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Please +see the transcriber’s note at the end of this text for details regarding +the handling of any textual issues encountered during its preparation. + +------------------------------------------------------------------------ + + + + + THE + + ENGLISH WORKS + + OF + + THOMAS HOBBES + + OF MALMESBURY; + + NOW FIRST COLLECTED AND EDITED + + BY + + SIR WILLIAM MOLESWORTH, BART. + + ------- + + VOL. VII. + + ------- + + + + + LONDON: + LONGMAN, BROWN, GREEN, AND LONGMANS, + PATERNOSTER-ROW. + + -- + + MDCCCXLV. + + + + + LONDON: + RICHARDS, PRINTER, 100, ST. MARTIN’S LANE. + + + + + + + + + CONTENTS. + + --- + + PAGE + SEVEN PHILOSOPHICAL PROBLEMS 1 + + DECAMERON PHYSIOLOGICUM 69 + + PROPORTION OF A STRAIGHT LINE TO HALF THE ARC OF A QUADRANT 178 + + SIX LESSONS TO THE SAVILIAN PROFESSORS OF THE MATHEMATICS 181 + + ΣΤΙΓΜΑΙ, OR MARKS OF THE ABSURD GEOMETRY ETC. OF DR. WALLIS 357 + + EXTRACT OF A LETTER FROM HENRY STUBBE 401 + + THREE PAPERS PRESENTED TO THE ROYAL SOCIETY AGAINST DR. WALLIS 429 + + CONSIDERATIONS ON THE ANSWER OF DR. WALLIS 443 + + LETTERS AND OTHER PIECES 449 + + + + + SEVEN + PHILOSOPHICAL PROBLEMS + AND + TWO PROPOSITIONS OF GEOMETRY. + + + BY + THOMAS HOBBES + OF MALMESBURY. + + WITH + AN APOLOGY FOR HIMSELF AND HIS WRITINGS. + DEDICATED TO THE KING IN THE YEAR 1662. + + TO THE KING. + +That which I do here most humbly present to your sacred Majesty, is the +best part of my meditations upon the natural causes of events, both of +such as are commonly known, and of such as have been of late +artificially exhibited by the curious. + +They are ranged under seven heads. 1. Problems of gravity. 2. Problems +of tides. 3. Problems of vacuum. 4. Problems of heat. 5. Problems of +hard and soft. 6. Problems of wind and weather. 7. Problems of motion +perpendicular and oblique, &c. To which I have added two propositions of +Geometry: one is, the duplication of the cube, hitherto sought in vain; +the other, a detection of the absurd use of arithmetic, as it is now +applied to geometry. + +The doctrine of natural causes hath not infallible and evident +principles. For there is no effect which the power of God cannot produce +by many several ways. + +But seeing all effects are produced by motion, he that supposing some +one or more motions, can derive from them the necessity of that effect +whose cause is required, has done all that is to be expected from +natural reason. And though he prove not that the thing was thus +produced, yet he proves that thus it may be produced when the materials +and the power of moving are in our hands: which is as useful as if the +causes themselves were known. And notwithstanding the absence of +rigorous demonstration, this contemplation of nature (if not rendered +obscure by empty terms) is the most noble employment of the mind that +can be, to such as are at leisure from their necessary business. + +This that I have done I know is an unworthy present to be offered to a +king: though considered, as God considers offerings, together with the +mind and fortune of the offerer, I hope will not be to your Majesty +unacceptable. + +But that which I chiefly consider in it is, that my writing should be +tried by your Majesty’s excellent reason, untainted with the language +that has been invented or made use of by men when they were puzzled; and +who is acquainted with all the experiments of the time; and whose +approbation, if I have the good fortune to obtain it, will protect my +reasoning from the contempt of my adversaries. + +I will not break the custom of joining to my offering a prayer; and it +is, that your Majesty will be pleased to pardon this following short +apology for my _Leviathan_ . Not that I rely upon apologies, but upon +your Majesty’s most gracious general pardon. + +That which is in it of theology, contrary to the general current of +divines, is not put there as my opinion, but propounded with submission +to those that have the power ecclesiastical. + +I did never after, either in writing or discourse, maintain it. + +There is nothing in it against episcopacy; I cannot therefore imagine +what reason any episcopal man can have to speak of me, as I hear some of +them do, as of an atheist, or man of no religion, unless it be for +making the authority of the Church wholly upon the regal power; which I +hope your Majesty will think is neither atheism nor heresy. + +But what had I to do to meddle with matters of that nature, seeing +religion is not philosophy, but law? + +It was written in a time when the pretence of Christ’s kingdom was made +use of for the most horrid actions that can be imagined; and it was in +just indignation of that, that I desired to see the bottom of that +doctrine of the kingdom of Christ, which divers ministers then preached +for a pretence to their rebellion: which may reasonably extenuate, +though not excuse the writing of it. + +There is therefore no ground for so great a calumny in my writing. There +is no sign of it in my life; and for my religion, when I was at the +point of death at St. Germain’s, the Bishop of Durham can bear witness +of it, if he be asked. Therefore I most humbly beseech your sacred +Majesty not to believe so ill of me upon reports, that proceed often, +and may do so now, from the displeasure which commonly ariseth from +difference in opinion; nor to think the worse of me, if snatching up all +the weapons to fight against your enemies, I lighted upon one that had a +double edge. + + Your Majesty’s poor and + + most loyal subject, + + THOMAS HOBBES. + + ========== + +[Illustration: _Seven Philosophical Problems. English Works, Vol. 7._] + + PHILOSOPHICAL PROBLEMS. + + --- + + + CHAPTER I. + PROBLEMS OF GRAVITY. + +_A._ What may be the cause, think you, that stones and other bodies +thrown upward, or carried up and left to their liberty, fall down again, +for aught a man can see, of their own accord? I do not think with the +old philosophers, that they have any love to the earth; or are sullen, +that they will neither go nor stay. And yet I cannot imagine, what body +there is above that should drive them back. + +_B._ For my part, I believe the cause of their descending is not in any +natural appetite of the bodies that descend; but rather that the globe +of the earth hath some special motion, by which it more easily casteth +off the air than it doth other bodies. And then this descent of those we +call heavy bodies must of necessity follow, unless there be some empty +spaces in the world to receive them. For when the air is thrown off from +the earth, somewhat must come into the place of it, in case the world be +full: and it must be those things which are hardliest cast off, that is, +those things which we say are heavy. + +_A._ But suppose there be no place empty, (for I will defer the question +till anon), how can the earth cast off either the air or anything else? + +_B._ I shall show you how, and that by a familiar example. If you lay +both your hands upon a basin with water in it, how little soever, and +move it circularly, and continue that motion for a while; and you shall +see the water rise upon the sides, and fly over. By which you may be +assured that there is a kind of circulating motion, which would cast off +such bodies as are contiguous to the body so moved. + +_A._ I know very well there is; and it is the same motion which country +people use to purge their corn; for the chaff and straws, by casting the +grain to the sides of the sieve, will come towards the middle. But I +would see the figure. + +_B._ Here it is. There is a circle pricked out, whose centre is A, and +three less circles, whose centres are B, C, D. Let every one of them +represent the earth, as it goeth from B to C, and from C to D, always +touching the uttermost circle and throwing off the air, as is marked at +E and F. And if the world were not full, there would follow by this +scattering of the air, a great deal of space left empty. But supposing +the world full, there must be a perpetual shifting of the air, one part +into the place of another. + +_A._ But what makes a stone come down, suppose from G? + +_B._ If the air be thrown up beyond G, it will follow that at the last, +if the motion be continued, all the air will be above G, that is, above +the stone; which cannot be, till the stone be at the earth. + +_A._ But why comes it down still with increasing swiftness? + +_B._ Because as it descends and is already in motion, it receiveth a new +impression from the same cause, which is the air, whereof as part +mounteth, part also must descend, supposing as we have done the +plentitude of the world. For, as you may observe by the figure, the +motion of the earth, according to the diameter of the uttermost circle, +is progressive; and so the whole motion is compounded of two motions, +one circular and the other progressive; and consequently the air ascends +and circulates at once. And because the stone descending receiveth a new +pressure in every point of its way, the motion thereof must needs be +accelerated. + +_A._ It is true; for it will be accelerated equally in equal times; and +the way it makes will increase in a double proportion to the times, as +hath heretofore been demonstrated by Galileo. I see the solution now of +an experiment, which before did not a little puzzle me. You know that if +two plummets hang by two strings of equal length, and you remove them +from the perpendicular equally, I mean in equal angles, and then let +them go, they will make their turns and returns together and in equal +times; and though the arches they describe grow continually less and +less, yet the times they spend in the greater arches will still be equal +to the time they spend in the lesser. + +_B._ It is true. Do you find any experiment to the contrary? + +_A._ Yes; for if you remove one of the plummets from the perpendicular, +so as, for example, to make an angle with the perpendicular of eighty +degrees, and the other so as to make an angle of sixty degrees; they +will not make their turns and returns in equal times. + +_B._ And what say you is the cause of this? + +_A._ Because the arches are the spaces which these two motions describe, +they must be in double proportion to their own times: which cannot be, +unless they be let go from equal altitudes, that is, from equal angles. + +_B._ It is right; and the experiment does not cross, but confirm the +equality of the times in all the arches they describe, even from ninety +degrees to the least part of one degree. + +_A._ But is it not too bold, if not extravagant an assertion, to say the +earth is moved as a man shakes a basin or a sieve? Does not the earth +move from west to east every day once, upon its own centre; and in the +ecliptic circle once a year? And now you give it another odd motion. How +can all these consist in one and the same body? + +_B._ Well enough. If you be a shipboard under sail, do not you go with +the ship? Cannot you also walk upon the deck? Cannot every drop of blood +move at the same time in your veins? How many motions now do you assign +to one and the same drop of blood? Nor is it so extravagant a thing to +attribute to the earth this kind of motion; but that I believe, if we +certainly knew what motion it is that causeth the descent of bodies, we +should find it either the same, or more extravagant. But seeing it can +be nothing above that worketh this effect, it must be the earth itself +that does it; and if the earth, then you can imagine no other motion to +do it withal but this. And you will wonder more, when by the same motion +I shall give you a probable account of the causes of very many other +works of nature. + +_A._ But what part of the heaven do you suppose the poles of your +pricked circle point to? + +_B._ I suppose them to be the same with the poles of the ecliptic. For, +seeing the axis of the earth in this motion and in the annual motion, +keeps parallel to itself, the axis must in both motions be parallel as +to sense. For the circle which the earth describes, is not of visible +magnitude at the distance it is from the sun. + +_A._ Though I understand well enough how the earth may make a stone +descend very swiftly under the ecliptic, or not far from it, where it +throws off the air perpendicularly; yet about the poles of the circle +methinks, it should cast off the air very weakly. I hope you will not +say, that bodies descend faster in places remote from the poles, than +nearer to them. + +_B._ No; but I ascribe it to the like motion in the sun and moon. For +such motions meeting, must needs cast the stream of the air towards the +poles; and then there will be the same necessity for the descent there, +that there is in other places, though perhaps a little more slowly. For +you may have observed, that when it snows in the south parts, the flakes +of snow are not so great as in the north: which is a probable sign they +fall in the south from a greater height, and consequently disperse +themselves more, as water does that falls down from a high and steep +rock. + +_A._ It is not improbable. + +_B._ In natural causes all you are to expect, is but probability; which +is better yet, than making gravity the cause, when the cause of gravity +is that which you desire to know; and better than saying the earth draws +it, when the question is, how it draws. + +_A._ Why does the earth cast off air more easily than it does water, or +any other heavy bodies? + +_B._ It is indeed the earth that casteth off that air which is next unto +it; but it is that air which casteth off the next air; and so +continually, air moveth air; which it can more easily do than any other +thing, because like bodies are more susceptible of one another’s +motions: as you may see in two lute-strings equally strained, what +motion one string being stricken communicates to the air, the same will +the other receive from the air; but strained to a differing note, will +be less or not at all moved. For there is no body but air, that hath not +some internal, though invisible, motion of its parts: and it is that +internal motion which distinguisheth all natural bodies one from +another. + +_A._ What is the cause why certain squibs, though their substance be +either wood or other heavy matter, made hollow and filled with +gunpowder, which is also heavy; do nevertheless, when the gunpowder is +kindled, fly upwards? + +_B._ The same that keeps a man that swims from sinking, though he be +heavier than so much water. He keeps himself up, and goes forward, by +beating back the water with his feet; and so does a squib, by beating +down the air with the stream of fired gunpowder, that proceeding from +its tail makes it recoil. + +_A._ Why does any brass or iron vessel, if it be hollow, float upon the +water, being so very heavy? + +_B._ Because the vessel and the air in it, taken as one body, is more +easily cast off than a body of water equal to it. + +_A._ How comes it to pass, that a fish, (especially such a broad fish as +a turbot or a plaice, which are broad and thin), in the bottom of the +sea, perhaps a mile deep, is not pressed to death with the weight of +water that lies upon the back of it? + +_B._ Because all heavy bodies descend towards one point, which is the +centre of the earth: and consequently the whole sea, descending at once, +does arch itself so, as that the upper parts cannot press the parts next +below them. + +_A._ It is evident; nor can there possibly be any weight, as some +suppose there is, of a cylinder of air or water or any other liquid +thing, while it remains in its own element, or is sustained and inclosed +in a vessel by which one part cannot press the other. + + + ========== + + + CHAPTER II. + PROBLEMS OF TIDES. + +_A._ What makes the flux and reflux of the sea, twice in a natural day? + +_B._ We must come again to our basin of water; wherein you have seen, +whilst it was moved, how the water mounteth up by the sides, and withal +goes circling round about. Now if you should fasten to the inside of the +basin some bar from the bottom to the top, you would see the water, +instead of going on, go back again from that bar ebbing, and the water +on the other side of the bar to do the same, but in counter-time; and +consequently to be highest where the contrary streams meet together; and +then return again, marking out four quarters of the vessel; two by their +meeting, which are the high waters; and two by their retiring, which are +the low waters. + +_A._ What bar is that you find in the ocean that stops the current of +the water, like that you make in the basin? + +_B._ You know that the main ocean lies east and west, between India and +the coast of America; and again on the other side, between America and +India. If therefore the earth have such a motion as I have supposed, it +must needs carry the current of the sea east and west: in which course, +the bar that stoppeth it, is the south part of America, which leaves no +passage for the water but the narrow strait of Magellan. The tide rises +therefore upon the coast of America; and the rising of the same in this +part of the world, proceedeth from the swelling chiefly of the water +there, and partly also from the North Sea; which lieth also east and +west, and has a passage out of the South Sea by the strait of Anian, +between America and Asia. + +_A._ Does not the Mediterranean Sea lie also east and west? Why are +there not the like tides there? + +_B._ So there are, proportionable to their lengths and quantity of +water. + +_A._ At Genoa, at Ancona, there are none at all, or not sensible. + +_B._ At Venice there are, and in the bottom of the straits, and a +current all along both the Mediterranean Sea and the Gulf of Venice: and +it is the current that makes the tides insensible at the sides; but the +check makes them visible at the bottom. + +_A._ How comes it about that the moon hath such a stroke in the +business, as so sensibly to increase the tides at full and change? + +_B._ The motion I have hitherto supposed but in the earth, I suppose +also in the moon, and in all those great bodies that hang in the air +constantly, I mean the stars, both fixed and errant. And for the sun and +moon, I suppose the poles of their motion to be the poles of the +equinoxial. Which supposed, it will follow (because the sun, the earth, +and the moon, at every full and change are almost in one straight line) +that this motion of the earth will then be made swifter than in the +quarters. For this motion of the sun and moon being communicated to the +earth, that hath already the like motion, maketh the same greater; and +much greater when they are all three in one straight line, which is only +at the full and change, whose tides are therefore called spring tides. + +_A._ But what then is the cause that the spring tides themselves are +twice a-year, namely, when the sun is in the equinoxial, greater than at +any other times? + +_B._ At other times of the year, the earth being out of the equinoxial, +the motion thereof, by which the tides are made, will be less augmented, +by so much as a motion in the obliquity of twenty-three degrees, or +thereabout, which is the distance between the equinoxial and ecliptic +circles, is weaker than the motion which is without obliquity. + +_A._ All this is reasonable enough, if it be possible that such motions +as you suppose in these bodies, be really there. But that is a thing I +have some reason to doubt of. For the throwing off of air, consequent to +these motions, is the cause, you say, that other things come to the +earth; and therefore the like motions in the sun and moon and stars, +casting off the air, should also cause all other things to come to every +one of them. From whence it will follow, that the sun, moon, and earth, +and all other bodies but air, should presently come together into one +heap. + +_B._ That does not follow. For if two bodies cast off the air, the +motion of that air will be repressed both ways, and diverted into a +course towards the poles on both sides; and then the two bodies cannot +possibly come together. + +_A._ It is true. And besides, this driving of the air on both sides, +north and south, makes the like motion of air there also. And this may +answer the question, how a stone could fall to the earth under the poles +of the ecliptic, by the only casting off of air? + +_B._ It follows from hence, that there is a certain and determinate +distance of one of these bodies, the stars, from another, without any +very sensible variation. + +_A._ All this is probable enough, if it be true that there is no vacuum, +no place empty in all the world. And supposing this motion of the sun +and moon to be in the plain of the equinoxial, methinks that this should +be the cause of the diurnal motion of the earth; and because this motion +of the earth is, you say, in the plain of the equinoxial, the same +should cause also a motion in the moon on her own centre, answerable to +the diurnal motion of the earth. + +_B._ Why not? What else can you think makes the diurnal motion of the +earth but the sun? And for the moon, if it did not turn upon its own +centre, we should see sometimes one, sometimes another face of the moon, +which we do not. + + + ========== + + + CHAPTER III. + PROBLEMS OF VACUUM. + +_A._ What convincing argument is there to prove, that in all the world +there is no empty place? + +_B._ Many; but I will name but one; and that is, the difficulty of +separating two bodies hard and flat laid one upon another. I say the +difficulty, not the impossibility. It is possible, without introducing +vacuum, to pull asunder any two bodies, how hard and flat soever they +be, if the force used be greater than the resistance of the hardness. +And in case there be any greater difficulty to part them, besides what +proceeds from their hardness, than there is to pull them further asunder +when they are parted, that difficulty is argument enough to prove there +is no vacuum. + +_A._ These assertions need demonstration. And first, how does the +difficulty of separation argue the plenitude of all the rest of the +world? + +_B._ If two flat polished marbles lie one upon another, you see they are +hardly separated in all points at one and the same instant; and yet the +weight of either of them is enough to make them slide off one from the +other. Is not the cause of this, that the air succeeds the marble that +so slides, and fills up the place it leaves? + +_A._ Yes, certainly. What then? + +_B._ But when you pull the whole superficies asunder, not without great +difficulty, what is the cause of that difficulty? + +_A._ I think, as most men do, that the air cannot fill up the space +between in an instant; for the parting is in an instant. + +_B._ Suppose there be vacuum in that air into which the marble you pull +off is to succeed, shall there be no vacuum in the air that was round +about the two marbles when they touched? Why cannot that vacuum come +into the place between? Air cannot succeed in an instant, because a +body, and consequently cannot be moved through the least space in an +instant. But emptiness is not a body, nor is moved, but is made by the +act itself of separation. There is therefore, if you admit vacuum, no +necessity at all for the air to fill the space left in an instant. And +therefore, with what ease the marble coming off presseth out the vacuum +of the air behind it, with the same ease will the marbles be pulled +asunder. Seeing then, if there were vacuum, there would be no difficulty +of separation, it follows, because there is difficulty of separation, +that there is no vacuum. + +_A._ Well, now, supposing the world full, how do you prove it possible +to pull those marbles asunder? + +_B._ Take a piece of soft wax; do not you think the one half touches the +other half as close as the smoothest marbles? Yet you can pull them +asunder. But how? Still as you pull, the wax grows continually more and +more slender; there being a perpetual parting or discession of the +outermost part of the wax one from another, which the air presently +fills; and so there is a continual lessening of the wax, till it be no +bigger than a hair, and at last separation. If you can do the same to a +pillar of marble, till the outside give way, the effect will be the +same, but much quicker, after it once begins to break in the +superficies; because the force that can master the first resistance of +the hardness, will quickly dispatch the rest. + +_A._ It seems so by the brittleness of some hard bodies. But I shall +afterward put some questions to you, touching the nature of hardness. +But now to return to our subject. What reason can you render (without +supposing vacuum) of the effects produced in the engine they use at +Gresham college? + +_B._ That engine produceth the same effects that a strong wind would +produce in a narrow room. + +_A._ How comes the wind in? You know the engine is a hollow round pipe +of brass, into which is thrust a cylinder of wood covered with leather, +and fitted to the cylinder so exactly as no air can possibly pass +between the leather and the brass? + +_B._ I know it; and that they may thrust it up, there is a hole left in +the cylinder to let the air out before it, which they can stop when they +please. There is also in the bottom of the cylinder a passage into a +hollow globe of glass, which passage they can also open and shut at +pleasure. And at the top of that globe there is a wide mouth to put in +what they please to try conclusions on, and that also to be opened and +shut as shall be needful. It is of the nature of a pop-gun which +children use, but great, costly, and more ingenious. They thrust forward +and pull back the wooden cylinder (because it requires much strength) +with an iron screw. What is there in all this to prove the possibility +of vacuum. + +_A._ When this wooden cylinder covered with leather, fit and close, is +thrust home to the bottom, and the holes in the hollow cylinder of brass +close stopped, how can it be drawn back, as with the screw they draw it, +but that the space it leaves must needs be empty: for it is impossible +that any air can pass into the place to fill it? + +_B._ Truly I think it close enough to keep out straw and feathers, but +not to keep out air, nor yet matter. For suppose they were not so +exactly close but that there were round about a difference for a small +hair to lie between; then will the pulling back of the cylinder of wood +force so much air in, as in retiring it forces back, and that without +any sensible difficulty. And the air will so much more swiftly enter as +the passage is left more narrow. Or if they touch, and the contact be in +some points and not in all, the air will enter as before, in case the +force be augmented accordingly. Lastly, though they touch exactly, if +either the leather yield, or the brass, which it may do, to the force of +a strong screw, the air will again enter. Do you think it possible to +make two superficies so exquisitely touch in all points as you suppose, +or leather so hard as not to yield to the force of a screw? The body of +leather will give passage both to air and water, as you will confess +when you ride in rainy and windy weather. You may therefore be assured +that in drawing out their wooden leather cylinder, they force in as much +air as will fill the place it leaves, and that with as much swiftness as +is answerable to the strength that drives it in. The effect therefore of +their pumping is nothing else but a vehement wind, a very vehement wind, +coming in on all sides of the cylinder at once into the hollow of the +brass pipe, and into the hollow of the glass globe joined to it. + +_A._ I see the reason already of one of their wonders, which is, that +the cylinder they pump with, if it be left to itself, after it is pulled +back, will swiftly go up again. You will say the air comes out again +with the same violence by reflection, and I believe it. + +_B._ This is argument enough that the place was not empty. For what can +fetch or drive up the sucker, as they call it, if the place within were +empty? For that there is any weight in the air to do it, I have already +demonstrated to be impossible. Besides, you know, when they have sucked +out, as they think, all the air from the glass globe, they can +nevertheless both see through it what is done, and hear a sound from +within when there is any made; which, if there were no other, but there +are many other, is argument enough that the place is still full of air. + +_A._ What say you to the swelling of a bladder even to bursting, if it +be a little blown when it is put into the receiver, for so they call the +globe of glass? + +_B._ The streams of air that from every side meeting together, and +turning in an infinite number of small points, do pierce the bladder in +innumerable places with great violence at once, like so many invisible +small wimbles, especially if the bladder be a little blown before it be +put in, that it may make a little resistance. And when the air has once +pierced it, it is easy to conceive, that it must afterward by the same +violent motion be extended till it break. If before it break you let in +fresh air upon it, the violence of the motion will thereby be tempered, +and the bladder be less extended; for that also they have observed. Can +you imagine how a bladder should be extended and broken by being too +full of emptiness? + +_A._ How come living creatures to be killed in this receiver, in so +little a time as three or four minutes of an hour? + +_B._ If they suck into their lungs so violent a wind thus made, you must +needs think it will presently stop the passage of their blood; and that +is death; though they may recover if taken out before they be too cold. +And so likewise will it put out fire; but the coals taken out whilst +they are hot will revive again. It is an ordinary thing in many +coal-pits, whereof I have seen the experience, that a wind proceeding +from the sides of the pit every way, will extinguish any fire let down +into it, and kill the workmen, unless they be quickly taken out. + +_A._ If you put a vessel of water into the receiver, and then suck out +the air, the water will boil; what say you to that? + +_B._ It is like enough it will dance in so great a bustling of the air; +but I never heard it would be hot. Nor can I imagine how vacuum should +make anything dance. I hope you are by this time satisfied, that no +experiment made with the engine at Gresham College, is sufficient to +prove that there is, or that there may be vacuum. + +_A._ The world you know is finite, and consequently, all that infinite +space without it is empty. Why may not some of that vacuum be brought +in, and mingled with the air here? + +_B._ I know nothing in matters without the world. + +_A._ What say you to Torricellio’s experiment in quicksilver, which is +this: there is a basin at A filled with quicksilver, suppose to B, and C +D a hollow glass pipe filled with the same, which if you stop with your +finger at B, and so set it upright, and then if you take away your +finger, the quicksilver will fall from C downwards but not to the +bottom, for it will stop by the way, suppose at D. Is it not therefore +necessary that that space between C and D be left empty? Or will you say +the quicksilver does not exactly touch the sides of the glass pipe? + +_B._ I will say neither. If a man thrust down into a vessel of +quicksilver a blown bladder, will not that bladder come up to the top? + +_A._ Yes, certainly, or a bladder of iron, or anything else but gold. + +_B._ You see then that air can pierce quicksilver. + +_A._ Yes, with so much force as the weight of quicksilver comes to. + +_B._ When the quicksilver is fallen to D, there is so much the more in +the basin, and that takes up the place which so much air took up before. +Whither can this air go if all the world without that glass pipe B C +were full? There must needs be the same or as much air come into that +space, which only is empty, between C and D: by what force? By the +weight of the quicksilver between D and B. Which quicksilver weigheth +now upward, or else it could never have raised that part higher, which +was at first in the basin. So you see the weight of quicksilver can +press the air through quicksilver up into the pipe, till it come to an +equality of force as in D, where the weight of the quicksilver is equal +to the force which is required in air to go through it. + +_A._ If a man suck a phial that has nothing in it but air, and presently +dip the mouth of it into water, the water will ascend into the phial. Is +not that an argument that part of the air had been sucked out, and part +of the room within the phial left empty? + +_B._ No. If there were empty space in the world, why should not there be +also some empty space in the phial before it was sucked? And then why +does not the water rise to fill that? When a man sucks the phial he +draws nothing out, neither into his belly, nor into his lungs, nor into +his mouth; only he sets the air within the glass into a circular motion, +giving it at once an endeavour to go forth by the sucking, and an +endeavour to go back by not receiving it into his mouth; and so with a +great deal of labour glues his lips to the neck of the phial. Then +taking it off, and dipping the neck of the phial into the water before +the circulation ceases, the air, with the endeavour it hath now gotten, +pierces the water and goes out: and so much air as goes out, so much +matter comes up into the room of it. + + + ========== + + + CHAPTER IV. + PROBLEMS OF HEAT AND LIGHT. + +_A._ What is the cause of heat? + +_B._ How know you, that any thing is hot but yourself? + +_A._ Because I perceive by sense it heats me. + +_B._ It is no good argument, the thing heats me; therefore it is hot. +But what alteration do you find in your body at any time by being hot? + +_A._ I find my skin more extended in summer than in winter; and am +sometimes fainter and weaker then ordinary, as if my spirits were +exhaled; and I sweat. + +_B._ Then that is it you would know the cause of. I have told you before +that by the motion I suppose both in the sun, and in the earth, the air +is dissipated, and consequently that there would be an infinite number +of small empty places, but that the world being full, there comes from +the next parts other air into the spaces they would else make empty. +When therefore this motion of the sun is exercised upon the superficies +of the earth, if there do not come out of the earth itself some corporal +substance to supply that tearing of the air, we must return again to the +admission of vacuum. If there do, then you see how by this motion fluid +bodies are made to exhale out of the earth. The like happens to a man’s +body or hand, which when he perceives, he says he is hot. And so of the +earth when it sendeth forth water and earth together in plants, we say +it does it by heat from the sun. + +_A._ It is very probable, and no less probable, that the same action of +the sun is that which from the sea and moist places of the earth, but +especially from the sea, fetcheth up the water into the clouds. But +there be many ways of heating besides the action of the sun or of fire. +Two pieces of wood will take fire if in turning they be pressed +together. + +_B._ Here again you have a manifest laceration of the air by the +reciprocal and contrary motions of the two pieces of wood, which +necessarily causeth a coming forth of whatsoever is aereal or fluid +within them, and (the motion pursued) a dissipation also of the other +more solid parts into ashes. + +_A._ How comes it to pass that a man is warmed even to sweating, almost +with every extraordinary labour of his body? + +_B._ It is easy to understand, how by that labour all that is liquid in +his body is tossed up and down, and thereby part of it also cast forth. + +_A._ There be some things that make a man hot without sweat or other +evaporation, as caustics, nettles, and other things. + +_B._ No doubt. But they touch the part they so heat, and cannot work +that effect at any distance. + +_A._ How does heat cause light, and that partially, in some bodies more, +in some less, though the heat be equal? + +_B._ Heat does not cause light at all. But in many bodies, the same +cause, that is to say, the same motion, causeth both together; so that +they are not to one another as cause and effect, but are concomitant +effects sometimes of one and the same motion. + +_A._ How? + +_B._ You know the rubbing or hard pressing of the eye, or a stroke upon +it, makes an apparition of light without and before it, which way soever +you look. This can proceed from nothing else but from the restitution of +the organ pressed or stricken, unto its former ordinary situation of +parts. Does not the sun by his thrusting back the air upon your eyes +press them? Or do not those bodies whereon the sun shines, though by +reflection, do the same, though not so strongly? And do not the organs +of sight, the eye, the heart, and brains, resist that pressure by an +endeavour of restitution outwards? Why then should there not be without +and before the eye, an apparition of light in this case as well as in +the other? + +_A._ I grant there must. But what is that which appears after the +pressing of the eye? For there is nothing without that was not there +before; or if there were, methinks another should see it better, or as +well as he; or if in the dark, methinks it should enlighten the place. + +_B._ It is a fancy, such as is the appearance of your face in a +looking-glass; such as is a dream; such as is a ghost; such as is a spot +before the eye that hath stared upon the sun or fire. For all these are +of the regiment of fancy, without any body concealed under them, or +behind them, by which they are produced. + +_A._ And when you look towards the sun or moon, why is not that also +which appears before your eyes at that time a fancy? + +_B._ So it is. Though the sun itself be a real body, yet that bright +circle of about a foot diameter cannot be the sun, unless there be two +suns, a greater and a lesser. And because you may see that which you +call the sun, both above you in the sky, and before you in the water, +and two suns, by distorting your eye, in two places in the sky, one of +them must needs be fancy. And if one, both. All sense is fancy, though +the cause be always in a real body. + +_A._ I see by this that those things which the learned call the +accidents of bodies, are indeed nothing else but diversity of fancy, and +are inherent in the sentient, and not in the objects, except motion and +quantity. And I perceive by your doctrine you have been tampering with +_Leviathan_ . But how comes wood with a certain degree of heat to shine, +and iron also with a greater degree; but no heat at all to be able to +make water shine? + +_B._ That which shineth hath the same motion in its parts that I have +all this while supposed in the sun and earth. In which motion there must +needs be a competent degree of swiftness to move the sense, that is, to +make it visible. All bodies that are not fluid will shine with heat, if +the heat be very great. Iron will shine and gold will shine; but water +will not, because the parts are carried away before they attain to that +degree of swiftness which is requisite. + +_A._ There are many fluid bodies whose parts evaporate, and yet they +make a flame, as oil, and wine, and other strong drinks. + +_B._ As for oil I never saw any inflamed by itself, how much soever +heated, therefore I do not think they are the parts of the oil, but of +the combustible body oiled that shine; but the parts of wine and strong +drinks have partly a strong motion of themselves, and may be made to +shine, but not with boiling, but by adding to them as they rise the +flame of some other body. + +_A._ How can it be known that the particles of wine have such a motion +as you suppose? + +_B._ Have you ever been so much distempered with drinking wine, as to +think the windows and table move? + +_A._ I confess, though you be not my confessor, I have; but very seldom; +and I remember the window seemed to go and come in a kind of circling +motion, such as you have described. But what of that? + +_B._ Nothing, but that it was the wine that caused it; which having a +good degree of that motion before, did, when it was heated in the veins, +give that concussion, which you thought was in the window, to the veins +themselves, and, by the continuation of the parts of man’s body, to the +brain; and that was it which made the window seem to move. + +_A._ What is flame? For I have often thought the flame that comes out of +a small heap of straw to be more, before it hath done flaming, than a +hundred times the straw itself. + +_B._ It was but your fancy. If you take a stick in your hand by one end, +the other end burning, and move it swiftly, the burning end, if the +motion be circular, shall seem a circle; if straight, a straight line of +fire, longer or shorter, according to the swiftness of the motion, or +the space it moves in. You know the cause of that. + +_A._ I think it is, because the impression of that visible object, which +was made at the first instant of the motion, did last till it was ended. +For then it will follow that it must be visible all the way, the +impressions in all points of the time being equal. + +_B._ The cause can be no other. The smallest spark of fire flying up +seems a line drawn upward; and again by that swift circular motion which +we have supposed for the cause of light, seems also broader than it is. +And consequently the flame of every thing must needs seem much greater +than it is. + +_A._ What are those sparks that fly out of the fire? + +_B._ They are small pieces of the wood or coals, or other fuel loosened +and carried away with the air that cometh up with them. And being +extinguished before their parts be quite dissipated into others, are so +much soot, and black, and may be fired again. + +_A._ A spark of fire may be stricken out of a cold stone. It is not +therefore heat that makes this shining. + +_B._ No it is the motion that makes both the heat and shining; and the +stroke makes the motion. For every of those sparks, is a little parcel +of the stone, which swiftly moved, imprinteth the same motion into the +matter prepared, or fit to receive it. + +_A._ How comes the light of the sun to burn almost any combustible +matter by refraction through a convex glass, and by reflection from a +concave? + +_B._ The air moved by the sun presseth the convex glass in such manner +as the action continued through it, proceedeth not in the same straight +line by which it proceeded from the sun, but tendeth more toward the +centre of the body it enters. Also when the action is continued through +the convex body, it bendeth again the same way. By which means the whole +action of the sun-beams are enclosed within a very small compass: in +which place therefore there must be a very vehement motion; and +consequently, if there be in that place combustible matter, such as is +not very hard to kindle, the parts of it will be dissipated, and receive +that motion which worketh on the eye as other fire does. + +The same reason is to be given for burning by reflection. For there also +the beams are collected into almost a point. + +_A._ Why may not the sun-beams be such a body as we call fire, and pass +through the pores of the glass so disposed as to carry them to a point, +or very near? + +_B._ Can there be a glass that is all pores? if there cannot, then +cannot this effect be produced by the passing of fire through the pores. +You have seen men light their tobacco at the sun with a burning glass, +or with a ball of crystal, held which way they will indifferently. Which +must be impossible, unless the ball were all pores. Again, neither you +nor I can conceive any other fire than we have seen, nor than such as +water will put out. But not only a solid globe of glass or crystal will +serve for a burning-glass, but also a hollow one filled with water. How +then does the fire from the sun pass through the glass of water without +being put out before it come to the matter they would have it burn? + +_A._ I know not. There comes nothing from the sun. If there did, there +is come so much from it already, that at this day we had had no sun. + + + ========== + + + CHAPTER V. + PROBLEMS OF HARD AND SOFT. + +_A._ What call you hard, and what soft? + +_B._ That body whereof no one part is easily put out of its place, +without removing the whole, is that which I and all men call hard; and +the contrary soft. So that they are but degrees one of another. + +_A._ What is the cause that makes one body harder than another, or, +seeing you say they are but degrees of one another, what makes one body +softer than another, and the same body sometimes harder, sometimes +softer? + +_B._ The same motion which we have supposed from the beginning for the +cause of so many other effects. Which motion not being upon the centre +of the part moved, but the part itself going in another circle to and +again, it is not necessary that the motion be perfectly circular. For it +is not circulation, but the reciprocation, I mean the to and again, that +does cast off, and lacerate the air, and consequently produce the +fore-mentioned effects. + +For the cause therefore of hardness, I suppose the reciprocation of +motion in those things which are hard, to be very swift, and in very +small circles. + +_A._ This is somewhat hard to believe. I would you could supply it with +some visible experience. + +_B._ When you see, for example, a cross-bow bent, do you think the parts +of it stir? + +_A._ No. I am sure they do not. + +_B._ How are you sure? You have no argument for it, but that you do not +see the motion. When I see you sitting still, must I believe there is no +motion in your parts within, when there are so many arguments to +convince me there is. + +_A._ What argument have you to convince me that there is motion in a +cross-bow when it stands bent? + +_B._ If you cut the string, or any way set the bow at liberty, it will +have then a very visible motion. What can be the cause of that? + +_A._ Why the setting of the bow at liberty. + +_B._ If the bow had been crooked before it was bent, and the string tied +to both ends, and then cut asunder, the bow would not have stirred. +Where lies the difference? + +_A._ The bow bent has a spring; unbent it has none, how crooked soever. + +_B._ What mean you by spring? + +_A._ An endeavour of restitution to its former posture. + +_B._ I understand spring as well as I do endeavour. + +_A._ I mean a principle or beginning of motion in a contrary way to that +of the force which bent it. + +_B._ But the beginning of motion is also motion, how insensible soever +it be. And you know that nothing can give a beginning of motion to +itself. What is it therefore that gives the bow (which you say you are +sure was at rest when it stood bent) its first endeavour to return to +its former posture? + +_A._ It was he that bent it. + +_B._ That cannot be. For he gave it an endeavour to come forward, and +the bow endeavours to go backward. + +_A._ Well, grant that endeavour be motion, and motion in the bow unbent, +how do you derive from thence, that being set at liberty it must return +to its former posture? + +_B._ Thus there being within the bow a swift (though invisible) motion +of all the parts, and consequently of the whole; the bending causeth +that motion, which was along the bow (that was beaten out when it was +hot into that length) to operate across the length in every part of it, +and the more by how much it is more bent; and consequently endeavours to +unbend it all the while it stands bent. And therefore when the force +which kept it bent is removed, it must of necessity return to the +posture it had before. + +_A._ But has that endeavour no effect at all before the impediment be +removed? For if endeavour be motion, and every motion have some effect +more or less, methinks this endeavour should in time produce something. + +_B._ So it does. For in time (in a long time) the course of this +internal motion will lie along the bow, not according to the former, but +to the new acquired posture. And then it well be as uneasy to return it +to its former posture, as it was before to bend it. + +_A._ That is true. For bows long bent lose their appetite to +restitution, long custom becoming nature. But from this internal +reciprocation of the parts, how do you infer the hardness of the whole +body? + +_B._ If you apply force to any single part of such a body, you must +needs disorder the motion of the next parts to it before it yield, and +there disordered, the motion of the next again must also be disordered; +and consequently no one part can yield without force sufficient to +disorder all: but then the whole body must also yield. Now when a body +is of such a nature as no single part can be removed without removing +the whole, men say that body is hard. + +_A._ Why does the fire melt divers hard bodies, and yet not all? + +_B._ The hardest bodies are those wherein the motion of the parts are +the most swift, and yet in the least circles. Wherefore if the fire, the +motion of whose parts are swift, and in greater circles, be made so +swift, as to be strong enough to master the motion of the parts of the +hard body, it will make those parts to move in a greater compass, and +thereby weaken their resistance, that is to say, soften them, which is a +degree of liquefaction. And when the motion is so weakened, as that the +parts lose their coherence by the force of their own weight, then we +count the body melted. + +_A._ Why are the hardest things the most brittle, insomuch that what +force soever is enough to bend them, is enough also to break them? + +_B._ In bending a hard body, as (for example) a rod of iron, you do not +enlarge the space of the internal motion of the parts of iron, as the +fire does; but you master and interrupt the motion, and that chiefly in +one place. In which place the motion that makes the iron hard being once +overcome, the prosecution of that bending must needs suddenly master the +motions of the parts next unto it, being almost mastered before. + +_A._ I have seen a small piece of glass, the figure whereof is this, A A +B C. Which piece of glass if you bend toward the top, as in C, the whole +body will shatter asunder into a million of pieces, and be like to so +much dust. I would fain see you give a probable reason of that. + +_B._ I have seen the experiment. The making of the glass is thus: they +dip an iron rod into the molten glass that stands in a vessel within the +furnace. Upon which iron rod taken out, there will hang a drop of molten +but tough metal of the figure you have described, which they let fall +into the water. So that the main drop comes first to the water, and +after it the tail, which though straight whilst it hung on the end of +the rod, yet by falling into the water becomes crooked. Now you know the +making of it, you may consider what must be the consequence of it. +Because the main drop A comes first to the water, it is therefore first +quenched, and consequently the motion of the parts of that drop, which +by the fire were made to be moved in a larger compass, is by the water +made to shrink into lesser circles towards the other end B, but with the +same or not much less swiftness. + +_A._ Why so? + +_B._ If you take any long piece of iron, glass, or other uniform and +continued body; and having heated one end thereof, you hold the other +end in your hand, and so quench it suddenly, though before you held it +easily enough, yet now it will burn your fingers. + +_A._ It will so. + +_B._ You see then how the motion of the parts from A toward C is made +more violent and in less compass by quenching the other parts first. +Besides, the whole motion that was in all the parts of the main drop A, +is now united in the small end B C. And this I take to be the cause why +that small part B C is so exceeding stiff. Seeing also this motion in +every small part of the glass, is not only circular, but proceeds also +all along the glass from A to B, the whole motion compounded will be +such as the motion of spinning any soft matter into thread, and will +dispose the whole body of the glass in threads, which in other hard +bodies are called the grain. Therefore if you bend this body (for +example) in C (which to do will require more force than a man would +think that has not tried) those threads of glass must needs be all bent +at the same time, and stand so, till by the breaking of the glass at C, +they be all at once set at liberty; and then all at once being suddenly +unbent, like so many brittle and overbent bows, their strings breaking, +be shivered in pieces. + +_A._ It is like enough to be so. And if nature have betrayed herself in +any thing, I think it is in this, and in that other experience of the +crossbow; which strongly and evidently demonstrates the internal +reciprocation of the motion, which you suppose to be in the internal +parts of every hard body. And I have observed somewhat in +looking-glasses which much confirms that there is some such motion in +the internal parts of glass, as you have supposed for the cause of +hardness. For let the glass be A B, and let the object at C be a candle, +and the eye at D. Now by divers reflections and refractions in the two +superficies of the glass, if the lines of vision be very oblique, you +shall see many images of the candle, as E, F, G, in such order and +position as is here described. But if you remove your eye to C, and the +candle to D, they will appear in a situation manifestly different from +this. Which you will yet more plainly perceive if the looking-glass be +coloured, as I have observed in red and blue glasses; and could never +conceive any probable cause of it, till now you tell me of this secret +motion of the parts across the grain of the glass, acquired by cooling +it this or that way. + +_B._ There be very many kinds of hard bodies, metals, stones, and other +kinds, in the bowels of the earth, that have been there ever since the +beginning of the world; and I believe also many different sorts of +juices that may be made hard. But for one general cause of hardness it +can be no other than such an internal motion of parts as I have already +described, whatsoever may be the cause of the several concomitant +qualities of their hardness in particular. + +_A._ We see water hardened every frosty day. It is likely therefore you +may give a probable cause of ice. What is the cause of freezing of the +ocean towards the poles of the earth? + +_B._ You know the sun being always between the tropics, and (as we have +supposed) always casting off the air; and the earth likewise casting it +off from itself, there must needs on both sides be a great stream of air +towards the poles, shaving the superficies of the earth and sea, in the +northern and southern climates. This shaving of the earth and sea by the +stream of air must needs contract and make to shrink those little +circles of the internal parts of earth and water, and consequently +harden them, first at the superficies, into a thin skin, which is the +first ice; and afterwards the same motion continuing, and the first ice +co-operating, the ice becomes thicker. And this I conceive to be the +cause of the freezing of the ocean. + +_A._ If that be the cause, I need not ask how a bottle of water is made +to freeze in warm weather with snow, or ice mingled with salt. For when +the bottle is in the midst of it, the wind that goeth out both of the +salt and of the ice as they dissolve, must needs shave the superficies +of the bottle, and the bottle work accordingly on the water without it, +and so give it first a thin skin, and at last thicken it into a solid +piece of ice. But how comes it to pass that water does not use to freeze +in a deep pit? + +_B._ A deep pit is a very thick bottle, and such as the air cannot come +at but only at the top, or where the earth is very loose and spungy. + +_A._ Why will not wine freeze as well as water? + +_B._ So it will when the frost is great enough. But the internal motion +of the parts of wine and other heating liquors is in greater circles and +stronger than the motion of the parts of water; and therefore less +easily to be frozen, especially quite through, because those parts that +have the strongest motion retire to the centre of the vessel. + + + ========== + + + CHAPTER VI. + PROBLEMS OF RAIN, WIND, AND OTHER WEATHER. + +_A._ What is the original cause of rain? And how is it generated? + +_B._ The motion of the air (such as I have described to you already) +tending to the disunion of the parts of the air, must needs cause a +continual endeavour (there being no possibility of vacuum) of whatsoever +fluid parts there are upon the face of the earth and sea, to supply the +place which would else be empty. This makes the water, and also very +small and loose parts of the earth and sea to rise, and mingle +themselves with the air, and to become mist and clouds. Of which the +greatest quantity arise there, where there is most water, namely, from +the large parts of the ocean; which are the South Sea, the Indian Sea, +and the sea that divided Europe and Africa from America; over which the +sun for the greatest part of the year is perpendicular, and consequently +raiseth a greater quantity of water; which afterwards gathered into +clouds, falls down in rain. + +_A._ If the sun can thus draw up the water, though but in small drops, +why can it not as easily hold it up? + +_B._ It is likely it would also hold them up, if they did not grow +greater by meeting together, nor were carried away by the air towards +the poles. + +_A._ What makes them gather together? + +_B._ It is not improbable that they are carried against hills, and there +stopt till more overtake them. And when they are carried towards the +North or South where the force of the sun is more oblique and thereby +weaker, they descend gently by their own weight. And because they tend +all to the centre of the earth, they must needs be united in their way +for want of room, and so grow bigger. And then it rains. + +_A._ What is the reason it rains so seldom, but snows so often upon very +high mountains? + +_B._ Because, perhaps, when the water is drawn up higher than the +highest mountains, where the course of the air between the equator and +the poles is free from stoppage, the stream of the air freezeth it into +snow. And it is in those places only where the hills shelter it from +that stream, that it falls in rain. + +_A._ Why is there so little rain in Egypt, and yet so much in other +parts nearer the equinoxial, as to make the Nile overflow the country? + +_B._ The cause of the falling of rain I told you was the stopping, and +consequently the collection of clouds about great mountains, especially +when the sun is near the equinoxial, and thereby draws up the water more +potently, and from greater seas. If you consider therefore that the +mountains in which are the springs of Nile, lie near the equinoxial and +are exceedingly great, and near the Indian Sea, you will not think it +strange there should be great store of snow. This as it melts makes the +rain of Nile to rise, which in April and May going on toward Egypt +arrive there about the time of the solstice, and overflow the country. + +_A._ Why should not the Nile then overflow that country twice a year, +for it comes twice a-year to the equinoxial. + +_B._ From the autumnal equinox, the sun goeth on toward the southern +tropic, and therefore cannot dissolve the snow on that side of the hills +that looks towards Egypt. + +_A._ But then there ought to be such another inundation southward. + +_B._ No doubt but there is a greater descent of water there in their +summer than at other times, as there must be wheresoever there is much +snow melted. But what should that inundate, unless it should overflow +the sea that comes close to the foot of those mountains? And for the +cause why it seldom rains in Egypt, it may be this, that there are no +very high hills near it to collect the clouds. The mountains whence Nile +riseth being near two thousand miles off. The nearest on one side are +the mountains of Nubia, and on the other side Sina and the mountains of +Arabia. + +_A._ Whence think you proceed the winds? + +_B._ From the motion, I think, especially of the clouds, partly also +from whatsoever is moved in the air. + +_A._ It is manifest that the clouds are moved by the winds; so that +there were winds before any clouds could be moved. Therefore I think you +make the effect before the cause. + +_B._ If nothing could move a cloud but wind, your objection were good. +But you allow a cloud to descend by its own weight. But when it so +descends, it must needs move the air before it, even to the earth, and +the earth again repel it, and so make lateral winds every way, which +will carry forward other clouds if there be any in their way, but not +the cloud that made them. The vapour of the water rising into clouds, +must needs also, as they rise, raise a wind. + +_A._ I grant it. But how can the slow motion of a cloud make so swift a +wind as it does? + +_B._ It is not one or two little clouds, but many and great ones that do +it. Besides, when the air is driven into places already covered, it +cannot but be much the swifter for the narrowness of the passage. + +_A._ Why does the south wind more often than any other bring rain with +it? + +_B._ Where the sun hath most power, and where the seas are greatest, +that is in the south, there is most water in the air; which a south wind +can only bring to us. But I have seen great showers of rain sometimes +also when the wind hath been north, but it was in summer, and came +first, I think, from the south or west, and was brought back from the +north. + +_A._ I have seen at sea very great waves when there was no wind at all. +What was it then that troubled the water? + +_B._ But had you not wind enough presently after? + +_A._ We had a storm within a little more than a quarter of an hour +after. + +_B._ That storm was then coming and had moved the water before it. But +the wind you could not perceive, for it came downwards with the +descending of the clouds, and pressing the water bounded above your sail +till it came very near. And that was it that made you think there was no +wind at all. + +_A._ How comes it to pass that a ship should go against the wind which +moves it, even almost point blank, as if it were not driven but drawn? + +_B._ You are to know first, that what body soever is carried against +another body, whether perpendicularly or obliquely, it drives it in a +perpendicular to the superficies it lighteth on. As for example, a +bullet shot against a flat wall, maketh the stone, or other matter it +hits, to retire in a perpendicular to that flat; or, if the wall be +round, towards the centre, that is to say, perpendicularly. For if the +way of the motion be oblique to the wall, the motion is compounded of +two motions, one parallel to the wall, and the other perpendicular. By +the former whereof the bullet is carried along the wall side, by the +other it approacheth to it. Now the former of these motions can have no +effect upon it; all the battery is from the motion perpendicular, in +which it approacheth, and therefore the part it hits must also retire +perpendicularly. If it were not so, a bullet with the same swiftness +would execute as much obliquely shot, as perpendicularly, which you know +it does not. + +_A._ How do you apply this to a ship? + +_B._ Let A B be the ship, the head of it A. If the wind blow just from A +towards B, it is true the ship cannot go forward howsoever the sail be +set. Let C D be perpendicular to the ship, and let the sail E C be never +so little oblique to it, and F C perpendicular to E C, and then you see +the ship will gain the space D F to the headward. + +_A._ It will so; but when it is very near to the wind it will go forward +very slowly, and make more way with her side to the leeward. + +_B._ It will indeed go slower in the proportion of the line A E to the +line C E. But the ship will not go so fast as you think sideward: one +cause is the force of that wind which lights on the side of the ship +itself; the other is the bellying of the sail; for the former, it is not +much, because the ship does not easily put from her the water with her +side; and bellying of the sail gives some little hold for the wind to +drive the ship astern. + +_A._ For the motion sideward I agree with you; but I had thought the +bellying of the sail had made the ship go faster. + +_B._ But it does not; only in a fore wind it hinders least. + +_A._ By this reason a broad thin board should make the best sail. + +_B._ You may easily foresee the great incommodities of such a sail. But +I have seen tried in little what such a wind can do in such a case. For +I have seen a board set upon four truckles, with a staff set up in the +midst of it for a mast, and another very thin and broad board fastened +to that staff in the stead of a sail, and so set as to receive the wind +very obliquely, I mean so as to be within a point of the compass +directly opposite to it, and so placed upon a reasonable smooth pavement +where the wind blew somewhat strongly. The event was first, that it +stood doubting whether it should stir at all or no, but that was not +long, and then it ran a-head extreme swiftly, till it was overthrown by +a rub. + +_A._ Before you leave the ship, tell me how it comes about that so small +a thing as a rudder can so easily turn the greatest ship. + +_B._ It is not the rudder only, there must also be a stream to do it; +you shall never turn a ship with a rudder in a standing pool, nor in a +natural current. You must make a stream from head to stern, either with +oars or with sails; when you have made such a stream, the turning of the +rudder obliquely holds the water from passing freely, and the ship or +boat cannot go on directly, but as the rudder inclines to the stern, so +will the ship turn; but this is too well known to insist upon. You have +observed that the rudders of the greatest ships are not very broad, but +go deep into the water, whereas western barges, though but small +vessels, have their rudders much broader, which argues that the holding +of water from passing is the true office of a rudder; and therefore to a +ship that draws much water the rudder is made deep accordingly; and in +barges that draw little water, the rudders being less deep, must so much +the more be extended in breadth. + +_A._ What makes snow? + +_B._ The same cause which, speaking of hardness, I supposed for the +cause of ice. For the stream of air proceeding from that both the earth +and the sun cast off the air, consequently maketh a stream of air from +the equinoxial towards the poles, passing amongst the clouds, shaving +those small drops of water whereof the clouds consist, and congeals them +as they do the water of the sea, or of a river. And these small frozen +drops are that which we call snow. + +_A._ But then how are great drops frozen into hailstones, and that +especially (as we see they are) in summer? + +_B._ It is especially in summer, and hot weather, that the drops of +water which make the clouds, are great enough; but it is then also that +clouds are sooner and more plentifully carried up. And therefore the +current of the air strengthened between the earth and the clouds, +becomes more swift; and thereby freezeth the drops of water, not in the +cloud itself, but as they are falling. Nor does it freeze them +thoroughly, the time of their falling not permitting it, but gives them +only a thin coat of ice; as is manifest by their sudden dissolving. + +_A._ Why are not sometimes also whole clouds when pregnant and ready to +drop, frozen into one piece of ice? + +_B._ I believe they are so whensoever it thunders. + +_A._ But upon what ground do you believe it? + +_B._ From the manner or kind of noise they make, namely a crack; which I +see not how it can possibly be made by water or any other soft bodies +whatsoever. + +_A._ Yes, the powder they call _aurum fulminans_, when thoroughly warm, +gives just such another crack as thunder. + +_B._ But why may not every small grain of that _aurum fulminans_ by +itself be heard, though a heap of them together be soft, as is any heap +of sand. Salts of all sorts are of the nature of ice. But gold is +dissolved into _aurum fulminans_ by nitre and other salts. And the least +grain of it gives a little crack in the fire by itself. And therefore +when they are so warmed by degrees, the crack cannot choose but be very +great. + +_A._ But before it be _aurum fulminans_ they use to wash away the salt +(which they call dulcifying it), and then they dry it gently by degrees. + +_B._ That is, they exhale the pure water that is left in the powder, and +leave the salt behind to harden with drying. Other powder made of salts +without any gold in them will give a crack as great as _aurum +fulminans_. A very great chemist of our times hath written, that salt of +tartar, saltpetre, and a little brimstone ground together into a powder, +and dried, a few grains of that powder will be made by the fire to give +as great a clap as a musket. + +_A._ Methinks it were worth your trial to see what effect a quart or a +pint of _aurum fulminans_ would produce, being put into a great gun made +strong enough on purpose, and the breech of the gun set in hot cinders, +so as to heat by degrees, till the powder fly. + +_B._ I pray you try it yourself; I cannot spare so much money. + +_A._ What is it that breaketh the clouds when they are frozen? + +_B._ In very hot weather the sun raiseth from the sea and all moist +places abundance of water, and to a great height. And whilst this water +hangs over us in clouds, or is again descending, it raiseth other +clouds, and it happens very often that they press the air between them, +and squeeze it through the clouds themselves very violently; which as it +passes shaves and hardens them in the manner declared. + +_A._ That has already been granted; my question is what breaks them? + +_B._ I must here take in one supposition more. + +_A._ Then your basin, it seems, holds not all you have need of. + +_B._ It may for all this, for the supposition I add is no more but this; +that what internal motion I ascribe to the earth, and the other concrete +parts of the world, is to be supposed also in every of their parts how +small soever; for what reason is there to think, in case the whole earth +have in truth the motion I have ascribed to it, that one part of it +taken away, the remaining part should lose that motion. If you break a +loadstone, both parts will retain their virtue, though weakened +according to the diminution of their quantity; I suppose therefore in +every small part of the earth the same kind of motion, which I have +supposed in the whole: and so I recede not yet from my basin. + +_A._ Let it be supposed, and withal, that abundance of earth, (which I +see you aim at), be drawn up together with the water. What then? + +_B._ Then if many pregnant clouds, some ascending and some descending +meet together, and make concavities between, and by the pressing out of +the air, as I have said before, become ice; those atoms, as I may call +them, of earth will, by the straining of the air through the water of +the clouds, be left behind, and remain in the cavities of the clouds, +and be more in number than for the proportion of the air therein. +Therefore for want of liberty they must needs justle one another, and +become, as they are more and more straightened of room, more and more +swift, and consequently at last break the ice suddenly and violently, +now in one place, and by and by in another; and make thereby so many +claps of thunder, and so many flashes of lightning. For the air +recoiling upon our eyes, is that which maketh those flashes to our +fancy. + +_A._ But I have seen lightning in a very clear evening, when there has +been neither thunder nor clouds. + +_B._ Yes, in a clear evening; because the clouds and the rain were below +the horizon, perhaps forty or fifty miles off; so that you could not see +the clouds nor hear the thunder. + +_A._ If the clouds be indeed frozen into ice, I shall not wonder if they +be sometimes also so situated, as, like looking-glasses, to make us see +sometimes three or more suns by refraction and reflection. + + + ========== + + + CHAPTER VII. + PROBLEMS OF MOTION PERPENDICULAR, AND OBLIQUE; + OF PRESSION AND PERCUSSION; REFLECTION AND + REFRACTION; ATTRACTION AND REPULSION. + +_A._ If a bullet from a certain point given, be shot against a wall +perpendicularly, and again from the same point obliquely, what will be +the proportion of the forces wherewith they urge the wall? For example, +let the wall be A B, a point given E, a gun C E, that carries the bullet +perpendicularly to F, and another gun D E, that carries the like bullet +with the same swiftness obliquely to G; in what proportion will their +forces be upon the wall? + +_B._ The force of the stroke perpendicular from E to F will be greater +then the oblique force from E to G, in the proportion of the line E G to +the line E F. + +_A._ How can the difference be so much? Can the bullet lose so much of +its force in the way from E to G? + +_B._ No; we will suppose it loseth nothing of its swiftness. But the +cause is, that their swiftness being equal, the one is longer in coming +to the wall than the other, in proportion of time, as E G to E F. For +though their swiftness be the same, considered in themselves, yet the +swiftness of their approach to the wall is greater in E F than in E G, +in proportion of the lines themselves. + +_A._ When a bullet enters not, but rebounds from the wall, does it make +the same angle going off, which it did falling on, as the sun-beams do? + +_B._ If you measure the angles close by the wall their difference will +not be sensible; otherwise it will be great enough, for the motion of +the bullet grows continually weaker. But it is not so with the sun-beams +which press continually and equally. + +_A._ What is the cause of reflection? When a body can go no further on, +it has lost its motion. Whence then comes the motion by which it +reboundeth? + +_B._ This motion of rebounding or reflecting proceedeth from the +resistance. There is a difference to be considered between the +reflection of light, and of a bullet, answerable to their different +motions, pressing and striking. For the action which makes reflection of +light, is the pressure of the air upon the reflecting body, caused by +the sun, or other shining body, and is but a contrary endeavour; as if +two men should press with their breasts upon the two ends of a staff, +though they did not remove one another, yet they would find in +themselves a great disposition to press backward upon whatsoever is +behind them, though not a total going out of their places. Such is the +way of reflecting light. Now, when the falling on of the sun-beams is +oblique, the action of them is nevertheless perpendicular to the +superficies it falls on. And therefore the reflecting body, by +resisting, turneth back that motion perpendicularly, as from F to E; but +taketh nothing from the force that goes on parallel in the line of E H, +because the motion never presses. And thus of the two motions from F to +E, and from E to H, is a compounded motion in the line F H, which maketh +an angle in B G, equal to the angle F G E. + +But in percussion (which is the motion of the bullet against a wall,) +the bullet no sooner goeth off than it loseth of its swiftness, and +inclineth to the earth by its weight. So that the angles made in falling +on and going off, cannot be equal, unless they be measured close to the +point where the stroke is made. + +_A._ If a man set a board upright upon its edge, though it may very +easily be cast down with a little pressure of one’s finger, yet a bullet +from a musket shall not throw it down, but go through it. What is the +cause of that? + +_B._ In pressing with your finger you spend time to throw it down. For +the motion you give to the part you touch is communicated to every other +part before it fall. For the whole cannot fall till every part be moved. +But the stroke of a bullet is so swift, as it breaks through, before the +motion of the part it hits can be communicated to all the other parts +that must fall with it. + +_A._ The stroke of a hammer will drive a nail a great way into a piece +of wood on a sudden. What weight laid upon the head of a nail, and in +how much time will do the same? It is a question I have heard propounded +amongst naturalists. + +_B._ The different manner of the operation of weight from the operation +of a stroke, makes it incalculable. The suddenness of the stroke upon +one point of the wood takes away the time of resistance from the rest. +Therefore the nail enters so far as it does. But the weight not only +gives them time, but also augments the resistance; but how much, and in +how much time, is, I think, impossible to determine. + +_A._ What is the difference between reflection and recoiling? + +_B._ Any reflection may, and not unproperly, be called recoiling; but +not contrariwise every recoiling reflection. Reflection is always made +by the reaction of a body pressed or stricken; but recoiling not always. +The recoiling of a gun is not caused by its own pressing upon the +gunpowder, but by the force of the powder itself, inflamed and moved +every way alike. + +_A._ I had thought it had been by the sudden re-entering of the air +after the flame and bullet were gone out. For it is impossible that so +much room as is left empty by the discharging of the gun, should be so +suddenly filled with the air that entereth at the touchhole. + +_B._ The flame is nothing but the powder itself, which scattered into +its smallest parts, seems of greater bulk by much, than in truth it is, +because they shine. And as the parts scatter more and more, so still +more air gets between them, entering not only at the touchhole, but also +at the mouth of the gun, which two ways being opposite, it will be much +too weak to make the gun recoil. + +_A._ I have heard that a great gun charged too much or too little, will +shoot, not above, nor below, but beside the mark; and charged with one +certain charge between both, will hit it. + +_B._ How that should be I cannot imagine. For when all things in the +cause are equal, the effects cannot be unequal. As soon as fire is +given, and before the bullet be out, the gun begins to recoil. If then +there be any unevenness or rub in the ground more on one side than on +the other, it shall shoot beside the mark, whether too much, or too +little, or justly charged; because if the line wherein the gun recoileth +decline, the way of the bullet will also decline to the contrary side of +the mark. Therefore I can imagine no cause of this event, but either in +the ground it recoils on, or in the unequal weight of the parts of the +breech. + +_A._ How comes refraction? + +_B._ When the action is in a line perpendicular to the superficies of +the body wrought upon, there will be no refraction at all. The action +will proceed still in the same straight line, whether it be pression as +in light, or percussion as in the shooting of a bullet. But when the +pression is oblique, then will the refraction be that way which the +nature of the bodies through which the action proceeds shall determine. + +_A._ How is light refracted? + +_B._ If it pass through a body of less, into a body of greater +resistance, and to the point of the superficies it falleth on, you draw +a line perpendicular to the same superficies, the action will proceed +not in the same line by which it fell on, but in another line bending +toward that perpendicular. + +_A._ What is the reason of that? + +_B._ I told you before, that the falling on worketh only in the +perpendicular; but as soon as the action proceedeth farther inward than +a mere touch, it worketh partly in the perpendicular, and partly +forward, and would proceed in the same line in which it fell on, but for +the greater resistance which now weakeneth the motion forward, and makes +it to incline towards the perpendicular. + +_A._ In transparent bodies it may be so; but there be bodies through +which the light cannot pass at all. + +_B._ But the action by which light is made, passeth through all bodies. +For this action is pression; and whatsoever is pressed, presseth that +which is next behind, and so continually. But the cause why there is no +light seen through it, is the unevenness of the parts within, whereby +the action is by an infinite number of reflections so diverted and +weakened, that before it hath proceeded through, it hath not strength +left to work upon the eye strongly enough to produce sight. + +_A._ If the body being transparent, the action proceed quite through, +into a body again of less resistance, as out of glass into the air, +which way shall it then proceed in the air? + +_B._ From the point where it goeth forth, draw a perpendicular to the +superficies of the glass, the action now freed from the resistance it +suffered, will go from that perpendicular, as much as it did before come +towards it. + +_A._ When a bullet from out of the air entereth into a wall of earth, +will that also be refracted towards the perpendicular. + +_B._ If the earth be all of one kind, it will. For the parallel motion, +will there also at the first entrance be resisted, which it was not +before it entered. + +_A._ How then comes a bullet, when shot very obliquely into any broad +water, and having entered, yet to rise again into the air? + +_B._ When a bullet is shot very obliquely, though the motion be never so +swift, yet the approach downwards to the water is very slow, and when it +cometh to it, it casteth up much water before it, which with its weight +presseth downwards again, and maketh the water to rise under the bullet +with force enough to master the weak motion of the bullet downwards, and +to make it rise in such manner as bodies use to rise by reflection. + +_A._ By what motion (seeing you ascribe all effects to motion) can a +loadstone draw iron to it? + +_B._ By the same motion hitherto supposed. But though all the smallest +parts of the earth have this motion, yet it is not supposed that their +motions are in equal circles; nor that they keep just time with one +another; nor that they have all the same poles. If they had, all bodies +would draw one another alike. For such an agreement of motion, of way, +of swiftness, and of poles, cannot be maintained, without the +conjunction of the bodies themselves in the centre of their common +motion, but by violence. If therefore the iron have but so much of the +nature of the loadstone as readily to receive from it the like motion, +as one string of a lute doth from another string strained to the same +note, (as it is like enough it hath, the loadstone being but one kind of +iron ore), it must needs after that motion received from it, unless the +greatness of the weight hinder, come nearer to it, because at distance +their motions will differ in time, and oppose each other, whereby they +will be forced to a common centre. If the iron be lifted up from the +earth, the motion of the loadstone must be stronger, or the body of it +nearer, to overcome the weight; and then the iron will leap up to the +loadstone as swiftly, as from the same distance it would fall down to +the earth; but if both the stone and the iron be set floating upon the +water, the attraction will begin to be manifest at a greater distance, +because the hindrance of the weight is in part removed. + +_A._ But why does the loadstone, if it float on a calm water, never fail +to place itself at last in the meridian just north and south. + +_B._ Not so, just in the meridian, but almost in all places with some +variations. But the cause I think is, that the axis of this magnetical +motion is parallel to the axis of the ecliptic, which is the axis of the +like motion in the earth, and consequently that it cannot freely +exercise its natural motion in any other situation. + +_A._ Whence may this consent of motion in the loadstone and the earth +proceed? Do you think, as some have written, that the earth is a great +loadstone? + +_B._ Dr. Gilbert, that was the first that wrote anything of this subject +rationally, inclines to that opinion. Descartes thought the earth, +excepting this upper crust of a few miles depth, to be of the same +nature with all other stars, and bright. For my part, I am content to be +ignorant; but I believe the loadstone hath been given its virtue by a +long habitude in the mine, the vein of it lying in the plane of some of +the meridians, or rather of some of the great circles that pass through +the poles of the ecliptic, which are the same with the poles of the like +motion supposed in the earth. + +_A._ If that be true, I need not ask why the filings of iron laid on a +loadstone equally distant from its poles will lie parallel to the axis, +but on each side will incline to the pole that is next. Nor why by +drawing a loadstone all along a needle of iron, the needle will receive +the same poles. Nor why when the loadstone and iron, or two loadstones, +are put together floating upon water, will fall one of them astern of +the other, that their like parts may look the same way, and their unlike +touch, in which action they are commonly said to repel one another. For +all this may be derived from the union of their motions. One thing more +I desire to know, and that is; what are those things they call spirits? +I mean ghosts, fairies, hobgoblins, and the like apparitions. + +_B._ They are no part of the subject of natural philosophy. + +_A._ That which in all ages, and all places is commonly seen (as those +have been, unless a great part of mankind be liars) cannot, I think, be +supernatural. + +_B._ All this that I have hitherto said, though upon better ground than +can be had for a discourse of ghosts, you ought to take but for a dream. + +_A._ I do so. But there be some dreams more like sense then others. And +that which is like sense pleases me as well in natural philosophy, as if +it were the very truth. + +_B._ I was dreaming also once of these things; but was wakened by their +noise. And they never came into any dream of mine since, unless +apparitions in dreams and ghost be all one. + + + ========== + + + CHAPTER VIII. + THE DELPHIC PROBLEM, OR DUPLICATION OF THE + CUBE. + +_A._ Have you seen a printed paper sent from Paris, containing the +duplication of the cube, written in French? + +_B._ Yes. It was I that writ it, and sent it thither to be printed, on +purpose to see what objections would be made to it by our professors of +algebra here. + +_A._ Then you have also seen the confutations of it by algebra. + +_B._ I have seen some of them; and have one by me. For there was but one +that was rightly calculated, and that is it which I have kept. + +_A._ Your demonstration then is confuted though but by one. + +_B._ That does not follow. For though an arithmetical calculation be +true in numbers, yet the same may be, or rather must be false, if the +units be not constantly the same. + +_A._ Is their calculation so inconstant, or rather so foolish as you +make it? + +_B._ Yes. For the same number is sometimes so many lines, sometimes so +many planes, and sometimes so many solids; as you shall plainly see, if +you will take the pains to examine first a demonstration I have to prove +the said duplication, and after that, the algebraic calculation which is +pretended to confute it. And not only that this one is false, but also +any other arithmetical account used in geometry, unless the numbers be +always so many lines, or always so many superficies, or always so many +solids. + +_A._ Let me see the geometrical demonstration. + +_B._ There it is. Read it. + + TO FIND A CUBE DOUBLE TO A CUBE GIVEN: + +Let the side of the cube given be V D. Produce V D to A, till A D be +double to D V. Then make the square of A D, namely A B C D. Divide A B +and C D in the middle at E and F. Draw E F. Draw also A C cutting E F in +I. Then in the sides B C and A D take B R and A S, each of them equal to +A I or I C. + +Lastly, divide S D in the middle at T, and upon the centre T, with the +distance T V, describe a semi-circle cutting A D in Y, and D C in X. + +I say the cube of D X is double to the cube of D V. For the three lines +D Y, D X, D V are in continual proportion. And continuing the +semi-circle V X Y till it cut the line R S, drawn and produced in Z, the +line S Z will be equal to D X. And drawing X Z it will pass through T. +And the four lines T V, T X, T Y and T Z will be equal. And therefore +joining Y X and Y Z, the figure V X Y Z will be a rectangle. + +[Illustration: + + _Delphic Problem. + Vol. VII. Eng. p. 60._ +] + +Produce C D to P so as D P be equal to A D. Now if Y Z produced fall on +P, there will be three rectangle equiangled triangles, D P Y, D Y X, and +D X V; and consequently four continual proportionals, D P, D Y, D X, and +D V, whereof D X is the least of the means. And therefore the cube of D +X will be double to the cube of D V. + +_A._ That is true; and the cube of D Y will be double to the cube of D +X; and the cube of D P double to the cube of D Y. But that Y Z produced, +falls upon P, is the thing they deny, and which you ought to +demonstrate. + +_B._ If Y Z produced fall not on P, then draw P Y, and from V let fall a +perpendicular upon P Y, suppose at _u._. Divide P V in the midst at +_a._, and join _a u._; which done _a u._ will be equal to _a._ V or _a._ +P. For because V _u._ P is a right angle, the point _u._ will be in the +semi-circle whereof P V is the diameter. + +Therefore drawing V R, the angle _u._ V R will be a right angle. + +_A._ Why so? + +_B._ Because T V and T Y are equal; and T D, T S equal; S Y will also be +equal to D V. And because D P and R S are equal and parallel, R Y will +be equal and parallel to P V. And therefore V R and P Y that join them +will be equal and parallel. And the angles P _u._ V, R V _u._ will be +alternate, and consequently equal. But P _u._ V is a right angle; +therefore also R V _u._ will be a right angle. + +_A._ Hitherto all is evident. Proceed. + +_B._ From the point Y raise a perpendicular cutting V R wheresoever in +_t._, and then (because P Y and V R are parallel) the angle Y _t_ V will +be a right angle. And the figure _u_ Y _t_ V a rectangle, and _u t_ +equal to Y V. But Y V is equal to Z X; and therefore Z X is equal to _u +t_. And _u t_ must pass through the point T (for the diameters of any +rectangle divide each other in the middle), therefore Z and _u_ are the +same point, and X and _t_ the same point. Therefore Y Z produced falls +upon P. And D X is the lesser of the two means between A D and D V. And +the cube of D X double to the cube of D V, which was to be demonstated + +_A._ I cannot imagine what fault there can be in this demonstration, and +yet there is one thing which seems a little strange to me. And it is +this. You take B R, which is half the diagonal, and which is the sine of +forty-five degrees, and which is also the mean proportional between the +two extremes; and yet you bring none of these proprieties into your +demonstration. So that though you argue from the construction, yet you +do not argue from the cause. And this perhaps your adversaries will +object, at least, against the art of your demonstration, or enquire by +what luck you pitched upon half the diagonal for your foundation. + +_B._ I see you let nothing pass. But for answer you must know, that if a +man argue from the negative of the truth, though he know not that it is +the truth which is denied, yet he will fall at last, after many +consequences, into one absurdity or another. For though false do often +produce truth, yet it produces also absurdity, as it hath done here. But +truth produceth nothing but truth. Therefore in demonstrations that tend +to absurdity, it is no good logic to require all along the operation of +the cause. + +_A._ Have you drawn from hence no corollaries? + +_B._ No. I leave that for others that will; unless you take this for a +corollary, that, what arithmetical calculation soever contradicts it, is +false. + +_A._ Let me see now the algebraical demonstration against it. + +_B._ Here it is: + + Let A B or A D be equal to 2 + Then D F or D V is equal to 1 + And B R or A S is equal to the square root of 2 + And D Y equal to 3 + want the square root of 2 + The cube of A B is equal to 8 + The cube of D Y is equal to 45 + want the square root of 1682 that is almost equal to 4 + For 45 want the square root of 1681 is equal to 4 + +Therefore D Y is a little less then the greater of the two means between +A D and D V. + +_A._ There is I see some little difference between this arithmetical and +your geometrical demonstration. And though it be insensible, yet if his +calculation be true, yours must needs be false, which I am sure cannot +be. + +_B._ His calculation is so true, that there is never a proposition in it +false, till he come to the conclusion, that the cube of D Y is equal to +45, want the square root of 1682. But that, and the rest, is false. + +_A._ I shall easily see that A D is certainly 2, whereof D V is 1, and A +V is certainly 3, whereof D V is 1. + +_B._ Right. + +_A._ And B R is without doubt the square root of 2. + +_B._ Why, what is 2? + +_A._ 2 is the line A D as being double to D V which is 1. + +_B._ And so, the line B R is the square root of the line A D. + +_A._ Out upon it, it is absurd. Why do you grant it to be true in +arithmetic? + +_B._ In arithmetic the numbers consist of so many units, and are never +considered there as nothings. And therefore every one line has some +latitude, and if you allow to B I, the semi-diagonal, the same latitude +you do to A B, or to B R, you will quickly see the square of half the +diagonal to be equal to twice the square of half A B. + +_A._ Well, but then your demonstration is not confuted; for the point Y +will have latitude enough to take in that little difference which is +between the root of 1681 and the root of 1682. This putting off an unit +sometimes for one line, sometimes for one square, must needs mar the +reckoning. Again he says, the cube of A B is equal to 8; but seeing A B +is 2, the cube of A B must be just equal to four of its own sides; so +that the unit which was before sometimes a line, sometimes a square, is +now a cube. + +_B._ It can be no otherwise when you so apply arithmetic to geometry, as +to number the lines of a plane, or the planes of a cube. + +_A._ In the next place, I find that the cube of D Y is equal to 45, want +the square root of 1682. What is that 45? Lines, or squares, or cubes? + +_B._ Cubes; cubes of D V. + +_A._ Then if you add to 45 cubes of D V the square root of 1682, the sum +will be 45 cubes of D V; and if you add to the cube of D Y the same root +of 1682, the sum will be the cube of D Y, plus the square root of 1682, +and these two sums must be equal. + +_B._ They must so. + +_A._ But the square root of 1682, being a line, adds nothing to a cube; +therefore the cube alone of D Y, which he says is equal almost to 4 +cubes of D V, is equal to 45 cubes of the same D V. + +_B._ All these impossibilities do necessarily follow the confounding of +arithmetic and geometry. + +_A._ I pray you let me see the operation by which the cube of D Y (that +is, the cube of 3, want the root of 2) is found equal to 45, want the +square root of 1682. + +_B._ Here it is. + + A DETECTION OF THE ABSURD USE OF ARITHMETIC AS IT IS NOW APPLIED TO + GEOMETRY. + + 3————√2 + 3————√2 + ——————— + —√18 + 2 + 9—√18 + ————————— + 9—√72 + 2 + 3——√2 + —————————————— + ——√162 + 12——√8 + 27——√648 + 6 + ——————————————————— + 27—√658—√162 + 18—√8 + + ====================== + +_A._ Why for two roots of 18 do you put the root of 72. + +_B._ Because 2 roots of 18 are equal to one root of four times 18, which +is 72. + +_A._ Next we have, that the root of 2 multiplied into 2 makes the root +of 8. How is that true? + +_B._ Does it not make 2 roots of two? And is not B R the root of 2, and +2 B R equal to the diagonal? And is not the square of the diagonal equal +to 8 squares of D V? + +_A._ It is true. But here the root of 8 is put for the cube of the root +of 2. Can a line be equal to a cube? + +_B._ No. But here we are in arithmetic again, and 8 is a cubic number. + +_A._ How does the root of 2 multiplied into the root of 72 make 12? + +_B._ Because it makes the root of 2 times 72, that is to say the root of +144 which is 12. + +_A._ How does 9 roots of 2 make the root of 162? + +_B._ Because it makes the root of 2 squares of 9, that is the root of +162. + +_A._ How does 3 roots of 72 make the root of 648? + +_B._ Because it makes the root of 9 times 72, that is of 648. + +_A._ For the total sum I see 27 and 18, which make 45. Therefore the +root of 648 together with the root of 162 and of 8, which are to be +deducted from 45, ought to be equal to the root of 1682. + + _B._ So they are. For 648 multiplied by 162 makes 104976, of + which the double root is 648 + and 648 and 162 added together make 810 + Therefore the root of 648, added to the root of 162, makes + the root of 1458 + Again 1458 into 8 is 11664. The double root whereof is 216 + The sum of 1458 and 8 added together 1466 + The sum of 1466 and 216 is 1682, and the root, the root of 1682 + +_A._ I see the calculation in numbers is right, though false in lines. +The reason whereof can be no other than some difference between +multiplying numbers into lines or planes, and multiplying lines into the +same lines or planes. + +_B._ The difference is manifest. For when you multiply a number into +lines, the product is lines; as the number 2 multiplied into 3 lines is +no more than 3 lines 2 times told. But if you multiply lines into lines +you make planes, and if you multiply lines into planes you make solid +bodies. In geometry there are but three dimensions, lengths, +superficies, and body. In arithmetic there is but one, and that is +number or length which you will. And though there be some numbers called +plane, other solids, others plano-solid, others square, others cubic, +others square-square, others quadrato-cubic, others cubi-cubic, &c., yet +are all these but one dimension, namely number, or a file of things +numbered. + +_A._ But seeing this way of calculation by numbers is so apparently +false, what is the reason this calculation came so near the truth? + +_B._ It is because in arithmetic units are not nothings, and therefore +have breadth. And therefore many lines set together make a superficies +though their breadth be insensible. And the greater the number is into +which you divide your line, the less sensible will be your error. + +_A._ Archimedes, to find a straight line equal to the circumference of a +circle, used this way of extracting roots. And it is the way also by +which the table of sines, secants, and tangents have been calculated. +Are they all out? + +_B._ As for Archimedes, there is no man that does more admire him than I +do: but there is no man that cannot err. His reasoning is good. But he, +as all other geometricians before and after him, have had two principles +that cross one another when they are applied to one and the same +science. One is, that a point is no part of a line, which is true in +geometry, where a part of a line when it is called a point, is not +reckoned; another is, that a unit is part of a number; which is also +true; but when they reckon by arithmetic in geometry, there a unit is +sometimes part of a line, sometimes a part of a square, and sometimes +part of a cube. As for the table of sines, secants, and tangents, I am +not the first that find fault with them. Yet I deny not but they are +true enough for the reckoning of acres in a map of land. + +_A._ What a deal of labour has been lost by them that being professors +of geometry have read nothing else to their auditors but such stuff as +this you have here seen. And some of them have written great books of it +in strange characters, such as in troublesome times, a man would suspect +to be a cypher. + +_B._ I think you have seen enough to satisfy you, that what I have +written heretofore concerning the quadrature of the circle, and of other +figures made in imitation of the parabola, has not been yet confuted. + +_A._ I see you have wrested out of the hands of our antagonists this +weapon of algebra, so as they can never make use of it again. Which I +consider as a thing of much more consequence to the science of geometry, +than either of the duplication of the cube, or the finding of two mean +proportionals, or the quadrature of a circle, or all these problems put +together. + + FINIS. + + + + + DECAMERON PHYSIOLOGICUM; + + OR, + + TEN DIALOGUES OF NATURAL PHILOSOPHY. + + BY + + THOMAS HOBBES + + OF MALMESBURY. + + TO WHICH IS ADDED + + THE PROPORTION OF A STRAIGHT LINE TO + HALF THE ARC OF A QUADRANT, + + BY THE SAME AUTHOR. + +[Illustration] + + DECAMERON PHYSIOLOGICUM. + + + CHAPTER I. + OF THE ORIGINAL OF NATURAL PHILOSOPHY. + +_A._ I have heard exceeding highly commended a kind of thing which I do +not well understand, though it be much talked of, by such as have not +otherwise much to do, by the name of philosophy; and the same again by +others as much despised and derided: so that I cannot tell whether it be +good or ill, nor what to make of it, though I see many other men that +thrive by it. + +_B._ I doubt not, but what so many do so highly praise must be very +admirable, and what is derided and scorned by many, foolish and +ridiculous. The honour and scorn falleth finally not upon philosophy, +but upon the professors. Philosophy is _the knowledge of natural +causes_. And there is no knowledge but of truth. And to know the true +causes of things, was never in contempt, but in admiration. Scorn can +never fasten upon truth. But the difference is all in the writers and +teachers. Whereof some have neither studied, nor care for it, otherwise +than as a trade to maintain themselves or gain preferment; and some for +fashion, and to make themselves fit for ingenious company: and their +study hath not been meditation, but acquiescence in the authority of +those authors whom they have heard commended. And some, but few, there +be, that have studied it for curiosity, and the delight which commonly +men have in the acquisition of science, and in the mastery of difficult +and subtil doctrines. Of this last sort I count Aristotle, and a few +others of the ancients, and some few moderns: and to these it is that +properly belong the praises which are given to philosophy. + +_A._ If I have a mind to study, for example natural philosophy, must I +then needs read Aristotle, or some of those that now are in request? + +_B._ There is no necessity of it. But if in your own meditation you +light upon a difficulty, I think it is no loss of time, to enquire what +other men say of it, but to rely only upon reason. For though there be +some few effects of nature, especially concerning the heavens, whereof +the philosophers of old time have assigned very rational causes, such as +any man may acquiesce in, as of eclipses of the sun and moon by long +observation, and by the calculation of their visible motions; yet what +is that to the numberless and quotidian phenomena of nature? Who is +there amongst them or their successors, that has satisfied you with the +causes of gravity, heat, cold, light, sense, colour, noise, rain, snow, +frost, winds, tides of the sea, and a thousand other things which a few +men’s lives are too short to go through, and which you and other curious +spirits admire (as quotidian as they are), and fain would know the +causes of them, but shall not find them in the books of naturalists; and +when you ask what are the causes of any of them, of a philosopher now, +he will put you off with mere words; which words, examined to the +bottom, signify not a jot more than I cannot tell, or because it is: +such as are intrinsical quality, occult quality, sympathy, antipathy, +antiperistasis, and the like. Which pass well enough with those that +care not much for such wisdom, though wise enough in their own ways; but +will not pass with you that ask not simply what is the cause, but in +what manner it comes about that such effects are produced. + +_A._ That is cozening. What need had they of that? When began they thus +to play the charlatans? + +_B._ Need had they none. But know you not that men from their very +birth, and naturally, scramble for every thing they covet, and would +have all the world, if they could, to fear and obey them? If by fortune +or industry one light upon a secret in nature, and thereby obtain the +credit of an extraordinary knowing man, should he not make use of it to +his own benefit? There is scarce one of a thousand but would live upon +the charges of the people as far as he dares. What poor geometrician is +there, but takes pride to be thought a conjurer? What mountebank would +not make a living out of a false opinion that he were a great physician? +And when many of them are once engaged in the maintenance of an error, +they will join together for the saving of their authority to decry the +truth. + +_A._ I pray, tell me, if you can, how and where the study of philosophy +first began. + +_B._ If we may give credit to old histories, the first that studied any +of the natural sciences were the astronomers of Ethiopia. My author is +Diodorus Siculus, accounted a very faithful writer, who begins his +history as high as is possible, and tells us that in Ethiopia were the +first astronomers; and that for their predictions of eclipses, and other +conjunctions and aspects of the planets, they obtained of their king not +only towns and fields to a third part of the whole land, but were also +in such veneration with the people, that they were thought to have +discourse with their gods, which were the stars; and made their kings +thereby to stand in awe of them, that they durst not either eat or drink +but what and when they prescribed; no nor live, if they said the gods +commanded them to die. And thus they continued in subjection to their +false prophets, till by one of their kings, called Ergamenes, (about the +time of the Ptolemies), they were put to the sword. But long before the +time of Ergamenes, the race of these astrologers (for they had no +disciples but their own children) was so numerous, that abundance of +them (whether sent for or no I cannot tell) transplanted themselves into +Egypt, and there also had their cities and lands allowed them, and were +in request not only for astronomy and astrology, but also for geometry. +And Egypt was then as it were an university to all the world, and +thither went the curious Greeks, as Pythagoras, Plato, Thales, and +others, to fetch philosophy into Greece. But long before that time, +abundance of them went into Assyria, and had their towns and lands +assigned them also there; and were by the Hebrews called Chaldees. + +_A._ Why so? + +_B._ I cannot tell; but I find in Martinius’s Lexicon they are called +Chasdim, and Chesdim, and (as he saith) from one Chesed the son of +Nachor; but I find no such man as Chesed amongst the issue of Noah in +the scripture. Nor do I find that there was any certain country called +Chaldæa; though a town where any of them inhabited were called a town of +the Chaldees. Martinius saith farther, that the same word Chasdim did +signify also Demons. + +_A._ By this reckoning I should conjecture they were called Chusdim, as +being a race of Ethiopians. For the land of Chus is Ethiopia; and so the +name degenerated first into Chuldim, and then into Chaldim; so that they +were such another kind of people as we call gipsies; saving that they +were admired and feared for their knavery, and the gipsies counted +rogues. + +_B._ Nay pray, except Claudius Ptolomæus, author of that great work of +astronomy, the Almegest. + +_A._ I grant he was excellent both in astronomy and geometry, and to be +commended for his _Almegest_ ; but then for his _Judiciar Astrologie_ +annexed to it, he is again a gipsy. But the Greeks that travelled, you +say, into Egypt, what philosophy did they carry home? + +_B._ The mathematics and astronomy. But for that sublunary physics, +which is commonly called natural philosophy, I have not read of any +nation that studied it earlier than the Greeks, from whom it proceeded +to the Romans. Yet both Greeks and Romans were more addicted to moral +than to natural philosophy; in which kind we have their writings, but +loosely and incoherently, written upon no other principles than their +own passions and presumptions, without any respect to the laws of +commonwealth, which are the ground and measure of all true morality. So +that their books tend rather to teach men to censure than to obey the +laws; which has been a great hindrance to the peace of the western world +ever since. But they that seriously applied themselves to natural +philosophy were but few, as Plato and Aristotle, whose works we have; +and Epicurus whose doctrine we have in Lucretius. The writings of +Philolaus and many other curious students being by fire or negligence +now lost: though the doctrines of Philolaus concerning the motion of the +earth have been revived by Copernicus, and explained and confirmed by +Galileo now of late. + +_A._ But methinks the natural philosophy of Plato, and Aristotle, and +the rest, should have been cultivated and made to flourish by their +disciples. + +_B._ Whom do you mean, the successors of Plato, Epicurus, Aristotle, and +the other first philosophers? It may be some of them may have been +learned and worthy men. But not long after, and down to the time of our +Saviour and his Apostles, they were for the most part a sort of needy, +ignorant, impudent, cheating fellows, who by the profession of the +doctrine of those first philosophers got their living. For at that time, +the name of philosophy was so much in fashion and honour amongst great +persons, that every rich man had a philosopher of one sect or another to +be a schoolmaster to his children. And these were they that feigning +Christianity, with their disputing and readiness of talking got +themselves into Christian commons, and brought so many heresies into the +primitive Church, every one retaining still a tang of what they had been +used to teach. + +_A._ But those heresies were all condemned in the first Council of Nice. + +_B._ Yes. But the Arian heresy for a long time flourished no less than +the Roman, and was upheld by divers Emperors, and never fully +extinguished as long as there were Vandals in Christendom. Besides, +there arose daily other sects, opposing their philosophy to the doctrine +of the Councils concerning the divinity of our Saviour; as how many +persons he was, how many natures he had. And thus it continued till the +time of Charlemagne, when he and Pope Leo the third divided the power of +the empire into temporal and spiritual. + +_A._ A very unequal division. + +_B._ Why? Which of them think you had the greater share? + +_A._ No doubt, the Emperor: for he only had the sword. + +_B._ When the swords are in the hands of men, whether had you rather +command the men or the swords? + +_A._ I understand you. For he that hath the hands of the men, has also +the use both of their swords and strength. + +_B._ The empire thus divided into spiritual and temporal, the freedom of +philosophy was to the power spiritual very dangerous. And for that cause +it behoved the Pope to get schools set up not only for divinity, but +also for other sciences, especially for natural philosophy. Which when +by the power of the Emperor he had effected, out of the mixture of +Aristotle’s metaphysics with the Scripture, there arose a new science +called School-divinity; which has been the principal learning of these +western parts from the time of Charlemagne till of very late. + +_A._ But I find not in any of the writings of the Schoolmen in what +manner, from the causes they assign, the effect is naturally and +necessarily produced. + +_B._ You must not wonder at that. For you enquire not so much, when you +see a change of anything, what may be said to be the cause of it, as how +the same is generated; which generation is the entire progress of nature +from the efficient cause to the effect produced. Which is always a hard +question, and for the most part impossible for a man to answer to. For +the alterations of the things we perceive by our five senses are made by +the motion of bodies, for the most part, either for distance, smallness, +or transparence, invisible. + +_A._ But what need had they then to assign any cause at all, seeing that +they could not show the effect was to follow from it? + +_B._ The Schools, as I said, were erected by the Pope and Emperor, but +directed by the Pope only, to answer and confute the heresies of the +philosophers. Would you have them then betray their profession and +authority, that is to say, their livelihood, by confessing their +ignorance? Or rather uphold the same, by putting for causes, strange and +unintelligible words; which might serve well enough not only to satisfy +the people whom they relied on, but also to trouble the philosophers +themselves to find a fault in. + +_A._ Seeing you say that alteration is wrought by the motion of bodies, +pray tell me first what I am to understand by the word body. + +_B._ It is a hard question, though most men think they can easily answer +it, as that it is whatsoever they can see, feel, or take notice of by +their senses. But if you will know indeed what is body, we must enquire +first what there is that is not body. You have seen, I suppose, the +effects of glasses, how they multiply and magnify the object of our +sight; as when a glass of a certain figure will make a counter or a +shilling seem twenty, though you be well assured there is but one. And +if you set a mark upon it, you will find the mark upon them all. The +counter is certainly one of those things we call bodies: are not the +others so too? + +_A._ No, without doubt. For looking through a glass cannot make them +really more than they are. + +_B._ What then be they but fancies, so many fancies of one and the same +thing in several places? + +_A._ It is manifest they are so many idols, mere nothings. + +_B._ When you have looked upon a star or candle with both your eyes, but +one of them a little turned awry with your finger, has not there +appeared two stars, or two candles? And though you call it a deception +of the sight, you cannot deny but there were two images of the object. + +_A._ It is true, and observed by all men. And the same I say of our +faces seen in looking-glasses, and of all dreams, and of all apparitions +of dead men’s ghosts; and wonder, since it is so manifest, I never +thought upon it before, for it is a very happy encounter, and such as +being by everybody well understood, would utterly destroy both idolatry +and superstition, and defeat abundance of knaves that cheat and trouble +the world with their devices. + +_B._ But you must not hence conclude that whosoever tells his dream, or +sometimes takes his direction from it, is therefore an idolater, or +superstitious, or a cheater. For God doth often admonish men by dreams +of what they ought to do; yet men must be wary in this case that they +trust not dreams with the conduct of their lives farther than by the +laws of their country is allowed: for you know what God says, Deut. +xiii: _If a prophet or a dreamer of dreams give thee a sign or a wonder, +and the sign come to pass, yet if he bid thee serve other Gods let him +be put to death_. Here by serving other Gods (since they have chosen God +for their King) we are to understand revolting from their King, or +disobeying of his laws. Otherwise I see no idolatry nor superstition in +following a dream, as many of the Patriarchs in the Old Testament, and +of the Saints in the New Testament did. + +_A._ Yes: their own dreams. But when another man shall dream, or say +that he has dreamed, and require me to follow that, he must pardon me if +I ask him by what authority, especially if he look I should pay him for +it. + +_B._ But if commanded by the laws you live under, you ought to follow +it. But when there proceed from one sound divers echoes, what are those +echoes? And when with fingers crossed you touch a small bullet, and +think it two; and when the same herb or flower smells well to one and +ill to another, and the same at several times, well and ill to yourself, +and the like of tastes, what are those echoes, feelings, odours, and +tastes? + +_A._ It is manifest they are all but fancies. But certainly when the sun +seems to my eye no bigger than a dish, there is behind it somewhere +somewhat else, I suppose a real sun, which creates those fancies, by +working, one way or other, upon my eyes, and other organs of my senses, +to cause that diversity of fancy. + +_B._ You say right; and that is it I mean by the word body, which +briefly I define to be any thing that hath a being in itself, without +the help of sense. + +_A._ Aristotle, I think, meaneth by body, _substance_, or _subjectum_, +wherein colour, sound, and other fancies are, as he says, inherent. For +the word essence has no affinity with substance. And Seneca says, he +understands it not. And no wonder: for essence is no part of the +language of mankind, but a word devised by philosophers out of the +copulation of two names, as if a man having two hounds could make a +third, if it were need, of their couples. + +_B._ It is just so. For having said in themselves, (for example): _a +tree is a plant_, and conceiving well enough what is the signification +of those names, knew not what to make of the word _is_, that couples +those names; nor daring to call it a body, they called it by a new name +(derived from the word _est_), _essentia_, and _substantia_, deceived by +the idiom of their own language. For in many other tongues, and namely +in the Hebrew, there is no such copulative. They thought the names of +things sufficiently connected, when they are placed in their natural +consequence; and were therefore never troubled with essences, nor other +fallacy from the copulative _est_. + + + ========== + + + CHAPTER II. + OF THE PRINCIPLES AND METHOD OF NATURAL + PHILOSOPHY. + +_A._ This history of the old philosophers has not put me out of love, +but out of hope of philosophy from any of their writings. I would +therefore try if I could attain any knowledge therein by my own +meditation: but I know neither where to begin, nor which way to proceed. + +_B._ Your desire, you say, is to know the causes of the effects or +phenomena of nature; and you confess they are fancies, and, +consequently, that they are in yourself; so that the causes you seek for +only are without you, and now you would know how those external bodies +work upon you to produce those phenomena. The beginning therefore of +your enquiry ought to be at; _What it is you call a cause?_ I mean an +efficient cause: for the philosophers make four kinds of causes, whereof +the efficient is one. Another they call the formal cause, or simply the +form or essence of the thing caused; as when they say, four equal angles +and four equal sides are the cause of a square figure; or that heaviness +is the cause that makes heavy bodies to descend; but that is not the +cause you seek for, nor any thing but this: _It descends because it +descends_. The third is the material cause, as when they say, the walls +and roof, &c. of a house are the cause of a house. The fourth is the +final cause, and hath place only in moral philosophy. + +_A._ We will think of final causes upon some other occasion; of formal +and material not at all: I seek only the efficient, and how it acteth +from the beginning to the production of the effect. + +_B._ I say then, that in the first place you are to enquire diligently +into the nature of motion. For the variations of fancies, or (which is +the same thing) of the phenomena of nature, have all of them one +universal efficient cause, namely the variety of motion. For if all +things in the world were absolutely at rest, there could be no variety +of fancy; but living creatures would be without sense of all objects, +which is little less than to be dead. + +_A._ What if a child new taken from the womb should with open eyes be +exposed to the azure sky, do not you think it would have some sense of +the light, but that all would seem unto him darkness? + +_B._ Truly, if he had no memory of any thing formerly seen, or by any +other sense perceived, (which is my supposition), I think he would be in +the dark. For darkness is darkness, whether it be black or blue, to him +that cannot distinguish. + +_A._ Howsoever that be, it is evident enough that whatsoever worketh is +moved: for action is motion. + +_B._ Having well considered the nature of motion, you must thence take +your principles for the foundation and beginning of your enquiry. + +_A._ As how? + +_B._ Explain as fully and as briefly as you can what you constantly mean +by motion; which will save yourself as well as others from being seduced +by equivocation. + +_A._ Then I say, motion is nothing but change of place for all the +effect of a body upon the organs of our senses is nothing but fancy. +Therefore we can fancy nothing from seeing it moved, but change of +place. + +_B._ It is right. But you must then tell me also what you understand by +place: for all men are not yet agreed on that. + +_A._ Well then; seeing we fancy a body, we cannot but fancy it +somewhere. And therefore I think place is the fancy of here or there. + +_B._ That is not enough. Here and there are not understood by any but +yourself, except you point towards it. But pointing is no part of a +definition. Besides, though it help him to find the place, it will never +bring him to it. + +_A._ But seeing sense is fancy, when we fancy a body, we fancy also the +figure of it, and the space it fills up. And then I may define place to +be the precise space within which the body is contained. For space is +also part of the image we have of the object seen. + +_B._ And how define you time? + +_A._ As place is to a body, so, I think, is time to the motion of it; +and consequently I take time to be our fancy or image of the motion. But +is there any necessity of so much niceness? + +_B._ Yes. The want of it is the greatest, if not the only, cause of all +the discord amongst philosophers, as may easily be perceived by their +abusing and confounding the names of things that differ in their nature; +as you shall see when there is occasion to recite some of the tenets of +divers philosophers. + +_A._ I will avoid equivocation as much as I can. And for the nature of +motion, I suppose I understand it by the definition. What is next to be +done? + +_B._ You are to draw from these definitions, and from whatsoever truth +else you know by the light of nature, such general consequences as may +serve for axioms, or principles of your ratiocination. + +_A._ That is hard to do. + +_B._ I will draw them myself, as many as for our present discourse of +natural causes we shall have need of; so that your part will be no more +than to take heed I do not deceive you. + +_A._ I will look to that. + +_B._ My first axiom then shall be this: Two bodies, at the same time, +cannot be in one place. + +_A._ That is true: for we number bodies as we fancy them distinct, and +distinguish them by their places. You may therefore add: nor one body at +the same time in two places. And philosophers mean the same, when they +say: there is no penetration of bodies. + +_B._ But they understand not their own words: for penetration signifies +it not. My second axiom is, that nothing can begin, change, or put an +end to, its own motion. For supposing it begin just now, or being now in +motion, change its way or stop; I require the cause why now rather than +before or after, having all that is necessary to such motion, change, or +rest, alike at all times? + +_A._ I do not doubt but the argument is good in bodies inanimate; but +perhaps in voluntary agents it does not hold. + +_B._ How it holds in voluntary agents we will then consider when our +method hath brought us to the powers and passions of the mind. A third +axiom shall be this: whatsoever body being at rest is afterwards moved, +hath for its immediate movement some other body which is in motion and +toucheth it. For, since nothing can move itself, the movent must be +external. And because motion is change of place, the movent must put it +from its place, which it cannot do till it touch it. + +_A._ That is manifest, and that it must more than touch it; it must also +follow it. And if more parts of the body are moved than are by the +movent touched, the movent is not immediate. And by this reason, a +continued body, though never so great, if the first superficies be +pressed never so little back, the motion will proceed through it. + +_B._ Do you think that to be impossible? I will prove it from your own +words: for you say that the movent does then touch the body which it +moveth. Therefore it puts it back; but that which is put back, puts back +the next behind, and that again the next; and so onward to any distance, +the body being continued. The same is also manifest by experience, +seeing one that walks with a staff can distinguish, though blind, +between stone and glass; which were impossible, if the parts of his +staff between the ground and his hand made no resistance. So also he +that in the silence of the night lays his ear to the ground, shall hear +the treading of men’s feet farther than if he stood upright. + +_A._ This is certainly true of a staff or other hard body, because it +keeps the motion in a straight line from diffusion. But in such a fluid +body as the air, which being put back must fill an orb, and the farther +it is put back, the greater orb, the motion will decrease, and in time, +by the resistance of air to air, come to an end. + +_B._ That any body in the world is absolutely at rest, I think not true: +but I grant, that in a space filled everywhere with body, though never +so fluid, if you give motion to any part thereof, that motion will by +resistance of the parts moved, grow less and less, and at last cease; +but if you suppose the space utterly void, and nothing in it, then +whatsoever is once moved shall go on eternally: or else that which you +have granted is not true, viz., that nothing can put an end to its own +motion. + +_A._ But what mean you by resistance? + +_B._ Resistance is the motion of a body in a way wholly or partly +contrary to the way of its movent, and thereby repelling or retarding +it. As when a man runs swiftly, he shall feel the motion of the air in +his face. But when two hard bodies meet, much more may you see how they +abate each other’s motion, and rebound from one another. For in a space +already full, the motion cannot, in an instant, be communicated through +the whole depth of the body that is to be moved. + +_A._ What other definitions have I need of? + +_B._ In all motion, as in all quantity, you must take the beginning of +your reckoning from the least supposed motion. And this I call the first +endeavour of the movent; which endeavour, how weak soever, is also +motion. For if it have no effect at all, neither will it do anything +though doubled, trebled, or by what number soever multiplied: for +nothing, though multiplied, is still nothing. Other axioms and +definitions we will take in, as we need them, by the way. + +_A._ Is this all the preparation I am to make? + +_B._ No, you are to consider also the several kinds and properties of +motion, viz., when a body being moved by one or more movents at once, in +what way it is carried, straight, circular, or otherwise crooked; and +what degree of swiftness; as also the action of the movent, whether +trusion, vection, percussion, reflection, or refraction; and farther you +must furnish yourself with as many experiments (which they call +phenomenon) as you can. And supposing some motion for the cause of your +phenomenon, try, if by evident consequence, without contradiction to any +other manifest truth or experiment, you can derive the cause you seek +for from your supposition. If you can, it is all that is expected, as to +that one question, from philosophy. For there is no effect in nature +which the Author of nature cannot bring to pass by more ways than one. + +_A._ What I want of experiments you may supply out of your own store, or +such natural history as you know to be true; though I can be well +content with the knowledge of the causes of those things which everybody +sees commonly produced. Let us therefore now enquire the cause of some +effect particular. + +_B._ We will begin with that which is the most universal, the universe; +and enquire in the first place, if any place be absolutely empty, that +is to say in the language of philosophers, whether there be any vacuum +in nature? + + + ========== + + + CHAPTER III. + OF VACUUM. + +_A._ It is hard to suppose, and harder to believe, that the infinite and +omnipotent Creator of all things should make a work so vast as is the +world we see, and not leave a few little spaces with nothing at all in +them; which put altogether in respect of the whole creation, would be +insensible. + +_B._ Why say you that? Do you think any argument can be drawn from it to +prove there is vacuum? + +_A._ Why not? For in so great an agitation of natural bodies, may not +some small parts of them be cast out, and leave the places empty from +whence they were thrown? + +_B._ Because He that created them is not a fancy, but the most real +substance that is; who being infinite, there can be no place empty where +He is, nor full where He is not. + +_A._ It is hard to answer this argument, because I do not remember that +there is any argument for the maintenance of vacuum in the writings of +divines: therefore I will quit that argument, and come to another. If +you take a glass vial with a narrow neck, and having sucked it, dip it +presently at the neck into a basin of water, you shall manifestly see +the water rise into the vial. Is not this a certain sign that you had +sucked out some of the air, and consequently that some part of the vial +was left empty? + +_B._ No; for when I am about to suck, and have air in my mouth, +contracting my checks I drive the same against the air in the glass, and +thereby against every part of the sides of the hard glass. And this +gives to the air within an endeavour outward, by which, if it be +presently dipped into the water, it will penetrate and enter into it. +For air if it be pressed will enter into any fluid, much more into +water. Therefore there shall rise into the vial so much water as there +was air forced into the basin. + +_A._ This I confess is possible, and not improbable. + +_B._ If sucking would make vacuum, what would become of those women that +are nurses? Should they not be in a very few days exhausted, were it not +that either the air which is in the child’s mouth penetrateth the milk +as it descends, and passeth through it, or the breast is contracted? + +_A._ From what experiment can you evidently infer that there is no +vacuum? + +_B._ From many, and such as to almost all men are known and familiar. If +two hard bodies, flat and smooth, be joined together in a common +superficies parallel to the horizontal plane, you cannot without great +force pull them asunder, if you apply your force perpendicularly to the +common superficies: but if you place that common superficies erect to +the horizon, they will fall asunder with their own weight. From whence I +argue thus: since their contiguity, in what posture soever, is the same, +and that they cannot be pulled asunder by a perpendicular force without +letting in the ambient air in an instant, which is impossible; or almost +in an instant, which is difficult: and on the other side, when the +common superficies is erect, the weight of the same hard bodies is able +to break the contiguity, and let in the air successively; it is manifest +that the difficulty of separation proceeds from this, that neither air +nor any other body can be moved to any, how small soever, distance in an +instant; but may easily be moved (the hardness at the sides once +mastered) successively. So that the cause of this difficulty of +separation is this, that they cannot be parted except the air or other +matter can enter and fill the space made by their diremption. And if +they were infinitely hard, not at all. And hence also you may understand +the cause why any hard body, when it is suddenly broken, is heard to +crack; which is the swift motion of the air to fill the space between. +Another experiment, and commonly known, is of a barrel of liquor, whose +tap-hole is very little, and the bung so stopped as to admit no air; for +then the liquor will not run: but if the tap-hole be large it will, +because the air pressed by a heavier body will pierce through it into +the barrel. The like reason holds of a gardener’s watering-pot, when the +holes in the bottom are not too great. A third experiment is this: turn +a thin brass kettle the bottom upwards, and lay it flat upon the water. +It will sink till the water rise within to a certain height, but no +higher: yet let the bottom be perforated, and the kettle will be full +and sink, and the air rise again through the water without. But if a +bell were so laid on, it would be filled and sink, though it were not +perforated, because the weight is greater than the weight of the same +bulk of water. + +_A._ By these experiments, without any more, I am convinced, that there +is not actually in nature any vacuum; but I am not sure but that there +may be made some little place empty, and this from two experiments, one +whereof is Toricellius’ experiment, which is this: take a cylinder of +glass, hollow throughout, but close at the end, in form of a sack. + +_B._ How long? + +_A._ As long as you will, so it be more than twenty-nine inches. + +_B._ And how broad? + +_A._ As broad as you will, so it be broad enough to pour into it +quicksilver. And fill it with quicksilver, and stop up the entrance with +your finger, so as to unstop it again at your pleasure. Then set down a +basin, or, if you will, a sea of quicksilver, and inverting the cylinder +full as it is, dip the end into the quicksilver, and remove your finger, +that the cylinder may empt itself. Do you conceive me? For there is so +many passing by, that I cannot paint it. + +_B._ Yes, I conceive you well enough. What follows? + +_A._ The quicksilver will descend in the cylinder, not till it be level +with that in the basin, according to the nature of heavy fluids, but +stay and stand above it, at the height of twenty-nine inches or very +near it, the bottom being now uppermost, that no air can get in. + +_B._ What do you infer from this? + +_A._ That all the cavity above twenty-nine inches is filled with vacuum. + +_B._ It is very strange that I, from this same experiment, should infer, +and I think evidently, that it is filled with air. I pray, tell me, when +you had inverted the cylinder, full as it was, and stopped with your +finger, dipped into the basin, if you had then removed your finger, +whether you think the quicksilver would not all have fallen out? + +_A._ No sure. The air would have been pressed upward through the +quicksilver itself: for a man with his hand can easily thrust a bladder +of air to the bottom of a basin of quicksilver. + +_B._ It is therefore manifest that quicksilver can press the air through +the same quicksilver. + +_A._ It is manifest; and also itself rise into the air. + +_B._ What cause then can there be, why it should stand still at twenty +nine inches above the level of the basin, rather than any place else? + +_A._ It is not hard to assign the cause of that. For so much quicksilver +as was above the twenty-nine inches, will rise the first level of that +in the basin, as much as if you had poured it on; and thereby bring it +to an equilibrium. So that I see plainly now, that there is no necessity +of vacuum from this experiment. For I considered only that naturally +quicksilver cannot ascend in air, nor air descend in quicksilver, though +by force it may. + +_B._ Nor do I think that Torricellius or any other vacuist thought of it +more than you. But what is the second experiment? + +_A._ There is a sphere of glass, which they call a recipient, of the +capacity of three or four gallons. And there is inserted into it the end +of a hollow cylinder of brass above a foot long; so that the whole is +one vessel, and the bore of the cylinder three inches diameter. Into +which is thrust by force a solid cylinder of wood, covered with leather +so just, as it may in every point exactly touch the concave superficies +of the brass. There is also, to let out the air which the wooden +cylinder as it enters (called the sucker) drives before it, a flap to +keep out the external air while they are pulling the sucker. Besides, at +the top of the recipient there is a hole to put into it anything for +experiment. The sucker being now forced up into the cylinder, what do +you think must follow? + +_B._ I think it will require as much strength to pull it back, as it did +to force it in. + +_A._ That is not it I ask, but what would happen to the recipient? + +_B._ I think so much air as would fill the place the sucker leaves, +would descend into it out of the recipient; and also that just so much +from the external air would enter into the recipient, between the brass +and the wood, at first very swiftly, but, as the place increased, more +leisurely. + +_A._ Why may not so much air rather descend into the place forsaken, and +leave as much vacuum as that comes to in the recipient? For otherwise no +air will be pumped out, nor can that wooden pestle be called a sucker. + +_B._ That is it I say. There is no air either pumped or sucked out. + +_A._ How can the air pass between the leather and the brass, or between +the leather and the wood, being so exactly contiguous, or through the +leather itself? + +_B._ I conceive no such exact contiguity, nor such fastness of the +leather: for I never yet had any that in a storm would keep out either +air or water. + +_A._ But how then could there be made in the recipient such strange +alteration both on animate and inanimate bodies? + +_B._ I will tell you how. The air descends out of the recipient, because +the air which the sucker removeth from behind itself, as it is pulling +out, has no place to retire into without, and therefore is driven into +the engine between the wood of the sucker and the brass of the cylinder, +and causes as much air to come into the place forsaken by the retiring +sucker; which causeth, by oft repetition of the force, a violent +circulation of the air within the recipient, which is able quickly to +kill anything that lives by respiration, and make all the alterations +that have appeared in the engine. + + + ========== + + + CHAPTER IV. + OF THE SYSTEM OF THE WORLD. + +_B._ You are come in good time; let us therefore sit down. There is ink, +paper, ruler, and compass. Draw a little circle to represent the body of +the sun. + +_A._ It is done. The centre is A, the circumference is L M. + +_B._ Upon the same centre A, draw a larger circle to stand for the +ecliptic: for you know the sun is always in the plane of the ecliptic. + +_A._ There it is. The diameters of it at right angles are B Z. + +_B._ Draw the diameter of the equator. + +_A._ How? + +_B._ Through the centre A (for the earth is also always in the plane of +the equator or of some of its parallels) so as to be distant from B +twenty-three degrees and a half. + +_A._ Let it be H I: and let C G be equal to B H; and so C will be one of +the poles of the ecliptic, suppose the north-pole; and then H will be +east, and I west. And C A produced to the circumference in E, makes E +the south-pole. + +_B._ Take C K equal to C G, and the chord G K will be the diameter of +the arctic circle, and parallel to H I, the diameter of the equator. +Lastly, upon the point B, draw a little circle wherein I suppose to be +the globe of the earth. + +_A._ It is drawn, and marked with _l m_. And B D and K G joined will be +parallel; and as H and I are east and west, and so are B and D, and G +and K. + +_B._ True; but producing Z B to the circumference _l m_ in _b_, the line +B _b_ will be in the diameter of the ecliptic of the earth, and B _m_ in +the diameter of the equator of the earth. In like manner, if you produce +K G cutting the circle, whose centre is G, in _d_ and _e_, and make an +angle _n_ G _d_ equal to _b_ B _m_, the line _n_ G will be in the +ecliptic of the earth, because G _d_ is in the equator of the earth. So +that in the annual motion of the earth through the ecliptic, every +straight line drawn in the earth, is perpetually kept parallel to the +place from whence it is removed. + +_A._ It is true; and it is the doctrine of Copernicus. But I cannot yet +conceive by what one motion this circle can be described otherwise than +we are taught by Euclid. And then I am sure that all the diameters shall +cross one another in the centre, which in this figure is A. + +_B._ I do not say that the diameters of a sphere or circle can be +parallel; but that if a circle of a lesser sphere be moved upon the +circumference of a great circle of a greater sphere, that the straight +lines that are in the lesser sphere may be kept parallel perpetually to +the places they proceed from. + +_A._ How? And by what motion? + +_B._ Take into your hand any straight line (as in this figure), the line +L A M, which we suppose to be the diameter of the sun’s body; and moving +it parallelly with the ends in the circumference, so as that the end M +may withal describe a small circle, as M _a_. It is manifest that all +the other points of the same line L M will, by the same motion, at the +same time, describe equal circles to it. Likewise if you take in your +hand any two diameters fastened together, the same parallel motion of +the line L M, shall cause all the points of the other diameter to make +equal circles to the same M _a_. + +_A._ It is evident; as also that every point of the sun’s body shall do +the like. And not only so, but also if one end describe any other +figure, all the other points of the body shall describe like and equal +figures to it. + +_B._ You see by this, that this parallel motion is compounded of two +motions, one circular upon the superficies of a sphere, the other a +straight motion from the centre to every point of the same superficies, +and beyond it. + +_A._ I see it. + +_B._ It follows hence, that the sun by this motion must every way repel +the air; and since there is no empty place for retiring, the air must +turn about in a circular stream; but slower or swifter according as it +is more or less remote from the sun; and that according to the nature of +fluids, the particles of the air must continually change place with one +another; and also that the stream of the air shall be the contrary way +to that of the motion, for else the air cannot be repelled. + +_A._ All this is certain. + +_B._ Well; then if you suppose the globe of the earth to be in this +stream which is made by the motion of the sun’s body from east to west, +the stream of air wherein is the earth’s annual motion will be from west +to east. + +_A._ It is certain. + +_B._ Well. Then if you suppose the globe of the earth, whose circle is +moved annually, to be _l m_, the stream of the air without the ecliptic +falling upon the superficies of the earth _l m_ without the ecliptic, +being slower, and the stream that falleth within swifter, the earth +shall be turned upon its own centre proportionally to the greatness of +the circles; and consequently their diameters shall be parallel; as also +are other straight lines correspondent. + +_A._ I deny not but the streams are as you say; and confess that the +proportion of the swiftness without, is to the swiftness within, as the +sun’s ecliptic to the ecliptic of the earth; that is to say, as the +angle H A B to the angle _m_ B _b_. And I like your argument the better, +because it is drawn from Copernicus his foundation. I mean the +compounded motion of straight and circular. + +_B._ I think I shall not offer you many demonstrations of physical +conclusions that are not derived from the motions supposed or proved by +Copernicus. For those conclusions in natural philosophy I most suspect +of falshood, which require most variety of suppositions for their +demonstrations. + +_A._ The next thing I would know, is how great or little you suppose +that circle _a_ M? + +_B._ I suppose it less than you can make it: for there appears in the +sun no such motion sensible. It is the first endeavour of the sun’s +motion. But for all that, as small as the circle is, the motion may be +as swift, and of as great strength as it is possible to be named. It is +but a kind of trembling that necessarily happeneth in those bodies, +which with great resistance press upon one another. + +_A._ I understand now from what cause proceedeth the annual motion. Is +the sun the cause also of the diurnal motion? + +_B._ Not the immediate cause. For the diurnal motion of the earth is +upon its own centre, and therefore the sun’s motion cannot describe it. +But it proceedeth as a necessary consequence from the annual motion. For +which I have both experience and demonstration. The experiment is this: +into a large hemisphere of wood, spherically concave, put in a globe of +lead, and with your hands hold it fast by the brim, moving your hand +circularly, but in a very small compass; you shall see the globe +circulate about the concave vessel, just in the same manner as the earth +doth every year in the air; and you shall see withal, that as it goes, +it turns perpetually upon its own centre, and very swiftly. + +_A._ I have seen it: and it is used in some great kitchens to grind +mustard. + +_B._ Is it so? Therefore take a hemisphere of gold, if you have it, the +greater the better, and a bullet of gold, and, without mustard, you +shall see the same effect. + +_A._ I doubt it not. But the cause of it is evident. For any spherical +body being in motion upon the sides of a concave and hard sphere, is all +the way turned upon its own centre by the resistance of the hard wood or +metal. But the earth is a bullet without weight, and meeteth only with +air, without any harder body in the way to resist it. + +_B._ Do you think the air makes no resistance, especially to so swift a +motion as is the annual motion of the earth? If it do make any +resistance, you cannot doubt but that it shall turn the earth +circularly, and in a contrary way to its annual motion; that is to say, +from east to west, because the annual motion is from west to east. + +_A._ I confess it. But what deduce you from these motions of the sun? + +_B._ I deduce, first, that the air must of necessity be moved both +circularly about the body of the sun according to the ecliptic, and also +every way directly from it. For the motion of the sun’s body is +compounded of this circular motion upon the sphere L M, and of the +straight motion of its semi-diameters from the centre A to the +superficies of the sun’s body, which is L M. And therefore the air must +needs be repelled every way, and also continually change place to fill +up the places forsaken by other parts of the air, which else would be +empty, there being no vacuum to retire unto. So that there would be a +perpetual stream of air, and in a contrary way to the motion of the +sun’s body, such as is the motion of water by the sides of a ship under +sail. + +_A._ But this motion of the earth from west to east is only circular, +such as is described by a compass about a centre; and cannot therefore +repel the air as the sun does. And the disciples of Copernicus will have +it to be the cause of the moon’s monthly motion about the earth. + +_B._ And I think Copernicus himself would have said the same, if his +purpose had been to have shown the natural causes of the motions of the +stars. But that was no part of his design; which was only from his own +observations, and those of former astronomers, to compute the times of +their motions; partly to foretel the conjunctions, oppositions, and +other aspects of the planets; and partly to regulate the times of the +Church’s festivals. But his followers, Kepler and Galileo, make the +earth’s motion to be the efficient cause of the monthly motion of the +moon about the earth; which without the like motion to that of the sun +in L M, is impossible. Let us therefore for the present take it in as a +necessary hypothesis; which from some experiment that I shall produce in +our following discourses, may prove to be a certain truth. + +_A._ But seeing A is the centre both of the sun’s body and of the annual +motion of the earth, how can it be (as all astronomers say it is) that +the orb of the annual motion of the earth should be eccentric to the +sun’s body? For you know that from the vernal equinox to the autumnal, +there be one hundred and eighty-seven days; but from the autumnal +equinox to the vernal, there be but one hundred and seventy-eight days. +What natural cause can you assign for this eccentricity? + +_B._ Kepler ascribes it to a magnetic virtue, viz. that one part of the +earth’s superficies has a greater kindness for the sun than the other +part. + +_A._ I am not satisfied with that. It is magical rather than natural, +and unworthy of Kepler. Tell me your own opinion of it. + +_B._ I think that the magnetical virtue he speaks of, consisteth in +this: that the southern hemisphere of the earth is for the greatest part +sea, and that the greatest part of the northern hemisphere is dry land. +But how it is possible that from thence should proceed the eccentricity +(the sun being nearest to the earth, when he is in the winter solstice), +I shall show you when we come to speak of the motions of air and water. + +_A._ That is time enough: for I intend it for our next meeting. In the +mean time I pray you tell me what you think to be the cause why the +equinoctial, and consequently the solstitial, points are not always in +one and the same point of the ecliptic of the fixed stars. I know they +are not, because the sun does not rise and set in points diametrically +opposite: for if it did, there would be no difference of the seasons of +the year. + +_B._ The cause of that can be no other, than that the earth, which is _l +m_, hath the like motion to that which I suppose the sun to have in L M, +compounded of straight and circular from west to east in a day, as the +annual motion hath in a year; so that, not reckoning the eccentricity, +it will be moved through the ecliptics in one revolution, as Copernicus +proveth, about one degree. Suppose then the whole earth moved from H to +I, (which is half the year) circularly, but falling from I to _i_ in the +same time about thirty minutes, and as much in the other hemisphere from +H to _k_; then draw the line _i k_, which will be equal and parallel to +H I, and be the diameter of the equator for the next year. But it shall +not cut the diameter of the ecliptic B Z in A, which was the equinoctial +of the former year, but in _o_ thirty-six seconds from the first degree +of Aries. Suppose the same done in the hemisphere under the plane of the +paper, and so you have the double of thirty-six seconds, that is +seventy-two seconds, or very near, for the progress of the vernal +equinox in a year. The cause why I suppose the arch I _i_ to be half a +degree in the ecliptic of the earth, is, that Copernicus and other +astronomers, and experience, agree in this, that the equinoctial points +proceed according to the order of the signs, Aries, Taurus, Gemini, &c. +from west to east every hundredth year one degree or very near. + +_A._ In what time do they make the whole revolution through the ecliptic +of the sky? + +_B._ That you may reckon. For we know by experience that it hath +proceeded about one degree, that is sixty minutes, constantly a long +time in a hundred years. But as one hundred years to one degree, so is +thirty-six thousand years to three hundred and sixty degrees. Also as +one hundred years to one degree, so is one year to the hundredth part of +one degree, or sixty minutes; which is (60)/(100), or thirty-six seconds +for the progress of one year; which must be somewhat more than a degree +according to Copernicus, who, (lib. iii. cap. 2) saith, that for four +hundred years before Ptolomy it was one degree almost constantly. Which +is well enough as to the natural cause of the precession of the +equinoctial points, which is the often-said compounded motion, though +not an exact astronomical calculation. + +_A._ And it is a great sign that his supposition is true. But what is +the cause that the obliquity of the ecliptic, that is, the distance +between the equinoctial and the solstice, is not always the same? + +_B._ The necessity of the obliquity of the ecliptic is but a consequence +to the precession of the equinoctial points. And therefore, if from C, +the north pole, you make a little circle, C _u_, equal to fifteen +minutes of a degree upon the earth, and another, _u s_, equal to the +same, which will appear like this figure 8, that is, (as Copernicus +calls it), a circle twined, the pole C will be moved half the time of +the equinoctial points, in the arc C _u_, and as much in the alternate +arc _u s_ descending to _s_. But in the arc _s u_, and its alternate +rising to C, the cause of the twining is the earth’s annual motion the +same way in the ecliptic, and makes the four quarters of it; and makes +also their revolution twice as slow as that of the equinoctial points. +And, therefore, the motion of it is the same compounded motion which +Copernicus takes for his supposition, and is the cause of the precession +of the equinoctial points, and consequently of the variation of the +obliquity, adding to it or taking from it somewhere more, somewhere +less; so as that one with another the addition is not much more, nor the +subtraction much less than thirty minutes. But as for the natural +efficient cause of this compounded motion, either in the sun, or the +earth, or any other natural body, it can be none but the immediate hand +of the Creator. + +_A._ By this it seems that the poles of the earth are always the same, +but make this 8 in the sphere of the fixed stars near that which is +called Cynosura. + +_B._ No: it is described on the earth, but the annual motion describes a +circle in the sphere of the fixed stars. Though I think it improper to +say a sphere of the fixed stars, when it is so unlikely that all the +fixed stars should be in the superficies of one and the same globe. + +_A._ I do not believe they are. + +_B._ Nor I, since they may seem less one than another, as well by their +different distances, as by their different magnitudes. Nor is it likely +that the sun (which is a fixed star) is the efficient cause of the +motion of those remoter planets, Mars, Jupiter, and Saturn; seeing the +whole sphere, whose diameter is the distance between the sun and the +earth, is but a point in respect of the distance between the sun and any +other fixed star. Which I say only to excite those that value the +knowledge of the cause of comets, to look for it in the dominion of some +other sun than that which moveth the earth. For why may not there be +some other fixed star, nearer to some planet than is the sun, and cause +such a light in it as we call a comet? + +_A._ As how? + +_B._ You have seen how in high and thin clouds above the earth, the +sun-beams piercing them have appeared like a beard; and why might not +such a beard have appeared to you like a comet, if you had looked upon +it from as high as some of the fixed stars? + +_A._ But because it is a thing impossible for me to know, I will proceed +in my own way of inquiry. And seeing you ascribe this compounded motion +to the sun and earth, I would grant you that the earth (whose annual +motion is from west to east) shall give the moon her monthly motion from +east to west. But then I ask you whether the moon have also that +compounded motion of the earth, and with it a motion upon its own +centre, as hath the earth? For seeing the moon has no other planet to +carry about her, she needs it not. + +_B._ I see reason enough, and some necessity, that the moon should have +both those motions. For you cannot think that the Creator of the stars, +when he gave them their circular motion, did first take a centre, and +then describe a circle with a chain or compass, as men do? No; he moved +all the parts of a star together and equally in the creation: and that +is the reason I give you. The necessity of it comes from this +phenomenon, that the moon doth turn one and the same face towards the +earth; which cannot be by being moved about the earth parallelly, unless +also it turn about its own centre. Besides, we know by experience, that +the motion of the moon doth add not a little to the motion of the sea: +which were impossible if it did not add to the stream of the air, and by +consequence to that of the water. + +_A._ If you could get a piece of the true and intimate substance of the +earth, of the bigness of a musket-bullet, do you believe that the bullet +would have the like compounded motion to that which you attribute to the +sun, earth, and moon? + +_B._ Yes, truly; but with less strength, according to its magnitude; +saving that by its gravity falling to the earth, the activity of it +would be unperceived. + +_A._ I will trouble you no more with the nature of celestial +appearances; but I pray you tell me by what art a man may find what part +of a circle the diameter of the sun’s body doth subtend in the ecliptic +circle? + +_B._ Kepler says it subtends thirty minutes, which is half a degree. His +way to find it is by letting in the sun-beams into a close room through +a small hole, and receiving the image of it upon a plane +perpendicularly. For by this means he hath a triangle, whose sides and +angles he can know by measure; and the vertical angle he seeks for, and +the substance of the arc of the sun’s body. + +_A._ But I think it impossible to distinguish where the part illuminate +toucheth the part not illuminate. + +_B._ Another way is this: upon the equinoctial day, with a watch that +shows the minutes standing by you, observe when the lower brim of the +sun’s setting first comes to the horizon, and set the index to some +minute of the watch; and observe again the upper brim when it comes to +the horizon: then count the minutes, and you have what you look for. +Other way I know none. + + + ========== + + + CHAPTER V. + OF THE MOTIONS OF WATER AND AIR. + +_A._ I have considered, as you bad me, this compounded motion with great +admiration. First, it is that which makes the difference between +_continuum_ and _contiguum_, which till now I never could distinguish. +For bodies that are but contiguous, with any little force are parted; +but by this compounded motion (because every point of the body makes an +equal line in equal time, and every line crosses all the rest) one part +cannot be separated from another, without disturbing the motion of all +the other parts at once. And is not that the cause, think you, that some +bodies when they are pressed or bent, as soon as the force is removed, +return again of themselves to their former figure? + +_B._ Yes, sure; saving that it is not of themselves that they return, +(for we were agreed that nothing can move itself), but it is the motion +of the parts which are not pressed, that delivers those that are. And +this restitution the learned now call the spring of a body. The Greeks +called it _antitypia_. + +_A._ When I considered this motion in the sun and the earth and planets, +I fancied them as so many bodies of the army of the Almighty in an +immense field of air, marching swiftly, and commanded (under God) by his +glorious officer the sun, or rather forced so to keep their order in +every part of every of those bodies, as never to go out from the +distance in which he had set them. + +_B._ But the parts of the air and other fluids keep not their places so. + +_A._ No: you told me that this motion is not natural in the air, but +received from the sun. + +_B._ True: but since we seek the natural causes of sublunary effects, +where shall we begin? + +_A._ I would fain know what makes the sea to ebb and flow at certain +periods, and what causeth such variety in the tides. + +_B._ Remember that the earth turneth every day upon its own axis from +west to east; and all the while it so turneth, every point thereof by +its compounded motion makes other circlings, but not on the same centre, +which is (you know) a rising in one part of the day, and a falling in +the other part. What think you must happen to the sea, which resteth on +it, and is a fluid body? + +_A._ I think it must make the sea rise and fall. And the same happeneth +also to the air, from the motion of the sun. + +_B._ Remember, also, in what manner the sea is situated in respect of +the dry land. + +_A._ Is not there a great sea that reacheth from the straits of Magellan +eastward to the Indies, and thence to the same straits again? And is not +there a great sea called the Atlantic sea that runneth northward to us? +And does not the great south sea run also up into the northern seas? But +I think the Indian and the South sea of themselves to be greater than +all the rest of the surface of the globe. + +_B._ How lieth the water in those two seas? + +_A._ East and west, and rises and falls a little, as it is forced to do +by this compounded motion, which is a kind of succussion of the earth, +and fills both the Atlantic and Northern seas. + +_B._ All this would not make a visible difference between high and low +water, because this motion being so regular, the unevenness would not be +great enough to be seen. For though in a basin the water would be thrown +into the air, yet the earth cannot throw the sea into the air. + +_A._ Yes; the basin, if gently moved, will make the water so move, that +you shall hardly see it rise. + +_B._ It may be so. But you should never see it rise as it doth, if it +were not checked. For at the straits of Magellan, the great South sea is +checked by the shore of the continent of Peru and Chili, and forced to +rise to a great height, and made to run up into the northern seas on +that side by the coast of China; and at the return is checked again and +forced through the Atlantic into the British and German seas. And this +is done every day. For we have supposed that the earth’s motion in the +ecliptic caused by the sun is annual; and that its motion in the +equinoctial is diurnal. It followeth therefore from this compounded +motion of the earth, the sea must ebb and flow twice in the space of +twenty-four hours, or thereabout. + +_A._ Has the moon nothing to do in this business? + +_B._ Yes. For she hath also the like motion. And is, though less swift, +yet much nearer to the earth. And therefore when the sun and moon are in +conjunction or opposition, the earth, as from two agents at once, must +needs have a greater succussion. And if it chance at the same time the +moon also be in the ecliptic, it will be yet greater, because the moon +then worketh on the earth less obliquely. + +_A._ But when the full or new moon happen to be then when the earth is +in the equinoctial points, the tides are greater than ordinary. Why is +that? + +_B._ Because then the force by which they move the sea, is at that time, +to the force by which they move the same at other times, as the +equinoctial circle to one of its parallels, which is a lesser circle. + +_A._ It is evident. And it is pleasant to see the concord of so many and +various motions, when they proceed from one and the same hypothesis. But +what say you to the stupendous tides which happen on the coasts of +Lincolnshire on the east, and in the river of Severn on the west? + +_B._ The cause of that, is their proper situation. For the current of +the ocean through the Atlantic sea, and the current of the south sea +through the northern seas, meeting together, rise the water in the Irish +and British seas a great deal higher than ordinary. Therefore the mouth +of the Severn being directly opposite to the current from the Atlantic +sea, and those sands on the coast of Lincolnshire directly opposite to +the current of the German sea, those tides must needs fall furiously +into them, by this succussion of the water. + +_A._ Does, when the tide runs up into a river, the water all rise +together, and fall together when it goes out? + +_B._ No: one part riseth and another falleth at the same time; because +the motion of the earth rising and falling, is that which makes the +tide. + +_A._ Have you any experiment that shows it? + +_B._ Yes. You know that in the Thames, it is high water at Greenwich +before it is high water at London-bridge. The water therefore falls at +Greenwich whilst it riseth all the way to London. But except the top of +the water went up, and the lower part downward, it were impossible. + +_A._ It is certain. It is strange that this one motion should salve so +many appearances, and so easily. But I will produce one experiment of +water, not in the sea, but in a glass. If you can show me that the cause +of it is this compounded motion, I shall go near to think it the cause +of all other effects of nature hitherto disputed of. The experiment is +common, and described by the Lord Chancellor Bacon, in the third page of +his natural history. Take, saith he, a glass of water, and draw your +finger round about the lip of the glass, pressing it somewhat hard; +after you have done so a few times, it will make the water frisk up into +a fine dew. After I had read this, I tried the same with all diligence +myself, and found true not only the frisking of the water to above an +inch high, but also the whole superficies to circulate, and withal to +make a pleasant sound. The cause of the frisking he attributes to a +tumult of the inward parts of the substance of the glass striving to +free itself from the pressure. + +_B._ I have tried and found both the sound and motion; and do not doubt +but the pressure of the parts of the glass was part of the cause. But +the motion of my finger about the glass was always parallel; and when it +chanced to be otherwise, both sound and motion ceased. + +_A._ I found the same. And being satisfied, I proceed to other +questions. How is the water, being a heavy body, made to ascend in small +particles into the air, and be there for a time sustained in form of a +cloud, and then fall down again in rain? + +_B._ I have shown already, that this compounded motion of the sun, in +one part of its circumlation, drives the air one way, and in the other +part, the contrary way; and that it cannot draw it back again, no more +than he that sets a stone a flying can pull it back. The air therefore, +which is contiguous to the water, being thus distracted, must either +leave a vacuum, or else some part of the water must rise and fill the +spaces continually forsaken by the air. But, that there is no vacuum, +you have granted. Therefore the water riseth into the air, and maketh +the clouds; and seeing they are very small and invisible parts of the +water, they are, though naturally heavy, easily carried up and down with +the wind, till, meeting with some mountain or other clouds, they be +pressed together into greater drops, and fall by their weight. So also +it is forced up in moist ground, and with it many small atoms of the +earth, which are either twisted with the rising water into plants, or +are carried up and down in the air incertainly. But the greatest +quantity of water is forced up from the great South and Indian Seas, +that lie under the tropic of capricorn. And this climate is that which +makes the sun’s perigæum to be always on the winter-solstice. And that +is the part of the terrestrial globe which Kepler says is kind to the +sun; whereas the other part of the globe, which is almost all dry land, +has an antipathy to the sun. And so you see where this magnetical virtue +of the earth lies. For the globe of the earth having no natural appetite +to any place, may be drawn by this motion of the sun a little nearer to +it, together with the water which it raiseth. + +_A._ Can you guess what may be the cause of wind? + +_B._ I think it manifest that the unconstant winds proceed from the +uncertain motion of the clouds ascending and descending, or meeting with +one another. For the winds after they are generated in any place by the +descent of a cloud, they drive other clouds this way and that way before +them, the air seeking to free itself from being pent up in a strait. For +when a cloud descendeth, it makes no wind sensible directly under +itself. But the air between it and the earth is pressed and forced to +move violently outward. For it is a certain experiment of mariners, that +if the sea go high when they are becalmed, they say they shall have more +wind than they would; and take in their sails all but what is necessary +for steering. They know, it seems, that the sea is moved by the descent +of clouds at some distance off: which presseth the water, and makes it +come to them in great waves. For a horizontal wind does but curl the +water. + +_A._ From whence come the rivers? + +_B._ From the rain, or from the falling of snow on the higher ground. +But when it descendeth under ground, the place where it again ariseth is +called the spring. + +_A._ How then can there be a spring upon the top of a hill? + +_B._ There is no spring upon the very top of a hill, unless some natural +pipe bring it thither from a higher hill. + +_A._ Julius Scaliger says, there is a river, and in it a lake, upon the +top of Mount Cenis in Savoy; and will therefore have the springs to be +ingendered in the caverns of the earth by condensation of the air. + +_B._ I wonder he should say that. I have passed over that hill twice +since the time I read that in Scaliger, and found that river as I +passed, and went by the side of it in plain ground almost two miles; +where I saw the water from two great hills, one on one side, the other +on the other, in a thousand small rillets of melting snow fall down into +it. Which has made me never to use any experiment the which I have not +myself seen. As for the conversion of air into water by condensation, +and of water into air by rarefaction, though it be the doctrine of the +Peripatetics, it is a thing incogitable, and the words are +insignificant. For by densum is signified only frequency and closeness +of parts; and by rarum the contrary. As when we say a town is thick with +houses, or a wood with trees, we mean not that one house or tree is +thicker than another, but that the spaces between are not so great. But, +since there is no vacuum, the spaces between the parts of air are no +larger than between the parts of water, or of any thing else. + +_A._ What think you of those things which mariners that have sailed +through the Atlantic Sea, called _spouts_, which pour down water enough +at once to drown a great ship? + +_B._ It is a thing I have not seen: and therefore can say nothing to it; +though I doubt not but when two very large and heavy clouds shall be +driven together by two great and contrary winds, the thing is possible. + +_A._ I think your reason good. And now I will propound to you another +experiment. I have seen an exceeding small tube of glass with both ends +open, set upright in a vessel of water, and that the superficies of the +water within the tube was higher a good deal than of that in the vessel; +but I see no reason for it. + +_B._ Was not part of the glass under water? Must not then the water in +the vessel rise? Must not the air that lay upon it rise with it? Whither +should this rising air go, since there is no place empty to receive it? +It is therefore no wonder if the water, pressed by the substance of the +glass which is dipt into it, do rather rise into a very small pipe, than +come about a longer way into the open air. + +_A._ It is very probable. I observed also that the top of the inclosed +water was a concave superficies; which I never saw in other fluids. + +_B._ The water hath some degree of tenacity, though not so great but +that it will yield a little to the motion of the air; as is manifest in +the bubbles of water, where the concavity is always towards the air. And +this I think the cause why the air and water meeting in the tube make +the superficies towards the air concave, which it cannot do to a fluid +of greater tenacity. + +_A._ If you put into a basin of water a long rag of cloth, first +drenched in water, and let the longer part of it hang out, it is known +by experience, that the water will drop out as long as there is any part +of the other end under water. + +_B._ The cause of it is, that water, as I told you, hath a degree of +tenacity. And therefore being continued in the rag till it be lower +without than within, the weight will make it continue dropping, though +not only because it is heavy (for if the rag lay higher without than +within, and were made heavier by the breadth, it would not descend), but +it is because all heavy bodies naturally descend with proportion of +swiftness duplicate to that of the time; whereof I shall say more when +we talk of gravity. + +_A._ You see how despicable experiments I trouble you with. But I hope +you will pardon me. + +_B._ As for mean and common experiments, I think them a great deal +better witnesses of nature, than those that are forced by fire, and +known but to very few. + + + ========== + + + CHAPTER VI. + OF THE CAUSES AND EFFECTS OF HEAT AND COLD. + +_A._ It is a fine day, and pleasant walking through the fields, but that +the sun is a little too hot. + +_B._ How know you that the sun is hot? + +_A._ I feel it. + +_B._ That is to say, you know that yourself, but not that the sun is +hot. But when you find yourself hot, what body do you feel? + +_A._ None. + +_B._ How then can you infer your heat from the sense of feeling? Your +walking may have made you hot: is motion therefore hot? No. You are to +consider the concomitants of your heat; as, that you are more faint, or +more ruddy, or that you sweat, or feel some endeavour of moisture or +spirits tending outward; and when you have found the causes of those +accidents, you have found the causes of heat, which in a living +creature, and especially in a man, is many times the motion of the parts +within him, such as happen in sickness, anger, and other passions of the +mind; which are not in the sun nor in fire. + +_A._ That which I desire now to know, is what motions and of what bodies +without me are the efficient causes of my heat. + +_B._ I showed you yesterday, in discoursing of rain, how by this +compounded motion of the sun’s body, the air was every way at once +thrust off west and east; so that where it was contiguous, the small +parts of the water were forced to rise, for the avoiding of vacuum. +Think then that your hand were in the place of water so exposed to the +sun. Must not the sun work upon it as it did upon the water? Though it +break not the skin, yet it will give to the inner fluids and looser +parts of your hand, an endeavour to get forth, which will extend the +skin, and in some climates fetch up the blood, and in time make the skin +black. The fire also will do the same to them that often sit with their +naked skins too near it. Nay, one may sit so near, without touching it, +as it shall blister or break the skin, and fetch up both spirits and +blood mixt into a putrid oily matter, sooner than in a furnace oil can +be extracted out of a plant. + +_A._ But if the water be above the fire in a kettle, what then will it +do? Shall the particles of water go toward the fire, as it did toward +the sun? + +_B._ No. For it cannot. But the motion of the parts of the kettle which +are caused by the fire, shall dissipate the water into vapour till it be +all cast out. + +_A._ What is that you call fire? Is it a hard or fluid body? + +_B._ It is not any other body but that of the shining coal; which coal, +though extinguished with water, is still the same body. So also in a +very hot furnace, the hollow spaces between the shining coals, though +they burn that you put into them, are no other body than air moved. + +_A._ Is it not flame? + +_B._ No. For flame is nothing but a multitude of sparks, and sparks are +but the atoms of the fuel dissipated by the incredible swift motion of +the movent, which makes every spark to seem a hundred times greater than +it is, as appears by this; that, when a man swings in the air a small +stick fired at one end, though the motion cannot be very swift, yet the +fire will appear to the eye to be a long, straight, or crooked line. +Therefore a great many sparks together flying upward, must needs appear +unto the sight as one continued flame. Nor are the sparks stricken out +of a flint any thing else but small particles of the stone, which by +their swift motion are made to shine. But that fire is not a substance +of itself, is evident enough by this, that the sun-beams passing through +a globe of water will burn as other fire does. Which beams, if they were +indeed fire, would be quenched in the passage. + +_A._ This is so evident, that I wonder so wise men as Aristotle and his +followers, for so long a time could hold it for an element, and one of +the primary parts of the universe. But the natural heat of a man or +other living creature, whence proceedeth it? Is there anything within +their bodies that hath this compounded motion? + +_B._ At the breaking up of a deer I have seen it plainly in his bowels +as long as they were warm. And it is called the peristaltic motion, and +in the heart of a beast newly taken out of his body; and this motion is +called systole and diastole. But they are both of them this compounded +motion, whereof the former causeth the food to wind up and down through +the guts, and the latter makes the circulation of the blood. + +_A._ What kind of motion is the cause of cold? Methinks it should be +contrary to that which causeth heat. + +_B._ So it is in some respect. For seeing the motion that begets heat, +tendeth to the separation of the parts of the body whereon it acteth, it +stands with reason, that the motion which maketh cold, should be such as +sets them closer together. But contrary motions are, to speak properly, +when upon two ends of a line two bodies move towards each other, the +effect whereof is to make them meet. But each of them, as to this +question, is the same. + +_A._ Do you think (as many philosophers have held and now hold), that +cold is nothing but a privation of heat? + +_B._ No. Have you never heard the fable of the satyr that dwelling with +a husbandman, and seeing him blow his fingers to warm them, and his +pottage to cool it, was so scandalized, that he ran from him, saying he +would no longer dwell with one that could blow both hot and cold with +one breath? Yet the cause is evident enough. For the air which had +gotten a calefactive power from his vital parts, was from his mouth and +throat gently diffused on his fingers, and retained still that power. +But to cool his pottage he straightened the passage at his lips, which +extinguished the calefactive motion. + +_A._ Do you think wind the general cause of cold? If that were true, in +the greatest winds we should have the greatest frosts. + +_B._ I mean not any of those uncertain winds which, I said, were made by +the clouds, but such as a body moved in the air makes to and against +itself; (for it is all one motion of the air whether it be carried +against the body, or the body against it); such a wind as is constant, +if no other be stirring, from east to west; and made by the earth +turning daily upon its own centre; which is so swift, as, except it be +kept off by some hill, to kill a man, as by experience hath been found +by those who have passed over great mountains, and specially over the +Andes which are opposed to the east. And such is the wind which the +earth maketh in the air by her annual motion, which is so swift, as +that, by the calculation of astronomers, to go sixty miles in a minute +of an hour. And therefore this must be the motion which makes it so cold +about the poles of the ecliptic. + +_A._ Does not the earth make the wind as great in one part of the +ecliptic as in another? + +_B._ Yes. But when the sun is in Cancer, it tempers the cold, and still +less and less, but least of all in the winter-solstice, where his beams +are most oblique to the superficies of the earth. + +_A._ I thought the greatest cold had been about the poles of the +equator. + +_B._ And so did I once. But the reason commonly given for it is so +improbable, that I do not think so now. For the cause they render of it +is only, that the motion of the earth is swiftest in the equinoctial, +and slowest about the poles; and consequently, since motion is the cause +of heat, and cold is but, as it was thought, a want of the same, they +inferred that the greatest cold must be about the poles of the +equinoctial. Wherein they miscounted. For not every motion causeth heat, +but this agitation only, which we call compounded motion; though some +have alleged experience for that opinion; as that a bullet out of a gun +will with its own swiftness melt. Which I never shall believe. + +_A._ It is a common thing with many philosophers to maintain their +fancies with any rash report, and sometimes with a lie. But how is it +possible that so soft a substance as water should be turned into so hard +a substance as ice? + +_B._ When the air shaves the globe of the earth with such swiftness, as +that of sixty miles in a minute of an hour, it cannot, where it meets +with still water, but beat it up into small and undistinguishable +bubbles, and involve itself in them as in so many bladders or skins of +water. And ice is nothing else but the smallest imaginable parts of air +and water mixed; which is made hard by this compounded motion, that +keeps the parts so close together, as not to be separated in one place +without disordering the motion of them all. For when a body will not +easily yield to the impression of an external movent in one place +without yielding in all, we call it hard; and when it does, we say it is +soft. + +_A._ Why is not ice as well made in a moved as in a still water? Are +there not great seas of ice in the northern parts of the earth? + +_B._ Yes, and perhaps also in the southern parts. But I cannot imagine +how ice can be made in such agitation as is always in the open sea, made +by the tides and by the winds. But how it may be made at the shore, it +is not hard to imagine. For in a river or current, though swift, the +water that adhereth to the banks is quiet, and easily by the motion of +the air driven into small insensible bubbles; and so may the water that +adhereth to those bubbles, and so forwards till it come into a stream +that breaks it, and then it is no wonder though the fragments be driven +into the open sea, and freeze together into greater lumps. But when in +the open sea, or at the shore, the tide or a great wave shall arise, +this young and tender ice will presently be washed away. And therefore I +think it evident, that as in the Thames the ice is first made at the +banks where the tide is weak or none, and, broken by the stream, comes +down to London, and part goes to the sea floating till it dissolve, and +part, being too great to pass the bridge, stoppeth there and sustains +that which follows, till the river be quite frozen over; so also the ice +in the northern seas begins first at the banks of the continent and +islands which are situated in that climate, and then broken off, are +carried up and down, and one against another, till they become great +bodies. + +_A._ But what if there be islands, and narrow inlets of the sea, or +rivers also about the pole of the equinoctial? + +_B._ If there be, it is very likely the sea may also there be covered +all over with ice. But for the truth of this, we must stay for some +farther discovery. + +_A._ When the ice is once made and hard, what dissolves it? + +_B._ The principal cause of it, is the weight of the water itself; but +not without some abatement in the stream of the air that hardeneth it; +as when the sunbeams are less oblique to the earth, or some contrary +wind resisteth the stream of the air. For when the impediment is +removed, then the nature of the water only worketh, and, being a heavy +body, downward. + +_A._ I forgot to ask you, why two pieces of wood rubbed swiftly one +against another, will at length set on fire. + +_B._ Not only at length, but quickly, if the wood be dry. And the cause +is evident, viz. the compounded motion which dissipates the external +small parts of the wood. And then the inner parts must of necessity, to +preserve the plentitude of the universe, come after; first the most +fluid, and then those also of greater consistence, which are first +erected, and the motion continued, made to fly swiftly out; whereby the +air driven to the eye of the beholder, maketh that fancy which is called +light. + +_A._ Yes; I remember you told me before, that upon any strong pressure +of the eye, the resistance from within would appear a light. But to +return to the enquiry of heat and cold, there be two things that beyond +all other put me into admiration. One is the swiftness of kindling in +gunpowder. The other is the freezing of water in a vessel, though not +far from the fire, set about with other water with ice and snow in it. +When paper or flax is flaming, the flame creeps gently on; and if a +house full of paper were to be burnt with putting a candle to it, it +will be long in burning; whereas a spark of fire would set on flame a +mountain of gunpowder in almost an instant. + +_B._ Know you not gunpowder is made of the powder of charcoal, +brimstone, and saltpetre? Whereof the first will kindle with a spark, +the second flame as soon as touched with fire; and the third blows it, +as being composed of many orbs of salt filled with air, and as it +dissolveth in the flame, furiously blowing increaseth it. And as for +making ice by the fireside; it is manifest that whilst the snow is +dissolving in the external vessel, the air must in the like manner break +forth, and shave the superficies of the inner vessel, and work through +the water till it be frozen. + +_A._ I could easily assent to this, if I could conceive how the air that +shaves, as you say, the outside of the vessel, could work through it. I +conceive well enough a pail of water with ice or snow dissolving in it, +and how it causeth wind. But how that wind should communicate itself +through the vessel of wood or metal, so as to make it shave the +superficies of the water which is within it, I do not so well +understand. + +_B._ I do not say the inner superficies of the vessel shaves the water +within it. But it is manifest that the wind made in the pail of water by +the melting snow or ice presseth the sides of the vessel that standeth +in it; and that the pressure worketh clean through, how hard soever the +vessel be; and that again worketh on the water within, by restitution of +its parts, and so hardeneth the water by degrees. + +_A._ I understand you now. The ice in the pail by its dissolution +transfers its hardness to the water within. + +_B._ You are merry. But supposing, as I do, that the ice in the pail is +more than the water in the vessel, you will find no absurdity in the +argument. Besides, the experiment, you know, is common. + +_A._ I confess it is probable. The Greeks have the word φρίκη (whence +the Latins have their word _frigus_) to signify the curling of the water +by the wind; and use the same also for horror, which is the passion of +one that cometh suddenly into a cold air, or is put into a sudden +affright, whereby he shrinks, and his hair stands upright. Which +manifestly shows that the motion which causeth cold, is that which +pressing the superficies of a body, sets the parts of it closer +together. But to proceed in my queries. Monsieur Des Cartes, whom you +know, hath written somewhere, that the noise we hear in thunder, +proceeds from breaking of the ice in the clouds; what think you of it? +Can a cloud be turned into ice? + +_B._ Why not? A cloud is but water in the air? + +_A._ But how? For he has not told us that. + +_B._ You know that it is only in summer, and in hot weather, that it +thunders; or if in winter, it is taken for a prodigy. You know also, +that of clouds, some are higher, some lower, and many in number, as you +cannot but have oftentimes observed, with spaces between them. +Therefore, as in all currents of water, the water is there swiftest +where it is straitened with islands, so must the current of air made by +the annual motion be swiftest there, where it is checked with many +clouds, through which it must, as it were, be strained, and leave behind +it many small particles of earth always in it, and in hot weather more +than ordinary. + +_A._ This I understand, and that it may cause ice. But when the ice is +made, how is it broken? And why falls it not down in shivers? + +_B._ The particles are enclosed in small caverns of the ice; and their +natural motion being the same which we have ascribed to the globe of the +earth, requires a sufficient space to move in. But when it is imprisoned +in a less room than that, then a great part of the ice breaks: and this +is the thunder-clap. The murmur following is from the settling of the +air. The lightning is the fancy made by the recoiling of the air against +the eye. The fall is in rain, not in shivers; because the prisons which +they break are extreme narrow, and the shivers being small, are +dissolved by the heat. But in less heat they would fall in drops of +hail, that is to say, half frozen by the shaving of the air as they +fall, and be in a very little time, much less than snow or ice, +dissolved. + +_A._ Will not that lightning burn? + +_B._ No. But it hath often killed men with cold. But this extraordinary +swiftness of lightning consisteth not in the expansion of the air, but +in a straight and direct stream from where it breaks forth; which is in +many places successively, according to the motion of the cloud. + +_A._ Experience tells us that. I have now done with my problems +concerning the great bodies of the world, the stars, and element of air +in which they are moved, and am therein satisfied, and the rather, +because you have answered me by the supposition of one only motion, and +commonly known, and the same with that of Copernicus, whose opinion is +received by all the learned; and because you have not used any of these +empty terms, sympathy, antipathy, antiperistasis, etc., for a natural +cause, as the old philosophers have done to save their credit. For +though they were many of them wise men, as Plato, Aristotle, Seneca, and +others, and have written excellently of morals and politics, yet there +is very little natural philosophy to be gathered out of their writings. + +_B._ Their ethics and politics are pleasant reading, but I find not any +argument in their discourses of justice or virtue drawn from the supreme +authority, on whose laws all justice, virtue, and good politics depend. + +_A._ Concerning this cover, or, as some have called it, the scurf or +scab of the terrestrial star, I will begin with you tomorrow. For it is +a large subject, containing animals, vegetables, metals, stones, and +many other kinds of bodies, the knowledge whereof is desired by most +men, and of the greatest and most general profit. + +_B._ And this is it, in which I shall give you the least satisfaction; +so great is the variety of motion, and so concealed from human senses. + + + ========== + + + CHAPTER VII. + OF HARD AND SOFT, AND OF THE ATOMS THAT FLY IN + THE AIR. + +_A._ Concerning this cover of the earth, made up of an infinite number +of parts of different natures, I had much ado to find any tolerable +method of enquiry. But I resolved at last to begin with the questions +concerning hard and soft, and what kind of motion it is that makes them +so. I know that in any pulsion of air, the parts of it go innumerable +and inexplicable ways; but I ask only if every point of it be moved? + +_B._ No. If you mean a mathematical point, you know it is impossible. +For nothing is movable but body. But I suppose it divisible, as all +other bodies, into parts divisible. For no substance can be divided into +nothings. + +_A._ Why may not that substance within our bodies, which are called +animal spirits, be another kind of body, and more subtile than the +common air? + +_B._ I know not why, no more than you or any man else knows why it is +not very air, though purer perhaps than the common air, as being +strained through the blood into the brain and nerves. But howsoever that +be, there is no doubt, but the least parts of the common air, +respectively to the whole, will easilier pierce, with equal motion, the +body that resisteth them, than the least parts of water. For it is by +motion only that any mutation is made in any thing; and all things +standing as they did, will appear as they did. And that which changeth +soft into hard, must be such as makes the parts not easily to be moved +without being moved all together; which cannot be done but by some +motion compounded. And we call hard, that whereof no part can be put out +of order without disordering all the rest; which is not easily done. + +_A._ How water and air beaten into extreme small bubbles is hardened +into ice, you have told me already, and I understand it. But how a soft +homogeneous body, as air or water, should be so hardened, I cannot +imagine. + +_B._ There is no hard body that hath not also some degree of gravity; +and consequently, being loose, there must be some efficient cause, that +is, some motion, when it is severed from the earth, to bring the same to +it again. And seeing this compounded motion gives to the air and water +an endeavour from the earth, the motion which must hinder it, must be in +a way contrary to the compounded motion of the earth. For whatsoever, +having been asunder, comes together again, must come contrary ways, as +those that follow one another go the same way, though both move upon the +same line. + +_A._ What experiment have you seen to this purpose? + +_B._ I have seen a drop of glass like that of the second figure, newly +taken out of the furnace, and hanging at the end of an iron rod, and yet +fluid, and let fall into the water and hardened. The club-end of it A A +coming first to the water, the tail B C following it. It is proved +before, that the motion that makes it is a compounded motion, and gives +an endeavour outward to every part of it; and that the motion which +maketh cold, is such as shaving the body in every point of contact, and +turning it, gives them all an endeavour inward. Such is this motion made +by the sinking of the hot and fluid glass into the water. It is +therefore manifest that the motion which hardeneth a soft body, must in +every point of contact be in the contrary way to that which makes a hard +body soft. And farther, that slender tail B C shall be made much more +hard than common glass. For towards the upper end, in C, you cannot +easily break it, as small as it is. And when you have broken it, the +whole body will fall into dust, as it must do, seeing the bending is so +difficult. For all the parts are bent with such force, that upon the +breaking at D, by their sudden restitution to their liberty, they will +break together. And the cause why the tail B C, being so slender, +becomes so hard, is, that all the endeavour in the great part A B, is +propagated to the small part B C, in the same manner as the force of the +sun-beams is derived almost to a point by a burning-glass. But the cause +why, when it is broken in D, it breaks also in so many other places, is, +that the endeavour in all the other parts, which is called the spring, +unbends it; from whence a motion is caused the contrary way, and that +motion continued bends it more the other way and breaks it, as a bow +over-bent is broken into shivers by a sudden breaking of the string. + +_A._ I conceive now how a body which having been hard and softened +again, may be rehardened; but how a fluid and mere homogeneous body, as +air or water, may be so, I see not yet. For the hardening of water is +making a hard body of two fluids, whereof one, which is the water, hath +some tenacity; and so a man may make a bladder hard with blowing into +it. + +_B._ As for mere air, which hath no natural motion of itself, but is +moved only by other bodies of a greater consistence, I think it +impossible to be hardened. For the parts of it so easily change places, +that they can never be fixed by any motion. No more I think can water, +which though somewhat less fluid, is with an insensible force very +easily broken. + +_A._ It is the opinion of many learned men, that ice, in long time, will +be turned into crystal; and they allege experience for it. For they say +that crystal is found hanging on high rocks in the Alps, like icicles on +the eaves of a house; and why may not that have formerly been ice, and +in many years have lost the power of being reduced? + +_B._ If that were so, it would still be ice, though also crystal: which +cannot be, because crystal is heavier than water, and therefore much +heavier than ice. + +_A._ Is there then no transubstantiation of bodies but by mixture? + +_B._ Mixture is no transubstantiation. + +_A._ Have you never seen a stone that seemed to have been formerly wood, +and some like shells, and some like serpents, and others like other +things? + +_B._ Yes. I have seen such things, and particularly I saw at Rome, in a +stone-cutter’s workhouse, a billet of wood, as I thought it, partly +covered with bark, and partly with the grain bare, as long as a man’s +arm, and as thick as the calf of a man’s leg; which handling I found +extreme heavy, and saw a small part of it which was polished, and had a +very fine gloss, and thought it a substance between stone and metal, but +nearest to stone. I have seen also a kind of slate painted naturally +with forest-work. And I have seen in the hands of a chemist of my +acquaintance at Paris, a broken glass, part of a retort, in which had +been the rosin of turpentine, wherein though there were left no rosin, +yet there appeared in the piece of glass many trees; and plants in the +ground about them, such as grow in woods; and better designed than they +could be done by any painter; and continued so for a long time. These be +great wonders of nature, but I will not undertake to show their causes. +But yet this is most certain, that nothing can make a hard body of a +soft, but by some motion of its parts. For the parts of the hardest body +in the world can be no closer together than to touch; and so close are +the parts of air and water, and consequently they should be equally +hard, if their smallest parts had not different natural motions. +Therefore if you ask me the causes of these effects, I answer, they are +different motions. But if you expect from me how and by what motions, I +shall fail you. For there is no kind of substance in the world now, that +was not at the first creation, when the Creator gave to all things what +natural and special motion he thought good. And as he made some bodies +wondrous great, so he made others wondrous little. For all his works are +wondrous. Man can but guess, nor guess farther, than he hath knowledge +of the variety of motion. I am therefore of opinion, that whatsoever +perfectly homogeneous is hard, consisteth of the smallest parts, or, as +some call them, atoms, that were made hard in the beginning, and +consequently by an eternal cause; and that the hardness of the whole +body is caused only by the contact of the parts by pressure. + +_A._ What motion is it that maketh a hard body to melt? + +_B._ The same compounded motion that heats, namely, that of fire, if it +be strong enough. For all motion compounded is an endeavour to +dissipate, as I have said before, the parts of the body to be moved by +it. If therefore hardness consist only in the pressing contact of the +least parts, this motion will make the same parts slide off from one +another, and the whole to take such a figure as the weight of the parts +shall dispose them to, as in lead, iron, gold, and other things melted +with heat. But if the small parts have such figures as they cannot +exactly touch, but must leave spaces between them filled with air or +other fluids, then this motion of the fire, will dissipate those parts +some one way, some another, the hard part still hard; as in the burning +of wood or stone into ashes or lime. For this motion is that which +maketh fermentation, scattering dissimilar parts, and congregating +similar. + +_A._ Why do some hard bodies resist breaking more one way than another? + +_B._ The bodies that do so, are for the most part wood, and receive that +quality from their generation. For the heat of the sun in the +spring-time draweth up the moisture at the root, and together with it +the small parts of the earth, and twisteth it into a small twig, by its +motion upwards, to some length, but to very little other dimensions, and +so leaves it to dry till the spring following; and then does the same to +that, and to every small part round about it; so that upward the +strength is doubled, and the next year trebled, &c. And these are called +the grain of the wood, and but touch one another, like sticks with +little or no binding, and therefore can hardly be broken across the +grain, but easily all-along it. Also some other hard bodies have this +quality of being more fragile one way than another, as we see in +quarrels of a glass window, that are aptest many times to break in some +crooked line. The cause of this may be, that when the glass, hot from +the furnace, is poured out upon a plain, any small stones in or under it +will break the stream of it into divers lines, and not only weaken it, +but also cause it falsely to represent the object you look on through +it. + +_A._ What is the cause why a bow of wood or steel, or other very hard +body, being bent, but not broken, will recover its former degree of +straightness? + +_B._ I have told you already, how the smallest parts of a hard body have +every one, by the generation of hardness, a circular, or other +compounded motion; such motion is that of the smallest parts of the bow. +Which circles in the bending you press into narrower figures, as a +circle into an ellipsis, and an ellipsis into a narrower but longer +ellipsis with violence; which turns their natural motion against the +outward parts of the bow so bent, and is an endeavour to stretch the bow +into its former posture. Therefore if the impediment be removed, the bow +must needs recover its former figure. + +_A._ It is manifest; and the cause can be no other but that, except the +bow have sense. + +_B._ And though the bow had sense, and appetite to boot, the cause will +be still the same. + +_A._ Do you think air and water to be pure and homogeneous bodies? + +_B._ Yes, and many bodies both hard and heavy to be so too, and many +liquors also besides water. + +_A._ Why then do men say they find one air healthy, another infectious? + +_B._ Not because the nature of the air varies, but because there are in +the air, drawn, or rather, beaten up by the sun, many little bodies, +whereof some have such motion as is healthful, others such as is hurtful +to the life of man. For the sun, as you see in the generation of plants, +can fetch up earth as well as water: and from the driest ground any kind +of body that lieth loose, so it be small enough, rather than admit any +emptiness. By some of these small bodies it is that we live; which being +taken in with our breath, pass into our blood, and cause it, by their +compounded motion, to circulate through the veins and arteries; which +the blood of itself, being a heavy body, without it cannot do. What kind +of substance these atoms are, I cannot tell. Some suppose them to be +nitre. As for those infectious creatures in the air, whereof so many die +in the plague, I have heard that Monsieur Des Cartes, a very ingenious +man, was of opinion, that they were little flies. But what grounds he +had for it, I know not, though there be many experiments that invite me +to believe it. For first, we know that the air is never universally +infected over a whole country, but only in or near to some populous +town. And therefore the cause must also be partly ascribed to the +multitude thronged together, and constrained to carry their excrements +into the fields round about and near to their habitation, which in time +fermenting breed worms, which commonly in a month or little more, +naturally become flies; and though engendered at one town, may fly to +another. Secondly, in the beginning of a plague, those that dwell in the +suburbs, that is to say, nearest to this corruption, are the poorest of +the people, that are nourished for the most part with the roots and +herbs which grow in that corrupted dirt; so that the same filth makes +both the blood of poor people, and the substance of the fly. And it is +said by Aristotle, that everything is nourished by the matter whereof it +is generated. Thirdly, when a town is infected, the gentlemen, and those +that live on wholsomest food, scarce one of five hundred die of the +plague. It seems therefore, whatsoever creatures they be that invade us +from the air, they can discern their proper nourishment, and do not +enter into the mouth and nostrils with the breath of every man alike, as +they would do if they were inanimate. Fourthly, a man may carry the +infection with him a great way into the country in his clothes, and +infect a village. Shall another man there draw the infection from the +clothes only by his breath? Or from the hangings of a chamber wherein a +man hath died? It is impossible. Therefore whatsoever killing thing is +in the clothes or hangings, it must rise and go into his mouth or +nostrils before it can do him hurt. It must therefore be a fly, whereof +great numbers get into the blood, and there feeding and breeding worms, +obstruct the circulation of blood, and kill the man. + +_A._ I would we knew the palate of those little animals; we might +perhaps find some medicine to fright them from mingling with our breath. +But what is that which kills men that lie asleep too near a +charcoal-fire? Is it another kind of fly? Or is charcoal venomous? + +_B._ It is neither fly nor venom, but the effect of a flameless glowing +fire, which dissipates those atoms that maintain the circulation of the +blood; so that for want of it, by degrees they faint, and being asleep +cannot remove, but in short time, there sleeping die; as is evident by +this, that being brought into the open air, without other help, they +recover. + +_A._ It is very likely. The next thing I would be informed of, is the +nature of gravity. But for that, if you please, we will take another +day. + + + ========== + + + CHAPTER VIII. + OF GRAVITY AND GRAVITATION. + +_B._ What books are those? + +_A._ Two books written by two learned men concerning gravity. I brought +them with me, because they furnish me with some material questions about +that doctrine; though of the nature of gravity, I find no more in either +of them than this, that gravity is an intrinsical quality, by which a +body so qualified descendeth perpendicularly towards the superficies of +the earth. + +_B._ Did neither of them consider that descending is local motion; that +they might have called it an intrinsical motion rather than an +intrinsical quality? + +_A._ Yes. But not how motion should be intrinsical to the special +individual body moved. For how should they, when you are the first that +ever sought the differences of qualities in local motion, except your +authority in philosophy were greater with them than it is? For it is +hard for a man to conceive, except he see it, how there should be motion +within a body, otherwise than as it is in living creatures. + +_B._ But it may be they never sought, or despaired of finding what +natural motion could make any inanimate thing tend one way rather than +another. + +_A._ So it seems. But the first of them inquires no farther than, why so +much water, being a heavy body, as lies perpendicularly on a fish’s back +in the bottom of the sea, should not kill it. The other, whereof the +author is Dr. Wallis, treateth universally of gravity. + +_B._ Well; but what are the questions which from these books you intend +to ask me? + +_A._ The author of the first book tells me, that water and other fluids +are bodies continued, and act, as to gravity, as a piece of ice would do +of the same figure and quantity. Is that true? + +_B._ That the universe, supposing there is no place empty, is one entire +body, and also, as he saith it is, a continual body, is very true. And +yet the parts thereof may be contiguous, without any other cohesion but +touch. And it is also true, that a vessel of water will descend in a +medium less heavy, but fluid, as ice would do. + +_A._ But he means that water in a tub would have the same effect upon a +fish in the bottom of the tub, as so much ice would have. + +_B._ That also would be true, if the water were frozen to the sides of +it. Otherwise the ice, if there be enough, will crush the fish to death. +But how applies he this, to prove that the water cannot hurt a fish in +the sea by its weight? + +_A._ It plainly appears that water does not gravitate on any part of +itself beneath it. + +_B._ It appears by experience, but not by this argument, though instead +of water the tub were filled with quicksilver. + +_A._ I thought so. But how it comes to pass that the fish remains +uncrushed, I cannot tell. + +_B._ The endeavour of the quicksilver downward is stopped by the +resistance of the hard bottom. But all resistance is a contrary +endeavour; that is, an endeavour upwards, which gives the like endeavour +to the quicksilver, which is also heavy, and thereby the endeavour of +the quicksilver is diverted to the sides round about, where stopped +again by the resistance of the sides, it receives an endeavour upwards, +which carries the fish to the top, lying all the way upon a soft bed of +quicksilver. This is the true manner how the fish is saved harmless. But +your author, I believe, either wanted age, or had too much business, to +study the doctrine of motion; and never considered that resistance is +not an impediment only, or privation, but a contrary motion; and that +when a man claps two pieces of wax together, their contrary endeavour +will turn both the pieces into one cake of wax. + +_A._ I know not the author; but it seems he has deeplier considered this +question than other men; for in the introduction to his book he saith, +“that men have pre-engaged themselves to maintain certain principles of +their own invention, and are therefore unwilling to receive anything +that may render their labour fruitless;” and, “that they have not +strictly enough considered the several interventions that abate, impede, +advance, or direct the gravitation of bodies.” + +_B._ This is true enough; and he himself is one of those men, in that he +considered not, that resistance is one of those interventions which +abate, impede, and direct gravitation. But what are his suppositions for +the questions he handles? + +_A._ His first is, that as in a pyramid of brick, wherein the bricks are +so joined that the uppermost lies everywhere over the joint or cement of +the two next below it, you may break down a part and leave a cavity, and +yet the bricks above will stand firm and sustain one another by their +cross posture: so also it is in wheat, hailshot, sand, or water; and so +they arch themselves, and thereby the fish is every way secured by an +arch of water over it. + +_B._ That the cause why fishes are not crushed nor hurt in the bottom of +the sea by the weight of the water, is the water’s arching itself, is +very manifest. For if the uppermost orb of the water should descend by +its gravity, it would tend toward the centre of the earth, and place +itself all the way in a less and lesser orb, which is impossible. For +the places of the same body are always equal. But that wheat, sand, +hailshot, or loose stones should make a firm arch, is not credible. + +_A._ The author therefore, it seems, quits it, and taketh a second +hypothesis for the true cause, though the former, he saith, be not +useless, but contributes its part to it. + +_B._ I see, though he depart from his hypothesis, he looks back upon it +with some kindness. What is his second hypothesis? + +_A._ It is, that air and water have an endeavour to motion upward, +downward, directly, obliquely, and every way. For air, he saith, will +come down his chimney, and in at his door, and up his stairs. + +_B._ Yes, and mine too; and so would water, if I dwelt under water, +rather than admit of vacuum. But what of that? + +_A._ Why then it would follow, that those several tendencies or +endeavours would so abate, impede, and correct one another, as none of +them should gravitate. Which being granted, the fish can take no harm; +wherein I find one difficulty, which is this: the water having an +endeavour to motion every way at once, methinks it should go no way, but +lie at rest; which, he saith, was the opinion of Stevinus, and rejecteth +it, saying, it would crush the fish into pieces. + +_B._ I think the water in this case would neither rest nor crush. For +the endeavour being, as he saith, intrinsical, and every way, must needs +drive the water perpetually outward; that is to say, as to this +question, upwards; and seeing the same endeavour in one individual body +cannot be more ways at once than one, it will carry it on perpetually +without limit, beyond the fixed stars; and so we shall never more have +rain. + +_A._ As ridiculous as it is, it necessarily follows. + +_B._ What are Dr. Wallis’s suppositions? + +_A._ He goes upon experiments. And, first, he allegeth this, that water +left to itself without disturbance, does naturally settle itself into a +horizontal plane. + +_B._ He does not then, as your author and all other men, take gravity +for that quality whereby a body tendeth to the centre of the earth. + +_A._ Yes, he defines gravity to be a natural propension towards the +centre of the earth. + +_B._ Then he contradicteth himself. For if all heavy bodies tend +naturally to one centre, they shall never settle in a plane, but in a +spherical superficies. But against this, that such an horizontal plane +is found in water by experience, I say it is impossible. For the +experiment cannot be made in a basin, but in half a mile at sea +experience visibly shows the contrary. According to this, he should +think also that a pair of scales should hang parallel. + +_A._ He thinks that too. + +_B._ Let us then leave this experiment. What says he farther concerning +gravity? + +_A._ He takes for granted, not as an experiment but an axiom, that +nature worketh not by election, but _ad ultimum virium_, with all the +power it can. + +_B._ I think he means, (for it is a very obscure passage), that every +inanimate body by nature worketh all it can without election; which may +be true. But it is certain that men, and beasts, work often by election, +and often without election; as when he goes by election, and falls +without it. In this sense I grant him, that nature does all it can. But +what infers he from it? + +_A._ That naturally every body has every way, if the ways oppose not one +another, an endeavour to motion; and consequently, that if a vessel have +two holes, one at the side, another at the bottom, the water will run +out at both. + +_B._ Does he think the body of water that runs out at the side, and that +which runs out at the bottom, is but one and the same body of water? + +_A._ No, sure; he cannot think but that they are two several parts of +the whole water in the vessel. + +_B._ What wonder is it then, if two parts of water run two ways at once, +or a thousand parts a thousand ways? Does it follow thence that one body +can go more than one way at once? Why is he still meddling with things +of such difficulty? He will find at last that he has not a genius either +for natural philosophy or for geometry. What other suppositions has he? + +_A._ My first author had affirmed, that a lighter body does not +gravitate on a heavier; against this Dr. Wallis thus argueth: Let there +be a siphon, A B C D, filled with quicksilver to the level A D; if then +you pour oil upon A as high as to E, he asketh if the oil in A E, as +being heavy, shall not press down the quicksilver a little at A, and +make it rise a little at D, suppose to F; and answers himself, that +certainly it will; so that it is neither an experiment nor an +hypothesis, but only his opinion. + +_B._ Whatsoever it be, it is not true; though the doctor may be +pardoned, because the contrary was never proved. + +_A._ Can you prove the contrary? + +_B._ Yes; for the endeavour of the quicksilver both from A and D +downward, is stronger than that of the oil downward. If, therefore, the +endeavour of the quicksilver were not resisted by the bottom B C, it +would fall so, by reason of the acceleration of heavy bodies in their +descending, as to leave the oil, so that it should not only not press, +but also not touch the quicksilver. It is true, in a pair of scales +equally charged with quicksilver, that the addition of a little oil to +either scale will make it preponderate. And that was it deceived him. + +_A._ It is evident. The last experiment he cites is the weighing of air +in a pair of scales, where it is found manifestly that it has some +little weight. For if you weigh a bladder, and put the weight into one +scale, and then blow the bladder full of air, and put it into the other +scale, the full bladder will outweigh the empty. Must not then the air +gravitate? + +_B._ It does not follow. I have seen the experiment just as you describe +it, but it can never be thence demonstrated that air has any weight. For +as much air as is pressed downward by the weight of the blown bladder, +so much will rise from below, and lay itself spherically at the altitude +of the centre of gravity of the bladder so blown. So that all the air +within the bladder above that centre is carried thither imprisoned, and +by violence: and the force that carries it up is equal to that which +presseth it down. There must, therefore, be allowed some little +counterpoise in the other scale to balance it. Therefore, the experiment +proves nothing to his purpose. And whereas they say there be small heavy +bodies in the air, which make it gravitate, do they think the force +which brought them thither cannot hold them there? + +_A._ I leave this question of the fish as clearly resolved, because the +water tending every way to one point, which is the centre of the earth, +must of necessity arch itself. And now tell me your own opinion +concerning the cause of gravity, and why all bodies descend or ascend +not all alike. For there can be no more matter in one place than another +if the places be equal. + +_B._ I have already showed you in general, that the difference of motion +in the parts of several bodies makes the difference of their natures. +And all the difference of motions consisteth either in swiftness, or in +the way, or in the duration. But to tell you in special why gold is +heaviest, and then quicksilver, and then, perhaps, lead, is more than I +hope to know, or mean to enquire; for I doubt not but that the species +of heavy, hard, opaque, and diaphanous, were all made so at their +creation, and at the same time separated from different species. So that +I cannot guess at any particular motions that should constitute their +natures, farther than I am guided by the experiments made by fire or +mixture. + +_A._ You hope not then to make gold by art? + +_B._ No, unless I could make one and the same thing heavier than it was. +God hath from the beginning made all the kinds of hard, and heavy, and +diaphanous bodies that are, and of such figure and magnitude as he +thought fit; but how small soever, they may by accretion become greater +in the mine, or perhaps by generation, though we know not how. But that +gold, by the art of man, should be made of not gold, I cannot +understand; nor can they that pretend to show how. For the heaviest of +all bodies, by what mixture soever of other bodies, will be made +lighter, and not to be received for gold. + +_A._ Why, when the cause of gravity consisteth in motion, should you +despair of finding it? + +_B._ It is certain that when any two bodies meet, as the earth and any +heavy body will, the motion that brings them to or towards one another, +must be upon two contrary ways; and so also it is when two bodies press +each other in order to make them hard; so that one contrariety of motion +might cause both hard and heavy, but it doth not, for the hardest bodies +are not always the heaviest; therefore I find no access that way to +compare the causes of different endeavours of heavy bodies to descend. + +_A._ But show me at least how any heavy body that is once above in the +air, can descend to the earth, when there is no visible movent to thrust +or pull it down. + +_B._ It is already granted, that the earth hath this compounded motion +supposed by Copernicus, and that thereby it casteth the contiguous air +from itself every way round about. Which air so cast off, must +continually, by its nature, range itself in a spherical orb. Suppose a +stone, for instance, were taken up from the ground, and held up in the +air by a man’s hand, what shall come into the place it filled when it +lay upon the earth? + +_A._ So much air as is equal to the stone in magnitude, must descend and +place itself in an orb upon the earth. But then I see that to avoid +vacuum, another orb of air of the same magnitude must descend, and place +itself in that, and so perpetually to the man’s hand; and then so much +air as would fill the place must descend in the same manner, and bring +the stone down with it. For the stone having no endeavour upward, the +least motion of the air, the hand being removed, will thrust it +downward. + +_B._ It is just so. And farther, the motion of the stone downward shall +continually be accelerated according to the odd numbers from unity; as +you know hath been demonstrated by Galileo. But we are nothing the +nearer, by this, to the knowledge of why one body should have a greater +endeavour downward than another. You see the cause of gravity is this +compounded motion with exclusion of vacuum. + +_A._ It may be it is the figure that makes the difference. For though +figure be not motion, yet it may facilitate motion, as you see commonly +the breadth of a heavy body retardeth the sinking of it. And the cause +of it is, that it makes the air have farther to go laterally, before it +can rise from under it. For suppose a body of quicksilver falling in the +air from a certain height, must it not, going as it does toward the +centre of the earth, as it draws nearer and nearer to the earth, become +more and more slender, in the form of a solid sector? And if it have far +to go, divide itself into drops? This figure of a solid sector is like a +needle with the point downward, and therefore I should think that +facilitating the motion of it does the same that would be done by +increasing the endeavour. + +_B._ Do not you see that this way of facilitating is the same in water, +and in all other fluid heavy bodies? Besides, your argument ought to be +applicable to the weighing of bodies in a pair of scales, which it is +not, for there they have no such figure; it should also hold in the +comparison of gravity in hard and fluid bodies. + +_A._ I had not sufficiently considered it. But supposing now, as you do, +that both heavy and hard bodies, in their smallest parts, were made so +in the creation; yet, because quicksilver is harder than water, a drop +of water shall in descending be pressed into a more slender sector than +a drop of quicksilver, and consequently the earth shall more easily cast +off any quantity of water than the same quantity of quicksilver. + +_B._ This one would think were true; as also that of simple fluid +bodies, those whose smallest parts, naturally, without the force of +fire, do strongliest cohere, are generally the heaviest. But why then +should quicksilver be heavier than stone or steel? Fluidity and hardness +are but degrees between greater fluidity and greater hardness. Therefore +to the knowledge of what it is that causeth the difference, in different +bodies, of their endeavour downward, there are required, if it can be +known at all, a great many more experiments than have been yet made. It +is not difficult to find why water is heavier than ice, or other body +mixed of air and water. But to believe that all bodies are heavier or +lighter according to the quantity of air within them, is very hard. + +_A._ I see by this, that the Creator of the world, as by his power he +ordered it, so by his wisdom he provided it should be never disordered. +Therefore leaving this question, I desire to know whether if a heavy +body were as high as a fixed star, it would return to the earth. + +_B._ It is hard to try. But if there be this compounded motion in the +great bodies so high, such as is in the earth, it is very likely that +some heavy bodies will be carried to them. But we shall never know it +till we be at the like height. + +_A._ What think you is the reason why a drop of water, though heavy, +will stand upon a horizontal plane of dry or unctuous wood, and not +spread itself upon it? For let A B, in the sixth figure, be the dry +plane, D the drop of water, and D C perpendicular to A B. The drop D, +though higher, will not descend and spread itself upon it. + +_B._ The reason I think is manifest. For those bodies which are made by +beating of water and air together, show plainly that the parts of water +have a great degree of cohesion. For the skin of the bubble is water, +and yet it can keep the air, though moved, from getting out. Therefore +the whole drop of water at D, hath a good deal of cohesion of parts. And +seeing A B is an horizontal plane, the way from the contact in D either +to A or B is upwards, and consequently there is no endeavour in D either +of those ways, but what proceeds from so much weight of water as is able +to break that cohesion, which so small a drop is too weak to do. But the +cohesion being once broken, as with your finger, the water will follow. + +_A._ Seeing the descent of a heavy body increaseth according to the odd +numbers 1, 3, 5, 7, &c. and the aggregates of those numbers, viz. of 1 +and 3; and 1 and 3 and 5; and of 1 and 3 and 5 and 7, &c. are square +numbers, namely 4, 9, 16; the whole swiftness of the descent will be, I +think, to the aggregate of so many swiftnesses equal to the first +endeavour, as square numbers are to their sides, 1, 2, 3, 4. Is it so? + +_B._ Yes, you know it hath been demonstrated by Galileo. + +_A._ Then if, for instance, you put into a pair of scales equal +quantities of quicksilver and water, seeing they are both accelerated in +the same proportion, why should not the weight of quicksilver to the +weight of water be in duplicate proportions to their first endeavours? + +_B._ Because they are in a pair of scales. For there the motion of +neither of them is accelerated. And therefore it will be, as the first +endeavour of the quicksilver to the first endeavour of the water, so the +whole weight to the whole weight. By which you may see, that the cause +which takes away the gravitation of liquid bodies from fish or other +lighter bodies within them, can never be derived from the weight. + +_A._ I have one question more to ask concerning gravity. If gravity be, +as some define it, an intrinsical quality, whereby a body descendeth +towards the centre of the earth, how is it possible that a piece of iron +that hath this intrinsical quality should rise from the earth, to go to +a loadstone? Hath it also an intrinsical quality to go from the earth? +It cannot be. The cause therefore must be extrinsical. And because when +they are come together in the air, if you leave them to their own +nature, they will fall down together, they must also have some like +extrinsical cause. And so this magnetic virtue will be such another +virtue as makes all heavy bodies to descend, in this our world, to the +earth. If therefore you can from this your hypothesis of compounded +motion, by which you have so probably salved the problem of gravity, +salve also this of the loadstone, I shall acknowledge both your +hypothesis to be true, and your conclusion to be well deduced. + +_B._ I think it not impossible. But I will proceed no farther in it now, +than, for the facilitating of the demonstrations, to tell you the +several proprieties of the magnet, whereof I am to show the causes. As +first, that iron, and no other body, at some little distance, though +heavy, will rise to it. Secondly, that if it be laid upon a still water +in a floating vessel, and left to itself, it will turn itself till it +lie in a meridian, that is to say, with one and the same line still +north and south. Thirdly, if you take a long slender piece of iron, and +apply the loadstone to it, and, according to the position of the poles +of the loadstone, draw it over to the end of the iron, the iron will +have the same poles with the magnet, so it be drawn with some pressure; +but the poles will lie in a contrary position; and also this long iron +will draw other iron to it as the magnet doth. + +Fourthly, this long iron, if it be so small as that poised upon a pin, +the weight of it have no visible effect, the navigators use it for the +needle of their compass, because it points north and south; saving that +in most places by particular accidents it is diverted; which diversion +is called the variation of the horizontal needle. Fifthly, the same +needle placed in a plane perpendicular to the horizon, hath another +motion called the inclination. Which that you may the better conceive, +draw a fourth figure; wherein let there be a circle to represent the +terrella, that is to say, a spherical magnet. + +_A._ Let this be it, whose centre is A, the north pole B, the south pole +C. + +_B._ Join B C, and cross it at right angles with the diameter D E. + +_A._ It is done. + +_B._ Upon the point D set the needle parallel to B C, with the cross of +the south pole, and the barb for the north; and describe a square about +the circle B D C E, and divide the arch D B into four equal parts in +_a_, _b_, _c._ + +_A._ It is done. + +_B._ Then place the middle of the needle on the points _a_, _b_, _c_, so +that they may freely turn; and set the barb which is at D towards the +north, and that which is at C towards the south. You see plainly by +this, that the angles of inclination through the arch D C taken +altogether, are double to a right angle. For when the south point of the +needle, looking north, as at D, comes to look south, as at C, it must +make half a circle. + +_A._ That is true. And if you draw the sine of the arch D _a_, which is +_d a_, and the sine of the arch B _a_, which is _a e_, and the sine of +the arch D _b_, which is _b f_, and the sine of the arch B _c_, which is +_c g_, the needle will lie upon _b f_ with the north-point downwards, so +that the needle will be parallel to A D. Then from _a_ draw the line _a +h_, making the angle _e a h_ equal to the angle D A _a_. And then the +needle at _a_ shall lie in the line _a h_ with the south point toward +_h_. Finally, draw the line _c h_, which, with _c g_, will also make a +quarter of a right angle; and therefore if the needle be placed on the +point _c_, it will lie in _c h_ with the south point toward _h_. And +thus you see by what degrees the needle inclines or dips under the +horizon more and more from D till it come to the north pole at B; where +it will lie parallel to the needle in D; but with their barbs looking +contrary ways. And this is certain by experience, and by none +contradicted. + +You see then why the degrees of the inclinatory needle, in coming from D +to B, are double to the degrees of a quadrant. It is found also by +experience, that iron both of the mine and of the furnace put into a +vessel so as to float, will lay itself (if some accident in the earth +hinder it not) exactly north and south. And now I am, from this +compounded motion supposed by Copernicus, to derive the causes why a +loadstone draws iron; why it makes iron to do the same; why naturally it +placeth itself in a parallel to the axis of the earth; why by passing it +over the needle it changes its poles; and what is the cause that it +inclines. But it is your part to remember what I told you of motion at +our second meeting; and what I told you of this compounded motion +supposed by Copernicus, at our fourth meeting. + + + ========== + + + CHAPTER IX. + OF THE LOADSTONE AND ITS POLES, AND WHETHER + THEY SHOW THE LONGITUDE OF PLACES ON THE EARTH. + +_A._ I come now to hear what natural causes you can assign of the +virtues of the magnet; and first, why it draws iron to it, and only +iron. + +_B._ You know I have no other cause to assign but some local motion, and +that I never approved of any argument drawn from sympathy, influence, +substantial forms, or incorporeal effluvia. For I am not, nor am +accounted by my antagonists for a witch. But to answer this question, I +should describe the globe of the earth greater than it is at B in the +first figure, but that the terrella in the fourth figure will serve our +turn. For it is but calling B and C the poles of the earth, and D E the +diameter of the equinoctial circle, and making D the east, and E the +west. And then you must remember that the annual motion of the earth is +from west to east, and compounded of a straight and circular motion, so +as that every point of it shall describe a small circle from west to +east, as is done by the whole globe. And let the circles about _a b c_ +be three of those small circles. + +_A._ Before you go any farther, I pray you show me how I must +distinguish east and west in every part of this figure. For wheresoever +I am on earth, suppose at London, and see the sun rise suppose in +Cancer, is not a straight line from my eye to the sun terminated in the +east? + +_B._ It is not due east, but partly east, partly south. For the earth, +being but a point compared to the sun, all the parallels to D E the +equator, such as are _e a_, _f b_, _c g_, if they be produced, will fall +upon the body of the sun. And therefore A _b_ is north-east; A _a_ east +north-east; and A _c_ north north-east. + +_A._ Proceed now to the cause of attraction. + +_B._ Suppose now that the internal parts of the loadstone had the same +motion with that of the internal parts of the sun which make the annual +motion of the earth from west to east, but in a contrary way, for +otherwise the loadstone and the iron can never be made to meet. Then set +the loadstone at a little distance from the earth, marked with _z_; and +the iron marked with _x_ upon the superficies of the earth. Now that +which makes _x_ rise to _z_, can be nothing else but air; for nothing +touches it but air. And that which makes the air to rise, can be nothing +but those small circles made by the parts of the earth, such are at _a b +c_, for nothing else touches the air. Seeing then the motion of each +point of the loadstone is from east to west in circles, and the motion +of each point of the iron from west to east; it follows, that the air +between the loadstone and the iron shall be cast off both east and west; +and consequently the place left empty, if the iron did not rise up and +fill it. Thus you see the cause that maketh the loadstone and the iron +to meet. + +_A._ Hitherto I assent. But why they should meet when some heterogeneous +body lies in the air between them, I cannot imagine. And yet I have seen +a knife, though within the sheath, attract one end of the needle of a +mariner’s compass; and have heard it will do the same though a +stone-wall were between. + +_B._ Such iron were indeed a very vigorous loadstone. But the cause of +it is the same that causeth fire or hot water, which have the same +compounded motion, to work through a vessel of brass. For though the +motion be altered by restraint within the heterogeneous body, yet being +continued quite through, it restores itself. + +_A._ What is the cause why the iron rubbed over by a loadstone will +receive the virtue which the loadstone hath of drawing iron to it? + +_B._ Since the motion that brings two bodies to meet must have contrary +ways, and that the motions of the internal parts of the magnet and of +the iron are contrary; the rubbing of them together does not give the +iron the first endeavour to rise, but multiplies it. For the iron +untouched will rise to a loadstone; but if touched, it becomes a +loadstone to other iron. For when they touch a piece of iron, they pass +the loadstone over it only one way, viz. from pole to pole; not back +again, for that would undo what before had been done; also they press it +in passing to the very end of the iron, and somewhat hard. So that by +this pressing motion all the small circles about the points _a b c_, are +turned the contrary way; and the halves of those small circles made on +the arch D B will be taken away and the poles changed, so as that the +north poles shall point south, and the south poles north, as in the +figure. + +_A._ But how comes it to pass, that when a loadstone hath drawn a piece +of iron, you may add to it another, as if they begat one another? Is +there the like motion in the generation of animals? + +_B._ I have told you that iron of itself will rise to the loadstone; +much more then will it adhere to it when it is armed with iron, and both +it and the iron have a plain superficies. For then not only the points +of contact will be many, which make the coherence stronger, but also the +iron wherewith it is armed is now another loadstone, differing a little, +which you perhaps think, as male and female. But whether this compounded +motion and confrication causeth the generation of animals, how should I +know, that never had so much leisure as to make any observation which +might conduce to that? + +_A._ My next question is, seeing you say the loadstone, or a needle +touched with it, naturally respecteth the poles of the earth, but that +the variation of it proceedeth from some accidents in the superficies of +the earth; what are those accidents? + +_B._ Suppose there be a hill upon the earth, for example, at _r_; then +the stream of the air which was between _z_ and _x_ westward, coming to +the hill, shall go up the hill’s side, and so down to the other side, +according to the crooked line which I have marked about the hill by +points; and this infallibly will turn the north point of the needle, +being on the east side, more towards the east, and that on the other +side more towards the west, than if there had been no hill. And where +upon the earth are there not eminences and depressions, except in some +wide sea, and a great way from land. + +_A._ But if that be true, the variation in the same place should be +always the same, for the hills are not removed. + +_B._ The variation of the needle at the same place is still the same; +but the variation of the variation is partly from the motion of the pole +itself, which by the astronomers is called _motus trepidationis_; and +partly from that, that the variation cannot be truly observed, for the +horizontal needle and the inclinatory needle incline alike, but cannot +incline in due quantity. For whether set upon a pin or an axis, their +inclination is hindered, in the horizontal needle, by the pin itself: if +upon an axis, if the axis be just, it cannot move; if slack, the weight +will hinder it; but chiefly because the north pole of the earth draws +away from it the north pole of the needle, for two like poles cannot +come together. And this is the cause why the variation in one place is +east, and another west. + +_A._ This is indeed the most probable reason why the variation varies +that ever I heard given; and I should presently acknowledge that this +parallel motion of the axis of the earth in the ecliptic, supposed by +Copernicus, is the true annual motion of the earth, but that there is +lately come forth a book called _Longitude Found_ , which makes the +magnetical poles distant from the poles of the earth eight degrees and a +half. + +_B._ I have the book. It is far from being demonstrated, as you shall +find, if you have the patience to see it examined. For wheresoever his +demonstration is true, the conclusion, if rightly inferred, will be +this, that the poles of the loadstone and the poles of the earth are the +same. And where, on the contrary, his demonstrations are fallacies, it +is because sometimes he fancieth the lines he hath drawn, not where they +are; sometimes because he mistakes his station; and sometimes because he +goes on some false principle of natural philosophy; and sometimes also +because he knoweth not sufficiently the doctrine of spherical triangles. + +_A._ I think that is the book there which lies at your elbow. Pray you +read. + +_B._ I find first (p. 4), that the grounds of his argument are the two +observations made by Mr. Burroughs, one at Vaygates, in 1576, where the +variation from the pole of the earth he found to be 11 deg. 15 min. +east; the other at Limehouse, near London, in 1580, where the variation +from the pole of the earth was 8 deg. 38 min. west, by which, he saith, +he might _find out the magnetical pole_. + +_A._ Where is Vaygates? + +_B._ In 70 degrees of north latitude; the difference of longitude +between London and it being 58 degrees. + +_A._ The longitude of places being yet to seek, how came he to know this +difference of 58 degrees, except the poles of the magnet and the earth +be the same? + +_B._ I believe he trusted to the globe for that. For the distance +between the places is not above 2000 miles the nearest way. But we will +pass by that, and come to his demonstration, and to his diagram, wherein +L is London, P the north-pole of the earth, V Vaygates. So that L P is +38 deg. 28 min.; P V 20 deg.; the angle L P V 58 deg. for the difference +between the longitudes of Vaygates and London. This is the construction. +But before I come to the demonstration, I have an inference to draw from +these observations, which is this. Because in the same year the +variation at London was 11 deg. 15 min. east, and at Vaygates 8 deg. 38 +min. west; if you subtract 11 deg. 15 min. from the arc L P; and 8 deg. +38 min. from the arc L V, the variation on both sides will be taken +away; so that P V being the meridian of Vaygates, and L P the meridian +of London, they shall both of them meet in P the pole of the earth. And +if the pole of the magnet be nearer to the zenith of London than is the +pole of the earth, it shall be just as much nearer to the zenith of +Vaygates in the meridian of Vaygates, which is P V; as is manifest by +the diurnal motion of the earth. + +_A._ All this I conceive without difficulty. Proceed to the +demonstration. + +_B._ Mark well now. His words are these (page 5): From P L V subtract 11 +deg. 15 min., and there remains the angle V L M. Consider now which is +the angle P L V, and which is the remaining angle V L M, and tell what +you understand by it. + +_A._ He has marked the angle P L V with two numbers, 11 deg. 15 min. and +21 deg. 50 min., which together make 33 deg. 5 min. And the angle 11 +deg. 15 min. being subtracted from P L V, there will remain 21 deg. 50 +min. for the angle V L M. I know not what to say to it. For I thought +the arc P V, which is 20 deg., had been the arc of the spherical angle P +L V; and that the arc L V had been 58 deg., because he says the angle L +P V is so; and that the arc L M had been 46 deg., because the angle L P +M is so; and lastly, that the angle P L M had been 8 deg. 30 min., +because the arc P M is so. + +_B._ And what you thought had been true, if a spherical angle were a +very angle. For all men that have written of spherical triangles take +for the ground of their calculation, as Regiomontanus, Copernicus, and +Clavius, that the arch of a spherical angle is the side opposite to the +angle. You should have considered also that he makes the angle V P M 12 +deg., but sets down no arc to answer it. But that you may find I am in +the right, look into the definitions which Clavius hath put down before +his treatise of spherical triangles, and amongst them is this; “the arc +of a spherical triangle is a part of a great circle intercepted between +the two sides drawn from the pole of the said great circle.” + +_A._ The book is nothing worth; for it is impossible to subtract an arc +of a circle out of a spherical angle. And I see besides that he takes +the superficies that lieth between the sides L P and L M for an arch, +which is the quantity of an angle; and is a line, and cannot be taken +out of a superficies. I wonder how any man that pretends to mathematics +could be so much mistaken. + +_B._ It is no great wonder. For Clavius himself striving to maintain +that a right angle is greater than the angle made by the diameter and +the circumference, fell into the same error. A corner, in vulgar speech, +and an angle, in the language of geometry, are not the same thing. But +it is easy even for a learned man sometimes to take them for the same, +as this author now has done; and proceeding he saith, subtract 8 deg. 38 +min. from the angle P V L, and there remains the angle L V M. + +_A._ That again is false, because impossible. What was it that deceived +him now? + +_B._ The same misunderstanding of the nature of a spherical angle. Which +appears farther in this, that when he knew the arc V P was part of a +great circle, he thought V M, which he maketh 8 deg. 30 min., were also +parts of a great circle; which is manifestly false. For two great +circles, because they pass through the centre, do cut each other into +halves. But V P is not half a circle. He sure thought himself at +Vaygates, and that P M V was equal to P V, although in the same +hemisphere. + +_A._ But how proves he that the arc P M is 8 degrees 30 minutes? + +_B._ Thus. We have in two triangles, P L M and P V M, two sides and one +angle included, to find P M the distance of the magnetical pole from the +pole of the earth 8 deg. 30 min. + +_A._ Is that all? It is very short for a demonstration of two so +difficult problems, as the quantity of 8 deg. 30 min.; and of the place +of the magnetical pole. But he has proved nothing till he has showed how +he found it. And though P M be 8 deg. 30 min., it follows not that M is +the magnetical pole. + +_B._ Nor is it true. For if P M be 8 deg. 30 min., and V M 8 deg. 38 +min., the whole arc P M V will be 17 deg. 8 min., which should be 20 +deg. Besides, whereas the variations were east and west, the subtracting +of them should be also east and west, but they are north and south. + +_A._ I am satisfied that the magnetical poles and the poles of the earth +are the same. But thus much I confess, if they were not the same, the +longitude were found. For the difference of the latitudes of the earth’s +equator and of the magnetical equator, is the difference of the +longitude. But proceed. + +_B._ “The earth being a solid body, and the magnetic sphere that +encompasseth the earth being a substance that hath not solidity to keep +pace with the earth, loseth in its motion: and that may be the cause of +the motion of the magnetic poles from east to west.” + +_A._ This is very fine and unexpected. The magnetic sphere, which I took +for a globe made of a magnet, has not solidity to keep pace with the +earth, though it be one of the hardest stones that are. It encompasseth +the earth; yet I thought nothing had encompassed the earth but air in +which I breath and move. By this also the whole earth must be a +loadstone. For two bodies cannot be in one place. So that he is yet no +farther than Dr. Gilbert whom he slights. And if the sphere be a magnet, +then the earth and loadstone have the same poles. See the force of +truth! which though it could not draw to it his reason, hath drawn his +words to it. + +_B._ But perhaps he meant that the magnetic virtue encompasseth the +earth, and not the magnetic body. + +_A._ But that helpeth him not. For if the body of the magnet be not +there, the virtue then is the virtue of the earth; and so again the +poles of the earth are magnetic poles. + +_B._ You see how unsafe it is to boast of doctrines as of God’s gifts, +till we are sure that they are true. For God giveth and denieth as he +pleaseth, not as ourselves wish; as now to him he hath given confidence +enough, but hath denied him, at least hitherto, the finding of the +longitudes. In the next place (p. 8) he seems much pleased that his +doctrine agrees with an opinion of Keplerus, that from the creation to +the year of our Lord, it is to the year 1657 now 5650 years; and with +that which he saith some divines have held in times past, that as this +world was created in six days, so it should continue six thousand years. +By which account the world will be at an end three hundred and fifty +years hence; though the Scripture tells us it shall come as a thief in +the night. O what advantage three hundred and forty years hence will +they have that know this, over them that know it not, by taking up money +at interest, or selling lands at twenty years’ purchase! + +_A._ But he says he will not meddle with that. + +_B._ Yes, when he had meddled with it too much already. + +_A._ But you have not told me wherein consisteth this agreement between +him and Keplerus. + +_B._ I forgot it. It is in the motion of the magnetic poles. For +precedently (p. 7), he had said “that their period or revolution was six +hundred years; their yearly motion thirty-six minutes; and (p. 8) that +their motion is by sixes. Six tenths of a degree in one year; six +degrees in ten years; sixty degrees in a hundred years; and six times +sixty degrees in six hundred years.” + +_A._ But what natural cause doth he assign of this revolution of six +hundred years? + +_B._ None at all. For the magnet lying upon the earth, can have no +motion at all but what the earth and the air give it. And because it is +always at 8 deg. 30 min. distance from the pole of the earth, the earth +can give it no other motion than what it gives to its own poles by the +precession of the equinoctial points. Nor can the air give it any motion +but by its stream; which must needs vary when the stream varieth. But +what a vast difference does he make between the period of the motion of +the equinoctial points, which is about or near thirty-six thousand years +according to Copernicus (lib. iii. cap. 6), which makes the annual +precession to be 36 seconds, and the period of the magnetical poles’ +motion, which is but six hundred years. + +_A_. Go on. + +_B_. He comes now (p. 15) to the inclinatory needle upon a spherical +loadstone. Where he shows, by diagram, that the needle and the +instrument together moved towards the magnetical pole, make the sum of +the inclinations equal to two quadrants, setting the north-point of the +needle southward: which I confess is true. But, in the same page, he +ascribeth the same motion to the earth in these words: “as the +horizontal needle hath a double motion about the round loadstone or +terrella, so also the inclinatory needle hath a double motion about the +earth.” What is this, but a confession that the poles of the magnet and +of the earth are the same? + +_A._ It is plain enough. + +_B._ Besides, seeing he placeth the magnetical pole at M in the meridian +of Vaygates, the needle being touched shall incline to the pole of the +earth which is P, as well there as at London, and make the north-pole of +the earth point south. + +_A._ It is certain, because he puts both the magnetical pole and the +pole of the earth in the same meridian of the earth. Nor see I any cause +why, the needle being the same, it should not be as subject to +variation, and to variation of variation, and to all accidents of the +earth there, as in any other part. + +_B._ He putteth (p. 16) a question, “at what distance from the earth are +the magnetic poles? and answers to it, they are very near the earth, +because the nearer the earth, the greater the strength.” What think you +of this? + +_A._ I think they are in the superficies of the magnet, as the pole of +the earth is in the superficies of the earth. And consequently, that +then the earth must be a part of the magnet, and their poles the same. +For the body of the magnet and the body of the earth, if they be two, +cannot be in one place. + +_B._ His next words are, “some things are to be considered concerning +those variations of the horizontal needle which are not according to the +situation of the place from the magnetic poles, but are contrary; as all +the West Indies according to the poles should be easterly, and they are +westerly. Which is by some accidental cause in the earth; and their +motion, as I formerly said, is a forced motion, and not natural.” + +_A._ He has clearly overthrown his main doctrine. For to say the motion +of the needle is forced and unnatural, is a most pitiful shift, and +manifestly false, no motion being more constant or less accidental, +notwithstanding the variation, to which the inclinatory needle is no +less subject than the horizontal needle. + +_B._ That which deceived him, was, that he thought them two sorts of +needles, forgetting what he had said of Norman’s invention of the +inclinatory needle by the inclining of the horizontal needle (p. 11). +For I will show you that what he says is easterly and should be +westerly, should be easterly as it is. Consider the fourth figure, in +which B is the north-pole, and B _c_ 11 deg. 15 min. easterly, which was +the variation at London in 1576 easterly. Suppose A _c_ to be the +needle, shall it not incline, as well here as at D _a_, and the +variation B _c_ be easterly? Again, D _a_ is 11 deg. 15 min., and the +needle in D parallel to A B, and at _a_ inclining also 11 deg. 15 min. +westerly. And is not the variation there D _a_ westerly, with the north +point of the needle in the line _a h_? + +_A._ But the West-Indies are not in this hemisphere B C D E. The +variation therefore will proceed in an arc of the opposite hemisphere, +which is westerly. + +_B._ I believe he might think so, forgetting that he and his compass +were on the superficies of the earth, and fancying them in the centre at +A. + +_A._ It is like enough. If we had a straight line exactly equal to the +arc of a quadrant, I think it would very much facilitate the doctrine of +spherical triangles. + +_B._ When you have done with your questions of natural philosophy, I +will give you a clear demonstration of the equality of a straight line +to the arc of a quadrant, which, if it satisfy you, you may carry with +you, and try thereby if you can find the angle of a spherical triangle +given. + +_A._ It is time now to give over. And at our next meeting I desire your +opinion concerning the causes of diaphaniety, and refraction. This +Copernicus has done much more than he thought of. For he has not only +restored to us astronomy, but also made the way open to physiology. + + + ========== + + + CHAPTER X. + OF TRANSPARENCE, REFRACTION, AND OF THE POWER + OF THE EARTH TO PRODUCE LIVING CREATURES. + +_A._ Thinking upon what you said yesterday, it looked like a generation +of living creatures. I saw the love between the loadstone and the iron +in their mutual attraction, their engendering in their close and +contrary motion, and their issue in the iron, which being touched, hath +the same attractive virtue. Now seeing they have the same internal +motion of parts with that of the earth, why should not their substance +be the same, or very near a-kin? + +_B._ The most of them, if not all, that have written on this subject, +when they call the loadstone a terrella, seem to think as you do. But I, +except I could find proof for it, will not affirm it. For the earth +attracteth all kind of bodies but air, and the loadstone none but iron. +The earth is a star, and it were too bold to pronounce any sentence of +its substance, especially of the planets, that are so lapt up in their +several coats, as that they cannot work on our eyes, or any organ of our +other senses. + +_A._ I come therefore now to the business of the day. Seeing all +generation, augmentation, and alteration is local motion, how can a body +not transparent be made transparent? + +_B._ I think it can never be done by the art of man. For as I said of +hard and heavy bodies in the creation, so I think of diaphanous, that +the very same individual body which was not transparent then, shall +never be made transparent by human art. + +_A._ Do not you see that every day men make glass, and other diaphanous +bodies not much inferior in beauty to the fairest gems? + +_B._ It is one thing to make one transparent of many by mixture, and +another to make transparent of not transparent. Any very hard stone, if +it be beaten into small sands, such as is used for hour-glasses, every +one of those sands, if you look upon it with a microscope, you will find +to be transparent; and the harder and whiter a stone is, so much the +more transparent, as I have seen in the stone of which are made +millstones, which stone is here called greet. And I doubt not but the +sands of white marble must be more transparent. But there are no sands +so transparent that they have not a scurf upon them, as hard, perhaps, +as the stone itself; which they whose profession it is to make glass, +have the art to scour and wash away. And therefore I think it no great +wonder to bring those sands into one lump, though I know not how they do +it. + +_A._ I know they do it with lye made with a salt extracted from the +ashes of an herb, of which salt they make a strong lye, and mingle it +with the sand, and then bake it. + +_B._ Like enough. But still it is a compound of two transparent bodies, +whereof one is the natural stone, the other is the mortar. This +therefore doth not prove, that one and the same body of not transparent +can be made transparent. + +_A._ Since they can make one transparent body of many, why do they not +of a great many small sparks of natural diamond compound one great one? +It would bear the charges of all the materials, and beside, enrich them. + +_B._ It is probable it would. But it may be they know no salt that +howsoever prepared, which, with how great a fire soever, can make them +melt. And, it may be, the true crystal of the mountain, which is found +in great pieces in the Alps, is but a compound of many small ones, and +made by the earth’s annual motion; for it is a very swift motion. +Suppose now that within a very small cavern of those rocks whose +smallest atoms are crystal, and the cavity filled with air; and consider +what a tumult would be made by the swift reciprocation of that air; +whether it would not in time separate those atoms from the rock, and +jumbling them together make them rub off their scurf from one another, +and by little and little to touch one another in polished planes, and +consequently stick together, till in length of time they become one lump +of clean crystal. + +_A._ I believe that the least parts of created substances lay mingled +together at first, till it pleased God to separate all dissimilar +natures, and congregate the similar, to which this annual motion is +proper. But they say that crystal is found in the open air hanging like +icicles upon the rocks, which, if true, defeats this supposition of a +narrow cavern, and therefore I must have some farther experience of it +before I make it my opinion. But howsoever, it still holds true that +diaphanous bodies of all sorts, in their least parts, were made by God +in the beginning of the world. But it may be true, notwithstanding those +icicles. For the force of the air that could break off those diaphanous +atoms in a cavern, can do the same in the open air. And I know that a +less force of air can break some bodies into small pieces, not much less +hard than crystal, by corrupting them. + +_B._ That which you now have said is somewhat. But I deny not the +possibility, but only doubt of the operation. You may therefore pass to +some other question. + +_A._ Well, I will ask you then a question about refraction. I know +already that for the cause of refraction, when the light falleth through +a thinner medium upon a thicker, you assign the resistance of the +thicker body; but you do not mean there, by _rarum_ and _densum_, two +bodies whereof in equal spaces one has more substance in it than the +other. + +_B._ No; for equal spaces contain equal bodies. But I mean by _densum_ +any body which more resisteth the motion of the air, and by _rarum_ that +which resisteth less. + +_A._ But you have not declared in what that resistance consisteth. + +_B._ I suppose it proceedeth from the hardness. + +_A._ But from thence it will follow, that all transparent bodies that +equally refract are equally hard, which I think is not true, because the +refraction of glass is not greater, at least in comparison of their +hardnesses, than that of water. + +_B._ I confess it. Therefore I think we must take in gravity to a share +in the production of this refraction. For I never considered refraction +but in glass, because my business then was only to find the causes of +the phenomena of telescopes and microscopes. Let therefore A B (in fig. +7) be a hard, and consequently, a heavy body; and from above, as from +the sun, let C A be the line of incidence, and produced to D; and draw A +E perpendicular to A B. It is manifest that the hardness in A B shall +turn the stream of the light inwards toward A E, suppose in the line A +_e_. It is also evident that the endeavour in B, which is, being heavy, +downward, shall turn the stream again inward, towards A E, as in A _b_. +Thus it is in refraction from the sun downwards. In like manner, if the +light come from below, as from a candle in the point D, the line of +incidence will be D A, and produced will pass to C. And the resistance +of the hardness in A will turn the stream A C inward, suppose into A +_l_, and make C _l_ equal to D _e_. For passing into a thinner medium, +it will depart from the perpendicular in an angle equal to the angle D A +_e_, by which it came nearer to it in A _e_. So also the resistance of +the gravity in the point A shall turn the stream of the light into the +line A _i_, and make the angle _l_ A _i_ equal to the angle _e_ A _b_. +And thus you see in what manner, though not in what proportion, hardness +and gravity conjoin their resistance in the causing of refraction. + +_A._ But you proved yesterday, that a heavy body does not gravitate upon +a body equally heavy. Now this A B has upper parts and lower parts; and +if the upper parts do not gravitate upon the lower parts, how can there +be any endeavour at all downward to contribute to the refraction? + +_B._ I told you yesterday, that when a heavy body was set upon another +body heavier or harder than itself, the endeavour of it downward was +diverted another way, but not that it was extinguished. But in this +case, where it lieth upon air, the first endeavour of the lowest part +worketh downward. For neither motion nor body can be utterly +extinguished by a less than an omnipotent power. All bodies, as long as +they are bodies, are in motion one way or other, though the farther it +be communicated, so much the less. + +_A._ But since you hold that motion is propagated through all bodies, +how hard or heavy soever they be, I see no cause but that all bodies +should be transparent. + +_B._ There are divers causes that take away transparency. First, if the +body be not perfectly homogeneous, that is to say, if the smallest parts +of it be not all precisely of the same nature, or do not so touch one +another as to leave no vacuum within it; or though they touch, if they +be not as hard in the contact as in any other line. For then the +refractions will be so changed both in their direction, and in their +strength, as that no light shall come through it to the eye; as in wood +and ordinary stone and metal. Secondly, the gravity and hardness may be +so great, as to make the angle refracted so great, as the second +refraction shall not direct the beam of light to the eye; as if the +angle of refraction were D A E, the refracted line would be +perpendicular to A B, and never come to the line A D, in which is the +eye. + +_A._ To know how much of the refraction is due to the hardness, and how +much to the gravity, I believe it is impossible, though the quantity of +the whole be easily measured in a diaphanous body given. And both you +and Mr. Warner have demonstrated, that as the sine of the angle +refracted in one inclination is to the sine of the angle refracted in +another inclination, so is the sine of one inclination to the sine of +the angle of the other inclination. Which demonstrations are both +published by Mersennus in the end of the first volume of his _Cogitata +Physico-Mathematica_. But since there be many bodies, through which +though there pass light enough, yet no object appear through them to the +eye, what is the reason of that? + +_B._ You mean paper. For paper windows will enlighten a room, and yet +not show the image of an object without the room. But it is because +there are in paper abundance of pores, through which the air passing +moveth the air within; by the reflections whereof anything within may be +seen. And in the same paper there are again as many parts not +transparent, through which the air cannot pass, but must be reflected +first to all parts of the object, and from them again to the paper; and +at the paper either reflected again or transmitted, according as it +falls upon pores or not pores; so that the light from the object can +never come together at the eye. + +_A._ There belongs yet to this subject the causes of the diversity of +colours. But I am so well satisfied with that which you have written of +it in the twenty-fourth chapter of your book _de Corpore_, that I need +not trouble you farther in it. And now I have but one question more to +ask you, which I thought upon last night. I have read in an ancient +historian, that living creatures after a great deluge were produced by +the earth, which being then very soft, there were bred in it, it may be +by the rapid motion of the sun, many blisters, which in time breaking, +brought forth, like so many eggs, all manner of living creatures great +and small, which since it is grown hard it cannot do. What think you of +it? + +_B._ It is true that the earth produced the first living creatures of +all sorts but man. For God said (Gen. i. 24), _Let the earth produce +every living creature, cattle, and creeping thing, &c._ But then again +(ver. 25) it is said that _God made the beast of the earth, &c._ So that +it is evident that God gave unto the earth that virtue. Which virtue +must needs consist in motion, because all generation is motion. But man, +though the same day, was made afterward. + +_A._ Why hath not the earth the same virtue now? Is not the sun the same +as it was? Or is there no earth now soft enough? + +_B._ Yes. And it may be the earth may yet produce some very small living +creatures: and perhaps male and female. For the smallest creatures which +we take notice of, do engender, though they do not all by conjunction; +therefore if the earth produce living creatures at this day, God did not +absolutely rest from all his works on the seventh day, but (as it is +chap. ii. 2) _he rested from all the work he had made_. And therefore it +is no harm to think that God worketh still, and when and where and what +he pleaseth. Beside, it is very hard to believe, that to produce male +and female, and all that belongs thereto, as also the several and +curious organs of sense and memory, could be the work of anything that +had not understanding. From whence, I think we may conclude, that +whatsoever was made after the creation, was a new creature made by God +no otherwise than the first creatures were, excepting only man. + +_A._ They are then in an error that think there are no more different +kinds of animals in the world now, than there were in the ark of Noah. + +_B._ Yes, doubtless. For they have no text of Scripture from which it +can be proved. + +_A._ The questions of nature which I could yet propound are innumerable. +And since I cannot go through them, I must give over somewhere, and why +not here? For I have troubled you enough, though I hope you will forgive +me. + +_B._ So God forgive us both as we do one another. But forget not to take +with you the demonstration of a straight line equal to an arc of a +circle. + + + + + THE PROPORTION OF A STRAIGHT LINE TO HALF THE ARC OF A QUADRANT. + + +[Illustration] + +Describe the square A B C D, and divide it by the diagonals A C and B D, +as also by the straight lines E G, F H, meeting in the centre I at right +angles, into four equal parts. Then with the radius A B describe the +quadrant B D cutting E G in K, and the diagonal A C in L; and so B L +will be half the arc B D, equal to which we are to find a straight line. +Divide I C into halves at M, and draw B M cutting E G in _a_. I say B M +is equal to the arc B L. For the demonstration whereof we are to assume +certain known truths and dictates of common-sense. + +1. That the arc B K is the third part of the arc B D, and consequently +two-thirds of the arc B L, and B K to K L as two to one. + +2. That if a straight line be equal to the arc B L, and one end in B, +the other will be somewhere in I C, and higher than the point L. + +3. That wheresoever it be, two-thirds of it must be equal to the arc B +K, and one-fifth to the arc K L. + +4. That the arc of a quadrant described in the third part of the radius, +or of E G, is equal to the third part of the arc B D, viz. to the arc B +K. I may therefore call a third part of E G, the radius of B K; and a +sixth part of E G, the radius of the arc K L, &c. + +5. And lastly, that any straight line drawn from B to I C, if it be +equal to the arc B L, it must cut the half radius I G, whose quadrantal +arc is B L, into the proportion of two to one. For as the whole arc to +the whole E G, so are the parts of it to the parts of E G. + +These premises granted, which I think cannot be denied, I say again, +that the straight line B M is equal to the arc B L. + + DEMONSTRATION. + +[Illustration] + +Because B I is to I M, by construction, as two to one, and the line I G +divides the angle B I C in the midst, B _a_ will be to _a_ M as two to +one, that is to say, as the arc B K to the arc K L. From the point M to +the side B C erect a perpendicular M N. And because C M is half C I, the +line M N will be half G C; and B N will be three-quarters of B C; and +the square of B M equal to ten squares of a quarter of B C; and because +B M is to B _a_ as three to two, M N will be to _a_ G as three to two. +But M N is a quarter of E G, therefore _a_ G is two-thirds of a quarter +of E G; that is, one-third of I G; that is, one-sixth of the whole E G. +And I _a_ one-third of E G. Therefore I _a_ is the radius of the arc B +K; and _a_ G the radius of the arc K L; and E G the radius of the whole +arc B L D. Lastly, if a straight line be drawn from B to any other point +of the line I C, though any line may be divided into the proportion of +two to one, it shall not pass through the point _a_, and therefore not +divide the radius of B L, which is I G, into the proportion of two to +one. Therefore no straight line can be drawn from B to I C, except B M, +so as to be equal to the arc B L. Therefore the straight line B M and +the arc B L are equal. + +Hence it follows, that seeing the square of B M is equal to ten squares +of a quarter of B C, that a straight line equal to the quadrantal arc B +L D is equal to ten squares of half the radius, as I have divers ways +demonstrated heretofore. + + + + + SIX LESSONS + TO THE + PROFESSORS OF THE MATHEMATICS, + + ONE OF GEOMETRY, THE OTHER OF ASTRONOMY, + IN THE CHAIRS SET UP BY THE NOBLE AND LEARNED SIR HENRY SAVILE, IN THE + UNIVERSITY OF OXFORD. + + TO THE RIGHT HONOURABLE + + HENRY LORD PIERREPONT, + + VISCOUNT NEWARK, EARL OF KINGSTON, AND + MARQUIS OF DORCHESTER. + + +MY MOST NOBLE LORD, + +Not knowing on my own part any cause of the favour your Lordship has +been pleased to express towards me, unless it be the principles, method, +and manners you have observed and approved in my writings; and seeing +these have all been very much reprehended by men, to whom the name of +public professors hath procured reputation in the university of Oxford, +I thought it would be a forfeiture of your Lordship’s good opinion, not +to justify myself in public also against them, which, whether I have +sufficiently performed or not in the six following Lessons addressed to +the same professors, I humbly pray your Lordship to consider. The volume +itself is too small to be offered to you as a present, but to be brought +before you as a controversy it is perhaps the better for being short. Of +arts, some are demonstrable, others indemonstrable; and demonstrable are +those the construction of the subject whereof is in the power of the +artist himself, who, in his demonstration, does no more but deduce the +consequences of his own operation. The reason whereof is this, that the +science of every subject is derived from a precognition of the causes, +generation, and construction of the same; and consequently where the +causes are known, there is place for demonstration, but not where the +causes are to seek for. Geometry therefore is demonstrable, for the +lines and figures from which we reason are drawn and described by +ourselves; and civil philosophy is demonstrable, because we make the +commonwealth ourselves. But because of natural bodies we know not the +construction, but seek it from the effects, there lies no demonstration +of what the causes be we seek for, but only of what they may be. + +And where there is place for demonstration, if the first principles, +that is to say, the definitions contain not the generation of the +subject, there can be nothing demonstrated as it ought to be. And this +in the three first definitions of Euclid sufficiently appeareth. For +seeing he maketh not, nor could make any use of them in his +demonstrations, they ought not to be numbered among the principles of +geometry. And Sextus Empiricus maketh use of them (misunderstood, yet so +understood as the said professors understand them) to the overthrow of +that so much renowned evidence of geometry. In that part therefore of my +book where I treat of geometry, I thought it necessary in my definitions +to express those motions by which lines, superficies, solids, and +figures, were drawn and described, little expecting that any professor +of geometry should find fault therewith, but on the contrary supposing I +might thereby not only avoid the cavils of the sceptics, but also +demonstrate divers propositions which on other principles are +indemonstrable. And truly, if you shall find those my principles of +motion made good, you shall find also that I have added something to +that which was formerly extant in geometry. + +For first, from the seventh chapter of my book _De Corpore_, to the +thirteenth, I have rectified and explained the principles of the +science; _id est_, I have done that business for which Dr. Wallis +receives the wages. In the seventh, I have exhibited and demonstrated +the proportion of the parabola and parabolasters to the parallelograms +of the same height and base; which, though some of the propositions were +extant without that demonstration, were never before demonstrated, nor +are by any other than this method demonstrable. + +In the eighteenth, as it is now in English, I have demonstrated, for +anything I yet perceive, equation between the crooked line of a parabola +or any parabolaster and a straight line. + +In the twenty-third I have exhibited the centre of gravity of any sector +of a sphere. + +Lastly, the twenty-fourth, which is of the nature of refraction and +reflection, is almost all new. + +But your Lordship will ask me what I have done in the twentieth, about +the quadrature of the circle. Truly, my Lord, not much more than before. +I have let stand there that which I did before condemn, not that I think +it exact, but partly because the division of angles may be more exactly +performed by it than by any organical way whatsoever; and I have +attempted the same by another method, which seemeth to me very natural, +but of calculation difficult and slippery. I call them only aggressions, +retaining nevertheless the formal manner of assertion used in +demonstration. For I dare not use such a doubtful word as _videtur_, +because the professors are presently ready to oppose me with a _videtur +quod non_. Nor am I willing to leave those aggressions out, but rather +to try if it may be made pass for lawful, (in spite of them that seek +honour, not from their own performances, but from other men’s failings), +amongst many difficult undertakings carried through at once to leave one +and the greatest for a time behind; and partly because the method is +such as may hereafter give farther light to the finding out of the exact +truth. + +But the principles of the professors that reprehend these of mine, are +some of them so void of sense, that a man at the first hearing, whether +geometrician or not geometrician, must abhor them. As for example: + +1. That two equal proportions are not double to one of the same +proportions. + +2. That a proportion is double, triple, &c. of a number, but not of a +proportion. + +3. That the same body, without adding to it, or taking from it, is +sometimes greater, and sometimes less. + +4. That a quantity may grow less and less eternally, so as at last to be +equal to another quantity; or, which is all one, that there is a last in +eternity. + +5. That the nature of an angle consisteth in that which lies between the +lines that comprehend the angle in the very point of their concourse, +that is to say, an angle is the superficies which lies between the two +points which touch, or, as they understand a point, the superficies that +lies between the two nothings which touch. + +6. That the quotient is the proportion of the division to the dividend. + +Upon these and some such other principles is grounded all that Dr. +Wallis has said, not only in his _Elenchus_ of my geometry, but also in +his treatises of the _Angle of Contact_, and in his _Arithmetica +Infinitorum_; which two last I have here in two or three leaves wholly +and clearly confuted. And I verily believe that since the beginning of +the world, there has not been, nor ever shall be, so much absurdity +written in geometry, as is to be found in those books of his; with which +there is so much presumption joined, that an ἀποκατάϛασις of the like +conjunction cannot be expected in less than a Platonic year. The cause +whereof I imagine to be this, that he mistook the study of _symbols_ for +the study of _geometry_, and thought symbolical writing to be a new kind +of method, and other men’s demonstrations set down in symbols new +demonstrations. The way of analysis by squares, cubes, &c., is very +ancient, and useful for the finding out whatsoever is contained in the +nature and generation of rectangled planes, which also may be found +without it, and was at the highest in Vieta; but I never saw anything +added thereby to the science of geometry, as being a way wherein men go +round from the equality of rectangled planes to the equality of +proportion, and thence again to the equality of rectangled planes, +wherein the symbols serve only to make men go faster about, as greater +wind to a windmill. + +It is in sciences as in plants; growth and branching is but the +generation of the root continued; nor is the invention of theorems +anything else but the knowledge of the construction of the subject +prosecuted. The unsoundness of the branches are no prejudice to the +roots, nor the faults of theorems to the principles. And active +principles will correct false theorems if the reasoning be good; but no +logic in the world is good enough to draw evidence out of false or +unactive principles. But I detain your Lordship too long. For all this +will be much more manifest in the following discourses, wherein I have +not only explained and rectified many of the most important principles +of geometry, but also by the examples of those errors which have been +committed by my reprehenders, made manifest the evil consequence of the +principles they now proceed on. So that it is not only my own defence +that I here bring before you, but also a positive doctrine concerning +the true grounds, or rather atoms of geometry, which I dare only say are +very singular, but whether they be very good or not, I submit to your +Lordship’s judgment. And seeing you have been pleased to bestow so much +time, with great success, in the reading of what has been written by +other men in all kinds of learning, I humbly pray your Lordship to +bestow also a little time upon the reading of these few and short +lessons; and if your Lordship find them agreeable to your reason and +judgment, let me, notwithstanding the clamour of my adversaries, be +continued in your good opinion, and still retain the honour of being, + + My most noble Lord, + Your Lordship’s most + humble and obliged servant, + THOMAS HOBBES. + +LONDON, _June 10, 1656_. + + + + + LESSONS + + OF + + THE PRINCIPLES OF GEOMETRY, &c. + + TO THE EGREGIOUS PROFESSORS OF THE MATHEMATICS, ONE OF + GEOMETRY, THE OTHER OF ASTRONOMY, IN THE CHAIRS SET + UP BY THE NOBLE AND LEARNED SIR HENRY SAVILE, + IN THE UNIVERSITY OF OXFORD. + + + LESSON I. + + +I suppose, most egregious professors, you know already that by geometry, +though the word import no more but the measuring of land, is understood +no less the measuring of all other quantity than that of bodies. And +though the definition of geometry serve not for proof, nor enter into +any geometrical demonstration, yet for understanding of the principles +of the science, and for a rule to judge by, who is a geometrician, and +who is not, I hold it necessary to begin therewith. + +Geometry is the science of determining the quantity of anything, not +measured, by comparing it with some other quantity or quantities +measured. Which science therefore whosoever shall go about to teach, +must first be able to tell his disciple what measuring or dimension is; +by what each several kind of quantity is measured; what quantity is, and +what are the several kinds thereof. Therefore as they, who handle any +one part of geometry, determine by definition the signification of every +word which they make the subject or predicate of any theorem they +undertake to demonstrate; so must he which intendeth to write a whole +body of geometry, define and determine the meaning of whatsoever word +belongeth to the whole science. The design of Euclid was to demonstrate +the properties of the five regular bodies mentioned by Plato; in which +demonstrations there was no need to allege for argument the definition +of quantity, which it may be was the cause he hath not anywhere defined +it, but done what he undertook without it. And though having perpetually +occasion to speak of measure, he hath not defined measure; yet instead +thereof he hath, in the beginning of his first elements, assumed an +axiom which serveth his turn sufficiently as to the measure of lines, +which is the eighth axiom; that those things which lie upon one another +all the way (called by him ἐφαρμόζοντα) are equal. Which axiom is +nothing else but a description of the art of measuring length and +superficies. For this ἐφάρμοσις can have no place in solid bodies, +unless two bodies could at the same time be in one place. But amongst +the principles of geometry universal, the definitions are necessary, +both of quantity and dimensions. + +Quantity is that which is signified by what we answer to him that +asketh, _how much_ any thing is? and thereby determine the magnitude +thereof. For magnitude being a word indefinite, if a man ask of a thing, +_quantum est?_ that is, _how much_ it is, we do not satisfy him by +saying it is magnitude or quantity, but by saying it is _tantum_, _so +much_. And they that first called it in Greek, πηλικότης, and in Latin +_quantity_, might more properly have called it in Latin _tantity_, and +in Greek τηλικότης; and we, if we allowed ourselves the eloquence of the +Greeks and Latins, should call it the _so-muchness_. + +There is therefore to everything concerning which a man may ask without +absurdity, _how much it is_, a certain quantity belonging, determining +the magnitude to be _so much_. Also wheresoever there is _more_ and +_less_, there is one kind of quantity or other. And first there is the +quantity of bodies, and that of three kinds: length, which is by one way +of measuring; superficies, made of the complication of two lengths, or +the measure taken two ways; and solid, which is the complication of +three lengths, or of the measure taken three ways, for breadth or +thickness are but other lengths. And the science of geometry, so far +forth as it contemplateth bodies only, is no more but by measuring the +length of one or more lines, and by the position of others known in one +and the same figure, to determine by ratiocination, how much is the +superficies; and by measuring length, breadth, and thickness, to +determine the quantity of the whole body. Of this kind of magnitudes and +quantities the subject is body. + +And because for the computing of the magnitudes of bodies, it is not +necessary that the bodies themselves should be present, the ideas and +memory of them supplying their presence, we reckon upon those imaginary +bodies, which are the quantities themselves, and say the length is so +great, the breadth so great, &c. which in truth is no better than to say +the length is so long, or the breadth so broad, &c. But in the mind of +an intelligent man it breedeth no mistake. + +Besides the quantity of bodies, there is a quantity of time. For seeing +men, without absurdity, do ask how much it is; by answering _tantum_, +_so much_, they make manifest there is a quantity that belongeth unto +time, namely, a length. And because length cannot be an accident of +time, which is itself an accident, it is the accident of a body; and +consequently the length of the time, is the length of the body; by which +length or line, we determine how much the time is, supposing some body +to be moved over it. + +Also we not improperly ask with _how much_ swiftness a body is moved; +and consequently there is a quantity of motion or swiftness, and the +measure of that quantity is also a line. But then again, we must suppose +another motion, which determineth the time of the former. Also of force, +there is a question of _how much_, which is to be answered by _so much_; +that is, by quantity. If the force consist in swiftness, the +determination is the same with that of swiftness, namely, by a line; if +in swiftness and quantity of body jointly, then by a line and a solid; +or if in quantity of body only, as weight, by a solid only. + +So also is number quantity; but in no other sense than as a line is +quantity divided into equal parts. + +Of an angle, which is of two lines whatsoever they be, meeting in one +point, the digression or openness in other points, it may be asked how +great is that digression? This question is answered also by quantity. An +angle therefore hath quantity, though it be not the subject of quantity; +for the body only can be the subject, in which body those straddling +lines are marked. + +And because two lines may be made to divaricate by two causes; one, when +having one end common and immoveable, they depart one from another at +the other ends circularly, and this is called simply an angle; and the +quantity thereof is the quantity of the arch, which the two lines +intercept. + +The other cause is the bending of a straight line into a circular or +other crooked line, till it touch the place of the same line, whilst it +was straight, in one only point. And this is called an angle of +contingence. And because the more it is bent, the more it digresseth +from the tangent, it may be asked _how much_ more? And therefore the +answer must be made by quantity; and consequently an angle of +contingence hath its quantity as well as that which is called simply an +angle. And in case the digression of two such crooked lines from the +tangent be uniform, as in circles, the quantity of their digression may +be determined. For, if a straight line be drawn from the point of +contact, the digression of the lesser circle will be to the digression +of the greater circle, as the part of the line drawn from the point of +contact, and intercepted by the circumference of the greater circle is +to the part of the same line intercepted by the circumference of the +lesser circle, or, which is all one, as the greater radius is to the +lesser radius. You may guess by this what will become of that part of +your last book, where you handle the question of the quantity of an +angle of contingence. + +Also there lieth a question of _how much comparatively_ one magnitude is +to another magnitude, as how much water is in a tun in respect of the +ocean, how much in respect of a pint; _little_ in the first respect, +_much_ in the latter. Therefore the answer must be made by some +respective quantity. This respective quantity is called _ratio_ and +proportion, and is determined by the quantity of their differences; and +if their differences be compared also with the quantities themselves +that differ, it is called simply proportion, or proportion geometrical. +But if the differences be not so compared, then it is called proportion +arithmetical. And where the difference is none, there is no quantity of +the proportion, which in this case is but a bare comparison. + +Also concerning heat, light, and divers other qualities, which have +degrees, there lieth a question of _how much_, to be answered by a _so +much_, and consequently they have their quantities, though the same with +the quantity of swiftness: because the intensions and remissions of such +qualities are but the intensions and remissions of the swiftness of that +motion by which the agent produceth such a quality. And as quantity may +be considered in all the operations of nature, so also doth geometry run +quite through the whole body of natural philosophy. + +To the principles of geometry the definition appertaineth also of a +_measure_, which is this, _one quantity is the measure of another +quantity, when it, or the multiple of it, is coincident in all points +with the other quantity_. In which definition, instead of that ἐφαρμογὴ +of Euclid, I put coincidence. For the superposition of quantities, by +which they render the word ἐφαρμογὴ, cannot be understood of bodies, but +only of lines and superficies. Nevertheless many bodies may be +coincident successively with one and the same place, and that place will +be their measure, as we see practised continually in the measuring of +liquid bodies, which art of measuring may properly be called ἐφάρμοσις, +but not superposition. + +Also the definitions of _greater_, _less_, and _equal_, are necessary +principles of geometry. For it cannot be imagined than any geometrician +should demonstrate to us the equality and inequality of magnitudes, +except he tell us first what those words do signify. And it is a wonder +to me, that Euclid hath not anywhere defined what are equals, or at +least, what are equal bodies, but serveth his turn throughout with that +forementioned ἐφάρμοσις, which hath no place in solids, nor in time, nor +in swiftness, nor in circular, or other crooked lines; and therefore no +wonder to me, why geometry hath not proceeded to the calculation neither +of crooked lines, nor sufficiently of motion, nor of many other things, +that have proportion to one another. + +Equal bodies, superficies, and lines, are those of which every one is +capable of being coincident with the place of every one of the rest: and +equal times, wherein with one and the same motion equal lines are +described. And equally swift are those motions by which we run over +equal spaces in any time determined by any other motion. And universally +all quantities are equal, that are measured by the same number of the +same measures. + +It is necessary also to the science of geometry, to define what +quantities are of one and the same kind, which they call _homogeneous_, +the want of which definitions hath produced those wranglings (which your +book _De Angulo Contactus_ will not make to cease) about the angle of +contingence. + +_Homogeneous_ quantities are those which may be compared by (ἐφάρμοσις) +application of their measures to one another; so that solids and +superficies are heterogeneous quantities, because there is no +coincidence or application of those two dimensions. + +No more is there of line and superficies, nor of line and solid, which +are therefore heterogeneous. But lines and lines, superficies and +superficies, solids and solids, are homogeneous. + +Homogeneous also are line, and the quantity of time; because the +quantity of time is measured by the application of a line to a line; for +though time be no line, yet the quantity of time is a line, and the +length of two times is compared by the length of two lines. + +Weight and solid have their quantity homogeneous, because they measure +one another by application, to the beam of a balance. Line and angle +simply so called, have their quantity homogeneous, because their measure +is an arch or arches of a circle applicable in every point to one +another. + +The quantity of an angle simply so called, and the quantity of an angle +of contingence are heterogeneous. For the measures by which two angles +simply so called are compared, are in two coincident arches of the same +circle; but the measure by which an angle of contingence is measured, is +a straight line intercepted between the point of contact and the +circumference of the circle; and therefore one of them is not applicable +to the other; and consequently of these two sorts of angles the +quantities are heterogeneous. The quantities of two angles of +contingence are homogeneous; for they may be measured by the ἐφάρμοσις +of two lines, whereof one extreme is common, namely, the point of +contact, the other extremes are in the arches of the two circles. + +Besides this knowledge of what is quantity and measure, and their +several sorts, it behoveth a geometrician to know why, and of what, they +are called principles. For not every proposition that is evident is +therefore a principle. A principle is the beginning of something. And +because definitions are the beginnings or first propositions of +demonstration, they are therefore called principles, principles, I say, +of demonstration. But there be also necessary to the teaching of +geometry other principles, which are not the beginnings of +demonstration, but of construction, commonly called petitions; as that +it may be granted _that a man can draw a straight line, and produce it; +and with any radius, on any centre describe a circle_, and the like. For +that a man may be able to describe a square, he must first be able to +draw a straight line; and before he can describe an equilateral +triangle, he must be able first to describe a circle. And these +petitions are therefore properly called principles, not of +demonstration, but of operation. As for the commonly received third sort +of principles, called _common notions_, they are principles, only by +permission of him that is the disciple; who being ingenuous, and coming +not to cavil but to learn, is content to receive them, though +demonstrable, without their demonstrations. And though definitions be +the only principles of demonstration, yet it is not true that every +definition is a principle. For a man may so precisely determine the +signification of a word as not to be mistaken, yet may his definition be +such as shall never serve for proof of any theorem, nor ever enter into +any demonstration, such as are some of the definitions of Euclid, and +consequently can be no beginnings of demonstration, that is to say, no +principles. + +All that hitherto hath been said, is so plain and easy to be understood, +that you cannot, most egregious professors, without discovering your +ignorance to all men of reason, though no geometricians, deny it. And +the same (saving that the words are all to be found in dictionaries) +new; also to him that means to learn, not only the practice, but also +the science of geometry necessary, and, though it grieve you, mine. And +now I come to the definitions of Euclid. + +The first is of a point: Σημεῖον, &c. “_Signum est, cujus est pars +nulla_,” that is to say, _a mark is that of which there is no part_. +Which definition, not only to a candid, but also to a rigid construer, +is sound and useful. But to one that neither will interpret candidly, +nor can interpret accurately, is neither useful nor true. Theologers say +the soul hath no part, and that an angel hath no part, yet do not think +that soul or angel is a point. A mark or as some put instead of it +ϛίγμη, which is a mark with a hot iron, is visible; if visible, then it +hath quantity, and consequently may be divided into parts innumerable. +That which is indivisible is no quantity; and if a point be not +quantity, seeing it is neither substance nor quality, it is nothing. And +if Euclid had meant it so in his definition, as you pretend he did, he +might have defined it more briefly, but ridiculously, thus, _a point is +nothing_. Sir Henry Savile was better pleased with the candid +interpretation of Proclus, that would have it understood respectively to +the matter of geometry. But what meaneth this _respectively to the +matter of geometry_? It meaneth this, that no argument in any +geometrical demonstration should be taken from the division, quantity, +or any part of a point; which is as much as to say, a point is that +whose quantity is not drawn into the demonstration of any geometrical +conclusion; or, which is all one, whose quantity is not considered. + +An accurate interpreter might make good the definition thus, _a point is +that which is undivided_; and this is properly the same with _cujus non +est pars_: for there is a great difference between _undivided_ and +_indivisible_, that is, between _cujus non est pars_, and _cujus non +potest esse pars_. Division is an act of the understanding; the +understanding is therefore that which maketh parts, and there is no part +where there is no consideration but of one. And consequently Euclid’s +definition of a point is accurately true, and the same with mine, which +is, that _a point is that body whose quantity is not considered_. And +_considered_ is that, as I have defined it chap. I. at the end of the +third article, which is not put to account in demonstration. + +Euclid therefore seemeth not to be of your opinion, that say a point is +nothing. But why then doth he never use this definition in the +demonstration of any proposition? Whether he useth it expressly or no, I +remember not; but the sixteenth proposition of the third book without +the force of this definition is undemonstrated. + +The second definition is of a line: γραμμὴ δὲ μῆκος ἂπλατες. “_Linea est +longitudo latitudinis expers_; _a line is length which hath no +breadth_;” and if candidly interpreted, sound enough, though rigorously +not so. For to what purpose is it to say _length not broad_, when there +is no such thing as a _broad length_. One path may be broader than +another path, but not one mile than another mile; and it is not the path +but the mile which is the way’s length. If therefore a man have any +ingenuity he will understand it thus, _that a line is a body whose +length is considered without its breadth_, else we must say absurdly a +_broad length_; or untruly, that there be bodies which have length and +yet no breadth; and this is the very sense which Apollonius, saith +Proclus, makes of this definition; “when we measure,” says he, “the +length of a way, we take not in the breadth or depth, but consider only +one dimension.” See this of Proclus cited by Sir Henry Savile, where you +shall find the very word _consider_. + +The fourth definition is of a straight line, thus Ἐυθεῖα γραμμή ἐϛιν, +&c. “_Recta linea est quæ ex æquo sua ipsius puncta inter jacet._” _A +straight line is that which lieth equally (or perhaps evenly) between +its own points._ This definition is inexcusable. Between what points of +its own can a straight line lie but between its extremes? And how lies +it evenly between them, unless it swerve no more from some other line +which hath the same extremes, one way than another? And then why are not +between the same points both the lines straight? How bitterly, and with +what insipid jests would you have reviled Euclid for this, if living now +he had written a _Leviathan_ . And yet there is somewhat in this +definition to help a man, not only to conceive the nature of a straight +line (for who doth not conceive it?) but also to express it. For he +meant perhaps to call a straight line that which is all the way from one +extreme to another, equally distant from any two or more such lines as +being like and equal have the same extremes. So the axis of the earth is +all the way equally distant from the circumference of any two or more +meridians. But then before he had defined a straight line, he should +have defined what lines are _like_, and what are _equal_. But it had +been best of all, first to have defined crooked lines, by the +possibility of a deduction or setting further asunder of their extremes; +and then straight lines, by the impossibility of the same. + +The seventh definition, which is that of a plain superficies, hath the +same faults. + +The eighth is of a plane angle, Ἑπὶπεδος γωνία ἑϛὶν ἡ ἐν ἐπιπέδω, &c. +“_Angulus planus est duarum linearum in plano se mutuo tangentium, et +non in directum jacentium, alterius ad alteram inclinatio._” _A plane +angle is the inclination one towards another of two lines that touch one +another in the same plane, and lie not in the same straight line._ +Besides the faults here observed by Sir Henry Savile, as the clause of +not lying in the same straight line, and the obscurity or equivocation +of the word _inclination_, there is yet another, which is, that by this +definition two right angles together taken, are no angle; which is a +fault which you somewhere (asking leave to use the word _angle_, +καταχριϛικώς acknowledge, but avoid not. For in geometry, where you +confess there is required all possible accurateness, every καταχρῆσις is +a fault. Besides you see by this definition, that Euclid is not of your, +but of Clavius’s opinion. For it is manifest that the two lines which +contain an angle of contact incline one towards another, and come +together in a point, and lie not in the same straight line, and +consequently make an angle. + +The thirteenth definition is exact, but makes against your doctrine, +that a point is nothing. Examine it. Ὅρος ἐϛὴν ὅ τινός ἐϛῖ πέρας. +“_Terminus est quod alicujus extremum est._” _A term or bound is that +which is the extreme of anything._ We had before, _the extremes of a +line are points_. But what is in a line the extreme, but the first or +last _part_, though you may make that part as small as you will? A point +is therefore a part, and nothing is no extreme. + +The fourteenth, Σχῆμα ἐϛὶ τὸ ὑπὸ τινος ἤ τινῶν ὅρων περιεχόμενον. +“_Figura est (subaudi quantitas) quæ ab aliquo, vel aliquibus terminis +undique continetur sive clauditur._” _A figure is quantity contained +within some bound or bounds._ Or shortly thus, _a figure is quantity +every way determined_, is in my opinion as exact a definition of a +figure as can possibly be given, though it must not be so in yours. For +this _determination_ is the same thing with _circumscription_; and +whatsoever is anywhere _(ubicunque) definitivè_ is there also +_circumscriptivè_; and by this means the distinction is lost, by which +theologers, when they deny God to be in any place, save themselves from +being accused of saying he is nowhere; for that which is nowhere is +nothing. This definition of Euclid cannot therefore possibly be embraced +by you that carry double, namely, mathematics and theology. For if you +reject it, you will be cast out of all mathematic schools; and if you +maintain it, from the society of all school-divines, and lose the thanks +of the favour you have shown (you the astronomer) to Bishop Bramhall. + +The fifteenth is of a circle. Κοὐκλος ἐστὶ σχῆμα ἐπίπεδον, &c. _A circle +is a plain figure comprehended by one line which is called the +circumference, to which circumference all the straight lines drawn from +one of the points within the figure are equal to one another._ This is +true. But if a man had never seen the generation of a circle by the +motion of a compass or other equivalent means, it would have been hard +to persuade him that there was any such figure possible. It had been +therefore not amiss first to have let him see that such a figure might +be described. Therefore so much of geometry is no part of philosophy, +which seeketh the proper passions of all things in the generation of the +things themselves. + +After the fifteenth till the last or thirty-fifth definition, all are +most accurate, but the last which is this, _parallel straight lines are +those which being in the same plane, though infinitely produced both +ways, shall never meet_. Which is less accurate. For how shall a man +know that there be straight lines which shall never meet, though both +ways infinitely produced? Or how is the definition of parallels, that +is, of lines perpetually equidistant, good, wherein the nature of +equidistance is not signified? Or if it were signified, why should it +not comprehend as well the parallelism of circular and other crooked +lines, as of straight, and as well of superficies, as of lines? By +parallels is meant equidistant both lines and superficies, and the word +is therefore not well defined without defining first equality of +distance. And because the distance between two lines or superficies, is +the shortest line that can join them, there either ought to be in the +definition the _shortest distance_, which is that of the perpendicular +and without inclination, or the distance in equal inclination, that is, +in equal angles. Therefore if parallels be defined to be those lines or +superficies, where the lines drawn from one to another in equal angles +be equal, the definition, as to like lines, or like superficies, will be +universal and convertible. And if we add to this definition, that the +equal angles be drawn not opposite ways, it will be absolute, and +convertible in all lines and superficies; and the definition will be +this: _parallels are those lines and superficies between which every +line drawn, in any angle, is equal to any other line drawn in the same +angle the same way_. For by this definition the distance between them +will perpetually be equal, and consequently they will never come nearer +together, how much, or which way soever they be produced. And the +converse of it will be also true, _if two lines, or two superficies be +parallel, and a straight line be drawn from one to the other, any other +straight line, drawn from one to the other in the same angle, and the +same way, will be equal to it_. This is manifestly true, and, most +egregious professors, new, at least to you. + +And thus much for the definitions placed before the first of Euclid’s +Elements. + +Before the third of his Elements is this definition: “_In circulo +æqualiter distare a centro rectæ lineæ dicuntur, cum perpendiculares quæ +a centro in ipsas ducuntur sunt æquales_.” _In a circle two straight +lines are said to be equally distant from the centre, upon which the +perpendiculars drawn from the centre are equal._ This is true; but it is +rather an axiom than a definition, as being demonstrable that the +perpendicular is the measure of the distance between a point and a +straight or a crooked line. + +Before the fifth Element the first definition is of a part: _Pars est +magnitudo magnitudinis, minor majoris, cum minor metitur majorem_. _A +part is one magnitude of another, the less of the greater, when the less +measureth the greater._ From which definition it followeth, that more +than a half is not a part of the whole. But because Euclid meaneth here +an aliquot part, as a half, a third, or a fourth, &c., it may pass for +the definition of a measure under the name of part, as thus: _a measure +is a part of the whole, when multiplied it may be equal to the whole_, +though properly a measure is external to the thing measured, and not the +aliquot part itself, but equal to an aliquot part. + +But the third definition is intolerable; it is the definition of λόγος, +in Latin _ratio_, in English, _proportion_, in this manner, λόγος ἐςὶ +δύο μεγεθῶν ὁμογενῶν ῆ κατὰ πηλικότητα προς ἄλληλα ποιὰ σχέσις. “_Ratio +est duarum magnitudinum ejusdem generis mutua quædam secundum +quantitatem habitudo._” _Proportion is a certain mutual habitude in +quantity, of two magnitudes of the same kind, one to another._ First, we +have here _ignotum per ignotius_; for every man understandeth better +what is meant by _proportion_ than by habitude. But it was the phrase of +the Greeks when they named like proportions, to say, the first to the +second, οὕτως ἔχει, _id est, ita se habet_, and in English, _is as_, the +third to the fourth. As for example, in the proportions of two to four, +and three to six, to say two to four, οὕτως ἔχει, _id est, ita se habet, +id est_, _is as_, three to six. From which phrase Euclid made this his +definition of proportion by ποιὰ σχέσις, which the Latins translate +_quædam habitudo_. _Quædam_ in a definition is a most certain note of +not understanding the word _defined_; and in Greek, ποιὰ σχέσις is much +worse; for to render rightly the Greek definition, we are to say in +English, that proportion is a what-shall-I-call-it-_isness_, or _soness_ +of two magnitudes, &c.; than which nothing can be more unworthy of +Euclid. It is as bad as anything was ever said in geometry by Orontius, +or by Dr. Wallis. That proportion is quantity compared, that is to say, +little or great in respect of some other quantity, as I have above +defined it, is I think intelligible. + +The fourth is, Ἀναλογία δέ ὲστιν ῆ των λόγων ὁμοιότης. “_Proportio vero +est rationum similitudo._” Here we have no one word by which to render +Ἀναλογία; for our word _proportion_ is already bestowed upon the +rendering of λόγος. Nevertheless the Greek may be translated into +English thus, _iterated proportions_. But iterated proportion is the +same with _eadem ratio_. To what purpose then serveth the sixth +definition, which is of _eadem ratio_? For Ἀναλογία and _eadem ratio_ +and _similitudo rationum_, are the same thing, as appeareth by Euclid +himself, where he defines those quantities, that are in the same +proportion by ἀνάλογον. Therefore the sixth definition is but a _lemma_, +and assumed without demonstration. + +The fourteenth, “_Compositio rationis est sumptio antecedentis cum +consequente, ceu unius, ad ipsum consequentem_,” _To compound +proportion, is to take both antecedent and consequent together as one +magnitude, and compare it to the consequent_, is good; though he might +have compared it as well with the antecedent; for both ways it had been +a composition of proportion. We are to note here, that the composition +defined in this place by Euclid is not adding together of proportions, +but of two quantities that have proportion. And therefore it is not the +same composition which he defineth in the fourth place before the sixth +element, for there he defineth the addition of one proportion to another +proportion in this manner: λόγος ἐκ λόγων συγκεῖσθαι λέγεται, &c. _A +proportion is said to be compounded of proportions, when their +quantities multiplied into one another make a proportion_; as when we +would compound or add together the proportions of three to two, and of +four to five, we must multiply three and four, which maketh twelve, and +two and five, which maketh ten. And then the proportion of twelve to ten +is the sum of the proportions of three to two, and of four to five, +which is true, but not a definition; for it may and ought to be +demonstrated. For to define what is addition of two proportions (which +are always in four quantities, though sometimes one of them be twice +named) we are to say, that they are then added together when we make the +second to another in the same proportion, which the third hath to the +fourth. + +And thus much of the definitions; of which some, very few, you see are +faulty; the rest either accurate, or good enough if well interpreted. +For the rest of the elements all are accurate, notwithstanding that you +allow not for good any definition in geometry that hath in it the word +_motion_, of which there be divers before the eleventh Element. But I +must here put you in mind, that geometry being a science, and all +science proceeding from a precognition of causes, the definition of a +sphere, and also of a circle, by the generation of it, that is to say, +by motion, is better than by the equality of distance from a point +within. + +The second sort of principles are those of construction, usually called +_postulata_, or petitions. As for those _notiones communes_, called +_axioms_, they are from the definitions of their terms demonstrable, +though they be so evident as they need not demonstration. These +petitions are by Euclid called Ἀιτήματα, such as are granted by favour, +that is, simply petitions, whereas by axiom is understood that which is +claimed as due. So that between Ἀξίωμα and Ἀίτημα there is this other +difference, that this latter is simply a petition, the former a petition +of right. + +Of petitions simply, the first is, _that from any point to any point may +be drawn a straight line_. The second, _that a finite straight line may +be produced_. The third, _that upon any centre at any distance may be +described a circle_. All which are both evident and necessary to be +granted. + +And by all these a man may easily perceive that Euclid in the +definitions of a point, a line, and a superficies, did not intend that a +point should be nothing, or a line be without latitude, or a superficies +without thickness; for if he did, his petitions are not only +unreasonable to be granted, but also impossible to be performed. For +lines are not drawn but by motion, and motion is of body only. And +therefore his meaning was, that the quantity of a point, the breadth of +a line, and the thickness of a superficies were not to be _considered_, +that is to say, not to be reckoned in the demonstration of any theorems +concerning the quantity of bodies, either in length, superficies, or +solid. + + ========== + + OF THE FAULTS THAT OCCUR IN + DEMONSTRATION. + + TO THE SAME EGREGIOUS PROFESSORS OF THE MATHEMATICS IN + THE UNIVERSITY OF OXFORD. + + + LESSON II. + +There be but two causes from which can spring an error in the +demonstration of any conclusion in any science whatsoever; and those are +ignorance or want of understanding, and negligence. For as in the adding +together of many and great numbers, he cannot fail that knoweth the +rules of addition, and is also all the way so careful, as not to mistake +one number or one place for another; so in any other science, he that is +perfect in the rules of logic, and is so watchful over his pen, as not +to put one word for another, can never fail of making a true, though not +perhaps the shortest and easiest demonstration. + +The rules of demonstration are but of two kinds: one, that the +principles be true and evident definitions; the other, that the +inferences be necessary. And of true and evident definitions, the best +are those which declare the cause or generation of that subject, whereof +the proper passions are to be demonstrated. For science is that +knowledge which is derived from the comprehension of the cause. But when +the cause appeareth not, then may, or rather must we define some known +property of the subject, and from thence derive some possible way, or +ways, of the generation. And the more ways of generation are explicated, +the more easy will be the derivation of the properties; whereof some are +more immediate to one, some to another generation. He therefore that +proceedeth from untrue, or not understood definitions, is ignorant of +that he goes about; which is an ill-favoured fault, be the matter he +undertaketh easy or difficult, because he was not forced to undergo a +greater charge than he could carry through. But he that from right +definitions maketh a false conclusion, erreth through human frailty, as +being less awake, more troubled with other thoughts, or more in haste +when he was in writing. Such faults, unless they be very frequent, are +not attended with shame, as being common to all men, or are at least +less ugly than the former, except then, when he that committeth them +reprehendeth the same in other men. For that is in every man +intolerable, which he cannot tolerate in another. But to the end that +the faults of both kinds may by every man be well understood, it will +not be amiss to examine them by some such demonstrations as are publicly +extant. And for this purpose I will take such as are in mine and in your +books, and begin with your _Elenchus_ of the geometry contained in my +book _De Corpore_; to which I will also join your book lately set forth +concerning the _Angle of Contact, Conic Sections_ , and your +_Arithmetica Infinitorum_; and then examine the rest of my philosophy, +and yours that oppugn it. For I will take leave to consider you both +everywhere as one author, because you publicly declare your approbation +of one another’s doctrine. + +My first definition is of a line, of length, and of a point. “The way,” +say I, “of a body moved, in which magnitude (though it always have some +magnitude) is not considered, is called a line; and the space gone over +by that motion, length, or one and a simple dimension.” To this +definition you say, first, “what mathematician did ever thus define a +line or length?” Whether you call in others for help or testimony, it is +not done like a geometrician; for they use not to prove their +conclusions by witnesses, but rely upon the strength of their own +reason; and when your witnesses appear, they will not take your part. +Secondly, you grant that what I say is true, but not a definition. But +to tell you truly what it is which we call a line, is to define a line. +Why then is not this a definition? “Because,” say you in the first +place, “it is not a reciprocal proposition.” But by your favour it is +reciprocal. For not only the way of a body whose quantity is not +considered is a line, but also every line is, or may be conceived to be, +the way of a body so moved. And if you object that there is a difference +between _is_ and _may be conceived to be_, Euclid, whom you call to your +aid, will be against you in the fourteenth definition before his +eleventh Element; where he defines a sphere just as convertibly as I +define a line; except you think the globes of the sun and stars cannot +be globes, unless they were made by the circumduction of a semicircle; +and again in the eighteenth definition, which is of a cone, unless you +admit no figure for a cone, which is not generated by the revolution of +a triangle; and again, in the twentieth definition, which is of a +cylinder, except it be generated by the circumvolution of a +parallelogram. Euclid saw that what proper passion soever should be +derived from these his definitions, would be true of any other cylinder, +sphere, or cone, though it were otherwise generated; and the description +of the generation of any one being by the imagination applicable to all, +which is equivalent to convertible, he did not believe that any rational +man could be misled by learning logic to be offended with it. Therefore +this exception proceedeth from want of understanding, that is, from +ignorance of the nature, and use of a definition. + +Again, you object and ask: “What need is there of motion, or of body +moved, to make a man understand what is a line? Are not lines in a body +at rest, as well as in a body moved? And is not the distance of two +resting points length, as well as the measure of the passage? Is not +length one and a simple dimension, and one and a simple dimension line? +Why then is not line and length all one?” See how impertinent these +questions are. Euclid defines a sphere to be a solid figure described by +the revolution of a semicircle about the unmoved diameter. Why do you +not ask, what need there is to the understanding of what a sphere is, to +bring in the motion of a semicircle? Is not a sphere to be understood +without such motion? Is not the figure so made a sphere without this +motion? And where he defines the axis of a sphere to be that unmoved +diameter, may not you ask, whether there be no axis of a sphere, when +the whole sphere, diameter and all, is in motion? But it is not to my +purpose to defend my definition by the example of that of Euclid. +Therefore first, I say, to me, howsoever it may be to others, it was fit +to define a line by motion. For the generation of a line is the motion +that describes it. And having defined philosophy in the beginning, to be +the knowledge of the properties from the generation, it was fit to +define it by its generation. And to your question, _is not distance +length?_ I answer, that though sometimes distance be equivalent to +length, yet certainly the distance between the two ends of a thread +wound up into a clue is not the length of the thread; for the length of +the thread is equal to all the windings whereof the clue is made. But if +you will needs have distance and length to be all one, tell me of what +the distance between any two points is the length. Is it not the length +of the way? And how is that called way, which is not defined by some +motion? And have not several ways between the same places, as by land +and by water, several lengths? But they have but one distance, because +the distance is the shortest way. Therefore between the length of the +path, and the distance of places, there is a real difference in this +case, and in all cases a difference of the consideration. Your +objection, that line is longitude, proceeds from want of understanding +English. Do men ever ask what is the line of a thread, or the line of a +table, or of any other body? Do they not always ask what is the length +of it? And why, but because they use their own judgments, not yet +corrupted by the subtlety of mistaken professors. Euclid defines a line +to _be length without breadth_. If those terms be all one, why said he +not that a _line is a line without breadth_? But what definition of a +line give you? None. Be contented then with such as you receive, and +with this of mine, which you shall presently see is not amiss. + +Your next objections are to my definition of a point. Which definition +adhereth to the former in these words, “and the body itself is called a +point.” Here again you call for help: “_Quis unquum mortalium, etc._ +What mortal man, what sober man, did ever so define a point?” It is +well, and I take it to be an honour to be the first that do so. But what +objection do you bring against it. This: “That a point added to a point, +if it have magnitude, makes it greater.” I say it doth so, but then +presently it loseth the name of a point, which name was given to signify +that it was not the meaning of him that used it in demonstration to add, +subtract, multiply, divide, or any way compute it. Then you come in +with, “perhaps you will say though it have magnitude, that magnitude is +not considered.” You need not say _perhaps_. You know I affirm it; and +therefore your argument might have been left out, but that it gave you +an occasion of a digression into scurvy language. + +And whereas you ask why I defined not a point thus: “_Punctum est corpus +quod non consideratur esse corpus, et magnum quod non consideratur esse +magnum_.” I will tell you why. First, because it is not Latin. Secondly, +because when I had defined it by _corpus_, there was no need to define +it again by _magnum_. I understand very well this language, “_punctum +est corpus, quod non consideratur ut corpus_.” A point is a body not +considered as body. But _punctum est corpus, quod non consideratur esse +corpus, vel esse magnum_, is not Latin; nor the version of it, _a point +is a body which is not considered to be a body_, English. My definition +was, that a point is that body whose magnitude is not considered, not +reckoned, not put to account in demonstration. And I exemplified the +same by the body of the earth describing the ecliptic line; because the +magnitude is not there reckoned nor chargeth the ecliptic line with any +breadth. But I perceive you understand not what the word _consideration_ +signifieth, but take it for comparison or relation; and say I ought to +define a point simply, and not by relation to a great body; as if to +reckon and to compare were the same thing. “_Omnia mihi_,” saith Cicero, +“_provisa et considerata sunt_.” I have provided and reckoned +everything. There is a great difference between reckoning and relation. + +Again, you ask, why _corpus motum_, a body moved? I will tell you; +because the motion was necessary for the generation of a line. And +though after the generation of the line the point should rest, yet it is +not necessary from this definition that it should be no more a point; +nor when Euclid defines a sphere by the circumduction of a semicircle +upon an axis that resteth, doth it follow from thence when the sphere, +axis, centre and all, as that of the earth, is moved from place to +place, that it is no more an axis. + +Lastly, you object “that motion is accidentary to a point, and +consequently not essential, nor to be put into the definition.” And is +not the circumduction of a semicircle accidentary to a sphere? Or do you +think the sphere of the sun was generated by the revolution of a +semicircle? And yet it was thought no fault in Euclid to put the motion +into the definition of a sphere. + +The conceit you have concerning definitions, that they must explicate +the essence of the thing defined, and must consist of a _genus_ and a +_difference_, is not so universally true as you are made believe, or +else there be very many insufficient definitions that pass for good with +you in Euclid. You are much deceived if you think these woful notions of +yours, and the language that doth everywhere accompany them, show +handsomely together. Or that such grounds as these be able to sustain so +many, and so haughty reproaches as you advance upon them, so as they +fall not, as you shall see immediately, upon your own head. I say a +point hath quantity, but not to be reckoned in demonstrating the +properties of lines, solids, or superficies; you say it hath no quantity +at all, but is plainly nothing. + +The first of the petitions of Euclid is, “that a line may be drawn from +point to point at any distance.” The second, “that a straight line may +be produced.” The third, “that on any centre a circle may be described +at any distance.” And the eighth axiom (which Sir H. Savile observes to +be the foundation of all geometry) is this, “_Quæ sibi mutuo congruunt, +etc._ Those things that are applied to one another in all points are +equal.” All or any of these principles being taken away, there is not in +Euclid one proposition demonstrated or demonstrable. If a point have no +quantity, a line can have no latitude; and because a line is not drawn +but by motion, by motion of a body, and body imprinteth latitude all the +way, it is impossible to draw or produce a straight line, or to describe +a circular line without latitude. Also if a line have no latitude, one +straight line cannot be applied to another. To them therefore that deny +a point to have quantity, that is, a line to have latitude, the +forenamed principles are not possible, and consequently no proposition +in geometry is demonstrated or demonstrable. You therefore that deny a +point to have quantity, and a line to have breadth, have nothing at all +of the science of geometry. The practice you may have, but so hath any +man that hath learned the bare propositions by heart; but they are not +fit to be professors either of geometry or of any other science that +dependeth on it. Some man perhaps may say that this controversy is not +much worth, and that we both mean the same thing. But that man, though +in other things prudent enough, knoweth little of science and +demonstration. For definitions are not only used to give us the notions +of those things whose appellations are defined, for many times they that +have no science have the ideas of things more perfect than such as are +raised by definitions. As who is there that understandeth not better +what a straight line is, or what proportion is, and what many other +things are, without definition, than some that set down the definitions +of them. But their use is, when they are truly and clearly made, to draw +arguments from them for the conclusions to be proved. And therefore you +that in your following censures of my geometry, take your argument so +often from this, that a point is nothing, and so often revile me for the +contrary, are not to be allowed such an excuse. He that is here +mistaken, is not to be called negligent in his expression, but ignorant +of the science. + +In the next place, you take exceptions to my definition of _equal +bodies_, which is this: “_Corpora æqualia sunt quæ eundem locum +possidere possunt_. Equal bodies are those which may have the same +place.” To which you object impertinently, that I may as well define a +man to be, _he that may be prince of Transylvania_, wittily, as you +count wit. Formerly in every definition, you exacted an explication of +the essence. You are therefore of opinion that the possibility of being +prince of Transylvania is no less essential to _a man_, than the +possibility of the being of two bodies successively in the same place, +is essential to _bodies equal_. + +You take no notice of the twenty-third article of this same chapter, +where I define what it is we call essence, namely, that accident for +which we give the thing its name. As the essence of a man is his +capacity of reasoning; the essence of a white body, whiteness, &c., +because we give the name of _man_ to such bodies as are capable of +reasoning, for that their capacity; and the name of _white_ to such +bodies as have that colour, for that colour. Let us now examine why it +is that men say bodies are one to another equal; and thereby we shall be +able to determine whether the _possibility of having the same place_ be +essential or not to _bodies equal_, and consequently whether this +definition be so like to the defining of a man by the _possibility of +being prince of Transylvania_ as you say it is. There is no man, besides +such egregious geometricians as yourselves, that inquireth the equality +of two bodies, but by measure. And for liquid bodies, or the aggregates +of innumerable small bodies, men (men, I say) measure them by putting +them one after another into the same vessel, that is to say, into the +same place, as Aristotle defines place, or into the space determined by +the vessel, as I define place. And the bodies that so fill the vessel, +they acknowledge and receive for equal. But though, when hard bodies +cannot be so measured, without the incommodity or trouble of altering +their figure, they then enquire, if the bodies are both of the same +kind, their equality by weight, knowing, without your teaching, that +equal bodies of the same nature weigh proportionably to their +magnitudes; yet they do it not for fear of missing of the equality, but +to avoid inconvenience or trouble. But you (you, I say), that have no +definition of equals, neither received from others, nor framed by +yourselves, out of your shallow meditation and deep conceit of your own +wits, contend against the common light of nature. So much is unheedy +learning a hinderance to the knowledge of the truth, and changeth into +elves those that were beginning to be men. + +Again, when men inquire the equality of two bodies in length, they +measure them by a common measure; in which measure they consider neither +breadth nor thickness, but how the length of it agreeth, first with the +length of one of the bodies, then with the length of the other. And both +the bodies whose lengths are measured, are successively in the same +place under their common measure. _Place_ therefore in lines also, is +the proper index and discoverer of equality and inequality. And as in +length, so it is in breadth and thickness, which are but lengths +otherwise taken in the same solid body. But now when we come from this +equality and inequality of lengths known by measure, to determine the +proportions of superficies and of solids, by ratiocination, then it is +that we enter into geometry; for the making of definitions, in +whatsoever science they are to be used, is that which we call +_philosophia prima_. It is not the work of a geometrician, as a +geometrician, to define what is equality, or proportion, or any other +word he useth, though it be the work of the same man, as a man. His +geometrical part is, to draw from them as many true and useful theorems +as he can. + +You object secondly, that a pyramis may be equal to a cube whilst it is +a pyramis. True. And so also whilst it is a pyramis it hath a +possibility by flexion and transposition of parts to become a cube, and +to be put into the place where another cube equal to it was before. This +is to argue like a child that hath not yet the perfect understanding of +any language. + +In the third and fourth objection, you teach me to define equal bodies +(if I will needs define them by place) by the _equality of place_, and +to say, _that bodies are equal that have equal places_. Teach others, if +you can, to measure their grain, not by the same, but equal bushels. + +In the fifth objection, you except against the the word _can_, in that I +say that bodies are equal which _can_ fill the same place. For the +greater body _can_, you say, fill the place of the less, though not +reciprocally the less of the greater. It is true, that though the place +of the less can never be the place of the greater, yet it may be filled +by a part of the greater. But it is not then the greater body that +filleth the place of the less, but a part of it, that is to say, a less +body. Howsoever, to take away from simple men this straw they stumble +at, I have now put the definition of equal bodies into these words: +_equal bodies are those whereof every one can fill the place of every +other_. And if my definition displease you, propound your own, either of +_equal bodies_, or of _equals_ simply. But you have none. Take therefore +this of mine. + +The sixth is a very admirable exception. “What,” say you, “if the same +body can sometimes take up a greater, sometimes a lesser place, as by +rarefaction and condensation?” I understand very well that bodies may be +sometimes thin and sometimes thick, as they chance to stand closer +together or further from one another. So in the mathematic schools, when +you read your learned lectures, you have a thick or thronging audience +of disciples, which in a great church would be but a very thin company. +I understand how thick and thin may be attributed to bodies in the +plural, as to a company; but I understand not how any one of them is +thicker in the school than in the church; or how any one of them taketh +up a greater room in the school, when he can get in, than in the street. +For I conceive the dimensions of the body, and of the place, whether the +place be filled with gold or with air, to be coincident and the same; +and consequently both the quantity of the air, and the quantity of the +gold, to be severally equal to the quantity of the place; and therefore +also, by the first axiom of Euclid, equal to one another; insomuch as if +the same air should be by condensation contained in a part of the place +it had, the dimensions of it would be the same with the dimensions of +part of the place, that is, should be less than they were, and by +consequence the quantity less. And then either the same body must be +less also, or we must make a difference between greater bodies and +bodies of greater quantity; which no man doth that hath not lost his +wits by trusting them with absurd teachers. When you receive salary, if +the steward give you for every shilling a piece of sixpence, and then +say, every shilling is condensed into the room of sixpence, I believe +you would like this doctrine of yours much the worse. You see how by +your ignorance you confound the affairs of mankind, as far forth as they +give credit to your opinions, though it be but little. For nature abhors +even empty words, such as are (in the meaning you assign them), +_rarefying_ and _condensing_. And you would be as well understood if you +should say (coining words by your own power), that the same body might +take up sometimes a greater, sometimes a lesser place, by wallifaction +and wardensation, as by rarefaction and condensation. You see how +admirable this your objection is. + +In the seventh objection you bewray another kind of ignorance, which is +the ignorance of what are the proper works of the several parts of +philosophy. “Though it were out of doubt,” say you, “that the same body +cannot have several magnitudes, yet seeing it is matter of natural +philosophy, nor hath anything to do with the present business, to what +purpose is it to mention it in a mathematical definition?” It seems by +this, that all this while you think it is a piece of the geometry of +Euclid, no less to make the definitions he useth, than to infer from +them the theorems he demonstrateth. Which is not true. For he that +telleth you in what sense you are to take the appellations of those +things which he nameth in his discourse, teacheth you but his language, +that afterwards he may teach you his art. But teaching of language is +not mathematic, nor logic, nor physic, nor any other science; and +therefore to call a definition, as you do, mathematical, or physical, is +a mark of ignorance, in a professor inexcusable. All doctrine begins at +the understanding of words, and proceeds by reasoning till it conclude +in science. He that will learn geometry must understand the terms before +he begin, which that he may do, the master demonstrateth nothing, but +useth his natural prudence only, as all men do when they endeavour to +make their meaning clearly known. For words understood are but the seed, +and no part of the harvest of philosophy. And this seed was it, which +Aristotle went about to sow in his twelve books of _metaphysics_, and in +his eight books concerning the hearing of _natural philosophy_. And in +these books he defineth time, place, substance or essence, quantity, +relation, &c., that from thence might be taken the definitions of the +most general words for principles in the several parts of science. So +that all definitions proceed from common understanding; of which, if any +man rightly write, he may properly call his writing _philosophia prima_, +that is, the seeds, or the grounds of philosophy. And this is the method +I have used, defining place, magnitude, and the other the most general +appellations in that part which I entitle _philosophia prima_. But you +now, not understanding this, talk of mathematical definitions. You will +say perhaps that others do the same as well as you. It may be so. But +the appeaching of others does not make your ignorance the less. + +In the eighth place you object not, but ask me _why I define equal +bodies apart_? I will tell you. Because all other things which are said +to be equal, are said to be so from the equality of bodies; as two lines +are said to be equal, when they be coincident with the length of one and +the same body; and equal times, which are measured by equal lengths of +body, by the same motion. And the reason is, because there is no subject +of quantity, or of equality, or of any other accident but body; all +which I thought certainly was evident enough to any uncorrupted +judgment; and therefore that I needed first to define equality in the +subject thereof, which is body, and then to declare in what sense it was +attributed to time, motion, and other things that are not body. + +The ninth objection is an egregious cavil. Having set down the +definition of _equal bodies_, I considered that some men might not allow +the attribute of equality to any things but those which are the subjects +of quantity, because there is no equality, but in respect of quantity. +And to speak rigidly, _magnum et magnitudo_ are not the same thing; for +that which is great, is properly a body, whereof greatness is an +accident. In what sense therefore, might you object, can an accident +have quantity? For their sakes therefore that have not judgment enough +to perceive in what sense men say the length is so long, or the +superficies so broad, &c. I added these words: “_Eadem ratione (qua +scilicet corpora dicuntur æqualia) magnitudo magnitudini æqualis +dicitur_,” that is, _in the same manner, as bodies are said to be equal, +their magnitudes also are said to be equal_. Which is no more than to +say, _when bodies are equal, their magnitudes also are called equal. +When bodies are equal in length, their lengths are also called equal. +And when bodies are equal in superficies, their superficies are also +called equal._ All which is common speech, as well amongst +mathematicians, as amongst common people; and, though improper, cannot +be altered, nor needeth to be altered to intelligent men. Nevertheless I +did think fit to put in that clause, that men might know what it is we +call equality, as well in magnitudes as in _magnis_, that is, in bodies. +Which you so interpret, as if it bore this sense, _that when bodies are +equal their superficies also must be equal_, contrary to your own +knowledge, only to take hold of a new occasion of reviling. How +unhandsome and unmanly this is, I leave to be judged by any reader that +hath had the fortune to see the world, and converse with honest men. + +Against the fourteenth article, where I prove that the same body hath +always the same magnitude, you object nothing but this, _that though it +be granted, that the same body hath the same magnitude, while it +resteth, yet I bring nothing to prove that when it changeth place, it +may not also change its magnitude by being enlarged or contracted_. +There is no doubt but to a body, whether at rest or in motion, more body +may be added, or part of it taken away. But then it is not the same +body, unless the whole and the part be all one. If the schools had not +set your wit awry, you could never have been so stupid as not to see the +weakness of such objections. That which you add in the end of your +objections to this eighth chapter, _that I allow not Euclid this axiom +gratis, that the whole is greater than a part_, you know to be untrue. + +At my eleventh chapter, you enter into dispute with me about the nature +of proportion. Upon the truth of your doctrine therein, and partly upon +the truth of your opinions concerning the definitions of a point, and of +a line, dependeth the question whether you have any geometry or none; +and the truth of all the demonstrations you have in your other books, +namely of the _Angle of Contact_ , and _Arithmetica Infinitorum_. Here I +say you enter, how you will get out, your reputation saved, we shall see +hereafter. + +When a man asketh what proportion one quantity hath to another, he +asketh how great or how little the one is comparatively to, or in +respect of the other. When a geometrician prefixeth before his +demonstrations a definition, he doth it not as a part of his geometry, +but of natural evidence, not to be demonstrated by argument, but to be +understood in understanding the language wherein it is set down; though +the matter may nevertheless, if besides geometry he have wit, be of some +help to his disciple to make him understand it the sooner. But when +there is no significant definition prefixed, as in this case, where +Euclid’s definition of proportion, that it is a _whatshicalt habitude of +two quantities, &c._, is insignificant, and you allege no other, every +one that will learn geometry, must gather the definition from observing +how the word to be defined is most constantly used in common speech. But +in common speech if a man shall ask how much, for example, is six in +respect of four, and one man answer that it is greater by two, and +another that it is greater by half of four, or by a third of six, he +that asked the question will be satisfied by one of them, though perhaps +by one of them now, and by the other another time, as being the only man +that knoweth why he himself did ask the question. But if a man should +answer, as you would do, that the proportion of six to two is of those +numbers a certain quotient, he would receive but little satisfaction. +Between the said answers to this question, how much is six in respect of +four? there is this difference. He that answereth that it is more by +two, compareth not two with four, nor with six, for two is the name of a +quantity absolute. But he that answereth it is more by half of four, or +by a third of six, compareth the difference with one of the differing +quantities. For halfs and thirds, &c. are names of quantity compared. + +From hence there ariseth two species or kinds of (_ratio_) proportion, +into which the general word _proportion_ may be divided. The one +whereof, namely, that wherein the difference is not compared with either +of the differing quantities, is called _ratio arithmetica_, arithmetical +proportion; the other _ratio geometrica_, geometrical proportion; and, +because this latter is only taken notice of by the name of proportion, +simply _proportion_. Having considered this, I defined proportion, +chapter II. article 3, in this manner: “_Ratio est relatio antecedentis +ad consequens secundum magnitudinem_:” _Proportion is the relation of +the antecedent to the consequent in magnitude_; having immediately +before defined relatives, antecedent, and consequent, in the same +article, and by way of explication added, that such relation was nothing +else but that one of the quantities was equal to the other, or exceeded +it by some quantity, or was by some quantity exceeded by it. And for +exemplification of the same, I added further, that the proportion of +three to two was, that three exceeded two by a unity; but said not that +the unity, or the difference whatsoever it were, was their proportion, +_for unity, and to exceed another by unity_, is not the same thing. This +is clear enough to others; let us therefore see why it is not so to you. +You say I make proportion to consist in that which remaineth after the +lesser quantity is subtracted out of the greater; and that you make it +to consist in the quotient, when one number is divided by the other. +Wherein you are mistaken; first, in that you say, I make the proportion +to consist in the remainder. For I make it to consist in the act of +exceeding, or of being exceeded, or of being equal; whereas the +remainder is always an absolute quantity, and never a proportion. To be +more or less than another number by two, is not the number two; likewise +to be equal to two, where the difference is _nothing_, is not that +_nothing_? Again, you mistake in saying the proportion consisteth in the +quotient. For divide twenty by five, the quotient is four. Is it not +absurd to say that the proportion of five to twenty, or of twenty to +five, is four? You may say the proportion of five to twenty, is the +proportion of one to four. And so say I. And you may therefore also say, +that the proportion of one to four is a measure of the proportion of +five to twenty, as being equal. And so say I. But that is only in +geometrical proportion, and not in proportion universally. For though +the _species_ obtain the denomination of the genus, yet it is not the +_genus_. But as the quotient giveth us a measure of the proportion of +the dividend to the divisor in geometrical proportion, so also the +remainder after subtraction is the measure of proportion arithmetical. + +You object in the next place, “that if the proportion of one quantity to +another be nothing but the excess or defect, then, wheresoever the +excess or defect is the same, there the proportion is the same.” This +you say follows in your logic, and from thence, that the proportion of +three to two, and five to four is the same. But is not three to two, and +five to four, where the excess is the same number, the same proportion +arithmetical? And is not arithmetical proportion, proportion? You take +here (_ratio_) proportion, which is the _genus_, for that _species_ of +it which is called geometrical, because usually this species has the +name of proportion simply. Also that the proportion of three to two, is +the same with that of nine to six; is it not because the excesses are +one and three, the same portions of three and nine, that is to say the +same excesses comparatively? I wonder you ask me not what is the _genus_ +of arithmetical and geometrical proportions, and what the _difference_? +The _genus_ is (_ratio_) proportion, or comparison in magnitude, and the +_difference_ is that one comparison is made by the absolute quantity, +the other by the comparative quantity, of the excess or defect, if there +be any. Can anything be clearer than this? You after come in with +_ignosce habitudini_ to no purpose. I am not so inhuman as not to pardon +dulness or madness: they are not voluntary faults. But when men +adventure voluntarily to talk of that they understand not censoriously +and scornfully, I may tell them of it. + +This difference between the excesses or defects, as they are simply or +comparatively reckoned, being thus explained, all the rest of that you +say in your objections to this eleventh chapter (saving that art. 5 for +_ratio binarii ad quinarium est superari ternario_, as it is in other +places, I have put too hastily _ratio binarii ad quinarium est +ternarius_), will be understood by every reader to be frivolous, and to +proceed from the ignorance of what proportion is. + +At the twelfth chapter you only note that I say, _that the proportion of +inequality is quantity, but the proportion of equality not quantity_, +and refer that which you have to say against it to the chapter +following; to which place I shall also come in the following lesson. + + + ========== + + + OF THE FAULTS THAT OCCUR IN + DEMONSTRATION. + + TO THE SAME EGREGIOUS PROFESSORS OF THE MATHEMATICS IN + THE UNIVERSITY OF OXFORD. + + + LESSON III. + +You begin your reprehension of my thirteenth chapter with a question; +whereas _I_ divide proportion into arithmetical and geometrical. You ask +me what _proportion it is I so divide_. Euclid divides an angle into +right, obtuse, and acute. I may ask you as pertinently, what angle it is +he so divides? Or, when you divide _animal_ into _homo_ and _brutum_, +what animal that is, which you so divide? You see by this, how absurd +your question is. But you say the definition of proportion which I make +at Chap. II. art. 3., namely, that proportion is the comparison of two +magnitudes, one to another, agreeth not, neither with arithmetical, nor +with geometrical proportion. I believe you thought so then, but having +read what I have said in the end of the last lesson, if you think so +still, your fault will be too great to be pardoned easily. But why did +you think so before? Is it not because there was no definition in Euclid +of proportion universal, and because he maketh no mention of proportion +arithmetical, and because you had not in your minds a sufficient notion +thereof yourselves to supply that defect? And is not this the cause +also, why you put in this parenthesis (if arithmetical proportion ought +to be called proportion)? Which is a confession that you know not +whether there be such a thing as arithmetical proportion or not, +notwithstanding that on all occasions you speak of arithmetical +proportionals. Yes, this was it that made you think that proportion +universally, and proportion geometrical, is the same, and yet to say you +cannot tell whether they be the same or not. It is no wonder, therefore, +if in such confusion of the understanding, you apprehend not that the +proportions of two to five, and nine to twelve, are the same; so you are +blinded by seeing that they are not the same proportions geometrical. +Nor doth it help you that I say the difference is the proportion; for by +difference you might, if you would, have understood the act of +differing. + +At the second article you note for a fault in method, that _after I had +used the words antecedent and consequent of a proportion in some of the +precedent chapters, I define them afterwards_. I do not believe you say +this against your knowledge, but that the eagerness of your malice made +you oversee; therefore go back again to the third article of chapter II. +where, having defined correlatives, I add these words, _of which the +first is called the_ antecedent, _the second the_ consequent. This is +but an oversight, though such as in me you would not have excused. + +At the thirteenth article you find fault with, that I say _that the +proportion of inequality, whether it be of excess or of defect, is +quantity, but the proportion of equality is not quantity_. Whether that +which you say, or that which I say, be the truth, is a question worthy +of a very strict examination. The first time I heard it argued, was in +Mersennus’ chamber at Paris, at such time as the first volume of his +_Cogitata Physico-Mathematica_ was almost printed; in which, because he +had not said all he would say of proportion, he was forced to put the +rest into a general preface, which, as was his custom, he did read to +his friends before he sent it to the press. In that general preface, +under the title _De Rationibus atque Proportionibus_, at the numbers +twelve, thirteen, fourteen, he maintaineth against Clavius, _that the +composition of proportion is_ (as of all other things) _a composition of +the parts to make a total_, and _that the proportion of equality +answereth in quantity to_ non-ens, _or nothing; the proportion of +excess, to_ ens, _or quantity; and the proportion of defect, to less +than nothing; because equality_ (he says) _is a term of middle +signification between excess and defect_. And there also he refuteth the +arguments which Clavius, at the end of the ninth Element of Euclid, +bringeth to the contrary. And though this were approved by divers good +geometricians then present, and never gainsaid by any since, yet do not +I say it upon the credit of them, but upon sufficient grounds. For it +hath been demonstrated by Eutocius, that _if there be three magnitudes, +the proportion of the first to the third is compounded of the +proportions of the first to the second, and of the second to the third_; +which also I demonstrate in this article. And if there were never so +many magnitudes ranked, it might be likewise demonstrated, that 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 of the +third to the fourth, and so on to the last. If, therefore, we put in +order any three numbers, whereof the two last be equal, as four, seven, +seven, the proportion of four the first to seven the last, will be +compounded of the proportions of four the first to seven the second, and +of seven the second to seven the third. Wherefore the proportion of +seven to seven (which is of equality) addeth nothing to the proportion +of four the first, to seven the second; and consequently the proportion +of seven to seven hath no quantity; but that the proportion of +inequality hath quantity, I prove it from this, that one inequality may +be greater than another. + +But for the clearing of this doctrine (which Mersennus calls intricate) +of the composition of proportions, I observed, first, that any two +quantities, being exposed to sense, their proportion was also exposed; +which is not intricate. Again, I observed that if besides the two +exposed quantities, there were exposed a third, so as the first were the +least, and the third the greatest, or the first the greatest, and the +third the least, that not only the proportions of the first to the +second, but also (because the differences and the quantities proceed the +same way) the proportion of the first to the last is exposed by +composition, or addition of the differences; nor is there any intricacy +in this. But when the first is less than the second, and the second +greater than the third, or the first greater than the second, and the +second less than the third, so that to make the first and second equal, +if we use addition, we must, to make the second and third equal, use +subtraction; then comes in the intricacy, which cannot be extricated, +but by such as know the truth of this doctrine which I now delivered out +of Mersennus, namely, that the proportions of excess, equality, and +defect, are as _quantity_, _not-quantity_, _nothing want quantity_; or +as symbolists mark them 0+1 . 0 . 0-1. And upon this ground I thought +depended the universal truth of this proposition, that in any rank of +magnitudes of the same kind, the proportion of the first to the last, +was compounded of all the proportions (in order) of the intermediate +quantities; the want of the proof thereof, Sir Henry Savile calls +(_nævus_) a mole in the body of geometry. This proposition is +demonstrated at the thirteenth article of this chapter. + +But before we come thither, I must examine the arguments you bring to +confute this proposition, that the _proportion of inequality is +quantity, of equality, not quantity_. + +And first, you object that equality and inequality are in the same +predicament: a pretty argument to flesh a young scholar in the logic +school, that but now begins to learn the predicaments. But what do you +mean by _æquale_ and _inequale_? Do you mean _corpus æquale_, and +_corpus inequale_? They are both in the predicament of substance, +neither of them in that of quantity. Or do you mean _æqualitas_ and +_inæqualitas_? They are both in the predicament of relation, neither of +them in that of quantity; and yet both _corpus_ and _inæqualitas_, +though neither of them be quantity, may be _quanta_, that is, both of +them have quantity. And when men say body is quantity, or inequality is +quantity, they are no otherwise understood, than if they had said +_corpus est tantum_, and _inæqualitas tanta_, not _tantitas_; that is, +bodies and inequalities are _so much_, not _somuchness_; and all +intelligent men are contented with that expression, and yourselves use +it. And the quantity of inequality is in the predicament of quantity, +because the measure of it is in that line by which one quantity exceeds +the other. But when neither exceedeth the other, then there is no line +of excess, or defect by which the equality can be measured, or said to +be _so much_, or be called quantity. Philosophy teacheth us how to range +our words; but Aristotle’s ranging them in his predicaments doth not +teach philosophy; and therefore no argument taken from thence, can +become a doctor and a professor of geometry. + +To prove that the proportion of inequality was quantity, but the +proportion of equality not quantity, my argument was this: that _because +one inequality may be greater or less than another, but one equality +cannot be greater nor less than another: therefore inequality hath +quantity, or is tanta, and equality not_. Here you come in again with +your predicaments, and object, that to be susceptible of _magis_ and +_minus_, belongs not to quantity, but to quality; but without any proof, +as if you took it for an axiom. But whether true or false, you +understand not in what sense it is true or false. It is true that one +inequality is inequality, _as well_ as another; as one heat is heat _as +well_ as another, but not _as great_. _Tam_, but not _tantus_. But so it +is also in the predicament of quantity; one line is as well a line as +another, but not so great. All degrees, intentions, and remissions of +quality, are greater or less quantity of force, and measured by lines, +superficies, or solid quantity, which are properly in the predicament of +quantity. You see how wise a thing it is to argue from the predicaments +of Aristotle, which you understand not; and yet you pretend to be less +addicted to the authority of Aristotle now than heretofore. + +In the next place you say, I may as well conclude from the not +susception of _greater_ and _less_, that a right angle is not quantity, +but an oblique one is. Very learnedly. As if to be _greater_ or _less_, +could be attributed to a quantity once determined. Number (that is, +number indefinitively taken) is susceptible of _greater_ and _less_, +because one number may be greater than another; and this is a good +argument to prove that number is quantity. And do you think the argument +the worse for this, that one six cannot be greater than another six? +After all these childish arguments which you have hitherto urged, can +you persuade any man, or yourselves, that you are logicians? + +To the fifth and sixth article you object, first, _that if I had before +sufficiently defined_ (ratio) _proportion, I needed not again define +what is_ (eadem ratio) _the same proportion_; and ask me _whether when I +have defined_ man, _I use to define anew what is the_ same man? You +think when you have the definition of _homo_, you have also the +definition of _idem homo_, when it is harder to conceive what _idem_ +signifies, than what _homo_. Besides, _idem_ hath not the same +signification always, and with whatsoever it be joined; it doth not +signify the same with _homo_, that it doth with _ratio_. For with _homo_ +it signifies the same _individual man_, but with _ratio_ it signifies a +like, or an equal proportion: and both (_ratio_) _proportion_ and +(_idem_) _the same_, being defined, there will still be need of another +definition for (_eadem ratio_) _the same proportion_; and this is enough +to defend both myself and Euclid, against this objection: for Euclid +also, after he had defined (_ratio_) _proportion_, and that +sufficiently, as he believed, yet he defines _the same proportion_ again +apart. I know you did not mean in this place to object anything against +Euclid, but you saw not what you were doing. There is within you some +special cause of intenebration, which you should do well to look to. + +In the next place you say, when I had defined arithmetical proportions +to be the same when the difference is the same; it was to be expected I +should define geometrical proportions to be then the same, when the +antecedents are of their consequents _totuple_ or _tantuple_, that is, +equimultiple (for _tantuplum_ signifies nothing). In plain words, you +expected, that as I defined one by subtraction, I should define the +other by the quotient in division. But why should you expect a +definition of the same proportion by the quotient? Neither reason nor +the authority of Euclid could move you to expect it. Or why should you +say _it was to be expected_? But it seems you have the vanity to place +the measure of truth in your own learning. In lines incommensurable +there may be the same proportion, when, nevertheless, there is no +quotient; for setting their symbols one above another doth not make a +quotient: for quotient there is none, but in _aliquot parts_. It is +therefore impossible to define proportion universally, by comparing +quotients. This incommensurability of magnitudes was it that confounded +Euclid in the framing of his definition of proportion at the fifth +Element. For when he came to numbers, he defined the _same proportion_ +irreprehensibly thus: _numbers are then proportional, when the first of +the second and the third of the fourth are equimultiple, or the same +part, or the same parts_; and yet there is in this definition no mention +at all of a quotient. For though it be true, that if in dividing two +numbers you make the same quotient, the dividends and the divisors are +proportional, yet that is not the definition of the same proportion, but +a theorem demonstrable from it. But this definition Euclid could not +accommodate to proportion in general, because of incommensurability. + +To supply this want, I thought it necessary to seek out some way, +whereby the proportion of two lines, commensurable or incommensurable, +might be continued perpetually the same. And this I found might be done +by the proportion of two lines described by some uniform motion, as by +an efficient cause both of the said lines, and also of their +proportions; which motions continuing, the proportions must needs be all +the way the same. And therefore I defined those magnitudes to have the +same geometrical proportion, _when some cause producing in equal times +equal effects, did determine both the proportions_. This, you say, needs +an Œdipus to make it understood. You are, I see, no Œdipus; but I do not +see any difficulty, neither in the definition nor in the demonstration. +That which you call perplexity in the explication, is your prejudice, +arising from the symbols in your fancy. For men that pretend no less to +natural philosophy than to geometry, to find fault with bringing motion +and time into a definition, when there is no effect in nature which is +not produced in time by motion, is a shame. But you swim upon other +men’s bladders in the superficies of geometry, without being able to +endure diving, which is no fault of mine; and therefore I shall, without +your leave, be bold to say, I am the first that hath made the grounds of +geometry firm and coherent. Whether I have added anything to the edifice +or not, I leave to be judged by the readers. You see, you that profess +with the pricking of bladders the letting out of their vapour, how much +you are deceived. You make them swell more than ever. + +For the corollaries that follow this sixth article, you say they contain +nothing new. Which is not true. For the ninth is new, and the +demonstrations of all the rest are new, being grounded upon a new +definition of proportion; and the corollaries themselves, for want of a +good definition of proportion, were never before exactly demonstrated. +For the truth of the sixth definition of the fifth Element of Euclid +cannot be known but by this definition of mine; because it requires a +trial in all numbers possible, that is to say, an infinite time of +trial, whether the quimultiples of the first and third, and of the +second and fourth, in all multiplications, do together exceed, together +come short, and are together equal; which trial is impossible. + +In objecting against the thirteenth and sixteenth article, I observe +that you bewray together, both the greatest ignorance and the greatest +malice; and it is well, for they are suitable to one another, and fit +for one and the same man. In the thirteenth article my proposition is +this: _If there be three magnitudes that have proportion one to another, +the proportions of the first to the second, and of the second to the +third, taken together_ (as one proportion), _are equal to the proportion +of the first to the third_. This demonstrated, there is taken away one +of those moles which Sir Henry Savile complaineth of in the body of +geometry. Let us see now what you say, both against the enunciation and +against the demonstration. + +Against the enunciation you object, _that other men would say_ (not the +proportions of the first to the second, and of the second to the third, +taken together, &c. but) _the proportion which is compounded of the +proportion of the first to the second, and of the second to the third_, +&c. Is not the compounding of any two things whatsoever the finding of +the sum of them both, or the taking of them together as one total? This +is that absurdity of which Mersennus, in the general preface to his +_Cogitata Physico-Mathematica_, hath convinced Clavius, who, at the end +of Euclid’s ninth Element, denieth the composition of proportion to be a +composition of parts to make a total; which, therefore, he denied, +because he did not observe, that the addition of a proportion of defect +to a proportion of excess, was a subtraction of magnitude; and because +he understood not that to say, composition is not the making a whole of +parts, was contradiction; which all but too learned men would as soon as +they heard abhor. Therefore, in saying that other men would not speak in +that manner, you say in effect they would speak absurdly. You do well to +mark what other geometricians say; but you would do better if you could +by your own meditation upon the things themselves, examine the truth of +what they say. But you have no mind, you say, to contend about the +phrase. Let us see, therefore, what it is you contend about. + +_The proportion_, you say, _which is compounded of double and triple +proportion, is not_, as I would have it, _quintuple, but sextuple_, as +in these numbers, six, three, one; where the proportion of six to three +is double, the proportion of three to one triple, and the proportion of +six to one sextuple, not quintuple. Tell me, egregious professors, how +is six to three double proportion? Is six to three the double of a +number, or the double of some proportion? All men know the number six is +double to the number three, and the number three triple to an unity. But +is the question here of compounding numbers, or of compounding +proportions? Euclid, at the last proposition of his ninth Element, says +indeed, that these numbers, one, two, four, eight, are ἐν διπλασίονι +ἀναλογία, in double proportion; yet there is no man that understands it +otherwise, than if he had said in proportion of the single quantity to +the double quantity; and after the same rate, if he had said three, +nine, twenty-seven, &c. had been in triple proportion, all men would +have understood it, of the proportion of any quantity to its triple. +Your instance, therefore, of six, three, one, is here impertinent, there +being in them no doubling, no tripling, no sextupling of proportions, +but of numbers. You may observe also, that Euclid never distinguished +between double and duplicate, as you do. One word διπλάσιον serves him +every where for either. Though, I confess, some curious grammarians take +διπλάσιον for duplicate in number, and διπλοῦν for double in quantity; +which will not serve your turn. Your geometry is not your own, but you +case yourselves with Euclid’s; in which, as I have showed you, there be +some few great holes; and you by misunderstanding him, as in this place, +have made them greater. Though the beasts that think your railing +roaring, have for a time admired you; yet now that through these holes +of your case I have showed them your ears, they will be less affrighted. +But to exemplify the composition of proportions, take these numbers, +thirty-two, eight, one, and then you shall see that the proportion of +thirty-two to one is the sum of the proportions of thirty-two to eight, +and of eight to one. For the proportion of thirty-two to eight is double +the proportion of thirty-two to sixteen; and the proportion of eight to +one, is triple the proportion of thirty-two to sixteen; and the +proportion of thirty-two to one is quintuple of thirty-two to sixteen; +but double and triple added together maketh quintuple. What can be here +denied? + +My demonstration consisteth of three cases: the first is when both the +proportions are of defect, which is then when the first quantity is the +least; as in these three quantities, A B, A C, A D. The first case I +demonstrated thus: (A B C D)/(a) Let it be supposed that the point A +were moved uniformly through the whole line A D. The proportions, +therefore, of A B to A C, and of A C to A D, are determined by the +difference of the times in which they are described. And the proportion +also of A B to A D, is that which is determined by the difference of the +times in which they are described; but the difference of the times in +which A B and A C are described, together with the difference of the +times wherein A C and A D are described, is the same with the difference +of the times wherein are described A B and A D. The same cause, +therefore, which determines both the 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_, article six, 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. + +Consider now your argumentation against it. “_Let there be taken_,” say +you, “_between A and B the point_ a; and then in your own words, I argue +thus: _The difference of the times wherein are described A B and A C, +together with the difference of the times wherein are described A C and +A D, is the same with the difference of the times in which are +described_ a _B and_ a _C (namely, B D, or B C + C D_); wherefore, 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_.” Let me ask +you here whether you suppose the motion from _a_ to B, or from _a_ to D, +to have the same swiftness with the motion from A to B, or from A to D? +If you do not, then you deny the supposition. If you do, then B C, which +is the difference of the times A B and A C, cannot be the difference of +the times in which are described _a_ B and _a_ C, except A B and _a_ B +are equal. Let any man judge now whether there be any paralogism in +Orontius that can equal this. And whether all that follows in the rest +of this, and the next two whole pages, be not all a kind of raving upon +the ignorance of what is the meaning of proportion, which you also make +more ill-favoured by writing it; not in language, but in _gambols_; I +mean in the symbols, which have made you call those demonstrations +short, which put into words so many as a true demonstration requires, +would be longer than any of those of Clavius upon the twelfth Element of +Euclid. + +To the sixteenth article you bring no argument, but fall into a loud +_oncethmus_ (the special figure wherewith you grace your oratory), +offended with my unexpected crossing of the doctrine you teach, that +proportion consisteth in a quotient. For that being denied you, your +_a/b - c/d + e/f - g/h + i/k_ comes to nothing, that is, to just as much +as it is worth. But are not you very simple men, to say that all +mathematicians speak so, when it is not speaking? When did you see any +man but yourselves publish his demonstrations by signs not generally +received, except it were not with intention to demonstrate, but to teach +the use of signs? Had Pappus no analytics? or wanted he the wit to +shorten his reckoning by signs? Or has he not proceeded analytically in +a hundred problems (especially in his seventh book), and never used +symbols? Symbols are poor unhandsome, though necessary, scaffolds of +demonstration; and ought no more to appear in public, than the most +deformed necessary business which you do in your chambers. “_But why_,” +say you, “_is this limitation to the proportion of the greater to the +less?_” I will tell you; because iterating of the proportion of the less +to the greater, is a making of the proportion less, and the defect +greater. And it is absurd to say that the taking of the same quantity +twice should make it less. And thence it is, that in quantities which +begin with the less, as one, two, four, the proportion of one to two is +greater than that of one to four, as is demonstrated by Euclid, Elem. 5, +prop. 8; and by consequent the proportion of one to four, is a +proportion of greater littleness than that of one to two. And who is +there, that when he knoweth that the respective greatness of four to +one, is double to that of the respective greatness of four to two, or of +two to one, will not presently acknowledge that the respective greatness +of one to two, or two to four, is double to the respective greatness of +one to four? But this was too deep for such men as take their opinions, +not from weighing, but from reading. + +Lastly you object against the corollary of art. 28; which you make +absurd enough by rehearsing it thus: _Si quantitas aliqua divisa +supponatur in partes aliquot æquales numero infinitas_, &c. Do you think +that of _partes aliquot_, or of _partes aliquotæ_, it can be said +without absurdity, that they are _numero infinitæ_? And then you say I +seem to mean, that if of the quantity A B, there be supposed a part C B, +infinitely little; and that between A C and A B be taken two means, one +arithmetical, A E, the other geometrical, A D, the difference between A +D and A E, will be infinitely little. My meaning is, and is sufficiently +expressed, that the said means taken everywhere (not in one place only) +will be the same throughout: and you that say there needed not so much +pains to prove it, and think you do it shorter, prove it not at all. For +why may not I pretend against your demonstration, that B E, the +arithmetical difference, is greater than B D, the geometrical +difference. You bring nothing to prove it; and if you suppose it, you +suppose the thing you are to prove. Hitherto you have proceeded in such +manner with your _Elenchus_, as that so many objections as you have +made, so many false propositions you have advanced. Which is a peculiar +excellence of yours, that for so great a stipend as you receive, you +will give place to no man living for the number and grossness of errors +you teach your scholars. + +At the fourteenth chapter your first exception is to the second article; +where I define a plane in this manner: _A plane superficies is that +which is described by a straight line so moved, as that every point +thereof describe a several straight line_. In which you require, first, +that instead of _describe_, I should have said _can describe_. Why do +you not require of Euclid, in the definition of a cone, instead of +_continetur_, _is contained_, he say _contineri potest_, _can be +contained_ ? If I tell you how one plane is generated, cannot you apply +the same generation to any other plane? But you object, that the plane +of a circle may be generated by the motion of the _radius_, whose every +point describeth, not a straight, but a crooked line, wherein you are +deceived; for you cannot draw a circle (though you can draw the +perimeter of a circle) but in a plane already generated. For the motion +of a straight line, whose one point resting, describeth with the other +points several perimeters of circles, may as well describe a conic +superficies, as a plane. The question, therefore, is, how you will, in +your definition, take in the plane which must be generated before you +begin to describe your circle, and before you know what point to make +your centre. This objection, therefore, is to no purpose; and besides, +that it reflecteth upon the perfect definitions of Euclid before the +eleventh Element, it cannot make good his definition (which is nothing +worth) of a plane superficies, before his first Element. + +In the next place, you reprehend briefly this _corollary, that two +planes cannot enclose a solid_. I should, indeed, have added, _with the +base on whose extremes they insist_: but this is not a fault to be +ashamed of; for any man, by his own understanding, might have mended my +expression without departing from my meaning. But from your doctrine, +_that a superficies has no thickness_, it is impossible to include a +solid, with any number of planes whatsoever, unless you say that solid +is included which nothing at all includes. + +At the third article, where I say _of crooked lines, some are everywhere +crooked, and some have parts not crooked_. You ask me what crooked line +has parts not crooked; and I answer, it is that line which with a +straight line makes a rectilineal triangle. But this, you say, cannot +stand with what I said before, namely, that a straight and crooked line +cannot be coincident; which is true, nor is there any contradiction; for +that part of a crooked line which is straight, may with a straight line +be coincident. + +To the fourth article, where I define _the centre of a circle to be that +point of the radius, which in the description of the circle is unmoved_; +you object as a contradiction, that I had before defined a point to be +the body which is moved in the description of a line: foolishly, as I +have already shown at your objection to Chap. VIII. art. 12. + +But at the sixth article, where I say, that _crooked and incongruous +lines touch one another but in one point_, you make a cavil from this, +that _a circle may touch a parabola in two points_. Tell me truly, did +you read and understand these words that followed? “_A crooked line +cannot be congruent with a straight line; because if it could, one and +the same line should be both straight and crooked._” If you did, you +could not but understand the sense of my words to be this: _when two +crooked lines which are incongruous, or a crooked and a straight line +touch one another, the contact is not in a line, but only in one point_; +and then your instance of a circle and a parabola was a wilful cavil, +not befitting a doctor. If you either read them not, or understood them +not, it is your own fault. In the rest that followeth upon this article, +with your diagram, there is nothing against me, nor anything of use, +novelty, subtlety, or learning. + +At the seventh article, where I define both an _angle_, simply so +called, and an _angle of contingence_, by their several generations; +namely, that the former is generated _when two straight lines are +coincident, and one of them is moved, and distracted from the other by +circular motion upon one common point resting, &c._; you ask me “_to +which of these kinds of angle I refer the angle made by a straight line +when it cuts a crooked line_?” I answer easily and truly, To that kind +of angle which is called simply an angle. This you understand not. “For +how”, will you say, “can that angle which is generated by the divergence +of two straight lines, be other than rectilineal? or how can that angle +which is not comprehended by two straight lines, be other than +curvilineal?” I see what it is that troubles you; namely, the same which +made you say before, that if the body which describes a line be a point, +then there is nothing which is not moved that can be called a point. So +you say here, “If an angle be generated by the motion of a straight +line, then no angle so generated can be curvilineal;” which is as well +argued, as if a man should say, the house was built by the carriage and +motion of stone and timber, therefore, when the carriage and that motion +is ended, it is no more a house. Rectilineal and curvilineal hath +nothing to do with the nature of an angle simply so called, though it be +essential to an angle of contact. The measure of an angle, simply so +called, is a circumference of a circle; and the measure is always the +same kind of quantity with the thing measured. The rectitude or curvity +of the lines, which drawn from the centre, intercept the arch, is +accidentary to the angle, which is the same, whether it be drawn by the +motion circular of a straight line or of a crooked. The diameter and the +circumference of a circle make a right angle, and the same which is made +by the diameter and the tangent. And because the point of contact is +not, as you think, nothing, but a line unreckoned, and common both to +the tangent and the circumference; the same angle computed in the +tangent is rectilineal, but computed in the circumference, not +rectilineal, but mixed: or, if two circles cut one another, curvilineal. +For every chord maketh the same angle with the circumference which it +maketh with the line that toucheth the circumference at the end of the +chord. And, therefore, when I divide an angle, simply so called, into +rectilineal and curvilineal, I respect no more the generation of it, +than when I divide it into right and oblique. I then respect the +generation, when I divide an angle into an angle simply so called, and +an angle of contact. This that I have now said, if the reader remember +when he reads your objections to this, and to the ninth article, he will +need no more to make him see that you are utterly ignorant of the nature +of an angle; and that if ignorance be madness, not I, but you, are mad: +and when an angle is comprehended between a straight and a crooked line +(if I may compute the same angle as comprehended between the same +straight line and the point of contact), that it is consonant to my +definition of a point by a _magnitude not considered_. But when you, in +your treatise, _De Angulo Contactus_ (chap. III. p. 6, l. 8) have these +words: “_Though the whole concurrent lines incline to one another, yet +they form no angle anywhere but in the very point of concourse_:” you, +that deny a point to be anything, tell me how two nothings can form an +angle; or if the angle be not formed, neither before the concurrent +lines meet, nor in the point of concourse, how can you apprehend that +any angle can possibly be framed? But I wonder not at this absurdity; +because this whole treatise of yours is but one absurdity, continued +from the beginning to the end, as shall then appear when I come to +answer your objections to that which I have briefly and fully said of +that subject in my 14th chapter. + +At the twelfth article, I confess your exception to my universal +definition of parallels to be just, though insolently set down. For it +is no fault of ignorance (though it also infect the demonstration next +it), but of too much security. The definition is this: _Parallels are +those lines or superficies, upon which two straight lines falling, and +wheresoever they fall, making equal angles with them both, are equal_; +which is not, as it stands, universally true. But inserting these words +_the same way_, and making it stand thus: _parallel lines or +superficies, are those upon which two straight lines falling the same +way, and wheresoever they fall, making equal angles, are equal_, it is +both true and universal; and the following consectary, with very little +change, as you may see in the translation, perspicuously demonstrated. +The same fault occurreth once or twice more; and you triumph +unreasonably, as if you had given therein a very great proof of your +geometry. + +The same was observed also upon this place by one of the prime +geometricians of Paris, and noted in a letter to his friend in these +words (Chap. XIV. art. 12): “_The definition of parallels wanteth +somewhat to be supplied_.” And of the consectary he says, “_It +concludeth not, because it is grounded on the definition of parallels_.” +Truly and severely enough, though without any such words as savour of +arrogance, or of malice, or of the clown. + +At the thirteenth article you recite the demonstration by which I prove +the perimeters of two circles to be proportional to their semidiameters; +and with _esto_, _fortasse_, _recte_, _omnino_, noddying to the several +parts thereof, you come at length to my last inference: _Therefore, by_ +Chap. XIII. art. 6, _the perimeters and semidiameters of circles are +proportional_; which you deny; and therefore deny, because you say it +followeth by the same ratiocination, that _circles also and spheres are +proportional to their semidiameters_. “_For the same distance_, you say, +_of the perimeter from the centre which determines the magnitude of the +semidiameter, determines also the magnitude both of the circle and of +the sphere_.” You acknowledge that perimeters and semidiameters have the +cause of their determination such as in equal times make equal spaces. +Suppose now a sphere generated by the semidiameters, whilst the +semicircle is turned about. There is but one _radius_ of an infinite +number of _radii_, which describes a great circle; all the rest describe +lesser circles parallel to it, in one and the same time of revolution. +Would you have men believe, that describing greater and lesser circles, +is according to the supposition (_temporibus æqualibus æqualia facere_) +to make equal spaces in equal times? Or, when by the turning about of +the semidiameter is described the plane of a circle, does it, think you, +in equal times make the planes of the interior circles equal to the +planes of the exterior? Or is the _radius_ that describes the inner +circles equal to the _radius_ that describes the exterior? It does not, +therefore, follow from anything I have said in this demonstration, that +either spheres or planes of circles, are proportional to their _radii_; +and consequently, all that you have said, triumphing in your own +incapacity, is said imprudently by yourselves to your own disgrace. They +that have applauded you, have reason by this time to doubt of all the +rest that follows, and if they can, to dissemble the opinion they had +before of your geometry. But they shall see before I have done, that not +only your whole _Elenchus_ , but also your other books of the _Angle of +Contact_ , &c. are mere ignorance and gibberish. + +To the fourteenth article you object, that (in the sixth figure) I +assume gratis, that F G, D E, B C, are proportional to A F, A D, A B; +and you refer it to be judged by the reader: and to the reader I refer +it also. The not exact drawing of the figure (which is now amended) is +it that deceived you. For A F, F D, D B, are equal by construction. +Also, A G, G E, E C, are equal by construction. And F G, D K, B H, K E, +H I, I C, are equal by parallelism. And because A F, F G, are as the +velocities wherewith they are described; also 2 A F (that is A D) and 2 +F G (that is D E) are as the same velocities. And finally, 3 A F (that +is A B) and 3 F G (that is B C) are as the same velocities. It is not +therefore assumed gratis, that F G, D E, B C are proportional to A F, A +D, A B, but grounded upon the sixth article of the thirteenth chapter; +and consequently your objection is nothing worth. You might better have +excepted to the placing of D E, first at adventure, and then making A D +two-thirds of A B; for that was a fault, though not great enough to +trouble a candid reader; yet great enough to be a ground, to a malicious +reader, of a cavil. + +That which you object to the third _corollary_ of art. 15, was certainly +a dream. There is no assuming of an angle C D E, for an angle H D E, or +B D E, neither in the demonstration, nor in any of the corollaries. It +may be you dreamt of somewhat in the twentieth article of chapter XVI. +But because that article, though once printed, was afterwards left out, +as not serving to the use I had designed it for, I cannot guess what it +is: for I have no copy of that article, neither printed nor written; but +am very sure, though it were not useful, it was true. + +Article the sixteenth. Here we come to the controversy concerning the +_angle of contact, which_, you say, _you have handled, in a special +treatise published; and that you have clearly demonstrated, in your +public lectures, that Peletarius was in the right. But that I agree not +sufficiently, neither with Peletarius nor with Clavius._ I confess I +agree not in all points with Peletarius, nor in all points with Clavius. +It does not thence follow that I agree not with the truth. I am not, as +you, of any faction, neither in geometry nor in politics. If I think +that you, or Peletarius, or Clavius, or Euclid, have erred, or been too +obscure, I see no cause for which I ought to dissemble it. And in this +same question I am of opinion that Peletarius did not well in denying +the _angle of contingence_ to be _an angle_. And that Clavius did not +well to say, _the angle of a semicircle_ was less than _a right-lined +right angle_. And that Euclid did not well to leave it so obscure what +he meant by _inclination_ in the definition of a _plane angle_, seeing +elsewhere he attributeth inclination only to acute angles; and scarce +any man ever acknowledged inclination in a straight line, to any other +line to which it was perpendicular. But you, in this question of what is +inclination, though you pretend not to depart from Euclid, are, +nevertheless, more obscure than he; and also are contrary to him. For +Euclid by inclination meaneth the inclination of one line _to_ another; +and you understand it of the inclination of one line _from_ another; +which is not inclination, but declination. For you make two straight +lines, when they lie one on another, to lie ἁκλινῶς, that is, without +any inclination (because it serves your turn); not observing that it +followeth thence, that inclination is a digression of one line _from_ +another. This is in your first argument in the behalf of Peletarius (p. +10, l. 22), and destroys his opinion. For, according to Euclid, the +greatest angle is the greatest inclination; and an angle equal to two +right angles by this ἀκλισία, should not be the greatest inclination, as +it is, but the least that can be. But if by the inclination of two +lines, we understand that proceeding of them to a common point, which is +caused by their generation, which, I believe, was Euclid’s meaning; then +will the _angle of contact_ be no less an _angle_ than a _rectilineal_ +angle, but only (as Clavius truly says it is) heterogeneous to it; and +the doctrine of Clavius more conformable to Euclid than that of +Peletarius. Besides, if it be granted you, that there is no inclination +of the circumference to the tangent, yet it does not follow that their +concourse doth not form some kind of angle; for Euclid defineth there +but one of the kinds of a plane angle. And then you may as much in vain +seek for the proportion of such angle to the angle of contact, as seek +for the _focus_ or _parameter of the parabola of Dives and Lazarus_. +Your first argument therefore is nothing worth, except you make good +that which in your second argument you affirm, namely, that all plane +angles, not excepting the angle of contact, are (_homogeneous_) of the +same kind. You prove it well enough of other curvilineal angles; but +when you should prove the same of an angle of contact, you have nothing +to say but (p. 17, l. 15), “_Unde autem illa quam somniet heterogenia +oriatur, neque potest ille ullatenus ostendere, neque ego vel +somniare_:” “_Whence should arise that diversity of kind which he dreams +of, neither can he at all show, nor I dream_;” as if you knew what he +could do if he were to answer you; or all were false which you cannot +dream of. So that besides your customary vanity, here is nothing +hitherto proved, neither for the opinion of Peletarius, nor against that +of Clavius. I have, I think, sufficiently explicated, in the first +lesson, that the angle of contact is quantity, namely, that it is the +quantity of that crookedness or flexion, by which a straight line is +bent into an arch of a circle equal to it; and that because the +crookedness of one arch may be greater than the crookedness of another +arch of another circle equal to it; therefore the question _quanta est +curvitas_, how much is the crookedness, is pertinent, and to be answered +by _quantity_. And I have also shown you in the same lesson, that the +quantity of one angle of contact is compared with that of another angle +of contact by a line drawn from the point of contact, and intercepted by +their circumferences; and that it cannot be compared by any measure with +a rectilineal angle. + +[Illustration] + +But let us see how you answer to that which Clavius has objected +already. “_They are heterogeneous_,” says he, “_because the angle of +contact, how oft soever multiplied, can never exceed a rectilineal +angle_.” To answer which, you allege _it is no angle at all; and that +therefore, it is no angle at all, because the lines have no inclination +one to another_. How can lines that have no inclination one to another, +ever come together? But you answer, _at least they have no inclination +in the point of contact_. And why have two straight lines inclination +before they come to touch, more than a straight line and an arch of a +circle? And in the point of contact itself, how can it be that there is +less inclination of the two points of a straight line and an arch of a +circle, than of the points of two straight lines? But the straight +lines, you say, will cut; which is nothing to the question; and yet this +also is not so evident, but that it may receive an objection. Suppose +two circles, A G B and C F B, to touch in B, and have a common tangent +through B. Is not the line C F B G A a crooked line? and is it not cut +by the common tangent D B E? What is the quantity of the two angles F B +E and G B D, seeing you say neither D B G nor E B F is an angle? It is +not, therefore, the cutting of a crooked line, and the touching of it, +that distinguisheth an angle simply, from an angle of contact. That +which makes them differ, and in kind, is, that the one is the quantity +of a _revolution_, and the other, the quantity of _flexion_. + +In the seventh chapter of the same treatise, you think you prove the +angle of contact, if it be an angle, and a rectilineal angle to be +(_homogeneous_) of the same kind; when you prove nothing but that you +understand not what you say. Those quantities which can be added +together, or subtracted one from another, are of the same kind; but an +angle of contact may be subtracted from a right angle, and the remainder +will be the angle of a semicircle, &c. So you say, but prove it not, +unless you think a man must grant you that the superficies contained +between the tangent and the arch, which is it you subtract, is the angle +of contact; and that the plane of the semicircle is the angle of the +semicircle, which is absurd; though, as absurd as it is, you say it +directly in your _Elenchus_ , p. 35, l. 14, in these words: “_When +Euclid defines a plane angle to be the inclination of two lines, he +meaneth not their aggregate, but that which lies between them_.” It is +true, he meaneth not the aggregate of the two lines; but that he means +that which lies between them, which is nothing else but an indeterminate +superficies, is false, or Euclid was as foolish a geometrician as either +of you two. + +Again, you would prove the angle of contact, if it be an angle, to be of +the same kind with a rectilineal angle, out of Euclid (III. 16); where +he says, _it is less than any acute angle_. And it follows well, that if +it be an angle, and less than any rectilineal angle, it is also of the +same kind with it. But, to my understanding, Euclid meant no more, but +that it was neither greater nor equal; which is as truly said of +heterogeneous, as of homogeneous quantities. If he meant otherwise, he +confirms the opinion of Clavius against you, or makes the quantity of an +angle to be a superficies, and indefinite. But I wonder how you dare +venture to determine whether two quantities be homogeneous or not, +without some definition of homogeneous (which is a hard word), that men +may understand what it meaneth. + +In your eighth chapter you have nothing but Sir H. Savile’s authority, +who had not then resolved what to hold; but esteeming the angle of +contact, first, as others falsely did, by the superficies that lies +between the tangent and the arch, makes the angle of contact and a +rectilineal angle homogeneous; and afterwards, because no multiplication +of the angle of contact can make it equal to the least rectilineal +angle, with great ingenuity returneth to his former uncertainty. + +In your ninth and tenth chapters you prove with much ado, that the +angles of like segments are equal; as if that might not have been taken +gratis by Peletarius, without demonstration. And yet your argument, +contained in the ninth chapter, is not a demonstration, but a +conjectural discourse upon the word _similitude_. And in the eleventh +chapter, wherein you answer to an objection, which might be made to your +argument in the precedent page, taken from the parallelism of two +concentric circles, though objection be of no moment, yet you have in +the same treatise of yours that which is much more foolish, which is +this, (p. 38, l. 12): “_Non enim magnitudo anguli_,” _&c._ _“_The_ +magnitude of an angle is not to be estimated by that straddling of the +legs, which it hath without the point of concourse, but by that +straddling which it hath in the point of the concourse itself._” I pray +you tell me what straddling there is of two coincident points, +especially such points as you say are nothing? When did you ever see two +nothings straddle? + +The arguments in your twelfth and thirteenth chapters are grounded all +on this untruth, that an angle is that which is contained between the +lines that make it; that is to say, is a plane superficies, which is +manifestly false; because the measure of an angle is an arch of a +circle, that is to say, a line; which is no measure of a superficies. +Besides this gross ignorance, your way of demonstration, by putting N +for a great number of sides of an equilateral polygon, is not to be +admitted; for, though you understand something by it, you demonstrate +nothing to anybody but those who understand your symbolic tongue, which +is a very narrow language. If you had demonstrated it in Irish or Welsh, +though I had not read it, yet I should not have blamed you, because you +had written to a considerable number of mankind, which now you do not. + +In your last chapters you defend Vitellio without need; for there is no +doubt but that whatsoever crooked line be touched by a straight line, +the angle of contingence will neither add anything to, nor take anything +from, a rectilineal right angle; but that it is because the angle of +contact is no angle, or no quantity, is not true. For it is therefore an +angle, because an angle of contact; and therefore quantity, because one +angle of contact may be greater than another; and therefore +heterogeneal, because the measure of an angle of contact cannot +(_congruere_) be applied to the measure of a rectilineal angle, as they +think it may, who affirm with you that the nature of an angle consisteth +in that which is contained between the lines that comprehend it, viz., +in a plane superficies. And thus you see in how few lines, and without +brachygraphy, your treatise of the angle of contingence is discovered +for the greatest part to be false, and for the rest, nothing but a +detection of some errors of Clavius grounded on the same false +principles with your own. To return now from your treatise of the angle +of contact back again to your _Elenchus_ . + +The fault you find at art. 18, is, that I understand not that Euclid +makes a _plane angle_ to be that which is contained between the two +lines that form it. It is true, that I do not understand that Euclid was +so absurd, as to think the nature of an angle to consist in superficies; +but I understand that you have not had the wit to understand Euclid. + +The nineteenth article of mine in this fourteenth chapter, is this: +“_All respect or variety of position of two lines, seemeth to be +comprehended in four kinds_. For they are either _parallel_, or (_being +if need be produced_) _make an angle_; or, (if drawn out far enough) +_touch_; or, lastly, they are _asymptotes_”; in which you are first +offended with the word _It seems_. But I allow you, that never err, to +be more peremptory than I am. For to me it seemed (I say again seemed) +that such a phrase, in case I should leave out something in the +enumeration of the several kinds of position, would save me from being +censured for untruth; and yet your instance of two straight lines in +divers planes, does not make my enumeration insufficient. For those +lines, though not parallels, nor cutting both the planes, yet being +moved parallelly from one plane to another, will fall into one or other +of the kinds of position by me enumerated; and consequently, are as much +that position, as two straight lines in the same plane, not parallel, +make the same angle, though not produced till they meet, which they +would make if they were so produced: for you have nowhere proved, nor +can prove, that two such lines do not make an angle. It is not the +actual concurrence of the lines, but the arch of a circle, drawn upon +that point for centre, in which they would meet if they were produced, +and intercepted between them, that constitutes the angle. + +Also your objection concerning asymptotes _in general_ is absurd. You +would have me add, that _their distance shall at last be less than any +distance that can be assigned_; and so make the definition of the +_genus_ the same with that of the _species_. But because you are not +professors of logic, it is not necessary for me to follow your counsel. +In like manner, if we understand one line to be moved towards another +always parallelly to itself, which is, though not actually, yet +potentially the same position, all the rest of your instances will come +to nothing. + +At the two-and-twentieth article you object to me the use of the word +_figure_, before I had defined it: wherein also you do absurdly; for I +have nowhere before made such use of the word _figure_, as to argue +anything from it; and therefore your objection is just as wise as if you +had found fault with putting the word figure in the titles of the +chapters placed before the book. If you had known the nature of +demonstration, you had not objected this. + +You add further, that by my definition of _figure_, a solid sphere, and +a sphere made hollow within, is the same figure; but you say not why, +nor can you derive any such thing from my definition. That which +deceived your shallowness, is, that you take those points that are in +the concave superficies of a hollowed sphere, not to be contiguous to +anything without it, because that whole concave superficies is within +the whole sphere. Lastly, for the fault you find with the definition of +_like figures in like positions_, I confess there wants the same word +which was wanting in the definition of parallels; namely, _ad easdem +partes_ (_the same way_) which should have been added in the end of the +definition of like figures, &c., and may easily be supplied by any +student of geometry, that is not otherwise a fool. + +At the fifteenth chapter, art. 1, number 6, you object as a +contradiction, that _I make motion to be the measure of time; and yet, +in other places, do usually measure motion and the affections thereof by +time_. If your thoughts were your own, and not taken rashly out of +books, you could not but, (with all men else that see time measured by +clocks, dials, hour-glasses, and the like), have conceived sufficiently, +that there cannot be of time any other measure besides motion; and that +the most universal measure of motion, is a line described by some other +motion; which line being once exposed to sense, and the motion whereby +it was described sufficiently explicated, will serve to measure all +other motions and their time: for time and motion (time being but the +mental image or remembrance of the motion) have but one and the same +dimension, which is a line. But you, that would have me measure +_swiftness_ and _slowness_ by longer and shorter motion, what do you +mean by _longer_ and _shorter motion_? Is _longer_ and _shorter_ in the +motion, or in the duration of the motion, which is time? Or is the +motion, or the duration of the motion, that which is exposed, or +designed by a line? Geometricians say often, _let the line A B be the +time_; but never say, _let the line A B be the motion_. There is no +unlearned man that understandeth not what is time, and motion, and +measure; and yet you, that undertake to teach it (most egregious +professors) understand it not. + +At the second article you bring another argument (which it seems in its +proper place you had forgotten), to prove that a point is not quantity +not considered, but absolutely nothing; which is this, _That if a point +be not nothing, then the whole is greater than its two halves_. How does +that follow? Is it impossible when a line is divided into two halves, +that the middle point should be divided into two halves also, being +quantity? + +At the seventh article, I have sufficiently demonstrated, that all +motion is infinitely propagated, as far as space is filled with body. +You allege no fault in the demonstration, but object from sense, _that +the skipping of a flea is not propagated to the Indies_. If I ask you +how you know it, you may wonder perhaps, but answer you cannot. Are you +philosophers, or geometricians, or logicians, more than are the simplest +of rural people? or are you not rather less, by as much as he that +standeth still in ignorance, is nearer to knowledge, than he that +runneth from it by erroneous learning? + +And, lastly, what an absurd objection is it which you make to the eighth +article, where I say that _when two bodies of equal magnitude fall upon +a third body, that which falls with greater velocity, imprints the +greater motion_? You object, _that not so much the magnitude is to be +considered as the weight_; as if the weight made no difference in the +velocity, when notwithstanding weight is nothing else but motion +downward. Tell me, when a weighty body thrown upwards worketh on the +body it meeteth with, do you not then think it worketh the more for the +greatness, and the less for the weight. + + + ========== + + + OF THE FAULTS THAT OCCUR IN + DEMONSTRATION. + + TO THE SAME EGREGIOUS PROFESSORS OF THE MATHEMATICS IN + THE UNIVERSITY OF OXFORD. + + + LESSON IV. + +Of twenty articles which you say (of nineteen which I say) make the +sixteenth chapter, you except but three, and confidently affirm the rest +are false. On the contrary, except three or four faults, such as any +geometrician may see proceed not from ignorance of the subject, or from +want of the art of demonstration, (and such as any man might have mended +of himself) but from security; I affirm that they are all true, and +truly demonstrated; and that all your objections proceed from mere +ignorance of the mathematics. + +The first fault you find is this, that I express not (art. 1) what +_impetus_ it is, which I would have to be multiplied into the time. + +The last article of my thirteenth chapter was this, “_If there be a +number of quantities propounded, howsoever equal or unequal to one +another; and there be another quantity which so often taken as there be +quantities propounded, is equal to their whole sum; that quantity I call +the mean arithmetical of them all_.” Which definition I did there insert +to serve me in the explication of those propositions of which the +sixteenth chapter consisted, but did not use it here as I intended. My +first proposition therefore as it standeth yet in the Latin, being this, +“_the velocity of any body moved during any time, is so much as is the +product of the impetus in one point of time, multiplied into the whole +time_;” to a man that hath not skill enough to supply what is wanting, +is not intelligible. Therefore I have caused it in the English to go +thus: “_the velocity of any body in whatsoever time moved, hath its +quantity determined by the sum of all the several_ (impetus) +_quicknesses, which it hath in the several points of the time of the +body’s motion_. And added, _that all the_ impetus _together taken +through the whole time is the same thing with the mean_ impetus (which +mean is defined (Chapter XIII. art. 29) _multiplied into the whole +time_.” To this first article, as it is uncorrected in the Latin, you +object, _that meaning by_ impetus _some middle_ impetus, _and assigning +none, I determine nothing_. And it is true. But if you had been +geometricians sufficient to be professors, you would have shewed your +skill much better, by making it appear that this middle _impetus_ could +be none but that, which being taken so often, as there be points in the +line of time, would be equal to the sum of all the several _impetus_ +taken in the points of time respectively; which you could not do. + +To the _corollary_, you ask first how _impetus_ can be ordinately +applied to a line; absurdly. For does not Archimedes sometimes say, and +with him many other excellent geometricians, _let such a line be the +time_? And do they not mean, that that line, or the motion over it, is +the measure of the time? And may not also a line serve to measure the +swiftness of a motion? _You thought_, you say, _only lines ought to be +said to be ordinately applied to lines_. Which I easily believe; for I +see you understand not that a line, though it be not the time itself, +may be the quantity of a time. You thought also, all you have said in +your _Elenchus_ , in your doctrine of the _angle of contact_, in your +_Arithmetica Infinitorum_, and in your _Conics_ , is true; and yet it is +almost all proved false, and the rest nothing worth. + +Secondly, you object, that _I design a parallelogram by one only side_. +It was indeed a great oversight, and argueth somewhat against the man, +but nothing against his art. For he is not worthy to be thought a +geometrician that cannot supply such a fault as that, and correct his +book himself. Though you could not do it, yet another from beyond sea +took notice of the same fault in this manner, “_He maketh a +parallelogram of but one side_; it should be thus: _vel denique per +parallelogrammum cujus unum latus est medium proportionale inter impetum +maximum (sive ultimo acquisitum) et impetus ejusdem maximi semissem; +alterum vero latus, medium proportionale, inter totum tempus, et ejusdem +totius temporis semissem_.” Which I therefore repeat, that you may learn +good manners; and know, that they who reprehend, ought also, when they +can, to add to their reprehension the correction. + +At the second article, you are pleased to advise me, instead of _in omni +motu uniformi_, to put in _in omnibus motibus uniformibus_. You have a +strange opinion of your own judgment, to think you know to what end +another man useth any word, better than himself. My intention was only +to consider motions uniform, and motions from rest uniformly, or +regularly accelerated, that I might thereby compute the lengths of +crooked lines, such as are described by any of those motions. And +therefore it was enough to prove this theorem to be true in all uniform +or uniformly accelerated _motion_, not _motions_; though it be true also +in the plural. It seems you think a man must write all he knows, whether +it conduce, or not, to his intended purpose. But that you may know that +I was not (as you think), ignorant how far it might be extended, you may +read it demonstrated at the same article in the English universally. +Against the demonstration itself you run into another article, namely, +the thirteenth, which is this problem: “_the length being given, which +is passed over in a given time by uniform motion, to find the length +which shall be passed over by motion uniformly accelerated in the same +time, so as that the_ impetus _last acquired be equal to the time_.” +Which you recite imperfectly, thereby to make it seem that such a length +is not determined. Whether you did this out of ignorance, or on purpose, +thinking it a piece of wit, as your pretended mystery which goes +immediately before, I cannot tell, for in neither place can any wit be +espied by any but yourselves. To imagine motions with their times and +ways, is a new business, and requires a steady brain, and a man that can +constantly read in his own thoughts, without being diverted by the noise +of words. The want of this ability, made you stumble and fall +unhandsomely in the very first place (that is in Chap. XIII. art. 13), +where you venture to reckon both motion and time at once; and hath made +you in this chapter to stumble in the like manner at every step you go. +As, for example, when I say, _as the product of the time, and impetus, +to the product of the time and impetus, so the space to the space when +the motion is uniform_; you come in with, _nay, rather as the time to +the time_; as if the parallelograms A I, and A H, were not also as the +times A B, and A F. Thus it is, when men venture upon ways they never +had been in before, without a guide. + +In the corollary, you are offended with the permutation of the +proportion of times and lines, because you think, (you that have scarce +one right thought of the principles of geometry), that line and time are +heterogeneous quantities. I know time and line are of divers natures; +and more, that neither of them is _quantity_. Yet they may be both of +them _quanta_, that is, they may _have quantity_; but that their +quantities are heterogeneous is false. For they are compared and +measured both of them by straight lines. And to this there is nothing +contrary in the place cited by you out of Clavius; or if there were, it +were not to be valued. And to your question, what is the proportion of +an _hour_ to an _ell_? I answer, it is the same proportion that _two +hours_ have to _two ells_. You see your question is not so subtle as you +thought it. By and bye you confess that in times and lines there is +_quid homogeneum_ (this _quid_ is an infallible sign of not fully +understanding what you say); which is false if you take it of the lines +themselves; though if you take it of their quantities, it is true +without a _quid_. Lastly, you tell m”e how I might have expressed myself +so as it might have been true. But because your expressions please me +not, I have not followed your advice. + +To the third article, which is this: “_In motu uniformiter a quiete +accelerato_,” _etc._ “_In motion uniformly accelerated from rest, that +is, when the impetus increaseth in proportion to the times, the length +run over in one time is to the length run over in another time, as the +product of the impetus multiplied by the time, to the product of the +impetus multiplied by the time_;” you object, “_that the lengths run +over are in that proportion which the impetus hath to the impetus; not +that which the impetus hath to the time, because impetus to time has no +proportion, as being heterogeneous_.” First, when you say the impetus, +do you mean some one impetus designed by some one of the unequal +straight lines parallel to the base B I? That is manifestly false. You +mean the aggregate of all those unequal parallels. But that is the same +thing with the time multiplied into the mean impetus. And so you say the +same that I do. Again, I ask, where it is that I say or dream that the +lengths run over are in the proportion of the impetus to the times? Is +it you or I that dream? And for your heterogeneity of the quantities of +time and of swiftness, I have already in divers places showed you your +error. Again, why do you make B I represent the lengths run over, which +I make to be represented by D E, a line taken at pleasure; and you also +a few lines before make the same B I to design the greatest acquired +impetus? These are things which show that you are puzzled and entangled +with the unusual speculation of time and motion, and yet are thrust on +with pride and spite to adventure upon the examination of this chapter. + +Secondly, you grant the demonstration to be good, supposing I mean it, +as I seem to speak, of one and the same motion. But why do I not mean it +of one and the same motion, when I say not in _motions_, but in _motion_ +uniform? _Because_, say you, _in that which follows, I draw it also to +different motions_. You should have given at least one instance of it; +but there is no such matter. And yet the proposition is in that case +also true; though then it must not be demonstrated by the similitude of +triangles, as in the case present. And therefore the objections you make +from different impetus acquired in the same time, and from other cases +which you mention, are nothing worth. + +At the fourth article, you allow the demonstration all the way (except +the faults of the third, which I have already proved to be none) till I +come to say, “_that because the proportion of F K to B I is double to +the proportion of A F to A B, therefore the proportion of A B to A F is +double to the proportion of B I to F K_.” This you deny, and wonder at +as strange, (for it is indeed strange to you), and in many places you +exclaim against it as extreme ignorance in geometry. In this place you +only say, “_no such matter; for though one proportion be double to +another, yet it does not follow that the converse is the double of the +converse_.” So that this is the issue to which the question is reduced, +whether you have any or no geometry. I say, if there be three quantities +in continual proportion, and the first be the least, the proportion of +the first to the second is double to the proportion of the first to the +third; and you deny it. The reason of our dissent consisteth in this, +that you think the doubling of a proportion to be the doubling of the +quantity of the proportion, as well in proportions of defect, as in +proportions of excess; and I think that the doubling of a proportion of +defect, is the doubling of the defect of the quantity of the same. As +for example in these three numbers, 1, 2, 4, which are in continual +proportion, I say the quantity of the proportion of one to two, is +double the quantity of the proportion of one to four. And the quantity +of the proportion of one to four, is half the quantity of the proportion +of one to two. And yet deny not but that the quantity of the defect in +the proportion of one to two is doubled in the proportion of one to +four. But because the doubling of defect makes greater defect, it maketh +the quantity of the proportion less. And as for the part which I hold in +this question, first, there is thus much demonstrated by Euclid, El. v. +prop. 8; that the proportion of one to two, is greater than the +proportion of one to four, though how much it is greater be not there +demonstrated. Secondly, I have demonstrated (Chap, XIII. art. 16); that +it is twice as great, that is to say, (to a man that speaks English), +double. The introducing of _duplicate_, _triplicate_, &c. instead of +_double_, _triple_, &c. (though now they be words well understood by +such as understand what proportion is), proceeded at first from such as +durst not for fear of absurdity, call the half of any thing double to +the whole, though it be manifest that the half of any defect is a double +quantity to the whole defect; for want added to want maketh greater +want, that is, a less positive quantity. This difference between +_double_ and _duplicate_, lighting upon weak understandings, has put men +out of the way of true reasoning in very many questions of geometry. +Euclid never used but one word both for _double_ and _duplicate_. It is +the same fault when men call half a quantity _subduplicate_, and a third +part _subtriplicate_ of the whole, with intention (as in this case) to +make them pass for words of signification different from the _half_ and +the _third part_. Besides, from my definition of proportion (which is +clear, and easy to be understood by all men, but such as have read the +geometry of others unluckily) I can demonstrate the same evidently and +briefly thus. My definition is this, _proportion is the quantity of one +magnitude taken comparatively to another_. Let there be therefore three +quantities, 1, 2, 4, in continual proportion. Seeing therefore the +quantity of four in respect of one, is twice as great as the quantity of +the same four in respect of 2, it followeth manifestly that the quantity +of 1 in respect of 4, is twice as little as the quantity of the same 1 +in respect of 2; and consequently the quantity of 1 in respect of 2, is +twice as great as the quantity of the same 1 in respect of 4; which is +the thing I maintain in this question. Would not a man that employs his +time at bowls, choose rather to have the advantage given him of three in +nine, than of one in nine? And why, but that three is a greater quantity +in respect of nine, than is one? Which is as much as to say, three to +nine hath a greater proportion than one to nine; as is demonstrated by +Euclid, El. v. prop. 8. Is it not therefore (you that profess +mathematics, and theology, and practise the depression of the truth in +both) well owled of you, to teach the contrary? But where you say “_that +the point K_ (in the second figure of the table belonging to this +sixteenth chapter) _is not in the parabolical line whose diameter is A +B, and base B I, but in the parabolical line of the complement of my +semiparabola_ (_as I may learn from the twenty-third proposition of +your_ Arithmetica Infinitorum) _whose diameter is A C, and base I C_.” +What line is that? Is it the same line with that of my semiparabola, or +not the same? If the same, why find you fault? If not the same, you +ought to have made a semiparabola on the diameter A C, and base I C, and +following my construction made it appear that K is not in the line +wherein I say it is; which you have not done, nor could do. + +Then again, running on in the same blindness of passion, you pretend I +make the proportion of B I to F K double to that of A B to A F, and then +confute it; when you knew I made the proportion of F K to B I, double to +that of F N, to B I, that is, of A F to A B; and this was it you should +have confuted. That which followeth is but a triumphing in your own +ignorance, wherein you also say, “_that all that I afterwards build upon +this doctrine is false_.” You see whether it be like to prove so or not. +As for your _Arithmetica Infinitorum_, I shall then read to you a piece +of a lesson on it when I come to your objections against the next +Chapter. In the mean time let me tell you, it is not likely you should +be great geometricians, that know not what is quantity, nor measure, nor +straight, nor angle, nor homogeneous, nor heterogeneous, nor proportion, +as I have already made appear in this and the former lessons. + +To the first corollary of this fourth article your exception I confess +is just, and (which I wonder at) without any incivility. But this argues +not ignorance, but security. For who is there that ever read any thing +in the Conics, that knows not that the parts of a parabola cut off by +lines parallel to the base, are in triplicate proportion to their bases? +But having hitherto designed the time by the diameter, and the impetus +by the base; and in the next chapter (where I was to calculate the +proportion of the parabola, to the parallelogram) intending to design +the time by the base, I mistook and put the diameter again for the time; +which any man but you might as easily have corrected as reprehended. + +To the second corollary, which is this, _that the lengths run over in +equal times by motion so accelerated, as that the impetus increase in +double proportion to their times, are as the differences of the cubic +numbers beginning at unity, that is, as seven, nineteen, thirty-seven, +&c._ you say it is false. But why? “_Because_” say you “_portions of the +parabola of equal altitude, taken from the beginning, are not as those +numbers seven, nineteen, thirty-seven, &c._” Does this, think you, +contradict any thing in this proposition of mine? Yes, because, you +think, the lengths gone over in equal times, are the same with the parts +of the diameter cut off from the vertex, and proportional to the numbers +one, two, three, &c. Whereas the lengths run over, are as the aggregates +of their velocities, that is, as the parts of the parabola itself, that +is, as the cubes of their bases, that is, as the numbers one, eight, +twenty-seven, sixty-four, &c., and consequently the lengths run over in +equal times, are as the differences of those cubic numbers, one, eight, +twenty-seven, sixty-four, whose differences are seven, nineteen, +thirty-seven, &c. The cause of your mistake was, that you cannot yet, +nor perhaps ever will, contemplate time and motion (which requireth a +steady brain) without confusion. + +The third corollary you also say is false, “_whether it be meant of +motion uniformly accelerated_ (as the words are) _or_ (_as perhaps_, you +say, _I meant it_) _of such motion as is accelerated in double +proportion to the time_.” You need not say perhaps I meant it. The words +of the proposition are enough to make the meaning of the corollary +understood. But so also you say it is false. Methinks you should have +offered some little proof to make it seem so. You think your authority +will carry it. But on the contrary I believe rather that they that shall +see how your other objections hitherto have sped, will the rather think +it true, because you think it false. The demonstration as it is, is +evident enough; and therefore I saw no cause to change a word of it. + +To the fifth article you object nothing, but that it dependeth on this +proposition (Chap. XIII. art. 16): “_That when three quantities are in +continual proportion, and the first is the least, as in these numbers, +four, six, nine, the proportion of the first to the second, is double to +the proportion of the same first to the last_;” which is there +demonstrated, and in the former lessons so amply explicated, as no man +can make any further doubt of the truth of it. And you will, I doubt +not, assent unto it. But in what estate of mind will you be then? A man +of a tender forehead after so much insolence, and so much contumelious +language grounded upon arrogance and ignorance, would hardly endure to +outlive it. In this vanity of yours, you ask me whether I be angry, or +blush, or can endure to hear you. I have some reason to be angry; for +what man can be so patient as not to be moved with so many injuries? And +I have some reason to blush, considering the opinion men will have +beyond sea, (when they shall see this in Latin) of the geometry taught +in Oxford. But to read the worst you can say against me, I can endure, +as easily at least, as to read any thing you have written in your +treatises of the _Angle of Contact_, of the _Conic Sections_, or your +_Arithmetica Infinitorum_. + +The sixth, seventh, eighth articles, you say are sound. True. But never +the more to be thought so for your approbation, but the less; because +you are not fit, neither to reprehend, nor praise; and because all that +you have hitherto condemned as false, hath been proved true. Then you +show me how you could demonstrate the sixth and seventh articles a +shorter way. But though there be your symbols, yet no man is obliged to +take them for demonstration. And though they be granted to be dumb +demonstrations, yet when they are taught to speak as they ought to do, +they will be longer demonstrations than these of mine. + +To the ninth article, which is this, “_If a body be moved by two movents +at once, concurring in what angle soever, of which, one is moved +uniformly, the other, with motion uniformly accelerated from rest, till +it acquire an impetus equal to that of the uniform motion, the line in +which the body is carried, shall be the crooked line of a +semiparabola_,” you lift up your voice again, and ask, _what latitude? +what diameter? what inclination_ of the diameter to the ordinate lines? +If your founder should see this, or the like objections of yours, he +would think his money ill bestowed. When I say, _in what angle soever_, +you ask, _in what angle?_ When I say _two movents, one uniform, the +other uniformly accelerated, make the body describe a semiparabolical +line_; you ask, _which is the diameter?_ as not knowing that the +accelerated motion describes the diameter, and the other a parallel to +the base. And when I say _the two movents meet in a point, from which +point both the motions begin, and one of them from rest_, you ask me +_what is the altitude?_ As if that point where the motion begins from +rest were not the vertex; or that the vertex and base being given, you +had not wit enough to see that the altitude of the parabola is +determined? When Galileo’s proposition, which is the same with this of +mine, supposed no more but a body moved by these two motions, to prove +the line described to be the crooked line of a semiparabola, I never +thought of asking him what altitude, nor what diameter, nor what angle, +nor what base, had his parabola. And when Archimedes said, let the line +A B be the time, I should never have said to him, _do you think time to +be a line_, as you ask me whether I think impetus can be the base of a +parabola. And why, but because I am not so egregious a mathematician, as +you are. In this giddiness of yours, caused by looking upon this +intricate business of motion, and of time, and the concourse of motion +uniform, and uniformly accelerated, you rave upon the numbers 1, 4, 9, +16, &c. without reference to any thing that I had said; insomuch as any +one that had seen how much you have been deceived in them before, in +your scurvy book of _Arithmetica Infinitorum_, would presently conclude, +that this objection was nothing else but a fit of the same madness which +possessed you there. + +My tenth article is like my ninth; and your objections to it are the +same which are to the former. Therefore you must take for answer just +the same which I have given to your objection there. + +To the eleventh, you say first, you have done it better at the +sixty-fourth article of your _Arithmetica Infinitorum_. But what you +have done there, shall be examined when I come to the defence of my next +chapter. And whereas I direct the reader for the finding of the +proportions of the complements of those figures to the figures +themselves, to the table of art. 3, Chap, XVII., you say that if the +increase of the _spaces_, were to the increase of the times, as one to +two, then the complement should be to the parallelogram as one to three, +and say you find not (1)/(3) in the table. Did you not see that the +table is only of those figures which are described by the concourse of a +motion uniform with a motion accelerated? You had no reason therefore to +look for (1)/(3) in that table; for your case is of motion uniform +concurring with motion retarded, because you make not the proportions of +the spaces to the proportions of the times as two to one, but the +contrary; so that your objection ariseth from want of observing what you +read. But I “_may learn_” you say, “_these, and greater matters than +these, in your twenty-third and sixty-fourth propositions of your_ +Arithmetica Infinitorum.” This, which you say here is a great absurdity; +but if you mean I shall find greater there, I will not say against you. +This (1)/(3) you looked for, belongs to the complements of the figures +calculated in that table; which because you are not able to find out of +yourselves, I will direct you to them. Your case is of (1)/(3) for the +complement of a parabola. Take the denominator of the fraction which +belongs to the parabola, namely three, and for numerator take the +numerator of the fraction which belongs to the triangle, namely one, and +you have the fraction sought. And in like manner for the complement of +any other figure. As, for example, of the second parabolaster, whose +fraction hath for denominator five, take the numerator of the fraction +of the same triangle which is one, and you have (1)/(3) for the fraction +sought for; and so of the rest, taking always one for the numerator. + +The twelfth article, which you say is miserably false, I have left +standing unaltered. For not comprehending the sense of the proposition, +you make a figure of your own, and fight against your own fancied +motions, different from mine. Other geometricians that understand the +construction better, find no fault. And if you had in your own fifth +figure drawn a line through N parallel to A E, and upon that line +supposed your accelerated motion, you would quickly have seen that in +the time A E, the body moved from rest in A, would have fallen short of +the diagonal A D; and that all your extravagant pursuing of your own +mistake had been absurd. + +My thirteenth article you say is ridiculous. But why? “_The impetus last +acquired cannot_” you say, “_be equal to a time_.” But the quantity of +the impetus may be equal to the quantity of a time, seeing they are both +measured by line. And when they are measured by the same described line, +each of their quantities is equal to that same line, and consequently to +one another. But when I meet with this kind of objection again, since I +have so often already shown it to be frivolous, and no less to be +objected against all the ancients that ever demonstrated any thing by +motion, than against me, I purpose to neglect it. + +Secondly, you object “_that motion uniformly accelerated does no more +determine swiftness, than motion uniform_.” True; you needed not have +used sixteen lines to set down that. But suppose I add, as I do, so as +the last acquired impetus be equal to the time. _But that_, you say, _is +not sense_; which is the objection I am to neglect. But, you say again, +supposing it sense, this limitation helps me nothing. Why? _Because_, +you say, _a parabola may be described upon a base given, and yet have +any altitude, or any diameter one will_. Who doubts it? But how follows +it from thence, that when a parabolical line is described by two +motions, one uniform, the other uniformly accelerated from rest, that +the determining of the base does not also determine the whole parabola? +But fifthly, you say, _that this equality of the impetus to the time +does not determine the base_. Why not? _Because_, you say, _it is an +error proceeding from this, that I understand not what is_ ratio +subduplicata. I looked for this. I have shown and inculcated +sufficiently before, but the error is on your side; and therefore must +tell you, that this objection, and also a great part of the rest of your +errors in geometry, proceedeth from this, that you know not what +proportion is. But see how wisely you argue about this duplication of +proportion. For thus you say _verbatim_. “_Stay a little. What +proportion has duplicate proportion to single proportion? Is it always +the same? I think not for example, duplicate proportion_ (4)/(1) = (2 in +2)/(1 in 1) _is double to the single_ (2)/(1). _Duplicate proportion_ +(9)/(1) = (3 in 3)/(1 in 1) _is triple to its single_ (3)/(1).” Let any +man, even of them that are most ready in your symbols, say in your +behalf (if he be not ashamed) that the proportion of nine to one is +triple to the proportion of three to one, as you do. + +In the fourteenth, fifteenth, and sixteenth articles, you bid me repeat +your objections to the thirteenth. I have done it; and find that what +you have objected to the thirteenth, may as well be objected to these; +and consequently, that my answer there will also serve me here. +Therefore, if you can endure it, read the same answer over again. + +But you have not yet done, you say, with these articles. Therefore +(after you had for a while spoken perplextly, conjecturing, not without +just cause, that I could not understand you) you say that to the end I +may the better perceive your meaning, I should take the example +following. “_Let a movent (in the first figure of this chapter) be moved +uniformly in the time A B, with the continual impetus A C, or B I, whose +whole velocity shall therefore be the parallelogram A C I B. And another +movent be uniformly accelerated, so as in the time A B it acquire the +same impetus B I. Now as the whole velocity, is to the whole velocity, +so is the length run over, to the length run over._” All this I +acknowledge to be according to my sense, saving that your putting your +word _movens_ instead of my word _mobile_ hath corrupted this article. +For in the first article, I meddle not with motion by concourse, wherein +only I have to do with two movents to make one motion; but in this I do, +wherein my word is not _movens_ but _mobile_; by which it is easy to +perceive you understand not this proposition. Then you proceed: “_But +the length run over by that accelerated motion is greater than the +length run over by that uniform motion._” Where do I say that? You +answer, “_in the ninth and thirteenth article, in making A B (in the +fifth figure) greater than A C; and A H (in the eighth figure) greater +than A B; and consequently, the triangle A B I, greater than the +parallelogram A C I B_.” That consequently is without consequence; for +it importeth nothing at all in this demonstration, whether A B, or A C +in the fifth figure be the greater. Besides I speak there of the +concourse of two movents, that describe the parabolical line A G D; +where the increasing impetus (because it increaseth as the times) will +be designed by the ordinate lines in the parabola A G D B. And if both +the motions in A B and A C were uniform, the aggregate of the impetus +would be designed by the triangle A B D, which is less than the +parallelogram A C D B. But you thought that the motion made by A C +uniformly, is the same with the motion made uniformly in the same time +by the motions in A B and A C concurring; so likewise, in the eighth +figure, there is nothing hinders A H from being greater than A B, unless +I had said that A B had been described in the time A C with the whole +impetus A C maintained entire; of which there is nothing in the +proposition, nor would at all have been pertinent to it. Therefore all +this new undertaking of the thirteenth, fourteenth, fifteenth, and +sixteenth articles, is to as little purpose as your former objections. +But I perceive that these new and hard speculations, though they turn +the edge of your wit, turn not the edge of your malice. + +At the seventeenth article, you show again the same confusion. Return to +the eighth figure: “_if in a time given a body run over two lengths, one +with uniform, the other with accelerated motion_”; as for example, if in +the same time A C, a body, run over the line A B with uniform motion, +and the line A H with motion accelerated; “_and again in a part of that +time it run over a part of the length A H, with uniform motion, and +another part of the same with motion accelerated_;” as for example, in +the time A M it run over with uniform motion the line A I, and with +motion accelerated the line A B. _I say the excess of the whole A H +above the part A B, is to the excess of the whole A B above the part A +I, as the whole A H to the whole A B._ But first you will say, that +these words _as the whole A H to the whole A B_, are left out in the +proposition. But you acknowledge that it was my meaning; and you see it +is expressed before I come to the demonstration. And therefore it was +absurdly done to reprehend it. Let us therefore pass to the +demonstration. Draw I K parallel to A C, and make up the parallelogram A +I K M. And supposing first the acceleration to be uniform, divide I K in +the midst at N; and between I N, and I K, take a mean proportional I L. +_And the straight line A L, drawn and produced, shall cut the line B D +in F, and the line C G in G_ (which lines C G, and B D, as also H G and +B F, are determined, though you could not carry it so long in memory, by +the demonstration of the thirteenth article). _For seeing A B is +described by motion uniformly accelerated, and A I by motion uniform in +the same time A M; and I L is a mean proportional between I N (the half +of I K) and I K; therefore by the demonstration of the thirteenth +article, A I is a mean proportional between A B and the half of A B, +namely A O. Again, because A B is described by uniform motion, and A H +by motion uniformly accelerated, both of them in the same time A C, B F +is a mean proportional between B D and half B D, namely B E; therefore +by the demonstration of the same thirteenth article, the straight line A +L F produced will fall on G; and the line A H will be to the line A B, +as the line A B to the line A I. And consequently as A H to A B, so H B +to B I; which was to be demonstrated._ And by the like demonstration the +same may be proved, where the acceleration is in any other proportion +that can be assigned in numbers, saving that whereas this demonstration +dependeth on the construction of the thirteenth article, if the motion +had been accelerated in double proportion to the times, it would have +depended on the fourteenth, where the lines are determined. Which +determinations being not repeated, but declared before, in the +thirteenth article, to which this diagram belongeth, you take no notice +of, but go back to a figure belonging to another article, where there +was no use of these determinations. But because I see that the words of +the proposition, are as of four motions, and not of two motions made by +twice two movents, I must pardon them that have not rightly understood +my meaning; and I have now made the proposition according to the +demonstration. Which being done, all that you have said in very near two +leaves of your _Elenchus_ comes to nothing; and the fault you find comes +to no more than a too much trusting to the skill and diligence of the +reader. And whereas after you had sufficiently troubled yourself upon +this occasion, you add, “_that if Sir H. Savile had read my Geometry, he +had never given that censure of Joseph Scaliger, in his lecture upon +Euclid, that he was the worst geometrician of all mortal men, not +exceptioning so much as Orontius, but that praise should have been kept +for me_.” You see by this time, at least others do, how little I ought +to value that opinion; and that though I be the least of geometricians, +yet my geometry is to yours as 1 to 0. I recite these words of yours, to +let the world see your indiscretion in mentioning so needlessly that +passage of your founder. It is well known that Joseph Scaliger deserved +as well of the state of learning, as any man before or since him; and +that though he failed in his ratiocination concerning the quadrature of +the circle, yet there appears in that very failing so much knowledge of +geometry, that Sir H. Savile could not but see that there were mortal +men very many that had less; and consequently he knew that that censure +of his in a rigid sense (without the license of an hyperbole) was +unjust. But who is there that will approve of such hyperboles to the +dishonour of any but of unworthy persons, or think Joseph Scaliger +unworthy of honour from learned men? Besides, it was not Sir H. Savile +that confuted that false quadrature, but Clavius. What honour was it +then for him to triumph in the victory of another? When a beast is slain +by a lion, is it not easy for any of the fowls of the air to settle +upon, and peck him? Lastly, though it were a great error in Scaliger, +yet it was not so great a fault as the least sin; and I believe that a +public contumely done to any worthy person after his death, is not the +least of sins. Judge therefore whether you have not done indiscreetly, +in reviving the only fault, perhaps that any man living can lay to your +founder’s charge; and yet this error of Scaliger’s was no greater than +one of your own of the like nature, in making the true spiral of +Archimedes equal to half the circumference of the circle of the first +revolution; and then thinking to cover your fault by calling it +afterwards an aggregate of arches of circles (which is no spiral at all +of any kind) you do not repair but double the absurdity. What would Sir +Henry Savile have said to this? + +The eighteenth article is this, “_in any parallelogram, if the two sides +that contain the angle be moved to their opposite sides, the one +uniformly, the other uniformly accelerated; the side that is moved +uniformly, by its concourse through all its longitude, hath the same +effect which it would have if the other motion were also uniform, and +the line described were a mean proportional between the whole length, +and the half of the same_.” + +To the proposition you object first, “_that it is all one whether the +other motion be uniform or not, because the effect of each of their +motions, is but to carry the body to the opposite side_.” But do you +think that whatsoever be the motions, the body shall be carried by their +concourse always to the same point of the opposite side? If not, then +the effect is not all one when a motion is made by the concourse of two +motions uniform and accelerated, and when it is made by the concourse of +two uniform or of two accelerated motions. + +Secondly, you say that these words, _and the line described were a mean +proportional between the whole length, and the half of the same_, have +no sense, or that you are deceived. True. For you are deceived; or +rather you have not understanding enough distinctly to conceive variety +of motions though distinctly expressed. For when a line is gone over +with motion uniformly accelerated, you cannot understand how a mean +proportional can be taken between it and its half; or if you can, you +cannot conceive that that mean can be gone over with uniform motion in +the same time that the whole line was run over by motion uniformly +accelerated. Yet these are things conceivable, and your want of +understanding must be made my fault. + +My demonstration is this, _in the parallelogram A B C D, (Fig. 11). Let +the side A B be conceived to be moved uniformly till it lie in C D; and +let the time of that motion be A C, or B D. And in the same time let it +be conceived that A C is moved with uniform acceleration, till it lie in +B D._ To which you object, _that then the acceleration last acquired +must be far greater than that wherewith A B is moved uniformly: else it +shall never come to the place you would have it in the same time_. What +proof bring you for this? None here. Where then? Nowhere that I +remember. On the contrary I have proved (Art. 9 of the chapter) that the +line described by the concourse of those two motions, namely, uniform +from A B to C D, and uniformly accelerated from A C to B D, is the +crooked line of the semiparabola A H D. And though I had not, yet it is +well known that the same is demonstrated by Galileo. And seeing it is +manifest that in what proportion the motion is accelerated in the line A +B, in the same proportion the impetus beginning from rest in A is +increased in the same times (which impetus is designed all the way by +the ordinate lines of the semiparabola), the greatest impetus acquired +must needs be the base of the semiparabola, namely B D, equal to A C, +which designs the whole time. I cannot therefore imagine what should +make you say without proof, that the greatest acquired impetus is +greater than that which is designed by the base B D. Next you say, “you +see not to what end I divide A B in the middle at E.” No wonder; for you +have seen nothing all the way. Others would see it is necessary for the +demonstration; as also that the point F is not to be taken arbitrarily; +and likewise that the thirteenth article, which you admit not for proof, +is sufficiently demonstrated, and your objections to it answered. By the +way you advise me, where I say _percursam eodem motu uniformi, cum +impetu ubique_, &c. to blot out _cum_; because the _impetus_ is not a +_companion_ in the way, but the _cause_. Pardon me in that I cannot take +your learned counsel; for the word _motu uniformi_ is the ablative of +the _cause_, and _impetu_ the ablative of the _manner_. But to come +again to your objections, you say, I make “_a greater space run over in +the same time by the slower motion than by the swifter_.” How does that +appear? _because there is no doubt, but the swiftness is greater where +the greatest impetus is always maintained, than where it is attained to +in the same time from rest_. True, but that is, when they are considered +asunder without concourse, but not then when by the concourse they +debilitate one another, and describe a third line different from both +the lines, which they would describe singly. In this place I compare +their effects as contributing to the description of the parabolical line +A H D. What the effects of their several motions are, when they are +considered asunder, is sufficiently shown before in the first article. +You should first have gotten into your minds the perfect and distinct +ideas of all the motions mentioned in this chapter, and then have +ventured upon the censure of them, but not before. And then you would +have seen that the body moved from A, describeth not the line A C, nor +the line A B, but a third, namely the semiparabolical line A H D. + +Again, where I say, _Wherefore, if the whole A B be uniformly moved to C +D, in the same time wherein A C is moved uniformly to F G_; you ask me +“_whether with the same impetus or not?_” How is it possible that in the +same time two unequal lengths should be passed over the same impetus? +“_But why_,” say you, “_do you not tell us with what impetus A C comes +to F G?_” What need is there of that, when all men know that in uniform +motion and the same time, impetus is to impetus, as length to length? +Which to have expressed had not been pertinent to the demonstration. +That which follows in the demonstration, _rursus suppono quod latus A +C_, &c. to these words, _ut ostensum est_, _Art. 12_, you confute with +saying you have proved that article to be false. But you may see now, if +you please, at the same place that I have proved your objection to be +frivolous. + +After this you run on without any argument against the rest of the +demonstration, showing nothing all the way, but that the variety and +concourse of motions, the speculations whereof you have not been used +to, have made you giddy. + +To the nineteenth article you apply the same objection which you made to +the eighteenth. Which having been answered, it appears that from the +very beginning of your Elenchus to this place all your objections +(except such as are made to three or four mistakes of small importance +in setting down my mind), are mere paralogisms, and such are less +pardonable than any paralogism in Orontius, both because the subject as +less difficult is more easily mastered, and because the same faults are +most shamefully committed by a reprehender than by any other man. + +I had once added to these nineteen articles a twentieth, which was this: +“_If from a point in the circumference there be drawn a cord, and a +tangent equal to it, the angle which they make shall be double to the +aggregate of all the angles made by the cords of all the equal arches +into which the arch given can possibly be divided_.” Which proposition +is true, and I did when I writ it think I might have use of it. But be +it, or the demonstration of it true or false, seeing it was not +published by me, it is somewhat barbarous to charge me with the faults +thereof. No doctor of humanity but would have thought it a poor and +wretched malice, publicly to examine and censure papers of geometry +never published, by what means soever they came into his hands. I must +confess that in these words, _in such kind of progression arithmetical_ +(that is, which begins with 0) _the sum of all the numbers taken +together, is equal to half the number that is made by multiplying the +greatest into the least_, there is a great error; for by this account +these numbers, 0, 1, 2, 3, 4, taken together, should be equal to +nothing. I should have said they are equal to that number which is made +by multiplying half the greatest into the number of the terms. There was +therefore, if those words were mine (for truly I have no copy of them, +nor have had since the book was printed, and I have no great reason, as +any man may see, to trust your faith) a great error in the writing, but +not an erroneous opinion in the writer. The demonstration so corrected +is true. And the angles that have the proportions of the numbers 1, 2, +3, 4, are in the table of your _Elenchus_ , fig. 12, the angles G A D, H +D E, I E F, K F B. And if the divisions were infinite, so that the first +were not to be reckoned but as a cypher, the angle C A B would be double +to them altogether. This mistake of mine, and the finding that I had +made no use of it in the whole book, was the cause why I thought fit to +leave it quite out. But your professorships, could not forbear to take +occasion thereby, to commend your zeal against _Leviathan_ to your +doctorships of divinity, by censuring it. + + + ========== + + + OF THE FAULTS THAT OCCUR IN + DEMONSTRATION. + + TO THE SAME EGREGIOUS PROFESSORS OF THE MATHEMATICS IN + THE UNIVERSITY OF OXFORD. + + + LESSON V. + + +At the seventeenth chapter, your first exception is to the definition of +proportional proportions, which is this: “_Four proportions are then +proportional, when the first is to the second, as the third to the +fourth_.” The reader will hardly believe that your exception is in +earnest. You say, I mean not by proportionality the “_quantity of the +proportions_.” Yes I do. Therefore I say again, that _four proportions +are then proportional, when the quantity of the first proportion, is to +the quantity of the second proportion, as the quantity of the third +proportion, to the quantity of the fourth proportion_. Is not my meaning +now plainly enough expressed? Or is it not the same definition with the +former. But what do I mean, you will say, by the quantity of a +proportion? I mean the determined greatness of it, that is, for example, +in these numbers, the quantity of the proportion of two to three, is the +same with the quantity of the proportion of four to six, or six to nine; +and again, the quantity of the proportion of six to four, is the same +with the quantity of the proportion of nine to six, or of three to two. +But now what do you mean by the quantity of a proportion? You mean that +two and three, are the quantities of the proportion of two to three (for +so Euclid calls them) and that six and four are the quantities of the +proportion of six to four, which is the same with the proportion of +three to two. And by this rule, one and the same proportion shall have +an infinite number of quantities; and consequently the quantity of a +proportion can never be determined. I call one proportion double to +another, when one is equal to twice the other; as the proportion of four +to one, is double to the proportion of two to one. You call that +proportion double where one number, line, or quantity absolute, is +double to the other; so that with you the proportion of two to one is a +double proportion. It is easy to understand how the number two is double +to one, but to what, I pray you, is double the proportion of two to one, +or of one to two? Is not every double proportion double to some +proportion? See whether this geometry of yours can be taken by any man +of sound mind for sense. “_But it is known_,” you say, “_that in +proportions, double is one thing, and duplicate another_;” so that it +seems to you, that in talking of proportion men are allowed to speak +senselessly. “_It is known_,” you say. To whom? It is indeed in use at +this day to call _double duplicate_, and _triple triplicate_. And it is +well enough; for they are words that signify the same thing, but that +they differ (in what subject soever) I never heard till now. I am sure +that Euclid, whom you have undertaken to expound, maketh no such +difference. And even there where he putteth these numbers, one, two, +four, eight, &c. for numbers in _double_ proportion (which is the last +proposition of the ninth element) he meaneth not that one to two, or two +to one, is a _double_ proportion, but that every number in that +progression is _double_ to the number next before it; and yet he does +not call it _analogia dupla_, but _duplicate_. This distinction in +proportions between _double_ and _duplicate_, proceeded long after from +want of knowledge that the proportion of one to two is _double_ to the +proportion of one to four; and this from ignorance of the different +nature of proportions of _excess_, and proportions of _defect_. And you +that have nothing but by tradition saw not the absurdities that did hang +thereon. + +In the second article I make E K, (fig. 1) the third part of L K, which +you say is false; and consequently the proposition undemonstrated. And +thus you prove it false: “_Let A C be to G C, or G K to G L, as eight to +one_ (_for seeing the point G is taken arbitrarily, we may place it +where we will, &c._)” and upon this placing of G arbitrarily, you prove +well enough that E K is not a third part of L K. But you did not then +observe, that I make _the altitude A G, less than any quantity given_, +and by consequence E K to differ from a third part by a less difference +than any quantity that can be given. Therefore as yet the demonstration +proceedeth well enough. But perceiving your oversight, you thought fit +(though before, you thought this confutation sufficient) to endeavour to +confute it another way; but with much more evidence of ignorance. For +when I come to say, _the proportion therefore between A C and G C is +triple, in arithmetical proportion, to the proportion between G K and G +E, &c._ you say, “_the proportion of A C to G C is the proportion of +identity, as also that of G K to G E.”_ But why? Does my construction +make it so? Do not I make G C less than A C, though with less difference +than any quantity that can be assigned? And then where I say, _therefore +E K is the third part of L K_, you come in, by parenthesis, with (_or a +fourth, or a fifth, &c._). Upon what ground? Because you think it will +pass for current, without proof, that a point is nothing. Which if it +do, geometry also shall pass for nothing, as having no ground nor +beginning but in nothing. But I have already in a former lesson +sufficiently showed you the consequence of that opinion. To which I may +add, that it destroys the method of _indivisibles_, invented by +Bonaventura; and upon which, not well understood, you have grounded all +your scurvy book of _Arithmetica Infinitorum_; where your indivisibles +have nothing to do, but as they are supposed to have quantity, that is +to say, to be _divisibles_. You allow, it seems, your own nothings to be +somethings, and yet will not allow my somethings to be considered as +nothing. The rest of your objections having no other ground than this, +“_that a point is nothing_,” my whole demonstration standeth firm; and +so do the demonstrations of all such geometricians, ancient and modern, +as have inferred any thing in the manner following, viz. _If it be not +greater nor less, then it is equal. But it is neither greater nor less. +Therefore, &c. If it be greater, say by how much. By so much. It is not +greater by so much. Therefore it is not greater. If it be less, say how +much, &c._ Which being good demonstrations are together with mine +overthrown by the nothingness of your _point_, or rather of your +understanding; upon which you nevertheless have the vanity of advising +me what to do, if I demonstrate the same again; meaning I should come to +your false, impossible, and absurd method of _Arithmetica Infinitorum_, +worthy to be gilded, I do not mean with gold. + +And for your question, why I set the base of my figure upwards, you may +be sure it was not because I was afraid to say, that the proportions of +the ordinate lines beginning at the vertex were triplicate, or otherwise +multiplicate of the proportions of the intercepted parts of the +diameter. For I never doubted to call double duplicate, nor triple +triplicate, &c., or if I had, I should have avoided it afterwards at the +tenth article of the same chapter. But because when I went about to +compare the proportions of the ordinate lines with those of their +contiguous diameters, the first thing I considered in them was in what +manner the base grew less and less till it vanished into a point. And +though the base had been placed below, it had not therefore required any +change in the demonstration. But I was the more apt to place the base +uppermost, because the motion began at the base, and ended at the +vertex. To proceed which way I pleased was in my own choice; and it is +of grace that I give you any account of it at all. + +To the third article, together with its table, you say, “_it falls in +the ruin of the second; and that the same is to be understood of the +sixth, seventh, eighth, and ninth_.” For confutation whereof I need to +say no more, but that they all stand good by the confutation of your +objections to the second. + +To the fourth article you say, “_the description of those curvilineal +figures is easy_.” True, to some men; and now that I have showed you the +way, it is easy enough for you also. For the way you propound is wholly +transcribed out of the figure of the second article, which article you +had before rejected. For seeing the lines H F, G E, A B, &c. are equal +to the lines C Q, C O, C D; and the lines Q F, O E, B D, equal to the +lines C H, C G, C A; the proportion of D B to O E, will be triple (that +is, triplicate) to the proportion of C O to G E; and the proportion of D +B to Q F, triple to the proportion of C D to C Q; and consequently, +because the complement B D C F E B is made by the decrease of A C in +triple proportion to that of the decrease of C D, it will be (by the +second article) a third part of the figure A B E F C A. So that it comes +all to one pass, whether we take triple proportion in decreasing to make +the complement, or triple proportion in increasing to make the figure; +for the proportion of H F to B A, is triple to the proportion of C H to +C A. Wherefore you have done no more but what you have seen first done, +saving that from your construction you prove not the figure to be triple +to the complement; perhaps because you have proved the contrary in your +_Arithmetica Infinitorum_. But your way differs from mine, in that you +call the proportion subtriplicate, which I call triplicate; as if the +divers naming of the same thing made it differ from itself. You might as +well have said briefly, the proposition is true, but ill proved, because +I call the proportion of one or two triple, or triplicate of that of one +to eight; which you say is false, and hath infected the fourth, fifth, +ninth, tenth, eleventh, thirteenth, fourteenth, fifteenth, sixteenth, +seventeenth, and nineteenth articles of the sixteenth chapter. But I +say, and you know now, that it is true; and that all those articles are +demonstrated. + +Lastly you add, “_Tu vero, in presente articulo, &c. id est, you bid +find as many mean proportionals as one will, between two given lines; as +if that could not be done by the geometry of planes, &c._” You might +have left out _Tu vero_ to seek an _Ego quidem_. But tell me, do you +think that you can find two mean proportionals (which is less than as +many as one will) by the geometry of planes? We shall see anon how you +go about it. I never said it was impossible, and if you look upon the +places cited by you more attentively, you will find yourself mistaken. +But I say, the way to do it has not been yet found out, and therefore it +may prove a solid problem for anything you know. + +The fifth article you reject, because it citeth the corollary of the +twenty-eighth article of the thirteenth chapter, where there is never a +word to that purpose. But there is in the twenty-sixth article; which +was my own fault, though you knew not but it might have been the +printer’s. + +To the tenth you object for almost three leaves together, against these +words of mine, _because_, in the sixth figure, _B C is to B F in +triplicate proportion of C D to F E, therefore inverting, F E is to C D +in triplicate proportion of B F to B C_. This you objected then. But now +that I have taught you so much geometry, as to know _that of three +quantities, beginning at the least, if the third be to the first in +triplicate proportion of the second to the first, also by conversion the +first to the second shall be in triplicate proportion of the first to +the third_; if it were to do again, you would not object it. + +My eleventh article you would allow for demonstrated, if my second had +been demonstrated, upon which it dependeth. Therefore seeing your +objections to that article are sufficiently answered, this article also +is to be allowed. + +The twelfth also is allowed upon the same reason. What falsities you +shall find in such following propositions as depend upon the same second +article, we shall then see when I come to the places where you object +against them. + +To the thirteenth article you object, “_that the same demonstration may +be as well applied to a portion of any conoeides, parabolical, +hyperbolical, elliptical, or any other, as to the portion of a sphere_.” +By the truth of this let any man judge of your and my geometry. Your +comparison of the sphere and conoeides, so far holds good, as to prove +that the superficies of the conoeides is greater than the superficies of +the cone described by the subtense of the parabolical, hyperbolical, or +elliptical line. But when I come to say, that _the cause of the excess +of the superficies of the portion of the sphere above the superficies of +the cone, consists in the angle D A B, and the cause of the excess of +the circle made upon the tangent A D, above the superficies of the same +cone, consists in the magnitude of the same angle D A B_, how will you +apply this to your conoeides? For suppose that the crooked line A B (in +the seventh figure) were not an arch of a circle, do you think that the +angles which it maketh with the subtense A B, at the points A and B, +must needs be equal? Or if they be not, does the excess of the +superficies of the circle upon A D above the superficies of the cone, or +the excess of the superficies of the portion of the conoeides above the +superficies of the same cone, consist in the angle D A B, or rather in +the magnitude of the two unequal angles D A B, and A B A? You should +have drawn some other crooked line, and made tangents to it through A +and B, and you would presently have seen your error. See how you can +answer this; for if this demonstration of mine stand firm, I may be bold +to say, though the same be well demonstrated by Archimedes, that this +way of mine is more natural, as proceeding immediately from the natural +efficient causes of the effect contained in the conclusion; and besides, +more brief and more easy to be followed by the fancy of the reader. + +To the fourteenth article you say that I “_commit a circle in that I +require in the fourth article the finding of two mean proportionals, and +come not till now to show how it is to be done_.” Nor now neither. But +in the mean time you commit two mistakes in saying so. The place cited +by you in the fourth article is, in the Latin, p. 215, line 26, in the +English, p. 255, line 24. Let any reader judge whether that be a +requiring it, or a supposing it to be done; this is your first mistake. +The second is, that in this place the proportion itself, which is, “_If +these deficient figures could be described in a parallelogram +exquisitely, there might be found thereby between any two lines given, +as many mean proportionals as one would_,” is a theorem, upon +supposition of these crooked lines exquisitely drawn; but you take it +for a problem. + +And proceeding in that error, you undertake the invention of two mean +proportionals, using therein my first figure, which is of the same +construction with the eighth that belongeth to this fourteenth article. +Your construction is, “_Let there be taken in the diameter C A, (fig. 1) +the two given lines, or two others proportional to them, as C H, C G, +and their ordinate lines H F, G E (which by construction are in +subtriplicate proportion of the intercepted diameters). These lines will +show the proportions which those four proportionals are to have._” But +how will you find the length of H F or G E, the ordinate lines? Will you +not do it by so drawing the crooked line C F E, as it may pass through +both the points F and E? You may make it pass through one of them, but +to make it pass through the other, you must find two mean proportionals +between G K and G L, or between H I and H P; which you cannot do, unless +the crooked line be exactly drawn; which it cannot be by the geometry of +planes. Go shew this demonstration of yours to Orontius, and see what he +will say to it. + +I am now come to an end of your objections to the seventeenth chapter, +where you have an epiphonema not to be passed over in silence. But +because you pretend to the demonstration of some of these propositions +by another method in your _Arithmetica Infinitorum_, I shall first try +whether you be able to defend those demonstrations as well as I have +done these of mine by the method of motion. + +The first proposition of your _Arithmetica Infinitorum_ is this lemma: +“_In a series, or row of quantities, arithmetically proportional, +beginning at a point or cypher, as 0, 1, 2, 3, 4, &c. to find the +proportion of the aggregate of them all, to the aggregate of so many +times the greatest, as there are terms_.” This is to be done by +multiplying the greatest into half the number of the terms. + +The demonstration is easy. But how do you demonstrate the same? “_The +most simple way_,” say you, “_of finding this and some other problems, +is to do the thing itself a little way, and to observe and compare the +appearing proportions, and then by induction to conclude it +universally_.” Egregious logicians and geometricians, that think an +induction, without a numeration of all the particulars sufficient, to +infer a conclusion universal, and fit to be received for a geometrical +demonstration! But why do you limit it to the natural consecution of the +numbers, 0, 1, 2, 3, 4, &c? Is it not also true in these numbers, 0, 2, +4, 6, &c. or in these, 0, 7, 14, 21, &c? Or in any numbers where the +difference of nothing and the first number is equal to the difference +between the first and second, and between the second and third, &c.? +Again, are not these quantities, 1, 3, 5, 7, &c. in continual proportion +arithmetical? And if you put before them a cypher thus, 0, 1, 3, 5, 7, +do you think that the sum of them is equal to the half of five times +seven? Therefore though your lemma be true, and by me (Chap. XIII. art. +5) demonstrated; yet you did not know why it is true; which also appears +most evidently in the first proposition of your _Conic Sections_ , where +first you have this, “_that a parallelogram whose altitude is infinitely +little, that is to say, none, is scarce anything else but a line_.” Is +this the language of geometry? How do you determine this word _scarce_? +The least altitude, is somewhat or nothing. If somewhat, then the first +character of your arithmetical progression must not be a cypher; and +consequently the first eighteen propositions of this your _Arithmetica +Infinitorum_ are all nought. If nothing, then your whole figure is +without altitude, and consequently your understanding nought. Again, in +the same proposition, you say thus: “_We will sometimes call those +parallelograms rather by the name of lines than of parallelograms, at +least when there is no consideration of a determinate altitude; but +where there is a consideration of a determinate altitude (which will +happen sometimes) there that little altitude shall be so far considered, +as that being infinitely multiplied it may be equal to the altitude of +the whole figure._” See here in what a confusion you are when you resist +the truth. When you consider no determinate altitude, that is no +quantity of altitude, then you say your parallelogram shall be called a +line. But when the altitude is determined, that is, when it is quantity, +then you will call it a parallelogram. Is not this the very same +doctrine which you so much wonder at and reprehend in me, in your +objections to my eighth chapter, and your word _considered_ used as I +used it? It is very ugly in one that so bitterly reprehendeth a doctrine +in another, to be driven upon the same himself by the force of truth +when he thinks not on it. Again, seeing you admit in any case those +infinitely little altitudes to be quantity, what need you this +limitation of yours, “_so far forth as that by multiplication they may +be made equal to the altitude of the whole figure_?” May not the half, +the third, the fourth, or the fifth part, &c. be made equal to the whole +by multiplication? Why could you not have said plainly, _so far forth as +that every one of those infinitely little altitudes be not only +something but an aliquot part of the whole_? So you will have an +_infinitely little_ altitude, that is to say, _a point to be both +nothing and something and an aliquot part_. And all this proceeds from +not understanding the ground of your profession. Well, the lemma is +true. Let us see the theorems you draw from it. The first is (p. 3) +“_that a triangle to a parallelogram of equal base and altitude is as +one to two_.” The conclusion is true, but how know you that? +“_Because_,” say you, “_the triangle consists as it were_ [_as it were_, +is no phrase of a geometrician] _of an infinite number of straight +parallel lines_.” Does it so? Then by your own doctrine, which is, that +“_lines have no breadth_,” the altitude of your triangle consisteth of +an infinite number of no altitudes, that is of an infinite number of +nothings, and consequently the area of your triangle has no quantity. If +you say that by the parallels you mean infinitely little parallelograms, +you are never the better; for if infinitely little, either they are +nothing, or if somewhat, yet seeing that no two sides of a triangle are +parallel, those parallels cannot be parallelograms. I see they may be +counted for parallelograms by not considering the quantity of their +altitudes in the demonstration. But you are barred of that plea, by your +spiteful arguing against it in your _Elenchus_ . Therefore this third +proposition, and with it the fourth, is undemonstrated. + +Your fifth proposition is, “_the spiral line is equal to half the circle +of the first revolution_.” But what spiral line? We shall understand +that by your construction, which is this: “_The straight line M A_ [in +your figure which I have placed at the end of the fifth lesson] _turned +round (the point M remaining unmoved) is supposed to describe with its +point A the circle A O A, whilst some point, in the same M A, whilst it +goes about, is supposed to be moved uniformly from M to A, describing +the spiral line_.” This therefore, is the spiral line of Archimedes; and +your proposition affirms it to be equal to the half of the circle A O A; +which you perceived not long after to be false. But thinking it had been +true, you go about to prove it, “_by inscribing in the circle an +infinite multitude of equal angles, and consequently an infinite number +of sectors, whose arches will therefore be in arithmetical proportion_;” +which is true. “_And the aggregate of those arches equal to half the +circumference A O A_;” which is true also. And thence you conclude +“_that the spiral line is equal to half the circumference of the circle +A O A_;” which is false. For the aggregate of that infinite number of +infinitely little arches, is not the spiral line made by your +construction, seeing by your construction the line you make is +manifestly the spiral of Archimedes; whereas no number, though infinite, +of arches of circles, how little soever, is any kind of spiral at all; +and though you call it a spiral, that is but a patch to cover your +fault, and deceiveth no man but yourself. Besides, you saw not how +absurd it was, for you that hold a point to be absolutely nothing, to +make an infinite number of equal angles (the radius increasing as the +number of angles increaseth) and then to say, “_that the arches of the +sectors whose angles they are, are as_ 0, 1, 2, 3, 4, &c.” For you make +the first angle 0, and all the rest equal to it; and so make 0, 0, 0, 0, +0, &c. to be the same progression with 0, 1, 2, 3, 4, &c. The influence +of this absurdity reacheth to the end of the eighteenth proposition. So +many are therefore false, or nothing worth. And you needed not to wonder +that the doctrine contained in them was omitted by Archimedes, who never +was so senseless as to think a spiral line was compounded of arches of +circles. + +Your nineteenth proposition is this other lemma: “_In a series, or a +row, of quantities, beginning from a point, or cypher, and proceeding +according to the order of the square numbers, as_ 0, 1, 4, 9, 16, _&c. +to find what proportion the whole series hath to so many times the +greatest_.” And you conclude “_the proportions to be that of 1 to 3_.” +Which is false, as you shall presently see. First, let the series of +squares with the prefixed cypher, and under every one of them the +greatest 4 be (0 . 1 . 4)/(4 . 4 . 4). And you have for the sum of the +squares 5, and for thrice the greatest 12, the third part whereof is 4. +But 5 is greater than 4, by 1, that is, by one twelfth of 12; which +quantity is somewhat, let it be called A. Again, let the row of squares +be lengthened one term further, and the greatepm divst set under every +one of them as (0 . 1 . 4 . 9)/(9 . 9 . 9 . 9). The sum of the squares +is 14, and the sum of four times the greatest is 36, whereof the third +part is 12. But 14 is greater than 12 by two unities, that is, by two +twelfths of 12, that is, by 2 A. The difference therefore between the +sum of the squares, and the sum of so many times the greatest square, is +greater, when the cypher is followed by three squares, than when by but +two. Again, let the row have five terms, as in these numbers (0 . 1 +. 4 . 9 . 16)/(16 . 16 . 16 . 16 . 16) with the greatest five times +described, and the sum of the squares will be 30, the sum of all the +greatest will be 80. The third part whereof is 26(2)/(3). But 30 is +greater than 26(2)/(3) by 3(1)/(3), that is, by three twelfths of +twelve, and (1)/(3) of a twelfth, that is, by 3(1)/(3) A. Likewise in +the series continued to six places with the greatest six times +subscribed, as ( 0 . 1 . 4 . 9 . 16 . 25)/(25 . 25 . 25 . 25 . 25 +. 25) the sum of the squares is 55, and the sum of the greatest six +times taken is 150, the third part whereof is 50. But 55 is greater than +50 by 5, that is, by five-twelfths of 12, that is by 5 A. And so +continually as the row groweth longer, the excess also of the aggregate +of the squares above the third part of the aggregate of so many times +the greatest square, growing greater. And consequently if the number of +the squares were infinite, their sum would be so far from being equal to +the third part of the aggregate of the greatest as often taken, as that +it would be greater than it by a quantity greater than any that can be +given or named. + +That which deceived you was partly this, that you think, as you do in +your _Elenchus_ , that these fractions (1)/(12) (1)/(18) (1)/(24) +(1)/(30) (1)/(36) &c. are proportions, as if (1)/(12) were the +proportion of one to twelve, and consequently (2)/(12) double the +proportion of one to twelve; which is as unintelligible as +school-divinity; and I assure you, far from the meaning of Mr. Ougthred +in the sixth chapter of his _Clavis Mathematica_, where he says that +4(3)/(7) is the proportion of 31 to 7; for his meaning is, that the +proportion of 4(3)/(7) to one, is the proportion of 31 to 7; whereas if +he meant as you do, then 8(6)/(7) should be double the proportion of 31 +to 7. Partly also because you think (as in the end of the twentieth +proposition) that if the proportion of the numerators of these fractions +(1)/(12) (1)/(18) (1)/(24) (1)/(30) (1)/(36) to their denominators +decrease eternally, they shall so vanish at last as to leave the +proportion of the sum of all the squares to the sum of the greatest so +often taken, (that is, an infinite number of times), as one to three, or +the sum of the greatest to the sum of the increasing squares, as three +to one; for which there is no more reason than for four to one, or five +to one, or any other such proportion. For if the proportions come +eternally nearer and nearer to the subtriple, they must needs also come +nearer and nearer to subquadruple; and you may as well conclude thence +that the upper quantities shall be to the lower quantities as one to +four, or as one to five, &c. as conclude they are as one to three. You +can see without admonition, what effect this false ground of yours will +produce in the whole structure of your _Arithmetica Infinitorum_; and +how it makes all that you have said unto the end of your thirty-eighth +proposition, undemonstrated, and much of it false. + +The thirty-ninth is this other lemma: “_In a series of quantities +beginning with a point or cypher, and proceeding according to the series +of the cubic numbers, as O. 1. 8. 27. 64, &c. to find the proportion of +the sum of the cubes to the sum of the greatest cube, so many times +taken as there be terms_.” And you conclude that “_they have a +proportion of 1 to 4_;” which is false. + +Let the first series be of three terms subscribed with the greatest (0. +1. 8.)/(8. 8. 8.); the sum of the cubes is nine; the sum of all the +greatest is 24; a quarter whereof is 6. But 9 is greater than 6 by three +unities. An unity is something. Let it be therefore A. Therefore the row +of cubes is greater than a quarter of three times eight, by three A. +Again, let the series have four terms, as (0. 1. 8. 27)/(27. 27. 27. +27); the sum of the cubes is 36; a quarter of the sum of all the +greatest is twenty-seven. But thirty-six is greater than twenty-seven by +nine, that is, by 9 A. The excess therefore of the sum of the cubes +above the fourth part of the sum of all the greatest, is increased by +the increase of the number of terms. Again, let the terms be five, as +(0. 1. 8. 27. 64)/(64. 64. 64. 64. 64), the sum of the cubes is one +hundred; the sum of all the greatest three hundred and twenty; a quarter +whereof is eighty. But one hundred is greater than eighty by twenty, +that is, by 20 A. So you see that this lemma also is false. And yet +there is grounded upon it all that which you have of comparing parabolas +and paraboloeides with the parallelograms wherein they are accommodated. +And therefore though it be true, that the parabola is (2)/(3) and the +cubical paraboloeides (3)/(4) of their parallelograms respectively, yet +it is more than you were certain of when you referred me, for the +learning of geometry, to this book of yours. Besides, any man may +perceive that without these two lemmas (which are mingled with all your +compounded series with their excesses) there is nothing demonstrated to +the end of your book: which to prosecute particularly, were but a vain +expense of time. Truly, were it not that I must defend my reputation, I +should not have showed the world how little there is of sound doctrine +in any of your books. For when I think how dejected you will be for the +future, and how the grief of so much time irrecoverably lost, together +with the conscience of taking so great a stipend, for mis-teaching the +young men of the University, and the consideration of how much your +friends will be ashamed of you, will accompany you for the rest of your +life, I have more compassion for you than you have deserved. Your +treatise of the _Angle of Contact_ , I have before confuted in a very +few leaves. And for that of your _Conic Sections_ , it is so covered +over with the scab of symbols, that I had not the patience to examine +whether it be well or ill demonstrated. + +Yet I observed thus much, that you find a tangent to a point given in +the section by a diameter given; and in the next chapter after, you +teach the finding of a diameter, which is not artificially done. + +I observe also, that you call the _parameter_ an imaginary line, as if +the place thereof were less determined than the diameter itself; and +then you take a mean proportional between the intercepted diameter, and +its contiguous ordinate line, to find it. And it is true, you find it: +but the parameter has a determined quantity, to be found without taking +a mean proportional. For the diameter and half the section being given, +draw a tangent through the vertex, and dividing the angle in the midst +which is made by the diameter and tangent, the line that so divideth the +angle, will cut the crooked line. From the intersection draw a line (if +it be a parabola) parallel to the diameter, and that line shall cut off +in the tangent from the vertex the parameter sought. But if the section +be an ellipsis, or an hyperbole, you may use the same method, saving +that the line drawn from the intersection must not be parallel, but must +pass through the end of the transverse diameter, and then also it shall +cut off a part of the tangent, which measured from the vertex is the +parameter. So that there is no more reason to call the parameter an +imaginary line than the diameter. + +Lastly, I observe that in all this your new method of conics, you show +not how to find the _burning points_, which writers call the _foci_ and +_umbilici_ of the section, which are of all other things belonging to +the conics most useful in philosophy. Why therefore were they not as +worthy of your pains as the rest, for the rest also have already been +demonstrated by others? You know the focus of the parabola is in the +axis distant from the vertex a quarter of the parameter. Know also that +the focus of an hyperbole, is in the axis, distant from the vertex, as +much as the hypotenusal of a rectangled triangle, whose one side is half +the transverse axis, the other side half the mean proportional between +the whole transverse axis and the parameter, is greater than half the +transverse axis. + +The cause why you have performed nothing in any of your books (saving +that in your _Elenchus_ you have spied a few negligences of mine, which +I need not be ashamed of) is this, that you understood not what is +_quantity_, _line_, _superficies_, _angle_, and _proportion_; without +which you cannot have the science of any one proposition in geometry. +From this one and first definition of Euclid, “_a point is that whereof +there is no part_,” understood by Sextus Empiricus, as you understand +it, that is to say misunderstood, Sextus Empiricus had utterly destroyed +most of the rest, and demonstrated, that in geometry there is no +science, and by that means you have betrayed the most evident of the +sciences to the sceptics. But as I understand it for _that whereof no +part is reckoned_, his arguments have no force at all, and geometry is +redeemed. If a line have no latitude, how shall a cylinder rolling on a +plane, which it toucheth not but in a line, describe a superficies? How +can you affirm that any of those things can be without quantity, whereof +the one may be greater or less than the other? But in the common contact +of divers circles the external circle maketh with the common tangent a +less angle of contact than the internal. Why then is it not quantity? An +angle is made by the concourse of two lines from several regions, +concurring, by their generation, in one and the same point. How then can +you say the angle of contact is no angle? One measure cannot be +applicable at once to the angle of contact, and angle of conversion. How +then can you infer, if they be both angles, that they must be +homogeneous? Proportion is the relation of two quantities. How then can +a quotient or fraction, which is quantity absolute, be a proportion? But +to come at last to your _Epiphonema_ , wherein, though I have perfectly +demonstrated all those propositions concerning the proportion of +parabolasters to their parallelograms, and you have demonstrated none of +them (as you cannot now but plainly see), but committed most gross +paralogisms, how could you be so transported with pride, as insolently +to compare the setting of them forth as mine, to the act of him that +steals a horse, and comes to the gallows for it. You have read, I think, +of the gallows set up by Haman. Remember therefore also who was hanged +upon it. + +After your dejection I shall comfort you a little, a very little, with +this, that whereas this eighteenth chapter containeth two problems, one, +“_the finding of a straight line equal to the crooked line of a +semi-parabola_;” the other, “_the finding of straight lines equal to the +crooked lines of the parabolasters, in the table of the third article of +the seventeenth chapter_;” you have truly demonstrated that they are +both false; and another hath also demonstrated the same another way. +Nevertheless, the fault was not in my method, but in a mistake of one +line for another and such as was not hard to correct; and is now so +corrected in the English as you shall not be able (if you can +sufficiently imagine motions) to reprehend. The fault was this, that in +the triangles which have the same base and altitude with the parabola +and parabolaster, I take for designation of the mean uniform impetus, a +mean proportional, in the first figure, between the whole diameter and +its half, and, in the second figure, a mean proportional between the +whole diameter and its third part; which was manifestly false, and +contrary to what I had shown in the sixteenth chapter. Whereas I ought +to have taken the half of the base, as now I have done, and thereby +exhibited the straight lines equal to those crooked lines, as I +undertook to do. Which error therefore proceeded not from want of skill, +but from want of care; and what I promised (as bold as you say the +promise was), I have now performed. + +The rest of your exceptions to this chapter, are to these words in the +end: “_There be some that say, that though there be equality between a +straight and crooked line, yet now, they say, after the fall of Adam, it +cannot be found without the especial help of divine grace_.” And you say +you think there be none that say so. I am not bound to tell you who they +are. Nevertheless, that other men may see the spirit of an ambitious +part of the clergy, I will tell you where I read it. It is in the +_Prolegomena_ of Lalovera, a Jesuit, to his Quadrature of the Circle, p. +13 and 14, in these words: “_Quamvis circuli tetragonismus sit_ φύσει +_possibilis, an tamen etiam_ πρός ἡμᾶς, _hoc est, post Adæ lapsum homo +ejus scientiam absque speciali divinæ gratiæ auxilio, possit comparare, +jure merito inquirunt theologi, pronunciantque; hanc veritatem tanta +esse caligine involutam ut illam videre nemo possit, nisi ignorantiæ ex +primi parentis prævaricatione propagatas tenebras indebitus divinæ lucis +radius dissipet; quod verissimum esse sentio_.” Wherein I observed that +he, supposing he had found that quadrature, would have us believe it was +not by the ordinary and natural help of God (whereby one man reasoneth, +judgeth and remembereth better than another), but by a special (which +must be a supernatural) help of God, that he hath given to him of the +order of Jesus above others that have attempted the same in vain. +Insinuating thereby, as handsomely as he could, a special love of God +towards the Jesuits. But you taking no notice of the word _special_, +would have men think I held, that human sciences might be acquired +without any help of God. And thereupon proceed in a great deal of ill +language to the end of your objections to this chapter. But I shall take +notice of your manners for altogether in my next lesson. + +At the nineteenth chapter you see not, you say, the method. Like enough. +In this chapter I consider not the cause of reflection, which consisteth +in the resistance of bodies natural; but I consider the consequences, +arising from the supposition of the equality of the angle of reflection, +to that of incidence; leaving the causes both of reflection, and of +refraction, to be handled together in the twenty-fourth chapter. Which +method, think what you will, I still think best. + +Secondly, you say I define not, here, but many chapters after, what an +angle of incidence, and what an angle of reflection is. Had you not been +more hasty than diligent readers, you had found that those definitions +of the angle of incidence, and of reflection, were here set down in the +first article, and not deferred to the twenty-fourth. Let not therefore +your own oversight be any more brought in for an objection. + +Thirdly, you say there is no great difficulty in the business of this +chapter. It may be so, now it is down; but before it was done, I doubt +not but you that are a professor would have done the same, as well as +you have done that of the _Angle of Contact_ , or the business of your +_Arithmetica Infinitorum_ . But what a novice in geometry would have +done I cannot tell. + +To the third, fourth, and fifth article, you object a want of +determination; and show it by instance, as to the third article. But +what those determinations should be, you determine not, because you +could not. The words in the third article, are first these, _if there +fall two straight lines parallel, &c._ which is too general. It should +be, _if there fall the same way two straight lines parallel, &c._ Next +these, _their reflected lines produced inwards shall make an angle, &c._ +This also is too general. I should have said, _their reflected lines +produced inwards, if they meet within, shall make an angle, &c._ Which +done, both this article and the fourth and fifth are fully demonstrated. +And without it, an intelligent reader had been satisfied, supplying the +want himself by the construction. + +To the eighth, you object only the too great length and labour of it, +because you can do it a shorter way. Perhaps so now, as being easy to +shorten many of the demonstrations both of Euclid, and other the best +geometricians that are or have been. And this is all you had to say to +my nineteenth chapter. Before I proceed, I must put you in mind that +these words of yours, “_adducis malleum, ut occidas muscam_,” are not +good Latin, _malleum affers_, _malleum adhibes_, _malleo uteris_, are +good. When you speak of bringing bodies animate, _ducere_ and _adducere_ +are good, for there _to bring_, is _to guide or lead_. And of bodies +inanimate, _adducere_ is good for _attrahere_, which is to draw to. But +when you bring a hammer, will you say _adduco malleum_, _I lead a +hammer_? A man may lead another man, and a ninny may be said to lead +another ninny, but not a hammer. Nevertheless, I should not have thought +fit to reprehend this fault upon this occasion in an Englishman, nor to +take notice of it, but that I find you in some places nibbling, but +causelessly, at my Latin. + +Concerning the twentieth chapter, before I answer to the objections +against the propositions themselves, I must answer to the exception you +first take to these words of mine, “_Quæ de dimensione circuli et +angulorum pronuntiata sunt tanquam exactè inventa, accipiat lector +tanquam dicta_ _problematicè._” To which you say thus: “_We are wont in +geometry to call some propositions theorems, others problems, &c. of +which a theorem is that wherein some assertion is propounded to be +proved; a problem that wherein something is commanded to be done_.” Do +you mean _to be done_, and not proved? By your favour, a problem in all +ancient writers signifies no more but a proposition uttered, to the end +to have it, by them to whom it is uttered, examined whether it be true +or not true, faisable or not faisable; and differs not amongst +geometricians from a theorem but in the manner of propounding. For this +proposition, _to make an equilateral triangle_, so propounded they call +a problem. But if propounded thus: _If upon the ends of a straight line +given be described two circles, whose radius is the same straight line, +and there be drawn from the intersection of the circles to their two +centres, two straight lines, there will be made an equilateral +triangle_, then they call it a theorem; and yet the proposition is the +same. Therefore these words, _accipiat lector tanquam dicta +problematicè_ signify plainly this, that I would have the reader, take +for propounded to him to examine, whether from my construction the +quadrature of the circle can be truly inferred or not; and this is not +to bid him, as you interpret it, to square the circle. And if you +believe that _problematicè_ signifies probably, you have been very +negligent in observing the sense of the ancient Greek philosophers in +the word problem. Therefore your _solemus in geometria_, &c. is nothing +to the purpose; nor had it been though you had spoken more properly, and +said _solent_, leaving out yourselves. + +[Illustration: + + _Six Lessons._ + _Vol. VII. Eng. p.310_II. 325_ +] + +My first article hath this title, “_from a false supposition, a false +quadrature of the circle_.” Seeing therefore you were resolved to show +where I erred, you should have proved either that the supposition was +true, and the conclusion falsely inferred, or contrarily, that though +the supposition be false, yet the conclusion is true; for else you +object nothing to my geometry, but only to my judgment, in thinking fit +to publish it; which nevertheless you cannot justly do, seeing it was +likely to give occasion to ingenious men (the practice of it being so +accurate to sense) to inquire wherein the fallacy did consist. And for +the problem as it was first printed, but never published, and +consequently ought to have passed for a private paper stolen out of my +study, your public objecting against it (in the opinion of all men that +have conversed so much with honest company as to know what belongs to +civil conversation), was sufficiently barbarous in divines. And seeing +you knew I had rejected that proposition, it was but a poor ambition to +take wing as you thought to do, like beetles from my egestions. But let +that be as it will, you will think strange now I should resume, and make +good, at least against your objection, that very same proposition. So +much of the figure as is needful you will find noted with the same +letters, and placed at the end of this fifth lesson. Wherein let B I, be +an arch not greater than the radius of the circle, and divided into four +equal parts, in L, N, O. Draw S N, the sine of the arch B N, and produce +it to T, so as S T be double to S N, that is, equal to the chord B I. +Draw likewise _a_ L, the sine of the arch B L, and produce it to _c_, so +as _a c_ be quadruple to _a_ L, that is, equal to the two chords B N, N +I. Upon the centre N with the radius N I, draw the arch I _d_, cutting B +U the tangent in _d_. Then will B N produced cut the arch I _d_, in the +midst at _o_. In the line B S produced take S _b_, equal to B S; then +draw and produce _b_ N, and it will fall on the point _d_. And B _d_, S +T, will be equal; and _d_ T joined and produced will fall upon _o_, the +midst of the arch I _d_. Join I T, and produce it to the tangent B U in +U. I say, that the straight line I T U shall pass through _c_. For +seeing B S, S _b_, are equal, and the angle at S a right angle, the +straight lines B N, and _b_ N, are also equal, and the triangles B N +_b_, _d_ N _o_ like and equal; and the lines _d_ T, T _o_ equal. Draw _o +i_ parallel to _d_ U, cutting I U in _i_; and the triangles _d_ T U, _o_ +T _i_ will also be like and equal. Produce S T to the arch _d o_ I in +_e_, and produce it further to _f_, so that the line _e f_ be equal to T +_e_; and then S _f_ will be equal to _a c_. Therefore _f c_ joined will +be parallel to B S. In _c f_ produced take _f g_ equal to _c f_; and +draw _g m_ parallel to _d_ U, cutting I U in _m_, and _d o_ in _n_; and +let the intersection of the two lines _a c_ and _d o_ be in _r_; which +being done, the triangles _m n_ T, _r c_ T will be like and equal. +Therefore _m n_ and _r c_ are equal; and consequently the straight line +I _m_ T U shall pass through _c_. Dividing therefore _a c_ in the midst +at _t_, and S N in the midst at _l_, and joining _t_ N, L _l_, the lines +L _l_, _t_ N, and _c_ T produced, will all meet in one and the same +point of B S produced; suppose at _q_. Therefore the point _q_ being +given by the two known points T and I, the lines drawn from _q_ through +equal parts of the sine of the arch B I, (for example through the points +P, Q, R, of the sine M I), shall cut off equal arches, as B L, L N, N O, +O I. And this is enough to make good that problem, as to your objection. + +The straight line therefore B U, for any thing you have said, is proved +equal to the arch B I, and the division of any angle given into any +proportion given, the quadrature of any sector, and the construction of +any equilateral polygon is also given. And though in this also I should +have erred, yet it cannot be denied but that I have used a more natural, +a more geometrical, and a more perspicuous method in the search of this +so difficult a problem, than you have done in your _Arithmetica +Infinitorum_. For though it be true that the aggregate of all the mean +proportionals between the radius, together with an infinitely little +part of the same, and the radius wanting an infinitely little part of +the same; and again, between the radius, together with two infinitely +little parts, and the radius wanting two infinitely little parts, and so +on eternally, will be equal to the quadrant (a thing which every mean +geometrician knew before); yet it was absurd to think those means could +be calculated in numbers by interpoling of a symbol; especially when you +make that symbol to stand for a number neither true nor surd; as if +there were a number that could neither be uttered in words, nor not be +uttered in words. For what else is surd, but that which cannot be +spoken? + +To the fifth article, though your discourse be long, you object but two +things. One is, that “_Whereas the spiral of Archimedes is made of two +motions, one straight, the other circular, both uniform, I taking the +motion compounded of them both for one of those that are compounded, +conclude falsely, that the generation of the spiral is like to the +generation of the parabola_.” What heed you use to take in your +reprehensions, appears most manifestly in this objection. For I say in +that demonstration of mine, that _the velocity of the point A in +describing the spiral increaseth continually in proportion to the +times_. For seeing it goes on uniformly in the semidiameter, it is +impossible it should not pass into greater and greater circles, +proportionally to the times, and consequently it must have a swifter and +swifter motion circular, to be compounded with the uniform motion in +every point of the radius as it turneth about. This objection therefore +is nothing but an effect of a will, without cause, to contradict. + +The other objection is, that “_Granting all to be true hitherto, yet +because it depends upon the finding of a straight line equal to a +parabolical line in the eighteenth chapter, where I was deceived, I am +also deceived here_.” True. But because in the eighteenth chapter of +this English edition I have found a straight line equal to the spiral +line of Archimedes. I must here put you in mind that by these words in +your objections to the fifth article at your number two, _Quatenus verum +est, etc._, _we have demonstrated prop. 10, 11, 13_, _Arithmetica +Infinitorum_; you make it appear that you thought your spiral (made of +arches or circles) was the true spiral of Archimedes; which is fully as +absurd as the quadrature of Joseph Scaliger, whose geometry you so much +despise. + +To the sixth article, which is a digression concerning the analytics of +geometricians, you deny _that the efficient cause of the construction +ought to be contained in the demonstration_. As if any problem could be +known to be truly done, otherwise than by knowing first how, that is to +say, by what efficient cause, and in what manner, it is to be done. +Whatsoever is done without that knowledge, cannot be demonstrated to be +done; as you see in your computation of the parabola, and paraboloeides, +in your _Arithmetica Infinitorum_. + +And whereas I said that _the ends of all straight lines drawn from a +straight line, and passing through one and the same point, if their +parts be proportional, shall be in a straight line_; is true and +accurate; as also, _if they begin in the circumference of a circle, they +shall also be in the circumference of another circle_. And so is this: +_if the proportion be duplicate, they shall be in a parabola_. All this +I say is true and accurately spoken. But this was no place for the +demonstration of it. Others have done it. And I perceive by that you put +in by parenthesis (“_Intelligis credo inter duas peripherias +concentricas_”) that you understand not what I mean. + +Hitherto reach your objections to my geometry: for the rest of your +book, it containeth nothing but a collection of lies, wherewith you do +what you can, to extenuate as vulgar, and disgrace as false, that which +followeth, and to which you have made no special objection. + +I shall therefore only add in this place concerning your _Analytica per +Potestates_, that it is no art. For the rule, both in Mr. Ougthred, and +in Des Cartes, is this: “_When a problem or question is propounded, +suppose the thing required done, and then using a fit ratiocination, put +A or some other vowel for the magnitude sought_.” How is a man the +better for this rule without another rule, how to know when the +ratiocination is fit? There may therefore be in this kind of analysis +more or less natural prudence, according as the analyst is more or less +wise, or as one man in choosing of the unknown quantity with which he +will begin, or in choosing the way of the consequences which he will +draw from the hypothesis, may have better luck than another. But this is +nothing to art. A man may sometimes spend a whole day in deriving of +consequences in vain, and perhaps another time solve the same problem in +a few minutes. + +I shall also add, that symbols, though they shorten the writing, yet +they do not make the reader understand it sooner than if it were written +in words. For the conception of the lines and figures (without which a +man learneth nothing) must proceed from words either spoken or thought +upon. So that there is a double labour of the mind, one to reduce your +symbols to words, which are also symbols, another to attend to the ideas +which they signify. Besides, if you but consider how none of the +ancients ever used any of them in their published demonstrations of +geometry, nor in their books of arithmetic, more than for the roots and +potestates themselves; and how bad success you have had yourself in the +unskilful using of them, you will not, I think, for the future be so +much in love with them as to demonstrate by them that first part you +promise of your _Opera Mathematica_. In which, if you make not amends +for that which you have already published, you will much disgrace those +mathematicians you address your epistles to, or otherwise have +commended; as also the Universities, as to this kind of learning, in the +sight of learned men beyond sea. And thus having examined your pannier +of Mathematics, and finding in it no knowledge, neither of quantity, nor +of measure, nor of proportion, nor of time, nor of motion, nor of any +thing, but only of certain characters, as if a hen had been scraping +there; I take out my hand again, to put it into your other pannier of +theology, and good manners. In the mean time I will trust the objections +made by you the astronomer (wherein there is neither close reasoning, +nor good style, nor sharpness of wit, to impose upon any man) to the +discretion of all sorts of readers. + + + ========== + + + OF MANNERS. + + TO THE SAME EGREGIOUS PROFESSORS OF THE MATHEMATICS IN + THE UNIVERSITY OF OXFORD. + + + LESSON VI. + +Having in the precedent lessons maintained the truth of my geometry, and +sufficiently made appear that your objections against it are but so many +errors of your own, proceeding from misunderstanding of the propositions +you have read in Euclid, and other masters of geometry; I leave it to +your consideration to whom belong, according to your own sentence, the +unhandsome attributes you so often give me upon supposition, that you +yourselves are in the right, and I mistaken; and come now to purge +myself of those greater accusations which concern my manners. It cannot +be expected that there should be much science of any kind in a man that +wanteth judgment; nor judgment in a man that knoweth not the manners due +to a public disputation in writing; wherein the scope of either party +ought to be no other than the examination and manifestation of the +truth. For whatsoever is added of contumely, either directly or +_scommatically_, is want of charity and uncivil, unless it be done by +way of reddition from him that is first provoked to it. I say unless it +be by way of reddition; for so was the judgment given by the emperor +Vespasian in a quarrel between a senator and a knight of Rome which had +given him ill language. For when the knight had proved that the first +ill language proceeded from the senator, the emperor acquitted him in +these words: “_Maledici senatoribus non oportere; remaledicere, fas et +civile esse_.” Nevertheless, now-a-days, uncivil words are commonly and +bitterly used by all that write in matter of controversy, especially in +divinity, excepting now and then such writers as have been more than +ordinarily well bred, and have observed how heinous and hazardous a +thing such contumely is amongst some sorts of men, whether that which is +said in disgrace be true or false. For evil words by all men of +understanding are taken for a defiance, and a challenge to open war. But +that you should have observed so much, who are yet in your mother’s +belly, was not a thing to be much expected. + +The faults in manners you lay to my charge are these: 1. _Self-conceit._ +2. _That I will be very angry with all men that do not presently submit +to my dictates._ 3. _That I had my doctrine concerning Vision, out of +papers which I had in my hands of Mr. Warner’s._ 4. _That I have injured +the universities._ 5. _That I am an enemy to religion._ These are great +faults; but such as I cannot yet confess. And therefore I must, as well +as I can, seek out the grounds upon which you build your accusation. +Which grounds (seeing you are not acquainted with my conversation) must +be either in my published writings, or reported to you by honest men, +and without suspicion of interest in reporting it. As for my +self-conceit and ostentation, you shall find no such matter in my +writings. That which you allege from thence is first, that in the +epistle dedicatory I say of my book _De Corpore_, “_though it be little, +yet it is full; and if good may go for great, great enough._” When a man +presenting a gift great or small to his betters, adorneth it the best he +can to make it the more acceptable; he that thinks this to be +ostentation and self-conceit, is little versed in the common actions of +human life. And in the same epistle, where I say of civil philosophy: +“_It is no ancienter than my book De Cive_;” these words are added: “_I +say it provoked, and that my detractors may see they lose their +labour_.” But that which is truly said, and upon provocation, is not +boasting, but defence. A short sum of that book of mine, now publicly in +French, done by a gentleman I never saw, carrieth the title of _Ethics +Demonstrated_ . The book itself translated into French, hath not only a +great testimony from the translator Sorberius, but also from Gassendus, +and Mersennus, who being both of the Roman religion had no cause to +praise it, or the divines of England have no cause to find fault with +it. Besides, you know that the doctrine therein contained is generally +received by all but those of the clergy, who think their interest +concerned in being made subordinate to the civil power; whose +testimonies therefore are invalid. Why therefore, if I commend it also +against them that dispraise it publicly, do you call it boasting? “_You +have heard_,” you say, “_that I had promised the quadrature of the +circle, &c._” You heard then that which was not true. I have been asked +sometimes, by such as saw the figure before me, what I was doing, and I +was not afraid to say I was seeking for the solution of that problem; +but not that I had done it. And afterwards being asked of the success, I +have said, I thought it done. This is not boasting; and yet it was +enough, when told again, to make a fool believe it was boasting. But +you, the astronomer, in the epistle before your philosophical essay, say +“_You had a great expectation of my philosophical and mathematical +works, before they were published_.” It may be so. Is that my fault? Can +a man raise a great expectation of himself by boasting? If he could, +neither of you would be long before you raised it of yourselves; saving +that what you have already published, has made it now too late. For I +verily believe there was never seen worse reasoning than in that +philosophical essay; which any judicious reader would believe proceeded +from a prevaricator, rather than from a man that believed himself; nor +worse principles, than those in your books of Geometry. The expectation +of that which should be written by me, was raised partly by the +_Cogitata Physica-Mathematica_ of Mersennus, wherein I am often named +with honour; and partly by others with whom I then conversed in Paris, +without any ostentation. That no man has a great expectation of any +thing that shall proceed from either of you two, I am content to let it +be your praise. + +Another argument of my self-conceit, you take from my contempt of the +writers of metaphysics and school-divinity. If that be a sign of +self-conceit, I must confess I am guilty; and if your geometry had then +been published, I had contemned that as much. But yet I cannot see the +consequence (unless you lend me your better logic) from despising +insignificant and absurd language, to self-conceit. + +And again, in your _Vindiciæ Academiarum_, you put for boasting, that in +my _Leviathan_ , page 331, I would have _that book by entire sovereignty +imposed upon the Universities_; and in my _Review_ , p. 713, that I say +of my _Leviathan_ , “_I think it may be profitably printed, and more +profitably taught in the University_.” The cause of my writing that +book, was the consideration of what the ministers before, and in the +beginning of, the civil war, by their preaching and writing did +contribute thereunto. Which I saw not only to tend to the abatement of +the then civil power, but also to the gaining of as much thereof as they +could (as did afterwards more plainly appear) unto themselves. I saw +also that those ministers, and many other gentlemen who were of their +opinion, brought their doctrines against the civil power from their +studies in the Universities. Seeing therefore that so much as could be +contributed to the peace of our country, and the settlement of sovereign +power without any army, must proceed from teaching; I had reason to +wish, that civil doctrine were truly taught in the Universities. And if +I had not thought that mine was such, I had never written it. And having +written it, if I had not recommended it to such as had the power to +cause it to be taught, I had written it to no purpose. To me therefore +that never did write anything in philosophy to show my wit, but, as I +thought at least, to benefit some part or other of mankind, it was very +necessary to commend my doctrine to such men as should have the power +and right to regulate the Universities. I say my doctrine; I say not my +_Leviathan_ . For wiser men may so digest the same doctrine as to fit it +better for a public teaching. But as it is, I believe it hath framed the +minds of a thousand gentlemen to a conscientious obedience to present +government, which otherwise would have wavered in that point. This +therefore was no vaunting, but a necessary part of the business I took +in hand. You ought also to have considered, that this was said in the +close of that part of my book which concerneth policy merely civil. +Which part, if you, the astronomer, that now think the doctrine unworthy +to be taught, were pleased once to honour with praises printed before +it, you are not very constant nor ingenuous. But whether you did so or +not, I am not certain, though it was told me for certain. If it were not +you, it was somebody else whose judgment has as much weight at least as +yours. + +And for anything you have to say from your own knowledge, I remember not +that I ever saw either of your faces. Yet you, the professor of +geometry, go about obliquely to make me believe that Vindex hath +discoursed with me, once at least, though I remember it not. I suppose +it therefore true; but this I am sure is false, that either he or any +man living did ever hear me brag of my science, or praise myself, but +when my defence required it. Perhaps some of our philosophers that were +at Paris at the same time, and acquainted with the same learned men that +I was acquainted with, might take for bragging the maintaining of my +opinions, and the not yielding to the reasons alledged against them. If +that be ostentation, they tell you the truth. But you that are so wise +should have considered, that even such men as profess philosophy are +carried away with the passions of emulation and envy (the sole ground of +this your accusation) as well as other men, and instanced in yourselves. +And this is sufficient to shake off your aspersions of ostentation and +self-conceit. For if I added, that my acquaintance know that I am +naturally of modest rather than of boasting speech, you will not believe +it; because you distinguish not between that which is said upon +provocation, and that which is said without provocation, from vain +glory. + +The next accusation is: “_That I will be very angry with all men that do +not presently submit to my dictates; and that for advancing the +reputation of my own skill, I care not what unworthy reflections I cast +on others_.” This is in the epistle placed before the _Vindiciæ +Academiarum_, subscribed by N S, as the plain song for H D in the rest +of the book to descant upon. I know well enough the authors’ names; and +am sorry that N S has lent his name to be abused to so ill a purpose. +But how does this appear? What argument, what witness is there of it? +You offer none; nor am I conscious of any. I begin to suspect since you, +the professor of geometry, have in your objections to the twentieth +chapter these words concerning “_Vindex, ocularis ille testis de quo hic +agitur, erat, ni fallor, ille ipse_,”--that Vindex himself, in other +company, has bestowed a visit on me. Seeing you will have me believe it, +let it be so; and, as it is likely, not long after my return into +England. At which time (for the reputation, it seems, I had gotten by my +boasting) divers persons that professed to love philosophy and +mathematics, came to see me; and some of them to let me see them, and +hear and applaud what they applauded in themselves. I see now it hath +happened to me with Vindex, as it happened to Dr. Harvey with Moranus. +Moranus, a jesuit, came out of Flanders hither, especially, as he says, +to see what learned men in divinity, ethics, physics, and geometry, were +here yet alive, to the end that by discoursing with them in these +sciences, he might correct either his own, or their errors. Amongst +others he was brought, he says, to that most civil and renowned old man +Dr. Harvey. That is very well. And in good earnest if he had made good +use of the time which was very patiently afforded him, he might have +learned of him (or of no man living) very much knowledge concerning the +circulation of the blood, the generation of living creatures, and many +other difficult points of natural philosophy. And if he had had anything +in him but common and childish learning, he could have showed it nowhere +more to his advantage, than before him that was so great a judge of such +matters. But what did he? That precious time (which was but little, +because he was to depart again presently for Flanders) he bestowed +wholly in venting his own childish opinions, not suffering the Doctor +scarce to speak; losing thereby the benefit he came for, and discovering +that he came not to hear what others could say, but to show to others +how learned he was himself already. Why else did he take so little time, +and so misspend it? Or why returned he not again? But when he had talked +away his time, and found (though patiently and civilly heard) he was not +much admired, he took occasion, writing against me, to be revenged of +Dr. Harvey, by slighting his learning publicly; and tells me that his +learning was only experiments; which he says I say have no more +certainty than civil histories. Which is false. My words are: “_Ante hos +nihil certi in physica erat præter experimenta cuique sua, et historias +naturales, si tamen et hæ dicendæ certæ sint, quæ civilibus historiis +certiores non sunt_.” Where I except expressly from uncertainty the +experiments that every man maketh to himself. But you see the near cut, +by which vain glory joined with ignorance passeth quickly over to envy +and contumely. + +Thus it seems by your own confession I was used by Vindex. He comes with +some of my acquaintance in a visit. What he said I know not, but if he +discoursed then, as in his philosophical essay he writeth, I will be +bold to say of myself, I was so far from morosity, or, to use his +phrase, from being tetrical, as I may very well have a good opinion of +my own patience. And if there passed between us the discourse you +mention in your _Elenchus_, page 116, it was an incivility in him so +great, that without great civility I could not have abstained from +bidding him be gone. That which passed between us you say was this: “_I +complained that whereas I made sense, nothing but a perception of motion +in the organ, nevertheless, the philosophy schools through all Europe, +led by the text of Aristotle, teach another doctrine, namely, that +sensation is performed by species_.” This is a little mistaken. For I do +glory, not complain, that whereas all the Universities of Europe hold +sensation to proceed from species, I hold it to be a perception of +motion in the organ. The answer of Vindex, you say, was: “_That the +other hypothesis, whereby sense was explicated by the principles of +motion, was commonly admitted here before my book came out, as having +been sufficiently delivered by Des Cartes, Gassendus, and Sir Kenelm +Digby, before I had published anything in this kind_.” This then, it +seems, was it that made me angry. Truly I remember not an angry word +that ever I uttered in all my life to any man that came to see me, +though some of them have troubled me with very impertinent discourse; +and with those that argued with me, how impertinently soever, I always +thought it more civility to be somewhat earnest in the defence of my +opinion, than by obstinate and affected silence to let them see I +contemned them, or hearkened not to what they said. If I were earnest in +making good, that the manner of sensation by such motion as I had +explicated in my _Leviathan_ , is in none of the authors by him named, +it was not anger, but a care of not offending him, with any sign of the +contempt which his discourse deserved. But it was incivility in him to +make use of a visit, which all men take for a profession of friendship, +to tell me that that which I had already published for my own, was found +before by Des Cartes, Gassendus, and Sir Kenelm Digby. But let any man +read Des Cartes; he shall find, that he attributeth no motion at all to +the object of sense, but an inclination to action, which inclination no +man can imagine what it meaneth. And for Gassendus, and Sir Kenelm +Digby, it is manifest by their writings, that their opinions are not +different from that of Epicurus, which is very different from mine. Or +if these two, or any of those I conversed with at Paris, had prevented +me in publishing my own doctrine, yet since it was there known and +declared for mine by Mersennus in the preface to his _Ballistica_ (of +which the three first leaves are employed wholly in the setting forth of +my opinion concerning sense, and the rest of the faculties of the soul) +they ought not therefore to be said to have found it out before me. And +consequently this answer which you say was given me by Vindex was +nothing else but untruth and envy; and, because it was done by way of +visit, incivility. But you have not alleged, nor can allege, any words +of mine, from which can be drawn that I am so angry as you say I am with +those that submit not to my dictates. Though the discipline of the +University be never so good; yet certainly this behaviour of yours and +his are no good arguments to make it thought so. But you the professor +of geometry, that out of my words spoken against Vindex in my twentieth +chapter, argue my angry humour, do just as well, as when (in your +_Arithmetica Infinitorum_) from the continual increase of the excess of +the row of squares above the third part of the aggregate of the +greatest, you conclude they shall at last be equal to it. For though you +knew that Vindex had given me first the worst words that possibly can be +given, yet you would have that return of mine to be a demonstration of +an angry humour; not then knowing what I told you even now in the +beginning of this lesson, of the sentence given by Vespasian. But to +this point I shall speak again hereafter. + +Your third accusation is: “_That I had my doctrine of vision, which I +pretended to be my own, out of papers which I had a long time in my +hands of Mr. Warner’s_.” I never had sight of Mr. Warner’s papers in all +my life, but that of _Vision by Refraction_ (which by his approbation I +carried with me to Park, and caused it to be printed under his own name, +at the end of Mersennus his _Cogitata Physico-Mathematica_, which you +may have there seen, and another treatise of the proportions of alloy in +gold and silver coin; which is nothing to the present purpose). In all +my conversation with him, I never heard him speak of anything he had +written, or was writing, _De penicillo optico_. And it was from me that +he first heard it mentioned that light and colour were but fancy. Which +he embraced presently as a truth, and told me it would remove a rub he +was then come to in the discovery of the place of the image. If after my +going hence he made any use of it (though he had it from me, and not I +from him), it was well done. But wheresoever you find my principles, +make use of them, if you can, to demonstrate all the symptoms of vision; +and I will do (or rather have done and mean to publish) the same; and +let it be judged by that, whether those principles be of mine, or other +men’s invention. I give you time enough, and this advantage besides, +that much of my optics hath been privately read by others. For I never +refused to lend my papers to my friends, as knowing it to be a thing of +no prejudice to the advancement of philosophy, though it be, as I have +found it since, some prejudice to the advancement of my own reputation +in those sciences; which reputation I have always postposed to the +common benefit of the studious. + +You say further (you the geometrician) that I had the proposition of the +spiral line equal to a parabolical line from Mr. Robervall: true. And if +I had remembered it, I would have taken also his demonstration; though +if I had published his, I would have suppressed mine. I was comparing in +my thoughts those two lines, spiral and parabolical, by the motions +wherewith they were described; and considering those motions as uniform, +and the lines from the centre to the circumference, not to be little +parallelograms, but little sectors, I saw that to compound the true +motion of that point which described the spiral, I must have one line +equal to half the perimeter, the other equal to half the diameter. But +of all this I had not one word written. But being with Mersennus and Mr. +Robervall in the cloister of the convent, I drew a figure on the wall, +and Mr. Robervall perceiving the deduction I made, told me that since +the motions which make the parabolical line, are one uniform, the other +accelerated, the motions that make the spiral must be so also; which I +presently acknowledged; and he the next day, from this very method, +brought to Mersennus the demonstration of their equality. And this is +the story mentioned by Mersennus, prop. 25, corol. 2, of his +_Hydraulica_; which I know not who hath most magnanimously interpreted +to you in my disgrace. + +The fourth accusation is: “_That I have injured the Universities_.” +Wherein? First, “_In that I would have the doctrine of my Leviathan by +entire sovereignty be imposed on them_.” You often upbraid me with +thinking well of my own doctrine; and grant by consequence, that I +thought this doctrine good; I desired not therefore that anything should +be imposed upon them, but what (at least in my opinion) was good both +for the Commonwealth and them. Nay more, I would have the state make use +of them to uphold the civil power, as the Pope did to uphold the +ecclesiastical. Is it not absurdly done to call this an injury? But to +question, you will say, whether the civil doctrine there taught be such +as it ought to be, or not, is a disgrace to the Universities. If that be +certain, it is certain also that those sermons and books, which have +been preached and published, both against the former and the present +government, directly or obliquely, were not made by such ministers and +others as had their breeding in the Universities; though all men know +the contrary. But the doctrine which I would have to be taught there, +what is it? It is this: “_That all men that live in a Commonwealth, and +receive protection of their lives and fortunes from the supreme governor +thereof, are reciprocally bound, as far as they are able, and shall be +required, to protect that governor_.” Is it, think you, an unreasonable +thing to impose the teaching of such doctrine upon the Universities? Or +will you say they taught it before, when you know that so many men which +came from the Universities to preach to the people, and so many others +that were not ministers, did stir the people up to resist the then +supreme civil power? And was it not truly therefore said, that the +Universities receiving their discipline from the authority of the pope, +were the shops and operatories of the clergy? Though the competition of +the papal and civil power be taken now away, yet the competition between +the ecclesiastical and the civil power hath manifestly enough appeared +very lately. But neither is this an upbraiding of an University (which +is a corporation or body artificial), but of particular men, that desire +to uphold the authority of a Church, as of a distinct thing from the +Commonwealth. How would you have exclaimed, if, instead of recommending +my _Leviathan_ to be taught in the Universities, I had recommended the +erecting of a new and lay-university, wherein lay-men should have the +reading of physics, mathematics, moral philosophy, and politics, as the +clergy have now the sole teaching of divinity? Yet the thing would be +profitable, and tend much to the polishing of man’s nature, without much +public charge. There will need but one house, and the endowment of a few +professions. And to make some learn the better, it would do very well +that none should come thither sent by their parents, as to a trade to +get their living by, but that it should be a place for such ingenuous +men, as being free to dispose of their own time, love truth for itself. +In the mean time divinity may go on in Oxford and Cambridge to furnish +the pulpit with men to cry down the civil power, if they continue to do +as they did. If I had, I say, made such a motion in my _Leviathan_ , +though it would have offended the divines, yet it had been no injury. +But it is an injury, you will say, to deny in general the utility of the +ancient schools, and to deny that we have received from them our +geometry. True, if I had not spoken distinctly of the schools of +philosophy, and said expressly, that the geometricians passed not then +under the name of philosophers; and that in the school of Plato (the +best of the ancient philosophers) none were received that were not +already in some measure geometricians. Euclid taught geometry; but I +never heard of a sect of philosophers called Euclidians, or +Alexandrians, or ranged with any of the other sects, as Peripatetics, +Stoics, Academics, Epicureans, Pyrrhonians, &c. But what is this to the +Universities of Christendom? Or why are we beholden for geometry to our +universities, more than to Gresham College, or to private men in London, +Paris, and other places, which never taught or learned it in a public +school? For even those men that living in our Universities have most +advanced the mathematics, attained their knowledge by other means than +that of public lectures, where few auditors, and those of unequal +proficiency, cannot make benefit by one and the same lesson. And the +true use of public professors, especially in the mathematics, being to +resolve the doubts, and problems, as far as they can, of such as come +unto them with desire to be informed. + +That the Universities now are not regulated by the Pope, but by the +civil power, is true, and well. But where say I the contrary? And thus +much for the first injury. + +Another, you say, is this, that in my _Leviathan_ , p. 670, I say: “_The +principal schools were ordained for the three professions of Roman +religion, Roman law, and the art of medicine_.” Thirdly, that I say: +“_Philosophy had no otherwise place there than as a hand-maid to Roman +religion_.” Fourthly: “_Since the authority of Aristotle was received +there, that study is not properly philosophy, but Aristotelity_.” +Fifthly: “_That for geometry, till of late times it had no place there +at all_.” As for the second, it is too evident to be denied; the +fellowships having been all ordained for those professions; and (saving +the change of religion) being so yet. Nor hath this any reflection upon +the Universities, either as they now are, or as they then were, seeing +it was not in their own power to endow themselves, or to receive other +laws and discipline than their founder and the state was pleased to +ordain. For the third, it is also evident. For all men know that none +but the Roman religion had any stipend or preferment in any university, +where that religion was established? No, nor for a great while, in their +commonwealths; but were everywhere persecuted as heretics. But you will +say, the words of my _Leviathan_ are not, philosophy “_had no place_,” +but “_hath no place_.” Are you not ashamed to lay to my charge a mistake +of the word _hath_ for _had_? which was either a mistake of the printer, +or if it were so in the copy, it could be no other than the mistake of a +letter in the writing, unless you think you can make men believe that +after fifty years being acquainted with what was publicly professed and +practised in Oxford and Cambridge, I knew not what religion they were +of. This taking of advantage from the mistake of a word, or of a letter, +I find also in the _Elenchus_, where for _prætendit se scire_, there is +_prætendit scire_, which you the geometrician sufficiently mumble, +mistaking it I think for an anglicism, not for a fault of the +impression. + +To the fourth, you pretend, that men are not now so tied to Aristotle as +not to _enjoy a liberty of philosophising, though it were otherwise when +I was conversant in Magdalen Hall_. Was it so then? Then am I absolved, +unless you can show some public act of the university made since that +time to alter it. For it is not enough to name some few particular +ingenuous men that usurp that liberty in their private discourses, or, +with connivance, in their public disputations. And your doctrine, that +even here you avow, of _abstracted essences_, _immaterial substances_, +and of _nunc-stans_; and your improper language in using the word (not +as mine, for I have it nowhere) _successive eternity_; as also your +doctrine of _condensation_, and your arguing from natural reason the +incomprehensible mysteries of religion, and your malicious writing, are +very shrewd signs that you yourselves are none of those which you say do +_freely philosophise_; but that both your philosophy and your language +are under the servitude, not of the Roman religion, but of the ambition +of some other doctors, that seek, as the Roman clergy did, to draw all +human learning to the upholding of their power ecclesiastical. Hitherto +therefore there is no injury done to the universities. For the fifth, +you grant it, namely, “_that till of late there was no allowance for the +teaching of geometry_.” But lest you should be thought to grant me +anything, you say, you the astronomer, “_geometry hath now so much place +in the universities, that when Mr. Hobbes shall have published his +philosophical and geometrical pieces, you assure yourself you shall be +able to find a greater number in the university who will understand as +much, or more, of them than he desired they should_,” &c. But though +this be true of the _now_, yet it maketh nothing against my _then_. I +know well enough that Sir Henry Savile’s lectures were founded and +endowed since. Did I deny _then_ that there were in Oxford many good +geometricians? But I deny _now_, that either of you is of the number. +For my philosophical and geometrical pieces are published, and you have +understood only so much in them, as all men will easily see by your +objections to them, and by your own published geometry, that neither of +you understand anything either in philosophy or in geometry. And yet you +would have those books of yours to stand for an argument, and to be an +index of the philosophy and geometry to be found in the universities. +Which is a greater injury and disgrace to them, than any words of mine, +though interpreted by yourselves. + +Your last and greatest accusation, or rather railing (for an accusation +should contain, whether true or false, some particular fact, or certain +words, out of which it might seem at least to be inferred), is, that I +am an enemy to religion. Your words are: “_It is said that Mr. Hobbes is +no otherwise an enemy to the Roman religion, saving only as it hath the +name of religion_.” This is said by Vindex. You, the geometrician, in +your epistle dedicatory, say thus: “_With what pride and imperiousness +he tramples on all things both human and divine, uttering fearful and +horrible words of God_, (peace), _of sin, of the holy Scripture, of all +incorporeal substances in general, of the immortal soul of man, and of +the rest of the weighty points of religion_ (down), _it is not so much +to be doubted as lamented_.” And at the end of your objections to the +eighteenth chapter, “_Perhaps you take the whole history of the fall of +Adam for a fable, which is no wonder, when you say the rules of +honouring and worshipping of God are to be taken from the laws_.” Down, +I say; you bark now at the supreme legislative power. Therefore it is +not I, but the laws which must rate you off. But do not many other men, +as well as you, read my _Leviathan_ , and my other books? And yet they +all find not such enmity in them against religion. Take heed of calling +them all atheists that have read and approved my _Leviathan_ . Do you +think I can be an atheist and not know it? Or knowing it, durst have +offered my atheism to the press? Or do you think him an atheist, or a +contemner of the Holy Scripture, that sayeth nothing of the Deity but +what he proveth by the Scripture? You that take so heinously that I +would have the rules of God’s worship in a Christian commonwealth taken +from the laws, tell me, from whom you would have them taken? From +yourselves? Why so, more than from me? From the bishops? Right, if the +supreme power of the commonwealth will have it so; if not, why from them +rather than from me? From a consistory of presbyters by themselves, or +joined with lay-elders, whom they may sway as they please? Good, if the +supreme governor of the commonwealth will have it so; if not, why from +them, rather than from me, or from any man else? They are wiser and +learneder than I. It may be so; but it has not yet appeared. Howsoever, +let that be granted. Is there any man so very a fool as to subject +himself to the rules of other men in those things which so nearly +concern himself, for the title they assume of being wise and learned, +unless they also have the sword which must protect them. But it seems +you understand the sword as comprehended. If so, do you not then receive +the rules of God’s worship from the civil power? Yes, doubtless; and you +would expect, if your consistory had that sword, that no man should dare +to exercise or teach any rules concerning God’s worship which were not +by you allowed. See therefore how much you have been transported by your +malice towards me, to injure the civil power by which you live. If you +were not despised, you would in some places and times, where and when +the laws are more severely executed, be shipped away for this your +madness to America, I would say, to Anticyra. What luck have I, when +this, of the laws being the rules of God’s public worship, was by me +said and applied to the vindication of the Church of England from the +power of the Roman clergy, it should be followed with such a storm from +the ministers, presbyterian and episcopal, of the Church of England? +Again, for those other points, namely, that I approve not of incorporeal +bodies, nor of other immortality of the soul, than that which the +Scripture calleth eternal life, I do but as the Scripture leads me. To +the texts whereof by me alleged, you should either have answered, or +else forborne to revile me for the conclusions I derived from them. +Lastly, what an absurd question is it to ask me whether it be in the +power of the magistrate, whether the world be eternal or not? It were +fit you knew it is in the power of the supreme magistrate to make a law +for the punishment of them that shall pronounce publicly of that +question anything contrary to that which the law hath once pronounced. +The truth is, you are content that the papal power be cut off, and +declaimed against as much as any man will; but the ecclesiastical power, +which of late was aimed at by the clergy here, being a part thereof, +every violence done to the papal power is sensible to them yet; like +that which I have heard say of a man, whose leg being cut off for the +prevention of a gangrene that began in his toe, would nevertheless +complain of a pain in his toe, when his leg was cut off. + +Thus much in my defence; which I believe if you had foreseen, this +accusation of yours had been left out. I come now to examine (though it +be done in part already) what manners those are which I find everywhere +in your writings. + +And first, how came it into your minds that a man can be an atheist, I +mean an atheist in his conscience? I know that David confesseth of +himself, upon sight of the prosperity of the wicked, that his feet had +almost slipped, that is, that he had slipped into a short doubtfulness +of the Divine Providence. And if anything else can cause a man to slip +in the same kind, it is the seeing such as you (who though you write +nothing but what is dictated to each of you by a doctor of divinity) do +break the greatest of God’s commandments, which is charity, in every +line before his face. And though such forgettings of God be somewhat +more than short doubtings, and sudden transportations incident to human +passion, yet I do not for that cause think you atheists and enemies of +religion, but only ignorant and imprudent Christians. But how, I say, +could you think me an atheist, unless it were because finding your +doubts of the Deity more frequent than other men do, you are thereby the +apter to fall upon that kind of reproach? Wherein you are like women of +poor and evil education when they scold; amongst whom the readiest +disgraceful word is whore: why not thief, or any other ill name, but +because, when they remember themselves, they think that reproach the +likeliest to be true? + +Secondly, tell me what crime it was which the Latins called by the name +of _scelus_? You think not, unless you be Stoics, that all crimes are +equal. _Scelus_ was never used but for a crime of greatest mischief, as +the taking away of life and honour; and besides, basely acted, as by +some clandestine way, or by such a way as might be covered with a lie. +But when you insinuate in a writing published that I am an atheist, you +make yourselves authors to the multitude, and do all you can to stir +them up to attempt upon my life; and if it succeed, then to sneak out of +it by leaving the fault on them that are but actors. This is to +endeavour great mischief basely, and therefore _scelus_. Again, to +deprive a man of the honour he hath merited, is no little wickedness; +and this you endeavour to do by publishing falsely that I challenge as +my own the inventions of other men. This is therefore _scelus_ publicly +to tell all the world that I will be angry with all men that do not +presently submit to my dictates; to deprive me of the friendship of all +the world; great damage, and a lie, and yours. For to publish any +untruth of another man to his disgrace, on hearsay from his enemy, is +the same fault as if he published it on his own credit. If I should say +I have heard that Dr. Wallis was esteemed at Oxford for a simple fellow, +and much inferior to his fellow-professor Dr. Ward (as indeed I have +heard, but do not believe it), though this be no great disgrace to Dr. +Wallis, yet he would think I did him injury. Therefore public accusation +upon hearsay is _scelus_. And whosoever does any of these things does +_sceleratè_. But you the professors of the mathematics at Oxford, by the +advice of two doctors of divinity have dealt thus with me. Therefore you +have done, I say not foolishly, though no wickedness be without folly, +but _sceleratè_, ὅπερ ἔδει δεῖξαι. + +Thirdly, it is ill manners, in reprehending truth, to send a man in a +boasting way to your own errors; as you the professor of geometry have +often sent me to your two tractates of the _Angle of Contact_ and +_Arithmetica Infinitorum_. + +Fourthly, it is ill manners, to diminish the just reputation of worthy +men after they be dead, as you the professor of geometry have done in +the case of Joseph Scaliger. + +Fifthly, when I had in my _Leviathan_ suffered the clergy of the Church +of England to escape, you did imprudently in bringing any of them in +again. An Ulysses upon so light an occasion would not have ventured to +return again into the cave of Polyphemus. + +Lastly, how ill does such levity and scurrility, which both of you have +shown so often in your writings, become the gravity and sanctity +requisite to the calling of the ministry? They are too many to be +repeated. Do but consider, you the geometrician, how unhandsome it is to +play upon my name, when both yours and mine are plebeian names; though +from Willis by Wallis, you go from yours in Wallisius. The jest of using +at every word _mi Hobbi_, is lost to them beyond sea. But this is not so +ill as some of the rest. I will write out one of them, as it is in the +fourth page of your _Elenchus_: “_Whence it appears that your Empusa was +of the number of those fairies which you call in English hob-goblins. +The word is made of_ ἕν and πους; _and thence comes the children’s play +called the play of Empusa, Anglicè_ (hitherto in Latin all but +_hob-goblins_, then follows in English) _fox, fox, come out of your +hole_ (then in Latin again), _in which the boy that is called the fox, +holds up one foot, and jumps with the other, which in English is to +hop_.” When a stranger shall read this, and hoping to find therein some +witty conceit, shall with much ado have gotten it interpreted and +explained to him, what will he think of our doctors of divinity at +Oxford, that will take so much pains as to go out of the language they +set forth in, for so ridiculous a purpose? You will say it is a pretty +_paranomasia_. How you call it there I know not, but it is commonly +called here a _clinch_; and such a one as is too insipid for a boy of +twelve years old, and very unfit for the sanctity of a minister, and +gravity of a doctor of divinity. But I pray you tell me where it was you +read the word _empusa_ for the boy’s play you speak of, or for any other +play amongst the Greeks? In this (as you have done throughout all your +other writings) you presume too much upon your first cogitations. There +be a hundred other scoffing passages, and ill-favoured attributes given +me in both your writings, which the reader will observe without my +pointing to them, as easily as you would have him; and which perhaps +some young students, finding them full of gall, will mistake for salt. +Therefore to disabuse those young men, and to the end they may not +admire such kind of wit, I have here and there been a little sharper +with you than else I would have been. If you think I did not spare you, +but that I had not wit enough to give you as scornful names as you give +me, are you content I should try? Yes (you the geometrician will say) +give me what names you please, so you call me not _Arithmetica +Infinitorum_. I will not. Nor _Angle of Contact_ ; nor _Arch Spiral_ ; +nor _Quotient_ . I will not. But I here dismiss you both together. So go +your ways, you _Uncivil Ecclesiastics, Inhuman Divines, Dedoctors of +morality, Unasinous Colleagues, Egregious pair of Issachars, most +wretched Vindices and Indices Academiarum_; and remember Vespasian’s +law, that it is uncivil to give ill language first, but civil and lawful +to return it. But much more remember the law of God, to obey your +sovereigns in all things; and not only not to derogate from them, but +also to pray for them, and as far as you can to maintain their +authority, and therein your own protection. And, do you hear? take heed +of speaking your mind so clearly in answering my _Leviathan_, as I have +done in writing it. You should do best not to meddle with it at all, +because it is undertaken, and in part published already, and will be +better performed, from term to term, by one Christopher Pike. + + + + + ΣΤΙΓΜΑΙ + + Αγεωμετρίας, Αγροικίας, Αντίπολιτείας, Αμαθείας, + + OR + + MARKS + + OF THE + + ABSURD GEOMETRY, RURAL LANGUAGE, SCOTTISH + CHURCH POLITICS, AND BARBARISMS + + OF + + JOHN WALLIS, + PROFESSOR OF GEOMETRY AND DOCTOR OF DIVINITY. + + BY + + THOMAS HOBBES, + + OF MALMESBURY. + + TO THE RIGHT HONOURABLE + + HENRY, LORD PIERREPONT, + + VISCOUNT NEWARK, EARL OF KINGSTON, AND + MARQUIS OF DORCHESTER. + + ========== + +MY MOST NOBLE LORD, + +I did not intend to trouble your Lordship twice with this contention +between me and Dr. Wallis. But your Lordship sees how I am constrained +to it; which, whatsoever reply the Doctor makes, I shall be constrained +to no more. That which I have now said of his Geometry, Manners, +Divinity, and Grammar, altogether is not much, though enough. As for +that which I here have written concerning his Geometry, which you will +look for first, is so clear, that not only your Lordship, and such as +have proceeded far in that science, but also any man else that doth but +know how to add and subtract proportions, (which is taught at the +twenty-third proposition of the sixth of Euclid), may see the Doctor is +in the wrong. That which I say of his ill language and politics is yet +shorter. The rest, which concerneth grammar, is almost all another +man’s, but so full of learning of that kind, as no man that taketh +delight in knowing the proprieties of the Greek and Latin tongues, will +think his time ill bestowed in the reading it. I give the Doctor no more +ill words, but am returned from his manners to my own. Your Lordship may +perhaps say, my compliment in my title-page is somewhat coarse; and it +is true. But, my Lord, it is since the writing of the title-page, that I +am returned from the Doctor’s manners to my own; which are such as I +hope you will not be ashamed to own me, my Lord, for one of + + Your Lordship’s most humble + + and obedient servants, + + THOMAS HOBBES. + + + + + + + + + ========== + + + + + TO + + DOCTOR WALLIS, + + IN ANSWER TO HIS + + SCHOOL DISCIPLINE + + --- + +SIR, + +When unprovoked you addressed unto me, in your _Elenchus_, your harsh +compliment with great security, wantonly to show your wit, I confess you +made me angry, and willing to put you into a better way of considering +your own forces, and to move you a little as you had moved me, which I +perceive my lessons to you have in some measure done; but here you shall +see how easily I can bear your reproaches, now they proceed from anger, +and how calmly I can argue with you about your geometry and other parts +of learning. + +I shall in the first part confer with you about your _Arithmetica +Infinitorum_, and afterwards compare our manner of elocution; then your +politics; and last of all your grammar and critics. Your spiral line is +condemned by him whose authority you use to prove me a plagiary, (that +is, a man that stealeth other men’s inventions, and arrogates them to +himself), whether it be Roberval or not that writ that paper, I am not +certain. But I think I shall be shortly; but whosoever it be, his +authority will serve no less to show that your doctrine of the spiral +line, from the fifth to the eighteenth proposition of your _Arithmetica +Infinitorum_, is all false; and that the principal fault therein (if all +faults be not principal in geometry, when they proceed from ignorance of +the science) is the same that I objected to you in my _Lessons_. And for +the author of that paper, when I am certain who it is, it will be then +time enough to vindicate myself concerning that name of plagiary. And +whereas he challenges the invention of your method delivered in your +_Arithmetica Infinitorum_, to have been his before it was yours, I +shall, I think, by and by say that which shall make him ashamed to own +it; and those that writ those encomiastic epistles to you ashamed of the +honour they meant to you. I pass therefore to the nineteenth +proposition, which in Latin is this: your geometry! + +“_Si proponatur series quantitatum in duplicata ratione arithmetice +proportionalium (sive juxta seriem numerorum quadraticorum) continue +crescentium, a puncto vel 0 inchoatarum, (puta ut 0. 1. 4. 9. 16. etc.), +propositum sit, inquirere quam habeat illa rationem ad seriem totidem +maximæ æqualium._ + +“_Fiat investigatio per modum inductionis ut_ (_in prop. 1_) + +_Eritque_, + +(0 + 1 = 1)/(1 + 1 = 2) = (1)/(3) + (1)/(6) + +(0 + 1 + 4 = 5)/(4 + 4 + 4 = 12) = (1)/(3) + (1)/(12) + +(0 + 1 + 4 + 9 = 14)/(9 + 9 + 9 + 9 = 36) = (1)/(3) + (1)/(18) _et sic +deinceps_. + +“_Ratio proveniens est ubique major quam subtripla seu (1)/(3); excessus +autem perpetuo decrescit prout numerus terminorum augetur (puta (1)/(6) +(1)/(12) (1)/(18) (1)/(24) etc.) aucto nimirum fractionis denominatore +sive consequente rationis in singulis locis numero senario (ut patet) ut +sit rationis provenientis excessus supra subtriplam, ea quam habet +unitas ad sextuplum numeri terminorum post 0; adeoque._” + +That is, if there be propounded a row of quantities in duplicate +proportion of the quantities arithmetically proportional (or proceeding +in the order of the square numbers) continually increasing; and +beginning at a point or 0; let it be propounded to find what proportion +the row hath; to as many quantities equal to the greatest; + +Let it be sought by induction (as in the first proposition). + +The proportion arising is everywhere greater than subtriple, or (1)/(3), +and the excess perpetually decreaseth as the number of terms is +augmented, as here, (1)/(6) (1)/(12) (1)/(18) (1)/(24) (1)/(30), &c. +denominator of the fraction being in every place augmented by the number +six, as is manifest; so that the excess of the rising proportion above +subtriple is the same which unity hath to six times the number of terms +after 0; and so. + +Sir, in these your characters I understand by the cross + that the +quantities on each side of it are to be added together and make one +aggregate; and I understand by the two parallel lines = that the +quantities between which they are placed are one to another equal; this +is your meaning, or you should have told us what you meant else; I +understand also, that in the first row 0 + 1 is equal to 1, and 1 + 1 +equal to 2; and that in the second row 0 + 1 + 4 is equal to 5; and 4 + +4 + 4 equal to 12; but (which you are too apt to grant) I understand +your symbols no further; but must confer with yourself about the rest. + +And first I ask you (because fractions are commonly written in that +manner) whether in the uppermost row (which is (0 + 1 = 1)/(1 + 1 = 2) = +(1)/(3) + (1)/(6))(0)/(1) be a fraction, (1)/(1) be a fraction, (1)/(2) +be a fraction, that is to say, a part of an unit, and if you will, for +the cypher’s sake, whether (0)/(1), be an infinitely little part of 1; +and whether (1)/(1) or 1 divided by 1 signify an unity? if that be your +meaning, then the fraction (0)/(1) added to the fraction (1)/(1) is +equal to the fraction (1)/(2): But the fraction (0)/(1) is equal to O; +therefore the fraction (0)/(1) + (1)/(1) is equal to the fraction +(1)/(1); and (1)/(1) equal to (1)/(2) which you will confess to be an +absurd conclusion, and cannot own that meaning. + +I ask you therefore again, if by (0)/(1) you mean the proportion of 0 to +1; and consequently by (1)/(1) the proportion of 1 to 1, and by (1)/(2) +the proportion of 1 to 2: if so, then it will follow, that if the +proportions of 0 to 1 and of 1 to 1 be compounded by addition, the +proportion arising will be the proportion of 1 to 2. But the proportion +of 0 to 1 is infinitely little, that is, none. Therefore the proposition +arising by composition will be that of 1 to 1, and equal (because of the +symbol =) to the proportion of 1 to 2, and so 1 = 2. This also is so +absurd that I dare say that you will not own it. + +There may be another meaning yet: perhaps you mean that the uppermost +quantity 0 + 1 is equal to the uppermost quantity 1; and the lowermost +quantity 1 + 1 equal to the lowermost quantity 2: which is true. But how +then in this equation (1)/(2) = (1)/(3) + (1)/(6)? Is the uppermost +quantity 1 equal to the uppermost quantity 1 + 1; or the lowermost +quantity 2 equal to the lowermost quantity 3 + 6? Therefore neither can +this be your meaning. Unless you make your symbols more significant, you +must not blame me for want of understanding them. + +Let us now try what better success we shall have where the places are +three, as here: + + (0 + 1 + 4 = 5)/(4 + 4 + 4 = 12) = (5)/(12) = (1)/(3) + (1)/(12): + +If your symbols be fractions, the compound of them by addition is +(5)/(4), for 0(1)/(4) and (4)/(4) make (5)/(4); and consequently +(because of the symbol = ) (5)/(4) equal to (5)/(12), which is not to be +allowed, and therefore that was not your meaning. If you meant that the +proportions of 0 to 4 and of 1 to 4 and of 4 to 4 compounded, is equal +to the proportion of 5 to 12, you will fall again into no less an +inconvenience. For the proportion arising out of that composition will +be the proportion of 1 to 4. For the proportion of 0 to 4 is infinitely +little. Then to compound the other two, set them in this order 1. 4. 4. +and you have a proportion compounded of 1 to 4 and of 4 to 4, namely, +the proportion of the first to the last, which is of 1 to 4, which must +be equal, by this your meaning, to the proportion of 5 to 12, and +consequently as 5 to 12, so is 1 to 4, which you must not own. Lastly, +if you mean that the uppermost quantities to the uppermost, and the +lowermost to the lowermost in the first equation are equal, it is +granted, but then again in the second equation it is false. It concerns +your fame in the mathematics to look about how to justify these +equations which are the premises to your conclusion following, namely, +that the proportion arising is every where greater than sub-triple, or a +third; and that the excess (that is, the excess above subtriple) +perpetually decreaseth as the number of terms is augmented, as here +(1)/(6) (1)/(12) (1)/(18) (1)/(24) (1)/(30), &c. which I will show you +plainly is false. + +But first I wonder why you were so angry with me for saying you made +proportion to consist in the quotient, as to tell me it was abominably +false, and to justify it, cite your own words _penes quotientem_; do not +you say here, the proportion is everywhere greater than subtriple, or +(1)/(3)? And is not (1)/(3) the quotient of 1 divided by 3? You cannot +say in this place that _penes_ is understood; for if it were expressed +you would not be able to proceed. + +But I return to your conclusion, that the excess of the proportion of +the increasing quantities above the third part of so many times the +greatest, decreaseth, as (1)/(6) (1)/(12) (1)/(18) (1)/(24) (1)/(30), +&c. For by this account in this row (0 + 1)/(1 + 1) = (1)/(2) where the +quantity above exceeds the third part of the quantities below by +(1)/(3), you make (1)/(3) equal to (1)/(6), which you do not mean. It +may be said your meaning is, that the proportion of 1 to the subtriple +of 2 which is (2)/(3), exceedeth what? I cannot imagine what, nor +proceed further where the terms be but two. Let us therefore take the +second row, that is, (0 + 1 + 4)/(4 + 4 + 4) = (5)/(12). The sum above +is 5, the sum below is 12, the third part whereof is 4; if you mean, +that the proportion of 5 to 4 exceeds the proportion of 4 to 12 (which +is subtriple) by (1)/(12), you are out again. For 5 exceeds 4 by unity, +which is (12)/(12). I do not think you will own such an equation as +(12)/(12) = (1)/(12) Therefore I believe you mean (and your next +proposition assures me of it), that the proportion of 5 to 4 exceeds +subtriple proportion by the proportion of 1 to 12; if you do so, you are +yet deceived. + +For if the proportion of 5 to 4 exceeds subtriple proportion by the +proportion of 1 to 12, then subtriple proportion, that is, of 4 to 12 +added to the proportion of 1 to 12 must make the proportion of 5 to 4. +But if you look on these quantities, 4, 12, 144, you will see, and must +not dissemble, that the proportion of 4 to 12 is subtriple, and the +proportion of 12 to 144 is the same with that of 1 to 12. Therefore by +your assertion it must be as 5 to 4 so 4 to 144, which you must not own. + +And yet this is manifestly your meaning, as appeareth in these words: +“_Ut sit rationis provenientis excessus supra subtriplam ea quam habet +unitas ad sextuplum numeri terminorum post 0, adeoque_,” which cannot be +rendered in English, nor need to be. For you express yourself in the +twentieth proposition very clearly; I noted it only that you may be more +merciful hereafter to the stumblings of a hasty pen. For _excessus ea +quam_ does not well, nor is to be well excused by _subauditur ratio_. +Your twentieth proposition is this: + +“_Si proponatur series quantitatum in duplicata ratione arithmetice +proportionalium (sive juxta seriem numerorum quadraticorum) continue +crescentium, a puncto vel 0 inchoatarum, ratio quam habet illa ad seriem +totidem maximæ æqualium subtriplam superabit; eritque excessus ea ratio +quam habet unitas ad sextuplum numeri terminorum post 0, sive quam habet +radix quadratica termini primi post 0 ad sextuplum radicis quadraticæ +termini maximi._” + +That is, if there be propounded a row of quantities in duplicate +proportion of arithmetically-proportionals (or according to the row of +square numbers) continually increasing, and beginning with a point or O. +The proportion of that row to a row of so many equals to the greatest, +shall be greater than subtriple proportion, and the excess shall be that +proportion which unity hath to the sextuple of the number of terms after +0, or the same which the square root of the first number after 0, hath +to the sextuple of the square root of the greatest. + +For proof whereof you have no more here than _patet ex præcedentibus_; +and no more before but _adeoque_. You do not well to pass over such +curious propositions so slightly; none of the ancients did so, nor, that +I remember, any man before yourself. The proposition is false, as you +shall presently see. + +Take, for example, any one of your rows: as (0 + 1 + 4)/(4 + 4 + 4). By +this proportion of yours 1 + 4, which makes 5, is to 12 in more than +subtriple proportion; by the proportion of 1 to the sextuple of 2 which +is 12. Put in order these three quantities 5, 4, 12, and you must see +the proportion of 5 to 12 is greater than the proportion of 4 to 12, +that is, subtriple proportion, by the proportion of 5 to 4. But by your +account the proportion of 5 to 4 is greater than that of 4 to 12 by the +proportion of 1 to 12. Therefore, as 5 to 4 so is 1 to 12, which is a +very strange paradox. + +After this you bring in this consectary: “_Cum autem crescente numero +terminorum excessus ille supra rationem subtriplam continue minuatur, ut +tandem quovis assignabili minor evadat (ut patet) si in infinitum +producatur, prorsus evaniturus est. Adeoque._” + +That is, seeing as the number of terms increaseth, that excess above +subtriple proportion continually decreaseth, so as at length it becomes +less than any assignable (as is manifest) if it be produced infinitely, +it shall utterly vanish, and so. And so what? + +Sir, this consequence of yours is false. For two quantities being given, +and the excess of the greater above the less, that excess may +continually be decreased, and yet never quite vanish. Suppose any two +unequal quantities differing by more than an unit, as 3 and 6, the +excess 3, let 3 be diminished, first by an unit, and the excess will be +2, and the quantities will be 3 and 5; 5 is greater than 4, the excess +1. Again, let 1 be diminished and made (1)/(2), the excess 4 and the +quantities 3 and 4(1)/(2), 4(1)/(2) is yet greater than 4. Again +diminish the excess to (1)/(4), the quantities will be 3 and 4(1)/(4), +yet still 4(1)/(4) is greater than 4. In the same manner you may proceed +to (1)/(8) (1)/(16) (1)/(32), &c. infinitely; and yet you shall never +come within an unit (though your unit stand for 100 miles) of the lesser +quantity propounded 3, if that 3 stands for 300 miles. The excesses +above subtriple proportion do not decrease in the manner you say it +does, but in the manner which I now shall show you. + +In the first row (0 + 1)/(1 + 1) a third of the quantities below is +(2)/(3), set in order these three quantities 1 (2)/(9) (2)/(3). The +first is 1, equal to the sum above, the last is (2)/(3), equal to the +subtriple of the sum below. The middlemost is (2)/(9) subtriple to the +last quantity (2)/(3). The excess of the proportion of 1 to (2)/(3) +above the subtriple proportion of (2)/(9) to (2)/(3) is the proportion +of 1 to (2)/(9) that is of 9 to 2, that is, of 18 to 4. + +Secondly, in the second row, which is (0 + 1 + 4)/(4 + 4 + 4), a third +of the sum below is 4, the sum above is 5. Set in order these +quantities, 1, 5, 4, 12. There the proportion of 15 to 12 is the +proportion of 5 to 4. The proportion of 4 to 12 is subtriple; the excess +is the proportion of 15 to 4, which is less than the proportion of 18 to +4, as it ought to be; but not less by the proportion of (1)/(6) to +(1)/(12) as you would have it. + +Thirdly, in the third row, which is (0 + 1 + 4 + 9)/(9 + 9 + 9 + 9). A +third of the sum below is 12, the sum above is 14. Set in order these +quantities, 42, 4, 12. There the proportion of 42 to 12 is the same with +that of 14 to 4. And the proportion of 4 to 12 subtriple, less than the +former excess of 15 to 4. And so it goes on decreasing all the way in +this manner, 18 to 4, 15 to 4, 14 to 4, &c. which differs very much from +your 1 to 6, 1 to 12, 1 to 18, &c. and the cause of your mistake is +this: you call the twelfth part of twelve (1)/(12), and the eighteenth +part of thirty-six you call (1)/(18), and so of the rest. But what need +of all those equations in symbols, to show that the proportion +decreases; is there any man can doubt, but that the proportion of 1 to 2 +is greater than that of 5 to 12, or that of 5 to 12 greater than that of +14 to 36, and so on continually forwards; or could you have fallen into +this error, unless you had taken, as you have done in very many places +of your _Elenchus_, the fractions (1)/(6) and (1)/(12), &c. which are +the quotients of 1 divided by 6 and 12, for the very proportions of 1 to +6 and 1 to 12. But notwithstanding the excess of the proportions of the +increasing quantities, to subtriple proportion decrease, still, as the +number of terms increaseth, and that what proportions soever I shall +assign, the decrement will in time (in time, I say, without proceeding +_in infinitum_) produce a less, yet it does not follow that the row of +increasing quantities shall ever be equal to the third part of the row +of so many equals to the last or greatest. For it is not, I hope, a +paradox to you, that in two rows of quantities the proportion of the +excesses may decrease, and yet the excesses themselves increase, and do +perpetually. + +For in the second and third rows, which are (0 + 1 + 4 = 5)/(4 + 4 + 4 += 12) and (0 + 1 + 4 + 9 = 14)/(9 + 9 + 9 + 9 = 36) 5 exceeds the third +part of 12 by a quarter of the square of 4, and 14 exceeds the third +part of 36 by 2 quarters of the square of 4, and proceeding on, the sum +of the increasing quantities where the terms are 5 (which sum is 30) +exceedeth the third part of those below, (those below are 80, and their +third part 26(2)/(3)) by 3 quarters and (1)/(2) a quarter of the square +of 4, and when the terms are 6, the quantities above will exceed the +third part of them below by 5 quarters of the square of 4. Would you +have men believe, that the further they go, the excess of the increasing +quantities above the third part of those below shall be so much the +less? And yet the proportions of those above, to the thirds of those +below, shall decrease eternally; and therefore your twenty-first +proposition is false, namely this: + +“_Si proponatur series infinita quantitatum in duplicata ratione +arithmetice proportionalium (sive juxta seriem numerorum quadraticorum), +continue crescentium a puncto sive 0 inchoatarum; erit illa ad seriem +totidem maximæ æqualium, ut 1 ad 3._” + +That is, if an infinite row of quantities be propounded in duplicate +proportion of arithmetically-proportionals (or according to the row of +quadratic numbers), continually increasing and beginning from a point or +0; that row shall be to the row of as many equals to the greatest, as 1 +to 3. This is false, _ut patet ex præcedentibus_; and, consequently, all +that you say in proof of the proportion of your _parabola_ to a +_parallelogram_, or of the _spiral_ (the true _spiral_) to a _circle_ is +in vain. + +But your spiral puts me in mind of what you have under-written to the +diagram of your proposition 5. _The spiral, in both figures, was to be +continued whole to the middle, but, by the carelessness of the graver, +it is in one figure_ manca, _in the other_ intercisa. + +Truly, Sir, you will hardly make your reader believe that a graver could +commit those faults without the help of your own copy, nor that it had +been in your copy, if you had known how to describe a spiral line then +as now. This I had not said, though truth, but that you are pleased to +say, though not truth, that I attributed to the printer some faults of +mine. + +I come now to the thirty-ninth proposition, which is this: + +“_Si proponatur series quantitatum in triplicata ratione arithmetice +proportionalium (sive juxta seriem numerorum cubicorum), continue +crescentium a puncto sive 0 inchoatarum (puta ut 0, 1, 8, 27, etc.), +propositum sit inquirere quam habeat series illa rationem ad seriem +totidem maximæ æqualium_: + +“_Fiat investigatio per modum inductionis_ (_ut in prop. 1, et prop. +19_): + + _Eritque_ + + (0 + 1 = 1)/(1 + 1 = 2) = (2)/(4) = (1)/(4) + (1)/(4) + + (0 + 1 + 8 = 9)/(8 + 8 + 8 = 24) = (1)/(4) + (1)/(8) + + (0 + 1 + 8 + 27 = 36)/(27 + 27 + 27 + 27 = 108) = (4)/(12) + = (1)/(4) + (1)/(12) + + _Et sic deinceps._ + +“_Ratio proveniens est ubique major quam subquadrupla, sive (1)/(4). +Excessus autem perpetuo decrescit, pro ut numerus terminorum augetur, +puta (1)/(4) (1)/(8) (1)/(12) (1)/(16) etc. Aucto nimirum fractionis +denominatore sive consequente rationis in singulis locis numero +quaternatio, ut patet, ut sit rationis provenientis excessus supra +subquadruplam ea quam habet unitas ad quadruplum numeri terminorum post +0 adeoque._” + +That is, if a row of quantities be propounded in triplicate proportion +of arithmetically proportionals (or according to the row of cubic +numbers), continually increasing, and beginning from a point or 0, as 0, +1, 8, 27, 64, &c., let it be propounded to inquire, what proportion that +row hath to a row of as many equals to the greatest. + +Be it sought by way of induction, as in proposition 1 and 19. + +The proposition arising is everywhere greater than subquadruple, or +(1)/(4), and the excess perpetually decreaseth as the number of terms +increaseth, as (1)/(4) (1)/(8) (1)/(12) (1)/(16) (1)/(20) &c. The +denominator of the fraction, or consequent of the proportion, being in +every place augmented by the number 4, as is manifest, so that the +excess of the arising proportion above subquadruple is the same with +that which an unit hath to the quadruple of the number of the terms +after 0, and so. Here are just the same faults which are in proposition +19. + +For, if (0)/(1) be a fraction, and (1)/(1) be a fraction, and (1)/(2) be +another fraction, then this equation (0 + 1 = 1)/(1 + 1 = 2) is false. +For this fraction (0)/(1) is equal to 0; and, therefore, we have (1)/(1) += (1)/(2), that is, the whole equal to half. But perhaps you do not mean +them fractions, but proportions; and, consequently, that the proportion +of 0 to 1, and of 1 to 1, compounded by addition (I say by addition, not +that I, but that you think there is a composition of proportions by +multiplication, which I shall show you anon is false), must be equal to +the proportion of 1 to 2, which cannot be. For the proportion of 0 to 1 +is infinitely little, that is, none at all; and, consequently, the +proportion of 1 to 1 is equal to the proportion of 1 to 2, which is +again absurd. There is no doubt but the whole number of 0 + 1 is equal +to 1, and the whole number of 1 + 1 equal to 2. But, reckoning them as +you do, not for whole numbers, but for fractions or proportions, the +equations are false. + +Again, your second equation, (2)/(4) = (1)/(4) + (1)/(4), though meant +of fractions, that is, of quotients, it be true, and serve nothing to +your purpose, yet, if it be meant of proportions, it is false. For the +proportion of 1 to 4, and of 1 to 4 being compounded, are equal to the +proportion of 1 to 16, and so you make the proportion of 2 to 4 equal to +the proportion of 1 to 16, where, as it is but subquaduplicate, as you +call it, or the quarter of it, as I call it. And, in the same manner, +you may demonstrate to yourself the same fault in all the other rows of +how many terms soever they consist. Therefore, you may give for lost +this thirty-ninth proposition, as well as all the other thirty-eight +that went before. As for the conclusion of it, which is, _that the +excess of the arising proportion_, &c. They are the words of your +fortieth proposition, where you express yourself better, and make your +error more easy to be detected. + +The proposition is this: + +“_Si proponatur series quantitatum in triplicata ratione arithmetice +proportionalium (sive juxta seriem numerorum cubicorum) continue +crescentium a puncto vel 0 inchoatarum, ratio quam habet illa ad seriem +totidem maximæ æqualium subquadruplam superabit; eritque excessus ea +ratio quam habet unitas ad quadruplum numeri terminorum post 0; sive +quam habet radix cubica termini primi post 0 ad quadruplum radicis +cubicæ termini maximi. Patet ex præcedente._ + +“_Quum autem crescente numero terminorum excessus ille supra rationem +subquadruplam ita continuo minuatur, ut tandem quolibet assignabili +minor evadat, ut patet, si in infinitum procedatur, prorsus evaniturus +est, adeoque._ + +“_Patet ex propositione_ _præcedente._” + +That is, if a row of quantities be propounded in triplicate proportion +of arithmetically proportionals (or according to the row of cubic +numbers), continually increasing, and beginning at a point or 0; the +proportion which that row hath to a row of as many equals to the +greatest, is greater than subquadruple proportion; and the excess is +that proportion which one unit hath to the quadruple of the number of +terms after 0; or, which the cubic root of the first term after 0 hath +to the quadruple of the root of the greatest term. + +It is manifest by the precedent propositions. + +And, seeing the number of terms increasing, that excess above quadruple +proportion doth so continually decrease, as that, at length, it becomes +less than any proportion that can be assigned, as is manifest, if the +proceeding be infinite, it shall quite vanish. And so + +This conclusion was annexed to the end of your thirty-ninth proposition, +as there proved. What cause you had to make a new proposition of it, +without other proof than _patet ex præcedente_, I cannot imagine. But, +howsoever, the proposition is false. + +For example, set forth any of your rows, as this of fewer terms: + + (0 + 1 + 8 + 27 = 36)/((27 + 27 + 27 + 27 = + 108) + +The row above is 36, the fourth part of the row below is 27. The +quadruple of the number of terms after 0 is 12. Then, by your account, +the proportion of 36 to 108 is greater than subquadruple proportion by +the proportion of 1 to 12. For trial whereof, set in order these three +quantities, 36, 27, 108. The proportion of 36 (the uppermost row) to 108 +(the lowermost row) is compounded by addition of the proportions 36 to +27, and 27 to 108. And the proportion of 36 to 108, exceedeth the +proportion of 27 to 108, by the proportion of 36 to 27. But the +proportion of 27 to 108 is subquadruple proportion. Therefore, the +proportion of 36 to 108 exceedeth subquadruple proportion, by the +proportion of 36 to 27. And, by your account, by the proportion of 1 to +12; and, consequently, as 36 to 27, so is 1 to 12. Did you think such +demonstrations as these should always pass? + +Then, for your inference from the decrease of the proportions of the +excess, to the vanishing of the excess itself, I have already showed it +to be false; and by consequence that your next proposition, namely, the +fortieth, is also false. + +The proposition is this: + +“_Si proponatur series infinita quantitatum in triplicata ratione +arithmetice proportionalium (sive juxta seriem numerorum cubicorum), +continue crescentium a puncto sive 0 inchoatarum, erit illa ad seriem +totidem maximæ æqualium, ut 1 ad 4, patet ex præcedente._” + +That is, if there be propounded an infinite row of quantities in +triplicate proportion of arithmetically proportionals (or according to +the row of cubic numbers), continually increasing, and beginning at a +point or 0; it shall be to the row of as many equals to the greatest as +1 to 4. Manifest out of the precedent proposition. + +Even as manifest as that 36, 27, 1, 12, are proportionals. Seeing, +therefore, your doctrine of the spiral lines and the spaces is given by +yourself for lost, and a vain attempt, your first forty-one propositions +are undemonstrated, and the grounds of your demonstrations all false. +The cause whereof is partly your taking quotient for proportion, and a +point for 0, as you do in the first, sixteenth, and fortieth +propositions, and in other places where you say, _beginning at a point +or 0_, though now you deny you ever said either. There be very many +places in your _Elenchus_, where you say both; and have no excuse for +it, but that, in one of the places, you say the proportion is _penes +quotientem_, which is to the same or no sense. + +Your forty-second proposition is grounded on the fortieth; and +therefore, though true, and demonstrated by others, is not demonstrated +by you. + +Your forty-third is this: + +“_Pari methodo invenietur ratio seriei infinitæ quantitatum arithmetice +proportionalium in ratione quadruplicata, quintuplicata, sextuplicata, +etc., arithmetice proportionalium a puncto seu 0 inchoatarum, ad seriem +totidem maximæ æqualium. Nempe in quadruplicata erit, ut 1 ad 5; in +quintuplicata, ut 1 ad 6; in sextuplicata, ut 1 ad 7. Et sic deinceps._” + +That is, by the same method will be found, the proportion of an infinite +row of arithmetically proportionals, in proportion quadruplicate, +quintuplicate, sextuplicate, &c., of arithmetically proportionals, +beginning at a point or 0, to the row of as many equals to the greatest; +namely, in quadruplicate, it shall be as 1 to 5; in quintuplicate, as 1 +to 6; in sextuplicate, as 1 to 7; and so forth. + +But by the same method that I have demonstrated, that the propositions +19, 20, 21, 39, 40, and 41, are false: any man else, that will examine +the forty-third may find it false also. And, because all the rest of the +propositions of your _Arithmetica Infinitorum_ depend on these, they may +safely conclude, that there is nothing demonstrated in all that book, +though it consist of 194 propositions. The proportions of your +parabolocides to their parallelograms are true, but the demonstrations +false, and infer the contrary. Nor were they ever demonstrated (at least +the demonstrations are not extant) but by me; nor can they be +demonstrated, but upon the same grounds, concerning the nature of +proportion, which I have clearly laid, and you not understood. For, if +you had, you could never have fallen into so gross an error as is this +your book of _Arithmetica Infinitorum_, or that of the angle of contact. +You may see by this, that your symbolic method is not only not at all +inventive of new theorems, but also dangerous in expressing the old. If +the best masters of symbolics think for all this you are in the right, +let them declare it. I know how far the analysis by the powers of the +lines extendeth, as well as the best of your half-learnt epistlers, that +approve so easily of such analogisms as those, 5, 4, 1, 12, and 36, 27, +1, 12, &c. + +It is well for you that they who have the disposing of the professors’ +places take not upon them to be judges of geometry. For, if they did, +seeing you confess you have read these doctrines in your school, you had +been in danger of being put out of your place. + +When the author of the paper wherein I am called Plagiary, and wherein +the honour is taken from you of being the first inventor of these fine +theorems, shall read this that I have here written, he will look to get +no credit by it; especially if it be Roberval, which methinks it should +not be. For he understands what proportion is, better than to make 5 to +4 the same with 1 to 12. Or to make, again, the proportion of 36 to 27 +the same with that of 1 to 12; and innumerable _disproportionalites_ +that may be inferred from the grounds you go on. But if it be Roberval +indeed, that snatches this invention from you, when he shall see this +burning coal hanging at it, he will let it fall again, for fear of +spoiling his reputation. + +But what shall I answer to the authority of the three great +mathematicians that sent you those encomiastic letters. For the first, +whom you say I use to praise, I shall take better heed hereafter of +praising any man for his learning whilst he is young, further than that +he is in a good way. But it seems he was in too ready a way of thinking +very well of himself, as you do of yourself. For the muddiness of my +brain I must confess it; but, Sir, ought not you to confess the same of +yours? No, men of your tenets use not to do so. He wonders, say you, you +thought it worth the while to foul your fingers about such a piece. It +is well; every man abounds in his own sense. If you and I were to be +compared by the compliments that are given us in private letters, both +you and your complimentors would be out of countenance; which +compliments, besides that which has been printed and published in the +commendations of my writings, if it were put together, would make a +greater volume than either of your libels. And truly, Sir, I had never +answered your Elenchus as proceeding from Dr. Wallis, if I had not +considered you also as the minister to execute the malice of that sort +of people that are offended with my _Leviathan_. + +As for the judgment of that public Professor that makes himself a +witness of the goodness of your geometry, a man may easily see by the +letter itself that he is a dunce. And for the English person of quality +whom I know not, I can say no more yet than I can say of all three, that +he is so ill a geometrician, as not to detect those gross paralogisms as +infer that 5 to 4 and 1 to 12 are the same proportion. He came into the +cry of those whom your title had deceived. + +And now I shall let you see that the composition of proportion by +multiplication, as it is in the fifth definition of the sixth element, +is but another way of adding proportions one to another. Let the +proportions be of 2 to 3, and of 4 to 5. Multiply 2 into 4 and 3 into 5, +the proportion arising is of 8 to 15. Put in order these three +quantities, 8, 12, 15. The proportion therefore of 8 to 15, compounded +of the proportions of 8 to 12, (that is, of 2 to 3) and of 12 to 15, +that is, of 4 to 5 by addition. Again, let the proportion be of 2 to 3, +and of 4 to 5, multiply 2 into 5 and 3 into 4, the proportions arising +is of 10 to 12. Put in order these three numbers, 10, 8, 12. The +proportion 10 to 12 is compounded of the proportions of 10 to 8, that is +of 5 to 4, and of 8 to 12, that is, of 2 to 3 by addition. I wonder you +know not this. + +I find not any more clamour against me for saying the proportion of 1 to +2 is double to that of 1 to 4. + +Your book, you speak of, concerning proportion against _Meibomius_ is +like to be very useful when neither of you both do understand what +proportion is. + +You take exceptions, as that I say, that _Euclid_ has but one word for +_double_ and _duplicate_; which nevertheless was said very truly, and +that word is sometimes διπλάσιος and sometimes διπλάσιων. And you think +you have come off handsomely with asking me whether διπλάσιος and +διπλασίων be one word. + +Nor are you now of the mind you were, that a point is not _quantity +unconsidered_, but that in an infinite series it may be safely +neglected. What is _neglected_ but unconsidered. + +Nor do you any more stand to it, that the _quotient_ is the +_proportion_. And yet were these the main grounds of your _Elenchus_. + +But you will say, perhaps, I do answer to the defence you have now made +in this your _School Discipline_: ’tis true. But ’tis not because you +answer never a word to my former objections against these propositions +19, 89; but because you do so shift and wriggle, and throw out ink, that +I cannot perceive which way you go, nor need I, especially in your +vindication of your _Arithmetica Infinitorum_. Only I must take notice +that in the end of it, you have these words, “Well, _Arithmetica +Infinitorum_ _is come off clear_” You see the contrary. For sprawling is +no defence. + +It is enough to me that I have clearly demonstrated both before +sufficiently, and now again abundantly, that your book of _Arithmetica +Infinitorum_ is all nought from the beginning to the end, and that +thereby I have effected that your authority shall never hereafter be +taken for a prejudice. And, therefore, they that have a desire to know +the truth in the questions between us, will henceforth, if they be wise, +examine my geometry, by attentive reading me in my own writings, and +then examine, whether this writing of yours confute or enervate mine. + +There is in my fifth lesson a proposition, with a diagram to it, to make +good, I dare say, at least against you, my twentieth chapter concerning +the dimension of a circle. If that demonstration be not shown to be +false, your objections to that chapter, though by me rejected, come to +nothing. I wonder why you pass it over in silence. But you are not, you +say, bound to answer it. True, nor yet to defend what you have written +against me. + +Before I give over the examination of your geometry, I must tell you +that your words, (p. 101 of your _School Discipline_), against the first +corollary are untrue. + +Your words are these: “_you affirm that the proportion of the parabola A +B I to the parabola A F K is triplicate to the proportion of the time A +B to A F, as it is in the English_.” This is not so. Let the reader turn +to the place and judge. And going on you say, “_or of the impetus B I to +F K as it is in the Latin_.” Nay, as it is in the English, and the other +in the Latin. It is but your mistake; but a mistake is not easily +excused in a false accusation. + +Your exception to my saying, “_that the differences of two quantities is +their proportion_,” (when they differ, as the no difference, when they +be equal), might have been put in amongst other marks of your not +sufficiently understanding the Latin tongue. _Differre_ and +_differentia_ differ no more than _vivere_ and _vita_, which is nothing +at all, but as the other words require that go with them, which other +words you do not much use to consider. But _differre_ and _the quantity +by which they differ_, are quite of another kind. _Differre_ (τὸ +διαφέρειν, τὸ ὑπερέχειν) _differing_, _exceeding_, is not quantity, but +relation. But the quantity by which they differ is always a certain and +determined quantity, yet the word _differentia_ serves for both, and is +to be understood by the coherence with that which went before. But I had +said before, and expressly to prevent cavil, that relation is nothing +but a comparison, and that proportion is nothing but relation of +quantities, and so defined them, and therefore I did there use the word +_differentia_ for _differing_, and not for the quantity which was left +by subtraction. For a quantity is not a differing. This I thought the +intelligent reader would of himself understand without putting me, +instead of _differentia_, to use (as some do, and I shall never do) the +mongrel word τὸ _differre_. And whereas in one only place for _differre +ternario_ I have writ _ternarius_, if you had understood what was +clearly expressed before, you might have been sure it was not my +meaning, and therefore the excepting against it was either want of +understanding, or want of candour, choose which you will. + +You do not yet clear your doctrine of _condensation_ and _rarefaction_. +But I believe you will by degrees become satisfied that they who say the +same numerical body may be sometimes greater, sometimes less, speak +absurdly, and that _condensation_ and _rarefaction_ here, and +_definitive_ and _circumscriptive_, and some other of your distinctions +elsewhere are but snares, such as school divines have invented + + ——ᾥσπερ άράχνης + Ὀυλόμενος χέζει ἀλύσεις μυίαις ἀθαρέσσι, + +to entangle shallow wits. + +And that that distinction which you bring here, “_that it is of the same +quantity while it is in the same place, but it may be of a different +quantity when it goes out of its place_,” (as if the place added to, or +took any quantity from the body placed), is nothing but mere words. It +is true that the body which swells changeth place, but it is not by +becoming itself a greater body, but by admixtion of air or other body, +as when water riseth up in boiling, it taketh in some parts of air. But +seeing the first place of the body is to the body equal, and the second +place equal to the same body, the places must also be equal to one +another, and consequently the dimensions of the body remain equal in +both places. + +Sir, when I said that such doctrine was taught in the Universities, I +did not speak against the Universities, but against such as you. I have +done with your geometry, which is one στιγμὴ. + + RURAL LANGUAGE. + +As for your eloquence, let the reader judge whether yours or mine be the +more _muddy_, though I in plain scolding should have outdone you, yet I +have this excuse which you have not, that I did but answer your +challenge at that weapon which you thought fit to choose. The catalogue +of the hard language which you put in at pages 3 and 4 of your _School +Discipline_, I acknowledge to be mine, and would have been content you +had put in all. The titles you say I give you of _fools_, _beasts_, and +_asses_, I do not give you, but drive back upon you, which is no more +than not to own them; for the rest of the catalogue, I like it so well +as you could not have pleased me better than by setting those passages +together to make them more conspicuous; that is all the defence I will +make to your accusations of that kind. + +And now I would have you to consider whether you will make the like +defence against the faults that I shall find in the language of your +_School Discipline_. + +I observe, first, the facetiousness of your title-page, “_Due correction +for Mr. Hobbes, or School Discipline, for not saying his Lessons +right_.” What a quibble is this upon the word lesson; besides, you know +it has taken wind; for you vented it amongst your acquaintance at Oxford +then when my _Lessons_ were but upon the press. Do you think if you had +pretermitted that piece of wit, the opinion of your judgment would have +been ere the less? But you were not content with this, but must make +this metaphor from the rod to take up a considerable part of your book, +in which there is scarce anything that yourself can think wittily said +besides it. Consider also these words of yours: “_It is to be hoped that +in time you may come to learn the language, for you be come to great_ A +_already_.” And presently after, “_were I great_ A, _before I would be +willing to be so used, I should wish myself little_ a _a hundred +times_.” Sir, you are a doctor of divinity and a professor of geometry, +but do not deceive yourself, this does not pass for wit in these parts, +no, nor generally at Oxford; I have acquaintance there that will blush +at the reading it. + +Again, in another place you have these words: “_Then you catechize us_, +‘_what is your name? Are you geometricians? Who gave you that name_,’” +&c. Besides in other places such abundance of the like insipid conceits, +as would make men think, if they were no otherwise acquainted with the +University but by reading your books, that the dearth there of salt were +very great. If you have any passage more like to salt than these are +(excepting _now and anon_) you may do well to show it to your +acquaintance, lest they despise you; for, since the detection of your +geometry, you have nothing left you else to defend you from contempt. +But I pass over this kind of eloquence, and come to somewhat yet more +rural. + +Page 27, line 1, you say I have given Euclid his _lurry_. And again, +page 129, line 11, “_and now he is left to learn his lurry_.” I +understand not the word _lurry_. I never read it before, nor heard it, +as I remember, but once, and that was when a clown threatening another +clown said he would give him _such a lurry come poop_, &c. Such words as +these do not become a learned mouth, much less are fit to be registered +in the public writings of a doctor of divinity. In another place you +have these words, “_just the same to a cow’s thumb_,” a pretty adage. + +Page 2, “_But prithee tell me_.” And again, page 95, “_prithee tell me, +why dost thou ask me such a question_,” and the like in many other +places. + +You cannot but know how easy it is and was for me to have spoken to you +in the same language. Why did I not? Because I thought that amongst men +that were civilly bred it would have redounded to my shame, as you have +cause to fear that this will redound to yours. But what moved you to +speak in that manner? Were you angry? If I thought that the cause, I +could pardon it the sooner, but it must be very great anger that can put +a man, that professeth to teach good manners, so much out of his wits as +to fall into such a language as this of yours. It was perhaps an +imagination that you were talking to your inferior, which I will not +grant you, nor will the heralds, I believe, trouble themselves to decide +the question. But, howsoever, I do not find that civil men use to speak +so to their inferiors. If you grant my learning but to be equal to +yours, (which you may certainly do without very much disparaging of +yourself abroad in the world), you may think it less insolence in me to +speak so to you in respect of my age, than for you to speak so to me in +respect of your young doctorship. You will find that for all your +doctorship, your elders, if otherwise of as good repute as you, will be +respected before you. But I am not sure that this language of yours +proceeded from that cause; I am rather inclined to think you have not +been enough in good company, and that there is still somewhat left in +your manners for which the honest youths of Hedington and Hincsey may +compare with you for good language, as great a doctor as you are. + +For my verses of the Peak, though they be as ill in my opinion as I +believe they are in yours, and made long since, yet they are not so +obscene as that they ought to be blamed by Dr. Wallis. I pray you, sir, +whereas you have these words in your _School Discipline_, page 96, +“_unless you will say that one and the same motion may be now and anon +too_.” What was the reason you put these words, _now and anon too_, in a +different character, that makes them to be more taken notice of? Do you +think that the story of the minister that uttered his affection (if it +be not a slander) not unlawfully but unseasonably, is not known to +others as well as to you? What needed you then, when there was nothing +that I had said could give the occasion, to use those words; there is +nothing in my verses that do _olere hircum_ so much as this of yours. I +know what good you can receive by ruminating on such ideas, or +cherishing of such thoughts. But I go on to other words of mine by you +reproached, “_you may as well seek the focus of the parabola of Dives +and Lazarus_,” which you say is mocking the Scripture; to which I answer +only, that I intended not to mock the Scripture, but you, and that which +was not meant for mocking was none. And thus you have a second στιγμὴ. + + GRAMMAR AND CRITIQUES. + +I come now to the comparison of our Grammar and Critiques. You object +first against the signification I give of στιγμὴ, and say thus: “_What +should come into your cap_ (that, if you mark it, in a man that wears a +square cap to one that wears a hat, is very witty) _to make you think +that_ στιγμὴ _signifies a mark or brand with a hot iron? I perceive +where the business lies, it was_ στίγμα _run in your mind when you +talked of_ στιγμὴ; _and because the words are somewhat alike you jumble +them both together_.” Sir, I told you once before, you presume too much +upon your first cogitations. Aristophanes, in _Ranis_, Act. V. Scen. 5, + + Κἄν μὴ ταχέως ἥκωσι + Νὴ τεν Ἀπόλλο στίξας ἀυτοὺς. + +The old commentator upon the word στίξας saith thus, ϛίξας ἀντὶ τοῦ +ϛιγματίσας, ἠν γάρ ξένος. That is, στίξας for ϛιγματίσας, for he +(Adimantus) was not a citizen. I hope the commentator does not here mock +Aristophanes for jumbling ϛίξας and ϛιγματίσας together, for want of +understanding Greek. No, ϛίξας and στιγματίσας signify the same, save +that for branding I seldom read ϛιγματίσας but ϛίξας. For ϛίγμα does no +more signify a brand with a hot iron, than ϛιγμὴ a point made also with +a hot iron. They have both one common theme ϛίζω, which does not signify +_pungo_, nor _interpungo_, nor _inuro_, for all your Lexicon, but _notam +imprimere_, or _pungendo notare_, without any restriction to burning or +punching. It is therefore no less proper to say that ϛιγμὴ is a mark +with a hot iron, than to say the same of στίγμα. The difference is only +this, that when they marked a slave, or a rascal, as you are not +ignorant is usually done here at the assizes in the hand or shoulder +with a hot iron, they called that ϛίγμα, not for the burning, but for +the mark. And as it would have been called ϛίγμα that was imprinted on a +slave, though made by staining or incision, so it is ϛιγμὴ, though done +with a hot iron. And therefore there was no jumbling of those two words +together, as for want of reading Greek authors, and by trusting too much +to your dictionaries, which you say are proofs good enough for such a +business, you were made to imagine. The use I have made thereof was to +show that a point, both by the word Σημεῖον in Euclid, and by the word +στιγμὴ in some others, was not _nothing_, but a _visible_ mark, the +ignorance whereof hath thrown you into so many paralogisms in geometry. + +But do you think you can defend your _Adducis Malleum_ as well as I have +now defended my ϛιγμὴ? You have brought, I confess, above a hundred +places of authors, where there is the word _duco_, or some of its +compounds, but none of them will justify _Adducis Malleum_, and, +excepting two of those places, you yourself seem to condemn them all, +comparing yours with none of the rest but with these two only, both out +of Plautus, by you not well understood. The first is in _Casina_, Act. +V. Scen. 2, “_Ubi intro hanc novam nuptam deduxi, via recta, clavem +abduxi_;” which you, presently presuming of your first thoughts, a +peculiar fault to men of your principles, assure yourself is right. But +if you look on the place as Scaliger reads it, cited by the commentator, +you will find it should be _obduxi_, and that _clavis_ is there used for +the bolt of the lock. Besides, he bolted it within. Whither then could +he carry away the key? The place is to be rendered thus, _when I had +brought in this new bride I presently locked the door_, and is this _as +bad every whit_ as _Adducis Malleum_? The second place is in +_Amphytruo_, Act. I. Scen. 1, “_Eam_ (cirneam), _ut a matre fuerat +natum, plenam vini eduxi meri_,” which you interpret _I brought out a +flagon of wine_, unlearnedly. They are the words of Mercury transformed +into Sosia. And to try whether Mercury were Sosia or not, Sosia asked +him where he was and what he did during the battle; to which Mercury +answered, who knew where Sosia then was and what he did, _I was in the +cellar, where I filled a cirnea, and brought it up full of wine, pure as +it came from its mother_. By the mother of the wine meaning the vine, +and alluding to the education of children, for _ebibi_ said _eduxi_, and +with an _emphasis_ in _meri_, because _cirnea_ (from Κφνάω, _misceo_) +was a vessel wherein they put water to temper to their wine. Intimating +that though the vessel was _cirnea_, yet the wine was _merum_. This is +the true sense of the place; but you will have _eduxi_ to be, _I brought +out_, though he came not out himself. You see, sir, that neither this is +so bad as _Adducis Malleum_. + +But suppose out of some one place in some one blind author you had +paralleled your _Adducis Malleum_, do you think it must therefore +presently be held for good Latin? Why more than _learn his lurry_ must +be therefore thought good English a thousand years hence, because it +will be read in Dr. Wallis’s long-lived works. But how do you construe +this passage (1 Tim. ii. 15) of the Greek Testament: Σωθήσεται δὲ διὰ +τῆς τεκνογονίας, ἐὰν μείνωσιν ἐν πίστει? You construe it thus: _she +shall be saved notwithstanding child-bearing, if (the woman) remain in +the faith_. Is child-bearing any obstacle to the salvation of women? You +might as well have translated the first verse of the fifth of Romans in +this manner, _Being then justified by faith, we have peace with God +notwithstanding our Lord Jesus Christ_. I let pass your not finding in +τεκνογονίας, as good a grammarian as you are, a nominative case to +μείνωσιν. If you had remembered the place, 1 Pet. iii. 20, εσώθησαν δὶ +ὑδατος, that is, _they were saved in the waters_, you would have thought +your construction justified then very well; but you had been deceived, +for διὰ does not there signify _causam, ablationem impedimenti_, but +_transitum_; not _cause or removing an impediment_, but _passage_. Being +come thus far, I found a friend that hath eased me of this dispute; for +he showed me a letter written to himself from a learned man, that hath +out of very good authors collected enough to decide all the grammatical +questions between you and me, both Greek and Latin. He would not let me +know his name, nor anything of him but only this, that he had better +ornaments than to be willing to go clad abroad in the habit of a +grammarian. But he gave me leave to make use of so much of the letter as +I thought fit in this dispute, which I have done, and have added it to +the end of this writing. But before I come to that, you must not take it +ill, though I have done with your _School Discipline_, if I examine a +little some other of your printed writings as you have examined mine; +for neither you in geometry, nor such as you in church politics, cannot +expect to publish any unwholesome doctrine without some antidotes from +me, as long as I can hold a pen. But why did you answer nothing to my +sixth _Lesson_? Because, you say, it concerned your colleague only. No, +sir, it concerned you also, and chiefly, for I have not heard that your +colleague holdeth those dangerous principles which I take notice of in +you, in my sixth _Lesson_, page 350, upon the occasion of these words, +not his but yours: “_Perhaps you take the whole history of the fall of +Adam for a fable, which is no wonder, seeing you say the rules of +honouring and worshipping of God are to be taken from the laws_.” In +answer to which I said thus: “_You that take so heinously, that I would +have the rule of God’s worship in a Christian commonwealth to be taken +from the laws, tell me from whom you would have them taken? From +yourself? Why so, more than from me? From the bishops? Right, if the +supreme power of the commonwealth will have it so; if not, why from them +rather than from me? From a consistory of presbyters themselves, or +joined with lay elders, whom they may sway as they please? Good, if the +supreme governor of the commonwealth will have it so. If not, why from +them rather than from me, or from any man else? They are wiser and +learneder than I; it may be so, but it has not yet appeared. Howsoever, +let that be granted. Is there any man so very a fool as to subject +himself to the rules of other men in those things which do so nearly +concern himself, for the title they assume of being wise and learned, +unless they also have the sword which must protect them? But it seems +you understand the sword as comprehended. If so, do not you then receive +the rules of God’s worship from the civil power? Yes, doubtless; and you +would expect, if your consistory had that sword, that no man should dare +to exercise or teach any rules concerning God’s worship which were not +by you allowed._” + +This will be thought strong arguing, if you do not answer it. But the +truth is, you could say nothing against it without too plainly +discovering your disaffection to the government. And yet you have +discovered it pretty well in your second _Thesis_, maintained in the Act +at Oxford, 1654, and since by yourself published. This _Thesis_ I shall +speak briefly to. + + SCOTCH CHURCH POLITICS. + +You define ministers of the Gospel to be those _to whom the preaching of +the Gospel by their office is enjoined by Christ_. Pray you, first, what +do you mean by saying preaching _ex officio is enjoined by Christ_? Are +they preachers _ex officio_, and afterwards enjoined to preach? _Ex +officio_ adds nothing to the definition; but a man may easily see your +purpose to disjoin yourself from the state by inserting it. + +Secondly, I desire to know in what manner you will be able out of this +definition to prove yourself a minister? Did Christ himself immediately +enjoin you to preach, or give you orders? No. Who then, some bishop, or +minister, or ministers? Yes; by what authority? Are you sure they had +authority immediately from Christ? No. How then are you sure but that +they might have none? At least, some of them through whom your authority +is derived might have none. And therefore if you run back for your +authority towards the Apostles’ times but a matter of sixscore years, +you will find your authority derived from the Pope, which words have a +sound very unlike to the voice of the laws of England. And yet the Pope +will not own you. There is no man doubts but that you hold that your +office comes to you by successive imposition of hands from the time of +the Apostles; which opinion in those gentle terms passeth well enough; +but to say you derive your authority from thence, not through the +authority of the sovereign power civil, is too rude to be endured in a +state that would live in peace. In a word, you can never prove you are a +minister, but by the supreme authority of the commonwealth. Why then do +you not put some such clause into your definition? As thus, _ministers +of the Gospel are those to whom the preaching of the Gospel is enjoined +by the sovereign power in the name of Christ_. What harm is there in +this definition, saving only it crosses the ambition of many men that +hold your principles? Then you define the power of a minister thus: +“_The power of a minister is that which belongeth to a minister of the +Gospel in virtue of the office he holds, inasmuch as he holds a public +station, and is distinguished from private Christians. Such as is the +power of preaching the Gospel, administering the sacrament, the use of +ecclesiastical censures, and ordaining of ministers_,” _&c._ + +Again, how will you prove out of this definition that you, or any man +else, hath the power of a minister, if it be not given him by him that +is the sovereign of the commonwealth? For seeing, as I have now proved, +it is from him that you must derive your ministry, you can have no other +power than that which is limited in your orders, nor that neither longer +than he thinks fit. For if he give it you for the instruction of his +subjects in their duty, he may take it from you again whensoever he +shall see you instruct them with undutiful and seditious principles. And +if the sovereign power give me command, though without the ceremony of +imposition of hands, to teach the doctrine of my _Leviathan_ in the +pulpit, why am not I, if my doctrine and life be as good as yours, a +minister as well as you, and as public a person as you are? For _public +person_, primarily, is none but the civil sovereign, and so secondarily, +all that are employed in the execution of any part of the public charge. +For all are his ministers, and therefore also Christ’s ministers because +he is so; and other ministers are but his vicars, and ought not to do or +say anything to his people contrary to the intention of the sovereign in +giving them their commission. + +Again, if you have in your commission a power to excommunicate, how can +you think that your sovereign who gave you that commission, intended it +for a commission to excommunicate himself? that is, as long as he stand +excommunicate, to deprive him of his kingdom. If all subjects were of +your mind, as I hope they will never be, they will have a very unquiet +life. And yet this has, as I have often heard, been practised in +Scotland, when ministers holding your principles had power enough, +though no right, to do it. + +And for administration of the sacraments, if by the supreme power of the +commonwealth it were committed to such of the laity as know how it ought +to be done as well as you, they would _ipso facto_ be ministers as good +as you. Likewise the right of ordination of ministers depends not now on +the imposition of hands of a minister or presbytery, but on the +authority of the Christian sovereign, Christ’s immediate vicar and +supreme governor of all persons and judge of all causes, both spiritual +and temporal, in his own dominions, which I believe you will not deny. + +This being evident, what acts are those of yours which you call +_authoritative_, and receive not from the authority of the civil power? +A constable does the acts of a constable _authoritatively_ in that +sense. Therefore you can no otherwise claim your power than a constable +claimeth his, who does not exercise his office in the constabulary of +another. But you forget that the Scribes and the Pharisees sit no more +in Moses’ chair. + +You would have every minister to be a minister of the universal Church, +and that it be lawful for you to preach your doctrine at Rome; if you +would be pleased to try, you would find the contrary. You bring no +argument for it that looks like reason. Examples prove nothing, where +persons, times, and other circumstances differ; as they differ very much +now when kings are Christians, from what they were then when kings +persecuted Christians. It is easy to perceive what you aim at. + +You would fain have market-day lectures set up by authority, (not by the +authority of the civil power, but by the authority of example of the +Apostles in the emission of preachers to the infidels), not knowing that +any Christian may lawfully preach to the infidels; that is to say, +proclaim unto them that Jesus is the Messiah, without need of being +otherways made a minister, as the deacons did in the Apostles’ time; nor +that many teachers, unless they can agree better, do anything else but +prepare men for faction, nay, rather you know it well enough, but it +conduces to your end upon the market-days to dispose at once both town +and country, under a false pretence of obedience to God, to a neglecting +of the commandments of the civil sovereign, and make the subject to be +wholly ruled by yourselves, wherein you have already found yourselves +deceived. You know how to trouble and sometimes undo a slack government, +and had need to be warily looked to, but are not fit to hold the reins. +And how should you, being men of so little judgment as not to see the +necessity of unity in the governor, and of absolute obedience in the +governed, as is manifest out of the place of your _Elenchus_ above +recited. The doctrine of the duty of private men in a commonwealth is +much more difficult, not only than the knowledge of your symbols, but +also than the knowledge of geometry itself. How then do you think, when +you err so grossly in a few equations, and in the use of most common +words, you should be fit to govern so great nations as England, Ireland, +and Scotland, or so much as to teach them? For it is not reading but +judgment that enables one man to teach another. + +I have one thing more to add, and that is the disaffection I am charged +withal to the universities. Concerning the Universities of Oxford and +Cambridge, I ever held them for the greatest and noblest means of +advancing learning of all kinds, where they should be therein employed, +as being furnished with large endowments and other helps of study, and +frequented with abundance of young gentlemen of good families and good +breeding from their childhood. On the other side, in case the same means +and the same wits should be employed in the advancing of the doctrines +that tend to the weakening of the public, and strengthening of the power +of any private ambitious party, they would also be very effectual for +that; and consequently that if any doctrine tending to the diminishing +of the civil power were taught there, not that the Universities were to +blame, but only those men that in the universities, either in lectures, +sermons, printed books, or theses, did teach such doctrine to their +hearers or readers. Now you know very well that in the time of the Roman +religion, the power of the Pope in England was upheld principally by +such teachers in the universities. You know also how much the divines +that held the same principles in Church government with you, have +contributed to our late troubles. Can I therefore be justly taxed with +disaffection to the universities for wishing this to be reformed? And it +hath pleased God of late to reform it in a great measure, and indeed as +I thought totally, when out comes this your _Thesis_ boldly maintained +to show the contrary. Nor can I yet call this your doctrine the doctrine +of the university; but surely it will not be unreasonable to think so, +if by public act of the university it be not disavowed, which done, and +that as often as there shall be need, there can be no longer doubt but +that the universities of England are not only the noblest of all +Christian universities, but also absolutely, and of the greatest benefit +to this commonwealth that can be imagined, except that benefit of the +head itself that uniteth and ruleth all. I have not here particularized +at length all the ill consequences that may be deduced from this +_Thesis_ of yours, because I may, when further provoked, have somewhat +to say that is new. So much for the third ϛιγμὴ. + + AN EXTRACT OF A LETTER CONCERNING THE + GRAMMATICAL PART OF THE CONTROVERSY + BETWEEN MR. HOBBES AND DR. WALLIS. + + +Mr. Hobbes hath these words: “_Longitudinem percursam motu uniformi, cum +impetu ubique ipsi B D æquali_.” Dr. Wallis saith _cum_ were better out, +unless you would have _impetus_ to be only a companion, not a _cause_. +Mr. Hobbes answered it was the _ablative case of the manner_. The truth +is the ablative case of the _manner_ and _cause_ both, may be used with +the conjunction _cum_, as may be justified. Cicero in Lib. II. _De Nat. +Deorum_: “_Moliri aliquid cum labore operoso ac molesto_;” and in his +oration for Cæcina: “_De se autem hoc prædicat, Antiocho Ebulii servo +imperasse ut in Cæcinam advenientem cum ferro invaderet_.” Let us see +then what Dr. Wallis objects against Tully, where a casualty is +imported, though we may use _with_ in English, yet not _cum_ in Latin; +to kill with a sword, importing this to have an instrumental or causal +influence, and not only that it hangs by the man’s side whilst some +other weapon is made use of, is not in Latin _occidere cum gladio_, but +_gladio occidere_. This shows that the Doctor hath not forgot his +grammar, for the subsequent examples as well as this rule are borrowed +thence. But yet he might have known that great personages have never +confined themselves to this pedantry, but have chosen to walk in a +greater latitude. Most of the elegancies and idioms of every language +are exceptions to his grammar. But since Mr. Hobbes saith it is the +ablative case of the manner, there is no doubt it may be expressed with +_cum_. The Doctor in the meantime knew no more than what Lilly had +taught him; Alvarez would have taught him more; and Vossius in his book, +_De Constructione_, _cap._ XLVII. expressly teacheth, “_Ablativos causæ, +instrumenti, vel modi, non a verbo regi sed a præpositione omissa, a vel +ab, de, e vel ex, præ, aut cum, ac præpositiones eas quandoque exprimi +nisi quod cum ablativis instrumenti haud temere invenias_;” and +afterwards he saith, “_non timere imitandum_.” If this be so, then did +Mr. Hobbes speak grammatically, and with Tully, but not _usually_. And +might not one retort upon the Doctor, that Vossius is as great a critic +as he? + +His next reflection is upon _prætendit scire_, this he saith is an +Anglicism. If this be all his accusation, upon this score we shall lose +many expressions that are used by the best authors, which I take to be +good Latinisms, though they be also Anglicisms, the latter being but an +imitation of the former. The Doctor therefore was too fierce to condemn +upon so general an account, that which was not to have been censured for +being an Anglicism, unless also it had been no Latinism. Mr. Hobbes +replies, that the printer had omitted _se_. He saith, this mends the +matter a little. It is very likely, for then it is just such another +Anglicism as that of Quintilian: “_Cum loricatus in foro ambularet, +prætendebat se id metu facere_.” The Doctor certainly was very +negligent, or else he could not have missed this in Robert Stephen. Or +haply he was resolved to condemn Quintilian for this and that other +Anglicism, “_Ignorantia prætendi non potest_,” as all those that have +used _prætendo_, which are many and as good authors as Dr. Wallis, that +makes his own encomiasts (not an Englishman amongst them) to write +Anglicisms. + +Then he blames “_Tractatus hujus partis tertiæ, in qua motus et +magnitudo per se et abstracte consideravimus, terminum hic statuo_.” +Here I must confess the exception is colourable, yet I can parallel it +with the like objection made by Erasmus against Tully, out of whom +Erasmus quotes this passage: “_Diutius commorans Athenis, quoniam venti +negabant solvendi facultatem, erat animus ad te scribere_;” and excuses +it thus, that Tully might have had at first in his thoughts _volebam_ or +_statuebam_, which he afterwards relinquished for _erat animus_, and did +not remember what he had antecedently written, which did not vary from +his succeeding thoughts, but words. And this excuse may pass with any +who know that Mr. Hobbes values not the study of words, but as it serves +to express his thoughts, which were the same whether he wrote _in qua +motus et magnitudo per se at abstracte considerati sunt_ or +_consideravimus_. And if the Doctor will make this so capital, he must +prove it _voluntary_, and show that it is greater than what is legible +in the puny letter of his encomiast, whom he would have to be beyond +exception. + +Now follows his ridiculous apology for _adducis malleum, ut occidas +muscam_. The cause why he did use that proverb, of his own phrasing, was +this. Mr. Hobbes had taken a great deal of pains to demonstrate what Dr. +Wallis thought he could have proved in short; upon this occasion he +objects, _adducis malleum ut occidas muscam_, which I shall suppose he +intended to English thus, _you bring a beetle to kill a fly_. Mr. Hobbes +retorted, that _adduco_ was not used in that sense. The Doctor +vindicates himself thus: _duco_, _deduco_, _reduco_, _perduco_, +_produco_, &c. signify the same thing, ergo, _adduco_ may be used in +that sense; which is a most ridiculous kind of arguing, where we are but +to take up our language from others, and not to coin new phrases. It is +not the grammar that shall secure the Doctor, nor weak analogies, where +elegance comes in contest. To justify his expression he must have showed +it _usu tritum_, or alleged the authority of some author of great note +for it. I have not the leisure to examine his impertinent citations +about those other compounds, nor yet of that simple verb _duco_; nay, to +justify his saying he hath not brought one parallel example. He talks +indeed very high, that _duco_, with its compounds, is a word of a large +signification, and amongst the rest _to bring_, _fetch_, _carry_, &c. is +so exceeding frequent in all authors, Plautus, Terence, Tully, Cæsar, +Tacitus, Pliny, Seneca, Virgil, Horace, Ovid, Claudian, &c. that he must +needs be either maliciously blind, or a very stranger to the Latin +tongue, that doth not know it, or can have the face to deny it. I read, +what will be my doom for not allowing his Latin; yet I must profess I +dare secure the Doctor for having read all authors, notwithstanding his +assertion, and I hope he will do the like for me. And for those which he +hath read, had he brought no better proofs than these, he had, I am +sure, been whipped soundly in Westminster School, for his impudence as +well as ignorance, by the learned master thereof at present. But I dare +further affirm, the Doctor hath not read in this point any, but only +consulted with Robert Stephen’s _Thesaurus Linguæ Latinæ_, whence he +hath borrowed his allegations in _adduco_; and for the other, I had not +so much idle time as to compare them. And, lest the fact might be +discovered, he hath sophisticated those authors whence Stephen cites the +expressions, and imposed upon them others. If it be not so, or that the +Doctor could not write it right when the copy was right before him, let +him tell me where he did ever read in Plautus, _adducta res in +fastidium_. I find the whole sentence in Pliny’s preface to Vespasian +(out of whom in the precedent paragraph he cites it) about the middle: +_alia vero ita multis prodita, ut in factidium sint adducta_, which is +the very example Stephanus useth, although he doth premise his _adducta +res in fastidium_. Let the Doctor tell where he ever did read in Horace, +_Ova noctuæ_, &c. _tædium vini adducunt_. Did he, or any else, with the +interposition of an &c. make Trochaics? I say, and Stephanus says so, +too, that it is in Pliny, lib. xiii. cap. 15, near the end; the whole +sentence runs thus: _Ebriosis Ova noctuæ per triduum data in vino, +tædium ejus adducunt_. I doubt not but these are the places he aimed at, +although he disguised and minced the quotations; if they be not, I +should be glad to augment my Stephanus with his additions. + +These things premised, I come to consider the Doctor’s proofs: _Res eo +adducta est_: _adducta vita in extremum_: _adducta res in fastidium_: +_rem ad mucrones et manus adducere_: _contractares et adducta in +augustum_: _res ad concordiam adduci potest_: _in ordinem adducerem_: +_adducere febres, sitim, tedium vini_ (all in Robert Stephen) betwixt +which and _adducere malleum_, what a vast difference there is, I leave +them to umpire _qui terretes et religiosas nacti sunt aures_, who are +the competent judges of elegancy, and only cast in the verdict of one or +two, who are in any place (where the purity of the Latin tongue +flourisheth) of great esteem. Losæus, in his _Scopæ Linguæ Latinæ, ad +purgandam Linguam a barbarie_, &c. (would any think that the Doctor’s +elegant expression, frequent in all authors, which none but the +malicious or ignorant can deny, should suffer so contumelious an +expurgation?) Losæus, I say, hath these words: _Adferre plerique minus +attenti utuntur pro adducere. Quod Plautus, in Pseudolo, insigni exemplo +notat_. + + _CA._--Attuli hunc. + _PS._--Quid attulisti? + _CA._--Adduxi volui dicere. + _PS._--Quis istic est? + _CA._--Charinus. + +_Satis igitur admonet discriminis inter ducere, reducere, adducere, et +abducere, quæ de persona; et ferre, adferre_, &c. _quæ de re dicuntur_. +Idem, _Demetrium, quem ego novi, adduce: argentum non moror quin feras_. +_Cavendum igitur est ne vulgi more_, (let the Doctor mark this, and know +that _this author is authentic amongst the Ciceronians_), _adferre de +persona, dicamus, sed adducere; licet et hoc de certis quibusdam rebus +non inepte dicatur_. In this last clause he saith as much as Mr. Hobbes +saith, and what the Doctor proves; but, that ever the Doctor brought an +example which might resemble _adducis malleum_, is denied; for I have +mentioned already his allegations, every one, of _adduco_. Another +author, (a fit antagonist for the elegant Doctor), is the _Farrago +sordidorum Verborum_, joined with the Epitome of L. Valla’s +_Elegancies_. He saith: _Accerse, adhuc Petrum, Latine dicitur, pro eo +quod pueri dicunt, adfer Petrum_. And this may suffice to justify Mr. +Hobbes’s exception who proceeded no further than this author to tell the +Doctor that _adduco_ was used of animals. But the Doctor replies, _this +signification is true, but so may the other be also_. I say if it never +have been used so, it cannot be so, for we cannot coin new Latin words, +no more than French or Spanish who are foreigners. Mr. Hobbes was upon +the negative, and not to disprove the contrary opinion. If the Doctor +would be believed, he must prove it by some example, (which is all the +proof of elegancy), and till he do so, not to believe him, it is +sufficient not to have cause. But, Doctor Wallis, _why not adduco for a +hammer as well as a tree?_ I answer yes, equally for either, and yet for +neither. Did ever anybody go about to mock his readers thus solemnly? I +do not find, to my best remembrance, any example of it in Stephen, and +the Doctor is not wiser than his book; if there be, it is strange the +Doctor should omit the only pertinent example, and trouble us with such +impertinences for three or four pages. In Stephen there are _adducere +habenas_ and _adducere lorum_, but in a different sense. It is not +impossible I may guess at the Doctor’s aim. In Tully _de Nat. Deor._ as +I remember, there is this passage: _Quum autem ille respondisset, in +agro ambulanti ramulum adductum ut remissus esset, in oculum suum +recidisse_, where it signifies nothing else but to be _bent_, _bowed_, +_pulled back_, and in that sense, _the hammer of a clock_, or that of a +_smith, when he fetcheth his stroke_, may be said _adduci_. And this, I +conceive, the Doctor would have us in the close think to have been his +meaning; else, what doth he drive at in these words? “When you have done +the best you can, you will not be able to find better words than +_adducere malleum_ and _reducere_, to signify the two contrary motions +of the _hammer_, the one when you strike with it (_excellently +trivial!_) the other when you take it back (_better and better_), _What +to do?_ to fetch another stroke. If any can believe that this was his +meaning, I shall justify his Latin, but must leave it to him to prove +its sense. If he intended no more, why did he go about to defend the +other meaning, and never meddle with this? Which yet might have been +proved by this one example of mine? May not, therefore, his own saying +be justly retorted upon him in this case, _Adducis malleum, ut occidas +muscam_? + +Another exception is, _Falsæ sunt, et multa istiusmodi_ +(_propositiones_). I wish the Doctor could bring so good parallels, and +so many, out of any author, for his _Adducis malleum_, as Tully affords +in this case. Take one for all, out of the beginning of his _Paradoxes_: +_Animadverti sæpe Catonem, cum in senatu sententiam diceret, Locos +graves ex Philosophia tractare, abhorrentes ab hoc usu forensi, et +publico, sed dicendo consequi tamen, ut illa etiam populo probabilia +viderentur_. This is but a _Solæcophanes_, and hath many precedents +more, as in the second book of his _Academical Questions_, &c. + +I cannot now stay upon each particular passage; I do not see any +necessity of tracing the Doctor in all his vagaries. Now, he disallows +_tanquam diceremus_, _as if we should say_. But why is that less +tolerable than _tanquam feceris_, _as if you had done_? “It should be +_quasi_, (forsooth!) or _ac si_, or _tanquam si_, which is Tully’s own +word.” What is _tanquam si_ become but one word? _Tanquam si tua res +agatur_, &c. Good Doctor, leave out Tully and all _Ciceronians_, or you +will for ever suffer for this, and your _Adducis malleum_. Is not this +to put yourself on their verdict when you oppose Mr. Hobbes with Tully? +But the Doctor gives his reason. And though he hath had the luck in his +_Adducis malleum_, to follow the first part of that saying, _Loquendum +cum vulgo_, yet now it is, _sentiendum cum sapientibus_. For _tanquam_ +without _si_ signifies but _as_, not _as if_. It is pity the Doctor +could not argue in symbols too, that so we might not understand him; but +suppose all his papers to carry evidence with them, because they are +_mathematically_ scratched. How does he construe this:-- + + “Plance tumes alto Drusorum sanguine, tanquam + Feceris ipse aliquid, propter quod nobilis esses.” + +So Cœlius, one much esteemed by Cicero, who hath inserted his Epistles +into his works, saith, in his fifth Epistle (Tul. Epist. Fam. lib. viii. +ep. 5), _Omnia desiderantur ab eo tanquam nihil denegatum sit ei quo +minus paratissimus esset qui publico negotio præpositus est_. But it was +not possible the Doctor should know this, it not being in Stephen, where +his examples for _tanquam si_ are. + +But, the Doctor having pitched upon this criticism, and penned it, +somebody, I believe, put him in mind of the absurdity thereof; and yet +the generous _Professor_, (who writes running hand and never transcribed +his papers, if I am not misinformed), presumed nobody else could be more +intelligent than he, who had perused Stephen. He would not retract +anything, but subjoins, “That he will allow it as passable, because +other modern writers, and some of the ancients, have so used it, as Mr. +Hobbes hath done.” I know not what authors the Doctor meant, for, if I +am not much mistaken, I do not find any in Stephen. His citation of +Columella is not right, (lib. v. cap. 5), nor can I deduce anything +thence till I have read the passage, but, if he take Juvenal and Cœlius +for modern authors, I hope he will admit of Accius, Nævius, and +Carmenta, for the only ancients. Let him think upon this criticism, and +never hope pardon for his _Adducis malleum_, which is not half so well +justified, and yet none but _madmen_ or _fools_ reject it. + +But certainly the Doctor should not have made it his business to object +_Anglicisms_, in whose Elenchus I doubt not but there may be found such +phrases as may serve to convince him that he is an Englishman, however +Scottified in his principles. If the Doctor doubt of it, or but desire a +catalogue, let him but signify his mind, and he shall be furnished with +a _Florilegium_. But I am now come to the main controversy about Empusa. +The Doctor saith nothing in defence of his _quibble_, nor gives any +reason why he jumbled languages to make a silly clinch, which will not +pass for wit either at Oxford or at Cambridge; no, nor at Westminster. + +It seems he had derived _Empusa_ from ἓν and ποῦς, and said it was a +kind of _Hobgoblin_ that hopped upon one leg: and hence it was that the +boys’ play (_Fox come out of thy hole_) came to be called _Empusa_. I +suppose he means _Ludus Empusæ_. This derivation he would have to be +good, and that we may know his reading, (though he hath scarce consulted +any of the authors), he saith Mr. Hobbes did laugh at it, until somebody +told him that it was in the Scholiast of Aristophanes (as good a critic +as Mr. Hobbes), Eustathius, Erasmus, Cœlius Rhodiginus, Stephanus, +Scapula, and Calepine. But sure he doth not think to scape so. To begin +with the last; Calepine doth indeed say, _uno incedit pede, unde et +nomen_. But he is a _Modern_, and I do not see why his authority should +outweigh mine if his author’s reasons do not. He refers to Erasmus and +Rhodiginus. Erasmus in the adage, _Proteo mutabilior_ hath these words +of Empusa: _Narrant autem uno videri pedi_--this is not to hop--_unde et +nomen inditum putant_, Ἔμπουσαν ὁιονεὶ ἑνίποδα. He doth not testify his +approbation of the derivation at all, only lets you know what +etymologies some have given before him. And doth anybody think that Dr. +Harmar was the first which began to show his wit, (or folly), in +etymologizing words? Cœlius Rhodiginus doth not own the derivation, only +saith, _Nominis ratio est, ut placet Eustathio, quia uno incedit +pede_;--is this to hop?--_sed nec desunt qui alterum interpretentur +habere æneum pedem, et inde appellatam Empusam; quod in Batrachis +Aristophanes expressit_. And then he recites the interpretation that +Aristophanes’s Scholiast doth give upon the text, of which by and by. If +any credit be to be attributed to this allegation, his last thoughts are +opposite to Dr. Wallis; and _Empusa_ must be so called, not because she +hopped upon one leg, but because she had but one, the other being brass. +But for the former derivation he refers to Eustathius. + +As to Eustathius, I do easily conjecture that the reader doth believe +that Rhodiginus doth mean Eustathius upon Homer, for that is the book of +most repute and fame, his other piece being no way considerable for bulk +or repute. But it is not that book, nor yet his History of Ismenias, but +his notes upon the 725th verse of Dionysius Περῖηγητής. The poet had +said of the stone _Jaspis_, that it was + + Ἐχθρηὶν Ἐμπούσησι καὶ ἄλλοις ἔιδώλοισιν, + +Upon which Eustathius thus remarks: Δοκεῖ γαρ ἀλεξίκακος εἶναι ἡ λίθος +ἅυτη, καὶ ἀποτροπιαςτικὴ φασμἀτων, ὧν ἕν ἐςτι καὶ ἡ Ἔμπουσα, δαιμονιόν +τι τερί τἰὼ Ἑκάτην, ἑνὶ ποδὶ δοκοῦν δἰήκεσθαι· (_fortè_ διερείδεσθαι +_Steph._) ὄθεν καὶ παρονομάζεται, ὡς ἔι τις ἔιπη μονόπους ποδι ζωοῦ· ὡς +τοῦ ἑτέρου ποδος χαλκοῦ ὄντος, κατὰ τὸν μῦθον. This testimony doth not +prove anything of _hopping_, and, as to the derivation, I cannot but say +that Eustathius had too much of the grammarian in him, and this is not +the first time, neither in this book, nor elsewhere, wherein he hath +trifled. It is observable out of the place, that there were more +_Empusas_ than one, as, indeed, the name is applied by several men to +any kind of frightful phantasm. And so it is used by several authors, +and for as much as phantasms are various, according as the persons +affrighted have been severally educated, &c. every man did impose this +name upon his own apprehensions. This gave men occasion to fain _Empusa_ +as such--for who will believe that she was not apprehended as having +four legs, when she appeared in the form of a cow, dog, &c.--but, as +apprehended by _Bacchus_ and his man at that time. I do not find that +she appeared in any shape but such as made use of legs in going, whence +I imagine that _Empusæ_ might be opposite to the θεοὶ νεποδες, which +appellation was anciently fixed upon the gods, (_propitious_) upon a +two-fold account; first, for that they were usually effigiated as having +no feet, which is evident from ancient sculpture, and secondly, for that +they are all said not to walk, but rather swim, if I may so express that +_non gradiuntur, sed fluunt_, which is the assertion of all the +commentators I have ever seen upon that verse of Virgil:-- + + “Et vera incessu patuit dea”---- + +This whole discourse may be much illustrated from a passage in +Heliodorus, Æthiop. lib. iii. sec. 12, 13. Calasiris told Cnemon that +the Gods Apollo and Diana did appear unto him; Cnemon replied, Ἀλλὰ τίνα +δὴ τρόπον ἒφασκες ἐνδεδεῖχθαἱ σοι τοῦς θεοῦς ὅτι μὴ ἐνύπνιον ἦλθον, ἀλλ’ +ἐναργῶς ἐφᾶνησαν; upon this the old priest answered, that both gods and +demons, when they appear to men, may be discovered by the curious +observer, both in that they never shut their eyes, καὶ τῳ βαδίσματι +πλέον, οὐ κατὰ διάστησιν τῶν ποδῶν οὐδέ μετάθεσιν ἀνυομένω, ἀλλὰ κατὰ +τινα ρὕμην ἀέριον, καὶ ὁρμὴν ἀπαραπόδιστον, τεμνόντων μᾶλλον τὸ περιεχον +ἢ διαπορευομένων. Δὶο δὴ καὶ τὰ ἀγάλματα τῶν θεῶν Ἀιγύπτιοι τὼ πὸδέ +ζευγνύντες καὶ ὥσπερ ἑνοῦντες ἵστᾶσιν. ἅ δὴ καὶ Ὅμηρος ἐιδῶς, ἅτε +Ἀιγύπτιος, καὶ τὴν ἱερὰν πάιδευσιν ἐκδιδαχθείς, συμβολικῶς τοῖς ἔπεσιν +ἐναπεθετο, τοῖς δυναμένοις συνιέναι γνωριζειν καταλιπών, ἐπι τοῦ +ποσειδῶνος, το + + Ἴχνια γὰρ μετόπισθε, ποδῶν ἠδέ κνημάων + Ῥεῖ ἔγνων ἀπιὸντος. + +οἴον ῥέοντος ἐν τῆ πορεία, τοῦτο γάρ εστι τό ῥεῖ ἀπιόντος, καὶ οῦχ ὥς +τινες ἠπάτηνται, ῥᾳδίως ἔγνων ὑπολαμβάνοντες. Farnaby, upon the place in +Virgil, observes, that _Deorum incessus est continuus et æqualis, non +dimotis pedibus, neque transpositis_, ἀλλὰ κατὰ ῥύμην ἀέριον. Cornelius +Schrevelius in the new Leyden notes saith, _Antiquissima quæque Deorum +simulachra, quod observarunt viri magni, erant_ τοῦς πόδας συμβεβηκότα, +_diique ipsi non gradiuntur sed fluunt_. Their statues were said to +stand rather upon columns than upon legs, for they seem to have been +nothing but columns shaped out into this or that figure, the base +whereof carrying little of the representation of a foot. These things +being premised, I suppose it easy for the intelligent reader to find out +the true etymology of _Empusa, quasi_ ἐν ποσιν οῦσα, or βάινουσα, from +going on her feet, whereas the other _gods_ and _demons_ had a different +gait. If any can dislike this deduction, and think her so named from +ἑνιπους, whereas she always went upon two legs, (if her shape permitted +it) though she might draw the one after her, as a man doth a wooden leg: +I say, if any, notwithstanding what hath been said, can join issue with +the Doctor, my reply shall be Σοὶ μὲν ταῦτα δοκοῦντ’ ἐστὶν, ἐμὸι δὲ +τάδε. + +Now, as to the words of Aristophanes upon which the Scholiast descants, +they are these:--speaking of an apparition strangely shaped, sometimes +like a camel, sometimes like an ox, a beautiful woman, a dog, &c. +Bacchus replies: + + Ἔμπουσα τοινὺν γ’ἐστι. + ΞΑ. πυρὶ γοῦν λάμπεται + ἅπαν το προσωπον, καὶ σκελος χαλκοῦν ἔχει. + ΔΙ. Νὴ τὸν Ποσειδῶ, καὶ βολιτινον θάτερον. + ΞΑ. Σἁφ’ ἵσθι. + +The Scholiast hereupon tells us that _Empusa_, was Φαντασμα δαιμονιῶδες +ὑπὸ Ἑκάτης ἐπιπεμπόμενον καὶ φαινόμενον τοἴς δυστυχοῦσιν, ὅ δοκεῖ πολλὰς +μορφας αλλασσεω καὶ ὁι μεν φασιν ἀυτην μονοποδα εῖναι, καὶ +ἐτυμολογοῦσιν’ ὁιονεὶ ἑνιποδα, διὰ το ἑνὶ ποδι κεχρῆσθαι. And this is +all that is material in the Scholiast, except that he adds by and by, +that βολιτινον σκελος is all one with the leg of an ass. And this very +text and Scholiast is that to which all the authors he names, and more, +do refer. + +I come now to Stephen, who, in his index, and in the word ποδίζω, gives +the derivation of _Empusa_. Ποδιζω, _gradior, incedo_, (not to hop) _sic +Suidas_ Ἔμπουσαν _dictam ait_ παρὰ το ἑνὶ ποδιζειν. In the index thus: +_sunt qui dictam putent_ παρὰ τὸ ἑνὶ ποδὶζειν, _quod uno incedat pedi, +quasi_ Ἔμπουσαν, _alterum enim pedem æneum habet_. But neither Stephen, +nor any else, except _Suidas_, whom the hypercritical Doctor had not +seen, no, not the Scholiast of Aristophanes (a better critic than Mr. +Hobbes) doth relate the etymology as their own. Nay, there is not one +that saith _Empusa_ hopped on one leg, which is to be proved out of +them. The great Etymological Dictionary deriveth it παρὰ τὸ ἐμποδιζειν, +to _hinder_, _let_, &c. its apparition being a token of ill luck. But, +as to the Doctor’s deduction, it saith, Ἔμπουσα Ψιλοῦπαι, εἰ καὶ δοκεῖ +παρὰ τὸ ἕνα συγκεῖσθαι. It doth only _seem_ so. And it is strange that +ἑν should not alter only its _aspiration_, but change its ν into μ, +which I can hardly believe admittable in Greek, least there should be no +difference betwixt its derivatives and those of ἐν. When I consider the +several μορμόνες which the Grecians had, some whereof did fly, some had +no legs, &c., I can think that the origin of this name may have been +thus: some amazed person saw a _spectrum_, and, giving another notice of +it, his companion might answer, it is Βριμὼ, Μορμὼ Ἡκὰτη, but he, +meeting with a new phantasm, cries, ἐν ποσὶ βαίνει or βαδίζει, for which +apprehension of his, somebody coined this expression of Ἔμποῦσα. It may +also be possibly deduced from Ἐμποδὶζω, so that τύχη ἐμποδιζουσα might +afterwards be reduced to the single term of _Empusa_. Nor do I much +doubt but that those who are conversant in languages, and know how that +several expressions are often jumbled together to make up one word upon +such like cases, will think this a probable origination. I believe, +then, that Mr. Hobbes’s friend did never tell him it was in Eustathius, +or that _Empusa_ was an _hopping phantasm_. It had two legs and went +upon both, as a man may upon a wooden leg. Ἔμποῦσα is also a name for +Lamia, and such was that which Menippus might have married, which, I +suppose, did neither hop nor go upon one leg, for he might have +discovered it. But Mr. Hobbes did not except against the derivation, +(although he might justly, derivations made afterwards carrying more of +fancy than of truth, and the Doctor is not excused for asserting what +others barely relate, none approve), but asked him where that is, in +what authors _he read that boys’ play to be so called_. To which +question, the Doctor, to show his reading and the good authors he is +conversant in, replies, _in Junius’s Nomenclator, Rider and Thomas’s +Dictionary, sufficient authors in such a business_, which, methinks, no +man should say that were near to so copious a library. It is to be +remembered that the trial now is in Westminster School, and amongst +Ciceronians, neither whereof will allow those to be sufficient authors +of any Latin word. Alas, they are but _Vocabularies_; and, if they bring +no author for their allegation, all that may be allowed them is, that, +by way of allusion, our modern play may be called _Ludus Empusæ_. But +that it is so called we must expect, till some author do give it the +name. These are so good authors, that I have not either of them in my +library. But I have taken the pains to consult, first, Rider; I looked +in him, (who was only author of the English Dictionary) and I could not +find any such thing. It is true, in the Latin Dictionary, which is +joined with Rider, but made by Holyoke; (O that the Doctor would but +mark!) in the index of obsolete words, there is _Ascoliasmus, Ludus +Empusæ_, _Fox to thy hole_, for which word, not signification, he +quoteth Junius. The same is in Thomasius, who refers to Junius in like +manner. But could the Doctor think the word obsolete, when the play is +still in fashion? Or, doth he think that this play is so ancient as to +have had a name so long ago, that it should now be grown obsolete? As +for Junius’s interpretation of _Empusa_, it is this: _Empusa, spectrum, +quod se infelicibus ingerit, uno pede ingrediens_. Had the Doctor ever +read him, he would have quoted him for his derivation of _Empusa_, I +suppose. In Ascoliasmus, he saith, _Ascoliasmus, Empusæ Ludus, fit ubi, +altero pede in aere librato, unico subsiliunt pede:_ ἀσκολιασμὸς +_Pollux; Almanicè, Hinckelen; Belgicè,_ _Op een been springhen; +Hinckepincken, Flandris_. But what is it in English he doth not tell, +although he doth so in other places often. What the Doctor can pick out +of the Dutch I know not; but, if that do not justify him, as I think it +doth not, he hath wronged Junius, and greatly imposed upon his readers. + +But, to illustrate this controversy further, I cannot be persuaded the +Doctor ever looked into Junius, for, if he had, I am confident, +according to his wonted accurateness, he would have cited Pollux’s +_Onomasticon_ into the bargain, for Junius refers to him, and I shall +set down his words, that so the reader may see what _Ascoliasmus_ was, +and all the Doctor’s authors say _Ludus Empusæ_ and _Ascoliasmus_ were +one and the same thing. Julius Pollux (lib. ix. cap. 7): Ὁ δε +Ἀσκολὶασμὸς, (old editions read it, Ἀ’σκολιασμὸς et ασκολιάζω) τοῦ +ἑτέρου ποδὸς αἰωρουμένου, κατὰ μόνου τοῦ ἑτέρου πηδᾶν ἔπόιει; ὅπερ +Ἀσκωλιάζὲιν ὠνόμαζον· ἤτοι εἰς μἢκος ἐνήλλαντο, ἢ ὁ μὲν ἐδίωκεν οὕτως, +οἱ δὲ ὑπέφευγον ἐπ’ ἀμφοῖν θὲοντες, ἕως τινὸς τῳ φερομένῳ ποδὶ ὁ διὼκων +δυνηθῇ τυχεῖν· ἤ καὶ στάντες ἐπήδων, ἀριθμοῦντες τὰ πηδήματα· προσέκειτο +γὰρ τῷ πλήθει τὸ νικᾶν. Ἀσκωλιάζειν δὲ ἐκαλεῖτο καὶ τὸ ἐπιπηδᾶν ἀσκῷ +κενῷ καὶ ὑποπλέω πνευματος, ἠλείμμένω, ἵναπερ ὀλισθάνοιεν περὶ τὴν +ἀλοιφὴν. “So that _Ascoliasmus_, and consequently, _Ludus Empusæ_, was a +certain sport which consisted in hopping, whether it were by striving +who could hop furthest, or whether only one did pursue the rest hopping, +and they fled before him on both legs, which game he was to continue +till he had caught one of his fellows, or whether it did consist in the +boys’ striving who could hop longest. Or, lastly, whether it did consist +in hopping upon a certain bladder, which, being blown up and well oiled +over, was placed upon the ground for them to hop upon, that so the +unctuous bladder might slip from under them and give them a fall.” And +this is all that Pollux holds forth. Now, of all these ways, there is +none that hath any resemblance with our _Fox to thy hole_; but the +second: and yet, in its description, there is no mention of beating him +with gloves, as they do now-a-days, and wherein the play consists as +well as in hopping. It might, notwithstanding, be called _Ludus Empusæ_, +but not in any sort our _Fox to thy hole_; so that the Doctor and his +authors are out, imposing that upon Junius and Pollux which they never +said. And thus much may suffice as to this point. I shall only add out +of Meursius’s _Ludi Græci_, that _Ascolia_ were not _Ludus Empusæ_ but +_Bacchisacra_, and he quotes Aristophanes’s Scholiast in Plutus, Ἀσκώλια +ἑορτὴ Διονύσου ἀσκὸν γαρ οἵνου πληροῦντες, ἑνὶ ποδὶ τοῦτον ἐπεπήδον, καὶ +ὁ πηδήσας ἆθλον εἶχε τὸν οἵνου. As also Hesychius, Ἀσκωλιάζειν, κυρίως +τὸ ἐπὶ τοῦς ἀσκοὺς ἅλλεσθαι. + +But I could have told the Doctor where he might have read of _Empusa_ as +being the name of a certain sport or game, and that is, _in Turnebus +Adversaria_, lib. xxvii. cap. 33. There he speaks of several games +mentioned by Justinian in his _Code_, at the latter end of the third +book, one of which he takes to be named _Empusa_; adding withal, _that +the other are games, it is indisputable_, only _Empusa in lite et causa +erit, quod nemo nobis facile assensurus sit Ludum esse, cum constet +spectrum quoddam fuisse formas, varie mutans. Sed quid vetat eo nomine +Ludum fuisse? Certe ad vestigia vitiatæ Scripturæ quam proximo accedit._ +Yet he only is satisfied in this conjecture, till somebody else shall +produce a better. And now what shall I say? Was not Turnebus as good a +critic, and of as great reading as Dr. Wallis, who had read over Pollux, +and yet is afraid that nobody will believe _Empusa_ to have been a game, +and all he allegeth for it is, _quid vetat_? Truly, all I shall say, and +so conclude this business, is, that he had read over an infinity of +books, yet, had not had the happiness, which the Doctor had, to consult +with _Junius’s Nomenclator, Thomasius and Rider’s Dictionary, authors +sufficient in such a case_. + +I now come to the Doctor’s last and greatest triumph, at which I cannot +but stand in admiration, when I consider he hath not got the victory. +Had the Doctor been pleased to have conversed with some of the fifth +form in Westminster School, (for he needed not to have troubled the +learned master), he might have been better informed than to have exposed +himself thus. + +Mr. Hobbes had said that στιγμὴ signified _a mark with a hot iron_; upon +which saying the Doctor is pleased to play the droll thus: “Prithee tell +me, good Thomas, before we leave this point, (O the wit of a divinity +doctor!) who it was told thee that στιγμὴ was a mark with an hot iron, +for it is a notion I never heard till now, and do not believe it yet. +Never believe him again that told thee that lie, for as sure as can be, +he did it to abuse thee; ϛιγμὴ signifies a distinctive point in writing, +made with a pen or quill, not a mark made with a _hot iron_, such as +they brand rogues withal; and, accordingly, ϛιζω δῖαϛιζω, _distinguo_, +_interstinguo_, are often so used. It is also used of a _mathematical_ +point, or somewhat else that is very small, στιγμὴ χρὸνου, a moment, or +the like. What should come in your cap, to make you think that ϛῖγμὴ +signifies a mark or brand with a _hot iron_? I perceive where the +business lies; it was ϛίγμα ran in your mind when you talked of ϛιγμὴ, +and, because the words are somewhat alike, you jumbled them both +together, according to your usual care and accurateness, as if they had +been the same.” + +When I read this I cannot but be astonished at the Doctor’s confidence, +and applaud him who said, ἀμάθεια θάρσὸς φέρει. That the Doctor should +never hear that ϛιγμὴ signifies _a mark with a hot iron_, is a manifest +argument of his ignorance. But, that he should advise Mr. Hobbes not to +believe his own readings, or any man’s else that should tell him it did +signify any such thing, is a piece of notorious impudence. That ϛιγμὴ +_signifies a distinctive point in writing made with a pen or quill_, (is +a pen one thing and a quill another to write with?) nobody denies. But, +it must be withal acknowledged it signifies many things else. I know the +Doctor is a _good historian_, else he should not presume to object the +want of history to another; let him tell us how long ago it is since men +have made use of pens or quills in writing; for, if that invention be of +no long standing, this signification must also be such, and so it could +not be that from any allusion thereunto the mathematicians used it for a +point. Another thing I would fain know of this great historian, how long +ago ϛίζω and διαϛίζω began to signify _interpungo_? For, if the +mathematics were studied before the mystery of printing was found out, +(as shall be proved whenever it shall please the Doctor, out of his no +reading, to maintain the contrary), then the _mathematical_ use thereof +should have been named before the _grammatical_. And, if this word be +translatitious, and that sciences were the effect of long contemplation, +the names used wherein are borrowed from talk, Mr. Hobbes did well to +say, that στιγμὴ precedaneously to that _indivisible_ signification +which it afterwards had, did signify a _visible mark_ made by a hot +iron, or the like. And, in this procedure, he did no more than any man +would have done, who considers that all our knowledge proceeds from our +senses; as also that words do, _primarily_, signify things obvious to +_sense_, and only _secondarily_, such as men call _incorporeal_. This +leads me to a further consideration of this word. Hesychius, (of whom it +is said that he is _Legendus non tanquam Lexicographus, sed tanquam +justus author_), interprets στιγμὴ, νυγμή, which is a point of a greater +or lesser size, made with any thing. So ϛίζω signifies to prick or mark +with anything in any manner, and hath no impropriated signification in +itself, but according to the writer that useth it. Thus, in a +_grammarian_ ϛίζω signifies to _distinguish_, by _pointing_ often; +sometimes, even in them, it is the same with ὀβελίζω; sometimes it +signifies to set a mark that something is wanting in that place, which +marks were called ϛιγμαί. In matters of policy, ϛίζω signifies to +_disallow_, because they used to put a ϛιγμὴ (not ϛίγμα) before his name +who was either disapproved or to be mulcted. In punishment it signifies +to _mark_ or _brand_, whereof I cannot at present remember any other +ways than that of an _hot iron_, which is most usual in authors, because +most practised by the ancients. But, that the mark which the _Turks_ and +others do imprint without burning may be said ϛίζεσθαι, I do not doubt, +no more than that Herodian did to give that term to the ancient Britons, +of whom he says, τὰ σώματα ἐϛίζοντο γραφαῖς ποικίλαις, καὶ ζώων +παντοδαπῶν εἰκόσι. Thus, horses that were branded with κάππα and σαν +(κοππἀτιαι and σαμφοραι) were said ϛίζεσθαι. Thus, in its origin, ϛιγμὴ +doth signify a _brand or mark with an hot iron_, or the like; and that +must be the proper signification of στιγμὴ, which is proper to ϛίζω, +none but such as Dr. Wallis can doubt. In its _descendants_ it is no +less evident, for, from στιγμὴ comes _stigmosus_, which signifies to be +branded; _Vitelliana cicatrice stigmosus_, not _stigmatosus_. So Pliny +in his Epistles, as Robert Stephen cites it. And στιγματιας (the +derivative of στιγμὴ, which signifies any mark, as well as a brand, even +such as remain after stripes, being black and blue), was a nickname +imposed upon the grammarian Nicanor, ὅτι περὶ στιγμῶν ἐπολυλόγησε. And, +though we had not any examples of στῖγμὴ being used in this sense, yet, +from thence, for any man to argue against it, (but he who knows no more +than Stephen tells him) is madness, unless he will deny that any word +hath lost its right signification, and is used only, by the authors we +have, although neither the Doctor nor I have read all them, in its +analogical signification. I have always been of opinion, that στιγμὴ +signified a _single point_, big or little, it matters not; and στίγμα, a +_composure of many_; as γραμμὴ signifies a _line_, and γράμμα a +_letter_, made of several lines. For στίγμα signified the _owl_, the +_sæmæna_, the letter K, yea, _whole words, lines, epigrams_ engraven in +men’s faces; and στιγμὴ, I doubt not, had signified _a single point_, +had such been used, and so it became translatitiously used by +grammarians and mathematicians. I could give grounds for this +conjecture, and not be so impertinent as the Doctor in his sermon, where +he told men that σοφός was not in Homer; that from ἄφρων came _ebrius_; +that _sobrietas_ was not bad Latin, and that _sobrius_ was once, as I +remember, in Tully. Is this to speak suitably to the oracles of God, or +rather to lash out into idle words? Hath the Doctor any ground to think +these are not impertinences? Or, are we, poor mortals, accountable for +such _idle_ words as fall from us in private discourses, whilst these +ambassadors from heaven _droll_ in the pulpit without any danger of an +after-reckoning? + +But I proceed to a further survey of the Doctor’s intolerable ignorance. +His charge in the end of the _school-master’s_ rant is, that he should +_remember_ στίγμα and στιγμὴ _are not all one_. I complained before that +he hath not cited Robert Stephen aright; now I must tell him he hath +been negligent in the reading of Henry Stephen: for in him he might have +found that στίγμα was sometimes all one with στιγμὴ, though there be no +example in him wherein στιγμὴ is used for στίγμα. Hath not Hesiod, (as +Stephen rightly citeth it), in his _Scutum_, 166-67. + + Στἴγματα δ’ ὥς ἐπέφαντο ἴδεῖν δεινοῖσι δράκουσι + Κυανέα κατὰ νῶτα + +_ubi scholiastes_ ὥσπερ δὲ στιγμαὶ ἦσαν ἐπάνω, τῶν ῥάχεων τῶν δρακόντων, +κατάστίκτοι γὰρ καὶ ποικίλοι ὁι ὄφεις. So Johannes Diaconus upon the +place, a man who (if I may use the Doctor’s phrase) was _as good a +critic as_ the Geometry Professor. + +Thus much for the _Doctor_. To the understanding _reader_, I say that +στιγμὴ is used for burning with a hot iron: _2 Macchab._ ix. 11, where +speaking of Antiochus’s lamentable death, his body putrefying and +breeding worms, he is said, ἐις ετίγνωσιν τοῦ θεοῦ ἔρχεθαι θείᾳ μάστιγι, +κατα στιγμὴν ἐπιτεινόμενος ταῖς ἀλγηδόσι; _being pained as if he had +been pricked or burned with hot irons_. And that this is the meaning of +that elegant writer, shall be made good against the Doctor, when he +shall please to defend the vulgar interpretation. Pausanias, in +_Bœoticis_, speaking of Epaminondas, who had taken a town belonging to +the Sicyonians, called Phœbia (Φουβία) wherein were many Bœotian +fugitives, who ought, by law, to have been put to death, saith he +dismissed them under other names, giving them only a _brand_ or _mark_. +Πόλισμα ἑλὼν Σικυωνἰων Φουβίαν, ἔνθὰ ἦσαν το πολὺ οἱ Βοιώτιοι φυγάδες, +στιγμήν ἀφίησι τοῦς ἐγκαταληφθέντας ἄλλην σφίσιν ἣν ετυχε πατρίδα +ἐπονομάζων ἐκάστω. It is true στιγμὴν is here put _adverbially_, but +that doth not alter the case. Again, Zonaras, in the third tome of his +History, in the life of the Emperor Theophilus, saith, that when +Theophanes and another monk had reproved the said emperor for +demolishing images, he took and _stigmatized_ each of them with twelve +_iambics_ in their faces: εἶτα καὶ τὰς ὄψεις ἀυτῶν κάτεστιξε καὶ ταῖς +στιγμαῖς μέλαν ἐπέχεε γράμματα δὲ ἐτύπουν τὰ στιγματα, τὰ δὲ ἦσαν ἴαμβοι +οὗτοι. A place so evident, that I know not what the Doctor can reply. +This place is just parallel to what the same author saith in the life of +Irene, τἀς ὄψέις σφών καταστιξας ἐν γράμμασι, μέλανος εγχεομένου τοῖς +στίγμασι. If the Doctor object that he is a modern author, he will never +be able to render him as inconsiderable as Adrianus Junius’s +_Nomenclator_, Thomasius and Rider. If any will deny that he writes good +Greek, Hieronymus Wolfius will tell them, his only fault is +περισσολογια, _redundancy_ in words, and not the use of _bad_ ones. + +Another example of στιγμὴ used in this sense, is in the collections out +of Diodorus Siculus, lib. xxxiv. as they are to be found at the end of +his works, and as Photius hath transcribed them into his _Bibliotheca_. +He saith that the Romans did buy multitudes of servants and employ them +in Sicily: Οἷς, ἐκ τῶν σωματοτροφείων ἀγεληδὸν απαχθεῖσιν, ἐυθύς +χαρακτῆρα ἐπέβαλλον, καὶ στιγμὰς τοἴς σώμασιν. These are the words but +of one author, but ought to pass for the judgment of two, seeing +Photius, by inserting them, hath made them his own. + +Besides, it is the judgment of a great _master_ of the Greek tongue, +that _stigmata non tam puncta ipsa quam punctis variatam superficiem +Græci vocaverunt_. I need not, I suppose, name him, so great a critic as +the Doctor cannot be ignorant of him. + +Nor, were στίγματα commonly, but upon extraordinary occasions, imprinted +with an hot iron. The letters were first made by incision, then the +blood _pressed_, and the place filled up with ink, the composition +whereof is to be seen in Aetius. And thus they did use to _matriculate_ +soldiers also in the hand. Thus, did the Grecian emperor, in the +precedent example of Zonaras. And if the Doctor would more, let him +repair to Vinetus’s comment upon the fifteenth Epigram of Ausonius. + +And now I conceive enough hath been said to vindicate Mr. Hobbes, and to +show the insufferable ignorance of the puny professor, and unlearned +critic. If any more shall be thought necessary, I shall take the pains +to collect more examples and authorities, though I confess I had rather +spend time otherwise, than in matter of so little moment. As for some +other passages in his book, I am no competent judge of _symbolic +stenography_. The Doctor (Sir Reverence) might have used a cleanlier +expression than that of a _shitten piece_, when he censures Mr. Hobbes’s +book. + +Hitherto the letter.[1] By which you may see _what came into my (not +square) cap to call_ στιγμὴ _a mark with a hot iron, and that they who +told me_ that, did no more tell me a lie than they told you a lie that +said the same of στίγμα; and, if στιγμὴ be not right as I use it now, +then call these notes not στιγρας, but στίγματα. I will not contend with +you for a trifle. For, howsoever you call them, you are like to be known +by them. Sir, the calling of a divine hath justly taken from you some +time that might have been employed in geometry. The study of algebra +hath taken from you another part, for algebra and geometry are not all +one; and you have cast away much time in practising and trusting to +symbolical writings; and for the authors of geometry you have read, you +have not examined their demonstrations to the bottom. Therefore, you +perhaps may be, but are not yet, a geometrician, much less a good +divine. I would you had but so much ethics as to be civil. But you are a +notable critic; so fare you well, and consider what honour you do, +either to the University where you are received for professor, or to the +University from whence you came thither, by your geometry; and what +honour you do to Emanuel College by your divinity; and what honour you +do to the degree of Doctor, with the manner of your language. And take +the counsel which you publish out of your encomiast his letter; think me +no more worthy of your pains, you see how I have fouled your fingers. + +----- + +Footnote 1: + + Written by Henry Stubbe, M.A. of Christ Church, Oxford, who was, + according to Anthony a Wood, “the most noted personage of his age that + these late times have produced.” + +----- + + + + + THREE PAPERS + + PRESENTED TO THE ROYAL SOCIETY + + AGAINST DR. WALLIS. + + + TOGETHER WITH + + CONSIDERATIONS + + ON DR. WALLIS’S ANSWER TO THEM, + + BY + + THOMAS HOBBES, + + OF MALMESBURY. + + + + + THREE PAPERS + + PRESENTED TO THE ROYAL SOCIETY. + + ========== + +TO THE RIGHT HONOURABLE AND OTHERS, THE LEARNED MEMBERS OF THE ROYAL + SOCIETY, FOR THE ADVANCEMENT OF SCIENCES. + +PRESENTETH _to your consideration, your most humble servant, Thomas +Hobbes, (who hath spent much time upon the same subject), two +propositions, whereof the one is lately published by Dr. Wallis, a +member of your Society, and Professor of Geometry; which if it should be +false, and pass for truth, would be a great obstruction in the way to +the design you have undertaken. The other is a problem, which, if well +demonstrated, will be a considerable advancement of geometry; and though +it should prove false, will in no wise be an impediment to the growth of +any other part of philosophy._ + + DR. WALLIS, + DE MOTU, _Cap._ v. _Prop._ 1. + +If there be understood an infinite row of quantities beginning with 0 or +(1)/(0), and increasing continually according to the natural order of +numbers, 0, 1, 2, 3, &c. or according to the order of their squares, as, +0, 1, 4, 9, &c. or according to the order of their cubes, as, 0, 1, 8, +27, &c. whereof the last is given; the proportion of the whole, shall be +to a row of as many, that are equal to the last, in the first case, as 1 +to 2; in the second case, as 1 to 3; in the third case, as 1 to 4, &c. + +This proposition is the ground of all his doctrine concerning the +centres of gravity of all figures. Wherein may it please you to +consider: + +First, whether there can be understood an infinite row of quantities, +whereof the last can be given. Secondly, whether a finite quantity can +be divided into an infinite number of lesser quantities, or a finite +quantity can consist of an infinite number of parts, which he buildeth +on as received from Cavallieri. Thirdly, whether (which in consequence +he maintaineth) there be any quantity greater than infinite. Fourthly, +whether there be, as he saith, any finite magnitude of which there is no +centre of gravity. Fifthly, whether there be any number infinite. For it +is one thing to say, that a quantity may be divided perpetually without +end, and another thing to say, that a quantity may be divided into an +infinite number of parts. Sixthly, if all this be false, whether that +whole book of _Arithmetica Infinitorum_, and that definition which he +buildeth on, and supposeth to be the doctrine of Cavallieri, be of any +use for the confirming or confuting of any propounded doctrine. + +Humbly praying you would be pleased to declare herein your judgment, the +examination thereof being so easy, that there needs no skill either in +geometry, or in the Latin tongue, or in the art of logic, but only of +the common understanding of mankind to guide your judgment by. + + THOMAS HOBBES, + ROSET. _Prop._ v. + + _To find a straight line equal to two-fifths of the arc of a + quadrant._ + +I describe a square A B C D, and in it a quadrant D A C. Suppose D T be +two-fifths of D C, then will the quadrantal arc T V be two-fifths of the +arc C A. Again let D R be a mean proportional between D C and D T; then +will the quadrantal arc R S be a mean proportional between the arc C A +and the arc T V. + +Suppose further a right line were given equal to the arc C A, and a +quadrantal arc therewith described; then will D C, C A, the arc on C A +be continually proportional. Set these proportionals in order by +themselves. + + D C, C A, arc on C A∺ + D R, R S, arc on R S∺ + D T, T V, arc on T V∺ + +which are in continual proportion of the semi-diameter of the arc. And D +C, D R, D T are in a continual proportion by construction, and therefore +also C A, R S, T V, and arc on C A, arc on R S, arc on T V, in continual +proportion. + +Therefore as D C to R S, so is R S to the arc on T V. And D C, R S, the +arc on T V will be continually proportional. And because D C, C A, the +arc on C A are also continually proportional, and have the first +antecedent D C common; the proportion of the arc on C A to the arc on T +V is (by Eucl. xiv. 28) duplicate of the proportion of C A to R S, and +the arc on R S a mean proportional between the arc on C A and the arc on +T V. + +Now if D C be greater than R S, also R S must be greater than the arc on +T V; and the arc C A greater than the arc on R S. Therefore seeing D C, +C A, arc on C A, are continually proportional; the arc on T V, the arc +on R S, the arc on C A cannot be continually proportional, which is +contrary to what has been demonstrated. Therefore D C is not greater +than R S. Suppose, then, R S to be greater than D C, then will the arc +on R S be a mean proportional between the arc on T V, and a greater arc +than that on C A; and so the inconvenience returneth. Therefore the +semidiameter D C is equal to the arc R S, and D R equal to T V, that is +to say to two-fifths of the arc C A, which was to be demonstrated. Nor +needeth there much geometry for examining of this demonstration. +Therefore I submit them both to your censure, as also the whole +_Rosetum_, a copy whereof I have caused to be delivered to the secretary +of your society. + +[Illustration] + + TO THE + + RIGHT HONOURABLE AND OTHERS, + + THE LEARNED MEMBERS + + OF + + THE ROYAL SOCIETY, + + FOR THE ADVANCEMENT OF THE SCIENCES. + + + ------- + + +Presenteth to your consideration, your most humble servant Thomas +Hobbes, a confutation of a theorem which hath a long time passed for +truth; to the great hinderance of Geometry, and also of Natural +Philosophy, which thereon dependeth. + + THE THEOREM. + +_The four sides of a square being divided into any number of equal +parts, for example into 10; and straight lines drawn through the +opposite points, which will divide the square into 100 lesser squares; +the received opinion, and which Dr. Wallis commonly useth, is, that the +root of those 100, namely 10, is the side of the whole square._ + + THE CONFUTATION. + +_The root 10 is a number of those squares, whereof the whole containeth +100, whereof one square is an unity; therefore the root 10, is 10 +squares: Therefore the root of 100 squares is 10 squares, and not the +side of any square; because the side of a square is not a superficies, +but a line. For as the root of 100 unities is 10 unities, or of 100 +soldiers 10 soldiers: so the root of 100 squares is 10 of those squares. +Therefore the theorem is false; and more false, when the root is +augmented by multiplying it by other greater numbers._ + +Hence it followeth, that no proposition can either be demonstrated or +confuted from this false theorem. Upon which, and upon the numeration of +infinites, is grounded all the geometry which Dr. Wallis hath hitherto +published. + +And your said servant humbly prayeth to have your judgment hereupon: and +that if you find it to be false, you will be pleased to correct the +same: and not to suffer so necessary a science as geometry to be +stifled, to save the credit of a professor. + + TO THE + + RIGHT HONOURABLE AND OTHERS, + + THE LEARNED MEMBERS + + OF + + THE ROYAL SOCIETY, + + FOR THE ADVANCEMENT OF THE SCIENCES. + + --- + +Your most humble servant Thomas Hobbes presenteth, that the quantity of +a line calculated by extraction of roots is not to be truly found. And +further presenteth to you the invention of a straight line equal to the +arc of a circle. + +A square root is a number which multiplied into itself produced a +number. + + DEFINITION. + +And the number so produced is called a square number. For example: +Because 10 multiplied by 10 makes 100; the root is 10, and the square +number 100. + + CONSEQUENT. + +In the natural row of numbers, as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, +13, 14, 15, 16, &c. every one is the square of some number in the same +row. But square numbers (beginning at 1) intermit first two numbers, +then four, then six, &c. So that none of the intermitted numbers is a +square number, nor has any square root. + + PROP. I. + +A square root (speaking of quantity) is not a line, such as Euclid +defines, without latitude, but a rectangle. + +[Illustration] + +Suppose A B C D be the square, and A B, B C, C D, D A, be the sides, and +every side divided into 10 equal parts, and lines drawn through the +opposite points of division; there will then be made 100 lesser squares, +which taken altogether are equal to the square A B C D. Therefore the +whole square is 100, whereof one square is an unit; therefore 10 units, +which is the root, is ten of the lesser squares, and consequently has +latitude; and therefore it cannot be the side of a square, which, +according to Euclid, is a line without latitude. + + CONSEQUENT. + +It follows hence, that whosoever taketh for a principle, that a side of +a square is a mere line without latitude, and that the root of a square +is such a line (as Dr. Wallis continually does) demonstrates nothing. +But if a line be divided into what number of equal parts soever, so the +line have breadth allowed it (as all lines must, if they be drawn), and +the length be to the breadth as number to an unit; the side and the roof +will be all of one length. + + PROP. II. + +[Illustration] + +Any number given is produced by the greatest root multiplied into +itself, and into the remaining fraction. Let the number given be two +hundred squares, the greatest root is 14(4)/(14) squares. I say that 200 +is equal to the product of 14 into itself, together with 14 multiplied +into (4)/(14). For 14 multiplied into itself makes 196. And 14 into +(4)/(14) makes (56)/(14) which is equal to 4. And 4 added to 196 maketh +200; as was to be proved. Or take any other number 8, the greatest root +is 2; which multiplied into itself is 4, and the remainder (2)/(4) +multiplied into 2, is 4, and both together 8. + + PROP. III. + +But the same square calculated geometrically by the like parts, +consisteth (by Euclid II. 4) of the same numeral great square 196, and +of the two rectangles under the greatest side 14, and the remainder of +the side, or (which is all one) of one rectangle under the greatest +side, and double the remainder of the side; and further of the square of +the less segment; which altogether make 200, and moreover (1)/(49) of +those 200 squares, as by the operation itself appeareth thus: + + The side of the greater segment is 14(4)/(14) + 14(4)/(14) + Which multiplied into itself makes 200. + +The product of 14, the greatest segment, into the two fractions +(4)/(14), that is, into (4)/(14) (or into twice (2)/(14)) is (56)/(14) +(that is 4); and that 4 added to 196 makes 200. + +Lastly, the product of (2)/(14) into (2)/(14) or (1)/(7) into (1)/(7) is +(1)/(49). And so the same square calculated by roots is less by (1)/(49) +of one of those two hundred squares, than by the true and geometrical +calculation; as was to be demonstrated. + + CONSEQUENT. + +It is hence manifest, that whosoever calculates the length of an arc or +other line by the extraction of roots, must necessarily make it shorter +than the truth, unless the square have a true root. + + ------- + +_The Radius of a Circle is a Mean Proportion between the Arc of a + Quadrant and two-fifths of the same._ + +Describe a square A B C D, and in it a quadrant D C A. In the side D C +take D T two-fifths of D C, and between D C and D T a mean proportional +D R, and describe the quadrantal arcs R S, T V. I say the arc R S is +equal to the straight line D C. For seeing the proportion of D C to D T +is duplicate of the proportion of D C to D R, it will be also duplicate +of the proportion of the arc C A to the arc R S, and likewise duplicate +of the proportion of the arc R S to the arc T V. + +Suppose some other arc, less or greater than the arc R S, to be equal to +D C, as for example _r s_: then the proportion of the arc _r s_ to the +straight line D T will be duplicate of the proportion of R S to T V, or +D R to D T. Which is absurd; because D _r_ is by construction greater or +less than D R. Therefore the arc R S is equal to the side D C, which was +to be demonstrated. + + COROL. + +[Illustration] + +Hence it follows that D R is equal to two-fifths of the arc C A. For R +S, T V, D T, being continually proportional, and the arc T V being +described by D T, the arc R S will be described by a straight line equal +to T V. But R S is described by the straight line D R. Therefore D R is +equal to T V, that is to two-fifths of C A. + +And your said servant most humbly prayeth you to consider, if the +demonstration be true and evident, whether the way of objecting against +it by square root, used by Dr. Wallis; and whether all his geometry, as +being built upon it, and upon his supposition of an infinite number, be +not false. + + + + + CONSIDERATIONS + + UPON THE ANSWER OF DOCTOR WALLIS + + TO THE + + THREE PAPERS OF MR. HOBBES. + + +Dr. Wallis says, all that is affirmed, is but _if we_ SUPPOSE _that, +this will follow_. + +But it seemeth to me, that if the supposition be impossible, then that +which follows will either be false, or at least undemonstrated. + +First, this proposition being founded upon his _Arithmetica +Infinitorum_, if there he affirm an absolute infiniteness, he must here +also be understood to affirm the same. But in his thirty-ninth +proposition he saith thus: “_Seeing that the number of terms increasing, +the excess above sub-quadruple is perpetually diminished, so at last it +becomes less than any proportion that can be assigned; if it proceed in +infinitum it must utterly vanish. And therefore if there be propounded +an infinite row of quantities in triplicate proportion of quantities +arithmetically proportioned (that is, according to the row of cubical +numbers) beginning from a point or 0; that row shall be to a row of as +many, equal to the greater, as 1 to 4._” It is therefore manifest that +he affirms, that in an infinite row of quantities the last is given; and +he knows well enough that this is but a shift. + +Secondly, he says, that usually in Euclid, and all after him, by +_infinite_ is meant but, more than any assignable _finite_, or the +greatest possible. I am content it be so interpreted. But then from +thence he must demonstrate those his conclusions, which he hath not yet +done. And when he shall have done it, not only the conclusions, but also +the demonstration, will be the same with mine in Cap. XIV. Art. 2, 3, +&c. of my book _De Corpore_. And so he steals what he once condemned. A +fine quality. + +Thirdly, he says, (by Euclid’s tenth proposition, but he tells not of +what book), that a line may be bisected, and the halves of it may again +be bisected, and so onwards infinitely; and that upon such supposed +section infinitely continued, the parts must be supposed infinitely +many. + +I deny that; for Euclid, if he says a line may be divisible into parts +perpetually divisible, he means that all the divisions, and all the +parts arising from those divisions, are perpetually finite in number. + +Fourthly, he says, that there may be supposed a row of quantities +infinitely many, and continually increasing, whereof the last is given. + +It is true, a man may say, (if that be supposing) that white is black: +but, if _supposing_ be _thinking_, he cannot suppose an infinite row of +quantities whereof the last is given. And if he say it, he can +demonstrate nothing from it. + +Fifthly, he says (for one absurdity begets another) _that a superficies +or solid may be supposed so constituted as to be_ infinitely long, _but_ +finitely great, _(the breadth continually decreasing in greater +proportion than the length increaseth), and so as to have no centre of +gravity. Such is Toricellio’s Solidum Hyperbolicum acutum, and others +innumerable, discovered by Dr. Wallis, Monsieur Fermat, and others. But, +to determine this, requires more of geometry and logic, (whatsoever it +do of the Latin tongue), than Mr. Hobbes is master of._ + +I do not remember this of Toricellio, and I doubt Dr. Wallis does him +wrong and Monsieur Fermat too. For, to understand this for sense, it is +not required that a man should be a geometrician or a logician, but that +he should be mad. + +In the next place, he puts to me a question as absurd as his answers are +to mine. Let him ask himself, saith he, if he be still of opinion, _that +there is no argument in natural philosophy to prove that the world had a +beginning_. First, whether, in case it had no beginning, there must not +have passed an infinite number of years before Mr. Hobbes was born. +Secondly, whether, at this time, there have not passed more, that is, +more than that infinite number. Thirdly, whether, in that infinite (or +more than infinite) number of years, there have not been a greater +number of days and hours, and of which, hitherto, the last is given. +Fourthly, whether, if this be an absurdity, we have not then, (contrary +to what Mr. Hobbes would persuade us), an argument in nature to prove +the world had a beginning. + +To this I answer, not willingly, but in service to the truth, that, by +the same argument, he might as well prove that God had a beginning. +Thus, in case he had not, there must have passed an infinite length of +time before Mr. Hobbes was born; but there hath passed at this day more +than that infinite length, by eighty-four years. And this day, which is +the last, is given. If this be an absurdity, have we not then an +argument in nature to prove that God had a beginning? Thus it is when +men entangle themselves in a dispute of that which they cannot +comprehend. But, perhaps, he looks for a solution of his argument to +prove that there is somewhat greater than infinite; which I shall do so +far as to show it is not concluding. If from this day backwards to +eternity be more than infinite, and from Mr. Hobbes his birth backwards +to the same eternity be infinite, then take away from this day backwards +to the time of Adam, which is more than from this day to Mr. Hobbes his +birth, then that which remains backwards must be less than infinite. All +this arguing of infinites is but the ambition of school-boys. + + TO THE LATTER PART OF THE FIRST PAPER. + +There is no doubt if we give what proportion we will of the radius to +the arc, but that the arc upon that arc will have the same proportion. +But that is nothing to my demonstration. He knows it, and wrongs the +Royal Society in presuming they cannot find the impertinence of it. + +My proof is this: that if the arc on T V, and the arc R S, and the +straight line C D, be not equal, then the arc on T V, the arc on R S, +and the arc on C A, cannot be proportional; which is manifest by +supposing in D C a less than the said D C, but equal to R S, and another +straight line, less than R S, equal to the arc on T V; and anybody may +examine it by himself. + +I have been asked by some that think themselves logicians, why I +proceeded upon ⅖ rather than any other part of the radius. The reason I +had for it was, that, long ago, some Arabians had determined, that a +straight line, whose square is equal to 10 squares of half the radius, +is equal to a quarter of the perimeter; but their demonstrations are +lost. From that equality it follows, that the third proportional to the +quadrant and radius, must be a mean proportional between the radius and +⅖ of the same. But, my answer to the logicians was, that, though I took +any part of the radius to proceed on, and lighted on the truth by +chance, the truth itself would appear by the absurdity arising from the +denial of it. And this is it that Aristotle means, where he +distinguishes between a direct demonstration and a demonstration leading +to an absurdity. Hence it appears that Dr. Wallis’s objections to my +_Rosetum_ are invalid as built upon roots. + + TO THE SECOND PAPER. + +First, he says that it concerns him no more than other men, which is +true. I meant it against the whole herd of them who apply their algebra +to geometry. Secondly, he says that a bare number cannot be the side of +a square figure. I would know what he means by a bare number. Ten lines +may be the side of a square figure. Is there any number so bare, as by +it we are not to conceive or consider anything numbered? Or, by 10 +nothings understands he bare 10? He struggles in vain, his conscience +puzzles him. Thirdly, he says 10 squares is the root of 100 square +squares. To which I answer, first, that there is no such figure as a +square square. Secondly, that it follows hence, that a root is a +superficies, for such is 10 squares. Lastly, he says that, neither the +number 10, nor 10 soldiers, is the root of 100 soldiers; because 100 +soldiers is not the product of 10 soldiers into 10 soldiers. This last I +grant, because nothing but numbers can be multiplied into one another. A +soldier cannot be multiplied by a soldier. But no more can a square +figure by a square figure, though a square number may. Again, if a +captain will place his 100 men in a square form, must he not take the +root of 100 to make a rank or file? And are not those 10 men? + + TO THE THIRD PAPER. + +He objects nothing here, but that _the side of a square is not a +superficies, but a line_, and that a _square root (speaking of quantity) +is not a line, but a rectangle_, is a contradiction. The reader is to +judge of that. + +To his scoffings I say no more, but that they may be retorted in the +same words, and are therefore childish. + +And now I submit the whole to the Royal Society, with confidence that +they will never engage themselves in the maintenance of these +unintelligible doctrines of Dr. Wallis, that tend to the suppression of +the sciences which they endeavour to advance. + + + + + LETTERS + AND OTHER PIECES. + + + + + LETTERS AND OTHER PIECES. + + ========== + + + I. + A LETTER FROM MR. HOBBS TO MY MR.[2] + + HONORABLE SIR, + +Though I may goe whither and when I will for anie necessity you have of +my service, yet there is a necessity of good manners that obliges me as +yo^r servant to lett you knowe att all times where to find me. Wee goe +out of Paris 3 weekes hence, or sooner, towards Venice, but by what way +I knowe not, because the ordinary high way through the territory of +Milan is encumbered with the warre betweene the French and the +Spaniards. Howsoever, wee have to be there in October next. If you +require anie service that I can doe there, it may please you to convey +your command by Devonshire house. But if you command me nothing, I have +forbidden my letters to look for answer: their busines being only to +informe and to lett you knowe that the image of your noblenes decayes +not in my memory, but abides fresh to keepe me eternally + + Your + THO. HOBBS. + +----- + +Footnote 2: + + This letter is to be found in the British Museum, amongst the + Lansdowne MSS. 238, entitled “a collection of letters to and from + persons of eminence in the reigns of Elizabeth, James I, and Charles + I, made by some person in the service of Sir Gervas Clifton”. It is + without date: but the allusion to the war between France and Spain, + and the passage in the VITA THO. HOBBES, “Anno sequente qui erat + Christi 1629, rogatus a nobilissimo viro domino Gervasio Clifton”, &c. + (p. xiv), show that it must have been written in either 1629 or 1630. + +----- + + + II. + TO A FRIEND IN ENGLAND. + + WORTHY SIR, + +I have been behind hand with you a long time for a letter I received of +yours at Angers, that place affording nothing wherewith to pay a debt of +that kind, all matter of news being sooner known in England than here: +and the news you writ me was of that kind, that none from England could +be more welcome, because it concerned the honour of Welbeck and Clifton, +two houses in which I am very much obliged. + +Monsieur having given the slip to the Spaniards at Bruxelles, came to +the King about ten days ago at St. Germains, where he was received with +great joy. The next day the Cardinal entertained him at Ruelle: and the +day after that he went to Limours, where he is now, and from thence he +goes away shortly to Bloys, to stay there this winter. The Cardinal of +Lyons is going to Rome to treat about the annulling of Monsieur’s +marriage, which is here by Parliament declared void, but yet they +require the sentence of the Pope. There goes somebody thither on the +part of his wife, to get the marriage approved: but who that is, I ’know +not. The Swedish party in Germany is in low estate, but the French +prepare a great army for those parts, pretending to defend the places +which the Swedes have put into the King of France his protection, +whereof Philipsbourgh is one; a place of importance for the Lower +Palatinate. This is all the French news. + +For your question, _why a man remembers less his own face, which he sees +often in a glass, than the face of a friend that he has not seen of a +great time_, my opinion in general is, that a man remembers best those +faces whereof he has had the greatest impressions, and that the +impressions are the greater for the oftener seeing them, and the longer +staying upon the sight of them. Now you know men look upon their own +faces but for short fits, but upon their friends’ faces long time +together, whilst they discourse or converse together; so that a man may +receive a greater impression from his friend’s face in a day, than from +his own in a year; and according to this impression, the image will be +fresher in his mind. Besides, the sight of one’s friend’s face two hours +together, is of greater force to imprint the image of it, than the same +quantity of time by intermissions. For the intermissions do easily +deface that which is but lightly imprinted. In general, I think that +lasteth longer in the memory which hath been stronglier received by the +sense. + +This is my opinion of the question you propounded in your letter. Other +new truths I have none, at least they appear not new to me. Therefore if +this resolution of your first question seems probable, you may propound +another, wherein I will endeavour to satisfy you, as also in any thing +of any other nature you shall command me, to my utmost power; taking it +for an honour to be esteemed by you, as I am in effect, + + Your humble and faithful servant + THO. HOBBES. + +_Paris, Oct. 21/31, 1634._ + +My Lord Fielding and his Lady came to Paris on Saturday night last. + + + III. + TO MY WORTHY FRIEND MR. GLEN.[3] + + WORTHY SIR, + +I received here in Florence, two days since, a letter from you of the +19th of January. It was long by the way; but when it came it did +thoroughly recompence that delay. For it was worth all the pacquets I +had received a great while together. All that passeth in these parts is +equally news, and therefore no news; else I would labour to requite your +letter in that point, though in the handsome setting down of it, I +should still be your inferior. + +I long infinitely to see those books of the Sabbaoth[4], and am of your +mind they will put such thoughts into the heads of vulgar people, as +will confer little to their good life. For when they see one of the ten +commandments to be _jus humanum_ merely, (as it must be if the Church +can alter it), they will hope also that the other nine may be so too. +For every man hitherto did believe that the ten commandments were the +moral, that is, the eternal law. + +I desire also to see Selden’s _Mare Clausum_, having already a great +opinion of it. + +You may perhaps, by some that go to Paris, send me those of the +Sabbaoth, for the other being in Latin, I doubt not to find it in the +Rue St. Jaques. + +We are now come hither from Rome, and hope to be in Paris by the end of +June. I thank you for your letter, and desire you to believe that I can +never grow strange to one, the goodness of whose acquaintance I have +found by so much experience. But I have to write to so many, that I +write to you seldomer than I desire; which I pray pardon, and esteem me + + Your most affectionate friend + and humble servant + THO. HOBBES. + +_Florence, Apr. (6)/(16) 1636._ + +My Lord and Mr. Nicholls, and all our company commend them to you. + +----- + +Footnote 3: + + Probably George Glen, who was installed Prebend of Worcester in 1660, + and died in 1669. + +Footnote 4: + + The History of the Sabbath. In two books. By Peter Heylyn. 4to. 1636. + +----- + + + IV. + LETTER TO SIR CHARLES CAVENDISH.[5] + + HONORABLE SIR, + +[Illustration] + +The last weeke I had the honor to receave two letters from you at once, +one of the 30 of Dec., the other of the 7^{th} of Jan., w^{ch} I +acknowledged, but could not answer in my last. In the first you begin +with a difficulty on the principle of Mons^r de Cartes, _that it is all +one to move a weight two spaces, or the double of that waight one +space_, and so on in other proportions: to w^{ch} you object the +difference of swiftnesse, w^{ch} is greater when a waight is moved two +spaces than when double waight is moved one space. Certenly de Cartes +his meaning was by force the same that mine, namely, a multiplication of +the weight of a body in to the swiftnesse wherew^{th} it is moved. So +that when I move a pound two foote at the rate of a mile an howre, I do +the same as if of 2 poundes I moved one pound a foote at y^e rate of a +mile an howre, the other pound another foote at the same rate, not in +directū, but parallell to the first pound. As if the wayt A B were moved +to C D at the rate of a mile an howre, ’tis all one as if the waight A E +were moved to F H at the same rate. Here is all the difference: this +swiftnesse or rate of a mile an howre is, in the first case, layd out in +the 2 spaces A G, G C, the latter, in the 2 spaces A G, E G. The first +case, as like as if a footman should run w^{th} double swiftnesse +endwayes, w^{ch} is y^e doubling of swiftnesse in one man: in the other, +it is as if you doubled the swiftnesse by doubling the man: for every +man has his owne swiftnesse. And so A H is the swiftnesse A G doubled, +as well as A D. For that, that Mons^r de Cartes will not have just twice +the force requisite to move the same weight twice as fast, I can say +nothing. The papers I have of his touching that are in my trunk, w^{ch} +hath bene taken by Dunkerkers, and taken againe from them by French, and +at length recovered by frends I made: but I shall not have it yet this +fortnight. In the meane time I am not in that opinion, but do assure +myself, the patient being the same, double force in the agent shall +worke upon it double effect. + +In the same letter you require a better explication of y^e proportion I +gather betweene wayght and swiftnesse: wherein, because you have not my +figure, I imagine you have mistaken me very much. And first, you thinke, +I suppose, D E equall to A B: w^{ch} I am sure is a mistake. For I put A +B for any line you will to expresse a _minutū secundum_. I will, +therefore, go over againe the demonstration I sent you before, and see +if I can do it cleerer. + +[Illustration] + +Let A B stand for the time knowne wherein the waight D descendeth to E. +And let there bee a cylinder of the same matter the waight D consisteth +of, and let the altitude of that cylinder be D C: w^{ch} I shew before +was the swiftnesse wherew^{th} that cylinder _presseth_, not wherew^{th} +it _falleth_. And wee are now to enquire how farre such matter as the +cylinder is made of must descend from D, before it attayne a swiftnesse +equall to this pressing swiftnesse D C. And I say it must fall to L. For +in the time A B it is knowne that the waight in D will fall to E: and it +is demonstrated by Gallileo, that when such waight comes to E, it shall +be able to go twice the space it hath fallen in the same time. Therefore +the waight D being in E, hath velocity to carry him the space D K +(w^{ch} I put double to D E) in the same time A B. But I put B F equall +to D K. Therefore, in the time A B, the waight’s velocity acquired in E +shall be such as to go from B to F without decrease of velocity by the +way. Hence I go on to finde in what point the waight in D comes to where +it getteth a velocity equall to C D. Therefore, I apply D C to G H, +parallel to B F: and then it is, as the time A B to the time A G, so the +velocity acquired at the end of the time A B to the velocity acquired at +the end of the time A G. For the swiftnesse acquired from time to time +(I say, not from place to place, but from time to time) are +proportionable to the times wherein they are acquired: w^{ch} is the +postulate on w^{ch} Galileo builds all his doctrine. And as A B to A G, +so the line B F to the line G H. But, at the end of the time A B, the +waight D is by supposition in E, in that degree of velocity as to go B F +or D K in the same time A B. The question therefore is, where the waight +D shall be at the end of the time A G. For there it hath the velocity of +going G H or D C in the same time, because the velocity G H is to the +velocity B F as the line G H to the line B F, or as the time of descent +A G to the time A B. But, because the spaces of the descent are in +double the proportion of the times of descent, make it as B F to G H, +that is, D K to D C, so D C to another, D L. The velocity, therefore, +acquired in the point of descent E, namely the velocity D K or B F, is +to the velocity acquired in the point L, namely, the velocity G H or D +C, (w^{ch} is the velocity of the cylinder’s waight), as D K to D L. And +therefore in L the waight D has acquired a velocity equal to the +velocity of the waight of the cylinder. + +In the same letter you desire to knowe, how any mediū, as water, +retardeth the motion of a stone that falls into it. To w^{ch} I answer +out of that you say afterwards, that nothing can hinder motion but +contrary motion: that the motion of the water, when a stone falls into +it, is point blanke contrary to the motion of the stone. For the stone +by descent causeth so much water to ascend as the bignesse of the stone +comes to. For imagine so much water taken out of the place w^{ch} the +stone occupies, and layd upon the superficies of the water: it presseth +downeward as the stone does, and maketh the water that is below to rise +upwards, and this rising upwards is contrary to the descent, and is no +other operation than we see in scales, when of two equal bullets in +magnitude that w^{ch} is of heavier metal maketh the other to rise. And +thus farre goes your letter of Dec. 30. + +For the first quære in your second letter, concerning how we see in the +time the lucide body contracts itselfe, I have no other solution but +that w^{ch} your selfe hath given: w^{ch} is, that the reciprocation is +so quicke, that the effect of the first motion lasteth till the next +comes, and longer. For by experience we observe that the end of a +firebrand swiftely moved about in circle, maketh a circle of fire: +w^{ch} could not be, if the impression made at the beginning of the +circulation did not last till the end of it. For if the same firebrand +be moved slower, there will appear but a peece of a circle, bigger or +lesser according to the swiftnesse or slownesse of y^e motion. For the +cause of such reciprocation, it is hard to guess what it is. It may well +be the reaction of the medium. For though the mediū yeld, yet it +resisteth to: for there can be no passion w^{th}out reaction. And if a +man could make an hypothesis to salve that contraction of y^e sun, yet +such is the nature of naturall thinges, as a cause may be againe +demanded of such hypothesis: and never should one come to an end +w^{th}out assigning the immediate hand of God. Whereas in mathematicall +sciences wee come at last to a definition, w^{ch} is a beginning or +principle, made true by pact and consent amongst ourselves. Further, you +conceave a difficulty how the medium can be continually driven on, if +there be such an alternate contraction. To w^{ch}, first I answer, that +the motion forward is propagated to the utmost distance in an instant, +and the first push is therefore enough, and in another instant is made +the returne back in y^e like manner. And though it were not done in an +instant, yet we see by experience in rivers, as in y^e Thames, that the +tide goes upward towards London pushed by the water below, and yet at +the same instant the water below is going backe to the sea. For seeing +it is high-water at Blackwall before ’tis so at Greenwich, the water +goes backe from Blackwall when it goes on at Greenwich. And so it would +happen, though Blackwall and Greenwich were nerer together then that any +quantity given could come betweene. + +[Illustration] + +In my letter from London, speaking of the refraction of a bullet, I +thinke I delivered my opinion to be, that a bullet falling out of a +thinner medium into a thicker, looseth in the entring nothing but motion +perpendicular: but being entred, he looseth proportionably both of one +and the other. For suppose a bullet, whose diameter is A B, be in the +thiner mediū, and enter at C into the thicker medium. The thicker +medium, at the first touch of B in the point C, worketh nothing upon the +line A B. And when the diameter A B is entred, suppose halfe way, yet +the thicker medium operates laterally but on one halfe of it. So that in +the somme there is a losse of velocity perpendicular (to the quantity +that the diameter A B requires) without any offence to the motion +laterall, but so much of the diameter as is within the thicker mediū is +retarded both wayes, and looses of his absolute motion, w^{ch} is +compounded of perpendicular and laterall, and that proportionally. +Suppose now that a bullet passe from A to D, and receave a peculiar +losse of his perpendicular motion by entring at D, so great that he +proceed in the perpendicular but halfe so farre, as for example from D +to I: and then being in, the thicknesse of the medium take away more of +his velocity both perpendicular and laterall, suppose halfe that w^{ch} +was left of the perpendicular motion and halfe of his first laterall +motion, so that the perpendicular motion is but D K, and the laterall +motion D E. Then will the line of refraction be D G. As for that +argumentation of Des Cartes, it is, in my opinion, as I have heretofore +endeavored to shew you, a mere paralogism. + +Lastly, you make this quære, why light hath not at severall inclinations +severall swiftnesses as well as a bullet. The bullet itself passeth +through the severall media: whereas in the motion of light, the body +moved, w^{ch} is the mediū, entreth not into the other medium, but +thrusteth it on: and so the parts of that medium thrust on one another, +whereby the laterall motion of the thicker medium hath nothing to worke +upon, because nothing enters, but stoppes onely and retardes, in oblique +_incidence_, that end w^{ch} comes first to it, and thereby causes a +refraction the contrary way to that of a bullet, in such manner as I set +forth to you in one of my letters from hence concerning the cause of +refraction. And this is all I can say for the present to the quæres of +y^r two last letters. + +I have enquired concerning perspectives after the manner of De Cartes. +Mydorgius tells me there is none that goes about them, as a thing too +hard to do. And I believe it. For here is one Mons^r de Bosne in towne, +that dwells at Bloys, an excellent workman, but by profession a lawier, +and is counsellor of Bloys, and a better philosopher in my opinion then +De Cartes, and not inferior to him in the analytiques. I have his +acquaintance by Pere Mersenne. He tells me he hath tryed De Cartes his +way, but cannot do it: and now he workes upon a crooked line of his owne +invention. He sayes he shall have made one w^{th}in a moneth after he +shall returne to Bloys: after that he will see what he can discover in +the heavens himselfe, and then if he discover any new thing he will let +his way be publique together w^{th} the effects. This is all the hope I +can give you yet. So w^{th} my prayers to God to keepe you in prosperity +this troublesome time, I rest + + Your most humble and obedient servant + TH. HOBBES. + +_Paris, Feb. 8, stile no. 1641._ + + To the Right Honorable + Sr CHARLES CAVENDYSSHE + present these + + / + / + / + / + at Wellinger. + +----- + +Footnote 5: + + Harleian MS. 6796. + +----- + + + V. + LETTER TO MR. BEALE.[6] + + SIR, + +The young woman at Over-Haddon hath been visited by divers persons of +this house. My Lord himself hunting the hare one day at the Town’s end, +with other gentlemen and some of his servants, went to see her on +purpose: and they all agree with the relation you say was made to +yourself. They further say on their own knowledge, that part of her +Belly touches her Back-bone. She began (as her Mother says) to loose her +appetite in December last, and had lost it quite in March following: +insomuch as that since that time she has not eaten nor drunk any thing +at all, but only wetts her lips with a feather dipt in water. They were +told also that her gutts (she alwayes keeps her bed) lye out by her at +her fundament shrunken. Some of the neighbouring ministers visit her +often: others that see her for curiosity give her mony, sixpence or a +shilling, which she refuseth, and her mother taketh. But it does not +appear they gain by it so much as to breed a suspition of a cheat. The +woman is manifestly sick, and ’tis thought she cannot last much longer. +Her talk (as the gentlewoman that went from this house told me) is most +heavenly. To know the certainty, there bee many things necessary which +cannot honestly be pryed into by a man. First, whether her gutts (as +’tis said) lye out. Secondly, whether any excrement pass that way, or +none at all. For if it pass, though in small quantity, yet it argues +food proportionable, which may, being little, bee given her secretly and +pass through the shrunken intestine, which may easily be kept clean. +Thirdly, whether no urine at all pass: for liquors also nourish as they +go. I think it were somewhat inhumane to examin these things too nearly, +when it so little concerneth the commonwealth: nor do I know of any law +that authoriseth a Justice of peace, or other subject, to restrain the +liberty of a sick person so farr as were needful for a discovery of this +nature. I cannot therefore deliver any judgment in the case. The +examining whether such a thing as this bee a miracle, belongs I think to +the Church. Besides, I myself in a sickness have been without all manner +of sustenance for more than six weeks together: which is enough to make +mee think that six months would not have made it a miracle. Nor do I +much wonder that a young woman of clear memory, hourely expecting death, +should bee more devout then at other times. ’Twas my own case. That +which I wonder at most, is how her piety without instruction should bee +so eloquent as ’tis reported. + + THO. HOBBES. + +_Chatsworth, Oct. 20. 68._ + +----- + +Footnote 6: + + Amongst the MSS. of the Royal Society. + +----- + + + VI. + LETTER TO MR. OLDENBURG.[7] + + + WORTHY S^R + +In the last Transactions for September and October I find a letter +addressed to you from D^r Wallis, in answer to my LUX MATHEMATICA. I +pray you tell me that are my old acquaintance, whether it be (his words +and characters supposed to be interpreted) intelligible. I know very +well you understand sense both in Latine, Greeke, and many other +languages. He shows you no ill consequence in any of my arguments. +Whereas I say there is no proportion of _infinite_ to _finite_. He +answers, he meant _indefinite_; but derives not his conclusion from any +other notion than simply _infinite_. I said the root of a square number +cannot be the length of the side of a square figure, because a root is +part of a square number, but length is no part of a square figure. To +which he answers nothing. In like manner, he shuffles off all my other +objections, though he know well enough that whatsoever he has written in +Geometry (except what he has taken from me and others) dependeth on the +truth of my objections. I perceive by many of his former writings that I +have reformed him somewhat as to the Principles of Geometry, though he +thanke me not. He shuffles and struggles in vaine, he has the hooke in +his guills, I will give him line enough: for (which I pray you tell him) +I will no more teach him by replying to any thing he shall hereafter +write, whatsoever they shall say that are confident of his Geometry. +_Qui volunt decipi, decipiuntur._ He tells you that I bring but _crambe +sæpe cocta_. For which I have a just excuse, and all men do the same; +they repeat the same words often when they talk with them that cannot +heare. + +I desire also this reasonable favour from you: that, if hereafter I +shall send you any paper tending to the advancement of physiques or +mathematiques, and not too long, you will cause it to be printed by him +that is printer to the Society, as you have done often for D^r Wallis: +it will save me some charges. + + I am, S^r, + + Your affectionate frend and humble seruant + + THOMAS HOBBES. + +_November the 26th, 1672._ + + ffor my worthy and much honoured + frend M^r HENRY OLDENBURGH, + Secretary to the Royal Society. + +----- + +Footnote 7: + + Amongst the MSS. of the Royal Society. + +----- + + + VII. + TO THE RIGHT HONOURABLE + THE MARQUIS OF NEWCASTLE.[8] + +The passions of man’s mind, except onely one, may bee observed all in +other living creatures. They have desires of all sorts, love, hatred, +feare, hope, anger, pitie, æmulation, and y^e like: onely of curiositie, +which is y^e desire to know y^e causes of thinges, I never saw signe in +any other living creature but in man. And where it is in man, I find +alwaies a defalcation or abatement for it of another passion, which in +beastes is commonly predominant, namely, a ravenous qualitie, which in +man is called _avarice_. The desire of knowledge and desire of needlesse +riches are incompatible, and destructive one of another. And therefore +as in the cognitive faculties reason, so in the motive curiositie, are +the markes that part y^e bounds of man’s nature from that of beastes. +Which makes mee, when I heare a man, upon the discovery of any new and +ingenious knowledge or invention, aske gravely, that is to say, +scornefully, _what ’tis good for_, meaning what monie it will bring in, +(when he knows as little, to one that hath sufficient what that overplus +of monie is good for), to esteeme that man not sufficiently removed bn +484.png from brutalitie. Which I thought fit to say by way of +anticipation to y^e grave scorners of philosophie, and that your +lordship, after having performed so noble and honourable acts for +defence of your countrie, may thinke it no dishonour in this unfortunate +leasure to have employed some thoughts in the speculation of the noblest +of the senses, _vision_. + +That which I have written of it is grounded especially upon that w^{ch} +about 16 yeares since I affirmed to your Lo^{PP} at Welbeck, that light +is a fancy in the minde, caused by motion in the braine, which motion +againe is caused by the motion of y^e parts of such bodies as we call +_lucid_: such as are the sunne and y^e fixed stars, and such as here on +earth is fire. By putting you in mind hereof, I doe indeed call you to +witnesse of it: because, the same doctrine having since been published +by another, I might bee challenged for building on another man’s ground. +Yett philosophical ground I take to be of such a nature, that any man +may build upon it that will, especially if the owner himselfe will nott. +But upon this ground, with the helpe of some other speculations drawne +from the nature of motion and action, I have, I thinke, derived y^e +reason of all the phænomena I have mett with concerning light and +vision, both solidly enough nott to be confuted, and withall easilie +enough to be understood by such as can give that attention thereto which +the figures, whereby such motion as causeth vision is described, do +require. All that I shall bee ever able to adde to it, is polishing: +for, being the first draught, it could nott bee so perfect as I hope +hereafter to make it in Latin. Butt as it is, it will sufficiently give +your Lo^{PP} satisfaction in those _quæres_ you were pleased to make +concerning this subject. I am content that it passe, in respect of some +drosse that yett cleaves to it, for ore: w^{ch} is much better than old +ends raked out of the kennell of sophisters’ bookes. And for such I +commend it to your Lo^p, and myselfe to your accustomed good opinion: +which hath beene hitherto so greate honour to mee, as I am nott known to +the world by any thing so much as by being, + + My most noble lord, + + Your Lo^{p’s} most humble + + and most obliged servant + + THO. HOBBES. + + +The treatise ends with the following passage:-- + + To conclude, I shall doe like those that build a new house where an + old one stood before, that is to say, carry away the rubbish. + + And first, away goes the old opinion that the _shewes_ (which they + call visible species) of all objects, are in all places, and all the + babble _de extramittendo et intromittendo_. For their species are + nothing else but fancie, made by the light proceeding directly or by + repurcussion or refraction made from the object to y^e eye, and so + moving the braine and other parts within. + + Secondly, the opinion which Vitellio takes for an axiome and + foundation of his _Catoptricques_, that y^e place of y^e image by + reflexion is in the perpendicular drawn from the object to the glasse. + For it is false both in plaine glasses and in sphæricall, whether + convexe or concave. + + Thirdly, the opinion that light is engendred faster in hard bodies, as + glasse, than in thin and fluid, as aire. + + Fourthly, that objects are seen by _penicilli_ that have their common + base in the pupills: for y^e center of y^e eye is in their common + base. + + Fifthly, the opinion that there bee other visuali lines by which wee + see distinctly besides y^e optique axis. + + Sixthly, the opinion that perspective glasses and amplifying glasses + are best made of hyperbolicall figures. + + Seventhly, the opinion that light is a bodie, or any other such thing + than such light as wee have in dreames. + + Eighthly, that y^e object appeares greater and lesse in ye same + proportion that y^e angles have under which they are seene. + + Lastly, is to be cast away the conceipt of millions of strings in ye + optique nerve, by which the object playes upon the braine, and makes + y^e soule listen unto it, and other innumerable such trash. + + How doe I feare that y^e attentive reader will find that which I have + delivered concerning y^e _Optiques_ fitt to bee cast outt as rubbish + among the rest. If hee doe, hee will recede from y^e authoritie of + experience, which confirmeth all I have said. Butt if it bee found + true doctrine, (though yett it wanteth polishing), I shall deserve the + reputation of having beene y^e first to lay the grounds of two + sciences; this of _Optiques_, y^e most curious, and y^t other of + _Natural Justice_, which I have done in my booke DE CIVE, y^e most + profitable of all other. + +----- + +Footnote 8: + + Harleian MS. 3360: a treatise on Optics, entitled “A minute or first + draught of the Optiques. In two parts. By Thomas Hobbes. At Paris, + 1646.” The second part, _On Vision_, we have in Latin, in the DE + HOMINE: the first, _On Illumination_, was never published. The + dedication to the Marquis of Newcastle, and the concluding paragraph, + is all that is here given of the treatise. + +----- + + + VIII. + TO THE KING’S MOST EXCELLENT MAJESTY + THE HUMBLE PETITION OF THOMAS HOBBES[9] + +Sheweth, that though your Majesty hath been pleased to take off the +restraint of late years laid upon the pensions payable out of your privy +purse, yet your Majesty’s Officers refuse to pay the pension of your +petitioner without your Majesty’s express command. + +And humbly beseaceth your Majesty, (considering his extreme age, +perpetual infirmity, frequent and long sickness, and the aptness of his +enemies to take any occasion to report that your petitioner by some ill +behaviour hath forfeited your wonted favour), that you would be pleased +to renew your order for the payment of it in such manner as to his great +comfort he hath for many years enjoyed it.[10] + +And daily prayeth to God Almighty to bless your Majesty with long life, +constant health, and happinesse. + +----- + +Footnote 9: + + Additional MSS. 4292. Brit. Mus. + +Footnote 10: + + Deinde redux mihi Rex concessit habere quotannis + Centum alias libras ipsius ex loculis: + Dulce mihi donum. + VITA _Carm. expres._ p. xcviii. + +----- + + + + + + + + + END OF VOL. VII. + + + + + + + + + RICHARDS, PRINTER, 100, ST. MARTIN’S LANE. + +------------------------------------------------------------------------ + + Transcriber’s Note + +Some Greek passages employ the stigma ligature (‘st’). The available +Unicode character (ϛ) is nearly indistinguishable from the final form of +sigma (ς). Occasionally, the ‘ου’ ligature is employed. The only +available character (ᴕ) is a Latin, not a Greek character. It is +rendered here as ‘ου’. There were also a number of instances of improper +placement of diacritical marks, particularly in cases where the +breathing mark and accents appear on the first rather than the second of +two leading vowels, e.g. ‘ὅυτως’ rather than ‘οὕτως’. + +At 93.4, the response ‘No sure’ is obviously incorrect. It is most +likely that it should have read ‘Not sure.’, but it may also have been +an unfinished line. + +Other errors deemed most likely to be the printer’s have been corrected, +and are noted here. Given the age of the text, any corrections were made +sparingly. The references are to the page and line in the original. + + 8.35 with increasing [s]wiftness? Restored. + 12.34 being so very heavy[?] Added. + 54.8 too much or too[l i/ li]ttle Replaced. + 62. was to be demonst[r]ated. Inserted. + 65.33 which is 72[.] Added. + 67.1 and multip[l]ying lines Inserted. + 72.5 and subtil doc[t]rines. Inserted. + 74.1 who begi[u/n]s his history Inverted. + 91.16 the space be[t]ween. Inserted. + 91.18 whose tap[-]hole is very little Inserted. + 96.5 and th[a/e]n H will be east Replaced. + 114.27 a spring[ ]upon the top Inserted. + 158.19 the air which[ which] was Redundant. + 163.11 8 degrees 30 minutes[.]? Removed. + 192.19 (called by him [ε/ἐ]φαρμόζοντα) Replaced. + 200.5 may so precisely deter[ter]mine Removed. + 206.1 lines whi[e/c]h shall never meet Replaced. + 208.10 [ὅυ/οὕ]τως ἔχει Replaced. + 208.13 [ὅυ/οὕ]τως ἔχει Replaced. + 225.8 as by raref[r]action and condensation. Removed. + 263.17 [“]_The magnitude of an angle_ Added. + 277.33 th[e/a]n of one in nine? Replaced. + 297.26 At the seventee[n]th chapter Inserted. + 300.22 _that of G K to G [E,/E.]_” Replaced. + 323.7 _tanquam dicta problematicè._[”] Added. + 338.31 and so mis[s]pend it? Inserted. + 351.16 by which you live[,/.] Replaced. + 376.4 _propositione præcedente._[”] Added. + 376.22 it shall quite vanish. [And so] Missing? + 342.12 to the present purpose[)]. Added (likely). + 382.12 whether διπλάσιος and διπλασ[ι]ίων be one Removed. + 391.5 by the word Σημ[~ε/εῖ]ον in Euclid Replaced. + 412.17 κα[ι/ὶ] ἡ Ἔμπουσα Replaced. + 413.26 τῶν θεῶν [Α/Ἀ]ιγύπτιοι Replaced. + 413.27 κα[ἱ/ὶ] ἅτε Ἀιγύπτιος, Replaced. + 413.30 ποδῶν [ὴ/ἠ]δέ κνημάων Replaced. + 414.1 ῥᾳδίως[ ]ἔγνων ὑπολαμβάνοντες. Inserted. + 414.4 ἀλλὰ κατὰ ῥύμην [ὰ/ἀ]έριον Replaced. + 415.23 deriveth it [τ/π]αρὰ τὸ ἐμποδιζειν Replaced. + 415.25 it saith, [Ἕ/Ἔ]μπουσα Ψιλοῦπαι Replaced. + 415.33 some had no legs, &c[.] Added. + 416.7 τύχη [ὲ/ἐ]μποδιζουσα Replaced. + 416.16 [Ἕ/Ἔ]μποῦσα is also a name for Lamia Replaced. + 418.16 [ὁι/οἱ] δὲ ὑπέφευγον Transposed. + 418.20 καὶ τὸ [ὲ/ἐ]πιπηδᾶν Replaced. + 419.13 γαρ [ὅι/οἵ]νου πληροῦντες, Replaced. + 419.14.1 εἶχε τὸν [ὅι/οἵ]νου. Replaced. + 419.14.2 [Α/Ἀ]σκωλιάζειν Replaced. + 423.1 καὶ ζώων παντοδαπῶν [ἐι/εἰ]κόσι Replaced. + 455.27 two spaces th[e/a]n when double Replaced. + 456.3 a pound two foote a[t] the rate of Added. + 456.6 rate of a mile an how[er/re] Transposed. + +*** END OF THE PROJECT GUTENBERG EBOOK 78674 *** |