therefore the length D E which is
transmitted in the time A B with impetus increasing to B I, to the
length D P which is transmitted in the time A F with impetus increasing
to F K, is as the product which is made of the time A B multiplied into
its mean impetus, to the product of the time A F multiplied also into
its mean impetus; which was to be proved.

Coroll. I. In motion uniformly accelerated, the proportion of the
lengths transmitted to that of their times, is compounded of the
proportions of their times to their times, and impetus to impetus.

Coroll. II. In motion uniformly accelerated, the lengths transmitted in
equal times, taken in continual succession from the beginning of motion,
are as the differences of square numbers beginning from unity, namely,
as 3, 5, 7, &c. For if in the first time the length transmitted be as 1,
in the first and second times the length transmitted will be as 4, which
is the square of 2, and in the three first times it will be as 9, which
is the square of 3, and in the four first times as 16, and so on. Now
the differences of these squares are 3, 5, 7, &c.

Coroll. III. In motion uniformly accelerated from rest, the length
transmitted is to another length transmitted uniformly in the same time,
but with such impetus as was acquired by the accelerated motion in the
last point of that time, as a triangle to a parallelogram, which have
their altitude and base common. For seeing the length D E (in fig. 1) is
passed through with velocity as the triangle A B I, it is necessary that
for the passing through of a length which is double to D E, the velocity
be as the parallelogram A I; for the parallelogram A I is double to the
triangle A B I.

4. In motion, which beginning from rest is so aclerated, that the
impetus thereof increases continually in proportion duplicate to the
proportion of the times in which it is made, a length transmitted in one
time will be to a length transmitted in another time, as the product
made by the mean impetus multiplied into the time of one of those
motions, to the product of the mean impetus multiplied into the time of
the other motion.

For let A B (in fig. 2) represent a time, in whose first instant A let
the impetus be as the point A; but as the time proceeds, so let the
impetus increase continually in duplicate proportion to that of the
times, till in the last point of time B the impetus acquired be B I;
then taking the point F anywhere in the time A B, let the impetus F K
acquired in the time A F be ordinately applied to that point F. Seeing
therefore the proportion of F K to B I is supposed to be duplicate to
that of A F to A B, the proportion of A F to A B will be subduplicate to
that of F K to B I; and that of A B to A F will be (by chap. XIII. art.
16) duplicate to that of B I to F K; and consequently the point K will
be in a parabolical line, whose diameter is A B and base B I; and for
the same reason, to what point soever of the time A B the impetus
acquired in that time be ordinately applied, the strait line designing
that impetus will be in the same parabolical line A K I. Wherefore the
mean impetus multiplied into the whole time A B will be the parabola A K
I B, equal to the parallelogram A M, which parallelogram has for one
side the line of time A B and for the other the line of the impetus A L,
which is two-thirds of the impetus B I; for every parabola is equal to
two-thirds of that parallelogram with which it has its altitude and base
common. Wherefore the whole velocity in A B will be the parallelogram A
M, as being made by the multiplication of the impetus A L into the time
A B. And in like manner, if F N be taken, which is two-thirds of the
impetus F K, and the parallelogram F O be completed, F O will be the
whole velocity in the time A F, as being made by the uniform impetus A O
or F N multiplied into the time A F. Let now the length transmitted in
the time A B and with the velocity A M be the strait line D E; and
lastly, let the length transmitted in the time A F with the velocity A N
be D P; I say that as A M is to A N, or as the parabola A K I B to the
parabola A K F, so is D E to D P. For as A M is to F L, that is, as A B
is to A F, so let D E be to D G. Now the proportion of A M to A N is
compounded of the proportions of A M to F L, and of F L to A N. But as A
M to F L, so by construction is D E to D G; and as F L is to A N (seeing
the time in both is the same, namely, A F), so is the length D G to the
length D P; for lengths transmitted in the same time are to one another
as their velocities are. Wherefore by ordinate proportion, as A M is to
A N, that is, as the mean impetus A L multiplied into its time A B, is
to the mean impetus A O multiplied into A F, so is D E to D P; which was
to be proved.

Coroll. I. Lengths transmitted with motion so accelerated, that the
impetus increase continually in duplicate proportion to that of their
times, if the base represent the impetus, are in triplicate proportion
of their impetus acquired in the last point of their times. For as the
length D E is to the length D P, so is the parallelogram A M to the
parallelogram A N, and so the parabola A K I B to the parabola A K F.
But the proportion of the parabola A K I B to the parabola A K F is
triplicate to the proportion which the base B I has to the base F K.
Wherefore also the proportion of D E to D P is triplicate to that of B I
to F K.

Coroll. II. Lengths transmitted in equal times succeeding one another
from the beginning, by motion so accelerated, that the proportion of the
impetus be duplicate to the proportion of the times, are to one another
as the differences of cubic numbers beginning at unity, that is as 7,
19, 37, &c. For if in the first time the length transmitted be as 1, the
length at the end of the second time will be as 8, at the end of the
third time as 27, and at the end of the fourth time as 64, &c.; which
are cubic numbers, whose differences are 7, 19, 37, &c.

Coroll. III. In motion so accelerated, as that the length transmitted be
always to the length transmitted in duplicate proportion to their times,
the length uniformly transmitted in the whole time, and with impetus all
the way equal to that which is last acquired, is as a parabola to a
parallelogram of the same altitude and base, that is, as 2 to 3. For the
parabola A K I B is the impetus increasing in the time A B; and the
parallelogram A I is the greatest uniform impetus multiplied into the
same time A B. Wherefore the lengths transmitted will be as a parabola
to a parallelogram, &c., that is, as 2 to 3.

5. If I should proceed to the explication of such motions as are made by
impetus increasing in proportion triplicate, quadruplicate,
quintuplicate, &c., to that of their times, it would be a labour
infinite and unnecessary. For by the same method by which I have
computed such lengths, as are transmitted with impetus increasing in
single and duplicate proportion, any man may compute such as are
transmitted with impetus increasing in triplicate, quadruplicate, or
what other proportion he pleases.

In making which computation he shall find, that where the impetus
increase in proportion triplicate to that of the times, there the whole
velocity will be designed by the first parabolaster (of which see the
next chapter); and the lengths transmitted will be in proportion
quadruplicate to that of the times. And in like manner, where the
impetus increase in quadruplicate proportion to that of the times, that
there the whole velocity will be designed by the second parabolaster,
and the lengths transmitted will be in quintuplicate proportion to that
of the times; and so on continually.

[Sidenote: If two bodies be moved with uniform motion through two
           lengths, the proportion of those lengths to one another, will
           be compounded of the proportions of time to time, and impetus
           to impetus, directly taken.]

6. If two bodies with uniform motion transmit two lengths, each with its
own impetus and time, the proportion of the lengths transmitted will be
compounded of the proportions of time to time, and impetus to impetus,
directly taken.

Let two bodies be moved uniformly (as in fig. 3), one in the time A B
with the impetus A C, the other in the time A D with the impetus A E. I
say the lengths transmitted have their proportion to one another
compounded of the proportions of A B to A D, and of A C to A E. For let
any length whatsoever, as Z, be transmitted by one of the bodies in the
time A B with the impetus A C; and any other length, as X, be
transmitted by the other body in the time A D with the impetus A E; and
let the parallelograms A F and A G be completed. Seeing now Z is to X
(by art. 2) as the impetus A C multiplied into the time A B is to the
impetus A E multiplied into the time A D, that is, as A F to A G; the
proportion of Z to X will be compounded of the same proportions, of
which the proportion of A F to A G is compounded; but the proportion of
A F to A G is compounded of the proportions of the side A B to the side
A D, and of the side A C to the side A E (as is evident by the Elements
of Euclid), that is, of the proportions of the time A B to the time A D,
and of the impetus A C to the impetus A E. Wherefore also the proportion
of Z to X is compounded of the same proportions of the time A B to the
time A D, and of the impetus A C to the impetus A E; which was to be
demonstrated.

Coroll. I. When two bodies are moved with uniform motion, if the times
and impetus be in reciprocal proportion, the lengths transmitted shall
be equal. For if it were as A B to A D (in the same fig. 3) so
reciprocally A E to A C, the proportion of A F to A G would be
compounded of the proportions of A B to A D, and of A C to A E, that is,
of the proportions of A B to A D, and of A D to A B. Wherefore, A F
would be to A G as A B to A B, that is, equal; and so the two products
made by the multiplication of impetus into time would be equal; and by
consequent, Z would be equal to X.

Coroll. II. If two bodies be moved in the same time, but with different
impetus, the lengths transmitted will be as impetus to impetus. For if
the time of both of them be A D, and their different impetus be A E and
A C, the proportion of A G to D C will be compounded of the proportions
of A E to A C and of A D to A D, that is, of the proportions of A E to A
C and of A C to A C; and so the proportion of A G to D C, that is, the
proportion of length to length, will be as A E to A C, that is, as that
of impetus to impetus. In like manner, if two bodies be moved uniformly,
and both of them with the same impetus, but in different times, the
proportion of the lengths transmitted by them will be as that of their
times. For if they have both the same impetus A C, and their different
times be A B and A D, the proportion of A F to D C will be compounded of
the proportions of A B to A D and of A C to A C; that is, of the
proportions of A B to A D and of A D to A D; and therefore the
proportion of A F to D C, that is, of length to length, will be the same
with that of A B to A D, which is the proportion of time to time.

[Sidenote: If two bodies pass through two lengths with uniform motion,
           the proportion of their times to one another, will be
           compounded of the proportions of length to length, and
           impetus to impetus reciprocally taken; also the proportion of
           their impetus to one another, will be compounded of the
           proportions of length to length, and time to time
           reciprocally taken.]

7. If two bodies pass through two lengths with uniform motion, the
proportion of the times in which they are moved will be compounded of
the proportions of length to length and impetus to impetus reciprocally
taken.

For let any two lengths be given, as (in the same fig. 3) Z and X, and
let one of them be transmitted with the impetus A C, the other with the
impetus A E. I say the proportion of the times in which they are
transmitted, will be compounded of the proportions of Z to X, and of A
E, which is the impetus with which X is transmitted, to A C, the impetus
with which Z is transmitted. For seeing A F is the product of the
impetus A C multiplied into the time A B, the time of motion through Z
will be a line, which is made by the application of the parallelogram A
F to the strait line A C, which line is A B; and therefore A B is the
time of motion through Z. In like manner, seeing A G is the product of
the impetus A E multiplied into the time A D, the time of motion through
X will be a line which is made by the application of A G to the strait
line A D; but A D is the time of motion through X. Now the proportion of
A B to A D is compounded of the proportions of the parallelogram A F to
the parallelogram A G, and of the impetus A E to the impetus A C; which
may be demonstrated thus. Put the parallelograms in order A F, A G, D C,
and it will be manifest that the proportion of A F to D C is compounded
of the proportions of A F to A G and of A G to D C; but A F is to D C as
A B to A D; wherefore also the proportion of A B to A D is compounded of
the proportions of A F to A G and of A G to D C. And because the length
Z is to the length X as A F is to A G, and the impetus A E to the
impetus A C as A G to D C, therefore the proportion of A B to A D will
be compounded of the proportions of the length Z to the length X, and of
the impetus A E to the impetus A C; which was to be demonstrated.

In the same manner it may be proved, that in two uniform motions the
proportion of the impetus is compounded of the proportions of length to
length and of time to time reciprocally taken.

For if we suppose A C (in the same fig. 3) to be the time, and A B the
impetus with which the length Z is passed through; and A E to be the
time, and A D the impetus with which the length X is passed through, the
demonstration will proceed as in the last article.

[Sidenote: If a body be carried on with uniform motion by two movents
           together, which meet in an angle, the line by which it passes
           will be a strait line, subtending the complement of that
           angle to 2 right angles.]

8. If a body be carried by two movents together, which move with strait
and uniform motion, and concur in any given angle, the line by which
that body passes will be a strait line.

Let the movent A B (in fig. 4) have strait and uniform motion, and be
moved till it come into the place C D; and let another movent A C,
having likewise strait and uniform motion, and making with the movent A
B any given angle C A B, be understood to be moved in the same time to D
B; and let the body be placed in the point of their concourse, A. I say
the line which that body describes with its motion is a strait line. For
let the parallelogram A B D C be completed, and its diagonal A D be
drawn; and in the strait line A B let any point E be taken; and from it
let E F be drawn parallel to the strait lines A C and B D, cutting A D
in G; and through the point G let H I be drawn parallel to the strait
lines A B and C D; and lastly, let the measure of the time be A C.
Seeing therefore both the motions are made in the same time, when A B is
in C D, the body also will be in C D; and in like manner, when A C is in
B D, the body will be in B D. But A B is in C D at the same time when A
C is in B D; and therefore the body will be in C D and B D at the same
time; wherefore it will be in the common point D. Again, seeing the
motion from A C to B D is uniform, that is, the spaces transmitted by it
are in proportion to one another as the times in which they are
transmitted, when A C is in E F, the proportion of A B to A E will be
the same with that of E F to E G, that is, of the time A C to the time A
H. Wherefore A B will be in H I in the same time in which A C is in E F,
so that the body will at the same time be in E F and H I, and therefore
in their common point G. And in the same manner it will be, wheresoever
the point E be taken between A and B. Wherefore the body will always be
in the diagonal A D; which was to be demonstrated.

Coroll. From hence it is manifest, that the body will be carried through
the same strait line A D, though the motion be not uniform, provided it
have like acceleration; for the proportion of A B to A E will always be
the same with that of A C to A H.

[Sidenote: If a body be carried by two movents together, one of them
           being moved with uniform, the other with accelerated motion,
           and the proportion of their lengths to their times being
           explicable in numbers, how to find out what line that body
           describes.]

9. If a body be carried by two movents together, which meet in any given
angle, and are moved, the one uniformly, the other with motion uniformly
accelerated from rest, that is, that the proportion of their impetus be
as that of their times, that is, that the proportion of their lengths be
duplicate to that of the lines of their times, till the line of greatest
impetus acquired by acceleration be equal to that of the line of time of
the uniform motion; the line in which the body is carried will be the
crooked line of a semiparabola, whose base is the impetus last acquired,
and vertex the point of rest.

Let the straight line A B (in fig. 5) be understood to be moved with
uniform motion to C D; and let another movent in the strait line A C be
supposed to be moved in the same time to B D, but with motion uniformly
accelerated, that is, with such motion, that the proportion of the
spaces which are transmitted be always duplicate to that of the times,
till the impetus acquired be B D equal to the strait line A C; and let
the semiparabola A G D B be described. I say that by the concourse of
those two movents, the body will be carried through the semiparabolical
crooked line A G D. For let the parallelogram A B D C be completed; and
from the point E, taken anywhere in the strait line A B, let E F be
drawn parallel to A C and cutting the crooked line in G; and lastly,
through the point G let H I be drawn parallel to the strait lines A B
and C D. Seeing therefore the proportion of A B to A E is by supposition
duplicate to the proportion of E F to E G, that is, of the time A C to
the time A H, at the same time when A C is in E F, A B will be in H I;
and therefore the moved body will be in the common point G. And so it
will always be, in what part soever of A B the point E be taken.
Wherefore the moved body will always be found in the parabolical line A
G D; which was to be demonstrated.

10. If a body be carried by two movents together, which meet in any
given angle, and are moved the one uniformly, the other with impetus
increasing from rest, till it be equal to that of the uniform motion,
and with such acceleration, that the proportion of the lengths
transmitted be every where triplicate to that of the times in which they
are transmitted; the line, in which that body is moved, will be the
crooked line of the first semiparabolaster of two means, whose base is
the impetus last acquired.

Let the strait line A B (in the 6th figure) be moved uniformly to C D;
and let another movent A C be moved at the same time to B D with motion
so accelerated, that the proportion of the lengths transmitted be
everywhere triplicate to the proportion of their times; and let the
impetus acquired in the end of that motion be B D, equal to the strait
line A C; and lastly, let A G D be the crooked line of the first
semiparabolaster of two means. I say, that by the concourse of the two
movents together, the body will be always in that crooked line A G D.
For let the parallelogram A B D C be completed; and from the point E,
taken anywhere in the strait line A B, let E F be drawn parallel to A C,
and cutting the crooked line in G; and through the point G let H I be
drawn parallel to the strait lines A B and C D. Seeing therefore the
proportion of A B to A E is, by supposition, triplicate to the
proportion of E F to E G, that is, of the time A C to the time A H, at
the same time when A C is in E F, A B will be in H I; and therefore the
moved body will be in the common point G. And so it will always be, in
what part soever of A B the point E be taken; and by consequent, the
body will always be in the crooked line A G D; which was to be
demonstrated.

11. By the same method it may be shown, what line it is that is made by
the motion of a body carried by the concourse of any two movents, which
are moved one of them uniformly, the other with acceleration, but in
such proportions of spaces and times as are explicable by numbers, as
_duplicate_, _triplicate_, &c., or such as may be designed by any broken
number whatsoever. For which this is the rule. Let the two numbers of
the length and time be added together; and let their sum be the
denominator of a fraction, whose numerator must be the number of the
length. Seek this fraction in the table of the third article of the
XVIIth chapter; and the line sought will be that, which denominates the
three-sided figure noted on the left hand; and the kind of it will be
that, which is numbered above over the fraction. For example, let there
be a concourse of two movents, whereof one is moved uniformly, the other
with motion so accelerated, that the spaces are to the times as 5 to 3.
Let a fraction be made whose denominator is the sum of 5 and 3, and the
numerator 5, namely the fraction 5⁄8. Seek in the table, and you will
find 5⁄8 to be the third in that row, which belongs to the three-sided
figure of four means. Wherefore the line of motion made by the concourse
of two such movents, as are last of all described, will be the crooked
line of the third parabolaster of four means.

12. If motion be made by the concourse of two movents, whereof one is
moved uniformly, the other beginning from rest in the angle of concourse
with any acceleration whatsoever; the movent, which is moved uniformly,
shall put forward the moved body in the several parallel spaces, less
than if both the movents had uniform motion; and still less and less, as
the motion of the other movent is more and more accelerated.

Let the body be placed in A, (in the 7th figure) and be moved by two
movents, by one with uniform motion from the strait line A B to the
strait line C D parallel to it; and by the other with any acceleration,
from the strait line A C to the strait line B D parallel to it; and in
the parallelogram A B D C let a space be taken between any two parallels
E F and G H. I say, that whilst the movent A C passes through the
latitude which is between E F and G H, the body is less moved forwards
from A B towards C D, than it would have been, if the motion from A C to
B D had been uniform.

For suppose that whilst the body is made to descend to the parallel E F
by the power of the movent from A C towards B D, the same body in the
same time is moved forwards to any point F in the line E F, by the power
of the movent from A B towards C D; and let the strait line A F be drawn
and produced indeterminately, cutting G H in H. Seeing therefore, it is
as A E to A G, so E F to G H; if A C should descend towards B D with
uniform motion, the body in the time G H, (for I make A C and its
parallels the measure of time,) would be found in the point H. But
because A C is supposed to be moved towards B D with motion continually
accelerated, that is, in greater proportion of space to space, than of
time to time, in the time G H the body will be in some parallel beyond
it, as between G H and B D. Suppose now that in the end of the time G H
it be in the parallel I K, and in I K let I L be taken equal to G H.
When therefore the body is in the parallel I K, it will be in the point
L. Wherefore when it was in the parallel G H, it was in some point
between G and H, as in the point M; but if both the motions had been
uniform, it had been in the point H; and therefore whilst the movent A C
passes over the latitude which is between E F and G H, the body is less
moved forwards from A B towards C D, than it would have been, if both
the motions had been uniform; which was to be demonstrated.

13. Any length being given, which is passed through in a given time with
uniform motion, to find out what length shall be passed through in the
same time with motion uniformly accelerated, that is, with such motion
that the proportion of the lengths passed through be continually
duplicate to that of their times, and that the line of the impetus last
acquired be equal to the line of the whole time of the motion.

Let A B (in the 8th figure) be a length, transmitted with uniform motion
in the time A C; and let it be required to find another length, which
shall be transmitted in the same time with motion uniformly accelerated,
so that the line of the impetus last acquired be equal to the strait
line A C.

Let the parallelogram A B D C be completed; and let B D be divided in
the middle at E; and between B E and B D let B F be a mean proportional;
and let A F be drawn and produced till it meet with C D produced in G;
and lastly, let the parallelogram A C G H be completed. I say, A H is
the length required.

For as duplicate proportion is to single proportion, so let A H be to A
I, that is, let A I be the half of A H; and let I K be drawn parallel to
the strait line A C, and cutting the diagonal A D in K, and the strait
line A G in L. Seeing therefore A I is the half of A H, I L will also be
the half of B D, that is, equal to B E; and I K equal to B F; for B D,
that is, G H, B F, and B E, that is, I L, being continual proportionals,
A H, A B and A I will also be continual proportionals. But as A B is to
A I, that is, as A H is to A B, so is B D to I K, and so also is G H,
that is, B D to B F; and therefore B F and I K are equal. Now the
proportion of A H to A I is duplicate to the proportion of A B to A I,
that is, to that of B D to I K, or of G H to I K. Wherefore the point K
will be in a parabola, whose diameter is A H, and base G H, which G H is
equal to A C. The body therefore proceeding from rest in A, with motion
uniformly accelerated in the time A C, when it has passed through the
length A H, will acquire the impetus G H equal to the time A C, that is,
such impetus, as that with it the body will pass through the length A C
in the time A C. Wherefore any length being given, &c., which was
propounded to be done.

14. Any length being given, which in a given time is transmitted with
uniform motion, to find out what length shall be transmitted in the same
time with motion so accelerated, that the lengths transmitted be
continually in triplicate proportion to that of their times, and the
line of the impetus last of all acquired be equal to the line of time
given.

Let the given length A B (in the 9th figure) be transmitted with uniform
motion in the time A C; and let it be required to find what length shall
be transmitted in the same time with motion so accelerated, that the
lengths transmitted be continually in triplicate proportion to that of
their times, and the impetus last acquired be equal to the time given.

Let the parallelogram A B D C be completed; and let B D be so divided in
E, that B E be a third part of the whole B D; and let B F be a mean
proportional between B D and B E; and let A F be drawn and produced till
it meet the strait line C D in G; and lastly, let the parallelogram A C
G H be completed. I say, A H is the length required.

For as triplicate proportion is to single proportion, so let A H be to
another line, A I, that is, make A I a third part of the whole A H; and
let I K be drawn parallel to the strait line A C, cutting the diagonal A
D in K, and the strait line A G in L; then, as A B is to A I, so let A I
be to another, A N; and from the point N let N Q be drawn parallel to A
C, cutting A G, A D, and F K produced in P, M, and O; and last of all,
let F O and L M be drawn, which will be equal and parallel to the strait
lines B N and I N. By this construction, the lengths transmitted A H, A
B, A I, and A N, will be continual proportionals; and, in like manner,
the times G H, B F, I L and N P, that is, N Q, N O, N M and N P, will be
continual proportionals, and in the same proportion with A H, A B, A I
and A N. Wherefore the proportion of A H, A B, A I and A N. Wherefore
the proportion of A H to A N is the same with that of B D, that is, of N
Q to N P; and the proportion of N Q to N P triplicate to that of N Q to
N O, that is, triplicate to that of B D to I K; wherefore also the
length A H is to the length A N in triplicate proportion to that of the
time B D, to the time I K; and therefore the crooked line of the first
three-sided figure of two means whose diameter is A H, and base G H
equal to A C, shall pass through the point O; and consequently, A H
shall be transmitted in the time A C, and shall have its last acquired
impetus G H equal to A C, and the proportions of the lengths acquired in
any of the times triplicate to the proportions of the times themselves.
Wherefore A H is the length required to be found out.

By the same method, if a length be given which is transmitted with
uniform motion in any given time, another length may be found out which
shall be transmitted in the same time with motion so accelerated, that
the lengths transmitted shall be to the times in which they are
transmitted, in proportion quadruplicate, quintuplicate, and so on
infinitely. For if B D be divided in E, so that B D be to B E as 4 to 1;
and there be taken between B D and B E a mean proportional F B; and as A
H is to A B, so A B be made to a third, and again so that third to a
fourth, and that fourth to a fifth, A N, so that the proportion of A H
to A N be quadruplicate to that of A H to A B, and the parallelogram N B
F O be completed, the crooked line of the first three-sided figure of
three means will pass through the point O; and consequently, the body
moved will acquire the impetus G H equal to A C in the time A C. And so
of the rest.

15. Also, if the proportion of the lengths transmitted be to that of
their times, as any number to any number, the same method serves for the
finding out of the length transmitted with such impetus, and in such
time.

For let A C (in the 10th figure) be the time in which a body is
transmitted with uniform motion from A to B; and the parallelogram A B D
C being completed, let it be required to find out a length in which that
body may be moved in the same time A C from A, with motion so
accelerated, that the proportion of the lengths transmitted to that of
the times be continually as 3 to 2.

Let B D be so divided in E, that B D be to B E as 3 to 2; and between B
D and B E let B F be a mean proportional; and let A F be drawn and
produced till it meet with C D produced in G; and making A M a mean
proportional between A H and A B, let it be as A M to A B, so A B to A
I; and so the proportion of A H to A I will be to that of A H to A B as
3 to 2; for of the proportions, of which that of A H to A M is one, that
of A H to A B is two, and that of A H to A I is three; and consequently,
as 3 to 2 to that of G H to B F, and (F K being drawn parallel to B I
and cutting A D in K) so likewise to that of G H or B D to I K.
Wherefore the proportion of the length A H to A I is to the proportion
of the time B D to I K as 3 to 2; and therefore if in the time A C the
body be moved with accelerated motion, as was propounded, till it
acquire the impetus H G equal to A C, the length transmitted in the same
time will be A H.

16. But if the proportion of the lengths to that of the times had been
as 4 to 3, there should then have been taken two mean proportionals
between A H and A B, and their proportion should have been continued one
term further, so that A H to A B might have three of the same
proportions, of which A H to A I has four; and all things else should
have been done as is already shown. Now the way how to interpose any
number of means between two lines given, is not yet found out.
Nevertheless this may stand for a general rule; _if there be a time
given, and a length be transmitted in that time with uniform motion; as
for example, if the time be_ A C, _and the length_ A B, _the strait
line_ A G, _which determines the length_ C G _or_ A H, _transmitted in
the same time_ A C _with any accelerated motion, shall so cut_ B D _in_
F, _that_ B F _shall be a mean proportional between_ B D _and_ B E, B E
_being so taken in_ B D, _that the proportion of length to length be
everywhere to the proportion of time to time, as the whole_ B D _is to
its part_ B E.

17. If in a given time two lengths be transmitted, one with uniform
motion, the other with motion accelerated in any proportion of the
lengths to the times; and again, in part of the same time, parts of the
same lengths be transmitted with the same motions, the whole length will
exceed the other length in the same proportion in which one part exceeds
the other part.

For example, let A B (in the 8th figure) be a length transmitted in the
time A C, with uniform motion; and let A H be another length transmitted
in the same time with motion uniformly accelerated, so that the impetus
last acquired be G H equal to A C; and in A H let any part A I be taken,
and transmitted in part of the time A C with uniform motion; and let
another part A B be taken and transmitted in the same part of the time A
C with motion uniformly accelerated; I say, that as A H is to A B, so
will A B be to A I.

Let B D be drawn parallel and equal to H G, and divided in the midst at
E, and between B D and B E let a mean proportional be taken as B F; and
the strait line A G, by the demonstration of art. 13, shall pass through
F. And dividing A H in the midst at I, A B shall be a mean proportional
between A H and A I. Again, because A I and A B are described by the
same motions, if I K be drawn parallel and equal to B F or A M, and
divided in the midst at N, and between I K and I N be taken the mean
proportional I L, the strait line A F will, by the demonstration of the
same art. 13, pass through L. And dividing A B in the midst at O, the
line A I will be a mean proportional between A B and A O. Where A B is
divided in I and O, in like manner as A H is divided in B and I; and as
A H to A B, so is A B to A I. Which was to be proved.

Coroll. Also as A H to A B, so is H B to B I; and so also B I to I O.

And as this, where one of the motions is uniformly accelerated, is
proved out of the demonstration of art. 13; so, when the accelerations
are in double proportion to the times, the same may be proved by the
demonstration of art. 14; and by the same method in all other
accelerations, whose proportions to the times are explicable in numbers.

18. If two sides, which contain an angle in any parallelogram, be moved
in the same time to the sides opposite to them, one of them with uniform
motion, the other with motion uniformly accelerated; that side, which is
moved uniformly, will affect as much with its concourse through the
whole length transmitted, as it would do if the other motion were also
uniform, and the length transmitted by it in the same time were a mean
proportional between the whole and the half.

Let the side A B of the parallelogram A B D C, (in the 11th figure) be
understood to be moved with uniform motion till it be coincident with C
D; and let the time of that motion be A C or B D. Also in the same time
let the side A C be understood to be moved with motion uniformly
accelerated, till it be coincident with B D; then dividing A B in the
middle in E, let A F be made a mean proportional between A B and A E;
and drawing F G parallel to A C, let the side A C be understood to be
moved in the same time A C with uniform motion till it be coincident
with F G. I say, the whole A B confers as much to the velocity of the
body placed in A, when the motion of A C is uniformly accelerated till
it comes to B D, as the part A F confers to the same, when the side A C
is moved uniformly and in the same time to F G.

For seeing A F is a mean proportional between the whole A B and its half
A E, B D will (by the 13th article) be the last impetus acquired by A C,
with motion uniformly accelerated till it come to the same B D; and
consequently, the strait line F B will be the excess, by which the
length, transmitted by A C with motion uniformly accelerated, will
exceed the length transmitted by the same A C in the same time with
uniform motion, and with impetus every where equal to B D. Wherefore, if
the whole A B be moved uniformly to C D in the same time in which A C is
moved uniformly to F G, the part F B, seeing it concurs not at all with
the motion of the side A C which is supposed to be moved only to F G,
will confer nothing to its motion. Again, supposing the side A C to be
moved to B D with motion uniformly accelerated, the side A B with its
uniform motion to C D will less put forwards the body when it is
accelerated in all the parallels, than when it is not at all
accelerated; and by how much the greater the acceleration is, by so much
the less it will put it forwards, as is shown in the 12th article. When
therefore A C is in F G with accelerated motion, the body will not be in
the side C D at the point G, but at the point D; so that G D will be the
excess, by which the length transmitted with accelerated motion to B D
exceeds the length transmitted with uniform motion to F G; so that the
body by its acceleration avoids the action of the part A F, and comes to
the side C D in the time A C, and makes the length C D, which is equal
to the length A B. Wherefore uniform motion from A B to C D in the time
A C, works no more in the whole length A B upon the body uniformly
accelerated from A C to B D, than if A C were moved in the same time
with uniform motion to F G; the difference consisting only in this, that
when A B works upon the body uniformly moved from A C to F G, that, by
which the accelerated motion exceeds the uniform motion, is altogether
in F B or G D; but when the same A B works upon the body accelerated,
that, by which the accelerated motion exceeds the uniform motion, is
dispersed through the whole length A B or C D, yet, so that if it were
collected and put together, it would be equal to the same F B or G D.
Wherefore, if two sides which contain an angle, &c.; which was to be
demonstrated.

19. If two transmitted lengths have to their times any other proportion
explicable by number, and the side A B be so divided in E, that A B be
to A E in the same proportion which the lengths transmitted have to the
times in which they are transmitted, and between A B and A E there be
taken a mean proportional A F; it may be shown by the same method, that
the side, which is moved with uniform motion, works as much with its
concourse through the whole length A B, as it would do if the other
motion were also uniform, and the length transmitted in the same time A
C were that mean proportional A F.

And thus much concerning motion by concourse.

------------------------------------------------------------------------

[Illustration:

  _Vol. 1. Lat. & Eng._
  C. XVI.
  _Fig. 1-11_
]

------------------------------------------------------------------------


                                -------


                              CHAP. XVII.

                         OF FIGURES DEFICIENT.

1. Definitions of a deficient figure; of a complete figure; of the
    complement of a deficient figure; and of proportions which are
    proportional and commensurable to one another.—2. The proportion of
    a deficient figure to its complement.—3. The proportions of
    deficient figures to the parallelograms in which they are described,
    set forth in a table.—4. The description and production of the same
    figures.—5. The drawing of tangents to them.—6. In what proportion
    the same figures exceed a strait-lined triangle of the same altitude
    and base.—7. A table of solid deficient figures described in a
    cylinder.—8. In what proportion the same figures exceed a cone of
    the same altitude and base.—9. How a plain deficient figure may be
    described in a parallelogram, so that it be to a triangle of the
    same base and altitude, as another deficient figure, plain or solid,
    twice taken, is to the same deficient figure, together with the
    complete figure in which it is described.—10. The transferring of
    certain properties of deficient figures described in a parallelogram
    to the proportions of the spaces transmitted with several degrees of
    velocity.—11. Of deficient figures described in a circle.—12. The
    proposition demonstrated in art. 2 confirmed from the elements of
    philosophy.—13. An unusual way of reasoning concerning the equality
    between the superficies of a portion of a sphere and a circle.—14.
    How from the description of deficient figures in a parallelogram,
    any number of mean proportionals may be found out between two given
    strait lines.


[Sidenote: Definition of a deficient figure.]

1. I call those _deficient figures_ which may be understood to be
generated by the uniform motion of some quantity, which decreases
continually, till at last it have no magnitude at all.

[Sidenote: Definitions of a complete figure; of the complement of a
           deficient figure; and of proportions which are proportional &
           commensurable to one another.]

And I call that a _complete figure_, answering to a deficient figure,
which is generated with the same motion and in the same time, by a
quantity which retains always its whole magnitude.

The _complement_ of a deficient figure is that which being added to the
deficient figure makes it complete.

Four proportions are said to be _proportional_, when the first of them
is to the second as the third is to the fourth. For example, if the
first proportion be duplicate to the second, and again, the third be
duplicate to the fourth, those proportions are said to be
_proportional_.

And _commensurable_ proportions are those, which are to one another as
number to number. As when to a proportion given, one proportion is
duplicate, another triplicate, the duplicate proportion will be to the
triplicate proportion as 2 to 3; but to the given proportion it will be
as 2 to 1; and therefore I call those three proportions _commensurable_.

[Sidenote: The proportion of a deficient figure to its complement.]

2. A deficient figure, which is made by a quantity continually
decreasing to nothing by proportions everywhere proportional and
commensurable, is to its complement, as the proportion of the whole
altitude to an altitude diminished in any time is to the proportion of
the whole quantity, which describes the figure, to the same quantity
diminished in the same time.

Let the quantity A B (in fig. 1), by its motion through the altitude A
C, describe the complete figure A D; and again, let the same quantity,
by decreasing continually to nothing in C, describe the deficient figure
A B E F C, whose complement will be the figure B D C F E. Now let A B be
supposed to be moved till it lie in G K, so that the altitude diminished
be G C, and A B diminished be G E; and let the proportion of the whole
altitude A C to the diminished altitude G C, be, for example, triplicate
to the proportion of the whole quantity A B or G K to the diminished
quantity G E. And in like manner, let H I be taken equal to G E, and let
it be diminished to H F; and let the proportion of G C to H C be
triplicate to that of H I to H F; and let the same be done in as many
parts of the strait line A C as is possible; and a line be drawn through
the points B, E, F and C. I say the deficient figure A B E F C is to its
complement B D C F E as 3 to 1, or as the proportion of A C to G C is to
the proportion of A B, that is, of G K to G E.

For (by art. 2, chapter XV.) the proportion of the complement B E F C D
to the deficient figure A B E F C is all the proportions of D B to B A,
O E to E G, Q F to F H, and of all the lines parallel to D B terminated
in the line B E F C, to all the parallels to A B terminated in the same
points of the line B E F C. And seeing the proportions of D B to O E,
and of D B to Q F &c. are everywhere triplicate of the proportions of A
B to G E, and of A B to H F &c. the proportions of H F to A B, and of G
E to A B &c. (by art. 16, chap. XIII.), are triplicate of the
proportions of Q F to D B, and of O E to D B &c. and therefore the
deficient figure A B E F C, which is the aggregate of all the lines H F,
G E, A B, &c. is triple to the complement B E F C D made of all the
lines Q F, O E, D B, &c.; which was to be proved.

It follows from hence, that the same complement B E F C D is 1⁄4 of the
whole parallelogram. And by the same method may be calculated in all
other deficient figures, generated as above declared, the proportion of
the parallelogram to either of its parts; as that when the parallels
increase from a point in the same proportion, the parallelogram will be
divided into two equal triangles; when one increase is double to the
other, it will be divided into a semiparabola and its complement, or
into 2 and 1.

The same construction standing, the same conclusion may otherwise be
demonstrated thus.

Let the strait line C B be drawn cutting G K in L, and through L let M N
be drawn parallel to the strait line A C; wherefore the parallelograms G
M and L D will be equal. Then let L K be divided into three equal parts,
so that it may be to one of those parts in the same proportion which the
proportion of A C to G C, or of G K to G L, hath to the proportion of G
K to G E. Therefore L K will be to one of those three parts as the
arithmetical proportion between G K and G L is to the arithmetical
proportion between G K and the same G K wanting the third part of L K;
and K E will be somewhat greater than a third of L K. Seeing now the
altitude A G or M L is, by reason of the continual decrease, to be
supposed less than any quantity that can be given; L K, which is
intercepted between the diagonal B C and the side B D, will be also less
than any quantity that can be given; and consequently, if G be put so
near to A in _g_, as that the difference between C _g_ and C A be less
than any quantity that can be assigned, the difference also between C
_l_ (removing L to _l_) and C B, will be less than any quantity that can
be assigned; and the line _g l_ being drawn and produced to the line B D
in _k_, cutting the crooked line in _e_, the proportion of G _k_ to G
_l_ will still be triplicate to the proportion of G _k_ to G _e_, and
the difference between _k_ and _e_, the third part of _k l_, will be
less than any quantity that can be given; and therefore the
parallelogram _e_ D will differ from a third part of the parallelogram A
_e_ by a less difference than any quantity that can be assigned. Again,
let H I be drawn parallel and equal to G E, cutting C B in P, the
crooked line in F, and O E in I, and the proportion of C _g_ to C H will
be triplicate to the proportion of H F to H P, and I F will be greater
than the third part of P I. But again, setting H in _h_ so near to _g_,
as that the difference between C _h_ and C _g_ may be but as a point,
the point P will also in _p_ be so near to _l_, as that the difference
between C _p_ and C _l_ will be but as a point; and drawing _h p_ till
it meet with B D in _i_, cutting the crooked line in _f_ and having
drawn _e o_ parallel to B D, cutting D C in _o_, the parallelogram _f o_
will differ less from the third part of the parallelogram _g f_, than by
any quantity that can be given. And so it will be in all other spaces
generated in the same manner. Wherefore the differences of the
arithmetical and geometrical means, which are but as so many points B,
_e_, _f_, &c. (seeing the whole figure is made up of so many indivisible
spaces) will constitute a certain line, such as is the line B E F C,
which will divide the complete figure A D into two parts, whereof one,
namely, A B E F C, which I call a deficient figure, is triple to the
other, namely, B D C F E, which I call the complement thereof. And
whereas the proportion of the altitudes to one another is in this case
everywhere triplicate to that of the decreasing quantities to one
another; in the same manner, if the proportion of the altitudes had been
everywhere quadruplicate to that of the decreasing quantities, it might
have been demonstrated that the deficient figure had been quadruple to
its complement; and so in any other proportion. Wherefore, a deficient
figure, which is made, &c. which was to be demonstrated.

The same rule holdeth also in the diminution of the bases of cylinders,
as is demonstrated in the second article of chapter XV.

[Sidenote: The proportion of deficient figures to the parallelograms in
           which they are described, set forth in a table.]

3. By this proposition, the magnitudes of all deficient figures, when
the proportions by which their bases decrease continually are
proportional to those by which their altitudes decrease, may be compared
with the magnitudes of their complements; and consequently, with the
magnitudes of their complete figures. And they will be found to be, as I
have set them down in the following tables; in which I compare a
parallelogram with three-sided figures; and first, with a strait-lined
triangle, made by the base of the parallelogram continually decreasing
in such manner, that the altitudes be always in proportion to one
another as the bases are, and so the triangle will be equal to its
complement; or the proportions of the altitudes and bases will be as 1
to 1, and then the triangle will be half the parallelogram. Secondly,
with that three-sided figure which is made by the continual decreasing
of the bases in subduplicate proportion to that of the altitudes; and so
the deficient figure will be double to its complement, and to the
parallelogram as 2 to 3. Then, with that where the proportion of the
altitudes is triplicate to that of the bases; and then the deficient
figure will be triple to its complement, and to the parallelogram as 3
to 4. Also the proportion of the altitudes to that of the bases may be
as 3 to 2; and then the deficient figure will be to its complement as 3
to 2, and to the parallelogram as 3 to 5; and so forwards, according as
more mean proportionals are taken, or as the proportions are more
multiplied, as may be seen in the following table. For example, if the
bases decrease so, that the proportion of the altitudes to that of the
bases be always as 5 to 2, and it be demanded what proportion the figure
made has to the parallelogram, which is supposed to be unity; then,
seeing that where the proportion is taken five times, there must be four
means; look in the table amongst the three-sided figures of four means,
and seeing the proportion was as 5 to 2, look in the uppermost row for
the number 2, and descending in the second column till you meet with
that three-sided figure, you will find 5⁄7; which shows that the
deficient figure is to the parallelogram as 5⁄7 to 1, or as 5 to 7.

                               | 1  |  2    3    4    5    6    7
Parallelogram                  | 1  |    :    :    :    :    :    :
                               +----+
Strait-sided triangle          | 1/2|    :    :    :    :    :    :
                               |----+
Three-sided figure of 1 mean   | 2/3|    :    :    :    :    :    :
                               |----+----+
Three-sided figure of 2 means  | 3/4| 3/5|    :    :    :    :    :
                               |----+----+----+
Three-sided figure of 3 means  | 4/5| 4/6| 4/7|    :    :    :    :
                               |----+----+----+----+
Three-sided figure of 4 means  | 5/6| 5/7| 5/8|5/9 |    :    :    :
                               |----+----+----+----+----+
Three-sided figure of 5 means  | 6/7| 6/8| 6/9|6/10|6/11|    :    :
                               |----+----+----+----+----+----+
Three-sided figure of 6 means  | 7/8| 7/9|7/11|7/11|7/12|7/13|   `` :
                               |----+----+----+----+----+----+----+
Three-sided figure of 7 means  | 8/9|8/10|8/11|8/12|8/13|8/14|8/15|
                               +----+----+----+----+----+----+----+

[Sidenote: Description & production of the same figures.]

4. Now for the better understanding of the nature of these three-sided
figures, I will show how they may be described by points; and first,
those which are in the first column of the table. Any parallelogram
being described, as A B C D (in figure 2) let the diagonal B D be drawn;
and the strait-lined triangle B C D will be half the parallelogram; then
let any number of lines, as E F, be drawn parallel to the side B C, and
cutting the diagonal B D in G; and let it be everywhere, as E F to E G,
so E G to another, E H; and through all the points H let the line B H H
D be drawn; and the figure B H H D C will be that which I call a
three-sided figure of one mean, because in three proportionals, as E F,
E G and E H, there is but one mean, namely, E G; and this three-sided
figure will be 2⁄3 of the parallelogram, and is called a _parabola_.
Again, let it be as E G to E H, so E H to another, E I, and let the line
B I I D be drawn, making the three-sided figure B I I D C; and this will
be 3⁄4 of the parallelogram, and is by many called a _cubic parabola_.
In like manner, if the proportions be further continued in E F, there
will be made the rest of the three-sided figures of the first column;
which I thus demonstrate. Let there be drawn strait lines, as H K and G
L, parallel to the base D C. Seeing therefore the proportion of E F to E
H is duplicate to that of E F to E G, or of B C to B L, that is, of C D
to L G, or of K M (producing K H to A D in M) to K H, the proportion of
B C to B K will be duplicate to that of K M to K H; but as B C is to B
K, so is D C or K M to K N, and therefore the proportion of K M to K N
is duplicate to that of K M to K H; and so it will be wheresoever the
parallel K M be placed. Wherefore the figure B H H D C is double to its
complement B H H D A, and consequently 2⁄3 of the whole parallelogram.
In the same manner, if through I be drawn O P I Q parallel and equal to
C D, it may be demonstrated that the proportion of O Q to O P, that is,
of B C to B O, is triplicate that of O Q to O I, and therefore that the
figure B I I D C is triple to its complement B I I D A, and consequently
¾ of the whole parallelogram, &c.

Secondly, such three-sided figures as are in any of the transverse rows,
may be thus described. Let A B C D (in fig. 3) be a parallelogram, whose
diagonal is B D. I would describe in it such figures, as in the
preceding table I call three-sided figures of three means. Parallel to D
C, I draw E F as often as is necessary, cutting B D in G; and between E
F and E G, I take three proportionals E H, E I and E K. If now there be
drawn lines through all the points H, I and K, that through all the
points H will make the figure B H D C, which is the first of those
three-sided figures; and that through all the points I, will make the
figure B I D C, which is the second; and that which is drawn through all
the points K, will make the figure B K D C the third of those
three-sided figures. The first of these, seeing the proportion of E F to
E G is quadruplicate of that E F to E H, will be to its complement as 4
to 1, and to the parallelogram as 4 to 5. The second, seeing the
proportion of E F to E G is to that of E F to E I as 4 to 2, will be
double to its complement, and 4⁄6 or 2⁄3 of the parallelogram. The
third, seeing the proportion of E F to E G is that of E F to E K as 4 to
3, will be to its complement as 4 to 3, and to the parallelogram as 4 to
7.

Any of these figures being described may be produced at pleasure, thus;
let A B C D (in fig. 4) be a parallelogram, and in it let the figure B K
D C be described, namely, the third three-sided figure of three means.
Let B D be produced indefinitely to E, and let E F be made parallel to
the base D C, cutting A D produced in G, and B C produced in F; and in G
E let the point H be so taken, that the proportion of F E to F G may be
quadruplicate to that of F E to F H, which may be done by making F H the
greatest of three proportionals between F E and F G; the crooked line B
K D produced, will pass through the point H. For if the strait line B H
be drawn, cutting C D in I, and H L be drawn parallel to G D, and
meeting C D produced in L; it will be as F E to F G, so C L to C I, that
is, in quadruplicate proportion to that of F E to F H, or of C D to C I.
Wherefore if the line B K D be produced according to its generation, it
will fall upon the point H.

[Sidenote: The drawing of tangents to them.]

5. A strait line may be drawn so as to touch the crooked line of the
said figure in any point, in this manner. Let it be required to draw a
tangent to the line B K D H (in fig. 4) in the point D. Let the points B
and D be connected, and drawing D A equal and parallel to B C, let B and
A be connected; and because this figure is by construction the third of
three means, let there be taken in A B three points, so, that by them
the same A B be divided into four equal parts; of which take three,
namely, A M, so that A B may be to A M, as the figure B K D C is to its
complement. I say, the strait line M D will touch the figure in the
point given D. For let there be drawn anywhere between A B and D C a
parallel, as R Q, cutting the strait line B D, the crooked line B K D,
the strait line M D, and the strait line A D, in the points P, K, O and
Q. R K will therefore, by construction, be the least of three means in
geometrical proportion between R Q and R P. Wherefore (by coroll. of
art. 28, chapter XIII.) R K will be less than R O; and therefore M D
will fall without the figure. Now if M D be produced to N, F N will be
the greatest of three means in arithmetical proportion between F E and F
G; and F H will be the greatest of three means in geometrical proportion
between the same F E and F G. Wherefore (by the same coroll. of art. 28,
chapter XIII.) F H will be less than F N; and therefore D N will fall
without the figure, and the strait line M N will touch the same figure
only in the point D.

[Sidenote: In what proportion the same figures exceed a strait-lined
           triangle of the same altitude and base.]

6. The proportion of a deficient figure to its complement being known,
it may also be known what proportion a strait-lined triangle has to the
excess of the deficient figure above the same triangle; and these
proportions I have set down in the following table; where if you seek,
for example, how much the fourth three-sided figure of five means
exceeds a triangle of the same altitude and base, you will find in the
concourse of the fourth column with the three-sided figures of five
means 2⁄10; by which is signified, that that three-sided figure exceeds
the triangle by two-tenths or by one-fifth part of the same triangle.

                                      1    2    3    4    5    6    7
       The triangle                 | 1      :    :    :    :    :    :
                                    +---+
      {A three-sided fig. of 1 mean |1/3|    :    :    :    :    :    :
      {                             +---+----+
      {A three-sided fig. of 2 means|2/4| 1/5|    :    :    :    :    :
      {                             +---+----+----+
The   {A three-sided fig. of 3 means|3/5| 2/6| 1/7|    :    :    :    :
      {                             +---+----+----+----+
excess{A three-sided fig. of 4 means|4/6| 3/7| 2/8| 1/9|    :    :    :
      {                             +---+----+----+----+----+
of    {A three-sided fig. of 5 means|5/7| 4/8| 3/9|2/10|1/11|    :    :
      {                             +---+----+----+----+----+----+
      {A three-sided fig. of 6 means|6/8| 5/9|4/10|3/11|2/12|1/13|    :
      {                             +---+----+----+----+----+----+----+
      {A three-sided fig. of 7 means|7/9|6/10|5/11|4/12|3/13|2/14|1/15|
                                    +---+----+----+----+----+----+----+

[Sidenote: A table of solid deficient figures described in a cylinder.]

7. In the next table are set down the proportion of a cone and the
solids of the said three-sided figures, namely, the proportions between
them and a cylinder. As for example, in the concourse of the second
column with the three-sided figures of four means, you have 5⁄9; which
gives you to understand, that the solid of the second three-sided figure
of four means is to the cylinder as 5⁄9 to 1, or as 5 to 9.

                                      | 1    2    3    4    5    6    7
      { A cylinder                    | 1  |    :    :    :    :    :    :
      {                               +----+    :    :    :    :    :    :
      { A cone                        |1/3 |    :    :    :    :    :    :
      {                               +----+----+    :    :    :    :    :
      { A three-sided fig. of 1 mean  |2/4 |    |    :    :    :    :    :
      {                               +----|----+    :    :    :    :    :
The   { A three-sided fig. of 2 means |3/5 |3/7 |    :    :    :    :    :
solids{                               +----+----+----+----+    :    :    :
of    { A three-sided fig. of 3 means |4/6 |4/8 |4/10|    |    :    :    :
      {                               +----+----+----+----+----+    :    :
      { A three-sided fig. of 4 means |5/7 |5/9 |5/11|5/13|    |    :    :
      {                               +----+----+----+----+----+----+    :
      { A three-sided fig. of 5 means |6/8 |6/10|6/12|6/14|6/16|    |    :
      {                               +----+----+----+----+----+----+----|
      { A three-sided fig. of 6 means |7/9 |7/11|7/13|7/15|7/17|7/19|    |
      {                               +----+----+----+----+----+----+----+
      { A three-sided fig. of 7 means |8/10|8/12|8/14|8/16|8/18|8/20|8/22|
                                      +----+----+----+----+----+----+----+

[Sidenote: In what proportion the same figures exceed a cone of the same
           altitude and base.]

8. Lastly, the excess of the solids of the said three-sided figures
above a cone of the same altitude and base, are set down in the table
which follows:

                                     1     2     3     4     5     6     7
          The Cone                | 1   :     :     :     :     :     :     :
                                  +-----+
        {Of the solid of a three-}| 6/12|     :     :     :     :     :     :
        {  sided figure of 1 mean}|     |
The ex- {                         +-----+-----+
cesses  {Ditto ditto, 2 means     |12/15| 6/21|     :     :     :     :     :
of the  {                         +-----+-----+-----+
solids  {Ditto ditto, 3 means     |18/18|12/24| 6/30|     :     :     :     :
of these{                         +-----+-----+-----+-----+
three-  {Ditto ditto, 4 means     |24/21|18/27|12/33| 6/39|     :     :     :
sided   {                         +-----+-----+-----+-----+-----+
figures {Ditto ditto, 5 means     |30/24|24/30|18/36|12/42| 6/48|     :     :
above a {                         +-----+-----+-----+-----+-----+-----+
cone.   {Ditto ditto, 6 means     |36/27|30/33|24/39|18/45|12/51| 6/57|     :
        {                         +-----+-----+-----+-----+-----+-----+-----+
        {Ditto ditto, 7 means     |42/30|36/36|30/42|24/48|18/54|12/60| 6/66|
                                  +-----+-----+-----+-----+-----+-----+-----+

[Sidenote: How a plain deficient figure may be described in a
           parallelogram, so that it be to a triangle of the same base
           and altitude, as another deficient figure, plain or solid,
           twice taken, is to the same deficient figure, together with
           the complete figure, in which it is described.]

9. If any of these deficient figures, of which I have now spoken, as A B
C D (in the 5th figure) be inscribed within the complete figure B E,
having A D C E for its complement; and there be made upon C B produced
the triangle A B I; and the parallelogram A B I K be completed; and
there be drawn parallel to the strait line C I, any number of lines, as
M F, cutting every one of them the crooked line of the deficient figure
in D, and the strait lines A C, A B and A I in H, G, and L; and as G F
is to G D, so G L be made to another, G N; and through all the points N
there be drawn the line A N I: there will be a deficient figure A N I B,
whose complement will be A N I K. I say, the figure A N I B is to the
triangle A B I, as the deficient figure A B C D twice taken is to the
same deficient figure together with the complete figure B E.

For as the proportion of A B to A G, that is, of G M to G L, is to the
proportion of G M to G N, so is the magnitude of the figure A N I B to
that of its complement A N I K, by the second article of this chapter.

But, by the same article, as the proportion of A B to A G, that is, of G
M to G L, is to the proportion of G F to G D, that is, by construction,
of G L to G N, so is the figure A B C D to its complement A D C E.

And by composition, as the proportion of G M to G L, together with that
of G L to G N, is to the proportion of G M to G L, so is the complete
figure B E to the deficient figure A B C D.

And by conversion, as the proportion of G M to G L is to both the
proportions of G M to G L and of G L to G N, that is, to the proportion
of G M to G N, which is the proportion compounded of both, so is the
deficient figure A B C D to the complete figure B E.

But it was, as the proportion of G M to G L to that of G M to G N, so
the figure A N I B to its complement A N I K. And therefore, A B C D. B
E :: A N I B. A N I K are proportionals. And by composition, A B C D + B
E. A B C D :: B K. A N I B are proportionals.

And by doubling the consequents, A B C D + B E. 2 A B C D :: B K. 2 A N
I B are proportionals.

And by taking the halves of the third and the fourth, A B C D + B E. 2 A
B C D :: A B I. A N I B are also proportionals; which was to be proved.

[Sidenote: The transferring of certain properties of deficient figures
           described in a parallelogram to the proportions of spaces
           transmitted with several degrees of velocity.]

10. From what has been said of deficient figures described in a
parallelogram, may be found out what proportions spaces, transmitted
with accelerated motion in determined times, have to the times
themselves, according as the moved body is accelerated in the several
times with one or more degrees of velocity.

For let the parallelogram A B C D, in the 6th figure, and in it the
three-sided figure D E B C be described; and let F G be drawn anywhere
parallel to the base, cutting the diagonal B D in H, and the crooked
line B E D in E; and let the proportion of B C to B F be, for example,
triplicate to that of F G to F E; whereupon the figure D E B C will be
triple to its complement B E D A; and in like manner, I E being drawn
parallel to B C, the three-sided figure E K B F will be triple to its
complement B K E I. Wherefore the parts of the deficient figure cut off
from the vertex by strait lines parallel to the base, namely, D E B C
and E K B F, will be to one another as the parallelograms A C and I F;
that is, in proportion compounded of the proportions of the altitudes
and bases. Seeing therefore the proportion of the altitude B C to the
altitude B F is triplicate to the proportion of the base D C to the base
F E, the figure D E B C to the figure E K B F will be quadruplicate to
the proportion of the same D C to F E. And by the same method, may be
found out what proportion any of the said three-sided figures has to any
part of the same, cut off from the vertex by a strait line parallel to
the base.

Now as the said figures are understood to be described by the continual
decreasing of the base, as of C D, for example, till it end in a point,
as in B; so also they may be understood to be described by the continual
increasing of a point, as of B, till it acquire any magnitude, as that
of C D.

Suppose now the figure B E D C to be described by the increasing of the
point B to the magnitude C D. Seeing therefore the proportion of B C to
B F is triplicate to that of C D to F E, the proportion of F E to C D
will, by conversion, as I shall presently demonstrate, be triplicate to
that B F to B C. Wherefore if the strait line B C be taken for the
measure of the time in which the point B is moved, the figure E K B F
will represent the sum of all the increasing velocities in the time B F;
and the figure D E B C will in like manner represent the sum of all the
increasing velocities in the time B C. Seeing therefore the proportion
of the figure E K B F to the figure D E B C is compounded of the
proportions of altitude to altitude, and base to base; and seeing the
proportion of F E to C D is triplicate to that of B F to B C; the
proportion of the figure E K B F to the figure D E B C will be
quadruplicate to that of B F to B C; that is, the proportion of the sum
of the velocities in the time B F, to the sum of the velocities in the
time B C, will be quadruplicate to the proportion of B F to B C.
Wherefore if a body be moved from B with velocity so increasing, that
the velocity acquired in the time B F be to the velocity acquired in the
time B C in triplicate proportion to that of the times themselves B F to
B C, and the body be carried to F in the time B F; the same body in the
time B C will be carried through a line equal to the fifth proportional
from B F in the continual proportion of B F to B C. And by the same
manner of working, we may determine what spaces are transmitted by
velocities increasing according to any other proportions.

It remains that I demonstrate the proportion of F E to C D to be
triplicate to that of B F to B C. Seeing therefore the proportion of C
D, that is, of F G to F E is subtriplicate to that of B C to B F; the
proportion of F G to F E will also be subtriplicate to that of F G to F
H. Wherefore the proportion of F G to F H is triplicate to that of F G,
that is, of C D to F E. But in four continual proportionals, of which
the least is the first, the proportion of the first to the fourth, (by
the 16th article of chapter XIII.), is subtriplicate to the proportion
of the third to the same fourth. Wherefore the proportion of F H to G F
is subtriplicate to that of F E to C D; and therefore the proportion of
F E to C D is triplicate to that of F H to F G, that is, of B F to B C;
which was to be proved.

It may from hence be collected, that when the velocity of a body is
increased in the same proportion with that of the times, the degrees of
velocity above one another proceed as numbers do in immediate succession
from unity, namely, as 1, 2, 3, 4, &c. And when the velocity is
increased in proportion duplicate to that of the times, the degrees
proceed as numbers from unity, skipping one, as 1, 3, 5, 7, &c. Lastly,
when the proportions of the velocities are triplicate to those of the
times, the progression of the degrees is as that of numbers from unity,
skipping two in every place, as 1, 4, 7, 10, &c., and so of other
proportions. For geometrical proportionals, when they are taken in every
point, are the same with arithmetical proportionals.

[Sidenote: Of deficient figures described in a circle.]

11. Moreover, it is to be noted that as in quantities, which are made by
any magnitudes decreasing, the proportions of the figures to one another
are as the proportions of the altitudes to those of the bases; so also
it is in those, which are made with motion decreasing, which motion is
nothing else but that power by which the described figures are greater
or less. And therefore in the description of _Archimedes' spiral_, which
is done by the continual diminution of the semidiameter of a circle in
the same proportion in which the circumference is diminished, the space,
which is contained within the semidiameter and the spiral line, is a
third part of the whole circle. For the semidiameters of circles,
inasmuch as circles are understood to be made up of the aggregate of
them, are so many sectors; and therefore in the description of a spiral,
the sector which describes it is diminished in duplicate proportions to
the diminutions of the circumference of the circle in which it is
inscribed; so that the complement of the spiral, that is, that space in
the circle which is without the spiral line, is double to the space
within the spiral line. In the same manner, if there be taken a mean
proportional everywhere between the semidiameter of the circle, which
contains the spiral, and that part of the semidiameter which is within
the same, there will be made another figure, which will be half the
circle. And to conclude, this rule serves for all such spaces as may be
described by a line or superficies decreasing either in magnitude of
power; so that if the proportions, in which they decrease, be
commensurable to the proportions of the times in which they decrease,
the magnitudes of the figures they describe will be known.

[Sidenote: The proposition demonstrated in art. 2 confirmed from the
           elements of philosophy.]

12. The truth of that proposition, which I demonstrated in art. 2, which
is the foundation of all that has been said concerning deficient
figures, may be derived from the elements of philosophy, as having its
original in this; _that all equality and inequality between two effects,
that is, all proportion, proceeds from, and is determined by, the equal
and unequal causes of those effects, or from the proportion which the
causes, concurring to one effect, have to the causes which concur to the
producing of the other effect_; and that therefore the proportions of
quantities are the same with the proportions of their causes. Seeing,
therefore, two deficient figures, of which one is the complement of the
other, are made, one by motion decreasing in a certain time and
proportion, the other by the loss of motion in the same time; the
causes, which make and determine the quantities of both the figures, so
that they can be no other than they are, differ only in this, that the
proportions by which the quantity which generates the figure proceeds in
describing of the same, that is, the proportions of the remainders of
all the times and altitudes, may be other proportions than those by
which the same generating quantity decreases in making the complement of
that figure, that is, the proportions of the quantity which generates
the figure continually diminished. Wherefore, as the proportion of the
times in which motion is lost, is to that of the decreasing quantities
by which the deficient figure is generated, so will the defect or
complement be to the figure itself which is generated.

[Sidenote: An unusual way of reasoning concerning the equality between
           the superficies of a portion of a sphere and a circle.]

13. There are also other quantities which are determinable from the
knowledge of their causes, namely, from the comparison of the motions by
which they are made; and that more easily than from the common elements
of geometry. For example, that the superficies of any portion of a
sphere is equal to that circle, whose radius is a strait line drawn from
the pole of the portion to the circumference of its base, I may
demonstrate in this manner. Let B A C (in fig. 7) be a portion of a
sphere, whose axis is A E, and whose base is B C; and let A B be the
strait line drawn from the pole A to the base in B; and let A D, equal
to A B, touch the great circle B A C in the pole A. It is to be proved
that the circle made by the radius A D is equal to the superficies of
the portion B A C. Let the plain A E B D be understood to make a
revolution about the axis A E; and it is manifest that by the strait
line A D a circle will be described; and by the arch A B the superficies
of a portion of a sphere; and lastly, by the subtense A B the
superficies of a right cone. Now seeing both the strait line A B and the
arch A B make one and the same revolution, and both of them have the
same extreme points A and B, the cause why the spherical superficies,
which is made by the arch, is greater than the conical superficies,
which is made by the subtense, is, that A B the arch is greater than A B
the subtense; and the cause why it is greater consists in this, that
although they be both drawn from A to B, yet the subtense is drawn
strait, but the arch angularly, namely, according to that angle which
the arch makes with the subtense, which angle is equal to the angle D A
B (for an angle of contingence adds nothing to an angle of a segment, as
has been shown in chapter XIV, article 16.) Wherefore the magnitude of
the angle D A B is the cause why the superficies of the portion,
described by the arch A B, is greater than the superficies of the right
cone described by the subtense A B.

Again, the cause why the circle described by the tangent A D is greater
than the superficies of the right cone described by the subtense A B
(notwithstanding that the tangent and the subtense are equal, and both
moved round in the same time) is this, that A D stands at right angles
to the axis, but A B obliquely; which obliquity consists in the same
angle D A B. Seeing therefore the quantity of the angle D A B is that
which makes the excess both of the superficies of the portion, and of
the circle made by the radius A D, above the superficies of the right
cone described by the subtense A B; it follows, that both the
superficies of the portion and that of the circle do equally exceed the
superficies of the cone. Wherefore the circle made by A D or A B, and
the spherical superficies made by the arch A B, are equal to one
another; which was to be proved.

[Sidenote: How from the description of deficient figures in a
           parallelogram, any number of mean proportionals may be found
           out between two given strait lines.]

14. If these deficient figures, which I have described in a
parallelogram, were capable of exact description, then any number of
mean proportionals might be found out between two strait lines given.
For example, in the parallelogram A B C D, (in figure 8) let the
three-sided figure of two means be described (which many call a _cubical
parabola_); and let R and S be two given strait lines; between which, if
it be required to find two mean proportionals, it may be done thus. Let
it be as R to S, so B C to B F; and let F E be drawn parallel to B A,
and cut the crooked line in E; then through E let G H be drawn parallel
and equal to the strait line A D, and cut the diagonal B D in I; for
thus we have G I the greatest of two means between G H and G E, as
appears by the description of the figure in article 4. Wherefore, if it
be as G H to G I, so R to another line, T, that T will be the greatest
of two means between R and S. And therefore if it be again as R to T, so
T to another line, X, that will be done which was required.

In the same manner, four mean proportionals may be found out, by the
description of a three-sided figure of four means; and so any other
number of means, &c.

------------------------------------------------------------------------

[Illustration:

  _Vol. 1. Lat. & Eng._
  C. XVII.
  _Fig. 1-8_
]

------------------------------------------------------------------------


                                -------


                             CHAPTER XVIII.

                OF THE EQUATION OF STRAIT LINES WITH THE
                  CROOKED LINES OF PARABOLAS AND OTHER
                FIGURES MADE IN IMITATION OF PARABOLAS.

1. To find the strait line equal to the crooked line of a
    semiparabola.—2. To find a strait line equal to the crooked line of
    the first semiparabolaster, or to the crooked line of any other of
    the deficient figures of the table of the 3d article of the
    precedent chapter.


[Sidenote: To find a strait line equal to the crooked line of a
           semiparabola.]

1. A parabola being given, to find a strait line equal to the crooked
line of the semiparabola.

Let the parabolical line given be A B C (in figure 1), and the diameter
found be A D, and the base drawn D C; and the parallelogram A D C E
being completed, draw the strait line A C. Then dividing A D into two
equal parts in F, draw F H equal and parallel to D C, cutting A C in K,
and the parabolical line in O; and between F H and F O take a mean
proportional F P, and draw A O, A P and P C. I say that the two lines A
P and P C, taken together as one line, are equal to the parabolical line
A B O C.

For the line A B O C being a parabolical line, is generated by the
concourse of two motions, one uniform from A to E, the other in the same
time uniformly accelerated from rest in A to D. And because the motion
from A to E is uniform, A E may represent the times of both those
motions from the beginning to the end. Let therefore A E be the time;
and consequently the lines ordinately applied in the semiparabola will
design the parts of time wherein the body, that describeth the line A B
O C, is in every point of the same; so that as at the end of the time A
E or D C it is in C, so at the end of the time F O it will be in O. And
because the velocity in A D is increased uniformly, that is, in the same
proportion with the time, the same lines ordinately applied in the
semiparabola will design also the continual augmentation of the impetus,
till it be at the greatest, designed by the base D C. Therefore
supposing uniform motion in the line A F, in the time F K the body in A
by the concourse of the two uniform motions in A F and F K will be moved
uniformly in the line A K; and K O will be the increase of the impetus
or swiftness gained in the time F K; and the line A O will be uniformly
described by the concourse of the two uniform motions in A F and F O in
the time F O. From O draw O L parallel to E C, cutting A C in L; and
draw L N parallel to D C, cutting E C in N, and the parabolical line in
M; and produce it on the other side to A D in I; and I N, I M and I L
will be, by the construction of a parabola, in continual proportion, and
equal to the three lines F H, F P and F O; and a strait line parallel to
E C passing through M will fall on P; and therefore O P will be the
increase of impetus gained in the time F O or I L. Lastly, produce P M
to C D in Q; and Q C or M N or P H will be the increase of impetus
proportional to the time F P or I M or D Q. Suppose now uniform motion
from H to C in the time P H. Seeing therefore in the time F P with
uniform motion and the impetus increased in proportion to the times, is
described the straight line A P; and in the rest of the time and
impetus, namely, P H, is described the line C P uniformly; it followeth
that the whole line A P C is described with the whole impetus, and in
the same time wherewith is described the parabolical line A B C; and
therefore the line A P C, made of the two strait lines A P and P C, is
equal to the parabolical line A B C; which was to be proved.

[Sidenote: To find a strait line equal to the crooked line of the first
           semiparabolaster or to the crooked line of any other of the
           deficient figures of the table of art. 3 of the preceding
           chapter.]

2. To find a strait line equal to the crooked line of the first
semiparabolaster.

[Sidenote: [Discussion of Figure 18.2]]

Let A B C be the crooked line of the first semiparabolaster; A D the
diameter; D C the base; and let the parallelogram completed be A D C E,
whose diagonal is A C. Divide the diameter into two equal parts in F,
and draw F H equal and parallel to D C, cutting A C in K, the crooked
line in O, and E C in H. Then draw O L parallel to E C, cutting A C in
L; and draw L N parallel to the base D C, cutting the crooked line in M,
and the strait line E C in N; and produce it on the other side to A D in
I. Lastly, through the point M draw P M Q parallel and equal to H C,
cutting F H in P; and join C P, A P and A O. I say, the two strait lines
A P and P C are equal to the crooked line A B O C.

For the line A B O C, being the crooked line of the first
semiparabolaster, is generated by the concourse of two motions, one
uniform from A to E, the other in the same time accelerated from rest in
A to D, so as that the impetus increaseth in proportion perpetually
triplicate to that of the increase of the time, or which is all one, the
lengths transmitted are in proportion triplicate to that of the times of
their transmission; for as the impetus or quicknesses increase, so the
lengths transmitted increase also. And because the motion from A to E is
uniform, the line A E may serve to represent the time, and consequently
the lines, ordinately drawn in the semiparabolaster, will design the
parts of time wherein the body, beginning from rest in A, describeth by
its motion the crooked line A B O C. And because D C, which represents
the greatest acquired impetus, is equal to A E, the same ordinate lines
will represent the several augmentations of the impetus increasing from
rest in A. Therefore, supposing uniform motion from A to F, in the time
F K there will be described, by the concourse of the two uniform motions
A F and F K, the line A K uniformly, and K O will be the increase of
impetus in the time F K; and by the concourse of the two uniform motions
in A F and F O will be described the line A O uniformly. Through the
point L draw the strait line L M N parallel to D C, cutting the strait
line A D in I, the crooked line A B C in M, and the strait line E C in
N; and through the point M the strait line P M Q parallel and equal to H
C, cutting D C in Q and F H in P. By the concourse therefore of the two
uniform motions in A F and F P in the time F P will be uniformly
described the strait line A P; and L M or O P will be the increase of
impetus to be added for the time F O. And because the proportion of I N
to I L is triplicate to the proportion of I N to I M, the proportion of
F H to F O will also be triplicate to the proportion of F H to F P; and
the proportional impetus gained in the time F P is P H. So that F H
being equal to D C, which designed the whole impetus acquired by the
acceleration, there is no more increase of impetus to be computed. Now
in the time P H suppose an uniform motion from H to C; and by the two
uniform motions in C H and H P will be described uniformly the strait
line P C. Seeing therefore the two strait lines A P and P C are
described in the time A E with the same increase of impetus, wherewith
the crooked line A B O C is described in the same time A E, that is,
seeing the line A P C and the line A B O C are transmitted by the same
body in the same time and with equal velocities, the lines themselves
are equal; which was to be demonstrated.

By the same method (if any of the semiparabolasters in the table of art.
3 of the precedent chapter be exhibited) may be found a strait line
equal to the crooked line thereof, namely, by dividing the diameter into
two equal parts, and proceeding as before. Yet no man hitherto hath
compared any crooked with any strait line, though many geometricians of
every age have endeavoured it. But the cause, why they have not done it,
may be this, that there being in Euclid no definition of equality, nor
any mark by which to judge of it besides congruity (which is the 8th
axiom of the first Book of his Elements) a thing of no use at all in the
comparing of strait and crooked; and others after Euclid (except
Archimedes and Apollonius, and in our time Bonaventura) thinking the
industry of the ancients had reached to all that was to be done in
geometry, thought also, that all that could be propounded was either to
be deduced from what they had written, or else that it was not at all to
be done: it was therefore disputed by some of those ancients themselves,
whether there might be any equality at all between crooked and strait
lines; which question Archimedes, who assumed that some strait line was
equal to the circumference of a circle, seems to have despised, as he
had reason. And there is a late writer that granteth that between a
strait and a crooked line there is equality; but now, says he, since the
fall of Adam, without the special assistance of Divine Grace it is not
to be found.

------------------------------------------------------------------------

[Illustration:

  _Vol. 1. Lat. & Eng._
  C. XVIII.
  _Fig. 1-2_
]

------------------------------------------------------------------------


                                -------


                              CHAPTER XIX.

                 OF ANGLES OF INCIDENCE AND REFLECTION,
                         EQUAL BY SUPPOSITION.

1. If two strait lines falling upon another strait line be parallel, the
    lines reflected from them shall also be parallel.—2. If two strait
    lines drawn from one point fall upon another strait line, the lines
    reflected from them, if they be drawn out the other way, will meet
    in an angle equal to the angle made by the lines of incidence.—3. If
    two strait parallel lines, drawn not oppositely, but from the same
    parts, fall upon the circumference of a circle, the lines reflected
    from them, if produced they meet within the circle, will make an
    angle double to that which is made by two strait lines drawn from
    the centre to the points of incidence.—4. If two strait lines drawn
    from the same point without a circle fall upon the circumference,
    and the lines reflected from them being produced meet within the
    circle, they will make an angle equal to twice that angle, which is
    made by two strait lines drawn from the centre to the points of
    incidence, together with the angle which the incident lines
    themselves make.—5. If two strait lines drawn from one point fall
    upon the concave circumference of a circle, and the angle they make
    be less than twice the angle at the centre, the lines reflected from
    them and meeting within the circle will make an angle, which being
    added to the angle of the incident lines will be equal to twice the
    angle at the centre.—6. If through any one point two unequal chords
    be drawn cutting one another, and the centre of the circle be not
    placed between them, and the lines reflected from them concur
    wheresoever, there cannot through the point, through which the two
    former lines were drawn, be drawn any other strait line whose
    reflected line shall pass through the common point of the two former
    lines reflected.—7. In equal chords the same is not true.—8. Two
    points being given in the circumference of a circle, to draw two
    strait lines to them, so that their reflected lines may contain any
    angle given.—9. If a strait line falling upon the circumference of a
    circle be produced till it reach the semidiameter, and that part of
    it, which is intercepted between the circumference and the
    semidiameter, be equal to that part of the semidiameter which is
    between the point of concourse and the centre, the reflected line
    will be parallel to the semidiameter.—10. If from a point within a
    circle, two strait lines be drawn to the circumference, and their
    reflected lines meet in the circumference of the same circle, the
    angle made by the reflected lines will be a third part of the angle
    made by the incident lines.


[Sidenote: Angles of incidence and reflection.]

Whether a body falling upon the superficies of another body and being
reflected from it, do make equal angles at that superficies, it belongs
not to this place to dispute, being a knowledge which depends upon the
natural causes of reflection; of which hitherto nothing has been said,
but shall be spoken of hereafter.

In this place, therefore, let it be supposed that the angle of incidence
is equal to the angle of reflection; that our present search may be
applied, not to the finding out of the causes, but some consequences of
the same.

I call an _angle of incidence_, that which is made between a strait line
and another line, strait or crooked, upon which it falls, and which I
call the _line reflecting_; and an _angle of reflection_ equal to it,
that which is made at the same point between the strait line which is
reflected and the line reflecting.

[Sidenote: If two strait lines falling upon another strait line be
           parallel, the lines reflected from them shall also be
           parallel.]

1. If two strait lines, which fall upon another strait line, be
parallel, their reflected lines shall be also parallel.

Let the two strait lines A B and C D (in fig. 1), which fall upon the
strait line E F, at the points B and D, be parallel; and let the lines
reflected from them be B G and D H. I say, B G and D H are also
parallel.

For the angles A B E and C D E are equal by reason of the parallelism of
A B and C D; and the angles G B F and H D F are equal to them by
supposition; for the lines B G and D H are reflected from the lines A B
and C D. Wherefore B G and D H are parallel.

[Sidenote: If two strait lines drawn from one point fall upon another
           strait line, the lines reflected from them, if they be drawn
           out the other way, will meet in an angle equal to the angle
           made by the lines of incidence.]

2. If two strait lines drawn from the same point fall upon another
strait line, the lines reflected from them, if they be drawn out the
other way, will meet in an angle equal to the angle of the incident
lines.

From the point A (in fig. 2) let the two strait lines A B and A D be
drawn; and let them fall upon the strait line E K at the points B and D;
and let the lines B I and D G be reflected from them. I say, I B and G D
do converge, and that if they be produced on the other side of the line
E K, they shall meet, as in F; and that the angle B F D shall be equal
to the angle B A D.

For the angle of reflection I B K is equal to the angle of incidence A B
E; and to the angle I B K its vertical angle E B F is equal; and
therefore the angle A B E is equal to the angle E B F. Again, the angle
A D E is equal to the angle of reflection G D K, that is, to its
vertical angle E D F; and therefore the two angles A B D and A D B of
the triangle A B D are one by one equal to the two angles F B D and F D
B of the triangle F B D; wherefore also the third angle B A D is equal
to the third angle B F D; which was to be proved.

Coroll. I. If the strait line A F be drawn, it will be perpendicular to
the strait line E K. For both the angles at E will be equal, by reason
of the equality of the two angles A B E and F B E, and of the two sides
A B and F B.

Coroll. II. If upon any point between B and D there fall a strait line,
as A C, whose reflected line is C H, this also produced beyond C, will
fall upon F; which is evident by the demonstration above.

[Sidenote: If two strait parallel lines, drawn not oppositely, but from
           the same parts, fall upon the circumference of a circle, the
           lines reflected from them, if produced they meet within the
           circle, will make an angle double to that which is made by
           two strait lines drawn from the centre to the points of
           incidence.]

3. If from two points taken without a circle, two strait parallel lines,
drawn not oppositely, but from the same parts, fall upon the
circumference; the lines reflected from them, if produced they meet
within the circle, will make an angle double to that which is made by
two strait lines drawn from the centre to the points of incidence.

Let the two strait parallels A B and D C (in fig. 3) fall upon the
circumference B C at the points B and C; and let the centre of the
circle be E; and let A B reflected be B F, and D C reflected be C G; and
let the lines F B and G C produced meet within the circle in H; and let
E B and E C be connected. I say the angle F H G is double to the angle B
E C.

For seeing A B and D C are parallels, and E B cuts A B in B, the same E
B produced will cut D C somewhere; let it cut it in D; and let D C be
produced howsoever to I, and let the intersection of D C and B F be at
K. The angle therefore I C H, being external to the triangle C K H, will
be equal to the two opposite angles C K H and C H K. Again, I C E being
external to the triangle C D E, is equal to the two angles at D and E.
Wherefore the angle I C H, being double to the angle I C E, is equal to
the angles at D and E twice taken; and therefore the two angles C K H
and C H K are equal to the two angles at D and E twice taken. But the
angle C K H is equal to the angles D and A B D, that is, D twice taken;
for A B and D C being parallels, the altern angles D and A B D are
equal. Wherefore C H K, that is the angle F H G is also equal to the
angle at E twice taken; which was to be proved.

Coroll. If from two points taken within a circle two strait parallels
fall upon the circumference, the lines reflected from them shall meet in
an angle, double to that which is made by two strait lines drawn from
the centre to the points of incidence. For the parallels A B and I C
falling upon the points B and C, are reflected in the lines B H and C H,
and make the angle at H double to the angle at E, as was but now
demonstrated.

[Sidenote: If two strait lines drawn from the same point without a circle fall upon the circumference, and the lines reflected from them being produced meet within
           the circle, they will make an angle equal to twice that angle, which is made by two strait lines drawn from the centre to the points of incidence, together with the angle which the incident lines themselves make.]

4. If two strait lines drawn from the same point without a circle fall
upon the circumference, and the lines reflected from them being produced
meet within the circle, they will make an angle equal to twice that
angle, which is made by two strait lines drawn from the centre to the
points of incidence, together with the angle which the incident lines
themselves make.

Let the two strait lines A B and A C (in fig. 4) be drawn from the point
A to the circumference of the circle, whose centre is D; and let the
lines reflected from them be B E and C G, and, being produced, make
within the circle the angle H; also let the two strait lines D B and D C
be drawn from the centre D to the points of incidence B and C. I say,
the angle H is equal to twice the angle at D together with the angle at
A.

For let A C be produced howsoever to I. Therefore the angle I C H, which
is external to the triangle C K H, will be equal to the two angles C K H
and C H K. Again, the angle I C D, which is external to the triangle C L
D, will be equal to the two angles C L D and C D L. But the angle I C H
is double to the angle I C D, and is therefore equal to the angles C L D
and C D L twice taken. Wherefore the angles C K H and C H K are equal to
the angles C L D and C D L twice taken. But the angle C L D, being
external to the triangle A L B, is equal to the two angles L A B and L B
A; and consequently C L D twice taken is equal to L A B and L B A twice
taken. Wherefore C K H and C H K are equal to the angle C D L together
with L A B and L B A twice taken. Also the angle C K H is equal to the
angle L A B once and A B K, that is, L B A twice taken. Wherefore the
angle C H K is equal to the remaining angle C D L, that is, to the angle
at D, twice taken, and the angle L A B, that is, the angle at A, once
taken; which was to be proved.

Coroll. If two strait converging lines, as I C and M B, fall upon the
concave circumference of a circle, their reflected lines, as C H and B
H, will meet in the angle H, equal to twice the angle D, together with
the angle at A made by the incident lines produced. Or, if the incident
lines be H B and I C, whose reflected lines C H and B M meet in the
point N, the angle C N B will be equal to twice the angle D, together
with the angle C K H made by the lines of incidence. For the angle C N B
is equal to the angle H, that is, to twice the angle D, together with
the two angles A, and N B H, that is, K B A. But the angles K B A and A
are equal to the angle C K H. Wherefore the angle C N B is equal to
twice the angle D, together with the angle C K H made by the lines of
incidence I C and H B produced to K.

[Sidenote: If two strait lines drawn from one point fall upon the
           concave circumference of a circle, and the angle they make be
           less than twice the angle at the centre, the lines reflected
           from them and meeting within the circle will make an angle,
           which being added to the angle of the incident lines will be
           equal to twice the angle at the centre.]

5. If two strait lines drawn from one point fall upon the concave
circumference of a circle, and the angle they make be less than twice
the angle at the centre, the lines reflected from them and meeting
within the circle will make an angle, which being added to the angle of
the incident lines, will be equal to twice the angle at the centre.

Let the two lines A B and A C (in fig. 5), drawn from the point A, fall
upon the concave circumference of the circle whose centre is D; and let
their reflected lines B E and C E meet in the point E; also let the
angle A be less than twice the angle D. I say, the angles A and E
together taken are equal to twice the angle D.

For let the strait lines A B and E C cut the strait lines D C and D B in
the points G and H; and the angle B H C will be equal to the two angles
E B H and E; also the same angle B H C will be equal to the two angles D
and D C H; and in like manner the angle B G C will be equal to the two
angles A C D and A, and the same angle B G C will be also equal to the
two angles D B G and D. Wherefore the four angles E B H, E, A C D and A,
are equal to the four angles D, D C H, D B G and D. If, therefore,
equals be taken away on both sides, namely, on one side A C D and E B H,
and on the other side D C H and D B G, (for the angle E B H is equal to
the angle D B G, and the angle A C D equal to the angle D C H), the
remainders on both sides will be equal, namely, on one side the angles A
and E, and on the other the angle D twice taken. Wherefore the angles A
and E are equal to twice the angle D.

Coroll. If the angle A be greater than twice the angle D, their
reflected lines will diverge. For, by the corollary of the third
proposition, if the angle A be equal to twice the angle D, the reflected
lines B E and C E will be parallel; and if it be less, they will concur,
as has now been demonstrated. And therefore, if it be greater, the
reflected lines B E and C E will diverge, and consequently, if they be
produced the other way, they will concur and make an angle equal to the
excess of the angle A above twice the angle D; as is evident by art. 4.

[Sidenote: If through any one point two unequal chords be drawn cutting
           one another, and the centre of the circle be not placed
           between them, and the lines reflected from them concur
           wheresoever, there cannot through the point, through which
           the two former lines were drawn, be drawn any other strait
           line whose reflected line shall pass through the common point
           of the two former lines reflected.]

6. If through any one point two unequal chords be drawn cutting one
another, either within the circle, or, if they be produced, without it,
and the centre of the circle be not placed between them, and the lines
reflected from them concur wheresoever; there cannot, through the point
through which the former lines were drawn, be drawn another strait line,
whose reflected line shall pass through the point where the two former
reflected lines concur.

Let any two unequal chords, as B K and C H (in fig. 6), be drawn through
the point A in the circle B C; and let their reflected lines B D and C E
meet in F; and let the centre not be between A B and A C; and from the
point A let any other strait line, as A G, be drawn to the circumference
between B and C. I say, G N, which passes through the point F, where the
reflected lines B D and C E meet, will not be the reflected line of A G.

For let the arch B L be taken equal to the arch B G, and the strait line
B M equal to the strait line B A; and L M being drawn, let it be
produced to the circumference in O. Seeing therefore B A and B M are
equal, and the arch B L equal to the arch B G, and the angle M B L equal
to the angle A B G, A G and M L will also be equal, and, producing G A
to the circumference in I, the whole lines L O and G I will in like
manner be equal. But L O is greater than G F N, as shall presently be
demonstrated; and therefore also G I is greater than G N. Wherefore the
angles N G C and I G B are not equal. Wherefore the line G F N is not
reflected from the line of incidence A G, and consequently no other
strait line, besides A B and A C, which is drawn through the point A,
and falls upon the circumference B C, can be reflected to the point F;
which was to be demonstrated.

It remains that I prove L O to be greater than G N; which I shall do in
this manner. L O and G N cut one another in P; and P L is greater than P
G. Seeing now L P. P G :: P N. P O are proportionals, therefore the two
extremes L P and P O together taken, that is L O, are greater than P G
and P N together taken, that is, G N; which remained to be proved.

[Sidenote: In equal chords the same is not true.]

7. But if two equal chords be drawn through one point within a circle,
and the lines reflected from them meet in another point, then another
strait line may be drawn between them through the former point, whose
reflected line shall pass through the latter point.

Let the two equal chords B C and E D (in the 7th figure) cut one another
in the point A within the circle B C D; and let their reflected lines C
H and D I meet in the point F. Then dividing the arch C D equally in G,
let the two chords G K and G L be drawn through the points A and F. I
say, G L will be the line reflected from the chord K G. For the four
chords B C, C H, E D and D I are by supposition all equal to one
another; and therefore the arch B C H is equal to the arch E D I; as
also the angle B C H to the angle E D I; and the angle A M C to its
verticle angle F M D; and the strait line D M to the strait line G M;
and, in like manner, the strait line A C to the strait line F D; and the
chords C G and G D being drawn, will also be equal; and also the angles
F D G and A C G, in the equal segments G D I and G C B. Wherefore the
strait lines F G and A G are equal; and, therefore, the angle F G D is
equal to the angle A G C, that is, the angle of incidence equal to the
angle of reflection. Wherefore the line G L is reflected from the
incident line C G; which was to be proved.

Coroll. By the very sight of the figure it is manifest, that if G be not
the middle point between C and D, the reflected line G L will not pass
through the point F.

[Sidenote: Two points being given in the circumference of a circle, to
           draw two strait lines to them so as that their reflected
           lines may contain any angle given.]

8. Two points in the circumference of a circle being given to draw two
strait lines to them, so as that their reflected lines may be parallel,
or contain any angle given.

In the circumference of the circle, whose centre is A, (in the 8th
figure) let the two points B and C be given; and let it be required to
draw to them from two points taken without the circle two incident
lines, so that their reflected lines may, first, be parallel.

Let A B and A C be drawn; as also any incident line D C, with its
reflected line C F; and let the angle E C D be made double to the angle
A; and let H B be drawn parallel to E C, and produced till it meet with
D C produced in I. Lastly, producing A B indefinitely to K, let G B be
drawn so that the angle G B K may be equal to the angle H B K, and then
G B will be the reflected line of the incident line H B. I say, D C and
H B are two incident lines, whose reflected lines C F and B G are
parallel.

For seeing the angle E C D is double to the angle B A C, the angle H I C
is also, by reason of the parallels E C and H I, double to the same B A
C; therefore also F C and G B, namely, the lines reflected from the
incident lines D C and H B, are parallel. Wherefore the first thing
required is done.

Secondly, let it be required to draw to the points B and C two strait
lines of incidence, so that the lines reflected from them may contain
the given angle Z.

To the angle E C D made at the point C, let there be added on one side
the angle D C L equal to half Z, and on the other side the angle E C M
equal to the angle D C L; and let the strait line B N be drawn parallel
to the strait line C M; and let the angle K B O be made equal to the
angle N B K; which being done, B O will be the line of reflection from
the line of incidence N B. Lastly, from the incident line L C, let the
reflected line C O be drawn, cutting B O at O, and making the angle C O
B. I say, the angle C O B is equal to the angle Z.

Let N B be produced till it meet with the strait line L C produced in P.
Seeing, therefore, the angle L C M is, by construction, equal to twice
the angle B A C, together with the angle Z; the angle N P L, which is
equal to L C M by reason of the parallels N P and M C, will also be
equal to twice the same angle B A C, together with the angle Z. And
seeing the two strait lines O C and O B fall from the point O upon the
points C and B; and their reflected lines L C and N B meet in the point
P; the angle N P L will be equal to twice the angle B A C together with
the angle C O B. But I have already proved the angle N P L to be equal
to twice the angle B A C together with the angle Z. Therefore the angle
C O B is equal to the angle Z; wherefore, two points in the
circumference of a circle being given, I have drawn, &c.; which was to
be done.

But if it be required to draw the incident lines from a point within the
circle, so that the lines reflected from them may contain an angle equal
to the angle Z, the same method is to be used, saving that in this case
the angle Z is not to be added to twice the angle B A C, but to be taken
from it.

[Sidenote: If a strait line falling upon the circumference of a circle
           be produced till it reach the semidiameter, and that part of
           it, which is intercepted between the circumference and the
           semidiameter, be equal to that part of the semidiameter which
           is between the point of concourse and the centre, the
           reflected line will be parallel to the semidiameter.]

9. If a strait line, falling upon the circumference of a circle, be
produced till it reach the semidiameter, and that part of it which is
intercepted between the circumference and the semidiameter be equal to
that part of the semidiameter which is between the point of concourse
and the centre, the reflected line will be parallel to the semidiameter.

Let any line A B (in the 9th figure) be the semidiameter of the circle
whose centre is A; and upon the circumference B D let the strait line C
D fall, and be produced till it cut A B in E, so that E D and E A may be
equal; and from the incident line C D let the line D F be reflected. I
say, A B and D F will be parallel.

Let A G be drawn through the point D. Seeing, therefore, E D and E A are
equal, the angles E D A and E A D will also be equal. But the angles F D
G and E D A are equal; for each of them is half the angle E D H or F D
C. Wherefore the angles F D G and E A D are equal; and consequently D F
and A B are parallel; which was to be proved.

Coroll. If E A be greater then E D, then D F and A B being produced will
concur; but if E A be less than E D, then B A and D H being produced
will concur.

[Sidenote: If from a point within a circle two strait lines be drawn to
           the circumference, and their reflected lines meet in the
           circumference of the same circle, the angle made by the
           reflected lines will be a third part of the angle made by the
           incident lines.]

10. If from a point within a circle two strait lines be drawn to the
circumference, and their reflected lines meet in the circumference of
the same circle, the angle made by the lines of reflection will be a
third part of the angle made by the lines of incidence.

From the point B (in the 10th figure) taken within the circle whose
centre is A, let the two strait lines B C and B D be drawn to the
circumference; and let their reflected lines C E and D E meet in the
circumference of the same circle at the point E. I say, the angle C E D
will be a third part of the angle C B D.

Let A C and A D be drawn. Seeing, therefore, the angles C E D and C B D
together taken are equal to twice the angle C A D (as has been
demonstrated in the 5th article); and the angle C A D twice taken is
quadruple to the angle C E D; the angles C E D and C B D together taken
will also be equal to the angle C E D four times taken; and therefore if
the angle C E D be taken away on both sides, there will remain the angle
C B D on one side, equal to the angle C E D thrice taken on the other
side; which was to be demonstrated.

Coroll. Therefore a point being given within a circle, there may be
drawn two lines from it to the circumference, so as their reflected
lines may meet in the circumference. For it is but trisecting the angle
C B D, which how it may be done shall be shown in the following chapter.

------------------------------------------------------------------------

[Illustration:

  _Vol. 1. Lat. & Eng._
  C. XIX.
  _Fig. 1-10_
]

------------------------------------------------------------------------


                                -------


                              CHAPTER XX.

                 OF THE DIMENSION OF A CIRCLE, AND THE
                     DIVISION OF ANGLES OR ARCHES.

1. The dimension of a circle never determined in numbers by Archimedes
    and others.—2. The first attempt for the finding out of the
    dimension of a circle by lines.—3. The second attempt for the
    finding out of the dimension of a circle from the consideration of
    the nature of crookedness.—4. The third attempt; and some things
    propounded to be further searched into.—5. The equation of the
    spiral of Archimedes with a strait line.—6. Of the analysis of
    geometricians by the powers of lines.


[Sidenote: The dimension of a circle never determined in numbers by
           Archimedes and others.]

1. In the comparing of an arch of a circle with a strait line, many and
great geometricians, even from the most ancient times, have exercised
their wits; and more had done the same, if they had not seen their
pains, though undertaken for the common good, if not brought to
perfection, vilified by those that envy the praises of other men.
Amongst those ancient writers whose works are come to our hands,
Archimedes was the first that brought the length of the perimeter of a
circle within the limits of numbers very little differing from the
truth; demonstrating the same to be less than three diameters and a
seventh part, but greater than three diameters and ten seventy-one parts
of the diameter. So that supposing the radius to consist of 10,000,000
equal parts, the arch of a quadrant will be between 15,714,285 and
15,704,225 of the same parts. In our times, Ludovicus Van Cullen and
Willebrordus Snellius, with joint endeavour, have come yet nearer to the
truth; and pronounced from true principles, that the arch of a quadrant,
putting, as before, 10,000,000 for radius, differs not one whole unity
from the number 15,707,963; which, if they had exhibited their
arithmetical operations, and no man had discovered any error in that
long work of theirs, had been demonstrated by them. This is the furthest
progress that has been made by the way of numbers; and they that have
proceeded thus far deserve the praise of industry. Nevertheless, if we
consider the benefit, which is the scope at which all speculation should
aim, the improvement they have made has been little or none. For any
ordinary man may much sooner and more accurately find a strait line
equal to the perimeter of a circle, and consequently square the circle,
by winding a small thread about a given cylinder, than any geometrician
shall do the same by dividing the radius into 10,000,000 equal parts.
But though the length of the circumference were exactly set out, either
by numbers, or mechanically, or only by chance, yet this would
contribute no help at all towards the section of angles, unless happily
these two problems, _to divide a given angle according to any proportion
assigned_, and _to find a strait line equal to the arch of a circle_,
were reciprocal, and followed one another. Seeing therefore the benefit
proceeding from the knowledge of the length of the arch of a quadrant
consists in this, that we may thereby divide an angle according to any
proportion, either accurately, or at least accurately enough for common
use; and seeing this cannot be done by arithmetic, I thought fit to
attempt the same by geometry, and in this chapter to make trial whether
it might not be performed by the drawing of strait and circular lines.

[Sidenote: The first attempt for the finding out of the dimension of a
           circle by lines.]

2. Let the square A B C D (in the first figure) be described; and with
the radii A B, B C, and D C, the three arches B D, C A, and A C; of
which let the two B D and C A cut one another in E, and the two B D and
A C in F. The diagonals therefore B D and A C being drawn will cut one
another in the centre of the square G, and the two arches B D and C A in
two equal parts in H and Y; and the arch B H D will be trisected in F
and E. Through the centre G let the two strait lines K G L and M G N be
drawn parallel and equal to the sides of the square A B and A D, cutting
the four sides of the same square in the points K, L, M, and N; which
being done, K L will pass through F, and M N through E. Then let O P be
drawn parallel and equal to the side B C, cutting the arch B F D in F,
and the sides A B and D C in O and P. Therefore O F will be the sine of
the arch B F, which is an arch of 30 degrees; and the same O F will be
equal to half the radius. Lastly, dividing the arch B F in the middle in
Q, let R Q, the sine of the arch B Q, be drawn and produced to S, so
that Q S be equal to R Q, and consequently R S be equal to the chord of
the arch B F; and let F S be drawn and produced to T in the side B C. I
say, the strait line B T is equal to the arch B F; and consequently that
B V, the triple of B T, is equal to the arch of the quadrant B F E D.

Let T F be produced till it meet the side B A produced in X; and
dividing O F in the middle in Z, let Q Z be drawn and produced till it
meet with the side B A produced. Seeing therefore the strait lines R S
and O F are parallel, and divided in the midst in Q and Z, Q Z produced
will fall upon X, and X Z Q produced to the side B C will cut B T in the
midst in α.

Upon the strait line F Z, the fourth part of the radius A B, let the
equilateral triangle _a_ Z F be constituted; and upon the centre _a_,
with the radius _a_ Z, let the arch Z F be drawn; which arch Z F will
therefore be equal to the arch Q F, the half of the arch B F. Again, let
the strait line Z O be cut in the midst in _b_, and the strait line _b_
O in the midst in _c_; and let the bisection be continued in this manner
till the last part O _c_ be the least that can possibly be taken; and
upon it, and all the rest of the parts equal to it into which the strait
line O F may be cut, let so many equilateral triangles be understood to
be constituted; of which let the last be _d_ O _c_. If, therefore, upon
the centre _d_, with the radius _d_ O, be drawn the arch O _c_, and upon
the rest of the equal parts of the strait line O F be drawn in like
manner so many equal arches, all those arches together taken will be
equal to the whole arch B F, and the half of them, namely, those that
are comprehended between O and Z, or between Z and F, will be equal to
the arch B Q or Q F, and in sum, what part soever the strait line O _c_
be of the strait line O F, the same part will the arch O _c_ be of the
arch B F, though both the arch and the chord be infinitely bisected. Now
seeing the arch O _c_ is more crooked than that part of the arch B F
which is equal to it; and seeing also that the more the strait line X
_c_ is produced, the more it diverges from the strait line X O, if the
points O and _c_ be understood to be moved forwards with strait motion
in X O and X _c_, the arch O _c_ will thereby be extended by little and
little, till at the last it come somewhere to have the same crookedness
with that part of the arch B F which is equal to it. In like manner, if
the strait line X _b_ be drawn, and the point _b_ be understood to be
moved forwards at the same time, the arch _c b_ will also by little and
little be extended, till its crookedness come to be equal to the
crookedness of that part of the arch B F which is equal to it. And the
same will happen in all those small equal arches which are described
upon so many equal parts of the strait line O F. It is also manifest,
that by strait motion in X O and X Z all those small arches will lie in
the arch B F, in the points B, Q and F. And though the same small equal
arches should not be coincident with the equal parts of the arch B F in
all the other points thereof, yet certainly they will constitute two
crooked lines, not only equal to the two arches B Q and Q F, and equally
crooked, but also having their cavity towards the same parts; which how
it should be, unless all those small arches should be coincident with
the arch B F in all its points, is not imaginable. They are therefore
coincident, and all the strait lines drawn from X, and passing through
the points of division of the strait line O F, will also divide the arch
B F into the same proportions into which O F is divided.

Now seeing X _b_ cuts off from the point B the fourth part of the arch B
F, let that fourth part be B _e_; and let the sine thereof, _f e_, be
produced to F T in _g_, for so _f e_ will be the fourth part of the
strait line _f g_, because as O _b_ is to O F, so is _f e_ to _f g_. But
B T is greater than _f g_; and therefore the same B T is greater than
four sines of the fourth part of the arch B F. And in like manner, if
the arch B F be subdivided into any number of equal parts whatsoever, it
may be proved that the strait line B T is greater than the sine of one
of those small arches, so many times taken as there be parts made of the
whole arch B F. Wherefore the strait line B T is not less than the arch
B F. But neither can it be greater, because if any strait line
whatsoever, less than B T, be drawn below B T, parallel to it, and
terminated in the strait lines X B and X T, it would cut the arch B F;
and so the sine of some one of the parts of the arch B F, taken so often
as that small arch is found in the whole arch B F, would be greater than
so many of the same arches; which is absurd. Wherefore the strait line B
T is equal to the arch B F; and the strait line B V equal to the arch of
the quadrant B F D; and B V four times taken, equal to the perimeter of
the circle described with the radius A B. Also the arch B F and the
strait line B T are everywhere divided into the same proportions; and
consequently any given angle, whether greater or less than B A F, may be
divided into any proportion given.

But the strait line B V, though its magnitude fall within the terms
assigned by Archimedes, is found, if computed by the canon of signs, to
be somewhat greater than that which is exhibited by the Rudolphine
numbers. Nevertheless, if in the place of B T, another strait line,
though never so little less, be substituted, the division of angles is
immediately lost, as may by any man be demonstrated by this very scheme.

Howsoever, if any man think this my strait line B V to be too great,
yet, seeing the arch and all the parallels are everywhere so exactly
divided, and B V comes so near to the truth, I desire he would search
out the reason, why, granting B V to be precisely true, the arches cut
off should not be equal.

But some man may yet ask the reason why the strait lines, drawn from X
through the equal parts of the arch B F, should cut off in the tangent B
V so many strait lines equal to them, seeing the connected straight line
X V passes not through the point D, but cuts the strait line A D
produced in _l_; and consequently require some determination of this
problem. Concerning which, I will say what I think to be the reason,
namely, that whilst the magnitude of the arch doth not exceed the
magnitude of the radius, that is, the magnitude of the tangent B C, both
the arch and the tangent are cut alike by the strait lines drawn from X;
otherwise not. For A V being connected, cutting the arch B H D in I, if
X C being drawn should cut the same arch in the same point I, it would
be as true that the arch B I is equal to the radius B C, as it is true
that the arch B F is equal to the strait line B T; and drawing X K it
would cut the arch B I in the midst in _i_; also drawing A _i_ and
producing it to the tangent B C in _k_, the strait line B _k_ will be
the tangent of the arch B _i_, (which arch is equal to half the radius)
and the same strait line B _k_ will be equal to the strait line _k_ I. I
say all this is true, if the preceding demonstration be true; and
consequently the proportional section of the arch and its tangent
proceeds hitherto. But it is manifest by the golden rule, that taking B
_h_ double to B T, the line X _h_ shall not cut off the arch B E, which
is double to the arch B F, but a much greater. For the magnitude of the
straight lines X M, X B, and M E, being known (in numbers), the
magnitude of the strait line cut off in the tangent by the strait line X
E produced to the tangent, may also be known; and it will be found to be
less than B _h_; Wherefore the strait line X _h_ being drawn, will cut
off a part of the arch of the quadrant greater than the arch B E. But I
shall speak more fully in the next article concerning the magnitude of
the arch B I.

And let this be the first attempt for the finding out of the dimension
of a circle by the section of the arch B F.

[Sidenote: The second attempt for the finding out of the dimension of a
           circle from the consideration of the nature of crookedness.]

3. I shall now attempt the same by arguments drawn from the nature of
the crookedness of the circle itself; but I shall first set down some
premises necessary for this speculation; and

First, if a strait line be bowed into an arch of a circle equal to it,
as when a stretched thread, which toucheth a right cylinder, is so bowed
in every point, that it be everywhere coincident with the perimeter of
the base of the cylinder, the flexion of that line will be equal, in all
its points; and consequently the crookedness of the arch of a circle is
everywhere uniform; which needs no other demonstration than this, that
the perimeter of a circle is an uniform line.

Secondly, and consequently: if two unequal arches of the same circle be
made by the bowing of two strait lines equal to them, the flexion of the
longer line, whilst it is bowed into the greater arch, is greater than
the flexion of the shorter line, whilst it is bowed into the lesser
arch, according to the proportion of the arches themselves; and
consequently, the crookedness of the greater arch is to the crookedness
of the lesser arch, as the greater arch is to the lesser arch.

Thirdly: if two unequal circles and a strait line touch one another in
the same point, the crookedness of any arch taken in the lesser circle,
will be greater than the crookedness of an arch equal to it taken in the
greater circle, in reciprocal proportion to that of the radii with which
the circles are described; or, which is all one, any strait line being
drawn from the point of contact till it cut both the circumferences, as
the part of that strait line cut off by the circumference of the greater
circle to that part which is cut off by the circumference of the lesser
circle.

For let A B and A C (in the second figure) be two circles, touching one
another, and the strait line A D in the point A; and let their centres
be E and F; and let it be supposed, that as A E is to A F, so is the
arch A B to the arch A H. I say the crookedness of the arch A C is to
the crookedness of the arch A H, as A E is to A F. For let the strait
line A D be supposed to be equal to the arch A B, and the strait line A
G to the arch A C; and let A D, for example, be double to A G.
Therefore, by reason of the likeness of the arches A B and A C, the
strait line A B will be double to the strait line A C, and the radius A
E double to the radius A F, and the arch A B double to the arch A H. And
because the strait line A D is so bowed to be coincident with the arch A
B equal to it, as the strait line A G is bowed to be coincident with the
arch A C equal also to it, the flexion of the strait line A G into the
crooked line A C will be equal to the flexion of the strait line A D
into the crooked line A B. But the flexion of the strait line A D into
the crooked line A B is double to the flexion of the strait line A G
into the crooked line A H; and therefore the flexion of the strait line
A G into the crooked line A C is double to the flexion of the same
strait line A G into the crooked line A H. Wherefore, as the arch A B is
to the arch A C or A H; or as the radius A E is to the radius A F; or as
the chord A B is to the chord A C; so reciprocally is the flexion or
uniform crookedness of the arch A C, to the flexion or uniform
crookedness of the arch A H, namely, here double. And this may by the
same method be demonstrated in circles whose perimeters are to one
another triple, quadruple, or in whatsoever given proportion. The
crookedness therefore of two equal arches taken in several circles are
in proportion reciprocal to that of their radii, or like arches, or like
chords; which was to be demonstrated.

Let the square A B C D be again described (in the third figure), and in
it the quadrants A B D, B C A and D A C; and dividing each side of the
square A B C D in the midst in E, F, G and H, let E G and F H be
connected, which will cut one another in the centre of the square at I,
and divide the arch of the quadrant A B D into three equal parts in K
and L. Also the diagonals A C and B D being drawn will cut one another
in I, and divide the arches B K D and C L A into two equal parts in M
and N. Then with the radius B F let the arch F E be drawn, cutting the
diagonal B D in O; and dividing the arch B M in the midst in P, let the
strait line E _a_ equal to the chord B P be set off from the point E in
the arch E F, and let the arch _a b_ be taken equal to the arch O _a_,
and let B _a_ and B _b_ be drawn and produced to the arch A N in _c_ and
_d_; and lastly, let the strait line A _d_ be drawn. I say the strait
line A _d_ is equal to the arch A N or B M.

I have proved in the preceding article, that the arch E O is twice as
crooked as the arch B P, that is to say, that the arch E O is so much
more crooked than the arch B P, as the arch B P is more crooked than the
strait line E _a_. The crookedness therefore of the chord E _a_, of the
arch B P, and of the arch E O, are as 0, 1, 2. Also the difference
between the arches E O and E O, the difference between the arches E O
and E _a_, and the difference between the arches E O and E _b_, are as
0, 1, 2. So also the difference between the arches A N and A N, the
difference between the arches A N and A _c_, and the difference between
the arches A N and A _d_, are as 0, 1, 2; and the strait line A _c_ is
double to the chord B P or E _a_, and the strait line A _d_ double to
the chord E _b_.

Again, let the strait line B F be divided in the midst in Q, and the
arch B P in the midst in R; and describing the quadrant B Q S (whose
arch Q S is a fourth part of the arch of the quadrant B M D, as the arch
B R is a fourth part of the arch B M, which is the arch of the
semiquadrant A B M) let the chord S _e_ equal to the chord B R be set
off from the point S in the arch S Q; and let B _e_ be drawn and
produced to the arch A N in _f_; which being done, the strait line A _f_
will be quadruple to the chord B R or S _e_. And seeing the crookedness
of the arch S _e_, or of the arch A _c_, is double to the crookedness of
the arch B R, the excess of the crookedness of the arch A _f_ above the
crookedness of the arch A _c_ will be subduple to the excess of the
crookedness of the arch A _c_ above the crookedness of the arch A N; and
therefore the arch N _c_ will be double to the arch _c f_. Wherefore the
arch _c d_ is divided in the midst in _f_, and the arch N _f_ is ¾ of
the arch N _d_. And in like manner if the arch B R be bisected in V, and
the strait line B Q in X, and the quadrant B X Y be described, and the
strait line Y _g_ equal to the chord B V be set off from the point Y in
the arch Y X, it may be demonstrated that the strait line B _g_ being
drawn and produced to the arch A N, will cut the arch _f d_ into two
equal parts, and that a strait line drawn from A to the point of that
section, will be equal to eight chords of the arch B V, and so on
perpetually; and consequently, that the strait line A _d_ is equal to so
many equal chords of equal parts of the arch B M, as may be made by
infinite bisections. Wherefore the strait line A _d_ is equal to the
arch B M or A N, that is, to half the arch of the quadrant A B D or B C
A.

Coroll. An arch being given not greater than the arch of a quadrant (for
being made greater, it comes again towards the radius B A produced, from
which it receded before) if a strait line double to the chord of half
the given arch be adapted from the beginning of the arch, and by how
much the arch that is subtended by it is greater than the given arch, by
so much a greater arch be subtended by another strait line, this strait
line shall be equal to the first given arch.

Supposing the strait line B V (in fig. 1) be equal to the arch of the
quadrant B H D, and A V be connected cutting the arch B H D in I, it may
be asked what proportion the arch B I has to the arch I D. Let therefore
the arch A Y be divided in the midst in _o_, and in the strait line A D
let A _p_ be taken equal, and A _q_ double to the drawn chord A _o_.
Then upon the centre A, with the radius A _q_, let an arch of a circle
be drawn cutting the arch A Y in _r_, and let the arch Y _r_ be doubled
at _t_; which being done, the drawn strait line A _t_ (by what has been
last demonstrated) will be equal to the arch A Y. Again, upon the centre
A with the radius A _t_ let the arch _t u_ be drawn cutting A D in _u_;
and the strait line A _u_ will be equal to the arch A Y. From the point
_u_ let the strait line _u s_ be drawn equal and parallel to the strait
line A B, cutting M N in _x_, and bisected by M N in the same point _x_.
Therefore the strait line A _x_ being drawn and produced till it meet
with B C produced in V, it will cut off B V double to B _s_, that is,
equal to the arch B H D. Now let the point, where the strait line A V
cuts the arch B H D, be I; and let the arch D I be divided in the midst
in _y_; and in the strait line D C, let D _z_ be taken equal, and D δ
double to the drawn chord D _y_; and upon the centre D with the radius D
δ let an arch of a circle be drawn cutting the arch B H D in the point
_n_; and let the arch _n m_ be taken equal to the arch I _n_; which
being done, the strait line D _m_ will (by the last foregoing corollary)
be equal to the arch D I. If now the strait lines D _m_ and C V be
equal, the arch B I will be equal to the radius A B or B C; and
consequently X C being drawn, will pass through the point I. Moreover,
if the semicircle B H D ϐ being completed, the strait lines ϐ I and B I
be drawn, making a right angle (in the semicircle) at I, and the arch B
I be divided in the midst at _i_, it will follow that A _i_ being
connected will be parallel to the strait line ϐ I, and being produced to
B C in _k_, will cut off the strait line B _k_ equal to the strait line
_k_ I, and equal also to the strait line A γ cut off in A D by the
strait line ϐ I. All which is manifest, supposing the arch B I and the
radius B C to be equal.

But that the arch B I and the radius B C are precisely equal, cannot
(how true soever it be) be demonstrated, unless that be first proved
which is contained in art. 1, namely, that the strait lines drawn from X
through the equal parts of O F (produced to a certain length) cut off so
many parts also in the tangent B C severally equal to the several arches
cut off; which they do most exactly as far as B C in the tangent, and B
I in the arch B E; insomuch that no inequality between the arch B I and
the radius B C can be discovered either by the hand or by ratiocination.
It is therefore to be further enquired, whether the strait line A V cut
the arch of the quadrant in I in the same proportion as the point C
divides the strait line B V, which is equal to the arch of the quadrant.
But however this be, it has been demonstrated that the strait line B V
is equal to the arch B H D.

[Sidenote: The third attempt; and some things propounded to be further
           searched into.]

4. I shall now attempt the same dimension of a circle another way,
assuming the two following lemmas.

Lemma I. If to the arch of a quadrant, and the radius, there be taken in
continual proportion a third line Z; then the arch of the semiquadrant,
half the chord of the quadrant, and Z, will also be in continual
proportion.

For seeing the radius is a mean proportional between the chord of a
quadrant and its semichord, and the same radius a mean proportional
between the arch of the quadrant and Z, the square of the radius will be
equal as well to the rectangle made of the chord and semichord of the
quadrant, as to the rectangle made of the arch of the quadrant and Z;
and these two rectangles will be equal to one another. Wherefore, as the
arch of a quadrant is to its chord, so reciprocally is half the chord of
the quadrant to Z. But as the arch of the quadrant is to its chord, so
is half the arch of the quadrant to half the chord of the quadrant.
Wherefore, as half the arch of the quadrant is to half the chord of the
quadrant (or to the sine of 45 degrees), so is half the chord of the
quadrant to Z; which was to be proved.

Lemma II. The radius, the arch of the semiquadrant, the sine of 45
degrees, and the semiradius, are proportional.

For seeing the sine of 45 degrees is a mean proportional between the
radius and the semiradius; and the same sine of 45 degrees is also a
mean proportional (by the precedent lemma) between the arch of 45
degrees and Z; the square of the sine of 45 degrees will be equal as
well to the rectangle made of the radius and semiradius, as to the
rectangle made of the arch of 45 degrees and Z. Wherefore, as the radius
is to the arch of 45 degrees, so reciprocally is Z to the semiradius;
which was to be demonstrated.

Let now A B C D (in fig. 4) be a square; and with the radii A B, B C and
D A, let the three quadrants A B D, B C A and D A C, be described; and
let the strait lines E F and G H, drawn parallel to the sides B C and A
B, divide the square A B C D into four equal squares. They will
therefore cut the arch of the quadrant A B D into three equal parts in I
and K, and the arch of the quadrant B C A into three equal parts in K
and L. Also let the diagonals A C and B D be drawn, cutting the arches B
I D and A L C in M and N. Then upon the centre H with the radius H F
equal to half the chord of the arch B M D, or to the sine of 45 degrees,
let the arch F O be drawn cutting the arch C K in O; and let A O be
drawn and produced till it meet with B C produced in P; also let it cut
the arch B M D in Q, and the strait line D C in R. If now the strait
line H Q be equal to the strait line D R, and being produced to D C in
S, cut off D S equal to half the strait line B P; I say then the strait
line B P will be equal to the arch B M D.

For seeing P B A and A D R are like triangles, it will be as P B to the
radius B A or A D, so A D to D R; and therefore as well P B, A D and D
R, as P B, A D (or A Q) and Q H are in continual proportion; and
producing H O to D C in T, D T will be equal to the sine of 45 degrees,
as shall by and by be demonstrated. Now D S, D T and D R are in
continual proportion by the first lemma; and by the second lemma D C. D
S:: D R. D F are proportionals. And thus it will be, whether B P be
equal or not equal to the arch of the quadrant B M D. But if they be
equal, it will then be, as that part of the arch B M D which is equal to
the radius, is to the remainder of the same arch B M D; so A Q to H Q,
or so B C to C P. And then will B P and the arch B M D be equal. But it
is not demonstrated that the strait lines H Q and D R are equal; though
if from the point B there be drawn (by the construction of fig. 1) a
strait line equal to the arch B M D, then D R to H Q, and also the half
of the strait line B P to D S, will always be so equal, that no
inequality can be discovered between them. I will therefore leave this
to be further searched into. For though it be almost out of doubt, that
the strait line B P and the arch B M D are equal, yet that may not be
received without demonstration; and means of demonstration the circular
line admitteth none that is not grounded upon the nature of flexion, or
of angles. But by that way I have already exhibited a strait line equal
to the arch of a quadrant in the first and second aggression.

It remains that I prove D T to be equal to the sine of 45 degrees.

In B A produced let A V be taken equal to the sine of 45 degrees; and
drawing and producing V H, it will cut the arch of the quadrant C N A in
the midst in N, and the same arch again in O, and the strait line D C in
T, so that D T will be equal to the sine of 45 degrees, or to the strait
line A V; also the strait line V H will be equal to the strait line H I,
or the sine of 60 degrees.

For the square of A V is equal to two squares of the semiradius; and
consequently the square of V H is equal to three squares of the
semiradius. But H I is a mean proportional between the semiradius and
three semiradii; and, therefore, the square of H I is equal to three
squares of the semiradius. Wherefore H I is equal to H V. But because A
D is cut in the midst in H, therefore V H and H T are equal; and,
therefore, also D T is equal to the sine of 45 degrees. In the radius B
A let B X be taken equal to the sine of 45 degrees; for so V X will be
equal to the radius; and it will be as V A to A H the semiradius, so V X
the radius to X N the sine of 45 degrees. Wherefore V H produced passes
through N. Lastly, upon the centre V with the radius V A let the arch of
a circle be drawn cutting V H in Y; which being done, V Y will be equal
to H O (for H O is, by construction, equal to the sine of 45 degrees)
and Y H will be equal to O T; and, therefore, V T passes through O. All
which was to be demonstrated.

I will here add certain problems, of which if any analyst can make the
construction, he will thereby be able to judge clearly of what I have
now said concerning the dimension of a circle. Now these problems are
nothing else (at least to sense) but certain symptoms accompanying the
construction of the first and third figure of this chapter.

Describing, therefore, again, the square A B C D (in fig. 5) and the
three quadrants A B D, B C A and D A C, let the diagonals A C and B D be
drawn, cutting the arches B H D and C I A in the middle in H and I; and
the strait lines E F and G L, dividing the square A B C D into four
equal squares, and trisecting the arches B H D and C I A, namely, B H D
in K and M, and C I A in M and O. Then dividing the arch B K in the
midst in P, let Q P the sine of the arch B P, be drawn and produced to
R, so that Q R be double to Q P; and, connecting K R, let it be produced
one way to B C in S, and the other way to B A produced in T. Also let B
V be made triple to B S, and consequently, (by the second article of
this chapter) equal to the arch B D. This construction is the same with
that of the first figure, which I thought fit to renew discharged of all
lines but such as are necessary for my present purpose.

In the first place, therefore, if A V be drawn, cutting the arch B H D
in X, and the side D C in Z, I desire some analyst would, if he can,
give a reason why the strait lines T E and T C should cut the arch B D,
the one in Y, the other in X, so as to make the arch B Y equal to the
arch Y X; or if they be not equal, that he would determine their
difference.

Secondly, if in the side D A, the strait line D _a_ be taken equal to D
Z, and V _a_ be drawn; why V _a_ and V B should be equal; or if they be
not equal, what is the difference.

Thirdly, drawing Z _b_ parallel and equal to the side C B, cutting the
arch B H D in _c_, and drawing the strait line A _c_, and producing it
to B V in _d_; why A _d_ should be equal and parallel to the strait line
_a_ V, and consequently equal also to the arch B D.

Fourthly, drawing _e_ K the sine of the arch B K, and taking (in _e_ A
produced) _e f_ equal to the diagonal A C, and connecting _f_ C; why _f_
C should pass through _a_ (which point being given, the length of the
arch B H D is also given) and _c_; and why _f e_ and _f c_ should be
equal; or if not, why unequal.

Fifthly, drawing _f_ Z, I desire he would show, why it is equal to B V,
or to the arch B D; or if they be not equal, what is their difference.

Sixthly, granting _f_ Z to be equal to the arch B D, I desire he would
determine whether it fall all without the arch B C A, or cut the same,
or touch it, and in what point.

Seventhly, the semicircle B D _g_ being completed, why _g_ I being drawn
and produced, should pass through X, by which point X the length of the
arch B D is determined. And the same _g_ I being yet further produced to
D C in _h_, why A _d_, which is equal to the arch B D, should pass
through that point _h_.

Eighthly, upon the centre of the square A B C D, which let be _k_, the
arch of the quadrant E _i_ L being drawn, cutting _e_ K produced in _i_,
why the drawn strait line _i_ X should be parallel to the side C D.

Ninthly, in the sides B A and B C taking _g l_ and B _m_ severally equal
to half B V, or to the arch B H, and drawing _m n_ parallel and equal to
the side B A, cutting the arch B D in _o_, why the strait line which
connects V _l_ should pass through the point _o_.

Tenthly, I would know of him why the strait line which connects _a_ H
should be equal to B _m_; or if not, how much it differs from it.

The analyst that can solve these problems without knowing first the
length of the arch B D, or using any other known method than that which
proceeds by perpetual bisection of an angle, or is drawn from the
consideration of the nature of flexion, shall do more than ordinary
geometry is able to perform. But if the dimension of a circle cannot be
found by any other method, then I have either found it, or it is not at
all to be found.

From the known length of the arch of a quadrant, and from the
proportional division of the arch and of the tangent B C, may be deduced
the section of an angle into any given proportion; as also the squaring
of the circle, the squaring of a given sector, and many the like
propositions, which it is not necessary here to demonstrate. I will,
therefore, only exhibit a strait line equal to the spiral of Archimedes,
and so dismiss this speculation.

[Sidenote: The equation of the spiral of Archimedes with a strait line.]

5. The length of the perimeter of a circle being found, that strait line
is also found, which touches a spiral at the end of its first
conversion. For upon the centre A (in fig. 6) let the circle B C D E be
described; and in it let Archimedes' spiral A F G H B be drawn,
beginning at A and ending at B. Through the centre A let the strait line
C E be drawn, cutting the diameter B D at right angles; and let it be
produced to I, so that A I be equal to the perimeter B C D E B.
Therefore I B being drawn will touch the spiral A F G H B in B; which is
demonstrated by Archimedes in his book _De Spiralibus_.

And for a strait line equal to the given spiral A F G H B, it may be
found thus.

Let the strait line A I, which is equal to the perimeter B C D E, be
bisected in K; and taking K L equal to the radius A B, let the rectangle
I L be completed. Let M L be understood to be the axis, and K L the base
of a parabola, and let M K be the crooked line thereof. Now if the point
M be conceived to be so moved by the concourse of two movents, the one
from I M to K L with velocity encreasing continually in the same
proportion with the times, the other from M L to I K uniformly, that
both those motions begin together in M and end in K; Galilæus has
demonstrated that by such motion of the point M, the crooked line of a
parabola will be described. Again, if the point A be conceived to be
moved uniformly in the strait line A B, and in the same time to be
carried round upon the centre A by the circular motion of all the points
between A and B; Archimedes has demonstrated that by such motion will be
described a spiral line. And seeing the circles of all these motions are
concentric in A; and the interior circle is always less than the
exterior in the proportion of the times in which A B is passed over with
uniform motion; the velocity also of the circular motion of the point A
will continually increase proportionally to the times. And thus far the
generations of the parabolical line M K, and of the spiral line A F G H
B, are like. But the uniform motion in A B concurring with circular
motion in the perimeters of all the concentric circles, describes that
circle, whose centre is A, and perimeter B C D E; and, therefore, that
circle is (by the coroll. of art. 1, chap, XVI) the aggregate of all the
velocities together taken of the point A whilst it describes the spiral
A F G H B. Also the rectangle I K L M is the aggregate of all the
velocities together taken of the point M, whilst it describes the
crooked line M K. And, therefore the whole velocity by which the
parabolical line M K is described, is to the whole velocity with which
the spiral line A F G H B is described in the same time, as the
rectangle I K L M is to the circle B C D E, that is to the triangle A I
B. But because A I is bisected in K, and the strait lines I M and A B
are equal, therefore the rectangle I K L M and the triangle A I B are
also equal. Wherefore the spiral line A F G H B, and the parabolical
line M K, being described with equal velocity and in equal times, are
equal to one another. Now, in the first article of chap. XVIII, a strait
line is found out equal to any parabolical line. Wherefore also a strait
line is found out equal to a given spiral line of the first revolution
described by Archimedes; which was to be done.

[Sidenote: Of the analysis of geometricians by the powers of lines.]

6. In the sixth chapter, which is of _Method_, that which I should there
have spoken of the analytics of geometricians I thought fit to defer,
because I could not there have been understood, as not having then so
much as named _lines_, _superficies_, _solids_, _equal_ and _unequal_,
_&c._ Wherefore I will in this place set down my thoughts concerning it.

_Analysis_ is continual reasoning from the definitions of the terms of a
proposition we suppose true, and again from the definitions of the terms
of those definitions, and so on, till we come to some things known, the
composition whereof is the demonstration of the truth or falsity of the
first supposition; and this composition or demonstration is that we call
_Synthesis_. _Analytica_, therefore, is that art, by which our reason
proceeds from something supposed, to principles, that is, to prime
propositions, or to such as are known by these, till we have so many
known propositions as are sufficient for the demonstration of the truth
or falsity of the thing supposed. _Synthetica_ is the art itself of
demonstration. Synthesis, therefore, and analysis, differ in nothing,
but in proceeding forwards or backwards; and _Logistica_ comprehends
both. So that in the analysis or synthesis of any question, that is to
say, of any problem, the terms of all the propositions ought to be
convertible; or if they be enunciated hypothetically, the truth of the
consequent ought not only to follow out of the truth of its antecedent,
but contrarily also the truth of the antecedent must necessarily be
inferred from the truth of the consequent. For otherwise, when by
resolution we are arrived at principles, we cannot by composition return
directly back to the thing sought for. For those terms which are the
first in analysis, will be the last in synthesis; as for example, when
in _resolving_, we say, these two rectangles are equal, and therefore
their sides are reciprocally proportional, we must necessarily in
_compounding_ say, the sides of these rectangles are reciprocally
proportional, and therefore the rectangles themselves are equal; which
we could not say, unless _rectangles have their sides reciprocally
proportional_, and _rectangles are equal_, were terms convertible.

Now in every analysis, that which is sought is the proportion of two
quantities; by which proportion, a figure being described, the quantity
sought for may be exposed to sense. And this _exposition_ is the end and
solution of the question, or the construction of the problem.

And seeing analysis is reasoning from something supposed, till we come
to principles, that is, to definitions, or to theorems formerly known;
and seeing the same reasoning tends in the last place to some equation,
we can therefore make no end of resolving, till we come at last to the
causes themselves of equality and inequality, or to theorems formerly
demonstrated from those causes; and so have a sufficient number of those
theorems for the demonstration of the thing sought for.

And seeing also, that the end of the analytics is either the
construction of such a problem as is possible, or the detection of the
impossibility thereof; whensoever the problem may be solved, the analyst
must not stay, till he come to those things which contain the efficient
cause of that whereof he is to make construction. But he must of
necessity stay, when he comes to prime propositions; and these are
definitions. These definitions therefore must contain the efficient
cause of his construction; I say of his construction, not of the
conclusion which he demonstrates; for the cause of the conclusion is
contained in the premised propositions; that is to say, the truth of the
proposition he proves is drawn from the propositions which prove the
same. But the cause of his construction is in the things themselves, and
consists in motion, or in the concourse of motions. Wherefore those
propositions, in which analysis ends, are definitions, but such as
signify in what manner the construction or generation of the thing
proceeds. For otherwise, when he goes back by synthesis to the proof of
his problem, he will come to no demonstration at all; there being no
true demonstration but such as is scientifical; and no demonstration is
scientifical, but that which proceeds from the knowledge of the causes
from which the construction of the problem is drawn. To collect
therefore what has been said into few words; ANALYSIS _is ratiocination
from the supposed construction or generation of a thing to the efficient
cause or coefficient causes of that which is constructed or generated.
And_ SYNTHESIS _is ratiocination from the first causes of the
construction, continued through all the middle causes till we come to
the thing itself which is constructed or generated_.

But because there are many means by which the same thing may be
generated, or the same problem be constructed, therefore neither do all
geometricians, nor doth the same geometrician always, use one and the
same method. For, if to a certain quantity given, it be required to
construct another quantity equal, there may be some that will inquire
whether this may not be done by means of some motion. For there are
quantities, whose equality and inequality may be argued from motion and
time, as well as from congruence; and there is motion, by which two
quantities, whether lines or superficies, though one of them be crooked,
the other strait, may be made congruous or coincident. And this method
Archimedes made use of in his book _De Spiralibus_. Also the equality or
inequality of two quantities may be found out and demonstrated from the
consideration of weight, as the same Archimedes did in his quadrature of
the parabola. Besides, equality and inequality are found out often by
the division of the two quantities into parts which are considered as
indivisable; as Cavallerius Bonaventura has done in our time, and
Archimedes often. Lastly, the same is performed by the consideration of
the powers of lines, or the roots of those powers, and by the
multiplication, division, addition, and subtraction, as also by the
extraction of the roots of those powers, or by finding where strait
lines of the same proportion terminate. For example, when any number of
strait lines, how many soever, are drawn from a strait line and pass all
through the same point, look what proportion they have, and if their
parts continued from the point retain everywhere the same proportion,
they shall all terminate in a strait line. And the same happens if the
point be taken between two circles. So that the places of all their
points of termination make either strait lines, or circumferences of
circles, and are called _plane places_. So also when strait parallel
lines are applied to one strait line, if the parts of the strait line to
which they are applied be to one another in proportion duplicate to that
of the contiguous applied lines, they will all terminate in a conical
section; which section, being the place of their termination, is called
a _solid place_, because it serves for the finding out of the quantity
of any equation which consists of three dimensions. There are therefore
three ways of finding out the cause of equality or inequality between
two given quantities; namely, first, by the computation of _motions_;
for by equal motion, and equal time, equal spaces are described; and
ponderation is motion. Secondly, by _indivisibles_: because all the
parts together taken are equal to the whole. And thirdly, by the
_powers_: for when they are equal, their roots also are equal; and
contrarily, the powers are equal, when their roots are equal. But if the
question be much complicated, there cannot by any of these ways be
constituted a certain rule, from the supposition of which of the unknown
quantities the analysis may best begin; nor out of the variety of
equations, that at first appear, which we were best to choose; but the
success will depend upon dexterity, upon formerly acquired science, and
many times upon fortune.

For no man can ever be a good analyst without being first a good
geometrician; nor do the rules of analysis make a geometrician, as
synthesis doth; which begins at the very elements, and proceeds by a
logical use of the same. For the true teaching of geometry is by
synthesis, according to Euclid's method; and he that hath Euclid for his
master, may be a geometrician without Vieta, though Vieta was a most
admirable geometrician; but he that has Vieta for his master, not so,
without Euclid.

And as for that part of analysis which works by the powers, though it be
esteemed by some geometricians, not the chiefest, to be the best way of
solving all problems, yet it is a thing of no great extent; it being all
contained in the doctrine of rectangles, and rectangled solids. So that
although they come to an equation which determines the quantity sought,
yet they cannot sometimes by art exhibit that quantity in a plane, but
in some conic section; that is, as geometricians say, not geometrically,
but mechanically. Now such problems as these, they call _solid_; and
when they cannot exhibit the quantity sought for with the help of a
conic section, they call it a _lineary_ problem. And therefore in the
quantities of angles, and of the arches of circles, there is no use at
all of the analytics which proceed by the powers; so that the ancients
pronounced it impossible to exhibit in a plane the division of angles,
except bisection, and the bisection of the bisected parts, otherwise
than mechanically. For Pappus, (before the 31st proposition of his
fourth book) distinguishing and defining the several kinds of problems,
says that "some are _plane_, others _solid_, and others _lineary_.
Those, therefore, which may be solved by strait lines and the
circumferences of circles, (that is, which may be described with the
rule and compass, without any other instrument), are fitly called
_plane_; for the lines, by which such problems are found out, have their
generation in a plane. But those which are solved by the using of some
one or more conic sections in their construction, are called _solid_,
because their construction cannot be made without using the superficies
of solid figures, namely, of cones. There remains the third kind, which
is called _lineary_, because other lines besides those already mentioned
are made use of in their construction, &c." And a little after he says,
"of this kind are the _spiral_ lines, the _quadratrices_, the
_conchoeides_, and the _cissoeides_, And geometricians think it no small
fault, when for the finding out of a plane problem any man makes use of
conics, or new lines." Now he ranks the trisection of an angle among
solid problems, and the quinquesection among lineary. But what! are the
ancient geometricians to be blamed, who made use of the quadratrix for
the finding out of a strait line equal to the arch of a circle? And
Pappus himself, was he faulty, when he found out the trisection of an
angle by the help of an hyperbole? Or am I in the wrong, who think I
have found out the construction of both these problems by the rule and
compass only? Neither they, nor I. For the ancients made use of this
analysis which proceeds by the powers; and with them it was a fault to
do that by a more remote power, which might be done by a nearer; as
being an argument that they did not sufficiently understand the nature
of the thing.

The virtue of this kind of analysis consists in the changing and turning
and tossing of rectangles and analogisms; and the skill of analysts is
mere logic, by which they are able methodically to find out whatsoever
lies hid either in the subject or predicate of the conclusion sought
for. But this doth not properly belong to algebra, or the analytics
specious, symbolical, or cossick; which are, as I may say, the
_brachygraphy_ of the analytics, and an art neither of teaching nor
learning geometry, but of registering with brevity and celerity the
inventions of geometricians. For though it be easy to discourse by
symbols of very remote propositions; yet whether such discourse deserve
to be thought very profitable, when it is made without any ideas of the
things themselves, I know not.

------------------------------------------------------------------------

[Illustration:

  C. XX.
  _of
  the translation_.
  _Fig. 1-2_
]

[Illustration:

  C. XX.
  _of
  the translation_.
  _Fig. 3_
]

[Illustration:

  C. XX.
  _of
  the translation_.
  _Fig. 4_
]

[Illustration:

  C. XX.
  _of
  the translation_.
  _Fig. 5-6_
]

------------------------------------------------------------------------


                                -------


                              CHAPTER XXI.

                          OF CIRCULAR MOTION.

1. In simple motion, every strait line taken in the body moved is so
    carried, that it is always parallel to the places in which it
    formerly was.—2. If circular motion be made about a resting centre,
    and in that circle there be an epicycle, whose revolution is made
    the contrary way, in such manner that in equal times it make equal
    angles, every strait line taken in that epicycle will be so carried,
    that it will always be parallel to the places in which it formerly
    was.—3. The properties of simple motion.—4. If a fluid be moved with
    simple circular motion, all the points taken in it will describe
    their circles in times proportional to the distances from the
    centre.—5. Simple motion dissipates heterogeneous and congregates
    homogeneous bodies.—6. If a circle made by a movent moved with
    simple motion be commensurable to another circle made by a point
    which is carried about by the same movent, all the points of both
    the circles will at some time return to the same situation.—7.. If a
    sphere have simple motion, its motion will more dissipate
    heterogeneous bodies by how much it is more remote from the
    poles.—8. If the simple circular motion of a fluid body be hindered
    by a body which is not fluid, the fluid body will spread itself upon
    the superficies of that body.—9. Circular motion about a fixed
    centre casteth off by the tangent such things as lie upon the
    circumference and stick not to it.—10. Such things, as are moved
    with simple circular motion, beget simple circular motion.—11. If
    that which is so moved have one side hard and the other side fluid,
    its motion will not be perfectly circular.


[Sidenote: In simple motion, every strait line taken in the body moved
           is so carried, that it is always parallel to the places in
           which it formerly was.]

1. I have already defined _simple motion_ to be that, in which the
several points taken in a moved body do in several equal times describe
several equal arches. And therefore in simple circular motion it is
necessary that every strait line taken in the moved body be always
carried parallel to itself; which I thus demonstrate.

First, let A B (in the first figure) be any strait line taken in any
solid body; and let A D be any arch drawn upon any centre C and radius
CA. Let the point B be understood to describe towards the same parts the
arch B E, like and equal to the arch A D. Now in the same time in which
the point A transmits the arch A D, the point B, which by reason of its
simple motion is supposed to be carried with a velocity equal to that of
A, will transmit the arch B E; and at the end of the same time the whole
A B will be in D E; and therefore A B and D E are equal. And seeing the
arches A D and B E are like and equal, their subtending strait lines AD
and BE will also be equal; and therefore the four-sided figure A B D E
will be a parallelogram. Wherefore A B is carried parallel to itself.
And the same may be proved by the same method, if any other strait line
be taken in the same moved body in which the strait line A B was taken.
So that all strait lines, taken in a body moved with simple circular
motion, will be carried parallel to themselves.

Coroll. I. It is manifest that the same will also happen in any body
which hath simple motion, though not circular. For all the points of any
strait line whatsoever will describe lines, though not circular, yet
equal; so that though the crooked lines A D and B E were not arches of
circles, but of parabolas, ellipses, or of any other figures, yet both
they, and their subtenses, and the strait lines which join them, would
be equal and parallel.

Coroll. II. It is also manifest, that the radii of the equal circles A D
and B E, or the axis of a sphere, will be so carried, as to be always
parallel to the places in which they formerly were. For the strait line
B F drawn to the centre of the arch B E being equal to the radius A C,
will also be equal to the strait line F E or C D; and the angle B F E
will be equal to the angle A C D. Now the intersection of the strait
lines C A and B E being at G, the angle C G E (seeing B E and A D are
parallel) will be equal to the angle D A C. But the angle E B F is equal
to the same angle D A C; and therefore the angles C G E and E B F are
also equal. Wherefore A C and B F are parallel; which was to be
demonstrated.

[Sidenote: If circular motion be made about a resting centre, and in
           that circle there be an epicycle whose revolution is made the
           contrary way, in such manner that in equal times it make
           equal angles, every strait line taken in that epicycle will
           be so carried, that it will always be parallel to the places
           in which it formerly was.]

2. Let there be a circle given (in the second figure) whose centre is A,
and radius A B; and upon the centre B and any radius B C let the
epicycle C D E be described. Let the centre B be understood to be
carried about the centre A, and the whole epicycle with it till it be
coincident with the circle F G H, whose centre is I; and let B A I be
any angle given. But in the time that the centre B is moved to I, let
the epicycle C D E have a contrary revolution upon its own centre,
namely from E by D to C, according to the same proportions; that is, in
such manner, that in both the circles, equal angles be made in equal
times. I say E C, the axis of the epicycle, will be always carried
parallel to itself. Let the angle F I G be made equal to the angle B A
I; I F and A B will therefore be parallel; and how much the axis A G has
departed from its former place A C (the measure of which progression is
the angle C A G, or C B D, which I suppose equal to it) so much in the
same time has the axis I G, the same with B C, departed from its own
former situation. Wherefore, in what time B C comes to I G by the motion
from B to I upon the centre A, in the same time G will come to F by the
contrary motion of the epicycle; that is, it will be turned backwards to
F, and I G will lie in I F. But the angles F I G and G A C are equal;
and therefore A C, that is, B C, and I F, (that is the axis, though in
different places) will be parallel. Wherefore, the axis of the epicycle
E D C will be carried always parallel to itself; which was to be proved.

Coroll. From hence it is manifest, that those two annual motions which
Copernicus ascribes to the earth, are reducible to this one circular
simple motion, by which all the points of the moved body are carried
always with equal velocity, that is, in equal times they make equal
revolutions uniformly.

This, as it is the most simple, so it is the most frequent of all
circular motions; being the same which is used by all men when they turn
anything round with their arms, as they do in grinding or sifting. For
all the points of the thing moved describe lines which are like and
equal to one another. So that if a man had a ruler, in which many pens'
points of equal length were fastened, he might with this one motion
write many lines at once.

[Sidenote: Properties of simple motion.]

3. Having shown what simple motion is, I will here also set down some
properties of the same.

First, when a body is moved with simple motion in a fluid medium which
hath no vacuity, it changes the situation of all the parts of the fluid
ambient which resist its motion; I say there are no parts so small of
the fluid ambient, how far soever it be continued, but do change their
situation in such manner, as that they leave their places continually to
other small parts that come into the same.

For (in the same second figure) let any body, as K L M N, be understood
to be moved with simple circular motion; and let the circle, which every
point thereof describes, have any determined quantity, suppose that of
the same K L M N. Wherefore the centre A and every other point, and
consequently the moved body itself, will be carried sometimes towards
the side where is K, and sometimes towards the other side where is M.
When therefore it is carried to K, the parts of the fluid medium on that
side will go back; and, supposing all space to be full, others on the
other side will succeed. And so it will be when the body is carried to
the side M, and to N, and every way. Now when the nearest parts of the
fluid medium go back, it is necessary that the parts next to those
nearest parts go back also; and supposing still all space to be full,
other parts will come into their places with succession perpetual and
infinite. Wherefore all, even the least parts of the fluid medium,
change their places, &c. Which was to be proved.

It is evident from hence, that simple motion, whether circular or not
circular, of bodies which make perpetual returns to their former places,
hath greater or less force to dissipate the parts of resisting bodies,
as it is more or less swift, and as the lines described have greater or
less magnitude. Now the greatest velocity that can be, may be understood
to be in the least circuit, and the least in the greatest; and may be so
supposed, when there is need.

[Sidenote: If a fluid be moved with simple circular motion, all the
           points taken in it will describe their circles in times
           proportional to the distances from the centre.]

4. Secondly, supposing the same simple motion in the air, water, or
other fluid medium; the parts of the medium, which adhere to the moved
body, will be carried about with the same motion and velocity, so that
in what time soever any point of the movent finishes its circle, in the
same time every part of the medium, which adheres to the movent, shall
also describe such a part of its circle, as is equal to the whole circle
of the movent; I say, it shall describe a part, and not the whole
circle, because all its parts receive their motion from an interior
concentric movent, and of concentric circles the exterior are always
greater than the interior; nor can the motion imprinted by any movent be
of greater velocity than that of the movent itself. From whence it
follows, that the more remote parts of the fluid ambient shall finish
their circles in times, which have to one another the same proportion
with their distances from the movent. For every point of the fluid
ambient, as long as it toucheth the body which carries it about, is
carried about with it, and would make the same circle, but that it is
left behind so much as the exterior circle exceeds the interior. So that
if we suppose some thing, which is not fluid, to float in that part of
the fluid ambient which is nearest to the movent, it will together with
the movent be carried about. Now that part of the fluid ambient, which
is not the nearest but almost the nearest, receiving its degree of
velocity from the nearest, which degree cannot be greater than it was in
the giver, doth therefore in the same time make a circular line, not a
whole circle, yet equal to the whole circle of the nearest. Therefore in
the same time that the movent describes its circle, that which doth not
touch it shall not describe its circle; yet it shall describe such a
part of it, as is equal to the whole circle of the movent. And after the
same manner, the more remote parts of the ambient will describe in the
same time such parts of their circles, as shall be severally equal to
the whole circle of the movent; and, by consequent, they shall finish
their whole circles in times proportional to their distances from the
movent; which was to be proved.

[Sidenote: Simple motion dissipates heterogeneous and congregates
           homogeneous bodies.]

5. Thirdly, the same simple motion of a body placed in a fluid medium,
congregates or gathers into one place such things as naturally float in
that medium, if they be homogeneous; and if they be heterogeneous, it
separates and dissipates them. But if such things as be heterogeneous do
not float, but settle, then the same motion stirs and mingles them
disorderly together. For seeing bodies, which are unlike to one another,
that is, heterogeneous bodies, are not unlike in that they are bodies;
for bodies, as bodies, have no difference; but only from some special
cause, that is, from some internal motion, or motions of their smallest
parts (for I have shown in chap. IX, art. 9, that all mutation is such
motion), it remains that heterogeneous bodies have their unlikeness or
difference from one another from their internal or specifical motions.
Now bodies which have such difference receive unlike and different
motions from the same external common movent; and therefore they will
not be moved together, that is to say, they will be dissipated. And
being dissipated they will necessarily at some time or other meet with
bodies like themselves, and be moved alike and together with them; and
afterwards meeting with more bodies like themselves, they will unite and
become greater bodies. Wherefore homogeneous bodies are congregated, and
heterogeneous dissipated by simple motion in a medium where they
naturally float. Again, such as being in a fluid medium do not float,
but sink, if the motion of the fluid medium be strong enough, will be
stirred up and carried away by that motion, and consequently they will
be hindered from returning to that place to which they sink naturally,
and in which only they would unite, and out of which they are
promiscuously carried; that is, they are disorderly mingled.

Now this motion, by which homogeneous bodies are congregated and
heterogeneous are scattered, is that which is commonly called
_fermentation_, from the Latin _fervere_; as the Greeks have their Ζύμη,
which signifies the same, from Ζέω _ferveo_. For seething makes all the
parts of the water change their places; and the parts of any thing, that
is thrown into it, will go several ways according to their several
natures. And yet all _fervour_ or seething is not caused by fire; for
new wine and many other things have also their fermentation and fervour,
to which fire contributes little, and sometimes nothing. But when in
fermentation we find heat, it is made by the fermentation.

[Sidenote: If a circle made by a movent moved with simple motion, be
           commensurable  to another circle made by a point which is
           carried about by  the same movent, all the points of both the
           circles will at some time  return to the same situation.]

6. Fourthly, in what time soever the movent, whose centre is A (in fig.
2) moved in K L N, shall, by any number of revolutions, that is, when
the perimeters B I and K L N be commensurable, have described a line
equal to the circle which passes through the points B and I; in the same
time all the points of the floating body, whose centre is B, shall
return to have the same situation in respect of the movent, from which
they departed. For seeing it is as the distance B A, that is, as the
radius of the circle which passes through B I is to the perimeter itself
B I, so the radius of the circle K L N is to the perimeter K L N; and
seeing the velocities of the points B and K are equal, the time also of
the revolution in I B to the time of one revolution in K L N, will be as
the perimeter B I to the perimeter K L N; and therefore so many
revolutions in K L N, as together taken are equal to the perimeter B I,
will be finished in the same time in which the whole perimeter B I is
finished; and therefore also the points L, N, F and H, or any of the
rest, will in the same time return to the same situation from which they
departed; and this may be demonstrated, whatsoever be the points
considered. Wherefore all the points shall in that time return to the
same situation; which was to be proved.

From hence it follows, that if the perimeters B I and L K N be not
commensurable, then all the points will never return to have the same
situation or configuration in respect of one another.

[Sidenote: If a sphere have simple motion, its motion will more
           dissipate heterogeneous bodies by how much it is more remote
           from  the poles.]

7. In simple motion, if the body moved be of a spherical figure, it hath
less force towards its poles than towards its middle to dissipate
heterogeneous, or to congregate homogeneous bodies.

Let there be a sphere (as in the third figure) whose centre is A and
diameter B C; and let it be conceived to be moved with simple circular
motion; of which motion let the axis be the strait line D E, cutting the
diameter B C at right angles in A. Let now the circle, which is
described by any point B of the sphere, have B F for its diameter; and
taking F G equal to B C, and dividing it in the middle in H, the centre
of the sphere A will, when half a revolution is finished, lie in H. And
seeing H F and A B are equal, a circle described upon the centre H with
the radius H F or H G, will be equal to the circle whose centre is A and
radius A B. And if the same motion be continued, the point B will at the
end of another half revolution return to the place from whence it began
to be moved; and therefore at the end of half a revolution, the point B
will be carried to F, and the whole hemisphere D B E into that
hemisphere in which are the points L, K and F. Wherefore that part of
the fluid medium, which is contiguous to the point F, will in the same
time go back the length of the strait line B F; and in the return of the
point F to B, that is, of G to C, the fluid medium will go back as much
in a strait line from the point C. And this is the effect of simple
motion in the middle of the sphere, where the distance from the poles is
greatest. Let now the point I be taken in the same sphere nearer to the
pole E, and through it let the strait line I K be drawn parallel to the
strait line B F, cutting the arch F L in K, and the axis H L in M; then
connecting H K, upon H F let the perpendicular K N be drawn. In the same
time therefore that B comes to F the point I will come to K, B F and I K
being equal and described with the same velocity. Now the motion in I K
to the fluid medium upon which it works, namely, to that part of the
medium which is contiguous to the point K, is oblique, whereas if it
proceeded in the strait line H K it would be perpendicular; and
therefore the motion which proceeds in I K has less power than that
which proceeds in H K with the same velocity. But the motions in H K and
H F do equally thrust back the medium; and therefore the part of the
sphere at K moves the medium less than the part at F, namely, so much
less as K N is less than H F. Wherefore also the same motion hath less
power to disperse heterogeneous, and to congregate homogeneous bodies,
when it is nearer, than when it is more remote from the poles; which was
to be proved.

Coroll. It is also necessary, that in planes which are perpendicular to
the axis, and more remote than the pole itself from the middle of the
sphere, this simple motion have no effect. For the axis D E with simple
motion describes the superficies of a cylinder; and towards the bases of
the cylinder there is in this motion no endeavour at all.

[Sidenote: If a simple circular motion of a fluid body be hindered by  a
           body which is not fluid, the fluid body will spread itself
           upon the superficies of that body.]

8. If in a fluid medium moved about, as hath been said, with simple
motion, there be conceived to float some other spherical body which is
not fluid, the parts of the medium, which are stopped by that body, will
endeavour to spread themselves every way upon the superficies of it. And
this is manifest enough by experience, namely, by the spreading of water
poured out upon a pavement. But the reason of it may be this. Seeing the
sphere A (in fig. 3) is moved towards B, the medium also in which it is
moved will have the same motion. But because in this motion it falls
upon a body not liquid, as G, so that it cannot go on; and seeing the
small parts of the medium cannot go forwards, nor can they go directly
backwards against the force of the movent; it remains, therefore, that
they diffuse themselves upon the superficies of that body, as towards O
and P; which was to be proved.

[Sidenote: Circular motion about a fixed centre casteth off by the
           tangent  such things as lie upon the circumference & stick
           not to it.]

9. Compounded circular motion, in which all the parts of the moved body
do at once describe circumferences, some greater, others less, according
to the proportion of their several distances from the common centre,
carries about with it such bodies, as being not fluid, adhere to the
body so moved; and such as do not adhere, it casteth forwards in a
strait line which is a tangent to the point from which they are cast
off.

For let there be a circle whose radius is A B (in fig. 4); and let a
body be placed in the circumference in B, which if it be fixed there,
will necessarily be carried about with it, as is manifest of itself. But
whilst the motion proceeds, let us suppose that body to be unfixed in B.
I say, the body will continue its motion in the tangent B C. For let
both the radius A B and the sphere B be conceived to consist of hard
matter; and let us suppose the radius A B to be stricken in the point B
by some other body which falls upon it in the tangent D B. Now,
therefore, there will be a motion made by the concourse of two things,
the one, endeavour towards C in the strait line D B produced, in which
the body B would proceed, if it were not retained by the radius A B; the
other, the retention itself. But the retention alone causeth no
endeavour towards the centre; and, therefore, the retention being taken
away, which is done by the unfixing of B, there will remain but one
endeavour in B, namely, that in the tangent B C. Wherefore the motion of
the body B unfixed will proceed in the tangent B C; which was to be
proved.

By this demonstration it is manifest, that circular motion about an
unmoved axis shakes off and puts further from the centre of its motion
such things as touch, but do not stick fast to its superficies; and the
more, by how much the distance is greater from the poles of the circular
motion; and so much the more also, by how much the things, that are
shaken off, are less driven towards the centre by the fluid ambient, for
other causes.

[Sidenote: Such things as are moved with simple circular motion, beget
           simple circular motion.]

10. If in a fluid medium a spherical body be moved with simple circular
motion, and in the same medium there float another sphere whose matter
is not fluid, this sphere also shall be moved with simple circular
motion.

Let B C D (in fig. 5) be a circle, whose centre is A, and in whose
circumference there is a sphere so moved, that it describes with simple
motion the perimeter B C D. Let also E F G be another sphere of
consistent matter, whose semidiameter is E H, and centre H; and with the
radius A H let the circle H I be described. I say, the sphere E F G
will, by the motion of the body in B C D, be moved in the circumference
H I with simple motion.

For seeing the motion in B C D (by art. 4 of this chapter) makes all the
points of the fluid medium describe in the same time circular lines
equal to one another, the points E, H and G of the strait line E H G
will in the same time describe with equal radii equal circles. Let E B
be drawn equal and parallel to the strait line A H; and let A B be
connected, which will therefore be equal and parallel to E H; and
therefore also, if upon the centre B and radius B E the arch E K be
drawn equal to the arch H I, and the strait lines A I, B K and I K be
drawn, B K and A I will be equal; and they will also be parallel,
because the two arches E K and H I, that is, the two angles K B E and I
A H are equal; and, consequently, the strait lines A B and K I, which
connect them, will also be equal and parallel. Wherefore K I and E H are
parallel. Seeing, therefore, E and H are carried in the same time to K
and I, the whole strait line I K will be parallel to E H, from whence it
departed. And, therefore, seeing the sphere E F G is supposed to be of
consistent matter, so as all its points keep always the same situation,
it is necessary that every other strait line, taken in the same sphere,
be carried always parallel to the places in which it formerly was.
Wherefore the sphere E F G is moved with simple circular motion; which
was to be demonstrated.

[Sidenote: If that which is so moved have one side hard and the other
           side fluid, its motion will not be perfectly circular.]

11. If in a fluid medium, whose parts are stirred by a body moved with
simple motion, there float another body, which hath its superficies
either wholly hard, or wholly fluid, the parts of this body shall
approach the centre equally on all sides; that is to say, the motion of
the body shall be circular, and concentric with the motion of the
movent. But if it have one side hard, and the other side fluid, then
both those motions shall not have the same centre, nor shall the
floating body be moved in the circumference of a perfect circle.

Let a body be moved in the circumference of the circle K L M N (in fig
2.) whose centre is A. And let there be another body at I, whose
superficies is either all hard or all fluid. Also let the medium, in
which both these bodies are placed, be fluid. I say, the body at I will
be moved in the circle I B about the centre A. For this has been
demonstrated in the last article.

Wherefore let the superficies of the body at I be fluid on one side, and
hard on the other. And first, let the fluid side be towards the centre.
Seeing, therefore, the motion of the medium is such, as that its parts
do continually change their places, (as hath been shown in art 5); if
this change of place be considered in those parts of the medium which
are contiguous to the fluid superficies, it must needs be that the small
parts of that superficies enter into the places of the small parts of
the medium which are contiguous to them; and the like change of place
will be made with the next contiguous parts towards A. And if the fluid
parts of the body at I have any degree at all of tenacity (for there are
degrees of tenacity, as in the air and water) the whole fluid side will
be lifted up a little, but so much the less, as its parts have less
tenacity; whereas the hard part of the superficies, which is contiguous
to the fluid part, has no cause at all of elevation, that is to say, no
endeavour towards A.

Secondly, let the hard superficies of the body at I be towards A. By
reason, therefore, of the said change of place of the parts which are
contiguous to it, the hard superficies must, of necessity, seeing by
supposition there is no empty space, either come nearer to A, or else
its smallest parts must supply the contiguous places of the medium,
which otherwise would be empty. But this cannot be, by reason of the
supposed hardness; and, therefore, the other must needs be, namely, that
the body come nearer to A. Wherefore the body at I has greater endeavour
towards the centre A, when its hard side is next it, than when it is
averted from it. But the body in I, while it is moving in the
circumference of the circle I B, has sometimes one side, sometimes
another, turned towards the centre; and, therefore, it is sometimes
nearer, sometimes further off from the centre A. Wherefore the body at I
is not carried in the circumference of a perfect circle; which was to be
demonstrated.

------------------------------------------------------------------------

[Illustration:

  _Vol. 1. Lat. & Eng._
  C. XXI.
  _Fig. 1-5_
]

------------------------------------------------------------------------


                                -------


                             CHAPTER XXII.

                      OF OTHER VARIETY OF MOTION.

1. Endeavour and pressure how they differ.—2. Two kinds of mediums in
    which bodies are moved.—3. Propagation of motion, what it is.—4.
    What motion bodies have, when they press one another.—5. Fluid
    bodies, when they are pressed together, penetrate one another.—6.
    When one body presseth another and doth not penetrate it, the action
    of the pressing body is perpendicular to the superficies of the body
    pressed.—7. When a hard body, pressing another body, penetrates the
    same, it doth not penetrate it perpendicularly, unless it fall
    perpendicularly upon it.—8. Motion sometimes opposite to that of the
    movent.—9. In a full medium, motion is propagated to any
    distance.—10. Dilatation and contraction what they are.—11.
    Dilatation and contraction suppose mutation of the smallest parts in
    respect of their situation.—12. All traction is pulsion.—13. Such
    things as being pressed or bent restore themselves, have motion in
    their internal parts.—14. Though that which carrieth another be
    stopped, the body carried will proceed.—15, 16. The effects of
    percussion not to be compared with those of weight.—17, 18. Motion
    cannot begin first in the internal parts of a body.—19. Action and
    reaction proceed in the same line.—20. Habit, what it is.


[Sidenote: Endeavour and pressure how they differ.]

1. I have already (chapter XV. art. 2) defined endeavour to be motion
through some length, though not considered as length, but as a point.
Whether, therefore, there be resistance or no resistance, the endeavour
will still be the same. For simply to endeavour is to go. But when two
bodies, having opposite endeavours, press one another, then the
endeavour of either of them is that which we call pressure, and is
mutual when their pressures are opposite.

[Sidenote: Two kinds of mediums in which bodies are moved.]

2. Bodies moved, and also the mediums in which they are moved, are of
two kinds. For either they have their parts coherent in such manner, as
no part of the moved body will easily yield to the movent, except the
whole body yield also, and such are the things we call _hard_: or else
their parts, while the whole remains unmoved, will easily yield to the
movent, and these we call _fluid_ or _soft_ bodies. For the words
_fluid_, _soft_, _tough_, and _hard_, in the same manner as _great_ and
_little_, are used only comparatively; and are not different kinds, but
different degrees of quality.

[Sidenote: Propagation of motion, what it is.]

3. To _do_, and to _suffer_, is to move and to be moved; and nothing is
moved but by that which toucheth it and is also moved, as has been
formerly shown. And how great soever the distance be, we say the first
movent moveth the last moved body, but mediately; namely so, as that the
first moveth the second, the second the third, and so on, till the last
of all be touched. When therefore one body, having opposite endeavour to
another body, moveth the same, and that moveth a third, and so on, I
call that action _propagation of motion_.

[Sidenote: What motion bodies have when they press one another.]

4. When two fluid bodies, which are in a free and open space, press one
another, their parts will endeavour, or be moved, towards the sides; not
only those parts which are there where the mutual contact is, but all
the other parts. For in the first contact, the parts, which are pressed
by both the endeavouring bodies, have no place either forwards or
backwards in which they can be moved; and therefore they are pressed out
towards the sides. And this expressure, when the forces are equal, is in
a line perpendicular to the bodies pressing. But whensoever the foremost
parts of both the bodies are pressed, the hindermost also must be
pressed at the same time; for the motion of the hindermost parts cannot
in an instant be stopped by the resistance of the foremost parts, but
proceeds for some time; and therefore, seeing they must have some place
in which they may be moved, and that there is no place at all for them
forwards, it is necessary that they be moved into the places which are
towards the sides every way. And this effect follows of necessity, not
only in fluid, but in consistent and hard bodies, though it be not
always manifest to sense. For though from the compression of two stones
we cannot with our eyes discern any swelling outwards towards the sides,
as we perceive in two bodies of wax; yet we know well enough by reason,
that some tumour must needs be there, though it be but little.

[Sidenote: Fluid bodies, when they are pressed together, penetrate one
           another.]

5. But when the space is enclosed, and both the bodies be fluid, they
will, if they be pressed together, penetrate one another, though
differently, according to their different endeavours. For suppose a
hollow cylinder of hard matter, well stopped at both ends, but filled
first, below with some heavy fluid body, as quicksilver, and above with
water or air. If now the bottom of the cylinder be turned upwards, the
heaviest fluid body, which is now at the top, having the greatest
endeavour downwards, and being by the hard sides of the vessel hindered
from extending itself sideways, must of necessity either be received by
the lighter body, that it may sink through it, or else it must open a
passage through itself, by which the lighter body may ascend. For of the
two bodies, that, whose parts are most easily separated, will be the
first divided; which being done, it is not necessary that the parts of
the other suffer any separation at all. And therefore when two liquors,
which are enclosed in the same vessel, change their places, there is no
need that their smallest parts should be mingled with one another; for a
way being opened through one of them, the parts of the other need not be
separated.

Now if a fluid body, which is not enclosed, press a hard body, its
endeavour will indeed be towards the internal parts of that hard body;
but being excluded by the resistance of it, the parts of the fluid body
will be moved every way according to the superficies of the hard body,
and that equally, if the pressure be perpendicular; for when all the
parts of the cause are equal, the effects will be equal also. But if the
pressure be not perpendicular, then the angles of the incidence being
unequal, the expansion also will be unequal, namely, greater on that
side where the angle is greater, because that motion is most direct
which proceeds by the directest line.

[Sidenote: When one body presseth another and doth not penetrate it, the
           action of the pressing body is perpendicular to the
           superficies of the body pressed.]

6. If a body, pressing another body, do not penetrate it, it will
nevertheless give to the part it presseth an endeavour to yield, and
recede in a strait line perpendicular to its superficies in that point
in which it is pressed.

Let A B C D (in fig. 1) be a hard body, and let another body, falling
upon it in the strait line E A, with any inclination or without
inclination, press it in the point A. I say the body so pressing, and
not penetrating it, will give to the part A an endeavour to yield or
recede in a strait line perpendicular to the line A D.

For let A B be perpendicular to A D, and let B A be produced to F. If
therefore A F be coincident with A E, it is of itself manifest that the
motion in E A will make A to endeavour in the line A B. Let now E A be
oblique to A D, and from the point E let the strait line E C be drawn,
cutting A D at right angles in D, and let the rectangles A B C D and A D
E F be completed. I have shown (in the 8th article of chapter XVI) that
the body will be carried from E to A by the concourse of two uniform
motions, the one in E F and its parallels, the other in E D and its
parallels. But the motion in E F and its parallels, whereof D A is one,
contributes nothing to the body in A to make it endeavour or press
towards B; and therefore the whole endeavour, which the body hath in the
inclined line E A to pass or press the strait line A D, it hath it all
from the perpendicular motion or endeavour in F A. Wherefore the body E,
after it is in A, will have only that perpendicular endeavour which
proceeds from the motion in F A, that is, in A B; which was to be
proved.

[Sidenote: When a hard body, pressing another body, penetrates the same,
           it doth not penetrate it perpendicularly, unless it fall
           perpendicularly upon it.]

7. If a hard body falling upon or pressing another body penetrate the
same, its endeavour after its first penetration will be neither in the
inclined line produced, nor in the perpendicular, but sometimes betwixt
both, sometimes without them.

Let E A G (in the same fig. 1) be the inclined line produced; and first,
let the passage through the medium, in which E A is, be easier than the
passage through the medium in which A G is. As soon therefore as the
body is within the medium in which is A G, it will find greater
resistance to its motion in D A and its parallels, than it did whilst it
was above A D; and therefore below A D it will proceed with slower
motion in the parallels of D A, than above it. Wherefore the motion
which is compounded of the two motions in E F and E D will be slower
below A D than above it; and therefore also, the body will not proceed
from A in E A produced, but below it. Seeing, therefore, the endeavour
in A B is generated by the endeavour in F A; if to the endeavour in F A
there be added the endeavour in D A, which is not all taken away by the
immersion of the point A into the lower medium, the body will not
proceed from A in the perpendicular A B, but beyond it; namely, in some
strait line between A B and A G, as in the line A H.

Secondly, let the passage through the medium E A be less easy than that
through A G. The motion, therefore, which is made by the concourse of
the motions in E F and F B, is slower above A D than below it; and
consequently, the endeavour will not proceed from A in E A produced, but
beyond it, as in A I. Wherefore, if a hard body falling, &c.; which was
to be proved.

This divergency of the strait line A H from the strait line A G is that
which, the writers of optics commonly called _refraction_, which, when
the passage is easier in the first than in the second medium, is made by
diverging from the line of inclination towards the perpendicular; and
contrarily, when the passage is not so easy in the first medium, by
departing further from the perpendicular.

[Sidenote: Motion sometimes opposite to that of the movent.]

8. By the 6th theorem it is manifest, that the force of the movent may
be so placed, as that the body moved by it may proceed in a way almost
directly contrary to that of the movent, as we see in the motion of
ships.

For let A B (in fig. 2) represent a ship, whose length from the prow to
the poop is A B, and let the wind lie upon it in the strait parallel
lines C B, D E and F G; and let D E and F G be cut in E and G by a
strait line drawn from B perpendicular to A B; also let B E and E G be
equal, and the angle A B C any angle how small soever. Then between B C
and B A let the strait line B I be drawn; and let the sail be conceived
to be spread in the same line B I, and the wind to fall upon it in the
points L, M and B; from which points, perpendicular to B I, let B K, M Q
and L P be drawn. Lastly, let E N and G O be drawn perpendicular to B G,
and cutting B K in H and K; and let H N and K O be made equal to one
another, and severally equal to B A. I say, the ship B A, by the wind
falling upon it in C B, D E, F G, and other lines parallel to them, will
be carried forwards almost opposite to the wind, that is to say, in a
way almost contrary to the way of the movent.

For the wind that blows in the line C B will (as hath been shown in art.
6) give to the point B an endeavour to proceed in a strait line
perpendicular to the strait line B I, that is, in the strait line B K;
and to the points M and L an endeavour to proceed in the strait lines M
Q and L P, which are parallel to B K. Let now the measure of the time be
B G, which is divided in the middle in E; and let the point B be carried
to H in the time B E. In the same time, therefore, by the wind blowing
in D M and F L, and as many other lines as may be drawn parallel to
them, the whole ship will be applied to the strait line H N. Also at the
end of the second time E G, it will be applied to the strait line K O.
Wherefore the ship will always go forward; and the angle it makes with
the wind will be equal to the angle A B C, how small soever that angle
be; and the way it makes will in every time be equal to the strait line
E H. I say, thus it would be, if the ship might be moved with as great
celerity sideways from B A towards K O, as it may be moved forwards in
the line B A. But this is impossible, by reason of the resistance made
by the great quantity of water which presseth the side, much exceeding
the resistance made by the much smaller quantity which presseth the prow
of the ship; so that the way the ship makes sideways is scarce sensible;
and, therefore, the point B will proceed almost in the very line B A,
making with the wind the angle A B C, how acute soever; that is to say,
it will proceed almost in the strait line B C, that is, in a way almost
contrary to the way of the movent; which was to be demonstrated.

But the sail in B I must be so stretched as that there be left in it no
bosom at all; for otherwise the strait lines L P, M Q and B K will not
be perpendicular to the plane of the sail, but falling below P, Q and K,
will drive the ship backwards. But by making use of a small board for a
sail, a little waggon with wheels for the ship, and of a smooth pavement
for the sea, I have by experience found this to be so true, that I could
scarce oppose the board to the wind in any obliquity, though never so
small, but the waggon was carried forwards by it.

By the same 6th theorem it may be found, how much a stroke, which falls
obliquely, is weaker than a stroke falling perpendicularly, they being
like and equal in all other respects.

Let a stroke fall upon the wall A B obliquely, as for example, in the
strait line C A (in fig. 3.) Let C E be drawn parallel to A B, and D A
perpendicular to the same A B and equal to C A; and let both the
velocity and time of the motion in C A be equal to the velocity and time
of the motion in D A. I say, the stroke in C A will be weaker than that
in D A, in the proportion of E A to D A. For producing D A howsoever to
F, the endeavour of both the strokes will (by art. 6) proceed from A in
the perpendicular A F. But the stroke in C A is made by the concourse of
two motions in C E and E A, of which that in C E contributes nothing to
the stroke in A, because C E and B A are parallels; and, therefore, the
stroke in C A is made by the motion which is in E A only. But the
velocity or force of the perpendicular stroke in E A, to the velocity or
force of the stroke in D A, is as E A to D A. Wherefore the oblique
stroke in C A is weaker than the perpendicular stroke in D A, in the
proportion of E A to D A or C A; which was to be proved.

[Sidenote: In a full medium, motion is propagated to any distance.]

9. In a full medium, all endeavour proceeds as far as the medium itself
reacheth; that is to say, if the medium be infinite, the endeavour will
proceed infinitely.

For whatsoever endeavoureth is moved, and therefore whatsoever standeth
in its way it maketh it yield, at least a little, namely, so far as the
movent itself is moved forwards. But that which yieldeth is also moved,
and consequently maketh that to yield which is in its way, and so on
successively as long as the medium is full; that is to say, infinitely,
if the full medium be infinite; which was to be proved.

Now although endeavour thus perpetually propagated do not always appear
to the senses as motion, yet it appears as action, or as the efficient
cause of some mutation. For if there be placed before our eyes some very
little object, as for example, a small grain of sand, which at a certain
distance is visible; it is manifest that it may be removed to such a
distance as not to be any longer seen, though by its action it still
work upon the organs of sight, as is manifest from that which was last
proved, that all endeavour proceeds infinitely. Let it be conceived
therefore to be removed from our eyes to any distance how great soever,
and a sufficient number of other grains of sand of the same bigness
added to it; it is evident that the aggregate of all those sands will be
visible; and though none of them can be seen when it is single and
severed from the rest, yet the whole heap or hill which they make will
manifestly appear to the sight; which would be impossible, if some
action did not proceed from each several part of the whole heap.

[Sidenote: Dilatation & contraction what they are.]

10. Between the degrees of hard and soft are those things which we call
_tough_, tough being that which may be bent without being altered from
what it was; and the bending of a line is either the adduction or
diduction of the extreme parts, that is, a motion from straitness to
crookedness, or contrarily, whilst the line remains still the same it
was; for by drawing out the extreme points of a line to their greatest
distance, the line is made strait, which otherwise is crooked. So also
the bending of a superficies is the diduction or adduction of its
extreme lines, that is, their dilatation and contraction.

[Sidenote: Dilatation & contraction suppose mutation of the smallest
           parts in respect of their situation.]

11. _Dilatation_ and _contraction_, as also all _flexion_, supposes
necessarily that the internal parts of the body bowed do either come
nearer to the external parts, or go further from them. For though
_flexion_ be considered only in the length of a body, yet when that body
is bowed, the line which is made on one side will be convex, and the
line on the other side will be concave; of which the concave, being the
interior line, will, unless something be taken from it and added to the
convex line, be the more crooked, that is, the greater of the two. But
they are equal; and, therefore, in flexion there is an accession made
from the interior to the exterior parts; and, on the contrary, in
tension, from the exterior to the interior parts. And as for those
things which do not easily suffer such transposition of their parts,
they are called brittle; and the great force they require to make them
yield, makes them also with sudden motion to leap asunder, and break in
pieces.

[Sidenote: All traction is pulsion.]

12. Also motion is distinguished into _pulsion_ and _traction_. And
pulsion, as I have already defined it, is when that which is moved goes
before that which moveth it. But contrarily, in traction the movent goes
before that which is moved. Nevertheless, considering it with greater
attention, it seemeth to be the same with pulsion. For of two parts of a
hard body, when that which is foremost drives before it the medium in
which the motion is made, at the same time that which is thrust forwards
thrusteth the next, and this again the next, and so on successively. In
which action, if we suppose that there is no place void, it must needs
be, that by continual pulsion, namely, when that action has gone round,
the movent will be behind that part, which at the first seemed not to be
thrust forwards, but to be drawn; so that now the body, which was drawn,
goes before the body which gives it motion; and its motion is no longer
traction, but pulsion.

[Sidenote: Such things as being pressed or bent restore themselves, have
           motion in their internal parts.]

13. Such things as are removed from their places by forcible compression
or extension, and, as soon as the force is taken away, do presently
return and restore themselves to their former situation, have the
beginning of their restitution within themselves, namely, a certain
motion in their internal parts, which was there, when, before the taking
away of the force, they were compressed, or extended. For that
restitution is motion, and that which is at rest cannot be moved, but by
a moved and a contiguous movent. Nor doth the cause of their restitution
proceed from the taking away of the force by which they were compressed
or extended; for the removing of impediments hath not the efficacy of a
cause, as has been shown at the end of the 3rd article of chap. XV. The
cause therefore of their restitution is some motion either of the parts
of the ambient, or of the parts of the body compressed or extended. But
the parts of the ambient have no endeavour which contributes to their
compression or extension, nor to the setting of them at liberty, or
restitution. It remains therefore that from the time of their
compression or extension there be left some endeavour or motion, by
which, the impediment being removed, every part resumes its former
place; that is to say, the whole restores itself.

[Sidenote: Though that which carrieth another be stopped, the body
           carried will proceed.]

14. In the carriage of bodies, if that body, which carries another, hit
upon any obstacle, or be by any means suddenly stopped, and that which
is carried be not stopped, it will go on, till its motion be by some
external impediment taken away.

For I have demonstrated (chap. VIII, art. 19) that motion, unless it be
hindered by some external resistance, will be continued eternally with
the same celerity; and in the 7th article of chap. IX, that the action
of an external agent is of no effect without contact. When therefore
that, which carrieth another thing, is stopped, that stop doth not
presently take away the motion of that which is carried. It will
therefore proceed, till its motion be by little and little extinguished
by some external resistance: which was to be proved; though experience
alone had been sufficient to prove this.

In like manner, if that body which carrieth another be put from rest
into sudden motion, that which is carried will not be moved forwards
together with it, but will be left behind. For the contiguous part of
the body carried hath almost the same motion with the body which carries
it; and the remote parts will receive different velocities according to
their different distances from the body that carries them; namely, the
more remote the parts are, the less will be their degrees of velocity.
It is necessary, therefore, that the body, which is carried, be left
accordingly more or less behind. And this also is manifest by
experience, when at the starting forward of the horse the rider falleth
backwards.

[Sidenote: The effects of percussion not to be compared with those of
           weight.]

15. In _percussion_, therefore, when one hard body is in some small part
of it stricken by another with great force, it is not necessary that the
whole body should yield to the stroke with the same celerity with which
the stricken part yields. For the rest of the parts receive their motion
from the motion of the part stricken and yielding, which motion is less
propagated every way towards the sides, than it is directly forwards.
And hence it is, that sometimes very hard bodies, which being erected
can hardly be made to stand, are more easily broken than thrown down by
a violent stroke; when, nevertheless, if all their parts together were
by any weak motion thrust forwards, they would easily be cast down.

16. Though the difference between _trusion_ and _percussion_ consist
only in this, that in trusion the motion both of the movent and moved
body begin both together in their very contact; and in percussion the
striking body is first moved, and afterwards the body stricken; yet
their effects are so different, that it seems scarce possible to compare
their forces with one another. I say, any effect of percussion being
propounded, as for example, the stroke of a beetle of any weight
assigned, by which a pile of any given length is to be driven into earth
of any tenacity given, it seems to me very hard, if not impossible, to
define with what weight, or with what stroke, and in what time, the same
pile may be driven to a depth assigned into the same earth. The cause of
which difficulty is this, that the velocity of the percutient is to be
compared with the magnitude of the ponderant. Now velocity, seeing it is
computed by the length of space transmitted, is to be accounted but as
one dimension; but weight is as a solid thing, being measured by the
dimension of the whole body. And there is no comparison to be made of a
solid body with a length, that is, with a line.

[Sidenote: Motion cannot begin first in the internal parts of a body.]

17. If the internal parts of a body be at rest, or retain the same
situation with one another for any time how little soever, there cannot
in those parts be generated any new motion or endeavour, whereof the
efficient cause is not without the body of which they are parts. For if
any small part, which is comprehended within the superficies of the
whole body, be supposed to be now at rest, and by and by to be moved,
that part must of necessity receive its motion from some moved and
contiguous body. But by supposition, there is no such moved and
contiguous part within the body. Wherefore, if there be any endeavour or
motion or change of situation in the internal parts of that body, it
must needs arise from some efficient cause that is without the body
which contains them; which was to be proved.

18. In hard bodies, therefore, which are compressed or extended, if,
that which compresseth or extendeth them being taken away, they restore
themselves to their former place or situation, it must needs be that
that endeavour or motion of their internal parts, by which they were
able to recover their former places or situations, was not extinguished
when the force by which they were compressed or extended was taken away.
Therefore, when the lath of a cross-bow bent doth, as soon as it is at
liberty, restore itself, though to him, that judges by sense, both it
and all its parts seem to be at rest; yet he, that judging by reason
doth not account the taking away of impediment for an efficient cause,
nor conceives that without an efficient cause any thing can pass from
rest to motion, will conclude that the parts were already in motion
before they began to restore themselves.

[Sidenote: Action and reaction proceed in the same line.]

19. _Action_ and _reaction_ proceed in the same line, but from opposite
terms. For seeing reaction is nothing but endeavour in the patient to
restore itself to that situation from which it was forced by the agent;
the endeavour or motion both of the agent and patient or reagent will be
propagated between the same terms; yet so, as that in action the term,
_from which_, is in reaction the term _to which_. And seeing all action
proceeds in this manner, not only between the opposite terms of the
whole line in which it is propagated, but also in all the parts of that
line, the terms _from which_ and _to which_, both of the action and
reaction, will be in the same line. Wherefore action and reaction
proceed in the same line, &c.

[Sidenote: Habit, what it is.]

20. To what has been said of motion, I will add what I have to say
concerning habit. _Habit_, therefore, is a generation of motion, not of
motion simply, but an easy conducting of the moved body in a certain and
designed way. And seeing it is attained by the weakening of such
endeavours as divert its motion, therefore such endeavours are to be
weakened by little and little. But this cannot be done but by the long
continuance of action, or by actions often repeated; and therefore
custom begets that facility, which is commonly and rightly called
_habit_; and it may be defined thus: HABIT _is motion made more easy and
ready by custom; that is to say, by perpetual endeavour, or by iterated
endeavours in a way differing from that in which the motion proceeded
from the beginning, and opposing such endeavours as resist_. And to make
this more perspicuous by example, we may observe, that when one that has
no skill in music first puts his hand to an instrument, he cannot after
the first stroke carry his hand to the place where he would make the
second stroke, without taking it back by a new endeavour, and, as it
were beginning again, pass from the first to the second. Nor will he be
able to go on to the third place without another new endeavour; but he
will be forced to draw back his hand again, and so successively, by
renewing his endeavour at every stroke; till at the last, by doing this
often, and by compounding many interrupted motions or endeavours into
one equal endeavour, he be able to make his hand go readily on from
stroke to stroke in that order and way which was at the first designed.
Nor are habits to be observed in living creatures only, but also in
bodies inanimate. For we find that when the lath of a cross-bow is
strongly bent, and would if the impediment were removed return again
with great force; if it remain a long time bent, it will get such a
habit, that when it is loosed and left to its own freedom, it will not
only not restore itself, but will require as much force for the bringing
of it back to its first posture, as it did for the bending of it at the
first.

------------------------------------------------------------------------

[Illustration:

  _Vol. 1. Lat. & Eng._
  C. XXII.
  _Fig. 1-3_
]

------------------------------------------------------------------------


                                -------


                              CHAP. XXIII.

                  OF THE CENTRE OF EQUIPONDERATION; OF
                  BODIES PRESSING DOWNWARDS IN STRAIT
                            PARALLEL LINES.

1. Definitions and suppositions.—2. Two planes of equiponderation are
    not parallel.—3. The centre of equiponderation is in every plane of
    equiponderation.—4. The moments of equal ponderants are to one
    another as their distances from the centre of the scale.—5, 6. The
    moments of unequal ponderants have their proportion to one another
    compounded of the proportions of their weights and distances from
    the centre of the scale.—7. If two ponderants have their weights and
    distances from the centre of the scale in reciprocal proportion,
    they are equally poised; and contrarily.—8. If the parts of any
    ponderant press the beams of the scale every where equally, all the
    parts cut off, reckoned from the centre of the scale, will have
    their moments in the same proportion with that of the parts of a
    triangle cut off from the vertex by strait lines parallel to the
    base.—9. The diameter of equiponderation of figures, which are
    deficient according to commensurable proportions of their altitudes
    and bases, divides the axis, so that the part taken next the vertex
    is to the other part of the complete figure to the deficient
    figure.—10. The diameter of equiponderation of the complement of the
    half of any of the said deficient figures, divides that line which
    is drawn through the vertex parallel to the base, so that the part
    next the vertex is to the other part as the complete figure to the
    complement.—11. The centre of equiponderation of the half of any of
    the deficient figures in the first row of the table of art. 3, chap.
    XVII, may be found out by the numbers of the second row.—12. The
    centre of equiponderation of the half of any of the figures of the
    second row of the same table, may be found out by the numbers of the
    fourth row.—13. The centre of equiponderation of the half of any of
    the figures in the same table being known, the centre of the excess
    of the same figure above a triangle of the same altitude and base is
    also known.—14. The centre of equiponderation of a solid sector is
    in the axis so divided, that the part next the vertex be to the
    whole axis, wanting half the axis of the portion of the sphere, as 3
    to 4.

                              DEFINITIONS.

[Sidenote: Definitions.]

I. A _scale_ is a strait line, whose middle point is immovable, all the
rest of its points being at liberty; and that part of the scale, which
reaches from the centre to either of the weights, is called the _beam_.

II. _Equiponderation_ is when the endeavour of one body, which presses
one of the beams, resists the endeavour of another body pressing the
other beam, so that neither of them is moved; and the bodies, when
neither of them is moved, are said to be _equally poised_.

III. _Weight_ is the aggregate of all the endeavours, by which all the
points of that body, which presses the beam, tend downwards in lines
parallel to one another; and the body which presses is called the
_ponderant_.

IV. _Moment_ is the power which the ponderant has to move the beam, by
reason of a determined situation.

V. _The plane of equiponderation_ is that by which the ponderant is so
divided, that the moments on both sides remain equal.

VI. _The diameter of equiponderation_ is the common section of the two
planes of equiponderation, and is in the strait line by which the weight
is hanged.

VII. _The centre of equiponderation_ is the common point of the two
diameters of equiponderation.

                             SUPPOSITIONS.

[Sidenote: Suppositions.]

I. When two bodies are equally poised, if weight be added to one of them
and not to the other, their equiponderation ceases.

II. When two ponderants of equal magnitude, and of the same species or
matter, press the beam on both sides at equal distances from the centre
of the scale, their moments are equal. Also when two bodies endeavour at
equal distances from the centre of the scale, if they be of equal
magnitude and of the same species, their moments are equal.

[Sidenote: Two planes
           of equiponderation
           are
           not parallel.]

2. No two planes of equiponderation are parallel.

Let A B C D (in fig. 1) be any ponderant whatsoever; and in it let E F
be a plane of equiponderation; parallel to which, let any other plane be
drawn, as G H. I say, G H is not a plane of equiponderation. For seeing
the parts A E F D and E B C F of the ponderant A B C D are equally
poised; and the weight E G H F is added to the part A E F D, and nothing
is added to the part E B C F, but the weight E G H F is taken from it;
therefore, by the first supposition, the parts A G H D and G B C H will
not be equally poised; and consequently G H is not a plane of
equiponderation. Wherefore, no two planes of equiponderation are
parallel; which was to be proved.

[Sidenote: The centre of equiponderation is in every plane of
           equiponderation.]

3. The centre of equiponderation is in every plane of equiponderation.

For if another plane of equiponderation be taken, it will not, by the
last article, be parallel to the former plane; and therefore both those
planes will cut one another. Now that section (by the 6th definition) is
the diameter of equiponderation. Again, if another diameter of
equiponderation be taken, it will cut that former diameter; and in that
section (by the 7th definition) is the centre of equiponderation.
Wherefore the centre of equiponderation is in that diameter which lies
in the said plane of equiponderation.

[Sidenote: The moments
           of equal ponderants
           are to one
           another as their
           distances from
           the centre of
           the scale.]

4. The moment of any ponderant applied to one point of the beam, to the
moment of the same or an equal ponderant applied to any other point of
the beam, is as the distance of the former point from the centre of the
scale, to the distance of the latter point from the same centre. Or
thus, those moments are to one another, as the arches of circles which
are made upon the centre of the scale through those points, in the same
time. Or lastly thus, they are as the parallel bases of two triangles,
which have a common angle at the centre of the scale.

Let A (in fig. 2) be the centre of the scale; and let the equal
ponderants D and E press the beam A B in the points B and C; also let
the strait lines B D and C E be diameters of equiponderation; and the
points D and E in the ponderants D and E be their centres of
equiponderation. Let A G F be drawn howsoever, cutting D B produced in
F, and E C in G; and lastly, upon the common centre A, let the two
arches B H and C I be described, cutting A G F in H and I. I say, the
moment of the ponderant D to the moment of the ponderant E is as A B to
A C, or as B H to C I, or as B F to C G. For the effect of the ponderant
D, in the point B, is circular motion in the arch B H; and the effect of
the ponderant E, in the point C, circular motion in the arch C I; and by
reason of the equality of the ponderants D and E, these motions are to
one another as the quicknesses or velocities with which the points B and
C describe the arches B H and C I, that is, as the arches themselves B H
and C I, or as the strait parallels B F and C G, or as the parts of the
beam A B and A C; for A B. A C:: B F. C G:: B H. C I. are proportionals;
and therefore the effects, that is, by the 4th definition, the moments
of the equal ponderants applied to several points of the beam, are to
one another as A B and A C; or as the distances of those points from the
centre of the scale; or as the parallel bases of the triangles which
have a common angle at A; or as the concentric arches B H and C I; which
was to be demonstrated.

[Sidenote: The moments of unequal ponderants have their proportion to
           one another compounded of the proportions of their weights
           and distances from the centre of the scale.]

5. Unequal ponderants, when they are applied to several points of the
beam, and hang at liberty, that is, so as the line by which they hang be
the diameter of equiponderation, whatsoever be the figure of the
ponderant, have their moments to one another in proportion compounded of
the proportions of their distances from the centre of the scale, and of
their weights.

Let A (in fig. 3) be the centre of the scale, and A B the beam; to which
let the two ponderants C and D be applied at the points B and E. I say,
the proportion of the moment of the ponderant C to the moment of the
ponderant D, is compounded of the proportions of A B to A E, and of the
weight C to the weight D; or, if C and D be of the same species, of the
magnitude C to the magnitude D.

Let either of them, as C, be supposed to be bigger than the other, D.
If, therefore, by the addition of F, F and D together be as one body
equal to C, the moment of C to the moment of F + D will be (by the last
article) as B G is to E H. Now as F + D is to D, so let E H be to
another E I; and the moment of F + D, that is of C, to the moment of D,
will be as B G to E I. But the proportion of B G to E I is compounded of
the proportions of B G to E H, that is, of A B to A E, and of E H to E
I, that is, of the weight C to the weight D. Wherefore unequal
ponderants, when they are applied, &c. Which was to be proved.

6. The same figure remaining, if I K be drawn parallel to the beam A B,
and cutting A G in K; and K L be drawn parallel to B G, cutting A B in
L, the distances A B and A L from the centre will be proportional to the
moments of C and D. For the moment of C is B G, and the moment of D is E
I, to which K L is equal. But as the distance A B from the centre is to
the distance A L from the centre, so is B G, the moment of the ponderant
C, to L K, or E I the moment of the ponderant D.

[Sidenote: If two ponderants have their weights and distances from the
           centre of the scale in reciprocal proportion, they are
           equally poised; and contrarily.]

7. If two ponderants have their weights and distances from the centre in
reciprocal proportion, and the centre of the scale be between the points
to which the ponderants are applied, they will be equally poised. And
contrarily, if they be equally poised, their weights and distances from
the centre of the scale will be in reciprocal proportion.

Let the centre of the scale (in the same third figure) be A, the beam A
B; and let any ponderant C, having B G for its moment, be applied to the
point B; also let any other ponderant D, whose moment is E I, be applied
to the point E. Through the point I let I K be drawn parallel to the
beam A B, cutting A G in K; also let K L be drawn parallel to B G, K L
will then be the moment of the ponderant D; and by the last article, it
will be as B G, the moment of the ponderant C in the point B, to L K the
moment of the ponderant D in the point E, so A B to A L. On the other
side of the centre of the scale, let A N be taken equal to A L; and to
the point N let there be applied the ponderant O, having to the
ponderant C the proportion of A B to A N. I say, the ponderants in B and
N will be equally poised. For the proportion of the moment of the
ponderant O, in the point N, to the moment of the ponderant C in the
point B, is by the 5th article, compounded of the proportions of the
weight O to the weight C, and of the distance from the centre of the
scale A N or A L to the distance from the centre of the scale A B. But
seeing we have supposed, that the distance A B to the distance A N is in
reciprocal proportion of the weight O to the weight C, the proportion of
the moment of the ponderant O, in the point N, to the moment of the
ponderant C, in the point B, will be compounded of the proportions of A
B to A N, and of A N to A B. Wherefore, setting in order A B, A N, A B,
the moment of O to the moment of C will be as the first to the last,
that is, as A B to A B. Their moments therefore are equal; and
consequently the plane which passes through A will (by the fifth
definition) be a plane of equiponderation. Wherefore they will be
equally poised; as was to be proved.

Now the converse of this is manifest. For if there be equiponderation
and the proportion of the weights and distances be not reciprocal, then
both the weights will always have the same moments, although one of them
have more weight added to it or its distance changed.

Coroll. When ponderants are of the same species, and their moments be
equal; their magnitudes and distances from the centre of the scale will
be reciprocally proportional. For in homogeneous bodies, it is as weight
to weight, so magnitude to magnitude.

[Sidenote: If the parts of any ponderant press the beams of the scale
           every where equally, all the parts cut off, reckoned from
           centre of the scale, will have their moments in the same
           proportion with that of the parts of a triangle cut off from
           the vertex by strait lines parallel to the base.]

8. If to the whole length of the beam there be applied a parallelogram,
or a parallelopipedum, or a prisma, or a cylinder, or the superficies of
a cylinder, or of a sphere, or of any portion of a sphere or prisma; the
parts of any of them cut off with planes parallel to the base will have
their moments in the same proportion with the parts of a triangle, which
has its vertex in the centre of the scale, and for one of its sides the
beam itself, which parts are cut off by planes parallel to the base.

First, let the rectangled parallelogram A B C D (in figure 4) be applied
to the whole length of the beam A B; and producing C B howsoever to E,
let the triangle A B E be described. Let now any part of the
parallelogram, as A F, be cut off by the plane F G, parallel to the base
C B; and let F G be produced to A E in the point H. I say, the moment of
the whole A B C D to the moment of its part A F, is as the triangle A B
E to the triangle A G H, that is, in proportion duplicate to that of the
distances from the centre of the scale.

For, the parallelogram A B C D being divided into equal parts, infinite
in number, by strait lines drawn parallel to the base; and supposing the
moment of the strait line C B to be B E, the moment of the strait line F
G will (by the 7th article) be G H; and the moments of all the strait
lines of that parallelogram will be so many strait lines in the triangle
A B E drawn parallel to the base B E; all which parallels together taken
are the moment of the whole parallelogram A B C D; and the same
parallels do also constitute the superficies of the triangle A B E.
Wherefore the moment of the parallelogram A B C D is the triangle A B E;
and for the same reason, the moment of the parallelogram A F is the
triangle A G H; and therefore the moment of the whole parallelogram to
the moment of a parallelogram which is part of the same, is as the
triangle A B E to the triangle A G H, or in proportion duplicate to that
of the beams to which they are applied. And what is here demonstrated in
the case of a parallelogram may be understood to serve for that of a
cylinder, and of a prisma, and their superficies; as also for the
superficies of a sphere, of an hemisphere, or any portion of a sphere.
For the parts of the superficies of a sphere have the same proportion
with that of the parts of the axis cut off by the same parallels, by
which the parts of the superficies are cut off, as Archimedes has
demonstrated; and therefore when the parts of any of these figures are
equal and at equal distances from the centre of the scale, their moments
also are equal, in the same manner as they are in parallelograms.

Secondly, let the parallelogram A K I B not be rectangled; the strait
line I B will nevertheless press the point B perpendicularly in the
strait line B E; and the strait line L G will press the point G
perpendicularly in the strait line G H; and all the rest of the strait
lines which are parallel to I B will do the like. Whatsoever therefore
the moment be which is assigned to the strait line I B, as here, for
example, it is supposed to be B E, if A E be drawn, the moment of the
whole parallelogram A I will be the triangle A B E; and the moment of
the part A L will be the triangle A G H. Wherefore the moment of any
ponderant, which has its sides equally applied to the beam, whether they
be applied perpendicularly or obliquely, will be always to the moment of
a part of the same in such proportion as the whole triangle has to a
part of the same cut off by a plane which is parallel to the base.

[Sidenote: The diameter of equiponderation of figures which are
           deficient according to commensurable proportions of their
           altitudes and bases, divides the axis, so that the part taken
           next the vertex is to the other part as the complete figure
           to the deficient figure.]

9. The centre of equiponderation of any figure, which is deficient
according to commensurable proportions of the altitude and base
diminished, and whose complete figure is either a parallelogram or a
cylinder, or a parallelopipedum, divides the axis, so, that the part
next the vertex, to the other part, is as the complete figure to the
deficient figure.

For let C I A P E (in fig. 5) be a deficient figure, whose axis is A B,
and whose complete figure is C D F E; and let the axis A B be so divided
in Z, that A Z be to Z B as C D F E is to C I A P E. I say, the centre
of equiponderation of the figure C I A P E will be in the point Z.

First, that the centre of equiponderation of the figure C I A P E is
somewhere in the axis A B is manifest of itself; and therefore A B is a
diameter of equiponderation. Let A E be drawn, and let B E be put for
the moment of the strait line C E; the triangle A B E will therefore (by
the third article) be the moment of the complete figure C D F E. Let the
axis A B be equally divided in L, and let G L H be drawn parallel and
equal to the strait line C E, cutting the crooked line C I A P E in I
and P, and the strait lines A C and A E in K and M. Moreover, let Z O be
drawn parallel to the same C E; and let it be, as L G to L I, so L M to
another, L N; and let the same be done in all the rest of the strait
lines possible, parallel to the base; and through all the points N, let
the line A N E be drawn; the three-sided figure A N E B will therefore
be the moment of the figure C I A P E. Now the triangle A B E is (by the
9th article of chapter XVII) to the three-sided figure A N E B, as A B C
D + A I C B is to A I C B twice taken, that is, as C D F E + C I A P E
is to C I A P E twice taken. But as C I A P E is to C D F E, that is, as
the weight of the deficient figure is to the weight of the complete
figure, so is C I A P E twice taken to C D F E twice taken. Wherefore,
setting in order C D F E + C I A P E. 2 C I A P E. 2 C D F E; the
proportion of C D F E + C I A P E to C D F E twice taken will be
compounded of the proportion of C D F E + C I A P E to C I A P E twice
taken, that is, of the proportion of the triangle A B E to the
three-sided figure A N E B, that is, of the moment of the complete
figure to the moment of the deficient figure, and of the proportion of C
I A P E twice taken to C D F E twice taken, that is, to the proportion
reciprocally taken of the weight of the deficient figure to the weight
of the complete figure.

Again, seeing by supposition A Z. Z B:: C D F E. C I A P E are
proportionals; A B. A Z:: C D F E + C I A P E. C D F E will also, by
compounding, be proportionals. And seeing A L is the half of A B, A L. A
Z:: C D F E + C I A P E. 2 C D F E will also be proportionals. But the
proportion of C D F E + C I A P E to 2 C D F E is compounded, as was but
now shown, of the proportions of moment to moment, &c., and therefore
the proportion of A L to A Z is compounded of the proportion of the
moment of the complete figure C D F E to the moment of the deficient
figure C I A P E, and of the proportion of the weight of the deficient
figure C I A P E to the weight of the complete figure C D F E; but the
proportion of A L to A Z is compounded of the proportions of A L to B Z
and of B Z to A Z. Now the proportion of B Z to A Z is the proportion of
the weights reciprocally taken, that is to say, of the weight C I A P E
to the weight C D F E. Therefore the remaining proportion of A L to B Z,
that is, of L B to B Z, is the proportion of the moment of the weight C
D F E to the moment of the weight C I A P E. But the proportion of A L
to B Z is compounded of the proportions of A L to A Z and of A Z to Z B;
of which proportions that of A Z to Z B is the proportion of the weight
C D F E to the weight C I A P E. Wherefore (by art. 5 of this chapter)
the remaining proportion of A L to A Z is the proportion of the
distances of the points Z and L from the centre of the scale, which is
A. And, therefore, (by art. 6) the weight C I A P E shall hang from O in
the strait line O Z. So that O Z is one diameter of equiponderation of
the weight C I A P E. But the strait line A B is the other diameter of
equiponderation of the same weight C I A P E. Wherefore (by the 7th
definition) the point Z is the centre of the same equiponderation; which
point, by construction, divides the axis so, that the part A Z, which is
the part next the vertex, is to the other part Z B, as the complete
figure C D F E is to the deficient figure C I A P E; which is that which
was to be demonstrated.

Coroll. I. The centre of equiponderation of any of those plane
three-sided figures, which are compared with their complete figures in
the table of art. 3, chap. XVII, is to be found in the same table, by
taking the denominator of the fraction for the part of the axis cut off
next the vertex, and the numerator for the other part next the base. For
example, if it be required to find the centre of equiponderation of the
second three-sided figure of four means, there is in the concourse of
the second column with the row of three-sided figures of four means this
fraction 5⁄7, which signifies that that figure is to its parallelogram
or complete figure as 5⁄7 to unity, that is, as 5⁄7 to 7⁄7, or as 5 to
7; and, therefore the centre of equiponderation of that figure divides
the axis, so that the part next the vertex is to the other part as 7 to
5.

Coroll. II. The centre of equiponderation of any of the solids of those
figures, which are contained in the table of art. 7 of the same chap.
XVII, is exhibited in the same table. For example, if the centre of
equiponderation of a cone be sought for, the cone will be found to be
1⁄3 of its cylinder; and, therefore, the centre of its equiponderation
will so divide the axis, that the part next the vertex to the other part
will be as 3 to 1. Also the solid of a three-sided figure of one mean,
that is, a parabolical solid, seeing it is 2⁄4, that is ½ of its
cylinder, will have its centre of equiponderation in that point, which
divides the axis, so that the part towards the vertex be double to the
part towards the base.

[Sidenote: The diameter of equiponderation of the complement of the half
           of any of the said deficient figures, divides that line which
           is drawn through the vertex parallel to the base, so that the
           part next the vertex is to the other part as the complete
           figure to the complement.]

10. The diameter of equiponderation of the complement of the half of any
of those figures which are contained in the table of art. 3, chap. XVII,
divides that line which is drawn through the vertex parallel and equal
to the base, so that the part next the vertex will be to the other part,
as the complete figure to the complement.

For let A I C B (in the same fig. 5) be the half of a parabola, or of
any other of those three-sided figures which are in the table of art. 3,
chap. XVII, whose axis is A B, and base B C, having A D drawn from the
vertex, equal and parallel to the base B C, and whose complete figure is
the parallelogram A B C D. Let I Q be drawn at any distance from the
side C D, but parallel to it; and let A D be the altitude of the
complement A I C D, and Q I a line ordinately applied in it. Wherefore
the altitude A L in the deficient figure A I C B is equal to Q I the
line ordinately applied in its complement; and contrarily, L I the line
ordinately applied in the figure A I C B is equal to the altitude A Q in
its complement; and so in all the rest of the ordinate lines and
altitudes the mutation is such, that that line, which is ordinately
applied in the figure, is the altitude of its complement. And,
therefore, the proportion of the altitudes decreasing to that of the
ordinate lines decreasing, being multiplicate according to any number in
the deficient figure, is submultiplicate according to the same number in
its complement. For example, if A I C B be a parabola, seeing the
proportion of A B to A L is duplicate to that of B C to L I, the
proportion of AD to A Q in the complement A I C D, which is the same
with that of B C to L I, will be subduplicate to that of C D to Q I,
which is the same with that of A B to A L; and consequently, in a
parabola, the complement will be to the parallelogram as 1 to 3; in a
three-sided figure of two means, as 1 to 4; in a three-sided figure of
three means, as 1 to 5, &c. But all the ordinate lines together in A I C
D are its moment; and all the ordinate lines in A I C B are its moment.
Wherefore the moments of the complements of the halves of deficient
figures in the table of art. 3 of chap. XVII, being compared, are as the
deficient figures themselves; and, therefore, the diameter of
equiponderation will divide the strait line A D in such proportion, that
the part next the vertex be to the other part, as the complete figure A
B C D is to the complement A I C D.

Coroll. The diameter of equiponderation of these halves may be found by
the table of art. 3 of chap. XVII, in this manner. Let there be
propounded any deficient figure, namely, the second three-sided figure
of two means. This figure is to the complete figure as 3⁄5 to 1, that is
3 to 5. Wherefore the complement to the same complete figure is as 2 to
5; and, therefore, the diameter of equiponderation of this complement
will cut the strait line drawn from the vertex parallel to the base, so
that the part next the vertex will be to the other part as 5 to 2. And,
in like manner, any other of the said three-sided figures being
propounded, if the numerator of its fraction found out in the table be
taken from the denominator, the strait line drawn from the vertex is to
be divided, so that the part next the vertex be to the other part, as
the denominator is to the remainder which that subtraction leaves.

[Sidenote: The centre of equiponderation of the half of any of the
           deficient figures in the first row of the table of art. 3,
           chapter xvii, may be found out by the numbers of the second
           row.]

11. The centre of equiponderation of the half of any of those
crooked-lined figures, which are in the first row of the table of art. 3
of chap. XVII, is in that strait line which, being parallel to the axis,
divides the base according to the numbers of the fraction next below it
in the second row, so that the numerator be answerable to that part
which is towards the axis.

For example, let the first figure of three means be taken, whose half is
A B C D (in fig. 6), and let the rectangle A B E D be completed. The
complement therefore will be B C D E. And seeing A B E D is to the
figure A B C D (by the table) as 5 to 4, the same A B E D will be to the
complement B C D E as 5 to 1. Wherefore, if F G be drawn parallel to the
base D A, cutting the axis so that A G be to G B as 4 to 5, the centre
of equiponderation of the figure A B C D will, by the precedent article,
be somewhere in the same F G. Again, seeing, by the same article, the
complete figure A B E D, is to the complement B C D E as 5 to 1,
therefore if B E and A D be divided in I and H as 5 to 1 the centre of
equiponderation of the complement B C D E will be somewhere in the
strait line which connects H and I. Let now the strait line L K be drawn
through M the centre of the complete figure, parallel to the base; and
the strait line N O through the same centre M, perpendicular to it; and
let the strait lines L K and F G cut the strait line H I in P and Q. Let
P R be taken quadruple to P Q; and let R M be drawn and produced to F G
in S. R M therefore will be to M S as 4 to 1, that is, as the figure A B
C D to its complement B C D E. Wherefore, seeing M is the centre of the
complete figure A B E D, and the distances of R and S from the centre M
be in proportion reciprocal to that of the weight of the complement B C
D E to the weight of the figure A B C D, R and S will either be the
centres of equiponderation of their own figures, or those centres will
be in some other points of the diameters of equiponderation H I and F G.
But this last is impossible. For no other strait line can be drawn
through the point M terminating in the strait lines H I and F G, and
retaining the proportion of M R to M S, that is, of the figure A B C D
to its complement B C D E. The centre, therefore, of equiponderation of
the figure A B C D is in the point S. Now, seeing P M hath the same
proportion to Q S which R P hath to R Q, Q S will be 5 of those parts of
which P M is four, that is, of which I N is four. But I N or P M is 2 of
those parts of which E B or F G is 6; and, therefore, if it be as 4 to
5, so 2 to a fourth, that fourth will be 2½. Wherefore Q S is 2½ of
those parts of which F G is 6. But F Q is 1; and, therefore, F S is 3½.
Wherefore the remaining part G S is 2½. So that F G is so divided in S,
that the part towards the axis is in proportion to the other part, as 2½
to 3½, that is as 5 to 7; which answereth to the fraction 5⁄7 in the
second row, next under the fraction 4⁄5 in the first row. Wherefore
drawing S T parallel to the axis, the base will be divided in like
manner.

By this method it is manifest, that the base of a semiparabola will be
divided into 3 and 5; and the base of the first three-sided figure of
two means, into 4 and 6; and of the first three-sided figure of four
means, into 6 and 8. The fractions, therefore, of the second row denote
the proportions, into which the bases of the figures of the first row
are divided by the diameters of equiponderation. But the first row
begins one place higher than the second row.

[Sidenote: The centre of equiponderation of the half of any of the
           figures of the second row of the same table may be found out
           by the numbers of the fourth row.]

12. The centre of equiponderation of the half of any of the figures in
the second row of the same table of art. 3, chap. XVII, is in a strait
line parallel to the axis, and dividing the base according to the
numbers of the fraction in the fourth row, two places lower, so as that
the numerator be answerable to that part which is next the axis.

Let the half of the second three-sided figure of two means be taken; and
let it be A B C D (in fig. 7); whose complement is B C D E, and the
rectangle completed A B E D. Let this rectangle be divided by the two
strait lines L K and N O, cutting one another in the centre M at right
angles; and because A B E D is to A B C D as 5 to 3, let A B be divided
in G, so that A G to B G be as 3 to 5; and let F G be drawn parallel to
the base. Also because A B E D is (by art. 9) to B C D E as 5 to 2, let
B E be divided in the point I, so that B I be to I E as 5 to 2; and let
I H be drawn parallel to the axis, cutting L K and F G in P and Q. Let
now P R be so taken, that it be to P Q as 3 to 2, and let R M be drawn
and produced to F G in S. Seeing, therefore, R P is to P Q, that is, R M
to M S, as A B C D is to its complement B C D E, and the centres of
equiponderation of A B C D and B C D E are in the strait lines F G and H
I, and the centre of equiponderation of them both together in the point
M; R will be the centre of the complement B C D E, and S the centre of
the figure A B C D. And seeing P M, that is I N, is to Q S, as R P is to
R Q; and I N or P M is 3 of those parts, of which B E, that is F G, is
14; therefore Q S is 5 of the same parts; and E I, that is F G, 4; and F
S, 9; and G S, 5. Wherefore the strait line S T being drawn parallel to
the axis, will divide the base A D into 5 and 9. But the fraction 5⁄9 is
found in the fourth row of the table, two places below the fraction ⅗ in
the second row.

By the same method, if in the same second row there be taken the second
three-sided figure of three means, the centre of equiponderation of the
half of it will be found to be in a strait line parallel to the axis,
dividing the base according to the numbers of the fraction 6⁄10, two
places below in the fourth row. And the same way serves for all the rest
of the figures in the second row. In like manner, the centre of
equiponderation of the third three-sided figure of three means will be
found to be in a strait line parallel to the axis, dividing the base, so
that the part next the axis be to the other part as 7 to 13, &c.

Coroll. The centres of equiponderation of the halves of the said figures
are known, seeing they are in the intersection of the strait lines S T
and F G, which are both known.

[Sidenote: The centre of equiponderation of the half of any of the
           figures in the same table being known, the centre of the
           excess of the same figure above a triangle of the same
           altitude and base is also known.]

13. The centre of equiponderation of the half of any of the figures,
which (in the table of art. 3, chap. XVII) are compared with their
parallelograms, being known; the centre of equiponderation of the excess
of the same figure above its triangle is also known.

For example, let the semiparabola A B C D (in fig. 8) be taken, whose
axis is A B; whose complete figure is A B E D; and whose excess above
its triangle is B C D B. Its centre of equiponderation may be found out
in this manner. Let F G be drawn parallel to the base, so that A F be a
third part of the axis; and let H I be drawn parallel to the axis, so
that A H be a third part of the base. This being done, the centre of
equiponderation of the triangle A B D will be I. Again, let K L be drawn
parallel to the base, so that A K be to A B as 2 to 5; and M N parallel
to the axis, so that A M be to A D as 3 to 8; and let M N terminate in
the strait line K L. The centre, therefore, of equiponderation of the
parabola A B C D is N; and therefore we have the centres of
equiponderation of the semiparabola A B C D, and of its part the
triangle A B D. That we may now find the centre of equiponderation of
the remaining part B C D B, let I N be drawn and produced to O, so that
N O be triple to I N; and O will be the centre sought for. For seeing
the weight of A B D to the weight of B C D B is in proportion reciprocal
to that of the strait line N O to the strait line I N; and N is the
centre of the whole, and I the centre of the triangle A B D; O will be
the centre of the remaining part, namely, of the figure B D C B; which
was to be found.

Coroll. The centre of equiponderation of the figure B D C B is in the
concourse of two strait lines, whereof one is parallel to the base, and
divides the axis, so that the part next the base be ⅗ or 9⁄15 of the
whole axis; the other is parallel to the axis, and so divides the base,
that the part towards the axis be ½, or 12⁄24 of the whole base. For
drawing O P parallel to the base, it will be as I N to N O, so F K to K
P, that is, so 1 to 3, or 5 to 15. But A F is 5⁄15, or ⅓ of the whole A
B; and A K is 6⁄15, or ⅖; and F K 1⁄15; and KP 3⁄15; and therefore A P
is 9⁄15 of the axis A B. Also A H is ⅓, or 8⁄24; and A M ⅜, or 9⁄24 of
the whole base; and therefore O Q being drawn parallel to the axis, M Q,
which is triple to H M, will be 3⁄24. Wherefore A Q is 12⁄24, or ½ of
the base A D.

The excesses of the rest of the three-sided figures in the first row of
the table of art. 3, chap. XVII, have their centres of equiponderation
in two strait lines, which divide the axis and base according to those
fractions, which add 4 to the numerators of the fractions of a parabola
9⁄15 and 12⁄24; and 6 to the denominators, in this manner:—

 In a parabola,                    the axis 9⁄15, the base 12⁄24.
 In the first three-sided figure,  the axis 13⁄21, the base 16⁄30.
 In the second three-sided figure, the axis 17⁄27, the base 20⁄36, &c.

And by the same method, any man, if it be worth the pains, may find out
the centres of equiponderation of the excesses above their triangles of
the rest of the figures in the second and third row, &c.

[Sidenote: The centre of equiponderation of a solid sector is in the
           axis so divided, that the part next the vertex be to the
           whole axis, wanting half the axis of the portion of the
           sphere, as 3 to 4.]

14. The centre of equiponderation of the sector of a sphere, that is, of
a figure compounded of a right cone, whose vertex is the centre of the
sphere, and the portion of the sphere whose base is the same with that
of the cone, divides the strait line which is made of the axis of the
cone and half the axis of the portion together taken, so that the part
next the vertex be triple to the other part, or to the whole strait line
as 3 to 4.

For let A B C (in fig. 9) be the sector of a sphere, whose vertex is the
centre of the sphere A; whose axis is A D; and the circle upon B C is
the common base of the portion of the sphere and of the cone whose
vertex is A; the axis of which portion is E D, and the half thereof F D;
and the axis of the cone, A E. Lastly, let A G be 3⁄4 of the strait line
A F. I say, G is the centre of equiponderation of the sector A B C.

Let the strait line F H be drawn of any length, making right angles with
A F at F; and drawing the strait line A H, let the triangle A F H be
made. Then upon the same centre A let any arch I K be drawn, cutting A D
in L; and its chord, cutting A D in M; and dividing M L equally in N,
let N O be drawn parallel to the strait line F H, and meeting with the
strait line A H in O.

Seeing now B D C is the spherical superficies of the portion cut off
with a plane passing through B C, and cutting the axis at right angles;
and seeing F H divides E D, the axis of the portion, into two equal
parts in F; the centre of equiponderation of the superficies B D C will
be in F (by art. 8); and for the same reason the centre of
equiponderation of the superficies I L K, K being in the strait line A
C, will be in N. And in like manner, if there were drawn, between the
centre of the sphere A and the outermost spherical superficies of the
sector, arches infinite in number, the centres of equiponderation of the
spherical superficies, in which those arches are, would be found to be
in that part of the axis, which is intercepted between the superficies
itself and a plane passing along by the chord of the arch, and cutting
the axis in the middle at right angles.

Let it now be supposed that the moment of the outermost spherical
superficies B D C is F H. Seeing therefore the superficies B D C is to
the superficies I L K in proportion duplicate to that of the arch B D C
to the arch I L K, that is, of B E to I M, that is, of F H to N O; let
it be as F H to N O, so N O to another N P; and again, as N O to N P, so
N P to another N Q; and let this be done in all the strait lines
parallel to the base F H that can possibly be drawn between the base and
the vertex of the triangle A F H. If then through all the points Q there
be drawn the crooked line A Q H, the figure A F H Q A will be the
complement of the first three-sided figure of two means; and the same
will also be the moment of all the spherical superficies, of which the
solid sector A B C D is compounded; and by consequent, the moment of the
sector itself. Let now F H be understood to be the semidiameter of the
base of a right cone, whose side is A H, and axis A F Wherefore, seeing
the bases of the cones, which pass through F and N and the rest of the
points of the axis, are in proportion duplicate to that of the strait
lines F H and N O, &c., the moment of all the bases together, that is,
of the whole cone, will be the figure itself A F H Q A; and therefore
the centre of equiponderation of the cone A F H is the same with that of
the solid sector. Wherefore, seeing A G is ¾ of the axis A F, the centre
of equiponderation of the cone A F H is in G; and therefore the centre
of the solid sector is in G also, and divides the part A F of the axis
so that A G is triple to G F; that is, A G is to A F as 3 to 4; which
was to be demonstrated.

Note, that when the sector is a hemisphere, the axis of the cone
vanisheth into that point which is the centre of the sphere; and
therefore it addeth nothing to half the axis of the portion. Wherefore,
if in the axis of the hemisphere there be taken from the centre ¾ of
half the axis, that is, ⅜ of the semidiameter of the sphere, there will
be the centre of equiponderation of the hemisphere.

------------------------------------------------------------------------

[Illustration:

  _Vol. 1. Lat. & Eng._
  C.XXIII.
  _Fig. 1-9_
]

------------------------------------------------------------------------




                             CHAPTER XXIV.

                     OF REFRACTION AND REFLECTION.

1. Definitions.—2. In perpendicular motion there is no refraction.—3.
    Things thrown out of a thinner into a thicker medium are so
    refracted that the angle refracted is greater than the angle of
    inclination.—4. Endeavour, which from one point tendeth every way,
    will be so refracted, as that the sine of the angle refracted will
    be to the sine of the angle of inclination, as the density of the
    first medium is to the density of the second medium, reciprocally
    taken.—5. The sine of the refracted angle in one inclination is to
    the sine of the refracted angle in another inclination, as the sine
    of the angle of that inclination is to the sine of the angle of this
    inclination.—6. If two lines of incidence, having equal inclination,
    be the one in a thinner, the other in a thicker medium, the sine of
    the angle of inclination will be a mean proportional between the two
    sines of the refracted angles.—7. If the angle of inclination be
    semirect, and the line of inclination be in the thicker medium, and
    the proportion of their densities be the same with that of the
    diagonal to the side of a square, and the separating superficies be
    plane, the refracted line will be in the separating superficies.—8.
    If a body be carried in a strait line upon another body, and do not
    penetrate the same, but be reflected from it, the angle of
    reflection will be equal to the angle of incidence.—9. The same
    happens in the generation of motion in the line of incidence.


                              DEFINITIONS.

[Sidenote: Definitions.]

I. Refraction is the _breaking_ of that strait line, in which a body is
moved or its action would proceed in one and the same medium, into two
strait lines, by reason of the different natures of the two mediums.

II. The former of these is called the _line of incidence_; the latter
the _refracted line_. III. The _point of refraction_ is the common point
of the line of incidence, and of the refracted line.

IV. The _refracting superficies_, which also is the _separating
superficies_ of the two mediums, is that in which is the point of
refraction.

V. The _angle refracted_ is that, which the refracted line makes in the
point of refraction with that line, which from the same point is drawn
perpendicular to the separating superficies in a different medium.

VI. The _angle of refraction_ is that which the refracted line makes
with the line of incidence produced.

VII. The _angle of inclination_ is that which the line of incidence
makes with that line, which from the point of refraction is drawn
perpendicular to the separating superficies.

VIII. The _angle of incidence_ is the complement to a right angle of the
angle of inclination.

And so, (in fig. 1) the refraction is made in A B F. The refracted line
is B F. The line of incidence is A B. The point of incidence and of
refraction is B. The refracting or separating superficies is D B E. The
line of incidence produced directly is A B C. The perpendicular to the
separating superficies is B H. The angle of refraction is C B F. The
angle refracted is H B F. The angle of inclination is A B G or H B C.
The angle of incidence is A B D.

IX. Moreover the _thinner medium_ is understood to be that in which
there is less resistance to motion, or to the generation of motion; and
the thicker that wherein there is greater resistance.

X. And that medium in which there is equal resistance everywhere, is a
_homogeneous medium_. All other mediums are _heterogeneous_.

[Sidenote: In perpendicular motion there is no refraction.]

 If a body pass, or there be generation of motion from one medium to
another of different density, in a line perpendicular to the separating
superficies, there will be no refraction.

For seeing on every side of the perpendicular all things in the mediums
are supposed to be like and equal, if the motion itself be supposed to
be perpendicular, the inclinations also will be equal, or rather none at
all; and therefore there can be no cause from which refraction may be
inferred to be on one side of the perpendicular, which will not conclude
the same refraction to be on the other side. Which being so, refraction
on one side will destroy refraction on the other side; and consequently
either the refracted line will be everywhere, which is absurd, or there
will be no refracted line at all; which was to be demonstrated.

Coroll. It is manifest from hence, that the cause of refraction
consisteth only in the obliquity of the line of incidence, whether the
incident body penetrate both the mediums, or without penetrating,
propagate motion by pressure only.

[Sidenote: Things thrown out of a thinner into a thicker medium are so
           refracted that the angle refracted is greater than the angle
           of inclination.]

3. If a body, without any change of situation of its internal parts, as
a stone, be moved obliquely out of the thinner medium, and proceed
penetrating the thicker medium, and the thicker medium be such, as that
its internal parts being moved restore themselves to their former
situation; the angle refracted will be greater than the angle of
inclination.

For let D B E (in the same first figure) be the separating superficies
of two mediums; and let a body, as a stone thrown, be understood to be
moved as is supposed in the strait line A B C; and let A B be in the
thinner medium, as in the air; and B C in the thicker, as in the water.
I say the stone, which being thrown, is moved in the line A B, will not
proceed in the line B C, but in some other line, namely, that, with
which the perpendicular B H makes the refracted angle H B F greater than
the angle of inclination H B C.

For seeing the stone coming from A, and falling upon B, makes that which
is at B proceed towards H, and that the like is done in all the strait
lines which are parallel to B H; and seeing the parts moved restore
themselves by contrary motion in the same line; there will be contrary
motion generated in H B, and in all the strait lines which are parallel
to it. Wherefore, the motion of the stone will be made by the concourse
of the motions in A G, that is, in D B, and in G B, that is, in B H, and
lastly, in H B, that is, by the concourse of three motions. But by the
concourse of the motions in A G and B H, the stone will be carried to C;
and therefore by adding the motion in H B, it will be carried higher in
some other line, as in B F, and make the angle H B F greater than the
angle H B C.

And from hence may be derived the cause, why bodies which are thrown in
a very oblique line, if either they be any thing flat, or be thrown with
great force, will, when they fall upon the water, be cast up again from
the water into the air.

For let A B (in fig. 2) be the superficies of the water; into which,
from the point C, let a stone be thrown in the strait line C A, making
with the line B A produced a very little angle C A D; and producing B A
indefinitely to D, let C D be drawn perpendicular to it, and A E
parallel to C D. The stone therefore will be moved in C A by the
concourse of two motions in C D and D A, whose velocities are as the
lines themselves C D and D A. And from the motion in C D and all its
parallels downwards, as soon as the stone falls upon A, there will be
reaction upwards, because the water restores itself to its former
situation. If now the stone be thrown with sufficient obliquity, that
is, if the strait line C D be short enough, that is, if the endeavour of
the stone downwards be less than the reaction of the water upwards, that
is, less than the endeavour it hath from its own gravity (for that may
be), the stone will by reason of the excess of the endeavour which the
water hath to restore itself, above that which the stone hath downwards,
be raised again above the superficies A B, and be carried higher, being
reflected in a line which goes higher, as the line A G.

[Sidenote: Endeavour, which from one point tendeth every way, will be so
           refracted, as that the sine of the angle refracted will be to
           the sine of the angle of inclination, as the density of the
           first medium is to the density of the second medium,
           reciprocally taken.]

4. If from a point, whatsoever the medium be, endeavour be propagated
every way into all the parts of that medium; and to the same endeavour
there be obliquely opposed another medium of a different nature, that
is, either thinner or thicker; that endeavour will be so refracted, that
the sine of the angle refracted, to the sine of the angle of
inclination, will be as the density of the first medium to the density
of the second medium, reciprocally taken.

First, let a body be in the thinner medium in A (fig. 3), and let it be
understood to have endeavour every way, and consequently, that its
endeavour proceed in the lines A B and A _b_; to which let B _b_ the
superficies of the thicker medium be obliquely opposed in B and _b_, so
that A B and A _b_ be equal; and let the strait line B _b_ be produced
both ways. From the points B and _b_, let the perpendiculars B C and _b
c_ be drawn; and upon the centres B and _b_, and at the equal distances
B A and _b_ A, let the circles A C and A _c_ be described, cutting B C
and _b c_ in C and _c_, and the same C B and _c b_ produced in D and
_d_, as also A B and A _b_ produced in E and _e_. Then from the point A
to the strait lines B C and _b c_ let the perpendiculars A F and A _f_
be drawn. A F therefore will be the sine of the angle of inclination of
the strait line A B, and A _f_ the sine of the angle of inclination of
the strait line A _h_, which two inclinations are by construction made
equal. I say, as the density of the medium in which are B C and _b c_ is
to the density of the medium in which are B D and _b d_, so is the sine
of the angle refracted, to the sine of the angle of inclination.

Let the strait line F G be drawn parallel to the strait line A B,
meeting with the strait line _b_ B produced in G.

Seeing therefore A F and B G are also parallels, they will be equal; and
consequently, the endeavour in A F is propagated in the same time, in
which the endeavour in B G would be propagated if the medium were of the
same density. But because B G is in a thicker medium, that is, in a
medium which resists the endeavour more than the medium in which A F is,
the endeavour will be propagated less in B G than in A F, according to
the proportion which the density of the medium, in which A F is, hath to
the density of the medium in which B G is. Let therefore the density of
the medium, in which B G is, be to the density of the medium, in which A
F is, as B G is to B H; and let the measure of the time be the radius of
the circle. Let H I be drawn parallel to B D, meeting with the
circumference in I; and from the point I let I K be drawn perpendicular
to B D; which being done, B H and I K will be equal; and I K will be to
A F, as the density of the medium in which is A F is to the density of
the medium in which is I K. Seeing therefore in the time A B, which is
the radius of the circle, the endeavour is propagated in A F in the
thinner medium, it will be propagated in the same time, that is, in the
time B I in the thicker medium from K to I. Therefore, B I is the
refracted line of the line of incidence A B; and I K is the sine of the
angle refracted; and A F the sine of the angle of inclination.
Wherefore, seeing I K is to A F, as the density of the medium in which
is A F to the density of the medium in which is I K; it will be as the
density of the medium in which is A F or B C to the density of the
medium in which is I K or B D, so the sine of the angle refracted to the
sine of the angle of inclination. And by the same reason it may be
shown, that as the density of the thinner medium is to the density of
the thicker medium, so will K I the sine of the angle refracted be to A
F the sine of the angle of inclination.

Secondly, let the body, which endeavoureth every way, be in the thicker
medium at I. If, therefore, both the mediums were of the same density,
the endeavour of the body in I B would tend directly to L; and the sine
of the angle of inclination L M would be equal to I K or B H. But
because the density of the medium, in which is I K, to the density of
the medium, in which is L M, is as B H to B G, that is, to A F, the
endeavour will be propagated further in the medium in which L M is, than
in the medium in which I K is, in the proportion of density to density,
that is, of M L to A F. Wherefore, B A being drawn, the angle refracted
will be C B A, and its sine A F. But L M is the sine of the angle of
inclination; and therefore again, as the density of one medium is to the
density of the different medium, so reciprocally is the sine of the
angle refracted to the sine of the angle of inclination; which was to be
demonstrated.

In this demonstration, I have made the separating superficies B _b_
plane by construction. But though it were concave or convex, the theorem
would nevertheless be true. For the refraction being made in the point B
of the plane separating superficies, if a crooked line, as P Q, be
drawn, touching the separating line in the point B; neither the
refracted line B I, nor the perpendicular B D, will be altered; and the
refracted angle K B I, as also its sine K I, will be still the same they
were.

[Sidenote: The sine of the refracted angle in one inclination is to the
           sine of the refracted angle in another inclination, as the
           sine of the angle of that inclination is to the sine of the
           angle of this inclination.]

5. The sine of the angle refracted in one inclination is to the sine of
the angle refracted in another inclination, as the sine of the angle of
that inclination to the sine of the angle of this inclination.

For seeing the sine of the refracted angle is to the sine of the angle
of inclination, whatsoever that inclination be, as the density of one
medium to the density of the other medium; the proportion of the sine of
the refracted angle, to the sine of the angle of inclination, will be
compounded of the proportions of density to density, and of the sine of
the angle of one inclination to the sine of the angle of the other
inclination. But the proportions of the densities in the same
homogeneous body are supposed to be the same. Wherefore refracted angles
in different inclinations are as the sines of the angles of those
inclinations; which was to be demonstrated.

[Sidenote: If two lines of incidence, having equal inclination, be one
           in a thinner the other in a thicker medium, the sine of the
           angle of inclination will be a mean proportional between the
           two sines of the refracted angles.]

6. If two lines of incidence, having equal inclination, be the one in a
thinner, the other in a thicker medium, the sine of the angle of their
inclination will be a mean proportional between the two sines of their
angles refracted.

For let the strait line A B (in fig. 3) have its inclination in the
thinner medium, and be refracted in the thicker medium in B I; and let E
B have as much inclination in the thicker medium, and be refracted in
the thinner medium in B S; and let R S, the sine of the angle refracted,
be drawn. I say, the strait lines R S, A F, and I K are in continual
proportion. For it is, as the density of the thicker medium to the
density of the thinner medium, so R S to A F. But it is also as the
density of the same thicker medium to that of the same thinner medium,
so A F to I K. Wherefore R S. A F :: A F. I K are proportionals; that
is, R S, A F, and I K are in continual proportion, and A F is the mean
proportional; which was to be proved.

[Sidenote: If the angle of inclination be semirect, and the line of
           inclination be in the thicker medium, and the proportion of
           their densities be the same with that of the diagonal to the
           side of a square, and the separating superficies be plain,
           the refracted line will be in the separating superficies.]

7. If the angle of inclination be semirect, and the line of inclination
be in the thicker medium, and the proportion of the densities be as that
of a diagonal to the side of its square, and the separating superficies
be plain, the refracted line will be in that separating superficies.

For in the circle A C (fig. 4) let the angle of inclination A B C be an
angle of 45 degrees. Let C B be produced to the circumference in D; and
let C E, the sine of the angle E B C, be drawn, to which let B F be
taken equal in the separating line B G. B C E F will therefore be a
parallelogram, and F E and B C, that is F E and B G equal. Let A G be
drawn, namely the diagonal of the square whose side is B G, and it will
be, as A G to E F so B G to B F; and so, by supposition, the density of
the medium, in which C is, to the density of the medium in which D is;
and so also the sine of the angle refracted to the sine of the angle of
inclination. Drawing therefore F D, and from D the line D H
perpendicular to A B produced, D H will be the sine of the angle of
inclination. And seeing the sine of the angle refracted is to the sine
of the angle of inclination, as the density of the medium, in which is
C, is to the density of the medium in which is D, that is, by
supposition, as A G is to F E, that is as B G is to D H; and seeing D H
is the sine of the angle of inclination, B G will therefore be the sine
of the angle refracted. Wherefore B G will be the refracted line, and
lye in the plain separating superficies; which was to be demonstrated.

Coroll. It is therefore manifest, that when the inclination is greater
than 45 degrees, as also when it is less, provided the density be
greater, it may happen that the refraction will not enter the thinner
medium at all.

[Sidenote: If a body be carried in a strait line upon another body, and
           do not penetrate it, but be reflected from it, the angle of
           reflection will be equal to the angle of incidence.]

8. If a body fall in a strait line upon another body, and do not
penetrate it, but be reflected from it, the angle of reflection will be
equal to the angle of incidence.

Let there be a body at A (in fig. 5), which falling with strait motion
in the line A C upon another body at C, passeth no further, but is
reflected; and let the angle of incidence be any angle, as A C D. Let
the strait line C E be drawn, making with D C produced the angle E C F
equal to the angle A C D; and let A D be drawn perpendicular to the
strait line D F. Also in the same strait line D F let C G be taken equal
to C D; and let the perpendicular G E be raised, cutting C E in E. This
being done, the triangles A C D and E C G will be equal and like. Let C
H be drawn equal and parallel to the strait line A D; and let H C be
produced indefinitely to I. Lastly let E A be drawn, which will pass
through H, and be parallel and equal to G D. I say the motion from A to
C, in the strait line of incidence A C, will be reflected in the strait
line C E.

For the motion from A to C is made by two coefficient or concurrent
motions, the one in A H parallel to D G, the other in A D perpendicular
to the same D G; of which two motions that in A H works nothing upon the
body A after it has been moved as far as C, because, by supposition, it
doth not pass the strait line D G; whereas the endeavour in A D, that is
in H C, worketh further towards I. But seeing it doth only press and not
penetrate, there will be reaction in H, which causeth motion from C
towards H; and in the meantime the motion in H E remains the same it was
in A H; and therefore the body will now be moved by the concourse of two
motions in C H and H E, which are equal to the two motions it had
formerly in A H and H C. Wherefore it will be carried on in C E. The
angle therefore of reflection will be E C G, equal, by construction, to
the angle A C D; which was to be demonstrated.

Now when the body is considered but as a point, it is all one whether
the superficies or line in which the reflection is made be strait or
crooked; for the point of incidence and reflection C is as well in the
crooked line which toucheth D G in C, as in D G itself.

[Sidenote: The same happens in the generation of motion in the line of
           incidence.]

9. But if we suppose that not a body be moved, but some endeavour only
be propagated from A to C, the demonstration will nevertheless be the
same. For all endeavour is motion; and when it hath reached the solid
body in C, it presseth it, and endeavoureth further in C I. Wherefore
the reaction will proceed in C H; and the endeavour in C H concurring
with the endeavour in H E, will generate the endeavour in C E, in the
same manner as in the repercussion of bodies moved.

If therefore endeavour be propagated from any point to the concave
superficies of a spherical body, the reflected line with the
circumference of a great circle in the same sphere will make an angle
equal to the angle of incidence.

For if endeavour be propagated from A (in fig. 6) to the circumference
in B, and the centre of the sphere be C, and the line C B be drawn, as
also the tangent D B E; and lastly if the angle F B D be made equal to
the angle A B E, the reflection will be made in the line B F, as hath
been newly shown. Wherefore the angles, which the strait lines A B and F
B make with the circumference, will also be equal. But it is here to be
noted, that if C B be produced howsoever to G, the endeavour in the line
G B C will proceed only from the perpendicular reaction in G B; and that
therefore there will be no other endeavour in the point B towards the
parts which are within the sphere, besides that which tends towards the
centre.

And here I put an end to the third part of this discourse; in which I
have considered motion and magnitude by themselves in the abstract. The
fourth and last part, concerning the _phenomena of nature_, that is to
say, concerning the motions and magnitudes of the bodies which are parts
of the world, real and existent, is that which follows.

------------------------------------------------------------------------

[Illustration:

  _Vol. 1. Lat. & Eng._
  C. XXIV.
  _Fig. 1-6_
]

------------------------------------------------------------------------


                                PART IV.
                                PHYSICS,
                      OR THE PHENOMENA OF NATURE.


                                -------


                              CHAPTER XXV.
                      OF SENSE AND ANIMAL MOTION.

1. The connexion of what hath been said with that which followeth.—2.
    The investigation of the nature of sense, and the definition of
    sense.—3. The subject and object of sense.—4. The organs of
    sense.—5. All bodies are not indued with sense.—6. But one phantasm
    at one and the same time.—7. Imagination the remains of past sense,
    which also is memory. Of sleep.—8. How phantasms succeed one
    another.—9. Dreams, whence they proceed.—10. Of the senses, their
    kinds, their organs, and phantasms proper and common.—11. The
    magnitude of images, how and by what it is determined.—12. Pleasure,
    pain, appetite and aversion, what they are.—13. Deliberation and
    will, what.


[Sidenote: The connexion of what hath been said with that which
           followeth.]

1. I have, in the first chapter, defined philosophy to be _knowledge of
effects acquired by true ratiocination, from knowledge first had of
their causes and generation; and of such causes or generations as may
be, from former knowledge of their effects or appearances_. There are,
therefore, two methods of philosophy; one, from the generation of things
to their possible effects; and the other, from their effects or
appearances to some possible generation of the same. In the former of
these the truth of the first principles of our ratiocination, namely
definitions, is made and constituted by ourselves, whilst we consent and
agree about the appellations of things. And this part I have finished in
the foregoing chapters; in which, if I am not deceived, I have affirmed
nothing, saving the definitions themselves, which hath not good
coherence with the definitions I have given; that is to say, which is
not sufficiently demonstrated to all those, that agree with me in the
use of words and appellations; for whose sake only I have written the
same. I now enter upon the other part; which is the finding out by the
appearances or effects of nature, which we know by sense, some ways and
means by which they may be, I do not say they are, generated. The
principles, therefore, upon which the following discourse depends, are
not such as we ourselves make and pronounce in general terms, as
definitions; but such, as being placed in the things themselves by the
Author of Nature, are by us observed in them; and we make use of them in
single and particular, not universal propositions. Nor do they impose
upon us any necessity of constituting theorems; their use being only,
though not without such general propositions as have been already
demonstrated, to show us the possibility of some production or
generation. Seeing, therefore, the science, which is here taught, hath
its principles in the appearances of nature, and endeth in the attaining
of some knowledge of natural causes, I have given to this part the title
of PHYSICS, or the _Phenomena of Nature_. Now such things as appear, or
are shown to us by nature, we call phenomena or appearances.

Of all the phenomena or appearances which are near us, the most
admirable is apparition itself, τὸ φαίνεσθαι; namely, that some natural
bodies have in themselves the patterns almost of all things, and others
of none at all. So that if the appearances be the principles by which we
know all other things, we must needs acknowledge sense to be the
principle by which we know those principles, and that all the knowledge
we have is derived from it. And as for the causes of sense, we cannot
begin our search of them from any other phenomenon than that of sense
itself. But you will say, by what sense shall we take notice of sense? I
answer, by sense itself, namely, by the memory which for some time
remains in us of things sensible, though they themselves pass away. For
he that perceives that he hath perceived, remembers.

In the first place, therefore, the causes of our perception, that is,
the causes of those ideas and phantasms which are perpetually generated
within us whilst we make use of our senses, are to be enquired into; and
in what manner their generation proceeds. To help which inquisition, we
may observe first of all, that our phantasms or ideas are not always the
same; but that new ones appear to us, and old ones vanish, according as
we apply our organs of sense, now to one object, now to another.
Wherefore they are generated, and perish. And from hence it is manifest,
that they are some change or mutation in the sentient.

[Sidenote: The investigation of the nature of sense, and the definition
           of sense.]

2. Now that all mutation or alteration is motion or endeavour (and
endeavour also is motion) in the internal parts of the thing that is
altered, hath been proved (in art. 9, chap. VIII) from this, that whilst
even the least parts of any body remain in the same situation in respect
of one another, it cannot be said that any alteration, unless perhaps
that the whole body together hath been moved, hath happened to it; but
that it both appeareth and is the same it appeared and was before.
Sense, therefore, in the sentient, can be nothing else but motion in
some of the internal parts of the sentient; and the parts so moved are
parts of the organs of sense. For the parts of our body, by which we
perceive any thing, are those we commonly call the organs of sense. And
so we find what is the subject of our sense, namely, that in which are
the phantasms; and partly also we have discovered the nature of sense,
namely, that it is some internal motion in the sentient.

I have shown besides (in chap. IX, art. 7) that no motion is generated
but by a body contiguous and moved: from whence it is manifest, that the
immediate cause of sense or perception consists in this, that the first
organ of sense is touched and pressed. For when the uttermost part of
the organ is pressed, it no sooner yields, but the part next within it
is pressed also; and, in this manner, the pressure or motion is
propagated through all the parts of the organ to the innermost. And thus
also the pressure of the uttermost part proceeds from the pressure of
some more remote body, and so continually, till we come to that from
which, as from its fountain, we derive the phantasm or idea that is made
in us by our sense. And this, whatsoever it be, is that we commonly call
_the object_, Sense, therefore, is some internal motion in the sentient,
generated by some internal motion of the parts of the object, and
propagated through all the media to the innermost part of the organ. By
which words I have almost defined what sense is.

Moreover, I have shown (art. 2, chap. XV) that all resistance is
endeavour opposite to another endeavour, that is to say, reaction.
Seeing, therefore, there is in the whole organ, by reason of its own
internal natural motion, some resistance or reaction against the motion
which is propagated from the object to the innermost part of the organ,
there is also in the same organ an endeavour opposite to the endeavour
which proceeds from the object; so that when that endeavour inwards is
the last action in the act of sense, then from the reaction, how little
soever the duration of it be, a phantasm or idea hath its being; which,
by reason that the endeavour is now outwards, doth always appear as
something situate without the organ. So that now I shall give you the
whole definition of sense, as it is drawn from the explication of the
causes thereof and the order of its generation, thus: SENSE _is a
phantasm, made by the reaction and endeavour outwards in the organ of
sense, caused by an endeavour inwards from the object, remaining for
some time more or less_.

[Sidenote: The subject and object of sense.]

3. The _subject_ of sense is the _sentient_ itself, namely, some living
creature; and we speak more correctly, when we say a living creature
seeth, than when we say the eye seeth. The object is the thing received;
and it is more accurately said, that we see the sun, than that we see
the light. For light and colour, and heat and sound, and other qualities
which are commonly called sensible, are not objects, but phantasms in
the sentients. For a phantasm is the act of sense, and differs no
otherwise from sense than _fieri_, that is, being a doing, differs from
_factum esse_, that is, being done; which difference, in things that are
done in an instant, is none at all; and a phantasm is made in an
instant. For in all motion which proceeds by perpetual propagation, the
first part being moved moves the second, the second the third, and so on
to the last, and that to any distance, how great soever. And in what
point of time the first or foremost part proceeded to the place of the
second, which is thrust on, in the same point of time the last save one
proceeded into the place of the last yielding part; which by reaction,
in the same instant, if the reaction be strong enough, makes a phantasm;
and a phantasm being made, perception is made together with it.

[Sidenote: The organs of sense.]

4. The _organs_ of sense, which are in the sentient, are such parts
thereof, that if they be hurt, the very generation of phantasms is
thereby destroyed, though all the rest of the parts remain entire. Now
these parts in the most of living creatures are found to be certain
spirits and membranes, which, proceeding from the _pia mater_, involve
the brain and all the nerves; also the brain itself, and the arteries
which are in the brain; and such other parts, as being stirred, the
heart also, which is the fountain of all sense, is stirred together with
them. For whensoever the action of the object reacheth the body of the
sentient, that action is by some nerve propagated to the brain; and if
the nerve leading thither be so hurt or obstructed, that the motion can
be propagated no further, no sense follows. Also if the motion be
intercepted between the brain and the heart by the defect of the organ
by which the action is propagated, there will be no perception of the
object.

[Sidenote: All bodies are not endued with sense.]

5. But though all sense, as I have said, be made by reaction,
nevertheless it is not necessary that every thing that reacteth should
have sense. I know there have been philosophers, and those learned men,
who have maintained that all bodies are endued with sense. Nor do I see
how they can be refuted, if the nature of sense be placed in reaction
only. And, though by the reaction of bodies inanimate a phantasm might
be made, it would nevertheless cease, as soon as ever the object were
removed. For unless those bodies had organs, as living creatures have,
fit for the retaining of such motion as is made in them, their sense
would be such, as that they should never remember the same. And
therefore this hath nothing to do with that sense which is the subject
of my discourse. For by sense, we commonly understand the judgment we
make of objects by their phantasms; namely, by comparing and
distinguishing those phantasms; which we could never do, if that motion
in the organ, by which the phantasm is made, did not remain there for
some time, and make the same phantasm return. Wherefore sense, as I here
understand it, and which is commonly so called, hath necessarily some
memory adhering to it, by which former and later phantasms may be
compared together, and distinguished from one another.

Sense, therefore, properly so called, must necessarily have in it a
perpetual variety of phantasms, that they may be discerned one from
another. For if we should suppose a man to be made with clear eyes, and
all the rest of his organs of sight well disposed, but endued with no
other sense; and that he should look only upon one thing, which is
always of the same colour and figure, without the least appearance of
variety, he would seem to me, whatsoever others may say, to see, no more
than I seem to myself to feel the bones of my own limbs by my organs of
feeling; and yet those bones are always and on all sides touched by a
most sensible membrane. I might perhaps say he were astonished, and
looked upon it; but I should not say he saw it; it being almost all one
for a man to be always sensible of one and the same thing, and not to be
sensible at all of any thing.

[Sidenote: But one phantasm at one and the same time.]

6. And yet such is the nature of sense, that it does not permit a man to
discern many things at once. For seeing the nature of sense consists in
motion; as long as the organs are employed about one object, they cannot
be so moved by another at the same time, as to make by both their
motions one sincere phantasm of each of them at once. And therefore two
several phantasms will not be made by two objects working together, but
only one phantasm compounded from the action of both.

Besides, as when we divide a body, we divide its place; and when we
reckon many bodies, we must necessarily reckon as many places; and
contrarily, as I have shown in the seventh chapter; so what number
soever we say there be of times, we must understand the same number of
motions also; and as oft as we count many motions, so oft we reckon many
times. For though the object we look upon be of divers colours, yet with
those divers colours it is but one varied object, and not variety of
objects.

Moreover, whilst those organs which are common to all the senses, such
as are those parts of every organ which proceed in men from the root of
the nerves to the heart, are vehemently stirred by a strong action from
some one object, they are, by reason of the contumacy which the motion,
they have already, gives them against the reception of all other motion,
made the less fit to receive any other impression from whatsoever other
objects, to what sense soever those objects belong. And hence it is,
that an earnest studying of one object, takes away the sense of all
other objects for the present. For _study_ is nothing else but a
possession of the mind, that is to say, a vehement motion made by some
one object in the organs of sense, which are stupid to all other motions
as long as this lasteth; according to what was said by Terence,
“_Populus studio stupidus in funambulo animum occuparat._” For what is
_stupor_ but that which the Greeks call ἀναισθησία, that is, a cessation
from the sense of other things? Wherefore at one and the same time, we
cannot by sense perceive more than one single object; as in reading, we
see the letters successively one by one, and not all together, though
the whole page be presented to our eye; and though every several letter
be distinctly written there, yet when we look upon the whole page at
once, we read nothing.

From hence it is manifest, that every endeavour of the organ outwards,
is not to be called sense, but that only, which at several times is by
vehemence made stronger and more predominant than the rest; which
deprives us of the sense of other phantasms, no otherwise than the sun
deprives the rest of the stars of light, not by hindering their action,
but by obscuring and hiding them with his excess of brightness.

[Sidenote: Imagination, the remains of past sense; which also is memory.
           Of sleep.]

7. But the motion of the organ, by which a phantasm is made, is not
commonly called sense, except the object be present. And the phantasm
remaining after the object is removed or past by, is called _fancy_, and
in Latin _imaginatio_; which word, because all phantasms are not images,
doth not fully answer the signification of the word _fancy_ in its
general acceptation. Nevertheless I may use it safely enough, by
understanding it for the Greek Φαντασία.

IMAGINATION therefore is nothing else but _sense decaying_, or
_weakened_, by the absence of the object. But what may be the cause of
this decay or weakening? Is the motion the weaker, because the object is
taken away? If it were, then phantasms would always and necessarily be
less clear in the imagination, than they are in sense; which is not
true. For in dreams, which are the imaginations of those that sleep,
they are no less clear than in sense itself. But the reason why in men
waking the phantasms of things past are more obscure than those of
things present, is this, that their organs being at the same time moved
by other present objects, those phantasms are the less predominant.
Whereas in sleep, the passages being shut up, external action doth not
at all disturb or hinder internal motion.

If this be true, the next thing to be considered, will be, whether any
cause may be found out, from the supposition whereof it will follow,
that the passage is shut up from the external objects of sense to the
internal organ. I suppose, therefore, that by the continual action of
objects, to which a reaction of the organ, and more especially of the
spirits, is necessarily consequent, the organ is wearied, that is, its
parts are no longer moved by the spirits without some pain; and
consequently the nerves being abandoned and grown slack, they retire to
their fountain, which is the cavity either of the brain or of the heart;
by which means the action which proceeded by the nerves is necessarily
intercepted. For action upon a patient, that retires from it, makes but
little impression at the first; and at last, when the nerves are by
little and little slackened, none at all. And therefore there is no more
reaction, that is, no more sense, till the organ being refreshed by
rest, and by a supply of new spirits recovering strength and motion, the
sentient awaketh. And thus it seems to be always, unless some other
preternatural cause intervene; as heat in the internal parts from
lassitude, or from some disease stirring the spirits and other parts of
the organ in some extraordinary manner.

[Sidenote: How phantasms succeed one another.]

8. Now it is not without cause, nor so casual a thing as many perhaps
think it, that phantasms in this their great variety proceed from one
another; and that the same phantasms sometimes bring into the mind other
phantasms like themselves, and at other times extremely unlike. For in
the motion of any continued body, one part follows another by cohesion;
and therefore, whilst we turn our eyes and other organs successively to
many objects, the motion which was made by every one of them remaining,
the phantasms are renewed as often as any one of those motions comes to
be predominant above the rest; and they become predominant in the same
order in which at any time formerly they were generated by sense. So
that when by length of time very many phantasms have been generated
within us by sense, then almost any thought may arise from any other
thought; insomuch that it may seem to be a thing indifferent and casual,
which thought shall follow which. But for the most part this is not so
uncertain a thing to waking as to sleeping men. For the thought or
phantasm of the desired end brings in all the phantasms, that are means
conducing to that end, and that in order backwards from the last to the
first, and again forwards from the beginning to the end. But this
supposes both appetite, and judgment to discern what means conduce to
the end, which is gotten by experience; and experience is store of
phantasms, arising from the sense of very many things. For φανταζεσθαι
and _meminisse_, _fancy_ and _memory_, differ only in this, that memory
supposeth the time past, which fancy doth not. In memory, the phantasms
we consider are as if they were worn out with time; but in our fancy we
consider them as they are; which distinction is not of the things
themselves, but of the considerations of the sentient. For there is in
memory something like that which happens in looking upon things at a
great distance; in which as the small parts of the object are not
discerned, by reason of their remoteness; so in memory, many accidents
and places and parts of things, which were formerly perceived by sense,
are by length of time decayed and lost.

The perpetual arising of phantasms, both in sense and imagination, is
that which we commonly call discourse of the mind, and is common to men
with other living creatures. For he that thinketh, compareth the
phantasms that pass, that is, taketh notice of their likeness or
unlikeness to one another. And as he that observes readily the
likenesses of things of different natures, or that are very remote from
one another, is said to have a good fancy; so he is said to have a good
judgment, that finds out the unlikenesses or differences of things that
are like one another. Now this observation of differences is not
perception made by a common organ of sense, distinct from sense or
perception properly so called, but is memory of the differences of
particular phantasms remaining for some time; as the distinction between
hot and lucid, is nothing else but the memory both of a heating, and of
an enlightening object.

[Sidenote: Dreams, whence they proceed.]

9. The phantasms of men that sleep, are _dreams_. Concerning which we
are taught by experience these five things. First, that for the most
part there is neither order nor coherence in them. Secondly, that we
dream of nothing but what is compounded and made up of the phantasms of
sense past. Thirdly, that sometimes they proceed, as in those that are
drowsy, from the interruption of their phantasms by little and little,
broken and altered through sleepiness; and sometimes also they begin in
the midst of sleep. Fourthly, that they are clearer than the
imaginations of waking men, except such as are made by sense itself, to
which they are equal in clearness. Fifthly, that when we dream, we
admire neither the places nor the looks of the things that appear to us.
Now from what hath been said, it is not hard to show what may be the
causes of these phenomena. For as for the first, seeing all order and
coherence proceeds from frequent looking back to the end, that is, from
consultation; it must needs be, that seeing in sleep we lose all thought
of the end, our phantasms succeed one another, not in that order which
tends to any end, but as it happeneth, and in such manner, as objects
present themselves to our eyes when we look indifferently upon all
things before us, and see them, not because we would see them, but
because we do not shut our eyes; for then they appear to us without any
order at all. The second proceeds from this, that in the silence of
sense there is no new motion from the objects, and therefore no new
phantasm, unless we call that new, which is compounded of old ones, as a
chimera, a golden mountain, and the like. As for the third, why a dream
is sometimes as it were the continuation of sense, made up of broken
phantasms, as in men distempered with sickness, the reason is manifestly
this, that in some of the organs sense remains, and in others it
faileth. But how some phantasms may be revived, when all the exterior
organs are benumbed with sleep, is not so easily shown. Nevertheless
that, which hath already been said, contains the reason of this also.
For whatsoever strikes the _pia mater_, reviveth some of those phantasms
that are still in motion in the brain; and when any internal motion of
the heart reacheth that membrane, then the predominant motion in the
brain makes the phantasm. Now the motions of the heart are appetites and
aversions, of which I shall presently speak further. And as appetites
and aversions are generated by phantasms, so reciprocally phantasms are
generated by appetites and aversions. For example, heat in the heart
proceeds from anger and fighting; and again, from heat in the heart,
whatsoever be the cause of it, is generated anger and the image of an
enemy, in sleep. And as love and beauty stir up heat in certain organs;
so heat in the same organs, from whatsoever it proceeds, often causeth
desire and the image of an unresisting beauty. Lastly, cold doth in the
same manner generate fear in those that sleep, and causeth them to dream
of ghosts, and to have phantasms of horror and danger; as fear also
causeth cold in those that wake. So reciprocal are the motions of the
heart and brain. The fourth, namely, that the things we seem to see and
feel in sleep, are as clear as in sense itself, proceeds from two
causes; one, that having then no sense of things without us, that
internal motion which makes the phantasm, in the absence of all other
impressions, is predominant; and the other, that the parts of our
phantasms which are decayed and worn out by time, are made up with other
fictitious parts. To conclude, when we dream, we do not wonder at
strange places and the appearances of things unknown to us, because
admiration requires that the things appearing be new and unusual, which
can happen to none but those that remember former appearances; whereas
in sleep, all things appear as present.

But it is here to be observed, that certain dreams, especially such as
some men have when they are between sleeping and waking, and such as
happen to those that have no knowledge of the nature of dreams and are
withal superstitious, were not heretofore nor are now accounted dreams.
For the apparitions men thought they saw, and the voices they thought
they heard in sleep, were not believed to be phantasms, but things
subsisting of themselves, and objects without those that dreamed. For to
some men, as well sleeping as waking, but especially to guilty men, and
in the night, and in hallowed places, fear alone, helped a little with
the stories of such apparitions, hath raised in their minds terrible
phantasms, which have been and are still deceitfully received for things
really true, under the names of _ghosts_ and _incorporeal substances_.

[Sidenote: Of the senses, their kinds, their organs and phantasms,
           proper and common.]

10. In most living creatures there are observed five kinds of senses,
which are distinguished by their organs, and by their different kinds of
phantasms; namely, _sight_, _hearing_, _smell_, _taste_, and _touch_;
and these have their organs partly peculiar to each of them severally,
and partly common to them all. The organ of sight is partly animate, and
partly inanimate. The inanimate parts are the three humours; namely, the
watery humour, which by the interposition of the membrane called uvea,
the perforation whereof is called the apple of the eye, is contained on
one side by the first concave superficies of the eye, and on the other
side by the ciliary processes, and the coat of the crystalline humour;
the crystalline, which, hanging in the midst between the ciliary
processes, and being almost of spherical figure, and of a thick
consistence, is enclosed on all sides with its own transparent coat; and
the vitreous or glassy humour, which filleth all the rest of the cavity
of the eye, and is somewhat thicker then the watery humour, but thinner
than the crystalline. The animate part of the organ is, first, the
membrane _choroeides_, which is a part of the _pia mater_, saving that
it is covered with a coat derived from the marrow of the optic nerve,
which is called the _retina_; and this _choroeides_, seeing it is part
of the _pia mater_, is continued to the beginning of the _medulla
spinalis_ within the scull, in which all the nerves which are within the
head have their roots. Wherefore all the animal spirits that the nerves
receive, enter into them there; for it is not imaginable that they can
enter into them anywhere else. Seeing therefore sense is nothing else
but the action of objects propagated to the furthest part of the organ;
and seeing also that animal spirits are nothing but vital spirits
purified by the heart, and carried from it by the arteries; it follows
necessarily, that the action is derived from the heart by some of the
arteries to the roots of the nerves which are in the head, whether those
arteries be the _plexus retiformis_, or whether they be other arteries
which are inserted into the substance of the brain. And, therefore,
those arteries are the complement or the remaining part of the whole
organ of sight. And this last part is a common organ to all the senses;
whereas, that which reacheth from the eye to the roots of the nerves is
proper only to sight. The proper organ of hearing is the tympanum of the
ear and its own nerve; from which to the heart the organ is common. So
the proper organs of smell and taste are nervous membranes, in the
palate and tongue for the taste, and in the nostrils for the smell; and
from the roots of those nerves to the heart all is common. Lastly, the
proper organ of touch are nerves and membranes dispersed through the
whole body; which membranes are derived from the root of the nerves. And
all things else belonging alike to all the senses seem to be
administered by the arteries, and not by the nerves.

The proper phantasm of sight is light; and under this name of light,
colour also, which is nothing but perturbed light, is comprehended.
Wherefore the phantasm of a lucid body is light; and of a coloured body,
colour. But the object of sight, properly so called, is neither light
nor colour, but the body itself which is lucid, or enlightened, or
coloured. For light and colour, being phantasms of the sentient, cannot
be accidents of the object. Which is manifest enough from this, that
visible things appear oftentimes in places in which we know assuredly
they are not, and that in different places they are of different
colours, and may at one and the same time appear in divers places.
Motion, rest, magnitude, and figure, are common both to the sight and
touch; and the whole appearance together of figure, and light or colour,
is by the Greeks commonly called εἴδος, and εἴδωλον, and ἱδέα; and by
the Latins, _species_ and _imago_; all which names signify no more but
appearance.

The phantasm, which is made by hearing, is sound; by smell, odour; by
taste, savour; and by touch, hardness and softness, heat and cold,
wetness, oiliness, and many more, which are easier to be distinguished
by sense than words. Smoothness, roughness, rarity, and density, refer
to figure, and are therefore common both to touch and sight. And as for
the objects of hearing, smell, taste, and touch, they are not sound,
odour, savour, hardness, &c., but the bodies themselves from which
sound, odour, savour, hardness, &c. proceed; of the causes of which, and
of the manner how they are produced, I shall speak hereafter.

But these phantasms, though they be effects in the sentient, as subject,
produced by objects working upon the organs; yet there are also other
effects besides these, produced by the same objects in the same organs;
namely certain motions proceeding from sense, which are called _animal
motions_. For seeing in all sense of external things there is mutual
action and reaction, that is, two endeavours opposing one another, it is
manifest that the motion of both of them together will be continued
every way, especially to the confines of both the bodies. And when this
happens in the internal organ, the endeavour outwards will proceed in a
solid angle, which will be greater, and consequently the idea greater,
than it would have been if the impression had been weaker.

[Sidenote: The magnitude of images, how and by what it is determined.]

11. From hence the natural cause is manifest, first, why those things
seem to be greater, which, _cæteris paribus_, are seen in a greater
angle: secondly, why in a serene cold night, when the moon doth not
shine, more of the fixed stars appear than at another time. For their
action is less hindered by the serenity of the air, and not obscured by
the greater light of the moon, which is then absent; and the cold,
making the air more pressing, helpeth or strengtheneth the action of the
stars upon our eyes; in so much as stars may then be seen which are seen
at no other time. And this may suffice to be said in general concerning
sense made by the reaction of the organ. For, as for the place of the
image, the deceptions of sight, and other things of which we have
experience in ourselves by sense, seeing they depend for the most part
upon the fabric itself of the eye of man, I shall speak of them then
when I come to speak of man.

[Sidenote: Pleasure, pain, appetite, and aversion, what they are.]

12. But there is another kind of sense, of which I will say something in
this place, namely, the sense of pleasure and pain, proceeding not from
the reaction of the heart outwards, but from continual action from the
outermost part of the organ towards the heart. For the original of life
being in the heart, that motion in the sentient, which is propagated to
the heart, must necessarily make some alteration or diversion of vital
motion, namely, by quickening or slackening, helping or hindering the
same. Now when it helpeth, it is pleasure; and when it hindereth, it is
pain, trouble, grief, &c. And as phantasms seem to be without, by reason
of the endeavour outwards, so pleasure and pain, by reason of the
endeavour of the organ inwards, seem to be within; namely, there where
the first cause of the pleasure or pain is; as when the pain proceeds
from a wound, we think the pain and the wound are both in the same
place.

[Sidenote: Pleasure, pain, appetite, and aversion, what they are.]

Now vital motion is the motion of the blood, perpetually circulating (as
hath been shown from many infallible signs and marks by Doctor Harvey,
the first observer of it) in the veins and arteries. Which motion, when
it is hindered by some other motion made by the action of sensible
objects, may be restored again either by bending or setting strait the
parts of the body; which is done when the spirits are carried now into
these, now into other nerves, till the pain, as far as is possible, be
quite taken away. But if vital motion be helped by motion made by sense,
then the parts of the organ will be disposed to guide the spirits in
such manner as conduceth most to the preservation and augmentation of
that motion, by the help of the nerves. And in animal motion this is the
very first endeavour, and found even in the embryo; which while it is in
the womb, moveth its limbs with voluntary motion, for the avoiding of
whatsoever troubleth it, or for the pursuing of what pleaseth it. And
this first endeavour, when it tends towards such things as are known by
experience to be pleasant, is called _appetite_, that is, an
approaching; and when it shuns what is troublesome, _aversion_, or
flying from it. And little infants, at the beginning and as soon as they
are born, have appetite to very few things, as also they avoid very few,
by reason of their want of experience and memory; and therefore they
have not so great a variety of animal motion as we see in those that are
more grown. For it is not possible, without such knowledge as is derived
from sense, that is, without experience and memory, to know what will
prove pleasant or hurtful; only there is some place for conjecture from
the looks or aspects of things. And hence it is, that though they do not
know what may do them good or harm, yet sometimes they approach and
sometimes retire from the same thing, as their doubt prompts them. But
afterwards, by accustoming themselves by little and little, they come to
know readily what is to be pursued and what to be avoided; and also to
have a ready use of their nerves and other organs, in the pursuing and
avoiding of good and bad. Wherefore appetite and aversion are the first
endeavours of animal motion.

Consequent to this first endeavour, is the impulsion into the nerves and
retraction again of animal spirits, of which it is necessary there be
some receptacle or place near the original of the nerves; and this
motion or endeavour is followed by a swelling and relaxation of the
muscles; and lastly, these are followed by contraction and extension of
the limbs, which is animal motion.

[Sidenote: Deliberation and will, what.]

13. The considerations of appetites and aversions are divers. For seeing
living creatures have sometimes appetite and sometimes aversion to the
same thing, as they think it will either be for their good or their
hurt; while that vicissitude of appetites and aversions remains in them,
they have that series of thoughts which is called _deliberation_; which
lasteth as long as they have it in their power to obtain that which
pleaseth, or to avoid that which displeaseth them. Appetite, therefore,
and aversion are simply so called as long as they follow not
deliberation. But if deliberation have gone before, then the last act of
it, if it be appetite, is called _will_; if aversion, _unwillingness_.
So that the same thing is called both will and appetite; but the
consideration of them, namely, before and after deliberation, is divers.
Nor is that which is done within a man whilst he willeth any thing,
different from that which is done in other living creatures, whilst,
deliberation having preceded, they have appetite.

Neither is the freedom of willing or not willing, greater in man, than
in other living creatures. For where there is appetite, the entire cause
of appetite hath preceded; and, consequently, the act of appetite could
not choose but follow, that is, hath of necessity followed (as is shown
in chapter IX, article 5). And therefore such a liberty as is free from
necessity, is not to be found in the will either of men or beasts. But
if by liberty we understand the faculty or power, not of willing, but of
doing what they will, then certainly that liberty is to be allowed to
both, and both may equally have it, whensoever it is to be had.

Again, when appetite and aversion do with celerity succeed one another,
the whole series made by them hath its name sometimes from one,
sometimes from the other. For the same deliberation, whilst it inclines
sometimes to one, sometimes to the other, is from appetite called
_hope_, and from aversion, _fear_. For where there is no hope, it is not
to be called fear, but _hate_; and where no fear, not hope, but
_desire_. To conclude, all the passions, called passions of the mind,
consist of appetite and aversion, except pure pleasure and pain, which
are a certain fruition of good or evil; as anger is aversion from some
imminent evil, but such as is joined with appetite of avoiding that evil
by force. But because the passions and perturbations of the mind are
innumerable, and many of them not to be discerned in any creatures
besides men; I will speak of them more at large in that section which is
concerning _man_. As for those objects, if there be any such, which do
not at all stir the mind, we are said to contemn them.

And thus much of sense in general. In the next place I shall speak of
sensible objects.


                                -------


                             CHAPTER XXVI.

                     OF THE WORLD AND OF THE STARS.

1. The magnitude and duration of the world, inscrutable.—2. No place in
    the world empty.—3. The arguments of Lucretius for vacuum,
    invalid.—4. Other arguments for the establishing of vacuum,
    invalid.—5. Six suppositions for the salving of the phenomena of
    nature.—6. Possible causes of the motions annual and diurnal; and of
    the apparent direction, station, and retrogradation of the
    planets.—7. The supposition of simple motion, why likely.—8. The
    cause of the eccentricity of the annual motion of the earth.—9. The
    cause why the moon hath always one and the same face turned towards
    the earth.—10. The cause of the tides of the ocean.—11. The cause of
    the precession of the equinoxes.


[Sidenote: The magnitude and duration of the world, inscrutable.]

1. Consequent to the contemplation of sense is the contemplation of
bodies, which are the efficient causes or objects of sense. Now every
object is either a part of the whole world, or an aggregate of parts.
The greatest of all bodies, or sensible objects, is the world itself;
which we behold when we look round about us from this point of the same
which we call the earth. Concerning the world, as it is one aggregate of
many parts, the things that fall under inquiry are but few; and those we
can determine, none. Of the whole world we may inquire what is its
magnitude, what its duration, and how many there be, but nothing else.
For as for place and time, that is to say, magnitude and duration, they
are only our own fancy of a body simply so called, that is to say, of a
body indefinitely taken, as I have shown before in chapter VII. All
other phantasms are of bodies or objects, as they are distinguished from
one another; as colour, the phantasm of coloured bodies; sound, of
bodies that move the sense of hearing, &c. The questions concerning the
magnitude of the world are whether it be finite or infinite, full or not
full; concerning its duration, whether it had a beginning, or be
eternal; and concerning the number, whether there be one or many; though
as concerning the number, if it were of infinite magnitude, there could
be no controversy at all. Also if it had a beginning, then by what cause
and of what matter it was made; and again, from whence that cause and
that matter had their being, will be new questions; till at last we come
to one or many eternal cause or causes. And the determination of all
these things belongeth to him that professeth the universal doctrine of
philosophy, in case as much could be known as can be sought. But the
knowledge of what is infinite can never be attained by a finite
inquirer. Whatsoever we know that are men, we learn it from our
phantasms; and of infinite, whether magnitude or time, there is no
phantasm at all; so that it is impossible either for a man or any other
creature to have any conception of infinite. And though a man may from
some effect proceed to the immediate cause thereof, and from that to a
more remote cause, and so ascend continually by right ratiocination from
cause to cause; yet he will not be able to proceed eternally, but
wearied will at last give over, without knowing whether it were possible
for him to proceed to an end or not. But whether we suppose the world to
be finite or infinite, no absurdity will follow. For the same things
which now appear, might appear, whether the Creator had pleased it
should be finite or infinite. Besides, though from this, that nothing
can move itself, it may rightly be inferred that there was some first
eternal movent; yet it can never be inferred, though some used to make
such inference, that that movent was eternally immoveable, but rather
eternally moved. For as it is true, that nothing is moved by itself; so
it is true also that nothing is moved but by that which is already
moved. The questions therefore about the magnitude and beginning of the
world, are not to be determined by philosophers, but by those that are
lawfully authorized to order the worship of God. For as Almighty God,
when he had brought his people into Judæa, allowed the priests the first
fruits reserved to himself; so when he had delivered up the world to the
disputations of men, it was his pleasure that all opinions concerning
the nature of infinite and eternal, known only to himself, should, as
the first fruits of wisdom, be judged by those whose ministry he meant
to use in the ordering of religion. I cannot therefore commend those
that boast they have demonstrated, by reasons drawn from natural things,
that the world had a beginning. They are contemned by idiots, because
they understand them not; and by the learned, because they understand
them; by both deservedly. For who can commend him that demonstrates
thus? "If the world be eternal, then an infinite number of days, or
other measures of time, preceded the birth of Abraham. But the birth of
Abraham preceded the birth of Isaac; and therefore one infinite is
greater than another infinite, or one eternal than another eternal;
which," he says, "is absurd." This demonstration is like his, who from
this, that the number of even numbers is infinite, would conclude that
there are as many even numbers as there are numbers simply, that is to
say, the even numbers are as many as all the even and odd together.
They, which in this manner take away eternity from the world, do they
not by the same means take away eternity from the Creator of the world?
From this absurdity therefore they run into another, being forced to
call eternity _nunc stans_, a standing still of the present time, or an
abiding now; and, which is much more absurd, to give to the infinite
number of numbers the name of unity. But why should eternity be called
an abiding now, rather than an abiding then? Wherefore there must either
be many eternities, or _now_ and _then_ must signify the same. With such
demonstrators as these, that speak in another language, it is impossible
to enter into disputation. And the men, that reason thus absurdly, are
not idiots, but, which makes the absurdity unpardonable, geometricians,
and such as take upon them to be judges, impertinent, but severe judges
of other men's demonstrations. The reason is this, that as soon as they
are entangled in the words _infinite_ and _eternal_, of which we have in
our mind no idea, but that of our own insufficiency to comprehend them,
they are forced either to speak something absurd, or, which they love
worse, to hold their peace. For geometry hath in it somewhat like wine,
which, when new, is windy; but afterwards though less pleasant, yet more
wholesome. Whatsoever therefore is true, young geometricians think
demonstrable; but elder not. Wherefore I purposely pass over the
questions of infinite and eternal; contenting myself with that doctrine
concerning the beginning and magnitude of the world, which I have been
persuaded to by the holy Scriptures and fame of the miracles which
confirm them; and by the custom of my country, and reverence due to the
laws. And so I pass on to such things as it is not unlawful to dispute
of.

[Sidenote: No place in the world empty.]

2. Concerning the world it is further questioned, whether the parts
thereof be contiguous to one another, in such manner as not to admit of
the least empty space between; and the disputation both for and against
it is carried on with probability enough. For the taking away of vacuum,
I will instance in only one experiment, a common one, but I think
unanswerable.

Let A B (in fig. 1) represent a vessel, such as gardeners use to water
their gardens withal; whose bottom B is full of little holes; and whose
mouth A may be stopped with one's finger, when there shall be need. If
now this vessel be filled with water, the hole at the top A being
stopped, the water will not flow out at any of the holes in the bottom
B. But if the finger be removed to let in the air above, it will run out
at them all; and as; soon as the finger is applied to it again, the
water will suddenly and totally be stayed again from running out. The
cause whereof seems to be no other but this, that the water cannot by
its natural endeavour to descend drive down the air below it, because
there is no place for it to go into, unless either by thrusting away the
next contiguous air, it proceed by continual endeavour to the hole A,
where it may enter and succeed into the place of the water that floweth
out, or else, by resisting the endeavour of the water downwards,
penetrate the same and pass up through it. By the first of these ways,
while the hole at A remains stopped, there is no possible passage; nor
by the second, unless the holes be so great that the water, flowing out
at them, can by its own weight force the air at the same time to ascend
into the vessel by the same holes: as we see it does in a vessel whose
mouth is wide enough, when we turn suddenly the bottom upwards to pour
out the water; for then the air being forced by the weight of the water,
enters, as is evident by the sobbing and resistance of the water, at the
sides or circumference of the orifice. And this I take for a sign that
all space is full; for without this, the natural motion of the water,
which is a heavy body, downwards, would not be hindered.

[Sidenote: The arguments of Lucretius for vacuum invalid.]

3. On the contrary, for the establishing of vacuum, many and specious
arguments and experiments have been brought. Nevertheless there seems to
be something wanting in all of them to conclude it firmly. These
arguments for vacuum are partly made by the followers of the doctrine of
Epicurus; who taught that the world consists of very small spaces not
filled by any body, and of very small bodies that have within them no
empty space, which by reason of their hardness he calls _atoms_; and
that these small bodies and spaces are every where intermingled. Their
arguments are thus delivered by Lucretius.

And first he says, that unless it were so, there could be no motion. For
the office and property of bodies is to withstand and hinder motion. If,
therefore, the universe were filled with body, motion would everywhere
be hindered, so as to have no beginning anywhere; and consequently there
would be no motion at all. It is true that in whatsoever is full and at
rest in all its parts, it is not possible motion should have beginning.
But nothing is drawn from hence for the proving of vacuum. For though it
should be granted that there is vacuum, yet if the bodies which are
intermingled with it, should all at once and together be at rest, they
would never be moved again. For it has been demonstrated above, in chap.
IX, art. 7, that nothing can be moved but by that which is contiguous
and already moved. But supposing that all things are at rest together,
there can be nothing contiguous and moved, and therefore no beginning of
motion. Now the denying of the beginning of motion, doth not take away
present motion, unless beginning be taken away from body also. For
motion may be either co-eternal, or concreated with body. Nor doth it
seem more necessary that bodies were first at rest, and afterwards
moved, than that they were first moved, and rested, if ever they rested
at all, afterwards. Neither doth there appear any cause, why the matter
of the world should, for the admission of motion, be intermingled with
empty spaces rather than full; I say full, but withal fluid. Nor,
lastly, is there any reason why those hard atoms may not also, by the
motion of intermingled fluid matter, be congregated and brought together
into compounded bodies of such bigness as we see. Wherefore nothing can
by this argument be concluded, but that motion was either coeternal, or
of the same duration with that which is moved; neither of which
conclusions consisteth with the doctrine of Epicurus, who allows neither
to the world nor to motion any beginning at all. The necessity,
therefore, of vacuum is not hitherto demonstrated. And the cause, as far
as I understand from them that have discoursed with me of vacuum, is
this, that whilst they contemplate the nature of fluid, they conceive it
to consist, as it were, of small grains of hard matter, in such manner
as meal is fluid, made so by grinding of the corn; when nevertheless it
is possible to conceive fluid to be of its own nature as homogeneous as
either an atom, or as vacuum itself.

The second of their arguments is taken from weight, and is contained in
these verses of Lucretius:

          Corporis officium est quoniam premere omnia deorsum;
          Contra autem natura manet sine pondere inanis;
          Ergo, quod magnum est æque, leviusque videtur,
          Nimirum plus esse sibi declarat inanis.--I. 363-66.

That is to say, _seeing the office and property of body is to press all
things downwards; and on the contrary, seeing the nature of vacuum is to
have no weight at all; therefore when of two bodies of equal magnitude,
one is lighter than the other, it is manifest that the lighter body hath
in it more vacuum than the other_.

To say nothing of the assumption concerning the endeavour of bodies
downwards, which is not rightly assumed, because the world hath nothing
to do with downwards, which is a mere fiction of ours; nor of this, that
if all things tended to the same lowest part of the world, either there
would be no coalescence at all of bodies, or they would all be gathered
together into the same place: this only is sufficient to take away the
force of the argument, that air, intermingled with those his atoms, had
served as well for his purpose as his intermingled vacuum.

The third argument is drawn from this, that lightning, sound, heat and
cold, do penetrate all bodies, except atoms, how solid soever they be.
But this reason, except it be first demonstrated that the same things
cannot happen without vacuum by perpetual generation of motion, is
altogether invalid. But that all the same things may so happen, shall in
due place be demonstrated.

Lastly, the fourth argument is set down by the same Lucretius in these
verses:

           Duo de concursu corpora lata
       Si cita dissiliant, nempe aer omne necesse est,
       Inter corpora quod fuerat, possidat inane.
       Is porro quamvis circum celerantibus auris
       Confluat, haud poterit tamen uno tempore totum
       Compleri spatium; nam primum quemque necesse est
       Occupet ille locum, deinde omnia possideantur.--I. 385-91.

That is, _if two flat bodies be suddenly pulled asunder, of necessity
the air must come between them to fill up the space they left empty. But
with what celerity soever the air flow in, yet it cannot in one instant
of time fill the whole space, but first one part of it, then
successively all_. Which nevertheless is more repugnant to the opinion
of Epicurus, than of those that deny vacuum. For though it be true, that
if two bodies were of infinite hardness, and were joined together by
their superficies which were most exactly plane, it would be impossible
to pull them asunder, in regard it could not be done but by motion in an
instant; yet, if as the greatest of all magnitudes cannot be given, nor
the swiftest of all motions, so neither the hardest of all bodies; it
might be, that by the application of very great force, there might be
place made for a successive flowing in of the air, namely, by separating
the parts of the joined bodies by succession, beginning at the outermost
and ending at the innermost part. He ought, therefore, first to have
proved, that there are some bodies extremely hard, not relatively as
compared with softer bodies, but absolutely, that is to say, infinitely
hard; which is not true. But if we suppose, as Epicurus doth, that atoms
are indivisible, and yet have small superficies of their own; then if
two bodies should be joined together by many, or but one only small
superficies of either of them, then I say this argument of Lucretius
would be a firm demonstration, that no two bodies made up of atoms, as
he supposes, could ever possibly be pulled asunder by any force
whatsoever. But this is repugnant to daily experience.

[Sidenote: Other arguments for the establishing of vacuum, invalid.]

4. And thus much of the arguments of Lucretius. Let us now consider the
arguments which are drawn from the experiments of later writers.

I. The first experiment is this: that if a hollow vessel be thrust into
water with the bottom upwards, the water will ascend into it; which they
say it could not do, unless the air within were thrust together into a
narrower place; and that this were also impossible, except there were
little empty places in the air. Also, that when the air is compressed to
a certain degree, it can receive no further compression, its small
particles not suffering themselves to be pent into less room. This
reason, if the air could not pass through the water as it ascends within
the vessel, might seem valid. But it is sufficiently known, that air
will penetrate water by the application of a force equal to the gravity
of the water. If therefore the force, by which the vessel is thrust
down, be greater or equal to the endeavour by which the water naturally
tendeth downwards, the air will go out that way where the resistance is
made, namely, towards the edges of the vessel. For, by how much the
deeper is the water which is to be penetrated, so much greater must be
the depressing force. But after the vessel is quite under water, the
force by which it is depressed, that is to say, the force by which the
water riseth up, is no longer increased. There is therefore such an
equilibration between them, as that the natural endeavour of the water
downwards is equal to the endeavour by which the same water is to be
penetrated to the increased depth.

II. The second experiment is, that if a concave cylinder of sufficient
length, made of glass, that the experiment may be the better seen,
having one end open and the other close shut, be filled with
quicksilver, and the open end being stopped with one's finger, be
together with the finger dipped into a dish or other vessel, in which
also there is quicksilver, and the cylinder be set upright, we shall,
the finger being taken away to make way for the descent of the
quicksilver, see it descend into the vessel under it, till there be only
so much remaining within the cylinder as may fill about twenty-six
inches of the same; and thus it will always happen whatsoever be the
cylinder, provided that the length be not less than twenty-six inches.
From whence they conclude that the cavity of the cylinder above the
quicksilver remains empty of all body. But in this experiment I find no
necessity at all of vacuum. For when the quicksilver which is in the
cylinder descends, the vessel under it must needs be filled to a greater
height, and consequently so much of the contiguous air must be thrust
away as may make place for the quicksilver which is descended. Now if it
be asked whither that air goes, what can be answered but this, that it
thrusteth away the next air, and that the next, and so successively,
till there be a return to the place where the propulsion first began.
And there, the last air thus thrust on will press the quicksilver in the
vessel with the same force with which the first air was thrust away; and
if the force with which the quicksilver descends be great enough, which
is greater or less as it descends from a place of greater or less
height, it will make the air penetrate the quicksilver in the vessel,
and go up into the cylinder to fill the place which they thought was
left empty. But because the quicksilver hath not in every degree of
height force enough to cause such penetration, therefore in descending
it must of necessity stay somewhere, namely, there, where its endeavour
downwards, and the resistance of the same to the penetration of the air,
come to an equilibrium. And by this experiment it is manifest, that this
equilibrium will be at the height of twenty-six inches, or thereabouts.

III. The third experiment is, that when a vessel hath as much air in it
as it can naturally contain, there may nevertheless be forced into it as
much water as will fill three quarters of the same vessel. And the
experiment is made in this manner. Into the glass bottle, represented
(in figure 2) by the sphere F G, whose centre is A, let the pipe B A C
be so fitted, that it may precisely fill the mouth of the bottle; and
let the end B be so near the bottom, that there may be only space enough
left for the free passage of the water which is thrust in above. Let the
upper end of this pipe have a cover at D, with a spout at E, by which
the water, when it ascends in the pipe, may run out. Also let H C be a
cock, for the opening or shutting of the passage of the water between B
and D, as there shall be occasion. Let the cover D E be taken off, and
the cock H C being opened, let a syringe full of water be forced in; and
before the syringe be taken away, let the cock be turned to hinder the
going out of the air. And in this manner let the injection of water be
repeated as often as it shall be requisite, till the water rise within
the bottle; for example, to G F. Lastly, the cover being fastened on
again, and the cock H C opened, the water will run swiftly out at E, and
sink by little and little from G F to the bottom of the pipe B.

From this phenomenon, they argue for the necessity of vacuum in this
manner. The bottle, from the beginning, was full of air; which air could
neither go out by penetrating so great a length of water as was injected
by the pipe, nor by any other way. Of necessity, therefore, all the
water as high as F G, as also all the air that was in the bottle before
the water was forced in, must now be in the same place, which at first
was filled by the air alone; which were impossible, if all the space
within the bottle were formerly filled with air precisely, that is,
without any vacuum. Besides, though some man perhaps may think the air,
being a thin body, may pass through the body of the water contained in
the pipe, yet from that other phenomenon, namely, that all the water
which is in the space B F G is cast out again by the spout at E, for
which it seems impossible that any other reason can be given besides the
force by which the air frees itself from compression, it follows, that
either there was in the bottle some space empty, or that many bodies may
be together in the same place. But this last is absurd; and therefore
the former is true, namely, that there was vacuum.

This argument is infirm in two places. For first, that is assumed which
is not to be granted; and in the second place, an experiment is brought,
which I think is repugnant to vacuum. That which is assumed is, that the
air can have no passage out through the pipe. Nevertheless, we see daily
that air easily ascends from the bottom to the superficies of a river,
as is manifest by the bubbles that rise; nor doth it need any other
cause to give it this motion, than the natural endeavour downwards of
the water. Why, therefore, may not the endeavour upwards of the same
water, acquired by the injection, which endeavour upwards is greater
than the natural endeavour of the water downwards, cause the air in the
bottle to penetrate in like manner the water that presseth it downwards;
especially, seeing the water, as it riseth in the bottle, doth so press
the air that is above it, as that it generateth in every part thereof an
endeavour towards the external superficies of the pipe, and consequently
maketh all the parts of the enclosed air to tend directly towards the
passage at B? I say, this is no less manifest, than that the air which
riseth up from the bottom of a river should penetrate the water, how
deep soever it be. Wherefore I do not yet see any cause why the force,
by which the water is injected, should not at the same time eject the
air.

And as for their arguing the necessity of vacuum from the rejection of
the water; in the first place, supposing there is vacuum, I demand by
what principle of motion that ejection is made. Certainly, seeing this
motion is from within outwards, it must needs be caused by some agent
within the bottle; that is to say, by the air itself. Now the motion of
that air, being caused by the rising of the water, begins at the bottom,
and tends upwards; whereas the motion by which it ejecteth the water
ought to begin above, and tend downwards. From whence therefore hath the
enclosed air this endeavour towards the bottom? To this question I know
not what answer can be given, unless it be said, that the air descends
of its own accord to expel the water. Which, because it is absurd, and
that the air, after the water is forced in, hath as much room as its
magnitude requires, there will remain no cause at all why the water
should be forced out. Wherefore the assertion of vacuum is repugnant to
the very experiment which is here brought to establish it.

Many other phenomena are usually brought for vacuum, as those of
_weather-glasses_, _æolipyles_, _wind-guns_, &c. which would all be very
hard to be salved, unless water be penetrable by air, without the
intermixture of empty space. But now, seeing air may with no great
endeavour pass through not only water, but any other fluid body though
never so stubborn, as quicksilver, these phenomena prove nothing.
Nevertheless, it might in reason be expected, that he that would take
away vacuum, should without vacuum show us such causes of these
phenomena, as should be at least of equal, if not greater probability.
This therefore shall be done in the following discourse, when I come to
speak of these phenomena in their proper places. But first, the most
general hypotheses of natural philosophy are to be premised.

And seeing that suppositions are put for the true causes of apparent
effects, every supposition, except such as be absurd, must of necessity
consist of some supposed possible motion; for rest can never be the
efficient cause of anything; and motion supposeth bodies moveable; of
which there are three kinds, _fluid_, _consistent_, and _mixed of both_.
_Fluid_ are those, whose parts may by very weak endeavour be separated
from one another; and _consistent_ those for the separation of whose
parts greater force is to be applied. There are therefore degrees of
consistency; which degrees, by comparison with more or less consistent,
have the names of _hardness_ or _softness_. Wherefore a fluid body is
always divisible into bodies equally fluid, as quantity into quantities;
and soft bodies, of whatsoever degree of softness, into soft bodies of
the same degree. And though many men seem to conceive no other
difference of _fluidity_, but such as ariseth from the different
magnitudes of the parts, in which sense dust, though of diamonds, may be
called fluid; yet I understand by _fluidity_, that which is made such by
nature equally in every part of the fluid body; not as dust is fluid,
for so a house which is falling in pieces may be called fluid; but in
such manner as water seems fluid, and to divide itself into parts
perpetually fluid. And this being well understood, I come to my
suppositions.

[Sidenote: Six suppositions for the salving of the phenomena of nature.]

5. First, therefore, I suppose that the immense space, which we call the
world, is the aggregate of all bodies which are either consistent and
visible, as the earth and the stars; or invisible, as the small atoms
which are disseminated through the whole space between the earth and the
stars; and lastly, that most fluid ether, which so fills all the rest of
the universe, as that it leaves in it no empty place at all.

Secondly, I suppose with Copernicus, that the greater bodies of the
world, which are both consistent and permanent, have such order amongst
themselves, as that the sun hath the first place, Mercury the second,
Venus the third, the Earth with the moon going about it the fourth, Mars
the fifth, Jupiter with his attendants the sixth, Saturn the seventh;
and after these, the fixed stars have their several distances from the
sun.

Thirdly, I suppose that in the sun and the rest of the planets there is
and always has been a simple circular motion.

Fourthly, I suppose that in the body of the air there are certain other
bodies intermingled, which are not fluid; but withal that they are so
small, that they are not perceptible by sense; and that these also have
their proper simple motion, and are some of them more, some less hard or
consistent.

Fifthly, I suppose with Kepler that as the distance between the sun and
the earth is to the distance between the moon and the earth, so the
distance between the moon and the earth is to the semidiameter of the
earth.

As for the magnitude of the circles, and the times in which they are
described by the bodies which are in them, I will suppose them to be
such as shall seem most agreeable to the phenomena in question.

[Sidenote: Possible causes of the motions annual and diurnal; and of the
           apparent direction, station, and retrogradation of the
           planets.]

6. The causes of the different seasons of the year, and of the several
variations of days and nights in all the parts of the superficies of the
earth, have been demonstrated, first by Copernicus, and since by Kepler,
Galileus, and others, from the supposition of the earth's diurnal
revolution about its own axis, together with its annual motion about the
sun in the ecliptic according to the order of the signs; and thirdly, by
the annual revolution of the same earth about its own centre, contrary
to the order of the signs. I suppose with Copernicus, that the diurnal
revolution is from the motion of the earth, by which the equinoctial
circle is described about it. And as for the other two annual motions,
they are the efficient cause of the earth's being carried about in the
ecliptic in such manner, as that its axis is always kept parallel to
itself. Which parallelism was for this reason introduced, lest by the
earth's annual revolution its poles should seem to be necessarily
carried about the sun, contrary to experience. I have, in art. 10, chap.
XXI, demonstrated, from the supposition of simple circular motion in the
sun, that the earth is so carried about the sun, as that its axis is
thereby kept always parallel to itself. Wherefore, from these two
supposed motions in the sun, the one simple circular motion, the other
circular motion about its own centre, it may be demonstrated that the
year hath both the same variations of days and nights, as have been
demonstrated by Copernicus.

For if the circle _a b c d_ (in fig. 3) be the ecliptic, whose centre is
_e_, and diameter _a e c_; and the earth be placed in _a_, and the sun
be moved in the little circle _f g h i_, namely, according to the order
_f_, _g_, _h_, and _i_, it hath been demonstrated, that a body placed in
_a_ will be moved in the same order through the points of the ecliptic
_a_, _b_, _c_, and _d_, and will always keep its axis parallel to
itself.

But if, as I have supposed, the earth also be moved with simple circular
motion in a plane that passeth through _a_, cutting the plane of the
ecliptic so as that the common section of both the planes be in _a c_,
thus also the axis of the earth will be kept always parallel to itself.
For let the centre of the earth be moved about in the circumference of
the epicycle, whose diameter is _l a k_, which is a part of the strait
line _l a c_; therefore _l a k_, the diameter of the epicycle, passing
through the centre of the earth, will be in the plane of the ecliptic.
Wherefore seeing that by reason of the earth's simple motion both in the
ecliptic and in its epicycle, the strait line _l a k_ is kept always
parallel to itself, every other strait line also taken in the body of
the earth, and consequently its axis, will in like manner be kept always
parallel to itself; so that in what part soever of the ecliptic the
centre of the epicycle be found, and in what part soever of the epicycle
the centre of the earth be found at the same time, the axis of the earth
will be parallel to the place where the same axis would have been, if
the centre of the earth had never gone out of the ecliptic.

Now as I have demonstrated the simple annual motion of the earth from
the supposition of simple motion in the sun; so from the supposition of
simple motion in the earth may be demonstrated the monthly simple motion
of the moon. For if the names be but changed, the demonstration will be
the same, and therefore need not be repeated.

[Sidenote: The supposition of simple motion, why likely.]

7. That which makes this supposition of the sun's simple motion in the
epicycle _f g h i_ probable, is first, that the periods of all the
planets are not only described about the sun, but so described, as that
they are all contained within the zodiac, that is to say, within the
latitude of about sixteen degrees; for the cause of this seems to depend
upon some power in the sun, especially in that part of the sun which
respects the zodiac. Secondly, that in the whole compass of the heavens
there appears no other body from which the cause of this phenomenon can
in probability be derived. Besides, I could not imagine that so many and
such various motions of the planets should have no dependance at all
upon one another. But, by supposing motive power in the sun, we suppose
motion also; for power to move without motion is no power at all. I have
therefore supposed that there is in the sun for the governing of the
primary planets, and in the earth for the governing of the moon, such
motion, as being received by the primary planets and by the moon, makes
them necessarily appear to us in such manner as we see them. Whereas,
that circular motion, which is commonly attributed to them, about a
fixed axis, which is called conversion, being a motion of their parts
only, and not of their whole bodies, is insufficient to salve their
appearances. For seeing whatsoever is so moved, hath no endeavour at all
towards those parts which are without the circle, they have no power to
propagate any endeavour to such bodies as are placed without it. And as
for them that suppose this may be done by magnetical virtue, or by
incorporeal and immaterial species, they suppose no natural cause; nay,
no cause at all. For there is no such thing as an incorporeal movent,
and magnetical virtue is a thing altogether unknown; and whensoever it
shall be known, it will be found to be a motion of body. It remains,
therefore, that if the primary planets be carried about by the sun, and
the moon by the earth, they have the simple circular motions of the sun
and the earth for the causes of their circulations. Otherwise, if they
be not carried about by the sun and the earth, but that every planet
hath been moved, as it is now moved, ever since it was made, there will
be of their motions no cause natural. For either these motions were
concreated with their bodies, and their cause is supernatural; or they
are coeternal with them, and so they have no cause at all. For
whatsoever is eternal was never generated.

I may add besides, to confirm the probability of this simple motion,
that as almost all learned men are now of the same opinion with
Copernicus concerning the parallelism of the axis of the earth, it
seemed to me to be more agreeable to truth, or at least more handsome,
that it should be caused by simple circular motion alone, than by two
motions, one in the ecliptic, and the other about the earth's own axis
the contrary way, neither of them simple, nor either of them such as
might be produced by any motion of the sun. I thought best therefore to
retain this hypothesis of simple motion, and from it to derive the
causes of as many of the phenomena as I could, and to let such alone as
I could not deduce from thence.

It will perhaps be objected, that although by this supposition the
reason may be given of the parallelism of the axis of the earth, and of
many other appearances, nevertheless, seeing it is done by placing the
body of the sun in the centre of that orb which the earth describes with
its annual motion, the supposition itself is false; because this annual
orb is eccentric to the sun. In the first place, therefore, let us
examine what that eccentricity is, and whence it proceeds.

[Sidenote: The cause of the eccentricity of the annual motion of the
           earth.]

8. Let the annual circle of the earth _a b c d_ (in fig. 3) be divided
into four equal parts by the strait lines _a c_ and _b d_, cutting one
another in the centre _e_; and let _a_ be the beginning of Libra, _b_ of
Capricorn, _c_ of Aries and _d_ of Cancer; and let the whole orb _a b c
d_ be understood, according to Copernicus, to have every way so great
distance from the zodiac of the fixed stars, that it be in comparison
with it but as a point. Let the earth be now supposed to be in the
beginning of Libra at _a_. The sun, therefore, will appear in the
beginning of Aries at _c_. Wherefore, if the earth be moved from _a_ to
_b_, the apparent motion of the sun will be from _c_ to the beginning of
Cancer in _d_; and the earth being moved forwards from _b_ to _c_, the
sun also will appear to be moved forwards to the beginning of Libra in
_a_; wherefore _c d a_ will be the summer arch, and the winter arch will
be _a b c_. Now, in the time, of the sun's apparent motion in the summer
arch, there are numbered 186¾ days; and, consequently, the earth makes
in the same time the same number of diurnal conversions in the arch _a b
c_; and, therefore, the earth in its motion through the arch _c d a_
will make only 178½ diurnal conversions. Wherefore the arch _a b c_
ought to be greater than the arch _c d a_ by 8¼ days, that is to say, by
almost so many degrees. Let the arch _a r_, as also _c s_, be each of
them an arch of two degrees and 1⁄16. Wherefore the arch _r b s_ will be
greater than the semicircle _a b c_ by 4⅛ degrees, and greater than the
arch _s d r_ by 8¼ degrees. The equinoxes, therefore, will be in the
points _r_ and _s_; and therefore also, when the earth is in _r_, the
sun will appear in _s_. Wherefore the true place of the sun will be in
_t_, that is to say, without the centre of the earth's annual motion by
the quantity of the sine of the arch _a r_, or the sine of two degrees
and 16 minutes. Now this sine, putting 100,000 for the radius, will be
near 3580 parts thereof. And so much is the eccentricity of the earth's
annual motion, provided that that motion be in a perfect circle; and _s_
and _r_ are the equinoctial parts. And the strait lines _s r_ and _c a_,
produced both ways till they reach the zodiac of the fixed stars, will
fall still upon the same fixed stars; because the whole orb _a b c d_ is
supposed to have no magnitude at all in respect of the great distance of
the fixed stars.

Supposing now the sun to be in _c_, it remains that I show the cause why
the earth is nearer to the sun, when in its annual motion it is found to
be in _d_, than when it is in _b_. And I take the cause to be this. When
the earth is in the beginning of Capricorn at _b_, the sun appears in
the beginning of Cancer at _d_; and then is the midst of summer. But in
the midst of summer, the northern parts of the earth are towards the
sun, which is almost all dry land, containing all Europe and much the
greatest part of Asia and America. But when the earth is in the
beginning of Cancer at _d_, it is the midst of winter, and that part of
the earth is towards the sun, which contains those great seas called the
South Sea and the Indian Sea, which are of far greater extent than all
the dry land in that hemisphere. Wherefore by the last article of
chapter XXI, when the earth is in _d_, it will come nearer to its first
movent, that is, to the sun which is in _t_; that is to say, the earth
is nearer to the sun in the midst of winter when it is in _d_, than in
the midst of summer when it is in _b_; and, therefore, during the winter
the sun is in its _Perigæum_, and in its _Apogæum_ during the summer.
And thus I have shown a possible cause of the eccentricity of the earth;
which was to be done.

I am, therefore, of Kepler's opinion in this, that he attributes the
eccentricity of the earth to the difference of the parts thereof, and
supposes one part to be affected, and another disaffected to the sun.
And I dissent from him in this, that he thinks it to be by magnetic
virtue, and that this magnetic virtue or attraction and thrusting back
of the earth is wrought by immateriate species; which cannot be, because
nothing can give motion but a body moved and contiguous. For if those
bodies be not moved which are contiguous to a body unmoved, how this
body should begin to be moved is not imaginable; as has been
demonstrated in art. 7, chap. IX, and often inculcated in other places,
to the end that philosophers might at last abstain from the use of such
unconceivable connexions of words. I dissent also from him in this, that
he says the similitude of bodies is the cause of their mutual
attraction. For if it were so, I see no reason why one egg should not be
attracted by another. If, therefore, one part of the earth be more
affected by the sun than another part, it proceeds from this, that one
part hath more water, the other more dry land. And from hence it is, as
I showed above, that the earth comes nearer to the sun when it shines
upon that part where there is more water, than when it shines upon that
where there is more dry land.

[Sidenote: The cause why the moon hath always one and the same face
           turned towards the earth.]

9. This eccentricity of the earth is the cause why the way of its annual
motion is not a perfect circle, but either an elliptical, or almost an
elliptical line; as also why the axis of the earth is not kept exactly
parallel to itself in all places, but only in the equinoctial points.

Now seeing I have said that the moon is carried about by the earth, in
the same manner that the earth is by the sun; and that the earth goeth
about the sun in such manner as that it shows sometimes one hemisphere,
sometimes the other to the sun; it remains to be enquired, why the moon
has always one and the same face turned towards the earth.

Suppose, therefore, the sun to be moved with simple motion in the little
circle _f g h i_, (in fig. 4) whose centre is _t_; and let ♈ ♋ ♎ ♑ be
the annual circle of the earth; and _a_ the beginning of Libra. About
the point _a_ let the little circle _l k_ be described; and in it let
the centre of the earth be understood to be moved with simple motion;
and both the sun and the earth to be moved according to the order of the
signs. Upon the centre _a_ let the way of the moon _m n o p_ be
described; and let _q r_ be the diameter of a circle cutting the globe
of the moon into two hemispheres, whereof one is seen by us when the
moon is at the full, and the other is turned from us.

The diameter therefore of the moon _q o r_ will be perpendicular to the
strait line _t a_. Wherefore the moon is carried, by reason of the
motion of the earth, from _o_ towards _p_. But by reason of the motion
of the sun, if it were in _p_ it would at the same time be carried from
_p_ towards _o_; and by these two contrary movents the strait line _q r_
will be turned about; and, in a quadrant of the circle _m n o p_, it
will be turned so much as makes the fourth part of its whole conversion.
Wherefore when the moon is in _p_, _q r_ will be parallel to the strait
line _m o_. Secondly, when the moon is in _m_, the strait line _q r_
will, by reason of the motion of the earth, be in _m o_. But by the
working of the sun's motion upon it in the quadrant _p m_, the same _q
r_ will be turned so much as makes another quarter of its whole
conversion. When, therefore, the moon is in _m_, _q r_ will be
perpendicular to the strait line _o m_. By the same reason, when the
moon is in _n_, _q r_ will be parallel to the strait line _m o_; and,
the moon returning to _o_, the same _q r_ will return to its first
place; and the body of the moon will in one entire period make also one
entire conversion upon her own axis. In the making of which, it is
manifest, that one and the same face of the moon is always turned
towards the earth. And if any diameter were taken in that little circle,
in which the moon were supposed to be carried about with simple motion,
the same effect would follow; for if there were no action from the sun,
every diameter of the moon would be carried about always parallel to
itself. Wherefore I have given a possible cause why one and the same
face of the moon is always turned towards the earth.

But it is to be noted, that when the moon is without the ecliptic, we do
not always see the same face precisely. For we see only that part which
is illuminated. But when the moon is without the ecliptic, that part
which is towards us is not exactly the same with that which is
illuminated.

[Sidenote: The cause of the tides of the ocean.]

10. To these three simple motions, one of the sun, another of the moon,
and the third of the earth, in their own little circles _f g h i_, _l
k_, and _q r_, together with the diurnal conversion of the earth, by
which conversion all things that adhere to its superficies are
necessarily carried about with it, may be referred the three phenomena
concerning the tides of the ocean. Whereof the first is the alternate
elevation and depression of the water at the shores, twice in the space
of twenty-four hours and near upon fifty-two minutes; for so it has
constantly continued in all ages. The second, that at the new and full
moons, the elevations of the water are greater than at other times
between. And the third, that when the sun is in the equinoctial, they
are yet greater than at any other time. For the salving of which
phenomena, we have already the four above-mentioned motions; to which I
assume also this, that the part of the earth which is called America,
being higher than the water, and extended almost the space of a whole
semicircle from north to south, gives a stop to the motion of the water.

This being granted, in the same 4th figure, where _l b k c_ is supposed
to be in the plane of the moon's monthly motion, let the little circle
_l d k e_ be described about the same centre _a_ in the plane of the
equinoctial. This circle therefore will decline from the circle _l b k
c_ in an angle of almost 28½ degrees; for the greatest declination of
the ecliptic is 23½, to which adding 5 for the greatest declination of
the moon from the ecliptic, the sum will be 28½ degrees. Seeing now the
waters, which are under the circle of the moon's course, are by reason
of the earth's simple motion in the plane of the same circle moved
together with the earth, that is to say, together with their own
bottoms, neither outgoing nor outgone; if we add the diurnal motion, by
which the other waters which are under the equinoctial are moved in the
same order, and consider withal that the circles of the moon and of the
equinoctial intersect one another; it will be manifest, that both those
waters, which are under the circle of the moon, and under the
equinoctial, will run together under the equinoctial; and consequently,
that their motion will not only be swifter than the ground that carries
them; but also that the waters themselves will have greater elevation
whensoever the earth is in the equinoctial. Wherefore, whatsoever the
cause of the tides may be, this may be the cause of their augmentation
at that time.

Again, seeing I have supposed the moon to be carried about by the simple
motion of the earth in the little circle _l b k c_; and demonstrated, at
the 4th article of chapter XXI, that whatsoever is moved by a movent
that hath simple motion, will be moved always with the same velocity; it
follows, that the centre of the earth will be carried in the
circumference _l b k c_ with the same velocity with which the moon is
carried in the circumference _m n o p_. Wherefore the time, in which the
moon is carried about in _m n o p_, is to the time, in which the earth
is carried about in _l b k c_, as one circumference to the other, that
is, as _a o_ to _a k_. But _a o_ is observed to be to the semidiameter
of the earth as 59 to 1; and therefore the earth, if _a k_ be put for
its semidiameter, will make fifty-nine revolutions in _l b k c_ in the
time that the moon makes one monthly circuit in _m n o p_. But the moon
makes her monthly circuit in little more than twenty-nine days.
Wherefore the earth shall make its circuit in the circumference _l b k
c_ in twelve hours and a little more, namely, about twenty-six minutes
more; that is to say, it shall make two circuits in twenty-four hours
and almost fifty-two minutes; which is observed to be the time between
the high-water of one day and the high-water of the day following. Now
the course of the waters being hindered by the southern part of America,
their motion will be interrupted there; and consequently, they will be
elevated in those places, and sink down again by their own weight, twice
in the space of twenty-four hours and fifty-two minutes. And thus I have
given a possible cause of the diurnal reciprocation of the ocean.

Now from this swelling of the ocean in those parts of the earth, proceed
the flowings and ebbings in the Atlantic, Spanish, British, and German
seas; which though they have their set times, yet upon several shores
they happen at several hours of the day. And they receive some
augmentation from the north, by reason that the shores of China and
Tartary, hindering the general course of the waters, make them swell
there, and discharge themselves in part through the strait of Anian into
the Northern Ocean, and so into the German Sea.

As for the spring tides which happen at the time of the new and full
moons, they are caused by that simple motion, which at the beginning I
supposed to be always in the moon. For as, when I showed the cause of
the eccentricity of the earth, I derived the elevation of the waters
from the simple motion of the sun; so the same may here be derived from
the simple motion of the moon. For though from the generation of clouds,
there appear in the sun a more manifest power of elevating the waters
than in the moon; yet the power of increasing moisture in vegetables and
living creatures appears more manifestly in the moon than in the sun;
which may perhaps proceed from this, that the sun raiseth up greater,
and the moon lesser drops of water. Nevertheless, it is more likely, and
more agreeable to common observation, that rain is raised not only by
the sun, but also by the moon; for almost all men expect change of
weather at the time of the conjunctions of the sun and moon with one
another and with the earth, more than in the time of their quarters.

In the last place, the cause why the spring tides are greater at the
time of the equinoxes hath been already sufficiently declared in this
article, where I have demonstrated, that the two motions of the earth,
namely, its simple motion in the little circle _l b k c_, and its
diurnal motion in _l d k e_, cause necessarily a greater elevation of
waters when the sun is about the equinoxes, than when he is in other
places. I have therefore given possible causes of the phenomenon of the
flowing and ebbing of the ocean.

[Sidenote: Cause of the precession of the equinoxes.]

11. As for the explication of the yearly _precession of the equinoctial
points_, we must remember that, as I have already shown, the annual
motion of the earth is not in the circumference of a circle, but of an
ellipsis, or a line not considerably different from that of an ellipsis.
In the first place, therefore, this elliptical line is to be described.

Let the ecliptic ♎ ♑ ♈ ♋ (in fig. 5) be divided into four equal parts by
the two strait lines _a b_ and ♑ ♋, cutting one another at right angles
in the centre _c_. And taking the arch _b d_ of two degrees and sixteen
minutes, let the strait line _d e_ be drawn parallel to _a b_, and
cutting ♑ ♋ in _f_; which being done, the eccentricity of the earth will
be _c f_. Seeing therefore the annual motion of the earth is in the
circumference of an ellipsis, of which ♑ ♋ is the greater axis, _a b_
cannot be the lesser axis; for _a b_ and ♑ ♋ are equal. Wherefore the
earth passing through _a_ and _b_, will either pass above ♑, as through
_g_, or passing through ♑, will fall between _c_ and _a_; it is no
matter which. Let it pass therefore through _g_; and let _g l_ be taken
equal to the strait line ♑ ♋; and dividing _g l_ equally in _i_, _g i_
will be equal to ♑ _f_, and _i l_ equal to _f_ ♋; and consequently the
point _i_ will cut the eccentricity _c f_ into two equal parts; and
taking _i h_ equal to _i f_, _h i_ will be the whole eccentricity. If
now a strait line, namely, the line ♎ _i_ ♈, be drawn through _i_
parallel to the strait lines _a b_ and _e d_, the way of the sun in
summer, namely, the arch ♎ _g_ ♈, will be greater than his way in
winter, by 8¼ degrees. Wherefore the true equinoxes will be in the
strait line ♎ _i_ ♈; and therefore the ellipsis of the earth's annual
motion will not pass through _a_, _g_, _b_, and _l_; but through ♎, _g_,
♈ and _l_. Wherefore the annual motion of the earth is in the ellipsis ♎
_g_ ♈ _l_; and cannot be, the eccentricity being salved, in any other
line. And this perhaps is the reason, why Kepler, against the opinion of
all the astronomers of former time, thought fit to bisect the
eccentricity of the earth, or, according to the ancients, of the sun,
not by diminishing the quantity of the same eccentricity, (because the
true measure of that quantity is the difference by which the summer arch
exceeds the winter arch), but by taking for the centre of the ecliptic
of the great orb the point _c_ nearer to _f_, and so placing the whole
great orb as much nearer to the ecliptic of the fixed stars towards ♋,
as is the distance between _c_ and _i_. For seeing the whole great orb
is but as a point in respect of the immense distance of the fixed stars,
the two strait lines ♎ ♈ and _a b_, being produced both ways to the
beginnings of Aries and Libra, will fall upon the same points of the
sphere of the fixed stars. Let therefore the diameter of the earth _m n_
be in the plane of the earth's annual motion. If now the earth be moved
by the sun's simple motion in the circumference of the ecliptic about
the centre _i_, this diameter will be kept always parallel to itself and
to the strait line _g l_. But seeing the earth is moved in the
circumference of an ellipsis without the ecliptic, the point _n_, whilst
it passeth through ♎ ♑ ♈, will go in a lesser circumference than the
point _m_; and consequently, as soon as ever it begins to be moved, it
will lose its parallelism with the strait line ♑ ♋; so that _m n_
produced will at last cut the strait line _g l_ produced. And
contrarily, as soon as _m n_ is past ♈, the earth making its way in the
internal elliptical line ♈ _l_ ♎, the same _m n_ produced towards _m_,
will cut _l g_ produced. And when the earth hath almost finished its
whole circumference, the same _m n_ shall again make a right angle with
a line drawn from the centre _i_, a little short of the point from which
the earth began its motion. And there the next year shall be one of the
equinoctial points, namely, near the end of ♍; the other shall be
opposite to it near the end of ♓. And thus the points in which the days
and nights are made equal do every year fall back; but with so slow a
motion, that, in a whole year, it makes but 51 first minutes. And this
relapse being contrary to the order of the signs, is commonly called the
_precession of the equinoxes_. Of which I have from my former
suppositions deduced a possible cause; which was to be done.

According to what I have said concerning the cause of the eccentricity
of the earth; and according to Kepler, who for the cause thereof
supposeth one part of the earth to be affected to the sun, the other
part to be disaffected; the apogæum and perigæum of the sun should be
moved every year in the same order, and with the same velocity, with
which the equinoctial points are moved; and their distance from them
should always be the quadrant of a circle; which seems to be otherwise.
For astronomers say, that the equinoxes are now, the one about 28
degrees gone back from the first star of Aries, the other as much from
the beginning of Libra; so that the apogæum of the sun or the aphelium
of the earth ought to be about the 28th degree of Cancer. But it is
reckoned to be in the 7th degree. Seeing, therefore, we have not
sufficient evidence of the ὁτί (that so it is,) it is in vain to seek
for the διότι (why it is so.) Wherefore, as long as the motion of the
apogæum is not observable by reason of the slowness thereof, and as long
as it remains doubtful whether their distance from the equinoctial
points be more or less than a quadrant precisely; so long it may be
lawful for me to think they proceed both of them with equal velocity.

Also, I do not at all meddle with the causes of the eccentricities of
Saturn, Jupiter, Mars, and Mercury. Nevertheless, seeing the
eccentricity of the earth may, as I have shewn, be caused by the unlike
constitution of the several parts of the earth which are alternately
turned towards the sun, it is credible also, that like effects may be
produced in these other planets from their having their superficies of
unlike parts.

And this is all I shall say concerning Sidereal Philosophy. And, though
the causes I have here supposed be not the true causes of these
phenomena, yet I have demonstrated that they are sufficient to produce
them, according to what I at first propounded.

[Illustration:

  _Vol. 1. Lat. & Eng._
  C. XXVI.
  _Fig. 1-5_
]


                                -------


                             CHAPTER XXVII.
                    OF LIGHT, HEAT, AND OF COLOURS.

1. Of the immense magnitude of some bodies, and the unspeakable
    littleness of others.—2. Of the cause of the light of the sun.—3.
    How light heateth.—4. The generation of fire from the sun.—5. The
    generation of fire from collision.—6. The cause of light in
    glow-worms, rotten wood, and the Bolognan stone.—7. The cause of
    light in the concussion of sea water.—8. The cause of flame, sparks,
    and colliquation.—9. The cause why wet hay sometimes burns of its
    own accord; also the cause of lightning.—10. The cause of the force
    of gunpowder; and what is to be ascribed to the coals, what to the
    brimstone, and what to the nitre.—11. How heat is caused by
    attrition.—12. The distinction of light into first, second, &c.—13.
    The causes of the colours we see in looking through a prisma of
    glass, namely, of red, yellow, blue, and violet colour.—14. Why the
    moon and the stars appear redder in the horizon than in the midst of
    the heaven.—15. The cause of whiteness.—16. The cause of blackness.


[Sidenote: Of the immense magnitude of some bodies, and the unspeakable
           littleness of others.]

1. Besides the stars, of which I have spoken in the last chapter,
whatsoever other bodies there be in the world, they may be all
comprehended under the name of intersidereal bodies. And these I have
already supposed to be either the most fluid æther, or such bodies whose
parts have some degree of cohesion. Now, these differ from one another
in their several _consistencies_, _magnitudes_, _motions_, and
_figures_. In consistency, I suppose some bodies to be harder, others
softer through all the several degrees of _tenacity_. In magnitude, some
to be greater, others less, and many unspeakably little. For we must
remember that, by the understanding, quantity is divisible into
divisibles perpetually. And, therefore, if a man could do as much with
his hands as he can with his understanding, he would be able to take
from any given magnitude a part which should be less than any other
magnitude given. But the Omnipotent Creator of the world can actually
from a part of any thing take another part, as far as we by our
understanding can conceive the same to be divisible. Wherefore there is
no impossible smallness of bodies. And what hinders but that we may
think this likely? For we know there are some living creatures so small
that we can scarce see their whole bodies. Yet even these have their
young ones; their little veins and other vessels, and their eyes so
small as that no microscope can make them visible. So that we cannot
suppose any magnitude so little, but that our very supposition is
actually exceeded by nature. Besides, there are now such microscopes
commonly made, that the things we see with them appear a hundred
thousand times bigger than they would do if we looked upon them with our
bare eyes. Nor is there any doubt but that by augmenting the power of
these microscopes (for it may be augmented as long as neither matter nor
the hands of workmen are wanting) every one of those hundred thousandth
parts might yet appear a hundred thousand times greater than they did
before. Neither is the smallness of some bodies to be more admired than
the vast greatness of others. For it belongs to the same Infinite Power,
as well to augment infinitely as infinitely to diminish. To make the
great orb, namely, that whose radius reacheth from the earth to the sun,
but as a point in respect of the distance between the sun and the fixed
stars; and, on the contrary, to make a body so little, as to be in the
same proportion less than any other visible body, proceeds equally from
one and the same Author of Nature. But this of the immense distance of
the fixed stars, which for a long time was accounted an incredible
thing, is now believed by almost all the learned. Why then should not
that other, of the smallness of some bodies, become credible at some
time or other? For the Majesty of God appears no less in small things
than in great; and as it exceedeth human sense in the immense greatness
of the universe, so also it doth in the smallness of the parts thereof.
Nor are the first elements of compositions, nor the first beginnings of
actions, nor the first moments of times more credible, than that which
is now believed of the vast distance of the fixed stars.

Some things are acknowledged by mortal men to be very great, though
finite, as seeing them to be such. They acknowledge also that some
things, which they do not see, may be of infinite magnitude. But they
are not presently nor without great study persuaded, that there is any
mean between infinite and the greatest of those things which either they
see or imagine. Nevertheless, when after meditation and contemplation
many things which we wondered at before are now grown more familiar to
us, we then believe them, and transfer our admiration from the creatures
to the Creator. But how little soever some bodies may be, yet I will not
suppose their quantity to be less than is requisite for the salving of
the phenomena. And in like manner I shall suppose their motion, namely,
their velocity and slowness, and the variety of their figures, to be
only such as the explication of their natural causes requires. And
lastly, I suppose, that the parts of the pure æther, as if it were the
first matter, have no motion at all but what they receive from bodies
which float in them, and are not themselves fluid.

[Sidenote: Of the cause of the light of the sun.]

2. Having laid these grounds, let us come to speak of causes; and in the
first place let us inquire what may be the cause of the light of the
sun. Seeing, therefore, the body of the sun doth by its simple circular
motion thrust away the ambient ethereal substance sometimes one way
sometimes another, so that those parts, which are next the sun, being
moved by it, do propagate that motion to the next remote parts, and
these to the next, and so on continually; it must needs be that,
notwithstanding any distance, the foremost part of the eye will at last
be pressed; and by the pressure of that part, the motion will be
propagated to the innermost part of the organ of sight, namely, to the
heart; and from the reaction of the heart, there will proceed an
endeavour back by the same way, ending in the endeavour outwards of the
coat of the eye, called the _retina_. But this endeavour outwards, as
has been defined in chapter XXV, is the thing which is called light, or
the phantasm of a lucid body. For it is by reason of this phantasm that
an object is called lucid. Wherefore we have a possible cause of the
light of the sun; which I undertook to find.

[Sidenote: How light heateth.]

3. The generation of the light of the sun is accompanied with the
generation of heat. Now every man knows what heat is in himself, by
feeling it when he grows hot; but what it is in other things, he knows
only by ratiocination. For it is one thing to grow hot, and another
thing to heat or make hot. And therefore though we perceive that the
fire or the sun heateth, yet we do not perceive that it is itself hot.
That other living creatures, whilst they make other things hot, are hot
themselves, we infer by reasoning from the like sense in ourselves. But
this is not a necessary inference. For though it may truly be said of
living creatures, that _they heat, therefore they are themselves hot_;
yet it cannot from hence be truly inferred that _fire heateth, therefore
it is itself hot_; no more than this, _fire causeth pain, therefore it
is itself in pain_. Wherefore, that is only and properly called hot,
which when we feel we are necessarily hot.

Now when we grow hot, we find that our spirits and blood, and whatsoever
is fluid within us, is called out from the internal to the external
parts of our bodies, more or less, according to the degree of the heat;
and that our skin swelleth. He, therefore, that can give a possible
cause of this evocation and swelling, and such as agrees with the rest
of the phenomena of heat, may be thought to have given the cause of the
heat of the sun.

It hath been shown, in the 5th article of chapter XXI, that the fluid
medium, which we call the air, is so moved by the simple circular motion
of the sun, as that all its parts, even the least, do perpetually change
places with one another; which change of places is that which there I
called fermentation. From this fermentation of the air, I have, in the
8th article of the last chapter, demonstrated that the water may be
drawn up into the clouds.

And I shall now show that the fluid parts may, in like manner, by the
same fermentation, be drawn out from the internal to the external parts
of our bodies. For seeing that wheresoever the fluid medium is
contiguous to the body of any living creature, there the parts of that
medium are, by perpetual change of place, separated from one another;
the contiguous parts of the living creature must, of necessity,
endeavour to enter into the spaces of the separated parts. For otherwise
those parts, supposing there is no vacuum, would have no place to go
into. And therefore that, which is most fluid and separable in the parts
of the living creature which are contiguous to the medium, will go first
out; and into the place thereof will succeed such other parts as can
most easily transpire through the pores of the skin. And from hence it
is necessary that the rest of the parts, which are not separated, must
altogether be moved outwards, for the keeping of all places full. But
this motion outwards of all parts together must, of necessity, press
those parts of the ambient air which are ready to leave their places;
and therefore all the parts of the body, endeavouring at once that way,
make the body swell. Wherefore a possible cause is given of heat from
the sun; which was to be done.

[Sidenote: The generation of fire from the sun.]

4. We have now seen how light and heat are generated; heat by the simple
motion of the medium, making the parts perpetually change places with
one another; and light by this, that by the same simple motion action is
propagated in a strait line. But when a body hath its parts so moved,
that it sensibly both heats and shines at the same time, then it is that
we say fire is generated.

Now by fire I do not understand a body distinct from matter combustible
or glowing, as wood or iron, but the matter itself, not simply and
always, but then only when it shineth and heateth. He, therefore, that
renders a cause possible and agreeable to the rest of the phenomena,
namely, whence, and from what action, both the _shining_ and _heating_
proceed, may be thought to have given a possible cause of the generation
of _fire_.

Let, therefore, A B C (in the first figure) be a sphere, or the portion
of a sphere, whose centre is D; and let it be transparent and
homogeneous, as crystal, glass, or water, and objected to the sun.
Wherefore, the foremost part A B C will, by the simple motion of the
sun, by which it thrusts forwards the medium, be wrought upon by the
sunbeams in the strait lines E A, F B, and G C; which strait lines may,
in respect of the great distance of the sun, be taken for parallels. And
seeing the medium within the sphere is thicker than the medium without
it, those beams will be refracted towards the perpendiculars. Let the
strait lines E A and G C be produced till they cut the sphere in H and
I; and drawing the perpendiculars A D and C D, the refracted beams E A
and G C will of necessity fall, the one between A H and A D, the other
between C I and C D. Let those refracted beams be A K and C L. And
again, let the lines D K M and D L N be drawn perpendicular to the
sphere; and let A K and C L be produced till they meet with the strait
line B D produced in O. Seeing, therefore, the medium within the sphere
is thicker than that without it, the refracted line A K will recede
further from the perpendicular K M than K O will recede from the same.
Wherefore K O will fall between the refracted line and the
perpendicular. Let, therefore, the refracted line be K P, cutting F O in
P; and for the same reason the strait line L P will be the refracted
line of the strait line C L. Wherefore, seeing the beams are nothing
else but the ways in which the motion is propagated, the motion about P
will be so much more vehement than the motion about A B C, by how much
the base of the portion A B C is greater than the base of a like portion
in the sphere, whose centre is P, and whose magnitude is equal to that
of the little circle about P, which comprehendeth all the beams that are
propagated from A B C; and this sphere being much less than the sphere A
B C, the parts of the medium, that is, of the air about P, will change
places with one another with much greater celerity than those about A B
C. If, therefore, any matter combustible, that is to say, such as may be
easily dissipated, be placed in P, the parts of that matter, if the
proportion be great enough between A C and a like portion of the little
circle about P, will be freed from their mutual cohesion, and being
separated will acquire simple motion. But vehement simple motion
generates in the beholder a phantasm of lucid and hot, as I have before
demonstrated of the simple motion of the sun; and therefore the
combustible matter which is placed in P will be made lucid and hot, that
is to say, will be fire. Wherefore I have rendered a possible cause of
fire; which was to be done.

[Sidenote: The generation of fire from collision.]

5. From the manner by which the sun generateth fire, it is easy to
explain the manner by which fire may be generated by the collision of
two flints. For by that collision some of those particles of which the
stone is compacted, are violently separated and thrown off; and being
withal swiftly turned round, the eye is moved by them, as it is in the
generation of light by the sun. Wherefore they shine; and falling upon
matter which is already half dissipated, such as is tinder, they
thoroughly dissipate the parts thereof, and make them turn round. From
whence, as I have newly shown, light and heat, that is to say fire, is
generated.

[Sidenote: The cause of light in glow-worms, rotten wood, and the
           Bolognan stone.]

6. The shining of glow-worms, some kinds of rotten wood, and of a kind
of stone made at Bologna, may have one common cause, namely, the
exposing of them to the hot sun. We find by experience that the Bologna
stone shines not, unless it be so exposed; and after it has been exposed
it shines but for a little time, namely, as long as it retains a certain
degree of heat. And the cause may be that the parts, of which it is
made, may together with heat have simple motion imprinted in them by the
sun. Which if it be so, it is necessary that it shine in the dark, as
long as there is sufficient heat in it; but this ceasing, it will shine
no longer. Also we find by experience that in the glow-worm there is a
certain thick humour, like the crystalline humour of the eye; which if
it be taken out and held long enough in one's fingers, and then be
carried into the dark, it will shine by reason of the warmth it received
from the fingers; but as soon as it is cold it will cease shining. From
whence, therefore, can these creatures have their light, but from lying
all day in the sunshine in the hottest time of summer? In the same
manner, rotten wood, except it grow rotten in the sunshine, or be
afterwards long enough exposed to the sun, will not shine. That this
doth not happen in every worm, nor in all kinds of rotten wood, nor in
all calcined stones, the cause may be that the parts, of which the
bodies are made, are different both in motion and figure from the parts
of bodies of other kinds.

[Sidenote: The cause of light in the concussion of sea water.]

7. Also the sea water shineth when it is either dashed with the strokes
of oars, or when a ship in its course breaks strongly through it; but
more or less, according as the wind blows from different points. The
cause whereof may be this, that the particles of salt, though they never
shine in the salt-pits, where they are but slowly drawn up by the sun,
being here beaten up into the air in greater quantities and with more
force, are withal made to turn round, and consequently to shine, though
weakly. I have, therefore, given a possible cause of this phenomenon.

[Sidenote: The cause of flame, sparks, & colliquation.]

8. If such matter as is compounded of hard little bodies be set on fire,
it must needs be, that, as they fly out in greater or less quantities,
the flame which is made by them will be greater or less. And if the
ethereal or fluid part of that matter fly out together with them, their
motion will be the swifter, as it is in wood and other things which
flame with a manifest mixture of wind. When, therefore, these hard
particles by their flying out move the eye strongly, they shine bright;
and a great quantity of them flying out together, they make a great
shining body. For flame being nothing but an aggregate of shining
particles, the greater the aggregate is, the greater and more manifest
will be the flame. I have, therefore, shown a possible cause of flame.
And from hence the cause appears evidently, why glass is so easily and
quickly melted by the small flame of a candle blown, which will not be
melted without blowing but by a very strong fire.

Now, if from the same matter there be a part broken off, namely, such a
part as consisteth of many of the small particles, of this is made a
spark. For from the breaking off it hath a violent turning round, and
from hence it shines. But though from this matter there fly neither
flame nor sparks, yet some of the smallest parts of it may be carried
out as far as to the superficies, and remain there as ashes; the parts
whereof are so extremely small, that it cannot any longer be doubted how
far nature may proceed in dividing.

Lastly, though by the application of fire to this matter there fly
little or nothing from it, yet there will be in the parts an endeavour
to simple motion; by which the whole body will either be melted, or,
which is a degree of melting, softened. For all motion has some effect
upon all matter whatsoever, as has been shown at art. 3, chap. XV. Now
if it be softened to such a degree, as that the stubbornness of the
parts be exceeded by their gravity, then we say it is melted; otherwise,
softened and made pliant and ductile.

Again, the matter having in it some particles hard, others ethereal or
watery; if, by the application of fire, these latter be called out, the
former will thereby come to a more full contact with one another; and,
consequently, will not be so easily separated; that is to say, the whole
body will be made harder. And this may be the cause why the same fire
makes some things soft, others hard.

[Sidenote: The cause why wet hay sometimes burns of its own accord; also
           the cause of lightning.]

9. It is known by experience that if hay be laid wet together in a heap,
it will after a time begin to smoke, and then burn as it were of itself.
The cause whereof seems to be this, that in the air, which is enclosed
within the hay, there are those little bodies, which, as I have
supposed, are moved freely with simple motion. But this motion being by
degrees hindered more and more by the descending moisture, which at the
last fills and stops all the passages, the thinner parts of the air
ascend by penetrating the water; and those hard little bodies, being so
thrust together that they touch and press one another, acquire stronger
motion; till at last by the increased strength of this motion the watery
parts are first driven outwards, from whence appears vapour; and by the
continued increase of this motion, the smallest particles of the dried
hay are forced out, and recovering their natural simple motion, they
grow hot and shine, that is to say, they are set on fire.

The same also may be the cause of lightning, which happens in the
hottest time of the year, when the water is raised up in greatest
quantity and carried highest. For after the first clouds are raised,
others after others follow them; and being congealed above, they happen,
whilst some of them ascend and others descend, to fall one upon another
in such manner, as that in some places all their parts are joined
together, in others they leave hollow spaces between them; and into
these spaces, the ethereal parts being forced out by the compressure of
the clouds, many of the harder little bodies are so pent together, as
they have not the liberty of such motion as is natural to the air.
Wherefore their endeavour grows more vehement, till at last they force
their way through the clouds, sometimes in one place, sometimes in
another; and, breaking through with great noise, they move the air
violently, and striking our eyes, generate light, that is to say, they
shine. And this shining is that we call lightning.

[Sidenote: The cause of the force of gunpowder; and what is to be
           ascribed to the coals, what to the brimstone, and what to the
           nitre.]

10. The most common phenomenon proceeding from fire, and yet the most
admirable of all others, is the force of gunpowder fired; which being
compounded of nitre, brimstone and coals, beaten small, hath from the
coals its first taking fire; from the brimstone its nourishment and
flame, that is to say, light and motion, and from the nitre the
vehemence of both. Now if a piece of nitre, before it is beaten, be laid
upon a burning coal, first it melts, and, like water, quencheth that
part of the coal it toucheth. Then vapour or air, flying out where the
coal and nitre join, bloweth the coal with great swiftness and vehemence
on all sides. And from hence it comes to pass, that by two contrary
motions, the one, of the particles which go out of the burning coal, the
other, of those of the ethereal and watery substance of the nitre, is
generated that vehement motion and inflammation. And, lastly, when there
is no more action from the nitre, that is to say, when the volatile
parts of the nitre are flown out, there is found about the sides a
certain white substance, which being thrown again into the fire, will
grow red-hot again, but will not be dissipated, at least unless the fire
be augmented. If now a possible cause of this be found out, the same
will also be a possible cause why a grain of gunpowder set on fire doth
expand itself with such vehement motion, and shine. And it may be caused
in this manner.

Let the particles, of which nitre consisteth, be supposed to be some of
them hard, others watery, and the rest ethereal. Also let the hard
particles be supposed to be spherically hollow, like small bubbles, so
that many of them growing together may constitute a body, whose little
caverns are filled with a substance which is either watery, or ethereal,
or both. As soon, therefore, as the hard particles are dissipated, the
watery and ethereal particles will necessarily fly out; and as they fly,
of necessity blow strongly the burning coals and brimstone which are
mingled together; whereupon there will follow a great expansion of
light, with vehement flame, and a violent dissipation of the particles
of the nitre, the brimstone and the coals. Wherefore I have given a
possible cause of the force of fired gunpowder.

It is manifest from hence, that for the rendering of the cause why a
bullet of lead or iron, shot from a piece of ordnance, flies with so
great velocity, there is no necessity to introduce such rarefaction, as,
by the common definition of it, makes the same matter to have sometimes
more, sometimes less quantity; which is inconceivable. For every thing
is said to be greater or less, as it hath more or less quantity. The
violence with which a bullet is thrust out of a gun, proceeds from the
swiftness of the small particles of the fired powder; at least it may
proceed from that cause without the supposition of any empty space.

[Sidenote: How heat is caused by attrition.]

11. Besides, by the attrition or rubbing of one body against another, as
of wood against wood, we find that not only a certain degree of heat,
but fire itself is sometimes generated. For such motion is the
reciprocation of pressure, sometimes one way, sometimes the other; and
by this reciprocation whatsoever is fluid in both the pieces of wood is
forced hither and thither; and consequently, to an endeavour of getting
out; and at last by breaking out makes fire.

[Sidenote: The distinction of light into first, second, &c.]

12. Now light is distinguished into, first, second, third, and so on
infinitely. And we call that first light, which is in the first lucid
body; as the sun, fire, &c.: second, that which is in such bodies, as
being not transparent are illuminated by the sun; as the moon, a wall,
&c.: and third, that which is in bodies not transparent, but illuminated
by second light, &c.

[Sidenote: The causes of the colours we see in looking through a prisma
           of glass, namely, of red, yellow, blue, & violet colour.]

13. _Colour_ is light, but troubled light, namely, such as is generated
by perturbed motion; as shall be made manifest by the red, yellow, blue
and purple, which are generated by the interposition of a diaphanous
prisma, whose opposite bases are triangular, between the light and that
which is enlightened.

[Sidenote: [Discussion of Figure 27.2]]

For let there be a prisma of glass, or of any other transparent matter
which is of greater density than air; and let the triangle A B C be the
base of this prisma. Also let the strait line D E be the diameter of the
sun's body, having oblique position to the strait line A B; and let the
sunbeams pass in the lines D A and E B C. And lastly, let the strait
lines D A and E C be produced indefinitely to F and G. Seeing therefore
the strait line D A, by reason of the density of the glass, is refracted
towards the perpendicular; let the line refracted at the point A be the
strait line A H. And again, seeing the medium below A C is thinner than
that above it, the other refraction, which will be made there, will
diverge from the perpendicular. Let therefore this second refracted line
be A I. Also let the same be done at the point C, by making the first
refracted line to be C K, and the second C L. Seeing therefore the cause
of the refraction in the point A of the strait line of A B is the excess
of the resistance of the medium in A B above the resistance of the air,
there must of necessity be reaction from the point A towards the point
B; and consequently the medium at A within the triangle A B C will have
its motion troubled, that is to say, the strait motion in A F and A H
will be mixed with the transverse motion between the same A F and A H,
represented by the short transverse lines in the triangle A F H. Again,
seeing at the point A of the strait line A C there is a second
refraction from A H in A I, the motion of the medium will again be
perturbed by reason of the transverse reaction from A towards C,
represented likewise by the short transverse lines in the triangle A H
I. And in the same manner there is a double perturbation represented by
the transverse lines in the triangles C G K and C K L. But as for the
light between A I and C G, it will not be perturbed; because, if there
were in all the points of the strait lines A B and A C the same action
which is in the points A and C, then the plane of the triangle C G K
would be everywhere coincident with the plane of the triangle A F H; by
which means all would appear alike between A and C. Besides, it is to be
observed, that all the reaction at A tends towards the illuminated parts
which are between A and C, and consequently perturbeth the first light.
And on the contrary, that all the reaction at C tends towards the parts
without the triangle or without the prisma A B C, where there is none
but second light; and that the triangle A F H shows that perturbation of
light which is made in the glass itself; as the triangle A H I shows
that perturbation of light which is made below the glass. In like
manner, that C G K shows the perturbation of light within the glass; and
C K L that which is below the glass. From whence there are four divers
motions, or four different illuminations or colours, whose differences
appear most manifestly to the sense in a prisma, whose base is an
equilateral triangle, when the sunbeams that pass through it are
received upon a white paper. For the triangle A F H appears red to the
sense; the triangle A H I yellow; the triangle C G K green, and
approaching to blue; and lastly, the triangle C K L appears purple. It
is therefore evident that when weak but first light passeth through a
more resisting diaphanous body, as glass, the beams, which fall upon it
transversely, make redness; and when the same first light is stronger,
as it is in the thinner medium below the strait line A C, the transverse
beams make yellowness. Also when second light is strong, as it is in the
triangle C G K, which is nearest to the first light, the transverse
beams make greenness; and when the same second light is weaker, as in
the triangle C K L, they make a purple colour.

[Sidenote: Why the moon and the stars appear redder in the horizon than
           in the midst of the heaven.]

14. From hence may be deduced a cause, why the moon and stars appear
bigger and redder near the horizon than in the mid-heaven. For between
the eye and the apparent horizon there is more impure air, such as is
mingled with watery and earthy little bodies, than is between the same
eye and the more elevated part of heaven. But vision is made by beams
which constitute a cone, whose base, if we look upon the moon, is the
moon's face, and whose vertex is in the eye; and therefore, many beams
from the moon must needs fall upon little bodies that are without the
visual cone, and be by them reflected to the eye. But these reflected
beams tend all in lines which are transverse to the visual cone, and
make at the eye an angle which is greater than the angle of the cone.
Wherefore, the moon appears greater in the horizon, than when she is
more elevated. And because those reflected beams go transversely, there
will be generated, by the last article, redness. A possible cause
therefore is shown, why the moon as also the stars appear greater and
redder in the horizon, than in the midst of heaven. The same also may be
the cause, why the sun appears in the horizon greater and of a colour
more degenerating to yellow, than when he is higher elevated. For the
reflection from the little bodies between, and the transverse motion of
the medium, are still the same. But the light of the sun is much
stronger than that of the moon; and therefore, by the last article, his
splendour must needs by this perturbation degenerate into yellowness.

But for the generation of these four colours, it is not necessary that
the figure of the glass be a prisma; for if it were spherical it would
do the same. For in a sphere the sunbeams are twice refracted and twice
reflected. And this being observed by Des Cartes, and withal that a
rainbow never appears but when it rains; as also, that the drops of rain
have their figures almost spherical; he hath shown from thence the cause
of the colours in the rainbow; which therefore need not be repeated.

[Sidenote: The cause of whiteness.]

15. _Whiteness_ is light, but light perturbed by the reflections of many
beams of light coming to the eye together within a little space. For if
glass or any other diaphanous body be reduced to very small parts by
contusion or concussion, every one of those parts, if the beams of a
lucid body be from any one point of the same reflected to the eye, will
represent to the beholder an idea or image of the whole lucid body, that
is to say, a phantasm of white. For the strongest light is the most
white; and therefore many such parts will make many such images.
Wherefore, if those parts lie thick and close together, those many
images will appear confusedly, and will by reason of the confused light
represent a white colour. So that from hence may be deduced a possible
cause, why glass beaten, that is, reduced to powder, looks white. Also
why water and snow are white; they being nothing but a heap of very
small diaphanous bodies, namely, of little bubbles, from whose several
convex superficies there are by reflection made several confused
phantasms of the whole lucid body, that is to say, whiteness. For the
same reason, salt and nitre are white, as consisting of small bubbles
which contain within them water and air; as is manifest in nitre, from
this, that being thrown into the fire it violently blows the same; which
salt also doth, but with less violence. But if a white body be exposed,
not to the light of the day, but to that of the fire or of a candle, it
will not at the first sight be easily judged whether it be white or
yellow; the cause whereof may be this, that the light of those things,
which burn and flame, is almost of a middle colour between whiteness and
yellowness.

[Sidenote: The cause of blackness.]

16. As whiteness is light, so _blackness_ is the privation of light, or
darkness. And, from hence it is, first, that all holes, from which no
light can be reflected to the eye, appear black. Secondly, that when a
body hath little eminent particles erected straight up from the
superficies, so that the beams of light which fall upon them are
reflected not to the eye but to the body itself, that superficies
appears black; in the same manner as the sea appears black when ruffled
by the wind. Thirdly, that any combustible matter is by the fire made to
look black before it shines. For the endeavour of the fire being to
dissipate the smallest parts of such bodies as are thrown into it, it
must first raise and erect those parts before it can work their
dissipation. If, therefore, the fire be put out before the parts are
totally dissipated, the coal will appear black; for the parts being only
erected, the beams of light falling upon them will not be reflected to
the eye, but to the coal itself. Fourthly, that burning glasses do more
easily burn black things than white. For in a white superficies the
eminent parts are convex, like little bubbles; and therefore the beams
of light, which fall upon them, are reflected every way from the
reflecting body. But in a black superficies, where the eminent particles
are more erected, the beams of light falling upon them are all
necessarily reflected towards the body itself; and, therefore, bodies
that are black are more easily set on fire by the sun beams, than those
that are white. Fifthly, that all colours that are made of the mixture
of white and black proceed from the different position of the particles
that rise above the superficies, and their different forms of asperity.
For, according to these differences, more or fewer beams of light are
reflected from several bodies to the eye. But in regard those
differences are innumerable, and the bodies themselves so small that we
cannot perceive them; the explication and precise determination of the
causes of all colours is a thing of so great difficulty, that I dare not
undertake it.

------------------------------------------------------------------------

[Illustration:

  __Vol. I. Lat. & Eng.__
  C. XXVII.
  _Fig. 1-2_]
]

                        │Fig 1     │Fig 2     │

------------------------------------------------------------------------


                                -------


                            CHAPTER XXVIII.

                OF COLD, WIND, HARD, ICE, RESTITUTION OF
                 BODIES BENT, DIAPHANOUS, LIGHTNING AND
                  THUNDER; AND OF THE HEADS OF RIVERS.

1. Why breath from the same mouth sometimes heats and sometimes
    cools.—2. Wind, and the inconstancy of winds, whence.—3 Why there is
    a constant, though not a great wind, from east to west, near the
    equator.—4. What is the effect of air pent in between the clouds.—5.
    No change from soft to hard, but by motion.—6. What is the cause of
    cold near the poles.—7.. The cause of ice; and why the cold is more
    remiss in rainy than in clear weather. Why water doth not freeze in
    deep wells as it doth near the superficies of the earth. Why ice is
    not so heavy as water; and why wine is not so easily frozen as
    water.—8. Another cause of hardness from the fuller contact of
    atoms; also, how hard things are broken.—9. A third cause of
    hardness from heat.—10. A fourth cause of hardness from the motion
    of atoms enclosed in a narrow space.—11. How hard things are
    softened.—12. Whence proceed the spontaneous restitution of things
    bent.—13. Diaphanous and opacous, what they are, and whence.—14. The
    cause of lightning and thunder.—15. Whence it proceeds that clouds
    can fall again after they are once elevated and frozen.—16. How it
    could be that the moon was eclipsed, when she was not diametrically
    opposite to the sun.—17. By what means many suns may appear at
    once.—18. Of the heads of rivers.


[Sidenote: Why breath from the same mouth sometimes heats and sometimes
           cools.]

1. As, when the motion of the ambient ethereal substance makes the
spirits and fluid parts of our bodies tend outwards, we acknowledge
heat; so, by the endeavour inwards of the same spirits and humours, we
feel cold. So that to cool is to make the exterior parts of the body
endeavour inwards, by a motion contrary to that of calefaction, by which
the internal parts are called outwards. He, therefore, that would know
the cause of cold, must find by what motion or motions the exterior
parts of any body endeavour to retire inwards. To begin with those
phenomena which are the most familiar. There is almost no man but knows,
that breath blown strongly, and which comes from the mouth with
violence, that is to say, the passage being strait, will cool the hand;
and that the same breath blown gently, that is to say, through a greater
aperture, will warm the same. The cause of which phenomenon may be this,
the breath going out hath two motions; the one, of the whole and direct,
by which the foremost parts of the hand are driven inwards; the other,
simple motion of the small particles of the same breath, which, (as I
have shown in the 3rd article of the last chapter, causeth heat.
According, therefore, as either of these motions is predominant, so
there is the sense sometimes of cold, sometimes of heat. Wherefore, when
the breath is softly breathed out at a large passage, that simple motion
which causeth heat prevaileth, and consequently heat is felt; and when,
by compressing the lips, the breath is more strongly blown out, then is
the direct motion prevalent, which makes us feel cold. For, the direct
motion of the breath or air is wind; and all wind cools or diminisheth
former heat.

[Sidenote: Wind, and the inconstancy of winds, whence.]

2. And seeing not only great wind, but almost any ventilation and
stirring of the air, doth refrigerate; the reason of many experiments
concerning cold cannot well be given without finding first what are the
causes of wind. Now, wind is nothing else but the direct motion of the
air thrust forwards; which, nevertheless, when many winds concur, may be
circular or otherwise indirect, as it is in whirlwinds. Wherefore, in
the first place we are to enquire into the causes of winds. Wind is air
moved in a considerable quantity, and that either in the manner of
waves, which is both forwards and also up and down, or else forwards
only.

Supposing, therefore, the air both clear and calm for any time how
little soever, yet, the greater bodies of the world being so disposed
and ordered as has been said, it will be necessary that a wind presently
arise somewhere. For, seeing that motion of the parts of the air, which
is made by the simple motion of the sun in his own epicycle, causeth an
exhalation of the particles of water from the seas and all other moist
bodies, and those particles make clouds; it must needs follow, that,
whilst the particles of water pass upwards, the particles of air, for
the keeping of all spaces full, be jostled out on every side, and urge
the next particles, and these the next; till having made their circuit,
there comes continually so much air to the hinder parts of the earth as
there went water from before it. Wherefore, the ascending vapours move
the air towards the sides every way; and all direct motion of the air
being wind, they make a wind. And if this wind meet often with other
vapours which arise in other places, it is manifest that the force
thereof will be augmented, and the way or course of it changed. Besides,
according as the earth, by its diurnal motion, turns sometimes the
drier, sometimes the moister part towards the sun, so sometimes a
greater, sometimes a less, quantity of vapours will be raised; that is
to say, sometimes there will be a less, sometimes a greater wind.
Wherefore, I have rendered a possible cause of such winds as are
generated by vapours; and also of their inconstancy.

From hence it follows that these winds cannot be made in any place,
which is higher than that to which vapours may ascend. Nor is that
incredible which is reported of the highest mountains, as the Peak of
Teneriffe and the Andes of Peru, namely, that they are not at all
troubled with these inconstant winds. And if it were certain that
neither rain nor snow were ever seen in the highest tops of those
mountains, it could not be doubted but that they are higher than any
place to which vapours use to ascend.

[Sidenote: Why there is a constant, though not a great wind, from east
           to west, near the equator.]

3. Nevertheless, there may be wind there, though not that which is made
by the ascent of vapours, yet a less and more constant wind, like the
continued blast of a pair of bellows, blowing from the east. And this
may have a double cause; the one, the diurnal motion of the earth; the
other, its simple motion in its own epicycle. For these mountains being,
by reason of their height, more eminent than all the rest of the parts
of the earth, do by both these motions drive the air from the west
eastwards. To which, though the diurnal motion contribute but little,
yet seeing I have supposed that the simple motion of the earth, in its
own epicycle, makes two revolutions in the same time in which the
diurnal motion makes but one, and that the semidiameter of the epicycle
is double to the semidiameter of the diurnal conversion, the motion of
every point of the earth in its own epicycle will have its velocity
quadruple to that of the diurnal motion; so that by both these motions
together, the tops of those hills will sensibly be moved against the
air; and consequently a wind will be felt. For whether the air strike
the sentient, or the sentient the air, the perception of motion will be
the same. But this wind, seeing it is not caused by the ascent of
vapours, must necessarily be very constant.

[Sidenote: What is the effect of air pent in between the clouds.]

4. When one cloud is already ascended into the air, if another cloud
ascend towards it, that part of the air, which is intercepted between
them both, must of necessity be pressed out every way. Also when both of
them, whilst the one ascends and the other either stays or descends,
come to be joined in such manner as that the ethereal substance be shut
within them on every side, it will by this compression also go out by
penetrating the water. But in the meantime, the hard particles, which
are mingled with the air and are agitated, as I have supposed, with
simple motion, will not pass through the water of the clouds, but be
more straitly compressed within their cavities. And this I have
demonstrated at the 4th and 5th articles of chapter XXII. Besides,
seeing the globe of the earth floateth in the air which is agitated by
the sun's motion, the parts of the air resisted by the earth will spread
themselves every way upon the earth's superficies; as I have shown at
the 8th article of chapter XXI.

[Sidenote: No change from soft to hard, but by motion.]

5. We perceive a body to be hard, from this, that, when touching it, we
would thrust forwards that part of the same which we touch, we cannot do
it otherwise than by thrusting forwards the whole body. We may indeed
easily and sensibly thrust forwards any particle of the air or water
which we touch, whilst yet the rest of its parts remain to sense
unmoved. But we cannot do so to any part of a stone. Wherefore I define
a _hard_ body to be that whereof no part can be sensibly moved, unless
the whole be moved. Whatsoever therefore is soft or fluid, the same can
never be made hard but by such motion as makes many of the parts
together stop the motion of some one part, by resisting the same.

[Sidenote: What it the cause of cold near the poles.]

6. Those things premised, I shall show a possible cause why there is
greater cold near the poles of the earth, than further from them. The
motion of the sun between the tropics, driving the air towards that part
of the earth's superficies which is perpendicularly under it, makes it
spread itself every way; and the velocity of this expansion of the air
grows greater and greater, as the superficies of the earth comes to be
more and more straitened, that is to say, as the circles which are
parallel to the equator come to be less and less. Wherefore this
expansive motion of the air drives before it the parts of the air, which
are in its way, continually towards the poles more and more strongly, as
its force comes to be more and more united, that is to say, as the
circles which are parallel to the equator are less and less; that is, so
much the more, by how much they are nearer to the poles of the earth. In
those places, therefore, which are nearer to the poles, there is greater
cold than in those which are more remote from them. Now this expansion
of the air upon the superficies of the earth, from east to west, doth,
by reason of the sun's perpetual accession to the places which are
successively under it, make it cold at the time of the sun's rising and
setting; but as the sun comes to be continually more and more
perpendicular to those cooled places, so by the heat, which is generated
by the supervening simple motion of the sun, that cold is again
remitted; and can never be great, because the action by which it was
generated is not permanent. Wherefore I have rendered a possible cause
of cold in those places that are near the poles, or where the obliquity
of the sun is great.

[Sidenote: The cause of ice; and why the cold is more remiss in rainy
           than in clear weather. Why water doth not freeze in deep
           wells, as it doth near the superficies of the earth. Why ice
           is not so heavy as water; and why wine is not so easily
           frozen as water.]

7. How water may be congealed by cold, may be explained in this manner.
Let A (in figure 1) represent the sun, and B the earth. A will therefore
be much greater than B. Let E F be in the plane of the equinoctial; to
which let G H, I K, and L C be parallel. Lastly, let C and D be the
poles of the earth. The air, therefore, by its action in those
parallels, will rake the superficies of the earth; and that with motion
so much the stronger, by how much the parallel circles towards the poles
grow less and less. From whence must arise a wind, which will force
together the uppermost parts of the water, and withal raise them a
little, weakening their endeavour towards the centre of the earth. And
from their endeavour towards the centre of the earth, joined with the
endeavour of the said wind, the uppermost parts of the water will be
pressed together and coagulated, that is to say, the top of the water
will be skinned over and hardened. And so again, the water next the top
will be hardened in the same manner, till at length the ice be thick.
And this ice, being now compacted of little hard bodies, must also
contain many particles of air received into it.

As rivers and seas, so also in the same manner may the clouds be frozen.
For when, by the ascending and descending of several clouds at the same
time, the air intercepted between them is by compression forced out, it
rakes, and by little and little hardens them. And though those small
drops, which usually make clouds, be not yet united into greater bodies,
yet the same wind will be made; and by it, as water is congealed into
ice, so will vapours in the same manner be congealed into snow. From the
same cause it is that ice may be made by art, and that not far from the
fire. For it is done by the mingling of snow and salt together, and by
burying in it a small vessel full of water. Now while the snow and salt,
which have in them a great deal of air, are melting, the air, which is
pressed out every way in wind, rakes the sides of the vessel; and as the
wind by its motion rakes the vessel, so the vessel by the same motion
and action congeals the water within it.

We find by experience, that cold is always more remiss in places where
it rains, or where the weather is cloudy, things being alike in all
other respects, than where the air is clear. And this agreeth very well
with what I have said before. For in clear weather, the course of the
wind which, as I said even now, rakes the superficies of the earth, as
it is free from all interruption, so also it is very strong. But when
small drops of water are either rising or falling, that wind is
repelled, broken, and dissipated by them; and the less the wind is, the
less is the cold.

We find also by experience, that in deep wells the water freezeth not so
much as it doth upon the superficies of the earth. For the wind, by
which ice is made, entering into the earth by reason of the laxity of
its parts, more or less, loseth some of its force, though not much. So
that if the well be not deep, it will freeze; whereas if it be so deep,
as that the wind which causeth cold cannot reach it, it will not freeze.

We find moreover by experience, that ice is lighter than water. The
cause whereof is manifest from that which I have already shown, namely,
that air is received in and mingled with the particles of the water
whilst it is congealing.

Lastly, wine is not so easily congealed as water, because in wine there
are particles, which, being not fluid, are moved very swiftly, and by
their motion congelation is retarded. But if the cold prevail against
this motion, then the outermost parts of the wine will be first frozen,
and afterwards the inner parts; whereof this is a sign, that the wine
which remains unfrozen in the midst will be very strong.

[Sidenote: Another cause of hardness from the fuller contact of atoms.
           Also how hard things are broken.]

8. We have seen one way of making things hard, namely, by congelation.
Another way is thus. Having already supposed that innumerable atoms,
some harder than others and that have several simple motions of their
own, are intermingled with the ethereal substance; it follows
necessarily from hence, that by reason of the fermentation of the whole
air, of which I have spoken in chapter XXI, some of those atoms meeting
with others will cleave together, by applying themselves to one another
in such manner as is agreeable to their motions and mutual contacts;
and, seeing there is no vacuum, cannot be pulled asunder, but by so much
force as is sufficient to overcome their hardness.

Now there are innumerable degrees of hardness. As for example, there is
a degree of it in water, as is manifest from this, that upon a plane it
may be drawn any way at pleasure by one's finger. There is a greater
degree of it in clammy liquors, which, when they are poured out, do in
falling downwards dispose themselves into one continued thread; which
thread, before it be broken, will by little and little diminish its
thickness, till at last it be so small, as that it seems to break only
in a point; and in their separation the external parts break first from
one another, and then the more internal parts successively one after
another. In wax there is yet a greater degree of hardness. For when we
would pull one part of it from another, we first make the whole mass
slenderer, before we can pull it asunder. And how much the harder
anything is which we would break, so much the more force we must apply
to it. Wherefore, if we go on to harder things, as ropes, wood, metals,
stones, &c., reason prompteth us to believe that the same, though not
always sensibly, will necessarily happen; and that even the hardest
things are broken asunder in the same manner, namely, by solution of
their continuity begun in the outermost superficies, and proceeding
successively to the innermost parts. In like manner, when the parts of
bodies are to be separated, not by pulling them asunder, but by breaking
them, the first separation will necessarily be in the convex superficies
of the bowed part of the body, and afterwards in the concave
superficies. For in all bowing there is in the convex superficies an
endeavour in the parts to go one from another, and in the concave
superficies to penetrate one another.

This being well understood, a reason may be given how two bodies, which
are contiguous in one common superficies, may by force be separated
without the introduction of vacuum; though Lucretius thought otherwise,
believing that such separation was a strong establishment of vacuum. For
a marble pillar being made to hang by one of its bases, if it be long
enough, it will by its own weight be broken asunder; and yet it will not
necessarily follow that there should be vacuum, seeing the solution of
its continuity may begin in the circumference, and proceed successively
to the midst thereof.

[Sidenote: A third cause of hardness, from heat.]

9. Another cause of hardness in some things may be in this manner. If a
soft body consist of many hard particles, which by the intermixture of
many other fluid particles cohere but loosely together, those fluid
parts, as hath been shown in the last article of chapter XXI, will be
exhaled; by which means each hard particle will apply itself to the next
to it according to a greater superficies, and consequently they will
cohere more closely to one another, that is to say, the whole mass will
be made harder.

[Sidenote: A fourth cause of hardness, from the motion of atoms enclosed
           in a narrow space.]

10. Again, in some things hardness may be made to a certain degree in
this manner. When any fluid substance hath in it certain very small
bodies intermingled, which, being moved with simple motion of their own,
contribute like motion to the parts of the fluid substance, and this be
done in a small enclosed space, as in the hollow of a little sphere, or
a very slender pipe, if the motion be vehement and there be a great
number of these small enclosed bodies, two things will happen; the one,
that the fluid substance will have an endeavour of dilating itself at
once every way; the other, that if those small bodies can nowhere get
out, then from their reflection it will follow, that the motion of the
parts of the enclosed fluid substance, which was vehement before, will
now be much more vehement. Wherefore, if any one particle of that fluid
substance should be touched and pressed by some external movent, it
could not yield but by the application of very sensible force. Wherefore
the fluid substance, which is enclosed and so moved, hath some degree of
hardness. Now, greater and less degree of hardness depends upon the
quantity and velocity of those small bodies, and upon the narrowness of
the place both together.

[Sidenote: How hard things are softened.]

11. Such things as are made hard by sudden heat, namely such as are
hardened by fire, are commonly reduced to their former soft form by
maceration. For fire hardens by evaporation, and therefore if the
evaporated moisture be restored again, the former nature and form is
restored together with it. And such things as are frozen with cold, if
the wind by which they were frozen change into the opposite quarter,
they will be unfrozen again, unless they have gotten a habit of new
motion or endeavour by long continuance in that hardness. Nor is it
enough to cause thawing, that there be a cessation of the freezing wind;
for the taking away of the cause doth not destroy a produced effect; but
the thawing also must have its proper cause, namely, a contrary wind, or
at least a wind opposite in some degree. And this we find to be true by
experience. For, if ice be laid in a place so well enclosed that the
motion of the air cannot get to it, that ice will remain unchanged,
though the place be not sensibly cold.

[Sidenote: Whence proceeds the spontaneous restitution of things bent.]

12. Of hard bodies, some may manifestly be bowed; others not, but are
broken in the very first moment of their bending. And of such bodies as
may manifestly be bended, some being bent, do, as soon as ever they are
set at liberty, restore themselves to their former posture; others
remain still bent. Now if the cause of this restitution be asked, I say,
it may be in this manner, namely, that the particles of the bended body,
whilst it is held bent, do nevertheless retain their motion; and by this
motion they restore it as soon as the force is removed by which it was
bent. For when any thing is bent, as a plate of steel, and, as soon as
the force is removed, restores itself again, it is evident that the
cause of its restitution cannot be referred to the ambient air; nor can
it be referred to the removal of the force by which it was bent; for in
things that are at rest the taking away of impediments is not a
sufficient cause of their future motion; there being no other cause of
motion, but motion. The cause therefore of such restitution is in the
parts of the steel itself. Wherefore, whilst it remains bent, there is
in the parts, of which it consisteth, some motion though invisible; that
is to say, some endeavour at least that way by which the restitution is
to be made; and therefore this endeavour of all the parts together is
the first beginning of restitution; so that the impediment being
removed, that is to say, the force by which it was held bent, it will be
restored again. Now the motion of the parts, by which this done, is that
which I called simple motion, or motion returning into itself. When
therefore in the bending of a plate the ends are drawn together, there
is on one side a mutual compression of the parts; which compression is
one endeavour opposite to another endeavour: and on the other side a
divulsion of the parts. The endeavour therefore of the parts on one side
tends to the restitution of the plate from the middle towards the ends;
and on the other side, from the ends towards the middle. Wherefore the
impediment being taken away, this endeavour, which is the beginning of
restitution, will restore the plate to its former posture. And thus I
have given a possible cause why some bodies, when they are bent, restore
themselves again; which was to be done.

As for stones, seeing they are made by the accretion of many very hard
particles within the earth; which particles have no great coherence,
that is to say, touch one another in small latitude, and consequently
admit many particles of air; it must needs be that, in bending of them,
their internal parts will not easily be compressed, by reason of their
hardness. And because their coherence is not firm, as soon as the
external hard particles are disjoined, the ethereal parts will
necessarily break out, and so the body will suddenly be broken.

[Sidenote: Diaphanous and opacous, what they are, and whence.]

13. Those bodies are called _diaphanous_, upon which, whilst the beams
of a lucid body do work, the action of every one of those beams is
propagated in them in such manner, as that they still retain the same
order amongst themselves, or the inversion of that order; and therefore
bodies, which are perfectly diaphanous, are also perfectly homogeneous.
On the contrary, an _opacous_ body is that, which, by reason of its
heterogeneous nature, doth by innumerable reflections and refractions in
particles of different figures and unequal hardness, weaken the beams
that fall upon it before they reach the eye. And of diaphanous bodies,
some are made such by nature from the beginning; as the substance of the
air, and of the water, and perhaps also some parts of stones, unless
these also be water that has been long congealed. Others are made so by
the power of heat, which congregates homogeneous bodies. But such, as
are made diaphanous in this manner, consist of parts which were formerly
diaphanous.

[Sidenote: The cause of lightning and thunder.]

14. In what manner clouds are made by the motion of the sun, elevating
the particles of water from the sea and other moist places, hath been
explained in chapter XXVI. Also how clouds come to be frozen, hath been
shown above at the 7th article. Now from this, that air may be enclosed
as it were in caverns, and pent together more and more by the meeting of
ascending and descending clouds, may be deduced a possible cause of
_thunder_ and _lightning_. For seeing the air consists of two parts, the
one ethereal, which has no proper motion of its own, as being a thing
divisible into the least parts; the other hard, namely, consisting of
many hard atoms, which have every one of them a very swift simple motion
of its own: whilst the clouds by their meeting do more and more straiten
such cavities as they intercept, the ethereal parts will penetrate and
pass through their watery substance; but the hard parts will in the
meantime be the more thrust together, and press one another; and
consequently, by reason of their vehement motions, they will have an
endeavour to rebound from each other. Whensoever, therefore, the
compression is great enough, and the concave parts of the clouds are,
for the cause I have already given, congealed into ice, the cloud will
necessarily be broken; and this breaking of the cloud produceth the
first clap of thunder. Afterwards the air, which was pent in, having now
broken through, makes a concussion of the air without, and from hence
proceeds the roaring and murmur which follows; and both the first clap
and the murmur that follows it make that noise which is called
_thunder_. Also, from the same air breaking through the clouds and with
concussion falling upon the eye, proceeds that action upon our eye,
which causeth in us a perception of that light, which we call
_lightning_. Wherefore I have given a possible cause of thunder and
lightning.

[Sidenote: Whence it proceeds that clouds can fall again, after they are
           once elevated and frozen.]

15. But if the vapours, which are raised into clouds, do run together
again into water or be congealed into ice, from whence is it, seeing
both ice and water are heavy, that they are sustained in the air? Or
rather, what may the cause be, that being once elevated, they fall down
again? For there is no doubt but the same force which could carry up
that water, could also sustain it there. Why therefore being once
carried up, doth it fall again? I say it proceeds from the same simple
motion of the sun, both that vapours are forced to ascend, and that
water gathered into clouds is forced to descend. For in chapter XXI,
article 11, I have shown how vapours are elevated; and in the same
chapter, article 5, I have also shown how by the same motion homogeneous
bodies are congregated, and heterogeneous dissipated; that is to say,
how such things, as have a like nature to that of the earth, are driven
towards the earth; that is to say, what is the cause of the descent of
heavy bodies. Now if the action of the sun be hindered in the raising of
vapours, and be not at all hindered in the casting of them down, the
water will descend. But a cloud cannot hinder the action of the sun in
making things of an earthly nature descend to the earth, though it may
hinder it in making vapours ascend. For the lower part of a thick cloud
is so covered by its upper part, as that it cannot receive that action
of the sun by which vapours are carried up; because vapours are raised
by the perpetual fermentation of the air, or by the separating of its
smallest parts from one another, which is much weaker when a thick cloud
is interposed, than when the sky is clear. And therefore, whensoever a
cloud is made thick enough, the water, which would not descend before,
will then descend, unless it be kept up by the agitation of the wind.
Wherefore I have rendered a possible cause, both why the clouds may be
sustained in the air, and also why they may fall down again to the
earth; which was propounded to be done.

[Sidenote: How it could be that the moon was eclipsed, when she was not
           diametrically opposite to the sun.]

16. Granting that the clouds may be frozen, it is no wonder if the moon
have been seen eclipsed at such time as she hath been almost two degrees
above the horizon, the sun at the same time appearing in the horizon;
for such an eclipse was observed by Mæstlin, at Tubingen, in the year
1590. For it might happen that a frozen cloud was then interposed
between the sun and the eye of the observer. And if it were so, the sun,
which was then almost two degrees below the horizon, might appear to be
in it, by reason of the passing of his beams through the ice. And it is
to be noted that those, that attribute such refractions to the
atmosphere, cannot attribute to it so great a refraction as this.
Wherefore not the atmosphere, but either water in a continued body, or
else ice, must be the cause of that refraction.

[Sidenote: By what means many suns may appear at once.]

17. Again, granting that there may be ice in the clouds, it will be no
longer a wonder that many suns have sometimes appeared at once. For
looking-glasses may be so placed, as by reflections to show the same
object in many places. And may not so many frozen clouds serve for so
many looking-glasses? And may they not be fitly disposed for that
purpose? Besides, the number of appearances may be increased by
refractions also; and therefore it would be a greater wonder to me, if
such phenomena as these should never happen.

And were it not for that one phenomenon of the new star which was seen
in Cassiopea, I should think comets were made in the same manner,
namely, by vapours drawn not only from the earth but from the rest of
the planets also, and congealed into one continued body. For I could
very well from hence give a reason both of their hair, and of their
motions. But seeing that star remained sixteen whole months in the same
place amongst the fixed stars, I cannot believe the matter of it was
ice. Wherefore I leave to others the disquisition of the cause of
comets; concerning which nothing that hath hitherto been published,
besides the bare histories of them, is worth considering.

[Sidenote: Of the heads of rivers.]

18. The heads of rivers may be deduced from rain-water, or from melted
snows, very easily; but from other causes, very hardly, or not at all.
For both rain-water and melted snows run down the descents of mountains;
and if they descend only by the outward superficies, the showers or
snows themselves may be accounted the springs or fountains; but if they
enter the earth and descend within it, then, wheresoever they break out,
there are their springs. And as these springs make small streams, so,
many small streams running together make rivers. Now, there was never
any spring found, but where the water which flowed to it, was either
further, or at least as far from the centre of the earth, as the spring
itself. And whereas it has been objected by a great philosopher, that in
the top of Mount Cenis, which parts Savoy from Piedmont, there springs a
river which runs down by Susa; it is not true. For there are above that
river, for two miles length, very high hills on both sides, which are
almost perpetually covered with snow; from which innumerable little
streams running down do manifestly supply that river with water
sufficient for its magnitude.


                                -------


                              CHAP. XXIX.

                  OF SOUND, ODOUR, SAVOUR, AND TOUCH.

1.. The definition of sound, and the distinctions of sounds.—2. The
    cause of the degrees of sounds.—3. The difference between sounds
    acute and grave.—4. The difference between clear and hoarse sounds,
    whence.—5. The sound of thunder and of a gun, whence it proceeds.—6.
    Whence it is that pipes, by blowing into them, have a clear
    sound.—7. Of reflected sound.—8. From whence it is that sound is
    uniform and lasting.—9. How sound may be helped and hindered by the
    wind.—10. Not only air, but other bodies how hard soever they be,
    convey sound.—11. The causes of grave and acute sounds, and of
    concent.—12. Phenomena for smelling.—13. The first organ and the
    generation of smelling.—14. How it is helped by heat and by
    wind.—15. Why such bodies are least smelt, which have least
    intermixture of air in them.—16. Why odorous things become more
    odorous by being bruised.—17. The first organ of tasting; and why
    some savours cause nauseousness.—18. The first organ of feeling; and
    how we come to the knowledge of such objects as are common to the
    touch and other senses.


[Sidenote: The definition of sound, and the distinction of sounds.]

1. SOUND _is sense generated by the action of the medium, when its
motion reacheth the ear and the rest of the organs of sense_. Now, the
motion of the medium is not the sound itself, but the cause of it. For
the phantasm which is made in us, that is to say, the reaction of the
organ, is properly that which we call _sound_.

The principal distinctions of sounds are these; first, that one sound is
stronger, another weaker. Secondly, that one is more grave, another more
acute. Thirdly, that one is clear, another hoarse. Fourthly, that one is
primary, another derivative. Fifthly, that one is uniform, another not.
Sixthly, that one is more durable, another less durable. Of all which
distinctions the members may be subdistinguished into parts
distinguishable almost infinitely. For the variety of sounds seems to be
not much less than that of colours.

As vision, so hearing is generated by the motion of the medium, but not
in the same manner. For sight is from pressure, that is, from an
endeavour; in which there is no perceptible progression of any of the
parts of the medium; but one part urging or thrusting on another
propagateth that action successively to any distance whatsoever; whereas
the motion of the medium, by which sound is made, is a stroke. For when
we hear, the drum of the ear, which is the first organ of hearing, is
stricken; and the drum being stricken, the _pia mater_ is also shaken,
and with it the arteries which are inserted into it; by which the action
being propagated to the heart itself, by the reaction of the heart a
phantasm is made which we call sound; and because the reaction tendeth
outwards, we think it is without.

[Sidenote: The cause of the degrees of sounds.]

2. And seeing the effects produced by motion are greater or less, not
only when the velocity is greater or less, but also when the body hath
greater or less magnitude though the velocity be the same; a sound may
be greater or less both these ways. And because neither the greatest nor
the least magnitude or velocity can be given, it may happen that either
the motion may be of so small velocity, or the body itself of so small
magnitude, as to produce no sound at all; or either of them may be so
great, as to take away the faculty of sense by hurting the organ.

From hence may be deduced possible causes of the strength and weakness
of sounds in the following phenomena.

The first whereof is this, that if a man speak through a trunk which
hath one end applied to the mouth of the speaker, and the other to the
ear of the hearer, the sound will come stronger than it would do through
the open air. And the cause, not only the possible, but the certain and
manifest cause is this, that the air which is moved by the first breath
and carried forwards in the trunk, is not diffused as it would be in the
open air, and is consequently brought to the ear almost with the same
velocity with which it was first breathed out. Whereas, in the open air,
the first motion diffuseth itself every way into circles, such as are
made by the throwing of a stone into a standing water, where the
velocity grows less and less as the undulation proceeds further and
further from the beginning of its motion.

The second is this, that if the trunk be short, and the end which is
applied to the mouth be wider than that which is applied to the ear,
thus also the sound will be stronger than if it were made in the open
air. And the cause is the same, namely, that by how much the wider end
of the trunk is less distant from the beginning of the sound, by so much
the less is the diffusion.

The third, that it is easier for one, that is within a chamber, to hear
what is spoken without, than for him, that stands without, to hear what
is spoken within. For the windows and other inlets of the moved air are
as the wide end of the trunk. And for this reason some creatures seem to
hear the better, because nature has bestowed upon them wide and
capacious ears.

The fourth is this, that though he, which standeth upon the sea-shore,
cannot hear the collision of the two nearest waves, yet nevertheless he
hears the roaring of the whole sea. And the cause seems to be this, that
though the several collisions move the organ, yet they are not severally
great enough to cause sense; whereas nothing hinders but that all of
them together may make sound.

[Sidenote: The difference between sounds acute and grave.]

3. That bodies when they are stricken do yield some a more grave, others
a more acute sound, the cause may consist in the difference of the times
in which the parts stricken and forced out of their places return to the
same places again. For in some bodies, the restitution of the moved
parts is quick, in others slow. And this also may be the cause, why the
parts of the organ, which are moved by the medium, return to their rest
again, sometimes sooner, sometimes later. Now, by how much the
vibrations or the reciprocal motions of the parts are more frequent, by
so much doth the whole sound made at the same time by one stroke consist
of more, and consequently of smaller parts. For what is acute in sound,
the same is subtle in matter; and both of them, namely acute sound and
subtle matter, consist of very small parts, that of time, and this of
the matter itself.

The third distinction of sounds cannot be conceived clearly enough by
the names I have used of _clear_ and _hoarse_, nor by any other that I
know; and therefore it is needful to explain them by examples. When I
say hoarse, I understand whispering and hissing, and whatsoever is like
to these, by what appellation soever it be expressed. And sounds of this
kind seem to be made by the force of some strong wind, raking rather
than striking such hard bodies as it falls upon. On the contrary, when I
use the word clear, I do not understand such a sound as may be easily
and distinctly heard; for so whispers would be clear; but such as is
made by somewhat that is broken, and such as is clamour, tinkling, the
sound of a trumpet, &c. and to express it significantly in one word,
noise. And seeing no sound is made but by the concourse of two bodies at
the least, by which concourse it is necessary that there be as well
reaction as action, that is to say, one motion opposite to another; it
follows that according as the proportion between those two opposite
motions is diversified, so the sounds which are made will be different
from one another. And whensoever the proportion between them is so
great, as that the motion of one of the bodies be insensible if compared
with the motion of the other, then the sound will not be of the same
kind; as when the wind falls very obliquely upon a hard body, or when a
hard body is carried swiftly through the air; for then there is made
that sound which I call a hoarse sound, in Greek συριγμος. Therefore the
breath blown with violence from the mouth makes a hissing, because in
going out it rakes the superficies of the lips, whose reaction against
the force of the breath is not sensible. And this is the cause why the
winds have that hoarse sound. Also if two bodies, how hard soever, be
rubbed together with no great pressure, they make a hoarse sound. And
this hoarse sound, when it is made, as I have said, by the air raking
the superficies of a hard body, seemeth to be nothing but the dividing
of the air into innumerable and very small files. For the asperity of
the superficies doth, by the eminences of its innumerable parts, divide
or cut in pieces the air that slides upon it.

[Sidenote: The difference between clear & hoarse sounds, whence.]

4. _Noise_, or that which I call clear sound, is made two ways; one, by
two hoarse sounds made by opposite motions; the other, by collision, or
by the sudden pulling asunder of two bodies, whereby their small
particles are put into commotion, or being already in commotion suddenly
restore themselves again; which motion, making impression upon the
medium, is propagated to the organ of hearing. And seeing there is in
this collision or divulsion an endeavour in the particles of one body,
opposite to the endeavour of the particles of the other body, there will
also be made in the organ of hearing a like opposition of endeavours,
that is to say, of motions; and consequently the sound arising from
thence will be made by two opposite motions, that is to say, by two
opposite hoarse sounds in one and the same part of the organ. For, as I
have already said, a hoarse sound supposeth the sensible motion of but
one of the bodies. And this opposition of motions in the organ is the
cause why two bodies make a noise, when they are either suddenly
stricken against one another, or suddenly broken asunder.

[Sidenote: The sound of thunder and of a gun, whence it proceeds.]

5. This being granted, and seeing withal that thunder is made by the
vehement eruption of the air out of the cavities of congealed clouds,
the cause of the great noise or clap may be the sudden breaking asunder
of the ice. For in this action it is necessary that there be not only a
great concussion of the small particles of the broken parts, but also
that this concussion, by being communicated to the air, be carried to
the organ of hearing, and make impression upon it. And then, from the
first reaction of the organ proceeds that first and greatest sound,
which is made by the collision of the parts whilst they restore
themselves. And seeing there is in all concussion a reciprocation of
motion forwards and backwards in the parts stricken; for opposite
motions cannot extinguish one another in an instant, as I have shown in
the 11th article of chapter VIII; it follows necessarily that the sound
will both continue, and grow weaker and weaker, till at last the action
of the reciprocating air grow so weak, as to be imperceptible. Wherefore
a possible cause is given both of the first fierce noise of the thunder,
and also of the murmur that follows it.

The cause of the great sound from a discharged piece of ordnance is like
that of a clap of thunder. For the gunpowder being fired doth, in its
endeavour to go out, attempt every way the sides of the metal in such
manner, as that it enlargeth the circumference all along, and withal
shorteneth the axis; so that whilst the piece of ordnance is in
discharging, it is made both wider and shorter than it was before; and
therefore also presently after it is discharged its wideness will be
diminished, and its length increased again by the restitution of all the
particles of the matter, of which it consisteth, to their former
position. And this is done with such motions of the parts, as are not
only very vehement, but also opposite to one another; which motions,
being communicated to the air, make impression upon the organ, and by
the reaction of the organ create a sound, which lasteth for some time;
as I have already shown in this article.

I note by the way, as not belonging to this place, that the possible
cause why a gun recoils when it is shot off, may be this; that being
first swollen by the force of the fire, and afterwards restoring itself,
from this restitution there proceeds an endeavour from all the sides
towards the cavity; and consequently this endeavour is in those parts
which are next the breech; which being not hollow, but solid, the effect
of the restitution is by it hindered and diverted into the length; and
by this means both the breech and the whole gun is thrust backwards; and
the more forcibly by how much the force is greater, by which the part
next the breech is restored to its former posture, that is to say, by
how much the thinner is that part. The cause, therefore, why guns
recoil, some more some less, is the difference of their thickness
towards the breech; and the greater that thickness is, the less they
recoil; and contrarily.

[Sidenote: Whence it is that pipes, by blowing into them, have a clear
           sound.]

6. Also the cause why the sound of a pipe, which is made by blowing into
it, is nevertheless clear, is the same with that of the sound which is
made by collision. For if the breath, when it is blown into a pipe, do
only rake its concave superficies, or fall upon it with a very sharp
angle of incidence, the sound will not be clear, but hoarse. But if the
angle be great enough, the percussion, which is made against one of the
hollow sides, will be reverberated to the opposite side; and so
successive repercussions will be made from side to side, till at last
the whole concave superficies of the pipe be put into motion; which
motion will be reciprocated, as it is in collision; and this
reciprocation being propagated to the organ, from the reaction of the
organ will arise a clear sound, such as is made by collision, or by
breaking asunder of hard bodies.

In the same manner it is with the sound of a man's voice. For when the
breath passeth out without interruption, and doth but lightly touch the
cavities through which it is sent, the sound it maketh is a hoarse
sound. But if in going out it strike strongly upon the _larynx_, then a
clear sound is made, as in a pipe. And the same breath, as it comes in
divers manners to the palate, the tongue, the lips, the teeth, and other
organs of speech, so the sounds into which it is articulated become
different from one another.

[Sidenote: Of reflected sound.]

7. I call that _primary sound_, which is generated by motion from the
sounding body to the organ in a strait line without reflection; and I
call that _reflected sound_, which is generated by one or more
reflections, being the same with that we call _echo_, and is iterated as
often as there are reflections made from the object to the ear. And
these reflections are made by hills, walls, and other resisting bodies,
so placed as that they make more or fewer reflections of the motion,
according as they are themselves more or fewer in number; and they make
them more or less frequently, according as they are more or less distant
from one another. Now the cause of both these things is to be sought for
in the situation of the reflecting bodies, as is usually done in sight.
For the laws of reflection are the same in both, namely, that the angles
of incidence and reflection be equal to one another. If, therefore, in a
hollow elliptic body, whose inside is well polished, or in two right
parabolical solids, which are joined together by one common base, there
be placed a sounding body in one of the burning points, and the ear in
the other, there will be heard a sound by many degrees greater than in
the open air; and both this, and the burning of such combustible things,
as being put in the same places are set on fire by the sun-beams, are
effects of one and the same cause. But, as when the visible object is
placed in one of the burning points, it is not distinctly seen in the
other, because every part of the object being seen in every line, which
is reflected from the concave superficies to the eye, makes a confusion
in the sight; so neither is sound heard articulately and distinctly when
it comes to the ear in all those reflected lines. And this may be the
reason why in churches which have arched roofs, though they be neither
elliptical nor parabolical, yet because their figure is not much
different from these, the voice from the pulpit will not be heard so
articulately as it would be, if there were no vaulting at all.

[Sidenote: From whence it is that sound is uniform and lasting.]

8. Concerning the _uniformity_ and _duration_ of sounds, both which have
one common cause, we may observe, that such bodies as being stricken
yield an unequal or harsh sound, are very heterogeneous, that is to say,
they consist of parts which are very unlike both in figure and hardness,
such as are wood, stones, and others not a few. When these are stricken,
there follows a concussion of their internal particles, and a
restitution of them again. But they are neither moved alike, nor have
they the same action upon one another; some of them recoiling from the
stroke, whilst others which have already finished their recoilings are
now returning; by which means they hinder and stop one another. And from
hence it is that their motions are not only unequal and harsh, but also
that their reciprocations come to be quickly extinguished. Whensoever,
therefore, this motion is propagated to the ear, the sound it makes is
unequal and of small duration. On the contrary, if a body that is
stricken be not only sufficiently hard, but have also the particles of
which it consisteth like to one another both in hardness and figure,
such as are the particles of glass and metals, which being first melted
do afterwards settle and harden; the sound it yieldeth will, because the
motions of its parts and their reciprocations are like and uniform, be
uniform and pleasant, and be more or less lasting, according as the body
stricken hath greater or less magnitude. The possible cause, therefore,
of sounds uniform and harsh, and of their longer or shorter duration,
may be one and the same likeness and unlikeness of the internal parts of
the sounding body, in respect both of their figure and hardness.

Besides, if two plane bodies of the same matter and of equal thickness,
do both yield an uniform sound, the sound of that body, which hath the
greatest extent of length, will be the longest heard. For the motion,
which in both of them hath its beginning from the point of percussion,
is to be propagated in the greater body through a greater space, and
consequently that propagation will require more time; and therefore also
the parts which are moved, will require more time for their return.
Wherefore all the reciprocations cannot be finished but in longer time;
and being carried to the ear, will make the sound last the longer. And
from hence it is manifest, that of hard bodies which yield an uniform
sound, the sound lasteth longer which comes from those that are round
and hollow, than from those that are plane, if they be like in all other
respects. For in circular lines the action, which begins at any point,
hath not from the figure any end of its propagation, because the line in
which it is propagated returns again to its beginning; so that the
figure hinders not but that the motion may have infinite progression.
Whereas in a plane, every line hath its magnitude finite, beyond which
the action cannot proceed. If, therefore, the matter be the same, the
motion of the parts of that body whose figure is round and hollow, will
last longer than of that which is plane.

Also, if a string which is stretched be fastened at both ends to a
hollow body, and be stricken, the sound will last longer than if it were
not so fastened; because the trembling or reciprocation which it
receives from the stroke, is by reason of the connection communicated to
the hollow body; and this trembling, if the hollow body be great, will
last the longer by reason of that greatness. Wherefore also, for the
reason above mentioned, the sound will last the longer.

[Sidenote: How sound may be helped and hindered by the wind.]

9. In hearing it happens, otherwise than in seeing, that the action of
the medium is made stronger by the wind when it blows the same way, and
weaker when it blows the contrary way. The cause whereof cannot proceed
from anything but the different generation of sound and light. For in
the generation of light, none of the parts of the medium between the
object and the eye are moved from their own places to other places
sensibly distant; but the action is propagated in spaces imperceptible;
so that no contrary wind can diminish, nor favourable wind encrease the
light, unless it be so strong as to remove the object further off or
bring it nearer to the eye. For the wind, that is to say the air moved,
doth not by its interposition between the object and the eye work
otherwise than it would do, if it were still and calm. For, where the
pressure is perpetual, one part of the air is no sooner carried away,
but another, by succeeding it, receives the same impression, which the
part carried away had received before. But in the generation of sound,
the first collision or breaking asunder beateth away and driveth out of
its place the nearest part of the air, and that to a considerable
distance, and with considerable velocity; and as the circles grow by
their remoteness wider and wider, so the air being more and more
dissipated, hath also its motion more and more weakened. Whensoever
therefore the air is so stricken as to cause sound, if the wind fall
upon it, it will move it all nearer to the ear, if it blow that way, and
further from it if it blow the contrary way; so that according as it
blows from or towards the object, so the sound which is heard will seem
to come from a nearer or remoter place; and the reaction, by reason of
the unequal distances, be strengthened or debilitated.

From hence may be understood the reason why the voice of such as are
said to speak in their bellies, though it be uttered near hand, is
nevertheless heard, by those that suspect nothing, as if it were a great
way off. For having no former thought of any determined place from which
the voice should proceed, and judging according to the greatness, if it
be weak they think it a great way off, if strong near. These
ventriloqui, therefore, by forming their voice, not as others by the
emission of their breath, but by drawing it inwards, do make the same
appear small and weak; which weakness of the voice deceives those, that
neither suspect the artifice nor observe the endeavour which they use in
speaking; and so, instead of thinking it weak, they think it far off.

[Sidenote: Not only air, but other bodies, how hard soever they be,
           convey sound.]

10. As for the medium, which conveys sound, it is not air only. For
water, or any other body how hard soever, may be that medium. For the
motion may be propagated perpetually in any hard continuous body; but by
reason of the difficulty, with which the parts of hard bodies are moved,
the motion in going out of hard matter makes but a weak impression upon
the air. Nevertheless, if one end of a very long and hard beam be
stricken, and the ear be applied at the same time to the other end, so
that, when the action goeth out of the beam, the air, which it striketh,
may immediately be received by the ear, and be carried to the tympanum,
the sound will be considerably strong.

In like manner, if in the night, when all other noise which may hinder
sound ceaseth, a man lay his ear to the ground, he will hear the sound
of the steps of passengers, though at a great distance; because the
motion, which by their treading they communicate to the earth, is
propagated to the ear by the uppermost parts of the earth which
receiveth it from their feet.

[Sidenote: The causes of grave and acute sounds, and of concent.]

11. I have shown above, that the difference between grave and acute
sounds consisteth in this, that by how much the shorter the time is, in
which the reciprocations of the parts of a body stricken are made, by so
much the more acute will be the sound. Now by how much a body of the
same bigness is either more heavy or less stretched, by so much the
longer will the reciprocations last; and therefore heavier and less
stretched bodies, if they be like in all other respects, will yield a
graver sound than such as be lighter and more stretched.

As for the concent of sounds, it is to be considered that the
reciprocation or vibration of the air, by which sound is made, after it
hath reached the drum of the ear, imprinteth a like vibration upon the
air that is inclosed within it; by which means the sides of the drum
within are stricken alternately. Now the concent of two sounds consists
in this, that the tympanum receives its sounding stroke from both the
sounding bodies in equal and equally frequent spaces of time; so that
when two strings make their vibrations in the same times, the concent
they produce is the most exquisite of all other. For the sides of the
tympanum, that is to say of the organ of hearing, will be stricken by
both those vibrations together at once, on one side or other. For
example, if the two equal strings A B and C D be stricken together, and
the latitudes of their vibrations E F and G H be also equal, and the
points E, G, F and H be in the concave superficies of the tympanum, so
that it receive strokes from both the strings together in E and G, and
again together in F and H, the sound, which is made by the vibrations of
each string, will be so like, that it may be taken for the same sound,
and is called _unison_; which is the greatest concord. Again, the string
A B retaining still its former vibration E F, let the string C D be
stretched till its vibration have double the swiftness it had before,
and let E F be divided equally in I. In what time therefore the string C
D makes one part of its vibration from G to H, in the same time the
string A B will make one part of its vibration from E to I; and in what
time the string C D hath made the other part of its vibration back from
H to G, in the same time another part of the vibration of the string A B
will be made from I to F. But the points F and G are both in the sides
of the organ, and therefore they will strike the organ both together,
not at every stroke, but at every other stroke. And this is the nearest
concord to unison, and makes that sound which is called an _eighth_.
Again, the vibration of the string A B remaining still the same it was,
let C D be stretched till its vibration be swifter than that of the
string A B in the proportion of 3 to 2, and let E F be divided into
three equal parts in K and L. In what time therefore the string C D
makes one third part of its vibration, which third part is from G to H,
the string A B will make one third part of its vibration, that is to
say, two-thirds of E F, namely, E L. And in what time the string C D
makes another third part of its vibration, namely H G, the string A B
will make another third part of its vibration, namely from L to F, and
back again from F to L. Lastly, whilst the string C D makes the last
third part of its vibration, that is from G to H, the string A B will
make the last third part of its vibration from L to E. But the points E
and H are both in the sides of the organ. Wherefore, at every third
time, the organ will be stricken by the vibration of both the strings
together, and make that concord which is called a _fifth_.

                               A-------B
                               C-------D
                                 G   E
                                 |   |
                                 |   | K
                                 |   |
                                 |   | I
                                 |   |
                                 |   | L
                                 |   |
                                 H   F

[Sidenote: Phenomena of smelling.]

12. For the finding out the cause of _smells_, I shall make use of the
evidence of these following phenomena. First, that smelling is hindered
by cold, and helped by heat. Secondly, that when the wind bloweth from
the object, the smell is the stronger; and, contrarily, when it bloweth
from the sentient towards the object, the weaker; both which phenomena
are, by experience, manifestly found to be true in dogs, which follow
the track of beasts by the scent. Thirdly, that such bodies, as are less
pervious to the fluid medium, yield less smell than such as are more
pervious; as may be seen in stones and metals, which, compared with
plants and living creatures, and their parts, fruits and excretions,
have very little or no smell at all. Fourthly, that such bodies, as are
of their own nature odorous, become yet more odorous when they are
bruised. Fifthly, that when the breath is stopped, at least in men,
nothing can be smelt. Sixthly, that the sense of smelling is also taken
away by the stopping of the nostrils, though the mouth be left open.

[Sidenote: The first organ and the generation of smelling.]

13. By the fifth and sixth phenomenon it is manifest, that the first and
immediate organ of smelling is the innermost cuticle of the nostrils,
and that part of it, which is below the passage common to the nostrils
and the palate. For when we draw breath by the nostrils we draw it into
the lungs. That breath, therefore, which conveys smells is in the way
which passeth to the lungs, that is to say, in that part of the nostrils
which is below the passage through which the breath goeth. For, nothing
is smelt, neither beyond the passage of the breath within, nor at all
without the nostrils.

And seeing that from different smells there must necessarily proceed
some mutation in the organ, and all mutation is motion; it is therefore
also necessary that, in smelling, the parts of the organ, that is to say
of that internal cuticle and the nerves that are inserted into it, must
be diversely moved by different smells. And seeing also, that it hath
been demonstrated, that nothing can be moved but by a body that is
already moved and contiguous; and that there is no other body contiguous
to the internal membrane of the nostrils but breath, that is to say
attracted air, and such little solid invisible bodies, if there be any
such, as are intermingled with the air; it follows necessarily, that the
cause of smelling is either the motion of that pure air or ethereal
substance, or the motion of those small bodies. But this motion is an
effect proceeding from the object of smell, and, therefore, either the
whole object itself or its several parts must necessarily be moved. Now,
we know that odorous bodies make odour, though their whole bulk be not
moved. Wherefore the cause of odour is the motion of the invisible parts
of the odorous body. And these invisible parts do either go out of the
object, or else, retaining their former situation with the rest of the
parts, are moved together with them, that is to say, they have simple
and invisible motion. They that say, there goes something out of the
odorous body, call it an effluvium; which effluvium is either of the
ethereal substance, or of the small bodies that are intermingled with
it. But, that all variety of odours should proceed from the effluvia of
those small bodies that are intermingled with the ethereal substance, is
altogether incredible, for these considerations; first, that certain
unguents, though very little in quantity, do nevertheless send forth
very strong odours, not only to a great distance of place, but also for
a great continuance of time, and are to be smelt in every point both of
that place and time; so that the parts issued out are sufficient to fill
ten thousand times more space, than the whole odorous body is able to
fill; which is impossible. Secondly, that whether that issuing out be
with strait or with crooked motion, if the same quantity should flow
from any other odorous body with the same motion, it would follow that
all odorous bodies would yield the same smell. Thirdly, that seeing
those effluvia have great velocity of motion (as is manifest from this,
that noisome odours proceeding from caverns are presently smelt at a
great distance) it would follow, that, by reason there is nothing to
hinder the passage of those effluvia to the organ, such motion alone
were sufficient to cause smelling; which is not so; for we cannot smell
at all, unless we draw in our breath through our nostrils. Smelling,
therefore, is not caused by the effluvium of atoms; nor, for the same
reason, is it caused by the effluvium of ethereal substance; for so also
we should smell without the drawing in of our breath. Besides, the
ethereal substance being the same in all odorous bodies, they would
always affect the organ in the same manner; and, consequently, the
odours of all things would be alike.

It remains, therefore, that the cause of smelling must consist in the
simple motion of the parts of odorous bodies without any efflux or
diminution of their whole substance. And by this motion there is
propagated to the organ, by the intermediate air, the like motion, but
not strong enough to excite sense of itself without the attraction of
air by respiration. And this is a possible cause of smelling.

[Sidenote: How smelling is helped by heat and by wind.]

14. The cause why smelling is hindered by cold and helped by heat may be
this; that heat, as hath been shown in chapter XXI, generateth simple
motion; and therefore also, wheresoever it is already, there it will
increase it; and the cause of smelling being increased, the smell itself
will also be increased. As for the cause why the wind blowing from the
object makes the smell the stronger, it is all one with that for which
the attraction of air in respiration doth the same. For, he that draws
in the air next to him, draws with it by succession that air in which is
the object. Now, this motion of the air is wind, and, when another wind
bloweth from the object, will be increased by it.

[Sidenote: Why such bodies are least smelt, which have the least
           intermixture of air in them.]

15. That bodies which contain the least quantity of air, as stones and
metals, yield less smell than plants and living creatures; the cause may
be, that the motion, which causeth smelling, is a motion of the fluid
parts only; which parts, if they have any motion from the hard parts in
which they are contained, they communicate the same to the open air, by
which it is propagated to the organ. Where, therefore, there are no
fluid parts as in metals, or where the fluid parts receive no motion
from the hard parts, as in stones, which are made hard by accretion,
there can be no smell. And therefore also the water, whose parts have
little or no motion, yieldeth no smell. But, if the same water, by seeds
and the heat of the sun, be together with particles of earth raised into
a plant, and be afterwards pressed out again, it will be odorous, as
wine from the vine. And as water passing through plants is by the motion
of the parts of those plants made an odorous liquor; so also of air,
passing through the same plants whilst they are growing, are made
odorous airs. And thus also it is with the juices and spirits, which are
bred in living creatures.

[Sidenote: Why odorous things become more odorous, when bruised.]

16. That odorous bodies may be made more odorous by contrition proceeds
from this, that being broken into many parts, which are all odorous, the
air, which by respiration is drawn from the object towards the organ,
doth in its passage touch upon all those parts, and receive their
motion. Now, the air toucheth the superficies only; and a body having
less superficies whilst it is whole, than all its parts together have
after it is reduced to powder, it follows that the same odorous body
yieldeth less smell whilst it is whole, than it will do after it is
broken into smaller parts. And thus much of smells.

[Sidenote: The first organ of tasting; and why some savours cause
           nauseousness.]

17. The taste follows; whose generation hath this difference from that
of the sight, hearing, and smelling, that by these we have sense of
remote objects; whereas, we taste nothing but what is contiguous, and
doth immediately touch either the tongue or palate, or both. From whence
it is evident, that the cuticles of the tongue and palate, and the
nerves inserted into them are the first organ of taste; and (because
from the concussion of the parts of these, there followeth necessarily a
concussion of the _pia mater_) that the action communicated to these is
propagated to the brain, and from thence to the farthest organ, namely,
the heart, in whose reaction consisteth the nature of sense.

Now, that savours, as well as odours, do not only move the brain but the
stomach also, as is manifest by the loathings that are caused by them
both; they, that consider the organ of both these senses, will not
wonder at all; seeing the tongue, the palate and the nostrils, have one
and the same continued cuticle, derived from the _dura mater_.

And that effluvia have nothing to do in the sense of tasting, is
manifest from this, that there is no taste where the organ and the
object are not contiguous.

By what variety of motions the different kinds of tastes, which are
innumerable, may be distinguished, I know not. I might with others
derive them from the divers figures of those atoms, of which whatsoever
may be tasted consisteth; or from the diverse motions which I might, by
way of supposition, attribute to those atoms; conjecturing, not without
some likelihood of truth, that such things as taste sweet have their
particles moved with slow circular motion, and their figures spherical;
which makes them smooth and pleasing to the organ; that bitter things
have circular motion, but vehement, and their figures full of angles, by
which they trouble the organ; and that sour things have strait and
reciprocal motion, and their figures long and small, so that they cut
and wound the organ. And in like manner I might assign for the causes of
other tastes such several motions and figures of atoms, as might in
probability seem to be the true causes. But this would be to revolt from
philosophy to divination.

[Sidenote: The first organ of feeling; and how we came to the knowledge
           of such objects as are common to the touch and to other
           senses.]

18. By the _touch_, we feel what bodies are cold or hot, though they be
distant from us. Others, as hard, soft, rough, and smooth, we cannot
feel unless they be contiguous. The organ of touch is every one of those
membranes, which being continued from the _pia mater_ are so diffused
throughout the whole body, as that no part of it can be pressed, but the
_pia mater_ is pressed together with it. Whatsoever therefore presseth
it, is felt as hard or soft, that is to say, as more or less hard. And
as for the sense of rough, it is nothing else but innumerable
perceptions of hard and hard succeeding one another by short intervals
both of time and place. For we take notice of rough and smooth, as also
of magnitude and figure, not only by the touch, but also by memory. For
though some things are touched in one point, yet rough and smooth, like
quantity and figure, are not perceived but by the flux of a point, that
is to say, we have no sense of them without time; and we can have no
sense of time without memory.


                                -------


                              CHAPTER XXX.

                              OF GRAVITY.

1. A thick body doth not contain more matter, unless also more place,
    than a thin.—2. That the descent of heavy bodies proceeds not from
    their own appetite, but from some power of the earth.—3. The
    difference of gravities proceedeth from the difference of the
    impetus with which the elements, whereof heavy bodies are made, do
    fall upon the earth.—4. The cause of the descent of heavy bodies.—5.
    In what proportion the descent of heavy bodies is accelerated.—6.
    Why those that dive do not, when they are under water, feel the
    weight of the water above them.—7. The weight of a body that
    floateth, is equal to the weight of so much water as would fill the
    space, which the immersed part of the body takes up within the
    water.—8. If a body be lighter than water, then how big soever that
    body be, it may float upon any quantity of water, how little
    soever.—9. How water may be lifted up and forced out of a vessel by
    air.—10. Why a bladder is heavier when blown full of air, than when
    it is empty.—11. The cause of the ejection upwards of heavy bodies
    from a wind-gun.—12. The cause of the ascent of water in a
    weather-glass.—13. The cause of motion upwards in living
    creatures.—14. That there is in nature a kind of body heavier than
    air, which nevertheless is not by sense distinguishable from it.—15.
    Of the cause of magnetical virtue.


[Sidenote: A thick body doth not contain more matter, unless also more
           place, than a thin.]

1. In chapter XXI I have defined thick and thin, as that place required,
so, as that by thick was signified a more resisting body, and by thin, a
body less resisting; following the custom of those that have before me
discoursed of refraction. Now if we consider the true and vulgar
signification of those words, we shall find them to be names collective,
that is to say, names of multitude; as _thick_ to be that, which takes
up more parts of a space given, and _thin_ that, which contains fewer
parts of the same magnitude in the same space, or in a space equal to
it. Thick therefore is the same with frequent, as a thick troop; and
thin the same with unfrequent, as a thin rank, thin of houses; not that
there is more matter in one place than in another equal place, but a
greater quantity of some named body. For there is not less matter or
body, indefinitely taken, in a desert, than there is in a city; but
fewer houses, or fewer men. Nor is there in a thick rank a greater
quantity of body, but a greater number of soldiers, than in a thin.
Wherefore the multitude and paucity of the parts contained within the
same space do constitute density and rarity, whether those parts be
separated by vacuum or by air. But the consideration of this is not of
any great moment in philosophy; and therefore I let it alone, and pass
on to the search of the causes of _gravity_.

[Sidenote: That the descent of heavy bodies proceeds not from their own
           appetite, but from some power of the earth.]

2. Now we call those bodies _heavy_, which, unless they be hindered by
some force, are carried towards the centre of the earth, and that by
their own accord, for aught we can by sense perceive to the contrary.
Some philosophers therefore have been of opinion, that the descent of
heavy bodies proceeded from some internal appetite, by which, when they
were cast upwards, they descended again, as moved by themselves, to such
place as was agreeable to their nature. Others thought they were
attracted by the earth. To the former I cannot assent, because I think I
have already clearly enough demonstrated that there can be no beginning
of motion, but from an external and moved body; and consequently, that
whatsoever hath motion or endeavour towards any place, will always move
or endeavour towards that same place, unless it be hindered by the
reaction of some external body. Heavy bodies, therefore, being once cast
upwards, cannot be cast down again but by external motion. Besides,
seeing inanimate bodies have no appetite at all, it is ridiculous to
think that by their own innate appetite they should, to preserve
themselves, not understanding what preserves them, forsake the place
they are in, and transfer themselves to another place; whereas man, who
hath both appetite and understanding, cannot, for the preservation of
his own life, raise himself by leaping above three or four feet from the
ground. Lastly, to attribute to created bodies the power to move
themselves, what is it else than to say that there be creatures which
have no dependance upon the Creator? To the latter, who attribute the
descent of heavy bodies to the attraction of the earth, I assent. But by
what motion this is done, hath not as yet been explained by any man. I
shall therefore in this place say somewhat of the manner and of the way
by which the earth by its action attracteth heavy bodies.

[Sidenote: The difference of gravities proceedeth from the difference of
           the impetus with which the elements, whereof heavy bodies are
           made, do fall upon the earth.]

3. That by the supposition of simple motion in the sun, homogeneous
bodies are congregated and heterogeneous dissipated, has already been
demonstrated in the 5th article of chapter XXI. I have also supposed,
that there are intermingled with the pure air certain little bodies, or,
as others call them, atoms; which by reason of their extreme smallness
are invisible, and differing from one another in consistence, figure,
motion, and magnitude; from whence it comes to pass that some of them
are congregated to the earth, others to other planets, and others are
carried up and down in the spaces between. And seeing those, which are
carried to the earth, differ from one another in figure, motion, and
magnitude, they will fall upon the earth, some with greater, others with
less impetus. And seeing also that we compute the several degrees of
gravity no otherwise than by this their falling upon the earth with
greater or less impetus; it follows, that we conclude those to be the
more heavy that have the greater impetus, and those to be less heavy
that have the less impetus. Our inquiry therefore must be, by what means
it may come to pass, that of bodies, which descend from above to the
earth, some are carried with greater, others with less impetus; that is
to say, some are more heavy than others. We must also inquire, by what
means such bodies, as settle upon the earth, may by the earth itself be
forced to ascend.

[Sidenote: The cause of the descent of heavy bodies.]

4. Let the circle made upon the centre C (in fig. 2) be a great circle
in the superficies of the earth, passing through the points A and B.
Also let any heavy body, as the stone A D, be placed anywhere in the
plane of the equator; and let it be conceived to be cast up from A D
perpendicularly, or to be carried in any other line to E, and supposed
to rest there. Therefore, how much space soever the stone took up in A
D, so much space it takes up now in E. And because all place is supposed
to be full, the space A D will be filled by the air which flows into it
first from the nearest places of the earth, and afterwards successively
from more remote places. Upon the centre C let a circle be understood to
be drawn through E; and let the plane space, which is between the
superficies of the earth and that circle, be divided into plane orbs
equal and concentric; of which let that be the first, which is contained
by the two perimeters that pass through A and D. Whilst therefore the
air, which is in the first orb, filleth the place A D, the orb itself is
made so much less, and consequently its latitude is less than the strait
line A D. Wherefore there will necessarily descend so much air from the
orb next above. In like manner, for the same cause, there will also be a
descent of air from the orb next above that; and so by succession from
the orb in which the stone is at rest in E. Either therefore the stone
itself, or so much air, will descend. And seeing air is by the diurnal
revolution of the earth more easily thrust away than the stone, the air,
which is in the orb that contains the stone, will be forced further
upwards than the stone. But this, without the admission of vacuum,
cannot be, unless so much air descend to E from the place next above;
which being done, the stone will be thrust downwards. By this means
therefore the stone now receives the beginning of its descent, that is
to say, of its gravity. Furthermore, whatsoever is once moved, will be
moved continually (as hath been shown in the 19th article of chapter
VIII) in the same way, and with the same celerity, except it be retarded
or accelerated by some external movent. Now the air, which is the only
body that is interposed between the earth A and the stone above it E,
will have the same action in every point of the strait line E A, which
it hath in E. But it depressed the stone in E; and therefore also it
will depress it equally in every point of the strait line E A. Wherefore
the stone will descend from E to A with accelerated motion. The possible
cause therefore of the descent of heavy bodies under the equator, is the
diurnal motion of the earth. And the same demonstration will serve, if
the stone be placed in the plane of any other circle parallel to the
equator. But because this motion hath, by reason of its greater
slowness, less force to thrust off the air in the parallel circles than
in the equator, and no force at all at the poles, it may well be thought
(for it is a certain consequent) that heavy bodies descend with less and
less velocity, as they are more and more remote from the equator; and
that at the poles themselves, they will either not descend at all, or
not descend by the axis; which whether it be true or false, experience
must determine. But it is hard to make the experiment, both because the
times of their descents cannot be easily measured with sufficient
exactness, and also because the places near the poles are inaccessible.
Nevertheless, this we know, that by how much the nearer we come to the
poles, by so much the greater are the flakes of the snow that falls; and
by how much the more swiftly such bodies descend as are fluid and
dissipable, by so much the smaller are the particles into which they are
dissipated.

[Sidenote: In what proportion the descent of heavy bodies is
           accelerated.]

5. Supposing, therefore, this to be the cause of the descent of heavy
bodies, it will follow that their motion will be accelerated in such
manner, as that the spaces, which are transmitted by them in the several
times, will have to one another the same proportion which the odd
numbers have in succession from unity. For if the strait line E A be
divided into any number of equal parts, the heavy body descending will,
by reason of the perpetual action of the diurnal motion, receive from
the air in every one of those times, in every several point of the
strait line E A, a several new and equal impulsion; and therefore also
in every one of those times, it will acquire a several and equal degree
of celerity. And from hence it follows, by that which Galileus hath in
his _Dialogues of Motion_ demonstrated, that heavy bodies descend in the
several times with such differences of transmitted spaces, as are equal
to the differences of the square numbers that succeed one another from
unity; which square numbers being 1, 4, 9, 16, &c. their differences are
3, 5, 7, &c.; that is to say, the odd numbers which succeed one another
from unity. Against this cause of gravity which I have given, it will
perhaps be objected, that if a heavy body be placed in the bottom of
some hollow cylinder of iron or adamant, and the bottom be turned
upwards, the body will descend, though the air above cannot depress it,
much less accelerate its motion. But it is to be considered that there
can be no cylinder or cavern, but such as is supported by the earth, and
being so supported is, together with the earth, carried about by its
diurnal motion. For by this means the bottom of the cylinder will be as
the superficies of the earth; and by thrusting off the next and lowest
air, will make the uppermost air depress the heavy body, which is at the
top of the cylinder, in such manner as is above explicated.

[Sidenote: Why those that dive, do not, when they are under water, feel
           the weight of the water above them.]

6. The gravity of water being so great as by experience we find it is,
the reason is demanded by many, why those that dive, how deep soever
they go under water, do not at all feel the weight of the water which
lies upon them. And the cause seems to be this, that all bodies by how
much the heavier they are, by so much the greater is the endeavour by
which they tend downwards. But the body of a man is heavier than so much
water as is equal to it in magnitude, and therefore the endeavour
downwards of a man's body is greater than that of water. And seeing all
endeavour is motion, the body also of a man will be carried towards the
bottom with greater velocity than so much water. Wherefore there is
greater reaction from the bottom; and the endeavour upwards is equal to
the endeavour downwards, whether the water be pressed by water, or by
another body which is heavier than water. And therefore by these two
opposite equal endeavours, the endeavour both ways in the water is taken
away; and consequently, those that dive are not at all pressed by it.

Coroll. From hence also it is manifest, that water in water hath no
weight at all, because all the parts of water, both the parts above, and
the parts that are directly under, tend towards the bottom with equal
endeavour and in the same strait lines.

[Sidenote: The weight of a body that floateth, is equal to the weight of
           so much water as would fill the space which the immersed part
           of the body takes up within the water.]

7. If a body float upon the water, the weight of that body is equal to
the weight of so much water as would fill the place which the immersed
part of the body takes up within the water.

Let E F (in fig. 3) be a body floating in the water A B C D; and let the
part E be above, and the other part F under the water. I say, the weight
of the whole body E F is equal to the weight of so much water as the
space F will receive. For seeing the weight of the body E F forceth the
water out of the space F, and placeth it upon the superficies A B, where
it presseth downwards; it follows, that from the resistance of the
bottom there will also be an endeavour upwards. And seeing again, that
by this endeavour of the water upwards, the body E F is lifted up, it
follows, that if the endeavour of the body downwards be not equal to the
endeavour of the water upwards, either the whole body E F will, by
reason of that inequality of their endeavours or moments, be raised out
of the water, or else it will descend to the bottom. But it is supposed
to stand so, as neither to ascend nor descend. Wherefore there is an
equilibrium between the two endeavours; that is to say, the weight of
the body E F is equal to the weight of so much water as the space F will
receive; which was to be proved.

[Sidenote: If a body be lighter than water, then how big soever that
           body be, it may float upon any quantity of water, how little
           soever.]

8. From hence it follows, that any body, of how great magnitude soever,
provided it consist of matter less heavy than water, may nevertheless
float upon any quantity of water, how little soever.

Let A B C D (in fig. 4) be a vessel; and in it let E F G H be a body
consisting of matter which is less heavy than water; and let the space A
G C F be filled with water. I say, the body E F G H will not sink to the
bottom D C. For seeing the matter of the body E F G H is less heavy than
water, if the whole space without A B C D were fall of water, yet some
part of the body E F G H, as E F I K, would be above the water; and the
weight of so much water as would fill the space I G H K would be equal
to the weight of the whole body E F G H; and consequently G H would not
touch the bottom D C. As for the sides of the vessel, it is no matter
whether they be hard or fluid; for they serve only to terminate the
water; which may be done as well by water as by any other matter how
hard soever; and the water without the vessel is terminated somewhere,
so as that it can spread no farther. The part therefore E F I K will be
extant above the water A G C F which is contained in the vessel.
Wherefore the body E F G H will also float upon the water A G C F, how
little soever that water be; which was to be demonstrated.

[Sidenote: How water may be lifted up and forced out of a vessel by
           air.]

9. In the 4th article of chapter XXVI, there is brought for the proving
of vacuum the experiment of water enclosed in a vessel; which water, the
orifice above being opened, is ejected upwards by the impulsion of the
air. It is therefore demanded, seeing water is heavier than air, how
that can be done. Let the second figure of the same, chapter XXVI be
considered, where the water is with great force injected by a syringe
into the space F G B. In that injection, the air (but pure air) goeth
with the same force out of the vessel through the injected water. But as
for those small bodies, which formerly I supposed to be intermingled
with air and to be moved with simple motion, they cannot, together with
the pure air, penetrate the water; but remaining behind are necessarily
thrust together into a narrower place, namely into the space which is
above the water F G. The motions therefore of those small bodies will be
less and less free, by how much the quantity of the injected water is
greater and greater; so that by their motions falling upon one another,
the same small bodies will mutually compress each other, and have a
perpetual endeavour of regaining their liberty, and of depressing the
water that hinders them. Wherefore, as soon as the orifice above is
opened, the water which is next it will have an endeavour to ascend, and
will therefore necessarily go out. But it cannot go out, unless at the
same time there enter in as much air; and therefore both the water will
go out, and the air enter in, till those small bodies which were left
within the vessel have recovered their former liberty of motion; that is
to say, till the vessel be again filled with air, and no water be left
of sufficient height to stop the passage at B. Wherefore I have shown a
possible cause of this phenomenon, namely, the same with that of
thunder. For as in the generation of thunder, the small bodies enclosed
within the clouds, by being too closely pent together, do by their
motion break the clouds, and restore themselves to their natural
liberty; so here also the small bodies enclosed within the space which
is above the strait line F G, do by their own motion expel the water as
soon as the passage is opened above. And if the passage be kept stopped,
and these small bodies be more vehemently compressed by the perpetual
forcing in of more water, they will at last break the vessel itself with
great noise.

[Sidenote: Why a bladder is heavier when blown full of air, than when it
           is empty.]

10. If air be blown into a hollow cylinder, or into a bladder, it will
increase the weight of either of them a little, as many have found by
experience, who with great accurateness have tried the same. And it is
no wonder, seeing, as I have supposed, there are intermingled with the
common air a great number of small hard bodies, which are heavier than
the pure air. For, the ethereal substance, being on all sides equally
agitated by the motion of the sun, hath an equal endeavour towards all
the parts of the universe; and, therefore, it hath no gravity at all.

[Sidenote: The cause of the ejection upwards of heavy bodies from a
           wind-gun.]

11. We find also by experience, that, by the force of air enclosed in a
hollow cannon, a bullet of lead may with considerable violence be shot
out of a gun of late invention, called the wind-gun. In the end of this
cannon there are two holes, with their valves on the inside, to shut
them close; one of them serving for the admission of air, and the other
for the letting of it out. Also, to that end which serves for the
receiving in of air, there is joined another cannon of the same metal
and bigness, in which there is fitted a rammer which is perforated, and
hath also a valve opening towards the former cannon. By the help of this
valve the rammer is easily drawn back, and letteth in air from without;
and being often drawn back and returned again with violent strokes, it
forceth some part of that air into the former cannon, so long, till at
last the resistance of the enclosed air is greater than the force of the
stroke. And by this means men think there is now a greater quantity of
air in the cannon than there was formerly, though it were full before.
Also, the air thus forced in, how much soever it be, is hindered from
getting out again by the aforesaid valves, which the very endeavour of
the air to get out doth necessarily shut. Lastly, that valve being
opened which was made for the letting out of the air, it presently
breaketh out with violence, and driveth the bullet before it with great
force and velocity.

As for the cause of this, I could easily attribute it, as most men do,
to condensation, and think that the air, which had at the first but its
ordinary degree of rarity, was afterwards, by the forcing in of more
air, condensed, and last of all, rarified again by being let out and
restored to its natural liberty. But I cannot imagine how the same place
can be always full, and, nevertheless, contain some times a greater,
sometimes a less quantity of matter; that is to say, that it can be
fuller than full. Nor can I conceive how fulness can of itself be an
efficient cause of motion. For both these are impossible. Wherefore we
must seek out some other possible cause of this phenomenon. Whilst,
therefore, the valve which serves for the letting in of air, is opened
by the first stroke of the rammer, the air within doth with equal force
resist the entering of the air from without; so that the endeavours
between the internal and external air are opposite, that is, there are
two opposite motions whilst the one goeth in and the other cometh out;
but no augmentation at all of air within the cannon. For there is driven
out by the stroke as much pure air, which passeth between the rammer and
the sides of the cannon, as there is forced in of air impure by the same
stroke. And thus, by many forcible strokes, the quantity of small hard
bodies will be increased within the cannon, and their motions also will
grow stronger and stronger, as long as the matter of the cannon is able
to endure their force; by which, if it be not broken, it will at least
be urged every way by their endeavour to free themselves; and as soon as
the valve, which serves to let them out, is opened, they will fly out
with violent motion, and carry with them the bullet which is in their
way. Wherefore, I have given a possible cause of this phenomenon.

[Sidenote: The cause of the ascent of water in a weather-glass.]

12. Water, contrary to the custom of heavy bodies, ascendeth in the
weather-glass; but it doth it when the air is cold: for when it is warm
it descendeth again. And this organ is called a thermometer or
thermoscope, because the degrees of heat and cold are measured and
marked by it. It is made in this manner. Let A B C D (in fig. 5) be a
vessel full of water, and E F G a hollow cylinder of glass, closed at E
and open at G. Let it be heated, and set upright within the water to F;
and let the open end reach to G. This being done, as the air by little
and little grows colder, the water will ascend slowly within the
cylinder from F towards E; till at last the external and internal air
coming to be both of the same temper, it will neither ascend higher nor
descend lower, till the temper of the air be changed. Suppose it,
therefore, to be settled anywhere, as at H. If now the heat of the air
be augmented, the water will descend below H; and if the heat be
diminished, it will ascend above it. Which, though it be certainly known
to be true by experience, the cause, nevertheless, hath not as yet been
discovered.

In the sixth and seventh articles of chapter XXVIII, where I consider
the cause of cold, I have shown, that fluid bodies are made colder by
the pressure of the air, that is to say, by a constant wind that
presseth them. For the same cause it is, that the superficies of the
water is pressed at F; and having no place, to which it may retire from
this pressure, besides the cavity of the cylinder between H and E, it is
therefore necessarily forced thither by the cold, and consequently it
ascendeth more or less, according as the cold is more or less increased.
And again, as the heat is more intense or the cold more remiss, the same
water will be depressed more or less by its own gravity, that is to say,
by the cause of gravity above explicated.

[Sidenote: Cause of motion upwards in living creatures]

13. Also living creatures, though they be heavy, can by leaping,
swimming and flying, raise themselves to a certain degree of height. But
they cannot do this except they be supported by some resisting body, as
the earth, the water and the air. For these motions have their beginning
from the contraction, by the help of the muscles, of the body animate.
For to this contraction there succeedeth a distension of their whole
bodies; by which distension, the earth, the water, or the air, which
supporteth them, is pressed; and from hence, by the reaction of those
pressed bodies, living creatures acquire an endeavour upwards, but such
as by reason of the gravity of their bodies is presently lost again. By
this endeavour, therefore, it is, that living creatures raise themselves
up a little way by leaping, but to no great purpose: but by swimming and
flying they raise themselves to a greater height; because, before the
effect of their endeavour is quite extinguished by the gravity of their
bodies, they can renew the same endeavour again.

That by the power of the soul, without any antecedent contraction of the
muscles or the help of something to support him, any man can be able to
raise his body upwards, is a childish conceit. For if it were true, a
man might raise himself to what height he pleased.

[Sidenote: That there is in nature a kind of body heavier than air,
           which nevertheless is not by sense distinguishable from it.]

14. The diaphanous medium, which surrounds the eye on all sides, is
invisible; nor is air to be seen in air, nor water in water, nor
anything but that which is more opacous. But in the confines of two
diaphanous bodies, one of them may be distinguished from the other. It
is not therefore a thing so very ridiculous for ordinary people to think
all that space empty, in which we say is air; it being the work of
reason to make us conceive that the air is anything. For by which of our
senses is it, that we take notice of the air, seeing we neither see, nor
hear, nor taste, nor smell, nor feel it to be anything? When we feel
heat, we do not impute it to the air, but to the fire: nor do we say the
air is cold, but we ourselves are cold; and when we feel the wind, we
rather think something is coming, than that any thing is already come.
Also, we do not at all feel the weight of water in water, much less of
air in air. That we come to know that to be a body, which we call air,
it is by reasoning; but it is from one reason only, namely, because it
is impossible for remote bodies to work upon our organs of sense but by
the help of bodies intermediate, without which we could have no sense of
them, till they come to be contiguous. Wherefore, from the senses alone,
without reasoning from effects, we cannot have sufficient evidence of
the nature of bodies.

For there is underground, in some mines of coals, a certain matter of a
middle nature between water and air, which nevertheless cannot by sense
be distinguished from air; for it is as diaphanous as the purest air;
and, as far as sense can judge, equally penetrable. But if we look upon
the effect, it is like that of water. For when that matter breaks out of
the earth into one of those pits, it fills the same either totally or to
some degree; and if a man or fire be then let down in it, it
extinguishes them in almost as little time as water would do. But for
the better understanding of this phenomenon, I shall describe the 6th
figure. In which let A B represent the pit of the mine; and let part
thereof, namely C B, be supposed to be filled with that matter. If now a
lighted candle be let down into it below C, it will as suddenly be
extinguished as if it were thrust into water. Also, if a grate filled
with coals thoroughly kindled and burning never so brightly, be let
down, as soon as ever it is below C, the fire will begin to grow pale,
and shortly after, losing its light, be extinguished, no otherwise than
if it were quenched in water. But if the grate be drawn up again
presently, whilst the coals are still very hot, the fire will, by little
and little, be kindled again, and shine as before. There is, indeed,
between this matter and water this considerable difference, that it
neither wetteth, nor sticketh to such things as are put down into it, as
water doth; which, by the moisture it leaveth, hindereth the kindling
again of the matter once extinguished. In like manner, if a man be let
down below C, he will presently fall into a great difficulty of
breathing, and immediately after into a swoon, and die unless he be
suddenly drawn up again. They, therefore, that go down into these pits,
have this custom, that as soon as ever they feel themselves sick, they
shake the rope by which they were let down, to signify they are not
well, and to the end that they may speedily be pulled up again. For if a
man be drawn out too late, void of sense and motion, they dig up a turf,
and put his face and mouth into the fresh earth; by which means, unless
he be quite dead, he comes to himself again, by little and little, and
recovers life by breathing out, as it were, of that suffocating matter,
which he had sucked in whilst he was in the pit; almost in the same
manner as they that are drowned come to themselves again by vomiting up
the water. But this doth not happen in all mines, but in some only; and
in those not always, but often. In such pits as are subject to it, they
use this remedy. They dig another pit, as D E, close by it, of equal
depth, and joining them both together with one common channel, E B, they
make a fire in the bottom E, which carries out at D the air contained in
the pit D E; and this draws with it the air contained in the channel E
B; which, in like manner, is followed by the noxious matter contained in
C B; and, by this means, the pit is for that time made healthful. Out of
this history, which I write only to such as have had experience of the
truth of it, without any design to support my philosophy with stories of
doubtful credit, may be collected the following possible cause of this
phenomenon; namely, that there is a certain matter fluid and most
transparent, and not much lighter than water, which, breaking out of the
earth, fills the pit to C; and that in this matter, as in water, both
fire and living creatures are extinguished.

[Sidenote: Of the cause of magnetical virtue.]

15. About the nature of heavy bodies, the greatest difficulty ariseth
from the contemplation of those things which make other heavy bodies
ascend to them; such as jet, amber, and the loadstone. But that which
troubles men most is the loadstone, which is also called _Lapis
Herculeus_; a stone, though otherwise despicable, yet of so great power
that it taketh up iron from the earth, and holds it suspended in the
air, as Hercules did Antæus. Nevertheless, we wonder at it somewhat the
less, because we see jet draw up straws, which are heavy bodies, though
not so heavy as iron. But as for jet, it must first be excited by
rubbing, that is to say, by motion to and fro; whereas the loadstone
hath sufficient excitation from its own nature, that is to say, from
some internal principle of motion peculiar to itself. Now, whatsoever is
moved, is moved by some contiguous and moved body, as hath been formerly
demonstrated. And from hence it follows evidently, that the first
endeavour, which iron hath towards the loadstone, is caused by the
motion of that air which is contiguous to the iron: also, that this
motion is generated by the motion of the next air, and so on
successively, till by this succession we find that the motion of all the
intermediate air taketh its beginning from some motion which is in the
loadstone itself; which motion, because the loadstone seems to be at
rest, is invisible. It is therefore certain, that the attractive power
of the loadstone is nothing else but some motion of the smallest
particles thereof. Supposing, therefore, that those small bodies, of
which the loadstone is in the bowels of the earth composed, have by
nature such motion or endeavour as was above attributed to jet, namely,
a reciprocal motion in a line too short to be seen, both those stones
will have one and the same cause of attraction. Now in what manner and
in what order of working this cause produceth the effect of attraction,
is the thing to be enquired. And first we know, that when the string of
a lute or viol is stricken, the vibration, that is, the reciprocal
motion of that string in the same strait line, causeth like vibration in
another string which hath like tension. We know also, that the dregs or
small sands, which sink to the bottom of a vessel, will be raised up
from the bottom by any strong and reciprocal agitation of the water,
stirred with the hand or with a staff. Why, therefore, should not
reciprocal motion of the parts of the loadstone contribute as much
towards the moving of iron? For, if in the loadstone there be supposed
such reciprocal motion, or motion of the parts forwards and backwards,
it will follow that the like motion will be propagated by the air to the
iron, and consequently that there will be in all the parts of the iron
the same reciprocations or motions forwards and backwards. And from
hence also it will follow, that the intermediate air between the stone
and the iron will, by little and little, be thrust away; and the air
being thrust away, the bodies of the loadstone and the iron will
necessarily come together. The possible cause therefore why the
loadstone and jet draw to them, the one iron, the other straws, may be
this, that those attracting bodies have reciprocal motion either in a
strait line, or in an elliptical line, when there is nothing in the
nature of the attracted bodies which is repugnant to such a motion.

But why the loadstone, if with the help of cork it float at liberty upon
the top of the water, should from any position whatsoever so place
itself in the plane of the meridian, as that the same points, which at
one time of its being at rest respect the poles of the earth, should at
all other times respect the same poles, the cause may be this; that the
reciprocal motion, which I supposed to be in the parts of the stone, is
made in a line parallel to the axis of the earth, and has been in those
parts ever since the stone was generated. Seeing therefore, the stone,
whilst it remains in the mine, and is carried about together with the
earth by its diurnal motion, doth by length of time get a habit of being
moved in a line which is perpendicular to the line of its reciprocal
motion, it will afterwards, though its axis be removed from the parallel
situation it had with the axis of the earth, retain its endeavour of
returning to that situation again; and all endeavour being the beginning
of motion, and nothing intervening that may hinder the same, the
loadstone will therefore return to its former situation. For, any piece
of iron that has for a long time rested in the plane of the meridian,
whensoever it is forced from that situation and afterwards left to its
own liberty again, will of itself return to lie in the meridian again;
which return is caused by the endeavour it acquired from the diurnal
motion of the earth in the parallel circles which are perpendicular to
the meridians.

If iron be rubbed by the loadstone drawn from one pole to the other, two
things will happen; one, that the iron will acquire the same direction
with the loadstone, that is to say, that it will lie in the meridian,
and have its axis and poles in the same position with those of the
stone; the other, that the like poles of the stone and of the iron will
avoid one another, and the unlike poles approach one another. And the
cause of the former may be this, that iron being touched by motion which
is not reciprocal, but drawn the same way from pole to pole, there will
be imprinted in the iron also an endeavour from the same pole to the
same pole. For seeing the loadstone differs from iron no otherwise than
as ore from metal, there will be no repugnance at all in the iron to
receive the same motion which is in the stone. From whence it follows,
that seeing they are both affected alike by the diurnal motion of the
earth, they will both equally return to their situation in the meridian,
whensoever they are put from the same. Also, of the latter this may be
the cause, that as the loadstone in touching the iron doth by its action
imprint in the iron an endeavour towards one of the poles, suppose
towards the North Pole; so reciprocally, the iron by its action upon the
loadstone doth imprint in it an endeavour towards the other pole, namely
towards the South Pole. It happens therefore in these reciprocations or
motions forwards and backwards of the particles of the stone and of the
iron betwixt the north and the south, that whilst in one of them the
motion is from north to south, and the return from south to north, in
the other the motion will be from south to north, and the return from
north to south; which motions being opposite to one another, and
communicated to the air, the north pole of the iron, whilst the
attraction is working, will be depressed towards the south pole of the
loadstone; or contrarily, the north pole of the loadstone will be
depressed towards the south pole of the iron; and the axis both of the
loadstone and of the iron will be situate in the same strait line. The
truth whereof is taught us by experience.

As for the propagation of this magnetical virtue, not only through the
air, but through any other bodies how hard soever, it is not to be
wondered at, seeing no motion can be so weak, but that it may be
propagated infinitely through a space filled with body of any hardness
whatsoever. For in a full medium, there can be no motion which doth not
make the next part yield, and that the next, and so successively without
end; so that there is no effect whatsoever, but to the production
thereof something is necessarily contributed by the several motions of
all the several things that are in the world.

[Sidenote: Conclusion.]

And thus much concerning the nature of body in general; with which I
conclude this my first section of the Elements of Philosophy. In the
first, second, and third parts, where the principles of ratiocination
consist in our own understanding, that is to say, in the legitimate use
of such words as we ourselves constitute, all the theorems, if I be not
deceived, are rightly demonstrated. The fourth part depends upon
hypotheses; which unless we know them to be true, it is impossible for
us to demonstrate that those causes, which I have there explicated, are
the true causes of the things whose productions I have derived from
them.

Nevertheless, seeing I have assumed no hypothesis, which is not both
possible and easy to be comprehended; and seeing also that I have
reasoned aright from those assumptions, I have withal sufficiently
demonstrated that they may be the true causes; which is the end of
physical contemplation. If any other man from other hypotheses shall
demonstrate the same or greater things, there will be greater praise and
thanks due to him than I demand for myself, provided his hypotheses be
such as are conceivable. For as for those that say anything may be moved
or produced by _itself_, by _species_, by _its own power_, by
_substantial forms_, by _incorporeal substances_, by _instinct_, by
_antiperistasis_, by _antipathy_, _sympathy_, _occult quality_, and
other empty words of schoolmen, their saying so is to no purpose.

And now I proceed to the phenomena of man's body; where I shall speak of
the _optics_, and of the _dispositions_, _affections_, and _manners_ of
men, if it shall please God to give me life, and show their causes.

[Illustration:

  _Vol. 1. Lat. & Eng._
  C. XXVIII & XXX.
  _Fig. 1-6_
]

                             END OF VOL. I.








                                LONDON:
             C. RICHARDS, ST. MARTIN'S LANE, CHARING CROSS.

------------------------------------------------------------------------

                           Transcriber’s Note

Figures 18.2 and 27.2 are not mentioned by name in the text. Figure
14.10 is mentioned only in Chapter XIII.

However, Figure 2 of Chapter XVIII would appear to be the subject of a
discussion on p. 270. A sidenote has been added, and includes a
reference to the figure.

Likewise Figure 2 of Chapter XXVII, depicting a ‘prisma’, is discussed
in detail on p. 459 _et seq._. Again, a sidenote has been interpolated
at the opening of that description, and includes a reference to the
figure.

The diacritics used in the Greek phrases is followed as printed, though
they sometimes seem non-standard (i.e., εἰναι rather than ειναὶ).

In each chapter's summary, the article numbers are prefixed with a dash
if they appeared midline. Dashes have been added to the text in the case
where the article number happened to appear in the first column.

On rare occasions, there are errors in the numbering of the articles in
the chapter summary. For example, in Chapter XVII, article 7 was
misnumbered as a second ‘6’.

In article 18 of Chapter XIII, a reference to article 10 of that chapter
used the Roman ‘X’ which is the only instance of that usage. It is now
referred to as article 10.

Other errors deemed most likely to be the printer’s have been corrected,
and are noted here. The references are to the page and line in the
original.

  52.17    _every man [i]s a living creature_             Restored.

  73.14    in the first place, we [we] are to search      Removed.

  255.11   seeing the [pro]proportion                     Removed.

  267.22   And therefore if it[ it] be again as R to T    Removed.

  307.24   the spiral of Arch[i/e]mides                   Replaced.

  329.29   with simple motion the[ the] perimeter         Removed.

  339.11   be cut in E and[ and] G by a strait line       Removed.

  372.12   to the base F H that[ that] can possibly be    Removed.
           drawn

  464.23   the sea appears b[l]ack when ruffled           Inserted.

  439.2    will make fifty-nine revo[lu]tions             Inserted.

                              Chapter XIV

[Illustration: Fig. 1.]

[Illustration: Fig. 2.]

[Illustration: Fig. 3.]

[Illustration: Fig. 4.]

[Illustration: Fig. 5.]

[Illustration: Fig. 6.]

[Illustration: Fig. 7.]

[Illustration: Fig. 8.]

[Illustration: Fig. 9.]

[Illustration: Fig. 10.]

                              CHAPTER XVI

[Illustration: Fig. 1.]

[Illustration: Fig. 2.]

[Illustration: Fig. 3.]

[Illustration: Fig. 4.]

[Illustration: Fig. 5.]

[Illustration: Fig. 6.]

[Illustration: Fig. 7.]

[Illustration: Fig. 8.]

[Illustration: Fig. 9.]

[Illustration: Fig. 10.]

[Illustration: Fig. 11.]

                              Chapter XVII

[Illustration: Fig. 1.]

[Illustration: Fig. 2.]

[Illustration: Fig. 3.]

[Illustration: Fig. 4.]

[Illustration: Fig. 5.]

[Illustration: Fig. 6.]

[Illustration: Fig. 7.]

[Illustration: Fig. 8.]

                             Chapter XVIII

[Illustration: Fig. 1.]

[Illustration: Fig. 2.]

                              Chapter XIX

[Illustration: Fig. 1.]

[Illustration: Fig. 2.]

[Illustration: Fig. 3.]

[Illustration: Fig. 4.]

[Illustration: Fig. 5.]

[Illustration: Fig. 6.]

[Illustration: Fig. 7.]

[Illustration: Fig. 8.]

[Illustration: Fig. 9.]

[Illustration: Fig. 10.]

                               Chapter XX

[Illustration: Fig. 1.]

[Illustration: Fig. 2.]

[Illustration: Fig. 3.]

[Illustration: Fig. 4.]

[Illustration: Fig. 5.]

[Illustration: Fig. 6.]

                              Chapter XXI

[Illustration: Fig. 1.]

[Illustration: Fig. 2.]

[Illustration: Fig. 3.]

[Illustration: Fig. 4.]

[Illustration: Fig. 5.]

                              Chapter XXII

[Illustration: Fig. 1.]

[Illustration: Fig. 2.]

[Illustration: Fig. 3.]

                             Chapter XXIII

[Illustration: Fig. 1.]

[Illustration: Fig. 2]

[Illustration: Fig. 3.]

[Illustration: Fig. 4]

[Illustration: Fig. 5.]

[Illustration: Fig. 6]

[Illustration: Fig. 7.]

[Illustration: Fig. 8]

[Illustration: Fig 9.]

                             CHAPTER XXIV.

[Illustration: Fig 1.]

[Illustration: Fig 2.]

[Illustration: Fig 3.]

[Illustration: Fig 4.]

[Illustration: Fig 5.]

[Illustration: Fig 6.]

                             CHAPTER XXVI.

[Illustration: Fig 1.]

[Illustration: Fig 2.]

[Illustration: Fig 3.]

[Illustration: Fig 4.]

[Illustration: Fig 5.]

                             Chapter XXVII.

[Illustration: Fig 1.]

[Illustration: Fig 2.]

                              Chapter XXX.

[Illustration: Fig 1.]

[Illustration: Fig 2.]

[Illustration: Fig 3.]

[Illustration: Fig 4.]

[Illustration: Fig 5.]

[Illustration: Fig 6.]



*** END OF THE PROJECT GUTENBERG EBOOK 73877 ***