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diff --git a/old/53161-0.txt b/old/53161-0.txt deleted file mode 100644 index 8689274..0000000 --- a/old/53161-0.txt +++ /dev/null @@ -1,14429 +0,0 @@ -Project Gutenberg's A View of Sir Isaac Newton's Philosophy, by Anonymous - -This eBook is for the use of anyone anywhere in the United States and most -other parts of the world at no cost and with almost no restrictions -whatsoever. You may copy it, give it away or re-use it under the terms of -the Project Gutenberg License included with this eBook or online at -www.gutenberg.org. If you are not located in the United States, you'll have -to check the laws of the country where you are located before using this ebook. - - - -Title: A View of Sir Isaac Newton's Philosophy - -Author: Anonymous - -Release Date: September 28, 2016 [EBook #53161] - -Language: English - -Character set encoding: UTF-8 - -*** START OF THIS PROJECT GUTENBERG EBOOK SIR ISAAC NEWTON'S PHILOSOPHY *** - - - - -Produced by Giovanni Fini, Markus Brenner, Irma Spehar and -the Online Distributed Proofreading Team at -http://www.pgdp.net (This file was produced from images -generously made available by The Internet Archive/Canadian -Libraries) - - - - - - - - - - TRANSCRIBER’S NOTES: - -—Obvious print and punctuation errors were corrected. - -—Bold text has been rendered as =bold text=. - -—Spaced out text (gesperrt) has been rendered as ~spaced text~. - -—Superscript letters have been rendered as a^b and a^{bc}. - - - - - A - ~VIEW~ - ~OF~ - Sir ~_ISAAC NEWTON_~’s - PHILOSOPHY. - -[Illustration] - - ~_LONDON_~: - - Printed by _S. PALMER_, 1728. - -[Illustration] - - To the Noble and Right Honourable - SIR _ROBERT WALPOLE._ - -_SIR,_ - -I Take the liberty to send you this view of Sir ~ISAAC NEWTON’S~ -philosophy, which, if it were performed suitable to the dignity of the -subject, might not be a present unworthy the acceptance of the greatest -person. For his philosophy operations of nature, which for so many -ages had imployed the curiosity of mankind; though no one before him -was furnished with the strength of mind necessary to go any depth in -this difficult search. However, I am encouraged to hope, that this -attempt, imperfect as it is, to give our countrymen in general some -conception of the labours of a person, who shall always be the boast -of this nation, may be received with indulgence by one, under whose -influence these kingdoms enjoy so much happiness. Indeed my admiration -at the surprizing inventions of this great man, carries me to conceive -of him as a person, who not only must raise the glory of the country, -which gave him birth; but that he has even done honour to human nature, -by having extended the greatest and most noble of our faculties, -reason, to subjects, which, till he attempted them, appeared to be -wholly beyond the reach of our limited capacities. And what can give us -a more pleasing prospect of our own condition, than to see so exalted -a proof of the strength of that faculty, whereon the conduct of our -lives, and our happiness depends; our passions and all our motives to -action being in such manner guided by our opinions, that where these -are just, our whole behaviour will be praise-worthy? But why do I -presume to detain you, SIR, with such reflections as these, who must -have the fullest experience within your own mind, of the effects of -right reason? For to what other source can be ascribed that amiable -frankness and unreserved condescension among your friends, or that -masculine perspicuity and strength of argument, whereby you draw the -admiration of the publick, while you are engaged in the most important -of all causes, the liberties of mankind? - - * * * * * - -I humbly crave leave to make the only acknowledgement within my power, -for the benefits, which I receive in common with the rest of my -countrymen from these high talents, by subscribing my self - - ~_SIR_~, - _Your most faithful_, - _and_ - _Most humble Servant_, - - ~HENRY PEMBERTON~. - - - - -~PREFACE~. - - -I _Drew up the following papers many years ago at the desire of some -friends, who, upon my taking care of the late edition of Sir_ ~ISAAC -NEWTON’S~ _Principia, perswaded me to make them publick. I laid hold -of that opportunity, when my thoughts were afresh employed on this -subject, to revise what I had formerly written. And I now send it -abroad not without some hopes of answering these two ends. My first -intention was to convey to such, as are not used to mathematical -reasoning, some idea of the philosophy of a person, who has acquired -an universal reputation, and rendered our nation famous for these -speculations in the learned world. To which purpose I have avoided -using terms of art as much as possible, and taken care to define such -as I was obliged to use. Though this caution was the less necessary at -present, since many of them are become familiar words to our language, -from the great number of books wrote in it upon philosophical subjects, -and the courses of experiments, that have of late years been given by -several ingenious men. The other view I had, was to encourage such -young gentlemen as have a turn for the mathematical sciences, to pursue -those studies the more chearfully, in order to understand in our -author himself the demonstrations of the things I here declare. And to -facilitate their progress herein, I intend to proceed still farther in -the explanation of Sir_ ~ISAAC NEWTON’S~ _philosophy. For as I have -received very much pleasure from perusing his writings, I hope it is -no illaudable ambition to endeavour the rendering them more easily -understood, that greater numbers may enjoy the same satisfaction._ - -_It will perhaps be expected, that I should say something particular -of a person, to whom I must always acknowledge my self to be much -obliged. What I have to declare on this head will be but short; for -it was in the very last years of Sir_ ~ISAAC~_’s life, that I had the -honour of his acquaintance. This happened on the following occasion. -Mr._ Polenus, _a Professor in the University of_ Padua, _from a -new experiment of his, thought the common opinion about the force -of moving bodies was overturned, and the truth of Mr._ Libnitz_’s -notion in that matter fully proved. The contrary of what Polenus had -asserted I demonstrated in a paper, which Dr._ ~MEAD~, _who takes all -opportunities of obliging his friends, was pleased to shew Sir_ ~ISAAC -NEWTON~ _This was so well approved of by him, that he did me the honour -to become a fellow-writer with me, by annexing to what I had written, -a demonstration of his own drawn from another consideration. When I -printed my discourse in the philosophical transactions, I put what Sir_ -~ISAAC~ _had written in a scholium by it self, that I might not seem to -usurp what did not belong to me. But I concealed his name, not being -then sufficiently acquainted with him to ask whether he was willing -I might make use of it or not. In a little time after he engaged me -to take care of the new edition he was about making if his Principia. -This obliged me to be very frequently with him, and as he lived at some -distance from me, a great number of letters passed between us on this -account. When I had the honour of his conversation, I endeavoured to -learn his thoughts upon mathematical subjects, and something historical -concerning his inventions, that I had not been before acquainted -with. I found, he had read fewer of the modern mathematicians, than -one could have expected; but his own prodigious invention readily -supplied him with what he might have an occasion for in the pursuit of -any subject he undertook. I have often heard him censure the handling -geometrical subjects by algebraic calculations; and his book of Algebra -he called by the name of Universal Arithmetic, in opposition to the -injudicious title of Geometry, which_ Des Cartes _had given to the -treatise, wherein he shews, how the geometer may assist his invention -by such kind of computations. He frequently praised_ Slusius, Barrow -_and_ Huygens _for not being influenced by the false taste, which then -began to prevail. He used to commend the laudable attempt of_ Hugo -de Omerique _to restore the ancient analysis, and very much esteemed -Apollonius’s book De sectione rationis for giving us a clearer notion -of that analysis than we had before. Dr._ Barrow _may be esteemed as -having shewn a compass of invention equal, if not superior to any of -the moderns, our author only excepted; but Sir_ ~ISAAC NEWTON~ _has -several times particularly recommended to me_ Huygens_’s stile and -manner. He thought him the most elegant of any mathematical writer of -modern times, and the most just imitator of the antients. Of their -taste, and form of demonstration Sir_ ~ISAAC~ _always professed -himself a great admirer: I have heard him even censure himself for -not following them yet more closely than he did; and speak with -regret of his mistake at the beginning of his mathematical studies, -in applying himself to the works of_ Des Cartes _and other algebraic -writers, before he had considered the elements of_ Euclide _with that -attention, which so excellent a writer deserves. As to the history -of his inventions, what relates to his discoveries of the methods of -series and fluxions, and of his theory of light and colours, the world -has been sufficiently informed of already. The first thoughts, which -gave rise to his Principia, he had, when he retired from_ Cambridge -_in 1666 on account of the plague. As he sat alone in a garden, he -fell into a speculation on the power of gravity: that as this power -is not found sensibly diminished at the remotest distance from the -center of the earth, to which we can rise, neither at the tops of the -loftiest buildings, nor even on the summits of the highest mountains; -it appeared to him reasonable to conclude, that this power must extend -much farther than was usually thought; why not as high as the moon, -said he to himself? and if so, her motion must be influenced by it; -perhaps she is retained in her orbit thereby. However, though the power -of gravity is not sensibly weakened in the little change of distance, -at which we can place our selves from the center of the earth; yet it -is very possible, that so high as the moon this power may differ much -in strength from what it is here. To make an estimate, what might be -the degree of this diminution, he considered with himself, that if the -moon be retained in her orbit by the force of gravity, no doubt the -primary planets are carried round the sun by the like power. And by -comparing the periods of the several planets with their distances from -the sun, he found, that if any power like gravity held them in their -courses, its strength must decrease in the duplicate proportion of the -increase of distance. This be concluded by supposing them to move in -perfect circles concentrical to the sun, from which the orbits of the -greatest part of them do not much differ. Supposing therefore the power -of gravity, when extended to the moon, to decrease in the same manner, -he computed whether that force would be sufficient to keep the moon -in her orbit. In this computation, being absent from books, he took -the common estimate in use among geographers and our seamen, before_ -Norwood _had measured the earth, that 60 English miles were contained -in one degree of latitude on the surface of the earth. But as this is -a very faulty supposition, each degree containing about 69½ of our -miles, his computation did not answer expectation; whence he concluded, -that some other cause must at least join with the action of the power -of gravity on the moon. On this account he laid aside for that time -any farther thoughts upon this matter. But some years after, a letter -which he received from Dr._ Hook, _put him on inquiring what was the -real figure, in which a body let fall from any high place descends, -taking the motion of the earth round its axis into consideration. -Such a body, having the same motion, which by the revolution of the -earth the place has whence it falls, is to be considered as projected -forward and at the same time drawn down to the center of the earth. -This gave occasion to his resuming his former thoughts concerning the -moon; and_ Picart _in_ France _having lately measured the earth, by -using his measures the moon appeared to be kept in her orbit purely by -the power of gravity; and consequently, that this power decreases as -you recede from the center of the earth in the manner our author had -formerly conjectured. Upon this principle he found the line described -by a falling body to be an ellipsis, the center of the earth being one -focus. And the primary planets moving in such orbits round the sun, he -had the satisfaction to see, that this inquiry, which he had undertaken -merely out of curiosity, could be applied to the greatest purposes. -Hereupon he composed near a dozen propositions relating to the motion -of the primary planets about the sun. Several years after this, some -discourse he had with Dr._ Halley, _who at Cambridge made him a -visit, engaged Sir_ ~ISAAC NEWTON~ _to resume again the consideration -of this subject; and gave occasion to his writing the treatise -which he published under the title of mathematical principles of -natural philosophy. This treatise, full of such a variety of profound -inventions, was composed by him from scarce any other materials than -the few propositions before mentioned, in the space of one year and an -half._ - -_Though his memory was much decayed, I found he perfectly understood -his own writings, contrary to what I had frequently heard in discourse -from many persons. This opinion of theirs might arise perhaps from his -not being always ready at speaking on these subjects, when it might -be expected he should. But as to this, it may be observed, that great -genius’s are frequently liable to be absent, not only in relation to -common life, but with regard to some of the parts of science they are -the best informed of. Inventors seem to treasure up in their minds, -what they have found out, after another manner than those do the same -things, who have not this inventive faculty. The former, when they -have occasion to produce their knowledge, are in some measure obliged -immediately to investigate part of what they want. For this they are -not equally fit at all times: so it has often happened, that such as -retain things chiefly by means of a very strong memory, have appeared -off hand more expert than the discoverers themselves._ - -_As to the moral endowments of his mind, they were as much to be -admired as his other talents. But this is a field I leave others to -exspatiate in. I only touch upon what I experienced myself during -the few years I was happy in his friendship. But this I immediately -discovered in him, which at once both surprized and charmed me: Neither -his extreme great age, nor his universal reputation had rendred him -stiff in opinion, or in any degree elated. Of this I had occasion -to have almost daily experience. The Remarks I continually sent him -by letters on his Principia were received with the utmost goodness. -These were so far from being any ways displeasing to him, that on -the contrary it occasioned him to speak many kind things of me to my -friends, and to honour me with a publick testimony of his good opinion. -He also approved of the following treatise, a great part of which we -read together. As many alterations were made in the late edition of -his Principia, so there would have been many more if there had been -a sufficient time. But whatever of this kind may be thought wanting, -I shall endeavour to supply in my comment on that book. I had reason -to believe he expected such a thing from me, and I intended to have -published it in his life time, after I had printed the following -discourse, and a mathematical treatise Sir_ ~ISAAC NEWTON~ _had written -a long while ago, containing the first principles of fluxions, for I -had prevailed on him to let that piece go abroad. I had examined all -the calculations, and prepared part of the figures; but as the latter -part of the treatise had never been finished, he was about letting me -have other papers, in order to supply what was wanting. But his death -put a stop to that design. As to my comment on the Principia, I intend -there to demonstrate whatever Sir_ ~ISAAC NEWTON~ _has set down without -express proof, and to explain all such expressions in his book, as -I shall judge necessary. This comment I shall forthwith put to the -press, joined to an english translation of his Principia, which I have -had some time by me. A more particular account of my whole design has -already been published in the new memoirs of literature for the month -of march 1727._ - -_I have presented my readers with a copy of verses on Sir_ ~ISAAC -NEWTON~, _which I have just received from a young Gentleman, whom I am -proud to reckon among the number of my dearest friends. If I had any -apprehension that this piece of poetry stood in need of an apology, -I should be desirous the reader might know, that the author is but -sixteen years old, and was obliged to finish his composition in a very -short space of time. But I shall only take the liberty to observe, that -the boldness of the digressions will be best judged of by those who are -acquainted with_ ~PINDAR~. - - - - - A - ~POEM~ - ON - Sir ~_ISAAC NEWTON_~. - - - TO ~NEWTON~’s genius, and immortal fame - Th’ advent’rous muse with trembling pinion soars. - Thou, heav’nly truth, from thy seraphick throne - Look favourable down, do thou assist - My lab’ring thought, do thou inspire my song. - NEWTON, who first th’ almighty’s works display’d, - And smooth’d that mirror, in whose polish’d face - The great creator now conspicuous shines; - Who open’d nature’s adamantine gates, - And to our minds her secret powers expos’d; - NEWTON demands the muse; his sacred hand - Shall guide her infant steps; his sacred hand - Shall raise her to the Heliconian height, - Where, on its lofty top inthron’d, her head - Shall mingle with the Stars. Hail nature, hail, - O Goddess, handmaid of th’ ethereal power, - Now lift thy head, and to th’ admiring world - Shew thy long hidden beauty. Thee the wise - Of ancient fame, immortal ~PLATO~’s self, - The Stagyrite, and Syracusian sage, - From black obscurity’s abyss to raise, - (Drooping and mourning o’er thy wondrous works) - With vain inquiry sought. Like meteors these - In their dark age bright sons of wisdom shone: - But at thy ~NEWTON~ all their laurels fade, - They shrink from all the honours of their names. - So glimm’ring stars contract their feeble rays, - When the swift lustre of ~AURORA~’s face - Flows o’er the skies, and wraps the heav’ns in light. - - THE Deity’s omnipotence, the cause, - Th’ original of things long lay unknown. - Alone the beauties prominent to sight - (Of the celestial power the outward form) - Drew praise and wonder from the gazing world. - As when the deluge overspread the earth, - Whilst yet the mountains only rear’d their heads - Above the surface of the wild expanse, - Whelm’d deep below the great foundations lay, - Till some kind angel at heav’n’s high command - Roul’d back the rising tides, and haughty floods, - And to the ocean thunder’d out his voice: - Quick all the swelling and imperious waves, - The foaming billows and obscuring surge, - Back to their channels and their ancient seats - Recoil affrighted: from the darksome main - Earth raises smiling, as new-born, her head, - And with fresh charms her lovely face arrays. - So his extensive thought accomplish’d first - The mighty task to drive th’ obstructing mists - Of ignorance away, beneath whose gloom - Th’ inshrouded majesty of Nature lay. - He drew the veil and swell’d the spreading scene. - How had the moon around th’ ethereal void - Rang’d, and eluded lab’ring mortals care, - Till his invention trac’d her secret steps, - While she inconstant with unsteady rein - Through endless mazes and meanders guides - In its unequal course her changing carr: - Whether behind the sun’s superior light - She hides the beauties of her radiant face, - Or, when conspicuous, smiles upon mankind, - Unveiling all her night-rejoicing charms. - When thus the silver-tressed moon dispels - The frowning horrors from the brow of night, - And with her splendors chears the sullen gloom, - While sable-mantled darkness with his veil - The visage of the fair horizon shades, - And over nature spreads his raven wings; - Let me upon some unfrequented green - While sleep sits heavy on the drowsy world, - Seek out some solitary peaceful cell, - Where darksome woods around their gloomy brows - Bow low, and ev’ry hill’s protended shade - Obscures the dusky vale, there silent dwell, - Where contemplation holds its still abode, - There trace the wide and pathless void of heav’n, - And count the stars that sparkle on its robe. - Or else in fancy’s wild’ring mazes lost - Upon the verdure see the fairy elves - Dance o’er their magick circles, or behold, - In thought enraptur’d with the ancient bards, - Medea’s baleful incantations draw - Down from her orb the paly queen of night. - But chiefly ~NEWTON~ let me soar with thee, - And while surveying all yon starry vault - With admiration I attentive gaze, - Thou shalt descend from thy celestial seat, - And waft aloft my high-aspiring mind, - Shalt shew me there how nature has ordain’d - Her fundamental laws, shalt lead my thought - Through all the wand’rings of th’ uncertain moon, - And teach me all her operating powers. - She and the sun with influence conjoint - Wield the huge axle of the whirling earth, - And from their just direction turn the poles, - Slow urging on the progress of the years. - The constellations seem to leave their seats, - And o’er the skies with solemn pace to move. - You, splendid rulers of the day and night, - The seas obey, at your resistless sway - Now they contract their waters, and expose - The dreary desart of old ocean’s reign. - The craggy rocks their horrid sides disclose; - Trembling the sailor views the dreadful scene, - And cautiously the threat’ning ruin shuns. - But where the shallow waters hide the sands, - There ravenous destruction lurks conceal’d, - There the ill-guided vessel falls a prey, - And all her numbers gorge his greedy jaws. - But quick returning see th’ impetuous tides - Back to th’ abandon’d shores impell the main. - Again the foaming seas extend their waves, - Again the rouling floods embrace the shoars, - And veil the horrours of the empty deep. - Thus the obsequious seas your power confess, - While from the surface healthful vapours rise - Plenteous throughout the atmosphere diffus’d, - Or to supply the mountain’s heads with springs, - Or fill the hanging clouds with needful rains, - That friendly streams, and kind refreshing show’rs - May gently lave the sun-burnt thirsty plains, - Or to replenish all the empty air - With wholsome moisture to increase the fruits - Of earth, and bless the labours of mankind. - O ~NEWTON~, whether flies thy mighty soul, - How shall the feeble muse pursue through all - The vast extent of thy unbounded thought, - That even seeks th’ unseen recesses dark - To penetrate of providence immense. - And thou the great dispenser of the world - Propitious, who with inspiration taught’st - Our greatest bard to send thy praises forth; - Thou, who gav’st ~NEWTON~ thought; who smil’dst serene, - When to its bounds he stretch’d his swelling soul; - Who still benignant ever blest his toil, - And deign’d to his enlight’ned mind t’ appear - Confess’d around th’ interminated world: - To me O thy divine infusion grant - (O thou in all so infinitely good) - That I may sing thy everlasting works, - Thy inexhausted store of providence, - In thought effulgent and resounding verse. - O could I spread the wond’rous theme around, - Where the wind cools the oriental world, - To the calm breezes of the Zephir’s breath, - To where the frozen hyperborean blasts. - To where the boist’rous tempest-leading south - From their deep hollow caves send forth their storms. - Thou still indulgent parent of mankind, - Left humid emanations should no more - Flow from the ocean, but dissolve away - Through the long series of revolving time; - And left the vital principle decay, - By which the air supplies the springs of life; - Thou hast the fiery visag’d comets form’d - With vivifying spirits all replete, - Which they abundant breathe about the void, - Renewing the prolifick soul of things. - No longer now on thee amaz’d we call, - No longer tremble at imagin’d ills, - When comets blaze tremendous from on high, - Or when extending wide their flaming trains - With hideous grasp the skies engirdle round, - And spread the terrors of their burning locks. - For these through orbits in the length’ning space - Of many tedious rouling years compleat - Around the sun move regularly on; - And with the planets in harmonious orbs, - And mystick periods their obeysance pay - To him majestick ruler of the skies - Upon his throne of circled glory fixt. - He or some god conspicuous to the view, - Or else the substitute of nature seems, - Guiding the courses of revolving worlds. - He taught great ~NEWTON~ the all-potent laws - Of gravitation, by whose simple power - The universe exists. Nor here the sage - Big with invention still renewing staid. - But O bright angel of the lamp of day, - How shall the muse display his greatest toil? - Let her plunge deep in Aganippe’s waves, - Or in Castalia’s ever-flowing stream, - That re-inspired she may sing to thee, - How ~NEWTON~ dar’d advent’rous to unbraid - The yellow tresses of thy shining hair. - Or didst thou gracious leave thy radiant sphere, - And to his hand thy lucid splendours give, - T’ unweave the light-diffusing wreath, and part - The blended glories of thy golden plumes? - He with laborious, and unerring care, - How different and imbodied colours form - Thy piercing light, with just distinction found. - He with quick sight pursu’d thy darting rays, - When penetrating to th’ obscure recess - Of solid matter, there perspicuous saw, - How in the texture of each body lay - The power that separates the different beams. - Hence over nature’s unadorned face - Thy bright diversifying rays dilate - Their various hues: and hence when vernal rains - Descending swift have burst the low’ring clouds, - Thy splendors through the dissipating mists - In its fair vesture of unnumber’d hues - Array the show’ry bow. At thy approach - The morning risen from her pearly couch - With rosy blushes decks her virgin cheek; - The ev’ning on the frontispiece of heav’n - His mantle spreads with many colours gay; - The mid-day skies in radiant azure clad, - The shining clouds, and silver vapours rob’d - In white transparent intermixt with gold, - With bright variety of splendor cloath - All the illuminated face above. - When hoary-headed winter back retires - To the chill’d pole, there solitary sits - Encompass’d round with winds and tempests bleak - In caverns of impenetrable ice, - And from behind the dissipated gloom - Like a new Venus from the parting surge - The gay-apparell’d spring advances on; - When thou in thy meridian brightness sitt’st, - And from thy throne pure emanations flow - Of glory bursting o’er the radiant skies: - Then let the muse Olympus’ top ascend, - And o’er Thessalia’s plain extend her view, - And count, O Tempe, all thy beauties o’er. - Mountains, whose summits grasp the pendant clouds, - Between their wood-invelop’d slopes embrace - The green-attired vallies. Every flow’r - Here in the pride of bounteous nature clad - Smiles on the bosom of th’ enamell’d meads. - Over the smiling lawn the silver floods - Of fair Peneus gently roul along, - While the reflected colours from the flow’rs, - And verdant borders pierce the lympid waves, - And paint with all their variegated hue - The yellow sands beneath. Smooth gliding on - The waters hasten to the neighbouring sea. - Still the pleas’d eye the floating plain pursues; - At length, in Neptune’s wide dominion lost, - Surveys the shining billows, that arise - Apparell’d each in Phœbus’ bright attire: - Or from a far some tall majestick ship, - Or the long hostile lines of threat’ning fleets, - Which o’er the bright uneven mirror sweep, - In dazling gold and waving purple deckt; - Such as of old, when haughty Athens power - Their hideous front, and terrible array - Against Pallene’s coast extended wide, - And with tremendous war and battel stern - The trembling walls of Potidæa shook. - Crested with pendants curling with the breeze - The upright masts high bristle in the air, - Aloft exalting proud their gilded heads. - The silver waves against the painted prows - Raise their resplendent bosoms, and impearl - The fair vermillion with their glist’ring drops: - And from on board the iron-cloathed host - Around the main a gleaming horrour casts; - Each flaming buckler like the mid-day sun, - Each plumed helmet like the silver moon, - Each moving gauntlet like the light’ning’s blaze, - And like a star each brazen pointed spear. - But lo the sacred high-erected fanes, - Fair citadels, and marble-crowned towers, - And sumptuous palaces of stately towns - Magnificent arise, upon their heads - Bearing on high a wreath of silver light. - But see my muse the high Pierian hill, - Behold its shaggy locks and airy top, - Up to the skies th’ imperious mountain heaves - The shining verdure of the nodding woods. - See where the silver Hippocrene flows, - Behold each glitt’ring rivulet, and rill - Through mazes wander down the green descent, - And sparkle through the interwoven trees. - Here rest a while and humble homage pay, - Here, where the sacred genius, that inspir’d - Sublime ~MÆONIDES~ and ~PINDAR’S~ breast, - His habitation once was fam’d to hold. - Here thou, O ~HOMER~, offer’dst up thy vows, - Thee, the kind muse ~CALLIOPÆA~ heard, - And led thee to the empyrean feats, - There manifested to thy hallow’d eyes - The deeds of gods; thee wise ~MINERVA~ taught - The wondrous art of knowing human kind; - Harmonious ~PHŒBUS~ tun’d thy heav’nly mind, - And swell’d to rapture each exalted sense; - Even ~MARS~ the dreadful battle-ruling god, - ~MARS~ taught thee war, and with his bloody hand - Instructed thine, when in thy sounding lines - We hear the rattling of Bellona’s carr, - The yell of discord, and the din of arms. - ~PINDAR~, when mounted on his fiery steed, - Soars to the sun, opposing eagle like - His eyes undazled to the fiercest rays. - He firmly seated, not like ~GLAUCUS’~ son, - Strides his swift-winged and fire-breathing horse, - And born aloft strikes with his ringing hoofs - The brazen vault of heav’n, superior there - Looks down upon the stars, whose radiant light - Illuminates innumerable worlds, - That through eternal orbits roul beneath. - But thou all hail immortalized son - Of harmony, all hail thou Thracian bard, - To whom ~APOLLO~ gave his tuneful lyre. - O might’st thou, ~ORPHEUS~, now again revive, - And ~NEWTON~ should inform thy list’ning ear - How the soft notes, and soul-inchanting strains - Of thy own lyre were on the wind convey’d. - He taught the muse, how sound progressive floats - Upon the waving particles of air, - When harmony in ever-pleasing strains, - Melodious melting at each lulling fall, - With soft alluring penetration steals - Through the enraptur’d ear to inmost thought, - And folds the senses in its silken bands. - So the sweet musick, which from ~ORPHEUS~’ touch - And fam’d ~AMPHION’S~, on the sounding string - Arose harmonious, gliding on the air, - Pierc’d the tough-bark’d and knotty-ribbed woods, - Into their saps soft inspiration breath’d - And taught attention to the stubborn oak. - Thus when great ~HENRY~, and brave ~MARLB’ROUGH~ led - Th’ imbattled numbers of ~BRITANNIA’S~ sons, - The trump, that swells th’ expanded cheek of fame, - That adds new vigour to the gen’rous youth, - And rouzes sluggish cowardize it self, - The trumpet with its Mars-inciting voice, - The winds broad breast impetuous sweeping o’er - Fill’d the big note of war. Th’ inspired host - With new-born ardor press the trembling ~GAUL~; - Nor greater throngs had reach’d eternal night, - Not if the fields of Agencourt had yawn’d - Exposing horrible the gulf of fate; - Or roaring Danube spread his arms abroad, - And overwhelm’d their legions with his floods. - But let the wand’ring muse at length return; - Nor yet, angelick genius of the sun, - In worthy lays her high-attempting song - Has blazon’d forth thy venerated name. - Then let her sweep the loud-resounding lyre - Again, again o’er each melodious string - Teach harmony to tremble with thy praise. - And still thine ear O favourable grant, - And she shall tell thee, that whatever charms, - Whatever beauties bloom on nature’s face, - Proceed from thy all-influencing light. - That when arising with tempestuous rage, - The North impetuous rides upon the clouds - Dispersing round the heav’ns obstructive gloom, - And with his dreaded prohibition stays - The kind effusion of thy genial beams; - Pale are the rubies on ~AURORA’S~ lips, - No more the roses blush upon her cheeks, - Black are Peneus’ streams and golden sands - In Tempe’s vale dull melancholy sits, - And every flower reclines its languid head. - By what high name shall I invoke thee, say, - Thou life-infusing deity, on thee - I call, and look propitious from on high, - While now to thee I offer up my prayer. - O had great ~NEWTON~, as he found the cause, - By which sound rouls thro’ th’ undulating air, - O had he, baffling times resistless power, - Discover’d what that subtle spirit is, - Or whatsoe’er diffusive else is spread - Over the wide-extended universe, - Which causes bodies to reflect the light, - And from their straight direction to divert - The rapid beams, that through their surface pierce. - But since embrac’d by th’ icy arms of age, - And his quick thought by times cold hand congeal’d, - Ev’n ~NEWTON~ left unknown this hidden power; - Thou from the race of human kind select - Some other worthy of an angel’s care, - With inspiration animate his breast, - And him instruct in these thy secret laws. - O let not ~NEWTON~, to whose spacious view, - Now unobstructed, all th’ extensive scenes - Of the ethereal ruler’s works arise; - When he beholds this earth he late adorn’d, - Let him not see philosophy in tears, - Like a fond mother solitary sit, - Lamenting him her dear, and only child. - But as the wise ~PYTHAGORAS~, and he, - Whose birth with pride the fam’d Abdera boasts, - With expectation having long survey’d - This spot their ancient seat, with joy beheld - Divine philosophy at length appear - In all her charms majestically fair, - Conducted by immortal ~NEWTON’S~ hand. - So may he see another sage arise, - That shall maintain her empire: then no more - Imperious ignorance with haughty sway - Shall stalk rapacious o’er the ravag’d globe: - Then thou, O ~NEWTON~, shalt protect these lines. - The humble tribute of the grateful muse; - Ne’er shall the sacrilegious hand despoil - Her laurel’d temples, whom his name preserves: - And were she equal to the mighty theme, - Futurity should wonder at her song; - Time should receive her with extended arms, - Seat her conspicuous in his rouling carr, - And bear her down to his extreamest bound. - - ~FABLES~ with wonder tell how Terra’s sons - With iron force unloos’d the stubborn nerves - Of hills, and on the cloud-inshrouded top - Of Pelion Ossa pil’d. But if the vast - Gigantick deeds of savage strength demand - Astonishment from men, what then shalt thou, - O what expressive rapture of the soul, - When thou before us, ~NEWTON~, dost display - The labours of thy great excelling mind; - When thou unveilest all the wondrous scene, - The vast idea of th’ eternal king, - Not dreadful bearing in his angry arm - The thunder hanging o’er our trembling heads; - But with th’ effulgency of love replete, - And clad with power, which form’d th’ extensive heavens. - O happy he, whose enterprizing hand - Unbars the golden and relucid gates - Of th’ empyrean dome, where thou enthron’d - Philosophy art seated. Thou sustain’d - By the firm hand of everlasting truth - Despisest all the injuries of time; - Thou never know’st decay when all around, - Antiquity obscures her head. Behold - Th’ Egyptian towers, the Babylonian walls, - And Thebes with all her hundred gates of brass, - Behold them scatter’d like the dust abroad. - Whatever now is flourishing and proud, - Whatever shall, must know devouring age. - Euphrates’ stream, and seven-mouthed Nile, - And Danube, thou that from Germania’s soil - To the black Euxine’s far remoted shore, - O’er the wide bounds of mighty nations sweep’st - In thunder loud thy rapid floods along. - Ev’n you shall feel inexorable time; - To you the fatal day shall come; no more - Your torrents then shall shake the trembling ground, - No longer then to inundations swol’n - Th’ imperious waves the fertile pastures drench, - But shrunk within a narrow channel glide; - Or through the year’s reiterated course - When time himself grows old, your wond’rous streams - Lost ev’n to memory shall lie unknown - Beneath obscurity, and Chaos whelm’d, - But still thou sun illuminatest all - The azure regions round, thou guidest still - The orbits of the planetary spheres; - The moon still wanders o’er her changing course, - And still, O ~NEWTON~, shall thy name survive: - As long as nature’s hand directs the world, - When ev’ry dark obstruction shall retire, - And ev’ry secret yield its hidden store, - Which thee dim-sighted age forbad to see - Age that alone could stay thy rising soul. - And could mankind among the fixed stars, - E’en to th’ extremest bounds of knowledge reach, - To those unknown innumerable suns, - Whose light but glimmers from those distant worlds, - Ev’n to those utmost boundaries, those bars - That shut the entrance of th’ illumin’d space - Where angels only tread the vast unknown, - Thou ever should’st be seen immortal there: - In each new sphere, each new-appearing sun, - In farthest regions at the very verge - Of the wide universe should’st thou be seen. - And lo, th’ all-potent goddess ~NATURE~ takes - With her own hand thy great, thy just reward - Of immortality; aloft in air - See she displays, and with eternal grasp - Uprears the trophies of great ~NEWTON~’s fame. - - R. GLOVER. - - THE - ~CONTENTS.~ - - _INTRODUCTION concerning Sir_ ~ISAAC NEWTON~’_s - method of reasoning in philosophy_ pag. 1 - - - BOOK I. - - ~CHAP. 1.~ _Of the laws of motion_ - _The first law of motion proved_ p. 29 - _The second law of motion proved_ p. 29 - _The third law of motion proved_ p. 31 - - ~CHAP. 2.~ _Further proofs of the laws of motion_ - _The effects of percussion_ p. 49 - _The perpendicular descent of bodies_ p. 55 - _The oblique descent of bodies in a straight line_ p. 57 - _The curvilinear descent of bodies_ p. 58 - _The perpendicular ascent of bodies_ ibid. - _The oblique ascent of bodies_ p. 59 - _The power of gravity proportional to the quantity of - matter in each body_ p. 60 - _The centre of gravity of bodies_ p. 62 - _The mechanical powers_ p. 69 - _The lever_ p. 71 - _The wheel and axis_ p. 77 - _The pulley_ p. 80 - _The wedge_ p. 83 - _The screw_ ibid. - _The inclined plain_ p. 84 - _The pendulum_ p. 86 - _Vibrating in a circle_ ibid. - _Vibrating in a cycloid_ p. 91 - _The line of swiftest descent_ p. 93 - _The centre of oscillation_ p. 94 - _Experiments upon the percussion of bodies made - by pendulums_ p. 98 - _The centre of percussion_ p. 100 - _The motion of projectiles_ p. 102 - _The description of the conic sections_ p. 106 - _The difference between absolute and relative motion, - as also between absolute and relative time_ p. 112 - - ~CHAP. 3.~ _Of centripetal forces_ p. 117 - - ~CHAP. 4.~ _Of the resistance of fluids_ p. 143 - _Bodies are resisted in the duplicate proportion of - their velocities_ p. 147 - _Of elastic fluids and their resistance_ p. 149 - _How fluids may be rendered elastic_ p. 150 - _The degree of resistance in regard to the proportion - between the density of the body and of the fluid_ - _In rare and uncompressed fluids_ p. 153 - _In compressed fluids_ p. 155 - _The degree of resistance as it depends upon the figure - of bodies_ - _In rare and uncompressed fluids_ p. 155 - _In compressed fluids_ p. 158 - - - BOOK II. - - ~CHAP. 1.~ _That the planets move in a space - empty of sensible matter_ p. 161 - _The system of the world described_ p. 162 - _The planets suffer no sensible resistance in their - motion_ p. 166 - _They are not kept in motion by a fluid_ p. 168 - _That all space is not full of matter without vacancies_ p. 169 - - ~CHAP. 2.~ _Concerning the cause that keeps in - motion the primary planets_ p. 171 - _They are influenced by a centripetal power directed to - the sun_ p. 171 - _The strength of this power is reciprocally in the - duplicate proportion of the distance_ ibid. - _The cause of the irregularities in the motions of the - planets_ p. 175 - _A correction of their motions_ p. 178 - _That the frame of the world is not eternal_ p. 180 - - ~CHAP. 3.~ _Of the motion of the moon and the other - secondary planets_ - _That they are influenced by a centripetal force - directed toward their primary, as the primary are - influenced by the sun_ p. 182 - _That the power usually called gravity extends to - the moon_ p. 189 - _That the sun acts on the secondary planets_ p. 190 - _The variation of the moon_ p. 193 - _That the circuit of the moons orbit is increased by the - sun in the quarters, and diminished in the - conjunction and opposition_ p. 198 - _The distance of the moon from the earth in the quarters - and in the conjunction and opposition is altered by - the sun_ p. 200 - _These irregularities in the moon’s motion varied by the - change of distance between the earth and sun_ p. 201 - _The period of the moon round the earth and her distance - varied by the same means_ ibid. - _The motion of the nodes and the inclination of the - moons orbit_ p. 202 - _The motion of the apogeon and change of the - eccentricity_ p. 218 - _The inequalities of the other secondary planets - deducible from these of the moon_ p. 229 - - - ~CHAP. 4.~ _Of comets_ - - _They are not meteors, nor placed totally without the - planetary system_ p. 230 - _The sun acts on them in the same manner as on the - planets_ p. 231 - _Their orbits are near to parabola’s_ p. 233 - _The comet that appeared at the end of the year 1680, - probably performs its period in 575 years, and - another comet in 75 years_ p. 234 - _Why the comets move in planes more different from - one another than the planets_ p. 235 - _The tails of comets_ p. 238 - _The use of them_ p. 243 244 - _The possible use of the comet it self_ p. 245 246 - - ~CHAP. 5.~ _Of the bodies of the sun and planets_ - - _That each of the heavenly bodies is endued with an - attractive power, and that the force of the same - body on others is proportional to the quantity of - matter in the body attracted_ p. 247 - _This proved in the earth_ p. 248 - _In the sun_ p. 250 - _In the rest of the planets_ p. 251 - _That the attractive power is of the same nature in - the sun and in all the planets, and therefore is - the same with gravity_ p. 252 - _That the attractive power in each of these bodies is - proportional to the quantity of matter in the body - attracting_ ibid. - _That each particle of which the sun and planets are - composed is endued with an attracting power, the - strength of which is reciprocally in the duplicate - proportion of the distance_ p. 257 - - _The power of gravity universally belongs to all matter_ p. 259 - - _The different weight of the same body upon the surface - of the sun, the earth, Jupiter and Saturn; the - respective densities of these bodies, and the - proportion between their diameters_ p. 261 - - ~CHAP. 6.~ _Of the fluid parts of the planets_ - - _The manner in which fluids press_ p. 264 - _The motion of waves on the surface of water_ p. 269 - _The motion of sound through the air_ p. 270 - _The velocity of sound_ p. 282 - _Concerning the tides_ p. 283 - _The figure of the earth_ p. 296 - _The effect of this figure upon the power of gravity_ p. 300 - _The effect it has upon pendulums_ p. 302 - _Bodies descend perpendicularly to the surface of - the earth_ p. 304 - _The axis of the earth changes its direction twice a - year, and twice a month_ p. 313 - _The figure of the secondary planets_ ibid. - - - BOOK III. - - ~CHAP. 1.~ _Concerning the cause of colours - inherent in the light_ - - _The sun’s light is composed of rays of different - colours_ p. 318 - _The refraction of light_ p. 319 320 - _Bodies appear of different colour by day-light, because - some reflect one kind of light more copiously than - the rest, and other bodies other kinds of light_ p. 329 - _The effect of mixing rays of different colours_ p. 334 - - ~CHAP. 2.~ _Of the properties of bodies whereon their - colours depend._ - - _Light is not reflected by impinging against the solid - parts of bodies_ p. 339 - _The particles which compose bodies are transparent_ p. 341 - _Cause of opacity_ p. 342 - _Why bodies in the open day-light have different - colours_ p. 344 - _The great porosity of bodies considered_ p. 355 - - ~CHAP. 3.~ _Of the refraction, reflection, and - inflection of light._ - - _Rays of different colours are differently refracted_ p. 357 - _The sine of the angle of incidence in each kind of rays - bears a given proportion to the sine of refraction_ p. 361 - _The proportion between the refractive powers in - different bodies_ p. 366 - _Unctuous bodies refract most in proportion to their - density_ p. 368 - _The action between light and bodies is mutual_ p. 369 - _Light has alternate fits of easy transmission and - reflection_ p. 371 - _The fits found to return alternately many thousand - times_ p. 375 - _Why bodies reflect part of the light incident upon them - and transmit another part_ ibid. - _Sir_ ~ISAAC NEWTON~_’s conjecture - concerning the cause of this alternate reflection - and transmission of light_ p. 376 - _The inflection of light_ p. 377 - - ~CHAP. 4.~ _Of optic glasses._ - - _How the rays of light are refracted by a spherical - surface of glass_ p. 378 - _How they are refracted by two such surfaces_ p. 380 - _How the image of objects is formed by a convex glass_ p. 381 - _Why convex glasses help the sight in old age, and - concave glasses assist short-sighted people_ p. 383 - _The manner in which vision is performed by the eye_ p. 385 - _Of telescopes with two convex glasses_ p. 386 - _Of telescopes with four convex glasses_ p. 388 - _Of telescopes with one convex and one concave glass_ ibid. - _Of microscopes_ p. 389 - _Of the imperfection of telescopes arising from the - different refrangibility of the light_ p. 390 - _Of the reflecting telescope_ p. 393 - - ~CHAP. 5.~ _Of the rainbow_ - _Of the inner rainbow_ p. 394 395 398 399 - _Of the outter bow_ p. 396 397 400 - _Of a particular appearance in the inner rainbow_ p. 401 - _Conclusion_ p. 405 - - - - -~ERRATA.~ - - PAGE 25. line 4. read _In these Precepts._ p. 40. l. 24. for _I_ - read _K_. p. 53. l. penult. f. Æ. r. F. p. 82. l. ult. f. 40. r. 41. - p. 83 l. ult. f. 43. r. 45. p. 91. l. 3. f. 48. r. 50. ibid. l. 25. - for 49. r. 51. p. 92. l. 18. f. _A G F E._ r. _H G F C._ p. 96. l. - 23. dele the comma after {⅓}. p. 140. l. 12. dele _and._ p. 144. l. - 15. f. _threefold._ r. _two-fold._ p. 162. l. 25. f. {⅓}. r. {⅞}. p. - 193. 1. 2. r. _always._ p. 199. l. penult. and p. 200. l. 3. 5. f. F. - r. C. p. 201. l. 8. f. _ascends._ r._ must ascend._ ibid. l. 10. f. - _it descends._ r. _descend._ p. 208. l. 14. f. _W T O._ r. _N T O._ - In _fig._ 110. draw a line from _I_ through _T_, till it meets the - circle _A D C B_, where place _W._ p. 216. l. penult. f. _action._ - r. _motion._ p. 221. l. 23. f. _A F._ r. _A H._ p. 232. l. 23. after - _invention_ put a full point. p. 253. l. penult. delete the comma - after _remarkable_. p. 255. l. ult. f. _D E._ r. _B E._ p. 278. l. - 17. f. ξ τ. r. ξ π. p. 299. l. 19 r. _the._ p. 361. l. 12. f. I. r. - t. p. 369. l. 2, 3. r. _Pseudo-topaz._ p. 378. l. 12. f. _that._ r. - _than._ p. 379. l. 15. f. _converge._ r. _diverge._ p. 384. l. 7. f. - _optic-glass._ r. _optic-nerve._ p. 391. l. 18. r. _as 50 to 78._ p. - 392. l. 18. after _telescope_ add _be about 100 feet long and the._ in - _fig. 161._ f. δ put ε. p. 399. l. 8. r. A n, A x. &c. p. 400. 1. 19. - r. A π, A ρ. A σ, A τ. A φ. p. 401. l. 14. r. _fig. 163._ The pages - 374, 375, 376 are erroneously numbered 375, 376, 377; and the pages - 382, 383 are numbered 381, 382. - - - - - A LIST of such of the - SUBSCRIBERS NAMES - As are come to the ~HAND~ of the - AUTHOR. - - A - - M_Onseigneur_ d’Aguesseau, _Chancelier de_ France - _Reverend_ Mr Abbot, _of_ Emanuel Coll. Camb. - _Capt._ George Abell - _The Hon. Sir_ John Anstruther, _Bar._ - Thomas Abney, _Esq;_ - Mr. Nathan Abraham - _Sir_ Arthur Acheson, Bart. - Mr William Adair - _Rev._ Mr John Adams, _Fellow of_ Sidney Coll. Cambridge - Mr William Adams - Mr George Adams - Mr William Adamson, _Scholar of_ Caius Coll. Camb. - Mr Samuel Adee, _Fell. of_ Corp. Chr. Coll. Oxon - Mr Andrew Adlam - Mr John Adlam - Mr Stephen Ainsworth - Mrs Aiscot - Mr Robert Akenhead, _Bookseller at_ Newcastle _upon_ Tyne - S. B. Albinus, M. D. Anatom. _and_ Chirurg _in_ Acad. L. B. Prof. - George Aldridge, _M. D._ - Mr George Algood - Mr Aliffe - Robert Allen, _Esq;_ - Mr Zach. Allen - _Rev._ Mr Allerton, _Fellow of_ Sidney Coll. Cambridge - Mr St. Amand - Mr John Anns - Thomas Anson, _Esq;_ - _Rev. Dr._ Christopher Anstey - Mr Isaac Antrabus - Mr Joshua Appleby - John Arbuthnot, _M. D._ - William Archer, _Esq;_ - Mr John Archer, _Merchant of_ Amsterdam - Thomas Archer, _Esq;_ - _Coll._ John Armstrong, Surveyor-General _of_ His Majesty’s Ordnance - Mr Armytage - Mr Street Arnold, _Surgeon_ - Mr Richard Arnold - Mr Ascough - Mr Charles Asgill - Richard Ash, _Esq; of_ Antigua - Mr Ash, _Fellow-Commoner of_ Jesus Coll. Cambridge - William Ashurst, _Esq; of_ Castle Henningham, Essex - Mr Thomas Ashurst - Mr Samuel Ashurst - Mr John Askew, _Merchant_ - Mr Edward Athawes, _Merchant_ - Mr Abraham Atkins - Mr Edward Kensey Atkins - Mr Ayerst - Mr Jonathan Ayleworth, _Jun._ - Rowland Aynsworth, _Esq;_ - - - B - - _His Grace the Duke of_ Bedford - _Right Honourable the Marquis of_ Bowmont - _Right Hon. the Earl of_ Burlington - _Right Honourable Lord Viscount_ Bateman - _Rt. Rev. Ld. Bp. of_ Bath _and_ Wells - _Rt. Rev. Lord Bishop of_ Bristol - _Right Hon. Lord_ Bathurst - Richard Backwell, _Esq;_ - Mr William Backshell, _Merch._ - Edmund Backwell, _Gent._ - _Sir_ Edmund Bacon - Richard Bagshaw, _of_ Oakes, _Esq;_ - Tho. Bagshaw, _of_ Bakewell, _Esq;_ - _Rev._ Mr. Bagshaw - _Sir_ Robert Baylis - _Honourable_ George Baillie, _Esq;_ - Giles Bailly, _M. 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Cambridge - Mr Henry Banyer, _of_ Wisbech, _Surgeon_ - Mr John Barber, _Apothecary in_ Coventry - Henry Steuart Barclay, _of_ Colairny, _Esq;_ - _Rev._ Mr Barclay, _Canon of_ Windsor - Mr David Barclay - Mr Benjamin Barker, _Bookseller in_ London - ---- Barker, _Esq;_ - Mr Francis Barkstead - _Rev._ Mr Barnard - Thomas Barrett, _Esq;_ - Mr Barrett - Richard Barret, _M. D._ - Mr Barrow, _Apothecary_ - William Barrowby, _M. D._ - Edward Barry, _M. D. of_ Corke - Mr Humphrey Bartholomew, _of_ University College, Oxon - Mr Benjamin Bartlett - Mr Henry Bartlett - Mr James Bartlett - Mr Newton Barton, _of_ Trinity College, Cambridge - _Rev._ Mr. Barton - William Barnsley, _Esq;_ - Mr Samuel Bateman - Mr Thomas Bates - Peter Barhurst, _Esq;_ - Mark Barr, _Esq;_ - Thomas Bast, _Esq;_ - Mr Batley, _Bookseller in_ London - Mr Christopher Batt, _jun._ - Mr William Batt, _Apothecary_ - Rev. Mr Battely, _M. A. 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Chief Justice of the Common Pleas, of_ - Ireland - Mr Joseph Wyeth - Thomas Wyndham, _Esq;_ - _Rev._ Mr John Wynne - - Y - - Mr John Yardley, _Surg. in_ Coven. - Mr Thomas Yates - Mrs Yeo, _of_ Exeter, _Bookseller_ - _Sir_ William Yonge - _Lady_ York - Nicholas Young, _of the_ Inner-Temple, _Esq;_ - Hitch Young, _Esq;_ - _Rev._ Edward Young, _L. L. D._ - -[Illustration] - - - - -INTRODUCTION. - -THE manner, in which Sir ~ISAAC NEWTON~ has published his philosophical -discoveries, occasions them to lie very much concealed from all, who -have not made the mathematics particularly their study. He once, -indeed, intended to deliver, in a more familiar way, that part of -his inventions, which relates to the system of the world; but upon -farther consideration he altered his design. For as the nature of -those discoveries made it impossible to prove them upon any other than -geometrical principles; he apprehended, that those, who should not -fully perceive the force of his arguments, would hardly be prevailed -on to exchange their former sentiments for new opinions, so very -different from what were commonly received[1]. He therefore chose -rather to explain himself only to mathematical readers; and declined -the attempting to instruct such in any of his principles, who, by -not comprehending his method of reasoning, could not, at the first -appearance of his discoveries, have been persuaded of their truth. But -now, since Sir ~ISAAC NEWTON~’s doctrine has been fully established -by the unanimous approbation of all, who are qualified to understand -the same; it is without doubt to be wished, that the whole of his -improvements in philosophy might be universally known. For this purpose -therefore I drew up the following papers, to give a general notion -of our great philosopher’s inventions to such, as are not prepared -to read his own works, and yet might desire to be informed of the -progress, he has made in natural knowledge; not doubting but there were -many, besides those, whose turn of mind had led them into a course of -mathematical studies, that would take great pleasure in tasting of this -delightful fountain of science. - -2. IT is a just remark, which has been made upon the human mind, that -nothing is more suitable to it, than the contemplation of truth; and -that all men are moved with a strong desire after knowledge; esteeming -it honourable to excel therein; and holding it, on the contrary, -disgraceful to mistake, err, or be in any way deceived. And this -sentiment is by nothing more fully illustrated, than by the inclination -of men to gain an acquaintance with the operations of nature; which -disposition to enquire after the causes of things is so general, that -all men of letters, I believe, find themselves influenced by it. Nor -is it difficult to assign a reason for this, if we consider only, that -our desire after knowledge is an effect of that taste for the sublime -and the beautiful in things, which chiefly constitutes the difference -between the human life, and the life of brutes. These inferior animals -partake with us of the pleasures, that immediately flow from the bodily -senses and appetites; but our minds are furnished with a superior -sense, by which we are capable of receiving various degrees of delight, -where the creatures below us perceive no difference. Hence arises -that pursuit of grace and elegance in our thoughts and actions, and -in all things belonging to us, which principally creates imployment -for the active mind of man. The thoughts of the human mind are too -extensive to be confined only to the providing and enjoying of what -is necessary for the support of our being. It is this taste, which -has given rise to poetry, oratory, and every branch of literature and -science. From hence we feel great pleasure in conceiving strongly, and -in apprehending clearly, even where the passions are not concerned. -Perspicuous reasoning appears not only beautiful; but, when set forth -in its full strength and dignity, it partakes of the sublime, and not -only pleases, but warms and elevates the soul. This is the source of -our strong desire of knowledge; and the same taste for the sublime -and the beautiful directs us to chuse particularly the productions of -nature for the subject of our contemplation: our creator having so -adapted our minds to the condition, wherein he has placed us, that all -his visible works, before we inquire into their make, strike us with -the most lively ideas of beauty and magnificence. - -3. BUT if there be so strong a passion in contemplative minds for -natural philosophy; all such must certainly receive a particular -pleasure in being informed of Sir ~ISAAC NEWTON~’s discoveries, who -alone has been able to make any great advancements in the true course -leading to natural knowledge: whereas this important subject had before -been usually attempted with that negligence, as cannot be reflected -on without surprize. Excepting a very few, who, by pursuing a more -rational method, had gained a little true knowledge in some particular -parts of nature; the writers in this science had generally treated of -it after such a manner, as if they thought, that no degree of certainty -was ever to be hoped for. The custom was to frame conjectures; and if -upon comparing them with things, there appeared some kind of agreement, -though very imperfect, it was held sufficient. Yet at the same time -nothing less was undertaken than intire systems, and fathoming at once -the greatest depths of nature; as if the secret causes of natural -effects, contrived and framed by infinite wisdom, could be searched -out by the slightest endeavours of our weak understandings. Whereas -the only method, that can afford us any prospect of success in this -difficult work, is to make our enquiries with the utmost caution, and -by very slow degrees. And after our most diligent labour, the greatest -part of nature will, no doubt, for ever remain beyond our reach. - -4. THIS neglect of the proper means to enlarge our knowledge, joined -with the presumption to attempt, what was quite out of the power of our -limited faculties, the Lord BACON judiciously observes to be the great -obstruction to the progress of science[2]. Indeed that excellent person -was the first, who expresly writ against this way of philosophizing; -and he has laid open at large the absurdity of it in his admirable -treatise, intitled NOVUM ORGANON SCIENTIARUM; and has there likewise -described the true method, which ought to be followed. - -5. THERE are, saith he, but two methods, that can be taken in the -pursuit of natural knowledge. One is to make a hasty transition -from our first and slight observations on things to general axioms, -and then to proceed upon those axioms, as certain and uncontestable -principles, without farther examination. The other method; (which he -observes to be the only true one, but to his time unattempted;) is to -proceed cautiously, to advance step by step, reserving the most general -principles for the last result of our inquiries[3]. Concerning the -first of these two methods; where objections, which happen to appear -against any such axioms taken up in haste, are evaded by some frivolous -distinction, when the axiom it self ought rather to be corrected[4]; -he affirms, that the united endeavours of all ages cannot make it -successful; because this original error in the first digestion of -the mind (as he expresses himself) cannot afterwards be remedied[5]: -whereby he would signify to us, that if we set out in a wrong way; no -diligence or art, we can use, while we follow so erroneous a course, -will ever bring us to our designed end. And doubtless it cannot prove -otherwise; for in this spacious field of nature, if once we forsake -the true path, we shall immediately lose our selves, and must for ever -wander with uncertainty. - -6. THE impossibility of succeeding in so faulty a method of -philosophizing his Lordship endeavours to prove from the many false -notions and prejudices, to which the mind of man is exposed[6]. And -since this judicious writer apprehends, that men are so exceeding -liable to fall into these wrong tracts of thinking, as to incur great -danger of being misled by them, even while they enter on the true -course in pursuit of nature[7]; I trust, I shall be excused, if, by -insisting a little particularly upon this argument, I endeavour to -remove whatever prejudice of this kind, might possibly entangle the -mind of any of my readers. - -7. HIS Lordship has reduced these prejudices and false modes of -conception under four distinct heads[8]. - -8. THE first head contains such, as we are subject to from the very -condition of humanity, through the weakness both of our senses, and of -the faculties of the mind[9]; seeing, as this author well observes, the -subtilty of nature far exceeds the greatest subtilty of our senses or -acutest reasonings[10]. One of the false modes of conception, which -he mentions under this head, is the forming to our selves a fanciful -simplicity and regularity in natural things. This he illustrates by -the following instances; the conceiving the planets to move in perfect -circles; the adding an orb of fire to the other three elements, and -the supposing each of these to exceed the other in rarity, just in a -decuple proportion[11]. And of the same nature is the assertion of -~DES CARTES~, without any proof, that all things are made up -of three kinds of matter only[12]. As also this opinion of another -philosopher; that light, in passing through different mediums, was -refracted, so as to proceed by that way, through which it would move -more speedily, than through any other[13]. The second erroneous turn -of mind, taken notice of by his Lordship under this head, is, that -all men are in some degree prone to a fondness for any notions, which -they have once imbibed; whereby they often wrest things to reconcile -them to those notions, and neglect the consideration of whatever will -not be brought to an agreement with them; just as those do, who are -addicted to judicial astrology, to the observation of dreams, and to -such-like superstitions; who carefully preserve the memory of every -incident, which serves to confirm their prejudices, and let slip out of -their minds all instances, that make against them[14]. There is also -a farther impediment to true knowledge, mentioned under the same head -by this noble writer, which is; that whereas, through the weakness -and imperfection of our senses, many things are concealed. from us, -which have the greatest effect in producing natural appearances; our -minds are ordinarily most affected by that, which makes the strongest -impression on our organs of sense; whereby we are apt to judge of -the real importance of things in nature by a wrong measure[15]. So, -because the figuration and the motion of bodies strike our senses more -immediately than most of their other properties, DES CARTES and his -followers will not allow any other explication of natural appearances, -than from the figure and motion of the parts of matter. By which -example we see how justly his Lordship observes this cause of error to -be the greatest of any[16]; since it has given rise to a fundamental -principle in a system of philosophy, that not long ago obtained almost -an universal reputation. - -9. THESE are the chief branches of those obstructions to knowledge, -which this author has reduced under his first head of false -conceptions. The second head contains the errors, to which particular -persons are more especially obnoxious[17]. One of these is the -consequence of a preceding observation: that as we are exposed to be -captivated by any opinions, which have once taken possession of our -minds; so in particular, natural knowledge has been much corrupted by -the strong attachment of men to some one part of science, of which -they reputed themselves the inventers, or about which they have spent -much of their time; and hence have been apt to conceive it to be of -greater use in the study of natural philosophy than it was: like -ARISTOTLE, who reduced his physics to logical disputations; and the -chymists, who thought, that nature could be laid open only by the -force of their fires[18]. Some again are wholly carried away by an -excessive veneration for antiquity; others, by too great fondness -for the moderns; few having their minds so well balanced, as neither -to depreciate the merit of the ancients, nor yet to despise the real -improvements of later times[19]. To this is added by his Lordship a -difference in the genius of men, that some are most fitted to observe -the similitude, there is in things, while others are more qualified to -discern the particulars, wherein they disagree; both which dispositions -of mind are useful: but to the prejudice of philosophy men are apt to -run into excess in each; while one sort of genius dwells too much upon -the gross and sum of things, and the other upon trifling minutenesses -and shadowy distinctions[20]. - -10. UNDER the third head of prejudices and false notions this writer -considers such, as follow from the lax and indefinite use of words in -ordinary discourse; which occasions great ambiguities and uncertainties -in philosophical debates (as another eminent philosopher has since -shewn more at large[21];) insomuch that this our author thinks a strict -defining of terms to be scarce an infallible remedy against this -inconvenience[22]. And perhaps he has no small reason on his side: for -the common inaccurate sense of words, notwithstanding the limitations -given them by definitions, will offer it self so constantly to the -mind, as to require great caution and circumspection for us not to be -deceived thereby. Of this we have a very eminent instance in the great -disputes, that have been raised about the use of the word attraction in -philosophy; of which we shall be obliged hereafter to make particular -mention[23]. Words thus to be guarded against are of two kinds. Some -are names of things, that are only imaginary[24]; such words are wholly -to be rejected. But there are other terms, that allude to what is real, -though their signification is confused[25]. And these latter must of -necessity be continued in use; but their sense cleared up, and freed, -as much as possible, from obscurity. - -11. THE last general head of these errors comprehends such, as follow -from the various sects of false philosophies; which this author divides -into three sorts, the sophistical, empirical, and superstitious[26]. By -the first of these he means a philosophy built upon speculations only -without experiments[27]; by the second, where experiments are blindly -adhered to, without proper reasoning upon them[28]; and by the third, -wrong opinions of nature fixed in mens minds either through false -religions, or from misunderstanding the declarations of the true[29]. - -12. THESE are the four principal canals, by which this judicious author -thinks, that philosophical errors have flowed in upon us. And he -rightly observes, that the faulty method of proceeding in philosophy, -against which he writes[30], is so far from assisting us towards -overcoming these prejudices; that he apprehends it rather suited to -rivet them more firmly to the mind[31]. How great reason then has his -Lordship to call this way of philosophizing the parent of error, and -the bane of all knowledge[32]? For, indeed, what else but mistakes can -so bold and presumptuous a treatment of nature produce? have we the -wisdom necessary to frame a world, that we should think so easily, -and with so slight a search to enter into the most secret springs of -nature, and discover the original causes of things? what chimeras, what -monsters has not this preposterous method brought forth? what schemes, -or what hypothesis’s of the subtilest wits has not a stricter enquiry -into nature not only overthrown, but manifested to be ridiculous and -absurd? Every new improvement, which we make in this science, lets us -see more and more the weakness of our guesses. Dr. HARVEY, by that -one discovery of the circulation of the blood, has dissipated all the -speculations and reasonings of many ages upon the animal oeconomy. -ASELLIUS, by detecting the lacteal veins, shewed how little ground all -physicians and philosophers had in conjecturing, that the nutritive -part of the aliment was absorbed by the mouths of the veins spread -upon the bowels: and then PECQUET, by finding out the thoracic duct, -as evidently proved the vanity of the opinion, which was persisted in -after the lacteal vessels were known, that the alimental juice was -conveyed immediately to the liver, and there converted into blood. - -13. AS these things set forth the great absurdity of proceeding in -philosophy on conjectures, by informing us how far the operations -of nature are above our low conceptions; so on the other hand, such -instances of success from a more judicious method shew us, that our -bountiful maker has not left us wholly without means of delighting -our selves in the contemplation of his wisdom. That by a just way of -inquiry into nature, we could not fail of arriving at discoveries -very remote from our apprehensions; the Lord ~BACON~ himself -argues from the experience of mankind. If, says he, the force of guns -should be described to any one ignorant of them, by their effects only, -he might reasonably suppose, that those engines of destruction were -only a more artificial composition, than he knew, of wheels and other -mechanical powers: but it could never enter his thoughts, that their -immense force should be owing to a peculiar substance, which would -enkindle into so violent an explosion, as we experience in gunpowder: -since he would no where see the least example of any such operation; -except perhaps in earthquakes and thunder, which he would doubtless -look upon as exalted powers of nature, greatly surpassing any art of -man to imitate. In the same manner, if a stranger to the original -of silk were shewn a garment made of it, he would be very far from -imagining so strong a substance to be spun out of the bowels of a small -worm; but must certainly believe it either a vegetable substance, like -flax or cotton; or the natural covering of some animal, as wool is -of sheep. Or had we been told, before the invention of the magnetic -needle among us, that another people was in possession of a certain -contrivance, by which they were inabled to discover the position of -the heavens, with vastly more ease, than we could do; what could have -been imagined more, than that they were provided with some fitter -astronomical instrument for this purpose than we? That any stone should -have so amazing a property, as we find in the magnet, must have been -the remotest from our thoughts[33]. - -14. BUT what surprizing advancements in the knowledge of nature may be -made by pursuing the true course in philosophical inquiries; when those -searches are conducted by a genius equal to so divine a work, will be -best understood by considering Sir ~ISAAC NEWTON~ discoveries. -That my’s reader may apprehend as just a notion of these, as can be -conveyed to him, by the brief account, which I intend to lay before -him; I have set apart this introduction for explaining, in the fullest -manner I am able, the principles, whereon Sir ~ISAAC NEWTON~ proceeds. -For without a clear conception of these, it is impossible to form any -true idea of the singular excellence of the inventions of this great -philosopher. - -15. THE principles then of this philosophy are; upon no consideration -to indulge conjectures concerning the powers and laws of nature, but -to make it our endeavour with all diligence to search out the real -and true laws, by which the constitution of things is regulated. The -philosopher’s first care must be to distinguish, what he sees to be -within his power, from what is beyond his reach; to assume no greater -degree of knowledge, than what he finds himself possessed of; but to -advance by slow and cautious steps; to search gradually into natural -causes; to secure to himself the knowledge of the most immediate cause -of each appearance, before he extends his views farther to causes -more remote. This is the method, in which philosophy ought to be -cultivated; which does not pretend to so great things, as the more -airy speculations; but will perform abundantly more: we shall not -perhaps seem to the unskilful to know so much, but our real knowledge -will be greater. And certainly it is no objection against this method, -that some others promise, what is nearer to the extent of our wishes: -since this, if it will not teach us all we could desire to be informed -of, will however give us some true light into nature; which no other -can do. Nor has the philosopher any reason to think his labour lost, -when he finds himself stopt at the cause first discovered by him, or -at any other more remote cause, short of the original: for if he has -but sufficiently proved any one cause, he has entered so far into the -real constitution of things, has laid a safe foundation for others to -work upon, and has facilitated their endeavours in the search after -yet more distant causes; and besides, in the mean time he may apply -the knowledge of these intermediate causes to many useful purposes. -Indeed the being able to make practical deductions from natural causes, -constitutes the great distinction between the true philosophy and the -false. Causes assumed upon conjecture, must be so loose and undefined, -that nothing particular can be collected from them. But those causes, -which are brought to light by a strict examination of things, will be -more distinct. Hence it appears to have been no unuseful discovery, -that the ascent of water in pumps is owing to the pressure of the -air by its weight or spring; though the causes, which make the air -gravitate, and render it elastic, be unknown: for notwithstanding -we are ignorant of the original, whence these powers of the air are -derived; yet we may receive much advantage from the bare knowledge of -these powers. If we are but certain of the degree of force, wherewith -they act, we shall know the extent of what is to be expected from -them; we shall know the greatest height, to which it is possible by -pumps to raise water; and shall thereby be prevented from making any -useless efforts towards improving these instruments beyond the limits -prescribed to them by nature; whereas without so much knowledge as -this, we might probably have wasted in attempts of this kind much time -and labour. How long did philosophers busy themselves to no purpose -in endeavouring to perfect telescopes, by forming the glasses into -some new figure; till Sir ~ISAAC NEWTON~ demonstrated, that -the effects of telescopes were limited from another cause, than was -supposed; which no alteration in the figure of the glasses could -remedy? What method Sir ~ISAAC NEWTON~ himself has found for -the improvement of telescopes shall be explained hereafter[34]. But -at present I shall proceed to illustrate, by some farther instances, -this distinguishing character of the true philosophy, which we have now -under consideration. It was no trifling discovery, that the contraction -of the muscles of animals puts their limbs in motion, though the -original cause of that contraction remains a secret, and perhaps may -always do so; for the knowledge of thus much only has given rise to -many speculations upon the force and artificial disposition of the -muscles, and has opened no narrow prospect into the animal fabrick. -The finding out, that the nerves are great agents in this action, -leads us yet nearer to the original cause, and yields us a wider -view of the subject. And each of these steps affords us assistance -towards restoring this animal motion, when impaired in our selves, -by pointing out the seats of the injuries, to which it is obnoxious. -To neglect all this, because we can hitherto advance no farther, is -plainly ridiculous. It is confessed by all, that ~GALILEO~ -greatly improved philosophy, by shewing, as we shall relate hereafter, -that the power in bodies, which we call gravity, occasions them to -move downwards with a velocity equably accelerated[35]; and that when -any body is thrown forwards, the same power obliges it to describe in -its motion that line, which is called by geometers a parabola[36]: -yet we are ignorant of the cause, which makes bodies gravitate. But -although we are unacquainted with the spring, whence this power in -nature is derived, nevertheless we can estimate its effects. When a -body falls perpendicularly, it is known, how long time it takes in -descending from any height whatever: and if it be thrown forwards, -we know the real path, which it describes; we can determine in what -direction, and with what degree of swiftness it must be projected, -in order to its striking against any object desired; and we can also -ascertain the very force, wherewith it will strike. Sir ~ISAAC -NEWTON~ has farther taught, that this power of gravitation extends -up to the moon, and causes that planet to gravitate as much towards -the earth, as any of the bodies, which are familiar to us, would, if -placed at the same distance[37]: he has proved likewise, that all -the planets gravitate towards the sun, and towards one another; and -that their respective motions follow from this gravitation. All this -he has demonstrated upon indisputable geometrical principles, which -cannot be rendered precarious for want of knowing what it is, which -causes these bodies thus mutually to gravitate: any more than we can -doubt of the propensity in all the bodies about us, to descend towards -the earth; or can call in question the forementioned propositions -of ~GALILEO~, which are built upon that principle. And as -~GALILEO~ has shewn more fully, than was known before, what -effects were produced in the motion of bodies by their gravitation -towards the earth; so Sir ~ISAAC NEWTON~, by this his -invention, has much advanced our knowledge in the celestial motions. -By discovering that the moon gravitates towards the sun, as well as -towards the earth; he has laid open those intricacies in the moon’s -motion, which no astronomer, from observations only, could ever find -out[38]: and one kind of heavenly bodies, the comets, have their motion -now clearly ascertained; whereof we had before no true knowledge at -all[39]. - -16. DOUBTLESS it might be expected, that such surprizing success should -have silenced, at once, every cavil. But we have seen the contrary. -For because this philosophy professes modestly to keep within the -extent of our faculties, and is ready to confess its imperfections, -rather than to make any fruitless attempts to conceal them, by seeking -to cover the defects in our knowledge with the vain ostentation of -rash and groundless conjectures; hence has been taken an occasion -to insinuate that we are led to miraculous causes, and the occult -qualities of the schools. - -17. BUT the first of these accusations is very extraordinary. If by -calling these causes miraculous nothing more is meant than only, that -they often appear to us wonderful and surprizing, it is not easy -to see what difficulty can be raised from thence; for the works of -nature discover every where such proofs of the unbounded power, and -the consummate wisdom of their author, that the more they are known, -the more they will excite our admiration: and it is too manifest to -be insisted on, that the common sense of the word miraculous can have -no place here, when it implies what is above the ordinary course of -things. The other imputation, that these causes are occult upon the -account of our not perceiving what produces them, contains in it great -ambiguity. That something relating to them lies hid, the followers -of this philosophy are ready to acknowledge, nay desire it should be -carefully remarked, as pointing out proper subjects for future inquiry. -But this is very different from the proceeding of the schoolmen in -the causes called by them occult. For as their occult qualities were -understood to operate in a manner occult, and not apprehended by us; so -they were obtruded upon us for such original and essential properties -in bodies, as made it vain to seek any farther cause; and a greater -power was attributed to them, than any natural appearances authorized. -For instance, the rise of water in pumps was ascribed to a certain -abhorrence of a vacuum, which they thought fit to assign to nature. And -this was so far a true observation, that the water does move, contrary -to its usual course, into the space, which otherwise would be left -void of any sensible matter; and, that the procuring such a vacuity -was the apparent cause of the water’s ascent. But while we were not in -the least informed how this power, called an abhorrence of a vacuum, -produced the visible effects; instead of making any advancement in the -knowledge of nature, we only gave an artificial name to one of her -operations: and when the speculation was pushed so beyond what any -appearances required, as to have it concluded, that this abhorrence -of a vacuum was a power inherent in all matter, and so unlimited as -to render it impossible for a vacuum to exist at all; it then became -a much greater absurdity, in being made the foundation of a most -ridiculous manner of reasoning; as at length evidently appeared, when -it came to be discovered, that this rise of the water followed only -from the pressure of the air, and extended it self no farther, than -the power of that cause. The scholastic stile in discoursing of these -occult qualities, as if they were essential differences in the very -substances, of which bodies consisted, was certainly very absurd; by -reason it tended to discourage all farther inquiry. But no such ill -consequences can follow from the considering of any natural causes, -which confessedly are not traced up to their first original. How -shall we ever come to the knowledge of the several original causes of -things, otherwise than by storing up all intermediate causes which we -can discover? Are all the original and essential properties of matter -so very obvious, that none of them can escape our first view? This is -not probable. It is much more likely, that, if some of the essential -properties are discovered by our first observations, a stricter -examination should bring more to light. - - -18. BUT in order to clear up this point concerning the essential -properties of matter, let us consider the subject a little distinctly. -We are to conceive, that the matter, out of which the universe of -things is formed, is furnished with certain qualities and powers, -whereby it is rendered fit to answer the purposes, for which it was -created. But every property, of which any particle of this matter is -in it self possessed, and which is not barely the consequence of the -union of this particle with other portions of matter, we may call an -essential property: whereas all other qualities or attributes belonging -to bodies, which depend on their particular frame and composition, are -not essential to the matter, whereof such bodies are made; because -the matter of these bodies will be deprived of those qualities, only -by the dissolution of the body, without working any change in the -original constitution of one single particle of this mass of matter. -Extension we apprehend to be one of these essential properties, and -impenetrability another. These two belong universally to all matter; -and are the principal ingredients in the idea, which this word matter -usually excites in the mind. Yet as the idea, marked by this name, -is not purely the creature of our own understandings, but is taken -for the representation of a certain substance without us; if we -should discover, that every part of the substance, in which we find -these two properties, should likewise be endowed universally with any -other essential qualities; all these, from the time they come to our -notice, must be united under our general idea of matter. How many -such properties there are actually in all matter we know not; those, -of which we are at present apprized, have been found out only by our -observations on things; how many more a farther search may bring to -light, no one can say; nor are we certain, that we are provided with -sufficient methods of perception to discern them all. Therefore, since -we have no other way of making discoveries in nature, but by gradual -inquiries into the properties of bodies; our first step must be to -admit without distinction all the properties, which we observe; and -afterwards we must endeavour, as far as we are able, to distinguish -between the qualities, wherewith the very substances themselves are -indued, and those appearances, which result from the structure only of -compound bodies. Some of the properties, which we observe in things, -are the attributes of particular bodies only; others universally belong -to all, that fall under our notice. Whether some of the qualities and -powers of particular bodies, be derived from different kinds of matter -entring their composition, cannot, in the present imperfect state -of our knowledge, absolutely be decided; though we have not yet any -reason to conclude, but that all the bodies, with which we converse, -are framed out of the very same kind of matter, and that their -distinct qualities are occasioned only by their structure; through -the variety whereof the general powers of matter are caused to produce -different effects. On the other hand, we should not hastily conclude, -that whatever is found to appertain to all matter, which falls under -our examination, must for that reason only be an essential property -thereof, and not be derived from some unseen disposition in the frame -of nature. Sir ~ISAAC NEWTON~ has found reason to conclude, -that gravity is a property universally belonging to all the perceptible -bodies in the universe, and to every particle of matter, whereof they -are composed. But yet he no where asserts this property to be essential -to matter. And he was so far from having any design of establishing -it as such, that, on the contrary, he has given some hints worthy of -himself at a cause for it[40]; and expresly says, that he proposed -those hints to shew, that he had no such intention[41]. - -19. IT appears from hence, that it is not easy to determine, what -properties of bodies are essentially inherent in the matter, out of -which they are made, and what depend upon their frame and composition. -But certainly whatever properties are found to belong either to any -particular systems of matter, or universally to all, must be considered -in philosophy; because philosophy will be otherwise imperfect. Whether -those properties can be deduced from some other appertaining to -matter, either among those, which are already known, or among such as -can be discovered by us, is afterwards to be sought for the farther -improvement of our knowledge. But this inquiry cannot properly have -place in the deliberation about admitting any property of matter or -bodies into philosophy; for that purpose it is only to be considered, -whether the existence of such a property has been justly proved or not. -Therefore to decide what causes of things are rightly received into -natural philosophy, requires only a distinct and clear conception of -what kind of reasoning is to be allowed of as convincing, when we argue -upon the works of nature. - -20. THE proofs in natural philosophy cannot be so absolutely -conclusive, as in the mathematics. For the subjects of that science -are purely the ideas of our own minds. They may be represented to our -senses by material objects, but they are themselves the arbitrary -productions of our own thoughts; so that as the mind can have a full -and adequate knowledge of its own ideas, the reasoning in geometry -can be rendered perfect. But in natural knowledge the subject of -our contemplation is without us, and not so compleatly to be known: -therefore our method of arguing must fall a little short of absolute -perfection. It is only here required to steer a just course between -the conjectural method of proceeding, against which I have so largely -spoke; and demanding so rigorous a proof, as will reduce all philosophy -to mere scepticism, and exclude all prospect of making any progress in -the knowledge of nature. - -21. THE concessions, which are to be allowed in this science, are by -Sir ~ISAAC NEWTON~ included under a very few simple precepts. - -22. THE first is, that more causes are not to be received into -philosophy, than are sufficient to explain the appearances of nature. -That this rule is approved of unanimously, is evident from those -expressions so frequent among all philosophers, that nature does -nothing in vain; and that a variety of means, where fewer would -suffice, is needless. And certainly there is the highest reason -for complying with this rule. For should we indulge the liberty of -multiplying, without necessity, the causes of things, it would reduce -all philosophy to mere uncertainty; since the only proof, which we can -have, of the existence of a cause, is the necessity of it for producing -known effects. Therefore where one cause is sufficient, if there really -should in nature be two, which is in the last degree improbable, we can -have no possible means of knowing it, and consequently ought not to -take the liberty of imagining, that there are more than one. - -23. THE second precept is the direct consequence of the first, that to -like effects are to be ascribed the same causes. For instance, that -respiration in men and in brutes is brought about by the same means; -that bodies descend to the earth here in EUROPE, and in AMERICA from -the same principle; that the light of a culinary fire, and of the sun -have the same manner of production; that the reflection of light is -effected in the earth, and in the planets by the same power; and the -like. - -24. THE third of these precepts has equally evident reason for it. -It is only, that those qualities, which in the same body can neither -be lessened nor increased, and which belong to all bodies that are -in our power to make trial upon, ought to be accounted the universal -properties of all bodies whatever. - -25. IN this precept is founded that method of arguing by induction, -without which no progress could be made in natural philosophy. For as -the qualities of bodies become known to us by experiments only; we -have no other way of finding the properties of such bodies, as are -out of our reach to experiment upon, but by drawing conclusions from -those which fall under our examination. The only caution here required -is, that the observations and experiments, we argue upon, be numerous -enough, and that due regard be paid to all objections, that occur, as -the Lord BACON very judiciously directs[42]. And this admonition is -sufficiently complied with, when by virtue of this rule we ascribe -impenetrability and extension to all bodies, though we have no sensible -experiment, that affords a direct proof of any of the celestial bodies -being impenetrable; nor that the fixed stars are so much as extended. -For the more perfect our instruments are, whereby we attempt to find -their visible magnitude, the less they appear; insomuch that all the -sensible magnitude, which we observe in them, seems only to be an -optical deception by the scattering of their light. However, I suppose -no one will imagine they are without any magnitude, though their -immense distance makes it undiscernable by us. After the same manner, -if it can be proved, that all bodies here gravitate towards the earth, -in proportion to the quantity of solid matter in each; and that the -moon gravitates to the earth likewise, in proportion to the quantity -of matter in it; and that the sea gravitates towards the moon, and -all the planets towards each other; and that the very comets have the -same gravitating faculty; we shall have as great reason to conclude by -this rule, that all bodies gravitate towards each other. For indeed -this rule will more strongly hold in this case, than in that of the -impenetrability of bodies; because there will more instances be had of -bodies gravitating, than of their being impenetrable. - -25. THIS is that method of induction, whereon all philosophy is -founded; which our author farther inforces by this additional precept, -that whatever is collected from this induction, ought to be received, -notwithstanding any conjectural hypothesis to the contrary, till such -times as it shall be contradicted or limited by farther observations on -nature. - -[Illustration] - -[Illustration] - - - - - ~BOOK I.~ - CONCERNING THE - MOTION of BODIES - IN GENERAL. - - - CHAP. I. - Of the LAWS of MOTION. - -HAVING thus explained Sir ~ISAAC NEWTON’s~ method of reasoning -in philosophy, I shall now proceed to my intended account of his -discoveries. These are contained in two treatises. In one of them, the -MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY, his chief design is to -shew by what laws the heavenly motions are regulated; in the other, -his OPTICS, he discourses of the nature of light and colours, and of -the action between light and bodies. This second treatise is wholly -confined to the subject of light: except some conjectures proposed -at the end concerning other parts of nature, which lie hitherto more -concealed. In the other treatise our author was obliged to smooth the -way to his principal intention, by explaining many things of a more -general nature: for even some of the most simple properties of matter -were scarce well established at that time. We may therefore reduce Sir -~ISAAC NEWTON~’s doctrine under three general heads; and I -shall accordingly divide my account into three books. In the first I -shall speak of what he has delivered concerning the motion of bodies, -without regard to any particular system of matter; in the second I -shall treat of the heavenly motions; and the third shall be employed -upon light. - -2. IN the first part of my design, we must begin with an account of the -general laws of motion. - -3. THESE laws are some universal affections and properties of matter -drawn from experience, which are made use of as axioms and evident -principles in all our arguings upon the motion of bodies. For as it -is the custom of geometers to assume in their demonstrations some -propositions, without exhibiting the proof of them; so in philosophy, -all our reasoning must be built upon some properties of matter, first -admitted as principles whereon to argue. In geometry these axioms are -thus assumed, on account of their being so evident as to make any -proof in form needless. But in philosophy no properties of bodies can -be in this manner received for self-evident; since it has been observed -above, that we can conclude nothing concerning matter by any reasonings -upon its nature and essence, but that we owe all the knowledge, we -have thereof, to experience. Yet when our observations on matter have -inform’d us of some of its properties, we may securely reason upon them -in our farther inquiries into nature. And these laws of motion, of -which I am here to speak, are found so universally to belong to bodies, -that there is no motion known, which is not regulated by them. These -are by Sir ~ISAAC NEWTON~ reduced to three[43]. - -4. THE first law is, that all bodies have such an indifference to rest, -or motion, that if once at rest they remain so, till disturbed by some -power acting upon them: but if once put in motion, they persist in -it; continuing to move right forwards perpetually, after the power, -which gave the motion, is removed; and also preserving the same degree -of velocity or quickness, as was first communicated, not stopping or -remitting their course, till interrupted or otherwise disturbed by some -new power impressed. - -5. THE second law of motion is, that the alteration of the state of -any body, whether from rest to motion, or from motion to rest, or -from one degree of motion to another, is always proportional to the -force impressed. A body at rest, when acted upon by any power, yields -to that power, moving in the same line, in which the power applied -is directed; and moves with a less or greater degree of velocity, -according to the degree of the power; so that twice the power shall -communicate a double velocity, and three times the power a threefold -velocity. If the body be moving, and the power impressed act upon the -body in the direction of its motion, the body shall receive an addition -to its motion, as great as the motion, into which that power would -have put it from a state of rest; but if the power impressed upon a -moving body act directly opposite to its former motion, that power -shall then take away from the body’s motion, as much as in the other -case it would have added to it. Lastly, if the power be impressed -obliquely, there will arise an oblique motion differing more or less -from the former direction, according as the new impression is greater -or less. For example, if the body A (in fig. 1.) be moving in the -direction A B, and when it is at the point A, a power be impressed upon -it in the direction A C, the body shall from henceforth neither move -in its first direction A B, nor in the direction of the adventitious -power, but shall take a course as A D between them: and if the power -last impressed be just equal to that, which first gave to the body -its motion; the line A D shall pass in the middle between A B and A -C, dividing the angle under B A C into two equal parts; but if the -power last impressed be greater than the first, the line A D shall -incline most to A C; whereas if the last impression be less than the -first, the line A D shall incline most to A B. To be more particular, -the situation of the line A D is always to be determined after this -manner. Let A E be the space, which the body would have moved through -in the line A B during any certain portion of time; provided that body, -when at A, had received no second impulse. Suppose likewise, that A F -is the part of the line A C, through which the body would have moved -during an equal portion of time, if it had been at rest in A, when it -received the impulse in the direction A C: then if from E be drawn a -line parallel to, or equidistant from A C, and from F another line -parallel to A B, those two lines will meet in the line A D. - -6. THE third and last of these laws of motion is, that when any body -acts upon another, the action of that body upon the other is equalled -by the contrary reaction of that other body upon the first. - -7. THESE laws of motion are abundantly confirmed by this, that all the -deductions made from them, in relation to the motion of bodies, how -complicated soever, are found to agree perfectly with observation. This -shall be shewn more at large in the next chapter. But before we proceed -to so diffusive a proof; I chuse here to point out those appearances of -bodies, whereby the laws of motion are first suggested to us. - -8. DAILY observation makes it appear to us, that any body, which we -once see at rest, never puts it self into fresh motion; but continues -always in the same place, till removed by some power applied to it. - -9. AGAIN, whenever a body is once in motion, it continues in that -motion some time after the moving power has quitted it, and it is left -to it self. Now if the body continue to move but a single moment, after -the moving power has left it, there can no reason be assigned, why it -should ever stop without some external force. For it is plain, that -this continuance of the motion is caused only by the body’s having -already moved, the sole operation of the power upon the body being the -putting it in motion; therefore that motion continued will equally be -the cause of its farther motion, and so on without end. The only doubt -that can remain, is, whether this motion communicated continues intire, -after the power, that caused it, ceases to act; or whether it does not -gradually languish and decrease. And this suspicion cannot be removed -by a transient and slight observation on bodies, but will be fully -cleared up by those more accurate proofs of the laws of motion, which -are to be considered in the next chapter. - -10. LASTLY, bodies in motion appear to affect a straight course without -any deviation, unless when disturbed by some adventitious power acting -upon them. If a body be thrown perpendicularly upwards or downwards, -it appears to continue in the same straight line during the whole time -of its motion. If a body be thrown in any other direction, it is found -to deviate from the line, in which it began to move, more and more -continually towards the earth, whither it is directed by its weight: -but since, when the weight of a body does not alter the direction of -its motion, it always moves in a straight line, without doubt in this -other case the body’s, declining from its first course is no more, -than what is caused by its weight alone. As this appears at first -sight to be unquestionable, so we shall have a very distinct proof -thereof in the next chapter, where the oblique motion of bodies will be -particularly considered. - -11. THUS we see how the first of the laws of motion agrees with -what appears to us in moving bodies. But here occurs this farther -consideration, that the real and absolute motion of any body is not -visible to us: for we are our selves also in constant motion along with -the earth whereon we dwell; insomuch that we perceive bodies to move -so far only, as their motion is different from our own. When a body -appears to us to lie at rest, in reality it only continues the motion, -it has received, without putting forth any power to change that motion. -If we throw a body in the course or direction, wherein we are carried -our selves; so much motion as we seem to have given to the body, so -much we have truly added to the motion, it had, while it appeared to us -to be at rest. But if we impel a body the contrary way, although the -body appears to us to have received by such an impulse as much motion, -as when impelled the other way; yet in this case we have taken from the -body so much real motion, as we seem to have given it. Thus the motion, -which we see in bodies, is not their real motion, but only relative -with respect to us; and the forementioned observations only shew us, -that this first law of motion has place in this relative or apparent -motion. However, though we cannot make any observation immediately on -the absolute motion of bodies, yet by reasoning upon what we observe -in visible motion, we can discover the properties and effects of real -motion. - -12. WITH regard to this first law of motion, which is now under -consideration, we may from the foregoing observations most truly -collect, that bodies are disposed to continue in the absolute motion, -which they have once received, without increasing or diminishing their -velocity. When a body appears to us to lie at rest, it really preserves -without change the motion, which it has in common with our selves: -and when we put it into visible motion, and we see it continue that -motion; this proves, that the body retains that degree of its absolute -motion, into which it is put by our acting upon it: if we give it such -an apparent motion, which adds to its real motion, it preserves that -addition; and if our acting on the body takes off from its real motion, -it continues afterwards to move with no more real motion, than we have -left it. - -13. AGAIN, we do not observe in bodies any disposition or power within -themselves to change the direction of their motion; and if they had any -such power, it would easily be discovered. For suppose a body by the -structure or disposition of its parts, or by any other circumstance in -its make, was indued with a power of moving it self; this self-moving -principle, which should be thus inherent in the body, and not depend on -any thing external, must change the direction wherein it would act, as -often as the position of the body was changed: so that for instance, -if a body was lying before me in such a position, that the direction, -wherein this principle disposes the body to move, was pointed directly -from me; if I then gradually turned the body about, the direction -of this self-moving principle would no longer be pointed directly -from me, but would turn about along with the body. Now if any body, -which appears to us at rest, were furnished with any such self-moving -principle; from the body’s appearing without motion we must conclude, -that this self-moving principle lies directed the same way as the -earth is carrying the body; and such a body might immediately be put -into visible motion only by turning it about in any degree, that this -self-moving principle might receive a different direction. - -14. FROM these considerations it very plainly follows, that if a body -were once absolutely at rest; not being furnished with any principle, -whereby it could put it self into motion, it must for ever continue in -the same place, till acted upon by something external: and also that -when a body is put into motion, it has no power within it self to make -any change in the direction of that motion; and consequently that the -body must move on straight forward without declining any way whatever. -But it has before been shewn, that bodies do not appear to have in -themselves any power to change the velocity of their motion: therefore -this first law of motion has been illustrated and confirmed, as much as -can be from the transient observations, which have here been discoursed -upon; and in the next chapter all this will be farther established by -more correct observations. - -15. BUT I shall now pass to the second law of motion; wherein, when it -is asserted, that the velocity, with which any body is moved by the -action of a power upon it, is proportional to that power; the degree of -power is supposed to be measured by the greatness of the body, which -it can move with a given celerity. So that the sense of this law is, -that if any body were put into motion with that degree of swiftness, as -to pass in one hour the length of a thousand yards; the power, which -would give the same degree of velocity to a body twice as great, would -give this lesser body twice the velocity, causing it to describe in the -same space of an hour two thousand yards. But by a body twice as great -as another, I do not here mean simply of twice the bulk, but one that -contains a double quantity of solid matter. - -16. WHY the power, which can move a body twice as great as another with -the same degree of velocity, should be called twice as great as the -power, which can give the lesser body the same velocity, is evident. -For if we should suppose the greater body to be divided into two equal -parts, each equal to the lesser body, each of these halves will require -the same degree of power to move them with the velocity of the lesser -body, as the lesser body it self requires; and therefore both those -halves, or the whole greater body, will require the moving power to be -doubled. - -17. THAT the moving power being in this sense doubled, should just -double likewise the velocity of the same body, seems near as evident, -if we consider, that the effect of the power applied must needs be -the same, whether that power be applied to the body at once, or in -parts. Suppose then the double power not applied to the body at -once, but half of it first, and afterwards the other half; it is not -conceivable for what reason the half last applied should come to have -a different effect upon the body, from that which is applied first; -as it must have, if the velocity of the body was not just doubled by -the application of it. So far as experience can determine, we see -nothing to favour such a supposition. We cannot indeed (by reason of -the constant motion of the earth) make trial upon any body perfectly at -rest, whereby to see whether a power applied in that case would have a -different effect, from what it has, when the body is already moving; -but we find no alteration in the effect of the same power on account of -any difference there may be in the motion of the body, when the power -is applied. The earth does not always carry bodies with the same degree -of velocity; yet we find the visible effects of any power applied to -the same body to be, at all times the very same: and a bale of goods, -or other moveable body lying in a ship is as easily removed from place -to place, while the ship is under sail, if its motion be steady, as -when it is fixed at anchor. - -18. NOW this experience is alone sufficient to shew to us the whole of -this law of motion. - -19. SINCE we find, that the same power will always produce the same -change in the motion of any body, whether that body were before moving -with a swifter or slower motion; the change wrought in the motion of -a body depends only on the power applied to it, without any regard to -the body’s former motion: and therefore the degree of motion, which the -body already possesses, having no influence on the power applied to -disturb its operation, the effects of the same power will not only be -the same in all degrees of motion of the body; but we have likewise no -reason to doubt, but that a body perfectly at rest would receive from -any power as much motion, as would be equivalent to the effect of the -same power applied to that body already in motion. - -20. AGAIN, suppose a body being at rest, any number of equal powers -should be successively applied to it; pushing it forward from time to -time in the same course or direction. Upon the application of the first -power the body would begin to move; when the second power was applied, -it appears from what has been said, that the motion of the body would -become double; the third power would treble the motion of the body; and -so on, till after the operation of the last power the motion of the -body would be as many times the motion, which the first power gave it, -as there are powers in number. and the effect of this number of powers -will be always the same, without any regard to the space of time taken -up in applying them: so that greater or lesser intervals between the -application of each of these powers will produce no difference at all -in their effects. Since therefore the distance of time between the -action of each power is of no consequence; without doubt the effect -will still be the same, though the powers should all be applied at -the very same instant; or although a single power should be applied -equal in strength to the collective force of all these powers. Hence -it plainly follows, that the degree of motion, into which any body -will be put out of a state of rest by any power, will be proportional -to that power. A double power will give twice the velocity, a treble -power three times the velocity, and so on. The foregoing reasoning will -equally take place, though the body were not supposed to be at rest, -when the powers began to be applied to it; provided the direction, in -which the powers were applied, either conspired with the action of -the body, or was directly opposite to it. Therefore if any power be -applied to a moving body, and act upon the body either in the direction -wherewith the body moves, so as to accelerate the body; or if it act -directly opposite to the motion of the body, so as to retard it: in -both these cases the change of motion will be proportional to the -power applied; nay, the augmentation of the motion in one case, and -the diminution thereof in the other, will be equal to that degree of -motion, into which the same power would put the body, had it been at -rest, when the power was applied. - -21. FARTHER, a power may be so applied to a moving body, as to act -obliquely to the motion of the body. And the effects of such an oblique -motion may be deduced from this observation; that as all bodies are -continually moving along with the earth, we see that the visible -effects of the same power are always the same, in whatever direction -the power acts: and therefore the visible effects of any power upon -a body, which seems only to be at rest, is always to appearance the -same as the real effect would be upon a body truly at rest. Now -suppose a body were moving along the line A B (in fig. 2.) and the eye -accompanied it with an equal motion in the line C D equidistant from A -B; so that when the body is at A, the eye shall be at C, and when the -body is advanced to E in the line A B, the eye shall be advanced to F -in the line C D, the distances A E and C F being equal. It is evident, -that here the body will appear to the eye to be at rest; and the line -F E G drawn from the eye through the body shall seem to the eye to be -immoveable; though as the body and eye move forward together, this -line shall really also move; so that when the body shall be advanced -to H and the eye to K, the line F E G shall be transferred into the -situation K H L, this line K H L being equidistant from F E G. Now -if the body when at E were to receive an impulse in the direction of -the line F E G; while the eye is moving on from F to K and carrying -along with it the line F E G, the body will appear to the eye to move -along this line F E G: for this is what has just now been said; that -while bodies are moving along with the earth, and the spectator’s eye -partakes of the same motion, the effect of any power upon the body -will appear to be what it would really have been, had the body been -truly at rest, when the power was applied. From hence it follows, that -when the eye is advanced to K, the body will appear somewhere in the -line K H L. Suppose it appear in M; then it is manifest, from what has -been premised at the beginning of this paragraph, that the distance H -M is equal to what the body would have run upon the line E G, during -the time, wherein the eye has passed from F to K, provided that the -body had been at rest, when acted upon in E. If it be farther asked, -after what manner the body has moved from E to M? I answer, through a -straight line; for it has been shewn above in the explication of the -first law of motion, that a moving body, from the time it is left to it -self, will proceed on in one continued straight line. - -22. IF E N be taken equal to H M and N M be drawn; since H M is -equidistant from E N, N M will be equidistant from E H. Therefore the -effect of any power upon a moving body, when that power acts obliquely -to the motion of the body, is to be determined in this manner. Suppose -the body is moving along the straight line A E B, if when the body is -come to E, a power gives it an impulse in the direction of the line E -G, to find what course the body will afterwards take we must proceed -thus. Take in E B any length E H, and in E G take such a length E N, -that if the body had been at rest in E, the power applied to it would -have caused it to move over E N in the same space of time, as it would -have employed in passing over E H, if the power had not acted at all -upon it. Then draw H L equidistant from E G, and N M equidistant from -E B. After this, if a line be drawn from E to the point M, where these -two lines meet, the line E M will be the course into which the body -will be put by the action of the power upon it at E. - -23. A MATHEMATICAL reader would here expect in some particulars more -regular demonstrations; but as I do not at present address my self to -such, so I hope, what I have now written will render my meaning evident -enough to those, who are unacquainted with that kind of reasoning. - -24. NOW as we have been shewing, that some actual force is necessary -either to put bodies out of a state of rest into motion, or to change -the motion, which they have once received; it is proper here to -observe, that this quality in bodies, whereby they preserve their -present state, with regard to motion or rest, till some active force -disturb them, is called the ~VIS INERTIAE~ of matter: and -by this property, matter, sluggish and unactive of it self, retains -all the power impressed upon it, and cannot be made to cease from -action, but by the opposition of as great a power, as that which first -moved it. By the degree of this ~VIS INERTIAE~, or power of -inactivity, as we shall henceforth call it, we primarily judge of the -quantity of solid matter in each body; for as this quality is inherent -in all the bodies, upon which we can make any trial, we conclude it to -be a property essential to all matter; and as we yet know no reason -to suppose, that bodies are composed of different kinds of matter, we -rather presume, that the matter of all bodies is the same; and that -the degree of this power of inactivity is in every body proportional -to the quantity of the solid matter in it. But although we have no -absolute proof, that all the matter in the universe is uniform, and -possesses this power of inactivity in the same degree; yet we can with -certainty compare together the different degrees of this power of -inactivity in different bodies. Particularly this power is proportional -to the weight of bodies, as Sir ~ISAAC NEWTON~ has demonstrated[44]. -However, notwithstanding that this power of inactivity in any body can -be more certainly known, than the quantity of solid matter in it; yet -since there is no reason to suspect that one is not proportional to -the other, we shall hereafter speak without hesitation of the quantity -of matter in bodies, as the measure of the degree of their power of -inactivity. - -25. THIS being established, we may now compare the effects of the -same power upon different bodies, as hitherto we have shewn the -effects of different powers upon the same body. And here if we limit -the word motion to the peculiar sense given to it in philosophy, we -may comprehend all that is to be said upon this head under one short -precept; that the same power, to whatever body it is applied, will -always produce the same degree of motion. But here motion does not -signify the degree of celerity or velocity with which a body moves, -in which sense only we have hitherto used it; but it is made use of -particularly in philosophy to signify the force with which a body -moves: as if two bodies A and B being in motion, twice the force -would be required to stop A as to stop B, the motion of A would be -esteemed double the motion of B. In moving bodies, these two things are -carefully to be distinguished; their velocity or celerity, which is -measured by the space they pass through during any determinate portion -of time; and the quantity of their motion, or the force, with which -they will press against any resistance. Which force, when different -bodies move with the same velocity, is proportional to the quantity of -solid matter in the bodies; but if the bodies are equal, this force is -proportional to their respective velocities, and in other cases it is -proportional both to the quantity of solid matter in the body, and also -to its velocity. To instance in two bodies A and B: if A be twice as -great as B, and they have both the same velocity, the motion of A shall -be double the motion of B; and if the bodies be equal, and the velocity -of A be twice that of B, the motion of A shall likewise be double that -of B; but if A be twice as large as B, and move twice as swift, the -motion of A will be four times the motion of B; and lastly, if A be -twice as large as B, and move but half as fast, the degree of their -motion shall be the same. - -26. THIS is the particular sense given to the word motion by -philosophers, and in this sense of the word the same power always -produces the same quantity or degree of motion. If the same power act -upon two bodies A and B, the velocities, it shall give to each of them, -shall be so adjusted to the respective bodies, that the same degree -of motion shall be produced in each. If A be twice as great as B, its -velocity shall be half that of B; if A has three times as much solid -matter as B, the velocity of A shall be one third of the velocity of B; -and generally the velocity given to A shall bear the same proportion to -the velocity given to B, as the quantity of solid matter contained in -the body B bears to the quantity of solid matter contained in A. - -27. THE reason of all this is evident from what has gone before. If -a power were applied to B, which should bear the same proportion to -the power applied to A, as the body B bears to A, the bodies B and -A would both receive the same velocity; and the velocity, which B -will receive from this power, will bear the same proportion to the -velocity, which it would receive from the action of the power applied -to A, as the former of these powers bears to the latter: that is, the -velocity, which A receives from the power applied to it, will bear -to the velocity, which B would receive from the same power, the same -proportion as the body B bears to A. - -28. FROM hence we may now pass to the third law of motion, where -this distinction between the velocity of a body and its whole motion -is farther necessary to be regarded, as shall immediately be shewn; -after having first illustrated the meaning of this law by a familiar -instance. If a stone or other load be drawn by a horse; the load -re-acts upon the horse, as much as the horse acts upon the load; for -the harness, which is strained between them, presses against the horse -as much as against the load; and the progressive motion of the horse -forward is hindred as much by the load, as the motion of the load is -promoted by the endeavour of the horse: that is, if the horse put forth -the same strength, when loosened from the load, he would move himself -forwards with greater swiftness in proportion to the difference between -the weight of his own body and the weight of himself and load together. - -29. THIS instance will afford some general notion of the meaning of -this law. But to proceed to a more philosophical explication: if a body -in motion strike against another at rest, let the body striking be -ever so small, yet shall it communicate some degree of motion to the -body it strikes against, though the less that body be in comparison -of that it impinges upon, and the less the velocity is, with which -it moves, the smaller will be the motion communicated. But whatever -degree of motion it gives to the resting body, the same it shall lose -it self. This is the necessary consequence of the forementioned power -of inactivity in matter. For suppose the two bodies equal, it is -evident from the time they meet, both the bodies are to be moved by the -single motion of the first; therefore the body in motion by means of -its power of inactivity retaining the motion first given it, strikes -upon the other with the same force, wherewith it was acted upon it -self: but now both the bodies being to be moved by that force, which -before moved one only, the ensuing velocity will be the same, as if the -power, which was applied to one of the bodies, and put it into motion, -had been applied to both; whence it appears, that they will proceed -forwards, with half the velocity, which the body first in motion had: -that is, the body first moved will have lost half its motion, and the -other will have gained exactly as much. This rule is just, provided -the bodies keep contiguous after meeting; as they would always do, if -it were not for a certain cause that often intervenes, and which must -now be explained. Bodies upon striking against each other, suffer an -alteration in their figure, having their parts pressed inwards by the -stroke, which for the most part recoil again afterwards, the bodies -endeavouring to recover their former shape. This power, whereby bodies -are inabled to regain their first figure, is usually called their -elasticity, and when it acts, it forces the bodies from each other, -and causes them to separate. Now the effect of this elasticity in the -present case is such, that if the bodies are perfectly elastic, so -as to recoil with as great a force as they are bent with, that they -recover their figure in the same space of time, as has been taken up -in the alteration made in it by their compression together; then this -power will separate the bodies as swiftly, as they before approached, -and acting upon both equally, upon the body first in motion contrary -to the direction in which it moves, and upon the other as much in the -direction of its motion, it will take from the first, and add to the -other equal degrees of velocity: so that the power being strong enough -to separate them with as great a velocity, as they approached with, the -first will be quite stopt, and that which was at rest, will receive all -the motion of the other. If the bodies are elastic in a less degree, -the first will not lose all its motion, nor will the other acquire -the motion of the first, but fall as much short of it, as the other -retains. For this rule is never deviated from, that though the degree -of elasticity determines how much more than half its velocity the body -first in motion shall lose; yet in every case the loss in the motion -of this body shall be transferred to the other, that other body always -receiving by the stroke as much motion, as is taken from the first. - -30. This is the case of a body striking directly against an equal body -at rest, and the reasoning here used is fully confirmed by experience. -There are many other cases of bodies impinging against one another: but -the mention of these shall be reserved to the next chapter, where we -intend to be more particular and diffusive in the proof of these laws -of motion, than we have been here. - - - - -CHAP. II. - -Farther proofs of the LAWS OF MOTION. - - -HAVING in the preceding chapter deduced the three laws of motion, -delivered by our great philosopher, from the most obvious observations, -that suggest them to us; I now intend to give more particular proofs -of them, by recounting some of the discoveries which have been made in -philosophy before Sir Isaac Newton. For as they were all collected by -reasoning upon those laws; so the conformity of these discoveries to -experience makes them so many proofs of the truth of the principles, -from which they were derived. - -2. LET us begin with the subject, which concluded the last chapter. -Although the body in motion be not equal to the body at rest, on which -it strikes; yet the motion after the stroke is to be estimated in the -same manner as above. Let A (in fig. 3.) be a body in motion towards -another body B lying at rest. When A is arrived at B, it cannot proceed -farther without putting B into motion; and what motion it gives to -B, it must lose it self, that the whole degree of motion of A and B -together, if neither of the bodies be elastic, shall be equal, after -the meeting of the bodies, to the single motion of A before the stroke. -Therefore, from what has been said above, it is manifest, that as soon -as the two bodies are met, they will move on together with a velocity, -which will bear the same proportion to the original velocity of A, as -the body A bears to the sum of both the bodies. - -3. IF the bodies are elastic, so that they shall separate after the -stroke, A must lose a greater part of its motion, and the subsequent -motion of B will be augmented by this elasticity, as much as the motion -of A is diminished by it. The elasticity acting equally between both -the bodies, it will communicate to each the same degree of motion; that -is, it will separate the bodies by taking from the body A and adding -to the body B different degrees of velocity, so proportioned to their -respective quantities of matter, that the degree of motion, wherewith -A separates from B, shall be equal to the degree of motion, wherewith -B separates from A. It follows therefore, that the velocity taken from -A by the elasticity bears to the velocity, which the same elasticity -adds to B, the same proportion, as B bears to A: consequently the -velocity, which the elasticity takes from A, will bear the same -proportion to the whole velocity, wherewith this elasticity causes the -two bodies to separate from each other, as the body B bears to the sum -of the two bodies A and B; and the velocity, which is added to B by the -elasticity, bears to the velocity, wherewith the bodies separate, the -same proportion, as the body A bears to the sum of the two bodies A -and B. Thus is found, how much the elasticity takes from the velocity -of A, and adds to the velocity of B; provided the degree of elasticity -be known, whereby to determine the whole velocity wherewith the bodies -separate from each other after the stroke[45]. - -4. AFTER this manner is determined in every case the result of a body -in motion striking against another at rest. The same principles will -also determine the effects, when both bodies are in motion. - -5. LET two equal bodies move against each other with equal swiftness. -Then the force, with which each of them presses forwards, being equal -when they strike; each pressing in its own direction with the same -energy, neither shall surmount the other, but both be stopt, if they -be not elastic: for if they be elastic, they shall from thence recover -new motion, and recede from each other, as swiftly as they met, if they -be perfectly elastic; but more slowly, if less so. In the same manner, -if two bodies of unequal bigness strike against each other, and their -velocities be so related, that the velocity of the lesser body shall -exceed the velocity of the greater in the same proportion, as the -greater body exceeds the lesser (for instance, if one body contains -twice the solid matter as the other, and moves but half as fast) two -such bodies will entirely suppress each other’s motion, and remain from -the time of their meeting fixed; if, as before, they are not elastic: -but, if they are so in the highest degree, they shall recede again, -each with the same velocity, wherewith they met. For this elastic -power, as in the preceding case, shall renew their motion, and pressing -equally upon both, shall give the same motion to both; that is, shall -cause the velocity, which the lesser body receives, to bear the same -proportion to the velocity, which the greater receives, as the greater -body bears to the lesser: so that the velocities shall bear the same -proportion to each other after the stroke, as before. Therefore if the -bodies, by being perfectly elastic, have the sum of their velocities -after the stroke equal to the sum of their velocities before the -stroke, each body after the stroke will receive its first velocity. -And the same proportion will hold likewise between the velocities, -wherewith they go off, though they are elastic but in a less degree; -only then the velocity of each will be less in proportion to the defect -of elasticity. - -6. IF the velocities, wherewith the bodies meet, are not in the -proportion here supposed; but if one of the bodies, as A, has a swifter -velocity in comparison to the velocity of the other; then the effect -of this excess of velocity in the body A must be joined to the effect -now mentioned, after the manner of this following example. Let A be -twice as great as B, and move with the same swiftness as B. Here A -moves with twice that degree of swiftness, which would answer to the -forementioned proportion. For A being double to B, if it moved but -with half the swiftness, wherewith B advances, it has been just now -shewn, that the two bodies upon meeting would stop, if they were not -elastic; and if they were elastic, that they would each recoil, so as -to cause A to return with half the velocity, wherewith B would return. -But it is evident from hence, that B by encountring A will annul half -its velocity, if the bodies be not elastic; and the future motion of -the bodies will be the same, as if A had advanced against B at rest -with half the velocity here assigned to it. If the bodies be elastic, -the velocity of A and B after the stroke may be thus discovered. As -the two bodies advance against each other, the velocity, with which -they meet, is made up of the velocities of both bodies added together. -After the stroke their elasticity will separate them again. The degree -of elasticity will determine what proportion the velocity, wherewith -they separate, must bear to that, wherewith they meet. Divide this -velocity, with which the bodies separate into two parts, that one of -the parts bear to the other the same proportion, as the body A bears to -B; and ascribe the lesser part to the greater body A, and the greater -part of the velocity to the lesser body B. Then take the part ascribed -to A from the common velocity, which A and B would have had after the -stroke, if they had not been elastic; and add the part ascribed to B to -the same common velocity. By this means the true velocities of A and B -after the stroke will be made known. - -7. IF the bodies are perfectly elastic, the great ~HUYGENS~ -has laid down this rule for finding their motion after concourse[46]. -Any straight line C D (in fig. 4, 5.) being drawn, let it be divided -in E, that C E bear the same proportion to E D, as the swiftness of A -bore to the swiftness of B before the stroke. Let the same line C D be -also divided in F, that C F bear the same proportion to F D, as the -body B bears to the body A. Then F G being taken equal to F E, if the -point G falls within the line C D, both the bodies shall recoil after -the stroke, and the velocity, wherewith the body A shall return, will -bear the same proportion to the velocity, wherewith B shall return, as -G C bears to G D; but if the point G falls without the line C D, then -the bodies after their concourse shall both proceed to move the same -way, and the velocity of A shall bear to the velocity of B the same -proportion, that G C bears to G D, as before. - -8. IF the body B had stood still, and received the impulse of the other -body A upon it; the effect has been already explained in the case, when -the bodies are not elastic. And when they are elastic, the result of -their collision is found by combining the effect of the elasticity with -the other effect, in the same manner as in the last case. - -9. WHEN the bodies are perfectly elastic, the rule of -~HUYGENS~[47] here is to divide the line C D (fig. 6.) in E as -before, and to take E G equal to E D. And by these points thus found, -the motion of each body after the stroke is determined, as before. - -10. IN the next place, suppose the bodies A and B were both moving the -same way, but A with a swifter motion, so as to overtake B, and strike -against it. The effect of the percussion or stroke, when the bodies are -not elastic, is discovered by finding the common motion, which the two -bodies would have after the stroke, if B were at rest, and A were to -advance against it with a velocity equal to the excess of the present -velocity of A above the velocity of B; and by adding to this common -velocity thus found the velocity of B. - -11. IF the bodies are elastic, the effect of the elasticity is to be -united with this other, as in the former cases. - -12. WHEN the bodies are perfectly elastic, the rule of HUYGENS[48] in -this case is to prolong C D (fig. 7.) and to take in it thus prolonged -C E in the same proportion to E D, as the greater velocity of A bears -to the lesser velocity of B; after which F G being taken equal to F E, -the velocities of the two bodies after the stroke will be determined, -as in the two preceding cases. - -13. THUS I have given the sum of what has been written concerning the -effects of percussion, when two bodies freely in motion strike directly -against each other; and the results here set down, as the consequence -of our reasoning from the laws of motion, answer most exactly to -experience. A particular set of experiments has been invented to make -trial of these effects of percussion with the greatest exactness. But -I must defer these experiments, till I have explained the nature of -pendulums[49]. I shall therefore now proceed to describe some of the -appearances, which are caused in bodies from the influence of the power -of gravity united with the general laws of motion; among which the -motion of the pendulum will be included. - -14. THE most simple of these appearances is, when bodies fall down -merely by their weight. In this case the body increases continually -its velocity, during the whole time of its fall, and that in the very -same proportion as the time increases. For the power of gravity acts -constantly on the body with the same degree of strength: and it has -been observed above in the first law of motion, that a body being once -in motion will perpetually preserve that motion without the continuance -of any external influence upon it: therefore, after a body has been -once put in motion by the force of gravity, the body would continue -that motion, though the power of gravity should cease to act any -farther upon it; but, if the power of gravity continues still to draw -the body down, fresh degrees of motion must continually be added to -the body; and the power of gravity acting at all times with the same -strength, equal degrees of motion will constantly be added in equal -portions of time. - -15. THIS conclusion is not indeed absolutely true: for we shall find -hereafter[50], that the power of gravity is not of the same strength at -all distances from the center of the earth. But nothing of this is in -the least sensible in any distance, to which we can convey bodies. The -weight of bodies is the very same to sense upon the highest towers or -mountains, as upon the level ground; so that in all the observations -we can make, the forementioned proportion between the velocity of a -falling body and the time, in which it has been descending, obtains -without any the least perceptible difference. - -16. FROM hence it follows, that the space, through which a body falls, -is not proportional to the time of the fall; for since the body -increases its velocity, a greater space will be passed over in the same -portion of time at the latter part of the fall, than at the beginning. -Suppose a body let fall from the point A (in fig. 8.) were to descend -from A to B in any portion of time; then if in an equal portion of time -it were to proceed from B to C; I say, the space B C is greater than A -B; so that the time of the fall from A to C being double the time of -the fall from A to B, A C shall be more than double of A B. - -17. THE geometers have proved, that the spaces, through which bodies -fall thus by their weight, are just in a duplicate or two-fold -proportion of the times, in which the body has been falling. That is, -if we were to take the line D E in the same proportion to A B, as the -time, which the body has imployed in falling from A to C, bears to the -time of the fall from A to B; then A C will be to D E in the same -proportion. In particular, if the time of the fall through A C be twice -the time of the fall through A B; then D E will be twice A B, and A C -twice D E; or A C four times A B. But if the time of the fall through -A C had been thrice the time of the fall through A B; D E would have -been treble of A B, and A C treble of D E; that is, A C would have been -equal to nine times A B. - -18. IF a body fall obliquely, it will approach the ground by slower -degrees, than when it falls perpendicularly. Suppose two lines A B, A -C (in fig. 9.) were drawn, one perpendicular, and the other oblique to -the ground D E: then if a body were to descend in the slanting line -A C; because the power of gravity draws the body directly downwards, -if the line A C supports the body from falling in that manner, it -must take off part of the effect of the power of gravity; so that -in the time, which would have been sufficient for the body to have -fallen through the whole perpendicular line A B, the body shall not -have passed in the line A C a length equal to A B; consequently the -line A C being longer than A B, the body shall most certainly take up -more time in passing through A C, than it would have done in falling -perpendicularly down through A B. - -19. THE geometers demonstrate, that the time, in which the body -will descend through the oblique straight line A C, bears the same -proportion to the time of its descent through the perpendicular A B, -as the line it self A C bears to A B. And in respect to the velocity, -which the body will have acquired in the point C, they likewise -prove, that the length of the time imployed in the descent through A -C so compensates the diminution of the influence of gravity from the -obliquity of this line, that though the force of the power of gravity -on the body is opposed by the obliquity of the line A C, yet the time -of the body’s descent shall be so much prolonged, that the body shall -acquire the very same velocity in the point C, as it would have got at -the point B by falling perpendicularly down. - -20. IF a body were to descend in a crooked line, the time of its -descent cannot be determined in so simple a manner; but the same -property, in relation to the velocity, is demonstrated to take place in -all cases: that is, in whatever line the body descends, the velocity -will always be answerable to the perpendicular height, from which the -body has fell. For instance, suppose the body A (in fig. 10.) were hung -by a string to the pin B. If this body were let fall, till it came to -the point C perpendicularly under B, it will have moved from A to C in -the arch of a circle. Then the horizontal line A D being drawn, the -velocity of the body in C will be the same, as if it had fallen from -the point D directly down to C. - -21. IF a body be thrown perpendicularly upward with any force, the -velocity, wherewith the body ascends, shall continually diminish, -till at length it be wholly taken away; and from that time the body -will begin to fall down again, and pass over a second time in its -descent the line, wherein it ascended; falling through this line with -an increasing velocity in such a manner, that in every point thereof, -through which it falls, it shall have the very same velocity, as it -had in the same place, when it ascended; and consequently shall come -down into the place, whence it first ascended, with the velocity which -was at first given to it. Thus if a body were thrown perpendicularly -up in the line A B (in fig. II.) with such a force, as that it should -stop at the point B, and there begin to fall again; when it shall have -arrived in its descent to any point as C in this line, it shall there -have the same velocity, as that wherewith it passed by this point C -in its ascent; and at the point A it shall have gained as great a -velocity, as that wherewith it was first thrown upwards. As this is -demonstrated by the geometrical writers; so, I think, it will appear -evident, by considering only, that while the body descends, the power -of gravity must act over again, in an inverted order, all the influence -it had on the body in its ascent; so as to give again to the body the -same degrees of velocity, which it had taken away before. - -22. AFTER the same manner, if the body were thrown upwards in the -oblique straight line C A (in fig. 9.) from the point C, with such a -degree of velocity as just to reach the point A; it shall by its own -weight return again through the line A C by the same degrees, as it -ascended. - -23. AND lastly, if a body were thrown with any velocity in a line -continually incurvated upwards, the like effect will be produced upon -its return to the point, whence it was thrown. Suppose for instance, -the body A (in fig. 12.) were hung by a string A B. Then if this body -be impelled any way, it must move in the arch of a circle. Let it -receive such an impulse, as shall cause it to move in the arch A C; and -let this impulse be of such strength, that the body may be carried from -A as far as D, before its motion is overcome by its weight: I say here, -that the body forthwith returning from D, shall come again into the -point A with the same velocity, as that wherewith it began to move. - -24. IT will be proper in this place to observe concerning the power of -gravity, that its force upon any body does not at all depend upon the -shape of the body; but that it continues constantly the same without -any variation in the same body, whatever change be made in the figure -of the body: and if the body be divided into any number of pieces, -all those pieces shall weigh just the same, as they did, when united -together in one body: and if the body be of a uniform contexture, -the weight of each piece will be proportional to its bulk. This has -given reason to conclude, that the power of gravity acts upon bodies -in proportion to the quantity of matter in them. Whence it should -follow, that all bodies must fall from equal heights in the same space -of time. And as we evidently see the contrary in feathers and such -like substances, which fall very slowly in comparison of more solid -bodies; it is reasonable to suppose, that some other cause concurs to -make so manifest a difference. This cause has been found by particular -experiments to be the air. The experiments for this purpose are made -thus. They set up a very tall hollow glass; within which near the top -they lodge a feather and some very ponderous body, usually a piece -of gold, this metal being the most weighty of any body known to us. -This glass they empty of the air contained within it, and by moving a -wire, which passes through the top of the glass, they let the feather -and the heavy body fall together; and it is always found, that as the -two bodies begin to descend at the same time, so they accompany each -other in the fall, and come to the bottom at the very same instant, -as near as the eye can judge. Thus, as far as this experiment can be -depended on, it is certain, that the effect of the power of gravity -upon each body is proportional to the quantity of solid matter, or -to the power of inactivity in each body. For in the limited sense, -which we have given above to the word motion, it has been shown, that -the same force gives to all bodies the same degree of motion, and -different forces communicate different degrees of motion proportional -to the respective powers[51]. In this case, if the power of gravity -were to act equally upon the feather, and upon the more solid body, -the solid body would descend so much slower than the feather, as to -have no greater degree of motion than the feather: but as both bodies -descend with equal swiftness, the degree of motion in the solid body is -greater than in the feather, bearing the same proportion to it, as the -quantity of matter in the solid body to the quantity of matter in the -feather. Therefore the effect of gravity on the solid body is greater -than on the feather, in proportion to the greater degree of motion -communicated; that is, the effect of the power of gravity on the solid -body bears the same proportion to its effect on the feather, as the -quantity of matter in the solid body bears to the quantity of matter -in the feather. Thus it is the proper deduction from this experiment, -that the power of gravity acts not on the surface of bodies only, but -penetrates the bodies themselves most intimately, and operates alike -on every particle of matter in them. But as the great quickness, with -which the bodies fall, leaves it something uncertain, whether they do -descend absolutely in the same time, or only so nearly together, that -the difference in their swift motion is not discernable to the eye; -this property of the power of gravity, which has here been deduced from -this experiment, is farther confirmed by pendulums, whose motion is -such, that a very minute difference would become sufficiently sensible. -This will be farther discoursed on in another place[52]; but here I -shall make use of the principle now laid down to explain the nature of -what is called the center of gravity in bodies. - -25. THE center of gravity is that point, by which if a body be -suspended, it shall hang at rest in any situation. In a globe of a -uniform texture the center of gravity is the same with the center of -the globe; for as the parts of the globe on every side of its center -are similarly disposed, and the power of gravity acts alike on every -part; it is evident, that the parts of the globe on each side of the -center are drawn with equal force, and therefore neither side can -yield to the other; but the globe, if supported at its center, must -of necessity hang at rest. In like manner, if two equal bodies A and -B (in fig. 13.) be hung at the extremities of an inflexible rod C D, -which should have no weight; these bodies, if the rod be supported at -its middle E, shall equiponderate; and the rod remain without motion. -For the bodies being equal and at the same distance from the point of -support E, the power of gravity will act upon each with equal strength, -and in all respects under the same circumstances; therefore the weight -of one cannot overcome the weight of the other. The weight of A can no -more surmount the weight of B, than the weight of B can surmount the -weight of A. Again, suppose a body as A B (in fig. 14.) of a uniform -texture in the form of a roller, or as it is more usually called a -cylinder, lying horizontally. If a straight line be drawn between C and -D, the centers of the extreme circles of this cylinder; and if this -straight line, commonly called the axis of the cylinder, be divided -into two equal parts in E: this point E will be the center of gravity -of the cylinder. The cylinder being a uniform figure, the parts on each -side of the point E are equal, and situated in a perfectly similar -manner; therefore this cylinder, if supported at the point E, must hang -at rest, for the same reason as the inflexible rod above-mentioned -will remain without motion, when suspended at its middle point. And it -is evident, that the force applied to the point E, which would uphold -the cylinder, must be equal to the cylinder’s weight. Now suppose two -cylinders of equal thickness A B and C D to be joined together at C B, -so that the two axis’s E F, and F G lie in one straight line. Let the -axis E F be divided into two equal parts at H, and the axis F G into -two equal parts at I. Then because the cylinder A B would be upheld -at rest by a power applied in H equal to the weight of this cylinder, -and the cylinder C D would likewise be upheld by a power applied in I -equal to the weight of this cylinder; the whole cylinder A D will be -supported by these two powers: but the whole cylinder may likewise be -supported by a power applied to K, the middle point of the whole axis -E G, provided that power be equal to the weight of the whole cylinder. -It is evident therefore, that this power applied in K will produce the -same effect, as the two other powers applied in H and I. It is farther -to be observed, that H K is equal to half F G, and K I equal to half -E F; for E K being equal to half E G, and E H equal to half E F, the -remainder H K must be equal to half the remainder F G; so likewise G -K being equal to half G E, and G I equal to half G F, the remainder I -K must be equal to half the remainder E F. It follows therefore, that -H K bears the same proportion to K I, as F G bears to E F. Besides, -I believe, my readers will perceive, and it is demonstrated in form -by the geometers, that the whole body of the cylinder C D bears the -same proportion to the whole body of the cylinder A B, as the axis F G -bears to the axis E F[53]. But hence it follows, that in the two powers -applied at H and I, the power applied at H bears the same proportion to -the power applied at I, as K I bears to K H. Now suppose two strings -H L and I M extended upwards, one from the point H and the other from -I, and to be laid hold on by two powers, one strong enough to hold up -the cylinder A B, and the other of strength sufficient to support the -cylinder C D. Here as these two powers uphold the whole cylinder, and -therefore produce an effect, equal to what would have been produced -by a power applied to the point K of sufficient force to sustain the -whole cylinder: it is manifest, that if the cylinder be taken away, -the axis only being left, and from the point K a string, as K N, be -extended, which shall be drawn down by a power equivalent to the weight -of the cylinder, this power shall act against the other two powers, as -much as the cylinder acted against them; and consequently these three -powers shall be upon a balance, and hold the axis H I fixed between -them. But if these three powers preserve a mutual balance, the two -powers applied to the strings H L and I M are a balance to each other; -the power applied to the string H L bearing the same proportion to -the power applied to the string I M, as the distance I K bears to the -distance K H. Hence it farther appears, that if an inflexible rod A B -(in fig. 15.) be suspended by any point C not in the middle thereof; -and if at A the end of the shorter arm be hung a weight, and at B -the end of the longer arm be also hung a weight less than the other, -and that the greater of these weights bears to the lesser the same -proportion, as the longer arm of the rod bears to the shorter; then -these two weights will equiponderate: for a power applied at C equal to -both these weights will support without motion the rod thus charged; -since here nothing is changed from the preceding case but the situation -of the powers, which are now placed on the contrary sides of the line, -to which they are fixed. Also for the same reason, if two weights A -and B (in fig. 16.) were connected together by an inflexible rod C D, -drawn from C the center of gravity of A to D the center of gravity of -B; and if the rod C D were to be so divided in E, that the part D E -bear the same proportion to the other part C E, as the weight A bears -to the weight B: then this rod being supported at E will uphold the -weights, and keep them at rest without motion. This point E, by which -the two bodies A and B will be supported, is called their common center -of gravity. And if a greater number of bodies were joined together, -the point, by which they could all be supported, is called the common -center of gravity of them all. Suppose (in fig. 17.) there were three -bodies A, B, C, whose respective centers of gravity were joined by the -three lines D E, D F, E F: the line D E being so divided in G, that D -G bear the same proportion to G E, as B bears to A; G is the center -of gravity common to the two bodies A and B; that is, a power equal -to the weight of both the bodies applied to G would support them, and -the point G is pressed as much by the two weights A and B, as it would -be, if they were both hung together at that point. Therefore, if a -line be drawn from G to F, and divided in H, so that G H bear the same -proportion to H F, as the weight C bears to both the weights A and -B, the point H will be the common center of gravity of all the three -weights; for H would be their common center of gravity, if both the -weights A and B were hung together at G, and the point G is pressed as -much by them in their present situation, as it would be in that case. -In the same manner from the common center of these three weights, you -might proceed to find the common center, if a fourth weight were added, -and by a gradual progress might find the common center of gravity -belonging to any number of weights whatever. - -26. AS all this is the obvious consequence of the proposition laid down -for assigning the common center of gravity of any two weights, by the -same proposition the center of gravity of all figures is found. In a -triangle, as A B C (in fig. 18.) the center of gravity lies in the line -drawn from the middle point of any one of the sides to the opposite -angle, as the line B D is drawn from D the middle of the line A C to -the opposite angle B[54]; so that if from the middle of either of the -other sides, as from the point E in the side A B, a line be drawn, as -E C, to the opposite angle; the point F, where this line crosses the -other line B D, will be the center of gravity of the triangle[55]. -Likewise D F is equal to half F B, and E F equal to half F C[56]. In a -hemisphere, as A B C (fig. 19.) if from D the center of the base the -line D B be erected perpendicular to that base, and this line be so -divided in E, that D E be equal to three fifths of B E, the point E is -the center of gravity of the hemisphere[57]. - -27. IT will be of use to observe concerning the center of gravity of -bodies; that since a power applied to this center alone can support -a body against the power of gravity, and hold it fixed at rest; the -effect of the power of gravity on a body is the same, as if that whole -power were to exert itself on the center of gravity only. Whence it -follows, that, when the power of gravity acts on a body suspended by -any point, if the body is so suspended, that the center of gravity -of the body can descend; the power of gravity will give motion to -that body, otherwise not: or if a number of bodies are so connected -together, that, when any one is put into motion, the rest shall, by -the manner of their being joined, receive such motion, as shall keep -their common center of gravity at rest; then the power of gravity -shall not be able to produce any motion in these bodies, but in all -other cases it will. Thus, if the body A B (in fig. 20, 21.) whose -center of gravity is C, be hung on the point A, and the center C be -perpendicularly under A (as in fig. 20.) the weight of the body will -hold it still without motion, because the center C cannot descend any -lower. But if the body be removed into any other situation, where the -center C is not perpendicularly under A (as in fig. 21.) the body by -its weight will be put into motion towards the perpendicular situation -of its center of gravity. Also if two bodies A, B (in fig. 22.) be -joined together by the rod C D lying in an horizontal situation, and -be supported at the point E; if this point be the center of gravity -common to the two bodies, their weight will not put them into motion; -but if this point E is not their common center of gravity, the bodies -will move; that part of the rod C D descending, in which the common -center of gravity is found. So in like manner, if these two bodies were -connected together by any more complex contrivance; yet if one of -the bodies cannot move without so moving the other, that their common -center of gravity shall rest, the weight of the bodies will not put -them in motion, otherwise it will. - -28. I SHALL proceed in the next place to speak of the mechanical -powers. These are certain instruments or machines, contrived for the -moving great weights with small force; and their effects are all -deducible from the observation we have just been making. They are -usually reckoned in number five; the lever, the wheel and axis, the -pulley, the wedge, and the screw; to which some add the inclined -plane. As these instruments have been of very ancient use, so the -celebrated ~ARCHIMEDES~ seems to have been the first, who discovered -the true reason of their effects. This, I think, may be collected -from what is related of him, that some expressions, which he used to -denote the unlimited force of these instruments, were received as very -extraordinary paradoxes: whereas to those, who had understood the cause -of their great force, no expressions of that kind could have appeared -surprizing. - -29. ALL the effects of these powers may be judged of by this one rule, -that, when two weights are applied to any of these instruments, the -weights will equiponderate, if, when put into motion, their velocities -will be reciprocally proportional to their respective weights. And what -is said of weights, must of necessity be equally understood of any -other forces equivalent to weights, such as the force of a man’s arm, -a stream of water, or the like. - -30. BUT to comprehend the meaning of this rule, the reader must know, -what is to be understood by reciprocal proportion; which I shall now -endeavour to explain, as distinctly as I can; for I shall be obliged -very frequently to make use of this term. When any two things are so -related, that one increases in the same proportion as the other, they -are directly proportional. So if any number of men can perform in a -determined space of time a certain quantity of any work, suppose drain -a fish-pond, or the like; and twice the number of men can perform twice -the quantity of the same work, in the same time; and three times the -number of men can perform as soon thrice the work; here the number -of men and the quantity of the work are directly proportional. On -the other hand, when two things are so related, that one decreases -in the same proportion, as the other increases, they are said to be -reciprocally proportional. Thus if twice the number of men can perform -the same work in half the time, and three times the number of men can -finish the same in a third part of the time; then the number of men -and the time are reciprocally proportional. We shewed above[58] how to -find the common center of gravity of two bodies, there the distances of -that common center from the centers of gravity of the two bodies are -reciprocally proportional to the respective bodies. For C E in fig. 16. -being in the same proportion to E D, as B bears to A; C E is so much -greater in proportion than E D, as A is less in proportion than B. - -31. NOW this being understood, the reason of the rule here stated will -easily appear. For if these two bodies were put in motion, while the -point E rested, the velocity, wherewith A would move, would bear the -same proportion to the velocity, wherewith B would move, as E C bears -to E D. The velocity therefore of each body, when the common center of -gravity rests, is reciprocally proportional to the body. But we have -shewn above[59], that if two bodies are so connected together, that the -putting them in motion will not move their common center of gravity; -the weight of those bodies will not produce in them any motion. -Therefore in any of these mechanical engines, if, when the bodies are -put into motion, their velocities are reciprocally proportional to -their respective weights, whereby the common center of gravity would -remain at rest; the bodies will not receive any motion from their -weight, that is, they will equiponderate. But this perhaps will be yet -more clearly conceived by the particular description of each mechanical -power. - -32. THE lever was first named above. This is a bar made use of to -sustain and move great weights. The bar is applied in one part to -some strong support; as the bar A B (in fig. 23, 24.) is applied at -the point C to the support D. In some other part of the bar, as E, is -applied the weight to be sustained or moved; and in a third place, as -F, is applied another weight or equivalent force, which is to sustain -or move the weight at E. Now here, if, when the level should be put in -motion, and turned upon the point C, the velocity, wherewith the point -F would move, bears the same proportion to the velocity, wherewith the -point E would move, as the weight at E bears to the weight or force at -F; then the lever thus charged will have no propensity to move either -way. If the weight or other force at F be not so great as to bear this -proportion, the weight at E will not be sustained; but if the force at -F be greater than this, the weight at E will be surmounted. This is -evident from what has been said above[60], when the forces at E and F -are placed (as in fig. 23.) on different sides of the support D. It -will appear also equally manifest in the other case, by continuing -the bar B C in fig. 24. on the other side of the support D, till C G -be equal to C F, and by hanging at G a weight equivalent to the power -at F; for then, if the power at F were removed, the two weights at G -and E would counterpoize each other, as in the former case: and it is -evident, that the point F will be lifted up by the weight at G with the -same degree of force, as by the other power applied to F; since, if the -weight at E were removed, a weight hung at F equal to that at G would -balance the lever, the distances C G and C F being equal. - -33. IF the two weights, or other powers, applied to the lever do -not counterbalance each other; a third power may be applied in any -place proposed of the lever, which shall hold the whole in a just -counterpoize. Suppose (in fig. 25.) the two powers at E and F did not -equiponderate, and it were required to apply a third power to the point -G, that might be sufficient to balance the lever. Find what power in -F would just counterbalance the power in E; then if the difference -between this power and that, which is actually applied at F, bear the -same proportion to the third power to be applied at G, as the distance -C G bears to C F; the lever will be counterpoized by the help of this -third power, if it be so applied as to act the same way with the power -in F, when that power is too small to counterbalance the power in E; -but otherwise the power in G must be so applied, as to act against the -power in F. In like manner, if a lever were charged with three, or any -greater number of weights or other powers, which did not counterpoize -each other, another power might be applied in any place proposed, -which should bring the whole to a just balance. And what is here said -concerning a plurality of powers, may be equally applied to all the -following cases. - -34. IF the lever should consist of two arms making an angle at the -point C (as in fig. 26.) yet if the forces are applied perpendicularly -to each arm, the same proportion will hold between the forces applied, -and the distances of the center, whereon the lever rests, from the -points to which they are applied. That is, the weight at E will be to -the force in F in the same proportion, as C F bears to C E. - -35. BUT whenever the forces applied to the lever act obliquely to the -arm, to which they are applied (as in fig. 27.) then the strength of -the forces is to be estimated by lines let fall from the center of -the lever to the directions, wherein the forces act. To balance the -levers in fig. 27, the weight or other force at F will bear the same -proportion to the weight at E, as the distance C E bears to C G the -perpendicular let fall from the point C upon the line, which denotes -the direction wherein the force applied to F acts: for here, if the -lever be put into motion, the power applied to F will begin to move in -the direction of the line F G; and therefore its first motion will be -the same, as the motion of the point G. - -36. WHEN two weights hang upon a lever, and the point, by which the -lever is supported, is placed in the middle between the two weights, -that the arms of the lever are both of equal length; then this lever is -particularly called a balance; and equal weights equiponderate as in -common scales. When the point of support is not equally distant from -both weights, it constitutes that instrument for weighing, which is -called a steelyard. Though both in common scales, and the steelyard, -the point, on which the beam is hung, is not usually placed just in the -same straight line with the points, that hold the weights, but rather -a little above (as in fig. 28.) where the lines drawn from the point -C, whereon the beam is suspended, to the points E and F, on which the -weights are hung, do not make absolutely one continued line. If the -three points E, C, and F were in one straight line, those weights, -which equiponderated, when the beam hung horizontally, would also -equiponderate in any other situation. - -[Illustration] - -But we see in these instruments, when they are charged with weights, -which equiponderate with the beam hanging horizontally; that, if -the beam be inclined either way, the weight most elevated surmounts -the other, and descends, causing the beam to swing, till by degrees -it recovers its horizontal position. This effect arises from the -forementioned structure: for by this structure these instruments are -levers composed of two arms, which make an angle at the point of -support (as in fig. 29, 30.) the first of which represents the case -of the common balance, the second the case of the steelyard. In the -first, where C E and C F are equal, equal weights hung at E and F will -equiponderate, when the points E and F are in an horizontal situation. -Suppose the lines E G and F H to be perpendicular to the horizon, then -they will denote the directions, wherein the forces applied to E and -F act. Therefore the proportion between the weights at E and F, which -shall equiponderate, are to be judged of by perpendiculars, as C I, C -K, let fall from C upon E G and F H: so that the weights being equal, -the lines C I, C K, must be equal also, when the weights equiponderate. -But I believe my readers will easily see, that since C E and C F are -equal, the lines C I and C K will be equal, when the points E and F are -horizontally situated. - -37. IF this lever be set into any other position (as in fig. 31.) then -the weight, which is raised highest, will outweigh the other. Here, -if the point F be raised higher than E, the perpendicular C K will be -longer than C I: and therefore the weights would equiponderate, if -the weight at F were less than the weight at E. But the weight at -F is equal to that at E; therefore is greater, than is necessary to -counterbalance the weight at E, and consequently will outweigh it, and -draw the beam of the lever down. - -38. IN like manner in the case of the steelyard (fig. 32.) if the -weights at E and F are so proportioned, as to equiponderate, when the -points E and F are horizontally situated; then in any other situation -of this lever the weight, which is raised highest, will preponderate. -That is, if in the horizontal situation of the points E and F the -weight at F bears the same proportion to the weight at E, as C I bears -to C K; then, if the point F be raised higher than E (as in fig. 32.) -the weight at F shall bear a greater proportion to the weight at E, -than C I bears to C K. - -39. FARTHER a lever may be hung upon an axis, and then the two arms -of the lever need not be continuous, but fixed to different parts of -this axis; as in fig. 33, where the axis A B is supported by its two -extremities A and B. To this axis one arm of the lever is fixed at the -point C, the other at the point D. Now here, if a weight be hung at E, -the extremity of that arm, which is fixed to the axis at the point C; -and another weight be hung at F, the extremity of the arm, which is -fixed on the axis at D; then these weights will equiponderate, when the -weight at E bears the same proportion to the weight at F, as the arm D -F bears to C E. - -40. THIS is the case, if both the arms are perpendicular to the axis, -and lie (as the geometers express themselves) in the same plane; or, in -other words, if the arms are so fixed perpendicularly upon the axis, -that, when one of them lies horizontally, the other shall also be -horizontal. If either arm stand not perpendicular to the axis; then, in -determining the proportion between the weights, instead of the length -of that arm, you must use the perpendicular let fall upon the axis from -the extremity of that arm. If the arms are not so fixed as to become -horizontal, at the same time; the method of assigning the proportion -between the weights is analogous to that made use of above in levers, -which make an angle at the point, whereon they are supported. - -41. FROM this case of the lever hung on an axis, it is easy to make a -transition to another mechanical power, the wheel and axis. - -42. THIS instrument is a wheel fixed on a roller, the roller being -supported at each extremity so as to turn round freely with the wheel, -in the manner represented in fig. 34, where A B is the wheel, C D the -roller, and E F its two supports. Now suppose a weight G hung by a cord -wound round the roller, and another weight H hung by a cord wound about -the wheel the contrary way: that these weights may support each other, -the weight H must bear the same proportion to the weight G, as the -thickness of the roller bears to the diameter of the wheel. - -43. SUPPOSE the line _k l_ to be drawn through the middle of the -roller; and from the place of the roller, where the cord, on which the -weight G hangs, begins to leave the roller, as at _m_, let the line_ -m n_ be drawn perpendicularly to _k l_; and from the point, where -the cord holding the weight H begins to leave the wheel, as at _o_, -let the line _o p_ be drawn perpendicular to _k l_. This being done, -the two lines _o p_ and _m n_ represent two arms of a lever fixed on -the axis _k l_; consequently the weight H will bear to the weight G -the same proportion, as _m n_ bears to _o p_. But _m n_ bears the -same proportion to _o p_, as the thickness of the roller bears to the -diameter of the wheel; for _m n_ is half the thickness of the roller, -and _o p_ half the diameter of the wheel. - -44. IF the wheel be put into motion, and turned once round, that the -cord, on which the weight G hangs, be wound once more round the axis; -then at the same time the cord, whereon the weight H hangs, will be -wound off from the wheel one circuit. Therefore the velocity of the -weight G will bear the same proportion to the velocity of the weight H, -as the circumference of the roller to the circumference of the wheel. -But the circumference of the roller bears the same proportion to the -circumference of the wheel, as the thickness of the roller bears to -the diameter of the wheel, consequently the velocity of the weight -G bears to the velocity of the weight H the same proportion, as the -thickness of the roller bears to the diameter of the wheel, which is -the proportion that the weight H bears to the weight G. Therefore as -before in the lever, so here also the general rule laid down above is -verified, that the weights equiponderate, when their velocities would -be reciprocally proportional to their respective weights. - -45. IN like manner, if on the same axis two wheels of different sizes -are fixed (as in fig. 35.) and a weight hung on each; the weights will -equiponderate, if the weight hung on the greater wheel bear the same -proportion to the weight hung on the lesser, as the diameter of the -lesser wheel bears to the diameter of the greater. - -46. IT is usual to join many wheels together in the same frame, which -by the means of certain teeth, formed in the circumference of each -wheel, shall communicate motion to each other. A machine of this nature -is represented in fig. 36. Here A B C is a winch, upon which is fixed -a small wheel D indented with teeth, which move in the like teeth of -a larger wheel E F fixed on the axis G H. Let this axis carry another -wheel I, which shall move in like manner a greater wheel K L fixed on -the axis M N. Let this axis carry another small wheel O, which after -the same manner shall turn about a larger wheel P Q fixed on the roller -R S, on which a cord shall be wound, that holds a weight, as T. Now -the proportion required between the weight T and a power applied to -the winch at A sufficient to support the weight, will most easily be -estimated, by computing the proportion, which the velocity of the point -A would bear to the velocity of the weight. If the winch be turned -round, the point A will describe a circle as A V. Suppose the wheel E F -to have ten times the number of teeth, as the wheel D; then the winch -must turn round ten times to carry the wheel E F once round. If wheel K -L has also ten times the number of teeth, as I, the wheel I must turn -round ten times to carry the wheel K L once round; and consequently -the winch A B C must turn round an hundred times to turn the wheel K L -once round. Lastly, if the wheel P Q has ten times the number of teeth, -as the wheel O, the winch must turn about one thousand times in order -to turn the wheel P Q, or the roller R S once round. Therefore here -the point A must have gone over the circle A V a thousand times, in -order to lift the weight T through a space equal to the circumference -of the roller R S: whence it follows, that the power applied at A will -balance the weight T, if it bear the same proportion to it, as the -circumference of the roller to one thousand times the circle A V; or -the same proportion as half the thickness of the roller bears to one -thousand times A B. - -47. I SHALL now explain the effect of the pulley. Let a weight hang by -a pulley, as in fig. 37. Here it is evident, that the power A, by which -the weight B is supported, must be equal to the weight; for the cord C -D is equally strained between them; and if the weight B move, the power -A must move with equal velocity. The pulley E has no other effect, than -to permit the power A to act in another direction, than it must have -done, if it had been directly applied to support the weight without the -intervention of any such instrument. - -48. AGAIN, let a weight be supported, as in fig. 38; where the weight -A is fixed to the pulley B, and the cord, by which the weight is -upheld, is annexed by one extremity to a hook C, and at the other end -is held by the power D. Here the weight is supported by a cord doubled; -insomuch that although the cord were not strong enough to hold the -weight single, yet being thus doubled it might support it. If the -end of the cord held by the power D were hung on the hook C, as well -as the other end; then, when both ends of the cord were tied to the -hook, it is evident, that the hook would bear the whole weight; and -each end of the string would bear against the hook with the force of -half the weight only, seeing both ends together bear with the force of -the whole. Hence it is evident, that, when the power D holds one end -of the weight, the force, which it must exert to support the weight, -must be equal to just half the weight. And the same proportion between -the weight and power might be collected from comparing the respective -velocities, with which they would move; for it is evident, that the -power must move through a space equal to twice the distance of the -pulley from the hook, in order to lift the pulley up to the hook. - -49. IT is equally easy to estimate the effect, when many pulleys are -combined together, as in fig. 39, 40; in the first of which the under -set of pulleys, and consequently the weight is held by six strings; and -in the latter figure by five: therefore in the first of these figures -the power to support the weight, must be one sixth part only of the -weight, and in the latter figure the power must be one fifth part. - -50. THERE are two other ways of supporting a weight by pulleys, which I -shall particularly consider. - -51. ONE of these ways is represented in fig. 41. Here the weight being -connected to the pulley B, a power equal to half the weight A would -support the pulley C, if applied immediately to it. Therefore the -pulley C is drawn down with a force equal to half the weight A. But if -the pulley D were to be immediately supported by half the force, with -which the pulley C is drawn down, this pulley D will uphold the pulley -C; so that if the pulley D be upheld with a force equal to one fourth -part of the weight A, that force will support the weight. But, for the -same reason as before, if the power in E be equal to half the force -necessary to uphold the pulley D; this pulley, and consequently the -weight A, will be upheld: therefore, if the power in E be one eighth -part of the weight A, it will support the weight. - -52. ANOTHER way of applying pulleys to a weight is represented in fig. -42. To explain the effect of pulleys thus applied, it will be proper to -consider different weights hanging, as in fig. 43. Here, if the power -and weights balance each other, the power A is equal to the weight B; -the weight C is equal to twice the power A, or the weight B; and for -the same reason the weight D is equal to twice the weight C, or equal -to four times the power A. It is evident therefore, that all the three -weights B, C, D together are equal to seven times the power A. But if -these three weights were joined in one, they would produce the case of -fig. 40: so that in that figure the weight A, where there are three -pulleys, is seven times the power B. If there had been but two pulleys, -the weight would have been three times the power; and if there had been -four pulleys, the weight would have been fifteen times the power. - -53. THE wedge is next to be considered. The form of this instrument is -sufficiently known. When it is put under any weight (as in fig. 44.) -the force, with which the wedge will lift the weight, when drove under -it by a blow upon the end A B, will bear the same proportion to the -force, wherewith the blow would act on the weight, if directly applied -to it; as the velocity, which the wedge receives from the blow, bears -to the velocity, wherewith the weight is lifted by the wedge. - -54. THE screw is the fifth mechanical power. There are two ways of -applying this instrument. Sometimes it is screwed into a hole, as -in fig. 45, where the screw A B is screwed through the plank C D. -Sometimes the screw is applied to the teeth of a wheel, as in fig. 46, -where the thread of the screw A B turns in the teeth of a wheel C D. In -both these cases, if a bar, as A E, be fixed to the end A of the screw; -the force, wherewith the end B of the screw in fig. 45 is forced down, -and the force, wherewith the teeth of the wheel C D in fig. 44 are -held, bears the same proportion to the power applied to the end E of -the bar; as the velocity, wherewith the end E will move, when the screw -is turned, bears to the velocity, wherewith the end B of the screw in -fig. 43, or the teeth of the wheel C D in fig. 46, will be moved. - -55. THE inclined plane affords also a means of raising a weight with -less force, than what is equal to the weight it self. Suppose it were -required to raise the globe A (in fig. 47.) from the ground B C up -to the point, whose perpendicular height from the ground is E D. If -this globe be drawn along the slant D F, less force will be required -to raise it, than if it were lifted directly up. Here if the force -applied to the globe bear the same proportion only to its weight, as -E D bears to F D, it will be sufficient to hold up the globe; and -therefore any addition to that force will put it in motion, and draw -it up; unless the globe, by pressing against the plane, whereon it -lies, adhere in some degree to the plane. This indeed it must always -do more or less, since no plane can be made so absolutely smooth as -to have no inequalities at all; nor yet so infinitely hard, as not to -yield in the least to the pressure of the weight. Therefore the globe -cannot be laid on such a plane, whereon it will slide with perfect -freedom, but they must in some measure rub against each other; and this -friction will make it necessary to imploy a certain degree of force -more, than what is necessary to support the globe, in order to give -it any motion. But as all the mechanical powers are subject in some -degree or other to the like impediment from friction; I shall here -only shew what force would be necessary to sustain the globe, if it -could lie upon the plane without causing any friction at all. And I -say, that if the globe were drawn by the cord G H, lying parallel to -the plane D F; and the force, wherewith the cord is pulled, bear the -same proportion to the weight of the globe, as E D bears to D F; this -force will sustain the globe. In order to the making proof of this, let -the cord G H be continued on, and turned over the pulley I, and let -the weight K be hung to it. Now I say, if this weight bears the same -proportion to the globe A, as D E bears to D F, the weight will support -the globe. I think it is very manifest, that the center of the globe A -will lie in one continued line with the cord H G. Let L be the center -of the globe, and M the center of gravity of the weight K. In the first -place let the weight hang so, that a line drawn from L to M shall lie -horizontally; and I say, if the globe be moved either up or down the -plane D F, the weight will so move along with it, that the center of -gravity common to both the weights shall continue in this line L M, and -therefore shall in no case descend. To prove this more fully, I shall -depart a little from the method of this treatise, and make use of a -mathematical proportion or two: but they are such, as any person, who -has read ~EUCLID’S ELEMENTS~, will fully comprehend; and are -in themselves so evident, that, I believe, my readers, who are wholly -strangers to geometrical writings, will make no difficulty of admitting -them. This being premised, let the globe be moved up, till its center -be at G, then will M the center of gravity of the weight K be sunk to -N; so that M N shall be equal to G L. Draw N G crossing the line M L -in O; then I say, that O is the common center of gravity of the two -weights in this their new situation. Let G P be drawn perpendicular to -M L; then G L will bear the same proportion to G P, as D F bears to D -E; and M N being equal to G L, M N will bear the same proportion to G -P, as D F bears to D E. But N O bears the same proportion to O G, as -M N bears to G P; consequently N O will bear the same proportion to -O G, as D F bears to D E. In the last place, the weight of the globe -A bears the same proportion to the other weight K, as D F bears to D -E; therefore N O bears the same proportion to O G, as the weight of -the globe A bears to the weight K. Whence it follows, that, when the -center of the globe A is in G, and the center of gravity of the weight -K is in N, O will be the center of gravity common to both the weights. -After the same manner, if the globe had been caused to descend, the -common center of gravity would have been found in this line M L. Since -therefore no motion of the globe either way will make the common center -of gravity descend, it is manifest, from what has been said above, that -the weights A and K counterpoize each other. - -56. I SHALL now consider the case of pendulums. A pendulum is made -by hanging a weight to a line, so that it may swing backwards and -forwards. This motion the geometers have very carefully considered, -because it is the most commodious instrument of any for the exact -measurement of time. - -57. I HAVE observed already[61], that if a body hanging perpendicularly -by a string, as the body A (in fig. 48.) hangs by the string A B, be -put so into motion, as to be made to ascend up the circular arch A C; -then as soon as it has arrived at the highest point, to which the -motion, that the body has received, will carry it; it will immediately -begin to descend, and at A will receive again as great a degree of -motion, as it had at first. This motion therefore will carry the -body up the arch A D, as high as it ascended before in the arch A C. -Consequently in its return through the arch D A it will acquire again -at A its original velocity, and advance a second time up the arch -A C as high as at first; by this means continuing without end its -reciprocal motion. It is true indeed, that in fact every pendulum, -which we can put in motion, will gradually lessen its swing, and -at length stop, unless there be some power constantly applied to -it, whereby its motion shall be renewed; but this arises from the -resistance, which the body meets with both from the air, and the string -by which it is hung: for as the air will give some obstruction to the -progress of the body moving through it; so also the string, whereon the -body hangs, will be a farther impediment; for this string must either -slide on the pin, whereon it hangs, or it must bend to the motion of -the weight; in the first there must be some degree of friction, and -in the latter the string will make some resistance to its inflection. -However, if all resistance could be removed, the motion of a pendulum -would be perpetual. - -58. BUT to proceed, the first property, I shall take notice of in this -motion, is, that the greater arch the pendulous body moves through, the -greater time it takes up: though the length of time does not increase -in so great a proportion as the arch. Thus if C D be a greater arch, -and E F a lesser, where C A is equal to A D, and E A equal to A F; -the body, when it swings through the greater arch C D, shall take up -in its swing from C to D a longer time than in swinging from E to F, -when it moves only in that lesser arch; or the time in which the body -let fall from C will descend through the arch C A is greater than the -time, in which it will descend through the arch E A, when let fall from -E. But the first of these times will not hold the same proportion to -the latter, as the first arch C A bears to the other arch E A; which -will appear thus. Let C G and E H be two horizontal lines. It has been -remarked above[62], that the body in falling through the arch C A will -acquire as great a velocity at the point A, as it would have gained by -falling directly down through G A; and in falling through the arch E A -it will acquire in the point A only that velocity, which it would have -got in falling through H A. Therefore, when the body descends through -the greater arch C A, it shall gain a greater velocity, than when it -passes only through the lesser; so that this greater velocity will in -some degree compensate the greater length of the arch. - -59. THE increase of velocity, which the body acquires in falling from -a greater height, has such an effect, that, if straight lines be drawn -from A to C and E, the body would fall through the longer straight -line C A just in the same time, as through the shorter straight line -E A. This is demonstrated by the geometers, who prove, that if any -circle, as A B C D (fig. 49.) be placed in a perpendicular situation; -a body shall fall obliquely through every line, as A B drawn from the -lowest point A in the circle to any other point in the circumference -just in the same time, as would be imployed by the body in falling -perpendicularly down through the diameter C A. But the time in which -the body will descend through the arch, is different from the time, -which it would take up in falling through the line A B. - -60. IT has been thought by some, that because in very small arches this -correspondent straight line differs but little from the arch itself; -therefore the descent through this straight line would be performed -in such small arches nearly in the same time as through the arches -themselves: so that if a pendulum were to swing in small arches, -half the time of a single swing would be nearly equal to the time, -in which a body would fall perpendicularly through twice the length -of the pendulum. That is, the whole time of the swing, according to -this opinion, will be four fold the time required for the body to fall -through half the length of the pendulum; because the time of the body’s -falling down twice the length of the pendulum is half the time required -for the fall through one quarter of this space, that is through half -the pendulum’s length. However there is here a mistake; for the whole -time of the swing, when the pendulum moves through small arches, bears -to the time required for a body to fall down through half the length of -the pendulum very nearly the same proportion, as the circumference of a -circle bears to its diameter; that is very nearly the proportion of 355 -to 113, or little more than the proportion of 3 to 1. If the pendulum -takes so great a swing, as to pass over an arch equal to one sixth part -of the whole circumference of the circle, it will swing 115 times, -while it ought according to this proportion to have swung 117 times; so -that, when it swings in so large an arch, it loses something less than -two swings in an hundred. If it swing through 1/10 only of the circle, -it shall not lose above one vibration in 160. If it swing in 1/20 of -the circle, it shall lose about one vibration in 690. If its swing be -confined to 1/40 of the whole circle, it shall lose very little more -than one swing in 2600. And if it take no greater a swing than through -1/60 of the whole circle, it shall not lose one swing in 5800. - -61. NOW it follows from hence, that, when pendulums swing in small -arches, there is very nearly a constant proportion observed between -the time of their swing, and the time, in which a body would fall -perpendicularly down through half their length. And we have declared -above, that the spaces, through which bodies fall, are in a two fold -proportion of the times, which they take up in falling[63]. Therefore -in pendulums of different lengths, swinging through small arches, the -lengths of the pendulums are in a two fold or duplicate proportion of -the times, they take in swinging; so that a pendulum of four times the -length of another shall take up twice the time in each swing, one of -nine times the length will make one swing only for three swings of the -shorter, and so on. - -62. THIS proportion in the swings of different pendulums not only holds -in small arches; but in large ones also, provided they be such, as the -geometers call similar; that is, if the arches bear the same proportion -to the whole circumferences of their respective circles. Suppose (in -fig. 48.) A B, C D to be two pendulums. Let the arch E F be described -by the motion of the pendulum A B, and the arch G H be described by -the pendulum C D; and let the arch E F bear the same proportion to the -whole circumference, which would be formed by turning the pendulum A -B quite round about the point A, as the arch G H bears to the whole -circumference, that would be formed by turning the pendulum C D quite -round the point C. Then I say, the proportion, which the length of the -pendulum A B bears to the length of the pendulum C D, will be two fold -of the proportion, which the time taken up in the description of the -arch E F bears to the time employed in the description of the arch G H. - -63. THUS pendulums, which swing in very small arches, are nearly an -equal measure of time. But as they are not such an equal measure to -geometrical exactness; the mathematicians have found out a method of -causing a pendulum so to swing, that, if its motion were not obstructed -by any resistance, it would always perform each swing in the same time, -whether it moved through a greater, or a lesser space. This was first -discovered by the great ~HUYGENS~, and is as follows. Upon the -straight line A B (in fig. 49.) let the circle C D E be so placed, as -to touch the straight line in the point C. Then let this circle roll -along upon the straight line A B, as a coach-wheel rolls along upon -the ground. It is evident, that, as soon as ever the circle begins to -move, the point C in the circle will be lifted off from the straight -line A B; and in the motion of the circle will describe a crooked -course, which is represented by the line C F G H. Here the part C H of -the straight line included between the two extremities C and H of the -line C F G H will be equal to the whole circumference of the circle C D -E; and if C H be divided into two equal parts at the point I, and the -straight line I K be drawn perpendicular to C H, this line I K will -be equal to the diameter of the circle C D E. Now in this line if a -body were to be let fall from the point H, and were to be carried by -its weight down the line H G K, as far as the point K, which is the -lowest point of the line C F G H; and if from any other point G a body -were to be let fall in the same manner; this body, which falls from -G, will take just the same time in coming to K, as the body takes up, -which falls from H. Therefore if a pendulum can be so hung, that the -ball shall move in the line A G F E, all its swings, whether long or -short, will be performed in the same time; for the time, in which the -ball will descend to the point K, is always half the time of the whole -swing. But the ball of a pendulum will be made to swing in this line by -the following means. Let K I (in fig. 52.) be prolonged upwards to L, -till I L is equal to I K. Then let the line L M H equal and like to K -H be applied, as in the figure between the points L and H, so that the -point which in this line L M H answers to the point H in the line K H -shall be applied to the point L, and the point answering to the point -K shall be applied to the point H. Also let such another line L N C be -applied between L and C in the same manner. This preparation being -made; if a pendulum be hung at the point L of such a length, that the -ball thereof shall reach to K; and if the string shall continually bend -against the lines H M L and L N C, as the pendulum swings to and fro; -by this means the ball shall constantly keep in the line C K H. - -64. NOW in this pendulum, as all the swings, whether long or short, -will be performed in the same time; so the time of each will exactly -bear the same proportion to the time required for a body to fall -perpendicularly down, through half the length of the pendulum, that is -from I to K, as the circumference of a circle bears to its diameter. - -65. IT may from hence be understood in some measure, why, when -pendulums swing in circular arches, the times of their swings are -nearly equal, if the arches are small, though those arches be of very -unequal lengths; for if with the semidiameter L K the circular arch O -K P be described, this arch in the lower part of it will differ very -little from the line C K H. - -66. IT may not be amiss here to remark, that a body will fall in this -line C K H (fig. 53.) from C to any other point, as Q or R in a shorter -space of time, than if it moved through the straight line drawn from -C to the other point; or through any other line whatever, that can be -drawn between these two points. - -67. BUT as I have observed, that the time, which a pendulum takes in -swinging, depends upon its length; I shall now say something concerning -the way, in which this length of the pendulum is to be estimated. If -the whole ball of the pendulum could be crouded into one point, this -length, by which the motion of the pendulum is to be computed, would -be the length of the string or rod. But the ball of the pendulum must -have a sensible magnitude, and the several parts of this ball will not -move with the same degree of swiftness; for those parts, which are -farthest from the point, whereon the pendulum is suspended, must move -with the greatest velocity. Therefore to know the time in which the -pendulum swings, it is necessary to find that point of the ball, which -moves with the same degree of velocity, as if the whole ball were to be -contracted into that point. - -68. THIS point is not the center of gravity, as I shall now endeavour -to shew. Suppose the pendulum A B (in fig. 54.) composed of an -inflexible rod A C and ball C B, to be fixed on the point A, and lifted -up into an horizontal situation. Here if the rod were not fixed to the -point A, the body C B would descend directly with the whole force of -its weight; and each part of the body would move down with the same -degree of swiftness. But when the rod is fixed at the point A, the -body must fall after another manner; for the parts of the body must -move with different degrees of velocity, the parts more remote from A -descending with a swifter motion, than the parts nearer to A; so that -the body will receive a kind of rolling motion while it descends. But -it has been observed above, that the effect of gravity upon any body -is the same, as if the whole force were exerted on the body’s center of -gravity[64]. - -[Illustration] - -Since therefore the power of gravity in drawing down the body must also -communicate to it the rolling motion just described; it seems evident, -that the center of gravity of the body cannot be drawn down as swiftly, -as when the power of gravity has no other effect to produce on the -body, than merely to draw it downward. If therefore the whole matter of -the body C B could be crouded into its center of gravity, so that being -united into one point, this rolling motion here mentioned might give -no hindrance to its descent; this center would descend faster, than it -can now do. And the point, which now descends as fast, as if the whole -matter or the body C B were crouded into it, will be farther removed -from the point A, than the center of gravity of the body C B. - -69. AGAIN, suppose the pendulum A B (in fig. 55.) to hang obliquely. -Here the power of gravity will operate less upon the ball of the -pendulum, than before: but the line D E being drawn so, as to stand -perpendicular to the rod A C of the pendulum; the force of gravity -upon the body C B, now it is in this situation, will produce the same -effect, as if the body were to glide down an inclined plane in the -position of D E. But here the motion of the body, when the rod is fixed -to the point A, will not be equal to the uninterrupted descent of the -body down this plane; for the body will here also receive the same -kind of rotation in its motion, as before; so that the motion of the -center of gravity will in like manner be retarded; and the point, which -here descends with that degree of swiftness, which the body would have, -if not hindered by being fixed to the point A; that is, the point, -which descends as fast, as if the whole body were crouded into it, will -be as far removed from the point A, as before. - -70. THIS point, by which the length of the pendulum is to be estimated, -is called the center of oscillation. And the mathematicians have laid -down general directions, whereby to find this center in all bodies. If -the globe A B (in fig. 56.) be hung by the string C D, whose weight -need not be regarded, the center of oscillation is found thus. Let the -straight line drawn from C to D be continued through the globe to F. -That it will pass through the center of the globe is evident. Suppose E -to be this center of the globe; and take the line G of such a length, -that it shall bear the same proportion to E D, as E D bears to E C. -Then E H being made equal to ⅖ of G, the point H shall be the center of -oscillation[65]. If the weight of the rod C D is too considerable to -be neglected, divide C D (fig. 57) in I, that D I be equal to ⅓, part -of C D; and take K in the same proportion to C I, as the weight of the -globe A B to the weight of the rod C D. Then having found H, the center -of oscillation of the globe, as before, divide I K in I, so that I L -shall bear the same proportion to L H, as the line C H bears to K; and -L shall be the center of oscillation of the whole pendulum. - -71. THIS computation is made upon supposition, that the center of -oscillation of the rod C D, if that were to swing alone without any -other weight annexed, would be the point I. And this point would be -the true center of oscillation, so far as the thickness of the rod is -not to be regarded. If any one chuses to take into consideration the -thickness of the rod, he must place the center of oscillation thereof -so much below the point I, that eight times the distance of the center -from the point I shall bear the same proportion to the thickness of the -rod, as the thickness of the rod bears to its length C D[66]. - -72. IT has been observed above, that when a pendulum swings in an -arch of a circle, as here in fig. 58, the pendulum A B swings in the -circular arch C D; if you draw an horizontal line, as E F, from the -place whence the pendulum is let fall, to the line A G, which is -perpendicular to the horizon: then the velocity, which the pendulum -will acquire in coming to the point G, will be the same, as any body -would acquire in falling directly down from F to G. Now this is to be -understood of the circular arch, which is described by the center of -oscillation of the pendulum. I shall here farther observe, that if the -straight line E G be drawn from the point, whence the pendulum falls, -to the lowest point of the arch; in the same or in equal pendulums the -velocity, which the pendulum acquires in G, is proportional to this -line: that is, if the pendulum, after it has descended from E to G, be -taken back to H, and let fall from thence, and the line H G be drawn; -the velocity, which the pendulum shall acquire in G by its descent from -H, shall bear the same proportion to the velocity, which it acquires -in falling from E to G, as the straight line H G bears to the straight -line E G. - -73. WE may now proceed to those experiments upon the percussion of -bodies, which I observed above might be made with pendulums. This -expedient for examining the effects of percussion was first proposed -by our late great architect Sir ~CHRISTOPHER WREN~. And it -is as follows. Two balls, as A and B (in fig. 59.) either equal or -unequal, are hung by two strings from two points C and D, so that, when -the balls hang down without motion, they shall just touch each other, -and the strings be parallel. Here if one of these balls be removed to -any distance from its perpendicular situation, and then let fall to -descend and strike against the other; by the last preceding paragraph -it will be known, with what velocity this ball shall return into its -first perpendicular situation, and consequently with what force it -shall strike against the other ball; and by the height to which this -other ball ascends after the stroke, the velocity communicated to this -ball will be discovered. For instance, let the ball A be taken up to -E, and from thence be let fall to strike against B, passing over in -its descent the circular arch E F. By this impulse let B fly up to G, -moving through the circular arch H G. Then E I and G K being drawn -horizontally, the ball A will strike against B with the velocity, -which it would acquire in falling directly down from I; and the ball -B has received a velocity, wherewith, if it had been thrown directly -upward, it would have ascended up to K. Likewise if straight lines be -drawn from E to F and from H to G, the velocity of A, wherewith it -strikes, will bear the same proportion to the velocity, which B has -received by the blow, as the straight line E F bears to the straight -line H G. In the same manner by noting the place to which A ascends -after the stroke, its remaining velocity may be compared with that, -wherewith it struck against B. Thus may be experimented the effects of -the body A striking against B at rest. If both the bodies are lifted -up, and so let fall as to meet and impinge against each other just upon -the coming of both into their perpendicular situation; by observing -the places into which they move after the stroke, the effects of their -percussion in all these cases may be found in the same manner as before. - -74. SIR ~ISAAC NEWTON~ has described these experiments; -and has shewn how to improve them to a greater exactness by making -allowance for the resistance, which the air gives to the motion of the -balls[67]. But as this resistance is exceeding small, and the manner -of allowing for it is delivered by himself in very plain terms, I need -not enlarge upon it here. I shall rather speak to a discovery, which -he made by these experiments upon the elasticity of bodies. It has -been explained above[68], that when two bodies strike, if they be not -elastic, they remain contiguous after the stroke; but that if they -are elastic, they separate, and that the degree of their elasticity -determines the proportion between the celerity wherewith they separate, -and the celerity wherewith they meet. Now our author found, that the -degree of elasticity appeared in the same bodies always the same, with -whatever degree of force they struck; that is, the celerity wherewith -they separated, always bore the same proportion to the celerity -wherewith they met: so that the elastic power in all the bodies, he -made trial upon, exerted it self in one constant proportion to the -compressing force. Our author made trial with balls of wool bound up -very compact, and found the celerity with which they receded, to bear -about the proportion of 5 to 9 to the celerity wherewith they met; and -in steel he found nearly the same proportion; in cork the elasticity -was something less; but in glass much greater; for the celerity, -wherewith balls of that material separated after percussion, he found -to bear the proportion of 15 to 16 to the celerity wherewith they -met[69]. - -75. I SHALL finish my discourse on pendulums, with this farther -observation only, that the center of oscillation is also the center -of another force. If a body be fixed to any point, and being put in -motion turns round it; the body, if uninterrupted by the power of -gravity or any other means, will continue perpetually to move about -with the same equable motion. Now the force, with which such a body -moves, is all united in the point, which in relation to the power of -gravity is called the center of oscillation. Let the cylinder A B C D -(in fig. 60.) whose axis is E F, be fixed to the point E. And supposing -the point E to be that on which the cylinder is suspended, let the -center of oscillation be found in the axis E F, as has been explained -above[70]. Let G be that center: then I say, that the force, wherewith -this cylinder turns round the point E, is so united in the point G, -that a sufficient force applied in that point shall stop the motion of -the cylinder, in such a manner, that the cylinder should immediately -remain without motion, though it were to be loosened from the point E -at the same instant, that the impediment was applied to G: whereas, if -this impediment had been applied to any other point of the axis, the -cylinder would turn upon the point, where the impediment was applied. -If the impediment had been applied between E and G, the cylinder would -so turn on the point, where the impediment was applied, that the end -B C would continue to move on the same way it moved before along with -the whole cylinder; but if the impediment were applied to the axis -farther off from E than G, the end A D of the cylinder would start out -of its present place that way in which the cylinder moved. From this -property of the center of oscillation, it is also called the center of -percussion. That excellent mathematician, Dr. BROOK TAYLOR, has farther -improved this doctrine concerning the center of percussion, by shewing, -that if through this point G a line, as G H I, be drawn perpendicular -to E F, and lying in the course of the body’s motion; a sufficient -power applied to any point of this line will have the same effect, as -the like power applied to G[71]: so that as we before shewed the center -of percussion within the body on its axis; by this means we may find -this center on the surface of the body also, for it will be where this -line H I crosses that surface. - -76. I SHALL now proceed to the last kind of motion, to be treated on -in this place, and shew what line the power of gravity will cause a -body to describe, when it is thrown forwards by any force. This was -first discovered by the great ~GALILEO~, and is the principle, -upon which engineers should direct the shot of great guns. But as in -this case bodies describe in their motion one of those lines, which in -geometry are called conic sections; it is necessary here to premise a -description of those lines. In which I shall be the more particular, -because the knowledge of them is not only necessary for the present -purpose, but will be also required hereafter in some of the principal -parts of this treatise. - -77. THE first lines considered by the ancient geometers were the -straight line and the circle. Of these they composed various figures, -of which they demonstrated many properties, and resolved divers -problems concerning them. These problems they attempted always to -resolve by the describing straight lines and circles. For instance, let -a square A B C D (fig. 61.) be proposed, and let it be required to make -another square in any assigned proportion to this. Prolong one side, -as D A, of this square to E, till A E bear the same proportion to A D, -as the new square is to bear to the square A C. If the opposite side B -C of the square A C be also prolonged to F, till B F be equal to A E, -and E F be afterwards drawn, I suppose my readers will easily conceive, -that the figure A B F E will bear to the square A B C D the same -proportion, as the line A E bears to the line A D. Therefore the figure -A B F E will be equal to the new square, which is to be found, but is -not it self a square, because the side A E is not of the same length -with the side E F. But to find a square equal to the figure A B F E -you must proceed thus. Divide the line D E into two equal parts in the -point G, and to the center G with the interval G D describe the circle -D H E I; then prolong the line A B, till it meets the circle in K; and -make the square A K L M, which square will be equal to the figure A B F -E, and bear to the square A B C D the same proportion, as the line A E -bears to A D. - -78. I SHALL not proceed to the proof of this, having only here set it -down as a specimen of the method of resolving geometrical problems -by the description of straight lines and circles. But there are some -problems, which cannot be resolved by drawing straight lines or circles -upon a plane. For the management therefore of these they took into -consideration solid figures, and of the solid figures they found that, -which is called a cone, to be the most useful. - -79. A CONE is thus defined by EUCLIDE in his elements of geometry[72]. -If to the straight line A B (in fig. 62.) another straight line, as A -C, be drawn perpendicular, and the two extremities B and C be joined by -a third straight line composing the triangle A C B (for so every figure -is called, which is included under three straight lines) then the two -points A and B being held fixed, as two centers, and the triangle A C B -being turned round upon the line A B, as on an axis; the line A C will -describe a circle, and the figure A C B will describe a cone, of the -form represented by the figure B C D E F (fig. 63.) in which the circle -C D E F is usually called the base of the cone, and B the vertex. - -80. NOW by this figure may several problems be resolved, which cannot -by the simple description of straight lines and circles upon a plane. -Suppose for instance, it were required to make a cube, which should -bear any assigned proportion to some other cube named. I need not here -inform my readers, that a cube is the figure of a dye. This problem -was much celebrated among the ancients, and was once inforced by the -command of an oracle. This problem may be performed by a cone thus. -First make a cone from a triangle, whose side A C shall be half the -length of the side B C Then on the plane A B C D (fig. 64.) let the -line E F be exhibited equal in length to the side of the cube proposed; -and let the line F G be drawn perpendicular to E F, and of such a -length, that it bear the same proportion to E F, as the cube to be -sought is required to bear to the cube proposed. Through the points E, -F, and G let the circle F H I be described. Then let the line E F be -prolonged beyond F to K, that F K be equal to F E, and let the triangle -F K L, having all its sides F K, K L, L F equal to each other, be hung -down perpendicularly from the plane A B C D. After this, let another -plane M N O P be extended through the point L, so as to be equidistant -from the former plane A B C D, and in this plane let the line Q L R -be drawn so, as to be equidistant from the line E F K. All this being -thus prepared, let such a cone, as was above directed to be made, be so -applied to the plane M N O P, that it touch this plane upon the line -Q R, and that the vertex of the cone be applied to the point L. This -cone, by cutting through the first plane A B C D, will cross the circle -F H I before described. And if from the point S, where the surface of -this cone intersects the circle, the line S T be drawn so, as to be -equidistant from the line E F; the line F T will be equal to the side -of the cube sought: that is, if there be two cubes or dyes formed, the -side of one being equal to E F, and the side of the other equal to F T; -the former of these cubes shall bear the same proportion to the latter, -as the line E F bears to F G. - -81. INDEED this placing a cone to cut through a plane is not a -practicable method of resolving problems. But when the geometers had -discovered this use of the cone, they applied themselves to consider -the nature of the lines, which will be produced by the intersection -of the surface of a cone and a plane; whereby they might be enabled -both to reduce these kinds of solutions to practice, and also to render -their demonstrations concise and elegant. - -82. WHENEVER the plane, which cuts the cone, is equidistant from -another plane, that touches the cone on the side; (which is the case of -the present figure;) the line, wherein the plane cuts the surface of -the cone, is called a parabola. But if the plane, which cuts the cone, -be so inclined to this other, that it will pass quite through the cone -(as in fig. 65.) such a plane by cutting the cone produces the figure -called an ellipsis, in which we shall hereafter shew the earth and -other planets to move round the sun. If the plane, which cuts the cone, -recline the other way (as in fig. 66.) so as not to be parallel to any -plane, whereon the cone can lie, nor yet to cut quite through the cone; -such a plane shall produce in the cone a third kind of line, which -is called an hyperbola. But it is the first of these lines named the -parabola, wherein bodies, that are thrown obliquely, will be carried -by the force of gravity; as I shall here proceed to shew, after having -first directed my readers how to describe this sort of line upon a -plane, by which the form of it may be seen. - -83. TO any straight line A B (fig. 67.) let a straight ruler C D be -so applied, as to stand against it perpendicularly. Upon the edge of -this ruler let another ruler E F be so placed, as to move along upon -the edge of the first ruler C D, and keep always perpendicular to it. -This being so disposed, let any point, as G, be taken in the line A B, -and let a string equal in length to the ruler E F be fastened by one -end to the point G, and by the other to the extremity F of the ruler E -F. Then if the string be held down to the ruler E F by a pin H, as is -represented in the figure; the point of this pin, while the ruler E F -moves on the ruler C D, shall describe the line I K L, which will be -one part of the curve line, whose description we were here to teach: -and by applying the rulers in the like manner on the other side of -the line A B, we may describe the other part I M of this line. If the -distance C G be equal to half the line E F in fig. 64, the line M I L -will be that very line, wherein the plane A B C D in that figure cuts -the cone. - -84. THE line A I is called the axis of the parabola M I L, and the -point G is called the focus. - -85. NOW by comparing the effects of gravity upon falling bodies, with -what is demonstrated of this figure by the geometers, it is proved, -that every body thrown obliquely is carried forward in one of these -lines, the axis whereof is perpendicular to the horizon. - -86. THE geometers demonstrate, that if a line be drawn to touch a -parabola in any point, as the line A B (in fig. 68.) touches the -parabola C D, whose axis is Y Z, in the point E; and several lines F -G, H I, K L be drawn parallel to the axis of the parabola: then the -line F G will be to H I in the duplicate proportion of E F to E H, -and F G to K L in the duplicate proportion of E F to E K; likewise -H I to K L in the duplicate proportion of E H to E K. What is to be -understood by duplicate or two-fold proportion, has been already -explained[73]. Accordingly I mean here, that if the line M be taken to -bear the same proportion to E H, as E H bears to E F, H I will bear -the same proportion to F G, as M bears to E F; and if the line N bears -the same proportion to E K, as E K bears to E F, K L will bear the -same proportion to F G, as N bears to E F; or if the line O bear the -same proportion to E K, as E K bears to E H, K L will bear the same -proportion to H I, as O bears to E H. - -87. THIS property is essential to the parabola, being so connected with -the nature of the figure, that every line possessing this property is -to be called by this name. - -88. NOW suppose a body to be thrown from the point A (in fig. 69.) -towards B in the direction of the line A B. This body, if left to -it self, would move on with a uniform motion through this line A B. -Suppose the eye of a spectator to be placed at the point C just under -the point A; and let us imagine the earth to be so put into motion -along with the body, as to carry the spectator’s eye along the line C D -parallel to A B; and that the eye would move on with the same velocity, -wherewith the body would proceed in the line A B, if it were to be -left to move without any disturbance from its gravitation towards the -earth. In this case if the body moved on without being drawn towards -the earth, it would appear to the spectator to be at rest. But if the -power of gravity exerted it self on the body, it would appear to the -spectator to fall directly down. Suppose at the distance of time, -wherein the body by its own progressive motion would have moved from A -to E, it should appear to the spectator to have fallen through a length -equal to E F: then the body at the end of this time will actually have -arrived at the point F. If in the space of time, wherein the body -would have moved by its progressive motion from A to G, it would have -appeared to the spectator to have fallen down the space G H: then the -body at the end of this greater interval of time will be arrived at -the point H. Now if the line A F H I be that, through which the body -actually passes; from what has here been said, it will follow, that -this line is one of those, which I have been describing under the name -of the parabola. For the distances E F, G H, through which the body -is seen to fall, will increase in the duplicate proportion of the -times[74]; but the lines A E, A G will be proportional to the times -wherein they would have been described by the single progressive motion -of the body: therefore the lines E F, G H will be in the duplicate -proportion of the lines A F, A G; and the line A F H I possesses the -property of the parabola. - -89. IF the earth be not supposed to move along with the body, the -case will be a little different. For the body being constantly drawn -directly towards the center of the earth, the body in its motion will -be drawn in a direction a little oblique to that, wherein it would be -drawn by the earth in motion, as before supposed. But the distance to -the center of the earth bears so vast a proportion to the greatest -length, to which we can throw bodies, that this obliquity does not -merit any regard. From the sequel of this discourse it may indeed -be collected, what line the body being thrown thus would be found -to describe, allowance being made for this obliquity of the earth’s -action[75]. This is the discovery of Sir IS. NEWTON; but has no use in -this place. Here it is abundantly sufficient to consider the body as -moving in a parabola. - -90. THE line, which a projected body describes, being thus known, -practical methods have been deduced from hence for directing the -shot of great guns to strike any object desired. This work was first -attempted by ~GALILEO~, and soon after farther improved by -his scholar ~TORRICELLI~; but has lately been rendred more -complete by the great Mr. ~COTES~, whose immature death is an -unspeakable loss to mathematical learning. If it be required to throw -a body from the point A (in fig. 70.) so as to strike the point B; -through the points A, B draw the straight line C D, and erect the line -A E perpendicular to the horizon, and of four times the height, from -which a body must fall to acquire the velocity, wherewith the body is -intended to be thrown. Through the points A and E describe a circle, -that shall touch the line C D in the point A. Then from the point -B draw the line B F perpendicular to the horizon, intersecting the -circle in the points G and H. This being done, if the body be projected -directly towards either of these points G or H, it shall fall upon -the point B; but with this difference, that, if it be thrown in the -direction A G, it shall sooner arrive at B, than if it were projected -in the direction A H. When the body is projected in the direction A -G; the time, it will take up in arriving at B, will bear the same -proportion to the time, wherein it would fall down through one fourth -part of A E, as A G bears to half A E. But when the body is thrown in -the direction of A H, the time of its passing to B will bear the same -proportion to the time, wherein it would fall through one fourth part -of A E, as A H bears to half A E. - -91. IF the line A I be drawn so as to divide the angle under E A D in -the middle, and the line I K be drawn perpendicular to the horizon; -this line will touch the circle in the point I, and if the body be -thrown in the direction A I, it will fall upon the point K: and this -point K is the farthest point in the line A D, which the body can be -made to strike, without increasing its velocity. - -92. THE velocity, wherewith the body every where moves, may be found -thus. Suppose the body to move in the parabola A B (fig. 71.) Erect A -C perpendicular to the horizon, and equal to the height, from which a -body must fall to acquire the velocity, wherewith the body sets out -from A. If you take any points as D and E in the parabola, and draw -D F and E G parallel to the horizon; the velocity of the body in D -will be equal to what a body will acquire in falling down by its own -weight through C F, and in E the velocity will be the same, as would -be acquired in falling through C G. Thus the body moves slowest at -the highest point H of the parabola; and at equal distances from this -point will move with equal swiftness, and descend from that highest -point through the line H B altogether like to the line A H in which it -ascended; abating only the resistance of the air, which is not here -considered. If the line H I be drawn from the highest point H parallel -to the horizon, A I will be equal to ¼ of B G in fig. 70, when the body -is projected in the direction A G, and equal to ¼ of B H, when the body -is thrown in the direction A H provided A D be drawn horizontally. - -93. THUS I have recounted the principal discoveries, which had been -made concerning the motion of bodies by Sir ~ISAAC NEWTON~’S -predecessors; all these discoveries, by being found to agree with -experience, contributing to establish the laws of motion, from whence -they were deduced. I shall therefore here finish what I had to say -upon those laws; and conclude this chapter with a few words concerning -the distinction which ought to be made between absolute and relative -motion. For some have thought fit to confound them together; because -they observe the laws of motion to take place here on the earth, which -is in motion, after the same manner as if it were at rest. But Sir -~ISAAC NEWTON~ has been careful to distinguish between the -relative and absolute consideration both of motion and time[76]. The -astronomers anciently found it necessary to make this distinction in -time. Time considered in it self passes on equably without relation to -any thing external, being the proper measure of the continuance and -duration of all things. But it is most frequently conceived of by us -under a relative view to some succession in sensible things, of which -we take cognizance. The succession of the thoughts in our own minds -is that, from whence we receive our first idea of time, but is a very -uncertain measure thereof; for the thoughts of some men flow on much -more swiftly, than the thoughts of others; nor does the same person -think equally quick at all times. The motions of the heavenly bodies -are more regular; and the eminent division of time into night and day, -made by the sun, leads us to measure our time by the motion of that -luminary: nor do we in the affairs of life concern our selves with any -inequality, which there may be in that motion; but the space of time -which comprehends a day and night is rather supposed to be always the -same. However astronomers anciently found these spaces of time not to -be always of the same length, and have taught how to compute their -differences. Now the time, when so equated as to be rendered perfectly -equal, is the true measure of duration, the other not. And therefore -this latter, which is absolutely true time, differs from the other, -which is only apparent. And as we ordinarily make no distinction -between apparent time, as measured by the sun, and the true; so we -often do not distinguish in our usual discourse between the real, and -the apparent or relative motion of bodies; but use the same words for -one, as we should for the other. Though all things about us are really -in motion with the earth; as this motion is not visible, we speak of -the motion of every thing we see, as if our selves and the earth stood -still. And even in other cases, where we discern the motion of bodies, -we often speak of them not in relation to the whole motion we see, but -with regard to other bodies, to which they are contiguous. If any body -were lying on a table; when that table shall be carried along, we say -the body rests upon the table, or perhaps absolutely, that the body is -at rest. However philosophers must not reject all distinction between -true and apparent motions, any more than astronomers do the distinction -between true and vulgar time; for there is as real a difference between -them, as will appear by the following consideration. Suppose all the -bodies of the universe to have their courses stopped, and reduced to -perfect rest. Then suppose their present motions to be again restored; -this cannot be done without an actual impression made upon some of them -at least. If any of them be left untouched, they will retain their -former state, that is, still remain at rest; but the other bodies, -which are wrought upon, will have changed their former state of rest, -for the contrary state of motion. Let us now suppose the bodies left -at rest to be annihilated, this will make no alteration in the state -of the moving bodies; but the effect of the impression, which was made -upon them, will still subsist. This shews the motion they received to -be an absolute thing, and to have no necessary dependence upon the -relation which the body said to be in motion has to any other body[77]. - -94. BESIDES absolute and relative motion are distinguishable by their -Effects. One effect of motion is, that bodies, when moved round any -center or axis, acquire a certain power, by which they forcibly -press themselves from that center or axis of motion. As when a body -is whirled about in a sling, the body presses against the sling, and -is ready to fly out as soon as liberty is given it. And this power -is proportional to the true, not relative motion of the body round -such a center or axis. Of this Sir ~ISAAC NEWTON~ gives the following -instance[78]. If a pail or such like vessel near full of water be -suspended by a string of sufficient length, and be turned about till -the string be hard twisted. If then as soon as the vessel and water -in it are become still and at rest, the vessel be nimbly turned about -the contrary way the string was twisted, the vessel by the strings -untwisting it self shall continue its motion a long time. And when the -vessel first begins to turn, the water in it shall receive little or -nothing of the motion of the vessel, but by degrees shall receive a -communication of motion, till at last it shall move round as swiftly -as the vessel it self. Now the definition of motion, which ~DES -CARTES~ has given us upon this principle of making all motion -meerly relative, is this: that motion, is a removal of any body from -its vicinity to other bodies, which were in immediate contact with -it, and are considered as at rest[79]. And if this be compared with -what he soon after says, that there is nothing real or positive in the -body moved, for the sake of which we ascribe motion to it, which is -not to be found as well in the contiguous bodies, which are considered -as at rest[80]; it will follow from thence, that we may consider the -vessel as at rest and the water as moving in it: and the water in -respect of the vessel has the greatest motion, when the vessel first -begins to turn, and loses this relative motion more and more, till -at length it quite ceases. But now, when the vessel first begins to -turn, the surface of the water remains smooth and flat, as before the -vessel began to move; but as the motion of the vessel communicates by -degrees motion to the water, the surface of the water will be observed -to change, the water subsiding in the middle and rising at the edges: -which elevation of the water is caused by the parts of it pressing from -the axis, they move about; and therefore this force of receding from -the axis of motion depends not upon the relative motion of the water -within the vessel, but on its absolute motion; for it is least, when -that relative motion is greatest, and greatest, when that relative -motion is least, or none at all. - -95. THUS the true cause of what appears in the surface of this water -cannot be assigned, without considering the water’s motion within the -vessel. So also in the system of the world, in order to find out the -cause of the planetary motions, we must know more of the real motions, -which belong to each planet, than is absolutely necessary for the uses -of astronomy. If the astronomer should suppose the earth to stand -still, he could ascribe such motions to the celestial bodies, as should -answer all the appearances; though he would not account for them in so -simple a manner, as by attributing motion to the earth. But the motion -of the earth must of necessity be considered, before the real causes, -which actuate the planetary system, can be discovered. - - - - -CHAP. III. - -Of CENTRIPETAL FORCES. - - -WE have just been describing in the preceding chapter the effects -produced on a body in motion, from its being continually acted upon -by a power always equal in strength, and operating in parallel -directions[81]. But bodies may be acted upon by powers, which in -different places shall have different degrees of force, and whose -several directions shall be variously inclined to each other. The most -simple of these in respect to direction is, when the power is pointed -constantly to one center. This is truly the case of that power, whose -effects we described in the foregoing chapter; though the center of -that power is so far removed, that the subject then before us is most -conveniently to be considered in the light, wherein we have placed it: -But Sir ISAAC NEWTON has considered very particularly this other case -of powers, which are constantly directed to the same center. It is upon -this foundation, that all his discoveries in the system of the world -are raised. And therefore, as this subject bears so very great a share -in the philosophy, of which I am discoursing, I think it proper in this -place to take a short view of some of the general effects of these -powers, before we come to apply them particularly to the system of the -world. - -2. THESE powers or forces are by Sir ~ISAAC NEWTON~ called centripetal; -and their first effect is to cause the body, on which they act, to quit -the straight course, wherein it would proceed if undisturbed, and to -describe an incurvated line, which shall always be bent towards the -center of the force. It is not necessary, that such a power should -cause the body to approach that center. The body may continue to -recede from the center of the power, notwithstanding its being drawn -by the power; but this property must always belong to its motion, that -the line, in which it moves, will continually be concave towards the -center, to which the power is directed. Suppose A (in fig. 72.) to be -the center of a force. Let a body in B be moving in the direction of -the straight line B C, in which line it would continue to move, if -undisturbed; but being attracted by the centripetal force towards A, -the body must necessarily depart from this line B C, and being drawn -into the curve line B D, must pass between the lines A B and B C. It is -evident therefore, that the body in B being gradually turned off from -the straight line B C, it will at first be convex toward the line B C, -and consequently concave towards the point A: for these centripetal -powers are supposed to be in strength proportional to the power of -gravity, and, like that, not to be able after the manner of an impulse -to turn the body sensibly out of its course into a different one in -an instant, but to take up some space of time in producing a visible -effect. That the curve will always continue to have its concavity -towards A may thus appear. In the line B C near to B take any point as -E, from which the line E F G may be so drawn, as to touch the curve -line B D in some point as F. Now when the body is come to F, if the -centripetal power were immediately to be suspended, the body would no -longer continue to move in a curve line, but being left to it self -would forthwith reassume a straight course; and that straight course -would be in the line F G: for that line is in the direction of the -body’s motion at the point F. But the centripetal force continuing its -energy, the body will be gradually drawn from this line F G so as to -keep in the line F D, and make that line near the point F to be convex -toward F G, and concave toward A. After the same manner the body may be -followed on in its course through the line B D, and every part of that -line be shewn to be concave toward the point A. - -3. THIS then is the constant character belonging to those motions, -which are carried on by centripetal forces; that the line, wherein the -body moves, is throughout concave towards the center of the force. In -respect to the successive distances of the body from the center there -is no general rule to be laid down; for the distance of the body from -the center may either increase, or decrease, or even keep always the -same. The point A (in fig. 73.) being the center of a centripetal -force, let a body at B set out in the direction of the straight line B -C perpendicular to the line A B drawn from A to B. It will be easily -conceived, that there is no other point in the line B C so near to A, -as the point B; that A B is the shortest of all the lines, which can -be drawn from A to any part of the line B C; all other lines, as A D, -or A E, drawn from A to the line B C being longer than A B. Hence it -follows, that the body setting out from B, if it moved in the line B -C, it would recede more and more from the point A. Now as the operation -of a centripetal force is to draw a body towards the center of the -force: if such a force act upon a resting body, it must necessarily put -that body so into motion, as to cause it to move towards the center -of the force: if the body were of it self moving towards that center, -the centripetal force would accelerate that motion, and cause it to -move faster down: but if the body were in such a motion, as being left -to itself it would recede from this center, it is not necessary, that -the action of a centripetal power upon it should immediately compel -the body to approach the center, from which it would otherwise have -receded; the centripetal power is not without effect, if it cause the -body to recede more slowly from that center, than otherwise it would -have done. Thus in the case before us, the smallest centripetal power, -if it act on the body, will force it out of the line B C, and cause it -to pass in a bent line between B C and the point A, as has been before -explained. When the body, for instance, has advanced to the line A D, -the effect of the centripetal force discovers it self by having removed -the body out of the line B C, and brought it to cross the line A D -somewhere between A and D: suppose at F. Now A D being longer than A B, -A F may also be longer than A B. The centripetal power may indeed be -so strong, that A F shall be shorter than A B; or it may be so evenly -balanced with the progressive motion of the body, that A F and A B -shall be just equal: and in this last case, when the centripetal force -is of that strength, as constantly to draw the body as much toward the -center, as the progressive motion would carry it off, the body will -describe a circle about the center A, this center of the force being -also the center of the circle. - -4. IF the body, instead of setting out in the line B C perpendicular -to A B, had set out in another line B G more inclined towards the -line A B, moving in the curve line B H; then as the body, if it were -to continue its motion in the line B G, would for some time approach -the center A; the centripetal force would cause it to make greater -advances toward that center. But if the body were to set out in the -line B I reclined the other way from the perpendicular B C, and were to -be drawn by the centripetal force into the curve line B K; the body, -notwithstanding any centripetal force, would for some time recede from -the center; since some part at least of the curve line B K lies between -the line B I and the perpendicular B C. - -5. THUS far we have explained such effects, as attend every centripetal -force. But as these forces may be very different in regard to the -different degrees of strength, wherewith they act upon bodies in -different places; I shall now proceed to make mention in general of -some of the differences attending these centripetal motions. - -6. TO reassume the consideration of the last mentioned case. Suppose a -centripetal power directed toward the point A (in fig. 74.) to act on -a body in B, which is moving in the direction of the straight line B -C, the line B C reclining off from A B. If from A the straight lines A -D, A E, A F are drawn at pleasure to the line C B; the line C B being -prolonged beyond B to G, it appears that A D is inclined to the line -G C more obliquely, than A B is inclined to it, A E is inclined more -obliquely than A D, and A F more than A E. To speak more correctly, the -angle under A D G is less than that under A B G, the angle under A E G -less than that under A D G, and the angle under A F G less than that -under A E G. Now suppose the body to move in the curve line B H I K. -Then it is here likewise evident, that the line B H I K being concave -towards A, and convex towards the line B C, it is more and more turned -off from the line B C; so that in the point H the line A H will be less -obliquely inclined to the curve line B H I K, than the same line A H -D is inclined to B C at the point D; at the point I the inclination -of the line A I to the curve line will be more different from the -inclination of the same line A I E to the line B C, at the point E; -and in the points K and F the difference of inclination will be still -greater; and in both the inclination at the curve will be less oblique, -than at the straight line B C. But the straight line A B is less -obliquely inclined to B G, than A D is inclined towards D G: therefore -although the line A H be less obliquely inclined towards the curve H B, -than the same line A H D is inclined towards D G; yet it is possible, -that the inclination at H may be more oblique, than the inclination at -B. The inclination at H may indeed be less oblique than the other, or -they may be both the same. This depends upon the degree of strength, -wherewith the centripetal force exerts it self, during the passage of -the body from B to H. After the same manner the inclinations at I and K -depend entirely on the degree of strength, wherewith the centripetal -force acts on the body in its passage from H to K: if the centripetal -force be weak enough, the lines A H and A I drawn from the center A to -the body at H and at I shall be more obliquely inclined to the curve, -than the line A B is inclined towards B G. The centripetal force may -be of that strength as to render all these inclinations equal, or if -stronger, the inclinations at I and K will be less oblique than at -B. Sir ~ISAAC NEWTON~ has particularly shewn, that if the -centripetal power decreases after a certain manner with the increase -of distance, a body may describe such a curve line, that all the -lines drawn from the center to the body shall be equally inclined to -that curve line.[82] But I do not here enter into any particulars, my -present intention being only to shew, that it is possible for a body to -be acted upon by a force continually drawing it down towards a center, -and yet that the body shall continue to recede from that center; for -here as long as the lines A H, A I, &c drawn from the center A to the -body do not become less oblique to the curve, in which the body moves; -so long shall those lines perpetually increase, and consequently the -body shall more and more recede from the center. - -7. BUT we may observe farther, that if the centripetal power, while -the body increases its distance from the center, retain sufficient -strength to make the lines drawn from the center to the body to become -at length less oblique to the curve; then if this diminution of the -obliquity continue, till at last the line drawn from the center to -the body shall cease to be obliquely inclined to the curve, and shall -become perpendicular thereto; from this instant the body shall no -longer recede from the center, but in its following motion it shall -again descend, and shall describe a curve line in all respects like to -that, which it has described already; provided the centripetal power, -every where at the same distance from the center, acts with the same -strength. So we observed in the preceding chapter, that, when the -motion of a projectile became parallel to the horizon, the projectile -no longer ascended, but forthwith directed its course downwards, -descending in a line altogether like that, wherein it had before -ascended[83]. - -8. THIS return of the body may be proved by the following proposition: -that if the body in any place, suppose at I, were to be stopt, and -be thrown directly backward with the velocity, wherewith it was -moving forward in that point I; then the body, by the action of the -centripetal force upon it, would move back again over the path I H B, -in which it had before advanced forward, and would arrive again at the -point B in the same space of time, as was taken up in its passage from -B to I; the velocity of the body at its return to the point B being -the same, as that wherewith it first set out from that point. To give -a full demonstration of this proposition, would require that use of -mathematics, which I here purpose to avoid; but, I believe, it will -appear in great measure evident from the following considerations. - -9. SUPPOSE (in fig. 75.) that a body were carried after the following -manner through the bent figure A B C D E F, composed of the straight -lines A B, B C, C D, D E, E F. First let it be moving in the line A B, -from A towards B, with any uniform velocity. At B let the body receive -an impulse directed toward some point, as G, taken within the concavity -of the figure. Now whereas this body, when once moving in the straight -line A B, will continue to move on in this line, so long as it shall be -left to it self; but being disturbed at the point B in its motion by -the impulse, which there acts upon it, it will be turned out of this -line A B into some other straight line, wherein it will afterwards -continue to move, as long as it shall be left to itself. Therefore -let this impulse have strength sufficient to turn the body into the -line B C. Then let the body move on undisturbed from B to C, but at C -let it receive another impulse pointed toward the same point G, and -of sufficient strength to turn the body into the line C D. At D let a -third impulse, directed like the rest to the point G, turn the body -into the line D E. And at E let another impulse, directed likewise to -the point G, turn the body into the line E F. Now, I say, if the body -while moving in the line E F be stopt, and turned back again in this -line with the same velocity, as that wherewith it was moving forward in -this line; then by the repetition of the former impulse at E the body -will be turned into the line E D, and move in it from E to D with the -same velocity as before it moved with from D to E; by the repetition of -the impulse at D, when the body shall have returned to that point, it -will be turned into the line D C; and by the repetition of the other -impulses at C and B the body will be brought back again into the line -B A, with the velocity, wherewith it first moved in that line. - -10. THIS I prove as follows. Let D E and F E be continued beyond E. In -D E thus continued take at pleasure the length E H, and let H I be so -drawn, as to be equidistant from the line G E. Then, by what has been -written upon the second law of motion[84], it follows, that after the -impulse on the body in E it will move through E I in the same time, as -it would have imployed in moving from E to H, with the velocity which -it had in the line D E. In F E prolonged take E K equal to E I, and -draw K L equidistant from G E. Then, because the body is thrown back in -the line F E with the same velocity as that wherewith it went forward -in that line; if, when the body was returned to E, it were permitted -to go straight on, it would pass through E K in the same time, as it -took up in passing through E I, when it went forward in the line E F. -But, if at the body’s return to the point E, such an impulse directed -toward the point D were to be given it, whereby it should be turned -into the line D E; I say, that the impulse necessary to produce this -effect must be equal to that, which turned the body out of the line D E -into E F; and that the velocity, with which the body will return into -the line E D, is the same, as that wherewith it before moved through -this line from D to E. Because E K is equal to E I, and K L and H I, -being each equidistant from G E, are by consequence equidistant from -each other; it follows, that the two triangular figures I E H and K -E L are altogether like and equal to each other. If I were writing to -mathematicians, I might refer them to some proportions in the elements -of EUCLID for the proof of this[85] but as I do not here address my -self to such, so I think this assertion will be evident enough without -a proof in form; at least I must desire my readers to receive it as a -proposition true in geometry. But these two triangular figures being -altogether like each other and equal; as E K is equal to E I, so E L is -equal to E H, and K L equal to H I. Now the body after its return to -E being turned out of the line F E into E D by an impulse acting upon -it in E, after the manner above expressed; the body will receive such -a velocity by this impulse, as will carry it through E L in the same -time, as it would have imployed in passing through E K, if it had gone -on in that line undisturbed. And it has already been observed, that the -time, in which the body would pass over E K with the velocity wherewith -it returns, is equal to the time it took up in going forward from E to -I; that is, equal to the time, in which it would have gone through E H -with the velocity, wherewith it moved from D to E. Therefore the time, -in which the body will pass through E L after its return into the line -E D, is the same, as would have been taken up by the body in passing -through E H with the velocity, wherewith the body first moved in the -line D E. Since therefore E L and E H are equal, the body returns into -the line D E with the velocity, which it had before in that line. Again -I say, the second impulse in E is equal to the first. By what has -been said on the second law of motion concerning the effect of oblique -impulses[86], it will be understood, that the impulse in E, whereby -the body was turned out of the line D E into the line E F, is of such -strength, that if the body had been at rest, when this impulse had -acted upon it, this impulse would have communicated so much motion to -the body, as would have carried it through a length equal to H I, in -the time wherein the body would have passed from E to H, or in the time -wherein it passed from E to I. In the same manner, on the return of the -body, the impulse in E, whereby the body is turned out of the line F -E into E D, is of such strength, that if it had acted on the body at -rest, it would have caused the body to move through a length equal to -K L, in the same time, as the body would imploy in passing through E K -with the velocity, wherewith it returns in the line F E. Therefore the -second impulse, had it acted on the body at rest, would have caused it -to move through a length equal to K L in the same space of time, as -would be taken up by the body in passing through a length equal to H I, -were the first impulse to act on the body when at rest. That is, the -effects of the first and second impulse on the body when at rest would -be the same; for K L and H I are equal: consequently the second impulse -is equal to the first. - -11. THUS if the body be returned through F E with the velocity, -wherewith it moved forward; we have shewn how by the repetition of the -impulse, which acted on it at E, the body will return again into the -line D E with the velocity, which it had before in that line. By the -same process of reasoning it may be proved, that, when the body is -returned back to D, the impulse, which before acted on the body at that -point, will throw the body into the line D C with the velocity, which -it first had in that line; and the other impulses being successively -repeated, the body will at length be brought back again into the line B -A with the velocity, wherewith it set out in that line. - -12. THUS these impulses, by acting over again in an inverted order -all their operation on the body, bring it back again through the -path, in which it had proceeded forward. And this obtains equally, -whatever be the number of the straight lines, whereof this curve -figure is composed. Now by a method of reasoning, which Sir ~ISAAC -NEWTON~ makes great use of, and which he introduced into geometry, -thereby greatly inriching that science[87]; we might make a transition -from this figure composed of a number of straight lines to a figure -of one continued curvature, and from a number of separate impulses -repeated at distinct intervals to a continual centripetal force, and -shew, that, because what has been here advanced holds universally -true, whatever be the number of straight lines, whereof the curve -figure A C F is composed, and howsoever frequently the impulses at -the angles of this figure are repeated; therefore the same will still -remain true, although this figure should be converted into one of a -continued curvature, and these distinct impulses should be changed -into a continual centripetal force. But as the explaining this method -of reasoning is foreign to my present design; so I hope my readers, -after what has been said, will find no difficulty in receiving the -proposition laid down above: that, if the body, which has moved through -the curve line B H I (in fig. 74.) from B to I, when it is come to I, -be thrown directly back with the same velocity as that, wherewith it -proceeded forward, the centripetal force, by acting over again all its -operation on the body, shall bring the body back again in the line I H -B: and as the motion of the body in its course from B to I was every -where in such a manner oblique to the line drawn from the center to -the body, that the centripetal power acted in some degree against the -body’s motion, and gradually diminished it; so in the return of the -body, the centripetal power will every where draw the body forward, and -accelerate its motion by the same degrees, as before it retarded it. - -13. THIS being agreed, suppose the body in K to have the line A K no -longer obliquely inclined to its motion. In this case, if the body -be turned back, in the manner we have been considering, it must be -directed back perpendicularly to A K. But if it had proceeded forward, -it would likewise have moved in a direction perpendicular to A K; -consequently, whether it move from this point K backward or forward, it -must describe the same kind of course. Therefore since by being turned -back it will go over again the line K I H B; if it be permitted to go -forward, the line K L, which it shall describe, will be altogether -similar to the line K H B. - -14. IN like manner we may determine the nature of the motion, if -the line, wherein the body sets out, be inclined (as in fig. 76.) -down toward the line B A drawn between the body and the center. If -the centripetal power so much increases in strength, as the body -approaches, that it can bend the path, in which the body moves, to -that degree, as to cause all the lines as A H, A I, A K to remain no -less oblique to the motion of the body, than A B is oblique to B C; -the body shall continually more and more approach the center. But if -the centripetal power increases in so much less a degree, as to permit -the line drawn from the center to the body, as it accompanies the -body in its motion, at length to become more and more erect to the -curve wherein the body moves, and in the end, suppose at K, to become -perpendicular thereto; from that time the body shall rise again. This -is evident from what has been said above; because for the very same -reason here also the body shall proceed from the point K to describe a -line altogether similar to the line, in which it has moved from B to K. -Thus, as it was observed of the pendulum in the preceding chapter[88], -that all the time it approaches towards being perpendicular to the -horizon, it more and more descends; but, as soon as it is come into -that perpendicular situation, it immediately rises again by the same -degrees, as it descended by before: so here the body more and more -approaches the center all the time it is moving from B to K; but thence -forward it rises from the center again by the same degrees, as it -approached by before. - -15. IF (in fig. 77.) the line B C be perpendicular to A B; then it has -been observed above[89], that the centripetal power may be so balanced -with the progressive motion of the body, that the body may keep moving -round the center A constantly at the same distance; as a body does, -when whirled about any point, to which it is tyed by a string. If the -centripetal power be too weak to produce this effect, the motion of -the body will presently become oblique to the line drawn from itself -to the center, after the manner of the first of the two cases, which -we have been considering. If the centripetal power be stronger, than -what is required to carry the body in a circle, the motion of the body -will presently fall in with the second of the cases, we have been -considering. - -16. IF the centripetal power so change with the change of distance, -that the body, after its motion has become oblique to the line drawn -from itself to the center, shall again become perpendicular thereto; -which we have shewn to be possible in both the cases treated of -above; then the body shall in its subsequent motion return again to -the distance of A B, and from that distance take a course similar -to the former: and thus, if the body move in a space free from all -resistance, which has been here all along supposed; it shall continue -in a perpetual motion about the center, descending and ascending -alternately therefrom. If the body setting out from B (in fig. 78.) in -the line B C perpendicular to A B, describe the line B D E, which in D -shall be oblique to the line A D, but in E shall again become erect to -A E drawn from the body in E to the center A; then from this point E -the body shall describe the line E F G altogether like to the line B D -E, and at G shall be at the same distance from A, as it was at B. But -likewise the line A G shall be erect to the body’s motion. Therefore -the body shall proceed to describe from G the line G H I altogether -similar to the line G F E, and at I have the same distance from the -center, as it had at E; and also have the line A I erect to its motion: -so that its following motion must be in the line I K L similar to I H -G, and the distance A L equal to A G. Thus the body will go on in a -perpetual round without ceasing, alternately inlarging and contracting -its distance from the center. - -[Illustration] - -17. IF it so happen, that the point E fall upon the line B A continued -beyond A; then the point G will fall on B, I on E, and L also on B; -so that the body will describe in this case a simple curve line round -the center A, like the line B D E F in fig. 79, in which it will -continually revolve from B to E and from E to B without end. - -18. IF A E in fig. 78 should happen to be perpendicular to A B, in this -case also a simple line will be described; for the point G will fall on -the line B A prolonged beyond A, the point I on the line A E prolonged -beyond A, and the point L on B: so that the body will describe a line -like the curve line B E G I in fig. 80, in which the opposite points B -and G are equally distant from A, and the opposite points E and I are -also equally distant from the same point A. - -19. IN other cases the line described will have a more complex figure. - -20. THUS we have endeavoured to shew how a body, while it is constantly -attracted towards a center, may notwithstanding by its progressive -motion keep it self from falling down to that center; but describe -about it an endless circuit, sometimes approaching toward that center, -and at other times as much receding from the same. - -21. BUT here we have supposed, that the centripetal power is of equal -strength every where at the same distance from the center. And this is -the case of that centripetal power, which will hereafter be shewn to be -the cause, that keeps the planets in their courses. But a body may be -kept on in a perpetual circuit round a center, although the centripetal -power have not this property. Indeed a body may by a centripetal -force be kept moving in any curve line whatever, that shall have its -concavity turned every where towards the center of the force. - -22. TO make this evident I shall first propose the case of a body -moving through the incurvated figure A B C D E (in fig. 81.) which is -composed of the straight lines A B, B C, C D, D E, and E A; the motion -being carried on in the following manner. Let the body first move in -the line A B with any uniform velocity. When it is arrived at the point -B, let it receive an impulse directed toward any point F taken within -the figure; and let the impulse be of that strength as to turn the body -out of the line A B into the line B C. The body after this impulse, -while left to itself, will continue moving in the line B C. At C let -the body receive another impulse directed towards the same point F, of -such strength, as to turn the body from the line B C into the line C D. -At D let the body by another impulse, directed likewise to the point F, -be turned out of the line C D into D E. And at E let another impulse, -directed toward the point F, turn the body from the line D E into E -A. Thus we see how a body may be carried through the figure A B C D E -by certain impulses directed always toward the same center, only by -their acting on the body at proper intervals, and with due degrees of -strength. - -23. BUT farther, when the body is come to the point A, if it there -receive another impulse directed like the rest toward the point F, and -of such a degree of strength as to turn the body into the line A B, -wherein it first moved; I say that the body shall return into this line -with the same velocity, as it had at first. - -24. LET A B be prolonged beyond B at pleasure, suppose to G; and from G -let G H be drawn, which if produced should always continue equidistant -from B F, or, according to the more usual phrase, let G H be drawn -parallel to B F. Then it appears, from what has been said upon the -second law of motion[90], that in the time, wherein the body would have -moved from B to G, had it not received a new impulse in B, by the means -of that impulse it will have acquired a velocity, which will carry it -from B to H. After the same manner, if C I be taken equal to B H, -and I K be drawn equidistant from or parallel to C F; the body will -have moved from C to K with the velocity, which it has in the line C -D, in the same time, as it would have employed in moving from C to I -with the velocity, it had in the line B C. Therefore since C I and B -H are equal, the body will move through C K in the same time, as it -would have taken up in moving from B to G with the original velocity, -wherewith it moved through the line A B. Again, D L being taken equal -to C K and L M drawn parallel to D F; for the same reason as before the -body will move through D M with the velocity, which it has in the line -D E, in the same time, as it would imploy in moving through B G with -its original velocity. In the last place, if E N be taken equal to D M, -and N O be drawn parallel to E F; likewise if A P be taken equal to E -O, and P Q be drawn parallel to A F: then the body with the velocity, -wherewith it returns into the line A B, will pass through A Q in the -same time, as it would have imployed in passing through B G with its -original velocity. Now as all this follows directly from what has above -been delivered, concerning the effect of oblique impulses impressed -upon bodies in motion; so we must here observe farther, that it can be -proved by geometry, that A Q will always be equal to E G. The proof of -this I am obliged, from the nature of my present design, to omit; but -this geometrical proportion being granted, it follows, that the body -has returned into the line A B with the velocity, which it had, when -it first moved in that line; for the velocity, with which it returns -into the line A B, will carry it over the line A Q in the same time, as -would have been taken up in its passing over an equal line B G with -the original velocity. - -25. THUS we have found, how a body may be carried round the figure A -B C D E by the action of certain impulses upon it which should all be -pointed toward one center. And we likewise see, that when the body is -brought back again to the point, whence it first set out; if it there -meet with an impulse sufficient to turn it again into the line, wherein -it moved at first, its original velocity will be again restored; and by -the repetition of the same impulses, the body will be carried again in -the same round. Therefore if these impulses, which act on the body at -the points B, C, D, E, and A, continue always the same, the body will -make round this figure innumerable revolutions. - -26. THE proof, which we have here made use of, holds the same in any -number of straight lines, whereof the figure A B D should be composed; -and therefore by the method of reasoning referred to above[91] we are -to conclude, that what has here been said upon this rectilinear figure, -will remain true, if this figure were changed into one of a continued -curvature, and instead of distinct impulses acting by intervals at the -angles of this figure, we had a continual centripetal force. We have -therefore shewn, that a body may be carried round in any curve figure -A B C ( fig. 82.) which shall every where be concave towards any one -point as D, by the continual action of a centripetal power directed to -that point, and when it is returned to the point, from whence it set -out, it shall recover again the velocity, with which it departed from -that point. It is not indeed always necessary, that it should return -again into its first course; for the curve line may have some such -figure as the line A B C D B E in fig. 83. In this curve line, if the -body set out from B in the direction B F, and moved through the line B -C D, till it returned to B; here the body would not enter again into -the line B C D, because the two parts B D and B C of the curve line -make an angle at the point B: so that the centripetal power, which at -the point B could turn the body from the line B F into the curve, will -not be able to turn the body into the line B C from the direction, in -which it returns to the point B; a forceable impulse must be given the -body in the point B to produce that effect. - -27. IF at the point B, whence the body sets out, the curve line return -into it self (as in fig. 82;) then the body, upon its arrival again at -B, may return into its former course, and thus make an endless circuit -about the center of the centripetal power. - -28. WHAT has here been said, I hope, will in some measure enable my -readers to form a just idea of the nature of these centripetal motions. - -29. I HAVE not attempted to shew, how to find particularly, what kind -of centripetal force is necessary to carry a body in any curve line -proposed. This is to be deduced from the degree of curvature, which -the figure has in each point of it, and requires a long and complex -mathematical reasoning. However I shall speak a little to the first -proportion, which Sir ~ISAAC NEWTON~ lays down for this -purpose. By this proposition, when a body is found moving in a curve -line, it may be known, whether the body be kept in its course by a -power always pointed toward the same center; and if it be so, where -that center is placed. The proposition is this: that if a line be drawn -from some fixed point to the body, and remaining by one extream united -to that point, it be carried round along with the body; then, if the -power, whereby the body is kept in its course, be always pointed to -this fixed point as a center, this line will move over equal spaces in -equal portions of time. Suppose a body were moving through the curve -line A B C D (in fig. 84.) and passed over the arches A B, B C, C D -in equal portions of time; then if a point, as E, can be found, from -whence the line E A being drawn to the body in A, and accompanying the -body in its motion, it shall make the spaces E A B, E B C, and E C D -equal, over which it passes, while the body describes the arches A B, B -C, and C D: and if this hold the same in all other arches, both great -and small, of the curve line A B C D, that these spaces are always -equal, where the times are equal; then is the body kept in this line by -a power always pointed to E as a center. - -30. THE principle, upon which Sir ~ISAAC NEWTON~ has -demonstrated this, requires but small skill in geometry to comprehend. -I shall therefore take the liberty to close the present chapter with -an explication of it; because such an example will give the clearest -notion of our author’s method of applying mathematical reasoning to -these philosophical subjects. - -31. HE reasons thus. Suppose a body set out from the point A (in fig. -85.) to move in the straight line A B; and after it had moved for some -time in that line, it were to receive an impulse directed to some point -as C. Let it receive that impulse at D; and thereby be turned into the -line D E; and let the body after this impulse take the same length of -time in passing from D to E, as it imployed in the passing from A to -D. Then the straight lines C A, C D, and C E being drawn, Sir ~ISAAC -NEWTON~ proves, that the and triangular spaces C A D and C D E are -equal. This he does in the following manner. - -32. LET E F be drawn parallel to C D. Then, from what has been said -upon the second law of motion[92], it is evident, that since the -body was moving in the line A B, when it received the impulse in the -direction D C; it will have moved after that impulse through the line -D E in the same time, as it would have taken up in moving through D -F, provided it had received no disturbance in D. But the time of the -body’s moving from D to E is supposed to be equal to the time of its -moving through A D; therefore the time, which the body would have -imployed in moving through D F, had it not been disturbed in D, is -equal to the time, wherein it moved through A D: consequently D F is -equal in length to A D; for if the body had gone on to move through -the line A B without interruption, it would have moved through all -parts thereof with the same velocity, and have passed over equal parts -of that line in equal portions of time. Now C F being drawn, since -A D and D F are equal, the triangular space C D F is equal to the -triangular space C A D. Farther, the line E F being parallel to C D, it -is proved by EUCLID, that the triangle C E D is equal to the triangle C -F D[93]: therefore the triangle C E D is equal to the triangle C A D. - -33. AFTER the same manner, if the body receive at E another impulse -directed toward the point C, and be turned by that impulse into the -line E G; if it move afterwards from E to G in the same space of time, -as was taken up by its motion from D to E, or from A to D; then C G -being drawn, the triangle C E G is equal to C D E. A third impulse at -G directed as the two former to C, whereby the body shall be turned -into the line G H, will have also the like effect with the rest. If the -body move over G H in the same time, as it took up in moving over E -G, the triangle C G H will be equal to the triangle C E G. Lastly, if -the body at H be turned by a fresh impulse directed toward C into the -line H I, and at I by another impulse directed also to C be turned into -the line I K; and if the body move over each of the lines H I, and I K -in the same time, as it imployed in moving over each of the preceding -lines A D, D E, E G, and G H: then each of the triangles C H I, and C -I K will be equal to each of the preceding. Likewise as the time, in -which the body moves over A D E, is equal to the time of its moving -over E G H, and to the time of its moving over H I K; the space C A D -E will be equal to the space C E G H, and to the space C H I K. In the -same manner as the time, in which the body moved over A D E G is equal -to the time of its moving over G H I K, so the space C A D E G will be -equal to the space C G H I K. - -34. FROM this principle Sir ~ISAAC NEWTON~ demonstrates the proposition -mentioned above, by that method of arguing introduced by him into -geometry, whereof we have before taken notice[94], by making according -to the principles of that method a transition from this incurvated -figure composed of straight lines, to a figure of continued curvature; -and by shewing, that since equal spaces are described in equal times -in this present figure composed of straight lines, the same relation -between the spaces described and the times of their description will -also have place in a figure of one continued curvature. He also deduces -from this proposition the reverse of it; and proves, that whenever -equal spaces are continually described; the body is acted upon by -a centripetal force directed to the center, at which the spaces -terminate. - - - - -CHAP. IV. - -Of the RESISTANCE of FLUIDS. - - -BEFORE the cause can be discovered, which keeps the planets in motion, -it is necessary first to know, whether the space, wherein they move, is -empty and void, or filled with any quantity of matter. It has been a -prevailing opinion, that all space contains in it matter of some kind -or other; so that where no sensible matter is found, there was yet a -subtle fluid substance by which the space was filled up; even so as -to make an absolute plenitude. In order to examine this opinion, Sir -~ISAAC NEWTON~ has largely considered the effects of fluids upon bodies -moving in them. - -2. THESE effects he has reduced under these three heads. In the -first place he shews how to determine in what manner the resistance, -which bodies suffer, when moving in a fluid, gradually increases in -proportion to the space, they describe in any fluid; to the velocity, -with which they describe it; and to the time they have been in motion. -Under the second head he considers what degree of resistance different -bodies moving in the same fluid undergo, according to the different -proportion between the density of the fluid and the density of the -body. The densities of bodies, whether fluid or solid, are measured by -the quantity of matter, which is comprehended under the same magnitude; -that body being the most dense or compact, which under the same bulk -contains the greatest quantity of solid matter, or which weighs most, -the weight of every body being observed above to be proportional -to the quantity of matter in it[95]. Thus water is more dense than -cork or wood, iron more dense than water, and gold than iron. The -third particular Sir ~IS. NEWTON~ considers concerning the -resistance of fluids is the influence, which the diversity of figure in -the resisted body has upon its resistance. - -3. FOR the more perfect illustration of the first of these heads, he -distinctly shews the relation between all the particulars specified -upon three different suppositions. The first is, that the same body -be resisted more or less in the simple proportion to its velocity; so -that if its velocity be doubled, its resistance shall become threefold. -The second is of the resistance increasing in the duplicate proportion -of the velocity; so that, if the velocity of a body be doubled, its -resistance shall be rendered four times; and if the velocity be -trebled, nine times as great as at first. But what is to be understood -by duplicate proportion has been already explained[96]. The third -supposition is, that the resistance increases partly in the single -proportion of the velocity, and partly in the duplicate proportion -thereof. - -4. IN all these suppositions, bodies are considered under two respects, -either as moving, and opposing themselves against the fluid by -that power alone, which is essential to them, of resisting to the -change of their state from rest to motion, or from motion to rest, -which we have above called their power of inactivity; or else, as -descending or ascending, and so having the power of gravity combined -with that other power. Thus our author has shewn in all those three -suppositions, in what manner bodies are resisted in an uniform fluid, -when they move with the aforesaid progressive motion[97]; and what the -resistance is, when they ascend or descend perpendicularly[98]. And -if a body ascend or descend obliquely, and the resistance be singly -proportional to the velocity, it is shewn how the body is resisted in -a fluid of an uniform density, and what line it will describe[99], -which is determined by the measurement of the hyperbola, and appears -to be no other than that line, first considered in particular by Dr. -~BARROW~[100], which is now commonly known by the name of the -logarithmical curve. In the supposition that the resistance increases -in the duplicate proportion of the velocity, our author has not given -us the line which would be described in an uniform fluid; but has -instead thereof discussed a problem, which is in some sort the reverse; -to find the density of the fluid at all altitudes, by which any given -curve line may be described; which problem is so treated by him, as -to be applicable to any kind of resistance whatever[101]. But here -not unmindful of practice, he shews that a body in a fluid of uniform -density, like the air, will describe a line, which approaches towards -an hyperbola; that is, its motion will be nearer to that curve line -than to the parabola. And consequent upon this remark, he shews how to -determine this hyperbola by experiment, and briefly resolves the chief -of those problems relating to projectiles, which are in use in the art -of gunnery, in this curve[102]; as ~TORRICELLI~ and others -have done in the parabola[103], whose inventions have been explained at -large above[104]. - -5. OUR author has also handled distinctly that particular sort of -motion, which is described by pendulums[105]; and has likewise -considered some few cases of bodies moving in resisting fluids round a -center, to which they are impelled by a centripetal force, in order to -give an idea of those kinds of motions[106]. - -6. THE treating of the resistance of pendulums has given him -an opportunity of inserting into another part of his work some -speculations upon the motions of them without resistance, which have -a very peculiar elegance; where in he treats of them as moved by a -gravitation acting in the law, which he shews to belong to the earth -below its surface[107]; performing in this kind of gravitation, where -the force is proportional to the distance from the center, all that -HUYGENS had before done in the common supposition of its being uniform, -and acting in parallel lines[108]. - -7. HUYGENS at the end of his treatise of the cause of gravity[109] -informs us, that he likewise had carried his speculations on the -first of these suppositions, of the resistance in fluids being -proportional to the velocity of the body, as far as our author. But -finding by experiment that the second was more conformable to nature, -he afterwards made some progress in that, till he was stopt, by not -being able to execute to his wish what related to the perpendicular -descent of bodies; not observing that the measurement of the curve -line, he made use of to explain it by, depended on the hyperbola. -Which oversight may well be pardoned in that great man, considering -that our author had not been pleased at that time to communicate to -the publick his admirable discourse of the QUADRATURE or MEASUREMENT -OF CURVE LINES, with which he has since obliged the world: for without -the use of that treatise, it is I think no injury even to our author’s -unparalleled abilities to believe, it would not have been easy for -himself to have succeeded so happily in this and many other parts of -his writings. - -8. WHAT HUYGENS found by experiment, that bodies were in reality -resisted in the duplicate proportion of their velocity, agrees with the -reasoning of our author[110], who distinguishes the resistance, which -fluids give to bodies by the tenacity of their parts, and the friction -between them and the body, from that, which arises from the power of -inactivity, with which the constituent particles of fluids are endued -like all other portions of matter, by which power the particles of -fluids like other bodies make resistance against being put into motion. - -9. THE resistance, which arises from the friction of the body -against the parts of the fluid, must be very inconsiderable; and the -resistance, which follows from the tenacity of the parts of fluids, is -not usually very great, and does not depend much upon the velocity of -the body in the fluid; for as the parts of the fluid adhere together -with a certain degree of force, the resistance, which the body receives -from thence, cannot much depend upon the velocity, with which the body -moves; but like the power of gravity, its effect must be proportional -to the time of its acting. This the reader may find farther explained -by Sir ~ISAAC NEWTON~ himself in the postscript to a discourse -published by me in THE PHILOSOPHICAL TRANSACTIONS, N^o 371. The -principal resistance, which most fluids give to bodies, arises from the -power of inactivity in the parts of the fluids, and this depends upon -the velocity, with which the body moves, on a double account. In the -first place, the quantity of the fluid moved out of place by the moving -body in any determinate space of time is proportional to the velocity, -wherewith the body moves; and in the next place, the velocity with -which each particle of the fluid is moved, will also be proportional -to the velocity of the body: therefore since the resistance, which -any body makes against being put into motion, is proportional both -to the quantity of matter moved and the velocity it is moved with; -the resistance, which a fluid gives on this account, will be doubly -increased with the increase of the velocity in the moving body; that -is, the resistance will be in a two-fold or duplicate proportion of the -velocity, wherewith the body moves through the fluid. - -10. FARTHER it is most manifest, that this latter kind of resistance -increasing with the increase of velocity, even in a greater degree than -the velocity it self increases, the swifter the body moves, the less -proportion the other species of resistance will bear to this: nay that -this part of the resistance may be so much augmented by a due increase -of velocity, till the former resistances shall bear a less proportion -to this, than any that might be assigned. And indeed experience shews, -that no other resistance, than what arises from the power of inactivity -in the parts of the fluid, is of moment, when the body moves with any -considerable swiftness. - -11. THERE is besides these yet another species of resistance, found -only in such fluids, as, like our air, are elastic. Elasticity belongs -to no fluid known to us beside the air. By this property any quantity -of air may be contracted into a less space by a forcible pressure, and -as soon as the compressing power is removed, it will spring out again -to its former dimensions. The air we breath is held to its present -density by the weight of the air above us. And as this incumbent -weight, by the motion of the winds, or other causes, is frequently -varied (which appears by the barometer;) so when this weight is -greatest, we breath a more dense air than at other times. To what -degree the air would expand it self by its spring, if all pressure -were removed, is not known, nor yet into how narrow a compass it is -capable of being compressed. Mr. BOYLE found it by experiment capable -both of expansion and compression to such a degree, that he could cause -a quantity of air to expand it self over a space some hundred thousand -times greater, than the space to which he could confine the same -quantity[111]. But I shall treat more fully of this spring in the air -hereafter[112]. I am now only to consider what resistance to the motion -of bodies arises from it. - -12. BUT before our author shews in what manner this cause of resistance -operates, he proposes a method, by which fluids may be rendered -elastic, demonstrating that if their particles be provided with a power -of repelling each other, which shall exert it self with degrees of -strength reciprocally proportional to the distances between the centers -of the particles; that then such fluids will observe the same rule in -being compressed, as our air does, which is this, that the space, into -which it yields upon compression, is reciprocally proportional to the -compressing weight[113]. The term reciprocally proportional has been -explained above[114]. And if the centrifugal force of the particles -acted by other laws, such fluids would yield in a different manner to -compression[115]. - -13. WHETHER the particles of the air be endued with such a power, -by which they can act upon each other out of contact, our author -does not determine, but leaves that to future examination, and to -be discussed by philosophers. Only he takes occasion from hence to -consider the resistance in elastic fluids, under this notion; making -remarks, as he passes along, upon the differences, which will arise, -if their elasticity be derived from any other fountain[116]. And this, -I think, must be confessed to be done by him with great judgment; -for this is far the most reasonable account, which has been given of -this surprizing power, as must without doubt be freely acknowledged -by any one, who in the least considers the insufficiency of all the -other conjectures, which have been framed; and also how little reason -there is to deny to bodies other powers, by which they may act upon -each other at a distance, as well as that of gravity; which we shall -hereafter shew to be a property universally belonging to all the bodies -of the universe, and to all their parts[117]. Nay we actually find -in the loadstone a very apparent repelling, as well as an attractive -power. But of this more in the conclusion of this discourse. - -14. BY these steps our author leads the way to explain the resistance, -which the air and such like fluids will give to bodies by their -elasticity; which resistance he explains thus. If the elastic power -of the fluid were to be varied so, as to be always in the duplicate -proportion of the velocity of the resisted body, it is shewn that -then the resistance derived from the elasticity, would increase in -the duplicate proportion of the velocity; in so much that the whole -resistance would be in that proportion, excepting only that small -part, which arises from the friction between the body and the parts -of the fluid. From whence it follows, that because the elastic power -of the same fluid does in truth continue the same, if the velocity of -the moving body be diminished, the resistance from the elasticity, and -therefore the whole resistance, will decrease in a less proportion, -than the duplicate of the velocity; and if the velocity be increased, -the resistance from the elasticity will increase in a less proportion, -than the duplicate of the velocity, that is in a less proportion, than -the resistance made by the power of inactivity of the parts of the -fluid. And from this foundation is raised the proof of a property of -this resistance, given by the elasticity in common with the others from -the tenacity and friction of the parts of the fluid; that the velocity -may be increased, till this resistance from the fluid’s elasticity -shall bear no considerable proportion to that, which is produced by the -power of inactivity thereof[118]. From whence our author draws this -conclusion; that the resistance of a body, which moves very swiftly in -an elastic fluid, is near the same, as if the fluid were not elastic; -provided the elasticity arises from the centrifugal power of the -parts of the medium, as before explained, especially if the velocity -be so great, that this centrifugal power shall want time to exert it -self[119]. But it is to be observed, that in the proof of all this our -author proceeds upon the supposition of this centrifugal power in the -parts of the fluid; but if the elasticity be caused by the expansion -of the parts in the manner of wool compressed, and such like bodies, -by which the parts of the fluid will be in some measure entangled -together, and their motion be obstructed, the fluid will be in a manner -tenacious, and give a resistance upon that account over and above what -depends upon its elasticity only[120]; and the resistance derived from -that cause is to be judged of in the manner before set down. - -15. IT is now time to pass to the second part of this theory; which -is to assign the measure of resistance, according to the proportion -between the density of the body and the density of the fluid. What -is here to be understood by the word density has been explained -above[121]. For this purpose as our author before considered two -distinct cases of bodies moving in mediums; one when they opposed -themselves to the fluid by their power of inactivity only, and another -when by ascending or descending their weight was combined with that -other power: so likewise, the fluids themselves are to be regarded -under a double capacity; either as having their parts at rest, and -disposed freely without restraint, or as being compressed together by -their own weight, or any other cause. - -16. IN the first case, if the parts of the fluid be wholly disingaged -from one another, so that each particle is at liberty to move all ways -without any impediment, it is shewn, that if a globe move in such -a fluid, and the globe and particles of the fluid are endued with -perfect elasticity; so that as the globe impinges upon the particles -of it, they shall bound off and separate themselves from the globe, -with the same velocity, with which the globe strikes upon them; then -the resistance, which the globe moving with any known velocity suffers, -is to be thus determined. From the velocity of the globe, the time, -wherein it would move over two third parts of its own diameter with -that velocity, will be known. And such proportion as the density of the -fluid bears to the density of the globe, the same the resistance given -to the globe will bear to the force, which acting, like the power of -gravity, on the globe without intermission during the space of time now -mentioned, would generate in the globe the same degree of motion, as -that wherewith it moves in the fluid[122]. But if neither the globe nor -the particles of the fluid be elastic, so that the particles, when the -globe strikes against them, do not rebound from it, then the resistance -will be but half so much[123]. Again, if the particles of the fluid and -the globe are imperfectly elastic, so that the particles will spring -from the globe with part only of that velocity wherewith the globe -impinges upon them; then the resistance will be a mean between the two -preceding cases, approaching nearer to the first or second, according -as the elasticity is more or less[124]. - -17. THE elasticity, which is here ascribed to the particles of the -fluid, is not that power of repelling one another, when out of -contact, by which, as has before been mentioned, the whole fluid may be -rendred elastic; but such an elasticity only, as many solid bodies have -of recovering their figure, whenever any forcible change is made in it, -by the impulse of another body or otherwise. Which elasticity has been -explained above at large[125]. - -18. THIS is the case of discontinued fluids, where the body, by -pressing against their particles, drives them before itself, while -the space behind the body is left empty. But in fluids which are -compressed, so that the parts of them removed out of place by the body -resisted immediately retire behind the body, and fill that space, which -in the other case is left vacant, the resistance is still less; for a -globe in such a fluid which shall be free from all elasticity, will -be resisted but half as much as the least resistance in the former -case[126]. But by elasticity I now mean that power, which renders -the whole fluid so; of which if the compressed fluid be possessed, -in the manner of the air, then the resistance will be greater than -by the foregoing rule; for the fluid being capable in some degree -of condensation, it will resemble so far the case of uncompressed -fluids[127]. But, as has been before related, this difference is most -considerable in slow motions. - -19. IN the next place our author is particular in determining the -degrees of resistance accompanying bodies of different figures; which -is the last of the three heads, we divided the whole discourse of -resistance into. And in this disquisition he finds a very surprizing -and unthought of difference, between free and compressed fluids. -He proves, that in the former kind, a globe suffers but half the -resistance, which the cylinder, that circumscribes the globe, will -do, if it move in the direction of its axis[128]. But in the latter -he proves, that the globe and cylinder are resisted alike[129]. And -in general, that let the shape of bodies be ever so different, yet if -the greatest sections of the bodies perpendicular to the axis of their -motion be equal, the bodies will be resisted equally[130]. - -20. PURSUANT to the difference found between the resistance of the -globe and cylinder in rare and uncompressed fluids, our author gives us -the result of some other inquiries of the same nature. Thus of all the -frustums of a cone, that can be described upon the same base and with -the same altitude, he shews how to find that, which of all others will -be the least resisted, when moving in the direction of its axis[131]. -And from hence he draws an easy method of altering the figure of any -spheroidical solid, so that its capacity may be enlarged, and yet the -resistance of it diminished[132]: a note which he thinks may not be -useless to ship-wrights. He concludes with determining the solid, which -will be resisted the least that is possible, in these discontinued -fluids[133]. - -21. THAT I may here be understood by readers unacquainted with -mathematical terms, I shall explain what I mean by a frustum of a cone, -and a spheroidical solid. A cone has been defined above. A frustum is -what remains, when part of the cone next the vertex is cut away by a -section parallel to the base of the cone, as in fig. 86. A spheroid is -produced from an ellipsis, as a sphere or globe is made from a circle. -If a circle turn round on its diameter, it describes by its motion a -sphere; so if an ellipsis (which figure has been defined above, and -will be more fully explained hereafter[134]) be turned round either -upon the longest or shortest line, that can be drawn through the middle -of it, there will be described a kind of oblong or flat sphere, as -in fig. 87. Both these figures are called spheroids, and any solid -resembling these I here call spheroidical. - -22. IF it should be asked, how the method of altering spheroidical -bodies, here mentioned, can contribute to the facilitating a ship’s -motion, when I just above affirmed, that the figure of bodies, which -move in a compressed fluid not elastic, has no relation to the -augmentation or diminution of the resistance; the reply is, that what -was there spoken relates to bodies deep immerged into such fluids, but -not of those, which swim upon the surface of them; for in this latter -case the fluid, by the appulse of the anterior parts of the body, is -raised above the level of the surface, and behind the body is sunk -somewhat below; so that by this inequality in the superficies of -the fluid, that part of it, which at the head of the body is higher -than the fluid behind, will resist in some measure after the manner -of discontinued fluids[135], analogous to what was before observed to -happen in the air through its elasticity, though the body be surrounded -on every side by it[136]. And as far as the power of these causes -extends, the figure of the moving body affects its resistance; for -it is evident, that the figure, which presses least directly against -the parts of the fluid, and so raises least the surface of a fluid -not elastic, and least compresses one that is elastic, will be least -resisted. - -23. THE way of collecting the difference of the resistance in rare -fluids, which arises from the diversity of figure, is by considering -the different effect of the particles of the fluid upon the body moving -against them, according to the different obliquity of the several -parts of the body upon which they respectively strike; as it is known, -that any body impinging against a plane obliquely, strikes with a less -force, than if it fell upon it perpendicularly; and the greater the -obliquity is, the weaker is the force. And it is the same thing, if the -body be at rest, and the plane move against it[137]. - -24. THAT there is no connexion between the figure of a body and its -resistance in compressed fluids, is proved thus. Suppose A B C D (in -fig. 88.) to be a canal, having such a fluid, water for instance, -running through it with an equable velocity; and let any body E, by -being placed in the axis of the canal, hinder the passage of the water. -It is evident, that the figure of the fore part of this body will have -little influence in obstructing the water’s motion, but the whole -impediment will arise from the space taken up by the body, by which it -diminishes the bore of the canal, and straightens the passage of the -water[138]. But proportional to the obstruction of the water’s motion, -will be the force of the water upon the body E[139]. Now suppose both -orifices of the canal to be closed, and the water in it to remain at -rest; the body E to move, so that the parts of the water may pass by -it with the same degree of velocity, as they did before; it is beyond -contradiction, that the pressure of the water upon the body, that -is, the resistance it gives to its motion, will remain the same; and -therefore will have little connexion with the figure of the body[140]. - -25. BY a method of reasoning drawn from the same fountain is determined -the measure of resistance these compressed fluids give to bodies, in -reference to the proportion between the density of the body and that of -the fluid. This shall be explained particularly in my comment on Sir -~IS. NEWTON~’s mathematical principles of natural philosophy; -but is not a proper subject to be insisted on farther in this place. - -26. WE have now gone through all the parts of this theory. There -remains nothing more, but in few words to mention the experiments, -which our author has made, both with bodies falling perpendicularly -through water, and the air[141], and with pendulums[142]: all which -agree with the theory. In the case of falling bodies, the times -of their fall determined by the theory come out the same, as by -observation, to a surprizing exactness; in the pendulums, the rod, by -which the ball of the pendulum hangs, suffers resistance as well as the -ball, and the motion of the ball being reciprocal, it communicates such -a motion to the fluid, as increases the resistance, but the deviation -from the theory is no more, than what may reasonably follow from these -causes. - -27. BY this theory of the resistance of fluids, and these experiments, -our author decides the question so long agitated among natural -philosophers, whether all space is absolutely full of matter. The -Aristotelians and Cartesians both assert this plenitude; the Atomists -have maintained the contrary. Our author has chose to determine this -question by his theory of resistance, as shall be explained in the -following chapter. - -[Illustration] - -[Illustration] - - - - - ~BOOK II.~ - CONCERNING THE - SYSTEM of the WORLD. - - - - -CHAP. I. - -That the Planets move in a space empty of all sensible matter. - - -I HAVE now gone through the first part of my design, and have -explained, as far as the nature of my undertaking would permit, what -Sir ~ISAAC NEWTON~ has delivered in general concerning the motion -of bodies. It follows now to speak of the discoveries, he has made -in the system of the world; and to shew from him what cause keeps -the heavenly bodies in their courses. But it will be necessary for -the use of such, as are not skilled in astronomy, to premise a brief -description of the planetary system. - -2. THIS system is disposed in the following manner. In the middle is -placed the sun. About him six globes continually roll. These are the -primary planets; that which is nearest to the sun is called Mercury, -the next Venus, next to this is our earth, the next beyond is Mars, -after him Jupiter, and the outermost of all Saturn. Besides these there -are discovered in this system ten other bodies, which move about some -of these primary planets in the same manner, as they move round the -sun. These are called secondary planets. The most conspicuous of them -is the moon, which moves round our earth; four bodies move in like -manner round Jupiter; and five round Saturn. Those which move about -Jupiter and Saturn, are usually called satellites; and cannot any of -them be seen without a telescope. It is not impossible, but there may -be more secondary planets, beside these; though our instruments have -not yet discovered any other. This disposition of the planetary or -solar system is represented in fig. 89. - -3. THE same planet is not always equally distant from the sun. But -the middle distance of Mercury is between ⅕ and ⅖ of the distance of -the earth from the sun; Venus is distant from the sun almost ¾ of the -distance of the earth; the middle distance of Mars is something more -than half as much again, as the distance of the earth; Jupiter’s -middle distance exceeds five times the distance of the earth, by -between ⅕ and 1/6 part of this distance; Saturn’s middle distance is -scarce more than 9½ times the distance between the earth and sun; but -the middle distance between the earth and sun is about 217⅛ times the -sun’s semidiameter. - -[Illustration] - -4. ALL these planets move one way, from west to east; and of the -primary planets the most remote is longest in finishing its course -round the sun. The period of Saturn falls short only sixteen days of 29 -years and a half. The period of Jupiter is twelve years wanting about -50 days. The period of Mars falls short of two years by about 43 days. -The revolution of the earth constitutes the year. Venus performs her -period in about 224½ days, and Mercury in about 88 days. - -5. THE course of each planet lies throughout in one plane or flat -surface, in which the sun is placed; but they do not all move in the -same plane, though the different planes, in which they move, cross each -other in very small angles. They all cross each other in lines, which -pass through the sun; because the sun lies in the plane of each orbit. -This inclination of the several orbits to each other is represented in -fig. 90. The line, in which the plane of any orbit crosses the plane of -the earth’s motion, is called the line of the nodes of that orbit. - -6. EACH planet moves round the sun in the line, which we have mentioned -above[143] under the name of ellipsis; which I shall here shew more -particularly how to describe. I have there said how it is produced in -the cone. I shall now shew how to form it upon a plane. Fix upon any -plane two pins, as at A and B in fig. 91. To these tye a string A C B -of any length. Then apply a third pin D so to the string, as to hold -it strained; and in that manner carrying this pin about, the point of -it will describe an ellipsis. If through the points A, B the straight -line E A B F be drawn, to be terminated at the ellipsis in the points -E and F, this is the longest line of any, that can be drawn within the -figure, and is called the greater axis of the ellipsis. The line G H, -drawn perpendicular to this axis E F, so as to pass through the middle -of it, is called the lesser axis. The two points A and B are called -focus’s. Now each planet moves round the sun in a line of this kind, so -that the sun is found in one focus. Suppose A to be the place of the -sun. Then E is the point, wherein the planet will be nearest of all to -the sun, and at F it will be most remote. The point E is called the -perihelion of the planet, and F the aphelion. In G and H the planet is -said to be in its middle or mean distance; because the distance A G or -A H is truly the middle between A E the least, and A F the greatest -distance. In fig. 92. is represented how the greater axis of each orbit -is situated in respect of the rest. The proportion between the greatest -and least distances of the planet from the sun is very different in the -different planets. - -[Illustration] - -In Saturn the proportion of the greatest distance to the least is -something less, than the proportion of 9 to 8, but much nearer to -this, than to the proportion of 10 to 9. In Jupiter this proportion -is a little greater, than that of 11 to 10. In Mars it exceeds the -proportion of 6 to 5. In the earth it is about the proportion of 30 to -29. In Venus it is near to that of 70 to 69. And in Mercury it comes -not a great deal short of the proportion of 3 to 2. - -[Illustration] - -7. EACH of these planets so moves through its ellipsis, that the line -drawn from the sun to the planet, by accompanying the planet in its -motion, will describe about the sun equal spaces in equal times, after -the manner spoke of in the chapter of centripetal forces[144]. There is -also a certain relation between the greater axis’s of these ellipsis’s, -and the times, in which the planets perform their revolutions through -them. Which relation may be expressed thus. Let the period of one -planet be denoted by the letter A, the greater axis of its orbit by -D; let the period of another planet be denoted by B, and the greater -axis of this planet’s orbit by E. Then if C be taken to bear the same -proportion to B, as B bears to A; likewise if F be taken to bear the -same proportion to E, as E bears to D; and G taken to bear the same -proportion likewise to F, as E bears to D; then A shall bear the same -proportion to C, as D bears to G. - -8. THE secondary planets move round their respective primary, much -in the same manner as the primary do round the sun. But the motions -of these shall be more fully explained hereafter[145]. And there is, -besides the planets, another sort of bodies, which in all probability -move round the sun; I mean the comets. The farther description of which -bodies I also leave to the place, where they are to be particularly -treated on[146]. - -9. FAR without this system the fixed stars are placed. These are all so -remote from us, that we seem almost incapable of contriving any means -to estimate their distance. Their number is exceeding great. Besides -two or three thousand, which we see with the naked eye, telescopes open -to our view vast numbers; and the farther improved these instruments -are, we still discover more and more. Without doubt these are luminous -globes, like our sun, and ranged through the wide extent of space; each -of which, it is to be supposed, perform the same office, as our sun, -affording light and heat to certain planets moving about them. But -these conjectures are not to be pursued in this place. - -10. I SHALL therefore now proceed to the particular design of this -chapter, and shew, that there is no sensible matter lodged in the space -where the planets move. - -11. THAT they suffer no sensible resistance from any such matter, is -evident from the agreement between the observations of astronomers in -different ages, with regard to the time, in which the planets have -been found to perform their periods. But it was the opinion of DES -CARTES[147], that the planets might be kept in their courses by the -means of a fluid matter, which continually circulating round should -carry the planets along with it. There is one appearance that may seem -to favour this opinion; which is, that the sun turns round its own -axis the same way, as the planets move. The earth also turns round its -axis the same way, as the moon moves round the earth. And the planet -Jupiter turns upon its axis the same way, as his satellites revolve -round him. It might therefore be supposed, that if the whole planetary -region were filled with a fluid matter, the sun, by turning round on -its own axis, might communicate motion first to that part of the fluid, -which was contiguous, and by degrees propagate the like motion to the -parts more remote. After the same manner the earth might communicate -motion to this fluid, to a distance sufficient to carry round the moon, -and Jupiter communicate the like to the distance of its satellites. -Sir ~ISAAC NEWTON~ has particularly examined what might be -the result of such a motion as this[148]; and he finds, that the -velocities, with which the parts of this fluid will move in different -distances from the center of the motion, will not agree with the motion -observed in different planets: for instance, that the time of one -intire circulation of the fluid, wherein Jupiter should swim, would -bear a greater proportion to the time of one intire circulation of the -fluid, where the earth is; than the period of Jupiter bears to the -period of the earth. But he also proves[149], that the planet cannot -circulate in such a fluid, so as to keep long in the same course, -unless the planet and the contiguous fluid are of the same density, -and the planet be carried along with the same degree of motion, as -the fluid. There is also another remark made upon this motion by -our author; which is, that some vivifying force will be continually -necessary at the center of the motion[150]. The sun in particular, by -communicating motion to the ambient fluid, will lose from it self as -much motion, as it imparts to the fluid; unless some acting principle -reside in the sun to renew its motion continually. If the fluid be -infinite, this gradual loss of motion would continue till the whole -should stop[151]; and if the fluid were limited, this loss of motion -would continue, till there would remain no swifter a revolution in the -sun, than in the utmost part of the fluid; so that the whole would turn -together about the axis of the sun, like one solid globe[152]. - -12. IT is farther to be observed, that as the planets do not move in -perfect circles round the sun; there is a greater distance between -their orbits in some places, than in others. For instance, the distance -between the orbit of Mars and Venus is near half as great again in one -part of their orbits, as in the opposite place. Now here the fluid, -in which the earth should swim, must move with a less rapid motion, -where there is this greater interval between the contiguous orbits; but -on the contrary, where the space is straitest, the earth moves more -slowly, than where it is widest[153]. - -13. FARTHER, if this our globe of earth swam in a fluid of equal -density with the earth it self, that is, in a fluid more dense than -water; all bodies put in motion here upon the earth’s surface must -suffer a great resistance from it; where as, by Sir ~ISAAC -NEWTON~’s experiments mentioned in the preceding chapter, bodies, -that fell perpendicularly down through the air, felt about 1/860 part -only of the resistance, which bodies suffered that fell in like manner -through water. - -14. Sir ~ISAAC NEWTON~ applies these experiments yet farther, -and examines by them the general question concerning the absolute -plenitude of space. According to the Aristotelians, all space was -full without any the least vacuities whatever. DESCARTES embraced the -same opinion, and therefore supposed a subtile fluid matter, which -should pervade all bodies, and adequately fill up their pores. The -Atomical philosophers, who suppose all bodies both fluid and solid to -be composed of very minute but solid atoms, assert that no fluid, how -subtile soever the particles or atoms whereof it is composed should be, -can ever cause an absolute plenitude; because it is impossible that -any body can pass through the fluid without putting the particles of -it into such a motion, as to separate them, at least in part, from one -another, and so perpetually to cause small vacuities; by which these -Atomists endeavour to prove, that a vacuum, or some space empty of -all matter, is absolutely necessary to be in nature. Sir ~ISAAC -NEWTON~ objects against the filling of space with such a subtile -fluid, that all bodies in motion must be unmeasurably resisted by a -fluid so dense, as absolutely to fill up all the space, through which -it is spread. And lest it should be thought, that this objection might -be evaded by ascribing to this fluid such very minute and smooth parts, -as might remove all adhesion or friction between them, whereby all -resistance would be lost, which this fluid might otherwise give to -bodies moving in it; Sir ~ISAAC NEWTON~ proves, in the manner -above related, that fluids resist from the power of inactivity of their -particles; and that water and the air resist almost entirely on this -account: so that in this subtile fluid, however minute and lubricated -the particles, which compose it, might be; yet if the whole fluid was -as dense as water, it would resist very near as much as water does; and -whereas such a fluid, whose parts are absolutely close together without -any intervening spaces, must be a great deal more dense than water, -it must resist more than water in proportion to its greater density; -unless we will suppose the matter, of which this fluid is composed, not -to be endued with the same degree of inactivity as other matter. But if -you deprive any substance of the property so universally belonging to -all other matter, without impropriety of speech it can scarce be called -by this name. - -15. Sir ~ISAAC NEWTON~ made also an experiment to try -in particular, whether the internal parts of bodies suffered any -resistance. And the result did indeed appear to favour some small -degree of resistance; but so very little, as to leave it doubtful, -whether the effect did not arise from some other latent cause[154]. - - - - -CHAP. II. - -Concerning the cause, which keeps in motion the primary planets. - - -SINCE the planets move in a void space and are free from resistance; -they, like all other bodies, when once in motion, would move on in a -straight line without end, if left to themselves. And it is now to be -explained what kind of action upon them carries them round the sun. -Here I shall treat of the primary planets only, and discourse of the -secondary apart in the next chapter. It has been just now declared, -that these primary planets move so about the sun, that a line extended -from the sun to the planet, will, by accompanying the planet in its -motion, pass over equal spaces in equal portions of time[155]. And -this one property in the motion of the planets proves, that they are -continually acted on by a power directed perpetually to the sun as a -center. This therefore is one property of the cause, which keeps the -planets in their courses, that it is a centripetal power, whose center -is the sun. - -2. AGAIN, in the chapter upon centripetal forces[156] it was observ’d, -that if the strength of the centripetal power was suitably accommodated -every where to the motion of any body round a center, the body might -be carried in any bent line whatever, whose concavity should be every -where turned towards the center of the force. It was farther remarked, -that the strength of the centripetal force, in each place, was to be -collected from the nature of the line, wherein the body moved[157]. -Now since each planet moves in an ellipsis, and the sun is placed in -one focus; Sir ~ISAAC NEWTON~ deduces from hence, that the -strength of this power is reciprocally in the duplicate proportion of -the distance from the sun. This is deduced from the properties, which -the geometers have discovered in the ellipsis. The process of the -reasoning is not proper to be enlarged upon here; but I shall endeavour -to explain what is meant by the reciprocal duplicate proportion. Each -of the terms reciprocal proportion, and duplicate proportion, has been -already defined[158]. Their sense when thus united is as follows. -Suppose the planet moved in the orbit A B C (in fig. 93.) about the sun -in S. Then, when it is said, that the centripetal power, which acts on -the planet in A, bears to the power acting on it in B a proportion, -which is the reciprocal of the duplicate proportion of the distance S -A to the distance S B; it is meant that the power in A bears to the -power in B the duplicate of the proportion of the distance S B to the -distance S A. The reciprocal duplicate proportion may be explained -also by numbers as follows. Suppose several distances to bear to each -other proportions expressed by the numbers 1, 2, 3, 4, 5; that is, let -the second distance be double the first, the third be three times, -the fourth four times, and the fifth five times as great as the -first. Multiply each of these numbers by it self, and 1 multiplied by -1 produces still 1, 2 multiplied by 2 produces 4, 3 by 3 makes 9, 4 -by 4 makes 16, and 5 by 5 gives 25. This being done, the fractions ¼, -1/9, 1/16, 1/25, will respectively express the proportion, which the -centripetal power in each of the following distances bears to the power -at the first distance: for in the second distance, which is double the -first, the centripetal power will be one fourth part only of the power -at the first distance; at the third distance the power will be one -ninth part only of the first power; at the fourth distance, the power -will be but one sixteenth part of the first; and at the fifth distance, -one twenty fifth part of the first power. - -3. THUS is found the proportion, in which this centripetal power -decreases, as the distance from the sun increases, within the compass -of one planet’s motion. How it comes to pass, that the planet can be -carried about the sun by this centripetal power in a continual round, -sometimes rising from the sun, then descending again as low, and from -thence be carried up again as far remote as before, alternately rising -and falling without end; appears from what has been written above -concerning centripetal forces: for the orbits of the planets resemble -in shape the curve line proposed in § 17 of the chapter on these -forces[159]. - -4. BUT farther, in order to know whether this centripetal force -extends in the same proportion throughout, and consequently whether -all the planets are influenced by the very same power, our author -proceeds thus. He inquires what relation there ought to be between -the periods of the different planets, provided they were acted -upon by the same power decreasing throughout in the forementioned -proportion; and he finds, that the period of each in this case would -have that very relation to the greater axis of its orbit, as I have -declared above[160] to be found in the planets by the observations -of astronomers. And this puts it beyond question, that the different -planets are pressed towards the sun, in the same proportion to their -distances, as one planet is in its several distances. And thence in the -last place it is justly concluded, that there is such a power acting -towards the sun in the foresaid proportion at all distances from it. - -5. THIS power, when referred to the planets, our author calls -centripetal, when to the sun attractive; he gives it likewise the -name of gravity, because he finds it to be of the same nature with -that power of gravity, which is observed in our earth, as will appear -hereafter[161]. By all these names he designs only to signify a power -endued with the properties before mentioned; but by no means would he -have it understood, as if these names referred any way to the cause of -it. In particular in one place where he uses the name of attraction, he -cautions us expressly against implying any thing but a power directing -a body to a center without any reference to the cause of it, whether -residing in that center, or arising from any external impulse[162]. - -6. BUT now, in these demonstrations some very minute inequalities in -the motion of the planets are neglected; which is done with a great -deal of judgment; for whatever be their cause, the effects are very -inconsiderable, they being so exceeding small, that some astronomers -have thought fit wholly to pass them by[163]. However the excellency -of this philosophy, when in the hands of so great a geometer as our -author, is such, that it is able to trace the least variations of -things up to their causes. The only inequalities, which have been -observed common to all the planets, are the motion of the aphelion and -the nodes. The transverse axis of each orbit does not always remain -fixed, but moves about the sun with a very slow progressive motion: -nor do the planets keep constantly the same plane, but change them, -and the lines in which those planes intersect each other by insensible -degrees. The first of these inequalities, which is the motion of the -aphelion, may be accounted for, by supposing the gravitation of the -planets towards the sun to differ a little from the forementioned -reciprocal duplicate proportion of the distances; but the second, -which is the motion of the nodes, cannot be accounted for by any -power directed towards the sun; for no such can give the planet any -lateral impulse to divert it from the plane of its motion into any new -plane, but of necessity must be derived from some other center. Where -that power is lodged, remains to be discovered. Now it is proved, as -shall be explained in the following chapter, that the three primary -planets Saturn, Jupiter, and the earth, which have satellites revolving -about them, are endued with a power of causing bodies, in particular -those satellites, to gravitate towards them with a force, which is -reciprocally in the duplicate proportion of their distances; and the -planets are in all respects, in which they come under our examination, -so similar and alike, that there is no reason to question, but they -have all the same property. Though it be sufficient for the present -purpose to have it proved of Jupiter and Saturn only; for these -planets contain much greater quantities of matter than the rest, and -proportionally exceed the others in power[164]. But the influence of -these two planets being allowed, it is evident how the planets come to -shift continually their planes: for each of the planets moving in a -different plane, the action of Jupiter and Saturn upon the rest will -be oblique to the planes of their motion; and therefore will gradually -draw them into new ones. The same action of these two planets upon -the rest will cause likewise a progressive motion of the aphelion; so -that there will be no necessity of having recourse to the other cause -for this motion, which was before hinted at[165]; viz, the gravitation -of the planets towards the sun differing from the exact reciprocal -duplicate proportion of the distances. And in the last place, the -action of Jupiter and Saturn upon each other will produce in their -motions the same inequalities, as their joint action produces in the -rest. All this is effected in the same manner, as the sun produces the -same kind of inequalities and many others in the motion of the moon and -the other secondary planets; and therefore will be best apprehended by -what shall be said in the next chapter. Those other irregularities in -the motion of the secondary planets have place likewise here; but are -too minute to be observable: because they are produced and rectified -alternately, for the most part in the time of a single revolution; -whereas the motion of the aphelion and nodes, which continually -increase, become sensible in a long series of years. Yet some of these -other inequalities are discernible in Jupiter and Saturn, in Saturn -chiefly; for when Jupiter, who moves faster than Saturn, approaches -near to a conjunction with him, his action upon Saturn will a little -retard the motion of that planet, and by the reciprocal action of -Saturn he will himself be accelerated. After conjunction, Jupiter will -again accelerate Saturn, and be likewise retarded in the same degree, -as before the first was retarded and the latter accelerated. Whatever -inequalities besides are produced in the motion of Saturn by the action -of Jupiter upon that planet, will be sufficiently rectified, by placing -the focus of Saturn’s ellipsis, which should otherwise be in the sun, -in the common center of gravity of the sun and Jupiter. And all the -inequalities in the motion of Jupiter, caused by Saturn’s action upon -him, are much less considerable than the irregularities of Saturn’s -motion[166]. - -7. THIS one principle therefore of the planets having a power, as well -as the sun, to cause bodies to gravitate towards them, which is proved -by the motion of the secondary planets to obtain in fact, explains -all the irregularities relating to the planets ever observed by -astronomers. - -8. Sir ~ISAAC NEWTON~ after this proceeds to make an -improvement in astronomy by applying this theory to the farther -correction of their motions. For as we have here observed the planets -to possess a principle of gravitation, as well as the sun; so it will -be explained at large hereafter, that the third law of motion, which -makes action and reaction equal, is to be applied in this case[167]; -and that the sun does not only attract each planet, but is it self -also attracted by them; the force, wherewith the planet is acted on, -bearing to the force, wherewith the sun it self is acted on at the same -time, the proportion, which the quantity of matter in the sun bears -to the quantity of matter in the planet. From the action between the -sun and planet being thus mutual Sir ISAAC NEWTON proves that the sun -and planet will describe about their common center of gravity similar -ellipsis’s; and then that the transverse axis of the ellipsis described -thus about the moveable sun, will bear to the transverse axis of the -ellipsis, which would be described about the sun at rest in the same -time, the same proportion as the quantity of solid matter in the sun -and planet together bears to the first of two mean proportionals -between this quantity and the quantity of matter in the sun only[168]. - -9. ABOVE, where I shewed how to find a cube, that should bear any -proportion to another cube[169], the lines F T and T S are two mean -proportionals between E F and F G; and counting from E F, F T is called -the first, and F S the second of those means. In numbers these mean -proportionals are thus found. - -[Illustration] - -Suppose A and B two numbers, and it be required to find C the first, -and D the second of the two mean proportionals between them. First -multiply A by it self, and the product multiply by B; then C will be -the number which in arithmetic is called the cubic root of this last -product; that is, the number C being multiplied by it self, and the -product again multiplied by the same number C, will produce the product -above mentioned. In like manner D is the cubic root of the product -of B multiplied by it self, and the produce of that multiplication -multiplied again by A. - -10. IT will be asked, perhaps, how this correction can be admitted, -when the cause of the motions of the planets was before found by -supposing the sun the center of the power, which acted upon them: for -according to the present correction this power appears rather to be -directed to their common center of gravity. But whereas the sun was -at first concluded to be the center, to which the power acting on the -planets was directed, because the spaces described round the sun in -equal times were found to be equal; so Sir ~ISAAC NEWTON~ -proves, that if the sun and planet move round their common center of -gravity, yet to an eye placed in the planet, the spaces, which will -appear to be described about the sun, will have the same relation to -the times of their description, as the real spaces would have, if the -sun were at rest[170]. I farther asserted, that, supposing the planets -to move round the sun at rest, and to be attracted by a power, which -every where should act with degrees of strength reciprocally in the -duplicate proportion of the distances; then the periods of the planets -must observe the same relation to their distances, as astronomers find -them to do. But here it must not be supposed, that the observations of -astronomers absolutely agree without any the least difference; and the -present correction will not cause a deviation from any one astronomer’s -observations, so much as they differ from one another. For in Jupiter, -where this correction is greatest, it hardly amounts to the 3000^{th} -part of the whole axis. - -11. UPON this head I think it not improper to mention a reflection made -by our excellent author upon these small inequalities in the planets -motions; which contains under it a very strong philosophical argument -against the eternity of the world. It is this, that these inequalities -must continually increase by slow degrees, till they render at length -the present frame of nature unfit for the purposes, it now serves[171]. -And a more convincing proof cannot be desired against the present -constitution’s having existed from eternity than this, that a certain -period of years will bring it to an end. I am aware this thought of -our author has been represented even as impious, and as no less than -casting a reflection upon the wisdom of the author of nature, for -framing a perishable work. But I think so bold an assertion ought to -have been made with singular caution. For if this remark upon the -increasing irregularities of the heavenly motions be true in fact, -as it really is, the imputation must return upon the asserter, that -this does detract from the divine wisdom. Certainly we cannot pretend -to know all the omniscient Creator’s purposes in making this world, -and therefore cannot undertake to determine how long he designed it -should last. And it is sufficient, if it endure the time intended by -the author. The body of every animal shews the unlimited wisdom of its -author no less, nay in many respects more, than the larger frame of -nature; and yet we see, they are all designed to last but a small space -of time. - -12. THERE need nothing more be said of the primary planets; the motions -of the secondary shall be next considered. - - - - -CHAP. III. - -Of the motion of the MOON and the other SECONDARY PLANETS. - - -THE excellency of this philosophy sufficiently appears from its -extending in the manner, which has been related, to the minutest -circumstances of the primary planets motions; which nevertheless -bears no proportion to the vast success of it in the motions of the -secondary; for it not only accounts for all the irregularities, by -which their motions were known to be disturbed, but has discovered -others so complicated, that astronomers were never able to distinguish -them, and reduce them under proper heads; but these were only to be -found out from their causes, which this philosophy has brought to -light, and has shewn the dependence of these inequalities upon such -causes in so perfect a manner, that we not only learn from thence in -general, what those inequalities are, but are able to compute the -degree of them. Of this Sir ~IS. NEWTON~ has given several -specimens, and has moreover found means to reduce the moon’s motion so -completely to rule, that he has framed a theory, from which the place -of that planet may at all times be computed, very nearly or altogether -as exactly, as the places of the primary planets themselves, which is -much beyond what the greatest astronomers could ever effect. - -2. THE first thing demonstrated of these secondary planets is, that -they are drawn towards their respective primary in the same manner -as the primary planets are attracted by the sun. That each secondary -planet is kept in its orbit by a power pointed towards the center of -the primary planet, about which the secondary revolves; and that the -power, by which the secondaries of the same primary are influenced, -bears the same relation to the distance from the primary, as the power, -by which the primary planets are guided, does in regard to the distance -from the sun[172]. This is proved in the satellites of Jupiter and -Saturn, because they move in circles, as far as we can observe, about -their respective primary with an equable course, the respective primary -being the center of each orbit: and by comparing the times, in which -the different satellites of the same primary perform their periods, -they are found to observe the same relation to the distances from -their primary, as the primary planets observe in respect of their mean -distances from the sun[173]. Here these bodies moving in circles with -an equable motion, each satellite passes over equal parts of its orbit -in equal portions of time; consequently the line drawn from the center -of the orbit, that is, from the primary planet, to the satellite, will -pass over equal spaces along with the satellite in equal portions of -time; which proves the power, by which each satellite is held in its -orbit, to be pointed towards the primary as a center[174]. It is also -manifest that the centripetal power, which carries a body in a circle -concentrical with the power, acts upon the body at all times with the -same strength. But Sir ~ISAAC NEWTON~ demonstrates that, when -bodies are carried in different circles by centripetal powers directed -to the centers of those circles, then, the degrees of strength of -those powers are to be compared by considering the relation between -the times, in which the bodies perform their periods through those -circles[175]; and in particular he shews, that if the periodical times -bear that relation, which I have just now asserted the satellites -of the same primary to observe; then the centripetal powers are -reciprocally in the duplicate proportion of the semidiameters of the -circles, or in that proportion to the distances of the bodies from the -centers[176]. Hence it follows that in the planets Jupiter and Saturn, -the centripetal power in each decreases with the increase of distance, -in the same proportion as the centripetal power appertaining to the -sun decreases with the increase of distance. I do not here mean that -this proportion of the centripetal powers holds between the power of -Jupiter at any distance compared with the power of Saturn at any other -distance; but only in the change of strength of the power belonging to -the same planet at different distances from him. Moreover what is here -discovered of the planets Jupiter and Saturn by means of the different -satellites, which revolve round each of them, appears in the earth by -the moon alone; because she is found to move round the earth in an -ellipsis after the same manner as the primary planets do about the sun; -excepting only some small irregularities in her motion, the cause of -which will be particularly explained in what follows, whereby it will -appear, that they are no objection against the earth’s acting on the -moon in the same manner as the sun acts on the primary planets; that -is, as the other primary planets Jupiter and Saturn act upon their -satellites. Certainly since these irregularities can be otherwise -accounted for, we ought not to depart from that rule of induction so -necessary in philosophy, that to like bodies like properties are to -be attributed, where no reason to the contrary appears. We cannot -therefore but ascribe to the earth the same kind of action upon the -moon, as the other primary planets Jupiter and Saturn have upon their -satellites; which is known to be very exactly in the proportion -assigned by the method of comparing the periodical times and distances -of all the satellites which move about the same planet; this abundantly -compensating our not being near enough to observe the exact figure of -their orbits. For if the little deviation of the moon’s orbit orbit -from a true permanent ellipsis arose from the action of the earth upon -the moon not being in the exact reciprocal duplicate proportion of the -distance, were another moon to revolve about the earth, the proportion -between the periodical times of this new moon, and the present, -would discover the deviation from the mentioned proportion much more -manifestly. - -3. BY the number of satellites, which move round Jupiter and Saturn, -the power of each of these planets is measured in a great diversity -of distance; for the distance of the outermost satellite in each of -these planets exceeds several times the distance of the innermost. In -Jupiter the astronomers have usually placed the innermost satellite -at a distance from the center of that planet equal to about 5⅔ of -the semidiameters of Jupiter’s body, and this satellite performs its -revolution in about 1 day 18½ hours. The next satellite, which revolves -round Jupiter in about 3 days 13⅕ hours, they place at the distance -from Jupiter of about 9 of that planet’s semidiameters. To the third -satellite, which performs its period nearly in 7 days 3¾ hours, they -assign the distance of about 14⅖ semidiameters. But the outermost -satellite they remove to 25⅓ semidiameters, and this satellite makes -its period in about 16 days 16½ hours[177]. In Saturn there is still -a greater diversity in the distance of the several satellites. By the -observations of the late ~CASSINI~, a celebrated astronomer -in France, who first discovered all these satellites, except one known -before, the innermost is distant about 4½ of Saturn’s semidiameters -from his center, and revolves round in about 1 day 21⅓ hours. The next -satellite is distant about 5¾ semidiameters, and makes its period in -about 2 days 17⅔ hours. The third is removed to the distance of about -8 semidiameters, and performs its revolution in near 4 days 12½ hours. -The fourth satellite discovered first by the great HUYGENS, is near -18⅔ semidiameters, and moves round Saturn in about 15 days 22⅔ hours. -The outermost is distant 56 semidiameters, and makes its revolution in -about 79 days 7⅘ hours[178]. Besides these satellites, there belongs -to the planet Saturn another body of a very singular kind. This is a -shining, broad, and flat ring, which encompasses the planet round. -The diameter of the outermost verge of this ring is more than double -the diameter of Saturn. ~HUYGENS~, who first described this -ring, makes the whole diameter thereof to bear to the diameter of -Saturn the proportion of 9 to 4. The late reverend Mr. POUND makes the -proportion something greater, viz. that of 7 to 3. The distances of the -satellites of this planet Saturn are compared by ~CASSINI~ to -the diameter of the ring. His numbers I have reduced to those above, -according to Mr. POUND’s proportion between the diameters of Saturn and -of his ring. As this ring appears to adhere no where to Saturn, so the -distance of Saturn from the inner edge of the ring seems rather greater -than the breadth of the ring. The distances, which have here been -given, of the several satellites, both for Jupiter and Saturn, may be -more depended on in relation to the proportion, which those belonging -to the same primary planet bear one to another, than in respect to the -very numbers, that have been here set down, by reason of the difficulty -there is in measuring to the greatest exactness the diameters of the -primary planets; as will be explained hereafter, when we come to treat -of telescopes[179]. By the observations of the forementioned Mr. POUND, -in Jupiter the distance of the innermost satellite should rather be -about 6 semidiameters, of the second 9-½, of the third 15, and of -the outermost 26⅔[180]; and in Saturn the distance of the innermost -satellite 4 semidiameters, of the next 6¼, of the third 8¾, of the -fourth 20⅓, and of the fifth 59[181]. However the proportion between -the distances of the satellites in the same primary is the only thing -necessary to the point we are here upon. - -4. BUT moreover the force, wherewith the earth acts in different -distances, is confirmed from the following consideration, yet more -expresly than by the preceding analogical reasoning. It will appear, -that if the power of the earth, by which it retains the moon in her -orbit, be supposed to act at all distances between the earth and moon, -according to the forementioned rule; this power will be sufficient to -produce upon bodies, near the surface of the earth, all the effects -ascribed to the principle of gravity. This is discovered by the -following method. Let A (in fig. 94.) represent the earth, B the moon, -B C D the moon’s orbit, which differs little from a circle, of which A -is the center. If the moon in B were left to it self to move with the -velocity, it has in the point B, it would leave the orbit, and proceed -right forward in the line B E, which touches the orbit in B. Suppose -the moon would upon this condition move from B to E in the space of -one minute of time. By the action of the earth upon the moon, whereby -it is retained in its orbit, the moon will really be found at the end -of this minute in the point F, from whence a straight line drawn to A -shall make the space B F A in the circle equal to the triangular space -B E A; so that the moon in the time wherein it would have moved from -B to E, if left to it self, has been impelled towards the earth from -E to F. And when the time of the moon’s passing from B to F is small, -as here it is only one minute, the distance between E and F scarce -differs from the space, through which the moon would descend in the -same time, if it were to fall directly down from B toward A without any -other motion. A B the distance of the earth and moon is about 60 of the -earth’s semidiameters, and the moon completes her revolution round the -earth in about 27 days 7 hours and 43 minutes: therefore the space E F -will here be found by computation to be about 16⅛ feet. Consequently, -if the power, by which the moon is retained in its orbit, be near the -surface of the earth greater, than at the distance of the moon in the -duplicate proportion of that distance; the number of feet, a body would -descend near the surface of the earth by the action of this power upon -it in one minute of time, would be equal to 16⅛ multiplied twice into -the number 60, that is, equal to 58050. But how fast bodies fall near -the surface of the earth may be known by the pendulum[182]; and by the -exactest experiments they are found to descend the space of 16⅛ feet in -a second of time; and the spaces described by falling bodies being in -the duplicate proportion of the times of their fall[183], the number of -feet, a body would describe in its fall near the surface of the earth -in one minute of time, will be equal to 16⅛ twice multiplied by 60, the -same as would be caused by the power which acts upon the moon. - -5. IN this computation the earth is supposed to be at rest, whereas -it would have been more exact to have supposed it to move, as well -as the moon, about their common center of gravity; as will easily be -understood, by what has been said in the preceding chapter, where it -was shewn, that the sun is subjected to the like motion about the -common center of gravity of it self and the planets. The action of -the sun upon the moon, which is to be explain’d in what follows, is -likewise here neglected: and Sir ISAAC NEWTON shews, if you take in -both these considerations, the present computation will best agree -to a somewhat greater distance of the moon and earth, viz. to 60½ -semidiameters of the earth, which distance is more conformable to -astronomical observations. - -6. THESE computations afford an additional proof, that the action of -the earth observes the same proportion to the distance, which is here -contended for. Before I said, it was reasonable to conclude so by -induction from the planets Jupiter and Saturn; because they act in -that manner. But now the same thing will be evident by drawing no other -consequence from what is seen in those planets, than that the power, -by which the primary planets act on their secondary, is extended from -the primary through the whole interval between, so that it would act in -every part of the intermediate space. In Jupiter and Saturn this power -is so far from being confined to a small extent of distance, that it -not only reaches to several satellites at very different distances, but -also from one planet to the other, nay even through the whole planetary -system[184]. Consequently there is no appearance of reason, why this -power should not act at all distances, even at the very surfaces of -these planets as well as farther off. But from hence it follows, that -the power, which retains the moon in her orbit, is the same, as causes -bodies near the surface of the earth to gravitate. For since the -power, by which the earth acts on the moon, will cause bodies near the -surface of the earth to descend with all the velocity they are found -to do, it is certain no other power can act upon them besides; because -if it did, they must of necessity descend swifter. Now from all this -it is at length very evident, that the power in the earth, which we -call gravity, extends up to the moon, and decreases in the duplicate -proportion of the increase of the distance from the earth. - -7. THIS finishes the discoveries made in the action of the primary -planets upon their secondary. The next thing to be shewn is, that the -sun acts upon them likewise: for this purpose it is to be observed, -that if to the motion of the satellite, whereby it would be carried -round its primary at rest, be superadded the same motion both in -regard to velocity and direction, as the primary it self has, it will -describe about the primary the same orbit, with as great regularity, -as if the primary was indeed at rest. The cause of this is that law -of motion, which makes a body near the surface of the earth, when let -fall, to descend perpendicularly, though the earth be in so swift a -motion, that if the falling body did not partake of it, its descent -would be remarkably oblique; and that a body projected describes in -the most regular manner the same parabola, whether projected in the -direction, in which the earth moves, or in the opposite direction, if -the projecting force be the same[185]. From this we learn, that if the -satellite moved about its primary with perfect regularity, besides its -motion about the primary, it would participate of all the motion of its -primary; have the same progressive velocity, with which the primary -is carried about the sun; and be impelled with the same velocity as -the primary towards the sun, in a direction parallel to that impulse -of its primary. And on the contrary, the want of either of these, -in particular of the impulse towards the sun, will occasion great -inequalities in the motion of the secondary planet. The inequalities, -which would arise from the absence of this impulse towards the sun are -so great, that by the regularity, which appears in the motion of the -secondary planets, it is proved, that the sun communicates, the same -velocity to them by its action, as it gives to their primary at the -same distance. For Sir ~ISAAC NEWTON~ informs us, that upon -examination he found, that if any of the satellites of Jupiter were -attracted by the sun more or less, than Jupiter himself at the same -distance, the orbit of that satellite, instead of being concentrical to -Jupiter, must have its center at a greater or less distance, than the -center of Jupiter from the sun, nearly in the subduplicate proportion -of the difference between the sun’s action upon the satellite, and upon -Jupiter; and therefore if any satellite were attracted by the sun but -1/1000 part more or less, than Jupiter is at the same distance, the -center of the orbit of that satellite would be distant from the center -of Jupiter no less than a fifth part of the distance of the outermost -satellite from Jupiter[186]; which is almost the whole distance of the -innermost satellite. By the like argument the satellites of Saturn -gravitate towards the sun, as much as Saturn it self at the same -distance; and the moon as much as the earth. - -8. THUS is proved, that the sun acts upon the secondary planets, as -much as upon the primary at the same distance: but it was found in the -last chapter, that the action of the sun upon bodies is reciprocally -in the duplicate proportion of the distance; therefore the secondary -planets being sometimes nearer to the sun than the primary, and -sometimes more remote, they are not alway acted upon in the same degree -with their primary, but when nearer to the sun, are attracted more, -and when farther distant, are attracted less. Hence arise various -inequalities in the motion of the secondary planets[187]. - -9. SOME of these inequalities would take place, though the moon, if -undisturbed by the sun, would have moved in a circle concentrical -to the earth, and in the plane of the earth’s motion; others depend -on the elliptical figure, and the oblique situation of the moon’s -orbit. One of the first kind is, that the moon is caused so to move, -as not to describe equal spaces in equal times, but is continually -accelerated, as she passes from the quarter to the new or full, and is -retarded again by the like degrees in returning from the new and full -to the next quarter. Here we consider not so much the absolute, as the -apparent motion of the moon in respect to us. - -10. THE principles of astronomy teach how to distinguish these two -motions. Let S (in fig. 95.) represent the sun, A the earth moving -in its orbit B C, D E F G the moon’s orbit, the place of the moon H. -Suppose the earth to have moved from A to I. Because it has been shewn, -that the moon partakes of all the progressive motion of the earth; -and likewise that the sun attracts both the earth and moon equally, -when they are at the same distance from it, or that the mean action -of the sun upon the moon is equal to its action upon the earth: we -must therefore consider the earth as carrying about with it the moon’s -orbit; so that when the earth is removed from A to I, the moon’s orbit -shall likewise be removed from its former situation into that denoted -by K L M N. But now the earth being in I, if the moon were found in O, -so that O I should be parallel to H A, though the moon would really -have moved from H to O, yet it would not have appeared to a spectator -upon the earth to have moved at all, because the earth has moved as -much it self; so that the moon would still appear in the same place -with respect to the fixed stars. But if the moon be observed in P, it -will then appear to have moved, its apparent motion being measured -by the angle under O I P. And if the angle under P I S be less than -the angle under H A S, the moon will have approached nearer to its -conjunction with the sun. - -11. TO come now to the explication of the mentioned inequality in -the moon’s motion: let S (in fig. 96.) represent the sun, A the -earth, B C D E the moon’s orbit, C the place of the moon, when in -the latter quarter. Here it will be nearly at the same distance from -the sun, as the earth is. In this case therefore they will both be -equally attracted, the earth in the direction A S, and the moon in -the direction C S. Whence as the earth in moving round the sun is -continually descending toward it, so the moon in this situation must in -any equal portion of time descend as much; and therefore the position -of the line A C in respect of A S, and the change, which the moon’s -motion produces in the angle under C A S, will not be altered by the -sun. - -12. BUT now as soon as ever the moon is advanced from the quarter -toward the new or conjunction, suppose to G, the action of the sun upon -it will have a different effect. Here, were the sun’s action upon the -moon to be applied in the direction G H parallel to A S, if its action -on the moon were equal to its action on the earth, no change would be -wrought by the sun on the apparent motion of the moon round the earth. -But the moon receiving a greater impulse in G than the earth receives -in A, were the sun to act in the direction G H, yet it would accelerate -the description of the space D A G, and cause the angle under G A D to -decrease faster, than otherwise it would. The sun’s action will have -this effect upon account of the obliquity of its direction to that, -in which the earth attracts the moon. For the moon by this means is -drawn by two forces oblique to each other, one drawing from G toward -A, the other from G toward H, therefore the moon must necessarily be -impelled toward D. Again, because the sun does not act in the direction -G H parallel to S A, but in the direction G S oblique to it, the sun’s -action on the moon will by reason of this obliquity farther contribute -to the moon’s acceleration. Suppose the earth in any short space of -time would have moved from A to I, if not attracted by the sun; the -point I being in the straight line C E, which touches the earth’s orbit -in A. Suppose the moon in the same time would have moved in her orbit -from G to K, and besides have partook of all the progressive motion of -the earth. Then if K L be drawn parallel to A I, and taken equal to it, -the moon, if not attracted by the sun, would be found in L. But the -earth by the sun’s action is removed from I. Suppose it were moved down -to M in the line I M N parallel to S A, and if the moon were attracted -but as much, and in the same direction, as the earth is here supposed -to be attracted, so as to have descended during the same time in the -line L O, parallel also to A S, down as far as P, till L P were equal -to I M; the angle under P M N would be equal to that under L I N, that -is, the moon will appear advanced no farther forward, than if neither -it nor the earth had been subject to the sun’s action. But this is upon -the supposition, that the action of the sun upon the moon and earth -were equal; whereas the moon being acted upon more than the earth, -did the sun’s action draw the moon in the line L O parallel to A S, -it would draw it down so far as to make L P greater than I M; whereby -the angle under P M N will be rendred less, than that under L I N. But -moreover, as the sun draws the earth in a direction oblique to I N, the -earth will be found in its orbit somewhat short of the point M; however -the moon is attracted by the sun still more out of the line L O, than -the earth is out of the line I N; therefore this obliquity of the sun’s -action will yet farther diminish the angle under P M N. - -13. THUS the moon at the point G receives an impulse from the sun, -whereby her motion is accelerated. And the sun producing this effect in -every place between the quarter and the conjunction, the moon will move -from the quarter with a motion continually more and more accelerated; -and therefore by acquiring from time to time additional degrees of -velocity in its orbit, the spaces, which are described in equal times -by the line drawn from the earth to the moon, will not be every where -equal, but those toward the conjunction will be greater, than those -toward the quarter. But now in the moon’s passage from the conjunction -D to the next quarter the sun’s action will again retard the moon, till -at the next quarter in E it be restored to the first velocity, which it -had in C. - -14. AGAIN as the moon moves from E to the full or opposition to the -sun in B, it is again accelerated, the deficiency of the sun’s action -upon the moon, from what it has upon the earth, producing here the same -effect as before the excess of its action. Consider the moon in Q, -moving from E towards B. Here if the moon were attracted by the sun in -a direction parallel to A S, yet being acted on less than the earth, -as the earth descends toward the sun, the moon will in some measure be -left behind. Therefore Q F being drawn parallel to S B, a spectator -on the earth would see the moon move, as if attracted from the point -Q in the direction Q F with a degree of force equal to that, whereby -the sun’s action on the moon falls short of its action on the earth. -But the obliquity of the sun’s action has also here an effect. In the -time the earth would have moved from A to I without the influence of -the sun, let the moon have moved in its orbit from Q to R. Drawing -therefore R T parallel to A I, and equal to the same, for the like -reason as before, the moon by the motion of its orbit, if not at all -attracted by the sun, must be found in T; and therefore, if attracted -in a direction parallel to S A, would be in the line T V parallel to -A S; suppose in W. But the moon in Q being farther off the sun than -the earth, it will be less attracted, that is, T W will be less than -I M, and if the line S M be prolonged toward X, the angle under X M -W will be less than that under X I T. Thus by the sun’s action the -moon’s passage from the quarter to the full would be accelerated, if -the sun were to act on the earth and moon in a direction parallel to A -S: and the obliquity of the sun’s action will still more increase this -acceleration. For the action of the sun on the moon is oblique to the -line S A the whole time of the moon’s passage from Q to T, and will -carry the moon out of the line T V toward the earth. Here I suppose the -time of the moon’s passage from Q to T so short, that it shall not pass -beyond the line S A. The earth also will come a little short of the -line I N, as was said before. From these causes the angle under X M W -will be still farther lessened. - -15. THE moon in passing from the opposition B to the next quarter will -be retarded again by the same degrees, as it is accelerated before -its appulse to the opposition. Because this action of the sun, which -in the moon’s passage from the quarter to the opposition causes it -to be extraordinarily accelerated, and diminishes the angle, which -measures its distance from the opposition; will make the moon slacken -its pace afterwards, and retard the augmentation of the same angle in -its passage from the opposition to the following quarter; that is, will -prevent that angle from increasing so fast, as otherwise it would. And -thus the moon, by the sun’s action upon it, is twice accelerated and -twice restored to its first velocity, every circuit it makes round the -earth. This inequality of the moon’s motion about the earth is called -by astronomers its variation. - -16. THE next effect of the sun upon the moon is, that it gives the -orbit of the moon in the quarters a greater degree of curvature, -than it would receive from the action of the earth alone; and on the -contrary in the conjunction and opposition the orbit is less inflected. - -17. WHEN the moon is in conjunction with the sun in the point D, the -sun attracting the moon more forcibly than it does the earth, the -moon by that means is impelled less toward the earth, than otherwise -it would be, and so the orbit is less incurvated; for the power, by -which the moon is impelled toward the earth, being that, by which it is -inflected from a rectilinear course, the less that power is, the less -it will be inflected. Again, when the moon is in the opposition in B, -farther removed from the sun than the earth is; it follows then, though -the earth and moon are both continually descending to the sun, that -is, are drawn by the sun toward it self out of the place they would -otherwise move into, yet the moon descends with less velocity than -the earth; insomuch that the moon in any given space of time from its -passing the point of opposition will have less approached the earth, -than otherwise it would have done, that is, its orbit in respect of -the earth will approach nearer to a straight line. In the last place, -when the moon is in the quarter in F, and equally distant from the -sun as the earth, we observed before, that the earth and moon would -descend with equal pace toward the sun, so as to make no change by -that descent in the angle under F A S; but the length of the line F -A must of necessity be shortned. Therefore the moon in moving from F -toward the conjunction with the sun will be impelled more toward the -earth by the sun’s action, than it would have been by the earth alone, -if neither the earth nor moon had been acted on by the sun; so that -by this additional impulse the orbit is rendred more curve, than it -would otherwise be. The same effect will also be produced in the other -quarter. - -18. ANOTHER effect of the sun’s action, consequent upon this we have -now explained, is, that though the moon undisturbed by the sun might -move in a circle having the earth for its center; by the sun’s action, -if the earth were to be in the very middle or center of the moon’s -orbit, yet the moon would be nearer the earth at the new and full, than -in the quarters. In this probably will at first appear some difficulty, -that the moon should come nearest to the earth, where it is least -attracted to it, and be farthest off when most attracted. Which yet -will appear evidently to follow from that very cause, by considering -what was last shewn, that the orbit of the moon in the conjunction -and opposition is rendred less curve; for the less curve the orbit of -the moon is, the less will the moon have descended from the place it -would move into, without the action of the earth. Now if the moon were -to move from any place without farther disturbance from that action, -since it would proceed in the line, which would touch its orbit in that -place, it would recede continually from the earth; and therefore if -the power of the earth upon the moon, be sufficient to retain it at the -same distance, this diminution of that power will cause the distance -to increase, though in a less degree. But on the other hand in the -quarters, the moon, being pressed more towards the earth than by the -earth’s single action, will be made to approach it; so that in passing -from the conjunction or opposition to the quarters the moon ascends -from the earth, and in passing from the quarters to the conjunction and -opposition it descends again, becoming nearer in these last mentioned -places than in the other. - -19. ALL these forementioned inequalities are of different degrees, -according as the sun is more or less distant from the earth; greater -when the earth is nearest the sun, and less when it is farthest off. -For in the quarters, the nearer the moon is to the sun, the greater is -the addition to the earth’s action upon it by the power of the sun; and -in the conjunction and opposition, the difference between the sun’s -action upon the earth and upon the moon is likewise so much the greater. - -20. This difference in the distance between the earth and the sun -produces a farther effect upon the moon’s motion; causing the orbit to -dilate when less remote from the sun, and become greater, than when at -a farther distance. For it is proved by Sir ~ISAAC NEWTON~, that the -action of the sun, by which it diminishes the earth’s power over the -moon, in the conjunction or opposition, is about twice as great, as -the addition to the earth’s action by the sun in the quarters[188]; so -that upon the whole, the power of the earth upon the moon is diminished -by the sun, and therefore is most diminished, when the action of the -sun is strongest: but as the earth by its approach to the sun has its -influence lessened, the moon being less attracted will gradually recede -from the earth; and as the earth in its recess from the sun recovers by -degrees its former power, the orbit of the moon must again contract. -Two consequences follow from hence: the moon will be most remote from -the earth, when the earth is nearest the sun; and also will take up a -longer time in performing its revolution through the dilated orbit, -than through the more contracted. - -21. THESE irregularities the sun would produce in the moon, if the -moon, without being acted on unequally by the sun, would describe a -perfect circle about the earth, and in the plane of the earth’s motion; -but though neither of these suppositions obtain in the motion of the -moon, yet the forementioned inequalities will take place, only with -some difference in respect to the degree of them; but the moon by not -moving in this manner is subject to some other inequalities also. -For as the moon describes, instead of a circle concentrical to the -earth, an ellipsis, with the earth in one focus, that ellipsis will be -subjected to various changes. It can neither preserve constantly the -same position, nor yet the same figure; and because the plane of this -ellipsis is not the same with that of the earth’s orbit, the situation -of the plane, wherein the moon moves, will continually change; neither -the line in which it intersects the plane of the earth’s orbit, nor the -inclination of the planes to each other, will remain for any time the -same. All these alterations offer themselves now to be explained. - -22. I SHALL first consider the changes which are made in the plane -of the moon’s orbit. The moon not moving in the same plane with the -earth, the sun is seldom in the plane of the moon’s orbit, viz. only -when the line made by the common intersection of the two planes, if -produced, will pass through the sun, as is represented in fig. 97. -where S denotes the sun; T the earth; A T B the earth’s orbit described -upon the plane of this scheme; C D E F the moon’s orbit, the part C -D E being raised above, and the part C F E depressed under the plane -of this scheme. Here the line C E, in which the plane of this scheme, -that is, the plane of the earth’s orbit and the plane of the moon’s -orbit intersect each other, being continued passes through the sun in -S. When this happens, the action of the sun is directed in the plane of -the moon’s orbit, and cannot draw the moon out of this plane, as will -evidently appear to any one that shall consider the present scheme: for -suppose the moon in G, and let a straight line be drawn from G to S, -the sun draws the moon in the direction of this line from G toward S: -but this line lies in the plane of the orbit; and if it be prolonged -from S beyond G, the continuation of it will lie on the plane C D E; -for the plane itself, if sufficiently extended, will pass through the -sun. But in other cases the obliquity of the sun’s action to the plane -of the orbit will cause this plane continually to change. - -23. SUPPOSE in the first place, the line, in which the two planes -intersect each other, to be perpendicular to the line which joins the -earth and sun. Let T (in fig. 98, 99, 100, 101.) represent the earth; S -the sun; the plane of this scheme the plane of the earth’s motion, in -which both the sun and earth are placed. Let A C be perpendicular to -S T, which joins the earth and sun; and let the line A C be that, in -which the plane of the moon’s orbit intersects the plane of the earth’s -motion. To the center T describe in the plane of the earth’s motion -the circle A B C D. And in the plane of the moon’s orbit describe the -circle A E C F, one half of which A E C will be elevated above the -plane of this scheme, the other half A F C as much depressed below it. - -24. NOW suppose the moon to set forth from the point A (in fig. 98.) in -the direction of the plane A E C. Here she will be continually drawn -out of this plane by the action of the sun: for this plane A E C, if -extended, will not pass through the sun, but above it; so that the sun, -by drawing the moon directly toward it self, will force it continually -more and more from that plane towards the plane of the earth’s motion, -in which it self is; causing it to describe the line A K G H I, which -will be convex to the plane A E C, and concave to the plane of the -earth’s motion. But here this power of the sun, which is said to draw -the moon toward the plane of the earth’s motion, must be understood -principally of so much only of the sun’s action upon the moon, as -it exceeds the action of the same upon the earth. For suppose the -preceding figure to be viewed by the eye, placed in the plane of that -scheme, and in the line C T A on the side of A, the plane A B C D will -appear as the straight line D T B, (in fig. 102.) and the plane A E C -F as another straight line F E; and the curve line A K G H I under the -form of the line T K G H I. - -[Illustration] - -Now it is plain, that the earth and moon being both attracted by the -sun, if the sun’s action upon both was equally strong, the earth T, -and with it the plane A E C F or line F T E in this scheme, would be -carried toward the sun with as great a pace as the moon, and therefore -the moon not drawn out of it by the sun’s action, excepting only from -the small obliquity of the direction of this action upon the moon -to that of the sun’s action upon the earth, which arises from the -moon’s being out of the plane of the earth’s motion, and is not very -considerable; but the action of the sun upon the moon being greater -than upon the earth, all the time the moon is nearer to the sun than -the earth is, it will be drawn from the plane A E C or the line T E by -that excess, and made to describe the curve line A G I or T G I. But it -is the custom of astronomers, instead of considering the moon as moving -in such a curve line, to refer its motion continually to the plane, -which touches the true line wherein it moves, at the point where at -any time the moon is. Thus when the moon is in the point A, its motion -is considered as being in the plane A E C, in whose direction it then -essaies to move; and when in the point K (in fig. 99.) its motion is -referred to the plane, which passes through the earth, and touches the -line A K G H I in the point K. Thus the moon in passing from A to I -will continually change the plane of her motion. In what manner this -change proceeds, I shall now particularly explain. - -25. LET the plane, which touches the line A K I in the point K (in fig. -99.) intersect the plane of the earth’s orbit in the line L T M. Then, -because the line A K I is concave to the plane A B C, it falls wholly -between that plane, and the plane which touches it in K; so that the -plane M K L will cut the plane A E C, before it meets with the plane of -the earth’s motion; suppose in the line Y T, and the point A will fall -between K and L. With a semidiameter equal to T Y or T L describe the -semicircle L Y M. Now to a spectator on the earth the moon, when in A, -will appear to move in the circle A E C F, and, when in K, will appear -to be moving in the semicircle L Y M. The earth’s motion is performed -in the plane of this scheme, and to a spectator on the earth the sun -will appear always moving in that plane. We may therefore refer the -apparent motion of the sun to the circle A B C D, described in this -plane about the earth. But the points where this circle, in which the -sun seems to move, intersects the circle in which the moon is seen at -any time to move, are called the nodes of the moon’s orbit at that -time. When the moon is seen moving in the circle A E C D, the points A -and C are the nodes of the orbit; when she appears in the semicircle -L Y M, then L and M are the nodes. Now here it appears, from what has -been said, that while the moon has moved from A to K, one of the nodes -has been carried from A to L, and the other as much from C to M. But -the motion from A to L, and from C to M, is backward in regard to the -motion of the moon, which is the other way from A to K, and from thence -toward C. - -26. FARTHER the angle, which the plane, wherein the moon at any time -appears, makes with the plane of the earth’s motion, is called the -inclination of the moon’s orbit at that time. And I shall now proceed -to shew, that this inclination of the orbit, when the moon is in K, is -less than when she was in A; or, that the plane L Y M, which touches -the line of the moon’s motion in K, makes a less angle with the plane -of the earth’s motion or with the circle A B C D, than the plane A E -C makes with the same. The semicircle L Y M intersects the semicircle -A E C in Y; and the arch A Y is less than L Y, and both together less -than half a circle. But it is demonstrated by the writers on that part -of astronomy, which is called the doctrine of the sphere, that when a -triangle is made, as here, by three arches of circles A L, A Y, and Y -L, the angle under Y A B without the triangle is greater than the angle -under Y L A within, if the two arches A Y, Y L taken together do not -amount to a semicircle; if the two arches make a complete semicircle, -the two angles will be equal; but if the two arches taken together -exceed a semicircle, the inner angle under Y L A is greater than the -other[189]. Here therefore the two arches A Y and L Y together being -less than a semicircle, the angle under A L Y is less, than the angle -under B A E. But from the doctrine of the sphere it is also evident, -that the angle under A L Y is equal to that, in which the plane of the -circle L Y K M, that is, the plane which touches the line A K G H I in -K, is inclined to the plane of the earth’s motion A B C; and the angle -under B A E is equal to that, in which the plane A E C is inclined to -the same plane. Therefore the inclination of the former plane is less -than the inclination of the latter. - -27. SUPPOSE now the moon to be advanced to the point G (in fig. 100.) -and in this point to be distant from its node a quarter part of the -whole circle; or in other words, to be in the midway between its two -nodes. And in this case the nodes will have receded yet more, and the -inclination of the orbit be still more diminished: for suppose the -line A K G H I to be touched in the point G by a plane passing through -the earth T: let the intersection of this plane with the plane of the -earth’s motion be the line W T O, and the line T P its intersection -with the plane L K M. In this plane let the circle N G O be described -with the semidiameter T P or N T cutting the other circle L K M in P. -Now the line A K G I is convex to the plane L K M, which touches it in -K; and therefore the plane N G O, which touches it in G, will intersect -the other touching plane between G and K; that is, the point P will -fall between those two points, and the plane continued to the plane of -the earth’s motion will pass beyond L; so that the points N and O, or -the places of the nodes, when the moon is in G, will be farther from A -and C than L and M, that is, will have moved farther backward. Besides, -the inclination of the plane N G O to the plane of the earth’s motion A -B C is less, than the inclination of the plane L K M to the same; for -here also the two arches L P and N P taken together are less than a -semicircle, each of these arches being less than a quarter of a circle; -as appears, because G N, the distance of the moon in G from its node N, -is here supposed to be a quarter part of a circle. - -28. AFTER the moon is passed beyond G, the case is altered; for then -these arches will be greater than quarters of the circle, by which -means the inclination will be again increased, tho’ the nodes still go -on to move the same way. Suppose the moon in H, (in fig. 101.) and that -the plane, which touches the line A K G I in H, intersects the plane of -the earth’s motion in the line Q T R, and the plane N G O in the line T -V, and besides that the circle Q H R be described in that plane; then, -for the same reason as before, the point V will fall between H and G, -and the plane R V Q will pass beyond the last plane O V N, causing -the points Q and R to fall farther from A and C than N and O. But the -arches N V, V Q are each greater than a quarter of a circle, N V the -least of them being greater than G N, which is a quarter of a circle; -and therefore the two arches N V and V Q together exceed a semicircle; -consequently the angle under B Q V will be greater, than that under B N -V. - -29. IN the last place, when the moon is by this attraction of the sun, -drawn at length into the plane of the earth’s motion, the node will -have receded yet more, and the inclination be so much increased, as -to become somewhat more than at first: for the line A K G H I being -convex to all the planes, which touch it, the part H I will wholly fall -between the plane Q V R and the plane A B C; so that the point I will -fall between B and R; and drawing I T W, the point W will be farther -remov’d from A than Q. But it is evident, that the plane, which passes -through the earth T, and touches the line A G I in the point I, will -cut the plane of the earth’s motion A B C D in the line I T W, and -be inclined to the same in the angle under H I B; so that the node, -which was first in A, after having passed into L, N and Q, comes at -last into the point W; as the node which was at first in C has passed -successively from thence through the points M, O and R to I: but the -angle under H I B, which is now the inclination of the orbit to the -plane of the ecliptic, is manifestly not less than the angle under E C -B or E A B, but rather something greater. - -30. THUS the moon in the case before us, while it passes from the -plane of the earth’s motion in the quarter, till it comes again into -the same plane, has the nodes of its orbit continually moved backward, -and the inclination of its orbit is at first diminished, viz. till it -comes to G in fig. 100, which is near to its conjunction with the sun, -but afterwards is increased again almost by the same degrees, till -upon the moon’s arrival again to the plane of the earth’s motion, the -inclination of the orbit is restored to something more than its first -magnitude, though the difference is not very great, because the points -I and C are not far distant from each other[190]. - -31. AFTER the same manner, if the moon had departed from the quarter in -C, it should have described the curve line C X W (in fig. 98.) between -the planes A F C and A D C, which would be convex to the former of -those planes, and concave to the latter; so that, here also, the nodes -should continually recede, and the inclination of the orbit gradually -diminish more and more, till the moon arrived near its opposition to -the sun in X; but from that time the inclination should again increase, -till it became a little greater than at first. This will easily -appear, by considering, that as the action of the sun upon the moon, -by exceeding its action upon the earth, drew it out of the plane A E -C towards the sun, while the moon passed from A to I; so, during its -passage from C to W, the moon being all that time farther from the sun -than the earth, it will be attracted less; and the earth, together with -the plane A E C F, will as it were be drawn from the moon, in such -sort, that the path the moon describes shall appear from the earth, as -it did in the former case by the moon’s being drawn away. - -32. THESE are the changes, which the nodes and the inclination of the -moon’s orbit undergo, when the nodes are in the quarters; but when the -nodes by their motion, and the motion of the sun together, come to -be situated between the quarter and conjunction or opposition, their -motion and the change made in the inclination of the orbit are somewhat -different. - -33. LET A G C H (in fig. 103.) be a circle described in the plane of -the earth’s motion, having the earth in T for its center. Let the -point opposite to the sun be A, and the point G a fourth part of the -circle distant from A. Let the nodes of the moon’s orbit be situated -in the line B T D, and B the node, falling between A, the place where -the moon would be in the full, and G the place where the moon would -be in the quarter. Suppose B E D F to be the plane, in which the moon -essays to move, when it proceeds from the point B. Because the moon -in B is more distant from the sun than the earth, it shall be less -attracted by the sun, and shall not descend towards the sun so fast -as the earth: consequently it shall quit the plane B E D F, which we -suppose to accompany the earth, and describe the line B I K convex -thereto, till such time as it comes to the point K, where it will be in -the quarter: but from thenceforth being more attracted than the earth, -the moon shall change its course, and the following part of the path -it describes shall be concave to the plane B E D or B G D, and shall -continue concave to the plane B G D, till it crosses that plane in L, -just as in the preceding case. Now I say, while the moon is passing -from B to K, the nodes, contrary to what was found in the foregoing -case, will proceed forward, or move the same way with the moon[191]; -and at the same time the inclination of the orbit will increase[192]. - -[Illustration] - -34. WHEN the moon is in the point I, let the plane M I N pass through -the earth T, and touch the path of the moon in I, cutting the plane of -the earth’s motion, in the line M T N, and the plane B E D in the line -T O. Because the line B I K is convex to the plane B E D, which touches -it in B, the plane N I M must cross the plane D E B, before it meets -the plane C G B; and therefore the point M will fall from B towards G, -and the node of the moon’s orbit being translated from B to M is moved -forward. - -35. I SAY farther, the angle under O M G, which the plane M O N makes -with the plane B G C, is greater than the angle under O B G, which -the plane B O D makes with the same. This appears from what has been -already explained; because the arches B O, O M are each less than the -quarter of a circle, and therefore taken both together are less than a -semicircle. - -36. AGAIN, when the moon is come to the point K in its quarter, the -nodes will be advanced yet farther forward, and the inclination of the -orbit also more augmented. Hitherto the moon’s motion has been referred -to the plane, which passing through the earth touches the path of the -moon in the point, where the moon is, according to what was asserted at -the beginning of this discourse upon the nodes, that it is the custom -of astronomers so to do. But here in the point K no such plane can be -found; on the contrary, seeing the line of the moon’s motion on one -side the point K is convex to the plane B E D, and on the other side -concave to the same, no plane can pass through the points T and K but -will cut the line B K L in that point. Therefore instead of such a -touching plane, we must here make use of what is equivalent, the plane -P K Q, with which the line B K L shall make a less angle than with any -other plane; for this plane does as it were touch the line B K in the -point K, since it so cuts it, that no other plane can be drawn so, -as to pass between the line B K and the plane P K Q. But now it is -evident, that the point P, or the node, is removed from M towards G, -that is, has moved yet farther forward; and it is likewise as manifest, -that the angle under K P G, or the inclination of the moon’s orbit in -the point K, is greater than the angle under I M G, for the reason so -often assigned. - -37. AFTER the moon has passed the quarter, the path of the moon being -concave to the plane A G C H, the nodes, as in the preceding case, -shall recede, till the moon arrives at the point L; which shews, that -considering the whole time of the moon’s passing from B to L, at the -end of that time the nodes shall be found to have receded, or to be -placed backwarder, when the moon is in L, than when it was in B. For -the moon takes a longer time in passing from K to L, than in passing -from B to K; and therefore the nodes continue to recede a longer time, -than they moved forwards; so that their recess must surmount their -advance. - -38. IN the same manner, while the moon is in its passage from K to L, -the inclination of the orbit shall diminish, till the moon comes to -the point, in which it is one quarter part of a circle distant from -its node; suppose in the point R; and from that time the inclination -shall again increase. Since therefore the inclination of the orbit -increases, while the moon is passing from B to K, and diminishes itself -again only, while the moon is passing from K to R, and then augments -again, till the moon arrive in L; while the moon is passing from B to -L, the inclination of the orbit is much more increased than diminished, -and will be distinguishably greater, when the moon is come to L, than -when it set out from B. - -39. IN like manner, while the moon is passing from L on the other side -the plane A G C H, the node shall advance forward, as long as the moon -is between the point L and the next quarter; but afterwards it shall -recede, till the moon come to pass the plane A G C H again in the point -V, between B and A: and because the time between the moon’s passing -from L to the next quarter is less, than the time between that quarter -and the moon’s coming to the point V, the node shall have more receded -than advanced; so that the point V will be nearer to A, than L is to C. -So also the inclination of the orbit, when the moon is in V, will be -greater, than when the moon was at L; for this inclination increases -all the time the moon is between L and the next quarter; it decreases -only while the moon is passing from this quarter to the mid way between -the two nodes, and from thence increases again during the whole passage -through the other half of the way to the next node. - -40. THUS we have traced the moon from her node in the quarter, and -shewn, that at every period of the moon the nodes will have receded, -and thereby will have approached toward a conjunction with the sun. -But this conjunction will be much forwarded by the visible motion of -the sun itself. In the last scheme the sun will appear to move from -S toward W. Suppose it appeared to have moved from S to W, while the -moon’s node has receded from B to V, then drawing the line W T X, -the arch V X will represent the distance of the line drawn between -the nodes from the sun, when the moon is in V; whereas the arch B A -represented that distance, when the moon was in B. This visible motion -of the sun is much greater, than that of the node; for the sun appears -to revolve quite round each year, and the node is near 19 years in -making one revolution. We have also seen, that when the node was in -the quadrature, the inclination of the moon’s orbit decreased, till -the moon came to the conjunction, or opposition, according to which -node it set out from; but that afterwards it again increased, till it -became at the next node rather greater than at the former. When the -node is once removed from the quarter nearer to a conjunction with the -sun, the inclination of the moon’s orbit, when the moon comes into the -node, is more sensibly greater, than it was in the node preceding; the -inclination of the orbit by this means more and more increasing till -the node comes into conjunction with the sun; at which time it has -been shewn above, that the sun has no power to change the plane of the -moon’s motion; and consequently has no effect either on the nodes, or -on the inclination of the orbit. - -41. AS soon as the nodes, by the action of the sun, are got out of -conjunction toward the other quarters, they begin again to recede as -before; but the inclination of the orbit in the appulse of the moon -to each succeeding node is less than at the preceding, till the nodes -come again into the quarters. This will appear as follows. Let A (in -fig. 104.) represent one of the moon’s nodes placed between the point -of opposition B and the quarter C. Let the plane A D E pass through -the earth T, and touch the path of the moon in A. Let the line A F G H -be the path of the moon in her passage from A to H, where she crosses -again the plane of the earth’s motion. This line will be convex toward -the plane A D E, till the moon comes to G, where she is in the quarter; -and after this, between G and H, the same line will be concave toward -this plane. All the time this line is convex toward the plane A D -E, the nodes will recede; and on the contrary proceed, while it is -concave to that plane. All this will easily be conceived from what has -been before so largely explained. But the moon is longer in passing -from A to G, than from G to H; therefore the nodes recede a longer -time, than they proceed; consequently upon the whole, when the moon is -arrived at H, the nodes will have receded, that is, the point H will -fall between B and E. The inclination of the orbit will decrease, till -the moon is arrived to the point F, in the middle between A and H. -Through the passage between F and G the inclination will increase, but -decrease again in the remaining part of the passage from G to H, and -consequently at H must be less than at A. The like effects, both in -respect to the nodes and inclination of the orbit, will take place in -the following passage of the moon on the other side of the plane A B E -C, from H, till it comes over that plane again in I. - -42. THUS the inclination of the orbit is greatest, when the line drawn -between the moon’s nodes will pass through the sun; and least, when -this line lies in the quarters, especially if the moon at the same time -be in conjunction with the sun, or in the opposition. In the first of -these cases the nodes have no motion, in all others, the nodes will -each month have receded: and this regressive motion will be greatest, -when the nodes are in the quarters; for in that case the nodes have -no progressive motion during the whole month, but in all other cases -the nodes do at some times proceed forward, viz. whenever the moon is -between either quarter, and the node which is less distant from that -quarter than a fourth part of a circle. - -43. IT now remains only to explain the irregularities in the moon’s -motion, which follow from the elliptical figure of the orbit. By -what has been said at the beginning of this chapter it appears, that -the power of the earth on the moon acts in the reciprocal duplicate -proportion of the distance: therefore the moon, if undisturbed by -the sun, would move round the earth in a true ellipsis, and the line -drawn from the earth to the moon would pass over equal spaces in equal -portions of time. That this description of the spaces is altered by -the sun, has been already declared. It has also been shown, that the -figure of the orbit is changed each month; that the moon is nearer the -earth at the new and full, and more remote in the quarters, than it -would be without the sun. Now we must pass by these monthly changes, -and consider the effect, which the sun will have in the different -situations of the axis of the orbit in respect of that luminary. - -44. THE action of the sun varies the force, wherewith the moon is drawn -toward the earth; in the quarters the force of the earth is directly -increased by the sun; at the new and full the same is diminished; and -in the intermediate places the influence of the earth is sometimes -aided, and sometimes lessened by the sun. In these intermediate -places between the quarters and the conjunction or opposition, the -sun’s action is so oblique to the action of the earth on the moon, as -to produce that alternate acceleration and retardment of the moon’s -motion, which I observed above to be stiled the variation. But besides -this effect, the power, by which the earth attracts the moon toward -itself, will not be at full liberty to act with the same force, as if -the sun acted not at all on the moon. And this effect of the sun’s -action, whereby it corroborates or weakens the action of the earth, is -here only to be considered. And by this influence of the sun it comes -to pass, that the power, by which the moon is impelled toward the -earth, is not perfectly in the reciprocal duplicate proportion of the -distance. Consequently the moon will not describe a perfect ellipsis. -One particular, wherein the moon’s orbit will differ from an ellipsis, -consists in the places, where the motion of the moon is perpendicular -to the line drawn from itself to the earth. In an ellipsis, after the -moon should have set out in the direction perpendicular to this line -drawn from itself to the earth, and at its greatest distance from the -earth, its motion would again become perpendicular to this line drawn -between itself and the earth, and the moon be at its nearest distance -from the earth, when it should have performed half its period; after -performing the other half of its period its motion would again become -perpendicular to the forementioned line, and the moon return into -the place whence it set out, and have recovered again its greatest -distance. But the moon in its real motion, after setting out as before, -sometimes makes more than half a revolution, before its motion comes -again to be perpendicular to the line drawn from itself to the earth, -and the moon is at its nearest distance; and then performs more than -another half of an intire revolution before its motion can a second -time recover its perpendicular direction to the line drawn from the -moon to the earth, and the moon arrive again to its greatest distance -from the earth. At other times the moon will descend to its nearest -distance, before it has made half a revolution, and recover again -its greatest distance, before it has made an intire revolution. The -place, where the moon is at its greatest distance from the earth, is -called the moon’s apogeon, and the place of the least distance the -perigeon. This change of the place, where the moon successively comes -to its greatest distance from the earth, is called the motion of the -apogeon. In what manner the sun causes the apogeon to move, I shall now -endeavour to explain. - -45. OUR author shews, that if the moon were attracted toward the -earth by a composition of two powers, one of which were reciprocally -in the duplicate proportion of the distance from the earth, and the -other reciprocally in the triplicate proportion of the same distance; -then, though the line described by the moon would not be in reality -an ellipsis, yet the moon’s motion might be perfectly explained by -an ellipsis, whose axis should be made to move round the earth; this -motion being in consequence, as astronomers express themselves, that -is, the same way as the moon itself moves, if the moon be attracted by -the sum of the two powers; but the axis must move in antecedence, or -the contrary way, if the moon be acted on by the difference of these -powers. What is meant by duplicate proportion has been often explained; -namely, that if three magnitudes, as A, B, and C, are so related, that -the second B bears the same proportion to the third C, as the first A -bears to the second B, then the proportion of the first A to the third -C, is the duplicate of the proportion of the first A to the second B. -Now if a fourth magnitude, as D, be assumed, to which C shall bear the -same proportion as A bears to B, and B to C, then the proportion of A -to D is the triplicate of the proportion of A to B. - -46. THE way of representing the moon’s motion in this case is thus. T -denoting the earth (in fig. 105, 106.) suppose the moon in the point -A, its apogeon, or greatest distance from the earth, moving in the -direction A F perpendicular to A B, and acted upon from the earth by -two such forces as have been named. By that power alone, which is -reciprocally in the duplicate proportion of the distance, if the moon -let out from the point A with a proper degree of velocity, the ellipsis -A M B may be described. But if the moon be acted upon by the sum of -the forementioned powers, and the velocity of the moon in the point -A be augmented in a certain proportion[193]; or if that velocity be -diminished in a certain proportion, and the moon be acted upon by the -difference of those powers; in both these cases the line A E, which -shall be described by the moon, is thus to be determined. Let the point -M be that, into which the moon would have arrived in any given space of -time, had it moved in the ellipsis A M B. Draw M T, and likewise C T D -in such sort, that the angle under A T M shall bear the same proportion -to the angle under A T C, as the velocity, with which the ellipsis A -M B must have been described, bears to the difference between this -velocity, and the velocity, with which the moon must set out from the -point A in order to describe the path A E. Let the angle A T C be taken -toward the moon (as in fig. 105.) if the moon be attracted by the sum -of the powers; but the contrary way (as in fig. 106.) if by their -difference. Then let the line A B be moved into the position C D, and -the ellipsis A M B into the situation C N D, so that the point M be -translated to L: then the point L shall fall upon the path of the moon -A E. - -47. THE angular motion of the line A T, wereby it is removed into the -situation C T, represents the motion of the apogeon; by the means of -which the motion of the moon might be fully explicated by the ellipsis -A M B, if the action of the sun upon it was directed to the center -of the earth, and reciprocally in the triplicate proportion of the -moon’s distance from it. But that not being so, the apogeon will not -move in the regular manner now described. However, it is to be observed -here, that in the first of the two preceding cases, where the apogeon -moves forward, the whole centripetal power increases faster, with the -decrease of distance, than if the intire power were reciprocally in -the duplicate proportion of the distance; because one part only is in -that proportion, and the other part, which is added to this to make up -the whole power, increases faster with the decrease of distance. On -the other hand, when the centripetal power is the difference between -these two, it increases less with the decrease of the distance, than -if it were simply in the reciprocal duplicate proportion of the -distance. Therefore if we chuse to explain the moon’s motion by an -ellipsis (as is most convenient for astronomical uses to be done, and -by reason of the small effect of the sun’s power, the doing so will -not be attended with any sensible error;) we may collect in general, -that when the power, by which the moon is attracted to the earth, by -varying the distance, increases in a greater than in the duplicate -proportion of the distance diminished, a motion in consequence must -be ascribed to the apogeon; but that when the attraction increases -in a less proportion than that named, the apogeon must have given -to it a motion in antecedence[194]. It is then observed by Sir IS. -NEWTON, that the first of these cases obtains, when the moon is in the -conjunction and opposition; and the latter, when the moon is in the -quarters: so that in the first the apogeon moves according to the order -of the signs; in the other, the contrary way[195]. But, as was said -before, the disturbance given to the action of the earth by the sun -in the conjunction and opposition being near twice as great as in the -quarters[196], the apogeon will advance with a greater velocity than -recede, and in the compass of a whole revolution of the moon will be -carried in consequence[197]. - -48. IT is shewn in the next place by our author, that when the line A B -coincides with that, which joins the earth and the sun, the progressive -motion of the apogeon, when the moon is in the conjunction or -opposition, exceeds the regressive in the quadratures more than in any -other situation of the line A B[198]. On the contrary, when the line -A B makes right angles with that, which joins the earth and sun, the -retrograde motion will be more considerable[199], nay is found so great -as to exceed the progressive; so that in this case the apogeon in the -compass of an intire revolution of the moon is carried in antecedence. -Yet from the considerations in the last paragraph the progressive -motion exceeds the other; so that in the whole the mean motion of the -apogeon is in consequence, according as astronomers find. Moreover, the -line A B changes its situation with that, which joins the earth and -sun, by such slow degrees, that the inequalities in the motion of the -apogeon arising from this last consideration, are much greater than -what arises from the other[200]. - -49. FARTHER, this unsteady motion in the apogeon is attended with -another inequality in the motion of the moon, that it cannot be -explained at all times by the same ellipsis. The ellipsis in general -is called by astronomers an eccentric orbit. The point, in which the -two axis’s cross, is called the center of the figure; because all -lines drawn through this point within the ellipsis, from side to side, -are divided in the middle by this point. But the center, about which -the heavenly bodies revolve, lying out of this center of the figure -in one focus, these orbits are said to be eccentric; and where the -distance of the focus from this center bears the greatest proportion -to the whole axis, that orbit is called the most eccentric: and in -such an orbit the distance from the focus to the remoter extremity of -the axis bears the greatest proportion to the distance of the nearer -extremity. Now whenever the apogeon of the moon moves in consequence, -the moon’s motion must be referred to an orbit more eccentric, than -what the moon would describe, if the whole power, by which the moon -was acted on in its passing from the apogeon, changed according to the -reciprocal duplicate proportion of the distance from the earth, and by -that means the moon did describe an immoveable ellipsis; and when the -apogeon moves in antecedence, the moon’s motion must be referred to an -orbit less eccentric. In the first of the two figures last referred -to, the true place of the moon L falls without the orbit A M B, to -which its motion is referred: whence the orbit A L E, truly described -by the moon, is less incurvated in the point A, than is the orbit A M -B; therefore the orbit A M B is more oblong, and differs farther from -a circle, than the ellipsis would, whose curvature in A were equal to -that of the line A L B, that is, the proportion of the distance of the -earth T from the center of the ellipsis to its axis will be greater in -the ellipsis A M B, than in the other; but that other is the ellipsis, -which the moon would describe, if the power acting upon it in the point -A were altered in the reciprocal duplicate proportion of the distance. -In the second figure, when the apogeon recedes, the place of the moon -L falls within the orbit A M B, and therefore that orbit is less -eccentric, than the immoveable orbit which the moon should describe. -The truth of this is evident; for, when the apogeon moves forward, the -power, by which the moon is influenced in its descent from the apogeon, -increases faster with the decrease of distance, than in the duplicate -proportion of the distance; and consequently the moon being drawn more -forcibly toward the earth, it will descend nearer to it. On the other -hand, when the apogeon recedes, the power acting on the moon increases -with the decrease of distance in less than the duplicate proportion of -the distance; and therefore the moon is less impelled toward the earth, -and will not descend so low. - -50. NOW suppose in the first of these figures, that the apogeon A is -in the situation, where it is approaching toward the conjunction or -opposition of the sun. In this case the progressive motion of the -apogeon is more and more accelerated. Here suppose that the moon, after -having descended from A through the orbit A E as far as F, where it -is come to its nearest distance from the earth, ascends again up the -line F G. Because the motion of the apogeon is here continually more -and more accelerating, the cause of its motion is constantly upon the -increase; that is, the power, whereby the moon is drawn to the earth, -will decrease with the increase of distance, in the moon’s ascent -from F, in a greater proportion than that wherewith it increased with -the decrease of distance in the moon’s descent to F. Consequently -the moon will ascend higher than to the distance A T, from whence it -descended; therefore the proportion of the greatest distance of the -moon to the least is increased. And when the moon descends again, the -power will yet more increase with the decrease of distance, than in the -last ascent it decreased with the augmentation of distance; the moon -therefore must descend nearer to the earth than it did before, and the -proportion of the greatest distance to the least yet be more increased. -Thus as long as the apogeon is advancing toward the conjunction or -opposition, the proportion of the greatest distance of the moon from -the earth to the least will continually increase; and the elliptical -orbit, to which the moon’s motion is referred, will be rendered more -and more eccentric. - -51. AS soon as the apogeon is passed the conjunction with the sun or -the opposition, the progressive motion thereof abates, and with it -the proportion of the greatest distance of the moon from the earth to -the least distance will also diminish; and when the apogeon becomes -regressive, the diminution of this proportion will be still farther -continued on, till the apogeon comes into the quarter; from thence this -proportion, and the eccentricity of the orbit will increase again. -Thus the orbit of the moon is most eccentric, when the apogeon is in -conjunction with the sun, or in opposition to it, and least of all when -the apogeon is in the quarters. - -52. THESE changes in the nodes, in the inclination of the orbit to the -plane of the earth’s motion, in the apogeon, and in the eccentricity, -are varied like the other inequalities in the motion of the moon, by -the different distance of the earth from the sun; being greatest, when -their cause is greatest, that is, when the earth is nearest to the sun. - -53. I SAID at the beginning of this chapter, that Sir ISAAC NEWTON -has computed the very quantity of many of the moon’s inequalities. -That acceleration of the moon’s motion, which is called the variation, -when greatest, removes the moon out of the place, in which it would -otherwise be found, something more than half a degree[201]. In the -phrase of astronomers, a degree is 1/360 part of the whole circuit of -the moon or any planet. If the moon, without disturbance from the sun, -would have described a circle concentrical to the earth, the sun will -cause the moon to approach nearer to the earth in the conjunction and -opposition, than in the quarters, nearly in the proportion of 69 to -70[202]. We had occasion to mention above, that the nodes perform their -period in almost 19 years. This the astronomers found by observation; -and our author’s computations assign to them the same period[203]. -The inclination of the moon’s orbit when least, is an angle about -1/18 part of that angle, which constitutes a perpendicular; and the -difference between the greatest and least inclination of the orbit is -determined by our author’s computation to be about 1/18 of the least -inclination[204]. And this also is agreeable to the observations -of astronomers. The motion of the apogeon, and the changes in the -eccentricity, Sir ~ISAAC NEWTON~ has not computed. The apogeon performs -its revolution in about eight years and ten months. When the moon’s -orbit is most eccentric, the greatest distance of the moon from the -earth bears to the least distance nearly the proportion of 8 to 7; when -the orbit is least eccentric, this proportion is hardly so great as -that of 12 to 11. - -54. SIR ~ISAAC NEWTON~ shews farther, how, by comparing the -periods of the motion of the satellites, which revolve round Jupiter -and Saturn, with the period of our moon round the earth, and the -periods of those planets round the sun with the period of our earth’s -motion, the inequalities in the motion of those satellites may be -derived from the inequalities in the moon’s motion; excepting only in -regard to that motion of the axis of the orbit, which in the moon makes -the motion of the apogeon; for the orbits of those satellites, as far -as can be discerned by us at this distance, appearing little or nothing -eccentric, this motion, as deduced from the moon, must be diminished. - - - - -~CHAP. IV.~ - -Of ~Comets~. - - -IN the former of the two preceding chapters the powers have been -explained, which keep in motion those celestial bodies, whose courses -had been well determined by the astronomers. In the last chapter we -have shewn, how those powers have been applied by our author to the -making a more perfect discovery of the motion of those bodies, the -courses of which were but imperfectly understood; for some of the -inequalities, which we have been describing in the moon’s motion, -were unknown to the astronomers. In this chapter we are to treat of -a third species of the heavenly bodies, the true motion of which was -not at all apprehended before our author writ; in so much, that here -Sir ~ISAAC NEWTON~ has not only explained the causes of the motion of -these bodies, but has performed also the part of an astronomer, by -discovering what their motions are. - -2. THAT these bodies are not meteors in our air, is manifest; because -they rise and set in the same manner, as the sun and stars. The -astronomers had gone so far in their inquiries concerning them, as to -prove by their observations, that they moved in the etherial spaces far -beyond the moon; but they had no true notion at all of the path, which -they described. The most prevailing opinion before our author was, -that they moved in straight lines; but in what part of the heavens was -not determined. DESCARTES[205] removed them far beyond the sphere of -Saturn, as finding the straight motion attributed to them, inconsistent -with the vortical fluid, by which he explains the motions of the -planets, as we have above related[206]. But Sir ISAAC NEWTON distinctly -proves from astronomical observation, that the comets pass through the -region of the planets, and are mostly invisible at a less distance, -than that of Jupiter[207]. - -3. AND from hence finding the comets to be evidently within the sphere -of the sun’s action, he concludes they must, necessarily move about the -sun, as the planets do[208]. The planets move in ellipsis’s; but it is -not necessary that every body, which is influenced by the sun, should -move in that particular kind of line. However our author proves, that -the power of the sun being reciprocally in the duplicate proportion of -the distance, every body acted on by the sun must either fall directly -down, or move in some conic section; of which lines I have above -observed, that there are three species, the ellipsis, parabola, and -hyperbola[209]. If a body, which descends toward the sun as low as the -orbit of any planet, move with a swifter motion than the planet does, -that body will describe an orbit of a more oblong figure, than that -of the planet, and have a longer axis at least. The velocity of the -body may be so great, that it shall move in a parabola, and having -once passed about the sun, shall ascend for ever without returning any -more: but the sun will be placed in the focus of this parabola. With a -velocity still greater the body will move in an hyperbola. But it is -most probable, that the comets move in elliptical orbits, though of a -very oblong, or in the phrase of astronomers, of a very eccentric form, -such as is represented in fig. 107, where S is the sun, C the comet, -and A B D E its orbit, wherein the distance of S and D far exceeds that -of S and A. Whence it is, that they sometimes are found at a moderate -distance from the sun, and appear within the planetary regions; at -other times they ascend to vast distances, far beyond the very orbit of -Saturn, and so become invisible. That the comets do move in this manner -is proved by our author, from computations built upon the observations, -which astronomers had made on many comets. These computations were -performed by Sir ~ISAAC NEWTON~ himself upon the comet, which appeared -toward the latter end of the year 1680, and at the beginning of the -year following[210]; but the learned Dr. HALLEY prosecuted the like -computations more at large in this, and also in many other comets[211]. -Which computations are made upon propositions highly worthy of our -author’s unparallel’d genius, such as could scarce have been discovered -by any one not possessed of the utmost force of invention; - -4. THOSE computations depend upon this principle, that the eccentricity -of the orbits of the comets is so great, that if they are really -elliptical, yet they approach so near to parabolas in that part of -them, where they come under our view, that they may be taken for such -without sensible error[212]: as in the preceding figure the parabola -F A G differs in the lower part of it about A very little from the -ellipsis D E A B. Upon which ground our great author teaches a method -of finding by three observations made upon any comet the parabola, -which nearest agrees with its orbit[213]. - -5. NOW what confirms this whole theory beyond the least room for -doubt is, that the places of the comets computed in the orbits, which -the method here mentioned assigns them, agree to the observations of -astronomers with the same degree of exactness, as the computations -of the primary planets places usually do; and this in comets, whose -motions are very extraordinary[214]. - -6. OUR author afterwards shews how to make use of any small deviation -from the parabola, that shall be observed, to determine whether the -orbits of the comets are elliptical or not, and so to discover if the -same comet returns at certain periods[215]. And upon examining the -comet in 1680, by the rule laid down for this purpose, he finds its -orbit to agree more exactly to an ellipsis than to a parabola, though -the ellipsis be so very eccentric, that the comet cannot perform its -period through it in the space of 500 years[216]. Upon this Dr. HALLEY -observed, that mention is made in history of a comet, with the like -eminent tail as this, having appeared three several times before; the -first of which appearances was at the death of JULIUS CESAR, and each -appearance was at the distance of 575 years from the next preceding. -He therefore computed the motion of this comet in such an elliptic -orbit, as would require this number of years for the body to revolve -through it; and these computations agree yet more perfectly with the -observations made on this comet, than any parabolical orbit will -do[217]. - -7. THE comparing together different appearances of the same comet, is -the only way to discover certainly the true form of the orbit: for -it is impossible to determine with exactness the figure of an orbit -so exceedingly eccentric, from single observations taken in one part -of it; and therefore Sir ~ISAAC NEWTON~[218] proposes to compare the -orbits, upon the supposition that they are parabolical, of such comets -as appear at different times; for if the same orbit be found to be -described by a comet at different times, in all probability it will -be the same comet which describes it. And here he remarks from Dr. -HALLEY, that the same orbit very nearly agrees to two appearances of -a comet about the space of 75 years distance[219]; so that if those -two appearances were really of the same comet, the transverse axis of -the orbit of the comet would be near 18 times the axis of the earth’s -orbit; and the comet, when at its greatest distance from the sun, will -be removed not less than 35 times as far as the middle distance of the -earth. - -8. AND this seems to be the shortest period of any of the comets. -But it will be farther confirmed, if the same comet should return a -third time after another period of 75 years. However it is not to be -expected, that comets should preserve the same regularity in their -periods, as the planets; because the great eccentricity of their orbits -makes them liable to suffer very considerable alterations from the -action of the planets, and other comets, upon them. - -9. IT is therefore to prevent too great disturbances in their motions -from these causes, as our author observes, that while the planets -revolve all of them nearly in the same plane, the comets are disposed -in very different ones; and distributed over all parts of the heavens; -that, when in their greatest distance from the sun, and moving slowest, -they might be removed as far as possible out of the reach of each -other’s action[220]. The same end is likewise farther answered in those -comets, which by moving slowest in the aphelion, or remotest distance -from the sun, descend nearest to it, by placing the aphelion of these -at the greatest height from the sun[221]. - -10. OUR philosopher being led by his principles to explain the motions -of the comets, in the manner now related, takes occasion from thence to -give us his thoughts upon their nature and use. For which end he proves -in the first place, that they must necessarily be solid and compact -bodies, and by no means any sort of vapour or light substance exhaled -from the planets or stars: because at the near distance, to which some -comets approach the sun, it could not be, but the immense heat, to -which they are exposed, should instantaneously disperse and scatter any -such light volatile substance[222]. In particular the forementioned -comet of 1680 descended so near the sun, as to come within a sixth -part of the sun’s diameter from the surface of it. In which situation -it must have been exposed, as appears by computation, to a degree of -heat exceeding the heat of the sun upon our earth no less than 28000 -times; and therefore might have contracted a degree of heat 2000 times -greater, than that of red hot iron[223]. Now a substance, which could -endure so intense a heat, without being dispersed in vapor, must needs -be firm and solid. - -11. IT is shewn likewise, that the comets are opake substances, shining -by a reflected light, borrowed from the sun[224]. This is proved from -the observation, that comets, though they are approaching the earth, -yet diminish in lustre, if at the same time they recede from the sun; -and on the contrary, are found to encrease daily in brightness, when -they advance towards the sun, though at the same time they move from -the earth[225]. - -12. THE comets therefore in these respects resemble the planets; that -both are durable opake bodies, and both revolve about the sun in conic -sections. But farther the comets, like our earth, are surrounded by -an atmosphere. The air we breath is called the earth’s atmosphere; -and it is most probable, that all the other planets are invested with -the like fluid. Indeed here a difference is found between the planets -and comets. The atmospheres of the planets are of so fine and subtile -a substance, as hardly to be discerned at any distance, by reason of -the small quantity of light which they reflect, except only in the -planet Mars. In him there is some little appearance of such a substance -surrounding him, as stars which have been covered by him are said to -look somewhat dim a small space before his body comes under them, as if -their light, when he is near, were obstructed by his atmosphere. But -the atmospheres which surround the comets are so gross and thick, as to -reflect light very copiously. They are also much greater in proportion -to the body they surround, than those of the planets, if we may judge -of the rest from our air; for it has been observed of comets, that the -bright light appearing in the middle of them, which is reflected from -the solid body, is scarce a ninth or tenth part of the whole comet, - -13. I SPEAK only of the heads of the comets, the most lucid part of -which is surrounded by a fainter light, the most lucid part being -usually not above a ninth or tenth part of the whole in breadth[226]. -Their tails are an appearance very peculiar, nothing of the same -nature appertaining in the least degree to any other of the celestial -bodies. Of that appearance there are several opinions; our author -reduces them to three[227]. The two first, which he proposes, are -rejected by him; but the third he approves. The first is, that they -arise from a beam of light transmitted through the head of the comet, -in like manner as a stream of light is discerned, when the sun shines -into a darkened room through a small hole. This opinion, as Sir ~ISAAC -NEWTON~ observes, implies the authors of it wholly unskilled in the -principles of optics; for that stream of light, seen in a darkened -room, arises from the reflection of the sun beams by the dust and motes -floating in the air: for the rays of light themselves are not seen, -but by their being reflected to the eye from some substance, upon -which they fall[228]. The next opinion examined by our author is that -of the celebrated DESCARTES, who imagins these tails to be the light -of the comet refracted in its passage to us, and thence affording an -oblong representation; as the light of the sun does, when refracted -by the prism in that noted experiment, which will have a great share -in the third book of this discourse[229]. But this opinion is at once -overturned from this consideration only, that the planets could be -no more free from this refraction than the comets; nay ought to have -larger or brighter tails, than they, because the light of the planets -is strongest. However our author has thought proper to add some farther -objections against this opinion: for instance, that these tails are not -variegated with colours, as is the image produced by the prism, and -which is inseparable from that unequal refraction, which produces that -disproportioned length of the image. And besides, when the light in -its passage from different comets to the earth describes the same path -through the heavens, the refraction of it should of necessity be in all -respects the same. But this is contrary to observation; for the comet -in 1680, the 28th day of December, and a former comet in the year 1577, -the 29th day of December, appear’d in the same place of the heavens, -that is, were seen adjacent to the same fixed stars, the earth likewise -being in the same place at both times; yet the tail of the latter comet -deviated from the opposition to the sun a little to the northward, and -the tail of the former comet declined from the opposition of the sun -five times as much southward[230]. - -14. THERE are some other false opinions, though less regarded than -these, which have been advanced upon this argument. These our -excellent author passes over, hastening to explain, what he takes to -be the true cause of this appearance. He thinks it is certainly owing -to steams and vapours exhaled from the body, and gross atmosphere of -the comets, by the heat of the sun; because all the appearances agree -perfectly to this sentiment. The tails are but small, while the comet -is descending to the sun, but enlarge themselves to an immense degree, -as soon as ever the comet has passed its perihelion; which shews the -tail to depend upon the degree of heat, which the comet receives from -the sun. And that the intense heat to which comets, when nearest the -sun, are exposed, should exhale from them a very copious vapour, is a -most reasonable supposition; especially if we consider, that in those -free and empty regions steams will more easily ascend, than here upon -the surface of the earth, where they are suppressed and hindered from -rising by the weight of the incumbent air: as we find by experiments -made in vessels exhausted of the air, where upon removal of the air -several substances will fume and discharge steams plentifully, which -emit none in the open air. The tails of comets, like such a vapour, -are always in the plane of the comet’s orbit, and opposite to the -sun, except that the upper part thereof inclines towards the parts, -which the comet has left by its motion; resembling perfectly the smoak -of a burning coal, which, if the coal remain fixed, ascends from it -perpendicularly; but, if the coal be in motion, ascends obliquely, -inclining from the motion of the coal. And besides, the tails of -comets may be compared to this smoak in another respect, that both -of them are denser and more compact on the convex side, than on the -concave. The different appearance of the head of the comet, after it -has past its perihelion, from what it had before, confirms greatly this -opinion of their tails: for smoke raised by a strong heat is blacker -and grosser, than when raised by a less; and accordingly the heads of -comets, at the same distance from the sun, are observed less bright and -shining after the perihelion, than before, as if obscured by such a -gross smoke. - -15. THE observations of HEVELIUS upon the atmospheres of comets still -farther illustrate the same; who relates, that the atmospheres, -especially that part of them next the sun, are remarkably contracted -when near the sun, and dilated again afterwards. - -16. TO give a more full idea of these tails, a rule is laid down by -our author, whereby to determine at any time, when the vapour in the -extremity of the tail first rose from the head of the comet. By this -rule it is found, that the tail does not consist of a fleeting vapour, -dissipated soon after it is raised, but is of long continuance; that -almost all the vapour, which rose about the time of the perihelion from -the comet of 1680, continued to accompany it, ascending by degrees, -being succeeded constantly by fresh matter, which rendered the tail -contiguous to the comet. From this computation the tails are found to -participate of another property of ascending vapours, that, when they -ascend with the greatest velocity, they are least incurvated. - -17. THE only objection that can be made against this opinion is the -difficulty of explaining, how a sufficient quantity of vapour can -be raised from the atmosphere of a comet to fill those vast spaces, -through which their tails are sometimes extended. This our author -removes by the following computation: our air being an elastic fluid, -as has been said before[231], is more dense here near the surface of -the earth, where it is pressed upon by the whole air above; than it is -at a distance from the earth, where it has a less weight incumbent. I -have observed, that the density of the air is reciprocally proportional -to the compressing weight. From hence our author computes to what -degree of rarity the air must be expanded, according to this rule, at -an height equal to a semidiameter of the earth: and he finds, that -a globe of such air, as we breath here on the surface of the earth, -which shall be one inch only in diameter, if it were expanded to the -degree of rarity, which the air must have at the height now mentioned, -would fill all the planetary regions even to the very sphere of Saturn, -and far beyond. Now since the air at a greater height will be still -immensly more rarified, and the surface of the atmospheres of comets is -usually about ten times the distance from the center of the comet, as -the surface of the comet it self, and the tails are yet vastly farther -removed from the center of the comet; the vapour, which composes those -tails, may very well be allowed to be so expanded, as that a moderate -quantity of matter may fill all that space, they are seen to take up. -Though indeed the atmospheres of comets being very gross, they will -hardly be rarified in their tails to so great a degree, as our air -under the same circumstances; especially since they may be something -condensed, as well by their gravitation to the sun, as that the parts -will gravitate to one another; which will hereafter be shewn to be the -universal property of all matter[232]. The only scruple left is, how so -much light can be reflected from a vapour so rare, as this computation -implies. For the removal of which our author observes, that the most -refulgent of these tails hardly appear brighter, than a beam of the -sun’s light transmitted into a darkened room through a hole of a single -inch diameter; and that the smallest fixed stars are visible through -them without any sensible diminution of their lustre. - -18. ALL these considerations put it beyond doubt, what is the true -nature of the tails of comets. There has indeed nothing been said, -which will account for the irregular figures, in which those tails -are sometimes reported to have appeared; but since none of those -appearances have ever been recorded by astronomers, who on the contrary -ascribe the same likeness to the tails of all comets, our author -with great judgment refers all those to accidental refractions by -intervening clouds, or to parts of the milky way contiguous to the -comets[233]. - -19. THE discussion of this appearance in comets has led Sir ~ISAAC -NEWTON~ into some speculations relating to their use, which I cannot -but extreamly admire, as representing in the strongest light -imaginable the extensive providence of the great author of nature, -who, besides the furnishing this globe of earth, and without doubt -the rest of the planets, so abundantly with every thing necessary -for the support and continuance of the numerous races of plants and -animals, they are stocked with, has over and above provided a numerous -train of comets, far exceeding the number of the planets, to rectify -continually, and restore their gradual decay, which is our author’s -opinion concerning them[234]. For since the comets are subject to such -unequal degrees of heat, being sometimes burnt with the most intense -degree of it, at other times scarce receiving any sensible influence -from the sun; it can hardly be supposed, they are designed for any -such constant use, as the planets. Now the tails, which they emit, -like all other kinds of vapour, dilate themselves as they ascend, and -by consequence are gradually dispersed and scattered through all the -planetary regions, and thence cannot but be gathered up by the planets, -as they pass through their orbs: for the planets having a power to -cause all bodies to gravitate towards them, as will in the sequel of -this discourse be shewn[235]; these vapours will be drawn in process of -time into this or the other planet, which happens to act strongest upon -them. And by entering the atmospheres of the earth and other planets, -they may well be supposed to contribute to the renovation of the face -of things, in particular to supply the diminution caused in the humid -parts by vegetation and putrefaction. For vegetables are nourished by -moisture, and by putrefaction are turned in great part into dry earth; -and an earthy substance always subsides in fermenting liquors; by which -means the dry parts of the planets must continually increase, and the -fluids diminish, nay in a sufficient length of time be exhausted, if -not supplied by some such means. It is farther our great author’s -opinion, that the most subtile and active parts of our air, upon which -the life of things chiefly depends, is derived to us, and supplied -by the comets. So far are they from portending any hurt or mischief -to us, which the natural fears of men are so apt to suggest from the -appearance of any thing uncommon and astonishing. - -20. THAT the tails of comets have some such important use seems -reasonable, if we consider, that those bodies do not send out those -fumes merely by their near approach to the sun; but are framed of a -texture, which disposes them in a particular manner to fume in that -sort: for the earth, without emitting any such steam, is more than half -the year at a less distance from the sun, than the comet of 1664 and -1665 approached it, when nearest; likewise the comets of 1682 and 1683 -never approached the sun much above a seventh part nearer than Venus, -and were more than half as far again from the sun as Mercury; yet all -these emitted tails. - -21. FROM the very near approach of the comet of 1680 our author draws -another speculation; for if the sun have an atmosphere about it, the -comet mentioned seems to have descended near enough to the sun to -enter within it. If so, it must have been something retarded by the -resistance it would meet with, and consequently in its next descent to -the sun will fall nearer than now; by which means it will meet with a -greater resistance, and be again more retarded. The event of which must -be, that at length it will impinge upon the sun’s surface, and thereby -supply any decrease, which may have happened by so long an emission of -light, or otherwise. And something like this our author conjectures -may be the case of those fixed stars which by an additional increase -of their lustre have for a certain time become visible to us, though -usually they are out of sight. There is indeed a kind of fixed stars, -which appear and disappear at regular and equal intervals: here some -more steady cause must be sought for; perhaps these stars turn round -their own axis’s, as our sun does[236], and have some part of their -body more luminous than the other, whereby they are seen, when the most -lucid part is next to us, and when the darker part is turned toward us, -they vanish out of sight. - - -22. WHETHER the sun does really diminish, as has been here suggested, -is difficult to prove; yet that it either does so, or that the earth -increases, if not both, is rendered probable from Dr. HALLEY’s -observation[237], that by comparing the proportion, which the -periodical time of the moon bore to that of the sun in former times, -with the proportion between them at present, the moon is found to be -something accelerated in respect of the sun. But if the sun diminish, -the periods of the primary planets will be lengthened; and if the earth -be encreased, the period of the moon will be shortened: as will appear -by the next chapter, wherein it shall be shewn, that the power of the -sun and earth is the result of the same power being lodg’d in all their -parts, and that this principle of producing gravitation in other bodies -is proportional to the solid matter in each body. - - - - -~CHAP~. V. - -Of the BODIES of the SUN and PLANETS. - - -OUR author, after having discovered that the celestial motions are -performed by a force extended from the sun and primary planets, follows -this power into the deepest recesses of those bodies themselves, and -proves the same to accompany the smallest particle, of which they are -composed. - - -2. PREPARATIVE hereto he shews first, that each of the heavenly bodies -attracts the rest, and all bodies, with such different degrees of -force, as that the force of the same attracting body is exerted on -others exactly in proportion to the quantity of matter in the body -attracted[238]. - -3. OF this the first proof he brings is from experiments made here -upon the earth. The power by which the moon is influenced was above -shewn to be the same, with that power here on the surface of the earth, -which we call gravity[239]. Now one of the effects of the principle -of gravity is, that all bodies descend by this force from the same -height in equal times. Which has been long taken notice of; particular -methods having been invented to shew that the only cause, why some -bodies were observed to fall from the same height sooner than others, -was the resistance of the air. This we have above related[240]; and -proved from hence, that since bodies resist to any change of their -state from rest to motion, or from motion to rest, in proportion to the -quantity of matter contained in them; the power that can move different -quantities of matter equally, must be proportional to the quantity. The -only objection here is, that it can hardly be made certain, whether -this proportion in the effect of gravity on different bodies holds -perfectly exact or not from these experiments; by reason that the -great swiftness, with which bodies fall, prevents our being able to -determine the times of their descent with all the exactness requisite. -Therefore to remedy this inconvenience, our author substitutes another -more certain experiment in the room of these made upon falling bodies. -Pendulums are caused to vibrate by the same principle, as makes -bodies descend; the power of gravity putting them in motion, as well -as the other. But if the ball of any pendulum, of the same length with -another, were more or less attracted in proportion to the quantity of -solid matter in the ball, that pendulum must accordingly move faster or -slower than the other. Now the vibrations of pendulums continue for a -great length of time, and the number of vibrations they make may easily -be determined without suspicion of error; so that this experiment may -be extended to what exactness one pleases: and our author assures us, -that he examined in this way several substances, as gold, silver, lead, -glass, sand, common salt, wood, water, and wheat; in all which he found -not the least deviation from the proportion mentioned, though he made -the experiment in such a manner, that in bodies of the same weight a -difference in the quantity of their matter less than a thousandth part -of the whole would have discovered it self[241]. It appears therefore, -that all bodies are made to descend by the power of gravity here, near -the surface of the earth, with the same degree of swiftness. We have -above observed this descent to be after the rate of 16⅛ feet in the -first second of time from the beginning of their fall. Moreover it -was also observed, that if any body, which fell here at the surface -of the earth after this rate, were to be conveyed up to the height of -the moon, it would descend from thence just with the same degree of -velocity, as that with which the moon is attracted toward the earth; -and therefore the power of the earth upon the moon bears the same -proportion to the power it would have upon those bodies at the same -distance, as the quantity of matter in the moon bears to the quantity -in those bodies. - -4. THUS the assertion laid down is proved in the earth, that the power -of the earth on every body it attracts is, at the same distance from -the earth, proportional to the quantity of solid matter in the body -acted on. As to the sun, it has been shewn, that the power of the sun’s -action upon the same primary planet is reciprocally in the duplicate -proportion of the distance; and that the power of the sun decreases -throughout in the same proportion, the motion of comets traversing the -whole planetary region testifies. This proves, that if any planet were -removed from the sun to any other distance whatever, the degree of -its acceleration toward the sun would yet remain reciprocally in the -duplicate proportion of its distance. But it has likewise been shewn, -that the degree of acceleration, which the sun gives to every one of -the planets, is reciprocally in the duplicate proportion of their -respective distances. All which compared together puts it out of doubt, -that the power of the sun upon any planet, removed into the place of -any ether, would give it the same velocity of descent, as it gives that -other; and consequently, that the sun’s action upon different planets -at the same distance would be proportional to the quantity of matter -in each. It has farther been shewn, that the sun attracts the primary -planets, and their respective secondary, when at the same distance, so -as to communicate to both the same degree of velocity; and therefore -the force, wherewith the sun acts on the secondary planet, bears -the same proportion to the force, wherewith at the same distance it -attracts the primary, as the quantity of solid matter in the secondary -planet bears to the quantity of matter in the primary. - -5. THIS property therefore is proved of both kinds of planets, in -respect of the sun. Therefore the sun possesses the quality found in -the earth, of acting on bodies with a degree of force proportional to -the quantity of matter in the body, which receives the influence. - -6. THAT the power of attraction, with which the other planets are -endued, should differ from that of the earth, can hardly be supposed, -if we consider the similitude between those bodies; and that it does -not in this respect, is farther proved from the satellites of Saturn -and Jupiter, which are attracted by their respective primary according -to the same law, that is, in the same proportion to their distances, as -the primary are attracted by the sun: so that what has been concluded -of the sun in relation to the primary planets, may be justly concluded -of these primary in respect of their secondary, and in consequence -of that, in regard likewise to all other bodies, viz. that they will -attract every body in proportion to the quantity of solid matter it -contains. - -7. HENCE it follows, that this attraction extends itself to every -particle of matter in the attracted body: and that no portion of matter -whatever is exempted from the influence of those bodies, to which we -have proved this attractive power to belong. - -8. BEFORE we proceed farther, we may here remark, that this attractive -power both of the sun and planets now appears to be quite of the same -nature in all; for it acts in each in the same proportion to the -distance, and in the same manner acts alike upon every particle of -matter. This power therefore in the sun and other planets is not of a -different nature from this power in the earth; which has been already -shewn to be the same with that, which we call gravity[242]. - -9. AND this lays open the way to prove, that the attracting power -lodged in the sun and planets, belongs likewise to every part of them: -and that their respective powers upon the same body are proportional to -the quantity of matter, of which they are composed; for instance, that -the force with which the earth attracts the moon, is to the force, with -which the sun would attract it at the same distance, as the quantity of -solid matter contained in the earth, to the quantity contained in the -sun[243]. - -10. THE first of these assertions is a very evident consequence from -the latter. And before we proceed to the proof, it must first be -shewn, that the third law of motion, which makes action and reaction -equal, holds in these attractive powers. The most remarkable attractive -force, next to the power of gravity, is that, by which the loadstone -attracts iron. Now if a loadstone were laid upon water, and supported -by some proper substance, as wood or cork, so that it might swim; -and if a piece of iron were caused to swim upon the water in like -manner: as soon as the loadstone begins to attract the iron, the iron -shall move toward the stone, and the stone shall also move toward -the iron; when they meet, they shall stop each other, and remain -fixed together without any motion. This shews, that the velocities, -wherewith they meet, are reciprocally proportional to the quantities -of solid matter in each; and that by the stone’s attracting the iron, -the stone itself receives as much motion, in the strict philosophic -sense of that word[244], as it communicates to the iron: for it has -been declared above to be an effect of the percussion of two bodies, -that if they meet with velocities reciprocally proportional to the -respective bodies, they shall be stopped by the concourse, unless their -elasticity put them into fresh motion; but if they meet with any other -velocities, they shall retain some motion after meeting[245]. Amber, -glass, sealing-wax, and many other substances acquire by rubbing a -power, which from its having been remarkable, particularly in amber, -is called electrical. By this power they will for some time after -rubbing attract light bodies, that shall be brought within the sphere -of their activity. On the other hand Mr. BOYLE found, that if a piece -of amber be hung in a perpendicular position by a string, it shall be -drawn itself toward the body whereon it was rubbed, if that body be -brought near it. Both in the loadstone and in electrical bodies we -usually ascribe the power to the particular body, whose presence we -find necessary for producing the effect. The loadstone and any piece of -iron will draw each other, but in two pieces of iron no such effect is -ordinarily observed; therefore we call this attractive power the power -of the loadstone: though near a loadstone two pieces of iron will also -draw each other. In like manner the rubbing of amber, glass, or any -such body, till it is grown warm, being necessary to cause any action -between those bodies and other substances, we ascribe the electrical -power to those bodies. But in all these cases if we would speak more -correctly, and not extend the sense of our expressions beyond what -we see; we can only say that the neighbourhood of a loadstone and a -piece of iron is attended with a power, whereby the loadstone and -the iron are drawn toward each other; and the rubbing of electrical -bodies gives rise to a power, whereby those bodies and other substances -are mutually attracted. Thus we must also understand in the power of -gravity, that the two bodies are mutually made to approach by the -action of that power. When the sun draws any planet, that planet also -draws the sun; and the motion, which the planet receives from the -sun, bears the same proportion to the motion, which the sun it self -receives, as the quantity of solid matter in the sun bears to the -quantity of solid matter in the planet. Hitherto, for brevity sake -in speaking of these forces, we have generally ascribed them to the -body, which is least moved; as when we called the power, which exerts -itself between the sun and any planet, the attractive power of the sun; -but to speak more correctly, we should rather call this power in any -case the force, which acts between the sun and earth, between the sun -and Jupiter, between the earth and moon, &c. for both the bodies are -moved by the power acting between them, in the same manner, as when -two bodies are tied together by a rope, if that rope shrink by being -wet, or otherwise, and thereby cause the bodies to approach, by drawing -both, it will communicate to both the same degree of motion, and cause -them to approach with velocities reciprocally proportional to the -respective bodies. From this mutual action between the sun and planet -it follows, as has been observed above[246], that the sun and planet do -each move about their common center of gravity. Let A (in fig. 108.) -represent the sun, B a planet, C their common center of gravity. If -these bodies were once at rest, by their mutual attraction they would -directly approach each other with such velocities, that their common -center of gravity would remain at rest, and the two bodies would at -length meet in that point. If the planet B were to receive an impulse, -as in the direction of the line D E, this would prevent the two bodies -from falling together; but their common center of gravity would be -put into motion in the direction of the line C F equidistant from B E. -In this case Sir ~ISAAC NEWTON~ proves[247], that the sun and planet -would describe round their common center of gravity similar orbits, -while that center would proceed with an uniform motion in the line C -F; and so the system of the two bodies would move on with the center -of gravity without end. In order to keep the system in the same place, -it is necessary, that when the planet received its impulse in the -direction B E, the sun should also receive such an impulse the contrary -way, as might keep the center of gravity C without motion; for if these -began once to move without giving any motion to their common center of -gravity, that center would always remain fixed. - -11. BY this may be understood in what manner the action between the sun -and planets is mutual. But farther, we have shewn above[248], that the -power, which acts between the sun and primary planets, is altogether of -the same nature with that, which acts between the earth and the bodies -at its surface, or between the earth and its parts, and with that which -acts between the primary planets and their secondary; therefore all -these actions must be ascribed to the same cause[249]. Again, it has -been already proved, that in different planets the force of the sun’s -action upon each at the same distance would be proportional to the -quantity of solid matter in the planet[250]; therefore the reaction -of each planet on the sun at the same distance, or the motion, which -the sun would receive from each planet, would also be proportional -to the quantity of matter in the planet; that is, these planets at -the same distance would act on the same body with degrees of strength -proportional to the quantity of solid matter in each. - -[Illustration] - -12. IN the next place, from what has been now proved, our great author -has deduced this farther consequence, no less surprizing than elegant; -that each of the particles, out of which the bodies of the sun and -planets are framed, exert their power of gravitation by the same law, -and in the same proportion to the distance, as the great bodies which -they compose. For this purpose he first demonstrates, that if a globe -were compounded of particles, which will attract the particles of any -other body reciprocally in the duplicate proportion of their distances, -the whole globe will attract the same in the reciprocal duplicate -proportion of their distances from the center of the globe; provided -the globe be of uniform density throughout[251]. And from this our -author deduces the reverse, that if a globe acts upon distant bodies by -the law just now specified, and the power of the globe is derived from -its being composed of attractive particles; each of those particles -will attract after the same proportion[252]. The manner of deducing -this is not set down at large by our author, but is as follows. The -globe is supposed to act upon the particles of a body without it -constantly in the reciprocal duplicate proportion of their distances -from its center; and therefore at the same distance from the globe, on -which side soever the body be placed, the globe will act equally upon -it. Now because, if the particles, of which the globe is composed, -acted upon those without in the reciprocal duplicate proportion of -their distances, the whole globe would act upon them in the same manner -as it does; therefore, if the particles of the globe have not all of -them that property, some must act stronger than in that proportion, -while others act weaker: and if this be the condition of the globe, -it is plain, that when the body attracted is in such a situation -in respect of the globe, that the greater number of the strongest -particles are nearest to it, the body will be more forcibly attracted; -than when by turning the globe about, the greater quantity of weak -particles should be nearest, though the distance of the body should -remain the same from the center of the globe. Which is contrary to what -was at first remarked, that the globe on all sides of it acts with the -same strength at the same distance. Whence it appears, that no other -constitution of the globe can agree to it. - -13. FROM these propositions it is farther collected, that if all -the particles of one globe attract all the particles of another in -the proportion so often mentioned, the attracting globe will act -upon the other in the same proportion to the distance between the -center of the globe which attracts, and the center of that which is -attracted[253]: and farther, that this proportion holds true, though -either or both the globes be composed of dissimilar parts, some rarer -and some more dense; provided only, that all the parts in the same -globe equally distant from the center be homogeneous[254]. And also, -if both the globes attract each other[255]. All which place it beyond -contradiction, that this proportion obtains with as much exactness -near and contiguous to the surface of attracting globes, as at greater -distances from them. - -14. THUS our author, without the pompous pretence of explaining the -cause of gravity, has made one very important step toward it, by -shewing that this power in the great bodies of the universe, is derived -from the same power being lodged in every particle of the matter which -composes them: and consequently, that this property is no less than -universal to all matter whatever, though the power be too minute to -produce any visible effects on the small bodies, wherewith we converse, -by their action on each other[256]. In the fixed stars indeed we have -no particular proof that they have this power; for we find no apperance -to demonstrate that they either act, or are acted upon by it. But -since this power is found to belong to all bodies, whereon we can make -observation; and we see that it is not to be altered by any change in -the form of bodies, but always accompanies them in every shape without -diminution, remaining ever proportional to the quantity of solid matter -in each; such a power must without doubt belong universally to all -matter. - -15. THIS therefore is the universal law of matter; which recommends -it self no less for its great plainness and simplicity, than for the -surprizing discoveries it leads us to. By this principle we learn the -different weight, which the same body will have upon the surfaces -of the sun and of diverse planets; and by the same we can judge of -the composition of those celestial bodies, and know the density of -each; which is formed of the most compact, and which of the most rare -substance. Let the adversaries of this philosophy reflect here, whether -loading this principle with the appellation of an occult quality, or -perpetual miracle, or any other reproachful name, be sufficient to -dissuade us from cultivating it; since this quality, which they call -occult, leads to the knowledge of such things, that it would have -been reputed no less than madness for any one, before they had been -discovered, even to have conjectured that our faculties should ever -have reached so far. - -16. SEE how all this naturally follows from the foregoing principles -in those planets, which have satellites moving about them. By the -times, in which these satellites perform their revolutions, compared -with their distances from their respective primary, the proportion -between the power, with which one primary attracts his satellites, and -the force with which any other attracts his will be known; and the -proportion of the power with which any planet attracts its secondary, -to the power with which it attracts a body at its surface is found, -by comparing the distance of the secondary planet from the center of -the primary, to the distance of the primary planet’s surface from the -same: and from hence is deduced the proportion between the power of -gravity upon the surface of one planet, to the gravity upon the surface -of another. By the like method of comparing the periodical time of a -primary planet about the sun, with the revolution of a satellite about -its primary, may be found the proportion of gravity, or of the weight -of any body upon the surface of the sun, to the gravity, or to the -weight of the same body upon the surface of the planet, which carries -about the satellite. - -17. BY these kinds of computation it is found, that the weight of the -same body upon the surface of the sun will be about 23 times as great, -as here upon the surface of the earth; about 10⅗ times as great, as -upon the surface of Jupiter; and near 19 times as great, as upon the -surface of Saturn[257]. - -18. THE quantity of matter, which composes each of these bodies, is -proportional to the power it has upon a body at a given distance. By -this means it is found, that the sun contains 1067 times as much matter -as Jupiter; Jupiter 158⅔ times as much as the earth, and 2-5/6 times as -much as Saturn[258]. The diameter of the sun is about 92 times, that of -Jupiter about 9 times, and that of Saturn about 7 times the diameter of -the earth. - -19. BY making a comparison between the quantity of matter in these -bodies and their magnitudes, to be found from their diameters, their -respective densities are readily deduced; the density of every body -being measured by the quantity of matter contained under the same bulk, -as has been above remarked[259]. Thus the earth is found 4¼ times -more dense than Jupiter; Saturn has between ⅔ and ¾ of the density of -Jupiter; but the sun has one fourth part only of the density of the -earth[260]. From which this observation is drawn by our author; that -the sun is rarified by its great heat, and that of the three planets -named, the more dense is nearer the sun than the more rare; as was -highly reasonable to expect, the densest bodies requiring the greatest -heat to agitate and put their parts in motion; as on the contrary, the -planets which are more rare, would be rendered unfit for their office, -by the intense heat to which the denser are exposed. Thus the waters -of our seas, if removed to the distance of Saturn from the sun, would -remain perpetually frozen; and if as near the sun as Mercury, would -constantly boil[261]. - -20. THE densities of the three planets Mercury, Venus, and Mars, which -have no satellites, cannot be expresly assigned; but from what is -found in the others, it is very probable, that they also are of such -different degrees of density, that universally the planet which is -nearest to the sun, is formed of the most compact substance. - - - - -~CHAP~. VI. - -Of the FLUID PARTS of the PLANETS. - - -THIS globe, that we inhabit, is composed of two parts; the solid earth, -which affords us a foundation to dwell upon; and the seas and other -waters, that furnish rains and vapours necessary to render the earth -fruitful, and productive of what is requisite for the support of life. -And that the moon, though but a secondary planet, is composed in like -manner, is generally thought, from the different degrees of light -which appear on its surface; the parts of that planet, which reflect a -dim light, being supposed to be fluid, and to imbibe the sun’s rays, -while the solid parts reflect them more copiously. Some indeed do not -allow this to be a conclusive argument: but whether we can distinguish -the fluid part of the moon’s surface from the rest or not; yet it is -most probable that there are two such different parts, and with still -greater reason we may ascribe the like to the other primary planets, -which yet more nearly resemble our earth. The earth is also encompassed -by another fluid the air, and we have before remarked, that probably -the rest of the planets are surrounded by the like. These fluid parts -in particular engage our author’s attention, both by reason of some -remarkable appearances peculiar to them, and likewise of some effects -they have upon the whole bodies to which they belong. - -2. FLUIDS have been already treated of in general, with respect to the -effect they have upon solid bodies moving in them[262]; now we must -consider them in reference to the operation of the power of gravity -upon them. By this power they are rendered weighty, like all other -bodies, in proportion to the quantity of matter, which is contained -in them. And in any quantity of a fluid the upper parts press upon -the lower as much, as any solid body would press on another, whereon -it should lie. But there is an effect of the pressure of fluids on -the bottom of the vessel, wherein they are contained, which I shall -particularly explain. The force supported by the bottom of such a -vessel is not simply the weight of the quantity of the fluid in the -vessel, but is equal to the weight of that quantity of the fluid, which -would be contained in a vessel of the same bottom and of equal width -throughout, when this vessel is filled up to the same height, as that -to which the vessel proposed is filled. Suppose water were contained -in the vessel A B C D (in fig. 109.) filled up to E F. Here it is -evident, that if a part of the bottom, as G H, which is directly under -any part of the space E F, be considered separately; it will appear -at once, that this part sustains the weight of as much of the fluid, -as stands perpendicularly over it up to the height of E F; that is, -the two perpendiculars G I and H K being drawn, the part G H of the -bottom will sustain the whole weight of the fluid included between -these two perpendiculars. Again, I say, every other part of the bottom -equally broad with this, will sustain as great a pressure. Let the -part L M be of the same breadth with G H. Here the perpendiculars -L O and M N being drawn, the quantity of water contained between -these perpendiculars is not so great, as that contained between the -perpendiculars G I and H K; yet, I say, the pressure on L M will be -equal to that on G H. This will appear by the following considerations. -It is evident, that if the part of the vessel between O and N were -removed, the water would immediately flow out, and the surface E F -would subside; for all parts of the water being equally heavy, it must -soon form itself to a level surface, if the form of the vessel, which -contains it, does not prevent. Therefore since the water is prevented -from rising by the side N O of the vessel, it is manifest, that it must -press against N O with some degree of force. In other words, the water -between the perpendiculars L O and M N endeavours to extend itself with -a certain degree of force; or more correctly, the ambient water presses -upon this, and endeavours to force this pillar or column of water into -a greater length. But since this column of water is sustained between -N O and L M, each of these parts of the vessel will be equally pressed -against by the power, wherewith this column endeavours to extend. -Consequently L M bears this force over and above the weight of the -column of water between L O and M N. To know what this expansive force -is, let the part O N of the vessel be removed, and the perpendiculars -L O and M N be prolonged; then by means of some pipe fixed over N O -let water be filled between these perpendiculars up to P Q an equal -height with E F. Here the water between the perpendiculars L P and M Q -is of an equal height with the highest part of the water in the vessel; -therefore the water in the vessel cannot by its pressure force it -up higher, nor can the water in this column subside; because, if it -should, it would raise the water in the vessel to a greater height than -itself. But it follows from hence, that the weight of water contained -between P O and Q N is a just balance to the force, wherewith the -column between L O and M N endeavours to extend. So the part L M of -the bottom, which sustains both this force and the weight of the water -between L O and M N, is pressed upon by a force equal to the united -weight of the water between L O and M N, and the weight of the water -between P O and Q N; that is, it is pressed on by a force equal to the -weight of all the water contained between L P and M Q. And this weight -is equal to that of the water contained between G I and H K, which is -the weight sustained by the part G H of the bottom. Now this being -true of every part of the bottom B C, it is evident, that if another -vessel R S T V be formed with a bottom R V equal to the bottom B C, and -be throughout its whole height of one and the same breadth; when this -vessel is filled with water to the same height, as the vessel A B C D -is filled, the bottoms of these two vessels shall be pressed upon with -equal force. If the vessel be broader at the top than at the bottom, -it is evident, that the bottom will bear the pressure of so much of -the fluid, as is perpendicularly over it, and the sides of the vessel -will support the rest. This property of fluids is a corollary from a -proposition of our author[263]; from whence also he deduces the effects -of the pressure of fluids on bodies resting in them. These are, that -any body heavier than a fluid will sink to the bottom of the vessel, -wherein the fluid is contained, and in the fluid will weigh as much as -its own weight exceeds the weight of an equal quantity of the fluid; -any body uncompressible of the same density with the fluid, will rest -any where in the fluid without suffering the least change either in -its place or figure from the pressure of such a fluid, but will remain -as undisturbed as the parts of the fluid themselves; but every body -of less density than the fluid will swim on its surface, a part only -being received within the fluid. Which part will be equal in bulk to -a quantity of the fluid, whose weight is equal to the weight of the -whole body; for by this means the parts of the fluid under the body -will suffer as great a pressure as any other parts of the fluid as much -below the surface as these. - -3. IN the next place, in relation to the air, we have above made -mention, that the air surrounding the earth being an elastic fluid, -the power of gravity will have this effect on it, to make the lower -parts near the surface of the earth more compact and compressed -together by the weight of the air incumbent, than the higher parts, -which are pressed upon by a less quantity of the air, and therefore -sustain a less weight[264]. It has been also observed, that our author -has laid down a rule for computing the exact degree of density in -the air at all heights from the earth[265]. But there is a farther -effect from the air’s being compressed by the power of gravity, which -he has distinctly considered. The air being elastic and in a state -of compression, any tremulous body will propagate its motion to the -air, and excite therein vibrations, which will spread from the body -that occasions them to a great distance. This is the efficient cause -of sound: for that sensation is produced by the air, which, as it -vibrates, strikes against the organ of hearing. As this subject was -extremely difficult, so our great author’s success is surprizing. - -4. OUR author’s doctrine upon this head I shall endeavour to explain -somewhat at large. But preliminary thereto must be shewn, what he has -delivered in general of pressure propagated through fluids; and also -what he has set down relating to that wave-like motion, which appears -upon the surface of water, when agitated by throwing any thing into it, -or by the reciprocal motion of the finger, &c. - -5. CONCERNING the first, it is proved, that pressure is spread -through fluids, not only right forward in a streight line, but also -laterally, with almost the same ease and force. Of which a very obvious -exemplification by experiment is proposed: that is, to agitate the -surface of water by the reciprocal motion of the finger forwards and -backwards only; for though the finger have no circular motion given -it, yet the waves excited in the water will diffuse themselves on each -hand of the direction of the motion, and soon surround the finger. Nor -is what we observe in sounds unlike to this, which do not proceed in -straight lines only, but are heard though a mountain intervene, and -when they enter a room in any part of it, they spread themselves into -every corner; not by reflection from the walls, as some have imagined, -but as far as the sense can judge, directly from the place where they -enter. - -6. HOW the waves are excited in the surface of stagnant water, may be -thus conceived. Suppose in any place, the water raised above the rest -in form of a small hillock; that water will immediately subside, and -raise the circumambient water above the level of the parts more remote, -to which the motion cannot be communicated under longer time. And -again, the water in subsiding will acquire, like all falling bodies, a -force, which will carry it below the level surface, till at length the -pressure of the ambient water prevailing, it will rise again, and even -with a force like to that wherewith it descended, which will carry it -again above the level. But in the mean time the ambient water before -raised will subside, as this did, sinking below the level; and in so -doing, will not only raise the water, which first subsided, but also -the water next without itself. So that now beside the first hillock, we -shall have a ring investing it, at some distance raised above the plain -surface likewise; and between them the water will be sunk below the -rest of the surface. After this, the first hillock, and the new made -annular rising, will descend; raising the water between them, which -was before depressed, and likewise the adjacent part of the surface -without. Thus will these annular waves be successively spread more -and more. For, as the hillock subsiding produces one ring, and that -ring subsiding raises again the hillock, and a second ring; so the -hillock and second ring subsiding together raise the first ring, and -a third; then this first and third ring subsiding together raise the -first hillock, the second ring, and a fourth; and so on continually, -till the motion by degrees ceases. Now it is demonstrated, that these -rings ascend and descend in the manner of a pendulum; descending with -a motion continually accelerated, till they become even with the plain -surface of the fluid, which is half the space they descend; and then -being retarded again by the same degrees as those, whereby they were -accelerated, till they are depressed below the plain surface, as much -as they were before raised above it: and that this augmentation and -diminution of their velocity proceeds by the same degrees, as that of -a pendulum vibrating in a cycloid, and whose length should be a fourth -part of the distance between any two adjacent waves: and farther, that -a new ring is produced every time a pendulum, whose length is four -times the former, that is, equal to the interval between the summits of -two waves, makes one oscillation or swing[266]. - -7. THIS now opens the way for understanding the motion consequent upon -the tremors of the air, excited by the vibrations of sonorous bodies: -which we must conceive to be performed in the following manner. - -8. LET A, B, C, D, E, F, G, H (in fig. 110.) represent a series of -the particles of the air, at equal distances from each other. I K L -a musical chord, which I shall use for the tremulous and sonorous -body, to make the conception as simple as may be. Suppose this chord -stretched upon the points I and L, and forcibly drawn into the -situation I K L, so that it become contiguous to the particle A in its -middle point K: and let the chord from this situation begin to recoil, -pressing against the particle A, which will thereby be put into motion -towards B: but the particles A, B, C being equidistant, the elastic -power, by which B avoids A, is equal to, and balanced by the power, by -which it avoids C; therefore the elastic force, by which B is repelled -from A, will not put B into any degree of motion, till A is by the -motion of the chord brought nearer to B, than B is to C: but as soon as -that is done, the particle B will be moved towards C; and being made -to approach C, will in the next place move that; which will upon that -advance, put D likewise into motion, and so on: therefore the particle -A being moved by the chord, the following particles of the air B, C, D, -&c. will successively be moved. Farther, if the point K of the chord -moves forward with an accelerated velocity, so that the particle A -shall move against B with an advancing pace, and gain ground of it, -approaching nearer and nearer continually; A by approaching will press -more upon B, and give it a greater velocity likewise, by reason that as -the distance between the particles diminishes, the elastic power, by -which they fly each other, increases. Hence the particle B, as well as -A, will have its motion gradually accelerated, and by that means will -more and more approach to C. And from the same cause C will more and -more approach D; and so of the rest. Suppose now, since the agitation -of these particles has been shewn to be successive, and to follow one -another, that E be the remotest particle moved, while the chord is -moving from its curve situation I K L into that of a streight line, as -I k L; and F the first which remains unaffected, though just upon the -point of being put into motion. Then shall the particles A, B, C, D, -E, F, G, when the point K is moved into k, have acquired the rangement -represented by the adjacent points _a, b, c, d, e, f, g_: in which _a_ -is nearer to _b_ than _b_ to _c_, and _b_ nearer to _c_ than _c_ to -_d_, and _c_ nearer to _d_ than _d_ to _e_ and _d_ nearer to _e_ than -_e_ to _f_, and lastly _e_ nearer to _f_ than _f_ to _g_. - -9. BUT now the chord having recovered its rectilinear situation I k L, -the following motion will be changed, for the point K, which before -advanced with a motion more and more accelerated, though by the force -it has acquired it will go on to move the same way as before, till it -has advanced near as far forwards, as it was at first drawn backwards; -yet the motion of it will henceforth be gradually lessened. The effect -of which upon the particles _a, b, c, d, e, f, g_ will be, that by the -time the chord has made its utmost advance, and is upon the return, -these particles will be put into a contrary rangement; so that _f_ -shall be nearer to _g_, than _e_ to _f_, and _e_ nearer to _f_ than _d_ -to _e_; and the like of the rest, till you come to the first particles -_a_, _b_, whose distance will then be nearly or quite what it was at -first. All which will appear as follows. The present distance between -_a_ and _b_ is such, that the elastic power, by which _a_ repels _b_, -is strong enough to maintain that distance, though a advance with the -velocity, with which the string resumes its rectilinear figure; and -the motion of the particle _a_ being afterwards slower, the present -elasticity between _a_ and _b_ will be more than sufficient to preserve -the distance between them. Therefore while it accelerates _b_ it will -retard _a_. The distance _b c_ will still diminish, till _b_ come about -as near to _c_, as it is from a at present; for after the distances -_a b_ and _b c_ are become equal, the particle _b_ will continue its -velocity superior to that of _c_ by its own power of inactivity, till -such time as the increase of elasticity between _b_ and _c_ more than -shall be between _a_ and _b_ shall suppress its motion: for as the -power of inactivity in _b_ made a greater elasticity necessary on the -side of a than on the side of _c_ to push _b_ forward, so what motion -_b_ has acquired it will retain by the same power of inactivity, till -it be suppressed by a greater elasticity on the side of _c_, than on -the side of _a_. But as soon as _b_ begins to slacken its pace the -distance of _b_ from c will widen as the distance _a b_ has already -done. Now as _a_ acts on _b_, so will _b_ on _c_, _c_ on _d_, &c. so -that the distances between all the particles _b, c, d, e, f, g_ will -be successively contracted into the distance of _a_ from _b_, and then -dilated again. Now because the time, in which the chord describes -this present half of its vibration, is about equal to that it took up -in describing the former; the particles _a_, _b_ will be as long in -dilating their distance, as before in contracting it, and will return -nearly to their original distance. And farther, the particles _b_, _c_, -which did not begin to approach so soon as _a_, _b_, are now about as -much longer, before they begin to recede; and likewise the particles -_c_, _d_, which began to approach after _b_, _c_, begin to separate -later. Whence it appears that the particles, whose distance began to be -lessened, when that of _a_, _b_ was first enlarged, viz. the particles -_f_, _g,_ should be about their nearest distance, when _a_ and _b_ have -recovered their prime interval. Thus will the particles _a, b, c, d, e, -f, g_ have changed their situation in the manner asserted. But farther, -as the particles _f_, _g_ or F, G gradually approach each other, they -will move by degrees the succeeding particles to as great a length, as -the particles A, B did by a like approach. So that, when the chord has -made its greatest advance, being arrived into the situation I ϰ L, the -particles moved by it will have the rangement noted by the points α, β, -γ, δ, ε, ζ, η, θ, λ, μ, ν, χ. Where α, β are at the original distance -of the particles in the line A H; ζ, η are the nearest of all, and the -distance ν χ is equal to that between α and β. - -10. BY this time the chord I ϰ L begins to return, and the distance -between the particles α and β being enlarged to its original magnitude, -α has lost all that force it had acquired by its motion, being now at -rest; and therefore will return with the chord, making the distance -between α and β greater than the natural; for β will not return so -soon, because its motion forward is not yet quite suppressed, the -distance β γ not being already enlarged to its prime dimension: but the -recess of α, by diminishing the pressure upon β by its elasticity, will -occasion the motion of β to be stopt in a little time by the action of -γ, and then shall β begin to return: at which time the distance between -γ and δ shall by the superior action of δ above β be enlarged to the -dimension of the distance β γ, and therefore soon after to that of α β. -Thus it appears, that each of these particles goes on to move forward, -till its distance from the preceding one be equal to its original -distance; the whole chain α, β, γ, δ, ε, ζ, η, having an undulating -motion forward, which is stopt gradually by the excess of the expansive -power of the preceding parts above that of the hinder. Thus are these -parts successively stopt, as before they were moved; so that when the -chord has regained its rectilinear situation, the expansion of the -parts of the air will have advanced so far, that the interval between -ζ η, which at present is most contracted, will then be restored to its -natural size: the distances between η and θ, θ and λ, λ and μ, μ and ν, -ν and χ, being successively contracted into the present distance of ζ -from η, and again enlarged; so that the same effect shall be produced -upon the parts beyond ζ η, by the enlargement of the distance between -those two particles, as was occasioned upon the particles α, β, γ, δ, -ε, ζ, η, θ, λ, μ, ν, χ, by the enlargement of the distance α β to its -natural extent. And therefore the motion in the air will be extended -half as much farther as at present, and the distance between ν and χ -contracted into that, which is at present between ζ and η, all the -particles of the air in motion taking the rangement expressed in figure -111. by the points α, β, γ, δ, ε, ζ, η, θ, λ, μ, ν, χ, ϰ, ρ, σ, τ, φ -wherein the particles from α to χ have their distances from each other -gradually diminished, the distances between the particles ν, χ being -contracted the most from the natural distance between those particles, -and the distance between α, β as much augmented, and the distance -between the middle particles ζ, η becoming equal to the natural. The -particles π, ρ, ω τ, φ which follow χ, have their distances gradually -greater and greater, the particles ν, χ, π, ρ, σ, τ, φ being ranged -like the particles _a, b, c, d, e, f, g_, or like the particles ζ, -η, θ, λ, μ, ν, χ in the former figure. Here it will be understood, -by what has been before explained, that the particles ζ, η being at -their natural distance from each other, the particle ζ is at rest, the -particles ε, δ, λ, β, ϰ between them and the string being in motion -backward, and the rest of the particles η, θ, λ, μ, ν, χ, π, ρ, σ, τ -in motion forward: each of the particles between η and χ moving faster -than that, which immediately follows it; but of the particles from χ -to φ, on the contrary, those behind moving on faster than those, which -precede. - -11. BUT now the string having recovered its rectilinear figure, though -it shall go on recoiling, till it return near to its first situation -I K L, yet there will be a change in its motion; so that whereas it -returned from the situation I ϰ L with an accelerated motion, its -motion shall from hence be retarded again by the same degrees, as -accelerated before. The effect of which change upon the particles of -the air will be this. As by the accelerated motion of the chord α -contiguous to it moved faster than β, γ, so as to make the interval α -β greater than the interval β γ, and from thence β was made likewise -to move faster than γ, and the distance between β and γ rendered -greater than the distance between γ and δ, and so of the rest; now the -motion of α being diminished, β shall overtake it, and the distance -between α and β be reduced into that, which is at present between β -and γ, the interval between β and γ being inlarged into the present -distance between α and β; but when the interval β γ is increased to -that, which is at present between α and β γ the distance between γ -and δ shall be enlarged to the present distance between γ and β, -and the distance between δ and ι inlarged into the present distance -between γ and δ; and the same of the rest. But the chord more and more -slackening its pace, the distance between α and β shall be more and -more diminished; and in consequence of that the distance between β and -γ shall be again contracted, first into its present dimension, and -afterwards into a narrower space; while the interval γ δ shall dilate -into that at present between α and β, and as soon as it is so much -enlarged, it shall contract again. Thus by the reciprocal expansion -and contraction of the air between α and ζ, by that time the chord is -got into the situation I K L, the interval ζ η shall be expanded into -the present distance between α and β; and by that time likewise the -present distance of α from β will be contracted into their natural -interval: for this distance will be about the same time in contracting -it self, as has been taken up in its dilatation; seeing the string will -be as long in returning from its rectilinear figure, as it has been in -recovering it from its situation I ϰ L. This is the change which will -be made in the particles between α and ζ. As for those between ζ and -χ, because each preceding particle advances faster than that, which -immediately follows it, their distances will successively be dilated -into that, which is at present between ζ and η. And as soon as any two -particles are arrived at their natural distance, the hindermost of them -shall be stopt, and immediately after return, the distances between -the returning particles being greater than the natural. And this -dilatation of these distances shall extend so far, by that time the -chord is returned into its first situation I K L, that the particles ι -χ shall be removed to their natural distance. But the dilatation of ν -χ shall contract the interval τ φ into that at present between ν and -χ, and the contraction of the distance between those two particles τ -and φ will agitate a part of the air beyond; so that when the chord is -returned into the situation I K L, having made an intire vibration, the -moved particles of the air will take the rangement expressed by the -points, _l, m, n, o, p, q, r, s, t, u, w, x, y, z_, 1, 2, 3, 4, 5, 6, -7, 8: in which _l m_, are at the natural distance of the particles, the -distance _m n_ greater than _l m_ and _n o_ greater than _m n_, and so -on, till you come to _q r_, the widest of all: and then the distances -gradually diminish not only to the natural distance, as _w x_, but till -they are contracted as much as χ τ was before; which falls out in the -points 2, 3, from whence the distances augment again, till you come to -the part of the air untouched. - -12. THIS is the motion, into which the air is put, while the chord -makes one vibration, and the whole length of air thus agitated in -the time of one vibration of the chord our author calls the length -of one pulse. When the chord goes on to make another vibration, it -will not only continue to agitate the air at present in motion, but -spread the pulsation of the air as much farther, and by the same -degrees, as before. For when the chord returns into its rectilinear -situation I _k_ L, _l m_ shall be brought into its most contracted -state, _q r_ now in the state of greatest dilatation shall be reduced -to its natural distance, the points _w_, _x_ now at their natural -distance shall be at their greatest distance, the points 2, 3 now most -contracted enlarged to their natural distance, and the points 7, 8 -reduced to their most contracted state: and the contraction of them -will carry the agitation of the air as far beyond them, as that motion -was carried from the chord, when it first moved out of the situation -I K L into its rectilinear figure. When the chord is got into the -situation I ϰ L, _l m_ shall recover its natural dimensions, _q r_ be -reduced to its state of greatest contraction, _w x_ brought to its -natural dimension, the distance 2 3 enlarged to the utmost, and the -points 7, 8 shall have recovered their natural distance; and by thus -recovering themselves they shall agitate the air to as great a length -beyond them, as it was moved beyond the chord, when it first came into -the situation I ϰ L. When the chord is returned back again into its -rectilinear situation, _l m_ shall be in its utmost dilatation, _q r_ -restored again to its natural distance, _w x_ reduced into its state of -greatest contraction, 2 3 shall recover its natural dimension, and 7 8 -be in its state of greatest dilatation. By which means the air shall be -moved as far beyond the points 7, 8, as it was moved beyond the chord, -when it before made its return back to its rectilinear situation; for -the particles 7, 8 have been changed from their state of rest and -their natural distance into a state of contraction, and then have -proceeded to the recovery of their natural distance, and after that to -a dilatation of it, in the same manner as the particles contiguous to -the chord were agitated before. In the last place, when the chord is -returned into the situation I K L, the particles of air from _l_ to δ -shall acquire their present rangement, and the motion of the air be -extended as much farther. And the like will happen after every compleat -vibration of the string. - -13. CONCERNING this motion of sound, our author shews how to compute -the velocity thereof, or in what time it will reach to any proposed -distance from the sonorous body. For this he requires to know the -height of air, having the same density with the parts here at the -surface of the earth, which we breath, that would be equivalent in -weight to the whole incumbent atmosphere. This is to be found by the -barometer, or common weatherglass. In that instrument quicksilver is -included in a hollow glass cane firmly closed at the top. The bottom is -open, but immerged into quicksilver contained in a vessel open to the -air. Care is taken when the lower end of the cane is immerged, that the -whole cane be full of quicksilver, and that no air insinuate itself. -When the instrument is thus fixed, the quicksilver in the cane being -higher than that in the vessel, if the top of the cane were open, the -fluid would soon sink out of the glass cane, till it came to a level -with that in the vessel. But the top of the cane being closed up, so -that the air, which has free liberty to press on the quicksilver in -the vessel, cannot bear at all on that, which is within the cane, the -quicksilver in the cane will be suspended to such a height, as to -balance the pressure of the air on the quicksilver in the vessel. Here -it is evident, that the weight of the quicksilver in the glass cane is -equivalent to the pressure of so much of the air, as is perpendicularly -over the hollow of the cane; for if the cane be opened that the air may -enter, there will be no farther use of the quicksilver to sustain the -pressure of the air without; for the quicksilver in the cane, as has -already been observed, will then subside to a level with that without. -Hence therefore if the proportion between the density of quicksilver -and of the air we breath be known, we may know what height of such air -would form a column equal in weight to the column of quicksilver within -the glass cane. When the quicksilver is sustained in the barometer -at the height of 30 inches, the height of such a column of air will -be about 29725 feet; for in this case the air has about 1/870 of the -density of water, and the density of quicksilver exceeds that of water -about 13⅔ times, so that the density of quicksilver exceeds that of the -air about 11890 times; and so many times 30 inches make 29725 feet. Now -Sir ~ISAAC NEWTON~ determines, that while a pendulum of the length of -this column should make one vibration or swing, the space, which any -sound will have moved, shall bear to this length the same proportion, -as the circumference of a circle bears to the diameter thereof; -that is, about the proportion of 355 to 113[267]. Only our author -here considers singly the gradual progress of sound in the air from -particle to particle in the manner we have explained, without taking -into consideration the magnitude of those particles. And though there -requires time for the motion to be propagated from one particle to -another, yet it is communicated to the whole of the same particle in an -instant: therefore whatever proportion the thickness of these particles -bears to their distance from each other, in the same proportion will -the motion of sound be swifter. Again the air we breath is not simply -composed of the elastic part, by which sound is conveyed, but partly of -vapours, which are of a different nature; and in the computation of the -motion of sound we ought to find the height of a column of this pure -air only, whose weight should be equal to the weight of the quicksilver -in the cane of the barometer, and this pure air being a part only of -that we breath, the column of this pure air will be higher than 29725 -feet. On both these accounts the motion of sound is found to be about -1142 feet in one second of time, or near 13 miles in a minute, whereas -by the computation proposed above, it should move but 979 feet in one -second. - -14. WE may observe here, that from these demonstrations of our author -it follows, that all sounds whether acute or grave move equally swift, -and that sound is swiftest, when the quicksilver stands highest in the -barometer. - -15. THUS much of the appearances, which are caused in these fluids from -their gravitation toward the earth. They also gravitate toward the -moon; for in the last chapter it has been proved, that the gravitation -between the earth and moon is mutual, and that this gravitation of -the whole bodies arises from that power acting in all their parts; so -that every particle of the moon gravitates toward the earth, and every -particle of the earth toward the moon. But this gravitation of these -fluids toward the moon produces no sensible effect, except only in the -sea, where it causes the tides. - -16. THAT the tides depend upon the influence of the moon has been the -receiv’d opinion of all antiquity; nor is there indeed the least shadow -of reason to suppose otherwise, considering how steadily they accompany -the moon’s course. Though how the moon caused them, and by what -principle it was enabled to produce so distinguish’d an appearance, -was a secret left for this philosophy to unfold: which teaches, that -the moon is not here alone concerned, but that the sun likewise has a -considerable share in their production; though they have been generally -ascribed to the other luminary, because its effect is greatest, and by -that means the tides more immediately suit themselves to its motion; -the sun discovering its influence more by enlarging or restraining the -moon’s power, than by any distinct effects. Our author finds the power -of the moon to bear to the power of the sun about the proportion of -4½ to 1. This he deduces from the observations made at the mouth of -the river Avon, three miles from Bristol, by Captain STURMEY, and at -Plymouth by Mr. COLEPRESSE, of the height to which the water is raised -in the conjunction and opposition of the luminaries, compared with the -elevation of it, when the moon is in either quarter; the first being -caused by the united actions of the sun and moon, and the other by the -difference of them, as shall hereafter be shewn. - -17. THAT the sun should have a like effect on the sea, as the moon, is -very manifest; since the sun likewise attracts every single particle, -of which this earth is composed. And in both luminaries since the power -of gravity is reciprocally in the duplicate proportion of the distance, -they will not draw all the parts of the waters in the same manner; -but must act upon the nearest parts stronger, than upon the remotest, -producing by this inequality an irregular motion. We shall now attempt -to shew how the actions of the sun and moon on the waters, by being -combined together, produce all the appearances observed in the tides. - -18. TO begin therefore, the reader will remember what has been said -above, that if the moon without the sun would have described an orbit -concentrical to the earth, the action of the sun would make the orbit -oval, and bring the moon nearer to the earth at the new and full, -than at the quarters[268]. Now our excellent author observes, that -if instead of one moon, we suppose a ring of moons, contiguous and -occupying the whole orbit of the moon, his demonstration would still -take place, and prove that the parts of this ring in passing from the -quarter to the conjunction or opposition would be accelerated, and be -retarded again in passing from the conjunction or opposition to the -next quarter. And as this effect does not depend on the magnitude of -the bodies, whereof the ring is composed, the same would hold, though -the magnitude of these moons were so far to be diminished, and their -number increased, till they should form a fluid[269]. Now the earth -turns round continually upon its own center, causing thereby the -alternate change of day and night, while by this revolution each part -of the earth is successively brought toward the sun, and carried off -again in the space of 24 hours. And as the sea revolves round along -with the earth itself in this diurnal motion, it will represent in some -sort such a fluid ring. - -19. BUT as the water of the sea does not move round with so much -swiftness, as would carry it about the center of the earth in the -circle it now describes, without being supported by the body of the -earth; it will be necessary to consider the water under three distinct -cases. The first case shall suppose the water to move with the degree -of swiftness, required to carry a body round the center of the -earth disingaged from it in a circle at the distance of the earth’s -semidiameter, like another moon. The second case is, that the waters -make but one turn about the axis of the earth in the space of a month, -keeping pace with the moon; so that all parts of the water should -preserve continually the same situation in respect of the moon. The -third case shall be the real one of the waters moving with a velocity -between these two, neither so swift as the first case requires, nor so -slow as the second. - -20. IN the first case the waters, like the body which they equalled -in velocity, by the action of the moon would be brought nearer the -center under and opposite to the moon, than in the parts in the middle -between these eastward or westward. That such a body would so alter -its distance by the moon’s action upon it, is clear from what has -been mentioned of the like changes in the moon’s motion caused by -the sun[270]. And computation shews, that the difference between the -greatest and least distance of such a body would not be much above 4½ -feet. But in the second case, where all the parts of the water preserve -the same situation continually in respect of the moon, the weight of -those parts under and opposite to the moon will be diminished by the -moon’s action, and the parts in the middle between these will have -their weight increased: this being effected just in the same manner, -as the sun diminishes the attraction of the moon towards the earth in -the conjunction and opposition, but increases that attraction in the -quarters. For as the first of these consequences from the sun’s action -on the moon is occasioned by the moon’s being attracted by the sun in -the conjunction more than the earth, and in the opposition less than -it, and therefore in the common motion of the earth and moon, the moon -is made to advance toward the sun in one case too fast, and in the -other is left as it were behind; so the earth will not have its middle -parts drawn towards the moon so strongly as the nearer parts, and yet -more forcibly than the remotest: and therefore since the earth and -moon move each month round their common center of gravity[271], while -the earth moves round this center, the same effect will be produced, -on the parts of the water nearest to that center or to the moon, as -the moon feels from the sun when in conjunction, and the water on the -contrary side of the earth will be affected by the moon, as the moon is -by the sun, when in opposition[272]; that is, in both cases the weight -of the water, or its propensity towards the center of the earth, will -be diminished. The parts in the middle between these will have their -weight increased, by being pressed towards the center of the earth -through the obliquity of the moon’s action upon them to its action -upon the earth’s center, just as the sun increases the gravitation -of the moon in the quarters from the same cause[273]. But now it is -manifest, that where the weight of the same quantity of water is least, -there it will be accumulated; while the parts, which have the greatest -weight, will subside. Therefore in this case there would be no tide or -alternate rising and falling of the water, but the water would form it -self into an oblong figure, whose axis prolonged would pass through the -moon. By Sir ~ISAAC NEWTON~’s computation the excess of this axis above -the diameters perpendicular to it, that is, the height of the waters -under and opposite to the moon above their height in the middle between -these places eastward or westward caused by the moon, is about 8⅔ feet. - -21. THUS the difference of height in this latter supposition is little -short of twice that difference in the preceding. But the case of the -sea is a middle between these two: for a body, which should revolve -round the center of the earth at the distance of a semidiameter without -pressing on the earth’s surface, must perform its period in less than -an hour and half, whereas the earth turns round but once in a day; and -in the case of the waters keeping pace with the moon it should turn -round but once in a month: so that the real motion of the water is -between the motions required in these two cases. Again, if the waters -moved round as swiftly as the first case required, their weight would -be wholly taken off by their motion; for this case supposes the body -to move so, as to be kept revolving in a circle round the earth by -the power of gravity without pressing on the earth at all, so that -its motion just supports its weight. But if the power of gravity had -been only 1/289 part of what it is, the body could have moved thus -without pressing on the earth, and have been as long in moving round, -as the earth it self is. Consequently the motion of the earth takes off -from the weight of the water in the middle between the poles, where -its motion is swiftest, 1/289 part of its weight and no more. Since -therefore in the first case the weight of the waters must be intirely -taken off by their motion, and by the real motion of the earth they -lose only 1/289 part thereof, the motion of the water will so little -diminish their weight, that their figure will much nearer resemble -the case of their keeping pace with the moon than the other. Upon -the whole, if the waters moved with the velocity necessary to carry -a body round the center of the earth at the distance of the earth’s -semidiameter without bearing on its surface, the water would be lowest -under the moon, and rise gradually as it moved on with the earth -eastward, till it came half way toward the place opposite to the moon; -from thence it would subside again, till it came to the opposition, -where it would become as low as at first; afterwards it would rise -again, till it came half way to the place under the moon; and from -hence it would subside, till it came a second time under the moon. But -in case the water kept pace with the moon, it would be highest where -in the other case it is lowest, and lowest where in the other it is -highest; therefore the diurnal motion of the earth being between the -motions of these two cases, it will cause the highest place of the -water to fall between the places of the greatest height in these two -cases. The water as it passes from under the moon shall for some time -rise, but descend again before it arrives half way to the opposite -place, and shall come to its least height before it becomes opposite -to the moon; then it shall rise again, continuing so to do till it has -passed the place opposite to the moon, but subside before it comes -to the middle between the places opposite to and under the moon; and -lastly it shall come to its lowest, before it comes a second time -under the moon. If A (in fig. 112, 113, 114.) represent the moon, B -the center of the earth, the oval C D E F in fig. 112. will represent -the situation of the water in the first case; but if the water kept -pace with the moon, the line C D E F in fig. 113. would represent -the situation of the water; but the line C D E F in fig. 114. will -represent the same in the real motion of the water, as it accompanies -the earth in its diurnal rotation: in all these figures C and E being -the places where the water is lowest, and D and F the places where it -is highest. Pursuant to this determination it is found, that on the -shores, which lie exposed to the open sea, the high water usually falls -out about three hours after the moon has passed the meridian of each -place. - -22. LET this suffice in general for explaining the manner, in which the -moon acts upon the seas. It is farther to be noted, that these effects -are greatest, when the moon is over the earth’s equator[274], that -is, when it shines perpendicularly upon the parts of the earth in the -middle between the poles. For if the moon were placed over either of -the poles, it could have no effect upon the water to make it ascend and -descend. So that when the moon declines from the equator toward either -pole, it’s action must be something diminished, and that the more, the -farther it declines. The tides likewise will be greatest, when the moon -is nearest to the earth, it’s action being then the strongest. - -23. THUS much of the action of the moon. That the sun should produce -the very same effects, though in a less degree, is too obvious to -require a particular explanation: but as was remarked before, this -action of the sun being weaker than that of the moon, will cause the -tides to follow more nearly the moon’s course, and principally shew it -self by heightening or diminishing the effects of the other luminary. -Which is the occasion, that the highest tides are found about the -conjunction and opposition of the luminaries, being then produced by -their united action, and the weakest tides about the quarters of the -moon; because the moon in this case raising the water where the sun -depresses it, and depressing it where the sun raises it, the stronger -action of the moon is in part retunded and weakened by that of the sun. -Our author computes that the sun will add near two feet to the height -of the water in the first case, and in the other take from it as much. -However the tides in both comply with the same hour of the moon. But at -other times, between the conjunction or opposition and quarters, the -time deviates from that forementioned, towards the hour in which the -sun would make high water, though still it keeps much nearer to the -moon’s hour than to the sun’s. - -24. AGAIN the tides have some farther varieties from the situation of -the places where they happen northward or southward. Let _p_ P (in fig. -115.) represent the axis, on which the earth daily revolves, let _h_ -_p_ H P represent the figure of the water, and let _n_ B N D be a globe -inscribed within this figure. Suppose the moon to be advanced from the -equator toward the north pole, so that _h_ H the axis of the figure of -the water _p_ A H P E _h_ shall decline towards the north pole N; take -any place G nearer to the north pole than to the south, and from the -center of the earth C draw C G F; then will G F denote the altitude -to which the water is raised by the tide, when the moon is above the -horizon: in the space of twelve hours, the earth having turned half -round its axis, the place G will be removed to _g_; but the axis _h_ -H will have kept its place preserving its situation in respect of the -moon, at least will have moved no more than the moon has done in that -time, which it is not necessary here to take into consideration. Now -in this case the height of the water will be equal to _g_ _f_, which -is not so great as G F. But whereas G F is the altitude at high water, -when the moon is above the horizon, _g_ _f_ will be the altitude of the -same, when the moon is under the horizon. The contrary happens toward -the south pole, for K L is less than _k_ _l_. Hence is proved, that -when the moon declines from the equator, in those places, which are on -the same side of the equator as the moon, the tides are greater, when -the moon is above the horizon, than when under it; and the contrary -happens on the other side of the equator. - -25. NOW from these principles may be explained all the known -appearances in the tides; only by the assistance of this additional -remark, that the fluctuating motion, which the water has in flowing -and ebbing, is of a durable nature, and would continue for some time, -though the action of the luminaries should cease; for this prevents -the difference between the tide when the moon is above the horizon, -and the tide when the moon is below it from being so great, as the -rule laid down requires. This likewise makes the greatest tides not -exactly upon the new and full moon, but to be a tide or two after; as -at Bristol and Plymouth they are found the third after. - -26. THIS doctrine farther shews us, why not only the spring tides fall -out about the new and full moon, and the neap tides about the quarters; -but likewise how it comes to pass, that the greatest spring tides -happen about the equinoxes; because the luminaries are then one of them -over the equator, and the other not far from it. It appears too, why -the neap tides, which accompany these, are the least of all, for the -sun still continuing over the equator continues to have the greatest -power of lessening the moon’s action, and the moon in the quarters -being far removed toward one of the poles, has its power thereby -weakned. - -27. MOREOVER the action of the moon being stronger, when near the -earth, than when more remote; if the moon, when new suppose, be at its -nearest distance from the earth, it shall when at the full be farthest -off; whence it is, that two of the very largest spring tides do never -immediately succeed each other. - -28. BECAUSE the sun in its passage from the winter solstice to the -summer recedes from the earth, and passing from the summer solstice -to the winter approaches it, and is therefore nearer the earth before -the vernal equinox than after, but nearer after the autumnal equinox -than before; the greatest tides oftner precede the vernal equinox than -follow it, and in the autumnal equinox on the contrary they oftner -follow it than come before it. - -29. THE altitude, to which the water is raised in the open ocean, -corresponds very well to the forementioned calculations; for as it was -shewn, that the water in spring tides should rise to the height of 10 -or 11 feet, and the neap tides to 6 or 7; accordingly in the Pacific, -Atlantic and Ethiopic oceans in the parts without the tropics, the -water is observed to rise about 6, 9, 12 or 15 feet. In the Pacific -ocean this elevation is said to be greater than in the other, as it -ought to be by reason of the wide extent of that sea. For the same -reason in the Ethiopic ocean between the tropics the ascent of the -water is less than without, by reason of the narrowness of the sea -between the coasts of Africa and the southern parts of America. And -islands in such narrow seas, if far from shore, have less tides than -the coasts. But now in those ports where the water flows in with great -violence upon fords and shoals, the force it acquires by that means -will carry it to a much greater height, so as to make it ascend and -descend to 30, 40 or even 50 feet and more; instances of which we have -at Plymouth, and in the Severn near Chepstow; at St. Michael’s and -Auranches in Normandy; at Cambay and Pegu in the East Indies. - -30. AGAIN the tides take a considerable time in passing through long -straits, and shallow places. Thus the tide, which is made on the west -coast of Ireland and on the coast of Spain at the third hour after the -moon’s coming to the meridian, in the ports eastward toward the British -channel falls out later, and as the flood passes up that channel still -later and later, so that the tide takes up full twelve hours in coming -up to London bridge. - -31. IN the last place tides may come to the same port from different -seas, and as they may interfere with each other, they will produce -particular effects. Suppose the tide from one sea come to a port at the -third hour after the moon’s passing the meridian of the place, but from -another sea to take up six hours more in its passage. Here one tide -will make high water, when by the other it should be lowest; so that -when the moon is over the equator, and the two tides are equal, there -will be no rising and falling of the water at all; for as much as the -water is carried off by one tide, it will be supplied by the other. -But when the moon declines from the equator, the same way as the port -is situated, we have shewn that of the two tides of the ocean, which -are made each day, that tide, which is made when the moon is above the -horizon, is greater than the other. Therefore in this case, as four -tides come to this port each day the two greatest will come on the -third, and on the ninth hour after the moon’s passing the meridian, and -the two least at the fifteenth and at the twenty first hour. Thus from -the third to the ninth hour more water will be in this port by the two -greatest tides than from the ninth to the fifteenth, or from the twenty -first to the following third hour, where the water is brought by one -great and one small tide; but yet there will be more water brought -by these tides, than what will be found between the two least tides, -that is, between the fifteenth and twenty first hour. Therefore in the -middle between the third and ninth hour, or about the moon’s setting, -the water will be at its greatest height; in the middle between the -ninth and fifteenth, as also between the twenty first and following -third hour it will have its mean height; and be lowest in the middle -between the fifteenth and twenty first hour, that is, at the moon’s -rising. Thus here the water will have but one flood and one ebb each -day. When the moon is on the other side of the equator, the flood will -be turned into ebb, and the ebb into flood; the high water falling -out at the rising of the moon, and the low water at the setting. Now -this is the case of the port of Batsham in the kingdom of Tunquin in -the East Indies; to which port there are two inlets, one between the -continent and the islands which are called the Manillas, and the other -between the continent and Borneo. - -32. THE next thing to be considered is the effect, which these fluids -of the planets have upon the solid part of the bodies to which they -belong. And in the first place I shall shew, that it was necessary upon -account of these fluid parts to form the bodies of the planets into a -figure something different from that of a perfect globe. Because the -diurnal rotation, which our earth performs about its axis, and the -like motion we see in some of the other planets, (which is an ample -conviction that they all do the like) will diminish the force, with -which bodies are attracted upon all the parts of their surfaces, except -at the very poles, upon which they turn. Thus a stone or other weighty -substance resting upon the surface of the earth, by the force which it -receives from the motion communicated to it by the earth, if its weight -prevented not, would continue that motion in a straight line from the -point where it received it, and according to the direction, in which it -was given, that is, in a line which touches the surface at that point; -insomuch that it would move off from the earth in the same manner, as -a weight fasten’d to a string and whirled about endeavours continually -to recede from the center of motion, and would forthwith remove it self -to a greater distance from it, if loosed from the string which retains -it. And farther, as the centrifugal force, with which such a weight -presses from the center of its motion, is greater, by how much greater -the velocity is, with which it moves; so such a body, as I have been -supposing to lie on the earth, would recede from it with the greater -force, the greater the velocity is, with which the part of the earth’s -surface it rests upon is moved, that is, the farther distant it is from -the poles. But now the power of gravity is great enough to prevent -bodies in any part of the earth from being carried off from it by this -means; however it is plain that bodies having an effort contrary to -that of gravity, though much weaker than it, their weight, that is, -the degree of force, with which they are pressed to the earth, will -be diminished thereby, and be the more diminished, the greater this -contrary effort is; or in other words, the same body will weigh heavier -at either of the poles, than upon any other part of the earth; and if -any body be removed from the pole towards the equator, it will lose of -its weight more and more, and be lightest of all at the equator, that -is, in the middle between the poles. - -33. THIS now is easily applied to the waters of the seas, and shews -that the water under the poles will press more forcibly to the earth, -than at or near the equator: and consequently that which presses -least, must give place, till by ascending it makes room for receiving -a greater quantity, which by its additional weight may place the whole -upon a ballance. To illustrate this more particularly I shall make -use of fig. 116 In which let A C B D be a circle, by whose revolution -about the diameter A B a globe should be formed, representing a globe -of solid earth. Suppose this globe covered on all sides with water to -the same height, suppose that of E A or B F, at which distance the -circle E G F H surrounds the circle A C B D; then it is evident, if the -globe of earth be at rest, the water which surrounds it will rest in -that situation. But if the globe be turned incessantly about its axis -A B, and the water have likewise the same motion, it is also evident, -from what has been explained, that the water between the circles E H -F G and A D B C will remain no longer in the present situation, the -parts of it between H and D, and between G and C being by this rotation -become lighter, than the parts between E and A and between B and F; so -that the water over the poles A and B must of necessity subside, and -the water be accumulated over D and C, till the greater quantity in -these latter places supply the defect of its weight. This would be the -case, were the globe all covered with water. And the same figure of the -surface would also be preserved, if some part of the water adjoining -to the globe in any part of it were turned into solid earth, as is too -evident to need any proof; because the parts of the water remaining -at rest, it is the same thing, whether they continue in the state of -being easily separable, which denominates them fluid, or were to be -consolidated together, so as to make a hard body: and this, though the -water should in some places be thus consolidated, even to the surface -of it. Which shews that the form of the solid part of the earth makes -no alteration in the figure the water will take: and by consequence in -order to the preventing some parts of the earth from being entirely -overflowed, and other parts quite deserted, the solid parts of the -earth must have given them much the same figure, as if the whole earth -were covered on all sides with water. - -34. FARTHER, I say, this figure of the earth is the same, as it would -receive, were it entirely a globe of water, provided that water were of -the same density as the substance of the globe. For suppose the globe A -C B D to be liquified, and that the globe E H F G, now entirely water, -by its rotation about its axis should receive such a figure as we have -been describing, and then the globe A C B D should be consolidated -again, the figure of the water would plainly not be altered, by such a -consolidation. - -35. BUT from this last observation our author is enabled to determine -the proportion between the axis of the earth drawn from pole to pole, -and the diameter of the equator, upon the supposition that all the -parts of the earth are of equal density; which he does by computing in -the first place the proportion of the centrifugal force of the parts -under the equator to the power of gravity; and then by considering -the earth as a spheroid, made by the revolution of an ellipsis about -its lesser axis, that is, supposing the line M I L K to be an exact -ellipsis, from which it can differ but little, by reason that the -difference between the lesser axis M L and the greater I K is but very -small. From this supposition, and what was proved before, that all the -particles which compose the earth have the attracting power explained -in the preceding chapter, he finds at what distance the parts under the -equator ought to be removed from the center, that the force, with which -they shall be attracted to the center, diminished by their centrifugal -force, shall be sufficient to keep those parts in a ballance with those -which lie under the poles. And upon the supposition of all the parts of -the earth having the same degree of density, the earth’s surface at the -equator must be above 17 miles more distant from the center, than at -the poles[275]. - -36. AFTER this it is shewn, from the proportion of the equatorial -diameter of the earth to its axis, how the same may be determined of -any other planet, whose density in comparison of the density of the -earth, and the time of its revolution about its axis, are known. And -by the rule delivered for this, it is found, that the diameter of the -equator in Jupiter should bear to its axis about the proportion of 10 -to 9[276], and accordingly this planet appears of an oval form to the -astronomers. The most considerable effects of this spheroidical figure -our author takes likewise into consideration; one of which is that -bodies are not equally heavy in all distances from the poles; but near -the equator, where the distance from the center is greatest, they are -lighter than towards the poles: and nearly in this proportion, that -the actual power, by which they are drawn to the center, resulting -from the difference between their absolute gravity and centrifugal -force, is reciprocally as the distance from the center. That this may -not appear to contradict what has before been said of the alteration -of the power of gravity, in proportion to the change of the distance -from the center, it is proper carefully to remark, that our author -has demonstrated three things relating hereto: the first is, that -decrease of the power of gravity as we recede from the center, which -has been fully explained in the last chapter, upon supposition that -the earth and planets are perfect spheres, from which their difference -is by many degrees too little to require notice for the purposes there -intended: the next is, that whether they be perfect spheres, or exactly -such spheroids as have now been mentioned, the power of gravity, as -we descend in the same line to the center, is at all distances as the -distance from the center, the parts of the earth above the body by -drawing the body towards them lessening its gravitation towards the -center[277]; and both these assertions relate to gravity alone: the -third is what we mentioned in this place, that the actual force on -different parts of the surface, with which bodies are drawn to the -center, is in the proportion here assigned[278]. - -38. THE next effect of this figure of the earth is an obvious -consequence of the former: that pendulums of the same length do not in -different distances from the pole make their vibrations in the same -time; but towards the poles, where the gravity is strongest, they move -quicker than near the equator, where they are less impelled to the -center; and accordingly pendulums, that measure the same time by their -vibrations, must be shorter near the poles than at a greater distance. -Both which deductions are found true in fact; of which our author has -recounted particularly several experiments, in which it was found, that -clocks exactly adjusted to the true measure of time at Paris, when -transported nearer to the equator, became erroneous and moved too slow, -but were reduced to their true motion by contracting their pendulums. -Our author is particular in remarking, how much they lost of their -motion, while the pendulums remained unaltered; and what length the -observers are said to have shortened them, to bring them to time. And -the experiments, which appear to be most carefully made, shew the earth -to be raised in the middle between the poles, as much as our author -found it by his computation[279]. - -39. THESE experiments on the pendulum our author has been very exact -in examining, inquiring particularly how much the extension of the -rod of the pendulum by the great heats in the torrid zone might make -it necessary to shorten it. For by an experiment made by PICART, and -another made by DE LA HIRE, heat, though not very intense, was found -to increase the length of rods of iron. The experiment of PICART was -made with a rod one foot long, which in winter, at the time of frost, -was found to increase in length by being heated at the fire. In the -experiment of DE LA HIRE a rod of six foot in length was found, when -heated by the summer sun only, to grow to a greater length, than it -had in the aforesaid cold season. From which observations a doubt has -been raised, whether the rod of the pendulums in the aforementioned -experiments was not extended by the heat of those warm climates to all -that excess of length, the observers found themselves obliged to lessen -them by. But the experiments now mentioned shew the contrary. For in -the first of them the rod of a foot long was lengthened no more than -1/9 part of what the pendulum under the equator must be diminished; -and therefore a rod of the length of the pendulum would not have been -extended above ⅓ of that length. In the experiment of DE LA HIRE, -where the heat was less, the rod of six foot long was extended no more -than 3/10 of what the pendulum must be shortened; so that a rod of the -length of the pendulum would not have gained above 3/20 or 1/7 of that -length. And the heat in this latter experiment, though less than in -the former, was yet greater than the rod of a pendulum can ordinarily -contract in the hottest country; for metals receive a great heat when -exposed to the open sun, certainly much greater than that of a human -body. But pendulums are not usually so exposed, and without doubt in -these experiments were kept cool enough to appear so to the touch; -which they would do in the hottest place, if lodged in the shade. Our -author therefore thinks it enough to allow about 1/10 of the difference -observed upon account of the greater warmth of the pendulum. - -40. THERE is a third effect, which the water has on the earth by -changing its figure, that is taken notice of by our author; for -the explaining of which we shall first prove, that bodies descend -perpendicularly to the surface of the earth in all places. The manner -of collecting this from observation, is as follows. The surfaces of -all fluids rest parallel to that part of the surface of the sea, which -is in the same place with them, to the figure of which, as has been -particularly shewn, the figure of the whole earth is formed. For if -any hollow vessel, open at the bottom, be immersed into the sea; it -is evident, that the surface of the sea within the vessel will retain -the same figure it had, before the vessel inclosed it; since its -communication with the external water is not cut off by the vessel. -But all the parts of the water being at rest, it is as clear, that if -the bottom of the vessel were closed, the figure of the water could -receive no change thereby, even though the vessel were raised out of -the sea; any more than from the insensible alteration of the power of -gravity, consequent upon the augmentation of the distance from the -center. But now it is clear, that bodies descend in lines perpendicular -to the surfaces of quiescent fluids; for if the power of gravity did -not act perpendicularly to the surface of fluids, bodies which swim on -them could not rest, as they are seen to do; because, if the power of -gravity drew such bodies in a direction oblique to the surface whereon -they lay, they would certainly be put in motion, and be carried to the -side of the vessel, in which the fluid was contained, that way the -action of gravity inclined. - -41. HENCE it follows, that as we stand, our bodies are perpendicular -to the surface of the earth. Therefore in going from north to south -our bodies do not keep in a parallel direction. Now in all distances -from the pole the same length gone on the earth will not make the same -change in the position of our bodies, but the nearer we are to the -poles, we must go greater length to cause the same variation herein. -Let M I L K (in fig. 117) represent the figure of the earth, M, L the -poles, I, K two opposite points in the middle between these poles. Let -T V and P O be two arches, T V being most remote from the pole L; draw -T W, V X, P Q, O R, each perpendicular to the surface of the earth, -and let T W, V X meet in Y, and P Q, O R in S. Here it is evident, -that in passing from V to T the position of a man’s body would be -changed by the angle under T Y V, for at V he would stand in the line -Y V continued upward, and at T in the line Y T; but in passing from O -to P the position of his body would be changed by the angle under O -S P. Now I say, if these two angles are equal the arch O P is longer -than T V: for the figure M I L K being oblong, and I K longer than M L, -the figure will be more incurvated toward I than toward L; so that the -lines T W and V X will meet in Y before they are drawn out to so great -a length as the lines P Q and O R must be continued to, before they -will meet in S. Since therefore Y T and Y V are shorter than P S and S -V, T V must be less than O P. If these angles under T Y V and O S P are -each 1/90 part of the angle made by a perpendicular line, they are said -each to contain one degree. And the unequal length of these arches O P -and V T gives occasion to the assertion, that in passing from north to -south the degrees on the earth’s surface are not of an equal length, -but those near the pole longer than those toward the equator. For the -length of the arch on the earth lying between the two perpendiculars, -which make an angle of a degree with each other, is called the length -of a degree on the earth’s surface. - -42. THIS figure of the earth has some effect on eclipses. It has been -observed above, that sometimes the nodes of the moon’s orbit lie in a -straight line drawn from the sun to the earth; in which case the moon -will cross the plane of the earth’s motion at the new and full. But -whenever the moon passes near the plane at the full, some part of the -earth will intercept the sun’s light, and the moon shining only with -light borrow’d from the sun, when that light is prevented from falling -on any part of the moon, so much of her body will be darkened. Also -when the moon at the new is near the plane of the earth’s motion, the -inhabitants on some part of the earth will see the moon come under -the sun, and the sun thereby be covered from them either wholly or in -part. Now the figure, which we have shewn to belong to the earth, will -occasion the shadow of the earth on the moon not to be perfectly round, -but cause the diameter from east to west to be somewhat longer than -the diameter from north to south. In eclipse of the sun this figure -of the earth will make some little difference in the place, where the -sun shall appear wholly or in any given part covered. Let A B C D (in -fig. 118.) represent the earth, A C the axis whereon it turns daily, -E the center. Let F A G C represent a perfect globe inscribed within -the earth. Let H I be a line drawn through the centers of the sun and -moon, crossing the surface of the earth in K, and the surface of the -globe inscribed in L. Draw E L, which will be perpendicular to the -surface of the globe in L: and draw likewise K M, so that it shall -be perpendicular to the surface of the earth in K. Now whereas the -eclipse would appear central at L, if the earth were the globe A G C -F, and does really appear so at K; I say, the latitude of the place K -on the real earth is different from the latitude of the place L on the -globe F A G C. What is called the latitude of any place is determined -by the angle which the line perpendicular to the surface of the earth -at that place makes with the axis; the difference between this angle, -and that made by a perpendicular line or square being called the -latitude of each place. But it might here be proved, that the angle -which K M makes with M C is less, than the angle made between L E and -E C: consequently the latitude of the place K is greater, than the -latitude, which the place L would have. - -43. THE next effect, which follows from this figure of the earth, -is that gradual change in the distance of the fixed stars from the -equinoctial points, which astronomers observe. But before this can be -explained, it is necessary to say something more particular, than has -yet been done, concerning the manner of the earth’s motion round the -sun. - -44. IT has already been said, that the earth turns round each day on -its own axis, while its whole body is carried round the sun once in a -year. How these two motions are joined together may be conceived in -some degree by the motion of a bowl on the ground, where the bowl in -rouling on continually turns upon its axis, and at the same time the -whole body thereof is carried straight on. But to be more express let -A (in fig. 119) represent the sun B C D E four different situations -of the earth in its orbit moving about the sun. In all these let F -G represent the axis, about which the earth daily turns. The points -F, G are called the poles of the earth; and this axis is supposed to -keep always parallel to it self in every situation of the earth; at -least that it would do so, were it not for a minute deviation, the -cause whereof will be explained in what follows. When the earth is in -B, the half H I K will be illuminated by the sun, and the other half -H L K will be in darkness. Now if on the globe any point be taken in -the middle between the poles, this point shall describe by the motion -of the globe the circle M N, half of which is in the enlightened part -of the globe, and half in the dark part. But the earth is supposed to -move round its axis with an equable motion; therefore on this point of -the globe the sun will be seen just half the day, and be invisible the -other half. And the same will happen to every point of this circle, in -all situations of the earth during its whole revolution round the sun. -This circle M N is called the equator, of which we have before made -mention. - -45. NOW suppose any other point taken on the surface of the globe -toward the pole F, which in the diurnal revolution of the globe -shall describe the circle O P. Here it appears that more than half -this circle is enlightned by the sun, and consequently that in any -particular point of this circle the sun will be longer seen than -lie hid, that is the day will be longer than the night. Again if we -consider the same circle O P on the globe situated in D the opposite -part of the orbit from B, we shall see, that here in any place of this -circle the night will be as much longer than the day. - -46. IN these situations of the globe of earth a line drawn from the -sun to the center of the earth will be obliquely inclined toward the -axis F G. Now suppose, that such a line drawn from the sun to the -center of the earth, when in C or E, would be perpendicular to the -axis F G; in which cases the sun will shine perpendicularly upon the -equator, and consequently the line drawn from the center of the earth -to the sun will cross the equator, as it passes through the surface -of the earth; whereas in all other situations of the globe this line -will pass through the surface of the globe at a distance from the -equator either northward or southward. Now in both these cases half the -circle O P will be in the light, and half in the dark; and therefore -to every place in this circle the day will be equal to the night. Thus -it appears, that in these two opposite situations of the earth the -day is equal to the night in all parts of the globe; but in all other -situations this equality will only be found in places situated in the -very middle between the poles, that is, on the equator. - -47. THE times, wherein this universal equality between the day and -night happens, are called the equinoxes. Now it has been long observed -by astronomers, that after the earth hath set out from either equinox, -suppose from E (which will be the spring equinox, if F be the north -pole) the same equinox shall again return a little before the earth has -made a compleat revolution round the sun. This return of the equinox -preceding the intire revolution of the earth is called the precession -of the equinox, and is caused by the protuberant figure of the earth. - -49. SINCE the sun shines perpendicularly upon the equator, when the -line drawn from the sun to the center of the earth is perpendicular to -the earth’s axis, in this case the plane, which should cut through -the earth at the equator, may be extended to pass through the sun; -but it will not do so in any other position of the earth. Now let us -consider the prominent part of the earth about the equator, as a solid -ring moving with the earth round the sun. At the time of the equinoxes, -this ring will have the same kind of situation in respect of the sun, -as the orbit of the moon has, when the line of the nodes is directed -to the sun; and at all other times will resemble the moon’s orbit in -other situations. Consequently this ring, which otherwise would keep -throughout its motion parallel to it self, will receive some change in -its position from the action of the sun upon it, except only at the -time of the equinox. The manner of this change may be understood as -follows. Let A B C D (in fig. 120) represent this ring, E the center of -the earth, S the sun, A F C G a circle described in the plane of the -earth’s motion to the center E. Here A and C are the two points, in -which the earth’s equator crosses the plane of the earth’s motion; and -the time of the equinox falls out, when the straight line A C continued -would pass through the sun. Now let us recollect what was said above -concerning the moon, when her orbit was in the same situation with this -ring. From thence it will be understood, if a body were supposed to -be moving in any part of this circle A B C D, what effect the action -of the sun on the body would have toward changing the position of the -line A C. In particular H I being drawn perpendicular to S E, if the -body be in any part of this circle between A and H, or between C and I, -the line A C would be so turned, that the point A shall move toward -B, and the point C toward D; but if it were in any other part of the -circle, either between H and C, or between I and A, the line A C would -be turned the contrary way. Hence it follows, that as this solid ring -turns round the center of the earth, the parts of this ring between A -and H, and between C and I, are so influenced by the sun, that they -will endeavour, so to change the situation of the line A C as to cause -the point A to move toward B, and the point C to move toward D; but all -the parts of the ring between H and C, and between I and A, will have -the opposite tendency, and dispose the line A C to move the contrary -way. And since these last named parts are larger than the other, they -will prevail over the other, so that by the action of the sun upon this -ring, the line A C will be so turned, that A shall continually be more -and more moving toward D, and C toward B. Thus no sooner shall the sun -in its visible motion have departed from A, but the motion of the line -A C shall hasten its meeting with C, and from thence the motion of this -line shall again hasten the sun’s second conjunction with A; for as -this line so turns, that A is continually moving toward D, so the sun’s -visible motion is the same way as from S toward T. - -49. THE moon will have on this ring the like effect as the sun, and -operate on it more strongly, in the same proportion as its force on -the sea exceeded that of the sun on the same. But the effect of the -action of both luminaries will be greatly diminished by reason of this -ring’s being connected to the rest of the earth; for by this means the -sun and moon have not only this ring to move, but likewise the whole -globe of the earth, upon whose spherical part they have no immediate -influence. Beside the effect is also rendred less, by reason that the -prominent part of the earth is not collected all under the equator, -but spreads gradually from thence toward both poles. Upon the whole, -though the sun alone carries the nodes of the moon through an intire -revolution in about 19 years, the united force of both luminaries on -the prominent parts of the earth will hardly carry round the equinox in -a less space of time than 26000 years. - -50. TO this motion of the equinox we must add another consequence of -this action of the sun and moon upon the elevated parts of the earth, -that this annular part of the earth about the equator, and consequently -the earth’s axis, will twice a year and twice a month change its -inclination to the plane of the earth’s motion, and be again restored, -just as the inclination of the moon’s orbit by the action of the sun -is annually twice diminished, and as often recovers its original -magnitude. But this change is very insensible. - -51. I SHALL now finish the present chapter with our great author’s -inquiry into the figure of the secondary planets, particularly of our -moon, upon the figure of which its fluid parts will have an influence. -The moon turns always the same side towards the earth, and consequently -revolves but once round its axis in the space of an entire month; for -a spectator placed without the circle, in which the moon moves, would -in that time observe all the parts of the moon successively to pass -once before his view and no more, that is, that the whole globe of the -moon has turned once round. Now the great slowness of this motion will -render the centrifugal force of the parts of the waters very weak, so -that the figure of the moon cannot, as in the earth, be much affected -by this revolution upon its axis: but the figure of those waters are -made different from spherical by another cause, viz. the action of -the earth upon them; by which they will be reduced to an oblong oval -form, whose axis prolonged would pass through the earth; for the same -reason, as we have above observed, that the waters of the earth would -take the like figure, if they had moved so slowly, as to keep pace with -the moon. And the solid part of the moon must correspond with this -figure of the fluid part: but this elevation of the parts of the moon -is nothing near so great as is the protuberance of the earth at the -equator, for it will not exceed 93 english feet. - -52. The waters of the moon will have no tide, except what will arise -from the motion of the moon round the earth. For the conversion of the -moon about her axis is equable, whereby the inequality in the motion -round the earth discovers to us at some times small parts of the moon’s -surface towards the east or west, which at other times lie hid; and -as the axis, whereon the moon turns, is oblique to her motion round -the earth, sometimes small parts of her surface toward the north, and -sometimes the like toward the south are visible, which at other times -are out of sight. These appearances make what is called the libration -of the moon, discovered by HEVELIUS. But now as the axis of the oval -figure of the waters will he pointed towards the earth, there must -arise from hence some fluctuation in them; and beside, by the change of -the moon’s distance from the earth, they will not always have the very -same height. - -[Illustration] - -[Illustration] - - - - -~BOOK III~. - - - - -~CHAP~ I. - -Concerning the cause of COLOURS inherent in the LIGHT. - - -AFTER this view which has been taken of Sir ISAAC NEWTON’S mathematical -principles of philosophy, and the use he has made of them, in -explaining the system of the world, &c. the course of my design directs -us to turn our eyes to that other philosophical work, his treatise of -Optics, in which we shall find our great author’s inimitable genius -discovering it self no less, than in the former; nay perhaps even -more, since this work gives as many instances of his singular force -of reasoning, and of his unbounded invention, though unassisted in -great measure by those rules and general precepts, which facilitate -the invention of mathematical theorems. Nor yet is this work inferior -to the other in usefulness; for as that has made known to us one great -principle in nature, by which the celestial motions are continued, and -by which the frame of each globe is preserved; so does this point out -to us another principle no less universal, upon which depends all those -operations in the smaller parts of matter, for whose sake the greater -frame of the universe is erected; all those immense globes, with which -the whole heavens are filled, being without doubt only design’d as so -many convenient apartments for carrying on the more noble operations of -nature in vegetation and animal life. Which single consideration gives -abundant proof of the excellency of our author’s choice, in applying -himself carefully to examine the action between light and bodies, so -necessary in all the varieties of these productions, that none of them -can be successfully promoted without the concurrence of heat in a -greater or less degree. - -2. ’TIS true, our author has not made so full a discovery of the -principle, by which this mutual action between light and bodies is -caused; as he has in relation to the power, by which the planets are -kept in their courses: yet he has led us to the very entrance upon it, -and pointed out the path so plainly which must be followed to reach it; -that one may be bold to say, whenever mankind shall be blessed with -this improvement of their knowledge, it will be derived so directly -from the principles laid down by our author in this book, that the -greatest share of the praise due to the discovery will belong to him. - -3. IN speaking of the progress our author has made, I shall distinctly -pursue three things, the two first relating to the colours of natural -bodies: for in the first head shall be shewn, how those colours are -derived from the properties of the light itself; and in the second upon -what properties of the bodies they depend: but the third head of my -discourse shall treat of the action of bodies upon light in refracting, -reflecting, and inflecting it. - -4. THE first of these, which shall be the business of the present -chapter, is contained in this one proposition: that the sun’s direct -light is not uniform in respect of colour, not being disposed in every -part of it to excite the idea of whiteness, which the whole raises; but -on the contrary is a composition of different kinds of rays, one sort -of which if alone would give the sense of red, another of orange, a -third of yellow, a fourth of green, a fifth of light blue, a sixth of -indigo, and a seventh of a violet purple; that all these rays together -by the mixture of their sensations impress upon the organ of sight -the sense of whiteness, though each ray always imprints there its own -colour; and all the difference between the colours of bodies when -viewed in open day light arises from this, that coloured bodies do not -reflect all the sorts of rays falling upon them in equal plenty, but -some sorts much more copiously than others; the body appearing of that -colour, of which the light coming from it is most composed. - -5. THAT the light of the sun is compounded, as has been said, is proved -by refracting it with a prism. By a prism I here mean a glass or other -body of a triangular form, such as is represented in fig. 121. But -before we proceed to the illustration of the proposition we have just -now laid down, it will be necessary to spend a few words in explaining -what is meant by the refraction of light; as the design of our present -labour is to give some notion of the subject, we are engaged in, to -such as are not versed in the mathematics. - -6. IT is well known, that when a ray of light passing through the air -falls obliquely upon the surface of any transparent body, suppose water -or glass, and enters it, the ray will not pass on in that body in the -same line it described through the air, but be turned off from the -surface, so as to be less inclined to it after passing it, than before. -Let A B C D (in fig. 122.) represent a portion of water, or glass, A -B the surface of it, upon which the ray of light E F falls obliquely; -this ray shall not go right on in the course delineated by the line -F G, but be turned off from the surface A B into the line F H, less -inclined to the surface A B than the line E F is, in which the ray is -incident upon that surface. - -7. ON the other hand, when the light passes out of any such body into -the air, it is inflected the contrary way, being after its emergence -rendred more oblique to the surface it passes through, than before. -Thus the ray F H, when it goes out of the surface C D, will be turned -up towards that surface, going out into the air in the line H I. - -8. THIS turning of the light out of its way, as it passes from one -transparent body into another is called its refraction. Both these -cases may be tried by an easy experiment with a bason and water. For -the first case set an empty bason in the sunshine or near a candle, -making a mark upon the bottom at the extremity of the shadow cast -by the brim of the bason, then by pouring water into the bason you -will observe the shadow to shrink, and leave the bottom of the bason -enlightned to a good distance from the mark. Let A B C (in fig. 123.) -denote the empty bason, E A D the light shining over the brim of it, -so that all the part A B D be shaded. Then a mark being made at D, if -water be poured into the bason (as in fig. 124.) to F G, you shall -observe the light, which before went on to D, now to come much short of -the mark D, falling on the bottom in the point H, and leaving the mark -D a good way within the enlightened part; which shews that the ray E A, -when it enters the water at I, goes no longer straight forwards, but is -at that place incurvated, and made to go nearer the perpendicular. The -other case may be tryed by putting any small body into an empty bason, -placed lower than your eye, and then receding from the bason, till you -can but just see the body over the brim. After which, if the bason be -filled with water, you shall presently observe the body to be visible, -though you go farther off from the bason. Let A B C (in fig. 125.) -denote the bason as before, D the body in it, E the place of your eye, -when the body is seen just over the edge A, while the bason is empty. -If it be then filled with water, you will observe the body still to be -visible, though you take your eye farther off. Suppose you see the body -in this case just over the brim A, when your eye is at F, it is plain -that the rays of light, which come from the body to your eye have not -come straight on, but are bent at A, being turned downwards, and more -inclined to the surface of the water, between A and your eye at F, than -they are between A and the body D. - -9. THIS we hope is sufficient to make all our readers apprehend, -what the writers of optics mean, when they mention the refraction -of the light, or speak of the rays of light being refracted. We -shall therefore now go on to prove the assertion advanced in the -forementioned proposition, in relation to the different kinds of -colours, that the direct light of the sun exhibits to our sense: which -may be done in the following manner. - -10. IF a room be darkened, and the sun permitted to shine into it -through a small hole in the window shutter, and be made immediately to -fall upon a glass prism, the beam of light shall in passing through -such a prism be parted into rays, which exhibit all the forementioned -colours. In this manner if A B (in fig. 126) represent the window -shutter; C the hole in it; D E F the prism; Z Y a beam of light coming -from the sun, which passes through the hole, and falls upon the prism -at Y, and if the prism were removed would go on to X, but in entring -the surface B F of the glass it shall be turned off, as has been -explained, into the course Y W falling upon the second surface of the -prism D F in W, going out of which into the air it shall be again -farther inflected. Let the light now, after it has passed the prism, be -received upon a sheet of paper held at a proper distance, and it shall -paint upon the paper the picture, image, or spectrum L M of an oblong -figure, whose length shall much exceed its breadth; though the figure -shall not be oval, the ends L and M being semicircular and the sides -straight. But now this figure will be variegated with colours in this -manner. From the extremity M to some length, suppose to the line _n -o_, it shall be of an intense red; from _n o_ to _p q_ it shall be an -orange; from _p q_ to _r s_ it shall be yellow; from thence to _t u_ it -shall be green; from thence to _w x_ blue; from thence to _y z_ indigo; -and from thence to the end violet. - -11. THUS it appears that the sun’s white light by its passage through -the prism, is so changed as now to be divided into rays, which exhibit -all these several colours. The question is, whether the rays while -in the sun’s beam before this refraction possessed these properties -distinctly; so that some part of that beam would without the rest have -given a red colour, and another part alone have given an..orange, -&c. That this is possible to be the case, appears from hence; that if -a convex glass be placed between the paper and the prism, which may -collect all the rays proceeding out of the prism into its focus, as a -burning glass does the sun’s direct rays; and if that focus fall upon -the paper, the spot formed by such a glass upon the paper shall appear -white, just like the sun’s direct light. - -[Illustration] - -The rest remaining as before, let P Q. (in fig. 127.) be the convex -glass, causing the rays to meet upon the paper H G I K in the point N, -I say that point or rather spot of light shall appear white, without -the least tincture of any colour. But it is evident that into this -spot are now gathered all those rays, which before when separate gave -all those different colours; which shews that whiteness may be made by -mixing those colours: especially if we consider, it can be proved that -the glass P Q does not alter the colour of the rays which pass through -it. Which is done thus: if the paper be made to approach the glass P -Q, the colours will manifest themselves as far as the magnitude of the -spectrum, which the paper receives, will permit. Suppose it in the -situation _h g i k_, and that it then receive the spectrum _l m_, this -spectrum shall be much smaller, than if the glass P Q were removed, -and therefore the colours cannot be so much separated; but yet the -extremity _m_ shall manifestly appear red, and the other extremity _l_ -shall be blue; and these colours as well as the intermediate ones shall -discover themselves more perfectly, the farther the paper is removed -from N, that is, the larger the spectrum is: the same thing happens, -if the paper be removed farther off from P Q than N. Suppose into the -position θ γ η ϰ, the spectrum λ μ painted upon it shall again discover -its colours, and that more distinctly, the farther the paper is -removed, but only in an inverted order: for as before, when the paper -was nearer the convex glass, than at N, the upper part of the image was -blue, and the under red; now the upper part shall be red, and the under -blue: because the rays cross at N. - -12. NAY farther that the whiteness at the focus N, is made by the union -of the colours may be proved without removing the paper out of the -focus, by intercepting with any opake body part of the light near the -glass; for if the under part, that is the red, or more properly the -red-making rays, as they are styled by our author, are intercepted, -the spot shall take a bluish hue; and if more of the inferior rays are -cut off, so that neither the red-making nor orange-making rays, and if -you please the yellow-making rays likewise, shall fall upon the spot; -then shall the spot incline more and more to the remaining colours. -In like manner if you cut off the upper part of the rays, that is the -violet coloured or indigo-making rays, the spot shall turn reddish, and -become, more so, the more of those opposite colours are intercepted. - -13. THIS I think abundantly proves that whiteness may be produced by a -mixture of all the colours of the spectrum. At least there is but one -way of evading the present arguments, which is, by asserting that the -rays of light after passing the prism have no different properties to -exhibit this or the other colour, but are in that respect perfectly -homogeneal, so that the rays which pass to the under and red part of -the image do not differ in any properties whatever from those, which -go to the upper and violet part of it; but that the colours of the -spectrum are produced only by some new modifications of the rays, made -at their incidence upon the paper by the different terminations of -light and shadow: if indeed this assertion can be allowed any place, -after what has been said; for it seems to be sufficiently obviated -by the latter part of the preceding experiment, that by intercepting -the inferior part of the light, which comes from the prism, the white -spot shall receive a bluish cast, and by stopping the upper part the -spot shall turn red, and in both cases recover its colour, when the -intercepted light is permitted to pass again; though in all these -trials there is the like termination of light and shadow. However our -author has contrived some experiments expresly to shew the absurdity of -this supposition; all which he has explained and enlarged upon in so -distinct and expressive a manner, that it would be wholly unnecessary -to repeat them in this place[280]. I shall only mention that of them, -which may be tried in the experiment before us. If you draw upon -the paper H G I K, and through the spot N, the straight line _w x_ -parallel to the horizon, and then if the paper be much inclined into -the situation _r s v t_ the line _w x_ still remaining parallel to -the horizon, the spot N shall lose its whiteness and receive a blue -tincture; but if it be inclined as much the contrary way, the same -spot shall exchange its white colour for a reddish dye. All which can -never be accounted for by any difference in the termination of the -light and shadow, which here is none at all; but are easily explained -by supposing the upper part of the rays, whenever they enter the eye, -disposed to give the sensation of the dark colours blue, indigo and -violet; and that the under part is fitted to produce the bright colours -yellow, orange and red: for when the paper is in the situation _r s t -u_, it is plain that the upper part of the light falls more directly -upon it, than the under part, and therefore those rays will be most -plentifully reflected from it; and by their abounding in the reflected -light will cause it to incline to their colour. Just so when the paper -is inclined the contrary way, it will receive the inferior rays most -directly, and therefore ting the light it reflects with their colour. - -14. IT is now to be proved that these dispositions of the rays of -light to produce some one colour and some another, which manifest -themselves after their being refracted, are not wrought by any action -of the prism upon them, but are originally inherent in those rays; and -that the prism only affords each species an occasion of shewing its -distinct quality by separating them one from another, which before, -while they were blended together in the direct beam of the sun’s light, -lay conceal’d. But that this is so, will be proved, if it can be shewn -that no prism has any power upon the rays, which after their passage -through one prism are rendered uncompounded and contain in them but one -colour, either to divide that colour into several, as the sun’s light -is divided, or so much as to change it into any other colour. This will -be proved by the following experiment[281]. The same thing remaining, -as in the first experiment, let another prism N O (in fig. 128.) be -placed either immediately, or at some distance after the first, in a -perpendicular posture, so that it shall refract the rays issuing from -the first sideways. Now if this prism could divide the light falling -upon it into coloured rays, as the first has done, it would divide the -spectrum breadthwise into colours, as before it was divided lengthwise; -but no such thing is observed. If L M were the spectrum, which the -first prism D E F would paint upon the paper H G I K; P Q lying in an -oblique posture shall be the spectrum projected by the second, and -shall be divided lengthwise into colours corresponding to the colours -of the spectrum L M, and occasioned like them by the refraction of the -first prism, but its breadth shall receive no such division; on the -contrary each colour shall be uniform from side to side, as much as in -the spectrum L M, which proves the whole assertion. - -15. THE same is yet much farther confirmed by another experiment. -Our author teaches that the colours of the spectrum L M in the first -experiment are yet compounded, though not so much as in the sun’s -direct light. He shews therefore how, by placing the prism at a -distance from the hole, and by the use of a convex glass, to separate -the colours of the spectrum, and make them uncompounded to any degree -of exactness[282]. And he shews when this is done sufficiently, if -you make a small hole in the paper whereon the spectrum is received, -through which any one sort of rays may pass, and then let that coloured -ray fall so upon a prism, as to be refracted by it, it shall in no case -whatever change its colour; but shall always retain it perfectly as at -first, however it be refracted[283]. - -16. NOR yet will these colours after this full separation of them -suffer any change by reflection from bodies of different colours; on -the other hand they make all bodies placed in these colours appear of -the colour which falls upon them[284]: for minium in red light will -appear as in open day light; but in yellow light will appear yellow; -and which is more extraordinary, in green light will appear green, in -blue, blue; and in the violet-purple coloured light will appear of a -purple colour; in like manner verdigrease, or blue bise, will put on -the appearance of that colour, in which it is placed; so that neither -bise placed in the red light shall be able to give that light the least -blue tincture, or any other different from red; nor shall minium in -the indigo or violet light exhibit the least appearance of red, or any -other colour distinct from that it is placed in. The only difference -is, that each of these bodies appears most luminous and bright in the -colour, which corresponds with that it exhibits in the day light, and -dimmed in the colours most remote from that; that is, though minium and -bise placed in blue light shall both appear blue, yet the bise shall -appear of a bright blue, and the minium of a dusky and obscure blue: -but if minium and bise be compared together in red light, the minium -shall afford a brisk red, the bise a duller colour, though of the same -species. - -17. AND this not only proves the immutability of all these simple -and uncompounded colours; but likewise unfolds the whole mystery, -why bodies appear in open day-light of such different colours, it -consisting in nothing more than this, that whereas the white light of -the day is composed of all sorts of colours, some bodies reflect the -rays of one sort in greater abundance than the rays of any other[285]. -Though it appears by the fore-cited experiment, that almost all these -bodies reflect some portion of the rays of every colour, and give the -sense of particular colours only by the predominancy of some sorts of -rays above the rest. And what has before been explained of composing -white by mingling all the colours of the spectrum together shews -clearly, that nothing more is required to make bodies look white, -than a power to reflect indifferently rays of every colour. But this -will more fully appear by the following method: if near the coloured -spectrum in our first experiment a piece of white paper be so held, as -to be illuminated equally by all the parts of that spectrum, it shall -appear white; whereas if it be held nearer to the red end of the image, -than to the other, it shall turn reddish; if nearer the blue end, it -shall seem bluish[286]. - -18. OUR indefatigable and circumspect author farther examined his -theory by mixing the powders which painters use of several colours, in -order if possible to produce a white powder by such a composition[287]. -But in this he found some difficulties for the following reasons. Each -of these coloured powders reflects but part of the light, which is -cast upon them; the red powders reflecting little green or blue, and -the blue powders reflecting very little red or yellow, nor the green -powders reflecting near so much of the red or indigo and purple, as -of the other colours: and besides, when any of these are examined in -homogeneal light, as our author calls the colours of the prism, when -well separated, though each appears more bright and luminous in its -own day-light colour, than in any other; yet white bodies, suppose -white paper for instance, in those very colours exceed these coloured -bodies themselves in brightness; so that white bodies reflect not only -more of the whole light than coloured bodies do in the day-light, but -even more of that very colour which they reflect most copiously. All -which considerations make it manifest that a mixture of these will not -reflect so great a quantity of light, as a white body of the same size; -and therefore will compose such a colour as would result from a mixture -of white and black, such as are all grey and dun colours, rather than a -strong white. Now such a colour he compounded of certain ingredients, -which he particularly sets down, in so much that when the composition -was strongly illuminated by the sun’s direct beams, it would appear -much whiter than even white paper, if considerably shaded. Nay he -found by trials how to proportion the degree of illumination of the -mixture and paper, so that to a spectator at a proper distance it -could not well be determined which was the more perfect colour; as he -experienced not only by himself, but by the concurrent opinion of a -friend, who chanced to visit him while he was trying this experiment. -I must not here omit another method of trying the whiteness of such a -mixture, proposed in one of our author’s letters on this subject[288]: -which is to enlighten the composition by a beam of the sun let into a -darkened room, and then to receive the light reflected from it upon a -piece of white paper, observing whether the paper appears white by that -reflection; for if it does, it gives proof of the composition’s being -white; because when the paper receives the reflection from any coloured -body, it looks of that colour. Agreeable to this is the trial he made -upon water impregnated with soap, and agitated into a froth[289]: -for when this froth after some short time exhibited upon the little -bubbles, which composed it, a great variety of colours, though these -colours to a spectator at a small distance discover’d themselves -distinctly; yet when the eye was so far removed, that each little -bubble could no longer be distinguished, the whole froth by the mixture -of all these colours appeared intensly white. - -19. OUR author having fully satisfied himself by these and many other -experiments, what the result is of mixing together all the prismatic -colours; he proceeds in the next place to examine, whether this -appearance of whiteness be raised by the rays of these different kinds -acting so, when they meet, upon one another, as to cause each of them -to impress the sense of whiteness upon the optic nerve; or whether each -ray does not make upon the organ of sight the same impression, as when -separate and alone; so that the idea of whiteness is not excited by the -impression from any one part of the rays, but results from the mixture -of all those different sensations. And that the latter sentiment is the -true one, he evinces by undeniable experiments. - -20. IN particular the foregoing experiment[290], wherein the convex -glass was used, furnishes proofs of this: in that when the paper is -brought into the situation θ γ η ϰ, beyond, beyond N the colours, that -at N disappeared, begin to emerge again; which shews that by mingling -at N they did not lose their colorific qualities, though for some -reason they lay concealed. This farther appears by that part of the -experiment, when the paper, while in the focus, was directed to be -enclined different ways; for when the paper was in such a situation, -that it must of necessity reflect the rays, which before their arrival -at the point N would have given a blue colour, those rays in this -very point itself by abounding in the reflected light tinged it with -the same colour; so when the paper reflects most copiously the rays, -which before they come to the point N exhibit redness, those same rays -tincture the light reflected by the paper from that very point with -their own proper colour. - -21. THERE is a certain condition relating to sight, which affords an -opportunity of examining this still more fully: it is this, that the -impressions of light remain some short space upon the eye; as when a -burning coal is whirl’d about in a circle, if the motion be very quick, -the eye shall not be able to distinguish the coal, but shall see an -entire circle of fire. The reason of which appearance is, that the -impression made by the coal upon the eye in any one situation is not -worn out, before the coal returns again to the same place, and renews -the sensation. This gives our author the hint to try, whether these -colours might not be transmitted successively to the eye so quick, -that no one of the colours should be distinctly perceived, but the -mixture of the sensations should produce a uniform whiteness; when the -rays could not act upon each other, because they never should meet, -but come to the eye one after another. And this thought he executed -by the following expedient[291]. He made an instrument in shape like -a comb, which he applied near the convex glass, so that by moving it -up and down slowly the teeth of it might intercept sometimes one and -sometimes another colour; and accordingly the light reflected from the -paper, placed at N, should change colour continually. But now when the -comb-like instrument was moved very quick, the eye lost all preception -of the distinct colours, which came to it from time to time, a perfect -whiteness resulting from the mixture of all those distinct impressions -in the sensorium. Now in this case there can be no suspicion of the -several coloured rays acting upon one another, and making any change in -each other’s manner of affecting the eye, seeing they do not so much as -meet together there. - -22. OUR author farther teaches us how to view the spectrum of colours -produced in the first experiment with another prism, so that it shall -appear to the eye under the shape of a round spot and perfectly -white[292]. And in this case if the comb be used to intercept -alternately some of the colours, which compose the spectrum, the round -spot shall change its colour according to the colours intercepted; but -if the comb be moved too swiftly for those changes to be distinctly -perceived, the spot shall seem always white, as before[293]. - -23. BESIDES this whiteness, which results from an universal composition -of all sorts of colours, our author particularly explains the effects -of other less compounded mixtures; some of which compound other colours -like some of the simple ones, but others produce colours different -from any of them. For instance, a mixture of red and yellow compound -a colour like in appearance to the orange, which in the spectrum lies -between them; as a composition of yellow and blue is made use of in all -dyes to make a green. But red and violet purple compounded make purples -unlike to any of the prismatic colours, and these joined with yellow -or blue make yet new colours. Besides one rule is here to be observed, -that when many different colours are mixed, the colour which arises -from the mixture grows languid and degenerates into whiteness. So when -yellow green and blue are mixed together, the compound will be green; -but if to this you add red and purple, the colour shall first grow dull -and less vivid, and at length by adding more of these colours it shall -turn to whiteness, or some other colour[294]. - -24. ONLY here is one thing remarkable of those compounded colours, -which are like in appearance to the simple ones; that the simple -ones when viewed through a prism shall still retain their colour, -but the compounded colours seen through such a glass shall be parted -into the simple ones of which they are the aggregate. And for this -reason any body illuminated by the simple light shall appear through -a prism distinctly, and have its minutest parts observable, as may -easily be tried with flies, or other such little bodies, which have -very small parts; but the same viewed in this manner when enlighten’d -with compounded colours shall appear confused, their smallest parts -not being distinguishable. How the prism separates these compounded -colours, as likewise how it divides the light of the sun into its -colours, has not yet been explained; but is reserved for our third -chapter. - -25. IN the mean time what has been said, I hope, will suffice to -give a taste of our author’s way of arguing, and in some measure to -illustrate the proposition laid down in this chapter. - -26. THERE are methods of separating the heterogeneous rays of the -sun’s light by reflection, which perfectly conspire with and confirm -this reasoning. One of which ways may be this. Let A B (in fig. 129) -represent the window shutter of a darkened room; C a hole to let in -the sun’s rays; D E F, G H I two prisms so applied together, that the -sides E F and G I be contiguous, and the sides D F, G H parallel; by -this means the light will pass through them without any separation -into colours: but if it be afterwards received by a third prism I K L, -it shall be divided so as to form upon any white body P Q the usual -colours, violet at _m_, blue at _n_, green at _o_, yellow at _r_, and -red at _s_. But because it never happens that the two adjacent surfaces -E F and G I perfectly touch, part only of the light incident upon the -surface E F shall be transmitted, and part shall be reflected. Let now -the reflected part be received by a fourth prism Δ Θ Λ, and passing -through it paint upon a white body Ζ Γ the colours of the prism, red -at _t_, yellow at _u_, green at _w_, blue at _x_, violet at _y_. If -the prisms D E F, G H I be slowly turned about while they remain -contiguous, the colours upon the body P Q shall not sensibly change -their situation, till such time as the rays become pretty oblique to -the surface E F; but then the light incident upon the surface E F shall -begin to be wholly reflected. And first of all the violet light shall -be wholly reflected, and thereupon will disappear at _m_, appearing -instead thereof at _y_, and increasing the violet light falling there, -the other colours remaining as before. If the prisms D E F, G H I be -turned a little farther about, that the incident rays become yet more -inclined to the surface E F, the blue shall be totally reflected, and -shall disappear in _n_, but appear at _x_ by making the colour there -more intense. And the same may be continued, till all the colours are -successively removed from the surface P Q to Ζ Γ. But in any case, -suppose when the violet and the blue have forsaken the surface P Q, -and appear upon the surface Ζ Γ, Ζ Γ, the green, yellow, and red only -remaining upon the surface P Q; if the light be received upon a paper -held any where in its whole passage between the light’s coming out of -the prisms D E F, G I H and its incidence upon the prism I K L, it -shall appear of the colour compounded of all the colours seen upon P -Q; and the reflected ray, received upon a piece of white paper held -any where between the prisms D E F and Δ Θ Σ shall exhibit the colour -compounded of those the surface P Q is deprived of mixed with the sun’s -light: whereas before any of the light was reflected from the surface -E F, the rays between the prisms G H I and I K L would appear white; -as will likewise the reflected ray both before and after the total -reflection, provided the difference of refraction by the surfaces D F -and D E be inconsiderable. I call here the sun’s light white, as I have -all along done; but it is more exact to ascribe to it something of a -yellowish tincture, occasioned by the brighter colours abounding in it; -which caution is necessary in examining the colours of the reflected -beam, when all the violet and blue are in it: for this yellowish turn -of the sun’s light causes the blue not to be quite so visible in it, -as it should be, were the light perfectly white; but makes the beam of -light incline rather towards a pale white. - - - - -~CHAP~. II. - -Of the properties of BODIES, upon which their COLOURS depend. - - -AFTER having shewn in the last chapter, that the difference between -the colours of bodies viewed in open day-light is only this, that some -bodies are disposed to reflect rays of one colour in the greatest -plenty, and other bodies rays of some other colour; order now requires -us to examine more particularly into the property of bodies, which -gives them this difference. But this our author shews to be nothing -more, than the different magnitude of the particles, which compose each -body: this I question not will appear no small paradox. And indeed -this whole chapter will contain scarce any assertions, but what will -be almost incredible, though the arguments for them are so strong and -convincing, that they force our assent. In the former chapter have -been explained properties of light, not in the least thought of before -our author’s discovery of them; yet are they not difficult to admit, -as soon as experiments are known to give proof of their reality; but -some of the propositions to be stated here will, I fear, be accounted -almost past belief; notwithstanding that the arguments, by which they -are established are unanswerable. For it is proved by our author, that -bodies are rendered transparent by the minuteness of their pores, and -become opake by having them large; and more, that the most transparent -body by being reduced to a great thinness will become less pervious to -the light. - -2. BUT whereas it had been the received opinion, and yet remains so -among all who have not studied this philosophy, that light is reflected -from bodies by its impinging against their solid parts, rebounding from -them, as a tennis ball or other elastic substance would do, when struck -against any hard and resisting surface; it will be proper to begin with -declaring our author’s sentiment concerning this, who shews by many -arguments that reflection cannot be caused by any such means[295]: some -few of his proofs I shall set down, referring the reader to our author -himself for the rest. - -3. IT is well known, that when light falls upon any transparent body, -glass for instance, part of it is reflected and part transmitted; -for which it is ready to account, by saying that part of the light -enters the pores of the glass, and part impinges upon its solid -parts. But when the transmitted light arrives at the farther surface -of the glass, in passing out of glass into air there is as strong a -reflection caused, or rather something stronger. Now it is not to -be conceived, how the light should find as many solid parts in the -air to strike against as in the glass, or even a greater number of -them. And to augment the difficulty, if water be placed behind the -glass, the reflection becomes much weaker. Can we therefore say, that -water has fewer solid parts for the light to strike against, than the -air? And if we should, what reason can be given for the reflection’s -being stronger, when the air by the air-pump is removed from behind -the glass, than when the air receives the rays of light. Besides the -light may be so inclined to the hinder surface of the glass, that it -shall wholly be reflected, which happens when the angle which the -ray makes with the surface does not exceed about 49⅓ degrees; but if -the inclination be a very little increased, great part of the light -will be transmitted; and how the light in one case should meet with -nothing but the solid parts of the air, and by so small a change of -its inclination find pores in great plenty, is wholly inconceivable. -It cannot be said, that the light is reflected by striking against -the solid parts of the surface of the glass; because without making -any change in that surface, only by placing water contiguous to it -instead of air, great part of that light shall be transmitted, which -could find no passage through the air. Moreover in the last experiment -recited in the preceding chapter, when by turning the prisms D E F, G -H I, the blue light became wholly reflected, while the rest was mostly -transmitted, no possible reason can be assigned, why the blue-making -rays should meet with nothing but the solid parts of the air between -the prisms, and the rest of the light in the very same obliquity find -pores in abundance. Nay farther, when two glasses touch each other, no -reflection at all is made; though it does not in the least appear, -how the rays should avoid the solid parts of glass, when contiguous -to other glass, any more than when contiguous to air. But in the last -place upon this supposition it is not to be comprehended, how the most -polished substances could reflect the light in that regular manner we -find they do; for when a polished looking glass is covered over with -quicksilver, we cannot suppose the particles of light so much larger -than those of the quicksilver that they should not be scattered as -much in reflection, as a parcel of marbles thrown down upon a rugged -pavement. The only cause of so uniform and regular a reflection must be -some more secret cause, uniformly spread over the whole surface of the -glass. - -4. BUT now, since the reflection of light from bodies does not depend -upon its impinging against their solid parts, some other reason must -be sought for. And first it is past doubt that the least parts of -almost all bodies are transparent, even the microscope shewing as -much[296]; besides that it may be experienced by this method. Take any -thin plate of the opakest body, and apply it to a small hole designed -for the admission of light into a darkened room; however opake that -body may seem in open day-light, it shall under these circumstances -sufficiently discover its transparency, provided only the body be very -thin. White metals indeed do not easily shew themselves transparent in -these trials, they reflecting almost all the light incident upon them -at their first superficies; the cause of which will appear in what -follows[297]. But yet these substances, when reduced into parts of -extraordinary minuteness by being dissolved in aqua fortis or the like -corroding liquors do also become transparent. - -5. SINCE therefore the light finds free passage through the least -parts of bodies, let us consider the largeness of their pores, and -we shall find, that whenever a ray of light has passed through any -particle of a body, and is come to its farther surface, if it finds -there another particle contiguous, it will without interruption pass -into that particle; just as light will pass through one piece of glass -into another piece in contact with it without any impediment, or any -part being reflected: but as the light in passing out of glass, or any -other transparent body, shall part of it be reflected back, if it enter -into air or other transparent body of a different density from that -it passes out of; the same thing will happen in the light’s passage -through any particle of a body, whenever at its exit out of that -particle it meets no other particle contiguous, but must enter into a -pore, for in this case it shall not all pass through, but part of it -be reflected back. Thus will the light, every time it enters a pore, -be in part reflected; so that nothing more seems necessary to opacity, -than that the particles, which compose any body, touch but in very -few places, and that the pores of it are numerous and large, so that -the light may in part be reflected from it, and the other part, which -enters too deep to be returned out of the body, by numerous reflections -may be stifled and lost[298]; which in all probability happens, as -often as it impinges against the solid part of the body, all the light -which does so not being reflected back, but stopt, and deprived of any -farther motion[299]. - -6. THIS notion of opacity is greatly confirmed by the observation, -that opake bodies become transparent by filling up the pores with any -substance of near the same density with their parts. As when paper -is wet with water or oyl; when linnen cloth is either dipt in water, -oyled, or varnished; or the oculus mundi stone steeped in water[300]. -All which experiments confirm both the first assertion, that light is -not reflected by striking upon the solid parts of bodies; and also -the second, that its passage is obstructed by the reflections it -undergoes in the pores; since we find it in these trials to pass in -greater abundance through bodies, when the number of their solid parts -is increased, only by taking away in great measure those reflections; -which filling the pores with a substance of near the same density -with the parts of the body will do. Besides as filling the pores of a -dark body makes it transparent; so on the other hand evacuating the -pores of a body transparent, or separating the parts of such a body, -renders it opake. As salts or wet paper by being dried, glass by being -reduced to powder or the surface made rough; and it is well known that -glass vessels discover cracks in them by their opacity. Just so water -itself becomes impervious to the light by being formed into many small -bubbles, whether in froth, or by being mixed and agitated with any -quantity of a liquor with which it will not incorporate, such as oyl -of turpentine, or oyl olive. - -7. A CERTAIN electrical experiment made by Mr. HAUKSBEE may not perhaps -be useless to clear up the present speculation, by shewing that -something more is necessary besides mere porosity for transmitting -freely other fine substances. The experiment is this; that a glass cane -rubbed till it put forth its electric quality would agitate leaf brass -inclosed under a glass vessel, though not at so great a distance, as if -no body had intervened; yet the same cane would lose all its influence -on the leaf brass by the interposition of a piece of the finest muslin, -whose pores are immensely larger and more patent than those of glass. - -8. THUS I have endeavoured to smooth my way, as much as I could, to -the unfolding yet greater secrets in nature; for I shall now proceed -to shew the reason why bodies appear of different colours. My reader -no doubt will be sufficiently surprized, when I inform him that the -knowledge of this is deduced from that ludicrous experiment, with which -children divert themselves in blowing bubbles of water made tenacious -by the solution of soap. And that these bubbles, as they gradually grow -thinner and thinner till they break, change successively their colours -from the same principle, as all natural bodies preserve theirs. - -9. OUR author after preparing water with soap, so as to render it very -tenacious, blew it up into a bubble, and placing it under a glass, -that it might not be irregularly agitated by the air, observed as -the water by subsiding changed the thickness of the bubble, making -it gradually less and less till the bubble broke; there successively -appeared colours at the top of the bubble, which spread themselves -into rings surrounding the top and descending more and more, till they -vanished at the bottom in the same order in which they appeared[301]. -The colours emerged in this order: first red, then blue; to which -succeeded red a second time, and blue immediately followed; after that -red a third time, succeeded by blue; to which followed a fourth red, -but succeeded by green; after this a more numerous order of colours, -first red, then yellow, next green, and after that blue, and at last -purple; then again red, yellow, green, blue, violet followed each other -in order; and in the last place red, yellow, white, blue; to which -succeeded a dark spot, which reflected scarce any light, though our -author found it did make some very obscure reflection, for the image of -the sun or a candle might be faintly discerned upon it; and this last -spot spread itself more and more, till the bubble at last broke. These -colours were not simple and uncompounded colours, like those which -are exhibited by the prism, when due care is taken to separate them; -but were made by a various mixture of those simple colours, as will -be shewn in the next chapter: whence these colours, to which I have -given the name of blue, green, or red, were not all alike, but differed -as follows. The blue, which appeared next the dark spot, was a pure -colour, but very faint, resembling the sky-colour; the white next to -it a very strong and intense white, brighter much than the white, which -the bubble reflected, before any of the colours appeared. The yellow -which preceded this was at first pretty good, but soon grew dilute; -and the red which went before the yellow at first gave a tincture of -scarlet inclining to violet, but soon changed into a brighter colour; -the violet of the next series was deep with little or no redness in -it; the blue a brisk colour, but came much short of the blue in the -next order; the green was but dilute and pale; the yellow and red were -very bright and full, the best of all the yellows which appeared among -any of the colours: in the preceding orders the purple was reddish, -but the blue, as was just now said, the brightest of all; the green -pretty lively better than in the order which appeared before it, though -that was a good willow green; the yellow but small in quantity, though -bright; the red of this order not very pure: those which appeared -before yet more obscure, being very dilute and dirty; as were likewise -the three first blues. - -10. NOW it is evident, that these colours arose at the top of the -bubble, as it grew by degrees thinner and thinner: but what the express -thickness of the bubble was, where each of these colours appeared upon -it, could not be determined by these experiments; but was found by -another means, viz. by taking the object glass of a long telescope, -which is in a small degree convex, and placing it upon a flat glass, -so as to touch it in one point, and then water being put between them, -the same colours appeared as in the bubble, in the form of circles or -rings surrounding the point where the glasses touched, which appeared -black for want of any reflection from it, like the top of the bubble -when thinnest[302]: next to this spot lay a blue circle, and next -without that a white one; and so on in the same order as before, -reckoning from the dark spot. And henceforward I shall speak of each -colour, as being of the first, second, or any following order, as it is -the first, second, or any following one, counting from the black spot -in the center of these rings; which is contrary to the order in which -I must have mentioned them, if I should have reputed them the first, -second, or third, &c. in order, as they arise after one another upon -the top of the bubble. - -11. But now by measuring the diameters of each of these rings, and -knowing the convexity of the telescope glass, the thickness of the -water at each of those rings may be determined with great exactness: -for instance the thickness of it, where the white light of the first -order is reflected, is about 3⅞ such parts, of which an inch contains -1000000[303]. And this measure gives the thickness of the bubble, where -it appeared of this white colour, as well as of the water between the -glasses; though the transparent body which surrounds the water in these -two cases be very different: for our author found, that the condition -of the ambient body would not alter the species of the colour at all, -though it might its strength and brightness; for pieces of Muscovy -glass, which were so thin as to appear coloured by being wet with -water, would have their colours faded and made less bright thereby; but -he could not observe their species at all to be changed. So that the -thickness of any transparent body determines its colour, whatever body -the light passes through in coming to it[304]. - -12. BUT it was found that different transparent bodies would not under -the same thicknesses exhibit the same colours: for if the forementioned -glasses were laid upon each other without any water between their -surfaces, the air itself would afford the same colours as the water, -but more expanded, insomuch that each ring had a larger diameter, and -all in the same proportion. So that the thickness of the air proper to -each colour was in the same proportion larger, than the thickness of -the water appropriated to the same[305]. - -13. IF we examine with care all the circumstances of these colours, -which will be enumerated in the next chapter, we shall not be -surprized, that our author takes them to bear a great analogy to the -colours of natural bodies[306]. For the regularity of those various and -strange appearances relating to them, which makes the most mysterious -part of the action between light and bodies, as the next chapter will -shew, is sufficient to convince us that the principle, from which -they flow, is of the greatest importance in the frame of nature; and -therefore without question is designed for no less a purpose than to -give bodies their various colours, to which end it seems very fitly -suited. For if any such transparent substance of the thickness -proper to produce any one colour should be cut into slender threads, -or broken into fragments, it does not appear but these should retain -the same colour; and a heap of such fragments should frame a body of -that colour. So that this is without dispute the cause why bodies are -of this or the other colour, that the particles of which they are -composed are of different sizes. Which is farther confirmed by the -analogy between the colours of thin plates, and the colours of many -bodies. For example, these plates do not look of the same colour when -viewed obliquely, as when seen direct; for if the rings and colours -between a convex and plane glass are viewed first in a direct manner, -and then at different degrees of obliquity, the rings will be observed -to dilate themselves more and more as the obliquity is increased[307]; -which shews that the transparent substance between the glasses does -not exhibit the same colour at the same thickness in all situations of -the eye: just so the colours in the very same part of a peacock’s tail -change, as the tail changes posture in respect of the sight. Also the -colours of silks, cloths, and other substances, which water or oyl can -intimately penetrate, become faint and dull by the bodies being wet -with such fluids, and recover their brightness again when dry; just -as it was before said that plates of Muscovy glass grew faint and dim -by wetting. To this may be added, that the colours which painters use -will be a little changed by being ground very elaborately, without -question by the diminution of their parts. All which particulars, and -many more that might be extracted from our author, give abundant -proof of the present point. I shall only subjoin one more: these -transparent plates transmit through them all the light they do not -reflect; so that when looked through they exhibit those colours, which -result from the depriving white light of the colour reflected. This -may commodiously be tryed by the glasses so often mentioned; which if -looked through exhibit coloured rings as by reflected light, but in a -contrary order; for the middle spot, which in the other view appears -black for want of reflected light, now looks perfectly white, opposite -to the blue circle; next without this spot the light appears tinged -with a yellowish red; where the white circle appeared before, it now -seems dark; and so of the rest[308]. Now in the same manner, the light -transmitted through foliated gold into a darkened room appears greenish -by the loss of the yellow light, which gold reflects. - -14. HENCE it follows, that the colours of bodies give a very probable -ground for making conjecture concerning the magnitude of their -constituent particles[309]. My reason for calling it a conjecture is, -its being difficult to fix certainly the order of any colour. The -green of vegetables our author judges to be of the third order, partly -because of the intenseness of their colour; and partly from the changes -they suffer when they wither, turning at first into a greenish or more -perfect yellow, and afterwards some of them to an orange or red; which -changes seem to be effected from their ringing particles growing denser -by the exhalation of their moisture, and perhaps augmented likewise -by the accretion of the earthy and oily parts of that moisture. How -the mentioned colours should arise from increasing the bulk of those -particles, is evident; seeing those colours lie without the ring -of green between the glasses, and are therefore formed where the -transparent substance which reflects them is thicker. And that the -augmentation of the density of the colorific particles will conspire -to the production of the same effect, will be evident; if we remember -what was said of the different size of the rings, when air was included -between the glasses, from their size when water was between them; which -shewed that a substance of a greater density than another gives the -same colour at a less thickness. Now the changes likely to be wrought -in the density or magnitude of the parts of vegetables by withering -seem not greater, than are sufficient to change their colour into those -of the same order; but the yellow and red of the fourth order are not -full enough to agree with those, into which these substances change, -nor is the green of the second sufficiently good to be the colour of -vegetables; so that their colour must of necessity be of the third -order. - -15. THE blue colour of syrup of violets our author supposes to be of -the third order; for acids, as vinegar, with this syrup change it red, -and salt of tartar or other alcalies mixed therewith turn it green. -But if the blue colour of the syrup were of the second order, the red -colour, which acids by attenuating its parts give it, must be of the -first order, and the green given it by alcalies by incrassating its -particles should be of the second; whereas neither of those colours is -perfect enough, especially the green, to answer those produced by these -changes; but the red may well enough be allowed to be of the second -order, and the green of the third; in which case the blue must be -likewise of the third order. - -16. THE azure colour of the skies our author takes to be of the -first order, which requires the smallest particles of any colour, -and therefore most like to be exhibited by vapours, before they have -sufficiently coalesced to produce clouds of other colours. - -17. THE most intense and luminous white is of the first order, if -less strong it is a mixture of the colours of all the orders. Of -the latter sort he takes the colour of linnen, paper, and such like -substances to be; but white metals to be of the former sort. The -arguments for it are these. The opacity of all bodies has been shewn -to arise from the number and strength of the reflections made within -them; but all experiments shew, that the strongest reflection is made -at those surfaces, which intercede transparent bodies differing most -in density. Among other instances of this, the experiments before us -afford one; for when air only is included between the glasses, the -coloured rings are not only more dilated, as has before been said, -than when water is between them; but are likewise much more luminous -and bright. It follows therefore, that whatever medium pervades the -pores of bodies, if so be there is any, those substances must be most -opake, the density of whose parts differs most from the density of the -medium, which fills their pores. But it has been sufficiently proved -in the former part of this tract, that there is no very dense medium -lodging in, at least pervading at liberty the pores of bodies. And it -is farther proved by the present experiments. For when air is inclosed -by the denser substance of glass, the rings dilate themselves, as -has been said, by being viewed obliquely; this they do so very much, -that at different obliquities the same thickness of air will exhibit -all sorts of colours. The bubble of water, though surrounded with the -thinner substance of air, does likewise change its colour by being -viewed obliquely; but not any thing near so much, as in the other case; -for in that the same colour might be seen, when the rings were viewed -most obliquely, at more than twelve times the thickness it appeared -at under a direct view; whereas in this other case the thickness was -never found considerably above half as much again. Now the colours of -bodies not depending only on the light, that is incident upon them -perpendicularly, but likewise upon that, which falls on them in all -degrees of obliquity; if the medium surrounding their particles were -denser than those particles, all sorts of colours must of necessity -be reflected from them so copiously, as would make the colours of all -bodies white, or grey, or at best very dilute and imperfect. But on -the other hand, if the medium in the pores of bodies be much rarer -than their particles, the colour reflected will be so little changed -by the obliquity of the rays, that the colour produced by the rays, -which fall near the perpendicular, may so much abound in the reflected -light, as to give the body their colour with little allay. To this -may be added, that when the difference of the contiguous transparent -substances is the same, a colour reflected from the denser substance -reduced into a thin plate and surrounded by the rarer will be more -brisk, than the same colour will be, when reflected from a thin plate -formed of the rarer substance, and surrounded by the denser; as our -author experienced by blowing glass very thin at a lamp furnace, -which exhibited in the open air more vivid colours, than the air does -between two glasses. From these considerations it is manifest, that -if all other circumstances are alike, the densest bodies will be most -opake. But it was observed before, that these white metals can hardly -be made so thin, except by being dissolved in corroding liquors, as -to be rendred transparent; though none of them are so dense as gold, -which proves their great opacity to have some other cause besides their -density; and none is more fit to produce this, than such a size of -their particles, as qualifies them to reflect the white of the first -order. - -18. FOR producing black the particles ought to be smaller than for -exhibiting any of the colours, viz. of a size answering to the -thickness of the bubble, where by reflecting little or no light it -appears colourless; but yet they must not be too small, for that will -make them transparent through deficiency of reflections in the inward -parts of the body, sufficient to stop the light from going through it; -but they must be of a size bordering upon that disposed to reflect -the faint blue of the first order, which affords an evident reason why -blacks usually partake a little of that colour. We see too, why bodies -dissolved by fire or putrefaction turn black: and why in grinding -glasses upon copper plates the dust of the glass, copper, and sand it -is ground with, become very black: and in the last place why these -black substances communicate so easily to others their hue; which is, -that their particles by reason of the great minuteness of them easily -overspread the grosser particles of others. - -19. I SHALL now finish this chapter with one remark of the exceeding -great porosity in bodies necessarily required in all that has here -been said; which, when duly considered, must appear very surprizing; -but perhaps it will be matter of greater surprize, when I affirm that -the sagacity of our author has discovered a method, by which bodies -may easily become so; nay how any the least portion of matter may be -wrought into a body of any assigned dimensions how great so ever, and -yet the pores of that body none of them greater, than any the smallest -magnitude proposed at pleasure; notwithstanding which the parts of the -body shall so touch, that the body itself shall be hard and solid[310]. -The manner is this: suppose the body be compounded of particles of -such figures, that when laid together the pores found between them -may be equal in bigness to the particles; how this may be effected, -and yet the body be hard and solid, is not difficult to understand; -and the pores of such a body may be made of any proposed degree of -smallness. But the solid matter of a body so framed will take up only -half the space occupied by the body; and if each constituent particle -be composed of other less particles according to the same rule, the -solid parts of such a body will be but a fourth part of its bulk; if -every one of these lesser particles again be compounded in the same -manner, the solid parts of the whole body shall be but one eighth of -its bulk; and thus by continuing the composition the solid parts of the -body may be made to bear as small a proportion to the whole magnitude -of the body, as shall be desired, notwithstanding the body will be by -the contiguity of its parts capable of being in any degree hard. Which -shews that this whole globe of earth, nay all the known bodies in the -universe together, as far as we know, may be compounded of no greater -a portion of solid matter, than might be reduced into a globe of one -inch only in diameter, or even less. We see therefore how by this means -bodies may easily be made rare enough to transmit light, with all -that freedom pellucid bodies are found to do. Though what is the real -structure of bodies we yet know not. - - - - -~CHAP. III.~ - -Of the REFRACTION, REFLECTION, and INFLECTION of LIGHT. - - -THUS much of the colours of natural bodies; our method now leads us -to speculations yet greater, no less than to lay open the causes of -all that has hitherto been related. For it must in this chapter be -explained, how the prism separates the colours of the sun’s light, as -we found in the first chapter; and why the thin transparent plates -discoursed of in the last chapter, and consequently the particles of -coloured bodies, reflect that diversity of colours only by being of -different thicknesses. - -2. FOR the first it is proved by our author, that the colours of the -sun’s light are manifested by the prism, from the rays undergoing -different degrees of refraction; that the violet-making rays, which -go to the upper part of the coloured image in the first experiment -of the first chapter, are the most refracted; that the indigo-making -rays are refracted, or turned out of their course by passing through -the prism, something less than the violet-making rays, but more than -the blue-making rays; and the blue-making rays more than the green; -the green-making rays more than the yellow; the yellow more than the -orange; and the orange-making rays more than the red-making, which are -least of all refracted. The first proof of this, that rays of different -colours are refracted unequally is this. If you take any body, and -paint one half of it red and the other half blue, then upon viewing -it through a prism those two parts shall appear separated from each -other; which can be caused no otherwise than by the prism’s refracting -the light of one half more than the light of the other half. But the -blue half will be most refracted; for if the body be seen through the -prism in such a situation, that the body shall appear lifted upwards -by the refraction, as a body within a bason of water, in the experiment -mentioned in the first chapter, appeared to be lifted up by the -refraction of the water, so as to be seen at a greater distance than -when the bason is empty, then shall the blue part appear higher than -the red; but if the refraction of the prism be the contrary way, the -blue part shall be depressed more than the other. Again, after laying -fine threads of black silk across each of the colours, and the body -well inlightened, if the rays coming from it be received upon a convex -glass, so that it may by refracting the rays cast the image of the body -upon a piece of white paper held beyond the glass; then it will be seen -that the black threads upon the red part of the image, and those upon -the blue part, do not at the same time appear distinctly in the image -of the body projected by the glass; but if the paper be held so, that -the threads on the blue part may distinctly appear, the threads cannot -be seen distinct upon the red part; but the paper must be drawn farther -off from the convex glass to make the threads on this part visible; and -when the distance is great enough for the threads to be seen in this -red part, they become indistinct in the other. Whence it appears that -the rays proceeding from each point of the blue part of the body are -sooner united again by the convex glass than the rays which come from -each point of the red parts[311]. But both these experiments prove that -the blue-making rays, as well in the small refraction of the convex -glass, as in the greater refraction of the prism, are more bent, than -the red-making rays. - -3. THIS seems already to explain the reason of the coloured spectrum -made by refracting the sun’s light with a prism, though our author -proceeds to examine that in particular, and proves that the different -coloured rays in that spectrum are in different degrees refracted; by -shewing how to place the prism in such a posture, that if all the rays -were refracted in the same manner, the spectrum should of necessity -be round: whereas in that case if the angle made by the two surfaces -of the prism, through which the light passes, that is the angle D F -E in fig. 126, be about 63 or 64 degrees, the image instead of being -round shall be near five times as long as broad; a difference enough -to shew a great inequality in the refractions of the rays, which -go to the opposite extremities of the image. To leave no scruple -unremoved, our author is very particular in shewing by a great number -of experiments, that this inequality of refraction is not casual, and -that it does not depend upon any irregularities of the glass; no nor -that the rays are in their passage through the prism each split and -divided; but on the contrary that every ray of the sun has its own -peculiar degree of refraction proper to it, according to which it is -more or less refracted in passing through pellucid substances always -in the same manner[312]. That the rays are not split and multiplied -by the refraction of the prism, the third of the experiments related -in our first chapter shews very clearly; for if they were, and the -length of the spectrum in the first refraction were thereby occasioned, -the breadth should be no less dilated by the cross refraction of the -second prism; whereas the breadth is not at all increased, but the -image is only thrown into an oblique posture by the upper part of the -rays which were at first more refracted than the under part, being -again turned farthest out of their course. But the experiment most -expressly adapted to prove this regular diversity of refraction is -this, which follows[313]. Two boards A B, C D (in fig. 130.) being -erected in a darkened room at a proper distance, one of them A B being -near the window-shutter E F, a space only being left for the prism G -H I to be placed between them; so that the rays entring at the hole M -of the window-shutter may after passing through the prism be trajected -through a smaller hole K made in the board A B, and passing on from -thence go out at another hole L made in the board C D of the same size -as the hole K, and small enough to transmit the rays of one colour -only at a time; let another prism N O P be placed after the board C -D to receive the rays passing through the holes K and L, and after -refraction by that prism let those rays fall upon the white surface -Q R. Suppose first the violet light to pass through the holes, and -to be refracted by the prism N O P to _s_, which if the prism N O P -were removed should have passed right onto W. If the prism G H I be -turned slowly about, while the boards and prism N O P remain fixed, -in a little time another colour will fall upon the hole L, which, if -the prism N O P were taken away, would proceed like the former rays -to the same point W; but the refraction of the prism N O P shall not -carry these rays to _s_, but to some place less distant from W as to -_t_. Suppose now the rays which go to _t_ to be the indigo-making rays. -It is manifest that the boards A B, C D, and prism N O P remaining -immoveable, both the violet-making and indigo-making rays are incident -alike upon the prism N O P, for they are equally inclined to its -surface O P, and enter it in the same part of that surface; which shews -that the indigo-making rays are less diverted out of their course by -the refraction of the prism, than the violet-making rays under an -exact parity of all circumstances. Farther, if the prism G H I be more -turned about, ’till the blue-making rays pass through the hole L, these -shall fall upon the surface Q R below I, as at _v_, and therefore are -subjected to a less refraction than the indigo-making rays. And thus -by proceeding it will be found that the green-making rays are less -refracted than the blue-making rays, and so of the rest, according to -the order in which they lie in the coloured spectrum. - -4. THIS disposition of the different coloured rays to be refracted -some more than others our author calls their respective degrees of -refrangibility. And since this difference of refrangibility discovers -it self to be so regular, the next step is to find the rule it observes. - -5. IT is a common principle in optics, that the sine of the angle of -incidence bears to the sine of the refracted angle a given proportion. -If A B (in fig. 131, 132) represent the surface of any refracting -substance, suppose of water or glass, and C D a ray of light incident -upon that face in the point D, let D E be the ray, after it has passed -the surface A B; if the ray pass out of the air into the substance -whose surface is A B (as in fig. 131) it shall be turned from the -surface, and if it pass out of that substance into air it shall be -bent towards it (as in fig. 132) But if F G be drawn through the point -D perpendicular to the surface A B, the angle under C D F made by the -incident ray and this perpendicular is called the angle of incidence; -and the angle under E D G, made by this perpendicular and the ray after -refraction, is called the refracted angle. And if the circle H F I G -be described with any interval cutting C D in H and D E in I, then -the perpendiculars H K, I L being let fall upon F G, H K is called -the sine of the angle under C D F the angle of incidence, and I L the -sine of the angle under E D G the refracted angle. The first of these -sines is called the sine of the angle of incidence, or more briefly the -sine of incidence, the latter is the sine of the refracted angle, or -the sine of refraction. And it has been found by numerous experiments -that whatever proportion the sine of incidence H K bears to the sine -of refraction I L in any one case, the same proportion shall hold in -all cases; that is, the proportion between these sines will remain -unalterably the same in the same refracting substance, whatever be the -magnitude of the angle under C D F. - -6. BUT now because optical writers did not observe that every beam of -white light was divided by refraction, as has been here explained, -this rule collected by them can only be understood in the gross of the -whole beam after refraction, and not so much of any particular part -of it, or at most only of the middle part of the beam. It therefore -was incumbent upon our author to find by what law the rays were parted -from each other; whether each ray apart obtained this property, and -that the separation was made by the proportion between the sines of -incidence and refraction being in each species of rays different; or -whether the light was divided by some other rule. But he proves by a -certain experiment that each ray has its sine of incidence proportional -to its sine of refraction; and farther shews by mathematical reasoning, -that it must be so upon condition only that bodies refract the light -by acting upon it, in a direction perpendicular to the surface of the -refracting body, and upon the same sort of rays always in an equal -degree at the same distances[314]. - -7. OUR great author teaches in the next place how from the refraction -of the most refrangible and least refrangible rays to find the -refraction of all the intermediate ones[315]. The method is this: -if the sine of incidence be to the sine of refraction in the least -refrangible rays as A to B C, (in fig. 133) and to the sine of -refraction in the most refrangible as A to B D; if C E be taken equal -to C D, and then E D be so divided in F, G, H, I, K, L, that E D, E -F, E G, E H, E I, E K, E L, E C, shall be proportional to the eight -lengths of musical chords, which found the notes in an octave, E D -being the length of the key, E F the length of the tone above that -key, E G the length of the lesser third, E H of the fourth, E I of the -fifth, E K of the greater sixth, E L of the seventh, and E C of the -octave above that key; that is if the lines E D, E F, E G, E H, E I, E -K, E L, and E C bear the same proportion as the numbers, 1, 9/8, 5/6, -¾, ⅓, ¾, 9/61, ½, respectively then shall B D, B F, be the two limits -of the sines of refraction of the violet-making rays, that is the -violet-making rays shall not all of them have precisely the same sine -of refraction, but none of them shall have a greater sine than B D, -nor a less than B F, though there are violet-making rays which answer -to any sine of refraction that can be taken between these two. In the -same manner B F and B G are the limits of the sines of refraction -of the indigo-making rays; B G, B H are the limits belonging to the -blue-making rays; B H, B I the limits pertaining to the green-making -rays, B I, B K the limits for the yellow-making rays; B K, B L the -limits for the orange-making rays; and lastly, B L and B C the extreme -limits of the sines of refraction belonging to the red-making rays. -These are the proportions by which the heterogeneous rays of light are -separated from each other in refraction. - -8. WHEN light passes out of glass into air, our author found A to B C -as 50 to 77, and the same A to B D as 50 to 78. And when it goes out -of any other refracting substance into air, the excess of the sine -of refraction of any one species of rays above its sine of incidence -bears a constant proportion, which holds the same in each species, to -the excess of the sine of refraction of the same sort of rays above -the sine of incidence into the air out of glass; provided the sines -of incidence both in glass and the other substance are equal. This -our author verified by transmitting the light through prisms of glass -included within a prismatic vessel of water; and draws from those -experiments the following observations: that whenever the light in -passing through so many surfaces parting diverse transparent substances -is by contrary refractions made to emerge into the air in a direction -parallel to that of its incidence, it will appear afterwards white at -any distance from the prisms, where you shall please to examine it; -but if the direction of its emergence be oblique to its incidence, in -receding from the place of emergence its edges shall appear tinged with -colours: which proves that in the first case there is no inequality -in the refractions of each species of rays, but that when any one -species is so refracted as to emerge parallel to the incident rays, -every sort of rays after refraction shall likewise be parallel to the -same incident rays, and to each other; whereas on the contrary, if the -rays of any one sort are oblique to the incident light, the several -species shall be oblique to each other, and be gradually separated by -that obliquity. From hence he deduces both the forementioned theorem, -and also this other; that in each sort of rays the proportion of the -sine of incidence to the sine of refraction, in the passage of the ray -out of any refracting substance into another, is compounded of the -proportion to which the sine of incidence would have to the sine of -refraction in the passage of that ray out of the first substance into -any third, and of the proportion which the sine of incidence would -have to the sine of refraction in the passage of the ray out of that -third substance into the second. From so simple and plain an experiment -has our most judicious author deduced these important theorems, by -which we may learn how very exact and circumspect he has been in this -whole work of his optics; that notwithstanding his great particularity -in explaining his doctrine, and the numerous collection of experiments -he has made to clear up every doubt which could arise, yet at the same -time he has used the greatest caution to make out every thing by the -simplest and easiest means possible. - -9. OUR author adds but one remark more upon refraction, which is, that -if refraction be performed in the manner he has supposed from the -light’s being pressed by the refracting power perpendicularly toward -the surface of the refracting body, and consequently be made to move -swifter in the body than before its incidence; whether this power act -equally at all distances or otherwise, provided only its power in the -same body at the same distances remain without variation the same in -one inclination of the incident rays as well as another; he observes -that the refracting powers in different bodies will be in the duplicate -proportion of the tangents of the lead angles, which the refracted -light can make with the surfaces of the refracting bodies[316]. -This observation may be explained thus. When the light passes into -any refracting substance, it has been shewn above that the sine of -incidence bears a constant proportion to the sine of refraction. -Suppose the light to pass to the refracting body A B C D (in fig. -134) in the line E F, and to fall upon it at the point F, and then to -proceed within the body in the line F G. Let H I be drawn through F -perpendicular to the surface A B, and any circle K L M N be described -to the center F. Then from the points O and P where this circle cuts -the incident and refracted ray, the perpendiculars O Q, P R being -drawn, the proportion of O Q to P R will remain the same in all the -different obliquities, in which the same ray of light can fall on the -surface A B. Now O Q is less than F L the semidiameter of the circle K -L M N, but the more the ray E F is inclined down toward the surface A -B, the greater will O Q be, and will approach nearer to the magnitude -of F L. But the proportion of O Q to P R remaining always the same, -when O Q, is largest, P R will also be greatest; so that the more the -incident ray E F is inclined toward the surface A B, the more the ray -F G after refraction will be inclined toward the same. Now if the line -F S T be so drawn, that S V being perpendicular to F I shall be to F L -the semidiameter of the circle in the constant proportion of P R to O -Q; then the angle under N F T is that which I meant by the least of all -that can be made by the refracted ray with this surface, for the ray -after refraction would proceed in this line, if it were to come to the -point F lying on the very surface A B; for if the incident ray came to -the point F in any line between A F and F H, the ray after refraction -would proceed forward in some line between F T and F I. Here if N W -be drawn perpendicular to F N, this line N W in the circle K L M N is -called the tangent of the angle under N F S. Thus much being premised, -the sense of the forementioned proposition is this. Let there be two -refracting substances (in fig. 135) A B C D, and E F G H. Take a point, -as I, in the surface A B, and to the center I with any semidiameter -describe the circle K L M. In like manner on the surface E F take -some point N, as a center, and describe with the same semidiameter -the circle O P Q. Let the angle under B I R be the least which the -refracted light can make with the surface A B, and the angle under F N -S the least which the refracted light can make with the surface E F. -Then if L T be drawn perpendicular to A B, and P V perpendicular to E -F; the whole power, wherewith the substance A B C D acts on the light, -will bear to the whole power wherewith the substance E F G H acts on, -the light, a proportion, which is duplicate of the proportion, which L -T bears to P V. - -10. UPON comparing according to this rule the refractive powers of a -great many bodies it is found, that unctuous bodies which abound most -with sulphureous parts refract the light two or three times more in -proportion to their density than others: but that those bodies, which -seem to receive in their composition like proportions of sulphureous -parts, have their refractive powers proportional to their densities; as -appears beyond contradiction by comparing the refractive power of so -rare a substance as the air with that of common glass or rock crystal, -though these substances are 2000 times denser than air; nay the same -proportion is found to hold without sensible difference in comparing -air with pseudo-topar and glass of antimony, though the pseudo-topar -be 3500 times denser than air, and glass of antimony no less than -4400 times denser. This power in other substances, as salts, common -water, spirit of wine, &c. seems to bear a greater proportion to their -densities than these last named, according as they abound with sulphurs -more than these; which makes our author conclude it probable, that -bodies act upon the light chiefly, if not altogether, by means of the -sulphurs in them; which kind of substances it is likely enters in some -degree the composition of all bodies. Of all the substances examined -by our author, none has so great a refractive power, in respect of its -density, as a diamond. - -11. OUR author finishes these remarks, and all he offers relating to -refraction, with observing, that the action between light and bodies is -mutual, since sulphureous bodies, which are most readily set on fire -by the sun’s light, when collected upon them with a burning glass, act -more upon light in refracting it, than other bodies of the same density -do. And farther, that the densest bodies, which have been now shewn to -act most upon light, contract the greatest heat by being exposed to the -summer sun. - -12. HAVING thus dispatched what relates to refraction, we must address -ourselves to discourse of the other operation of bodies upon light -in reflecting it. When light passes through a surface, which divides -two transparent bodies differing in density, part of it only is -transmitted, another part being reflected. And if the light pass out -of the denser body into the rarer, by being much inclined to the -foresaid surface at length no part of it shall pass through, but be -totally reflected. Now that part of the light, which suffers the -greatest refraction, shall be wholly reflected with a less obliquity -of the rays, than the parts of the light which undergo a less degree -of refraction; as is evident from the last experiment recited in the -first chapter; where, as the prisms D E F, G H I, (in fig. 129.) were -turned about, the violet light was first totally reflected, and then -the blue, next to that the green, and so of the rest. In consequence -of which our author lays down this proportion; that the sun’s light -differs in reflexibility, those rays being most reflexible, which are -most refrangible. And collects from this, in conjunction with other -arguments, that the refraction and reflection, of light are produced -by the same cause, compassing those different effects only by the -difference of circumstances with which it is attended. Another proof -of this being taken by our author from what he has discovered of -the passage of light through thin transparent plates, viz. that any -particular species of light, suppose, for instance, the red-making -rays, will enter and pass out of such a plate, if that plate be of -some certain thicknesses; but if it be of other thicknesses, it will -not break through it, but be reflected back: in which is seen, that -the thickness of the plate determines whether the power, by which that -plate acts upon the light, shall reflect it, or suffer it to pass -through. - -13. BUT this last mentioned surprising property of the action between -light and bodies affords the reason of all that has been said in the -preceding chapter concerning the colours of natural bodies; and must -therefore more particularly be illustrated and explained, as being what -will principally unfold the nature of the action of bodies upon light. - -14. TO begin: The object glass of a long telescope being laid upon a -plane glass, as proposed in the foregoing chapter, in open day-light -there will be exhibited rings of various colours, as was there related; -but if in a darkened room the coloured spectrum be formed by the prism, -as in the first experiment of the first chapter, and the glasses be -illuminated by a reflection from the spectrum, the rings shall not -in this case exhibit the diversity of colours before described, but -appear all of the colour of the light which falls upon the glasses, -having dark rings between. Which shews that the thin plate of air -between the glasses at some thicknesses reflects the incident light, -at other places does not reflect it, but is found in those places to -give the light passage; for by holding the glasses in the light as -it passes from the prism to the spectrum, suppose at such a distance -from the prism that the several sorts of light must be sufficiently -separated from each other, when any particular sort of light falls -on the glasses, you will find by holding a piece of white paper at a -small distance beyond the glasses, that at those intervals, where the -dark lines appeared upon the glasses, the light is so transmitted, -as to paint upon the paper rings of light having that colour which -falls upon the glasses. This experiment therefore opens to us this -very strange property of reflection, that in these thin plates it -should bear such a relation to the thickness of the plate, as is here -shewn. Farther, by carefully measuring the diameters of each ring it -is found, that whereas the glasses touch where the dark spot appears -in the center of the rings made by reflexion, where the air is of -twice the thickness at which the light of the first ring is reflected, -there the light by being again transmitted makes the first dark ring; -where the plate has three times that thickness which exhibits the -first lucid ring, it again reflects the light forming the second lucid -ring; when the thickness is four times the first, the light is again -transmitted so as to make the second dark ring; where the air is five -times the first thickness, the third lucid ring is made; where it has -six times the thickness, the third dark ring appears, and so on: in -so much that the thicknesses, at which the light is reflected, are -in proportion to the numbers 1, 3, 5, 7, 9, &c. and the thicknesses, -where the light is transmitted, are in the proportion of the numbers -0, 2, 4, 6, 8, &c. And these proportions between the thicknesses which -reflect and transmit the light remain the same in all situations of the -eye, as well when the rings are viewed obliquely, as when looked on -perpendicularly. We must farther here observe, that the light, when it -is reflected, as well as when it is transmitted, enters the thin plate, -and is reflected from its farther surface; because, as was before -remarked, the altering the transparent body behind the farther surface -alters the degree of reflection as when a thin piece of Muscovy glass -has its farther surface wet with water, and the colour of the glass -made dimmer by being so wet; which shews that the light reaches to the -water, otherwise its reflection could not be influenced by it. But -yet this reflection depends upon some power propagated from the first -surface to the second; for though made at the second surface it depends -also upon the first, because it depends upon the distance between the -surfaces; and besides, the body through which the light passes to the -first surface influences the reflection: for in a plate of Muscovy -glass, wetting the surface, which first receives the light, diminishes -the reflection, though not quite so much as wetting the farther surface -will do. Since therefore the light in passing through these thin plates -at some thicknesses is reflected, but at others transmitted without -reflection, it is evident, that this reflection is caused by some -power propagated from the first surface, which intermits and returns -successively. Thus is every ray apart disposed to alternate reflections -and transmissions at equal intervals; the successive returns of which -disposition our author calls the fits of easy reflection, and of easy -transmission. But these fits, which observe the same law of returning -at equal intervals, whether the plates are viewed perpendicularly or -obliquely, in different situations of the eye change their magnitude. -For what was observed before in respect of those rings, which appear -in open day-light, holds likewise in these rings exhibited by simple -lights; namely, that these two alter in bigness according to the -different angle under which they are seen: and our author lays down a -rule whereby to determine the thicknesses of the plate of air, which -shall exhibit the same colour under different oblique views[317]. And -the thickness of the aereal plate, which in different inclinations of -the rays will exhibit to the eye in open day-light the same colour, is -also varied by the same rule[318]. He contrived farther a method of -comparing in the bubble of water the proportion between the thickness -of its coat, which exhibited any colour when seen perpendicularly, -to the thickness of it, where the same colour appeared by an oblique -view; and he found the same rule to obtain here likewise[319]. But -farther, if the glasses be enlightened successively by all the several -species of light, the rings will appear of different magnitudes; in -the red light they will be larger than in the orange colour, in that -larger than in the yellow, in the yellow larger than in the green, -less in the blue, less yet in the indigo, and least of all in the -violet: which shew that the same thickness of the aereal plate is not -fitted to reflect all colours, but that one colour is reflected where -another would have been transmitted; and as the rays which are most -strongly refracted form the least rings, a rule is laid down by our -author for determining the relation, which the degree of refraction of -each species of colour has to the thicknesses of the plate where it is -reflected. - -15. FROM these observations our author shews the reason of that great -variety of colours, which appears in these thin plates in the open -white light of the day. For when this white light falls on the plate, -each part of the light forms rings of its own colour; and the rings -of the different colours not being of the same bigness are variously -intermixed, and form a great variety of tints[320]. - -16. IN certain experiments, which our author made with thick glasses, -he found, that these fits of easy reflection and transmission returned -for some thousands of times, and thereby farther confirmed his -reasoning concerning them[321]. - -17. UPON the whole, our great author concludes from some of the -experiments made by him, that the reason why all transparent bodies -refract part of the light incident upon them, and reflect another -part, is, because some of the light, when it comes to the surface of -the body, is in a fit of easy transmission, and some part of it in -a fit of easy reflection; and from the durableness of these fits he -thinks it probable, that the light is put into these fits from their -first emission out of the luminous body; and that these fits continue -to return at equal intervals without end, unless those intervals be -changed by the light’s entring into some refracting substance[322]. He -likewise has taught how to determine the change which is made of the -intervals of the fits of easy transmission and reflection, when the -light passes out of one transparent space or substance into another. -His rule is, that when the light passes perpendicularly to the surface, -which parts any two transparent substances, these intervals in the -substance, out of which the light passes, bear to the intervals in the -substance, whereinto the light enters, the same proportion, as the sine -of incidence bears to the sine of refraction[323]. It is farther to be -observed, that though the fits of easy reflection return at constant -intervals, yet the reflecting power never operates, but at or near a -surface where the light would suffer refraction; and if the thickness -of any transparent body shall be less than the intervals of the fits, -those intervals shall scarce be disturbed by such a body, but the light -shall pass through without any reflection[324]. - -18. WHAT the power in nature is, whereby this action between light and -bodies is caused, our author has not discovered. But the effects, which -he has discovered, of this power are very surprising, and altogether -wide from any conjectures that had ever been framed concerning it; and -from these discoveries of his no doubt this power is to be deduced, -if we ever can come to the knowledge of it. Sir ISAAC NEWTON has in -general hinted at his opinion concerning it; that probably it is -owing to some very subtle and elastic substance diffused through the -universe, in which such vibrations may be excited by the rays of -light, as they pass through it, that shall occasion it to operate so -differently upon the light in different places as to give rise to these -alternate fits of reflection and transmission, of which we have now -been speaking[325]. He is of opinion, that such a substance may produce -this and other effects also in nature, though it be so rare as not to -give any sensible resistance to bodies in motion[326]; and therefore -not inconsistent with what has been said above, that the planets move -in spaces free from resistance[327]. - -19. IN order for the more full discovery of this action between light -and bodies, our author began another set of experiments, wherein he -found the light to be acted on as it passes near the edges of solid -bodies; in particular all small bodies, such as the hairs of a man’s -head or the like, held in a very small beam of the sun’s light, cast -extremely broad shadows. And in one of these experiments the shadow -was 35 times the breadth of the body[328]. These shadows are also -observed to be bordered with colours[329]. This our author calls the -inflection of light; but as he informs us, that he was interrupted -from prosecuting these experiments to any length, I need not detain my -readers with a more particular account of them. - - - - -~CHAP. IV.~ - -Of OPTIC GLASSES. - - -SIR ~ISAAC NEWTON~ having deduced from his doctrine of light and -colours a surprising improvement of telescopes, of which I intend -here to give an account, I shall first premise something in general -concerning those instruments. - -2. IT will be understood from what has been said above, that when light -falls upon the surface of glass obliquely, after its entrance into -the glass it is more inclined to the line drawn through the point of -incidence perpendicular to that surface, than before. Suppose a ray of -light issuing from the point A (in fig. 136) falls on a piece of glass -B C D E, whose surface B C, whereon the ray falls, is of a spherical -or globular figure, the center whereof is F. Let the ray proceed in -the line A G falling on the surface B C in the point G, and draw F G -H. Here the ray after its entrance into the glass will pass on in some -line, as G I, more inclined toward the line F G H that the line A G is -inclined thereto; for the line F G H is perpendicular to the surface B -C in the point G. By this means, if a number of rays proceeding from -any one point fall on a convex spherical surface of glass, they shall -be inflected (as is represented in fig. 137,) so as to be gathered -pretty close together about the line drawn through the center of the -glass from the point, whence the rays proceed; which line henceforward -we shall call the axis of the glass: or the point from whence the rays -proceed may be so near the glass, that the rays shall after entring the -glass still go on to spread themselves, but not so much as before; so -that if the rays were to be continued backward (as in fig. 138,) they -should gather together about the axis at a place more remote from the -glass, than the point is, whence they actually proceed. In these and -the following figures A denotes the point to which the rays are related -before refraction, B the point to which they are directed afterwards, -and C the center of the refracting surface. Here we may observe, that -it is possible to form the glass of such a figure, that all the rays -which proceed from one point shall after refraction be reduced again -exactly into one point on the axis of the glass. But in glasses of a -spherical form though this does not happen; yet the rays, which fall -within a moderate distance from the axis, will unite extremely near -together. If the light fall on a concave spherical surface, after -refraction it shall spread quicker than before (as in fig. 139,) unless -the rays proceed from a point between the center and the surface of the -glass. If we suppose the rays of light, which fall upon the glass, not -to proceed from any point, but to move so as to tend all to some point -in the axis of the glass beyond the surface; if the glass have a convex -surface, the rays shall unite about the axis sooner, than otherwise -they would do (as in fig. 140,) unless the point to which they tended -was between the surface and the center of that surface. But if the -surface be concave, they shall not meet so soon: nay perhaps converge. -(See fig. 141 and 142.) - -5. FARTHER, because the light in passing out of glass into the air is -turned by the refraction farther off from the line drawn through the -point of incidence perpendicular to the refracting surface, than it was -before; the light which spreads from a point shall by parting through -a convex surface of glass into the air be made either to spread less -than before (as in fig. 143,) or to gather about the axis beyond the -glass (as in fig. 144.) But if the rays of light were proceeding to a -point in the axis of the glass, they should by the refraction be made -to unite sooner about that axis (as in fig. 145.) If the surface of -the glass be concave, rays which proceed from a point shall be made to -spread faster (as in fig 146,) but rays which are tending to a point in -the axis of the glass, shall be made to gather about the axis farther -from the glass (as in fig. 147) or even to diverge (as in fig. 148,) -unless the point, to which the rays are directed, lies between the -surface of the glass and its center. - -4. THE rays, which spread themselves from a point, are called -diverging; and such as move toward a point, are called converging rays. -And the point in the axis of the glass, about which the rays gather -after refraction, is called the focus of those rays. - -5. IF a glass be formed of two convex spherical surfaces (as in fig. -149,) where the glass AB is formed of the surfaces A C B and A D B, the -line drawn through the centers of the two surfaces, as the line E F, is -called the axis of the glass; and rays, which diverge from any point of -this axis, by the refraction of the glass will be caused to converge -toward some part of the axis, or at least to diverge as from a point -more remote from the glass, than that from whence they proceeded; for -the two surfaces both conspire to produce this effect upon the rays. -But converging rays will be caused by such a glass as this to converge -sooner. If a glass be formed of two concave surfaces, as the glass A -B (in fig. 150,) the line C D drawn through the centers, to which the -two surfaces are formed, is called the axis of the glass. Such a glass -shall cause diverging rays, which proceed from any point in the axis -of the glass, to diverge much more, as if they came from some place in -the axis of the glass nearer to it than the point, whence the rays -actually proceed. But converging rays will be made either to converge -less, or even to diverge. - -[Illustration] - -6. IN these glasses rays, which proceed from any point near the axis, -will be affected as it were in the same manner, as if they proceeded -from the very axis it self, and such as converge toward a point at a -small distance from the axis will suffer much the same effects from -the glass, as if they converged to some point in the very axis. By -this means any luminous body exposed to a convex glass may have an -image formed upon any white body held beyond the glass. This may be -easily tried with a common spectacle-glass. For if such a glass be held -between a candle and a piece of white paper, if the distances of the -candle, glass, and paper be properly adjusted, the image of the candle -will appear very distinctly upon the paper, but be seen inverted; the -reason whereof is this. Let A B (in fig. 151) be the glass, C D an -object placed cross the axis of the glass. Let the rays of light, which -issue from the point E, where the axis of the glass crosses the object, -be so refracted by the glass, as to meet again about the point F. The -rays, which diverge from the point C of the object, shall meet again -almost at the same distance from the glass, but on the other side of -the axis, as at G; for the rays at the glass cross the axis. In like -manner the rays, which proceed from the point D, will meet about H on -the other side of the axis. None of these rays, neither those which -proceed from the point E in the axis, nor those which issue from C or -D, will meet again exactly in one point; but yet in one place, as is -here supposed at F, G, and H, they will be crouded so close together, -as to make a distinct image of the object upon any body proper to -reflect it, which shall be held there. - -7. IF the object be too near the glass for the rays to converge after -the refraction, the rays shall issue out of the glass, as if they -diverged from a point more distant from the glass, than that from -whence they really proceed (as in fig. 152,) where the rays coming from -the point E of the object, which lies on the axis of the glass A B, -issue out of the glass, as if they came from the point F more remote -from the glass than E; and the rays proceeding from the point C issue -out of the glass, as if they proceeded from the point G; likewise the -rays which issue from the point D emerge out of the glass, as if they -came from the point H. Here the point G is on the same side of the -axis, as the point C; and the point H on the same side, as the point -D. In this case to an eye placed beyond the glass the object should -appear, as if it were in the situation G F H. - -8. IF the glass A B had been concave (as in, fig. 153,) to an eye -beyond the glass the object C D would appear in the situation G H, -nearer to the glass than really it is. Here also the object will not be -inverted; but the point G is on the same side the axe with the point C, -and H on the same side as D. - -9. HENCE may be understood, why spectacles made with convex glasses -help the sight in old age: for the eye in that age becomes unfit to -see objects distinctly, except such as are remov’d to a very great -distance; whence all men, when they first stand in need of spectacles, -are observed to read at arm’s length, and to hold the object at a -greater distance, than they used to do before. But when an object is -removed at too great a distance from the sight, it cannot be seen -clearly, by reason that a less quantity of light from the object will -enter the eye, and the whole object will also appear smaller. Now by -help of a convex glass an object may be held near, and yet the rays of -light issuing from it will enter the eye, as if the object were farther -removed. - -10. AFTER the same manner concave glasses assist such, as are short -sighted. For these require the object to be brought inconveniently near -to the eye, in order to their seeing it distinctly; but by such a glass -the object may be removed to a proper distance, and yet the rays of -light enter the eye, as if they came from a place much nearer. - -11. WHENCE these defects of the sight arise, that in old age -objects cannot be seen distinct within a moderate distance, and in -short-sightedness not without being brought too near, will be easily -understood, when the manner of vision in general shall be explain’d; -which I shall now endeavour to do, in order to be better understood in -what follows. The eye is form’d, as is represented in fig. 154. It is -of a globular figure, the fore part whereof scarce more protuberant -than the rest is transparent. Underneath this transparent part is a -small collection of an humour in appearance like water, and it has also -the same refractive power as common water; this is called the aqueous -humour, and fills the space A B C D in the figure. Next beyond lies the -body D E F G; this is solid but transparent, it is composed with two -convex surfaces, the hinder surface E F G being more convex, than the -anterior E D G. Between the outer membrane A B C, and this body E D G -F is placed that membrane, which exhibits the colours, that are seen -round the sight of the eye; and the black spot, which is called the -sight or pupil, is a hole in this membrane, through which the light -enters, whereby we see. This membrane is fixed only by its outward -circuit, and has a muscular power, whereby it dilates the pupil in -a weak light, and contracts it in a strong one. The body D E F G is -called the crystalline humour, and has a greater refracting power than -water. Behind this the bulk of the eye is filled up with what is called -the vitreous humor, this has much the same refractive power with water. -At the bottom of the eye toward the inner side next the nose the optic -glass enters, as at H, and spreads it self all over the inside of the -eye, till within a small diftance from A and C. Now any object, as I K, -being placed before the eye, the rays of light issuing from each point -of this object are so refracted by the convex surface of the aqueous -humour, as to be caused to converge; after this being received by the -convex surface E D G of the crystalline humour, which has a greater -refractive power than the aqueous, the rays, when they are entered into -this surface, still more converge, and at going out of the surface E F -G into a humour of a less refractive power than the crystalline they -are made to converge yet farther. By all these successive refractions -they are brought to converge at the bottom of the eye, so that a -distinct image of the object as L M is impress’d on the nerve. And by -this means the object is seen. - -11. IT has been made a difficulty, that the image of the object -impressed on the nerve is inverted, so that the upper part of the image -is impressed on the lower part of the eye. But this difficulty, I -think, can no longer remain, if we only consider, that upper and lower -are terms merely relative to the ordinary position of our bodies: and -our bodies, when view’d by the eye, have their image as much inverted -as other objects; so that the image of our own bodies, and of other -objects, are impressed on the eye in the same relation to one another, -as they really have. - -12. THE eye can see objects equally distinct at very different -distances, but in one distance only at the same time. That the eye may -accomodate itself to different distances, some change in its humours -is requir’d. It is my opinion, that this change is made in the figure -of the crystalline humour, as I have indeavoured to prove in another -place. - -13. IF any of the humours of the eye are too flat, they will refract -the light too little; which is the case in old age. If they are too -convex, they refract too much; as in those who are short-sighted. - -14. THE manner of direct vision being thus explained, I proceed to give -some account of telescopes, by which we view more distinctly remote -objects; and also of microscopes, whereby we magnify the appearance of -small objects. In the first place, the most simple sort of telescope -is composed of two glasses, either both convex, or one convex, and the -other concave. (The first sort of these is represented in fig. 155, the -latter in fig. 156.) - -15. IN fig. 155 let A B represent the convex glass next the object, C -D the other glass more convex near the eye. Suppose the object-glass A -B to form the image of the object at E F; so that if a sheet of white -paper were to be held in this place, the object would appear. Now -suppose the rays, which pass the glass A B, and are united about F, to -proceed to the eye glass C D, and be there refracted. Three only of -these rays are drawn in the figure, those which pass by the extremities -of the glass A B, and that which passes its middle. If the glass C D -be placed at such a distance from the image E F, that the rays, which -pass by the point F, after having proceeded through the glass diverge -so much, as the rays do that come from an object, which is at such a -distance from the eye as to be seen distinctly, these being received -by the eye will make on the bottom of the eye a distinct representation -of the point F. In like manner the rays, which pass through the object -glass A B to the point E after proceeding through the eye-glass C D -will on the bottom of the eye make a distinct representation of the -point E. But if the eye be placed where these rays, which proceed from -E, cross those, which proceed from F, the eye will receive the distinct -impression of both these points at the same time; and consequently -will also receive a distinct impression from all the intermediate -parts of the image E F, that is, the eye will see the object, to which -the telescope is directed, distinctly. The place of the eye is about -the point G, where the rays H E, H F cross, which pass through the -middle of the object-glass A B to the points E and F; or at the place -where the focus would be formed by rays coming from the point H, and -refracted by the glass C D. To judge how much this instrument magnifies -any object, we must first observe, that the angle under E H F, in which -the eye at the point H would see the image E F, is nearly the same as -the angle, under which the object appears by direct vision; but when -the eye is in G, and views the object through the telescope, it sees -the same under a greater angle; for the rays, which coming from E and F -cross in G, make a greater angle than the rays, which proceed from the -point H to these points E and F. The angle at G is greater than that at -H in the proportion, as the distance between the glasses A B and C D is -greater than the distance of the point G from the glass C D. - -16. THIS telescope inverts the object; for the rays, which came from -the right-hand side of the object, go to the point E the left side of -the image; and the rays, which come from the left side of the object, -go to F the right side of the image. These rays cross again in G, so -that the rays, which come from the right side of the object, go to the -right side of the eye; and the rays from the left side of the object go -to the left side of the eye. Therefore in this telescope the image in -the eye has the same situation as the object; and seeing that in direct -vision the image in the eye has an inverted situation, here, where -the situation is not inverted, the object must appear so. This is no -inconvenience to astronomers in celestial observations; but for objects -here on the earth it is usual to add two other convex glasses, which -may turn the object again (as is represented in fig. 157,) or else to -use the other kind of telescope with a concave eye-glass. - -17. IN this other kind of telescope the effect is founded on the same -principles, as in the former. The distinctness of the appearance is -procured in the same manner. But here the eye-glass C D (in fig. 156) -is placed between the image E F, and the object glass A B. By this -means the rays, which come from the right-hand side of the object, -and proceed toward E the left side of the image, being intercepted by -the eye-glass are carried to the left side of the eye; and the rays, -which come from the left side of the object, go to the right side of -the eye; so that the impression in the eye being inverted the object -appears in the same situation, as when view’d by the naked eye. The -eye must here be placed close to the glass. The degree of magnifying in -this instrument is thus to be found. Let the rays, which pass through -the glass A B at H, after the refraction of the eye-glass C D diverge, -as if they came from the point G; then the rays, which come from the -extremities of the object, enter the eye under the angle at G; so -that here also the object will be magnified in the proportion of the -distance between the glasses, to the distance of G from the eye-glass. - -18. THE space, that can be taken in at one view in this telescope, -depends on the breadth of the pupil of the eye; for as the rays, which -go to the points E, F of the image, are something distant from each -other, when they come out of the glass C D, if they are wider asunder -than the pupil, it is evident, that they cannot both enter the eye -at once. In the other telescope the eye is placed in the point G, -where the rays that come from the points E or F cross each other, and -therefore must enter the eye together. On this account the telescope -with convex glasses takes in a larger view, than those with concave. -But in these also the extent of the view is limited, because the -eye-glass does not by the refraction towards its edges form so distinct -a representation of the object, as near the middle. - -18. MICROSCOPES are of two sorts. One kind is only a very convex glass, -by the means of which the object may be brought very near the eye, and -yet be seen distinctly. This microscope magnifies in proportion, as -the object by being brought near the eye will form a broader impression -on the optic nerve. The other kind made with convex glasses produces -its effects in the same manner as the telescope. Let the object A B -(in fig. 158) be placed under the glass C D, and by this glass let an -image be formed of this object. Above this image let the glass G H be -placed. By this glass let the rays, which proceed from the points A and -B, be refracted, as is expressed in the figure. In particular, let the -rays, which from each of these points pass through the middle of the -glass C D, cross in I, and there let the eye be placed. Here the object -will appear larger, when seen through the microscope, than if that -instrument were removed, in proportion as the angle, in which these -rays cross in I, is greater than the angle, which the lines would make, -that should be drawn from I to A and B; that is, in the proportion made -up of the proportion of the distance of the object A B from I, to the -distance of I from the glass G H; and of the proportion of the distance -between the glasses, to the distance of the object A B from the glass C -D. - - -19. I SHALL now proceed to explain the imperfection in these -instruments, occasioned by the different refrangibility of the light -which comes from every object. This prevents the image of the object -from being formed in the focus of the object glass with perfect -distinctness; so that if the eye-glass magnify the image overmuch, -the imperfections of it must be visible, and make the whole appear -confused. Our author more fully to satisfy himself, that the different -refrangibility of the several sorts of rays is sufficient to produce -this irregularity, underwent the labour of a very nice and difficult -experiment, whose process he has at large set down, to prove, that the -rays of light are refracted as differently in the small refraction -of telescope glasses as in the larger of the prism; so exceeding -careful has he been in searching out the true cause of this effect. -And he used, I suppose, the greater caution, because another reason -had before been generally assigned for it. It was the opinion of all -mathematicians, that this defect in telescopes arose from the figure, -in which the glasses were formed; a spherical refracting surface not -collecting into an exact point all the rays which come from any one -point of an object, as has before been said[330]. But after our author -has proved, that in these small refractions, as well as in greater, -the sine of incidence into air out of glass, to the sine of refraction -in the red-making rays, is as 50 to 77, and in the blue-making rays 50 -to 78; he proceeds to compare the inequalities of refraction arising -from this different refrangibility of the rays, with the inequalities, -which would follow from the figure of the glass, were light uniformly -refracted. For this purpose he observes, that if rays issuing from -a point so remote from the object glass of a telescope, as to be -esteemed parallel, which is the case of the rays, which come from the -heavenly bodies; then the distance from the glass of the point, in -which the least refrangible rays are united, will be to the distance, -at which the most refrangible rays unite, as 28 to 27; and therefore -that the least space, into which all the rays can be collected, will -not be less than the 55th part of the breadth of the glass. For if A -B (in fig. 159) be the glass, C D its axis, E A, F B two rays of the -light parallel to that axis entring the glass near its edges; after -refraction let the least refrangible part of these rays meet in G, -the most refrangible in H; then, as has been said, G I will be to I -H, as 28 to 27; that is, G H will be the 28th part of G I, and the -27th part of H I; whence if K L be drawn through G, and M N through H, -perpendicular to C D, M N will be the a 28th part of A B, the breadth -of the glass, and K L the 27th part of the same; so that O P the least -space, into which the rays are gathered, will be about half the mean -between these two, that is the 55th part of A B. - -20. THIS is the error arising from the different refrangibility -of the rays of light, which our author finds vastly to exceed the -other, consequent upon the figure of the glass. In particular, if the -telescope glass be flat on one side, and convex on the other; when the -flat side is turned towards the object, by a theorem, which he has laid -down, the error from the figure comes out above 5000 times less than -the other. This other inequality is so great, that telescopes could -not perform so well as they do, were it not that the light does not -equally fill all the space O P, over which it is scattered, but is much -more dense toward the middle of that space than at the extremities. And -besides, all the kinds of rays affect not the sense equally strong, the -yellow and orange being the strongest, the red and green next to them, -the blue indigo and violet being much darker and fainter colours; and -it is shewn that all the yellow and orange, and three fifths of the -brighter half of the red next the orange, and as great a share of the -brighter half of the green next the yellow, will be collected into a -space whose breadth is not above the 250th part of the breadth of the -glass. - -[Illustration] - -And the remaining colours, which fall without this space, as they are -much more dull and obscure than these, so will they be likewise much -more diffused; and therefore call hardly affect the sense in comparison -of the other. And agreeable to this is the observation of astronomers, -that telescopes between twenty and sixty feet in length represent the -fixed stars, as being about 5 or 6, at most about 8 or 10 seconds in -diameter. Whereas other arguments shew us, that they do not really -appear to us of any sensible magnitude any otherwise than as their -light is dilated by refraction. One proof that the fixed stars do not -appear to us under any sensible angle is, that when the moon passes -over any of them, their light does not, like the planets on the same -occasion, disappear by degrees, but vanishes at once. - -21. OUR author being thus convinced, that telescopes were not -capable of being brought to much greater perfection than at present -by refractions, contrived one by reflection, in which there is no -separation made of the different coloured light; for in every kind of -light the rays after reflection have the same degree of inclination -to the surface, from whence they are reflected, as they have at their -incidence, so that those rays which come to the surface in one line, -will go off also in one line without any parting from one another. -Accordingly in the attempt he succeeded so well, that a short one, not -much exceeding six inches in length, equalled an ordinary telescope -whose length was four feet. Instruments of this kind to greater -lengths, have of late been made, which fully answer expectation[331]. - - - - -~CHAP. V.~ - -Of the RAINBOW. - - -I SHALL now explain the rainbow. The manner of its production was -understood, in the general, before Sir ~ISAAC NEWTON~ had discovered -his theory of colours; but what caused the diversity of colours in it -could not then be known, which obliges him to explain this appearance -particularly; whom we shall imitate as follows. The first person, who -expressly shewed the rainbow to be formed by the reflection of the -sun-beams from drops of falling rain, was ANTONIO DE DOMINIS. But this -was afterwards more fully and distinctly explained by DESCARTES. - -2. THERE appears most frequently two rainbows; both of which are caused -by the foresaid reflection of the sun-beams from the drops of falling -rain, but are not produced by all the light which falls upon and are -reflected from the drops. The inner bow is produced by those rays only -which enter the drop, and at their entrance are so refracted as to -unite into a point, as it were, upon the farther surface of the drop, -as is represented in fig. 160; where the contiguous rays _a b_, _c d_, -_e f_, coming from the sun, and therefore to sense parallel, upon -their entrance into the drop in the points _b, d, f_, are so refracted -as to meet together in the point _g_, upon the farther surface of -the drop. Now these rays being reflected nearly from the same point -of the surface, the angle of incidence of each ray upon the point g -being equal to the angle of reflection, the rays will return in the -lines _g h, g k, g l_, in the same manner inclined to each other, as -they were before their incidence upon the point _g_, and will make -the same angles with the surface of the drop at the points _b, k, -l_, as at the points _b, d, f_, after their entrance; and therefore -after their emergence out of the drop each ray will be inclined to the -surface in the same angle, as when it first entered it; whence the -lines _b m, k n, l o_, in which the rays emerge, must be parallel to -each other, as well as the lines _a b, c d, e f_, in which they were -incident. But these emerging rays being parallel will not spread nor -diverge from each other in their passage from the drop, and therefore -will enter the eye conveniently situated in sufficient plenty to cause -a sensation. Whereas all the other rays, whether those nearer the -center of the drop, as _p q, r s_, or those farther off, as _t u, w -x_, will be reflected from other points in the hinder surface of the -drop; namely, the ray _p q_ from the point _y, r s_ from _z, t v_ -from α, and _w x_ from β. And for this reason by their reflection and -succeeding refraction they will be scattered after their emergence from -the forementioned rays and from each other, and therefore cannot enter -the eye placed to receive them copious enough to excite any distinct -sensation. - -3. THE external rainbow is formed by two reflections made between the -incidence and emergence of the rays; for it is to be noted, that the -rays _g h, g k, g l_, at the points _h, k, l_, do not wholly pass -out of the drop, but are in part reflected back; though the second -reflection of these particular rays does not form the outer bow. For -this bow is made by those rays, which after their entrance into the -drop are by the refraction of it united, before they arrive at the -farther surface, at such a distance from it, that when they fall -upon that surface, they may be reflected in parallel lines, as is -represented in fig. 161; where the rays _a b, c d, e f_, are collected -by the refraction of the drop into the point _g_, and passing on from -thence strike upon the surface of the drop in the points _h, k, l_, and -are thence reflected to _m, n, o_, passing from _h_ to _m_, from _k_ -to _n_, and from _l_ to _o_ in parallel lines. For these rays after -reflection at _m, n, o_, will meet again in the point _p_, at the same -distance from these points of reflection _m, n, o_, as the point _g_ is -from the former points of reflection _h, k, l_. Therefore these rays -in passing from _p_ to the surface of the drop will fall upon that -surface in the points _q, r, s_ in the same angles, as these rays made -with the surface in _b, d, f_, after refraction. Consequently, when -these rays emerge out of the drop into the air, each ray will make -with the surface of the drop the same angle, as it made at its first -incidence; so that the lines _q t, r v, s w_, in which they come from -the drop, will be parallel to each other, as well as the lines _a b, -c d, e f_, in which they came to the drop. By this means these rays -to a spectator commodiously situated will become visible. But all the -other rays, as well those nearer the center of the drop _x y_, _z_ -α, as those more remote from it β γ, δ ε, will be reflected in lines -not parallel to the lines _h m, k n, l o_; namely, the ray _x y_, in -the line ζ η, the ray ϰ α in the line θ ϰ, the ray β γ in the line -λ μ, and the ray δ ε in the line ν χ. Whence these rays after their -next reflection and subsequent refraction will be scattered from the -forementioned rays, and from one another, and by that means become -invisible. - -4. IT is farther to be remarked, that if in the first case the incident -rays _a b, c d, e f_, and their correspondent emergent rays _h m, k -n, l o_, are produced till they meet, they will make with each other -a greater angle, than any other incident ray will make with its -corresponding emergent ray. And in the latter case, on the contrary, -the emergent rays _q t, r v, s w_ make with the incident rays an acuter -angle, than is made by any other of the emergent rays. - -5. OUR author delivers a method of finding each of these extream angles -from the degree of refraction being given; by which method it appears, -that the first of these angles is the less, and the latter the greater, -by how much the refractive power of the drop, or the refrangibility of -the rays is greater. And this last consideration fully compleats the -doctrine of the rainbow, and shews, why the colours of each bow are -ranged in the order wherein they are seen. - -6. SUPPOSE A (in fig. 162.) to be the eye, B, C, D, E, F, drops of -rain, M _n_, O _p_, Q _r_, S _t_, V _w_ parcels of rays of the sun, -which entring the drops B, C, D, E, F after one reflection pass out to -the eye in A. Now let M _n_ be produced to η till it meets with the -emergent ray likewise produced, let O _p_ produced meet its emergent -ray produced in ϰ, let Q _r_ meet its emergent ray in λ, let S _t_ meet -its emergent ray in μ, and let V _w_ meet its emergent ray produced -in ν. If the angle under M η A be that, which is derived from the -refraction of the violet-making rays by the method we have here spoken -of, it follows that the violet light will only enter the eye from -the drop B, all the other coloured rays passing below it, that is, -all those rays which are not scattered, but go out parallel so as to -cause a sensation. For the angle, which these parallel emergent rays -makes with the incident in the most refrangible or violet-making rays, -being less than this angle in any other sort of rays, none of the rays -which emerge parallel, except the violet-making, will enter the eye -under the angle M η A, but the rest making with the incident ray M η -a greater angle than this will pass below the eye. In like manner if -the angle under O ϰ A agrees to the blue-making rays, the blue rays -only shall enter the eye from the drop C, and all the other coloured -rays will pass by the eye, the violet-coloured rays passing above, the -other colours below. Farther, the angle Q λ A corresponding to the -green-making rays, those only shall enter the eye from the drop D, the -violet and blue-making rays passing above, and the other colours, that -is the yellow and red, below. And if the angle S μ A answers to the -refraction of the yellow-making rays, they only shall come to the eye -from the drop E. And in the last place, if the angle V ν A belongs to -the red-making and least refrangible rays, they only shall enter the -eye from the drop F, all the other coloured rays passing above. - -7. BUT now it is evident, that all the drops of water found in any of -the lines A ϰ, A λ, A μ, A ν, whether farther from the eye, or nearer -than the drops B, C, D, E, F, will give the same colours as these do, -all the drops upon each line giving the same colour; so that the light -reflected from a number of these drops will become copious enough to be -visible; whereas the reflection from one minute drop alone could not be -perceived. But besides, it is farther manifest, that if the line A Ξ be -drawn from the sun through the eye, that is, parallel to the lines M -_n_, O _p_, Q _r_, S _t_, V _w_, and if drops of water are placed all -round this line, the same colour will be exhibited by all the drops at -the same distance from this line. Hence it follows, that when the sun -is moderately elevated above the horizon, if it rains opposite to it, -and the sun shines upon the drops as they fall, a spectator with his -back turned to the sun must observe a coloured circular arch reaching -to the horizon, being red without, next to that yellow, then green, -blue, and on the inner edge violet; only this last colour appears faint -by being diluted with the white light of the clouds, and from another -cause to be mentioned hereafter[332]. - -8. THUS is caused the interior or primary bow. The drops of rain at -some distance without this bow will cause the exterior or secondary -bow by two reflections of the sun’s light. Let these drops be G, H, I, -K, L; X _y_, Z α, Γ β, Δ ι, Θ ζ denoting parcels of rays which enter -each drop. Now it has been remarked, that these rays make with the -visible refracted rays the greatest angle in those rays, which are -most refrangible. Suppose therefore the visible refracted rays, which -pass out from each drop after two reflections, and enter the eye in -A, to intersect the incident rays in π, ρ, σ, τ, φ respectively. It -is manifest, that the angle under Θ φ A is the greatest of all, next -to that the angle under Δ τ A, the next in bigness will be the angle -under Γ σ A, the next to this the angle under Z ρ A, and the least of -all the angle under X π A. From the drop L therefore will come to the -eye the violet-making, or most refrangible rays, from K the blue, from -I the green, from H the yellow, and from G the red-making rays; and -the like will happen to all the drops in the lines A π, A ρ, A τ, A φ, -and also to all the drops at the same distances from the line A Ξ all -round that line. Whence appears the reason of the secondary bow, which -is seen without the other, having its colours in a contrary order, -violet without and red within; though the colours are fainter than in -the other bow, as being made by two reflections, and two refractions; -whereas the other bow is made by two refractions, and one reflection -only. - -9. THERE is a farther appearance in the rainbow particularly described -about five years ago[333], which is, that under the upper part or -the inner bow there appears often two or three orders of very faint -colours, making alternate arches of green, and a reddish purple. At the -time this appearance was taken notice of, I gave my thoughts concerning -the cause of it[334], which I shall here repeat. Sir ~ISAAC NEWTON~ has -observed, that in glass, which is polished and quick-silvered, there is -an irregular refraction made, whereby some small quantity of light is -scattered from the principal reflected beam[335]. If we allow the same -thing to happen in the reflection whereby the rainbow is caused, it -seems sufficient to produce the appearance now mentioned. - -10. LET A B (in fig. 162.) represent a globule of water, B the point -from whence the rays of any determinate species being reflected to C, -and afterwards emerging in the line C D, would proceed to the eye, and -cause the appearance of that colour in the rainbow, which appertains to -this species. Here suppose, that besides what is reflected regularly, -some small part of the light is irregularly scattered every way; so -that from the point B, besides the rays that are regularly reflected -from B to C, some scattered rays will return in other lines, as -in B E, B F, B G, B H, on each side the line B C. Now it has been -observed above[336], that the rays of light in their passage from one -superficies of a refracting body to the other undergo alternate fits -of easy transmission and reflection, succeeding each other at equal -intervals; insomuch that if they reach the farther superficies in one -sort of those fits, they shall be transmitted; if in the other kind -of them, they shall rather be reflected back. Whence the rays that -proceed from B to C, and emerge in the line C D, being in a fit of -easy transmission, the scattered rays, that fall at a small distance -without these on either side (suppose the rays that pass in the lines -B E, B G) shall fall on the surface in a fit of easy reflection, and -shall not emerge; but the scattered rays, that pass at some distance -without these last, shall arrive at the surface of the globule in a fit -of easy transmission, and break through that surface. Suppose these -rays to pass in the lines B F, B H; the former of which rays shall have -had one fit more of easy transmission, and the latter one fit less, -than the rays that pass from B to C. Now both these rays, when they -go out of the globule, will proceed by the refraction of the water -In the lines F I, H K, that will be inclined almost equally to the -rays incident on the globule, which come from the sun; but the angles -of their inclination will be less than the angle, in which the rays -emerging in the line C D are inclined to those incident rays. And after -the same manner rays scattered from the point B at a certain distance -without these will emerge out of the globule, while the intermediate -rays are intercepted; and these emergent rays will be inclined to the -rays incident on the globule in angles still less than the angles, in -which the rays F I and H K are inclined to them; and without these rays -will emerge other rays, that shall be inclined to the incident rays in -angles yet less. - -[Illustration] - -Now by this means may be formed of every kind of rays, besides the -principal arch, which goes to the formation of the rainbow, other -arches within every one of the principal of the same colour, though -much more faint; and this for divers successions, as long as these weak -lights, which in every arch grow more and more obscure, shall continue -visible. Now as the arches produced by each colour will be variously -mixed together, the diversity of colours observ’d in these secondary -arches may very possibly arise from them. - -11. IN the darker colours these arches may reach below the bow, and -be seen distinct. In the brighter colours these arches are lost in -the inferior part of the principal light of the rainbow; but in all -probability they contribute to the red tincture, which the purple of -the rainbow usually has, and is most remarkable when these secondary -colours appear strongest. However these secondary arches in the -brightest colours may possibly extend with a very faint light below the -bow, and tinge the purple of these secondary arches with a reddish hue. - -12. THE precise distances between the principal arch and these fainter -arches depend on the magnitude of the drops, wherein they are formed. -To make them any degree separate it is necessary the drop be exceeding -small. It is most likely, that they are formed in the vapour of the -cloud, which the air being put in motion by the fall of the rain may -carry down along with the larger drops; and this may be the reason, why -these colours appear under the upper part of the bow only, this vapour -not descending very low. As a farther confirmation of this, these -colours are seen strongest, when the rain falls from very black clouds, -which cause the fiercest rains, by the fall whereof the air will be -most agitated. - -13. TO the like alternate return of the fits of easy transmission and -reflection in the passage of light through the globules of water, which -compose the clouds, Sir ISAAC NEWTON ascribes some of those coloured -circles, which at times appear about the sun and moon[337]. - -[Illustration] - -[Illustration] - - - - -CONCLUSION. - - -SIR ~ISAAC NEWTON~ having concluded each of his philosophical treatises -with some general reflections, I shall now take leave of my readers -with a short account of what he has there delivered. At the end of -his mathematical principles of natural philosophy he has given us his -thoughts concerning the Deity. Wherein he first observes, that the -similitude found in all parts of the universe makes it undoubted, that -the whole is governed by one supreme being, to whom the original is -owing of the frame of nature, which evidently is the effect of choice -and design. He then proceeds briefly to state the best metaphysical -notions concerning God. In short, we cannot conceive either of space -or time otherwise than as necessarily existing; this Being therefore, -on whom all others depend, must certainly exist by the same necessity -of nature. Consequently wherever space and time is found, there God -must also be. And as it appears impossible to us, that space should be -limited, or that time should have had a beginning, the Deity must be -both immense and eternal. - -2. AT the end of his treatise of optics he has proposed some thoughts -concerning other parts of nature, which he had not distinctly searched -into. He begins with some farther reflections concerning light, which -he had not fully examined. In particular he declares his sentiments at -large concerning the power, whereby bodies and light act on each other. -In some parts of his book he had given short hints at his opinion -concerning this[338], but here he expressly declares his conjecture, -which we have already mentioned[339], that this power is lodged in -a very subtle spirit of a great elastic force diffused thro’ the -universe, producing not only this, but many other natural operations. -He thinks it not impossible, that the power of gravity itself should be -owing to it. On this occasion he enumerates many natural appearances, -the chief of which are produced by chymical experiments. From numerous -observations of this kind he makes no doubt, that the smallest parts of -matter, when near contact, act strongly on each other, sometimes being -mutually attracted, at other times repelled. - -3. THE attractive power is more manifest than the other, for -the parts of all bodies adhere by this principle. And the name of -attraction, which our author has given to it, has been very freely -made use of by many writers, and as much objected to by others. He has -often complained to me of having been misunderstood in this matter. -What he lays upon this head was not intended by him as a philosophical -explanation of any appearances, but only to point out a power in nature -not hitherto distinctly observed, the cause of which, and the manner of -its acting, he thought was worthy of a diligent enquiry. To acquiesce -in the explanation of any appearance by asserting it to be a general -power of attraction, is not to improve our knowledge in philosophy, but -rather to put a stop to our farther search. - - FINIS. - -[Illustration] - - - - - FOOTNOTES: - -[1] Philosoph. Nat. princ. math. L. iii. introduct. - -[2] Nov. Org. Scient. L. i. Aphorism. 9. - -[3] Nov. Org. L. i. Aph. 19. - -[4] Ibid. Aph. 25. - -[5] Aph. 30. Errores radicales & in prima digestione mentis ab -excellentia functionum & remediorum sequentium non curantur. - -[6] Aph. 38. - -[7] Ibid. - -[8] Aph. 39. - -[9] Aph. 41. - -[10] Aph. 10, 24. - -[11] Aph. 45. - -[12] De Cartes Princ. Phil. Part. 3. §. 52. - -[13] Fermat, in Oper. pag. 156, &c. - -[14] Nov. Org. Aph. 46. - -[15] Aph. 50. - -[16] Ibid. - -[17] Aph 53. - -[18] Aph. 54. - -[19] Aph. 56. - -[20] Aph. 55. - -[21] Locke, On human understanding, B. iii. - -[22] Nov. Org. Aph. 59. - -[23] In the conclusion. - -[24] Nov. Org. L. i. Aph. 59. - -[25] Ibid. Aph. 60. - -[26] Ibid. Aph. 62. - -[27] Aph. 63. - -[28] Aph. 64. - -[29] Aph. 65. - -[30] See above, § 4, 5. - -[31] Nov. Org. L. i. Aph. 69. - -[32] Ibid. - -[33] Ibid. Aph. 109. - -[34] Book III. Chap. iv. - -[35] Book I. Chap. 2. § 14. - -[36] Ibid. § 85, &c. - -[37] See Book II. Ch. 3. § 3, 4. of this treatise. - -[38] See Book II. Ch. 3. of this treatise. - -[39] See Chap. 4. - -[40] At the end of his Optics. in Qu. 21. - -[41] See the same treatise, in Advertisement 2. - -[42] Nov. Org. Lib. i. Ax. 105. - -[43] Princip. philos. pag. 13, 14. - -[44] Princ. Philos. L. II. prop. 24. corol. 7. See also B. II. Ch. 5. § -3. of this treatise. - -[45] How this degree of elasticity is to be found by experiment, will -be shewn below in § 74. - -[46] In oper. posthum de Motu corpor. ex percussion. prop. 9. - -[47] In the above-cited place. - -[48] In the place above-cited. - -[49] These experiments are described in § 73. - -[50] Book II. Chap. 5. - -[51] Chap. 1. § 25, 26, 27, compared with § 15, &c. - -[52] Book II. Chap. 5. § 3. - -[53] See Euclid’s Elements, Book XII. prop. 13. - -[54] Archimed. de æquipond. prop. 11. - -[55] Ibid. prop. 12. - -[56] Lucas Valerius De centr. gravit. solid. L. I. prop. 2. - -[57] Idem L. II. prop. 2. - -[58] § 25. - -[59] § 27. - -[60] Pag. 65, 68. - -[61] § 23. - -[62] § 20 - -[63] § 17. - -[64] § 27. - -[65] Hugen. Horolog. oscillat. pag. 141, 142. - -[66] See Hugen. Horolog. Oscillat. p. 142. - -[67] Princip. Philos. pag. 22. - -[68] Chap. 1. § 29. - -[69] Princip. Philos. pag. 25. - -[70] § 71. - -[71] See Method. Increment. prop. 25. - -[72] Lib. XI. Def. - -[73] Chap. 2. § 17. - -[74] See above Ch. 2. § 17. - -[75] From B II. Ch. 3. - -[76] Prin. Philos. pag. 7, &c. - -[77] See Newton, princip. philos. pag. 9. lin. 30. - -[78] Princip. Philos. pag. 10. - -[79] Renat. Des Cart. Princ. Philos. Part. II. § 25. - -[80] Ibid. § 30. - -[81] § 85, &c. - -[82] Princip. Philos. Lib. I. prop. 9. - -[83] § 92. - -[84] Ch. II. § 22. - -[85] Viz. L. I. prop. 30, 29, & 26. - -[86] Ch. II. § 21, 22. - -[87] viz. His doctrine of prime and ultimate ratios. - -[88] § 57 - -[89] § 3. - -[90] Ch. 2. § 22. - -[91] § 12. - -[92] Ch. 1. sect. 21, 22. - -[93] Elem. Book I. p. 37. - -[94] § 12. - -[95] Ch 1 § 24. - -[96] Ch 2 select. 17. - -[97] Newt. Princ. L. II. prop. 2; 5, 6, 7; 11, 12. - -[98] Prop. 3; 8, 9; 13, 14. - -[99] Prop. 4. - -[100] Prælect. Geometr. pag. 123. - -[101] Newton. Princ. Lib. II. prop. 10. - -[102] Newton. Princ. Lib II. prop 10. in schol. - -[103] Torricelli de motu gravium. - -[104] Ch. 2 § 85, &c. - -[105] Newt. Princ L. II. sect 6. - -[106] L. II. sect. 4. - -[107] See B. II. Ch 6. § 7. of this treatise. - -[108] Lib. I. sect. 10. - -[109] De la Pesanteur, pag. 169, and the following. - -[110] Newton. Princ. L. II. prop 4. schol. - -[111] See his Tract on the admirable rarifaction of the air. - -[112] Book II. Ch. 6. - -[113] Princ. philos. Lib. II. prop. 23. - -[114] Book I. Ch. 2. § 30. - -[115] Princ. philos. Lib. II. prop. 23, in schol. - -[116] Princ. philos. Lib. II. prop. 33. coroll. - -[117] Lib. II. Ch. 5. - -[118] Ibid. Prop. 35. coroll. 2. - -[119] Ibid. coroll. 3. - -[120] Vid. ibid. coroll. 6. - -[121] In § 2. - -[122] Princ. philos. Lib. II. Prop. 35. - -[123] Ibid. - -[124] Id. - -[125] h. 1. § 29. - -[126] Princ. philos. Lib. II. Prop. 38, compared with coroll. 1 of -prop. 35. - -[127] L. II. Lem. 7. schol. pag. 341. - -[128] Lib. II. Prop. 34. - -[129] Lib. II. Lem. 7. p. 341. - -[130] Schol. to Lem. 7. - -[131] Prop. 34. schol. - -[132] Ibid. - -[133] Ibid. - -[134] Book II. Ch. I. § 6. - -[135] Vid. Newt. princ. in schol. to Lem. 7, of Lib. II. pag. 341. - -[136] Sect. 17. of this chapter. - -[137] See Princ. philos. Lib. II. prop. 34. - -[138] Vid. Princ. philos. Lib. II. Lem. 5. p. 314. - -[139] Lemm. 6. - -[140] Ibid. 7. - -[141] Newt. Princ. Lib. II. prop. 40, in schol. - -[142] Lib. II. in schol. post prop. 31. - -[143] Book I. ch. 2 § 82. - -[144] Book I. Ch. 3 § 29. - -[145] Ch. 3. of this present book. - -[146] Ch. 4. - -[147] In Princ. philos. part. 3. - -[148] Philos. princ. mathem. Lib. II. prop. 2. & schol. - -[149] Ibid. prop 53. - -[150] Philos. princ. prop. 52. coroll. 4. - -[151] Ibid. - -[152] Coroll. 11. - -[153] See ibid. schol. post prop. 53. - -[154] Princ. philos. pag. 316, 317. - -[155] Ch. I. § 7. - -[156] Book I. Ch. 3. - -[157] Book I. Ch. 3. § 29. - -[158] Ibid. Ch. 2. § 30, 17. - -[159] Book I. Ch. 3. - -[160] Ch. 1. § 7. - -[161] Chap. 5. § 8. - -[162] Princ. pag. 60. - -[163] Street, in Astron. Carolin. - -[164] See Chap. 5. §9, &c. - -[165] In the foregoing page. - -[166] See Newton. Princ. Lib. III. prop. 13. - -[167] Chap. 5. § 10. - -[168] Princ. Lib. I. prop. 60. - -[169] Book I, Chap. 2. § 80. - -[170] Princ. philos. Lib. I. prop. 58. coroll. 3. - -[171] Newt. Optics. pag. 378. - -[172] Newton. Princ. Lib. III. prop. 1. - -[173] Newton, Princ. Lib. III. pag. 390,391. compared with pag. 393. - -[174] Book I. Ch. 3. § 29. - -[175] Princ. philos. Lib. I. prop. 4. - -[176] Ibid. coroll. - -[177] Newt. Princ. philos. Lib. III. pag. 390. - -[178] Newt. Princ. philos. Lib. III. pag. 391, 392. - -[179] Book III. Ch. 4. - -[180] Newt. Princ. philos. Lib. III. pag. 391. - -[181] Ibid. pag. 392. - -[182] See Book I. Ch. 2. § 60, 64. - -[183] Book I. Ch. 2. § 17. - -[184] See Ch. II. § 6. - -[185] The second of the laws of motion laid down in Book I. Ch. 1. - -[186] Newton. Princ. philos. Lib. III. prop. 6. pag. 401. - -[187] Newton’s Princ. philos. Lib. III. prop. 22, 23. - -[188] Newton. Princ. Lib. I. prop. 66. coroll. 7. - -[189] Menelai Sphaeric. Lib. I. prop. 10. - -[190] Vid. Newt. Princ. Lib. I. prop. 66. coroll. 10. - -[191] Vid. Newt. Princ. Lib. III prop. 30. p. 440. - -[192] Ibid. Lib. I. prop. 66. coroll. 10. - -[193] What this proportion is, may be known from Coroll. 2 prop. 44. -Lib. I. Princ. philos. Newton. - -[194] Princ. Phil. Newt. Lib. I. prop. 45. Coroll. 1. - -[195] Pr. Phil. Newt. Lib. I. prop. 66. Coroll. 7. - -[196] See § 19 of this chapter. - -[197] Phil. Nat. Pr. Math Lib. I. prop. 66. cor. 8. - -[198] Ibid. Coroll. 8. - -[199] Ibid. - -[200] Ibid. - -[201] Newt. Princ. Lib. III. prop. 29. - -[202] Ibid. prop. 28. - -[203] Ibid. prop. 31. - -[204] Newt. Princ. pag. 459. - -[205] In Princ. philos. part. 3. § 41. - -[206] Chap. 1. § 11. - -[207] Newton. Princ. philos. Lib. III. Lemm. 4. pag. 478. - -[208] Princ. philos. Lib. III. prop. 40. - -[209] Book I. chap. 2. § 82. - -[210] Princ. philos. Lib. III. pag. 499, 500. - -[211] Ibid. pag. 500, and 520, &c. - -[212] Princ. Philos. Lib. III. prop. 40. - -[213] Ibid. prop. 41. - -[214] Ibid. pag. 522. - -[215] Ibid. prop. 42. - -[216] Newt. Princ. philos. edit. 2. p. 464, 465. - -[217] Ibid. edit. 3. p 501, 502. - -[218] Ibid. pag. 519. - -[219] Ibid. pag. 524. - -[220] Newt. Princ. philos. p. 525. - -[221] Ibid. - -[222] Ibid. pag. 508. - -[223] Ibid. - -[224] Ibid. pag. 484. - -[225] Ibid. pag. 482, 483. - -[226] Ibid. pag. 481. - -[227] Ibid. pag. 509. - -[228] See the fore-cited place. - -[229] Ibid. and Cartes. Princ. Phil. part. 3. § 134, &c. - -[230] Vid. Phil. Nat. princ. Math. p. 511. - -[231] Book I. Ch. 4. § 11. - -[232] Ch. 5. - -[233] All these arguments are laid down in Philos. Nat. Princ. Lib. -III. from p. 509, to 517. - -[234] Philos. Nat. Princ. Lib. III. p. 515. - -[235] Ch. 5. - -[236] See Ch. 1. § 11. - -[237] Newt. Princ. Philos. pag. 525, 526. An account of all the stars -of both these kinds, which have appeared within the last 150 years may -be seen in the Philosophical transactions, vol. 29. numb. 346. - -[238] Newt. Princ. Philos. Nat. Lib. III. prop. 6. - -[239] Ch. 3. § 6. - -[240] Book I. Ch. 2. § 24. - -[241] Newt. Princ. Lib. III. prop. 6. - -[242] Ch. 3. § 6. - -[243] Newt. Princ. philos. Lib. III. prop. 7. cor. 1. - -[244] See Book I. Ch. 1. § 15. - -[245] Ibid. § 5, 6. - -[246] Chap. 2. § 8. - -[247] Newt. Princ. Lib. I. prop. 63. - -[248] § 8. - -[249] See Introd. § 23. - -[250] § 4, 5. - -[251] Newt. Princ. philos. Lib. I. prop. 74. - -[252] Ibid. coroll. 3. - -[253] Lib. I. Prop. 75. and Lib. III. prop. 8. - -[254] Lib. I. Prop. 76. - -[255] Ibid. cor. 5. - -[256] Vid. Lib. III. Prop. 7. coroll. 1 - -[257] Newt. Princ. Lib. III. prop. 8. coroll. 1. - -[258] Ibid. coroll. 2. - -[259] Book I. Ch. 4. § 2. - -[260] Newt. Princ. Lib. III. prop. 8. coroll. 3. - -[261] Ibid. coroll. 4. - -[262] Book I. Ch. 4. - -[263] Lib. II. prop. 20. cor. 2. - -[264] Chap. 4. § 17. - -[265] Ibid. - -[266] Vid. Newt. Princ. Lib. II. prop. 46. - -[267] Princ. philos. Lib. II. prop. 49. - -[268] Chap. 3. § 18. - -[269] Newt. Princ. philos. Lib. I. prop. 66. coroll. 18. - -[270] § 8. - -[271] Ch. 3. § 5. - -[272] Ch. 3 § 17. - -[273] Ibid. - -[274] See below § 44. - -[275] Newton Princ. Lib. III. prop. 19. - -[276] Lib. III. prop. 19. - -[277] Lib. I. prop. 73. - -[278] Lib. III. prop. 20. - -[279] Ibid. - -[280] Opt. B. I. part. 2. prop. 1. - -[281] Newt. Opt. B. 1. part 1. experim. 5. - -[282] Ibid. prop. 4. - -[283] Newt. Opt. B. 1. part 2. exper. 5. - -[284] Ibid exper. 6. - -[285] Newton Opt. B. I. prop. 10. - -[286] Ibid exp. 9. - -[287] Newt. Opt. B. I. part 1. exp 15. - -[288] Philos. Transact. N. 88, p. 5099. - -[289] Opt B. I. par. 2. exp. 14. - -[290] Ibid. exp. 10. - -[291] Opt. pag. 122. - -[292] Opt. B. I. part 2. exp. 11. - -[293] Ibid prop. 4, 6. - -[294] Opt. pag. 51. - -[295] Opt. Book II. prop. 8. - -[296] Opt. Book II. par. 3. prop. 2. - -[297] § 17. - -[298] Opt. Book II. par. 3. prop. 4. - -[299] Opt. Book II. pag. 241. - -[300] Ibid. pag. 224. - -[301] Ibid. Obs. 17. &c. - -[302] Ibid. Obs. 10. - -[303] Ibid. pag. 206. - -[304] Obser. 21. - -[305] Observ. 5. compared with Observ. 10 - -[306] Ibid. prop. 5. - -[307] Observ. 7. - -[308] Observ. 9. - -[309] Ibid prop. 7. - -[310] Opt. pag. 243. - -[311] Newt. Opt. B. I. part. 1. prop. I. - -[312] Opt. B. I. part. 1. prop. 2. - -[313] Opt. B. I. part 1. Expec. 6. - -[314] Opt. pag. 67, 68, &c. - -[315] Ibid. B. 1. par. 2. prop. 3. - -[316] Opt. B. II. par. 3. prop. 10. - -[317] Opt. B. II. par. 3. prop. 15. - -[318] Ibid. par. 1. observ. 7. - -[319] Ibid. Observ. 19. - -[320] Opt. B. II. par. 2. pag. 199. &c. - -[321] Ibid. par. 4 - -[322] Ibid. part. 3. prop. 13. - -[323] Ibid. prop. 17. - -[324] Ibid. prop. 13. - -[325] Opt. Qu. 18, &c. - -[326] See Concl. S. 2. - -[327] B. II. Ch. 1. - -[328] Opt. B. III. Obs. 1. - -[329] Ibid. Obs. 2. - -[330] § 2. - -[331] Philos. Trans. No. 378. - -[332] § 11. - -[333] Philos. Transact No. 375. - -[334] Ibid. - -[335] Opt. B. II. part 4. - -[336] Ch. 3. § 14. - -[337] Opt. B. II. part 4. obs. 13. - -[338] Opt. pag. 255. - -[339] Ch. 3. § 18. - - - - - - -End of the Project Gutenberg EBook of A View of Sir Isaac Newton's Philosophy, by -Anonymous - -*** END OF THIS PROJECT GUTENBERG EBOOK SIR ISAAC NEWTON'S PHILOSOPHY *** - -***** This file should be named 53161-0.txt or 53161-0.zip ***** -This and all associated files of various formats will be found in: - http://www.gutenberg.org/5/3/1/6/53161/ - -Produced by Giovanni Fini, Markus Brenner, Irma Spehar and -the Online Distributed Proofreading Team at -http://www.pgdp.net (This file was produced from images -generously made available by The Internet Archive/Canadian -Libraries) - - -Updated editions will replace the previous one--the old editions will -be renamed. - -Creating the works from print editions not protected by U.S. copyright -law means that no one owns a United States copyright in these works, -so the Foundation (and you!) can copy and distribute it in the United -States without permission and without paying copyright -royalties. 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