li UNIVERSITY OF CALIFORNIA AT LOS ANGELES To the Noble and Right Honourable Sir ROBERT W A L P L E. SIR, Take the liberty to fend you this view of Sir I s a A c N e w- ton's philofophy, which, if It were performed fuitable to the dignity of the fubje6l, might not be a prefent unworthy the acceptance of the greateit perfon. For his phi- lofophy affords us the only true account of the A 2 opera- 403700 DEDICATION. Operations of nature, which for fo many ages had imployed the curiofity of mankind ; though no one before him was fornillied with the ftrength of mind neceffary to go any depth in this difficult fearch. However, I am encouraged to hope, that this attempt, imperfect as it is, to give our countrymen in general fome concep- tion of the labours of a perfon, who lliall al- ways be the boaft of this nation, may be re- ceived with indulgence by one, under whofe influence thefe kingdoms enjoy fo much hap- pinefs. Indeed my admiration at the furprizing inventions of this great man, carries me to con- ceive of him as a perfon, who not only muft raife the glory of the country, which gave him birth ; but that he has even done honour to hu- man nature, by having extended the greateft and mofl noble of our faculties, reafon, to fub- ie6ls, which, till he attempted them, appeared to be wholly beyond the reach of our limi- ted capacities. And what can give us a more DEDICATION. more pleafiiig profpecl of our own condition, than to fee fo exalted a proof of the ftrength of that faculty, whereon the condu6l of our lives, and our happinefs depends ; ourpaflions and all our motives to a6lion being in fuch manner guided by our opinions, that where thefe are jufi:, our whole behaviour will be praife-worthy ? But why do I prefume to de- tain you, Sir, with fuch reflections as thele who muft have the fulled experience within your own mind, of the efle^ts of right reafon ? For to what other fource can be afcribed that amiable franknefs and unreferved condefcenfion among your friends, or that mafculine perfpicuity and ftrength of argument, whereby you draw the admiration, of the publick, Avhile you are engaged in the moft important of all caufes the liberties of mankind? I humbly crave leave to make the only ac- knowledgement within my power, for the benefits, Avhich DEDICATION. which I receive in common with the reft of my countrymen from thefe high talents, by fub- fcribing my felf o I jR., Your moft faithfuh and Mojl humble Servant] Henry Pemberton. ^ P R E F A C TireiD lip the follo-wIng papers many years ago at the defire of feme friends^ who^ upon my taking care of the late edition of. Sir IsaacNewton 'sTrincipia^perfwaded me to make them- pnhlich I laid hold of that opportunity^ when my thoughts, •were afreJJj employed on this fiih\eU^ to revife what I had formerly, written. And I now [end it abroad not withota fome hopes of an- fwering thefe two ends. My firji intentionwasto convey to [iich^asare not fifed to mathematical reafoningyfome idea of the phikfophy of a perfon^. who has acquired an unicerfal reputation^ and rendered our nation • famous for thefe fpeculations in the learned world. To which pur—- pofe I have avoided iifing terms of art aS' much as poffihle^y and ta- ken care to define fuch as I was oblige d to ufe. Though this cautioiv was the lefs neceffary at prefent, Jince many of them are become fa- miliar words to our language, from the great number of hooks wrote in it upon philofophical fubjecis, and the courfes of experiments, that have of late years been given by fever al ingenious men. The other view I had, was to encourage fuch young gentlemen as have a turn for the mathematical fciences, to pur foe thofejiudies the more chcarfully^ in order to underftand in our author himfelfthe demonfirations of t ba- things I here declare. And to facilitate their progrefs herein^ 1 in- tend to proceed fti II farther in the explanation of Sir Isaac New- ton 'j philofophy. For as J have received very much pleafure from penifmg his writings^ I hope it ism illaudahle ambition to endeavour the rendering them more eafily imderjiood^ that greater numbers may- enjoy the fame fatisfaUion. It will perhaps be expefHed^ that I flmild fay fomething particu- lar ofaperfon^ to whom Imuft always acknowledge my felfto hermich- obliged. What I have to declare on this head will be hut fhort ^ . for^ it was in the very laji years of Sir I s a a cV life, that I had the ho. ^ z noun- PREFACE. iwtir of his acquaintance. This happened on the following occa/ion^ Mr. Polenus, a 'Profeffor in the TJni'verfny of Padua, from a new ex- periment of his^ thought the common opinion about the force of moving bodies was overturned, and the truth of Mr. LibnitzV notion in that matter f idly prot'ed. The contrary of what Polenus had aferted I demonjirated in a paper, which 1)r. Mead, who takes all opportii~ vities of obliging his friends, was pleafed to JJoew Sir Isaac New- ton. This was fo well approved of by him^ that he did me the ho- nour to become a fellow-writer with me, hy annexing to what I had written, a demonftration of his own drawn from another confidera- tion. When I printed my difconrfe in the philofophical tranfaUions, I put what Sir Isaac had written in a fcholium by it [elf, that I might not feem to ufurp what did not belong to me. "Bnt I concealed his name, net being then fufficiently acquainted with him to ash whe- ther he was willing I might make nfe of it or not. In a little time after he engaged me to take care of the new edition he was about making of his Trincipia. This obliged me to be very frequently with him, and as he lived at fome dijiance from me, a great number of letters paffed between lis on this account. When I had the honour of his converfation, I endeavoured to learn his thoughts upon mathema- tical fnbjeUs, and fomething hijiorical concerning his inventions, that I had not been before acquainted with. I found, he had re ad fewer of the modern mathematicians, than one could have expeBed -, but his own prodigious invention readily fupp lied him with what he might have an occafan for in the purfuit of anyfubje'ci he undertook. I have often heard him cenfure the handling geometrical fubje'cis by algebraic calculations', and his book of Algebra he called by the name ofUniverfal Arithmetic, in oppojition to tl^e in'pidicious title of Geometry, which Des Cai-tes had given to the treatife, wherein he floews, hew the geonteter'inay aj]ifi his invention byfuch kind of computations. He frequently prAifed Slufius, Barrow ^//^ Huygens/t>r not being, influenced by the falfe tafte, which then began to prevail. He tifed to commend the laudable attempt of HPiigo deOmerique torcftore the ancient analvjts, and very much ejicemed k.^^o.' loniusV book De fedioiie rationis for giviilg us a clearer notion of that Mnalyjis than we had before. T)r. Barrow may be ejieemed as hav- ing PREFACE. ing JJ-iewn a compafs of im'ention equals if not fnpericr to any of the moderns y onr author only excepted; hut Sir Isaac Newton has fei'eral times particularly recommended to me HuygensV ft He and manner. He thought him the moft elegant of any mathematical wri- ter of modern times ^ and the moji juji imitator of the antients. Of their tafte^ and form of demonjiration Sir Isaac always profejfed hiwfelf a great admirer : Ihace heard him eoen cenfure himfelf for not following them yet more clcfely than he did ; and [peak with re- gret of his miftake at the heginning of his mathematical findies^ in applying himfelf to the works of Des Cartes and other algebraic wri- ters^ before he had confidered the elements cf Euclide with that atten- tion^ which fo excellent a writer deferces. As to the hiftory of his im'cntions^ what relates to his difccceries of the methods of feries and fluxions^ and of his theory of light and colours , the world has been fuf- Jiciently informed of already. The frft thoughts^ which ga-ve rife to his Trincipia^ he had^when he retired from Cambridge in \ 6 66 on account of the plague. As he [at alone in a garden, he fell into a [peculation on the power of gravity: that as this power is not found fenfibly diminifl^ed at the remotefl difiance from the center of the earthy to which we can rife, neither at the tops of the loftiefi buildings^ nor even on the fnmmits of the high eft mountains ; it appeared to him reafouable to conclude, that this power mi ft extend much farther than was ifually thought -, why not as high as the moon, faid he to him- felf 1 and iffo, her motion muft be influenced by it-^ perhaps Jl.ie is re- tained in her orbit thereby. Hcwet'er, tho2igh the power of graz'ity is not fenfibly weakened in the little change of diftance^ at which we can place our [ekes from the center of the earth ; yet it is eery poffible-> that [o high as the moon this power may differ much in firength from what it is here. To make an eftimate, what might be the degree of this diminution, he confidered with himfelf, that if the moon he re- tained in her crbit by the force of gravity, no doubt the prijnary pla- nets are carried round the fun by the like power. And b'^ comparing the periods of the [eceral planets with their diftances fromthe [tin, he found, that if any pi wer like gravity held them in their courfes, its firength mtiji decreafe in the duplicate proportion of the increafe of difiance. This [a] he PREFACE. he concluded hy fnppcfing them to mot'e in peifeB circles concentrical to the fun., from which the orbits of the great ejl part of them do i:ot much differ. Stippofmg therefore the power of graiity, when extended to the moon, to decreafc in the fame manner, he computed whether that force would he fufficient to keep the moon in her orhit. In this computation, being abfent from hooks, he took the common efiimate in life among geographers and our feamen, before Norwood had mea-- fared the earth, that 60 Engl ifJj miles were contained in one degree of latitude on the fuiface of the earth. Sut as this is a eery faulty fuppofition, each degree containing about 69 { of our miles, his com- putation did not anfwer expeUation-, whence he concluded, that fome- other caufe muft at leaji join with the aUion of the power of gravity on the moon. On this account be laid aft de for that time any farther thoughts upon this matter. TBut fome years after, a letter which he receiced from T)r. Hool^, put him on inquiring what was the real JJgtire, in which a body let fall frojn any high place defcends, taking the mition of the earth round its axis into confideration. Such a body, hating the fame motion, which by the reiwlution of the earth the place has whence it falls, is to be confidered as projeUed forward and at the fame time drawn down to the center of the earth. This face occafion to his refiiming his former thoughts concerning the moon ; and Picart //; France hadng lately meafured the earth, by tifing his meafures the Tnoon appeared to be kept in her orbit purely by the power of gravity -, and confequently, that this power decreafes as you recede ftom the center of the earth in the manner our author had formerly conjectured. Upon this principle he found the line de- fer Ihed by a falling body to be an ellipfis, the center of the earth be- in^ one focus. And the primary planets moving in fuch orbits round the fun, he had the fatisfaUion to fee, that this inquiry, which he had undertaken merely out of curiofity, could be applied to the greateft pu^rpofcs^ Hereupon he compofed near a dozen proportions relating to the motion of the primary planets about the fun. Several years after this, fome difcourfe he had with 'Dr. Halley, who at Cambridge made him a vifit, engaged Sir Isaac Newton to nfumc again the confideration of this fitbjecf -, and gave occafion . to PREFACE. to his miting the treat I fc -which he piihiiJJoed wider the title of nta' thematic al principles of natural phikfophy. This treat ife, full of fiich a 'variety of profound int'entions, was compofed hy him front fcarce any other materials than the few proportions before mentioned^ in the [pace of one year and an half. Ithongh his memory was mnch decayed^ I found he perfeUly mi^ derjiood his own writings^ contrary to what I had frequently heard in difconrfe from many perfons. This opinion of theirs might arife perhaps from his not being always ready at [peaking on thefe fiih- jeUs^ when it might be expeUed he flmdd. 'But as to this^ it may be obferi'ed, that great genius'' s are frequently liable to be ahfent, not only in relation to common life^but with regard tofome of the parts offcience they are the beji informed of. In-ventors feemto treajure up in their minds^ what they hace found out^ after another manner than thofe do the fame things, who have not this im'entit'e faculty. The former-, when they hace occafwn to produce their knowledge:, are in fome mea- fure obliged immediately to incejii gate part of what they want. For this they are not equally Jit at all tiynes : fo it has often happened^ that fuch as retain things chiefly by weans of a 'very firong memory^ hai'e appeared offhand more expert than the difcoverers themfekes. Js to the moral endowments of his mind, they were as much to be admired as his other talents. "But this is a feld Ilea've others to exfpatiate in. I only touch upon what I experienced my [elf during the few years I was happy in his friendJJjip. But this I immediately difcovered in him, which at once both furprized and charmed me : Neither his extreme great age, nor his iini'verfal reputation had rendred him jliff in opinion, or in any degree elated. Of this I had occafwn to have almcji daily experience. The Remarks I continually fent him ly letters on his Trincipia were received with the utmoft goodnefs. Thefe were fo far from beiu^^ any ways d/fplea- fing to him, that on the contrary it occafioned him to f peak many kind things of me to my friends, and to honour me with a puhlick teftir.iony of his good opinion. He alfo approved of the following treatife, a great part of which we read together. Js many alterations were [] a 2 ] made PREFACE. .maJe in the late edition of bis Trincipia^ fo there ':^oiild h ace been many more if there had been a fuficient time, ^ut whatet^er of this kind may bethought iDanting^ lOjallendeacoitrtoftipply in my com- ment on that booh I had reafon to believe he expeUed fitch a thing from me^ and I intended to have puUiflied it in his life time^ after I had printed the following difcoiirfe^ and a mathematical treatife Sir Isaac Ne-\vton had written a long while ago, containing the Jirfi 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 treatife had never been finiJJjed, he was about letting me have other papers, in order to Supply what was wanting. "But his death put a flop to that de- ■fign. As to my comment on the 'Principia, I intend there to de- monjirate whatever Sir Isaac Newton has fet down without exprefs proof, and to explain allfuch expreffions in his book, as Iflmll judge neceffary. T'his comment 1 floall forthwith put to the prefs-^ joined to an englifJo tranfiation of his Principia, which I have had fome time by me. J more particular account of tny whole de- fign has already been publifoed in the new memoirs of literature for the month of march 1727. I have prcfentcd my readers with a copy of verfes on Sir Isaac N E w T o N, which I have juji received from a young Gentleman, whom I am proud to reckon among the number of my dear eft friends^ If I had any apprehevfion that this piece of poetry flood in need of an apology, I JJjotdd be deflrous the reader might know, that the tiuthor is but fixteen years old, and was obliged to finiflj his compofi- tion in a very fliort f pace of time. But 1 fJjall only take the liberty to ohferve, that the boldnefs of the digreffions will be befl judged cf ly thofe who are acquainted with Pindar. A PO E M POEM O N Sir ISAAC NEWTON. TO N E w T o n's genius, and immortal fame Th' advent'rous mule with trembling pinion Ibars. Thou, heav'nly truth, from thy feraphick throne Look favourable down, do thou aflift My lab'ring thought, do thou inlpire my long. Newton, who lirft th' almighty's works dilplay'd, And fmooth'd that mirror, in whole polilh'd face The great creator now confpicuous fhines ; Who open'd nature's adamantine gates. And to our minds her fecret powers expos'd ; Newton demands the mule j his facred hand Shall guide her infant fteps ; his lacred hand Shall raile her to the Heliconian height, Where, on its lofty top inthron'd, her head Shall mingle with the Stars. Hail nature, hail, O Goddels, handmaid of th' ethereal power, Now lift thy head, and to th' admiring world Shew thy long hidden beauty. Thee the wile Of ancient fame, immortal P l a T o's lelf, The Stagyrite, and Syracufian fage. From A Poem on Sir Isaac Newton. From black obfcurlty's abyls to raife, (Drooping and mourning o'er thy wondrous works) With vain inquiry fought. Like meteors thefe In their dark age bright Ions of wiidom fhone : But at thy Newton all their laurels fade, They fhrink from all the honours of their names. So glimm'ring ftars contract their feeble rays, When the fwift luftre of A u r o R a's free Flows o'er the skies, and wraps the heav'ns in light. The Deity's omnipotence, the caule, Th' original of things long lay unknown. Alone the beauties prominent to fight (Of the celeftial power the outward form) Drew praife and wonder from the gazing world. As ^^•hen the deluge overlpread the earth, Whilft yet the mountains only rear'd their heads Above the furface of the wild expanfe, Whelm'd deep below the great foundations lay, Till fome kind angel at heav'n's liigh command Roul'd back the rifmg tides, and haughty floods, And to the ocean thunder'd out his voice : Quick all the fwelling and imperious waves, The foaming billows and obfcuring furge, Back to their channels and their ancient feats Recoil affrighted : from the darkfome main Earth raifes fmiling, as new-born, her head, And with frefti charms her lovely face arrays. So his extenfive thought accomplifh'd lirll The mighty task to drive th' obftmcling mifts Of ignorance away, beneath whofe gloom Th' inihrouded majefty of Nature lay. He drew the veil and fwell'd the fpreading fcenc. How had the moon around th' ethereal void Rang'd, A Poem on Sir Isaac Newton. Rang'd, and eluded lab'ring mortals care, Till his invention trac'd her lecret fteps, While Ihe inconftant with unfteady rein Through endlefs mazes and meanders guides In its unequal courfe her changing carr : Whether behind the fun's fuperior light She hides the beauties of her radiant face, Or, when conlpicuous, fmiles upon mankind, Unveiling all her night-rejoicing charms. When thus the filver-trefled moon dilpels The frowning horrors from the brow of night. And with her Iplendors chears the fullen gloom, W'hile fable-mantled darknels with his veil The vifage of the fair horizon Jhades, And over nature Ipreads his raven wings ; Let me upon fome unfrequented green While fleep fits heavy on the drowfy world. Seek out Ibme Iblitary peaceful cell, Where darklbme woods around their gloomy brows Bow low, and ev'ry hill's protended fiiade Oblcures the dusky vale, there filent dwell. Where contemplation holds its ftill abode, There trace the wide and pathlels void of heav'n. And count the ftars that Iparkle on its robe. Or elfe in fancy's wild'ring mazes loft Upon the verdure fee 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 N e \v t o n let me foar with thee, And while furveying all yon ftarry vault With admiration I attentive gaze. Thou Ihalt defcend from thy celeftial feat. Aiid A Poem on Sir Isaac Newton. And waft aloft my high-afpiring mind, Shalt fhew me there how nature has ordain'd Her fundamental laws, Ihalt lead my thought Through all the wand'rings of th' uncertain moon, And teach me all her operating powers. She and the fun with influence conjoint Wield the huge axle of the whirling earth, And from their juft diredion turn the poles. Slow urging on the progrefs of the years. The conftellations feem to leave their feats, And o'er the skies with folemn pace to move. You, fplendid rulers of the day and night. The feas obey, at your refiftlefs fway Now they contract their waters, and expofe The dreary defart of old ocean's reign. The craggy rocks their horrid fides difclofe ; Trembling the lailor views the dreadful fcene, And cautioufly the threat'ning ruin ftiuns. But where the fhallow waters hide the fands. There ravenous deftrudlion lurks conceal'd. There the ill-guided velfel falls a prey, And all her numbers gorge his greedy jaws. But quick returnmg fee th' impetuous tides Back to th' abandon'd ftiores impell the main. Again the foaming feas extend their waves. Again the rouling floods embrace the Ihoars, And veil the horrours of the empty deep. Thus the obfequious feas your power confefs, While from the furfice healthful vapours rife plenteous throughout the atmofphere difi"us'd, • Or to iupply the mountain's heads with fprings, Or fill the hanging clouds with needful rains. That friendly ftreams, and kind refrelhing fhow'rs May gently lave the fun-burnt thirfty plains, Or A Poem on Sir Isaac Newton". Or to replenilh all the empty air With whollbme moifture to increale the fruits Of earth, and blefs the labours of mankind. O Newton, whether flies thy mighty foul, How fhall the feeble mufe purlue through all The vaft extent of thy unbounded thought, That even leeks th' unfeen recefles dark To penetrate of providence immenfe. And thou the great difpenfer of the world Propitious, who with infpiration taught'ft Our greateft bard to lend thy praifes forth ; Thou, who gav'lt Newton thought j who fniil'dft ferenc, When to its bounds he ftretch'd his fwelling foul j Who ftill benignant ever blefi: his toil, And deign'd to his enlight'ned mind t' appear Confefs'd around th' interminated world : To me O thy divine infufion grant (O thou in all fo infinitely good) That I may fing thy everlafting works. Thy inexhaufted ftore of providence. In thought effulgent and rclbunding verfo. O could I fpread 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 blafts. To where the boift'rous tempeft-leading fouth From their deep hollow caves lend fortii their ftorms. Thou ftill indulgent parent of mankind, Left humid emanations Ihould no more Flow from the ocean, but diflTolve away Through the long feries of revolving time j And left the vital principle decay. By which the air lupplies the fprings of life ; Thou haft the fiery vilag'd comets form'd [b] With A Poem on Sir Isaac Newton. With vivifying Iplrits all replete, Which they abundant breathe about the void, Renewing the prolifick Ibul of things. No longer now on thee aniaz'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 gralp the skies engirdle round. And Ipread the terrors of their burning locks. For thele through orbits in the length'ning fpace Of many tedious rouling years compleat Around the fun move regularly on j And with the planets in harmonious orbs. And my flick periods their obeyfance pay To him majeftick ruler of the skies Upon his throne of circled glory fixr. He or Ibme god conlpicuous to the view. Or elfe the lubftitute of nature feems, Guiding the courfes of revolving worlds. He taught great Newton the all-potent laws Of gravitation, by whole fimple power The univerle exifts. Nor here the fage Big with invention ftill renewing ftaid. But O bright angel of the lamp of day, How Ihall the mule difplay his greateft toil ? Let her plunge deep in Aganippe's waves, Or in Caftalia's ever-flowing ftream, That re-infpired Ihe may fing to thee, How Newton dar'd advent'rous to unbraid The yellow trelTes of thy fhining hair. Or didft thou gracious leave thy radiant fphere. And to his hand thy lucid fplendours give, T' unweave the light-diffuling wreath, and part The A Poem on Sir Isaac Newton. The blended glories of thy golden plumes ? He with laborious, and unerring care, How difF'rent and imbodied colours form Thy piercing light, with juft diftindion found. He with quick fight purfu'd thy darting rays, When penetrating to th' obfcure recels Of Iblid matter, there perlpicuous faw, How in the texture of each body lay The power that feparates the diff'rent beams. Hence over nature's unadorned face Thy bright diverfifying rays dilate Their various hues : and hence when vernal rains Delcending fwift have burft the low'ring clouds. Thy fplendors through the diflipating mifts In its fair vefture of unnumber'd hues Array the Ihow'ry bow. At thy approach The morning rilen from her pearly couch With rofy blufhes decks her virgin cheek j The ev'ning on the frontifpiccc of heav'n His mantle Ipreads with many colours gay ; The mid-day skies in radiant azure clad, The Ihining clouds, and filver vapours rob'd In white tranfparent intermixt with gold, With bright variety of fplendor cloath All the illuminated face above. When hoary-headed winter back retires To the chill'd pole, there Iblitary fits Encompals'd round with winds and tempefts bleak In caverns of impenetrable ice, And from behind the diflipated gloom Like a new Venus from the parting furge The gay-apparcll'd fpring advances on ; When thou in thy meridian brightncls fitt'ft. And from thy throne pure emanations flow [b a] Of A Poem on Sir Isaac Newton. of glory burfting o'er the radiant skies : Then let the mule Olympus' top afcend, And o'er Thcflalia's plain extend her view, And count, O Tempe, all thy beauties o'er. Mountains, whole fummits grafp the pendant clouds, Between their wood-invelop'd Hopes embrace The green-attired vallies. Every flow'r Here in the pride of bounteous nature clad Smiles on the bofom of th' enamell'd meads. Over the fmiling lawn the filver floods Of fair Pcneus gently roul along, While the reflcded colours from the flow'rs. And verdant borders pierce the lympid waves, And paint with all their variegated hue The yellow lands beneath. Smooth gliding on The waters haften to the neighbouring fea. Still the pleas'd eye the floating plain purfues; At length, in Neptune's wide dominion loft. Surveys the Ihining billows, that arife Apparell'd each in Phoebus' bright attire :' Or from a far fome tall majeftick Ihip, Or the long hoftile lines of threat'ning fleets, Which o'er the bright uneven mirror fweep. In dazling gold and waving purple deckt j Such as of old, when haughty Athens power Their hideous front, and terrible array Againft Fallene's coaft extended wide, And with tremendous war and battel ftern The trembling walls of Potidsea Ihook. Crefted with pendants curling with the breeze The upright mafts high briftlc in the air. Aloft exalting proud their gilded heads. The filvcr waves againft the painted prows Raife their rcfplcndent bofoms, aud impcarl The A Poem on Sir Isaac Newton. The fair vermilliou with their glift'ring drops : And from on board the iron-cloathed hoft Around the main a gleaming horrour cafls ; Each flaming buckler like the mid-day fun, Each plumed helmet like the filvcr moon, Each moving gauntlet like the light'ning's blaze, . And like a ftar each brazen pointed Ipear. But lo the facred high-ereded fanes, Fair citadels, and marble-crowned towers, And liimptuous palaces of ftately towns Magnificent arife, upon their heads Bearing on high a wreath of filver light, . But lee my mule the high Pierian hill, Behold its Ihaggy locks and airy top,,-. Up to the skies th' imperious mountain heaves The fhining verdure of the nodding woods. See where the filver Hippocrene flows. Behold each glitt'ring rivulet, and rill Through mazes wander down the green defcent, . And fparkle through the interwoven trees. Here reft a while and humble homage pay. Here, where the facred genius, that infpir'd Sublime M^t o n i d e s and P i n d a r's breaft. His habitation once was fam'd to hold. Here thou, O Homer, offer'dft up thy vows j Thee, the kind mule Calliop^a heard, And led thee to the empyrean feats. There manifefted to thy hallow'd eyes The deeds of gods; thee wile Minerva taught ^. The wondrous art of knowing human kind j Harmonious Ph OE B u s tun'd thy heav'nly mind, And fwell'd to rapture each exalted lenfe ; • ' Even Mars the dreadful battle-ruling god. Mars taiight thee war, and with his bloody hand Inftrudted .A Poem on Sir Isaac Newton. Inftructed thine, when in thy founding lines We hear the rattling of Bellona's carr, The yell of dilcord, and the din of arms. Pindar, when mounted on his fiery fteed, Soars to the fun, oppofing eagle like His eyes undazled to the fierceft rays. He firmly feated, not like Glaucus' fon, Strides his fwift-winged and fire-breathing horle, And born aloft ftrikes with his ringing hoofs The brazen vault of heav'n, fuperior there Looks down upon the ftars, whofe radiant light Illuminates innumerable worlds, That through eternal orbits roul beneath. But thou all hail immortalized Ion Of harmony, all hail thou Thracian bard, To whom Apollo gave his tuneful lyre. O might'ft thou, Orpheus, now again revive, And Newton fhould inform thy lift'ning ear How the foft notes, and Ibul-inchanting ftrains Of thy own lyre were on the wind convey 'd. He taught the mule, how found progreffive floats Upon the waving particles of air, When harmony in ever-pleafing ftrains, Melodious melting at each lulling fall, With foft alluring penetration fteals Through the enraptur'd ear to inmoft thought, And folds the lenfes in its filken bands. So the fwcct mufick, which from O R p h e u s' touch And fam'd A m r h i o n's, on the founding ftring Arofe harmonious, gliding on the air, Pierc'd the tough-bark'd and knotty-ribbed woods, Into their faps foft inlpiration breath'd And taught attention to the ftubborn oak. Thus when great Henry, and brave Marlb'rough I^ Th' A Poem on Sir Isaac Newton; Th' imbattled numbers of Britannia's Ions, The trump, that Iwells th' expanded cheek of fame, That adds new vigour to the gen'rous youth. And rouzes fluggilh cowardize it felf, The trumpet with its Mars-inciting voice, The winds broad breaft impetuous fweeping o'er Fill'd the big note of war. Th' iiifpired hoft With new-born ardor prels the trembling G a u l j Nor greater throngs had reach'd eternal night. Not if the fields of Agencourt had yawn'd Expofing horrible the gulf of fate ; Or roaring Danube Ipread his arms abroad, And overwhelm'd their legions with his floods. But let the wand'ring mufe at length return j Nor yet, angelick genius of the fun, In worthy lays her high-attempting long Has blazon'd forth thy venerated name. Then let her fweep the loud-refounding lyre Again, again o'er each melodious firing Teach harmony to tremble with thy praile. And ftill thine ear O favourable grant, And fhe Ihall tell thee, that whatever charms, Whatever beauties bloom on nature's face, Proceed from thy all-influencing light. That when arifing with tempeftuous rage, The North impetuous rides upon the clouds Difperfing round the heav'ns obftrudive gloom, And with his dreaded prohibition ftays The kind etfufion of thy genial beams j Pale are the rubies on Aurora's lips. No more the rofes blulh upon her cheeks, Black are Peneus' ftreams and golden fands In Tempe's vale dull melancholy fits, And every flower reclines its languid head. By •A Poem on Sir Isaac Newtok. ..' By what high name fhall I invoke thee, fay, Thou life-infufing 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 caufe, * By which found rouls thro' th' undulating air, f O had he, baffling times refiftlels power, , Difcover'd what that fubtle Ipirit is. Or whatlbe'er diffufive elfe is Ipread Over the wide-extended univerle. Which caufes bodies to refled the light. And from their ftraight diredion to divert The rapid beams, that through their furfice pierce. But fince cmbrac'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 feled Some other worthy of an angel's care, With inlpiration animate his breaft, .And him inftrucl in tliefe thy iecrct laws. O let not Newton, to whole Ipacious view, Now unobftrudcd, all th' extenfive fcenes Of the ethereal ruler's works arife ; When he beholds this earth he late adorn'd, Let him not fee philolbphy in tears. Like a fond mother folitary fit. Lamenting him her dear, and only child. But as the wiie Pythagoras, and he. Whole birth with pride the fam'd Abdera boafts, With expeftation having long liirvey'd This Ipot their antient leat, with joy beheld Divine philolbphy at length appear In all her charms majeftically fair. Conduced by immortal N e w t o n's hand : So A Poem on Sir Isaac Newton. So may he fee another fage aiile, That fhall maintain her empire : then no more Imperious ignorance with haughty Iway Shall ftalk rapacious o'er the ravag'd globe : Then thou, O Newton, fhalt proted thele lines, The humble tribute of the grateful mule ; Ne'er ftiall the facrilegious hand defpoil Her laurel'd temples, whom his name preferves : And were fhe equal to the mighty theme. Futurity Ihould wonder at her long ; Time Ihould receive her with extended arms, Seat her confpicuous in his rouling carr, And bear her down to his extreamcft bound. Fables with wonder tell how Terra's Ions With iron force unloos'd the ftubborn nerves Of hills, and on the cloud-infhrouded top Of Pelion Olfa pil'd. But if the vaft Gigantick deeds of lavage ftrength demand Aftonilhment from men, what then fhalt thou, O what expreflive rapture of the Ibul, When thou before us, N e w T o n, doft dilplay The labours of thy great excelling mind j W^hen thou unveileft all the woiidrous fcene, The vaft 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' extenfive heavens. O happy he, whofe enterprizing hand Unbars the golden and relucid gates Of th' empyrean dome, where thou enthron'd Philolbphy art leated. Thou fuftain'd By the firm hand of everlafting truth [c] Defpifeft A Poem on Sir Isaac Newton, Defpifeft all the injuries of time ; Thou never know'ft decay when all around, Antiquity oblcures her head. Behold Th' Egyptian towers, the Babylonian walls. And Thebes with all her hundred gates of brafi, Behold them fcatter'd like the dull abroad. Whatever now is flouriftiing and proud. Whatever Ihall, muft know devouring age. Euphrates' ftream, and feven-mouthed Nile, And Danube, thou that from Germania's foil To the black Euxine's far remoted fhore. O'er the wide bounds of mighty nations fweep'fi: In thunder loud thy rapid floods along. Ev'n you fhall feel inexorable time ; To you the fatal day Ihall come ; no more Your torrents then fhall fhake the trembling ground, No longer then to inundations fwoFn Th' imperious waves the fertile paftures drench, But fhrunk within a narrow channel glide j Or through the year's reiterated courle When time himfelf grows old, your wond'rous ftreams Loft ev'n to memory Ihall lie unknown Beneath obfcurity, and Chaos whelm'd. But ftill thou fun illuminateft all The azure regions round, thou guideft ftill The orbits of the planetary fpheres j The moon ftill wanders o'er her changing courfe,. And ftill, O Newton, ftiall thy name lurvivc : As long as nature's hand drrefts the world, When ev'ry dark obftrudion fliall retire. And ev'ry fecret yield its hidden ftore, Which thee dim-fighted age forbad to lee Age that alone could ftay thy riling foul. And could mankind among the fixed ftars, E'en A Poem on Sir Isaac Newton. E'en to th' extremcfl bounds of knowledge reacli, To thole unknown innumerable liins, Whole light but glimmers from thole dillant worlds, Ev'n to thofe utmoft boundaries, thole bars That Ihut the entrance of th' illumin'd Ipace Where angels only tread the vaft unknown, Thou ever fhould'ft be leen immortal there : In each new Iphere, each new-appearing lun, In fartheft regions at the very verge Of the wide univerfe Ihould'ft thou be leen. And lo, th' all-potent goddels Nature takes With her own hand thy great, thy juft reward Of immortality j aloft in air See Ihe dilplays, and with eternal gralp Uprears the trophies of great N e w t o n's fame. R. Glover. [c i] CON' THE C ONTENTS, I NTRODUCTWN concerning Sir Isaac NewtonV method of reafoning in philofophy pag. i Book I. Chap, i . Of the lanvs of motion The fir Jl laiv of motion proved p. 51 The fecond la'vj of motion pro'ved P- 3 ^ The third lanjo of motion fronjed p. 45 Chap. 2 . Further proofs of the Janns of motion The ejfeBs of percujjion p. 4;) 7he perpendicular defcent of bodies p. j y The oblique defcent of bodies in a jlrai^t line p. j 7 The cur-vi linear defcent of bodies p. ^K The perpendicular afcent of bodies ibid. jhe oblique afcent of bodies p. j*? Tide poiuer of gra'vity proportional to the quantity of\ matter in each body )^' 7he centre of gra'vity of bodies p. 6z The mechanical poivers p. 6p The leaver p. 71 The 'wheel and axis p. 77 The pulley p. 80 The nvedge p- 85 The fcreiv ibid. The inclined plam P* ^4 The CONTENTS. the pendulum p, 8 5 The centre of ofcillation p. 5? 4 'Experiments upon the percujjton of bodies made hy pendulums p . 5? 8 The centre of percuj/ton p, 100 The motion of projeUiles p. i o r The defcription of the conic feBions p. io<> The difference hetiveen ahfolute and relative motion^ as alfo bet'ween ahfolute and relative time ^ "' Chap. 3. Of centripetal forces p. 117 Chap. 4. Of the refijlance of fluids p. 145 Bodies are rejifled in the duplicate proportion of their} 'velocities j of elaflic fluids and their refjlance p. 145? Ho'W fluids may he rendered elajlic p. j ^ o The degree of rejijlance in regard to the proportion het-ixieen the denjity of the body and of the fluid In rare and uncompreffed fluids p. i j j In compreffed fluids p. i y ^ The degree of refijiance as it depends upon thefgure of bodies In rare and uncompreffed fluids p. i ^ j In compreffed fluids p. i j § Book II. 161 Chap, i . That the planets move in a fpace empty ofl^ fenfible matter S " The fyflem of the loorld defer ibed p. i6z The planets fuffer no fenfible refiflance in their motion p. 166 'ihey are not kept in motion by a fluid p. 167 That all fpace is not full of matter ^jjithout vacancies p. i p. i ;j 8 conjunUion and oppojition ■ j The dijlatice of the moon from the earth in the quarters'^ and in the conjunBion and oppojition is altered > p. zoo hy the fun j Thefe irregularities in the moons motion 'varied hy the? change of dijlance hetiveen the earth and fun S ^ The period of the moon round the earth and her dijlance") .. . , 'varied hy the fame means j The motion of the nodes and the inclination of the\ , \. ^ f p. 201 mooyi s orbit J ^ The motion cf the apogeon and change of the} „ eccentricity j i^* The CONTENTS. The inequalities of the Other fecondary planets deducthle\ from thefe of the moon j P' ^^^ C H A p. 4. of comets They are not meteors ^ nor placed totally 'without the\ planetary fyjlem j P* ^^*^ The fun aBs on them in the fame manner as on the? planets ^ P* ^ 3 '' Their orbits are near to parabola's P« i 5 ? The comet that appeared at the end of the year i ^. ij4 another comet i?i 7 5 years j Why the comets mo've in planes more different froml one another than the planets j P* ^5 5 The tails of comets p, 2. ? g The ufe of them p. 243, 244 The pofftble ufe of the comet it felf p. 245, i4 p. zjz. the fame nvith gra'vity j That the attraBive ponver in each of thefe bodies is proportional to the quantity of matter in the bodyC ibi J, attraBing Tlat Itfl CONTENTS. 'j'hat each particle of nvhich the fun and planets are") compojedis endued luith a7i attra&h?(r ponver, thX fJrengih of 'which is reciprocally in the dupli-C ^' cate proportion of the dijlance ^ the ponver of gravity univerfally belongs to all matter p. 255? 'tloe different 'weight of the fame body upon the furface- of the fun, the earth, Jupiter and Saturn j the re fpe&ive denftties of thefe bodies, and the propor- tion betnveen their diameters Chap. 6. Of the fluid parts of the planets The manner in ichich fluids prefs p. z6^ The motion of nva'ves on the fur face of 'water p. z6^ The motion of found through the air p, 270 The velocity of found p. iSz Concerning the tides p. 283 Tlje fgure of the earth p, 2,5J<> The effeU of this fgure upon the po'wer of gravity p. 501 The e^eB it has upon pendulums p. 30^ Bodies defcend perpendicularly to the fur face of the earth p. 304 The axis of the earth changes its direUion fwice a"]^ year, and fwice a month j r' ? > The fgure of the fecondary planets ibid. Book III. Chap, i . Concerning the caufe of colours ittherent in the light The funs light is compofed of rays of different colours p. 318 The refra&ion of light p. 315', 3^0 Bodies appear of different colour by day- light, becaufe'l fame reflet one kind of light more copioufy than > p. 3 li? the refl, and other bodies other kinds of light j The efeB of mixing rays of different colours p. 354 Chap. A LIST of fuch of the SUBSCRIBERS NAMES As are come to the Hand of the AUTHOR. MOn/eigneur d'Aguefleau, Cbimcelier r. Cotes, of Pomfret Caleb Cotefworth, M. D. Peter Cottingham, Efqi Mr John Cottington Sir John Hinde Cotton Mr James Coulter George Courthop, »/ Whiligh<» SulTcX, Efq; Mr Peter Courthope Mr John Coufsmaker, j»a, Mr Henry Coward, Merchant Anthony Afli'ey Cowptr, Efq; The Hon. Spencer Cowpc-r, Efqi One of the Rupees of the Court cf Common I'leas Mr Edward Cowper Rev. Mr John Cowper Sir Charles Cox Samuel Cox, Efq; Mr Cox, of ■ New Coll. Oxoa Mr Thomas Cox Mr Thomas Cradock, M. A. Rev. Mr John Craig Rev. Mr John Cranfton, ArclM deacon c/'Cloghor John Craller, Efq; Mr John Creech James Crcsd, Efq; Rev. Mr William Crery John Crew, of Crew Hal), in Chefhlre, Efq\ Thomas Crifp, Efq; Mr Richard Crifpe Rev. Mr Samuel Cufwick Tobias Croft, of Trinity Col- lege, Cambridge Mr John Crook Rev. Dr Crofle, Mapr cf Ka- thcrine Hall Chriftopher Crowe, Efq; George Crawl, Efq; Hon. Nathaniel Crump, Efq; of Antigua Mrs Mary Cud worth Alexander Cunningham, Eft]-, Hcorjr Subscribers Names. Henry Cunningham, Efqi Mr Cunningham Br Curtis of Scvenoak Mr William Curtis Henry Cur wen, Efi; Mr John Cafwall, of London, Merchant Dr Jacob de Caftro Sarraento Hh Grace the Duke of Devonfliire His Grace the Duke of Dorfet Right Rev. LJ. Bi/Jjop of Darham Right Rev. Ld. BiJJjop pf St. Da- vid Right Hon. Lord Delaware - Right Hon. Lord VJohy Right Rev. Lord Bi/liof of Derry Right Rev. Lord Bifiop cf Donne Rr. Rev. Lord Biflmpof Dromore Right Hon. Dalhn, Lord Chief Baron of Ireland Mr Thomas Dade Caff. John Da;;ge Mr Timothy Dallowe Mr Jarnes Danzey, Surgeon Rev. Dr Richard Daniel, Dean «/" Armagh Mr Danvers Sir Coniers Darcy, Knight of the Bath Mr Serjeant Darnel Mr Jofcph Dalh Peter Davall, Efq; Henry Davensnt, Efi; Davies Davenport, of the Inner- Temple, Efq; Sir Jermyn Davers, Bart. Capt. Thomas Davers Alexander Davie, Efq; Rev. Dr. Davies , Majler of Queen's College, Cambridge Mr John Davies, of Chrift- Church, Oxon Mr Davies, rillomiy at Law Mr Wiiliam Dawliins, Merch. Rowland Dawkin, (/Glamor- gan (hire, Efq, Mr John Dawfon Edward Dawfon, Efq; Mr Richard Dawfon William Dawfonne, Efqj, Thomas Djy, E/q; Mr |ohn Day Mr Nathaniel Day Mr Deacon Mr William Deane Mr James Dearden, of Trinity College, Cambridge Sir Matthew Deckers, Biirt. Edward Decring, Efq; Simon Degge, Efq; Mr Staunton Dcgge, ^. B. cf Trinity Col. Cambridge Rev: Dr Patrick Delaney Mr Ddhammon Rev. Mr Denne Mr William Denne Capt. Jonathan Dennis Daniel Deringi Efq; Jacob Desboverie, Efq; Mr James Devetdl, [Surgeoa^ in Briftol Rev. Mr John Diaper Mr Rivers Dickenibn Dr. George Dickens, of Liver- pool Hon . Edward Digby, Efq; Mr Dillingham Mr Thomas Dinely Mr Samuel Difney, cf Bennet College, Cambridge Robert Dixoiit^Sfq; . Pierce Dodd, M. D. Right Hon. Geo. Doddinton, Eff, Rev. 5(r John Doiben, ofFindon, Bart. Nehemiah Donellan, EJq; Paul Doranda, Efq; iamcs Douglas, M D. Ir Richard Dovey, ^. B. of Wadham College, Oxon John. Dawdal, Efq; William Mac Dowell, Efq; Mr Peter Downer Mr James Dowoes Sir Francis Henry Drake, Knt. WilliamjDrake, o/^Barnoidfwick- Cotes, Efq; Mr Rich. Drewett, of Fareham Mr Chriftopher Driffield, of Chrift-Church, Oxon Edmund Dris, A. M. Fellow 0/ Trinity Coll. Cambridge George Drummond, Efq; Lord Prozojl of Edtnburgh Mr Colin Drummond, Ptofeffor cf Phdofophy m the Umverjity of Edinburgh Henry Dry, Efq; Richard Ducane Efq; Rev. Dr Pafchal DucaflTe, Dean. cf Ferns George Ducket, Efq; Mr Daniel DutTclnay Mr Thomas Dug 'ale Mr Humphry Dii ualfe, Merchtmt Mr James Duncan iohn Duncombe, Efq; Tr WiJliam Duncombe John Dundafs, jun. of Duddifl- ftown, Efq; William Dui.ftar, Efq; James Dupont, cf Trinity Coll Cambridge E ■ R:igk Rev. and Right fftnt. Lord Erskine Theophilus, Lord Biflicp o/Elphisa Mr Thomas Eames Rev. Mr. Jabcz Earle - Mr William Eaft t:ir Peter Eaton Mr Jolin Ecclefton James Eckerfall, Efq; — — Edgecumbe, Efqj Rev. Mr Edgley Rev.'Dr Edraundfon, Rrefidenr of St. John's Coll. Carabridgs Arthur Edwards, Efq; Thomas Edwards, Efq; Vigerus Edwards, Efq; Capt. Arthur Edwards Mr Edwards Mr William Eldertoa Mrs Elizabeth Elgar Sir Gilbert Eliot, of Minto , Barf, one of the Lords of SelTion Mr John Elliot, Merchant George Ellis, of Barbadoes, Efq; Mr John EJlifoo, of Sheffield Sir Richard Ellys, Bart. Library of Emaauel - College, Cambridge Francis Emerton, Gent. Thomas Emmerfon, Efq; ^Jr Henry Emmet Mr John Emmet Thomas Empfon, cf the Middle- Temple, Efq;, Mr Thomas Engeir Mr Robert England Mr Nathaniel Englifli Rev. Mr En fly, Minifler of ihs Scotch Church m Rotterdam JohnEflington, Efq; Rev. Subscribers Name s. 5«v. Mr Charles Efte, of Clmd- Churcb, Oxon Mr Hugh Etherfey, apothecary Henry Evans of Surry, £/^' Ifaac Ewer, Eff, Mr Charles Ewer Rev. Mr Richard Exton Sir John Eyles, B^r. Sir Jofeph Eyies Ei^hi Hon. Sir Robert Eyre, Lord Chief Jii/lice of the Common Fleas. Edward Eyre, Eff, Henry Samuel Eyre, Efq, KingfmiJI Eyre, E/ji Mr Eyre Right Rev. Jofiab, Lord B'l/ljsp of Femes and Loghlin Den Heer Fagcl Mr Thomas Fairchild Thomas Fairfax, of the Mlddls Temple, Efq^ Mr John Falconer, Merchant Daniel Falkiner, Efq; Charles Farewell, Efq; Mr Thomas Farnaby, of Merton College, Oxon Mr William Farrel ]ames Farrel, Efq; Thomas Farrer, Efqi Dennis Farrer, Efq; ]ohn Farrington, Eff, Mr Faukener Mr Edward Faulkner Francis Fauquiere, E/^; Charles De la Fay, Efq; Thomas De lay Fay, Efqi Capt. Lewis De la Fay Nicholas Fazakerly, Efqi Governoitr Feake Mr John Fell, p/AttercIiffe Marcyn Fellowes, Eff, Coflon Fellows, E/qi Mr Thomas Fellows Mr Francis Fcnnell Mr Michael Fenwick John Ferdinand, of the Inner- Temple, Efq, Mr James Feme, Surgeon Mr John Ferrand, of Trinity College, Cambridge Mr Daniel MufTaphia Fidalgo Mr Fidler Hen. Mrs Celia Fiennes Hon. and Rev. Mr. Finch, Dean of York Ho'). Edward Finch, Efqi Mr John Finch Philip Fincher Efq; Mr Michael Firth, of Trinity College, Cambridge Hon. John Fita-Morris, Efqi Mr Fletcher Martin Folkes, Efq; Dr Foot Mr Francis Forerter John Forefter, Efqi Mrs Alice Forth .'Mr John Forthe Mr Jofeph Foskett Mr Edward Fofter Mr Peter Fofler Peter Foulkes, D. V>. Canon of Chrift-Churcfa, Oxon Rev. Dr. Robert Foulkes Rev. Mr Robert Foulks, M. ^. Eelloro of Magdalen College, Cambridge Mr Abel Founereau, Merchant Mr Chriflopher Fowler Mr John Fowler, ij/'Northamp. Mr Jofeph Fowler Hon. Sir William Fownes, Bar. George Fox, Efqi Edward Foy, Efq; Rev. Dr. Frankland, Dean of Gloucefter Frederick Frankland, Efq; Mr Jofeph Franklin Mr Abraham Franks Thomas Frederick, Efq; Gentle- man Commoner oftiew College, Oxon Thomas Freeke, Eff, Mr Jofeph Freame Richard Freeman, Efq; Mr Francis Freeman, of Briflol Ralph Freke, Efq; Patrick French, Efq; Edward French, M. D. Dr. Frewin John Freind, M. D. Mr Thomas Froft Thomas Fry, c/Hanham, Glou- cefterfliire, Efq; Mr Rowland Fry, Merchant Francis Fuljani, Efq; Rev. Mr Fuller, Fellovo of Ema- nuel College, Cambridge Mr John Fuller Thomas Fuller, M. D. Mr William Full wood, «/ Hun- tingdon Rev. James Fynney, D. D. Pre- bendary of Durham Capt. Fylhe Mr Francis Fayrana, BookfelUr in London His Grace the Duke of Grafton Right Hon. Earl of Godolphin Right Hon. Lady Betty Germain Right Hon. Lord Garlet R'ght Rev. Bijljop of Gloucefter Right Hon. Lord St. George Rt. Hon. Lord Chief Baron Gilbert Mr Jonathan Gale, c/' Jamaica Roger Gale, Efqi His Excellency Monfieur Galrao, Envoy of Portugal James Gambier, Efqi Mr Jofeph Gambol, o/Barbadoes Mr Jofeph Gamonfon Mr Henry Garbrand Rfv.Mr Gardiner Mr Nathaniel Garland Mr Nathaniel Garland, jun, Mr Joas Garland Mr James Garland Mrs Anne Garland Mr Edward Garlick Mr Alexander Garrett Mr John Gafcoygne, Merchant Rev. Dr Gasketh Mr Henry Gatham Mr John Gay Thomas Gearing, Efq; Coll. Gee Mr Edward Gee, of Queen's College, Cambridge Mr Jofhua Gee, fen. Mr Jofliua Gee, jun. Richard Fitz-Gerald, cf Cny's- Inn, Efq Mr Thomas Gerrard Edward Gibbon, Efqi John Gibbon, Efqi Mr Harry Gibbs Rev. Mr Philip Gibbs Thomas Gibfon, Efqi Mr John Gibfon Mr Samuel Gideon Rev. Dr Clandifh Gilbert, ef Trinity College, Dublin Mr John Gilbert John Girardot, Efq; Mr John Girl, Surgeon Rev. Dr, Gilbert, Dean of Exe- ter, 4. Books Mr Subscribers Names. Mr Gisby, Apothecary Mr Ricb'arJ G.'.uivilic iohn Glover, Efq, If John Glover, Merchant Mr i'homas Giover, Merchant John Goddard, Mtrchunt, m Rot- terdam Peter Godfrey, Efqi Mr Jofeph Godfrey Capt. Jolin Godiee Joleph Godnian, J5/^; Caft. Harry GofF Mr Thomas Goldney Jonatinn Goldfmyth, M. D, Rev. Mr William Goldwin Gooday, E/q. John Goodrick, Efq; Fellow Commoner of Trinity Coll. Cambridge Sir Henry Goodrick, Bart. Mr Thomas Goodwin Sir William Gordon, Bar. Right Hon. Sir Ralph Gore, Bart. Arthur Gore, Efq^ Mr Francis Gore Mr John Charles Goris Rev. Mr William GoQing, M.A. William Goflin, Efq; Mr William Goflip, A. B. of Trin. Coll. Cambridge John Gould, jun. Efq; Nathaniel Gould, Efqi Mr Thomas Gould Rev. Mr Gowan, of Leyden Richard Graham, jun. Efqi Mr George Graham Mr Thomas Grainger Mr Walter Grainger Mr John Grant Monfieur S' Gravefande, Frofef- fsr of Aftronomy and Ex- perim. Philofophy in Leyden Dr Gray Mr Charles Gray of Colchefter Mr John Greaves Mr Francis Green Dr Green, Frofejfor of Phyfick in Cambridge Samuel Green, Gent. Mr George Green, B. D. Mr Peter Green Mr Matthew Green Mr Nathaniel Green, Apothecary Mr Stephen Greenhill, of Jefus College, Cambridge Mr Arthur Greenhill Mr jofeph Greenup Mr Randolph Greenway, of Thavics Ina Mr Thomas Gregg, of the Mid- dle Temple Mr Gregory, ProfelT. of Mo- dern Hift. in Oxon Mrs Katlierine Gregory Samuel Gray, Efq; Mr Richard Gray, Merchant in Rotterdam Thomas Griffiths, M.D, Mr Stephen Griggman Mr Rere Grillet Mr Richard Grimes Johannes Groeneveld, J. U. 8c M.D. «». Mr Thomas Hollis Mr John Hollifter Mr Edward Holloway Mr Tiiomas Holmes Rev. Mr Holmes, Vello-xi of En\anuel College, Cambridge Rev. Mr Samuel Holt Matthew Holworthy,-JB/5i Mr John Hook Mr Le Hook Mrs Elizabeth Hooke John Hooker, Efip Sslx John Hoole Mr Samuel Hoole Mr Thomas Hope Thsmas Hopgood, Gent. Sir Richard Hopkins . Richard Hopwood, M. D. Mr Henry Home Rev. Mr John Horfeley Sarautl Horfem n, M. D. Mr Suphcn H-nlcniao Mr Thomas Houghton Mr Thomss Houlding James How, E/q; John Hjw, of Hans Cope, E/g; Mr John Howe Mr Richard How ^Hon- Edward Howard, EJq; William Howard, E/q; -Rev. Dean Robert Howard Thomas Hucks, E/q; Mr Hudsford, cf Trinity Coll. Oxon Capt. Robert Hudfcn, jun. Mr John Hughes Edward Hulfe, M. D. Sir Guftavus Humes Rev. Mr David Humphreys, S.T. B. Fetlom of Trin. Coll. Cambridge Maurice Hunt, E/q; Mr Hunt, of Hart-Hall, Oxon Mr John Hunt iames Hunter, E/q; ■Ir William Hunter Mr John Hufley, 0/ Sheffield Ignatius Huffey, E/q; Rev. Mr Chriftopher Hufley; M. A. Rccior of Weft- Wick- ham in Kent Thomas Hutchinfon; Efq; Fel' low Commoner of Sidney. College, Cambridge Rev. Mr Hutchinfon, of Hart- Hall, Oxon Mr Sandys Hutchinfon, cf Tri- nity College, Cambridge Mr Huxley, M. A. of Brazen Nofe College, Oxon Mr Thomas Hyam, Merchant Mr John Hyde Mr Hyett, Gent. Commoner of Pembroke College, Oxon Right Hon. the Earl of Ihy Edward Jackfon, E/q; Mr Stephen Jackfon, Merchant Mr Cuthbcrt Jackfon Rev. Mr. Peter Jackfon Mr Jofhua Jackfon John Jacob, Efq; Mr CONTENTS. Chap, i . Of the properties of bodies 'whereon their colours depend. ■Light is not refle&ed hy impinging againjl the folid^ parts of bodies j ''' ^ ' ^ The particles nvhich compofe bodies are tranfparent p; 541 Caufe of opacity . p. 542 Why bodies tn the open day -light hmje different colours p. 3 44 The great porojity of bodies confidered p. 355 Chap. 3 . Of the refra&ion, refleBion^ and infleBion of light. Rays of different colours are differently refraBed p. 357 The fine of the angle of incidence in each kind of raysj ., bears a giqjen proportion to the fine of refraBion^ ^' The proportiofi betiveen the refraBi've ponvers in diffe^^ rent bodies j r* ^ / UnBuous bodies refraBmoJl in proportion to their denjity p. 3 The infleBion of light p. 377 Chap. 4. Of optic glares. How the rays of light are rejr-aBed hy a fpherical fur-l faceofglafs' f P' 3^^" How they are refraBed by two fuchfurf aces p, 5 8ci How the image of ohjeBs is formed hy a conuex glafs p. 381 Why con^vex glaffes help thejrght in old age, and cofi-l canje gl^ffes ajfifi port-fighted people j" P* 5 ^ Ihi manner in which wifion is performed hy the eye p. 384 CONTENTS. Of telefcopes ^iuith tnvo con'vex glajfes of telef copes with four con'vex glaffes of teJefcopes nvith one con'vex and o?ie coma've glafs of microfcopes of the imperfeBion oftelefcopes arifmg from the diffe-' rent rejrangihility of the light ^ of the refle&ing telefcope Chap. 5 . of the rainhoiv of the inner rainhonu p. 5 5)4, 3^5, and 358, is>9 Of the outter hoiu p. 3 5><», 35)7, and 400 . of a particular appearance in the inner rainhow p. 40 i Conclujton p. 40^ p- 38Theodore Janflen, Bart. Mr John Jarvis, Surgeon at Dartford in Kent Mr Edward Jafper Edward Jauncy, »/ the MidJle- Temple Efq; Rev. 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Mr Benjamin Woodroof, Prebendary of Worcefter Mr Jofeph Woodward Jofiah Woolafton, Efq; Mr Woolball, Merchant Francis Woollafton, Efq; Charlton Woollafton, Efqi Mr William Woollafton Wight Woolly, Efq; Library of theCi'.hed. of WorceRer JoCas Wordfworth, jun. Efcj; Mr John Worfter, Merchant Rev. Dr. William Wotton Mr John Wowen Edward Wright, of the Middle- Temple, Efqi Henry Wright, of Molberly, in Cheftiire, Efq; Samuel Wright, Efqi William Wright, of Offerton," in Cheftiire, Efq; Mr Wright Mr William Wright, »/Baldock, Hertfordftiire Rev. Mr Wriglcy, Fellow of St. John's College, Cambridge Rt. Hon. Thomas Wyndham, Ld. Chief Jtiflice of the Common Fleas, of Ireland Mr Jofeph Wyeth Thomas Wyndham, Efq; Rev. Mr John Wynne Mr John Y3rdley,5'«rg.»'» Coven. Mr Thomas Yates Mrs Yeo, of Exeter, BoohfelUr Sir William Yonge Lady York Nicholas Young, ef the Inner- Temple, Efq; Hitch Young, Efq; Rev. Edward Young, L. L. D, INTRODUCTION. HE manner, in which Sir Isaac Newton has publifhed his philofophical difcove- ries, occafions them to lie very much concealed from all, who have not made the mathematics particularly their jflu- dy. He once, indeed, intended to de- Hver, in a more familiar way, that part of his inventions, which relates to the fyftem of the world ; but upon farther confideration he altered his defign. For as the nature of thofe difcoveries made it impoflible to prove them upon any other than geometrical principles ; he appre- hended, that thofe, who fhould not fully perceive the force of his arguments, would hardly be prevailed on to exchange their former fentiments for new opinions, fo very different from B what z INTRODUCTION. what were commonly received *. He therefore chofe rather to explain himfelf only to mathematical readers ; and decHned the attempting to infl:ru6l fuch in any of his principles, who, by not comprehending his method of reafoning, could not, at tlie firft appearance of his difcoveries, have been perfuaded of their truth. But now, lince Sir I s a a c N e w t o n's dodrine has been fully eftabHfhedby the unanimous approbation of all, who are qualified to underfland the fame ; it is without doubt to be wifhed, that the whole of his improvements in philofo- phy might be univerfally known. For this purpofe therefore I drew up the following papers, to give a general notion of our great philofopher's inventions to liich, as are not prepared to read his own works, and yet might defire to be informed of the progrefs, he has made in natural knowledge ; not doubting but there were many, befides tliofe, whofe turn of mind had led them into a courfe of mathematical ftudies, tliat would take great pleafure in tailing of this deHghtful fountain of fcience. 1. It is a juft remark, which has been made upon the hu- man mind, that nothing is more fuitable to it, than the con- templation of truth ; and that all men are moved with a ftrong defire after knowledge ; efteeming it honourable to excel therein ; and holding it, on the contrary, difgraceful to mi- ftake, err, or be in any way deceived. And this ientiment is by nothing more fully illuftrated, than by the inclination of men to gain an acquaintance with the operations of na- ture y which difpofition to enquire after the caufes of things is ^ Pliilofoph. N.U. princ. math. L. iii. introdudl. fo INTRODUCTION. 3 fo general, that all men of letters, I believe, find themfelves influenced by it. Nor is it difficult to aflign a reafon for diis, if we confider only, that our defire after knowledge is an ef- fed of that tafte for the fublime and the beautiful in things , which chiefly conftitutes the difference between the human life, and die life of brutes. Thefe inferior animals partake with us of the pleafures, that immediately flow from die bo- dily fenfes and appetites ; but our minds are furnifhed with a fuperior fenfe, by which we are capable of receiving various degrees of delight, where the creatures below us perceive no difference. Hence arifes that purfuit of grace and elegance in our thoughts and adlions, and in all things belonging to us, which principally creates imployment for the adlive mind of man. The thoughts of the human mind are too extenfive to be confined only to the providing and enjoying of what is neceffary for the fupport of our being. It is this tafte, which has given rife to poetry, oratory, and every branch of litera- ture and fcience. From hence we feel great plealure in con- ceiving fl:rongly, and in apprehending clearly, even where the paffions are not concerned. Perlpicuous reafoning ap- pears not only beautiful ; but, when fet forth in its full ftrength and dignity, it partakes of the fublime, and not only pleafes, but warms and elevates the foul. This is the fource of our fl:rong defire of knowledge; and the lame tafte for the fublime and the beautifiil diredls us to chufe particularly the productions of nature for the fubjedi of our- contemplation : our creator having fo adapted our minds to the condition, wherein he has placed us, that all his vifible B X works, 4 IKTRODUCTION. works, before we inquire into their make, ftrike us with. the moft lively ideas of beauty and magnificence. 9 . But if there be fo ftrong a pafTion in contemplative minds for natural philofophy ; all fuch muft certainly receive a particular pleallire in being informed of Sir Isaac Newton's difcoveries, who alone has been able to make any great advancements in the true courfe leading to natural know- ledge : whereas this important fubjed had before been u- fually attempted with that negligence, as cannot be re- fleded on without furprize. Excepting a very few, who, by purfuing a more rational method, had gained a little true knowledge in fbme particular parts of nature ; the writers in this fcience had generally treated of it after fuch a manner, as if they thought, that no degree of certainty was ever to be ho- ped for. The cuftom was to frame conjedures; and if upon comparing them with things, there appeared fome kind of a- greement, though very imperfed:, it was held fufficient. Yet at the lame time nothing lefs was undertaken than intire iy- ftems, and fathoming at once the greateft depths of nature ; as if the fecret caules of natural effedls, contrived and framed by infinite v^ifdom, could be fearched out by the flighteft endeavours of our weak underftandings. Whereas the only method, that can afford us any profped of fuccefs in this difficult work, is to make our enquiries with the utmoft caution, and by very flow degrees. And after our moft dili- gent labour, the greateft part of nature will, no doubt, for e- ver remain beyond our reach. 4. This INTRODUCTION. 4. This neglect of the proper means to cnlar"-e our knowledge, joined with the prefumption to attempt, what was quite out of the power of our limited faculties, the Lord Bacon judicioufly obferves to be the great obftrudlion to the progrefs of fcience \ Indeed that excellent perfon was the firft, who exprefly writ againft this way of philofophizing ; and he has laid open at large the abfurdity of it in his admirable treatife, intitled Novum organon scientiarum; and has there likewife de.fcribed the true method, which ought to be followed. y. There are, faith he, but two methods, that can be taken in the purfuit of natural knowledge. One is to make a hafty tranfition from our firft and flight obfervations on things to general axioms, and then to proceed upon thofe axioms, as certain and unconteftable principles, without far- ther examination. The other method ; ( which he obferves to be the only true one, but to his time unattempted ; ) is to proceed cautioufly, to advance ftep by ftep, referving the moft general principles for the laft relult of our inquiries \ Concerning the firft of thefe two methods ; where objedions, which happen to appear againft any fuch axioms taken up in hafte, are evaded by fome frivolous diftindion, when the ax- iom it felf ought rather to be corredted " ; he affirms, that the united endeavours of all ages cannot make it fuccefsfiil ; becaule this original error in the firft digeftion of the mind ( as he expreftes himfelf ) cannot afterwards be remedied ^ : whereby he would fignify to us, that if we fet out in, a " Nov. Org. Sclent. L. i. Aphorifm. 9. I" Nov. Org. L. I. Aph. 19, t Ibid. Aph. 25. •* Aph. 30. Errores radicales & in prima di- gcftione mentis ab cxcellentia fundionum & rc- mediorum fequentium non curantur. wrong 6 INTRODUCTION. wrong way ; no diligence or art, we can ufe, while we follow fo erroneous a courfe, will ever bring us to our de- ligned end. And doubtlefs it cannot prove otherwife j for in this fpacious field of nature, if once we forfake the true path, we fhall immediately lofe our felves, and muft for ever wander widi uncertainty. 6^. T H E impoflibility of fucceeding in fo faulty a method of philofophizing his Lordfliip endeavours to prove from the many falfe notions and prejudices, to which the mind of man is expofed *. And fince this judicious writer apprehends, that men are fo exceeding liable to fall into thele wrong tracts of thinking, as to incur great danger of being milled by them, even while they enter on the true courfe in purfuit of na- ture ^ ; I truft, I fhall be excufed, if, by infifting a little par- ticularly upon this argument, I endeavour to remove what- ever prejudice of this kind, might poflibly entangle the mind of any of my readers. 7. His Lordfhip has reduced thefe prejudices and filfe modes of conception under four diftindl heads ". 8. T H E firft head contains fuch, as we are flibjed to from the very condition of humanity, through the weaknefs both of our fenfes, and of the faculties of the mind ^ ; feeing, as this author well obferves, the fubtilty of nature far exceeds the greateft fubtilty of our fenfes or acuteft reafonings ^ One » Aph. 38. 1 <> Aph. 41. •> Ibid. I e Aph. 10, 24. <: Aph. 39. of INTRODUCTION. 7 of the falfe modes of conception, which he mentions un- der this head, is the forming to our felves a fanciful fnn- plicity and regularity in natural things. This he illuflrates by the following inftances ; the conceiving the planets to move in perfed circles ; the adding an orb of fire to the o- ther three elements, and the fuppofing each of theie to ex- ceed the odier in rarity, juft in a decuple proportion \ And of the fame nature is the aflertion of Des Cartes, without any proof, that all things are made up of three kinds of matter only ^. As alfo this opinion of another philofopher ; that light, in pafling through different me- diums, was refraded, fo as to proceed by that way, through which it would move more Ipeedily, than through any o-~ ther ". The fecond erroneous turn of mind, taken notice of by his Lordfhip under this head, is, that all men are in fome degree prone to a fondnels for any notions, which they have once imbibed ; whereby they often wreft things to reconcile them to thofe notions, and negled: the confideration of what- ever will not be brought to an agreement with them ; juft as thofe do, who are addidled to judicial aftrology, to the obler- vation of dreams, and to flich-like fuperftitions ; who care- fully preferve the memory of every incident, which ferves to confirm their prejudices, and let flip out of dieir minds all in- ftances, that make againft them ^. There is alio a farther impe- diment to true knowledge, mentioned under the fame head by this noble writer, which is ; that whereas, through the weak- nefs and imperfedion of our fenfes, many things are concealed. » Aph. 45. J c FeiTnat, in Oper. pag. 156, Src. > Dcs Cartes Princ. Phil. Part.3, §. 52. j ^ Nqv. Org. Aph. 46, fi-om B INTRODUCTION. from us, which have the greateft effed in producing natural appearances ; our minds are ordinarily moft afteded by that, which makes the ftrongeft impreffion on our organs of fenfe ; whereby we are apt to judge of the real impor- tance of things in nature by a wrong meafure \ So, becaufe the figuration and the motion of bodies ftrike our fenfes more immediately than moft of their other properties, Des Cartes and his followers will not allow any other expHcation of natu- ral appearances, than from the figure and motion of the parts of matter. By which example we fee how juftly his Lord- fhip obferves this caufe of error to be the greateft of any '° ; fince it has given rife to a fundamental principle in a fyftem of philofophy, that not long ago obtained almoft an univer- lal reputation. 9. T H E s E are the chief branches of thofe obftrudions to knowledge, which this author has reduced under his firft head of falfe conceptions. The fecond head contains the errors, to which particular perfons are more elpecially obno- xious *". One of thefe is the confequence of a preceding ob- fervation : that as we are expofed to be captivated by any opi- nions, which have once taken pofleffion of our minds ; fo in particular, natural knowledge has been much corrupted by the ftrong attachment of men to fome one part of fcience, of which they reputed themfelves the inventers , or about which they have fpent much of their time ; and hence have been apt to conceive it to be of greater ufe in the ftudy of na- > Aph. 50. I ' Aph. 53. ^ J bid. tural INTRODUCTION. 9 tural philofophy than it was : like Aristotle, who redu- ced his phyfics to logical difputations ; and the chymifts, who thought, that nature could be laid open only by the force of their fires \ Some again are wholly carried away by an exceflive veneration for antiquity ; others, by too great fond- jiefs for the moderns ; few having their minds fo well balanced, as neither to depreciate the merit of the ancients, nor yet to defpife the real improvements of later times ^. To this is added by his Lordfhip a difference in the genius of men, that fome are moft fitted to obferve the fimiUtude, diere is in things, while others are more qualified to difcern the par- ticulars, wherein they difagree ; both which difpofitions of mind are ufeful : but to the prejudice of philofophy men are apt to nm into excefs in each ; while one fort of genius dwells too much upon the grofs and fum of things, and the other upon trifling minuteneffes and fliadowy difl:in6lions ^ 10. Under the third head of prejudices and falfe notions this writer confiders fuch, as follow from the lax and indefi- nite ufe of words in ordinary difcourfe ; which occafions great ambiguities and uncertainties in philofophical debates (as ano- ther eminent philofopher has fince fliewn more at large ^ ;) in- fomuch that this our author diinks a ftrid defining of terms to be fcarce an infallible remedy againfl; this inconvenience ^ And perhaps he has no fmall reafon on his fide : for the common inaccurate fenfe of words, notwithftanding the Hmitations given them by definitions, will offer it felf fo conftandy to " ^r'^- 54- '' Locke. On human uiiderfhr.ding,B.iii, *■ Aph. 56, « Nov. Org. Aph. 55. ' Aph. 55. C the 10 INTRODUCTION. the mind, as to require great caution and circumfpedlion for us not to be deceived diereby. Of this we have a very eminent inftance in the great difputes, that have been raifed about the ufe of the word attradlion in philofophy ; of which we fhall be obHged hereafter to make particular mention ^ Words thus to be guarded againft are of two kinds. Some are names of things, that are only imaginary '° ; fuch words iare wholly to be rejected. But there are other terms, that al- lude to what is real, though their fignification is confufed ". And thefe latter muft of neceflity be continued in ufe ; but their fenfe cleared up, and freed, as much as poflible,. from obfcurity. II. The laft general head of thefe errors comprehends fuch, as follow from the various fe6ls of falfe philofophies ; which this author divides into three forts, the fophiftical, em- pirical, and fuperftitious ^. By the firft of thefe he means a philofophy built upon {peculations only without experi- ments ^ ; by the fecond, where experiments are blindly ad- hered to, without proper reafoning upon them ^ ; and by the third, wrong opinions of nature fixed in mens minds ei- ther through falfe religions, or from mifunderftanding the declarations of the true ^. II. T H E s E are the four principal canals, by which this ju- dicious author thinks, that philofophical errors have flowed in upon us. And he rightly obferves, that the faulty method of » In the conclufion. * Nov. Org. L. i. AjA. 59. ' Ibid. Aph. 60. o Iba. Aph. 64, « Aph. 63. « Aph. 64. ^ Aph. 65. proceeding INTRODUCTION. ii proceeding In philofophy, agalnft which he writes \ is fo far from affifting us towards overcoming thefe prejudices ; that he apprehends it rather fuited to rivet them more firmly to the mind ^. How great reafon then has his Lordfhip to call this way of philofophizing the parent of error, and the bane of all knowledge " ? For, indeed, what elfe but miftakes can fb bold and prefumptuous a treatment of nature produce ? have we the wifdom neceilary to frame a world, that we fhould think fo eafily, and with fo flight a fearch to enter into the moft fecret fprings of nature, and difcover the original caufes of things ? what chimeras, what monfters has not this prepofle- rous method brought forth ? what fchemes, or what hypotlie- fis's of the fubtileft wits has not a ftrifter enquiry into nature not only overthrown, but manifefted to be ridiculous and ablurd ? Ever)'' new improvement, which we make in this fcience, lets us fee more and more the weaknefs of ourguelles. Dr.HARVEV, by that one difcovery of tlie circulation of the blood, has diflipated all the fpeculations and reafonings of many ages up- on the animal oeconomy. A s e l l r u s, by deted:ing the la- dieal veins, fliewed how little ground all phyficians and phi- lofophers had in conjeduring, that the nutritive part of the aliment was abforbed by the mouths oi the veins fpread upon the bowels : and then Pecquet, by finding out the thora- cic dud, as evidently proved the vanity of the opinion, which was perfifted in after the ladeal vefTeis were known, that the alimental juice was conveyed immediately to the liver, and there converted into blood. ' Sec above, § 4, 5. ] c Ibid. '' Nov. Org.L. i. Aph. 69. 13. As 2x INTRODUCTION. 15. As thefe things fet forth the great abfurdity of pro— eeeding in philofophy on conjedlures, by informing us how far^ the operations of nature are above our low conceptions ; fo- on the other hand, Rich inftances of fuccefs from a more judicious method fliew us, that our bountiful maker has not left us wholly without means of delighting our felves in the contemplation of his wifdom. That by a juil; way of inquiry into nature, we could not fail of arriving at difcoveries- very remote from our apprehenfions ; the Lord Bacon him- felf argues from the experience of mankind. If, fays he, the force of guns fhould be defcribed to any one ignorant of them, by their effedls only ; he might reafonably fuppofe, that tliofe engines of deftrudlion were only a more artificial com- pofition, than he knew, of wheels and other mechanical powers : but it could never enter his thoughts, that their immenfe force fhould be owing to a pecuHar fubflance, .. which would enkindle into fo violent an explofion, as we: experience in gunpowder : fince he would no where fee. the leafl: example of any fuch operation ; except perhaps in earthquakes and thunder, which he would doubtlels look, upon as exalted powers of nature, greatly furpafTmg any art of man to imitate. In the fame manner, if a ftranger to the ori- ginal of filk were fhewn a garment made of it, he would be very far from imagining fo ftrong a fubflance to be fpun out of the bowels of a fmall worm ; but mufl certainly believe it either a vegetable fubftance, Hke flax or cotton ; or the na- tural covering of fome animal, as wool is of fheep. Or had we been told, before the invention of the magnetic needle among us, that another people was in pofTefTion of a certain contrivance INTRODUCTION. 15 contrivance, by which they were inabled to difcover the po- rtion of the heavens, with vaftly more eafe, than we could do ; what could have been imagined more, than that they were provided with fome fitter aflronomical inftrument for tliis purpofe than we ? That any ftone fhould have lb amaz- ing a property, as we find in the magnet, muft have been . the remoteft from our thoughts ^ 14. But what furprizing advancements in the knowledge of nature may be made by purfuing the true courfe in philo- fophical inquiries ; when thofe fearches are conduced by a genius equal to fo divine a work, will be beft underftood by confidering Sir Isaac N e w t o n's difcoveries. That my reader may apprehend as juft a notion of thele, as can be con- veyed to him, by the brief account, which I intend to lay be- fore him ; I have fet apart this introdu(£lion for explaining, in the fiilleft manner I am able, the principles, whereon Sir Isaac Newton proceeds. For without a clear concep- tion of thefe, it is impofTible to form any true idea of the fingular excellence of the inventions of this great philofopher. u I y. The principles then of diis philofophy are ; upon no con- fideration to indulge conjectures concerning the powers and laws of nature, but to make it our endeavour with all diligence to fearch out the real and true laws, by which the conftitution of things is regulated. The philofopher 's firft care muft be to diftinguifh, what he fees to be within his power, from what » Ibid. Aph. 109. 4''. is 14 INTRODUCTION. is beyond his reach ; to afTume no greater degree of know- ledge, than what he finds himfelf poffeffed of; but to advance by flow and cautious fteps ; to fearch gradually into natural cauf- es ; to lecure to himfelf the knowledge of the moft immediate caufe of each appearance, before he extends his views farther to caufes more remote. This is the method, in which philofo- phy ought to be cultivated ; which does not pretend to fo great things, as the more airy fpeculations ; but will perform abun- dantly more : we fliall not perhaps feem to the unskilful to know fo much, but our real knowledge will be greater. And certainly it is no objedion againft this method, that fome o- thers promife, what is nearer to die extent of our wiflies : fince this, if it will not teach us all we could defire to be informed of, will however give us fome true light into nature ; which no other can do. Nor Has the philofopher any reafon to think his labour loft, when he finds himfelf ftopt at the caufe firft difcovered by him, or at any other more remote caufe, fliort of the original : for if he has but fufficiently proved any one caufe, he has entered fo far into the real conftitution of things, has laid a fafe foundation for others to work upon, and has facilitated their endeavours in the fearch after yet more diftant caufes ; and befides, in the mean time he may apply the knowledge of thefe intermediate caufes to many ufeful purpofes. Indeed the being able to make pradrical dedu- dions from natural caufes, conftitutes the great diflindion between the true philofophy and the falfe. Caufes af- fumed upon conjediure, muft be fo loofe and undefined, diat nodiing particular can be colleded from them. But thofe caufes, which are brought to light by a ftrift examination .of INTRODUCTION. ly of things, will be more diftind. Hence it appears to have been no unufeful difcovery, that the afcent of water in pumps is owing to the preffure of the air by its weight or fpring ; thoush die caufes, which make the air gravitate, and render it elaftic, be unknown : for notwithftanding we are igno- rant of the original, whence thefe powers of the air are de- rived ; yet we may receive much advantage from the bare knowledge of thefe powers. If we are but certain of the de- gree of force, wherewith they ad:, we fhall know the extent of what is to be expeded from them ; we iTiall know the greateft height, to which it is poiTible by pumps to raife water; and fhall thereby be prevented from making any ufelels ejfforts towards improving diefe inftruments beyond the limits pre- fcribed to them by nature ; whereas without fo much know- ledge as this, we might probably have wafted in attempts of this kind much time and labour. How long did philofo- phers bufy themfelves to no purpofe in endeavouring to perfedl telefcopes, by forming the glafles into fome new figure ; till Sir Isaac Newton demonftrated, that the effedls of tele- fcopes were limited from another caufe, than was fuppofed ; which no alteration in the figure of the glafies could remedy ? What method Sir Isaac Newton himfelf has found for the improvement of telefcopes fhall be explained hereafter *. But at prefent I fhall proceed to illuftrate, by fome farther inftan- ces, this diftinguifhing charader of the true philofophy, which we have now under confideration. It was no trifling difcove- ry, that the contradion of the mufcles of animals puts their limbs in motion, though the original caufe of that contradion > Book III. Ch.ip. iv. remains i6 INTRODUCTION. remains a lecret, and perhaps may always do fo ; for the knowledge of thus much only has given rife to many fpecu- lations upon the force and artificial difpofition of the mufcles, and has opened no narrow profped: into the animal fabrick. The finding out, that the nerves are great agents in this a- dlion, leads us yet nearer to the original caufe, and yields us a wider view of the fubjed:. And each of thele fteps affords us afiiftance towards reftoring this animal motion, when impair- ed in our felves, by pointing out the feats of the injuries, to which it is obnoxious. To neglect all this, becaufe we can hitherto advance no farther, is plainly ridiculous. It is confefTed by all, that Galileo greatly improved philofo- phy, by fhewing, as we fhall relate hereafter, that the power in bodies, which we call gravity, occafions them to move downwards with a velocity equably accelerated '^ ; and that when any body is thrown forwards, the fame power obliges it to defcribe in its motion that line, which is called by geometers a parabola ^ : yet we are ignorant of the caufe, which makes bodies gravitate. But although we are unacquainted with the fpring, whence this power in nature is derived, neverthe- iefs we can eftimate its effeds. When a body falls perpendicu- larly, it is known, how long time it takes in defcending from any height whatever : and if it be thrown forwards, we know the real path, which it defcribes ; we can determine in what diredlion, and with what degree of fwiftnefs it muft be pro- jected , in order to its ftriking againft any objedl defired; and we can alfo afcertain the very force, wherewith it will ftrike. > BookL Ch,ip, 2. §14. i> IbiJ. § 85, &c. Sir INTRODUCTION. 17 Sir IsaacNewton has farther taught, that this power of gravitation extends up to the moon, and caufes that planet to gravitate as much towards the earth, as any of the bodies, which are famiHar to us, would, if placed at the fame diftance ^ : he has proved Hkewife, that all the planets gravitate towards the fun, and towards one anodier ; and that their refpedive motions follow from this gravitation. All this he has demon- ftrated upon indifputable geometrical principles, which cannot be rendered precarious for want of knowing what it is, which caufes thefe bodies thus mutually to gravitate ; any more than we can doubt of die propenfity in all the bodies about us, to defcend towards the earth ; or can call in queftion the fore- mentioned proportions of Galileo, which are built upon that principle. And as Galileo has f hewn more fully, than was known before, what effeds were produced in the motion of bodies by their gravitation towards the earth ; fo Sir I s A A c N E w T o N, by this his invention, has much advan- ced our knowledge in the celeftial motions. By difcovering that the moon gravitates towards the fun , as well as towards the earth; he has laid open thofe intricacies in the moon's motion, which no aftronomer, from obfervations only, could ever find out^: and one kind of heavenly bodies, the comets, have their motion now clearly afcertained ; whereof we had before no true knowledge at all ". 1(5. Doubtless it might be expected, that fuch furprizing fuccefs iliould have filenced, at once, every cavil. But we » See Book n. Ch. 3. § 3,4. of this treatife. | ' See Chap. 4. * See Book II. Ch. 3. of diis treatife. | D have i8 INTRODUCTION. have feen the contrary. For becaufe this philofophy profefl^a modeftly to keep within the extent of our faculties, and is ready to confeis its imperfedions, rather than to make any fruitlefs attempts to conceal them, by fceking to cover the de- feats in our knowledge with the vain oflentation of rafh and groundlefs conjedtures ; hence has been taken an occalion to infinuate that we are led to miraculous caufes, and die occult qualities of the fchools. 17. But the firft of thefe accufations is very extraordina^^ ry. If by calling thefe caufes miraculous nothing more is meant than only, that they often appear to us wonderftil and furprizing, it is not eafy to fee what difficulty can be railed from thence ; for the works of nature difcover every where fuch proofs of the unbounded power, and the conlummate wifdom of their author, that the more they are known, the more they will excite our admiration : and it is too manifefb to be infifled on, tliat the common fenfe of the word mira- culous can have no place here, when it implies what is above the ordinary courfe of diings. The other imputation, that thefe caufes are occiilt upon the account of our not perceiving what produces them, contains in it great ambiguity. That fomething relating to them lies hid, the followers of this philofophy are ready to acknowledge, nay defire it fhould be carefully remarked, as pointing out proper fubjeds for fu- ture inquiry.. But this is very different from the proceeding of the fchoolmen in the caufes called by diem occult. For as their occult qualities were underftood to operate in a man- ner occult, and not apprehended by us j fo diey were ob- t truded: INTRODUCTION. i^ tnided upon ns for fuch original and efTential properties in bo- dies, as made it vain to feek any farther caufe ; and a great- er power was attributed to them, than any natural appearances authorized. For inftance, the rife of water in pumps was afcri- bed to a certain abhorrence of a vacuum, which they thought fit to aflign to nature. And this was fo far a true obfervation, that the water does move, contrary to its ufual courie, into the fpace, which otherwife would be left void of any fenlible matter ; and, that the procuring fuch a vacuity was the appa- rent caufe of the water's afcent. But while we were not in the leaft informed how this power, called an abhorrence of a vacuum, produced the vifible effeds ; inftead of making a- ny advancement in the knowledge of nature, we only gave an artificial name to one of her operations : and when the fpeculation was pufhed fo beyond what any appearances re- quired, as to have it concluded, that this abhorrence of a va- cuum was a power inherent in all matter, and fo unlimited as to render it impofTible for a vacuum to exift at all ; it dien became a much greater abfurdity, in being made the foun- dation of a moft ridiculous manner of reafoning; as at length evidently appeared, when it came to be difcovered, that this rife of the water followed only from the preffure of the air, and extended it felf no farther, than the power of that caufe. The fcholaftic ftile in difcourfing of thefe occult qualities, as if they were effential differences in the very fubftances, of which bodies conlifted, was certainly very abfurd; by reafon it tended to difcourage all fartlier inquiry. But no fuch ill confequences can follow^ from the considering of any natural caufes, which confefTedly are not traced up to D X their 20 INTRODUCTION. their firfl original. How fliall we ever come to the know- ledge of the feveral original caiifes of things, otherwife than by ftoring up all intermediate caufes which we can difcover ? Are all the original and effential properties of matter fo very obvious, that none of them can efcape our firft view ? This is not probable. It is much more likely, that, if fome of the effential properties are difcovered by our firft obfervations, a ftrider examination fhould bring more to light. 1 8. But in order to clear up this point concerning the efi'ential properties of matter, let us confider the fiibje^t a lit- tle diftindly. We are to conceive, that the matter, out of which the univerfe of things is formed, is fiirnifhed with ccTr- tain qualities and powers, whereby it is rendered fit to anfwer the purpofes, for which it was created. But every property, of which any particle of this matter is in it felf pofl'effed, and which is not barely the confequence of the union of this part>- cle with- other portions of matter, we may call an efiential pro- perty : whereas all other qualities or attributes belonging to bodies, which depend on their particular frame and compofi- tion, are not effential to the matter, whereof fuch bodies are made ; becaufe the matter of thefe bodies will be deprived of thofe qualities, only by the diffolution of the body, with- out working any change in the original conftitution of one fingle particle of this mafs of matter. Extenfion we appre- hend to be one of thefe effential properties, and impenetrabi- lity another. Thefe two belong univerfally to all matter ; and are the principal ingredients in the idea, which this word. matter ufually excites in the mind, Yet as the idea, marked by INTRODUCTION. xr by this name, is not purely the creature of our own un- derftandings, but is taken for the reprefentation of a certain, fubftance wddiout us ; if we fhould difcaver, that every part of the fubftance, in wliich we find thefe two properties, fhould Hkewife be endowed univerfally widi any other effen- tiial quahties ; all thefe, from the time they come to our no- tice, muft be united under our general idea of matter. How many fuch properties there are adually in all matter we know not; thofe, of which we are at prefent apprized, have beert found out only by our obfervations on things ; how many more a farther fearch may bring to light, no one can fay ;. nor are we certain, that we are provided with fufficient me- thods of perception to difcern them all. Therefore, fince w^e have no other way of making difcoveries in nature, but by gradual inquiries into the properties of bodies.; our firft ftep muft be to admit widiout diftincElion all the properties, which we obferve ; and afterwards we muft endeavour, as far as we are able, to diftinguilli between the qualities, wherewith the very fubftances themfelves are indued, and thofe appearancesj which refult from the ftrudlure only of compound bodies* Some of the properties, which we obferve in things, are ths attributes of particular bodies only ; others univerfally belong to all, that fall under our notice. Whether fome of the qualities and powers of particular bodies, be derived from dif- ferent kinds oi matter entring their compofition, cannot, in the prefent imperfed ftate of our knowledge, abfolutely be decided; though we have not yet any reafon to conclude, but that. all ';he bodies, with which we converfe, are fi-amed out of the very fame kmd of matter, and daat tkeir diftind quali- 2L^ I N T R D U C T I K qualities are occafloned only by their flrudlure ; through the va- riety whereof the general powers of matter are caufed to pro- See the fame treatife, in ia Qu. 21. J Advertifement 2. inquiry INTRODUCTION. 15 inquiry cannot properly have place in the deliberation about ad- mitting any property of matter or bodies into philofophy ; for- that purpofe it is only to be confidered, whether the exiftence- of fuch a property has been juftly proved or not. Therefore. to decide what caufes of things are rightly received into na- tural philofophy, requires only a diftind: and clear conception, of what kind of reafoning is to be allowed of as convincing, , when we argue upon the works of naturCo. zo. The proofs in natural philofophy cannot be Co abfo- lutely conclufive, as in the mathematics. For the fubje£ls oF that fcience are purely die ideas of our own minds. ^ They may be reprefented to our fenfes by material objedls, but they are themfelves the arbiti-ary produdlions of our own thoughts ; fo that as the mind can have a full and adequate knowledge of its own ideas, the reafoning in geometry can be rendered perfect. But in natural knowledge the fubjedl of our con- templation is without us, and not fo compleatly to be known z: therefore our micthod of arguing muft fall a little fhort of ab- folute perfedion. It is only here required to fleer a juft courfe between the conjectural metliod of proceeding, againft which I have fo largely Ipoke ; and demanding fo rigorous a proof, as will reduce all philofophy to mere foepticifm, and exclude all profped of making any progrefs in the knowledge of nature. - a I. T KTE conceflions, which are to be allowed in this fci- ence, are by Sir Isaac Newton included under a very few Umple precepts, 2:a,. Ths- 44 INTRODUCTION. 11. The firft is, tliat more caufes are not to be received into philofophy, than are fufficient to explain the appearances of nature. That this rule is approved of unanimoufly, is e- vident from thofe exprefTions fo frequent among all philofo- phers, that nature does notliing in vain ; and that a varie- ty of means, where fewer would fuffice, is needlefs. And certainly there is the higheft reafon for complying with this rule. For fhould we indulge the liberty of multiplying, ■without necefllty, the caufes of tilings, it would reduce all philofophy to mere uncertainty ; lince the only proof, which we can have, of the exiftence of a caufe, is the ne-" ceflity of it for producing known effeds. Therefore where one caufe is fufficient, if there really fhould in nature be two, wliich is in the laft degree improbable, we can have no poffible means of knowing it, and confequently ought not to take the liberty of imagining, that there are more than one. 1 g . The fecond precept is the diredl confequence of the firft, that to like effedls are to be afcribed the fame caufes. For inflance, that refpiration in men and in brutes is brought about by the fame means ; that bodies defcend to the earth here in E u r o p e, and in A m e r i c a from the fame principle ; that the light of a culinary fire, and of the fun have the fame manner of produdlion ; that the reflection of light is efFeded in the earth, and in the planets by the fame power ; and the like. 14. The third of thefe precepts has equally evident rea- fon for it. It is only, that thofe qualities, which in the fame body can neither be lefTened nor increafed, and which belong to INTRODUCTION. ^s to all bodies that are in our power to make trial upon, ought to be accounted the univerfal properties of all bodies what- ever. If. In this precept is founded that method of arguing by indudion, without wliich no progrefs could be made in na- tural philofophy. For as the qualities of bodies become known to us by experiments only ; we have no other way of finding the properties of fuch bodies, as are out of our reach to experiment upon, but by drawing conclufions from thole which fall under our examination. The only caution here required is, that the obfervations and experiments, we argue upon, be numerous enough, and that due regard be paid to all objedions, that occur, as the Lord Bacon very judi- cioufly directs \ And this admonition is fufficiently com- plied with, when by virUie of this rule we afcribe impene- trability and extenfion to all bodies, though we have no fen- fible experiment, that aftords a dired proof of any of the ce- leflial bodies being impenetrable ; nor that the fixed ftars are fo much as extended. For the more perfed our inftru- ments are, whereby we attempt to find their vifible magni- tude, the lefs they appear ; infomuch that all the fenfible magnitude, which we obferve in them, feems only to be an optical deception by the fcattering of their light. However? I fuppofe no one wiU imagine they are without any magni- tude, though their immenfe diftance makes it undifcernable by us. After the fame manner, if it can be proved, that all » Nov. Org. Lib. i. Ax. icj. E bodies r6 INTRODUCTION. bodies here gravitate towards the earth, in proportion to the quantity of foHd matter in each ; and that the moon gravitates to the earth Hkewife, in proportion to the quantity of matter in it ; and that the fea gravitates towards the moon, and all the planets towards each other; and that the very comets have the fame gravitating faculty ; we fhall have as great reafon to conclude by this rule, that all bodies gravitate towards each other. For indeed this rule will more ftrongly hold in this cafe, than in that of the impenetrability of bodies ; becaufe there will more inftances be had of bodies gravitating, than of their being impenetrable. 1.S- T H I s is that method of indudion, whereon all phi- lofophy is founded ; which our author farther inforces by this additional precept, that whatever is colleded from this indudion, ought to be received, notwithflanding any conje- ctural hypothecs to the contrary, till fuch times as it fhall bsL contradidted or limited by ferther oblervations on nature^ BOOK -^fyrx^^ ^£^: ■i^.^tfu ^/m^^ BOOK I. Concerning the MOTION of BODIES IN GENERAL. C H A p. I. Of the LAWS of MOTION. A V I N G thus explained Sir Isaac Newton's method of realbning in philofophy, I {hall now proceed to my intended account of his diicove- ries. Thefe are contained in two trea- tifes. In one of them, the Mathema- tical PRINCIPLES OF NATURAL PHILOSO- PHY, his chief defign is to fliew by what laws the heavenly E % motions 28 Sir Is A AC Newton's Book I. motions are regulated ; in the other, his Optics, he difcourfes of the nature of Hght and colours, and of the adlion between light and bodies. This fecond treatife is wholly confined to the fubjed: of light : except fome conjedures propofed at die end concerning other parts of nature, which lie hitherto more concealed. In the other treatife our author was obliged to fmooth the way to his principal intention, by explaining ma- ny things of a more general nature : for even fome of the moft fimple properties of matter were fcarce well eftabUfhed at that time. We may therefore reduce Sir I s a a c N l w t o n's do- £lrine under three general heads ; and I fhall accordingly di- vide my account into three books. In the firft I fhall Ipeak of what he has delivered concerning the motion of bodies, without regard to any particular fyftem of matter ; in the fe- cond I fhall treat of the heavenly motions ; and the third Ihall be employed upon light. a. In the firft part of my defign, we muft begin with an account of the general laws of motion. 3 . These laws are fome univerfal affedions and proper- ties of matter drawn from experience, which are made ufe of as axioms and evident principles in all our arguings upon the motion of bodies. For as it is the cuftom of geometers to affume in their demonftrations fome propofitions, without exhibiting the proof of them ; fo in philofophy, all our rea- foning muft be built upon fome properties of matter, firft ad- mitted as principles whereon to argue. In geometry thefe ax- ioms are thus aflumed, on account of their being fo evident as Chap. I. PHILOSOPHY. 29 as to make any proof in form needlcfs. But in philofophy no properties of bodies can be in this manner received for felf- e\ident ; iince it has been obferved above, that we can con- clude nothing concerning matter by any reafonings upon its nature and effence, but that we owe all the knowledge, we have thereof, to experience. Yet when our obfervations on matter have inform'd us of fome of its properties, we may fe- curely reafon upon tliem in our fartlier inquiries into nature. And thefe laws of motion, of which I am here to Ipeak, are found fo univerfally to belong to bodies, that there is no mo- tion known, which is not regulated by them. Thefe are by Sir Isaac Newton reduced to three \ 4. T H E firft law is, that all bodies have fuch an indifference to reft, or motion, that if once at reft they remain fo, till di- fturbed by fome power adting upon them : but if once put in motion, they perftft in it ; continuing to move right for- wards perpetually, after the power, which gave the motion, is removed ; and alfo preferving the fame degree of velocity or quicknefs, as was firft communicated, not flopping or re- mitting their courfe, till interrupted or otherwife difturbed by fome new power imprefted. 5*. T H E fecond law of motion is, that the alteration of the ftate of any body, whether from reft to motion, or from mo- tion to reft, or from one degree of motion to another, is al- ways proportional to the force imprefted. A body at reft, when » Princip. philof. p3g.i3, 14. ad:ed ^o Sir Isaac N e w t o n's Book I. a6led upon by any power, yields to that power, moving in the lame line, in which the power applied is directed ; and moves with a lefs or greater degree of velocity, according to the degree of the power ; fo that twice the power fhall com- municate a double velocity, and three times the power a threefold velocit)''. If the body be moving, and the power impreffed a6l upon the body in the direction of its motion, the body fhall receive an addition to its motion, as great as the motion, into which that power would have put it from a ftate of reft ; but if the power impreffed upon a moving bo- dy ad; diredly oppolite to its former motion, that power fhall then take away from the body's motion, as much as in the o- ther cafe it would have added to it. Laftly, if the power be impreffed obliquely, there will arife an oblique motion dif- fering more or lefs from the former direftion, according as the new impreflion is greater or lefs. For example, if the bo- dy A (in fig. I .) bemoving in the diredlion A B, and when it is at the point A, a power be imprefled upon it in the direction A C, the body fhall from henceforth neitlier move in its firft diredion A B, nor in the diredion of the adventitious power, but fhall take a courfe as A D between them : and if the power laft impreffed be juft equal to that, which firft gave to the body its motion ; the line A D fhall pafs in the middle between A B and A C, dividing the angle under BAG into two equal parts ; but if the power laft impreffed be greater than the firft, the line AD fhall incline moft to AC ; whereas if the laft impreftion be lefs than the firft, the line A D fhall incline moft to AB. To be more particular, the fituation of the Chap. I. PHILOSOPHY. qi the line A D is always to be determined after this manner. Let AE be die fpace, 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 fecond im- pulfe. Suppofe Hkewife, that AF is the part of the line AC, through which die body would have moved during an equal portion of time, if it had been at reft in A, when it received the impulfe in the direction A C : then if from E be drawn a line parallel to, or equidiftant from A C, and from F an- other line parallel to AB, thofe two Hues will meet in the Hne A D. 6. The third and laft of thefe laws of motion is, that when any body ad:s upon another, the adion of that body upon the other is equalled by the contrary readlion of that other body upon the iirft. 7. These laws of motion are abundantly confirmed by this, that all the dedudions made from them, in relation to the motion of bodies, how complicated foever, are found to agree perfedily with obfervation. This fhall be fhewn more at large in the next chapter. But before we proceed to fo diffufive a proof ; I chufe here to point out thofe appearan- ces of bodies, whereby the laws of motion are firft fiiggefted to us.. 8. Daily obfervation- makes it appear to us, that any body, which we once fee at reft, never puts it felf into frefh motion : 32 Sir Is A AC Newton's BookL motion ; but continues always in the fame place, till removed by fome power applied to it. 9. A G A r N, whenever a body is once in motion, it continues in that motion fome time after the moving power has quitted it, and it is left to it felf Now if the body continue to move but a fingle moment, after the moving power has left it, there can no reafon be affigned, why it rtiould ever flop without fome external force. For it is plain, that this continuance of the motion is caufed only by the body's having already mov- ed, the fole operation of the power upon the body being the putting it in motion ; therefore that motion continued will e- qually be the caufe of its farther motion, and fo on without end. The only doubt that can remain, is, whether this motion communicated continues intire, after the power, that caufed it, ceafes to a(5l ; or whether it does not gradually languilli and decreafe. And this fufpicion cannot be removed by a tranfi- ent and flio-ht obfervation on bodies, but will be fullv cleared up by thofe more accurate proofs of the laws of motion, which are to be confidered in the next chapter. 10. Lastly, bodies in motion appear to affedl a flraight courfe without any deviation, unlefs when difturbed by fome adventitious power ading upon them. If a body be thrown perpendicularly upwards or downwards, it appears to continue in the fame ftraight line during the whole time of its motion. If a body be thrown in any other diredion, it is found to de- viate from the line, in which it began to move, more and more Ghap. I. P H I L O S O P H Y. 33 more continually towards the earth, whither it is directed by its weight : but fince, when the weight of a body does not alter the direction of its motion, it always moves in a ftraight line, without doubt in this other cafe the body's declining from its firft courfe is no more, than what is cau- fed by its weight alone. As this appears at firft fight to be unqueftionable, fo we fl:iall have a very diftind proof thereof in the next chapter, where the oblique motion of bodies will . be particularly confidered* II. Thus we fee how the firft of the laws of motion agrees with what appears to us in moving bodies. But here occurs this farther confideration, that the real and ab- folute motion of any body is not vifible to us : for we are our felves alfo in conftant motion along with the earth whereon we dwell ; infomuch that we perceive bo-^ dies to move fo far only, as their motion is different from our own. When a body appears to us to lie at reft, 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 courfe or diredion, wherein we are carried our felves , fo much motion as we feem to have given to the body, fo much we have truly added to the motion, it had, while it appeared to us to be at reft. But if we impel a body the contrary way, although the body appears to us to have received by fuch an impulfe as much motion, as when impelled the other way ; yet in this cafe we have taken from the body fo much real motion, as we feem to have given it. Thus the motion, which wc fee in bodiesj F is. \ ■ 54. Sir I s A A c N E w T n's Book I. is not their real motion, but only relative with refped to tis ; and the forementioned obfervations only fhew us, that this firft law of motion has place in this relative or apparent motion. However, though we cannot make any obferva- tion immediately on the abfolute motion of bodies, yet by reafoning upon what we obferve in vifible motion, we can difcover the properties and efFedts of real motion. II. With regard to this firft law of motion, which is now under consideration, we may from the foregoing ob- fervations moft truly collect, that bodies are difpofed to con- tinue in the abfolute motion, which they have once received, without increafing or diminifhing their velocity. When a body appears to us to lie at reft, it really preferves without change the motion, which it has in common with our felves : and when we put it into vifible motion, and we fee it conti- nue that motion ; this proves, that the body retains that de- gree of its abfolute motion, into which it is put by our a<3:ing upon it : if we give it fuch an apparent motion, which adds. to its real motion, it preferves that addition ; and if our ad:- ing 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. A G A IN, we do not obferve in bodies any dilpofition or power within themfelves to change the direction of their mo- tion ; and if they had any fuch power, it would ealily be dis- covered. For fuppofe a body by the ftruAure or difpofition of its parts, or by any other circumftance in its make, was in- dued Chap. I. PHILOSOPHY. 35 dued with a power of moving it felf j this felf-moving prin- ciple, which fhoidd be thus inherent in the body, and not depend on any thing external, muft change the direction wherein it would ad, as often as the polition of the body was changed : fo that for inftance, if a body was lying be- fore me in fuch a pofition, that the direction, wherein this principle difpofes the body to move, was pointed directly from me ; if I then gradually turned the body about, the diredion of this felf-moving principle would no longer be pointed di- redly from me, but would turn about along with the body. Now if any body, wliich appears to us at reft, were furnifti- ed with any fuch felf-moving principle ; from the body's ap- pearing without motion we muft conclude, that this felf-mov- ing principle lies direded die fame way as the earth is car- rying the body ; and fuch a body might immediately be put into viftble motion only by turning it about in any degree> that this felf-moving principle might receive a different di- redion. 14. From thefe conftderations it very plainly follows, that if a body were once abfolutely at reft ; not being fur- niftied with any principle, whereby it could put it felf into motion, it muft for ever continue in the lame place, till adled upon by fomething external : and alfo that when a body is put into motion, it has no power widiin it felf to make any change in the diredlion of that motion ; and confequently that the body muft move on ftraight forward without declin- ing any way whatever. But it has before been ftiewn, that bodies do not appear to have in themfelves any power to F X change 36 Sir I s A A c N E w T o n's Book I. change the velocity of their motion : therefore this firft law of motion has been illuftrated and confirmed, as much as caii' be from the tranfient obfervations, which have here been dif- courfed upon^ and in the next chapter all this will be farther eftablifhed by more corred: obfervations. If. But I fliall now pafs to the fecond law of motion ;• wherein, when it is afferted, that the velocity, with which any body is moved by the adlion of a power upon it, is pro- portional to that power ; the degree of power is fuppofed to be meafured by the greatnefs of the body, which it can move with a given celerity. So that the fenfe of this law is, that if any body were put into motion widi that degree of fwift- nefs, as to pafs in one hour the length of a thoufand yards ; the power, which would give the fame degree of velocity to a body twice as great, would give this leffer body twice the velocity, caufing it to defcribe in the fame fpace of an hour two thoufand yards. But by a body twice as great as another, I do not here mean fimply of twice the bulk, but one that contains a double quantity of folid matter. 1 6. Why the power, which can move a body twice as great as another with the fame degree of velocity, fhould be called twice as great as the power, which can give the lefler body the fame velocity, is evident. For if we fhould fuppofe the greater body to be divided into two equal parts, each equal to the leffer body, each of thefe halves will require the fame degree of power to move them with die velocity of the lefler body, as the lefler body it felf requires ; and therefore both 5" thofe Chap. I. PHILOSOPHY, 37 tliofe halves, or the whole greater body, will require the moving power to be doubled. 17. That die moving power being in this fenfe doubled, fhould juft double likewife the velocity of the fame body, , feems near as evident, if we confider, that the efFe6t of the power applied muft needs be the fame, whether that power be applied to the body at once, or in parts. Suppofe then the double power not applied to the body at once, but half of it firft, and afterwards the other half ; it is not conceivable for what reafon the half laft applied fhould come to have a dif- ferent effedl upon the body, from that which is applied firft ; as it muft have, if the velocity of the body was not juft dou- bled by tlie appHcation of it. So far as experience can deter- mine, we fee nothing to favour fuch a fuppofition. We can- not indeed ( by reafon of the conftant motion of the earth ) make trial upon any body perfectly at reft, whereby to fee whether a power applied in that cafe would have a different. effedt, from what it has, when the body is already moving ; but we find no alteration in the effedl of the fame power on account of any difference there may be in the motion of the body, when die power is applied. The earth does not al- ways carry bodies with the fame degree of velocity ; yet we- find the vifible effedls of any power applied to the fame bo- dy to be at all times the very fame : and a bale of goods, or other moveable body lying in a fhip is as eafily removed: from place to place, while the fhip is under fail, if its motioiii be fteady, as when it is fixed at anchor. ^O3'?00 38 Sir I s A A c N E w T o n's Book I. 18. Now this experience is alone fufficient to fhew to us the whole of tliis law of motion. 19. S I N c E we find, that the fame power will always pro- duce the fame change in the motion of any body, whether that body were before moving with a fwifter or flower mo- tion ; the change wrought in the motion of a body depends only on the power appHed to it, without any regard to the body's former motion : and therefore the degree of motion, which the body already poffeffes, having no influence on the power applied to difturb its operation, the effe([ls of the fame power will not only be the fame in all degrees of mo- tion of the body ; but we have likewife no reafon to doubt, but that a body perfedly at reft would receive from any pow- er as much motion, as would be equivalent to the efFe6l of the fame power applied to that body already in motion. ao. Again, fappofe a body being at reft, any number of equal powers fhould be fuccefTively applied to it ; pufhing it forward from time to time in the fame courfe or diredion. Upon the appHcation of the firft power the body would begin to move ; when the fecond power was applied, it appears from what has been faid, that the motion of the body would be- come double; the third power would treble the motion of the body ; and fo on, till after the operation of the laft power the motion of the body would be as many times the motion, which the firft power gave it, as there are powers in number. And the effect of this number of powers will be always the , fame. Chap. I. PHILOSOPHY. 39 fame, without any regard to the fpace of time taken un in applying them : (o that greater or lePier intervals betv/eeir the appHcation of each of thefe powers will produce no dif- ference at all in tlieir efFeds. Since therefore the diftance of time between the action of each power is of no confeqiience i widiout doubt the effect will ftill be the fame, though the powers fliould all be applied at the very fame inftant ; or al- though a fingle power fhould be applied equal in ftrength tO' the colledlive force of all thefe powers. Hence it plainly fol- lows, that the degree of modon, into which any body will' be put out of a ftate of reft by any power, will be proportio- nal to that power. A double power will give twice the velo- city, a treble power three times the velocity, and fb on. The foregoing reafoning will equally take place, though the bo- dy were not fuppofed to be at reft, when the powers began to be applied to it ; provided the diredion, in which the powers were applied, either confpired with the adion of the body, or was diredlly oppoiite to it. Therefore if any power be ap- plied to a moving body, and ad upon the body either in the diredion wherewith the body moves, fo as to accelerate the body ; or if it ad diredly oppofite to the motion of tlic body, fo as to retard it : in both thefe cafes the change of motion will be proportional to the power applied ; nay, the augmentation of the motion in one cafe, and the dimi- nution thereof in the other, will be equal to that degree of motion, into which the fame power would put the body, hadl it been at reft, when the power was applied. xr. Fart.h.e;r; Ao Sir Is A AC N E w T o n's BookL 21. Farther, a power may be fo applied to a moving bod}'', as to a.S: obliquely to the motion of the body. And the effefts of fuch an oblique motion may be deduced from this obfervation ; that as all bodies are continually moving a- long with the earth, we fee that the vilible effefts of the fame power are always the fame, in whatever direction the power afts : and therefore the vifible efFe(fts of any power upon a body, which feems only to be at reft, is always to appearance the lame as the real effed: would be upon a body truly at reft. Now fuppofe a body were moving along the line A B (in fig. i.) and the eye accompanied it with an equal motion in the line C D equidiftant from AB ; fo that when the body is at A, the eye fliall be at C, and when the body is advanced to E in the line A B, tlie eye fliall be advanced to F in the line C D, the diftances A E and C F being equal. It is evident, that here the body will appear to the eye to be at reft ; and the line F E G drawn from the eye through the body fhall feem to the eye to be immoveable ; though as the body and eye move forward together, this line fhall really alfo move ; fo that when the body fhall be advanced to H and the eye to K, the line FEG fhall be transferred into the fituation KHL, this line KHL being equidiftant from FEG. Now if the bo- dy \vhen at E \^^ere to receive an impulfe in the direction of the line FEG; while the eye is moving on from F to I, and carrying along with it the line FEG, the body will appear to the eye to move along this line FEG: for this is what has juft now been faid ; that while bodies are moving along with the earth, and the fpedator's eye partakes of the fame motion, the eftcct of any power upon the body will appear to be what it Chap. I. PHILOSOPHY. 41 it would really have been, had die body been truly at refl> when the power was applied. From hence it follows, that when the eye is advanced to K, the body will appear fome- where in the line K H L. Suppofe it appear in M ; then it is manifeft, from what has been premiied at the beginning of this paragraph, diat the diftance HM is equal to what the body would have run upon the line E G, during the time, wherein the eye has pafTed from F to K, provided that the bo- dy had been at reft, when a£led upon in E. If it be farther asked, after what manner the body has moved from E to M ? I anfwer, through a ftraight Hne ; for it has been fhewn above in the explication of the firft law of motion, that a mov- ing body, from the time it is left to it felf, will proceed on in one continued ftraight line. XI. If en be taken equal to HM and NM be drawn ; ftnceHMis equidiftant from EN, NM will be equidiftant from E H. Therefore the effed: of any power upon a moving body, when that power adls obliquely to the motion of the body, is to be determined in this manner. Suppofe the bo- dy is moving along the ftraight line A E B, if when the body is come to E, a power gives it an impulfe in the diredion of the line EG, to find what courfe the body will afterwards take wc muft proceed thus. Take in E B any length E H, and in E G take fuch a length E N, that if the body had been at reft in E, the power applied to it would have caufed it to move over E N in the fame fpace of time, as it would have employed in paftmg over EH, if the power had not aded at all upon it. Then draw H L equidiftant from E G, and N M equidiftant G from 42 Sir Isaac N e w t o n's Book L from EB. After this, if a line be drawn from E to the point M, where thefe two lines meet, the line EM will be the courfe into which the body will be put by the adion of the power upon it at E. zg. A MATHEMATICAL reader would here exped in fome particulars more regular demonftrations j but as I do not at prefent addrefs my felf to fuch, fo I hope, what I have now written will render my meaning evident enough to thofe, who are unacquainted with that kind of reafoning. 14. Now as we have been ITiewing, that fome aftual force is necellary either to put bodies out of a ftate of reft in- to motion, or to change the motion, which they have once received ; it is proper here to obferve, that this quality in bo- dies, whereby they preferve their prefent ftate, with regard to motion or reft, till fome acflive force difturb them, is cal- led the vis iNERTiAEof matter : and by this property, mat- ter, ftuggilli and unaftive of it felf, retains all the power im- prefl'ed upon it, and cannot be made to ceafe from adion, but by the oppoHtion of as great a power, as that which firft mov- ed it. By the degree of this vis inertia e, or power of inac- tivity, as we fliall henceforth call it, we primarily judge of the quantity of folid 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 eflential to all matter ; and as we yet know no reafon to fuppofe, that bodies are compo- fed of different kinds of matter, we rather prefume, that the matter of all bodies is the fame ; and that the degree of this Chap. I. PHILOSOPHY. 43 this power of inadivity Is in e\Try body proportional to the quantity of the foHd matter in it. But although we have no abfolute proof, that all the matter in the univerfe is uniform, and pofleffes this power of inadlivity in the fame degree ; yet we can with certainty compare together the different degrees of this power of inadivity in different bodies. Particularly this power is proportional to the weight of bodies, as Sir Isaac Newton has demonftrated *. However, notwithftanding that this power of inactivity in any body can be more certain- ly known, than the quantity of folid matter in it ; yet fince there is no reafon to fufped: that one is not proportional to the other, we fhall hereafter fpeak without hefitation of the quan- tity of matter in bodies, as the meafure of the degree of their power of inadlivity. 1 5". This being eftablifhed, we may now compare the effects of the lame power upon different bodies, as liither- to we have fhewn the effeds of different powers upon the iame body. And here if we limit the word motion to the pecuHar fenfe given to it in philofophy, we may comprehend all that is to be faid upon this head under one fliort precept ; that the fame power, to whatever body it is applied, will al- ways produce the fame degree of motion. But here motion does not lignify the degree of celerity or velocity with which a body moves, in which fenfe only we have hitherto ufed it ; but it is made ufe of particularly in philofophy to Hgnify the force with which a body moves : as if two bodies A and B be- » Princ. Philof. L. II. prop. 24. corol. 7. See alfo B.II. Oh. 5. § 3. of this treatife. G X ing 44 Sir I s A A c N E w T o n's B ook I. ing in motion, twice the force would be required to ftop A as to ftop B, the motion of A would be efteemed double the motion of B. In moving bodies, thefe two things are care- fully to be diftingviifhed ; their velocity or celerity, which is meafured by the fpace they pals through during any determi- nate portion of time ; and the quantity of their motion, or the force, with which they will prefs againft any refiftance. Which force, when different bodies move with the fame ve- locity, is proportional to the quantity of folid matter in the bodies ; but if the bodies are equal, this force is proportio- nal to their refpedive velocities, and in other cafes it is pro- portional both to the quantity of folid matter in the body, and alio to its velocity. To inftance in two bodies A and B : if A be twice as great as B, and they have both the fame velocity, the motion of A fhall be double the motion of B ; and if the bo- dies be equal, and the velocity of A be twice that of B, the motion of A fhall likewife be double that of B ; but if A be twice as large as B, and move twice as fwift, the motion of A will be four times the motion of B ; and laftly, if A be twice as large as B, and move but half as faft, the degree of their motion fhall be the fame» 16. Th I s is the particular fenfe given to tlie word motion by philofophers, and in this fenfe of the word the fame pow- er always produces the fame quantity or degree of motion. If the fame power aft upon two bodies A and B, the velocities, it fhall give to each of them, fhall be fo adjufted to the refpec- tive bodies, that the fame degree of motion fhall be produced in each. If Abe twice as great as B, its velocity fhall be half that C«Ap. I. PHILOSOPHY. 45 that of B ; if A has three times as much foHd matter as B, the velocity of A fhall be one third of the velocity of B ; and ge- nerally the velocity given to A fhall bear the fame proportion to the velocity given to B, as the quantity of foUd matter con- tained in the body B bears to the quantity of folid matter con- tained in A. 17. The reafon of all this is evident from what has gone before. If a power were applied to B, which fhould bear the fame proportion to the power applied to A, as the body B bears to A, the bodies B and A would both receive the fame velocity ; and the velocit)^, which B will receive from this power, will bear the fame proportion to the velocity, which it would receive from the adlion of the power applied to A> as the former of thefe 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 fame power, the fame proportion as the body B bears to A. 18. From hence we may now pafs to the third law o£ " motion, where this diftindion between the velocity of a bo- dy and its whole motion is farther neceflary to be regarded, as.. iliall immediately be fhewn ; after having firft illuftrated the: meaning of this law by a familiar inftance, If a ftone or Or-^ ther load be drawn by a horfe ; the load re-adts upon the horfe, :, as much as the horfe ads upon the load ; for the harness, which is ftrained between them, preffes againft the horfe. as. much as againft the load; and the progrellive motion of the. horfe,:. ^6 Sir I s A A c N E w T n's Book I. Iiorfe fonvard :j kindred as much by the load, as the motion of the load is f imoted by the endeavour of the horfe : that is, if the hcrfe put forth the fame ftrength, when loofened from the load, lie would move himfelf forwards with greater fwiftnefs in proportion to the difference between the weight of his own body and the weight of himfelf and load to- gether. 19. This inftance will afford fome general notion of the meaning of this law. But to proceed to a more pliilofophi- cal explication : if a body in motion ftrike againft another at reft, let the body ftriking be ever fo fmall, yet fhall it com- municate fome degree of motion to the body it ftrikes againft, though the lefs that body be in comparlfon of that it impin- ges upon, and the lefs the velocity is, with which it moves, the fmaller will be the motion communicated. But whatever degree of motion it gives to the refting body, the fame it fhall lofe it felf This is the neceffary coniequence of the forementioned power of inadivity in matter. For fuppofe the two bodies equal, it is evident from the time they meet, both the bodies are to be moved by the ftngle motion of the firft ; therefore the body in motion by means of its power of inadlivity retaining the motion firft given it, ftrikes upon the other with the fame force, wherewith it was a6ted upon it ielf : but now both the bodies being to be moved by that force, which before moved one only, the enfuing velocity will be the fame, as if the power, which was applied to one of the bodies, and put it into motion, had been applied to both J whence it appears, that they will proceed forwards, 3 with Chap. f. P H I L O S O P H T 47 with half the velocity, which the body firft in motion had j that is, the body firft moved will have lofl hal fits motion, and the other will have gained exadly as much. rule is juft, provided the bodies keep contiguous after meeting ; as they would always do, if it were not for a certain caufe that often intervenes, and which muft now be explained. BodiV-g upon ftriking againft each other, fuffer an alteration in their figure, having their parts preffed inwards by the fboke, which for the moft part recoil again afterwards, the bodies endea- vouring to recover their former fliape. This power, whereby bodies are inabled to regain their firft figure, is ufiially called dieir elafticity, and when it ad:s, it forces the bodies from each other, and caufes them to feparate. Now the effedt of this elafticity in the prefent cafe is fuch, that if the bodies are perfectly elaftic, fo as to recoil with as great a force as they are bent with, that they recover their figure in the fame Ipace of time, as has been taken up in the alteration made in it by their comprefiion together ; then this power will feparate the bodies as fwiftly, as they before approached, and ading up- on both equally, upon the body firft in motion contrary to . the diredion in which it moves, and upon the other as much in the diredion of its motion, it will take from the firft, and add to the other equal degrees of velocity : fo that the power being ftrong enough to feparate them with as great a velocity, . as they approached with, the firft will be quite ftopt, and that which was at reft, will receive all the motion of the other. If the bodies are elaftic in a lefs degree, the firft will not lofe all its motion, nor will the other acquire the motion of the firft, but fall as much fhort of it, as the other retains. For 4-8 Sir Isaac N e w t o n's Book I. For this rule is never deviated from, that though the degree of elafticity determines how much more than half its veloci- ty the body firft in motion lliall lofe ; yet in every cafe the lofs in the motion of this body fliall be transferred to the other, that other body always receiving by the ftroke as much mo- tion, as is taken from tlie firft. 30. T H I s is the cafe of a body ftriking diredly againfl an equal body at reft, and the reafoning here ufed is fully con- firmed by experience. There are many other cafes of bodies impinging againft one another : but the mention of thele fhall be referved to the next chapter, where we intend to be more particular and diffufive in the proof of thefe laws of mo- tion, 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 philofopher, trom the moft obvious obfervations, that fuggeft them to us ; I now intend to give more particular proofs of them, by re- counting fome of die difcoveries which have been made in philofophy before Sir I s a a c Newton. For as tliey were all colleded by reafoning upon tliofe laws ; fo the conformity of thefe difcoveries to experience makes them fo many proofs of the truth of the principles, from which they were derived. 1. Let Chap.2. philosophy. 49 1. Let us begin with the fubjed, which concluded the laft chapter. Although the body in motion be not equal to the body at reft, on which it ftrikes ; yet the motion after the ftroke is to be eftimated in the fame manner as above. Let A ( in fig. g .) be a body in motion towards another body B lying at reft. When A is arrived at B, it cannot proceed farther without putting B into motion ; and what motion it gives to B, it muft lofe it felf, that the whole degree of mo- tion of A and B together, if neither of the bodies be elaftic, (hall be equal, after die meeting of the bodies, to the fingle motion of A before the ftroke. Therefore, from what has been faid above, it is manifeft, that as foon as the two bodies are met, they will move on together with a velocity, which will bear the fame proportion to the original velocity of A, as the body A bears to the fum of both the bodies. g. If the bodies are elaftic, fo that they (hall feparate af- ter the ftroke, A muft lofe a greater part of its motion, and the fubfequent motion of B will be augmented by this elafti- city, as much as the motion of A is diminifhed by it. The elafticity ading equally between both the bodies, it will com- municate to each the fame degree of motion ; that is, it will feparate the bodies by taking from the body A and adding to the body B different degrees of velocity, fo proportioned to their refpe^ive quantities of matter, that the degree of mo- tion, wherewith A feparates from B, fhall be equal to the de- gree of motion, wherewith B feparates from A. It follows therefore , that the velocity taken from A by the elafticity bears to the velocity, which the fame elafticity adds to B, the H fame 50 Sir Isaac Newton's BookL fame proportion, as B bears to A : confequently the velocity, which the elafticity takes from A, will bear the fame propor- tion to the whole velocity, wherewith this elafticity caufes the two bodies to feparate from each other, as the body B bears to the fum of the two bodies A and B ; and the velocity, which is added to B by the elafticity, bears to the velocity, where- with the bodies feparate, the fame proportion, as the body A bears to the fum of the two bodies A and B. Thus is found, how much the elafticity takes from the velocity of A, and adds to the velocity of B ; provided the degree of elafticity be known, whereby to determine the whole velocity wherewith the bodies feparate from each other after the ftroke ^ 4,. After this manner is determined in every cafe the re- fult of a body in motion ftriking againft another at reft. The fame principles will alfo determine the effedls, when both bodies are in motion. 5*. L ET two equal bodies move againft each other with e- qual fwiftnels. Then the force, with which each of them prefles forwards, being equal when they ftrike ; each prefix ing in its own direction with the fame energy, neither fhall furmount the other, but both be ftopt, if they be not elaftic : for if they be elaftic, they fhall from thence recover new mo- tion, and recede from each other, as fwiftly as they met, if they be perfedlly elaftic ; but more ftowly, if lefs fo. In the fame manner, if two bodies of unequal bignefs ftrike againft each other, and their velocities be fo related, that the velocity » How this degree ofelaiUcity is to be found by experiment, will be fliewn below in § 74. of Chap. 2. PHILOSOPHY. 51 of the lefTer body fhall exceed the velocity of the greater in the fame proportion, as the greater body exceeds the leflcr (for inftance, if one body contains twice the folid matter as the o- ther, and moves but half as fall) two fuch bodies will entire- ly fupprefs each other's motion, and remain from the time of their meeting fixed ; if, as before> they are not elaftic : but, if they are fo in the highefl: degree, they fhall recede again, each with the fame velocity, wherewith they met. For this elaftic power, as in the preceding cafe, fhall renew their mo- tion, and prefling equally upon both, fhall give the fame mo- tion to both j that is, fhall caufe the velocity, which the leller body receives, to bear the fame proportion to the velocity, which the greater receives, as the greater body bears to the leller : fo that the velocities fhall bear the lame proportion to each other after the ftroke, as before. Therefore if the bodies, by being perfedly elaftic, have the fum of their velocities after the ftroke equal to the fiim of their velocities before the ftroke, each body after the ftroke will receive its firft veloci- ty. And the fame proportion will hold likewife between the velocities, wherewith they go off, tliough they are elaftic but in a lefs degree ; only then the velocity of each will be lefs in proportion to the defedt of elafticity. 6. If the velocities, wherewith the bodies meet, are not in the proportion here fuppofed ; but if one of the bodies, as A, has a fwifter velocity in comparifon to the velocity of the other; then the effedl of this excefs of velocity in the body A muft be joined to the effed: now mentioned, after the manner of this following example. Let A be twice as great as B, and H 2. move 52 Sir Is A AC Newton's Book I. move with the fame fwiftncfs as B. Here A moves with twice that degree of fwiftnefs, which would anfwer to the foremen- tioned proportion. For A being double to B, if it moved but with half the fwiftnefs, wherewith B advances, it has been juft now fhewn, that the two bodies upon meeting would flop, if they were not elaflic ; and if they were elaflic, that they would each recoil, fo as to caufe 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 velo- city, if the bodies be not elaftic; and the future motion of the bodies will be the fame, as if A had advanced againft B at reft with half the velocity here afligned to it. If the bodies be elaftic, the velocity of A and B after the ftroke may be thus difcovered. As the two bodies advance againft each other, the velocity, with which they meet, is made up of the velo- cities of both bodies added together. After the ftroke their elafticity will feparate them again. The degree of elafticity will determine what proportion the velocity, wherewith they feparate, muft bear to that, wherewith they meet. Divide this velocity, with which the bodies feparate into two parts, that one of the parts bear to the other the fame proportion, as the body A bears to B ; and afcribe the lefler part to the great- er body A, and the greater part of the velocity to die lelfer body B. Then take the part afcribed to A from the common velocity, which A and B would have had after the ftroke, if they had not been elaftic ; and add the part afcribed to B to the fame common velocity. By this means the true velocities of A and B after the ftroke wiU be made known. 7, If Chap. 2. PHILOSOPHY. 53 7. I F the bodies are perfedly elaftic, the great H u y c e n s has laid down this rule for finding their motion after con- courfe \ Any ftraight line CD (in fig. 4, y. ) being drawn, let it be divided in E, that CE bear the fame proportion to E D, as the fwiftnefs of A bore to the fwiftnefs of B before the ftroke. Let the fame line C D be alfo divided in F, that C F bear the fame 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 fhall recoil after the ftroke, and the velocity, wherewith the body A fhall return, will bear the fame proportion to the velocity, wherewith B fhall return, as G C bears to G D ; but if the point G falls with- out the Une C D, then the bodies after their concourfc fhall both proceed to move the fame way, and the velocity of A fhall bear to the velocity of B the fame proportion, that G C bears to G D, as before. 8. If the body B had ftood ftill, and received the impuKe of the other body A upon it ; the effedl has been already ex^ plained in the cafe, when the bodies are not elaftic. And when they are elaftic, the reftilt of their collifion is found by combining the effed of the elafticity with the other effe Archimed . dc asquipond. prop. 1 1 . ^ Ibid. prop. 1. 2. c Luc.is Valerius De cemr. gr.;vit. folid. L. I. prop. 1. " Idem L. IT. prop. 2. K z hold 68 Sir Isaac Newton's Book I. hold it fixed at reft ; the effed of the power of gravity on a body is the fame, as if that whole power were to exert itfelf on the center of gravity only. Whence it follows, that, when the power of gravity adls on a body fufpended by any point, if the body is fo fufpended, that the center of gravity of the body can defcend ; the power of gravity will give motion to that body, otherwife not : or if a number of bodies are fo connected together, that, when any one is put into motion, the reft fhall, by the manner of their being joined, receive fuch motion, as fhall keep their common center of gravity at reft \ then the power of gravity fhall not be able to produce any motion in thefe bodies, but in all other cafes it will. Thus, ifthebodyAB (infig. xo,ti.) whofe center of gra- vity is C, be hung on the point A, and the center C be per- pendicularly under A (as in fig. lo. ) the weight of the bo- dy will hold it ftill without motion, becaufe the center C cannot defcend any lower. But if the body be removed in- to any other fituation, where the center C is not perpendi- cularly under A ( as in fig. XI.) the body by its weight will be put into motion towards the perpendicular fituation of its center of gravity. Alfo if two bodies A, B (in fig. n.) be joined together by the rod C D lying in an horizontal fitua- tion, and be fupported at tlie point E ; if this point be the center of gravity common to the two bodies, tlieir 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 defcending, in which the common center of gravity is found. So in like manner, if diefe two bodies were conneded together by any more complex contrivance \ yet if Chap. 2. PHILOSOPHY. 69 if one of the bodies cannot move without fo moving the- other, that their common center of gravity fhall reft, the weight of the bodies will not put them in motion, otherwife it will i8. I s H A L L proceed in the next place to fpeak of the me^ chanical powers. Thefe are certain inftruments or machines, contrived for the moving great weights with fmall force ; and their effeds are all deducible from the obfervation we have juft been making. They are ufually reckoned in number five ; the lever, die wheel and axis, the pulley, the wedge, and the fcrew ; to which fome add the inclined plane. As thefe inftruments have been of very ancient ufe, fo the cele- brated Archimedes feems to have been the firft, who dif- covered the true reafbn of their effedts. This, I think, may be colledled from what is related of him, that Ibme expreftions, which he ufed to denote the urdimited force of diefe in- ftruments, were received as very extraordinary paradoxes: whereas to thofe, who had underftood the caule of their great force, no expreftions of that kind could have appeared furprizing. 19. All the effe<9:s of thefe powers may be judged of by this one rule, that, when two weights are applied to any of thefe inftruments, the weights will equiponderate, if, when put into motion, their velocities will be reciprocally propor- rional to their refpedive weights. And what is faid of weights, niuft of neceftity be equally underftood of any other forces equi- yo Sir Isaac Newton's Book! e'quivalent to weights, fuch as the force of a man's arm, a ftream of water, or the Hke. go. But to comprehend the meaning of this rule, the reader miift know, what is to be underftood by reciprocal proportion ; which I iliall now endeavour to explain, as di- ftinftly as I can ; for I fhall be obliged very frequently to make ufe of this term. When any two things are fo related, that one increafes in the {ame proportion as the other, they are diredlly proportional. So if any number of men can perform in a determined fpace of time a certain quantity of any work, luppofe drain a fifh-pond, or the like ; and twice the num- ber of men can perform twice the quantity of the fame work, in the fame time ; and three times the number of men can perform as foon thrice the work ; here the number of men and the quantity of the work are diredly proportional. On the other hand, when two things are fo related, that one de- creafes in the fame proportion, as the other increafes, they are faid to be reciprocally proportional. Thus if twice the number of men can perform the fame work in half the time, and three times the number of men can finifh the fame in a third part of the time ; then the number of men and the time arc reciprocally proportional. We fliev/ed above ' how to find the common center of gravity of two bodies, there the diftances of that common center from the centers of gra- vity of the tvv'o bodies are reciprocally proportional to the re- fpedive bodies. For C E in fig. j 6. being in the fime pro- ' § 25. portion Chap. 2. PHILOSOPHY. 71 portion to ED, as B bears to A ; C E is fo much greater in proportion than ED, as A is lefs in proportion than B. 31. Now this being underftood, the reafon of the rule here ftated will ealily appear. For if thefe two bodies were put in motion, while the point E refted, the velocity, where- with A would move, would bear the fame 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 refts, is reciprocally proportional to die body. But we have fhewn above *, that if two bodies are fo connected to- gether, that the putting them in motion will not move their common center of gravity ; the weight of thofe bodies v/ill not produce in diem any motion. Therefore in any of thele mechanical engines, if, when the bodies are put into motion, their velocities are reciprocally proportional to their refpeftive weights, whereby the comm.on center of gravity would re- main at reft ; 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 delcrip- tion of each mechanical power. 31. The lever was firft named above. This is a bar made ufe of to fuftain and move great weights. The bar is ap- plied in one part to fome ftrong fupport ; as the bar A B ( in fig. ig, 14.. ) is applied at the point C to the fupport D. In fome other part of the bar, as E, is appHed the weight to be fuftained or moved ; and in a third place, as F, is applied ano- ther weight or equivalent force, which is to fuftain or move » § 27. the 72 Sir Isaac Newton's Book I. the weight at E. Now here, if, when the lever fhould be put in motion, and turned upon the point C, tlie velocity, wherewith the point F would move, bears the fame propor- tion 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 propeniity to move eidier way. If the weight or other force at F be not fo great as to bear this proportion, the weight at E will not be fuftained ; but if the force at F be greater than this, the weight at E will be furmounted. This is evident from what has been faid above % when die forces at E and F are placed (as in fig. 2 g .) on different fides of the fupport D. It will appear alfo equal- ly manifeft in the other cafe, by continuing the bar B C in fig. 14. on the other fide of the fupport 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 cafe : and it is evident, diat the point F will be lifted up by the weight at G with the fame degree of force, as by the other power applied to F ; fince, if the weight at E were removed, a weight hung at F equal to that at G would balance the lever, the diftances CG 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 propofed of the lever, wliich fhall « Pag. 65, 68. hold Chap. 2. PHILOSOPHY. 73 hold the whole in a juft counterpoize. Suppofe (in fig. if.) the two powers at E and F did not equiponderate, and it were required to apply a third powxr to the point G, that inight be fuiiicient to balance the lever. Find what power in F would juft counterbalance the power in E ; then if the difference between this power and that, which is adually applied at F, bear the fame proportion to the third power to be applied at G, as the diftance C G bears to C F ; the lever will be coun- terpoized by the help of this tliird power, if it be fo applied as to ad; die fame way with the power in F, when that power is too fmall to counterbalance the power in E ; but other- wife the power in G muft be fo appHed, as to a6i againft the power in F. In like manner, il 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 propofed, which Ihould bring the whole to a juft balance. And what is here faid concerning a plurality of powers, may be equally applied to all the following cafes. 34. If the lever ftiould confift of two arms making an angle at the point C (as in fig. r6.) yet if the forces are ap- plied perpendicularly to each arm, the fame proportion will hold between tlie forces applied, and the diftances of the cen- ter, whereon the lever refts, from the points to which they are applied. That is, the weight at E will be to the force in F in the fame proportion, as C F bears to C E. 5 i". But whenever the forces applied to the lever ad ob- liquely to the arm, to which they are applied ( as in fig. 17. ) L then 74 Sir IsA.^c N.£n.l(5N's Bc^aL tlien the fkengdi* of the fisr^es is to be eftamttsd bjs liassikt fdl from the C€F>te? of ^e letter to the diredions^ wherein, dbsa. forces ad. Ta balance fthe levers in fig, X7, the weigkt or other fotce aM F wUl beai th& farR« proportion to- die weight at E, as the diftance GE bsats t© G G the perpendicular let M from the point G upon the iine, which denotes the dir«^tt©». wherein the force applied- to F ads : for- here, if the l«ver be put into mtotion, the power applied to F will b^n to mo9« i& the diredion of the line F G; and therefore k» fe-ft motion will be the fame, as the motion of the point G. 36. When two weights hang upon a lever, and the pointy by which the lever is fupported, is placed in tlite middle be- tween the two weights, that the arms of the lever are both of equal lengtli ^ then this lever is particularly called a ba- lance ; and equal weights equipo;ii half the diameter of the wheel. -4^4,. Tf the "wheel be put into motion, and turned once round, tliat the cord, on which the -weight G hangs, be wound once more round the axis ; then at the fame time the cord, whereon the weight H hangs, will be wound off from the wheel ofle circuit. Therefore the velocity df the weight % will bear the fame 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 fame ' jproportion to the circumference of the wheel, as the thick- nefs of the roller bears" to the diameter of the wheel, corife- '■quently the velocity of the weight G bears to the velocity '•^of the weight H the fameproporticsn, as the thicknefs of ''^he roller bears' to the diameter of the vdicel, which is the proportion that the weight H bears to the weight G. There- fore as before in the lever, fo here alfo the general rule laid down Ch-w. 4. PHILOSOPHY. 79. down above is veiihedv that the weights equipondcrajt-e, when their velo<;ities wouJd be reciptocaily proportional- to. thcin refpej^tive weights. ^ j?. In Hke r^ianner, if on the fame axis two wheels of dif- ferent fiz^ ape fixed ( as in iig. 5 y. ) and a weight hung on each ;^ the weights will equiponderate, if the weight hung oji the greater wheel bear die lame proportion to the weight hung on the leffer, as the diameter or the lefler wheel bears to the diameter of the greater. 4^. I T is ufual to join many wheels together in the faine. frame, which by the means or certain teeth, formed in the cir-r. cumference of each wheel, fliall communicate motion to eacl^ other. A machine of this nature is reprefented in fig. g 6. Here A B C is a winch, upon which is fixed a fmall wheel D indent- ed with teeth, which mo.ve in the hke teedi of a larger wheel E F fixed on the axis G H. Let this axis carry another wheel I, which fliall move in like manner a greater wheel K L fixed on the axis M N. Let this axis carry another fmall wheel O, which after the fame manner fhall turn about a larger wheel P Q. fixed on the roller R S, on which a cord fhall be wound, that holds a weight, as T. Now the proportion required be- tween the weight T and a power applied to the winch at A fufficient to fupport the weight, will mofl: eafily be eftimated, 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 defcribe a circle as A V. Sup- pofe the wheel E F to have tqn times ijhie |iij.raber of ^e? tli, as the 8o Sir Isaac Newton's Book!. the wheel D; then the winch miift turn round ten times to carry the wheel E F once round. If the wheel K L has alfo ten times the number of teeth, as I, the wheel I muft turn round ten times to carry the wheel KL once round ; and confe- quently the winch ABC muft turn round an hundred times to turn the wheel KL once round. Laftly, if the wheel PQ^ has ten times the number of teeth, as the wheel O, the winch muft turn about one thoufand times in order to turn the wheel PQj or the roller RS once round. Therefore here the point A muft have gone over the circle A V a thoufand times, in or- der to lift the weight T through a fpace equal to the circum- ference of the roller R S : whence it follows, that the power applied at A will balance the weight T, if it bear the fame proportion to it, as the circumference of the roller to one thoufand times die circle A V ; or the fame proportion as half the thicknefs of the roller bears to one thoufand times AB. ^7. I SHALL now explain the efFed of the pulley. Let a w^eight hang by a pulley, as in fig. g 7. Here it is evi- dent, that the power A, by which the weight B is fupported, muft be equal to the weight ; for the cord C D is equally ftrained between them ; and if the weight B move, the power A muft move with equal velocity. The pulley E has no other cffeift, than to permit the power A to a6l in another diredion, than it muft have done, if it had been diredly applied to fupport the weight without the intervention of any fuch inftrument. 4.8. Again, let a weight be fupported, as in fig. 38; where the weight A is fixed to the pulley B, and the cord, by which Chap. 2. PHILOSOPHY. 8r 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 fupported by a cord doubled ; infomuch tliat although the cord were not ftrong enough to hold the weight fingle, yet being thus doubled it might fupport it. If the end of the cord held by the power D were hung on the hook C, as well as the otlier end ; then, when both ends of the cord were tied to the hook, it is evident, that the hook would bear the whole weight j and each end of the firing would bear againft the hook with tlie force of half the weight only, feeing both ends together bear with the force of die whole. Hence it is evident, that, when the power D holds one end of the weight, the force, which it muft exert to fupport the weight, muft be equal to juft half die weight. And the fame proportion between the weight and power might be colled:- ed from comparing the refpeftive velocities, with which they would move ; for it is evident, that the power mull: move through a fpace equal to twice the difcance of the pulley from the hook, in order to lift the pulley up to the hook. 4.9. It is equally eafy to eftimate the effe^fl:, when many pulleys are combined together, as in fig. 3 9, 4.0 ; in the firft of which the under fet ot pulleys , and confequently the weight is held by fix firings ; and in the latter figure by fi\'e : therefore in the firft of tliefe figures the power to fiapport the weight, muft be one fi.xth part only of the weight, and in the latter figure the power muft be one fifth part. M ^o. There §2 Sir I s A A c N E w T o n's Book I. so. There are two other ways of fupporting a weight by pulleys, which I fhall particularly conlider. SI. One of thefe ways is reprefented in fig. 41. Here the weight being connected to the pulley B, a power equal to half the weight A would fupport the pulley C, if applied im- mediately 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 fupported by half the force, with which the pulley C is drawn down, this pulley D will uphold the pulley C ; fo that if the pulley D be upheld with a force equal to one fourth part of the weight A, that force will fup- port the weight. But, for the fame reafon as before, if the power in E be equal to half the force neceffary to uphold the pulley D ; this pulley, and confequently the weight A, will be upheld : therefore, if the power in E be one eighth part of the weight A, it will fupport the weight. SZ. Another way of applying pulleys to a weight is reprefented in fig. 41. To explain the effed of pulleys thus applied, it will be proper to confider different weights hang- ing, as in fig.4 g . Here, if the power and weights balance each other, the power A is equal to the weight B ; the weight C is €qual to twice the power A, or the weight B ; and for the lame reafon the weight D is equal to twice the weight C, or equal to four times the power A. It is evident therefore, tliat all the three weights B, C, D together are equal to feven times the power A. But if thefe three weights were joined in one, they would produce the cafe of fig. 40 : fo that in that figure the weight Chap.2. philosophy. 83 weight A, where there are three pulleys, is feven times the power B. If there had been but two pulleys, the weight would have been three times the power ; and if there had ben four pulleys, tlie weight would have been fifteen times the power. y 5 . The wedge is next to be confidered. The form of this inftrument is fufficiently known. When it is put under any weight (as in fig. 44. ) the force, with which the wedge will Hft the weight, when drove under it by a blow upon the end A B, will bear the fame proportion to the force, where- with the blow would a£l on the weight, if direftly 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. 5-4. The fcrew is the fifth mechanical power. There are two ways of applying this inftrument. Sometimes it is fcrewed into a hole, as in fig. 45-, where the fcrew AB is fcrewed through the plank C D. Sometimes the fcrew is applied to the teeth of a wheel, as in fig. 45, where the thred of the fcrew A B turns in the teeth of a wheel C D. In both thefe cafes, if a bar, as AE, be fixed to the end A of the fcrew ; the force, wherewith the end B of the fcrew in fig. 45- is forced down, and the force, wherewith the teeth of tlie wheel C D in fig. 44 are held, bears the fame proportion to the power applied to the end E of the bar ; as the velocity, wherewith the end E will move, when the fcrew is turned, bears to the velocity, wherewith the end B of the fcrew in fig. 43, or die teedi of the wheel C D in fig. 4(>, will be moved. M z 5-5-. The 84 Sir Isaac Newton's Book I. 5" 5". The inclined plane affords alfo a means of railing a weight with lefs force, than what is equal to the weight it felf Siippofe it were required to raife the globe A ( in fig. 4.7. ) from the ground B C up to the point, whofe perpendi- cular height from the ground is E D. If this globe be drawn along the flant D F, lefs force will be required to raife it, thon if it were lifted diredly up. Here if the force applied to the globe bear the fame proportion only to its weight, as E D bears to F D, it will be fufficient to hold up the globe ; and there- fore any addition to that force will put it in motion, and draw it up ; unlefs the globe, by prefling againft the plane, where- on it lies, adhere in fome degree to the plane. This indeed it muft always do more or lefs, {ince no plane can be made fo abfolutely fmooth as to have no inequalities at all ; nor yet (6 infinitely hard, as not to yield in the leaft to the prefllire of the weight. Therefore the globe cannot be laid on fuch a plane, whereon it will Aide with perfed freedom, but they muft in fome meafure rub againft each other ; and this fridion will make it neceffary to imploy a certain degree of force more than what is neceffary to fupport the globe, in order to give it any motion. But as all the mechanical powers are fubjedl in fome degree or other to the like impediment from fridion ; I fhall here only fliew what force would be neceffary to fu- ftain the globe, if it could lie upon the plane without cauf- ing any fridion at all. And I fay, 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 fame proportion to the weight of the globe, as E D bears to D F ; this Chap. 2. PHILOSOPHY. 85 this force will fuftain the globe. In order to the making proof of this, let the cord GH be continued on, and turned over the pulley I, and let the weight K be hung to it- Now I lay, if this weight bears the fame proportion to die globe A, as D E bears to D F, die weight will fupport the globe. I think it is very manifefl-, 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 tlie weight K. In the firft place let the weight hang fo, that a line drawn from L to M ihall lie horizontally ; and I fay, if the globe be moved either up or down the plane D F, the weight will fo move along with it, that the center of gravity common to both the weights fhall continue in this line L M, and therefore fhall in no cafe defcend. To prove this more fully, I Hiall depart a little from the method of this treatile, and make ufe of a mathematical propolition or two : but they are fuch, as any perfon, who has read Euclid's Elements, will fully comprehend j and are in themfelves fo evident, that, I believe, my readers, who are wholly ftrangers to geometri- cal writings, will make no difRculty of admitting them. This being premifed, let the globe be moved up, till its center be at G, then will M the center of gravity of the weight K be funk to N ; fo that M N fhall be equal to G L. Draw N G crofling the line M L in O ; then I fay, that O is the common center of gravity of the two weights in this their new fitua- tion. Let G P be drawn perpendicular to M L ; then G L will bear the fame proportion to G P, as D F bears to D E ; and M N being equal to G L, M N will bear the fame proportion to. 86 Sir I s A A c N E w T o n's Book I. to G P, as D F bears to DE. But N O bears the fame propor- tion to OG, as MN bears to GP; confeqiie itly NO will bear the fame proportion to OG, as DF bears to )E. In the laft place, the weight of the globe A bears the fame proportion to the other weight K, as D F bears to D E ; therefore N O bears the fame 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 die weight K is in N, O will be the center of gravity common to both the weights. After the fame manner, if the globe had been caufed to defcend, 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 defcend, it is manifeft, from what has been faid above, that the weights A and K counterpoize each other. f6. I SHALL now confider the cafe of pendulums. A pendulum is made by hanging a weight to a line, fo that it may Rving backwards and forwards. This motion the geo- meters have very carefully confidered, becaufe it is the moft commodious inftrument of any for the exa6l meafurement of time. 5*7. I HAVE obferved already ^ that if a body hanging perpendicularly by a firing, as the body A (in ng. 4.8. ) hangs by the firing A B, be put fo into motion, as to be made to a- fcend up the circular arch A C ; then as foon as it has arrived » § 23, at Chap. 2. PHILOSOPHY, 87 at the higheft point, to which the motion, that the body has received, will carry it; it will immediately begin to defcend, and at A will receive again as great a degree of motion, as it had at firfl. This motion therefore will carry the body up the arch AD, as high as it afcended before in the arch AC. Confequently in its return through the arch D A it will acquire again at A its original velocity, and advance a fecond time up the arch A C as high as at firfl ; by this means continuing with- out end its reciorocal motion. It is true indeed, that in fad: every pendulum, which we can put in motion, will gradual- ly leffen its fwing, and at length ftop, unlefs diere be fome power conilantly applied to it, whereby its motion fliall be renewed ; but this arifes from the reliftance, wliich the body meets with both from the air, and the firing by which it is hung : for as the air will give fome obfl:ru6lion to the progrels of the body moving through it ; fo alfo the firing, whereon the body hangs, will be a farther impediment ; for this firing muft either Aide on tlie pin, whereon it hangs, or it muft bend to the motion of the weight ; in the firft there muft be fome degree of fridion, and in the latter the ftring will make fome reliftance to its inflexion. However, if all reiiftance could be removed, the motion of a pendulum would be perpetual, 5-8. But to proceed, the firft property, I fhall take no- tice 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 increafe in fo great a proportion as the arch. Thus if CD be a ffreater arch, and EF a lefTer, where C A is equal to A D, and E A equal to A F ; the body, when 88 Sir Is A AC Newton's BookI. when it fwings through the greater arch C D, fhall take up in its fwing from C to D a longer time than in Twinging from E to F, when it moves only in that leffer arch ; or the time in which the body let fall from C will defcend through the arch C A is greater than the time, in which it will defi:end through the arch E A, when let fall from E. But the firfl: of thefe times will not hold the fame proportion to the latter, as the firft 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 ^, 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 diredly down through G A ; and in falling through die arch E A it will acquire in the point A on- ly that velocity, which it would have got in falling through H A. Therefore, when the body defcends through the great- er arch C A, it fhall gain a greater velocity, than when it paf- fes only through the leffer ; fo that this greater velocity will in fome degree compenfate the greater length of the arch. ^i). The increafe of velocity, which the body acquires in falling from a greater height, has fuch an effed, that, if fcraight lines be drawn from A to C and E, the body would fall through the longer ftraight line C A juft in the fame time, as through the Ihorter ftraight line E A. This is demonftrat- ed by the geometers, who prove, that if any circle, as ABCD (fig. 49.) be placed in a perpendicular fituation; a body lliall fall obliquely through every line, as A B drawn from the lowefl: point A in the circle to any other point in the circum- » § 20. ference Chap. 2. P H I L O S O P fe[ Y. 89 fcrence jiifl in the fame time, as would be imploycd by die body in falling perpendicularly down through the diameter C A. But the time in which the body will defcend through the arch, is dilTerent from the time, which it would take up in falling through the line A B. 60. It has been thought by fome, that becaufe in very fmall arches this correfpondent ftraight Hne differs but little from the arch itfelf; therefore the defcent through this ftraight line would be performed in fuch fmall arches nearly in the fame time as through the arches themfelves : fo that if a pendulum were to fwing in fmall arches, half the time of a fingle fwing would be nearly equal to the time, in which a body would fall perpendicularly through twice the lengdi of the pendulum. That is, the whole time of the fwing, ac- cording to this opinion, will be four fold the time required for the body to fall through half the length of the pendu- lum ; becaufe 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 fpace, that is through half the pendulum's length. However there is here a miftake ; for the whole time of the fwing, when the pendulum moves through fmall arches, bears to the time required for a body to fall dovvn through half the length of tlie pendulum very nearly the fame proportion, as the circumference of a circle bears to its diameter j that is very nearly the proportion of 3 yy to 115, or little more than the proportion of 5 to i. If the pendulum takes fo great a fwing, as to pafs over an arch equal to one fixth part of the whole circumference of the N circle, po Sir Isaac Newton's Book I. circle, it will fwing 1 1 y times, while it ought according to this proportion to have fwung 117 times ; fo that, when it fwings in fo large an arch, it lofes fomething lefs than two fwings in an hundred. If it fwing through ~ only of the circle, it fhall not lofe above one vibration in 160. If it Iwing in ^^ of the circle, it fhall lofe about one vibration in 6^00. If its fwing be confined to ^^ of the whole circle, it fhall lofe very Httle more than one fwing in ^6oo. And if it take no greater a fwing than through ^ of the whole cir- cle, it fhall not lofe one fwing in _j-8oo. 61. N o w it follows from hence, that, when pendulums fwing in fmall arches, there is very nearly a conftant propor- tion obferved between the time of their fwing, and the time^ in which a body would fall perpendicularly down through half their length. And we have declared above, that the Ipaces, through which bodies fall, are in a two fold propor- tion of the times, which they take up in falling \ There- fore in pendulums of different lengths, fwinging tliroug hfmall arches, the lengths of the pendulums are in a two fold or duplicate proportion of the times, they take in fwinging ; fb that a pendulum of four times the length of another fhall take up twice the time in each fwing, one of nine times the length will make one fwing only for tliree fwings of the {hotter, and. fo on. 62. This proportion in the fwings of different pendu- lidiiis act oniy holds in fmall arches ; but in large ones alfb, S '7.' provided Chap. 2. PHILOSOPHY. 91 provided they be fuch, as the geometers call limilar ; that is, if die arches bear the fame proportion to the whole cir- cumferences of their refpedlive circles. Suppofe ( in fig. 4,8. ) A B, C D to be two pendulums. Let the arch E F be defcrib- ed by the motion of the pendulum A B, and the arch G H be defcribed by the pendulum C D ; and let the arch E F bear the fame proportion to the whole circumference, which would be formed by turning the pendulum A B quite round about the point A, as the arch GH bears to the whole cir- cumference, that would be formed by turning the pendu- lum C D quite round the point C. Then I fay, the propor- tion, which the length of the pendulum AB bears to the length of the pendulum CD, will be two fold of the propor- tion, which die time taken up in the defcription of the arch E F bears to the rime employed in die defcription of the arch GH. <^g. Thus pendulums, which fwing in very fmall arches, are nearly an equal meafure of time. But as they are not fuch an equal meahire to geometrical exa6lnefs ; the mathematicians have found out a method of caufing a pendulum fo to fwing, that, if its motion were not obftrufted by any refiftance, it would always perform each fwing in the fame rime, whether it moved through a greater, or a leffer fpace. This was firft difcovered by the great HuY GENS, and is as follows. Up- on the ftraight line A B ( in fig. 49. ) let the circle C D E be fo placed, as to touch the ftraight line in the point C Then let this circle roll along upon the ftraight line A B, as a coach- wheel rolls along upon the ground. It is evident, that, as N X foon p2 Sir Isaac Newton's BookL foon as ever the circle begins to move, the point C in the cir- cle will be lifted off from the ftraight line A B ; and in the motion of the circle will defcribe a crooked courfe, which is reprefented by the line C F G H. Here the part C H of the ftraight line included between the two extremities C and H of the Hne 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 ftraight Hne I K be drawn per- pendicular to CH, this line IK 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 loweft point of the line C F G H ; and if from any other point G a body were to be let fall in the fame manner ; this body, which falls from G, will take juft the fame time in coming to K, as the body takes up, which falls from H. Therefore if a pendulum can be fo hung, that the ball fhall move in the line AGFE, all its fwings, whether long or fhort, will be per- formed in the fame time ; for the time, in which the ball will defcend to the point K, is always half the time of the whole fwing. But the ball of a pendulum will be made to fwing in this line by the following means. Let K I ( in fig. ^•l.) be prolonged upwards to L, till IL is equal to IK. Then let the line LMH equal and Hke to KH be applied, as in the figure between the points L and H, fo that the point which in this line LMH anfwers to the point H in the line KH fhall be applied to the point L, and the point anfwering to the point K fhall be applied to the point H. Alfo let fuch anodier line L N C be appHed between L and C in the fame manner. Chap. 2. PHILOSOPHY. 93 manner. This preparation being made ; if a pendulum be hung at the point L of fuch a length, that the ball thereof {hall reach to K ; and if the ftring fhall continually bend a- gainft the lines H M L and L N C, as the pendulum fwings to and fro ; by this means the ball fhall conftantly keep in the line CKH. 6^4. Now in this pendulum, as all the fwings, whether long or fhort, will be performed in the fame time ; fo the time of each will exadlly bear the fame proportion to the time re- quired for a body to fall perpendicularly down, through half the length of the pendulum, that is from I to K, as the cir- cumference of a circle bears to its diameter. 6S' It niay from hence be underflood in fome meaflire, why, when pendulums fwing in circular arches, the times of their fwings are nearly equal, if the arches are fmall, though thofe arches be of very unequal lengths ; for if with the fe- midiameter L K the circular arch O K P be defcribed, this arch in the lower part of it will differ very little from the line CKH. 66. It may not be amifs here to remark, that a body will fall in this line CKH (fig. yg.) from C to any other point, as Q_or R in a fhorter {pace of time, than if it moved through the ftraight line drawn from C to the other point ; or through any other line whatever, that can be drawn be- tween thefe two points. 3 6^]. But 94 Sir Isaac Newton's Book I. (^7. But as I have obferved, that die time, wliich a pen- dulum takes in Twinging, depends upon its length ; I fhall now fay fomething concerning the way, in which this length of the pendulum is to be eftimated. If the whole ball of the pendulum could be crouded into one point, this length, by w^hich the motion of the pendulum is to be computed, would be the length of the firing or rod. But the ball of the pen- dulum mufl: have a fenlible magnitude^ and the feveral parts of this ball v/ill not move with the fame degree of fwiftnefs ; for thofe parts, which are fartheft from the point, whereon the pendulum is fufpended, muft move with the greateft ve- locity. Therefore to know the time in which the pendulum fwings, it is necelTary to find that point of the ball, which moves with the fame degree ol velocity, as if the whole ball were to be contraded into that point. 68. This point is not the center of gravity, as I fhall now endeavour to fhew. Suppofe the pendulum A B ( in fig. y^,. ) compofcd of an inflexible rod A C and ball C B, to be fixed on the point A, and lifted up into an horizontal fituation. Here if die rod were not fixed to the point A, the body C B would defcend diredly with the whole force of its weight ; and each part of the body would move down widi the fame degree of fwiftnefs. But when the rod is fixed at the point A, the body muft fall after another manner ; for the parts of the body muft move with different degrees of velocity, the parts more remote from A defcending with a fwifter mo- '• tion, than the parts nearer to A ; fo that the body will re- • ceive a kind of rolHng motion while it defcends. But it has been Q Chap. 2. PHILOSOPHY. 95 been obferved above, that the effedV of gravity upon any bo- dy is the fame, as if the whole force were exerted on the bo- dy's center of gravity \ Since therefore the power of gravity in drawing down the body muft alfo communicate to it the rolHng motion jufl: defcribed ; it feems evident, that the cen- ter of gravity oi the body cannot be drawn down as fwiftly, as when the power of gravity has no other effed; to produce on the body, than merely to draw it downward. If there- fore the whole matter of the body C B could be crouded into its center of gravity, fo that being united into one point, this rolling motion here mentioned might give no hindrance to its defcent ; this center would defcend fafter, tlian it can now do. And the point, which now defcends as faft, as if the whole matter of the body C B were crouded into it, will be farther removed from the point A, than the center of gravity of the body CB. 6^9. Again, fuppofe the pendulum A B (in fig. y^'.) to hang obliquely. Here the power of gravity will operate lefe upon the ball of the pendulum, than before : but the lineDE being drawn fo, as to ftand perpendicular to the rod A C of the pendulum ; the force of gravity upon the body CB, now it is in this fituation, will produce the fame effedt, as; if the body were to glide down an inclined plane in the po- (iticai 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 uninter- rupted defcent of the body down this plane ; for the body wim p(5 Sir I s A A c N E w T o n's Book I. will here alfo receive the fame kind of rotation in its motion, as before ; fo that the motion of the center of gravity will in like manner be retarded ; and the point, which here de- fcends with that degree of fwiftnefs, which the body would have, if not hindered by being fixed to the point A ; that is, the point, which defcends as faft, 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 eftimated, is called the center of ofcillation. And the mathematicians have laid down general direftions, whereby to find this center in all bodies. If the globe A B ( in fig. ^6. ) be hung by the firing C D, whofe weight need not be re- garded, the center of ofcillation is found thus. Let the ftraight line drawn from C to D be continued through the globe to F. That it will pafs through the center of the globe is evident. Suppofe E to be this center of the globe ; and take the line G of fuch a length, that it fhall bear the fame proportion to ED, as E D bears to E C. Then E H being made equal to ; of G, the point H fhall be the center of of- cillation \ If the weight of the rod C D is too confiderable to be negleded, divide C D ( fig. 5-7 ) in I, that D I be equal to ^, part of C D ; and take K in the fame proportion to C I, as the weight of the globe AB to the weight of the rod CD. Then having found H, the center of olcillation of the globe, as before, divide I K in L> fo that I L fhall bear the fame pro- » Hugen. Horolog. ofcilkt. pag. 141, 142. portion Chap. 2. PHILOSOPHY. 97 portion to LH, as the line CH bears to K ; and L fliall be the center of ofcillation of die whole pendulum. 71. This computation is made upon fuppo{ition,tliat the center of ofcillation of the rod C D, if that were to fwing alone without any other weight annexed, would be the point I. And this point would be the true center of ofcillation, fo far as the thicknefs of the rod is not to be regarded. If any one chufes to take into confideration the thicknefs of the rod, he muft place the center of ofcillation thereof fo much below the point I, that eight times the diftance of the center from the point I fhall bear the fame proportion to the tliicknefs of the rod, as the diicknefs of the rod bears to its length CD*. 7a. It has been obferved above, that when a pendulum fwings in an arch of a circle, as here in fig. 5-8, the pendu- lum A B fwings in the circular arch CD; if you draw an ho- rizontal line, as E F, from the place whence the pendulum is let fall, to the line AG, which is perpendicular to the horizon: then the velocity, which die pendulum will acquire in com- ing to the point G, will be the fame, as any body would ac- quire in falling diredly down from F to G. Now this is to be underftood of die circular arch, which is defcribed by the cen- ter of ofcillation of the pendulum. I fhall here farther ob- ferve, that if the flraight line E G be drawn from the point, whence the pendulum falls, to the lowcft point of the arch ; in the fame or in equal pendulums the velocity, which the » Sec Hugen. Horolcg. Ofcilht. p. 14:. O pendulum pS Sir I s A AC N E w T o n's BookI. pendulum acquires in G, is proportional to this line : that is, if the pendulum, after it has defcended 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 fliall acquire in G by its defcent from H, fhall bear the fame proportion to the velocity, which it acquires in falling from E to G, as the ftraight line H G bears to the ftraight line E G. 73. We may now proceed to thofe experiments upon the percuflion of bodies, which I obferved above might be made with pendulums. This expedient for examining the effeds of percufTion was firft propofed by our late great architedl SirCHRisTOPHERWREN. And it is as follows. Two balls, as A and B (in fig. yp.) either equal or ime- qual, are hung by two ftrings from two points C and D, fo that, when the balls hang down without motion, they fhall juft touch each other, and the ftrings be parallel. Here if one of thefe balls be removed to any diftance from its perpen- dicular fituation, and then let fall to defcend and ftrike a- gainft the other ; by the laft preceding paragraph it will be known, with what velocity this ball fhall return into its iirft perpendicular fituation, and confequently with what force it fhall ftrike againft the other ball ; and by the height to which diis other ball afcends after the ftroke, the velocity commu- nicated to this ball will be difcovered. For inftance, let the ball A be taken up to E, and from thence be let fall to ftrike againft B, pafling over in its defcent the circular arch EF. By this impulfe let B fly up to G, moving through the circu- lar arch HG. Then EI and GK being drawn horizontally, the. Chap. 2. PHILOSOPHY. ^^ die ball A will ftrike againft B with the velocity, which it would acquire in falling diredly down from I ; and the ball B has received a velocity, wherewith, if it had been thrown diredly upward, it would have afcended up to K. Likewife if ftraight lines be drawn from E to F and from H to G, the velocity of A, wherewith it ftrikes, will bear the fame pro- portion to the velocity, which B has received by the blow, as the ftraight line E F bears to the ftraight line H G. In the fame manner by noting the place to which A afcends after the fboke, its remaining velocity may be compared with that, wherewith it ftruck againft B. Thus may be experimented the effedts of the body A ftriking againft B at reft. If botli the bodies are lifted up, and fo let fall as to meet and impinge againft each other juft upon the coming of both into their perpendicular fttuation ; by obfenang the places into which they move after the ftroke, the effects of their percuftion in all thefe cafes may be found in the fame manner as before. 74.. SirlsAAcNEWToN has defcribed thefe experiments ; and has fhewn how to improve diem to a greater exa£lnefs by making allowance for the refiftance, which the air gives to the motion of die balls ^ But as this refiftance is exceeding fmall, and the manner of allowing for it is deli\'ered by him- felf in very plain terms, I need not enlarge upon it here. I fliall rather fpeak to a difcovery, which he made by thefe ex- periments upon the elafticity of bodies. It has been explained abo\'e ^', that when two bodies ftrike, if they be not elaftic., » Princip. PhJlo!". pag, 22. '' Chap. I, $ 29, O X they I oo Sir Isaac N e w t o n's Book L they remain contiguous after the ftroke ; but that if they are elaflic, they feparate, and that the degree of their elafticity determines the proportion between the celerity wherewith they feparate, and the celerity wherewith they meet. Now our author found, that the degree of elafticity appeared in the fame bodies always the fame, with whatever degree of force they flruck ; that is, the celerity wherewith they fe- parated, always bore the fame proportion to the celerity wherewith they met : fo that the elaftic power in all the bo- dies, he made trial upon, exerted it fclf in one conftant pro- portion to the comprefTing 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 propor- tion of 5" to 9 to the celerity wherewith they met ; and in fteel he found nearly the fame proportion ; in cork the elafti- city was fomething lefs ; but in glafs much greater ; for the celerity, wherewith balls of that material feparated after per- cufTion, he found to bear the proportion of 1 5- to i d to the celerity wherewith they met ^. 75*. I SHALL finifh my difcourfe on pendulums, with this farther obfervation only, that the center of ofcillation is alfo 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 fame equa- ble motion. Now the force, with which fucb a body * Princip. Philof. png. 25, 4 movesj Chap. 2. PHILOSOPHY. loi moves, is all united in the point, which in relation to the power of gravity is called the center of ofcillation. Let the cylinder A B C D (in fig. 60. ) whofe axis is E F, be fixed to the point E. And fuppofing the point E to be that on which the cylinder is fufpended, let the center of ofcillation be found in the axis E F, as has been explained above ^ Let G be that center : then I fay, that the force, wherewith this cy- linder turns round the point E, is fo united in the point G, that a {lifficient force applied in that point iliall flop the motion of the cylinder, in fuch a manner, that the cylinder fhould im- mediately remain without motion, though it were to be loof^ ened from tlie point E at the fame inftant, that the impedi- ment 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 appHed between E and G, the cy- linder would fo turn on the point, where the impediment was applied, tliat tlie end B C would continue to move on the fame way it moved before along w^ith 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 ftart out of its prefent place that way in which the cylinder moved. From, this property of the center of ofcillation, it is alfo called the center of percuflion. That excellent mathematician, Dr.BRooK Taylor, has farther improved this dodrine concerning the center of percuflion, by fhev/ing, that if through this point G a line, as GHI, be drawn perpendicular to EF, and lying => § 71. ill 102 Sir Isaac Newton's Book I. in die courfe of the body's motion ; a fufficient power appli- ed to any point of this Une will have the fame effed, as the like power applied to G ^ : fo that as we before fhewed the center of percuflion within the body on its axis ; by this means we may find this center on the furface of the body alfo, for it will be where this line H I crofTes that furface. 76^. I SHALL now proceed to the lafl kind of motion, to be treated on in this place, and fhew what line the power of gravity will caufe a body to defcribe, when it is thrown for- wards by any force. This was firft difcovered by the great Galileo, and is the principle, upon which engineers fhould diredl the fhot of great guns. But as in this cafe bo- dies defcribe in their motion one of thofe Unes, which in geo- metry are called conic fedions ; it is neceflary here to pre- mife a defcription of thofe hnes. In which I fhall be the more particular, becaufe the knowledge of them is not only neceflary for the prefent purpofe, but will be alfo required hereafter in fome of the principal parts of this treatile. 77, The firft lines confidered by the ancient geometers •were the ftraight line and the circle. Of thefe they compof- ed various figures, of which they demoriftrated many proper- ties, and refolved divers problems concerning them. Thefe problems they attempted always to refolve by the defcribing ftraight lines and circles. For inftance, let a fquare A B C D ( fig. 61.) be propofed, and let it be required to make ano- » Sec Mcdiod. Increment, prop. 25. ther Chap. 2. PHILOSOPHY. 103 tlier fquare in any afligned proportion to this. Prolong one fide, as D A, of this fquare to E, till A E bear the fame propor- tion to A D, as the new fquare is to bear to tlie fquare A C. If the oppofite fide B C of the fquare A C be alfo prolono-ed to F, till BF be equal to AE, and EF be afterwards drawn, I fuppofe my readers will eafily conceive, diat the figure ABFE will bear to the fquare A B C D the fame proportion, as the line AE bears to the Une AD. Therefore the figure ABFE will be equal to the new fquare, which is to be found, but is not it felf a fquare, becaufe the fide AE is not of the fame lengdi with the fide E F. But to find a fquare equal to the figure ABFE you muft proceed thus. Divide the line D E into two equal parts in the point G, and to the center G with the inter- val G D defcribe the circle D H E I ; then prolong the Kne A B, till it meets die circle in K -, and make the fquare AKLM, which fquare will be equal to the figure ABFE, and bear to the fquare ABCD the fame proportion, as the Hne AE bears to AD* 78. I SHALL not proceed to the proof of this, ha:\ing only here fet it down as a fpecimen of the method of refolv- ing geometrical problems by the defcription of ftraight lines and circles. But there are fome problems, which cannot be refolved by drawing ftraight lines or circles upon a plane. For the management therefore of thefe they took into confidera- tion folid figures, and of the folid figures they found diat, which is called a cone, to be the moft ufefiil, 7.9' • -^- 1 04 Sir I s A A c N E w T o n's Book L 79. A CONE is thus defined by Eu glide in his ele- ments of geometry '. Ifto the ftraightline AB (in fig.dl.) another ftraight line, as A C, be drawn perpendicular, and the two extremities B and C be joined by a third ftraight line compofing the triangle A C B ( for fo every figure is called, which is included under three ftraight lines : ) then the two points A and B being held fixed, as two centers, and the trian- gle A C B being turned round upon the line A B, as on an axis ; the line A C will defcribe a circle, and the figure A C B will defcribe a cone, of the form reprefented by the figure BCDEF ( fig. (Jg.) in which the circle CDEF is ufually called the bafe of the cone, and B the vertex. 80. N o w by this figure may feveral problems be refolved, which cannot by the fimple defcription of ftraight lines and circles upon a plane. Suppofe for inftance, it were required to make a cube, which ftiould bear any aftigned proportion to fome other cube named. I need not here inform my read- ers, 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 per- formed by a cone thus. Firft make a cone from a triangle, whofe fide AC fhall be half the length of the fide BC Then on the plane ABCD ( fig. (54- ) let the line EF be exhibited equal in length to the fide of the cube propofed ; and let the line F G be drawn perpendicular to E F, and of fuch a lengdi, that it bear the fame proportion to E F, as the =" Lib. XL Dcf. cube Chap. 2. PHILOSOPHY. 105 cube to be fought is required to bear to the cube propofed. Through the points E, F, and G let the circle F H I be defcribed. Then let the line EF be prolonged beyond F to K, that FK be equal to F E, and let the triangle F KL, having all its fides FK, KL, LF equal to each other, be hung down perpendi- cularly from the plane A B C D. After diis, let another plane M N P be extended through the point L, fo as to be equi- diftant from the former plane A B C D, and in this plane let the line Q^L R be drawn fo, as to be equidiftant from the line E F K. All this being thus prepared, let fuch a cone, as was above direded to be made, be fo 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 firft plane A B C D, will crofs the circle F H I be- fore defcribed. And if from the point S, where the furface of this cone interfedls the circle, the line S T be drawn fo, as to be equidiftant from the line E F ; the line F T will be equal to the frde of the cube fought : that is, if there be two cubes or dyes formed, the fide of one being equal to E F, and the fide of the other equal to F T ; the former of thefe cubes fhall bear the fame proportion to the latter, as the line E F bears to EG. 81. Indeed this placing a cone to cut through a plane is not a practicable method of refolving problems. But when the geometers had difcovcred this ufe of the cone, they ap- plied themfelves to confider the nature of the lines, which will be produced by the interfedion of the furface of a cone P and 1 06 Sir Isaac N e vv t o n's Book I. and a plane ; whereby they might be enabled both to reduce thefe kinds of folutions to pradice, and alfo to render their demonftrations concife and elegant. 8 2. Wh e n e V e r the plane, which cuts the cone, is equi- diftant from another plane, that touches the cone on the fide; ( which is the cafe of the prefent figure ; ) the line, wherein the plane cuts the furface of the cone, is called a parabola. But if the plane, which cuts the cone, be fo inclined to this other, that it will pafs quite through the cone (as in fig. 6y.) filch a plane by cutting the cone produces the figure called an cUipfis, in which we fhall hereafter fhew the earth and other planets to move round the fijn. If the plane, which cuts the cone, recline the other way (as in £.g.66. ) fo as not to be parallel to any plane, whereon the cone can lie, nor yet to cut quite through the cone ; fiich a plane (hall produce in tlie cone a third kind of line, which is called an hyperbola. But it is the firfl: of thefe lines named the parabola, wherein bodies, that are thrown obliquely, will be carried by the force of gravity ; as I fliall here proceed to fhew, after having firft dire6led my readers how to defcribe this fort of line upon a plane, by which the form of it may be feen. 85. To any ftraight line A B (fig. 6 J.) let a ftraight ruler CDbe fo applied, as to ftand againft it perpendicularly. Upon the edge of this ruler let another ruler E F be fo placed, as to move along upon the edge of the firft ruler C D, and keep al- ways perpendicular to it. This being fo difpofcd, let any point, as G, be taken in the Hne A B, and let a ftring equal ia Chap. 2. PHILOSOPHY. 107 in length to the ruler E F be faftened by one end to the point G, and by the other to the extremity F of the ruler E F. Then if the firing be held down to the ruler E F by a pin H, as is reprefented in the figure ; the point of this pin, while the ruler EF moves on the ruler CD, fhall defcribe die line IKL? which will be one part of the curve line, whofe defcription we were here to teach : and by applying the rulers in the like manner on the other fide of the line A B, we may defcribe the other part I M of this line. If the diftance C G be equal to half the line EF in fig. 6'4, the hne MIL 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 MIL, and the point G is called the focus. 85-. N o w by comparing the effeSis of gravity upon falling bodies, with what is demonftrated of this figure by the geo- meters, it is proved, that every body thrown obliquely is carried forward in one of thefe lines, the axis whereof is per- pendicular to the horizon. S6. The geometers demonftrate, that if a line be drawn to touch a parabola in any point, as the Hne A B (infig.dS.) touches the parabola C D, whofe axis is YZ, in the point E; andfeveral 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 EF to EH, and EG to KL in the duplicate proportion of EF to E K ; likewife H I to K L in the duplicate proportion of E H to EK. What is to be underftood by duplicate or two-fold P z ^ pro- io8 Sir Isaac Newton's Book I. proportion, has been already explained *. Accordingly I mean here, that if the line M be taken to bear the fame pro- portion to E H, as E H bears to E F, HI will bear the fame proportion to EG, as M bears to EF ; and if the line N bears the fime proportion to E K, as E K bears to E F, K L will bear the fame proportion to EG, as N bears to E F ; or if the line O bear the fame proportion to EK, as EK bears to EH, KL will bear the fame proportion to HI, as O bears to E H. 87. This property is effential to the parabola, being fo conneded with the nature of the figure, that every Hne poffeirmg this property is to be called by this name. 88. Now fuppofe a body to be thrown from the point A ( in fig. 69. ) towards B in the diredion of the line A B. This body, if left to it felf, would move on with a uniform mo- tion through this line A B. Suppofe the eye of a fpedator to be placed at the point C jufl under the point A ; and let us imagine the earth to be fo put into motion along with the body, as to carry the fpedator's eye along the Hne C D parallel to A B i and tliat die eye ihould move on with the fame velo- city, wherewith the body would proceed in the line A B, if it were to be left to move without any difturbance from its gravitation towards the earth. In this cafe if the body mov- ed on without being drawn towards the earth , it would ap- pear to the fpedator to be at reft. But if the power of gra- vity exerted it felf on the body, it would appear to the Ipe- » Chap. a,. § 17. dator Chap. 2. PHILOSOPHY. 109 ctator to fall diredly down. Suppofe at the diftaiice of time, wherein the body by its own progreflive motion would have moved from A to E, it fhould appear to the fpedator to have fallen through a length equal to EF : then the body at the end of this time will actually have arrived at the point F. If in the fpace of time, wherein the body would have mov- ed by its progreflive motion from A to G, it would have ap- peared to the fpedator to have fallen down the fpace G H : then the body at the end of this greater interval of time will be arrived at the point H. Now if tlie line AFHI be diat, through wliich the body adlually paffes ; from what has here been faid, it will follow, that this line is one of thole, which I have been defcribing under the name of the parabo- la. For the diftances E F, G H, through wliich the body is feen to fall, will increafe in the duplicate proportion of the times * ; but the lines A E, AG will be proportional to the times wherein they would have been defcribed by the fingle progreflive motion of the body : therefore die lines E F, G H will be in the duplicate proportion of the lines AF, AGj and the line AFHI poffelTes the property of the parabola. 89. If the earth be not fuppofed to move along with the body, the cafe will be a little different. For the body be- ing conftantly drawn direftly towards the center of the earth, the body in its motion will be drawn in a dired:ion a little ob- lique to that, wherein it would be drawn by the earth in mo- tion, as before fuppofed. But the diftance to the center of the » Seeabove Ch.2. § 17. eartk no Sir I s A A c N E w T o n's Book L earth bears fo vafl: a proportion to the greateft length, to which we can throw bodies, that this obHquity does not merit any regard. From the feqiiel of this difcourfe it may indeed be colleded, what Kne the body being thrown thus would be found to defcribe, allowance being made for this obliquity of the earth's aftion ^. This is the difcovery of Sir I s. N e w t on ; but has no ufe in this place. Here it is abundantly fufEcient to confider the body as moving in a parabola. ^o. The line, which a projedled body defcribes, being thus known, practical methods have been deduced from hence for direding the fliot of great guns to ftrike any ob- jed; deiired. This work was firft attempted by Galileo, and foon after farther improved by his fcholar Torricelli; but has lately been rendred more complete by the great Mr. Cotes, whofe immature death is an unfpeakable lofs to mathematical learning. If it be required to throw a body from the point A ( in fig. 70. ) fo as to ftrike the point B ; through the points A, B draw the ftraight line C D, and ered the line AE perpendicular to the horizon, and of four times the height, from which a body muft £ill to acquire the velo- city, wherewith the body is intended to be thrown. Through the points A and E defcribe a circle, that fhall touch the line C D in the point A. Then from the point B draw the line B F perpendicular to the horizon, interfering tlie circle in the points G and H. This being done, if the body be projected diredly towards either of thefe points G or H, it fhall fall up- on the point B j but with this difference, that, if it be thrown » FromB.II. Ch. 3. • ^ ^ €hap.2. philosophy. m in the diredlon AG, it fliall fooner arrive at B, than if it were projedled in the diredion A H. When the body is projefted in the direction A G ; the time, it will take up in arriving at B, will bear the fame proportion to the time, wherein it would fan down through one fourth part of A E, as AG bears tO' half A E. But when the body is thrown in the diredion of A H, the time of its paffing to B will bear the fame proportion to the time, wherein it would fall through one fourth part of AE, as AH bears to half AE. 91. If the line A I be drawn fo as to divide the angle un- der E A D in the middle, and the line I K be drawn perpen- dicular to the horizon ; this line will touch the circle in the point I, and if the body be thrown in the diredion A I, it will fall upon the point K: and this point K is the fartheft point in the line A D, which the body can be made to ftrike, without increaling its velocity. 92. The velocity, wherewith the body every where moves, may be found thus. Suppole the body to move in the parabola AB (fig. 71-) Ered AC perpendicular to the horizon, and equal to the height, from which a body muft fall to acquire the velocity, wherewith the body fets 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 fame, as would be acquired in falling through CG. Thus the body moves floweft at the higheft point H of the parabola ; and at equal diflances from this point will move 112 Sir I s A A c N E w T o n's Book I. move with equal fwiftnefs, and defcend from that higheft point through the Hne H B altogether like to the line A H in which it afcended j abating only the refiftance of the air, which is not here confidered. If the line HI be drawn frojn the higheft point H parallel to the horizon, A I will be equll to 4 of BG in fig. 70, when the body is projected in the direc- - tion A G, and equal to 4 of B H, when the body is thrown in the diredion AH provided AD be drawn horizontally. ^3. Thus I have recounted the principal difcoveries, - which had been made concerning the motion of bodies by Sir I s A A c N E w T o n's predecelTors ; all thefe difcoveries, by being found to agree with experience, contributing to effca- blifli the laws of motion, from whence they were deduced. I fhall therefore here finifh what I had to fay upon thofe laws ; and conclude this chapter with a {ew words concern- ing the diftindion which ouo;ht to be made between abfolute and relative motion. For fome have thought fit to confound them together ; becaufe they obferve the laws of motion to take place here on the earth, which is in motion, after the fame manner as if it were at reft. But Sir Isaac Newton has been careful to diftinguifh between the relative and abfolute confideration both of motion and time^ The aftronomers anciently found it neceflary to make this diftindion in time. Time confidered in it fell paffes on equably without relation to any thing external, being the proper meafure of the continu- ance and duration of all things. But it is moft frequently con- « Princip. Philof. pig. 10. | c Ibid.fi jo. >> Rciut. Des Cart. Piinc. PhUof.Part. II. § 25. \ Q^ 2, and ii6 Sir Isaac Newton's Book I. - and the water as moving in it : and the water in refpeft of the veffel has the greateft motion, when the v^ffel firft begins to turn, and lofes this relative motion more and more, till at length it quite ceafes. But now, when the veffel firft begins to turn, the furface of the water remains fmooth and flat, as before the vefl'el began to move ; but as the motion of the veffel communicates by degrees motion to the water, the fur- face of the water will be obferved to change, the water fub- fidins in the middle and rifino- at the edo;es: which elevation of the water is caufed by the parts of it preflmg 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 veffel, but on its abfolute motion ; for it is leaft, when that relative motion is greateft, and greateft, when that relative motion is leaft, or none at all. 95-. Thus the true caufe of what appears in the furface of this water cannot be afligned, without confidering the water's motion within the veffel. So alfo in the fyftem of the world, in order to find out the caufe of the planetary mo- tions, we muft know more of the real motions, which be- long to each planet, than is abfolutely neceffary for the ufes of aftronomy. If the aftronomer fhould fuppofe the earth to ftand ftill, he could afcribe fuch motions to the celeftial bo- dies, as ftiould anfwer all the appearances ; though he would not account for them in fo fimple a manner, as by attributing motion to the earth. But the motion of the earth muft of neceffity be confidered, before the real caufes, which actuate the planetary fyftem, can be difcovercd. Chap. Chap. 3- PHILOSOPHY, 117 Chap. III. Of CENTRIPETAL FORCES. WE have jufl been defcribing in the preceding chapter the effeds produced on a body in motion, from its being continually a6led upon by a power always equal in ftrengdi, and operating in parallel diredlions \ But bodies may be adled upon by powers, which in different places (hall have different degrees of force, and whofe feveral diredlions fhall be varioufly inclined to each other. The moft fim- ple of thefe in refped: to diredion is, when the power is pointed conftantly to one center. This is truly the cafe of that power, whole effects we defcribed in the foregoing chap- ter; though the center of that power is fb far removed, that the fubjed then before us is moft conveniently to be condder- ed in. the light, wherein we have placed it : But Sir Isaac Newton has confidered very particularly this other cafe oF powers, which are conftandy diredled to the lame center. It is upon this foundation, that all his difcoveries in the lyftem of the world are raifed. And therefore, as this lubjedl bears fo very great a Ihare in the philofophy, of which I am dif- courfing, I think it proper in this place to take a Ihort view ot fome of the general effedls of thefe powers, before we- come to apply them particularly to the lyftem of the world. = § 85, &c. %, Thesf, 1 1 8 Sir I s A A c N E vv T o n's Book I. z. These powers or forces are by Sir I s a a c Newton called centripetal ; and their firft effect is to caufe the body, on which they ad, to quit the ftraight courfe, wherein it would proceed if undifturbed, and to defcribe an incurvated line, which fhall always be bent towards the center of the force. It is not neceffary, that fuch a power fhould caufe the body to approach that center. The body may continue to recede from the center of the power, notwithftanding its being drawn by the power; but this property muft always belong to its motion, that the Hne, in which it moves, will continually be concave towards the center, to which the power is direded. Suppofe A ( in fig. 71.) to be the center of a force. Let a body in B be moving in the diredlion of the ftraight line B C, in which Hne it would continue to move, if undifturbed ; but being attracted by the centripetal force towards A, the body muft neceflarily depart from this line B C, and being drawn into the curve line B D, muft pafs between the lines A B and B C. It is evident therefore, that the body in B being gra- dually turned off from the ftraight line B C, it will at firft be convex toward the line BC, and confequently concave to- wards the point A: for thefe centripetal powers are fuppofed to be in ftrength proportional to the power of gravity, and, like that, not to be able after the manner of an impulfe to turn the body fenfibly out of its courfe into a different one in an in- ftant, but to take up fome Ipace of time in producing a vifi- ble effed:. 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 Hne E F G may be fo drawn Chap. 3- PHILOSOPHY. 119 drawn, as to touch the curve line B D in fome point as F. Now when the body is come to F, if the centripetal power were im- mediately to be fufpended, the body would no longer conti- nue to move in a curv^e line, but being left to it felf would forthwith reaflume a ftraight courfe ; and that ftraight courfe would be in the line F G : for that line is in the direction of the body's motion at the point F. But die centripetal force continuing its energy, the body will be gradually drawn from this line F G fo 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 fame manner the body may be followed on in its courfe through the line B D, and every part of that line be fhewn to be concave toward the point A. 3 . This then is the conftant charafter belonging to thole 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 reipeft to the fuccefTive diftances of the body from the center there is no general rule to be laid down ; for the diftance of the body from the center may ei- ther increafe, or decreale, or even keep always die fame. The point A ( in fig. 75.) being the center of a centripetal force, let a body at B fet out in the direction of the ftraight line B C perpendicular to die line A B drawn from A to B. It will be ealily conceived, that there is no other point in the line B C fo near to A, as the point B; that AB is the fliorteft of all the lines, which can be drawn from A to any part of the line B C j all other lines, as AD, or AE, drawn from A to the Hne BC being longer than A B. Hence it follows, that the body fet- ting:: I20 Sir Is A AC Newton's Book I. ting out from B, if it moved in the line BC, 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 fuch a force ad upon a refting body, it muft neceffarily put that body fo into motion, as to caufe it to move towards the center of the force : if the body were of it felf moving towards that center, the centripetal force would accelerate that motion, and caufe it to move fafter down : but if the body were in fuch a motion, as being left to itfelf it would recede from this center, it is not necef- lary, that the action of a centripetal power upon it fhould immediately compel the body to approach the center, from which it would otlierwife have receded ; the centripetal power is not without effeft, if it caufe die body to recede more ilovsay from that center, than odierwife it would have done. Thus in t]^e cafe before us, the fmalleft centripetal power, if it ad on tlie body, will force it out of the line B C, and caufe it to pafs in a bent line between B C and the point A, as has been before explained. When the body, for in- ftancc, has advanced to the line A D, the effed of the cen- tripetal force diibovers it felf by having rem.oved the body out of die line B C, and brought it to crofs the line A D fome- where between A and D: fuppofe at F. Now AD being longer than A B, A F m.ay alfo be longer than A B. The cen- tripetal power may indeed be fo ftrong, that AF fhall be fhorter than A B ; or it may be fo evenly balanced with the progreffive motion of the body, that A F and A B fhall be juft equal : and in this laft cafe, when tlie centripetal force is of tliat ftrength, as conflantly to draw the body as mucli toward the Chap. 3- PHILOSOPHY. 121 the center, as the progrefTu'e motion would carry it off, the body will defcribe a circle about the center A, this center of die force being alfo the center of the circle. 4. I F the body, inftead of fetting out in the line B C per- pendicular to A B, had fet out in another line B G more in- cHned 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 Hne B G, would for fome time approach the center A ; the centri- petal force would caufe it to make greater advances toward that center. But if the body were to fet out in the line B I re- clined the other w^ay from the perpendicular B C, and were to be drawn by the centripetal force into the curve line B K ; die body, notwithftanding any centripetal force, would for fome time recede from the center ; fince fome part at leaft of the curve Hne B K lies between the line B I and the perpendicular BC. y. Thus far we have explained fuch effc£ls, as attend every centripetal force. But as thefe forces may be \'ery diffe- rent in regard to the different degrees of flrength, where- with they a6t upon bodies in different places; I fhall now pro- ceed to make mention in general of fome of the differences attending thefe centripetal motions. 6. T o reaffume the confideration of the laft mentioned cafe. Suppofe a centripetal power direded toward the point A ( in fig. 74. ) to a<£i: on a body in B, which is moving in the diredlion of the ftraight Hne B C, die line B C recHning off from A B. If from A the ftraight lines AD, A E, A F are R drawn 12 2 Sir I s A A c N E w T o n's Book I. drawn at pleafiire to the line C B ; tlie line C B being prolong- ed 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 inclin- ed more obliquely than A D, and A F more than A E. To fpeak more corredly, the angle under A D G is leis than that under A B G, the angle under AEG lefs than that under A D G, and the angle under A F G h(5 than that under A E G. Now fuppofe the body to move in the curve line BHIK. Then it is here likewife evident, that the line BHIK be ing concave towards A, and convex towards the line B C, it is more and more turned off from the line B C ; fo that in the point H the line A H will be lefs obliquely inclin- ed to the curve line BHIK, than the fame line A H D is inclin- ed 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 in- cHnation of the fame Hne A I E to the Hne B C, at the point E y and in the points K and F the difference of inclination will be ftill greater ; and in both the inclination at tlie curve will be lefs oblique, than at the ftraight line B C. But the ftraight line A B is lefs obliquely incHned to B G, than A D is inclined towards D G : therefore although the line A H be lefs oblique- ly inclined towards the curve H B, than the fame line A H D is incHned towards D G ; yet it is poffible, that the inclination at H may be more oblique, than the incHnation at B. The in- clination at H may indeed be lefs oblique than the other, or they may be both the fame. This depends upon the degree of ftrength, wherewith the centripetal force exerts it felfj during the paffage of the body from B to H. After the fame manner the inclinatioGs at I and K depend entirely on Uie de- gree Chap.3. philosophy. 123 oree of ftrength, wherewith the centripetal force a£ls on the body in its paffage from H to K : if the centripetal force be weak enough, the lines AH and A I drawn from the center A to the body at H and at I fhall be more obliquely inclined to tlie curve, than the Hne A B is inclined towards B G. The cen- tripetal force may be of tliat ftrength as to render all thefe in- clinations equal, or if ftronger, the inclinations at I and K will be lefs oblique than at B. Sir I s a a c N e w t o n has par- ticularly fhewn, that if the centripetal power decreafes after a certain manner with the increale of diftance, a body may defcribe fuch a curve line, that all the lines drawn from the center to the body fhall be equally inclined to that curve line \ But I do not here enter into any particulars, my prefent inten- tion being only to fhew, that it is pofTible for a body to be ad:ed upon by a force continually drawing it down towards a center, and yet that the body fhall continue to recede from that center ; for here as long as the lines AH, A I, &c drawn from the center A to the body do not become lefs oblique to the curve, in which the body moves ; fo long fhall thofe lines perpetually increafe, and confequently the body fhall more and more recede from the center. 7. B u T we may obferve farther, that if the centripetal power, while the body increafes its diftance from the center, retain fufficient ftrength to make the lines drawn from the center to the body to become at length lefs oblique to tlie curve J then if this diminution of the obliquity continue, till f Princip. Philof. Lib. I. prop. 9. R X at 124- Sir I s A A c N E w T n's Book L at laft the line drawn from the center to the body fhall ceafe to be obliquely inclined to the curve, and fhall become per- pendicular thereto ; from this inftant the body fhall no longer recede from the center, but in its following motion it (liall again defcend, and fhall defcribe a curve line in all relpedis like to that, which it has defcribed already ; provided the centripetal power, every where at the fame diftance from the center, ads with the fame ftrengdi. So we obferved in the preceding chapter, that, when the motion of a projectile be- came parallel to the horizon, the projectile no longer afcend- ed, but forthwith direded its courfe downwards, defcending in a line altogether like that, wherein it had before afcended \ 8. This return of the body may be proved by the fol- lowing proportion : that if the body in any place, fuppofe at I, were to be ftopt, and be thrown diredlly backward with the velocity, wherewith it was moving forward in that point I; then the body, by the adion 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 fame fpace of time, as was taken up in its paffage from B to I ; the velocity of the body at its return to the point B being the fame, as that wherewith it firft fet out from that point. To give a full demonftration of this proportion, would require that ufe of mathematics, which I here pur- pofe to avoid ; but, I believe, it will appear in great meafure: evident from the following conliderations.. Q. Sup- Chap. 3- PHILOSOPHY. 125 9. Suppose (in fig. 75". ) that a body were carried after the following manner through the bent figure A B C D E F, compofed of the ftraight lines AB, B C, C D, D E, E F. Firft let it be moving in the line A B, from A towards B, with any- uniform velocity. At B let the body receive an impulfe di- reded tov/ard fome point, as G, taken within the concavity of the figure. Now whereas tliis body, when once moving in the ftraight line A B, will continue to move on in this line, fo long as it fhall be left to it felf ; but being difturbed at the point B in its motion by the impulfe, which there a6ts upon it, it will be turned out of this Hne A B into fome other ftraight line, wherein it will afterwards continue to move, as long as it fhall be left to itfelf Therefore let this impulfe have ftrength fufiicient to turn the body into the line B C Then let the body move on undifturbed from B to C, but at C let it receive another impulfe pointed toward the fame point G, and of flif- ficient ftrength to turn the body into the line CD. At D let a third impulfe, direded like the reft to the point G, turn the body into die line D E. And at E let another impulfe, direct- ed Hkewife to the point G, turn the body into the line E Fo . Now, i fay, if the body while moving in the line E F be ftopt, and turned back again in this line with the fame velocity, as that wherewith it was moving forward in this line ; then by the repetition of the former impulfe at E the body will be turned into the line E D, and move in it from E to D widi the fame velocity as before it moved with from D to E ; by the repetition of the impulfe at D, when the body fhall have returned to that point, it will be turned into the line 0C ; and by the repetition of the other impulfes at C and B 3 the 126 Sir Isaac Newton's Book I. tlie body will be brought back again into the line B A, with the velocity, wherewith it firfl moiled in that line. lo. T H I s I prove as follows. Let D E and F E be conti- rmed beyond E. Jn D E thus continued take at pleafure the length E H, and let H I be fo drawn, as to be equidiftant from the line G E. Then, by what has been written upon the fe- cond law of motion % it follows, that after the impulfe on the body in E it will move through EI in the fame time, as it would have imployed in moving from E to H, with the ve- locity which it had in the line D E. In F E prolonged take E K equal to E I, and draw K L equidiftant from G E. Then, bec^ufe the body is thrown back in the line F E with the lame velocity as that wherewith it went forward in that line ; if, when the body was returned to E, it were permitted to go ftraight on, it would pafs through E K in the fame time, as it took up in pafQng through E I, when it went forward in the line E F. But, if at the body's return to the point E, fuch an impulfe directed toward die point D were to be given it, where- by it fhould be turned into the line D E ; I fay, tliat the impulfe neceffary to produce this effect muft 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 fame, as that wherewith it before moved through this line from D to E. Becaufe E K is equal to E I, and KL and HI, being each equidiftant from GE, are by conle- ^uence equidiftant from each other ; it follows, that the two « Ch. II. § 22. • trian- Chap. 3- PHILOSOPHY. 127 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 fome propofitions in the elements of Euclid for the proof of this '"^ : but as I do not here addrefs my felf to fuch, fo I think this alTcrtion will be evident enough without a proof in form ; at leaft I muft defire my readers to receive it as a propolition true in geometry. But thefe two triangu- lar figures being altogetlier like each other and equal ; as E K is equal to EI, fo EL is equal to EH, and KL equal to HI. Now the body after its return to E being turned out of the line F E into E D by an impulfe ading upon it in E, after the man- ner above exprefled ; the body will receive fuch a velocity by this impulfe, as will carry it through E L in the fame time, as it would have imployed in paifing through E K, if it had gone on in that line undifturbed. And it has already been obferv- ed, that the time, in which the body would pafs over EK 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 tlirough E H with the ve- locity, wherewith it moved from D to E. Therefore the time, in which the body will pafs through E L after its return into the line E D, is the lame, as would have been taken up by the body in pafling through E H with the velocity, where- with the body firft moved in the line D E. Since therefore EL and EH are equal, the body returns into the Hne D E with the velocity, which it had before in that line. Again I lay, th© fecond impulfe in E is equal to the firft. By what has ' Viz. L, I. prop. 30j 29, & 26. been f 128 Sir I s A A c N E w T o n's Book I. beenTaid on the fecond law of motion concerning the effect of obHqiie impulfes% it will be underftood, that the impulfeinE, whereby the body was turned out of the line D E into the line EF, is of fuch ftrength, that if the body had been at reft, when this impulfe had adled upon it, this impulfe would have communicated fo 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 pa (Ted from E to H, or in the time wherein it paffed from E to I. In the fame manner, on the re- turn of the body, the impulfe in E, whereby the body is turn- ed out of the line F E into E D, is of fuch ftrength, that if it had a6led on the body at reft, it would have caufed the body •to move tlirough a length equal to K L, in the fime time, as the body would imploy in pafting through E K with the velo- city, wherewith it retvirns in the line F E. Therefore the fe- cond impulfe, had it aded on the body at reft, would have caufed it to move through a length equal to K L in the fame fpace of time, as would be taken up by the body in pafting through a length equal to H I, were the firft impulfe to ad on the body when at reft. That is, the cffedis of the firft and fecond impulfe on the body when at reft would be the fame; for K L and H I are equal : confequently the fecond impulfe is equal to the firft. II. Thus if the body be returned through FE with the velocity, wherewith it m.oved forward ; we ha\^e fhewn how by the repetition of the impulfe, which aded on it at E, the • Ch. II. §-;i,22. body Chap. 3- PHILOSOPHY. 129 body will return again into the line DE with the velocity, which it had before in that line. By the fame procefs of rea- foning it may be proved, that, when the body is returned back to D, the impulfe, which before adled on the body at that point, will throw the body into the line DC with the ve- locit)^, which it hrft had in that line ; and the other impuifes being fucceflively repeated, the body will at length be brought back again into the line B A with tlie velocity, wherewith it fet out in that line. 1 1. T H u s thefe impuifes, by ailing over again in an invert- ed order all their operation on die body, bring it back again through the path, in which it had proceeded forward. And this obtains equally, whatever be the number of the ftraight lines, whereof this curve figure is compofed. Now by a me- thod of reafoning, wliich Sir Isaac Newton makes great ufe of, and which he introduced into geometry, thereby greatly inriching that fcience '"; we might make a tranfition from this figure compofed of a number of ftraight lines to a figure of one continued curvature, and from a number of fe- parate impuifes repeated at diftindl intervals to a continual centripetal force, and fhew, that, bccaufe what has been here advanced holds univerfally true, whatever be the num- ber of ftraight lines, whereof the curve figure A C F is com- pofed, and howfoever frequently the impuifes at the angles of this figure are repeated ; therefore the fame will ftill remain true, although this figure fhould be converted into one of a continued curvature, and thefe diftind impuifes fhould be * viz. His doftrine of prime and ultimate ntios. S changed 1 50 sir I s A A c N E w T o n's Book I. changed into a continual centripetal force. But as the explain- ing this method of reafoning is foreign to my prefent defign j fo I hope my readers, after what has been faid, will find no difficulty in receiving the propofition 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 diredly back with the fame velocity as that, wherewith it proceeded forward, the centripetal force, by ading over again all its operation on the body, fhall bring the body back again in the line I H B : and as the motion of the body in its courfe from B to I was eve- ry where in fuch a manner oblique to the line drawn from the center to the body, that the centripetal power adled in fome degree againft the body's motion, and gradually diminifhed it ; fo in the return of the body, the centripetal power v^^ill every where draw the body forward, and accelerate its motion by the fame degrees, as before it retarded it. 15. This being agreed, fuppofe the body in K to have the line A K no longer obliquely inclined to its motion. In this cafe, if the body be turned back, in the manner we have been con- fidering, it muft be diredled back perpendicularly to AK. But if it had proceeded foi-ward, it would likewife have mov- ed in a diredion perpendicular to A K ; confequently, whe- ther it move from this point K backward or forward, it muft defcribe the fime kind of courfe. Therefore fince by being turned back it will go over again the line KIHB; if it be per- mitted to go forward, the line KL, which it fliall defcribe^ will he altogether fimilar to die line K H B. 4 In Chap. 3- PHILOSOPHY. 131 14. In like manner we may determine the nature of the motion, if the line, wherein the body fets out, be inclined (as in fig. 7(5.) down toward the line BA drawn between the body and the center. If the centripetal power fo much in- creafes in ftrength, as the body approaches, that it can bend the path, in which the body moves, to that degree, as to caufe all the Hnes as AH, A I, AK to remain no lefs oblique to the motion of the body, than A B is oblique to B C ; the body fliall continually more and more approach the center. But if the centripetal power increafes in fo much lels a degree, as to permit the line drawn from the center to the body, as it ac- companies the body in its motion, at length to become more and more eredl to the curve wherein the body moves, and in the end, fuppole at K, to become perpendicular tliereto ; from that time the body fliall rife again. This is evident from what has been faid above ; becaufe for the very fame reafon here alfo the body fhall proceed from the point K to defcribe a Hne alto- gether fimilar to the line, in which it has moved from B to K. Thus, as it was obferved of the pendulum in the preceding chap- ter *, that all the time it approaches towards being perpendicu- lar to the horizon, it more and more defcends ; but, as foon as it is come into that perpendicular fituation, it immediately rifes again by the fame degrees, as it defcended by before : fo here tlie body more and more approaches the center all the time it is moving from B to K ; but tlience forward it riles from the center again by the fame degrees, as it approached by before. § 57- S z ly. If 132 Sir Is A AC Newton's BookI. I 5". I F (in fig.7 7.) the line B C be perpendicular to A B ; then It has been obferved above % that the centripetal power may be (o balanced with the progrefTive motion of the body, that the body may keep moving round the center A conftantly at the fame diftance ; as a body does, when whirled about any point, to which it is tyed by a firing. If tlie centripetal power be too weak to produce this efFed:, the motion of the body will prefently become oblique to the line drawn from itfelf to the center, after tlie manner of the firfl: ot the two cafes, which we have been conlidering. If the centripetal power be ftronger, than what is required to carry the body in a cir- cle, the motion of the body will prefently fall in with the fe- cond of the cafes, we have been confidering. 16. I F the centripetal power fo change with the change of diftance, that the body, after its motion has become oblique to the line drawn from itfelf to the center, fhall again become perpendicular thereto ; which we have fhewn to be poflible in both the cafes treated of above ; then the body (hall in its fubfequent motion return again to the diftance of AB, and from that diftance take a courfe fimilar to the former : and thus, if the body move in a fpace free from all reftftance, which has been here all along fuppofed ; it ftiall continue in. a perpetual motion about the center, defcending and amend- ing alternately therefrom. If the body fetting out from B (in fig.7 8.) in the line BC perpendicular to AB, defcribe the line B D E, which in D fhall be oblique to the line A D, but in E §3. fhall Chap. 3- PHILOSOPHY. 133 fh:ill again become ere£l to AE drawn from the body in E to the center A ; then from this point E the body fhail defcribe the line E F G altogether like to the line B D E, and at G fhall be at the fam€ diftance from A, as it was at B. But likewilc the line A G Hiall be eredt to the body's motion. Therefore the body fhall proceed to defcribe from G the line GHI altoge- ther fimilar to the line G F E, and at I have the fame diftancc from the center, as it had at E ; and alfo have the line A I ered: to its motion : fo that its following motion muft be in the Hne I K L fmiilar to I H G, and the diftance A L equal to A G. Thus the body will go on in a perpetual round without ceafing, al- ternately inlarging and contrading its diftance from the center, 17. I F it fo happen, tliat 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 alfo on B ; fo that the body will defcribe in this cafe a fimple 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 AE in fig. 78 iliould happen to be perpendicular: to A B, in this cafe alfo a fimple line will be defcribed ; 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 : fo that the body will defcribe a line like the curve line BEG I in fig. 80, in which the oppofite points B and G. are equally diftant from A, and the oppofite points E and I are alfo equally diftant from tlie fame point A. . 134- Sir Isaac Newton's Book I. 19. In other cafes die line defcribed will have a more complex figure. ao. Thus we have endeavoured to fhew how a body, while it is conftantly attradled towards a center, may notwith- ftanding by its progreflive motion keep it felf from falling down to that center ; but defcribe about it an endlefs circuit, fometimes approaching toward that center, and at other times as much receding from the fame. 1 1 . B u T here we have fuppofed, that the centripetal power is of equal ftrength every where at the fame diftance from the center. And this is the cafe of that centripetal power, which will hereafter be fhewn to be the caufe, that keeps the planets in their courfes. 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 fhall have its concavity turned every where towards the center of the force. ^2. T o make this evident I (hall firft propofe the cafe of a body moving through the incurvated figure ABCDE (in fig. 8 1 .) which is compofed of the ftraight lines A B, B C, C D, D E, and EA; the motion being carried on in the following manner. Let the body firft move in the line A B with any uniform velo- city. When it is arrived at the point B, let it receive an im- pulfe diredcd toward any point F taken within the figure ; and let the impulfe be of that ftrength as to turn the body out of m /"y. J^-f '^fi.fi7- Chap. 3- PHILOSOPHY. 135 of the line A B into the Hne B C. The body after this im- piilfe, while left to itfelf, will continue moving in the hne B C-. At C let the body receive another impulfe direded towards the fame point F, of fuch ftrength, as to turn the body from the line B C into the hne CD. At D let the body by another impulfe, direded likewife to the point F,be turned out of the line C D into D E. And at E let another impulfe, directed to- ward tlie point F, turn the body from the line DE into EA. Thus we fee how a body may be carried through the figure ABCDE by certain impulfes direded always toward the fame center, only by their afting on the body at proper intervals,, and with due degrees of ftrength.. ag . B u T farther, when the body is come to the point A, if it there receive another impulfe direded like the reft toward the point F, and of fuch a degree of ftrength as to turn the body into the line A B, wherein it firft moved ; I fay that the body fhall return into this line with the fame velocity, as it had at firft . 14.. Let AB be prolonged beyond B at pleafure, fuppofe to G ; and from G let G H be drawn, which if produced fhould always continue equidiftant from B F, or, according to the more ufiial phrafe, let G H be drawn parallel to B F. Then it appeals, from what has been faid upon the fecond law of motion % that in the time, wherein the body would have moved . from B to G, had it not received a new impulfe in B-, by the means of that impulfe it will have acquired a velocity, which will carry it from B to H. After the fame manner, if C I be I 136 Sir I s A A c N E w T G n's Book L taken equal to B H, and I K be drawn equidiftant from or pa- rallel 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 fame time, as it would have employed in moving from C to I with the velocity, it had in the line B C. Therefore fince C I and B H are equal, the body will move through C K in the fame time, as it would have taken up in moving from B to G with the original velo- city, v/herewith it moved through the Hne AB. Again, DL being taken equal to CK and L M drawn parallel to D F ; for the fame reafon as before the body v/ili move through DM with the velocity, which it has in the Hne DE, in the fame time, as it would imploy in moving through B G with its original ve- locity. In the laft place, if E N be taken equal to D M, and N O be drawn parallel to E F ; likewife 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, v/ill pais through ACVin the fame time, as it would have imployed in" pafling through B G with its original velocity. Now as all this follows diredly from what has above been delivered, con- cerning the effcft of oblique impulfes impreffed upon bodies in motion ; fo we mufl here obferve farther, that it can be proved by geometry, tfiat A Q, will always be equal to B G. The proof of this I am obliged, from the nature of my pre- fent defign, 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 firft moved in that line ; for the velocity, with which it returns into the line AB, will carry it over the line A Q,in the fam^ time, as would 3 have Chap. 3- PHILOSOPHY. 137 have been taken up in its pafTmg over an equal line B G with the original velocity. 2 y. T H u s we have found, how a body may be carried round tlic hgure A B C D E by the adion of certain inipulfes upon it, wliich iliould all be pointed toward one center. And we like- wife fee, that when the body is brought back again to the point, whence it firft fet out ; it it there meet with an im- pi:lfe fufficient to turn it again into the line, wherein it mov- ed at hrft, its original velocity will be again reftored ; and by the repetition of the fame impulfes, the body will be carried again in the fame round. Therefore if tliefe im.pulfes, which adt on the body at the points B, C, D, E, and A, continue al- \vays the fame, the body will make round this figure innu- merable revolutions. 7.6. The proof, which we have here made ufe of, holds the lame in any number of ftraight lines, v/hereof the figure A B D fhould be compofed ; and therefore by the method of reafoning referred to above ^ we are to conclude, that what has here been faid upon tliis red;ilinear figure, will remain true, if this figure were changed into one of a continued curvature, and inftead of diftind: impulfes adiing by intervals at the angles of this figure, we had a continual centripetal force. We have therefore fhewn, that a body may be carried round in any curve figure ABC ( fig. 8l.) which fiiall every where be concave towards any one point as D, by the continual adion 3 T cf 138 Sir I s A A c N E w T o n's Book I. of a centripetal power direfted to that point, and when it is returned to the point, from whence it fet out, it fhall recover again the velocity, with which it departed from that point. It is not indeed always neceffary, that it fhould return again into its firft courfe ; for the curve line may have fonie fuch figure as the line A B C D B E in fig. 8 g . In this curve Hne, if the body fet out from B in the diredion B F, and moved through the line BCD, till it returned to B ; here the body would not enter again into the line BCD, becaufe the two. parts B D and B C of the curve line make an angle at the point B : fo 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 diredlion, in which it returns to the point B ; a forceable impulfe muft be given the body in the point B to produce that effed. 2,7. If kt the point B, whence the body lets out, the curve line return into it felf ( as in fig. 8x ; ) then the body, up- on its arrival again at B, may return into its former courfe, and thus make an endlefs circuit about the center of the cen- tripetal power. t8. What has here been faid, I hope, will in fome mea- lure enable my readers to form a jufi: idea of the nature of thefe centripetal motions. 2. 9 , I H A V E not attempted to fhew, how to find particular- ly, wha of centripetal force is necefiary to carry a body in. any curve line propofed. This is to be deduced Irom the de- gree. Chap. 3- PHILOSOPHY. 139 gree of curvature, whicli the figure has in each point of it, and requires a long and complex mathematical reafoning. However T (liall fpeak a little to the firft propofition, which Sir Isaac Newton lays down for this purpofe. By this propofition, when a body is found moving in a curve line, it may be known, whether the body be kept in its courfe by a power always pointed toward the fame center; and if it be foj where that center is placed. The propofition is this : that if a line be drawn from fome fixed point to the body, and re- maining 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 courfe, be always pointed to this fixed point as a center, this line will move over equal fpaces in equal portions of time. Suppofe a body were moving through the curve line A B C D (in fig. 84.) and palTed over the arches A B, BC, 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 fhall make the fpaces E A B, E B C, and E C D equal, over which it paf- fes, while the body defcribes the arches A B, B C, and C D : and if this hold the fame in all other arches, both great and fmall, of the curve line A B C D, that thefe fpaces 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 I s a a c N e w t o n has demonftrated this, requires but fmall skill in geometry to com- prehend. I fhall therefore take the liberty to clofe the pre- T X fent 140 Sir Isaac Newton's Book I. fent chapter with an explication of it ; becaufe fuch an exam- ple will give the cleareft notion of our author's method of ap- plying mathematical reafoning to thefc philofophical fubjedls. 51. He reafons thus. Suppofe a body fet out from the point A ( in fig. 85-.) to move in the ftraight line A B ; and after it had moved for forne time in that line, it were to receive an impulfe directed to fome point as C. Let it receive that im- pulfe at D ; and thereby be turned into the line D E ; and let the body after this impulfe take the fame length of time in pafling from. D to E, as it imployed in the pafTmg from A to D. Then the ftraight lines C A, CD, and C E being drawn, Sir Isaac Newton proves, that the and triangular fpaces CAD and CDE are equal. This he does in the following manner. 3 T. L E T E F be drawn parallel to C D. Then, from what has been faid upon the fecond law of motion % it is evident, that ilnce the body was moving in the line A B, when it received the impulfe in the diredion DC; it will have moved after that impulfe through the line D E in the fame time, as it woul Ch I § 14. b Chi fclca 17. againfl: Chap. 4- PHILOSOPHY. 14.5 againft die fluid by tliat power alone, which is eflcntial to tliem, of refilling to the change of their ftate from reft to motion, or from motion to reft, which we have above cal- led their power of inadivity ; or elfe, as defending or af- ccnding, and fo having the power of gravity combined with that other power. Thus our author has fliewn in all tlioie three fuppolitions, in what manner bodies are relifted in an uniform fluid, when they move with tlie aforelaid progreflive motion '' ; and what the reilftance is, when tliey afcend or defcend perpendicularly K And if a body alcend or defcend obliquely, and the reflftance be flngly proportional to the ve- locity, it is fliewn how the body is reflfted in a fluid of an uni- form denflty, and what line it will defcribe ", which is de- termined by the meafurement of tlie hyperbola, and ap- pears to be no other than that line, flrft confldered in par- ticular by Dr. B .\ r r o w '■\ which is now commonly known by the name of the logarithmical cun^. In the fuppo- fition that the reflftance increafes in die dupHcate propor- tion of the velocity, our author has not given us the line which would be defcribed in an uniform fluid ; but has in- ftead thereof difcufi'ed a problem, which is in fome fort the reverfe ; to find the denflty of the fluid at all altitudes, by which any given curve line may be defcribed ; which pro- blem is fo treated by him, as to be applicable to any kind of •reflftance whatever ^ But here not unmindful of pradlice, he fliews that a body in a fluid of uniform denflty, like die » Newt. Princ.L. II. prop. 2:5,6,7; II, 12. I * Prxk:1. G.-iiTietr. p.i.r. 123. •> Prop. J ; 8, 9; 13, 14. I < Newton, Princ. Lib. I J. prop. 10. •= Prop. 4. . I V air. 1^6 Sir Is AAc Newton's BookI. air, will defcribe a line, which approaches towards an hy- perbola ; diat is, its motion will be nearer to that curve line than to the parabola. And confeqiient upon this remark, he {hews how to determine this hyperbola by experiment, and briefly refolves the chief of thofe problems relating to proje- d:iles, which are in ufe in the art of gunnery, in this curve ^; as ToRRicELLi and odiers have done in die parabola ^ , whofe inventions have been explained at large above ". y. Our author has alfo handled diftindlly that particu- lar fort of motion, which is defcribed by pendulums '^ ; and has likewife confidered fome few cafes of bodies moving in refifling fluids round a center, to which they are impelled by a centripetal force, in order to give an idea of thofe kinds of motipns ^ 6^. T H E treating of the reflfliance of pendulums has giv- en him an opportunity of inferting into another part of his work fome fpeculations upon the motions of them with- out reflftance, which have a very peculiar elegance ; where in he treats of them as moved by a gravitation adling in the law, which he fhews to belong to the earth below its furface ^ ; performing in this kind of gravitation, where the force is proportional to the diftance from the center, all that H u y G E N s had before done in the common fuppofltion of its being uniform, and ading in parallel lines ^. » Newton. Princ. Lib 11 prop. lo. in fchol. ^ Torriccllide motu gravium. ' Ch 1 § 8j-, 8cc. * Newt. Princ. L.il, fe> Ibid. when Chap.4. philosophy, 155 when out of contact, by which, as has before been men- tioned, the whole fluid may be rendred elaftic ; but fuch an elafticity only, as many foHd bodies have of recovering their figure, whenever any forcible change is made in it, by the impulfe of another body or otherwife. Which elafticity has been explained above at large ''. 18. T H I s is the cafe of difcontinued fluids, where the bo- dy, by prefling againft their particles, drives them before itfelf, while the fpace behind the body is left empty. But in fluids which are compreflTed, fo that the parts of them re- moved out of place by the body reftfted immediately retire behind the body, and fill that fpace, which in the other cafe is left vacant, the refiftance is ftill lefs ; for a globe in fuch a fluid which fliall be free from all elafticity, will be refifted but half as much as the leaft refiftance in the former cafe ''. But by elafticity I now mean that power, which renders the whole fluid fo; of which if the compreflTed fluid be poflTefl^ed, in the manner of the air, then the refiftance will be greater than by the foregoing rule ; for the fluid being capable in fome degree of condenfation, it will refemble fo far the cafe of un*. comprefied fluids '^. But, as has been before related, this dif- ference is moft confiderable in flow motions. 19. In the next place our author is particular in deter- mining the degrees of refiftance accompanying bodies of diflerent figures ; which is the laft of the three heads, we = h. I. § 29. I coroll. I. of prop.5f. *• Princ.phil0f.Lib.II.Prop.3S, compared with [ ' L.II. Lem.7. fchol, pag. 541. X z divided ic^6 Sir Isaac Newton's Book! divided the whole difcoiirfe of refiftance into. And in this dilquifition he finds a very furprizing and unthought of dif- ference, between free and compreffed fluids. He pro^'es, that in the former kind, a globe fuflers but half the refin- ance, which the cylinder, that circumfcribes the globe, will do, if it move in the djre<^ion of its axis ". But in the lat- ter he proves, that the 'g^obe and cylinder are refifted a- Ijke ''. And in general, that let the fliape of. bodies be eyer fo different, yet if the greateft fedtions of the bodies perpendicular to the axis of their motioa be equal, the. bodies will be refifled equally ''. 2.0. Pursuant to the difference found between the re- fiflance of the globe and cylinder in rare and uncompreffed fluids, our aut^ior gives us the refult of fom.e other inquiries cf the fame nature. Thus of all the fruftums of a cone, that can be defcribed upon the fame bafe and with tlie fame altitude, he fhews how to find that, which of all others will be die leaft refifled, wlien moving in the direction of its axis ''. And from hence he draws an eafy method of al- tering the figure of any fpheroidical folid, fo that its capa- city may be enlarged, and yet the refiftance of it diminifli- ed *": a note which he tliinks may not be ufelcls to fhip- wrights. He concludes with determining the folid, which will be refifled the leafl that is poifible, in thefe difcontinued fluid f ' Lib. IT. Prop. 54." I "^ Prop. 34. fcho!. •> Lib. n. L<'m.7. p. 341; I •■ Ibid, t Schol. to Lcm,7. » Mbid. XI. That Ckap.4. philosophy. 157 Zl. That I may here be undGrflood by readers uriac- quainted widi madiematlcal terms, I fliall explain wliat I mean by a fruflum of a cone, and a Spheroidical folid. A cone has been defined above. A firuftum is what remains,^ wlien part of the cone next the vertex is cut'aw?iy by- a. lec- tion parallel to the bafe of the cone, as in fig. 8 6^. A fpheroid^ is produced from an ellipfis, as a fphere or globe is made from a circle. If a circle turn round on its diameter, it de- fcribes by its motion a fphere ; fo if an ellipfis ( which figure has been defined above, and will be more fully explained' hereafter M be turned round either upon die longeft or* fliorteft line, that can be drawn through the middle of it, there will be dcfcribed a kind of oblong or fiat fphere, as in fig. 87. Both thefe figures are called fpheroids, and any folid refembling thefe I here call fpheroidical. Tl. If it fiiould be asked, how the method of altering; ipheroidical bodies, here mentioned, can contribute to the facilitating a fhip's motion, when T juft above affirmed, that the figure of bodies, which move in a comprefied fiuid not elaftic, has no relation to the augmentation or di- minution of the refiftance ; the reply is, that \\^hat was there fpoken relates to bodies deep immerged into fuch flu- ids, but not of thofe, which fwim upon the fiirface of them ; for in this latter cafe the fluid, by the appulfe of the an- terior parts of the body, is raifed above the level of the . fiirface, and behind the body is funk fomewhat below; (qhj » Book II. Ch. I. §,<5. that I $8 Sir Isaac N e w t o n^s Book I. :that by this inequality in the fuperficies of the fluid, that part of it, which at the head of the body is higher tlian the fluid behind, will reflft in fome meafure after the manner of dilcontinued fluids *, analogous to what was be- fore obferved to happen in the air through its elafticity, though the body be furrounded on every fide by it ^. And as far as the power of thefe caufes extends, the figure of the moving body affeds its refiftance ; for it is evident, that the figure, which prefixes leaft diredly againfl die parts of the fluid, and fo raifes leaft the furface of a fluid not elaftic, and leafl; comprefljbs one that is elaftic, will be leaft refifted. 13. The way of collecting the difference of the refifl:ance in rare fluids, which arifes from the diverfity of figure, is by confidering the different effed; of the particles of the fluid upon the body moving againft them, according to the diffe- rent obliquity of the feveral parts of the body upon which they refpedtively ftrike ; as it is known, that any body im- pinging againft a plane obliquely, ftrikes with a lefs force, than if it fell upon it perpendicularly; and the greater the obliquity is, the weaker is the force. And it is the lame thing, if the body be at reft, and the plane move againft it ^ 14. That there is no connexion between the figure •of a body and its refiftance in comprefl"ed fluids, is proved thus. Suppofe A B C D (in fig. 8 8.) to be a canal, having fuch a fluid, water for inftance, running through it widi an equable =1 Vid. Ncwf. princ. in fchol. to Lem. 7, of I ^ Scft. 17. of tliis chapter. Lib. II. pag. 341. I "^ See Prmc. philof. Lib. II. prop. 34.. -elocity Chap. I. PHILOSOPHY i6i . '^ i'-' tjtvr- ./>/. V»i\ I ', 'ff/^/ ■' BOOK II. CONdERNlNG THE SYSTEM of the WORLD. Chap. L That the Planets move in a (pace empty of all fenfible matter. HAVE now gone through the firft part of my defign, and have ex- plained, 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 fpeak of the difcoveries, he has made in the fyflem of the world ; Y and i62 Sir Isaac Newton^s BookII, and to fhew from Kirn what caufe keeps the heavenly bo- dies in their courfes. But it will be neceflary for the ufe of fuch, as are not skilled in aftronomy, to premife a brief de- fcription of the planetary fyftem. 1. This fyftem is diljx)fed in the following manner. In the middle is placed the fun. About him iix globes con- tinually roll. Thele are the primary planets; that which is neareft to the {im is called -Mercury, the next Venus, next to this is our earth, the next beyond is Mars, after him Jupiter, and the outermoft of all Saturn. Befides thele there are difcovered in this fyftem ten other bodies, which, move about fome of thefe primary planets in the lame maimer, as they move round the fun. Thefe are called fecondary planets. The moft confpicuous of them is the moon, which moves round our earth ; four bodies move irr like manner round Jupiter ; and five round Saturn. Thole which move about Jupiter and Saturn, are ufually called fatellites ; and cannot any of them be feen without a te- lefcope. It is not impofTible, but there may be more fe- condary planets, befide thefe ; though our inftruments have not yet difcovered any other. This difpofition of the planetary or folar fyftem is reprefented in fig. 89. J. The fame planet is not always equally diftant from the ftin. But the middle diftance of Mercury is between - and ; of the diftance of the earth from the fun ; Venus is diftant from the fun almoft f of the diftance of the sarth; the. middle diftance of Mars is fomething more thai^ half iT Chap, i : PHILOSOPHY. r(S j half as much again, as the diftance of the earth ; Jupiter*s middle diftance exceeds five times the diftance of the earth, by between y and I part of this diftance ; Saturn*s middle diftance is fcarce more than 9^ times the diftance between the earth and ftin ; but the middle diftance between the earth and fun is about 1 1 7j- times the fun's femidi- ameter. 4. All thefe planets move one way, fi-om weft to eaft ; and of the primary planets the moft remote is long- eft in finiftiing its courfc round the ftin. The perioci of Saturn falls fhort only fixteen days of 19 years and a half The period of Jupiter is twelve years wanting a- bout so days. The period of Mars falls fliort of two years by about 45 days. The revolution of the earth conftikutes the year. Venus performs her period in about Xl^i days> and mercury in about 88 days, j-. The courfe of each planet lies throughout in one plane or flat furface, in which the fun is placed ; but they do not all move in the fame plane, though the different planes, in which they move, crofs each other in very ftnall angles. They all crofs each other in lines, which pais through the fun ; becaufe the fun lies in the plane of each orbit. This inclination of the feveral orbits to each other is reprefented in fig. 90. The line, in which the plane of any orbit crofies the plane of the earth's motion, is called the line of the nodes of that orbit, Y 1' . d. Each 1 6^ Sir I s A A c N E w T o n's Book II. 6. Each planet moves round the fun in the line, which we have mentioned above ^ under the name of ellipfis ; which I fhall here fhew more particularly how to defcribe. I have there laid how it is produced in the cone. I fhall now fliew how to form it upon a plane. Fix upon any plane two pins> as at A and B in fig. 9 1 . To thefe tye a firing A C B of any kngth. Then apply a third pin D fo to the ftring, as to hold it ftrained ; and in that manner carrying this pin about, th& point of it will defcribe an ellipfis. If through the points A, B the flraight line EABF be drawn, to be terminated at the ellipfis in the points E and F, this is the longeft line of any, that can be drawn within the figure, and is call- ed the greater axis of the ellipfis. The line GH, drawn perpendicular to this axis EF, fo as to pafs through the middle of it, is called the lefl~er axis. The two points A and B are called focus's. Now each planet moves round the fun in a line of this kind, fo that the fun is found in one focus. Suppofe A to be the place of the fun. Then E is the point, wherein the planet will be neareft of all to the fun, and at F it will be moft remote. The point E is call- ed the perihelion of the planet, and F the aphelion. In G and H the planet is faid to be in its middle or mean diftance ; becaufe the diftance AG or AH is truly the middle be- tween AE the leaft, and AF the greateft diftance. In fig. 91. is reprefented how the greater axis of each orbit is fituated ia refpe^l of the reft. The proportion between the greateft and leaft diftances of the planet from the fun is very different in the different planets. In Saturn the proportion of the »Book.i. ch.2. §8i.- greateft J-l^ JO (♦/•/• J 'A ,^a/7//-//. Chap. I. PHILOSOPHY. i6c. greatefl: diftance to the leaftis fbmething lefs, thah the propor- tion of 9 to 8 ; but much nearer to this, than to the propor- tion of ID to 9. In Jupiter this proportion is a little greater, than that of 1 1 to 10. In Mars it exceeds the proportion of 6 to y. In the earth it is about the proportion of 3 o to ig. In Venus it is near to that of 70 to 6^9. And in Mercury it comes not a great deal fhort of the proportion of 3 to i. 7. E A c H of thefe planets fo m.oves through its ellipfis, that ' the line drawn from the fun to the planet, by accompanying the planet in its motion, will defcribe about the fun equal j(pa- ces in equal times, after the manner fpoke of in the chapter of centripetal forces *. There is alfo a certain relation between the greater axis's of thefe ellipfis's, and the times, in which die planets perform their revolutions through them. Which relation may be expreffed thus. Let the period of one planet be denoted by the letter A, the A D- greater axis of its orbit by D ; let the period B E . of another planet be denoted by B, and the C F greater axis of this planet's orbit by E. Then G if C be taken to bear the fame proportion to B, as B bears to A ; likewife if F be taken to bear the fame pro- portion to E, as E bears to D ; and G taken to bear the fame proportion likewife to F, as E bears to D ; then A fhall bear, the fame proportion to C, as D bears to G. . 8. The fecondary planets move round their refpedlive primary, much in the fame manner as the primary do round'. * Book I. Ch.3,§ 19. tha 1 66 Sir I s A A c N E w 1 n's Book IL the fun. But the motions of thefe fhall be more fully ex- plained hereafter \ And there is, beddes the planets, another fort of bodies, which in all probability move round the fun ; I mean the comets. The farther defcription of which bodies I alfo leave to the place, where they are to be particularly treated on ^. 9. Far without this fyftem the fixed ftars are placed. Thefe are all fo remote from us, that we feem almoft incapa- ble of contriving any means to eflimate their diftance. Their number is exceeding great. Befrdes two or three thoufand, which we fee with the naked eye, telefcopes open to our view vaft numbers ; and the farther improved thefe inftruments are, we Hill difcover more and more. Without doubt theie are luminous globes, like our fun, and ranged through the wide extent of fpace ; each of which, it is to be fuppofed, perform the fame office, as our flm, affording light and heat to certain planets moving about them. But thefe conjedures are not to be purfued in this place. 10. I SHALL therefore now proceed to the particular de- (ign of this chapter, and (hew, that there is no fenfible mat- ter lodged in the fpace where the planets move. 11. That they fuffer no fenfible refiflance from any fuch matter, is evident from the agreement between the obfer- vations of aflronomers in different ages, with regard to the time, in which the planets have been found to perform their » Ch. jiofthisprefentbook," ^ Ch.4. periods. HAP. I. PHILOSOPHY, 167 periods. But it was the opinion of D e s C a r t e s % that the planets might be kept in their courfes by the means of a fluid matter, which continually circulating round fhould carry the planets along with it. There is one appearance that may feem to favour this opinion ; which is, that the flin turns round its own axis the fame way, as the planets move. The earth alfo turns round its axis the fame way, as the moon moves round the earth. And the planet Jupiter turns upon its axis the fame way, as his fatellites re\^olve round him. It might therefore be fuppofed, that if the whole planetary region were filled with a fluid matter, the fun, by turning round on its own axis, might communicate motion firft to that part of the fluid, which was contiguous, and by degrees propagate the like motion to the parts more remote. After the lame manner the earth might communicate motion to this fluid, to a diftance fuflieient to carry round the moon,and Jupiter comf municate. the like to the difl:ance of its fatellites. Sir I s a a c Newton has particularly examined what might be the reliilt of fuch a motion as this ^; and he finds, that the velocitie^j , with which the parts of this fluid will move in different df- ftances from the center of the motion, will notagfee with the morion obferved in different planets : for inftance, that the time of one intire circularion of the fluid, wherein Jupiter fhould fwim, would bear a greater proportion to the time of one intire circulation of die fluid, where the earth is ; than the period of Jupiter bears to the period of the earth. But he alfo proves % that the planet cannot circulate in fuch a fluid, » InPrSnc. phiW.part. 5. I &fchol. ^ Philof. princ. mathera. Lib. II, prop. 2. | ' Ibid, prop.f 3. fo 1 68 Sir Isaac Newton's Book II. fb as to keep long in the fame courfe, iinlefs the planet and the contiguous fluid are of the fame denflty, and the planet be carried along with the fame degree of motion, as the fluid. There is alfo another remark made upon this motion by our author; which is, that fome vivifying force will be continual- ly neceflary at the center of the motion \ The fun in par- ticular, by communicating motion to the ambient fluid, will lofe from it feif as much motion, as it imparts to the fluid ; unlefs fome ading principle refide in the fun to renew its motion continually. If the fluid be infinite, this gradual lols of motion would continue till the whole fhould flop ^ j and if the fluid were limited, this lofs of motion would continue, till there would remain no fwifter a revolution in the fun, than in the utmoft part of the fluid ; fo that the whole would turn together about the axis of the fun, like one folid globe ^ II. I T is farther to be obferved, that as the planets do not move in pcrfed circles round the fun ; there is a greater dift- ance between their orbits in fome places, than in others. For inftance, the diftance between the orbit of Mars and Venus is near half as great again in one part of their orbits, as in the oppofite place. Now here the fluid, in which the earth fhould fwim, mufl: move with a lefs rapid motion, where there is this greater interval between the contiguous orbits ; but on the contrary, where the fpace is ftraiteft, the earth moves more flowly, than where it is wideft \ » Philof.princ. prop. fi. coroll.4. I 'Coroll. 11, '' Ibid. I * Secibid.fchd.poftprop, fj. 13. Farther Chap. I. PHILOSOPHY. 169 I g . Farther, if this our globe of earth fwam in a fluid of equal denfity with the earth it felf, that is, in a fluid more denfe than water ; all bodies put in motion here upon the earth's furface muft fufiJer a great reflftance from it ; where asj by Sir Isaac Newton's experiments mentioned in the preceding chapter, bodies, that fell perpendicularly down through the air, felt about ^^ part only of the reflftance, which bodies fuffered that fell in Hke manner through water. 14. Sir I s A A c N E w T o N applies thefe experiments yet farther, and examines by them the general queftion concern- ing the abfolute plenitude of Ipace. According to the Ariftote- lians, all fpace was full without any the leaft vacuities whate- ver. DesCartes embraced the fame opinion, and therefore fuppofed a fubtile fluid matter, which fhould pervade all bo- dies, and adequately fill up their pores. The Atomical philo- fophers, who fuppofe all bodies both fluid and folid to be com- pofed of very minute but folid atoms, afl^ert that no fluid, how fubtile fbever the particles or atoms whereof it is compofed fhould be, can ever caufe an abfolute plenitude ; becaufe it is impoflible that any body can pafs through the fluid widi- out putting the particles of it into fuch a motion, as to fepa- rate them, at leaft in part, from one another, and fo perpetu- ally to caufe fmall vacuities ; by which thefe Atomifts endea- vour to prove, that a vacuum, or fome Ipace empty of all matter, is abfolutely necefl'ary to be in nature. Sir I s a a c Newton objects againft the filling of fpace with fuch a fub- tile fluid, fhat all bodies in motion muft be unmeafurably re- Z fifted 170 Sir Is A AC Newton's BookII. fifted by a fluid fo denfe, as abfolutely to fill up all the fpace, through which it is fpread. And left it fhould be thought, that this objedion might be evaded by afcribing to this fluid fuch very minute and fmooth parts, as might remove all ad- heflon or fridiion between them, whereby all reflftance would be loft, which this fluid might otherwife give to bo- dies moving in it ; Sir I s a a c N e w t o n proves, in the manner above related, that fluids reflft from the power of inactivity of their particles ; and that water and the air re- flft almoft entirely on this account : fo that in this fubtile fluid, however minute and lubricated the particles, which compofe it, might be ; yet if the whole fluid was as denfe as water, it would reflft very near as much as water does ; and whereas fiach a fluid, whofe parts are abfolutely clofe toge- ther without any intervening fpaces, muft be a great deal more denfe than water, it muft reflft more than water in proportion to its greater denflty ; unlefs we will fuppofe the matter, of which this fluid is compofed, not to be endued with the fame degree of inadivity as other matter. But if you deprive any fubftance of the property fo univerfally be- longing to all other matter, without impropriety of fpeech it can fcarce be called by this name. 1 5^. Sir I s A A c N E w T o N made alfo an experiment to try in particular, whether the internal parts of bodies fuflered any re- flftance. And the refult did indeed appear to favour fome fmall degree of rcflftance; but fo very little, as to leave it doubtful, whether the effed: did not arife from fome other latent caufe'\ ' Princ. philof. paT. 3 16, 317. Chapi Chap.2: philosophy. 171 C H A p. II. Concerning the caufe, which keeps in motion the primary planets. SINCE the planets move in a void fpace and are free from refiftance ; they, Hke all other bodies, vi^hen once in motion, would move on in a ftraight line without end, if left to themfelves. And it is now to be explained what kind of adlion upon them carries them round the fun. Here I fhall treat of the primary planets only, and dif- courfe of the fecondary apart in the next chapter. It has been juft now declared, that thele primary planets move fo about the fun, that a line extended from die ilin to the planet, will, by accompanying the planet in its motion, pafs over equal fpa- ces in equal portions of time ^ And this one property in the motion of the planets proves, that they are continually ad:ed on by a power directed perpetually to the fun as a center. This therefore is one property of the caufe, which keeps the planets in their courfes, that it is a centripetal power, whofe center is the fun. 1. Again, in the chapter upon centripetal forces ^ it was obferv'd, that if the ftrength of the centripetal power was fuitably accommodated every where to the motion of any body round a center, the body might be carried in » Ch. I. §7. I'Bookl. Ch. ;, Z z an/ 172 Sir Isaac Newton's BookIL any bent line whatever, whofe concavity fliould be every where turned towards tlie center of the force. It was far- ther remarked, that the flrength of the centripetal force, in each place, was to be colledted from the nature of the line, wherein the body moved ^ Now fmce each planet moves in an ellipiis, and the fun is placed in one focus ; Sir Isaac Newton deduces from hence, that the ftrength of this power is reciprocally in the duplicate proportion of the diftance from the fun. This is deduced from the properties, which the geometers have difcovered in the ellipfis. The pro- cefs of the reafoning is not proper to be enlarged upon here • but I fhall endeavour to explain what is meant by the recipro- cal duplicate proportion. Each of the term s reciprocal pro- portion, and duplicate proportion, has been already defined \ Their fenfe when thus united is as follows. Suppofe the planet inoved in the orbit ABC (in fig. 95.) about the fun in S. Then, when it is faid, that the centripetal power, which ad:s on the planet in A, bears to the power ading on it in B a propor- tion, which is the reciprocal of the duplicate proportion of the diftance S A to the diftance SB; it is meant that the power in A bears to the power in B the duplicate of the proportion of the diftance S B to the diftance S A. The reciprocal du- plicate proportion may be explained alfo by numbers as fol- lows. Suppofe feveral diftances to bear to each other propor- tions exprefied by the numbers i, i, 3, 4, y ; that is, let the fecond diftance be double the firft, the third be three times, the fourth four times, and the fifth five times as great as the => Book I. Ch, 3. § i^j. b Ibki. Ch. 2. § 30, 17. firft, Chap. 2. PHILOSOPHY. lyy firft. Multiply each of thefe numbers by it felf, and i multi- plied by I produces ftill i , z multiplied by^ i produces 4, 5 by g makes 9, 4 by 4 makes id, and 5" by y gives zf. This being done, the fradions -^^j, f,, ~, will refpedively exprefs the proportion, which the centripetal power in each of the following diftances bears to the power at the firft diftance : for in the fecond diftance, which is double the firft, the centri- petal power will be one fourth part only of the power at the firft diftance ; at the third diftance the power will be one ninth part only of the firft power ; at the fourth diftance, the power will be but one fixteenth part of the firft ; and at die fifth diftance,, one twenty fifth part of the firft power. 3. Thus is found the proportion, in which this centripetal power decreafes, as the diftance from the fun increafes, within the compafs of one planet's motion. How it comes to pafs, that the planet can be carried about the ilin by this centripetal power in a continual round, fometimes rifing from the fun, then defcending again as low, and from thence be carried ' up again as far remote as before, alternately rifing and falling - without end ; appears from what has been written above con- cerning centripetal forces: for the orbits of the planets re- femble in fhape the curve line propofed in § 17 of the chapter on thefe forces \ 4. B u T farther, in order to know whether this centripetal « force extends in the fame proportion throughout, and confe- queiitly whether all the planets are influenced by the \'ery fame » Book I. Ch.3: power.. \ ij^ Sir Isaac Newton's BookII. power, our author proceeds thus. He inquires what relation there ought to be between the periods of the different planets, provided they were a6ted upon by the fame power decrealing throughout in the forementioned proportion ; and he finds, that the period of each in this cafe would have that very rela- tion to the greater axis of its orbit, as I have declared above •"* to be found in the planets by the obfervations of aftronomers. And this puts it beyond queftion, that the different planets are preffed towards the fun, in the fame proportion to their diftan- ces, as one planet is in its feveral diftances. And thence in the laft place it is juftly concluded, that there is fuch a power a6ling towards the fun in the forefaid proportion at all diftances from it. y. This power, when referred to the planets, our author calls centripetal, when to the fun attradive ; he gives it Hke- wife the name of gravity, becaufe he finds it to be of the fame nature with that power of gravity, which is obferved in our earth, as will appear hereafter ^. By all thefe names he defigns only to fignify a power endued with the properties before mentioned ; but by no means would he have it underftood, as if thefe names referred any way to the caufe of it. In particular in one place where he ufes the name of attradion, he cauti- ons us exprefsly againft implying any thing but a power di- reding a body to a center without any reference to the caufe of it, whether refiding in that center, or arifing from any external impulfe ". » Ch. i.§ 7. *> Chap. 5-. §8. ' Princpag. 60. d. But Chap. 2. PHILOSOPHY. 175 6. But now, in thefe demonftrations fbme very minute in- equalities in the motion of the planets are negleded ; which is done with a great deal of judgment ; for whatever be their caufe, the effedls are very inconiiderable, they being fo exceed- ing fmall, that fome aftronomers have thought fit wholly to pafs them by •\ However the excellency of this philofophy, when in the hands of fo great a geometer as our author, is fuch, that it is able to trace die leaft variations of things up to their caufes. The only inequalities, which have been obferved common to all the planets, are the motion of the aphelion and the nodes The tranfverfe axis of each orbit does not always remain fix- ed, but moves about the fun with a very flow progrefiive motion : nor do the planets keep conftantly the fame plane, but change them, and the lines in which thofe planes inter- {cSl each other by infenfible degrees. The firft of thefe inequalities, which is the motion of the aphelion, may be ac- counted for, by fuppofing the gravitation of the planets to- wards the fun to differ a litde from the forementioned re- ciprocal duplicate proportion of the diftances ; but the fe- cond, which is the motion of the nodes, cannot be account- ed for by any power dired:ed towards tlie fun ; for no fuch can give the planet any lateral impulfe to divert it from the plane of its motion into any 'new plane, but of neceflity mull be derived from fome other center. Where that power is lodged, remains to be difcovered. Now it is proved, as fhall be explained in the following chapter, that the three primary planets Saturn, Jupiter, and the earth, which have fatellites revolving about them, are endued with a power of * Street, in Aftron. Carolin. caufm" 176 Sir I s A Ac N E w T o n's Book II. caufing bodies, in particular tliofe fatellites, to gravitate to- wards them with a force, which is reciprocally in the duplicate proportion of their diftances; and the planets are in all re- Ipedls, in which they come under our examination, fo fimilar and alike, that there is no reafon to queftion, but they have all the fame property. Though it be fufficient for the prefent purpofe to have it proved of Jupiter and Saturn only ; for thefe planets contain much greater quantities of matter than the reft, and proportionally exceed the others in power ^ But the influence of thefe two planets being allowed, it is evi- vident how die planets come to fhift continually their planes : for each of the planets moving in a difierent plane, the adtion of Jupiter and Saturn upon the reft will be oblique to the planes of their motion ; and therefore will gradually draw them into new ones. The fame adion of thefe two planets up- on the reft will caufe likewife a progreflive motion of the auhelion ; fo that there will be no neceflity of having recourfe to the other caufe for this motion, which was before hinted at^; viz, the gravitation of the planets towards the fun differing from the exaft reciprocal duplicate proportion of the diftan- ces. And in the laft place, the ad:ion of Jupiter and Saturn upon each other will produce in their motions the fame ine- qualities, as their joint adion produces in the reft. All this is effedled in the fame manner, as the fun produces the fame kind of inequalities and many others in the motion of the moon and the other fecondary planets ; and therefore will be left apprehended by what fliall be faid in the next chapter. ^ Sec Ciiap. f . § 9, Sec. •" In the foregoing page. 2 Thofe Chap. 2. PHILOSOPHY. 177 Tliofe Other Irregularities in the motion of the fecondary planets have place likewife here ; but are too minute to be obfervable : becaufe they are produced and reftified alternate- ly, for the mod part in the time of a lingle revolution ; whereas the motion of the aphelion and nodes, which conti- nually increafe, become fenfible in a long feries of years. Yet fome of thefe other inequalities are difcernible in Jupiter and Saturn, in Saturn chiefly j for when Juprter, who movies faPter than Saturn, approaches near to a conjunclion with him, his action upon Saturn will a little retard the motion of that pla- net, and by the reciprocal adion of Saturn he will himfelf be accelerated. After conjundion, Jupiter will again accelerate Saturn, and be Hkewife retarded in die fame degree, as before die firft was retarded and the latter accelerated. Whatever inequalities befides are produced in the motion of Saturn by the adion of Jupiter upon that planet, will be fufficiently rec- tified, by placing the focus of Saturn's ellipfis, which fhould otherwife be in the fun, in the common center of gravity of the fun and Jupiter. And all die inequalities in the mo- tion of Jupiter, caufed by Saturn^s adion upon him, are much lefs confiderable than the irregularities of Saturn's motion*. 7. This one principle therefore of the planets having a power, as well as .the fun, to caufe bodies to gravitate tov/ards them, which is proved by the motion of the fecondary pla- nets to obtain in fad, explains all the irregularities relating to •the planets ever obfcrved by aftronomers. * See Newton. Princ. Lib.III. prop. 13; A a 8. Sir 178 Sir Isaac Newton's Book It. 8. Sir Isaac Newton after tliis proceeds to make an improvement in aftronomy by applying this tlieory to the far- ther correftion of their motions. For as we have here obferv- ed the planets to poffefs a principle of gravitation, as well as the fun ; fo it will be explained at large hereafter, that the third law of motion, which makes adion and reaftion equal, is to be applied in this cafe "* ; and that the fun does not only attradl each planet, but is it felf alfo attracted by them ; the force, wherewith the planet is aded on, bearing to the force, wherewith the llin it felf is adled on at the fame time, the proportion, which die quantity of matter in the fun bears to the quantity of matter in the planet. From the adion be- tween the fun and planet being thus mutual Sir Isaac Newton proves that the fun and planet will delcribe about their common center of gravity fimilar ellipfis's ; and then that the tranfverfe axis of the ellipfis defcribed thus about the move- able fun, will bear to the tranfverfe axis of the ellipfis, which would be defcribed about the fun at reft in the fame time, the fame proportion as the quantity of folid matter in the fun and planet together bears to the firft of two mean proportionals be- tween this quantity and the quantity of matter in the fun only K p. Above, where I fhewed how to find a cube, that fliould bear any proportion to another cube " , the lines F T and T S are two mean proportionals between E F and F G ; and counting from EF, F T is called die firft, and F S the fe- cond of thofe means. In numbers thefe mean proportionals fChap, ;-. § 10. fc Princ.LibJ. proji. 60, f Book L Chap. 2, § So. ars Chap. 2, PHILOSOPHY. 179 are tlius found. Suppofe A and B two numbers, and it be required to find C the firfl:, and D the fecond of the two mean proportionals between them. Firft A C multiply A by it felf, and the produd: multiply b D - by B ; then C will be the number which in arith- metic is called the cubic root of tliis laft product ; that is, the number C being multiplied by it felf, and the product again multiplied by the fame number C, will produce the produdt above mentioned. In like manner D is the cubic root of the produd of B multiplied by it felf, and the pro- duce of that multiplication multiplied again by A. 10. It will be asked, perhaps, how tliis ccrredion can be admitted, when the caufe of the motions of the planets was before found by fuppofing the fun the center of the power,* which acted upon them : for according to the prelent correc- tion this power appears rather to be direfted to their common center of gravity. But whereas the fun was at firft conclu- ded to be the center, to which the power acting on the planets was diredled, becaufe the fpaces defcribed round the fun in equal times were found to be equal ; fo Sir I s a a c Newton proves, that if the fun and planet move round their common center of gravity, yet to an eye placed in tlie planet, the fpa- ces, which will appear to be defcribed about the fun, will have the fame relation to the times of their dcfcription, as the real {paces would have, if the fun were at reft \ I farther aflerted, tlia.t, fuppofing the planets to move round die fun at reft, » Princ. phik)f. Lib. T. prop. ;S. coroU. 3. A a X and i8o Sir Isaac Newton's BookIL and to be attraclcd by a power, which every where iLould adt with degrees of ftrength reciprocally in the duplicate proportion of the diftances ; then the periods ot the planets muft obfcrve the fame relation to their diflances, as aftrono- mers find them to do. But here it muft not be fuppofed, that the obfervations of aflronomers abfolutely agree without any the leall difference ; and the prefent corredlion will not caufe a deviation from any one aftronomer's obfervations, fo much as they differ from one another. For in Jupiter, where this corredion is greateft, it hardly amounts to the gooo*'^ part of the whole axis. II. Upon this head I think it not improper ta mention a refle6lion made by our excellent author upon thefe fmall -in- equaHties in the planets motions ; which contains under it a very ftrong philofophical argument againft the eternity of the world. It is this, that thefe inequalities mufl: continually in- creafe by flow degrees, till they render at length the prefent frame of nature unfit for the purpofes, it now ferves \ And a more convincing proof cannot be defired againfl the pre- fent conftitution's having exifted 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 reprefented even as impious, and as no lefs than cafling a refledion upon the wifdom of the author of nature, for framing a perifii- able work. But I think fo bold an affertion ought to have been made with fingular caution. For if this remark upon the increafing irregularities of the heavenly motions " Newt, Optics, png. 37 S, i be Chap; 3. P H I L O S O P H Y. tSr be true in fa£l, as it really is, the imputation muft return' upon the afferter, that this does detradl from the divine vvifdom. Certainly we cannot pretend to know all the- omnifcient Creator's purpofes in making this world, and tlierefore cannot undertake to determine how lono; he de- figned it fliould laft. And it is fufficient, if it endure: die time intended by the author. The body of every ani- mal fhews the unlimited wifdom of its author no lefs, nay- in many refpedts more, than the larger frame of nature ;; and yet we fee, they are all defigned to laft but a fmalli fpace of time. 1 1. There need nothing more be faid of the primary pla- nets ; the motions of the. fecondary fhall be next confidered;. Chap. IIL Of the motion of the MOON and the other SECONDARY PLANETS. THE excellency of this philofophy fufficiently appears from its extending in the manner, which has been re- lated, to the minuteft circumftances of the primary planets motions ; which neverthelefs bears no proportion to the vaft fuccefs of It in the motions of the fecondary ; for it not only accounts for all the irregularities, by which their motions were knowji to be difturbed, but has difcovered others fo complicat- ed, that aftronomers were never able to diflinguifli them, and reduce them under proper heads j but thefe were only to be found 1 8 2 Sir I s A A c N E w T ON^'s Book II. found out from their caufes, which this philofophy has brought to hght, aiid has fhewn the dependence of thefe inequalities upon fuch caufes in fo perfed a manner, that we not only learn from thence in general, what thofe inequalities are, but are able to compute the degree of them. Of this Sir I s. N e w t o n has given feveral fpecimens, and has moreover found means to reduce the moon's motion fo 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 ex- adly, as the places of the primary planets themfelves, which is much beyond what the greateft aitronomers could ever effect, i. T H E firft thing demonflrated of thefe fecondary planets is, that they are drawn towards their refpedive primary in the fame manner as the primary planets are attraded by the fun. That each fecondary planet is kept in its orbit by a power pointed towards the center of the primary planet, about which the fecondary revolves ; and that the power, by which the fecondaries of the fame primary are influenced, bears the fame relation to the diftance from the primary, as the power, fcy which the primary planets are guided, does in regard to the diftance from the fun ^ This is proved in the fatellites of Jupiter and Saturn, becaule they move in circles, as far as we can obferve, about their refpedive primary with an equable courfe, the refpedive primary being tlie center of each or- bit : and by comparing the times, in which the different la- tellites of the fame primary perform their periods, they are * Newton. Price. Lib. IIL prop. j. found Chap. 3- PHILOSOPHY. 189 found to obferve the fame relation to tlie diftances from their primary, as the primary planets obferve in rcfpect of their mean diftances from the fun \ Here thefe bodies moving in circles with an equable motion, each fitcUite paifes over e- qual parts of its orbit in equal portions of time ; confequent- ly tlie line drawn from tlie center of the orbit, that is, fronis the primary planet, to the fatellite, v/ill pals over equal fpa- ces along with the fatellite in equal portions of time; which; proves the power, by which each fatellite is held in its orbit^- to be pointed towards the primary as a center ^ It is alfo ma^- nifeft that tlie centripetal power, which carries a body in a^ circle concentrical with tlie power, a6ls upon the body at alL times with the fame flrength. But Sir I s a a c Newton de^> monftrates that, when bodies are carried in different circles by centripetal powers diredled to the centers of thofe circles, theri- the degrees of ftrength of thofe powers are to be compared by confidering the relation between the times, in which the bo- dies perform their periods through thofe circles '^ ; and in par- ticular he (hews, that if the periodical times bear that relation^, which I have jufl: now afferted the fatellites of the fame pri- mary to obferve ; then the centripetal powers are reciprocally in the duplicate proportion of the femidiameters of the circles, . or ir; Jiat proportion to the diftances of the bodies from die centers '^. Hence it follows that in the planets Jupiter and Saturn, the. centripetal power in each decreases with the in- creafe of difiance, in the fam.e proportion as the centripetal ■' Newton. Funs. Lib. III. pag.}9o, 39 3. com- I <" Princ. philof. Lib, L prop.4; pared with pag. 393. d Ibid, coroll, tBookl. Ch. ?, §15; i ■' power 1 84 Sir I s A A c N E w T o n's Book IL power appertaining to the fun decreafes with the increafe of diftance. I do not here mean that this proportion of the cen- tripetal powers holds between the power of Jnp'ter at any di- ftance compared widi the power of Saturn at any other di- ftance ; but only in the change of ftrength of the power be- longing to the fame planet at different diftances from him. Moreover what is here dilJcovered of die planets Jupiter and Saturn by means of the different fatellites, which revolve round each of them, appears in the earth by the mcon alone ; becaufe fhe is found to move round the earth in an ellipHs af- ter the fame manner as the primary planets do about the fun ; excepting only fome fmall irregularities in her motion, the caufe of which will be particularly explained in what follows, whereby it will appear, that they are no objedion againft the earth's ading on the moon in the fame manner as the fun ads on the primary planets ; that is, as the other primary planet? Tupiter and Saturn ad upon their fatellites. Certain- ly iince.thefe irregularities canbe otherwife accounted for, we .ought not to depart from that rule of indudion fo neceffary in philofophy, that to like bodies like properties are to be at- tributed, where no reafon to the contrary appears. We can- not therefore but afcribe to the earth the fame kind of adion upon the moon, as the other primary planets Jupiter and Sa- turn have upon their fatellites ; which is known to be yery exadly in the proportion ailigned by the metiiod of compar- ing the periodical times and diftances of all the fatellites, which move about the fame planet ; this abundantly compenfating -Gur not being near -enough to obferve tlie exad figure of their orbits. For if the little deviation ot the moon's 01 bit from Chap. 3. PHILOSOPHY. 185 orbit from a true permanent ellipiis arofe from the adlion of the earth upon the moon not being in the exa<3; reciprocal dupH- cate proportion of the diftance, were another moon to revolve about the earth, the proportion between the periodical tim.es of this new moon, and the prefent, would difcover the devia- tion from the mentioned proportion much more manifeftly. 3. By the number of latellites, which move round Jupiter and Saturn, the power of each of thefe planets is meafured in a great diverlity of diftance ; for the diftance of the outermoft fatellite in each of thefe planets exceeds feveral times the dift- ance of the innermoft. In Jupiter the aftronomers have ufually placed the innermoft fatellite at a diftance from the center of that planet equal to about ^~ of the femidiameters of Jupiter's body, and this fatellite performs its revolution in about i day 18^ hours. The next fatellite, which revolves round Jupiter in about 3 days 13^ hours, they place at the diftance from Jupiter •of about 9 of that planet's femidiameters. To the third fa- tellite, which performs its period nearly in 7 days 3 - hours, they aflign the diftance of about 14 j femidiameters. But the outermoft fatellite they remove to zs\ femidiameters, and this fatellite makes its period in about 1 6 days 16'- hours ^ In Saturn there is ftill a greater diverftty in the diftance of the feveral fatellites. By the obfervations of the late C a s s i n r, a celebrated aftronomer in France, who firft difcovered all theie fatellites, except one known before, the innermoft is diftant about 4-^ of Saturn's femidiameters h-om his center, and re- ^ Newt. Pr!nc. philof. Li'o. III. psg 390. B b solves i86 Sir Isaac Newton'^s BookIL volves round in about i day 1 1 j- hours. The next fatellite is diftant about y ^ femidiameters, and makes its period in a- bout z days 17-^ hours. The third is removed to the dift- ance of about 8 femidiameters, and performs its revolution in near 4, days 1 1 ^ hours. The fourth fatelHte difcovered firft by the great Huygens, is near 18 j femidiameters, and> moves round Saturn in about i s days zzj hours. The out- ermoft is diftant ^6 femidiameters, and makes its revolution^ in about 79 days 7^ hours \ Befides thefe fatellites, there, belongs to the planet Saturn another body of a very fingular kind. This is a fhining, broad, and flat ring, which encom- pafles the planet round. The diameter of the outermoft verge of this ring is more than double the diameter of Saturn. Huygens, who firft defcribed this ring, makes the whole diameter thereof to bear to the diameter of Saturn the pro- portion of 9 to 4. The late reverend Mr. Pound makes the proportion fomething greater, viz. that of 7 to 3 . The di- ftances of the fatellites of this planet Saturn are compared by C A s s I N I. to the diameter, of the ring. His numbers I have • reduced to thofe above, according to Mr. Pound's propor- tion between the diameters of Saturn and of his ring. As this ring appears to adhere no where to Saturn, fo the dift- ance of Saturn from the inner edge of the ring feems rather greater than die breadth of the ring. The diftances, which have here been given, of the feveral fatellites, both for Jupiter and Saturn, may be more depended on in relation to the proportion,, which thofe belonging to the fame primary planet; » Newt. Pnnc. phCof. Lib. III. pag. 351, 391. bear. Chap. 3. PHILOSOPHY. 187 bear one to another, than in refpecfl: to the very numbers, that have been here fet down, by reafon of the difficulty there is in meafuring to the greateft exad:nefs the diameters of the pri- mary planets ; as will be explained hereafter, when we come to treat of telefcopes *. By the obfervations of the foremen- tioned Mr. Pound, in Jupiter the diftance of the innermoft fatellite fhould rather be about (5 femidiajneters, of the fccond ^{y of the third 15*, and of the outermoft x6~'; and in Sa- turn the diftance of the innermoft fatellite 4 femidiametersj of the next 67. of the third 8^, of the fourth io4, and of the fifth 5-9 "". However the proportion between the diftances of the fatellites in the fame primary is the only thing necef- fary to the point we are here upon. 4. B u T moreover the force, wherewith the earth afts in different diftances, is confirmed from the following confider^- ation, yet more exprefly than by the preceding analogical reafoning. It will appear, that if the power of the earth, by which it retains the moon in her orbit, be fuppofed to adl at all diftances between the earth and moon, according to the fore- mentioned rule ; this power will be fufficient to produce up- on bodies, near the furface of the earth, all the effects afcribcd to the principle of gravity. This is difcovered by tlie fol- lowing method. Let A ( in fig. 94. ) reprefent 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 felf to move with die velocity, it has in the point B, it » Book III, Ch.4, * Nc,\t, Princ. phi'.of. Lib. III. pag. ^gu * Ibid. pa^'. 392. Bb X would 1 88 Sir Isaac Newton's BookII. would leave the orbit, and proceed right forward in the line BE, which touches the orbit in B. Suppofe the moon would upon this condition move from B to E in the fpace of one mi- nute of time. By the adion of the earth upon the moon, where- by it is retained in its orbit, the moon will really be found at the end of this m.inute in the point F, from whence a ftraight line drawn to Afhall make the fpace BF A in the circle equal to the triangular fpace B E A ; fo that the moon in the time wherein it would have moved from B to E, if left to it felf, has b een impelled towards the earth from E to F. And when the time of the moon's pafling from B to F is fmall, as here it is only one minute, the diftance between E and F fcarce differs from the fpace, through which the moon would defcend in the fame time, if it were to fall diredly down from B toward A without any other motion. A B the diftance of the earth and moon is about do of the earth's femidiameters, and the moon completes her revolution round the earth in about 17 days 7 hours and 45 minutes: therefore the fpace EF will here be found by computation to be about 1 6-^ feet. Confequently, if the power, by which the moon is retained in its orbit, be near the furface of the earth greater, than at the diftance of the moon in the duplicate proportion of that diftance ; the number of feet, a body would defcend near the furface of the earth by the a6lion of this power upon it in one minute of time, would be equal to 16 1 multiplied twice into the num- ber do, that is, equal to ySo ^^o. But how faft bodies fall near the furface of the earth may be known by the pendulum ' ; and a See Book I. Ch.i, § 60,64. by Chap. 3- PHILOSOPHY. 189 by the exadleft; experiments they are found to defcend the fpace of i6i feet in a fecond of time; and the fpaces defcribed by falHng bodies being in the duplicate proportion of the times of their fall *, the number of feet, a body would defcribe in its. fall near the furface of the earth in one minute of time, will be equal to 1 6 7 twice multiplied by (5o, the lame as would, be caufed by the power which ads upon the moon. y. In this computation the earth is fuppofed to be at reft, whereas it would have been more exa6t to have fup- pofed it to move, as well as the moon, about their com- mon center of gravity ; as will eafily be underftood, by what has been faid in the preceding chapter, where it vi^as fhewn, that the fun is fubjedted to the Hke motion about the com- mon center of gravity of it felf and the planets. The ac- tion of the fun upon the moon, which is to be explain'd in what follows, is likewife here negledled : and Sir I s a a c Newton fhews, if you take in both thefe confiderations, the prefent computation will beft agree to a fomewhat great- er diftance of the moon and earth, viz. to 60 { femidiame- ters of the earth, which diftance is more conformable to aftronomical obfer\'ations. 6. These computations afford an additional proof, that the a6lion of the earth obferves the fame proportion to the- diftance, which is here contended for. Before I faid, it was reafonable to conclude fo by indudion from the pla- » Bopkl. Ch,2. § 17, nets I po Sir I s A A c N E w T o n's Book IL nets Jupiter and Saturn; becaufe they adt in that manner. But now the fame thing will be evident by drawing no other confequence from what is feen in thofe planets, than that the power, by which the primary planets adt on tlieir fecondary, IS extended from the primary through the whole interval be- tween, fo that it would a£t in every part of the intermediate Jpace. In Jupiter and Saturn this power is fb far from being confined to a fmall extent of diftance, that it not only reaches to feveral fatellites at very different diftances, but alfo from one planet to the other, nay even through the whole plane- tary fyftem '. Confequently there is no apper-rance of reafon, why this power fhould not ad: at all diftances, even at the very furfaces of thefe planets as wel as farther off. But from hence it follows, that the power, which retains the moon in her orbit, is the fame, as caufes bodies near the furface of the earth to gravitate. For lince the power, by which the earth adls on die moon, wUl caufe bodies near the furface of the earth to defcend with all the velocity they are found to do, it is certain no other power can ad upon them befides; becaufe if it did, they muft of neceflity defcend fwifter. Now from all this it is at length very evident, that the power in the earth, which we call gravity, ex- tends up to the moon, and decreafes in the duplicate pro- -portion of the increafe -of the diftance from the earth. 1. This finiflies the difcovcries made in the adion of 5the primary planets upon their fecondary. The next thing » SeeCh, II. § 6. to Chap. 3- PHILOSOPHY. 191 to be fliewn is, that the fun ads upon them likewife : for this purpofe it is to be obferved, that if to the motion of the fatelUte, whereby it would be carried round its primary at reft, be fuperadded the fame motion both in regard to velocity and diredion, as the primary it felf has, it will defcribe about the primary the fame orbit, with as great regularity, as if the primary was indeed at reft. The caule of this is that law of motion, which makes a body near the furface of tlie earth, when let fall, to defcend perpendicularly, though the earth be in fo fwift a motion, that if the falling body did not partake of it, its defcent would be remarkably oblique; and that a bo- dy projeded deicribes in the moft regular manner the fame parabola, whether projeded in the diredion, in which the earth moves, or in the oppoftte diredion, if the projed- ing force be the fame ^ . From this we learn , that if the fatellite moved about its primary with perfed re- gularity, befides its motion about the primary, it would participate of all the motion of its primary; have the lame progreflive velocity, with which the primary is car- ried about the fun; and be impelled with die lame velo- city as the primary towards the fun, in a diredion parallel to that impulfe of its primary. And on the contrary, the want of either of thefe, in particular of the impulle to- wards the fun, will occafton great inequalities in the mo- tion of the fecondary planet. The inequalities, which would.: arife from the abfence of this impulfe towards the fun arft- * The fccond of the laws of motion laid down in Book I. Ch.J. fo 192 Sir Is A AC Newton's BookIL fo great, that by the regularity, which appears in the mo- tion of the fecondary planets, it is proved, that the fun com- municates the fame velocity to them by its adtion, as it gives to their primary at the fame diftance. For Sir I s a a c N e w- TON informs us, that upon examination he found, that if any of the fatellites of Jupiter were attracted by the fun more or lefs, than Jupiter himfelf at the fame diftance, the orbit of that fatellite, inftead of being concentrical to Ju- piter, muft have its center at a greater or lefs diftance, than the center of Jupiter from the fun, nearly in the fubduplicate proportion of the difference between the fun's adion upon the fatellite, and upon Jupiter ; and therefore if any fatel- lite were attraded by the fun but ~ part more or lels, than Jupiter is at the fime diftance, the center of the orbit of that fatellite would be diftant from the center of Jupiter no lefs than a fifth part of the diftance of the out- ermoft fatellite from Jupiter ^ ; which is almoft the whole diftance of the innermoft fatellite. ;By the like argument the fatelHtes of Saturn gravitate towards the fun, as much -as Saturn it felf at the fame diftance; and the moon as -much as the eartli. 8. Thus is proved, that the fun ai3:s upon the fecon- ,dary planets, as much as upon the primary at the fame .diftance : but it was found in the laft chapter, that the •adlion of the ftin upon bodies is reciprocally in the du- jilicate proportion of the diftance ; therefore the fecondary * NcwtohrPiinc. philof. Lib. HI. prop. <3. pag. 401. planets Chap. 3. PHILOSOPHY. 193 planets being fometimes nearer to the fun than the pri- mary, and fometimes more remote, they are not alway aded upon in the fame degree with their primary, but when nearer to the fun, are attraAed more, and when far- ther diftant, are attracted lefs. Hence arife various inequali- ties in the motion of the fecondary planets \ 9. S o M E of thefe inequalities would ta ke place, tliough the moon, if undifturbed by the fun, 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 ob- hque lituation of the moon's orbit. One of the firfl kind is, that the moon is caufed fo to move, as not to defcribe equal fpaces in equal times, but is continually accelerated, as fhe paiTes 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 conlider not fo much the ab- Iblute, as the apparent motion of the moon in refpedl to us. 10. The principles of aftronomy teach how to diftin- guifh thefe two motions. Let S (in fig. 9 f.) reprefent the fun, A the earth moving in its orbit B C, D E F G the moon's orbit, the place of the moon H. Suppofe the earth to have moved from A to I. Becaufe it has been fhewn, that the moon partakes of all the progrefTive motion of the earth ; and likewife that the fun attradls both the earth and moon equal- ly, when they are at the fame diftance from it, or that the mean adlion of the fun upon the moon is equal to its adioii * Newton's Princ. philof. Lib. III. prop. 22, 2j. C c upon ip4 Sir Isaac Newton's Book IL upon the earth : we muft therefore confider the earth as car- rying about with it the moon's orbit ; fo that when the earth is removed from A to !_, the moon's orbit fhall like- wife be removed from its former lituation into that denoted by KLMN. But now the earth being in I, if the moon were found in O, {6 that 01 fhould be parallel to HA, though the moon would really have moved from H to 0, yet it would not have appeared to a fpedator upon the earth to have moved at all, becaufe the earth has moved as much it felf ; fo that the moon would ftill appear in the fame place with refpedl to the fixed ftars. But if the moon be obferved in P, it will then appear to have moved, its apparent motion being meafured by the angle under DIP. And if the angle under PIS be lefs than the angle under HAS, the moon will have approached nearer to its conjundion with the fun. II. To come now to the explication of the mentioned inequality in the moon's motion : let S (in fig. pd.) reprefent the fun, 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 fame diftance from the fun, as the earth is. In this cafe there- fore they will both be equally attracted, the earth in the dire- d:ion A S, and the moon in the diredion C S. Whence as the earth in moving round the fun is continually defcending to- ward it, fo the moon in this fituation muft in any equal por- tion of time defcend as much ; and therefore the pofition of the line AC in refpeft of AS, and the change, which the moon's motion produces in the angle under CAS, will not be altered by the fun. II. But Chap 3: PHILOSOPHY. 195 JZ. But now as fbon as ever the moon is advanced from the quarter toward the new or conjundion, fuppofc to G, the adion of the fun upon it will have a different effed:. Here, were the fun's adlion upon the moon to be appHed in the di- redion G H parallel to A S, if its adion on the moon were equal to its adion on the earth, no change would be wrought by the fun on the apparent motion of the moon round the earth. But the moon receiving a greater impulfe in G than the earth receives in A, were the fun to ad in the diredion GH, yet it would accelerate the defcription of the fpace DAG, and caufe the angle under G A D to decreafe fafter, than otherwife it would. The fun's adion will have this effed upon account of the obliquity of its diredion to that, in which the earth attrads 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 muft neceffarily be impelled toward D. Again, becaufe the fun does not ad in the diredion G H pa- rallel to S A, but in the diredion GS oblique to it, the fiin's adion on the moon will by reafon of this obliquity farther con- tribute to the moon's acceleration. Suppofe the earth in any fhort fpace of time would have moved from A to I, if not attradedby the fun ; the point I being in the ftraight line CE, which touches the earth's orbit in A. Suppofe the moon in the fame time would have moved in her orbit from G to K, and befides have partook of all the progreffive motion of the earth. Then if K L be drawn parallel to A I, and taken equal to it, the moon, if not attraded by the fun, would be found C c i in ^ 196 Sir Isaac Newton's Book II. in L. But the earth by the fun's adlion is removed from I. Sup- pofe it were moved dow^n to M in the hne IMN parallel to S A, and if the moon w^ere attrad;ed but as much, and in the fame direction, as the earth is here fuppoffd to be attraded, fo as to have defcended during the fame time in the line L O, parallel alfo to AS, down as far as P^ till LP were equal to IM, the angle under PMN would be equal to that under LIN, that is, the moon will appear advanced no far- ther forward, than if neither it nor the earth had been fubjedl to the fun's adlion. But this is upon the fuppolition, that the adlion of the fun upon the moon and earth were equal ; whereas the moon being adled upon more than the earth, did the fun's adion draw the moon in the line L O parallel to A S, it would draw it down fo far as to make L P greater than I M ; whereby the angle under PMN will be rendred lefs, than that under LIN. But moreover, as the fun draws the earth in a direction oblique to I N, the earth will be found in its orbit fomewhat fhort of the point M ; however the moon is attracted by the fun ftill more out of the line LO, than the earth is out of the line I N ; therefore this obli- quity of the fun's adion will yet farther diminifh the angle under PMN. 13. Thus the moon at the point G receives animpulfe from the fun, whereby her motion is accelerated. And the fun producing this effed in every place between the quarter and the conjundion, 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 Ghap. 3. PHILOSOPHY. 197 of v^elocity in its orbit, the fpaces, which are defcribed in equal times by the Hne drawn from the earth to the moon, will not be every where equal, but thofe toward the conjun<3:ion . will be greater, than thofe toward the quarter. But now in the moon's paffage from the conjundion D to the next quarter the fun's adion will again retard the moon, till at the next quarter in E it be reftored to the firft velocity, which it had inc. 14.. Again as the moon moves from E to the full or op- polition to the fun in B, it is again accelerated, the deficiency of the fun's adion upon the moon, from what it has upon the earth, producing here the fame effed as before the excefs of its adion, - Confi,der the moon in Q^ moving from E towards B. Here if the moon were attraded by the fun in a diredion parallel to AS, yet being aded on lefs than the earth, as. the earth defcends toward the fun, the moon will in fome meafure be left behind. Therefore Q^F being ■ drawn pa- rallel to S B, a fpedator on the earth would lee the moon move, as if attraded from the point Q^ in the diredion Q_F with a degree of force equal to that, whereby the fun's adion on the moon falls fhort of its adion on the earth. But - the obliquity of the fun's adion has alfo here an effed. In the time the earth would have moved from A to I without the influence of the fun, let the moon have moved in its orbit from Q_ to R. Drawing therefore R T parallel to A I, and equal to the-: fame, for the like reafon as before, the moon by the motion of its orbit, if not at all attraded by the fun, muft be found in T ; and therefore_, if attraded in a diredion parallel to S A, would be ipS Sir Isaac N e w t o n's Book II. be in the line T V parallel to A S ; fuppofe in W. But the moon in Q^ being farther off the fun than the earth, it will be lefs attraded, that is, T \V will be lefs than I M, and if the line SM be prolonged toward X, the angle under XMW will be lefs than that under XIT. Thus by the fun's adion the moon's palTage from the quarter to the full would be ac- celerated, if the fun were to ad on the earth and moon in a diredion parallel to A S : and the obliquity of the fun's ac- tion will ftill more increafe this acceleration. For the adion of the fun on the moon is oblique to the line SA the whole time of the moon's paflage from Q_ to T, and will carry the moon out of the line T V toward the earth. Here I fup- pofe the time of the moon's paffage from Q. to T fo fliort, that it fhall not pafs beyond the line S A. The earth alfo will come a little fhort of the line I N, as was faid before. From thefe caufes the angle under XMW will be flill farther leffened. I j-. The moon in pafiing from the oppofition B to the next quarter will be retarded again by the lame degrees, as it is accelerated before its appulfe to the oppofition. Bccaufe this adion of the fun, which in the moon's paffage from the quarter to the oppofition caufes it to be extraordinarily accele- rated, and diminifhes the angle, which meafures its diftance from the oppofition ; will make the moon flacken its pace af- terwards, and retard the augmentation of the fame angle in its paffage from the oppofition to the following quarter ; that is, will prevent that angle from increafing fo faff, as otherwife it would. And thus the moon, by the fun's adicn upon it, is twice accelerated and twice reftored to its firft velocity, every circuit Chap. 3- PHILOSOPHY. 199 circuit it makes round the earth. This inequality of the moon's motion about the earth is called by aftronomcrs its variation. 16. The next effect of the fun upon the moon isj tliat it gives the orbit of the moon in the quarters a greater de- gree of curvature, than it would receive from the aftion of the earth alone ; and on the contrary in the conjunction and oppofition the orbit is lefs infleded, 17. When the moon is in conjundion with the flm in the point D, the fun attrading the moon more forcibly than it dues the earth, the moon by that means is impelled lefs to- ward the earth, than otherwife it would be, and fo the orbit is lefs incurvated ; for the power, by which the moon is im- pelled toward the earth, being that, by which it is infleded from a rectilinear courfe, the lefs that power is, the lefs it will be infleded. Again, when the moon is in the oppofi- tion in B, farther removed from the flin than the earth is ; it follows then, though the earth and moon are both conti- nually defccnding to the fun, that is, are drawn by the fun toward it felf out of the place they would otherwile move into, yet the moon defcends with lefs velocity than the earth ; infomuch that the moon in any given fpace of time from its paffing the point of oppofition will have lefs approached the earth, than otherwife it would have done, that is, its orbit in refped of the earth will ap- proach nearer to a ftraight line. In the laft place, when the moon is in the quarter in F, and equally diftant from the fun as the earth, we obfervcd before, that the 200 Sir Isaac Newton's Book II. the earth and moon would defcend with equal pace to- ward the fun, fo as to make no change by that defcent in the angle under FAS; but the length of the line F A muft of necefTity be fhortned. Therefore the moon in moving from F toward the conjundion with the fun will be impelled more toward the earth by the fun's adlion, than it would have been by the earth alone, if neither the earth nor moon had been ad:ed on by the fun ; fo that by this additional impulfe the orbit is rendred more curve, than it would otherwife be. The fame effect will alfo be produced in the other quarter. 1 8. Another effect of the fun^s action, confequent upon this we have now explained, is, that though the moon un- difturbed by the fun might move in a circle having the earth for its center ; by the fun'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 firft appear fome difficulty, that the moon fliould come neareft to the earth, where it is leaft attracted to it, and be fartheft off when moft attracted. Which yet will appear evidently to follow from that very caufe, by confidering what was laft fhewn, that the orbit of the moon in the conjunction and oppofition is ren- dred lefs curve ; for the lefs curve the orbit of the moon is, the lefs w^ill the moon have defcended- 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 difturbance from that action, fince it would proceed in the line, which would touch its orbit in that place, it would recede Chap. 3- PHILOSOPHY. 201 recede continually from the earth ; and therefore if the power of the earth upon the moon, be fufficient to retain it at the fame diftance, this diminution of that power will caufe the diftance to increafe, though in a lefs degree. But on the other hand in the quarters, the moon, being preffed more to- wards the earth than by the earth's fingle action, will be made to approach it ; fo that in paffing from the conjunction or oppofition to the quarters the moon afcends from the earth, and in pafling from the quarters to the conjunction a nd oppofition it defcends again, becoming nearer in thefe laft mentioned places than in the other. 19. All thefe forementioned inequalities are of different degrees, according as the fun is more or lefs diftant from the earth ; greater when the earth is neareft the fun, and lels when it is farthefl: off. For in the quarters, the nearer the moon is to the fun, the greater is the addition to the earth's adion upon it by the power of the fun ; and in the conjun- (Slion and oppofition, the difference between the fun's a6lion upon the earth and upon the moon is likewife fo much the greater. 10. This difference in the diftance between the earth and the fun produces a farther effed; upon the moon's mo- tion ; caufing the orbit to dilate when lefs remote from the fun, and become greater, than when at a farther diftance. For it is proved by Sir I s a a c Newton, that the adion of the fun, by which it diminifties the earth's power over the moon, in the conjundion or oppofttion, is about twice as D d great 202 Sir I s A A c N E w T o n's Book IL great, as the addition to the eartli's adlion by the fun in the quarters ^ ; fo that upon the whole, the power of the earth upon the moon is diminiflied by the fun, and therefore is moft diminifhed, when the a6lion of the fun is ftrongeft : but as the earth by its approach to the fun has its influence lefl^en- edj die moon being lefs attracted will gradually recede from the earth ; and as the earth in its recefs from the fun recovers by degrees its former power, the orbit of the moon mufl: a- gain contrail. Two confequences follow from hence : the moon will be moft remote from the earth, when the earth is neareft the fun ; and alfo will take up a longer time in per- forming its revolution through the dilated orbit, than through the more contraded. a I. These ii-regularities the fun would produce in the moon, if the moon, without being adled on unequally by the fun, would defcribe a perfedl circle about the earth, and in the plane of the earth's motion ; but though neither of thefe fuppofitions obtain in the motion of the mioon, yet the fore- mentioned inequalities will take place, only with fome diffe- rence in refped to the degree of them ; but the moon by not moving in this manner is fubjed: to fome other inequalities at- fo. For as the moon defcribes, inftead of a circle concentri- cal to the earth, an ellipfis, with die earth in one focus, that ellipfis will be fubjedled to various changes. It can neither preferve conftandy the fame pofltion, nor yet the fame H-^ gure ; and becaufe the plane of this ellipfis is not the fame » Newton. Princ. Lib.I. prop. 66. coroU. 7, with Chap. 3- PHILOSOPHY. 203 with that of the earth's orbit, the Situation of the plane, where- in the moon moves, will continually change ; neither the line in wliich it interfeds the plane of the earth's orbit, nor the inclination of the planes to each other, will remain for any time the fame. All thefe alterations offer themfelves now to be explained. 11. I SHALL firft conlider the changes which are made in the plane of the moon's orbit. The moon not moving in the fame plane with the earth, the fun is feldom in the plane of the moon's orbit, viz. only when the Hne made by the common interfedion of the two planes , if produced* will pafs through the fun, as is reprefented in fig. 97. where S denotes the fun ; T the earth ; A T B the earth's orbit de- fcribed upon the plane of this fcheme ; C D E F the moon's orbit, the part C D E being raifed above, and the part C F E depreffed under the plane of this fcheme. Here the line C E> in which the plane of this fcheme, that is, the plane of the earth's orbit and the plane of the moon's orbit interfed: each other, being continued paffes through the fun in S. When this happens, die adlion of the fun is direded 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 fhall confider the prefent fcheme : for fuppofe the moon in G, and let a ftraight line be drawn from G to S, the fun draws the moon in the diredion 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 itfelf, if fufEciendy extended, will pafs dirough the fun. D d 1 But 204- Sir Isaac Newton's Book II. But in other cafes the obliquity of the fun's aftion to the plane of the orbit will caufe this plane continually to change. a^. Suppose in the firfl place, the line, in which the two planes interfed: each other, to be perpendicular to the line which joins the earth and fun. Let T (iniig.^8599,100,101.) reprefent the earth ; S the fun ; the plane of this fcheme the plane of the earth's motion, in which both the fun and earth are placed. Let A C be perpendicular to S T, which joins the earth and fun ; and let the line A C be that, in which the plane of the moon's orbit interfedls the plane of the earth's motion. To the center T defcribe in the plane of the earth's motion the circle A B C D. And in the plane of the moon's orbit defcribe the circle AECF, one half of which AEC will be elevated above the plane of this fcheme, the other half AF C as much deprefled below it. 24. Now fuppofe the moon to fet forth from the point A (in £g. 98. ) in the Jiredlion of the plane AEC. Here flie v/ill be continually drawn out of this plane by the adrion of the fim : for this plane AEC, if extended, will not pafs through the fun, but above it ; fo that the fun, by drawing the moon direftly toward it lelf, will force it continually more and more from that plane towards the plane of the earth's motion, in which it felf is ; caufing it to defcribe the line A K G H I, which will be convex to the plane AEC, and concave to the plane of the earth's motion. But here this power of tlie fun, which is faid to draw the moon toward the plane of the earth's motion, muft be underftood principally of fo much only of tlie m Chap. 3. PHILOSOPHY 205 the fun's a£lion upon the moon, as it exceeds the adion of the fame upon the earth. For fuppofe the preceding figure to be viewed by the eye, placed in the plane of that fcheme, and in tlie line C T A on the fide of A, the plane A B C D will appear as the ftraight line D T B, (in fig. i o 1.) and the plane A E C F as an- other ftraight line F E ; and the curve line A K G H I under the. form of the line T K G H I. Now it is plain, that the earth and moon being both attracted by the fun, if the fun's action up- on both was equally flrong, the earth T_, and with it the plane A E C F or line F T E in this fcheme, would be carried toward the fun with as great a pace as the moon, and therefore the moon not drawn out of it by the fun's adion, excepting only from the fmall obliquity of the dircdlion of this action upon the moon to that of the fun's a6lion upon the earth, , which arifes from the moon's being out of the plane of the earth's motion, and is not very confiderable ; but the adion of the fun upon the moon being greater than upon the earth, all the time the moon is nearer to the hui than the earth is, it will be drawn from the plane AEC or the line TE by that excefs, and made to defcribe the curve line AG I or T G I. But it is the cufLom of aftronomers, inftead of con- fidering the moon as moving in fuch 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 tlie moon is in the point A, its m.otion - is confidered as being in the plane AEC, in v/hofe direction it then eflaies to move ; and when in the point K ( in fig. 99. ) its motion is referred to the plane, which paffes through the earth, and touches the line AKGHI in the po'nt-K. Thus ths 2o6 Sir Isaac N e w t o n's Book II. the moon in pafTing from A to I will continually change the plane of her motion. In what manner this change proceeds, I fhall now particularly explain. ay. L ET the plane, which touches the line AKI in the point K (in fig. 9 9.) interfed the plane of tlie earth's orbit in the line L T M. Then, becaufe the line A K I is concave to the plane ABC, it falls wholly between that plane, and tLc plane which touches it in K ; fo that the plane M K L will cut the plane AEC, before it meets with the plane of the earth's motion ; fuppofe in the Une Y T, and the point A will fall between K and L. With a femidiameter equal to T Y or T L defcribe the femi- circle L Y M. Now to a fpeftator 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 femicircle L Y M. The earth's motion is performed in the plane of this fcheme, and to a fpeclator on the earth the fun will appear always moving in that plane. We may therefore refer the apparent motion of the fun to the circle A B C D, defcribed in this plane about the earth. Eut die points where this circle, in which the fun feems to move, interfedls the circle in which the moon is feen at any time to move, are called the nodes of the moon's orbit at that time. When the moon is feen moving in the cir- cle AEC D, the points A and C are the nodes of the orbit ; when ihe appears in the femicircle L Y M, then L and M are the nodes. Now here it appears, from what has been faid, tliat 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 4 M, is Chap. 3. PHILOSOPHY. 207 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. 1(5. Farther the angle, which tlie plane, wherein the moon at any time appears, makes with the plane of the eardi's motion, is called the inclination of the moon's orbit at that time. And I fhall now proceed to fhew, that this inclina- tion of the orbit, when the moon is in K, is lefs than when (he was in A ; or, that the plane L Y M, which touches the line of the moon's motion in K, makes a lefs angle with the plane of the earth's motion or with the circle A B C D, than the plane AE C makes with the fame. The femicircle L Y M interfedls the femicircle AEC in Y; and the arch AY is lefs- than L Y, and both together lefs than half a circle. But it is de- monftrated by the writers on that part of aftronomy, which is called the dodrine of the fphere, that when a triangle is made, as here, by three arches of circles AL, A Y, and YL, the angle under YAB without the triangle is greater than tlie angle under YLA within, if the two arches AY, YL taken together do not amount to a femicircle ; if the two arches make a com- plete femicircle, the two angles will be equal ; but if the two arches taken together exceed a {emicircle, the inner angle un- der YLA is greater than the other \ Here therefore the two arches AY and LY together being lefs than a femicircle, the- angle under ALY is lefs, than the angle under B AE. But from the dodrine of the fphere it is alfo evident, that the an- gle under ALY is equal to that, in which the plane of the » Menelai'Sphaeric. Lib. I. prop. 10, circle :2o8 Sir Isaac N e w t o n's Book II. circle L Y K M, that is, the plane which touches the line A K G H I in K, is incHned to the plane of the earth's motion ABC; and the angle under B A E is equal to that, in which the plane A E C is inclined to the fame plane. Therefore the inclina- tion of the former plane is lefs than the inclination of the latter. 17. Suppose now the moon to be advanced to the point iG (in fig. 100.) and in this point to be diftant 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 cafe the nodes will have receded yet more, and the inclination of the orbit be ftill more diminifhed : for fuppofe the line A K G H I to be touched in the point G by a plane pafTmg through the earth T : let the interfedion of this plane with the plane of the earth's motion be the line W T O, and the line T P its in- terfedion with the plane L K M. In this plane let the circle NGO be defcribcd with the femidiameter TP or NT cutting tlie 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 NGO, which touches it in G, will interfed the other touching plane between G and K ; that is, the point P will fall between thofe two points, and the plane continued to the plane of the earth's motion will pafs beyond L ; fo 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. Befides, the inclination of the plane NGO to the plane of the earth's motion ABC is lefs, than the inclination of the plane E K M to the fame ; for liere alfo the two arches L P and N P taken together are lefs than Chap. 3. PHILOSOPHY. 209 than a femicircle, each of tliefe arches being lefs than a quar- ter of a circle ; as appears, becaufe G N> the diftance of the moon in G from its node Nj is here fuppofed to be a quarter part of a circle. 28. After the moon is paffed beyond G, the cafe is altered ; for then thefe arches will be greater than quarters of the circle, by which means the inclination will be again increafed , tho' the nodes ftill go on to move the fame way. Suppofe the moon in H, (in fig. 1 o I. ) and that the plane, which touches the line A K G I in H, interfeds the plane of the earth's mo- tion in the Hne Q^TR, and the plane NGO in the line TV, and befides that the circle Q^H R be defcribed in that plane ; then, for the fame reafon as before, the point V will fall be- tween H and G, and the plane R V Q. will pafs beyond the laft plane O V N, caufing 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 leaft 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 fe- micircle ; confequently the angle under B Q^V will be greater, than that under B N ^^ 29. In the laft place, when the moon is by this attra- ction of the fun, drawn at lengdi into the plane of the earth's motion, the node will have receded yet more, and the incli- nation be fo much increafed, as to become fomewhat more than at firft : 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 E e .the 21 o Sir Is A AC Newton's Book II. the plane Q^V R and the plane A B C ; fo that the point I will fall between B and R \ and drawing I T W, the point W will be far- ther remov'd from A than Q. But it is evident, that the plane, which paffes through the earth T, and touches the line A G I in the point I, will cut die plane of the earth's motion ABCD in the line I T W, and be inclined to the fame in the angle im- der H I B ; fo that the node, which was firfl: in A, after having paffed into L, N and Q_, comes at laft into the point \V ; as the node which was at firft in C has paffed fucceffively from thence through the points M, O and R to I : but the angle under H I B, which is now the incHnation of the orbit to the plane of the ecliptic, is manifeftly not lefs than the angle under E C B or EAB, but rather fomething greater. 30. Thus the moon in the cafe before us, while it paf^ fes from the plane of the earth's motion in the quarter, till it comes again into the fame plane, has the nodes of its orbit continually moved backward, and the inclination of its orbit is at firfl: diminifhed, viz. till it comes to Gin fig. 1 00, which is near to its conjund:ion with the fun, but afterwards is increal^ ed again almoft by the fame degrees, till upon the moon's arrival again to the plane of the earth's motion, the inclina- tion of the orbit is reftored to fomething more than its firft magnitude, though the difference is not very great, because the points I and C are not far diftant from each other \ ^ Vid. Newt, Princ. Lib. I. prop. 66, coroU. lo. ^I. After. Chap. 3. PHILOSOPHY. an 3 r. After the fame manner, if the moon had depart- ed from the quarter in C, it fliould have defcribed the curve line C X W (in fig. 98.) between the planes AFC and ADC, which would be convex to the former of thofe planes, and concave to the latter ; fo that, here alfo, the nodes ihould continually recede, and the inclination of the orbit gradually diminifii more and more, till the moon arrived near its oppo- fition to the fun in X ; but from tliat time the inclination fhould again increafe, till it became a litde greater than at firft. This will eafily appear, by confidering, that as the adion of the fun upon the moon, by exceeding its adion upon the earth, drew it out of the plane A E C towards the llin, while the moon pafied from A to I ; fo, during its paflage from C to W, the moon being all that time farther from the fun dian the earth, it will be attracted lels ; and the earth, together with the plane AECF, will as it were be drawn from the moon, in fuch fort, that the path the moon defcribes fliall appear from the earth, as it did in the former cafe by the moon's being drawn away. 51. T H E s E are the changes, which the nodes and the in- clination of the moon's orbit undergo, when the nodes are in die quarters ; but when die nodes by their motion, and the motion of the fun together, come to be fituated between the quarter and conjunction or oppofition, their motion and the change made in the incHnation of the orbit are fomewhat dif- ferent. E e 1 3 3 . L E T 212 Sir Isaac Newton's BookIL 33. Let AGCH (in fig. 103.) be a circle defcribed in the plane of the earth's motion, having the earth in T for its center. Let the point oppofite to the fun be A, and the point G a fourth part of the circle diftant from A. Let the nodes of the moon's orbit be fituated in the line B T D, and B die node, falHng be- tween A, the place where the moon would be in the full> and G the place where the moon would be in the quarter. Suppofe B E D F to be the plane, in which the moon eilays to move, when it proceeds from the point B. Becaufe the moon in B is more diftant from the fun than the eardi, it fhall be lefs attraded by the fun, and fhall not defcend towards the fun fo faft as the earth : confequently it fhall quit the plane B E D F, which we fuppofe to accompany the earth, and de- fcribe the Hne BIK convex thereto, till fuch 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 fhall change its courfe, and the following part of the path it defcribes fhall be concave to the plane BED or BGD, and fhall continue concave to the plane B G D, till it crofTes that plane in L, juft as in tlie preceding cafe. Now I fay, while the moon is pafling from B to K, the nodes, contrary to what was found in the foregoing cafe, will proceed for- ward, or move the fame way with the moon ^ ; and at the fame time the inclination of the orbit will increafe K 54. When the moon is in the point I, let the plans MINpafs through the earth T, and touch the path of the * Vid. Newt. Princ. Li'j.III. prop, jc, p, 44?: * Ibid. Lib. I, prop. 66, coroll. 10; mooni Chap. 3- PHILOSOPHY. 213 moon in I, cutting the plane of the earth's motion in the Hne M T N, and the plane B E D in the line T O. Becaufe the line B I K is convex to the plane B R D, which touches it in B, the plane NIM mufl crofs the plane DEB, before it meets the plane C G B ; and therefore the point M will fall from B to- wards G, and the node of the mocti's orbit being tranflated from B to M is moved forward. g ^. I SAY farther, the angle under OMG, 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 BOD makes with the fame. This appears from what has been already explained ; becaufe the arches BO, O M are each lefs than the quarter of a circle, and therefore taken both together are lefs than a fe- micircle. 3d. Again, when the moon is come to the point K in its quarter, die nodes will be advanced yet farther forward,, and the incHnation of the orbit alfo more augmented. Hi- therto the moon's motion has been referred to the plane,, which paffing through tlie earth touches the path of the moon in die point, where the moon is, according to what was aflerted at the beginning of this difcourle upon the nodes, that it is the cuftom of aftronomers fo to do. But here in the point K no fuch plane can be found ; on the contra- ry, feeing the line of the mooa's motion on one fide the point K is convex to the plane BED, and on the other fide con-- cave to the fame, no plane can pafs through the points T and: Kj, but will cut the line B K L in that point. Therefore inftead 214 Sir Isaac Newton's BookIL of Rich a touching plane, we muft here make life of what is equivalent, the plane P K Q,, with which the line B K L fhall make a lels angle than with any other plane ; for this plane does as it were touch the line B K in the point K, fmce it fo cuts it, that no other plane can be drawn fo, as to pafs be- tween the line B K and the plane P K Q. But now it is evi- dent, that the point P, or the node, is removed from M to- wards G, that is, has moved yet farther forward j and it is likewife as manifeft, that the angle under K P G, or the in- clination of the moon's orbit in the point K, is greater than the angle under IMG, for the reafon fo often affigned. 37. After the moon has palled the quarter, the path of the moon being concave to the plane A G C H, the nodes, as in the preceding cafe, fhall recede, till the moon arrives at the point L ; which fhews, that confidering the whole time of the moon's pafTing from B to L, at the end of that time the nodes fhall be found to have receded, or to be placed back- warder, when the moon is in L, than when it was in B. For the moon takes a longer time in pafling from K to L, than in pailing from B to K ; and therefore the nodes continue to recede a longer time, than they moved forwards j fo tliat their recefs muft furmount their advance. 38. In the fame manner, while the moon is in its paflage from K to L, the inclination of the orbit fhall diminifh, till the moon comes to the point, in which it is one quarter part of a circle diftant from its node , fuppofe in the point R J and from that time the inclination fliall again increafe. Since Chap.3. philosophy. 215 Since therefore the inclination of the orbit increafes, wliile the moon is pailing from B to K, and diminiflies itfclf a- gain only, while the moon is paiTmg from K to R, and then augments again, till die moon arrive in L ; while the moon is pafling from B to L, the inclination of the orbit is much more increafed than diminifhed, and will be diflinguifhably greater,, when the moon is come to L, than when it fet out from B. 30. In like manner, while the moon is paffing from L on the other {ide the plane A G C H, the node fball advance for- ward, as long as the moon is between the point L and the next quarter ; but afterv/ards it fhall recede, till the moon come to pafs the plane A G C H again in the point V, between B and A : and becaufe the time between the moon's paiTmg from L to the next quarter is lefs, than the time between diat quar- ter and the moon's coming to the point V, the node fhall have more receded than advanced ; fo that the point V will be nearer to A, than L is to C. So alfo the incHnation of the orbit, when the moon is in V, will be greater, than when the moon was at L ; for this inclination increafes all the time the moon is between L and the next quarter -, it decreafes only w^hile the moon is pafling from this quarter to the mid way between the two nodes, and from thence increafes again du- ring the whole paffage through the other half of the way to the next node. 4,0. Thus we have traced the moon from her node in the quarter, and fhewn, that at every period of the moon the nodes will have receded, and thereby will have approached toward 21 6 Sir I s A A c N E w T o n's Book II. toward a conjundion with the fun. But this conjundion will be much forwarded by the viiible motion of the fun itfelf. In the laft fcheme the fun will appear to move from S to- w^ard W. Suppofe 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 reprefent the diftance of the line drawn between the nodes from the fun, when the moon is in V ; whereas the arch B A reprefented that diftance, when the moon was in B. This vifible motion of the fun is much greater, than that of the node ; for the fun appears to revolve quite round each year, and the node is near 1 9 years in mak- ing one revolution. We have alfo feen, that when the node was in the quadrature, the inclination of the moon's orbit de- creafed, till the moon came to the conjundion, or oppoH- tion, according to which node it fet out from ; but that af- terwards it again increafed, till it became at the next node ra- ther greater than at the former. When the node is once re- moved from the quarter nearer to a conjuncSlion with the fun, the inclination of the moon's orbit, when the moon comes into the node, is more fenfibly greater, than it was in the node preceding ; the inclination of the orbit by this means more and more increaling till the node comes into conjundion with the fun ; at which time it has been fhewn above, that the fun has no power to change the plane of the moon's motion ; and confequently has no effect either on the nodes, or on the in- clination of the orbit. 41. As foon as the nodes, by the adion of the fun, are got out of conjundion toward the other quarters, they begin again CHAP.ci. PHILOSOPHY. 217 again to recede as before ; but the inclination ol' the orbit in the appulfe of the moon to each fucceeding node is lefs than at the preceding, till the nodes come again into the quar- ters. This will appear as follows. Let A (in fig. 1 04.) re- prefent one of the moon's nodes placed between the point of oppofition B and the quarter C. Let the plane A D E pals through the earth T, and touch the path of the moon in A. Let the line AFGH be the path of the moon in her paffage from A to H, where fhe croffes 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 fhe is in the quarter ; and after this, between G and H, the fame 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 eafily be con- ceived from what has been before fo largely explained. But the moon is longer in pafTing from A to G, than from G to H ; therefore the nodes recede a longer time, than they proceed ; confequently 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 decreafe, till the moon is arrived to the point F, in the middle between A and H. Through the paffage between F and G the incli- nation will increafe, but decreafe again in the remaining part of the paffage from G to H, and confequently at H muff be lefs than at A. The Uke effeds, both in refpecl to the nodes and inclination of the orbit, will take place in the following paffage of the moon on the other fide of the plane A B E C, from H, till it comes over that plane again in I. F f 41. Thus 2, 1 8 Sir Isaac N e w t o n's Book II. 4,1, Thus the inclination of the orbit is greateil, when the Hne drawn between the moon's nodes will pafs through the fun -J and leaft, when this line lies in the quarters, efpecial- ly if the moon at the fame time be in conjunction with the fun, or in the oppolition. In the firfl: of thefe cafes the nodes have no motion, in all others, the nodes will each montli have receded : and this regrefTive motion will be greateft, when the nodes are in the quarters ; for in that cafe the nodes have no progreffive motion during the whole month, but in all other cafes the nodes do at fome times proceed forward, viz. whenever the moon is between either quarter, and the node which is lefs diftant from that quarter than a fourth part of a circle. 4ag. 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 faid at the beginning of this chapter it appears, that the power of the earth on the moon a£ts in the reciprocal duplicate proportion of the diftance : therefore the moon, if undifturbed by the fun, would mov^e round the earth in a true ellipfis, and the line drawn from the earth to the moon would pafs over equal fpaces in equal portions of time. That this defcription of the fpaces is altered by the fun, has been already declared. It has alfo been fhewn, 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 with- out the fun. Now we muft pafs by thefe monthly changes, and confider the efied, which the fun will have in the diifer- ent Chap.3. philosophy. 2i«) ent finiations of the axis of the orbit in rcfped of that Ki- minary. 44. The aftion of the fun varies the force, wherewith tlie moon is drawn toward the earth ; in the quarters tlie force of the earth is diredly increafcd by the fun ; at the new and full the fame is diminiflied ; and in the interme- diate places the influence of the earth is fometimes aided, and fometimes leflened by the fun. In thefe intermediate places between the quarters and the conjunction or oppofition, the fun's adion is fo oblique to the adtion of the earth on the moon, as to produce that alternate acceleration and re- tardment of the moon's motion, which I obferved above to be ftiled the variation. But befides this effedl, the power, by which the earth attrad;s the moon toward itfelf, will not be at full liberty to ad: with the fame force, as if the fun '*' adled not at all on the moon. And tliis effe6l of the fun's ,/ a6lion, whereby it corroborates or weakens the adlion of the earth, is here only to be confidered. And by this influence of the fun it comes to pafs, that the power, by which the moon is impelled toward the earth, is not perfe6lly in the re- ciprocal duplicate proportion of the diftance. Confequently the moon will not defcribe a perfedl ellipfis. One particular, wherein the moon's orbit will differ from an ellipfls, con- fifts in the places, where the motion of the moon is perpen- dicular to the line drawn from itfelf to the earth. In an ellipfls, after the moon fliould have fet out in the direction perpendicular to this line drawn from itfelf to the earth, and at its greateft diftance from the earth, its motion would F f X again 220 Sir Isaac Newton's Book II. again become perpendicular to this line drawn between it- felf and the earth, and the moon be at its neareft diflance from the earth, when it fhould have performed half its pe- riod ; after performing the other half of its period its mo- tion would again become perpendicular to the foremention- ed line, and the moon return into the place whence it fet out, and have recovered again its greateft diftance. But the moon in its real motion, after fetting out as before, fometimes makes more than half a revolution, before its motion comes again to be perpendicular to the line drawn from itfelf to the earth, and the moon is at its neareft diftance ; and then performs more than another half of an intire revolution before its mo- tion can a fecond time recover its perpendicular diredion to the line drawn from the moon to the earth, and the moon arrive again to its greateft diftance from the earth. At other times the moon will defcend to its neareft diftance, before it has made half a revolution, and recover again its greateft di- ftance, before it has made an intire revolution. The place, where the moon is at its greateft diftance from the earth, is call- ed the moon's apogeon, and the place of the leaft diftance the perigeon. This change of the place, where the moon fucceffively comes to its greateft diftance from the earth, is called the motion of the apogeon. In what manner the fun caufes the apogeon to move, I lliall now endeavour to explain. 4^. Our author fhews, that if the moon were attract- ed toward the eartli by a composition of two powers, one of which were reciprocally in the duplicate proportion of the diftance from the earth, and the other reciprocally in Chap. 3- PHILOSOPHY. 221 in the triplicate proportion of the fame diftance ; then, though the line defcribed by the moon would not be in reality an ellipiis, yet the moon's motion might be perfedly explained by an ellipfis, whofe axis fhould be made to move round the earth ; this motion being in confequence, as aftro- nomers exprefs themfelves, that is, the fame way as the moon itfelf moves, if the moon be attracted by the fum of the two powers ; but the axis mufl: move in antecedence, or the con- trary way, if the moon be afted on by the difference of thefe powers. What is meant by duplicate proportion has been often explained ; namely, that if three magnitudes, as A, B, and C, are fo related, that the fecond B bears the fame pro- portion to the third C, as the firft A bears to the fecond B, then the proportion of the firft A to the third C, is the duplicate of the proportion of the firft A to the fecond B. Now if a fourth magnitude, as D,. be afllimed, to which C fhall bear the fame proportion as A bears to B, and B to C, then the proportion of A to D is the triplicate of the pro- portion of A to B. 46^. The way of reprefenting the moon's motion in this cafe is thus. T denoting the earth ( in fig. 105-, I o (5. ) fiippofe the moon in tlie point A, its apogeon, or greateft diftance from the earth, moving in the diredlion AF per- pendicular to A B, and aded upon from the earth by two fuch forces as have been named. By that power alone, which is reciprocally in the duplicate proportion of the diftance, if the moon fet out from the point A •>\'ith a proper degree of velocity, the ellipfis AMB may be de- fcribed. 222 Sir Is AAc Newton's BookIL fcribed. But if the moon be a6led upon by the fum of the forementioned powers, and the velocity of the moon in the point A be augmented in a certain proportion =" ; or if that velocity be diminiflied in a certain proportion, and the moon be adled upon by the difference of thofe powers; in both thefe cafes the line AE, which fhall be defcribed by the moon, is thus to be determined. Let the point M be that, into which the moon would have arrived in any given fpace of time, had it moved in the ellipfis A M B. Draw M T, and likewife CTD in fuch fort, that the angle under ATM fhall bear the fame proportion to the angle under AT C, as the velocity, with which the ellipfis A M B muft have been de- fcribed, bears to the difference between this velocity, and the velocity, with which the moon muft fet out from the point A in order to defcribe the path A E. Let die angle A T C be ta- ken toward the moon ( as in fig. I o 5". ) ii the moon be attrad- ed by the fum of the powers ; but the contrary way ( as in fig. 106^.) if by their difference. Then let the line AB be moved into the pofition C D, and the ellipfis A M B into the fituation C N D, fo that the point M be tranflated to L : then the point L fhall fall upon the path of the moon A E. 4,7. The angular motion of the line A T, wereby it is removed into the fituation CT, reprefents the motion of the xipogeon ; by the means of which the motion of the moon might be fully explicated by the ellipfis A MB, if the adlion of the fun upon it was directed to the center ot the earth, and ^ Wliat this proportion i", may be known from Coroil. i prop 44. Lib. I. Trine, philof. Newton. 2 reci- Chap. 3. PHILOSOPHY. 223 reciprocally In the triplicate proportion of the moon's diftance from it. But that not being fo, the apogeon will not move in the regular manner now defcribed. However, it is to be ob- ferved here, that in the firft of the two preceding cafes, where the apogeon moves forward, the whole centripetal power increafes fafter, with the decreafe of diftance, than if the intire power were reciprocally in the duplicate proportion of the diftance ; becaufe one part only is in that proportion, and the other part, which is added to this to make up the whole power, increafes fafter with the decreafe of diftance- On the other hand, when the centripetal power is the differ- ence between thefe two, it increafes lels with the decreafe of the diftance, than if it were {imply in the reciprocal dupU- cate proportion of the diftance. Therefore if we chufe to ex- plain the moon's motion by an ellipfis ( as is moft convenient for aftronomical ufes to be done, and by reafon of the fmall effect of the fun's power, the doing fo will not be attended with any fenfible error;) we may colled: in general, that when the power, by which the moon is attracted to the earth, by varying the diftance, increafes in a greater than in the du- plicate proportion of the diftance diminifhed, a motion in con- lequence muft be afcribed to the apogeon ; but that when the attraction increafes in a lefs proportion than that named, the apogeon muft have given to it a motion in antecedence '. It is then obferved by Sir I s. N e w t o n, that die firft of thefe cafes obtains, when the moon is in the conjunction and oppofitlcvii ; and the latter, when the moon is in the quarters : fo that in the firft the apogeon moves according to the order of the » Princ. Phil. Newt. Lib.I. ji»p. 4J-. Coroll. i» fign^3^ '2 24- Sir I s A A c N E w T o n's Book II. figns ■■, in the other, the contrary way '\ But, as was faid before, the difturbance given to the atlion of the earth by the fun in the conjundion and oppofition being near twice as great as in the quarters '', the apogeon will advance with a greater velocity than recede, and in the compafs of a whole revo- lution of the moon will be carried in confequence ". 4,8. It is fhewn in the next place by our author, that when the line A B coincides with that, which joins the earth and the fun, the progreflive motion of the apogeon, when the moon is in the conjundion or oppolition, exceeds the regrefliv^e in the quadratures more than in any other fitua- tion of the line A B ^. On the contrary, when the line A B makes right angles with that, which joins the earth and fun, the retrograde motion will be more confiderable " , nay is found fo great as to exceed the progrefiive ; fo that in this cafe the apogeon in the compafs of an intire revolution of the moon is carried in antecedence. Yet from the confi- derations in the lafi: paragraph the progreflive motion ex- ceeds the other ; fo that in the whole the mean motion of the apogeon is in confequence, according as aftronomers find. Moreover, the line A B changes its fituation with that, which joins the earth and fun, by fuch flow degrees, that the inequalities in the motion of the apogeon arifing from this laft confederation, are much greater than what arifes from the other'. » Pr. Phil. Newt. T.ib.I. prop. 66. Coroll. 7. ^ See§ 19. oF ihi.'; chapter. c PhiJ.Nat.Pr.Math Lib.I.prop.(S 5 cor.8. ■I IbiJ. Coroll. ' Ibid. f Ibid. 40. Far- Chap. 3. PHILOSOPHY. 22$ 49. F A R T H E Rj this unfteady motion in die apogeon is at- tended with another inequality in the motion of the moon, that it cannot be explained at all times by the fame ellipfis. The ellipfis in general is called by aftronomers an eccentric orbit. The point, in which the two axis's crofs, is called the center of the figure ; becaufe all lines drawn through this point within the ellipfis, from fide to fide, 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, thefe orbits are faid to be eccentric ; and where the diftance of the focus from this center bears the greateft proportion to the whole axis, that orbit is called the moft eccentric : and in fuch an orbit the diftance from the focus to the remoter ex- tremity of the axis bears the greateft proportion to the di- ftance of the nearer extremity. Now whenever the apo- geon of the moon moves in confequence, the moon's motion muft be referred to an orbit more eccentric, than what the moon would defcribe, if the whole power, by which the moon was adled on in its pafling from the apogeon, changed according to the reciprocal duplicate proportion of the di- ftance from the earth, and by that means the moon did de- fcribe an immoveable ellipfis j and when the apogeon moves in antecedence, die moon's motion muft be referred to an orbit lefs eccentric. In the firft of the two figures laft re- ferred to, the true place of the moon L falls without the orbit A M B, to which its motion is referred : whence the orbit ALE, truly defcribed by the moon, is lefs incurvated in the point A, than is the orbit A M B ; therefore the orbit A M B is more ob- long, and differs farther from a circle, than tlie ellipfis would, G g wliofe 22(5 Sir I s A A c N E w T o n's Book II. whofe curvature in A were equal to that of the Hne ALB, that is, the proportion of the diftance of the earth T fronx the center of the elHpfis to its axis will be greater in the el- lipfis A MB, than in the other ; but that other is the ellipfis, which the moon would defcribe, if the power ading upon it in the point A were altered in the reciprocal duplicate pro- portion of the diftance. In the fecond jfigure, when the apo which have appeared within the laft ifo years may be feen in the Philofbphical tranfa^ions, vol, zp. numb. 3-f<>. the Chap. 5- PHILOSOPHY. 247 the proportion, which the periodical time of the moon bore to that of the fun in former times, witli the proportion be- tween them at prefent, the moon is found to be fomething accelerated in refped of the fun. But if the fun diminifh, the periods ol the primary planets will be lengthened ; and if the earth be encreafed, the period of the moon will be fhortened : as will appear by the next chapter, wherein it fhall be fhewn, that the power of the fun and earth is the refult of the fame power being lodg'd in all their parts, and that this principle of producing gravitation in other bo-- dies is proportional to the folid matter in each body. C H A P. V. Of the B O D I E S of the S U N and PLANETS. OUR author, after having difcovered that the celeftiaE motions are performed by a force extended from the fun and primary planets, follows this power into the deep- eft receffes of thofe bodies themfelves, and proves the fame to accompany the fmalleft particle, of which they are com-- pofed. 2. Preparative hereto he fhews firft, that each of the heavenly bodies attrads the reft, and all bodies, with fuch different degrees of force, as that the force of the fame at-- tra)diRg; 248 Sir I s A A c N E w T o n's Book IL tra> Ibid, cor .j-. ' Vid Lib.III. Prop. 7. coro:i. 1. Ll X i;".This 2 6o Sir Isaac N e w t o n's Book II. ly. This therefore is the univerfal law of matter ; wliich recommends it fcif no leG for its great plainnefs and iim- plicity, than for the fiirprizing difcoveries it leads us to. By this principle we learn the diilerent weight, which the fame body will have upon the furfaces of the fun and of di- verfe planets; and by the fame we can judge of the com.po- fition of thofe celeftial bodies, and know the denfity of each ; which is formed of the moft compact., and which of the moft rare fubftance. Let the adverfaries of this philofo- phy refled: here, whether loading this principle with the appellation of an occult quality, or perpetual miracle, or any other reproachful name, be fufficient to dilihade us from cultivating it ; lince this quality, which they call occult, leads to the knowledge of fuch things, that it would have been re- puted no lefs than madnefs for any one, before they had been difcovered, even to have conjedured that our faculties fliould ever have reached fo far. j6. See how all this naturally follows from the forego- ing principles in thofe planets, which have fatellites mov- ing about them. By the times, in which thefe fatellites perform their rev^olutions, compared with their diftances from their refpeftive primary, the proportion between the power, with which one primary attrads his fatellites, and the force with which any other attracts his will be known ; and the proportion of the power with wliich any planet attrads its fecondary, to the power with which it attracts a body at its furface is found, by comparing the diflance of the fecondary planet from the center of the primary, to the C JAp. 5. PHILOSOPHY. 161 tlie diftance of the primary planet's furfacc from the fame : and from hence is deduced the proportion between the power of gravity upon the furface of one planet, to the gravity upon the furface of another. By the like method of comparing the periodical time of a primary planet about the fun, with the revolution of a fatellite about its primary, may be found the proportion of gravity, or of the weight of any body up- on the furface of the fun, to the gravity, or to the w^eight of the fame body upon the furface of the planet, which carries about the fatellite. 17. By thefe kinds of computation it is found, that the weiglit of the fame body upon the furface of the fun will be about ig times as great, as here upon the furface of the earth ; about 10^. times as great, as upon the furface ol Jupi- ter ; and near 1 9 times as great, as upon the furface of Saturn '\ 18. The quantity of matter, which compofes each of thefe bodies, is proportional to the power it has upon a body at a given diftance. By this means it is found, that the fun contains iO(57 times as much matter as Jupiter ; Jupi- ter lyB-j times as much as the earth, and af times as much as Saturn ^ The diameter of the fun is about 9 1 times, that of Jupiter about 9 times, and that of Saturn about 7 times the diameter of the earth. » Newt.Princ. Lib. III. prop. 8. coroll. i . '' Ibid, coroll. 2, 19. Bv 262 Sir I s A A c N E w T o n's Book II. I p. By making a comparifon between the quantity of matter in thefe bodies and their magnitudes, to be found from their diameters, their refpeftive deniities are readily deduced ; the denfity of every body being meafured by the quantity of matter contained under the fame bulk, as has been above remarked \ Thus the earth is found ^'- times more denfe than Jupiter ; Saturn has between ^ and - of the denlity of Jupiter ; but the fun has one fourth part only of the denfity of the earth *". From which this obfervation is drawn by our author ; that the fun is rarificd by its great heat, and that of the three planets named, the more denfe is nearer the fun than the more rare ; as was highly reafonable to exped, the denfeft bodies requiring the greateft 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 intenfe heat to which the denfer are expofed. Thus the waters of our feas, if removed to the diftance of Saturn from the fun, would remain perpetually frozen ; and if as near the fun as Mercury, would conftantly boil ". 10. T II E denfities of the three planets Mercury, Venus, and Mars, which have no fatellites, cannot be exprefly alTign- ed ; but from what is found in the others, it is very proba- ble, that they alfo are of fuch different degrees of denfity, that univerfilly the planet which is neareft to the fun, is formed of the mofl: compadl fubflance. a BockI.CIi.4.§i. *> Newt. Piinc. Lib. III. prop. ?. coroll, 3. ' Hi'd, coroll. 4. C li A P. C;iAp. 6. PHILOSOPHY. 253 Chap. VI. Of the FLUID PARTS of the PLANETS. THIS globe, that we inhabit, is compofed of two parts ; the foUd earth, which affords us a foundation to dwell upon ; and the feas and other waters, that furnifh rains and vapours neceffary to render the earth fruitful, and produdive- of what is requifite for the fupport of life. And that the moon, though but a fecondary planet, is compofed in like manner, is generally thought, from the different degrees of light which appear on its furface ; the parts of that planet, which refled: a dim light, being fuppofed to be fluid, and to imbibe the fun's rays, while the folid parts refle£l them more copioufly. Some indeed do not allow this to be a concluffve argument : but whether we can diffinguiffi the fluid part of the moon's furface from the reft or not; yet it is moft proba- ble that there are two fuch different parts, and with ftill great- er reafon we may afcribe the like to the other primary planets, which yet more nearly refemble our earth. The earth is alfo encompaffed by anotlier fluid the air, and we have before re- marked, that probably the reft of the planets are furrounded by the like. Thefe fluid parts in particular engage our au- thor's attention, both by reafon of fome remarkable appear- ances peculiar to them, and likewife of fome effcdls they have upon the whole bodies to which they belong. 4. a. Fluids 264 Sir I s A A c N E vv T o n's Book II. 1. Fluids have been already treated of in general, with refpedl to the ef/e6l they have upon foHd bodies moving in them ^ ; now we muft confider them in reference to the ope- ration of the power of gravity upon them. By this power they are rendered weighty, Hke 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 prefs upon the lower as much, as any folid body would prefs on another, whereon it fhould lie. But there is an effed: of the preffure of fluids on the bottom of the veffel, wherein they are contained, which I fhall particularly explain. The force fupported by the bot- tom of fuch a veffel is not fimply the weight of the quantity of the fluid in the veflel, but is equal to the weight of that quantity of the fluid, which would be contained in a veflel of the fame bottom and of equal width throughout, when this veffel is filled up to the fame height, as that to which the vef- fel propofed is filled. Suppofe water were contained in the veffel A B C D (in fig. 1 09.) filled up to E F. Here it is evident, that if a part of the bottom, as G H, which is direftly under any part of the fpace EF, be confidered feparately ; it will ap- pear at once, that this part fuftains the weight ot as much of the fluid, as ftands perpendicularly over it up to the height of F F ; that is, the two perpendiculars G I and H K being drawn, the part G H of the bottom will fuflain the whole weight of the fluid included between thefe two perpendiculars. Again, I fay, every other part of the bottom equally broad with this, will fuflain as great a preflure. Let the part L M be of the » Book I Ch 4. fame Chap. 6. PHILOSOPHY. 265 ■fame breadth with G H. Here the perpendiculars L O and M N being drawn, the quantity of water contained between thefe perpendiculars is not fo great, as that contained between the perpendiculars G I and H K ; yet, I fiy , the prefTure on L M will be equal to that on G H. This will appear by the fol- lowing conliderations. It is evident, that if the part of the vefTel between O and N were removed, the water would im- mediately flow out, and the furface EF would fubfi.de ; for all parts of the water being equally heavy, it mufl: foon form itfelf to a level furface, if the form of the veffel, which con- tains it, does not prevent. Therefore flnce the water is pre- vented from rifmg by the fide N O of the vefl'el , it is mani- feft, that it muft prefs againft N O with fome degree of force. In other words, the water between the perpendiculars L O and MN endeavours to extend itfelf with a certain degree of force; or more correftly, the ambient water prefles upon this, and endeavours to force this pillar or column of water into a grea- ter length. But fince this column of water is fuftained be- tween N O and L M, each of thefe parts of the veffel will be equally preflTed againft by the power, wherewith this column endeavours to extend. Confequently LM bears this force over and above the weight of the column of water between L O and M N. To know what this expanfiive force is, let the part O N of the veffel be removed, and the perpendiculars L O and M N be prolonged ; then by means of fome pipe fixed over NO let water be filled betv/een thefe perpendiculars up to P Q.an equal height with E F. Here the water between the per- pendiculars LP and M (^ is of an equal height with the high- eft part of the water in the veffel ; therefore the water in the M m veffel o6(5 Sir Is A AC Newton's Book II. veflel cannot by its prefiure force it up higher, nor can the water in this column fubfide ; becaufe, if it fliould, it would raife the water in the veffel to a greater height than itfelf. But it follows from hence, that the weight of water contained between P O ajid Q.N is a juft balance to the force, wherewith the column between LO and MN endeavours to extend. So the part L M of the bottom, which fuftains both this force and the weight of the water between L O and M N, is pref- fed 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 prelled on by a force equal to the weight of all the water contained between LP and MQ. And this weight is equal to that of the water contained between. G I and H K, which is the weight fuftained 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 veffel 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 fame breadth ; when this veffel is filled with water to the fame height, as the veffel A B C D is filled, the bottoms of thefe two veffels fhall be preffed upon with equal force. If the veffel be broader at the top than at the bottom, it is evident, that the bottom will bear the preffure of fo much of the fluid, as is perpen- dicularly over it, and the fides of the veffel will fupport die reft. This property of fluids is a corollary from a propofi- tion of our author * ; firom whence alfo he deduces the ef- fects of the preffure of fluids on bodies refting in them, ? Lib. II. prop. 20. cor. a.' Thele Chap. 6. PHILOSOPHY. 267 Tliefc are, diat any body heavier than a fluid will fink to the bottom of the vefl'el, wherein the fluid is contained, and in the fluid will weigh as much as its own weight ex- ceeds the weight of an equal quantity of the fluid ; any body uncompreflible of the fame denflty with the fluid, will reft any where in the fluid without fuflering the leaft change ei- ther in its place or figure from the prefliire of fuch a fluid, but will remain as undifturbed as the parts of the fluid them- felves ; but every body of lefs denflty than the fluid will fwim on its furface, a part only being received within the fluid. Which part will be equal in bulk to a quantity of the fluid, whofe weight is equal to the weight of the whole bo- dy ; for by this means the parts of the fluid under the bo- dy will fufter as great a preflure as any other parts of tlie fluid as much below the furface as thefe. 3 . In the next place, in relation to the air, we have a- bove made mention, that the air furrounding the earth being an elaftic fluid, the power of gravity will have this efl'ed; on it, to make the lower parts near the liirface of the earth more compact and comprefled together by the weight of the air incumbent, than the higher parts, which are pref- fed upon by a lefs quantity of the air, and therefore fu- ftain a lefs weight \ It has been alfo obferved, that our au- thor has laid down a rule for computing the exa6l degree of denflty in the air at all heights from the earth ^. But there is a farther efled from the air's being comprefiTed by * Chap. 4. § 17. '' IbiJ. M m X the 2(58 Sir Isaac Newton's Book II. the power of gravity, which he has difl:in6lly confidered. The air being elaftic and in a ftate of compreflion, any tre- mulous body will propagate its motion to the air, and excite therein vibrations, which will fpread from the body that occafions them to a great diftance. This is the efficient caufe of found : for that fenfation is produced by the air, which, as it vibrates, ftrikes againft the organ of hearing. As this fubjed: was extremely difficult, fo our great author's fuccefe, is furprizing. 4^. Our author's dodrine upon this head I fhall endea- vour to explain fomewhat at large. But preliminary thereto muft be fhewn, what he has delivered in general of pref- fure propagated through fluids ; and alfo what he has fet down relating to that wave-like motion, which appears up- on the furface of water, when agitated by throwing any thing into it, or by the reciprocal motion of the finger, &c. y. Concerning the firft, it is proved, that preiTure is fpread through fluids, not only right forward in a flreight line, but alfo laterally, with aim oft the fame cafe and force. Of which a very obvious exemplification by experiment is propofed : that is, to agitate the furface of water by the re- ciprocal 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 diflufe themfelves on each hand of the diredion of the motion, and foon furround the finger. Nor is what we obferve in founds unlike to this, which, do not proceed in (Iraight lines only, but are heard though a I mountain Chap. 6, PHILOSOPHY. 269 mountain intervene, and when they enter a room in any- part of it, they fpread themfch'es into every corner ; not by reflection from the w^alls, as fome have imagined, but as^. far as the fenfe can judge, diredly from the place where they enter. 6. How the waves are excited in the flirface of ftagnant water, may be thus conceived. Suppofe in any place, the water raifed above the reft in form of a fmall hillock ; that water will immediately iublide, and raife the circumambient, water above the level of the parts more remote, to which tha motion cannot be communicated under longer time. And- again, the water in fubflding will acquire, like all falling bo- dies, a force, which will carry it below the level furface, till at length the prelTure of the ambient water prevailing, it will rife again, and even with a force like to that wherewith it de- fcended, which will carry it again above the level. But in the mean time the ambient water before raifed will fubfide,. as this did, finking below the level ; and in fo doing, will not only raife the water, which firft fubflded, but alfo the wa- ter next without itfelf. So that now befide the firft hillock,. we fliall have a ring invefting it, at fome diftance raifed above: the plain furface likewife ; and between them die water willl be funk below the reft of the furface. After this, thefirft hil- lock, and the new made annular riling, will defcend ; railing ; the water between them, which was before depreffed, and like- wife the adjacent part of the furface without. Thus will thele annular waves be luccefiively fpread more and more. For, , as the hillock fubftding produces one ring, and that ring fub-- fic 270 Sir Isaac New t on's Book TL fiding raifes again the hillock, and a fecond ring ; fo the hil- lock and fecond ring fubfiding together raife the firft ring, and a third ; then this firft and third ring fubiiding together raife the firft hillock, the fecond ring, and a fourth ; and fo •on continually, till the motion by degrees ceafes. Now it is de- monftrated, that thefe rings afcend and defcend in the manner of a pendulum ; defcending with a motion continually acce- lerated, till they become even with the plain furface of the flu- id, which is half the fpace they defcend ; and then being re- tarded again by the fime degrees as thofe, whereby they were accelerated, till they are depreffed below the plain furface, as much as they were before raifed above it: and that this augmen- tation and diminution of their velocity proceeds by the fame degrees, as that of a pendulum vibrating in a cycloid, and whofe length fhould be a fourth part of the diftance between any two adjacent waves : and farther, that a new ring is produced every time a pendulum, whofe length is four times the former, that is, equal to the interval between the fum- mits of two waves, makes one ofcillation or fwing \ 7. This now opens the way for underftanding the mo- tion confequent upon the tremors of the air, excited by the vibrations of fonorous bodies: which we muft conceive to be performed in the following manner. 8. L E T A, B, C, D, F, F, G, H ( in fig. 1 1 o. ) reprefent a fe- Ties of the particles of the air, at equal diftances from each other, I K L a mufical chord, which I fliall ufe for the tre- • Vid.Newt. Princ. Lil\ II. prop. 4.6. iiiulous Chap. 6. PHILOSOPHY. 271 milieus and fonoroiis body, to make the conception as fim- ple as may be. Suppofe this chord llretched upon the points I and L, and forcibly drawn into the fituation IK L, fo that it become contiguous to the particle A in its middle point K - and let the chord from this fituation begin to recoil, preinng againft the particle A, which will thereby be put into motion towards B : but the particles A, B, C being equidiftant, the elaftic power, by wliich B avoids A, is equal to, and balan- ced by the power, by which it avoids C ; therefore the elaftic 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 foon as that is done, the particle B will be moved towards C ; and being made, to ap- - proach C, will in the next place move that ; which will up- on that advance, put D likewife into motion, and fo on t therefore the particle A being moved by the chord, the fol- lowing particles of the air B, C, D, &c. will fuccefiively be moved. Farther, if the point K of the chord moves for- ward with an accelerated velocity, fo that the particle A (hall move againft B with an advancing pace, and gain ground of it, approaching nearer and nearer continually ; A by approach- ing will prefs more upon B, and give it a greater velocity likewife, by reafon that as the diftance between the particles diminifhes, the elaftic power, by which they fly each other, increafes. 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 fame caufe C will more and more approach D ; and fo of the reft. Suppofe now, ftnce the agitation of thefe particles has been fliewn to be ■fi'^Q, Sir Isaac N e w t o n's Book IL te fucceflive, and to follow one another, that E be tlie re- nioteft particle moved, while the chord is moving from its curve lituation I K L into that of a ftreight line, as I /t L ; and F the iirfl: which remains unaffeded, though juft upon the point of being put into motion. Then fhall the particles A, B, C, D, E, F, G, when the point K is moved into k^ have acquired the rangement reprefented by the adjacent points a^ h-, Cn, dy e, f, g : in which a is nearer to h than b to 6", and 3? nearer to c than c to ^/, and c nearer to d than d to e, and .//nearer to e than e to/j and laftly e nearer toythany'to^. o. But now the chord having recovered its rectilinear li- tuation I /^ L, the following motion will be changed, for the point K, which before advanced with a motion more aad more accelerated, though by the force it has acquired it will go on to move the lame way as before, till it has advanced near as far forwards, as it was at firfl drawn backwards ; yet the motion of it will henceforth be gradually leiTened. The effed; of which upon the particles a^ h-, c, d^ e, f^ g will be, that by the time the chord has made its utmoft advance, and is upon the return, thefe particles will be put into a contrary rangement ; fo that f fhall be nearer to ^, than e to f^ and ■e nearer to f than dtoe; and the like of the reft, till you come to the firft particles a, Z?, whofe diftance will then be nearly or quite what it was at firft. All which will appear as follows. The prefent diflance between a and b is flich, that the elaftic power, by which a repels by is ftrong enough to maintain that diftance, though a advance with the velocity, with which the firing refumes its redilinear figure ; and the motion CHAP.d PHILOSOPHY. 273 motion of the particle a being afterwards flower, the prefent elafticity between a and I will be more than fufficient to preferve the diftance between them. There- fore while it accelerates h it will retard a. The di- ftance he will ftill diminiih, till h come about as near to c, as it is from a at prefent ; for after the diftances ah and he are become equal, the particle h will continue its velocity fuperior to that of c by its own power of in- adlivity, till fuch time as the increafe of elafticity between h and c more than {hall be between a and h fhall fup- prefs its motion : for as the power of inadlivity in h made a greater elafticity necefiary on the fide of a than on the lide of c to pufh h forward, fo what motion h has acquired it will retain by the fame power of inadivity, till it be fupprefled by a greater elafticity on the ftde of £", than on the fide of a. But as foon as h begins to flacken its pace the diftance of h from c will widen as the diftance ah has already done. Now as a ads on ^, fo will ^ on c, c on d, Is^c fo that the diftan- ces between all the particjes ^, ^, c, d^ ^jf,g will be fucceflively contraded into the diftance ot a from h, and then dilated again. Now becaufe the time, in which the chord defcribes this prefent half of its vibration, is about equal to that it took up in defcribing the former ; the particles a, h will be as long in dilating their diftance, as before in contracting it, and will return nearly to their original diftance. And farther, the particles ^, c, which did not begin to approach fo foon as a^ h, are now about as much longer, before they begin to recede ; and likewife the particles c, dj which began to ap- proach after h, C> M) 6, \, /Mr, V, ^. Where a, 8, are at the original diftance of the particles in the line AH ; ^,v, are the neareft of all, and the diftance »| is equal to that between a and /3. 10. By this time the chord Iz L begins to return, and the (.diftance between the particles a and j8 being enlarged to its (Original magnitude, oc has loft all that force it had acqui- Ted by its motion, being now at reft ; and therefore will Tetum with the chord, making the diftance between « and ;@ greater than the natural; for Q, will not return fo foon, ibecanfe its motion forward is not yet quite fupprefled, the ^diftance ^y not being already enlarged to its prime dimcn- iion : but the recefs of a, by diminifhing the prefture up- on |8 by its elafticity, will occafton the motion of S> to be ftopt in a Httle time by the aftion of y, and then fhall /S begin to return: at which time the diftance between y and J\ Ihall by the fuperior acHrion of /> above /S be en- larged to the dimenfton of the diftance /Sy, and therefore foon after to that of et/g. Thus it appears, that each oi thefe particles goes on to move forward, till its diftance from the Chap. 6. PHILOSOPHY. 275 the preceding one be equal to its original diflancc ; the whole chain et, 6, y, o'^, <, (^, «, having an undulating motion forward, which is ftopt gradually by the excefs of the cx- paniive power of the preceding parts above that of the hinder. Thus are tliefe parts fucccfiively ftopt, as before they were moved ; fo that when the chord has regained its redlilinear fituation, the expanfion of tlie parts of the air will have advanced fo far, that the interval between t, »t, which at prefent is moft contraded, will then be reftored to its natural fize : the diftances between » and 9, 9 and a, \ and <<, a and v, » and |, being fucceffively contraded into the prefent diftance of { from «, and again enlarged ; (o that the fame effedt fliall be produced upon die parts beyond ^ «, by the enlargement of the diftance between thofe two particles, as was occafioned upon the particles «t, 5, y, «n, t, C >ij 8) A, (i) », ^, by the enlargement of the diftance a /3 to its natural extent. And therefore the motion in the air will be extended half as much fardber as at prefent, and the diftance between » and g contracled into that, which is at prefent between { and «, all the particles of the air in motion taking the rangement expreffed in figure III. by the points «, iS, y, J^, «, (, », 9, a, ^, v, ^, w, p, o-, t, ?) ; wherein the particles from <» to | have their diftances from each other gradually diminifhed, the diftances between the particles », ^ being contraded the moft from the natural dift- ance between thofe particles, and the diftance between , ^, 1 , 1^ g , 4, T, 6, 7, 8 : in which / m^ are at the natural diftance of the particles, the diftance m n greater than Im, and n greater than m n, and fo on^ till you come to qr, the wideft of all: and then the diftances gradually diminifti not only to the natural diftance, as wXy but till they are contraded as much as ^ t was before j which falls out in the points 1,5, from whence the diftances augment again, till you come to the part of the air untouched. IX. T H I s 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 pulfe. When the chord goes on to make another vibration, it will not only continue to agitate the air at prefent in motion, but fpread the pulfation of the air as much farther, and by the fame degrees, as be- fore. For when the chord returns into its rectilinear fitu- ation ly^L, Im fhall be brought into its moft contraded ftate Chap. 6. PHILOSOPHY. 279 ftate, qr now in the ftate of greateft dilatation {liall be re- duced to its natural diftancc, the points -z^, x now at their natural diftance fhall be at their greateft diftance, the points 1, 3 now moft contraded enlarged to their natural diftance, and the points y, 8 reduced to their moft contraded ftate : 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 firft moved out of the fttuation I K L into its redilinear figure. When the chord is got into the fitu- ation Ix.L, Ini fhall recover its natural dimenfions, ^r be reduced to its ftate of greateft contradiion, 7^ X brought to its natural dimenfion, the diftance i g enlarged to the ut- moft, and the points y, 8 fhall have recovered their na- tural diftance ; and by thus recovering themfelves they fhall agitate the air to as great a length beyond them, as it was moved beyond the chord, when it firft came into the fitu- ation I»L. When the chord is returned back again into its redihnear fituation, Im fhall be in its utmoft dilatation, qr reftored again to its natural diftance, 74/ x reduced into its ftate of greateft contraction, i g fhall recover its natu- ral dimenfion, and 7 8 be in its ftate of greateft dilatation. By which means the air fhall 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 redilinear fituation ; for the particles. 7,8 have been changed from their ftate of reft and their natural diftance into a ftate of contradlion, and then have proceeded to the recovery of their natural diftance, and af- ter that to a dilatation of it, in the fame manner as the- particles contiguous to the chord were agitated before. In the t'So Sir Isaac N e w t o n*s Book IL the lafl place, when the chord is returned into the Situation i K L, the particles of air from / to of\ fliall acquire their pre- fent rangement, and the motion of the air be extended as much farther. And the like will happen after every com- pleat vibration of the ftring. 13. Concerning this motion of found, our author fhews how to compute the velocity thereof, or in what time it will reach to any propofed diftance from the fonorous body. For this he requires to know the height of air, hav- ing the fame deniity with the parts here at the furface of the earth, which we breath, that would be equivalent in weight to the whole incumbent atmofphere. This is to be found by die barometer, or common weatlierglafs. In that inftrument quickfilver is included in a hollow glafs cane firmly clofed at the top. The bottom is open, but immerged into quickfilver contained in a veiTel open to the air. Care is taken when the lower end of the cane is immer- ged, that the whole cane be full of quickfilver, and that no air infinuate itfelf When the inftrument is thus fixed, the quick- filver in the cane being higher than that in the vefl'el, if the top of the cane were open, the fluid would foon fink out of the glais cane, till it came to a level with that in the veffel. But the top of the cane being clofed up, fo that the air, which has free liberty to prefs on the quick- filver in the veffel, cannot bear at all on that, which is with- in the cane, the quickfilver in the cane will be fufpended to fuch a height, as to balance the prefliire of the air on the quickfilver in the veffel. Here it is evident, that the 1 weight Chap. 6. PHILOSOPHY. 281 weight of the qulckfilver in the glafs cane is equivalent to the preffure of fo 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 ufe of the quickfil- ver to fuftain the preffure of the air without ; for the quick- filver in the cane, as has already been obferved, will then fub- lide to a level with that without. Hence tlierefore if the pro- portion between the denfity of quickfilver and of the air we breath be known, we may know what height of fuch air would form a column equal in weight to the column of quickfil- ver within the glals cane. When the quickfilver is fuftain- ed in the barometer at the height of g o inches, the height of fuch a column of air will be about 29715" feet; for in this cafe the air has about g^ of the denfity of water, and the denfity of quickfilver exceeds that of water about 13-^ times, fo that the deniity of quickfilver exceeds that of the air about 11890 times; and fo many times go in- ches make 19715- feet. Now Sir Isaac Newton de- termines, that while a pendulum of die length of this column fliould make one vibration or fwing, the fpace, which any found will have moved, fhall bear to this length the fame proportion, as the circumference of a circle bears to the di- ameter thereof ; that is, about the proportion of 3 5* 5* to 113*. Only our author here confiders fingly the gradual progrefs of found in the air from particle to particle in the manner v/e have explained, witliout taking into confidera- tion the magnitude of thofe particles. And though there requires time for the motion to be propagated from one par- » Princ. philof Lib. II. prop. 49. O o tide ' 282 Sir Is A AC Newton's BookIL tide to another, yet it is communicated to tlie whole of the fame particle in an inftant: therefore whatever propor- tion the thicknefs of thefe particles bears to their diftance from each other, in the fame proportion will the motion of found be fwifter. Again the air we breath is not fim- ply compofed of the elaftic part, by which found is con- veyed, but partly of vapours, which are of a different na- ture J and in the computation of the motion of found we ought to find the height of a column of this pure air on- ly, whofe weight fhould be equal to die weight of the quick- iilver 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 19715- feet. On both thefe accounts the motion of found is found to be about 1 14.1 feet in one fe- cond of time, or near i g miles in a minute, whereas by the computation propofed above, it fhould move but 979 feet in one fecond. 14. We may obferve here, that from thefe demonftra- tions of our audior it follows, that all founds whether a- cute or grave move equally fwift, and that found is fwifteft, when the quickfilver flands higheft in the barometer. I S". Thus much of the appearances, which are caufed in thefe fluids firom dieir gravitation toward the earth. They alfo gravitate toward the moon ; for in the laft chapter it has been proved, that the gravitation between the earth and moon is mutual, and that this gravitation of the whole bo- dies arifes from that power acting in all their parts ; fo that every Chap. 5. PHILOSOPHY. 283 every particle of the moon gravitates toward the earth, and every particle of the earth toward the moon. But this gravitation of thefe fluids toward the moon produces no feniible effed, except only in the fea, where it caufes 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 leaft fhadow of reafon to fuppofe otherwife, confldering how fteadily they accompany the moon's courfe. Though how the moon caufed them, and by what princi- ple it was enabled to produce fo diftinguifli'd an appearance, was a fecret left for this philofophy to unfold : which teaches, that the moon is not here alone concerned, but that the fun likewife has a conflderable fhare in their produdlion; though they have been generally afcribed to the other lu- minary, becaufe its effed: is greateft, and by that means the tides more immediately fuit themfelves to its motion; the fun difcovering its influence more by enlarging or re- Jdraining the moon's power, than by any diftind: effeds. Our author finds the power of the moon to bear to the power of the fun about the proportion of 4 7 to i. This he deduces from the obfervations made at the mouth of the river Avon, three miles from Briftol, by Captain S t u r- M E Y, and at Plymouth by Mr. C o l e ? r e s s e, of the height to which the water is raifed in the conjunction and oppo- fition of the luminaries, compared with the elevation ot it, when the moon is in either quarter; the firft being caufed O o z by 284 Sir I s A AC Newton's Book II. by the united actions of the fun and moon, and the o- ther by the difference of them, as fliall hereafter be fhewn. 17. That the fun fhould have a like effeA on the fea, as the moon, is very manifeft ; fmce the fun likewife attracts every fingle particle, of which this earth is compofed. And in both luminaries (ince the power of gravity is reciprocally in the duplicate proportion of the diftance, they will not draw all the parts of the waters in the fame manner ; but muft ad; upon tlie neareft parts ftronger, than upon the re- motefl, producing by this inequality an irregular motion. We fhall now attempt to fhew how the adions of the Rin and moon on the waters, by being combined together, pro- duce all the appearances obferved in the tides. 18. To begin therefore, the reader will remember what has been faid above, that if the moon without the fun would have defcribed an orbit concentrical to the earth, the ad:ion of the fun would make the orbit oval, and bring the moon nearer to the earth at the new and full, than at the quarters^ Now our excellent author obferves, that if inftead of one moon, we fuppofe a ring of moons, contiguous and occupying the whole orbit of the moon, his demonftration would ftill take place, and prove that the parts of this ring in pafTing from the quarter to the conjundion or oppofition would be accelerated, and be retarded again in pafling from the conjunction or op- pofition to the next quarter. And as this effedl does not de- » Chap. J. § iS. pend Chap. 6. PHILOSOPHY. 285 pend on the magnitude of the bodies, whereof the ring is compofed, the fame would hold, though the magnitude of thele moons were fo far to be diminiflied, and their num- ber increafed, till they fhould form a fluid \ Now the earth turns round continually upon its own center, cauf- ing thereby the alternate change of day and night, while by this revolution each part of the earth is fucceffively brought toward the fim, and carried off again in the fpace of 24 hours. And as the fea revolves round along with the earth itfelf in this diurnal motion, it will reprefent in fome fort fuch a fluid ring. 19. But as the water of the fea does not move round with fo much fwiftnefs, as would carry it about the center of the earth in the circle it now defcribes, without being fupported by the body of the earth ; it will be neceflary to conflder the water under three diftinft cafes. The firft cafe fhall fuppofe the water to move with the degree of fwiftnefs, required to carry a body round the center of the earth dif- ingaged from it in a circle at die difl:ance of the earth's femidiameter, like another moon. The fecond cafe is, that the waters make but one turn about the axis of the earth in the fpace of a month, keeping pace with the moon ;. fo that all parts of the water fhould preferve continually the fame fltuation in refpedt of the moon. The third cafe fhall be the real one of the waters moving with a ve- locity between thcfe two, neither fo fwift as the flrll: calc requires, nor fo flow as the fecond, » Newt. Princ. philof. Lib. 1. prop. 66. corol], iS. 10. In .286 Sir I s A A c N E w T o n's Book II. 10. In the firfl: cafe the waters, Uke the body which they equalled in velocity, by the adlion of the moon would be brought nearer the center under and oppofite to the moon, than in the parts in the middle between thefe eaftward or weftward. That fuch a body would fo alter its diftance by the moon's action upon it, is clear from what has been mentioned of the like changes in the moon's motion caufed by the fun \ And computation fhews, that the differ- ence between the greateft and leaft diftance of fuch a body would not be much above 4 ~ feet. But in the fecond cafe, where all the parts of the water preferve the fame iitua- tion continually in refped of the moon, the weight of thofe parts under and oppofite to the moon will be diminifhed by the moon's a6lion, and the parts in the middle between thefe will have their weight increafed : this being effeded jufl: in the fame manner, as the fun diminifhes the attradi- on of the moon towards the earth in the conjun6lion and oppofition , but increafes that attradion in the quarters. For as the firfl of thefe confequences from the fun's ac- tion on the moon is occafioned by the moon's being at- traded by the fun in the conjundion more than the earth, and in the oppofition lefs than it, and therefore in the common motion of the earth and moon, the moon is made to advance toward the fun in one cafe too faft, and in the other is left as it wxre behind ; fo the earth will not have its middle parts drawn towards the moon fo ftrong- ly as the nearer parts, and yet more forcibly than the re- raoteft : and therefore fince the earth and moon move each » § s. month Chap. 6. PHILOSOPHY. 287 month round their common center of gravity ' , while the earth moves round this center, the fame effed: will be produced, on the parts of the water neareft to that cen- ter or to the moon, as the moon feels from the fun when in conjundlion, and the water on the contrary fide of the earth will be affeded by the moon, as the moon is by the fun, when in oppoiition '' ; that is, in both cafes the weight of the water, or its propenftty towards the center of the earth, will be diminished. The parts in the middle between thefe will have their weight incrcafed, by being prefTed towards the center of the earth through the obliquity of the moon's adion upon them to its adion upon the earth's center, juft as the fun increafes the gravitation of the moon in the quarters from the fame caufe ". But now it is mani- feft, that where the weight of the fame quantity of water is leaf!:, there it will be accumulated ; while the parts, which have the greateft weight, will fublide. Therefore in this cafe there would be no tide or alternate riling and falling of the water, but die water would form it felf into an oblong figure, whofe axis prolonged would pafs through the moon. By Sir Isaac Newton's computation the excefs of this axis above the diameters perpendicular to it> that is, the height of the waters under and oppofite to the moon above their height in the middle between thefe pkr ces eaftward or weft ward cauftd bv the moon, is about dj feet. » ch. 5. § ;-. 'Ch. 3 § 17. ' im. 21. Thus- 3 288 Sir I s A A c N E vv T o n's Book II. 'LI. Thus the difference of height in this latter fup- pofition is Httle fhort of twice that difference in the pre- ceding. But the cafe of the fea is a middle between thefe two : for a body, which fhould revolve round the center of the earth at the diftance of a femidiameter without preffing on the earth's furface, muff perform its period in lefs than an hour and half, whereas the earth turns round but once in a day ; and in the cafe of the waters keeping pace with the moon it fhould turn round but once in a month : fo that the real motion of the water is between the motions re- quired in thefe two cafes. Again, if the waters moved round as fwiftly as the firft cafe required, their v/eight would be wholly taken otF by their motion ; for this cafe fuppofes the body to move fo, as to be kept revolving in a circle round the earth by the power of gravity without preflmg on the earth at all, fo that its motion juft fupports its weight. But if the power of gravity had been only ^ part of what it is, the body could have moved thus without pref- fing on the earth, and have been as long in moving round, as the earth it felf is. Confequently the motion of the earth takes off from the weight of the water in the mid- dle between the poles, where its motion is fwifteft, ^1^ part of its weight and no more. Since therefore in the firft cafe the weight of the waters muft be intirely taken off by their motion, and by the real motion of the earth they lofe only 7^ part thereof, the motion of the water will fo little diminifh their weight, that their figure will much nearer re- femble the cafe of their keeping pace with the moon than the other. Upon the whole, if the waters moved with the I velo- Chap. 6. PHILOSOPHY. 289 velocity neccffary to carry a body round the center of the earth at the diftancc of the earth's femidiametcr without hearing on its furface, the water would be loweft under the moon, and rife gradually as it moved on with the earth eaftward, till it came half way toward the place oppofite to the moon ; from thence it would fublide again, till it came to the oppofition, where it would become as low as at firfl ; afterwards it would rife again, till it came half way to the place under the moon ; and from hence it would fubiide, till it came a fecond time under the moon. But in cafe the water kept pace with the moon, it would be higheft where in the other cafe it is loweft, and loweft where in the other it is higheft ; therefore the diurnal motion of the earth being between the motions of thefe two cafes, it will caufe the higheft place of the water to fall between the places of the greateft height in thefe two cafes. The water as it paffes from under the moon fhall for fome time rife, but defcend again before it arrives half way to the oppoiite place, and fliall come to its leaft height before it becomes oppofite to the moon; then it fhall rife again, continuing lb to do till it has pajfifed the place oppofite to the moon, but fubfide before it comes to the middle between the places oppofite to and under the moon j and laftly it fliall come to its loweft, before it comes a le- cond time under the moon. If A (in fig. Ill, 1 1 g, 1 1 4.) reprefent the moon, B the center of the earth, the oval C D E F in fig. 112. will reprefent the fituation of the water in the firft cafe ; but if the water kept pace with the moon, the line CDEF in fig. iig. would reprefent thp fitua- P p tion 290 Sir I s A A c N E w T o nV Book II. tion of the water ; but the Hne CDEF in fig. 1 14,. will re- prefent the fame in the real motion of the water, as it accompanies the earth in its diurnal rotation : in all thefe figures C and E being the places where the water is low- eft, and D and F the places where it is higheft. Purfu- ant to this determination it is found, that on the fhores, which lie expofed to the open fea, the high water ufually falls out about three hours after the moon has paffed the meridian of each place. ZZ' Let this fuffice in general for explaining the man- ner, in which the moon ads upon the feas. It is farther to be noted, that thefe effefts are greateft, when the moon is over the earth's equator \ that is, when it fhines perpen- dicularly 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 effed: upon the water to make it afcend and defcend. So that when the moon declines from the e- quator toward either pole, it's adion muft be fometliing diminifhed, and that the more, the farther it decHnes. The tides likewife will be greateft, when the moon is. neareft to the earth, it'^s action being then die ftrongeft. ig. Thus much of the adion of the moon. That the fun, fliould produce the veiy fame efteds, though m. a lefs degree, is too obvious to require a particular expla- Qatioa: but as was remarked before, this adion of the; » Sec below §. 4.4. fun; Chap. (5. PHILOSOPHY. 291 fun being weaker than that of the moon, will caufe tlic tides to follow more nearly the moon's courfe, and princi- pally fhew it felf by heightening or diminifhing the ef- fects of the other luminary. Which is the occalion, that the higheft tides are found about the conjundion and oppo- fition of the luminaries, being then produced by their uni- ted aftion, and the weakeft tides about the quarters of the moon ; becaufe the moon in this cafe raifing the water where the fun deprelTes it, and depre fling it where the fun raifes it, the ftronger adiion of the moon is in part retunded and weakened by that of the fun. Our author computes that the fun will add near two feet to the height of the water in the firft cafe, and in the other take from it as much. However the tides in both comply with the iame hour of the moon. But at other times , between the conjunction or oppoHtion and quarters, the time de- viates from that forementioned, towards the hour in which the liin would make high water , tliough ftill it keeps much nearer to the moon's hour than to the fun's. 14. Again the tides have fome farther varieties from the fltuation of the places where they happen northward orfouthward. Let /> P (in fig. Iiy.) reprefent the axis, on which the earth daily revolves, let h pVL^ reprefent the figure of the water, and let ;/ B N D be a globe infcri- bed within this figure. Suppofe the moon to be advanced from the equator toward the north pole, fo that h H the axis of the figure of the water /)AHPE/^ fliall decline towards the north pole N ; take any place G nearer to P p X the 2p2 Sir Is A AC Newton's BookIL the north pole than to the Ibuth, and from the center of the earth C draw C G F ; then will G F denote the altitude to which the water is raifed by the tide, when the moon is above the horizon : in the fpace of twelve hours, the earth having turned half round its axis, the place G will be removed to^; but the axis h H will have kept its place preferving its fituation in refped: of the moon, at leafl: will have moved no more than the moon has done in that time, which it is not neceflary here to take into confederation. Now in this cafe the height of the water will be equal to g f\ which is not fo 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 altitiide of the fame, when the moon is under the horizon. The contrary happens toward the fouth pole, for KL is lefs than lil. Hence is proved, that when the moon declines from the equator, in thofe places, which are. on the fame fide of the equator as the moon, the tides are greater, when the moon is above the horizon, than when tinder it ; and the contrary happens on the other, fide of the equator. 1)-. Now from thefe principles may be explained aU the knov/n appearances in the tides ; only by the afiifiir ance of this additional remark, that the fluduating jnoti- on, which the water has in flowing and ebbing, is ot a durable nature, and would continue for fome time, though the adion of the luminaries fhould ceafe ; for this prevents the difference between the tide when the moon is above the Chap. 6. PHILOSOPHY. 293 the horizon, and the tide when the moon is below it from being lb great, as the rule laid down requires. This likcwife makes the greateft tides not exadly upon the new and full moon, but to be a tide or two after ; as at Briftol and Ply- mouth they are found the third after. r6. This dodrine farther fhews us, why not only the fpring tides fall out about the new and full moon, and the neap tides about the quarters ; but likewife how it comes to pafs, that the greatefl fpring tides happen about the equi- noxes ; becaufe 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 thefe, are the lead of all; for the iun ftill continuing over the equator continues to have the greatei-t power of lelTening the moon's adion, and the moon in the quarters being far removed toward one of the poles, has its power thereby weakned, 17. Moreover the adlion of the moon being ftronger, when near the earth, than when more remote; if the moon, when new fuppole, be at its neareft diftance from the eartfi, it fhall when at the full be fartheft off; whence it is, that two of the very largeft fpring tides do never immediately fucceed each other. 18. Because the fun in its paffage from the winter folftice to the fummer recedes from the earth, and pailing from the fummer folftice to the winter approaches it, and is therefore nearer the earth before the vernal equinox than after. •^94- Sir Is AAc Newton's BookII. after, but nearer after the autumnal equinox than before ; the greatefl: tides oftner precede the vernal equinox than follow itj and in the autumnal equinox on the contrary lliey oftner follow it than come before it. 29. The altitucJe, to which the water is raifed in the open ocean, correfponds very well to the forementioned calcu- lations ; for as it was fhewn, that the water in fpring tides {hould rife 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 obierved to rife about d, 9, II or ly feet. In the Pacific ocean this elevation is faid to be greater than in the other, as it ought to be by reafon of the wide extent of that fea. For tha fame reafon in the Ethiopic ocean be- tween the tropics the afcent of the water is lefs than with- out, by reafon of the narrownefs of the fea between the coafts of Africa and the fouthern parts of America. And ifiands in fuch narrow feas, if far from fliore, have lefs tides than the coafb. But now in thofe ports where the water flows in with great violence upon fords and fhoals, the force it acquires by that means will carry it to a much greater height, fo as to make it afcend and defcend to 30, 4.0 or even yo feet and more ; inftances of which we have at Plymouth , and in the Severn near Chepftow ; at St. Michael's and Auranches in Normandy ; at Cambay and Pegu in tlie Eaft Indies. 50. Again die tides take a confiderable time in pafling through long flraits, and fhallow places. Thus the tide, 3 which Chap. 6. PHILOSOPHY. 25*5 which is made on the weft coaft of Ireland and on the coaft of Spain at the third hour after the moon's coming to the meridian, in the ports eaftward toward the Britifh channel falls out later, and as the flood pafTes up that chan- nel ftill later and later, fo that the tide takes up full twelve- hours in coming up to London bridge. 31. In the laft place tides may come to the iame port from different feas, and as they may interfere with each other, they will produce particular effeds. Suppofe die tide from one fea come to a port at the third hour after the moon'^s pafling the meridian of the place, but from another fea to take up fix hours more in its paffage. Here one tide will make high water , when by the other it fliould be loweft ; fo that when the moon is over the equator, and the two tides are equal, there will be no rifing and falling of the water at all ; for as much as the water is carried ofF by one tide, it will be fupplied by the other. But when the moon declines from the equator, the fame way as the poit; is fituated , we have fhewn 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 cafe, as four tides come to this port each day the two greateft will come on the third, and on the ninth hour after the moon's pafTing the meridian, and the two leaft at the fifteenth and at the twenty firll hour. Thus from the third to the ninth hour more water will be in this port by the two greateft tides than from^ the ninth to the fifteenth, or from the twenty firft to the following.^ 2p(5 Sir Isaac N e w t o n's Book IL following third hour, where the water is brought by one ■great and one fmall tide ; but yet there will be more .water brought by thefe tides, than what will be found be- tween the two leaft tides, that is, between the fifteenth and twenty firft hour. Therefore in the middle between the third and ninth hour, or about the moon's fetting, the wa- ter will be at its greateft height ; in the middle between the ninth and fifteenth, as alfo between the twenty firft and following third hour it will have its mean height ; and be loweft in "the middle between the fifteenth and twenty firft hour, that is, at the moon's rifing. Thus here the water will have but one flood and one ebb each day. When the moon is on the other fide of the equator, the flood will be turned into ebb, and the ebb into flood ; the high water fall- ing out at the rifing of the moon, and the low water at the fetting. Now this is the cafe of the port of Batfham in the kingdom of Tunquin in the Eaft Indies ; to which port there are two inlets , one between the continent and the iftands which are called the Manillas, and the other between the continent and Borneo. 31. The next thing to be confidered is the effed, which thefe fluids of the planets have upon the folid part of the bodies to which they belong. And in the firft place I fhall fhew, that it was necefi^ary upon account of thefe fluid parts to form the bodies of the planets into a figure fome- thing different from that of a perfed: globe. Becaufe the diurnal rotation, which our earth performs about its axis, and the like motion we fee in fome of the other planets, which Chap. 6. PHILOSOPHY. 297 (which is an ample convidlion that they all do the like) will diminifh the force, with which bodies are attracted upon all the parts of their fiirfaces, except at the very poles, upon which they turn. Thus a ftone or other weighty fubftance refting upon the furface of the earth, by the force which it receives from the motion cgmmunicated to it by the earth, if its weight prevented not , would con- tinue that motion in a ftraight line from the point where it received it, and according to the diredlion, in which it was given, that is, in a line which touches the furface at that point ; infomuch that it would move off from the earth in the fame manner, as a weight faften'd to a ftring and whirled about endeavours continually to recede from the center of motion , and would forthwith remove it felf to a greater diftance from it, if loofed from the ftring which retains it. And farther, as the centrifugal force, with which fuch a weight preffes from the center of its motion, is greater, by how much greater the velocity is, with which it moves ; fo fuch a body, as I have been fuppofing 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 furface it refts upon is moved, that is, the farther diftant 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 v/eaker than it, their weight, that is, the de- gree of force, with which they are preffed to the earth, will be diniiniflied thereby, and be the more diminiflicdy Q^q ths 298 Sir Isaac N e w t o n's Book II. the greater this contrary effort is j or in other words, the fame 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 lofe of its weight more and more , and be lighteft of all at the equator, that is, in the middle between the poles. 33. This now is eafily applied to the waters of the feas, and fhews that the water under the poles will prefs more forci- bly to the eardi, than at or near the equator : and confequent- ly that which preffes leaft, muft give place, till by afcend- ing it makes room for receiving a greater quantity, which by its additional weight may place the whole upon a ballance. To illuftrate this more particularly I fhall make ufe of fig. 116 In which let A C B D be a circle, by whofe revolution about the diameter A B a globe fhould be formed, reprefenting a globe of folid earth. Suppofe this globe covered on all fides with water to the fame height, fuppofe that of E A or B F, at which diftance tlie circle EGFH furrounds the circle A C B D ; then it is evident, if the globe of earth be at reft, the water which furrounds it will reft in that fituation. But if the globe be turned inceffantly about its axis AB, and the water have likewife the fame motion, it is alfo evident, from what has been explained, that tlie water be- tween the circles EHFG and ADBC will remain no longer in the prefent fituation, 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 ; fo that the water over the poles A and B mufl of nccefllty fubfide, and the water Chap.^ the angle under OS P. Now I fay, if thefe two angles are equal the arch OP is longer than TV: for the figure MILK being oblong, and I K longer than M L, the figure will be more incurvated toward I than toward L ; fo that the lines T W and V X will meet in Y before they are drawn out to fo great a length as the lines PQ. and OR muft be continued to, before they will meet in S. Since therefore YT and y V are fhorter than P S and S V, TV muft be lefs than OP. If thefe angles under TYV and OSP are each ~ part of the angle made by a perpendicular line, they are faid each to contain one degree. And the unequal length of thefe arches O P and V T gives occafion to the affertion, that in pafTmg from north to fouth the degrees on the earth's fur- face are not of an equal length, but thofe near the pole longer than thofe 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 furface. 42. This figure of the earth has fome effcdi: on eclipfes. It has been obferved above, that fometimes the nodes of the moon's orbit lie in a ftraight line drawn from the fun to the earth; in which cafe the moon will crofs the plane of the earth's motion at the new and full. But whenever the moon pafTcs near the plane at the full, fome part of the earth will intercept the fun's light, and the moon fhining only with light borrow'd from the fun, when that light is prevented from falling on any part of the moon, fo much cf her body will be darkened. Alfo when the moon at the new. Chap. 6. PHILOSOPHY. 307 new is near the plane of the earth's motion, the inhabitants on fome part of the earth will fee the moon come under the fun, and the fun thereby be covered from them either wholly or in part. Now the figure, which we have fhewn to belong to the earth, will occafion the fhadow of the earth on the moon not to be perfedly round, but caufe the diameter from eaft to weft to be fome what longer than the diameter from north to fouth. In eclipfes of the fun this figure of the earth will make fome little difference in the place, where the fun fhall appear wholly or in any given part covered. Let ABCD (in fig. ii8.) reprefent the earth, A C the axis whereon it turns daily, E the center. Let F A G C reprefent a perfeft globe infcribed within the earth. Let H I be a line drawn through the centers of the fun and moon, crof- fing the furface of the earth in K, and the furface of the globe infcribed in L. Draw E L, which will be perpendicular to the furface of the globe in L : and draw Hkewife K M, fo that it fhall be perpendicular to the furface of the earth in K. Now whereas the eclipfe would appear central at L, if the earth were the globe A G C F, and does really appear fo at K ; I fay, the latitude of the place K on th« 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 furface of the earth at that place makes with the axis ; the difference between this angle, and that made by a perpendicular line or •fquare being called the latitude of each place. But it might here be proved, that the angle which K M makes with M C is lefs, than the angle made between L E and E C : confe- R r X quently 308 Sir Isaac Newton's BookIL quently the latitude of the place K is greater, than the la- titude, which the place L would have. 4g. The next effed", which follows from this figure of the earth, is that gradual change in the diftance of the fix- ed fiars from the equinoftial points, which aftronomers ob- ferve. But before this can be explained, it is neceflary to- fay fomething more particular, than has yet been done, concerning the manner of the earth's motion round the fun, 44.. I T has already been laid, that the earth turns round each day on its own axis, while its whole body is carried round the fun once in a year. How thefe two motions are joined together may be conceived in fome degree by the motion of a bowl on the ground, where the bowl in rouling on continually turns upon its axis, and at the lame time the whole body thereof is carried flxaight on. But to be more exprefs let A (in fig. lip) reprefent the fun BCDE four difi'erent fituations of the earth in its orbit moving about the fun. In all thefe let FG reprefent the axis, about which the earth daily turns. The points F, G are called the poles of the earth; and this axis is fuppo- fed to keejaalways parallel to it felf in every fituation of the earth ;\at leaft that it would do fo, were it not for a. minute deviation, the caufe whereof will be explained in what follows. When the eardi is in B, the half H I K will be illuminated by the fun, and the other half HLK will be in darknefs. Now if on the globe any point be taken in Chap. 6. PHILOSOPHY. gop. in the middle between the poles, this point fhall defcriber 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 fuppofed to move round its axis with an equable motion ; therefore on this point of the- globe the fun will be feen juft half the day, and be invlfi- ble the other half. And the fame will happen to every point of this circle, in all fituations of the earth during its whole revolution round the fun. This circle M N is calledC the equator, of which we have before made mention. 45-. Now fuppofe any other point taken on the furface: of the globe toward the pole F, which in the diurnal re- volution of the globe fhall defcribe the circle OP. Here: it appears that more than half this circle is enlightned by the fun, and confequently that in any particular point of this circle the fun will be longer feen than lie hid, that is the day will be longer than the night. Again if we con- iider the fame circle OP on the globe fituated in D the op- pofite part of the orbit from B, we fhall fee, that here in any place of tliis circle the night will be as much longer, than the day. 46. I N thefe Situations of the globe of earth a line drawn from the fun to the center of the earth will be obliquely incHned toward the axjs F G. Now fuppofe, that ftich a line drawn from the fun to the center of the earthj, when in C or E, would be perpendicular to the. axis F.Gj , 3 1 o Sir I s A A c N E w T o n's Book II. in which cafes the i'un will fhine perpendicularly upon the equator, and confequently the line drawn from the center of the earth to the fun will crofs the equator, as it pafies through the furface of the earth ; whereas in all other fitu- ations of the globe this Hne will pafs through the furface of the globe at a diftance from the equator either north- ward or fouthward. Now in both thefe cafes half the cir- cle O P will be in the light, and half in the dark ; and there- fore to every place in this circle the day will be equal to the night. Thus it appears, that in thefe two oppodte fituations of the earth the day is equal to the night in all parts of the globe ; but in all other fituations this equality will only be found in places fituated in the very middle •between the poles, that is, on the equator. 4,7. The times, wherein this univerfal equality between the day and night happens, are called the equinoxes. Now it has been long obferved by aflronomers, that after the earth hath fet out from either equinox, fuppofe from E (which will be the fpring equinox, if F be the north pole) the fame equinox fhall again return a little before the earth has made a compleat revolution round the fun. This re- turn of the equinox preceding the intire revolution of the £arth is called the preceiTion of the equinox, and is caufed hj the protuberant figure of the earth. 4,8. Since the fun fhines perpendicularly upon the e- (t^uator, when the line drawn from the fun to the center cjf the earth is perpendicular to the earth's axis, in this cafe the Chap. 6. PHILOSOPHY. 311 the plane, which fhould cut through the earth at the e- quator, may be extended to pafs through the fun; but it will not do fo in any other pofition of the earth. Now let us confider the prominent part of the earth about the equator, as a folid ring moving with the earth round the fun. At the time of the equinoxes, this ring will have the fame kind of fituation in refpeft of the fun, as die orbit of the moon has, when the line of the nodes is di- reded to the iun ; and at all other times will refemble the moon's orbit in other fituations. Confequently this ring, which otherwife would keep throughout its motion paral- lel to it felf, will receive fome change in its pofition from the adlion of the fun upon it, except only at the time of the equinox. The manner of this change may be under- ftood as follows. Let A B C D (in fig. no) reprefent this ring, E the center of the earth, S the fun , A F C G a circle de- fcribed 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 e- quator crofies the plane of the earth's motion ; and the time- of the equinox tails out, when the firaight line A C con- tinued would pafs through the fun. Now let us recollect what was faid above concerning the moon, when her or- bit was in the fame fituation with this ring. From thence" it will be underftood, if a body were fuppofed to be mo- ving in any part of this circle A B C D, what effed the adi- on of the fun on the body would have toward changing the pofition of the line AC. In particular HI 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 AC 3 would; ^i 2 Sir I s A A c N E w T n's Book IL would be fo turned, that the point A iliall move toward B, and the jx)int 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 AC would be turned the contrary way. Hence it follows, that as this folid ring turns round the center of the earth, the parts of this ring between A and H, and between C and I, are fo influenced by the fun, that they will endeavour, fo to change the Situation of the line AC as to caufe 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 oppoflte tendency, and difpofe the line AC to move the contrary way. And flnce thefe laft named parts are larger than the other, tliey will prevail over the other, fo that by the adion of the fun upon this ring, the line AC will be fo turned, that A fhall continually be more and more moving toward D, and C toward B. Thus no fooner fhall the fun in its viflble motion have departed from A, but the mo- tion of the line AC fliall haften its meeting with C, and from thence the motion of this line (hall again haften the fun's fecond conjunction v/ith A ; for as this line fo turns, that A is continually moving toward D, fo the fun's vifible motion is the fame way as from S toward T. 49. The moon will have on this ring the like efFedl as the fun, and operate on it more ftrongly, in the fame pro- portion as its force on the fea exceeded that of the fun on the fame. But the effed of the adion of both luminaries will be gi'eady diminifhed by reafon of tliis ring's being conned- ed r Chap. 6. PHILOSOPHY. 313 cd to the reft of the earth ; for by this means the fun and moon have not only this ring to move, but Hkewife the whole globe of the earth, upon whofe fpherical part they have no immediate influence. Befide the effed: is alfo rendred lels, by reafon that the prominent part of the earth is not colleded all under the equator, but fprcads gradually from thence toward both poles. Upon the whole, though the fun alone carries the nodes of the moon through an intire revolution in about 19 years, the united force of both lu- minaries on the prominent parts of the earth will hardly carry round the equinox in a lefs fpace of time than z6ooo years. SO. To this motion of the equinox we mufl: add ano- ther confequence of this action of the fun and moon up- on the elevated parts of the earth, that this annular part of the earth about the equator, and confequently 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 reftored, juft as the inclination of the moon's orbit by the adion of the fun is annually twice diminifhed, and as often recovers its original magnitude. But this change is very infenfible. 5"!. I fliall now finiih the prefent chapter with our great author's inquiry into the figure of the fecondary planets, par- ticularly of cur moon, upon the figure of which its fluid parts will have an influence. The moon turns always die fame fide towards the earth, and confequently revolves but once round its axis in the fpace of an entire month; S f for Q^ I ^|. Sir I s A A c N E w T o n's Book II. for a fpedator placed without the circle, in which the moon moves, would in that time obferve all the parts of the moon fucceflively to pafs once before his view and no more, that is, that the whole globe of the moon has turned once round. Now the great flownefs of this motion v/ill render the cen- trifugal force of the parts of the waters very weak, fo that the figure of the moon cannot, as in the earth, be much af- fed:ed by this revolution upon its axis : but the figure of thofe waters are made different from fpherical by another caufe, viz. the adion of the earth upon them ; by which tliey will be reduced to an oblong oval form, whofe axis prolonged would pafs through the earth ; for the fame reafon, as v/e have above obferved, that the waters of the earth would take the like figure, if they had moved fo flowly, as to keep pace with the moon. And the folid part of the moon rauft correfpond with this figure of the fluid part : but this ele- vation of the parts of the moon is nothing near fo great as is the protuberance of the earth at the equator, for it will not exceed 93 engHfli feet. 5-1. The waters of the moon will have no tide, except what will arife from the motion of the moon round the earth. For the converfion of the moon about her axis is e- quable, whereby the inequality in the motion round the earth difcovers to us at fome times fmall parts of the moon's furface towards the eafl: or weft, which at other times lie hid ; and as the axis, whereon the moon turns, is oblique to her motion round the earth, fometimes fmall parts of her furface Chap. 6. PHILOSOPHY. 315 furface toward the north, and fometimes the like toward the fouth are vifible, which at other times are out of fight* Thefe appearances make what is called the libration of the: moon, difcovered by Hevehus. But now as the axis of the oval figure of the waters will be pointed towards the earth, there muft arife fi-om hence fome fluduation in them ; and befide, by the change of the moon's diftance from the earth, they will not always have the very fame height. S f 2 BOOK III. 31(5 Sir I s A A c N E w T o n's Book ' /' tn^ Jt4t//t. BOOK III. Chap I, Concerning the caufe of COLOURS inhe- rent in the LIGHT. F TE R this view which has been taken of Sir I s A A c N E w T o n's mathema- tical principles of philofophy, and the ufe he has made of them, in explain- ing the fyftem of the world, &c. the courfe of my defign direds us to turn our eyes to that other philofophical work, his treatife of Optics, in which we Hiall find our great author's inimitable genius difcovering it felf no lefs, than in the Chap. I. PHILOSOPHY. 317 the former ; nay perliaps even more, fince this work rrjves as many inftances of his fingular force of reafoning, and of his unbounded invention, though unaflifted in great meafure by thofe rules and general precepts, which facili- tate the invention of mathematical theorems. Nor yet is this work inferior to the other in ufefulnefs ; for as that has made knov/n to us one great principle in nature, by which the celeflial motions are continued, and by which the frame of each globe is preferved ; fo does this point out to us another principle no lefs univerfal, upon which de- pends all thofe operations in the fmaller parts of matter, for whofe fake the greater frame of the univerfe is ereded ; all thofe immenfe globes, with which the whole heavens are filled, being without doubt only defign'd as fo many con- venient apartments for carrying on the more noble opera- tions of nature in vegetation and animal life. Which fin- gle confideration gives abundant proof of the excellency of our author's choice, in applying himfelf carefully to ex- amine the adion between light and bodies, fo neceffary in all the varieties of thefe produdions, that none of them can be fuccefsfully promoted without the concurrence of heat in a greater or lefs degree. 1. ' T I s true, our author has not made fb full a diicovery of the principle, by which this mutual aftion between light and bodies is caufed ; as he has in relation to the power, by which the planets are kept in their courfes: yet he has led us to the very entrance upon it, and pointed out the path fo plainly which muft be followed to reach it ; that one may he ^i8 Sir Isaac N e w t o n's Book III, be bold to fay, whenever mankind fliall be blefied with this improvement of their knowledge, it will be derived fo di- redlly from the principles laid down by our author in this book, that the greateft fhare of the praife due to the dif- covery will belong to him. 5. In fpeaking of the progrefs our author has made, I fhall diftinftly purfue three things, the two firft relating to the colours of natural bodies : for in the firft head fhall be fhewn, how thofe colours are derived from the proper- ties of the light itfelf ; and in the fecond upon what properties of the bodies they depend : but the third head of my difcourfe fliall treat of the adion of bodies upon light in refrading, refleding, and infleding it. 4,. The firft of thefe, which Ihall be the bufinels of the prefent chapter, is contained in this one propofition : that the fun's dired light is not uniform in refped of colour, not being difpofed in every part of it to excite the idea of white- nels, which the whole raifes ; but on the contrary is a com- pofition of different kinds of rays, one fort of which if a- lone would give the fenfe of red, another of orange, a third of yellow, a fourth of green, a filth of light blue, a fixth of indigo, and a feventh of a violet purple ; that all thefe rays together by the mixture of their fenfations imprefs upon the organ of fight the fenfe of whitenefs, though each ray always imprints there its own colour j and all the difference between the colours of bodies when view- ed in open day light arifes from this, that coloured bodies do Chap. I : PHILOSOPHY. 319 do not reflcd all the forts of rays falling upon tlicm in e- qual plenty, but fome forts much more copiouHy than o- thers ; the body appearing of that colour, of which die light coming from it is mod compofed. 5-. That the light of the fun is compounded, as has been faidj is proved by reirading it with a prifm. By a prifm I here mean a glafs or other body of a triangular form, fuch as is re- prefented in fig. 1 1 1 . But before we proceed to the illuflration of the propofition we have juft now laid down, it will be ne- ceflary to fpend a few words in explaining what is meant by the refradion of light j as the defign of our prefent labour is to give fome notion ol the lubjecl, we are engaged in, , to fuch as are not verfed in the mathematics. 6. It is well knov/n, that w^hen a ray of light pafling through the air falls obliquely upon the furface of any tranf- parent body, fuppofe water or glafs, and enters it, the ray will not pafs on in that body in the fame line it delcribed through the air, but be turned off from the furface, fo as to be lefs inclined to it after pafling it, than before. Let ABCD (in fig. ill.) reprefent a portion of water, or glais, A B the furface of it, upon which the ray of light E F fall* obliquely ; this ray fiiall not go right on in the courfe de- lineated by the line FG, but be turned off from the fur- face A B into the line F H, lefs inclined to tlie furface A B than the line EF is, in wliich the ray is incident upon that furface^ 4 7, Oh 320 Sir Isaac N e w t o n's Book III. 7. O N the other hand, when the Hght paffes out of a- ny fuch body into the air, it is inflected the contrary way, being after its emergence rendred more oblique to the fur- face it paffes through, than before. Thus the ray F H, when it goes out of the furface C D, will be turned up towards that furface, going out into the air in the line HI. 8. This turning of the Hght out of its way, as it paffes from one tranfparent body into another is called its refradlion. Both thefe cafes may be tried by an eafy experiment with a bafon and water. For the firft cafe fet an empty bafon in the funfhine or near a candle, making a mark upon the bottom at the extremity of the fhadov/ cad by the brim of the bafon, then by pouring water into the bafon you v/ill obferve the fhadow to flirink, and leave the bottom of the bafon enlightned to a good diftance from the mark. Let ABC (in fig. 1 13.) denote the empty bafon, EAD the light fliining over the brim of it, fo tliat all the part A B D be fhaded. Then a mark being made at D, if water be poured into the bafon (as in fig. 1x4.) to FG, you fhall ob- ferve tlie light, which before went on to D, now to come much fhort 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 fhews that the ray E A, when it enters the water at I, goes no longer ftraight forwards, but is at that place incurvated, and made to go nearer die perpendicular. The other cafe may be tryed by putting any fmall body into an empty bafon, placed lower than your eye, and then receding from the bafon, till you can but juft fee Chap. I. PHILOSOPHY. 321 fee the body over the brim. After which, if the bafon be filled with water, you fliall picfcntly obfcrve the body to be vifiblc, though you go farther off from the bafon. Let ABC (ill fig- iiy.) denote the bafon as before, D the body in it, E the place of your eye, when the body is feen juft over the edge A, while the bafon is empty. If it be then filled with water, you will obferve the body ftill to be vifible, though you take your eye farther off. Suppofe you fee the body in this cafe juft 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 ftraight on, but are bent at A, being turned downwards, and more inclined to the furface of the water, between A and your eye at F, than they are between A and the body D. 9. This we hope is fufiicient to make all our readers apprehend, what the writers of optics mean, when they mention the refradion of the light, or fpeak of the rays of light being refracfted. We fhall therefore now go on to prove the afiertion advanced in the forementioned propofition, in relation to the different kinds of colours, that the dired hght of the fun exhibits to our fenfe ; which may be done in the following manner. 10. If a room be darkened, and the fun permitted to fhine into it through a fmall hole in the window fliutter, and be made immediately to fall upon a glals prifm , the beam of light fhall in pafiing through fuch a prifm be parted into rays, which exhibit all the forementioned colours. In this man- T t Kcr c^22 Sir Is A AC Newton's Book III. ner If AB(in fig. iz6) reprefent the window fliutter ; C the hole in it ; D E F the prifm ; Z Y a beam of Hght coming from the fun, which pafTes through the hole, and falls up- on the prifm at Y, and if the prifm were removed would go on to Xj but in entring the furface BF of the glafs it fhall be turned off, as has been explained, into the courle Y W falling upon the fecond furface of the prifm DF in W, going out of which into the air it fhall be again farther in- flected. Let the light now, after it has palTed the prifm, be received upon a fheet of paper held at a proper diflance, and it fhall paint upon the paper the pidure, image, or fpecl- rum L M of an oblong figure, whofe length fhall much ex- ceed its breadth j though the figure fhall not be oval, the ends L and M being femicircular and the fides ftraight. But now this figure will be variegated with colours in this manner. From the extremity M to fome length, fuppofe to the line nOy it fhall be of an intenfe red; from no to pq it fhall be an orange ; from pq to rs it fliall be yel- low; from thence to ^u it fhall be green; from thence to w X blue ; from thence to j^ z indigo ; and from thence to the end violet. II. Thus it appears that the fun's white light by its paf- lage through the prifm, is fo changed as now to be divi- ded into rays, which exhibit all thefe feveral colours. The queftion is, whether the rays while in the fun's beam be- fore this refraction poffeffed thefe properties diftindily; fo that fome part of that beam would without the reft have given a red colour, and another part alone have given an orange. />/>. E 9. OA ^ 0-- c Chap. I. PHILOSOPHY. 323 orange, &c. That this is pofTible to be the cafe, appears from hence; that if a convex glafs be placed between the paper and the prifm, which may colled all the rays proceeding out of tlie prifm into its focus, as a burning glafs does the fun's diredl rays ; and if that focus fall upon the paper, the fpot formed by fuch a glafs upon the paper fhall appear white, juft like the fun's diredl light. The reft remaining as before, let PQ_ (in fig. 117.) be the convex glals, cau- fing the rays to meet upon the paper HGIK in the point Nj I fay that point or rather fpot of light fhall appear white, without the leaft tinfture of any colour. But it is evident that into this fpot are now gathered all thofe rays, which be- fore when feparate gave all thofe different colours ; which fhews that whitenefs may be made by mixing thofe colours : efpecially if we confider, it can be proved that the glafs PQ. does not alter the colour of the rays which pafs through it. Which is done thus : if the paper be made to approach the glafs PQ^, the colours will manifeft them- felves as far as the magnitude of the fpedirum, which the paper receives, will permit. Suppofe it in the fituation /jg ik, and that it then receive the fpedrum /w, this fpedtrum fliall be much fmaller, than if the glals PQ^ were removed, and therefore the colours cannot be (o much feparated ; but yet the extremity m fliall manifeflly appear red , and the other extremity / fhall be blue; and thefe colours as well as the intermediate ones fhall difcover themfelves more perfedly, the farther the paper is removed from N, that is, the larger the fpeftrum is : the fame thing happens, if the paper be removed farther ofF from P Q. than N. Sup- T t X pof: 324 Sirls A AC Nk vvton's Book III. pofe into the pofition ^y-fix^ the fpedriim x^i painted upon it iliall again difcover its colours, and that more diflin^tly, the farther the paper is removed, but only in an inverted or- der: for as before, when the paper was nearer the con- vex glafs, than at N, the upper part of the image was blue, and the under red ; now the upper part fhall be red, and the under blue : becaufe the rays crofs at N. I a. Nay farther that the whitenefs at the focus Nj is made by the union of the colours may be proved without re- moving the paper out of the focus, by intercepting with any opake body pait of the light near the glafs; for if the under part, that is the red, or more properly the red-making rays, as they are ftyled by our author, are intercepted, the fpot fhall take a bluifh hue ; and if more of the inferior rays are cut off, fo that neither the red-making nor orange- making rays, and if you pleafe the yellow-making rays like- wife, fhall fall upon the fpot ; then fliall the fpot 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 fpot fhall turn reddifh, and become, more fo, the more of thofe oppofite colours are intercepted 13. T H I s I think abundantly proves that whitenefs may be produced by a mixture of all the colours of the fpec- tr.um. At leaft there is but one way of evading the pre- fcnt arguments, which is, by afferting that the rays of light after paffmg the prifm have no different properties to ex- hibit this or the other colour, but are in that refpedl per- fealy Chap. I. PHILOSOPHY. 325 l^edly homogeneal, fo that the rays which pafs to the un- der and red part of the image do not differ in any pro- perties whatever from thofe, which go to the upper and violet part of it ; but that the colours of the fpedrum are produced only by fome new modifications of the rays, made at their incidence upon the paper by the different termi- nations of light and fliadow : if indeed this affertion can be allowed any place, after what has been faid ; for it feems to be fufficiently obviated by the latter part of the pre- ceding experiment, that by intercepting the inferior part of the light, which comes from the prifm, die white fpot fhall receive a bluifh caft, and by flopping the upper part the' fpot fhall turn red, and in both cafes recover its colour, when the intercepted light is permitted to pafs again ; though in all thefe trials there is the like termination of light and' fhadow. However our author has contrived fome experi- ments exprefly to fhew the abfurdity of this fuppoHtion ; all which he has explained and enlarged upon in fo dif- tinft and expreflive a manner, that it would be wholly un- neceffary to repeat ^them in this place \ I fhall only men- tion 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 fpot N, the ftraight line w x parallel to the horizon, and then if the paper be much inclined into the fituation rji;/ the line 71/ x ftill remaining parallel to the horizon, the fpot N fliall lofe its whitenefs and receive a blue tin- cture ; but if it be inclined as much the contrary way, the: fame fpot fhall exchange its white colour for a reddifh dye., ' Cpr, B. I. part i. prop- i. All -^2(5 Sir Isaac N e w t o n's Book III. A 11 which can never be accounted for by any difference in the termination of the light and fhadow, which here is ^one at all ; but are eafily explained by fuppoiing the upper part of the rays, whenever they enter the eyc^ difpofed to give the fenfation 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 ill the fituation ri'/^^, it is plain that the upper part of the light falls more diredly upon it, than the under part, and therefore thofe rays will be moft plentifully reflecled from it ; and by their abounding in the reflected light will caufe it to incline to their colour. Juft fo v/hen the paper is inclined the contrary way, it will receive the inferior rays moft diredly, and tlierefore ting the light it refleds with their colour. 14.. It is now to be proved that thefe difpofitions of the Tays of light to produce fome one colour and fome another, which manifeft themfelves after their being refraded, are not wrought by any adion of the pmfm upon them, but are originally inherent in thofe rays ; and that the prifm only affords each fpecies an occafion (^ fhewing its diftind qua- lity by feparating them one from another, which before, while they were blende^ together in the dired beam of the fun's light, lay conceal'd. But that this is ^o, will be pro- ved, if it can be fhewn that no prifm has any power upon the rays, which after their paffage through one prifm are rendered uncompounded and contain in them but one co- lour, either to divide that colour into feveral, as the fun's hght Chap. I. PHILOSOPHY. 327 light is divided, or fo much as to change it into any other colour. This will be proved by the following experiment \ The fame thing remaining, as in the firft experiment, let another prifm NO (in fig. ix8.) be placed either immedi- ately, or at fome diftance after the firft, in a perpendiculai pofture, fo that it fliall refrad: the rays iffuing from the firft fideways. Now if this prifm could divide the light falling upon it into coloured rays, as the firft has done, it would divide the fpedrum breadth wife into colours , as ■ before it was divided lengthwiie ; but no fiich thing is ob- ferved. If LM were the fpedirum, which the firft prifm: DEF would paint upon the paper HGIK; P(^ lyhig in an obhque pofture fliall be the fpeftrum projeded by the fecond, and ihall be divided lengthwife into colours cor- refponding to the colours of the fpedrum LM, and occa- fioned like them by the refraction of the firft prifm, but its breadth fhall receive no fuch divifion.; on the contrary each colour fhall be uniform from fide to fide, as much as in the fpedlrum L M, which proves the whole aftertioii. ly. The fame is yet much farther confirmed by ano- ther experiment. Our author teaches that the colours of the fpedrum L M in the firft experiment are yet compound- ed, though not fo much as in the fun's dired; light.. He fhews therefore how, by placing the prifm at a diftance from the hole, and by the ufe of a convex glafs, to feparate'die- colours of the fpedrum, and make them uncompounded to any degree of exaftnefs '\ And he fhews when tliij » Newt. Opt. B. I. part i, expe.'im. <•. ^ Ibid prop. 4, is. ^28 Sir Is AAc Newton's BookIII. is done fufficiently, if you make a fmall hole in the paper whereon the fped:rum is received, through which any one fort of rays may pafs, and then let that coloured ray fall fo upon a prifm , as to be refraded by it, it fhall in no cafe whatever change its colour ; but fhall always retain it perfedly as at firft, however it be refraded \ id. Nor yet will thefe colours after this full feparation of them fuffer any change by refledion from bodies of dif- ferent colours; on the other hand they make all bodies pla- ced in thefe colours appear of the colour which falls upon them^: f-^r 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 verdigreafe, or blue bife, will put on the appearance of that colour, in which it is placed : fo that neither bife placed in the red light fhall be able to give that light the leaft blue tindure, or any other dijffe- rent from red; nor fhall minium in the indigo or violet light exhibit the leaft appearance of red, or any other co- lour diftind from that it is placed in. The only difference is, that each of thefe bodies appears moft luminous and bright in the colour, which correfponds with that it exhibits in the day light, and dimmeft in the colours moft remote from that ^ that is, though minium and bife placed in blue light iliall both appear blue, yet the bife fhall appear of a bright blue, and the minium of a dusky and obfcure blue: but » Newt. Opt. B. I. part i. exper. /. •> Ibidexper. (J. 3 if Chap. i. PH I L O S O P H Y. 329 if minium and bife be compared together in red light, the minium fhali afford a brisk red, the bile a duller colour, though of the fame (pecies. 17. And this not only proves the immutability of all thefe fimple and uncompounded colours ; but likewife un- folds the whole myflery, why bodies appear in open day- light of fuch different colours, it confifting in nothing more than this, that whereas the white light of the day is com- pofed of all forts of colours, fome bodies refled; the rays of one fort in greater abundance than the rays of any other *. Though it appears by the forecited experiment, that almoft all thefe bodies rcfled: fome portion of the rays of every colour, and give the fenfe of particular colours only by the predominancy of fome forts of rays above the reft. And what has before been explained of compofing white by mingling all the colours of the fpedrum together ihews clearly, that nothing more is required to make bodies look white, than a power to refled: indifferently rays of every colour. But this will more fully appear by the following method: if near the coloured fpeftrum in our firft experiment a piece of white paper be fo held, as to he illuminated equally by all the parts of that fpedlrum, it fhall appear white ; where- as if it be held nearer to the red end of the image, than to the other, it fhall turn reddifh j if nearer tlie blue end, it fhall feem bluifh ^ ^ . ' Newton Opt. B. I. prop. 10. * Ibid exp. 9. U u 18. Our g ^o Sir Isaac N e w t o n's Book III. 1 8. Our indefatigable and circumfped author farther examined his theory by mixing the powders which paint- ers ufe of feveral colours, in order it poflible to produce a white powder by fuch a compofition ^ But in this he found fome difficulties tor the following reafons. Each of thefe coloured powders reflects but part of the light, which is caft upon them; the red powders refleding little green or blue, and the blue powders refledling very little red or yelloWj nor the green powders refleding near fo nmch of the red or indigo and purple, as of the other colours : and befides, when any of thefe are examined in homogeneal light, as our author calls the colours of the prifm, when well fe- parated, though each appears more bright and luminous in its own day-light colour^ than in any other ; yet white bo- dies, fuppofe white paper for inftance, in thofe very colours exceed thefe coloured bodies themfelves in brightnefs ; fo that white bodies refled not only more of the whole light than coloured bodies do in the day-light, but even more of that very colour which they lefledl moft copioufly. All which confiderations make it manifefl: that a mixture of thefe will not refle<5t fo great a quantity of light, as a white body of the fame fize ; and therefore will compofe fuch a colour as would refult from a mixture of white and black, fuch as are all grey and dun colours, rather than a ftrong white.. Now fuch a colour he compounded of certain ingredients which he particularly fets down, in fo much that when the compofition was ftrongly illuminated by the fun's diredt beams, it v^^ould appear much whiter than even white pa- » Newt. Opt. B. I. part i. cxp if, per,, Chaf.j: philosophy. 531 per, if confiderably fliadcd. Nay he found by trials Jiow to proportion the degree ot ilkimination of the mixture and paper, fo that to a fpectator at a proper diftance it could not well be determined which was the more perfect colour ; as he experienced not only by himfelf, but by the concurrent opinion of a friend, who chanced to vifit him while he was trying this experiment. I muft not here o- mit another method of trying the whitenefs of fuch a mix- ture, propofed in one of our author's letters on this fub- jed*: which is to enlighten the compoiition by a beam of the fun let into a darkened room , and then to receive the light refle6led from it upon a piece of white paper, obfer- ving whether the paper appears white by that refledion ; for if it does, it gives proof of the compoiition's being white ; becaufe when the paper receives the reflexion from any coloured body, it looks of that colour. Agreeable to this is the trial he made upon water impregnated with foap, and agitated into a froth ^ : for when this froth after fomc fliort time exhibited upon the little bubbles, which compo- fed it, a great variety of colours, though thefe colours to a fpedator at a fmall diilance difcover'd themfclves diftinftly ; yet when the eye was fo far removed, that each little bub- ble could no longer be diftinguifhed, the whole froth by the mixture of all thefe colours appeared intenfly white. 19. Our author having fully fatisfied himfelf by thefe and many other experiments, what the refult is of mixing » Philof. Tranfad. N. 88, p. J099. >> Opt. B. I. par. 2. exp. 14. U u 1 together 532 Sir Isaac Newton's BookIII, together all the prifmatic colours ; he proceeds in the next place to examine, whether this appearance of whitenefs be raifed by the rays of thefe different kinds adling fo, when thev meet, upon one another, as to caufe each of them to imprefs the fenfe of whitenefs upon the optic nerve ; or. whe- ther each ray does not make upon the organ of light the fame impreflion, as when feparate and alone ; fo that the idea of whitenefs is not excited by the impreflion from any one part of the rays, but refults from the mixture of all thole different fenfations. And that the latter fentiment is the true one, he evinces by undeniable experiments. ao. I N particular the foregoing experiment ^ , wherein the convex glafs was ufed, furnifhes proofs of this : in that when the paper is brought into the lituation 8y)tx., beyond N the colours, that at N difappeared, begin to emerge again ; which fhews that by mingling at N they did not lofe their colorific qualities, though for fome reafon tliey lay conceal- ed. This farther appears by diat part of the experiment, when the paper, while in the focus, was directed to be en- clined different ways ; for when the paper was in fuch a lituation, that it muft of neceflity refleft the rays, which before their arrival at the point N would have given a blue colour, thofe rays in this very point itfelf by abounding in the refleded light tinged it with the fame colour ; fo when the paper relleds moft copioufly the rays, which before they come to the point N exhibit rednefs, thofe fame rays tia- ' Ibid. cxp. to. dure Chap. I. PHILOSOPHY. 333 dure the light refleded by the paper from that very poinc with their own proper colour. II. There is a certain condition relating to iight, which affords an opportunity of examining this ftill more fully: it is this, that the imprefTions of light remain fome fliort Ipace upon the eye ; as when a burning coal is whirl'd about in a circle, if the motion be very quick, the eye fhall not be able to diftinguifli the coal, but fhall fee an entire circle of fire. The reafon of which appearance is, that the im- prefTion made by the coal upon the eye in any one fituation is not worn out, before the coal returns again to the fame place, and renews the fenfation. This gives our author the hint to trjj whether thefe colours might not be tranfmitted fucceilively to the eye fo quick, that no one of the colours fhould be diflindlly perceived, but the mixture of the fen- fations fliould produce a uniform whitenefs ; when the rays could not ad: upon each other, becaufe they never fhould meet, but come to the eye one after another. And this thought he executed by the following expedient ^ He made an in- flrument in fhape like a comb, which he applied near the convex glafs, fo that by moving it up and down flowly the teeth of it might intercept fometimes one and fometimes. another colour ; and accordingly the light refleded from the paper, placed at N, fhould change colour continuaiiy. But now when the comb-like inftrument was moved verj' quick^ the eye loft all preception of the diflind colours, which came to it from time to time, a perfed whitenefs refulting from the • Opt. pag. I iii mixture 334- ^^^ Isaac Newton's Book III. mixture of all thofe diftindl imprefiions in the fenforium. Now in this cafe there can be no fufpicion of the feveral coloured rays ading upon one another, and making any change in each other's manner of aifeding the eye, feeing they do not (o much as meet together there. ii. Our author farther teaches us how to view the fpec- trum of colours produced in the firft experiment with a- nother prifm , fo that it fhall appear to the eye under the fhape of a round fpot and perfedly white *. And in this cafe if the comb be ufed to intercept alternately fome of the colours, which compofe the fpedrum, the round fpot fhall change its colour according to the colours intercepted ; but if the comb be moved too fwiftly for thofe changes to be diftindly 'perceived, the fpot fhall feem always white, as before *". ag. Besides this whitenefs, which refults from an uni- verfal compolition of all forts of colours, our author par- ticularly explains the effeds of other lefs compounded mix- tures; fome of which compound other colours like fome of the fimple ones, but others produce colours different from any of them. For inftance, a mixture of red and yellow compound a colour like in appearance to the orange, w^hich in the fpedlrum lies between them; as a compofition of yel- low and blue is made ufe of in all dyes to make a green. But red and violet purple compounded make purples un- like to any of the prifmatic colours, and thefe joined with » Opt, B. I. J art 1. exp. ii. •• Ibid prop. 4, 6, yellow Chap. I. PHILOSOPHY. 335 yellow or blue make yet new colours. Befides one rule is here to be obferved, that when many different colours are mixed, the colour which arifes from the mixture grows languid and dege- nerates into whitenefs. 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 fhall firflgrow dull and lefs vivid, and at length by adding more of thefe colours it (hall turn to whitenefs, or fome other colour \ 14. Only here is one thing remarkable of thofe com- pounded colours, which are like in appearance to the fimple ones ; that the fimple ones when viewed through a prifm fhall flill retain their colour, but the compounded colours (cen through fuch a glafs fliall be parted into the fimple ones of which they are the aggregate. And for this reafon any body illuminated by the fimple light fliall appear through a prifm diftindlly, and have its minuteft parts obfervable, as may ea- fily be tried with flies, or other fuch little bodies, which have very fmall parts ; but the fame viewed in this manner when enlighten'd with compounded colours fhall appear confufed, their fmallefl; parts not being diflinguifliable. How the prifm feparates thefe compounded colours, as like wife how it divides the light of the fun into its colours, has not yet been explained , but is referved for our third chapter. If. In the mean time what has been laid, I hope, wifl". fufiice to give a tafte of our author's way of arguing, and m 33^ Sir Isaac Newton's Book III. in fome meafure to illuftrate the propofition laid down in this chapter. z6. There are methods of feparating the heterogene- ous rays of the fun's Hght by refiedion, which perfedly confpire with and coniii-m this reafoning. One of which ways may be this. Let AB (in fig. 119) reprefent the win- dow fhutter of a darkened room ; C a hole to let in the fun's rays ; D E F, G H I two prifms fo applied together, that the iides EF and GI be contiguous, and the fides DF, GH parallel ; by tliis means the light will pafs through them with- out any feparation into colours : but if it be afterwards re- ceived by a third prifm IKL, it fhall be divided fo as to form upon any white body P Q. the ufual colours , violet at W, blue at n^ green at which happens when the angle which the ray makes with the furface does not exceed about 49 -^ degrees ; but if the inclination be a very little increafed, great part of the light will be tranfmitted ; and how the light in one cafe fliould meet with nothing but the folid parts of the air, and by ib fmall a change of its inclination find pores in great plenty, is wholly inconceivable. It cannot be faid, that the light is reflected by ftriking againfl: the folid parts of the lurface of the glafs ; becaufe without making any change in that furface, only by placing water contiguous to it inflead of air, great part of that light fhall be tranfmitted, which could find no paffage through the air. Moreover in the laft ex- periment recited in the preceding chapter, when by turn- ing the prifms DEF, GHI, the blue light became wholly reflected, while the reft was moftly tranfmitted, no pofiible reafon can be afilgned, why the blue-making rays fhould meet with nothing but the folid parts of the air between the prifms, and the reft of the light in the very fame obli- quity find pores in abundance. Nay farther, when two glaff- Cj touch each other, no refledion at all is madej though k Chap. 2. PHILOSOPHY. 34.Z it does not in the leaft appear, how the rays Hiould avoid the folid parts of glafs, when contiguous to other glafs, any more than when contiguous to air. But in the lafl place upon this fuppolition it is not to be comprehended, how the mod: poli£hed fubftances could refled the light in that regular manner we find they do; for when a polifhed look- ing glafs is covered over with quick filver, we cannot fuppofe the particles of light fo much larger than thofe of the quick- filver, that they fhould not be fcattered as much in refled;ion> as a parcel of marbles thrown down upon a rugged pavement. The only caufe of fo uniform and regular a refledion muft be fome more fecret caufe, uniformly fpread over the whole fur- face of the glals. 4. B u T now, Unce the reflexion 6f light from bodies does not depend upon its impinging againft their folid parts, fome other reafon muft be fought for. And firft it is paft doubt diat the leaft parts of almoft all bodies are tranfpa- rent, even the microfcope {hewing as much ^ ; befides that it may be experienced by this method. Take any thin plate of the opakeft body, and apply it to a fmall hole defigned for the admifTion o^ light into a darkened room ; however opake that body may fcem in open day-light, it fliall un- der thefe circumftances fufficiently difcover its traufparency, provided only the body be very thin. White metals indeed do not eafily fhevv themfelves tranfparent in thefe trials, they lefleAing almoft all the light incident upon them at their firft fuperiicies ; the caufe of which will appear in what ' Opt. Book ir. par. 3. prop, u follows 34- Sir Is A AC Newton's Book III. follows \ But yet thefe fubflances, when reduced into parts of extraordinary minutenefs by being diflblved in aqua fortis or the like corroding liquors do alio become tranfparent. y. Since tlierefore the light finds free pafiage through the leafl parts of bodies, let us confider the largenefs' of their pores, and we fhall find, that whenever a ray of light has paffed through any particle of a body, and is come to its farther furface, if it finds there another particle con- tiguous, it will without interruption pafs into that particle; juil as light will pafs through one piece of glafs into ano- ther piece in contadl with it without any impediment, or any part being refleded : but as the light in pafling out of glafsj or any other tranfparent body, fliall part of it be re- fleded back, if it enter into air or other tranfparent body ■ of a different denfity from that it paffes out of ; the fame thing will happen in the light's paffage through any parti >. cle of a body, whenever at its exit out of that particle it meets no other particle contiguous, but mufl; enter into a pore, for in this cafe it (hall not all pafs through, but part of it be refledled back. Thus will the light, every time it enters a pore, be in part refle6ted \ fo that nothing more feems neceffary to opacity, than that the particles, which com- pofe any body, touch but in very few places, and that the pores of it are numerous and large, fo that the light may in part be refleded from it, and the other part, which en- ters too deep to be returned out of the body, by numerous i-dfle^tions may be fiifled and loft ^ ; which in all probabi- » 5 '?• *■ Opt. Book II. pir. 3. prop. 4. lity Chap. 2. PHILOSOPHY. 343 lity happens, as often as it impinges againft the foHd part of the body, all the light which does fo not being refiedt- ed back, but ftopt, and deprived of any farther motion V 6. This notion of opacity is greatly confirmed by the obferv^ation, that opake bodies become tranfparent by fill- ing up the pores with any fubftance of near the fame den- iity with their parts. As when paper is wet with water or oyl ; when linnen cloth is either dipt in water, oyled, or varniflied ; or the oculus mundi ftone fteeped in water ^ All which experiments confirm both the firft affertion, that light is not refleded by ftriking upon the folid parts of bodies ; and alfo the fecond, that its paflage is obftruded by the reflexions it undergoes in the pores ; fince we find . it in thefe trials to pafs in greater abundance through bo- dies, wlien the number of their folid parts is increafed, on- ly by taking away in great meafure thofe refled:ions ; whicli. filling the pores with a fubftance of near the fame denfi- ty with the parts of the body will do... Befides as filling the pores of a dark body makes it tranfparent ; fo on the other hand evacuating the pores of a body tranfparent, or. feparating the parts of fuch a body, renders it opake. As. falts or wet paper by being dried, glafs by being reduced to powder or the furface made rough ; and it is well known that glafs veflels diicover cracks in them by their opacity. Juffc lo water itfclf becomes impervious to the light by being formed into many fmall bubbles , whether in froth, or by being mixed and agitated widi any quantity of a hqucrr » Opt. Book Il.pag. 14.1, ^ Ibid, pig. ti^... 4 witite 344- Sir Is A AC Newton's Book III. with which it will not incorporate, fiich as oyl of turpentine, or oyl olive. 7. A CERTAIN ele£lrical experiment made by Mr. Hauks- BEE may not perhaps be ufelefs to clear up the prefent fpe- culation, by {hewing that fomething more is neceffary be- lides mere porofity for tranfmitting freely other fine fub- ftances. The experiment is this ; that a glafs cane rubbed till it put forth its ele6lric quality would agitate leaf brafs inclofed under a glafs vefTel, though not at fo great a dift- ance, as if no body had intervened ; yet the fame cane would lofe all its influence on the leaf brafs by the inter- pofition of a piece of the finefl: muflin , whofe pores are immenfely larger and more patent than thofe of glafs. 8. Thus I have endeavoured to fmooth my way, as much as I could, to the unfolding yet greater fecrets in nature ; for I fhall now proceed to fhew the reafon why bodies ap- pear of different colours. My reader no doubt will be fufficiently furprized, when I inform him that the knowledge of this is deduced from that ludicrous experiment, with which children divert themfelves in blowing bubbles of water made tenacious by the folution of foap. And that thefc bubbles, as they gradually grow thinner and thinner till they break, change fucceffively their colours from the fame principle, as all natural bodies preferve theirs. 9. Our author after preparing water with foap, fo as to render it very tenacious, blew it up into a bubble, and plac- 3 H Chap. 2. PHILOSOPHY. 345 ino- it under a glafs, that it might not be irregularly agitated by the air, obferved as the water by fubfiding changed the thicknefs of the bubble, making it gradually lels and lefs till the bubble broke ; there fucceflively appeared colours at the top of the bubble, which fpread themfelves into rings furround- ing the top and defcending more and more, till they vanifhed at the bottom in the fame order in which they appeared'. The colours emerged in this order: firft red, then blue ; to which fucceeded red a fecond time, and blue immediately follow- ed ; after that red a third time, fucceeded by blue; to which followed a fourth red, but fucceeded by green ; after this a more numerous order of colours, firft red, then yellow, next green, and after that blue, and at laft purple ; then again red, yellow, green, blue, violet followed each other in order ; and in the laft place red, yellow, white, blue ; to which fucceeded a dark fpot, which reflected fcarce any light, though our author found it did make fome very ob- fcure refteftion, for the image of the fun or a candle might be faintly difcerned upon it ; and this laft fpot fpread itfelf more and more, till the bubble at laft broke. Thefe co- lours were not fimple and uncompounded colours, like thole which are exhibited by the prifm, when due care is taken to feparate them ; but v/ere made by a various mixture of thofe fimple colours, as will be fhewn in the next chapter : whence thefe colours, to which I have given the name of blue, green, or red, were not all alike, but differed as fol- lows. The blue, which appeared next the dark fpot, v/as a pure colour, but very faint, refembling the sky-colour , the = Ibid. Ohf. I :. gcc. Y y white 34-6 Sir I s A A c N E w T o n's Book III. white next to it a very {Irong and intenfe white, bright- er much than the white, which the bubble reflected, before any of the colours appeared. The yellow which preced- ed this was at iirft pretty good, but foon grew dilute; and the red which went betore the yellow at firft gave a tin- cture of fcarlet inclining to violet, but foon changed into a brighter colour ; the violet of the next feries was deep with Uttle or no rednefs in it ; the blue a brisk colour, but ■ came much fhort of the blue in the next order ; the green was but dilute and pale ; the yellow and red were very bright and full, the beft of all the yellows which appeared among any of the colours : in the preceding orders the pur- ple wasreddifh, but the blue, as was jufl; nowfaid, the bright- eft of all ; the green pretty lively better than in the order which appeared before it, though that was a good willow orreen ; the yellow but fmall in quantity, though bright ; the red of this order not very pure : thofe which appeared be- fore yet more obfcure, being very dilute and dirty ; as were likewife the three firft blues. 10. Now it is evident, that thefe colours arofe at the top of the bubble, as it grew by degrees thinner and thin- ner : but what the exprefs thicknefs of the bubble was, where each of thefe colours appeared upon it, could not be de- termined by thefe experiments; but was found by another means, viz. by taking the objed glafs of a long telefcope, which is in a fmall degree convex, and placing it upon a flat glafs, fo as to touch it in one point, and then water be- ing put between them, the fame colours appeared as in the. (^ bubble. Chap. 2: PHILOSOPHY. 347 bubble, in the form of circles or rings furrounding the point where the glaffcs touched, which appeared black for want of any refledion from it, like the top of the bubble when thinneft ' : next to this fpot lay a blue circle, and next without that a white one ; and fo on in the fame or- der as before, reckoning from the dark fpot. And hencefor- ward I fliall fpeak of each colour, as being of the firft, fe- cond, or any following order, as it is the firft, fecond, or a- ny following one, counting from the black fpot in the cen- ter of thefe rings ; which is contrary to the order in which I muft have mentioned them, if I fliould have reputed them the firft, fecond, or third, &c. in order, as they arife after one another upon the top of the bubble. II. But now by meafuring the diameters of each of thefe rings, and knowing the convexity of the telefcope glals, the thicknefs of the water at each of thofe rings may be determi- ned with great exacftnefs : for inftance the thicknefs of it, where the white light of the firft order is refledled, is a- bout 3 I fuch parts, of which an inch contains loooooo''* And this meafure gives the thicknefs of the bubble, where it appeared of this white colour, as well as of the water between the glaiTes j though the transparent body which furrounds the water in thefe two cafes be very different : for .our author found, that the condition of the ambient body would not alter the fpecies of the colour at all, though jt might its ftrength and brightnefs ; for pieces of Mufcovy glafs, which were fo thin as to appear coloured by being ' Ibid. Obf. 10. >> Ibid. pag. 106. Y y X wet 348 Sir I s A A c N E w T o n's Book III. W€t with water, would have their colours faded and made lefs bright thereby ; but he could not obferve their fpecies at all to be changed. So that the thicknefs of any tranf- parent body determines its colour, whatever body the light paiTes through in coming to it '. I a. But it was found that different tranfparent bodies would not under the fame thickneffes exhibit the fame co- lours : for if the forementioned glailes were laid upon each other without any water between their furfaces, the air it- felf would afford the fame colours as the water, but more expanded, infomuch that each ring had a larger diameter, and all in the fame proportion. So that the thicknefs of the air proper to each colour was in the fame proportion largerj than tlie thicknefs of the water appropriated to the fame ^. 13. If we examine with care all the circumftances of thele colours, which will be enumerated in the next chap- ter, we Ihall not be iurprized, that our author takes them to bear a great analogy to the colours of natural bodies ^ For the regularity of thofe various and flrange appearances relating to them, which makes the moft myfterious part of the adi- on between light and bodies, as the next chapter will fhew^ is fiifficient to convince us that the principle, from which they llov/, is of the greatelt importance in the frame of nature ; and therefore without qucftion is defigned for no lefs a purpofe than to give bodies their various colours, to which end it fecms very fitly fuited. For if any fuch tranf- » Obftr. II. *" Obferv, j, compared with O'olirv. 10.. c Ibid prop. y. parent Chap. 2. PHILOSOPHY. 349 parent fubftance of the thickncfs proper to produce any one colour fliould be cut into flender threads , or broken into fragments , it does not appear but thefe fhould retain the fame colour ; and a heap of fuch fragments fhould frame a body of that colour. So that this is without difpute the caufe why bodies are of this or the other colour, that the particles of which they are compo- fed are of different fizes. Which is farther confirmed by the analogy between the colours of thin plates, and the co- lours of many bodies. For example, thefe plates do not look of the fame colour when viewed obliquely, as when feen direct ; for if the rings and colours between a convex and plane glafs are viewed firfl: in adired: manner, and then at different degrees of obliquity, the rings will be obferv^d to di- late themfelves more and more as the obliquity is increafed ^ j which fhews that the tranfparent fubftance between the glafles does not exhibit the fame colour at the fame thicknefs in all {ituations of the eye : juft fo the colours in the very fame part of a peacock's tail change, as the tail changes pofture in refped ot the fight. Alfo the colours of filks, cloths^. and other fubllances, which water or oyl can intimately penetrate, become faint and dull by the bodies being ^^'ec with fuch fluids, and recover their brightnefs again when dry ; juft as it was before faid that plates ol Mufcovy glafs grew faint and dim by v/etting. To this may be added, that the colours which painters ule will be a little changed by being- ground very elaborately, without qiieftion by the diminution of their parts. All which particulars, and many more that » Obfer?. 7. might 5^0 Sir Isaac N e w t o n's Book IIL might be extrafted from our author, give abundant proof of the prefent point. I fhall only fubjoin one more: thefe tranfpa- rent plates tranfmit through them all the light they do not re- fled; ; fo that when looked through they exhibit thofe colours, which refult from the depriving white light of the colour re- flefted. This may commodioufly be tryed by the glaffes fo often mentioned ; which if looked through exhibit coloured rings as by refleded light, but in a contrary order ; for the mid- dle fpot, w^hich in the other view appears black for want of refleded light, now looks perfedly white, oppoiite to the blue circle; next without this fpot the light appears tinged with a yellowilli red; where the white circle ap- peared before, it now feems dark ; and fo of the reft \ Now in die fame manner, the light tranlmitted through fo- liated gold into a darkened room appears greenifh by the lofs of the yellow light, which gold refleds. 1 4. H E N c E it follows, that the colours of bodies give a very probable ground for making conjcdure concerning the magnitude of their conftituent particles ^. My reafon for calling it a conjedure is, its being difficult to fix certainly the order of any colour. The green of vegetables our au- thor judges to be of the third order, partly becaulc of the in- tenfenefs of tlieir colour ; and partly from the changes they fuffer when they wither, turning at firft into a greenifli or more perfed yellow, and afterwards fome of them to an o- j-inge or red ; which changes feem to be elfeded from their tinging particles grov/ing denfer by the exhalation of their » Obfcrv. 9. ^ Ibid prop. 7. 1 nioifture Chap. 2. PHILOSOPHY. 35,^ moifturc, and perhaps augmented likewife by the accretion, of the earthy and oily parts of that moifturc. How the men- tioned colours fhould a rife from increafing the bulk of thofe particles, is evident ; fcemg thofe colours lie without the ring of green between the glaii'es, and are therefore formed where the tranfparent fubflance which refleds tliem is thicker- And that the augmentation of the denlity of the colorific particles will confpire to the production of the fame effedl, will be evident ; if we remember what was faid of the dif- ferent fize of the rings, when air was included between the glaffes, from their fize when water was between them ; which fhewed that a fubftance of a greater denfity than another gives the fame colour at a lefs thicknefs. Now the changes likely to be wrought in the denfity or magni- tude of the parts of vegetables by withering feem not greater, than are fufficient to change their colour into thofe of the lame order; but the yellow and red of the fourth order are not Rill enough to agree widi thofe, into which thefe flib- ftances change, nor is the green of the fecond fufficiently good to be the colour of vegetables ; fo that their colour muft of necefilty be of the third order. I y. The blue colour of fyrup of violets our author fuppofes to be of the third order; for acids, as vinegar, with this fyrup change it red, and fait of tartar or other alca- lies mixed therewith turn it green. But if the blue colour of the fyrup were of the fecond order, the red colour, which acids by attenuating its parts give it^ muft be of the. firft order, and the green given it by alcalies by incraflating 5^2 Sir I s A A c N E w T o n's Book IIL its particles fliculd be of the fecond ; whereas neither of thofe colours is perfect enough, efpecially the green, to anfwer thofe produced by thefe changes ; but the red may well e- nough be allowed to be of the fecond order, and the green of the third ; in which cafe the blue muft be likewife of the third order. J 6. The azure colour of the skies our author takes to be of the firft order, which requires the fmalleft particles of any colour, and therefore moft like to be exhibited by vapours, before they have fufficiently coalefced to produce clouds of other colours. 17. The moft intenfe and luminous white is of the firft order, if lefs ftrong it is a mixture of the colours of all the orders. Of the latter fort he takes the colour of lin- nen, paper, and fuch like fubftances to be; but white me- tals to be of the former fort. The arguments for it are thefe. The opacity of all bodies has been fhewn to arile from the number and ftrength of the reflexions made with- in them ; but all experiments fhew, that the ftrongeft re- fledion is made at thofe furfaces, which intercede tranfpa- rent bodies differing moft in denfity. Among other in- ftances of this, the experiments before us aiiord one ; for when air only is included between the glaffes, the coloured rings are not only more dilated, as has before been faid, than when water is between them ; but are likewife much more luminous and bright. It follows therefore, that what- ever medium per\'ades the pores oi bodies^ if fo be there is Chap. 2. PH I L O S O P H Y. 353 is any, thofe fubftances mufl be moft opakc, the denfity of whofe parts differs moft from the denfity of the medium, which fills their pores. But it has been fufficiently proved in the former part of this trad:, that there is no very denlc medium lodging in, at leaft pervading at liberty the pores of bodies. And it is farther proved by the prefcnt expe- riments. For when air is inclofed by the denier fubftance of glals, the rings dilate themfelves, as has been laid, by be- ing viewed obliquely; this they do fo very much, that at different obliquities the fame thicknels of air will exhibit all forts of colours. The bubble of water, though furrounded with the thinner fubftance of air, does likewife chanse its colour by being viewed obliquely j but not any thing near fo much, as in the other cafe ; for in that the fame colour might be feen, when the rings were viewed moft obliquely, at more than twelve times the thicknefs it appeared at un- der a dired view; whereas in this other cafe the thicknels was never found confiderably above half as much again. Now the colours of bodies not depending only on tlie light> that is incident upon them perpendicularly, but likewife upon that, which falls on them in all degrees of obliquity; if the medium furrounding their particles were denfer than thofe particles, all forts of colours muft of neceflity be reflected from them fo copioufty, as would make the colours of all bodies white, or grey, or at beft very dilute and imperfedt. But on tlie other hand, if the medium in the pores of bodies be much rarer than their particles, the colour refledled will be lo little changed by the obliquity of the rays, that the colour pro- duced by the rays, which fall near the perpendicular, may Z z fo 354 Sir Isaac Newton's BookIII. fo much abound in the refledled light, as to give the body their colour with little allay. To this may be added, that when the difference of the contiguous tranlparent fubftances is the fame, a colour refleded from the denfer fubftance reduced into a thin plate and furrounded by the rarer will be more brisk, than the fame colour will be, when refledled from a thin plate formed of the rarer fubftance, and llir- rounded by the denfer ; as our author experienced by blowing glals very thin at a lamp furnace, which exhibited in the open air more vivid colours, than the air does be- tween two glaffes. From thefe confiderations it is manifeft, that if all other circumftances are alike, the denfeft bodies will be moft opake. But it was obferved before, that thefe white metals can hardly be made fo thin, except by being diffolved in corroding liquors, as to be rendred tranfparent ; though none of them are fo denie as gold, which proves their great opacity to have fome other caufe befides their denfity ; and none is more fit to produce this, than fuch a fize of their particksj as qualifies them to reflect the white of the firft order. 1 8. For producing black the particles ought to be fmaller than for exhibiting any of the colours,, viz. of a fize anfwering to the thicknefs of the bubble, where by re^ fieding little or no light it appears colourlcfs ; but yet they muft not be too fmall, for that will make them tranfparent through deficiency of refiedions in the inward parts of thje body, fuffkicnt to flop the light from going through it -, but tliey muft be of a fize bordering upon that difpofcd. Chap. 2. PHILOSOPHY. 355 difpofcd to refled the faint blue of tlie firft order, which affords an evident reafon why blacks ufually partake a Httle of that colour. We fee too, why bodies diflblved by fire or putreflidion turn black : and why in grinding glafies up- on copper plates the duft of the glafs, copper, and fand it is ground with, become very black : and in the lafl: place why thefe black fubftances communicate fo eafily to others their hue; which is, that their particles by reafon of the great minutenefs of diem eafily overfpread the grofler par- ticles of others. * 19. I SHALL now finiili this chapter with one remark of the exceeding great poroHty in bodies neceffarily requi- red in all that has here been faid , which, when duly confidered, mufl: appear very furprizing ; but perhaps it will be matter of greater furprizc, when I affirm that the fagacity of our author has diicovered a method, by which bodies may eafily become fo j nay how any the leaft portion of matter may be wrought into a body of any affigned dimeniions how great fo ever , and yet the pores of that body none of them greater, than any the fmalleft magnitude propofed at pleafure ; notwithftanding which the parts of the body fhall fo touch, that the body itfclt fliall be hard and folid \ The manner is this : fuppofe the body be compounded of particles of fuch figures, that when laid together the pores found be- tween them may be equal in bignefs to the particles ; how this may be effefted, and yet the body be hard and folid, is not difficult to underftand ; and the pores of fuch a bo- »Opt. pag 14?, L z 1 tj Q^c^6 Sir Isaac Newton's Book III. dv may be made of any propofed degree of fmallnefs. But the folid matter of a body fo framed will take up only half the fpace occupied by the body; and if each conftituent particle be compofed of other lefs particles according to the fame rule, the folid parts of fuch a body will be but a fourth Dart of its bulk ; if every one of thele leffer parti- cles again be compounded in the fome manner, the folid parts of the whole body fhall be but one eighth of its bulk ; and thus by continuing the compofition the folid parts of the body may be made to bear as fmall a proportion to the whole magnitude of the body, as fhall be delired, notwith- ftanding the body will be by the contiguity of its parts ca- pable of being in any degree hard. Which fhews that this whole globe of earth, nay all the known bodies in the u- niverfe together, as far as we know, may be compounded of no greater a porUon of fohd matter, than might be re- duced into a globe of one inch only in diameter, or even lefs. We fee therefore how by this means bodies may ea- fily be made rare enough to tranfmit light, with all that freedom pellucid bodies are found to do. Though what is the real ftrudure 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 fpeculations yet greater, no lefs Chap. 3. PHILOSOPHY. 357 lefs than to lay open the caiifes of all that has hitherto been related. For it mufl: in this chapter be explained, how the prifm feparates the colours of the fun's light, as we found in the firfl: chapter ; and why the thin tranfparent plates diicourfed of in the laft chapter, and confequently the par- ticles of coloured bodies, relied: that diverfity of colours only by being of different thickneiTcs. a. F o R the firfl it is proved by our author, that the colours of the fun's light are manifefted by the prifm, from the rays undergoing different degrees of refradion ; that the violet- making rays, which go to the upper part of the coloured image in the firft experiment of the firft chapter, are the moft refraded ; that the indigo-making rays are refracted, or turned out of their courfe by pafling through the prifm, fomething lefs 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 leaft of all re- fraded. The firft proof of this, that rays of different co- lours 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 prifm thofe two parts fhall ap- pear feparated from each other ; which can be caufed no otherwife than by the prifm's refrading the light of one half more than the light of the other half But the blue half will be moft refraded ; for if the body be feen through the prifm in fuch a fituation, that the body fhall appear .lifted 3 =, B Sir I s A A c N E w T o n's Book III. lifted upwards by the refradion, as a body within a bafon of water, in the experiment mentioned in the firft chapter, a].)peared to be Ufted up by the refracftion of the water, fo as to be feen at a greater diflance than when the bafon is empty, then fhall the blue part appear higher than the red ; but if the refraction of the prifm be the contrary way, the blue part fliall be deprefled more than the other. Again, after laying iine threads of black filk acrofs each of the co- lours, and the body well inlightened, if the rays com- ing from it be received upon a convex glafs, fo that it may by refrading the rays caft the image of the body upon a piece of white paper held beyond the glafs ; then it will be (ken tliat the black threads upon the red part of the image, and thofe upon the blue part, do not at the fame time appear diftindly in the image of the body projeded by the glafs j but if the paper be held fo, that the threads on the blue part may diftindly appear, the threads can- not be feen diftincl upon the red part ; but the paper muft be drawn farther off from the convex glafs to make the threads on this part viiible ; and when the diftance is great e- nough for the threads to be feen in this red part, they become indiftindl in the other. Whence it appears that the rays pro- ceeding from each point of the blue part of the body are fooner united again by the convex glafs than the rays which tome from each point of the red parts ^ But both thefe ex- periments prove that the blue-making rays, as well in the fmall rcfradion of the convex glafs, as in the greater refraction (Oi the prifm, are more bent, than the red-making rays. » Newt. Opt. B. I. part. i. jrop. I, 3. Thi S Chap. 2.' PHILOSOPHY. 359 5 . This feems already to explain the reafon of the co- loured fpedrum made by refrafting the fun's light withaprifm'; though our author proceeds to examine that in particular, and proves that the different coloured rays in that Ipedrum are in different degrees refraded j by fhewing how to place the prifm in fuch a poilure, that if all the rays were re- fraded in the fame manner, tlie fpedrum fhould of neceffi- ty be round : whereas in that cafe if the angle made by the two furfaces of the prifm, through which the Hght paffes, that is the angle DFE in fig. iz6, be about (5 5 or 6^ degrees, the image inftead of being round fliall be near five times as long as broad ; a difierence enough to fhew a great inequality in the refradions of the rays, which go to the oppofite extremities of the image. To leave no fcruple unremoved, our author is very particular in fhewing by a great number of experiments, that this inequality of refra- dion is not cafual, and that it does not depend upon any ir- regularities of the glafs ; no nor that the rays are in their paffage through the prifm each fplit and divided j but on: the contrary that every ray of tlie fun has its own peculiar degree of refradion proper to it, according to which it is more or lefs refraded in paffmg through pellucid fubftances always in the fame manner \ That the rays are not fplit and multiplied by the refradion of the priiin, the third of the experiments related' in our firft chapter lliews very clear- ly ; for if they were, and the length of the Ipedrum irr the firft refradion were thereby occafioned , the breadth- fliould be no, iefs dilated by the crof': refradion of th«2L ie.- » Of.*. B. I. parf. I. pjop..i, &ond[ 360 Sir Isaac Newton's Book III. cond prifm ; whereas the breadth is not at all increafed , but tlie image is only thrown into an oblique pofture by the upper part of the rays which were at iirft more refraded than the under part, being again turned fartheil: out of their courfe. But the experiment mod exprefsly adapted to prove this regular diveriity of refraftion is this, which follows*. Two boards AB, CD(in fig. 130.) being ereded in a dar- kened room at a proper diftance, one of them A B being near the window-fhutter E F, a fpace only being left for the prifm G H I to be placed between them .; fo that the rays entring at the hole M of the window-fhutter may af- ter paflmg through the prifm be trajeded through a fmal- ler hole K made in the board A B, and pafling on from thence go out at another hole L made in the board C D of the fame fize as the hole K, and fmall enough to tranfmit the rays of one colour only at a time ; let another prifm NOP be placed after the board CD to receive the rays paf- fing through the holes K and L, and after refraction by that prifm let thofe rays fall upon the white furface Q^R. Sup- pofe firft the violet light to pafs through the holes, and to be refraded by the prifm NOP to j, which if the prifm j!SI O P were removed fliould have pafied right on to W. If the prifm G H I be turned flowly about, while the boards and prifm NOP remain fixed, in a little time another colour will fall upon the hole L^ which, if the prifm NOP were taken away, would proceed like the former rays to the fame point \V ; but the refradion of the prifm NOP fhall not car- ry thefe rays to J", but to fome place lefs diflant from W as » Opt. B, I, part I. Expec. 6. ■ " to Chap. 3. PHILOSOPHY. gdt to t. Suppofe now the rays which go to / to be the indigo- making rays. It is manifeft that the boards AB, CD, and prifm NOP remaining immoveable, both the violet-making and indigo-making rays are incident alike upon the prifm NOP, for they are equally inclined to its furface O P, and enter it in the fime part of that furface ; which fliews that the indigo- making rays are lefs diverted out of their courfe by the re- fraction of the prifm, than the violet-making rays under an exad: parity of all circumftances. Farther, if the prifm G H I be more turned about, 'till the blue-making rays pals tlirough the hole L, thefe fhall fall upon the furface Q^R below I, as at i;, and therefore are fubjed:ed to a lefs re- fradion than the indigo-making rays. And thus by pro- ceeding it will be found that the green-making rays are lefs refrad:ed than the blue-making rays, and fo of the refl, according to the order in which they lie in tlie coloured ipedrum. 4. This difpofition of the different coloured rays to be refraded fome more than others our author calls their refpedive degrees of refrangibility. And fince this differ- ence of refrangibility difcovers it felf to be fo regular, the next ftep is to find the rule it obferves. y. It is a common principle in optics, that the fine of the angle of incidence bears to the fine of the refraded an- gle a given proportion. If AB (in fig. Igl, Igi) rc- prelent die furface of any refrading fubftance, fuppofe of water or glafs, and C D a ray of light incident upon that fur- A a a face 3^2 Sir Is A AC Newton's Book III. face in the point D, let D E be the ray, after it has paffed the furface A B ; if the ray pafs out of the air into the fubftance whofe furface is A B (as in fig. 1 3 1 ) it fhall be turned trom the furface, and if it pafs out of that fubftance into air it fhall be bent towards it (as in fig. I 5 1 ) But if F G be drawn through the point D perpendicular to the furface A B, the angle under CDF made by the incident ray and this perpendicular is called the angle of incidence; and the angle under B D G,made by this perpendicular and the ray after refradion, is called the refradled angle. And if the circle H F I G be defcribed with any interval cutting C D in H and D E in I, then the perpendiculars H K, I L be- ing let fall upon F G, H K is called the fine of the angle under CDF the angle of incidence, and I L the fine of the angle under E D G the refradled angle. The firft of thefe fines is called the fine of the angle of incidence, or more briefly the fine of incidence, the latter is the fine of the refradled angle, or the fine of refradion. And it has been found by numerous experiments that whatever proportion the fine of incidence H K bears to the fine of refradion I L in any one cafe, the fame proportion fhall hold in all cafes ; that is, the proportion between thefe fines will remain unalterably the fame in the fame refrading fub- ftance, whatever be the magnitude of the angle under CDF. 6. But now becaufe optical writers did not oblerve that every beam of white light was divided by refradion, as has been here explained, this rule coUedled by them can on- ly be underftood in the grofs of the whole beam after re- fradion Chap.3. philosophy. 363 fradion, and not fo much of any particular part of it, or at moft 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 feparation was made by the proportion between the fines of incidence and refradi- on being in each fpecies of rays different ; or whether the light was divided by fome other rule. But he proves by a certain experiment that each ray has its fine of incidence "proportional to its fine of refradion ; and farther fhews by mathematical reafoning, that it muft be fo upon condition only that bodies refradl tlie Hght by ading upon it, in a diredion perpendicular to the furface of the refrading body, and upon the fame fort of rays always in an equal degree at the fame difl;ances\ 7. O u R great author teaches in the next place how from the refraclion of the moft refrangible and leall refrangible rays to find the refraftion of all the intermediate ones ^ The method is this: if the fine of incidence be to the fine of re- fraction in the leaft refrangible rays as A to B C, (in fig. 133) and to the fine of refradion in the mof} refrangible as A to B D; if C E be taken equal to C D, and then E D be fb divided in F, G, H, I, K, L, that ED, E F, EG, EH, EI, EK, EL, E C, fliall be proportional to the eight lengths of mufical chords, which found the notes in an oda^e, ED being; the length ot the key, E F the length of the tone above > Opt. pag. 67,6s, e^f. ^ roid.B. I, par. i. prop. 5. A a a 1 that 364. Sir Isaac Newton's Book III. that key, E G the length of the lefler third, E H of the fourth, E I of the fifth, E K of the greater fixth, E L of the fcventh, and E C of the oftave above that key ; that is if the Hnes E D, E F, E G, E H, E I, E K, E L, and E C bear the fame proportion as the numbers, T, I, f-, j, r, ^ ^f' r> refpe6lively then Ihall B D, BF, be the two Hmits of the fines of refradion of the violet-making rays, that is the violet-making rays fiiall not all of them have precifely the fame fine of refradion , but none of them fliall have a greater fine than B D, nor a lefs than B F, though there are violet- making raysA^hijh anfwer to any fine of refradion that can be taken be- tween thefe two. In the fame manner B F and B G are the limits of the fines of refradion of the indigo-ma- king rays ; B G, B H are the limits belonging to the blue- making rays; B H, B I the limits pertaining to the green-ma- king rays, B I, B K the limits for the yellow-making rays; B K, B L the limits for the orange-making rays ; and laftly, B L and B C the extreme limits of the fines of refradion belonging to the red- making rays. Thefe are the propor- tions by which the heterogeneous rays of light are feparated from each other in refradion, 8. When light pafix^s out of glafs into air, our author found A to B C as 5-0 to 77, and the fame A to B D as yo to 78. And v/hen it goes out of any other refrading fub- fiance into air, the excefs of the fine of refradion of any one fpecies of rays above its fine of incidence bears a con- (tant proportion, which holds the fame in each Ijiecies, to tlie cxccfs of the fine of refradion of the fame fort of rays above Chap. 3. PHILOSOPHY. gd^ above the iine of incidence into the air out of glafs ; provided the fines of incidence both in glafs and the other fiibftancc are equal. This our author \criiied by tranfmitting the light through prifms of glafs included within a prifmatic vefTel of water; and draws from thofe experiments the fol- lowing obfervations : that whenever the light in pafTmg through fo many furfaces parting diverfe tranfparent fub- ftances is by contrary refractions made to emerge into the air in a diredion parallel to that of its incidence, it will appear afterwards white at any difiance from the prifms, where you iliall pleafe to examine it ; but if the diredion of its emergence be oblique to its incidence, in receding from the place of emergence its edges fhall appear tinged with colours: which pro\Ts that in the lirft cafe there is no inequality in the refradions of each fpecies of rays, but that when any one fpecies is fo retraced as to emerge pa- rallel to the incident rays, every fort of rays after refracti- on fliall likewife be parallel to the fame incident rays, and to each other ; whereas on the contrary, if the rays of any one fort are oblique to the incident light, the feveral {pecies fnall be oblique to each other, and be gradually feparated by that obliquity. From hence he de- duces both the foremen tioned theorem, and alio this other; that in each fort of rays the proportion of the fine of in- cidence to the fine of refraction, in the pafHige of the ray out of any refrading fubflance into another, is compounded of the proportion to which the fine of incidence would have to the fine of refradlion in the pafTage of that ray out of the firft fubftance into any third, and of the proportion which the 366 Sir I s A A c N E w T o n's Book III. the line of incidence would have to the fine of refradion in the paffage of the ray out of that third fubftance into the fecond. From fo fimple and plain an experiment has our moft judicious author deduced thefe important theo- rems, by which we may learn how very exadt and circum- fped: he has been in this whole work of his optics ; that notwithftanding his great particularity in explaining his dodrine, and the numerous collection of experiments he has made to clear up every doubt which could arife, yet at the fame time he has ufed the greateft caution to make out every thing by the fimpleft and ealieft means poUible. 9. Our author adds but one remark more upon refra- ction, which is, that if refradlion be performed in the man- ner he has fuppofed from the light's being prefled by the refrading power perpendicularly toward the furface of the refracting body, and confequently be made to move fwifter in the body than before its incidence ; whether this power aft equally at all diftances or other\\'ife, provided only its power in the fame body at the lame diftances remain with- out variation the fime in one inclination of the incident rays as well as another ; he obferves that the refracting po- wers in different bodies will be in the duplicate proportion of the tangents of the leaft angles, which the retraCted light can make with the furfaces of the rcfraCting bodies •'. This obfervation may be explained thus. When the light paffes into any refraCting fubftance, it has been Ihewn above that die fine of incidence bears a conftant proportion to the fine » Opt. B. II. par. 3. prop. 10. of Chap. 3. PHILOSOPHY. 357 of refradion. Suppofe the light to pals to the refrading body A B C D (In fig. I g 4) in the line E F, and to fall upon it at the point F, and then to proceed within the body in the line FG. Let HI be drawn through F perpendicular to the fur- face AB, and any circle KLMN be defcribed to the cen- ter F. Then from the points O and P where this circle cuts the incident and refraded ray, the perpendiculars OQ^ PR being drawn, the proportion of 0Q_ to PR will remain the fame in all the different obHquities, in which the fame ray of light can fall on the furface AB. Now OQ, is lefs than FL the femidiameter of the circle KLMN, but the more the ray E F is inclined down toward the furface A B, the greater will OQ. be, and will approach nearer to the ma- gnitude of F L. But the proportion of O Q. to P R remain- ing always the fame, when OQ, is largeft, PR will alfo be greatefl: ; fo that the more the incident ray EF is inclined toward the furface A B, the more the ray F G after refracti- on will be inclined toward the fame. Now if the line F S T be fo drawn, that S V being perpendicular to F I fliall be to F L the femidiameter of the circle in the conftant pro- portion of PR to OQ,; then the angle under NET is that wliich I meant by the leaft of all that can be made by the refradled ray with this furface, for the ray after refradion would proceed in this line, if it were to come to the point F lying on the very furface 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 fome line between FT and FI. Here if NW be drawn perpendicu- lar to FN, this line NW in the circle KLMN is called the 368 Sir Isaac Newton's Book III. the tangent of the angle under NFS. Thus much being premi- fedjthe fenfe of the forementioned propofition is this. Let there be two refrading fubftances (in fig. 1 3 y) A B C D , and E F G H. Take a point, as I, in the furface AB, and to the center I with any femidiameter defcribe the circle KLM. In like manner on the furtace EF take fome point N, as a center, and defcribe with the fame femidiameter the circle OPQ.. Let the angle under B I R be the leaft which the refraded light can make with the furface A B, and the angle under FNS the leaft which the refradled light can make with the furface EF. Then if LT be drawn perpendicular to A B, and P V perpendicular to E F ; the whole power, where- with the fubftance A B C D ads on the Hght, will bear to the whole power wherewith the fubftance E F G H ads on, the light, a proportion, which is duplicate of the proporti- on, which L T bears to P V. 10. Upon comparing according to this rule the refra- dive powers of a great many bodies it is found, that undu- ous bodies which abound moft with fulphureous parts refrad the light two or three times more in proportion to their denfity than others : but that thofe bodies, which feem to receive in their compofition like proportions of fulphu- reous parts, have their refradive powers proportional to their denfities; as appears beyond contradidion by comparing the refradive power of fo rare a fubftance as the air with that of common glafs or rock cryftal , though thefe fub- ftances arc 2000 times denfer than air; nay the fame pro- portion Chap. 3; P H I L O S O P H Y: r^6^ portion is found to hold without fcnfiblc difference in com- paring air with pfeudo-topar and glafs of antimony, though the pfeudo-topar be 3 5*00 times denfer than air, and glafs of antimony no Icfs than 4400 times denfer. This pow- er in other fubftances, as- falts, common water, fpirit of wine, Sec. feems to bear a greater proportion to their den- fities than thefe lafl: named, according as they abound with fulphurs more than thefe; which makes our author conclude it probable, that bodies aft upon the light chiefly, if not altogether, by means ot the fulphurs in them ; which kind of fubftances it is likely enters in fome degree the compo- fition ot all bodies. Of all the fubftances examined by our author, none has fo great a refradive power, in relpedt of its denlity, as a diamond. II. Our author finiflies thefe remarks, and all he offers relating to refradion, with obferving, that the adlion be- tween light and bodies is mutual, fince fulphureous bodies, which are moft readily fet on fire by the fun's light, when colleded upon them with a burning glafs, adl more upon Iio;ht in refradrino: it, than other bodies of the fame denfi- ty do. And farther, that the denfeft bodies, which have been now fliewn to ad mofl upon light, contracl the greateft heat by being expofed to the fummer fun. II. Having thus difpatched what relates to refraction, we muft addrefs ourfelves to difcourfe of the other opera- tion of bodies upon light in refleding it. When light paffes through a furface, which divides two tranfparent bo- B b b dies 370 Sir I s A A c N E w T o N*s Book III, dies differing in denfity, part of it only is tranfmitted, another part being refleded. And if the Hght pafs out of the denfer body into the rarer, by being much incHned to the forefaid furface at length no part of it fliall pafs through, but be totally reflected. Now th^t part of the light, which fuffers the greateft refradion, fhall be wholly refleded with a lefs obliquity of the rays , than the parts of the light which undergo a lefs degree of refradlion ; as is evident from the laft experiment recited in the firft chapter; where, as the prifms D E F, G H I, (in fig. 119.) were turned a- bout, the violet light was firft totally refleded, and then the blue, next tothat the green, and fo of the reft. In con- {equence of which our author lays down this propolition ; that the fun's light differs in reflexibility, thofe rays being moft re- flexible, which are moft refrangible. And colleds from this, in conjundion with other arguments , that the refradion and refledion of light are produced by the fame caufe> compaiflng thofe different effeds only by the difference of circumftances with which it is attended. Another proof of this being taken by our author from what he has dif- covered of the paffage of light through thin tranfparent plates, viz. that any particular fpecies of light, fuppofe, for infl;ance, the red-making rays, will enter and pafs out of fuch a plate, if that plate be of fome certain thicknef- {es ; but if it be of other thicknefles, it will not break through it, but be refleded back: in which is feen, that the thick- iiefs of the plate determines whether the power, by which that plate ads upon the light, fhall refled it, or fufter it to pals through. 13. But Chap. 3- PHILOSOPHY. 371 13. But this laft mentioned furprifing property of the aftion between light and bodies affords the reafon of all that has been faid in the preceding chapter concerning the colours of natural bodies j and muft therefore more parti- cularly be illuftrated and explained , as being what will principally unfold the nature of the adion of bodies upon light. 14. T o begin: The object glafs of a long telefcope being laid upon a plane glafs, as propofed in the foregoing chap- ter, in open day-light there will be exhibited rings of va- rious colours, as was there related ; but if in a darkened room the coloured fpedtrum be formed by the prifm, as in the firft experiment of the firft chapter, and the glaffes be illuminated by a refledion from the fpedrum, the rings fhall not in this cafe exhibit the diverfity of colours before defcribed, but appear all of the colour of the light which falls upon the glaffes, having dark rings between. Which fhews that the thin plate of air between the glaffes at fome thickneffes refleds the incident light, at other places does not refledt it , but is found in thofe pla- ces to give the light paffage ; for by holding the glaffes in the light as it paffes from the prifm to the fpctSrum, fup- pofe at fuch a diftance from the prifm that the feveral forts of Hght muft be fufficiently feparated from each other , when any particular fort of light falls on the glaffes, you will find by holding a piece of white paper at a fmall diftance be- yond the glaffes, that at thofe intervals, where the dark rings appeared upon the glafies, the light is fo tranfmittcd, B b b 2 us 372> Sir Isaac N e \V t o n's Book TIL as to paint upon the paper rings of light having that co- lour which falls upon the glalles. This experiment there- fore opens to us this very ftrange property of refledion, that in thcfe thin plates it fliould bear fuch a relation to the thicknefs of the plate, as is here fhewn. Farther, by care- fully meafin-ing the diameters of each ring it is found, that whereas the glaffes touch where the dark fpot appears in the center of the rings made by refleftion, where the air is of twice the thicknefs at which the light of the firft ring is refleded, there the light by being again tranfmitted ma- kes the firft dark ring ; where the plate has three times that thicknefs which exhibits the firft lucid ring, it a- o-ain reflefts the light forming the fecond lucid ring ; when the thicknefs is four times the firft, the light is again tranf- mitted fo as to make the fecond dark ring; where the air is five times the firft thicknefs, the third lucid ring is made ;; where it has fix times the thicknefs, the third dark ring ap- pears, and fo on : in fo much that the thicknefles, at which the light is refleded, are in proportion to the numbers i, 3, 5") I1 9j ^^- ^^^^ ^^'^ thicknefles, where the light is tranf- mitted, are in the proportion of the numbers o, 1, 4, (5, 8y, &Ci And thefe proportions between the thicknefles which rcfled and tranfmit the light remain the fame in all fitua- tions of the eye, as well when- the rings are viewed obliquely, as when looked on perpendicularly. We muft farther here obferve, that the light, when it is refleded, ss well as when it is tranfmitted, enters the thin plate, and is reflcded from its far- ther furface; becaufe, as was before remarked, the altering the tranfparent body bcliind the farther furface alters the de- gree Chap. 3." PHILOSOPHY. 373 gree of refledion as when a thin piece of Mufcovy glafs has its farther furface wet with water, and the colour of the glafs made dimmer by being fo wet ; which fhews that the light reaches to the water, otherwife its refled:ion could not be influenced by it. But yet this reflexion depends upon fome power propagated from the firft furface to the fecond ; for though made at the fecond furface it de- pends alfo upon the firft, becaufe it depends upon the di" ftance between the furfaces; and befldes, the body through which the light paffes to the firfl; furface influences the re- fledion : for in a plate of Mufcovy glafs, wetting the furfa- ce, which flrft receives the light, diminifhes the reflection, though not quite fo much as wetting the farther furface wilf do. Since therefore the light in pafling through thefe thin plates at fome thicknefi'es is reflected, but at others tranfmit- ted without refledion, it is evident, that this refledion is caufed by fome power propagated from the firfl: furface, which intermits and returns fucceflively. Thus is every ray apart difpcfed to alternate refledions and tranfmiflions at equal intervals; the fucceflive returns of which difpofition our author calls the fits of eafy refledion, and of eafy tranf- miflion. But thefe fits, which obferve the fame law of returning at equal intervals, whether the plates are viewed perpendicularly or obliquely , in different fituations of the. eye change their magnitude. For what was obferved before: iri refpedl of thofe rings, which appear in open day-light,^ holds likewife in thefe rings exhibited by fimple lights ; name- ly, that thefe two alter in bignefs according to the dif- ferent angle under which they are feen : and our author- qy^ Sir I s A A c N E w T n's Book III. lays down a rule whereby to determine the thickneffes of \ the plate of air, which fhall exhibit the fame colour under different oblique views \ And the thicknefs of the aereal plate, which in different inclinations of the rays will exhi- bit to the eye in open day-light the fame colour, is alfo vr- ried by the lame rule ''. He contrived farther a method of comparing in the bubble of water the proportion be- tween the thicknefs of its coat, which exhibited any colour when feen perpendicularly, to the thicknefs of it, where the fame colour appeared by an oblique viev/ ; and he found .die lame rule to obtain here likewife "". But farther, if the glaffes be enlightened fucceffively by all the feveral fpecies 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, lefs in the blue, lefs yet in the indigo, and leaft of all in the violet : which fhew sthat the fame diicknefs of the aereal plate is not fitted to refled all colours, but that one colour is refleded where another would have been tranfmitted; and as the rays which are moft ftrongly re- fraded form the leaft rings, a rule is laid down by our au- thor for determining the relation, which the degree of re- fradion of each fpecies of colour has to the thickneffes of the plate where it is refleded. I j-. From thefe obfervations our author fh^ws the rcafon of that great variety of colours, which appears in thefe thin plates in the open white light of the day. For when this white * Qp . B. II. par. 3. prop. 15-. " .bd. par. i, ohktv. ^. ' Ibid. Obferv. 19 light Chap. 3. PHILOSOPHY. 375 light falls on tlie plate, each part of the light forms rings of its own colour ; and the rings of the different colours not being of the fame bignefs are varioufly intermixed, and form a great variety of tints \ 16. In certain experiments, which our author made with thick glaffes, he found, that thefe fits of eafy refledlion and tranfmifiion returned for fome thoufands of times, and there- by farther confirmed his reafoning concerning them ^ 17. Upon the whole, our great author concludes from fome of the experiments made by him, that the reafon why all tranfparent bodies refrad part of the light incident upon them, and refled another part, is, becaufe fome of the light, when it comes to the furface of the body, is in a fit of eafy tranfmif- fion, and fome part of it in a fit of eafy refledion ; and from the durablenefs of thefe fits he thinks it probable, that the light is put into thefe fits from their firft emifiion out of the luminous body ; and that thefe fits continue to return at e- qual intervals without end, unlefs thofe intervals be changed by the light's entring into fome refrading fubftance ". He likewife has taught how to determine the change which is made of the intervals of the fits of eafy tranfmiflion and re- flexion, when the Hgbt paffes out of one tranfparent Ipace or fubftance into another. His rule is, that when the light paf- fes' perpendicularly to the furface, which parts any two tranf- parent fubftances, thefe intervq.ls in the fubftance, out of » Opt B. IL par, 1. pag. 195. See, ^ I'jid. par. 4 ' Ibid. part. 3. prop, i j. whidi 37<^ Sir Isaac Newton's Book III. which the light pafies, bear to the intervals in the fubftance, -vvhereinto the light enters, the fame proportion, as the fine of incidence bears to the fine of refradrion '. It is farther to be obferved, that though the fits of eafy reflexion return at con- flant intervals, yet the refleding power never operates, but at or near a furface where the light would fuffer refradion ; and if the thicknefs of any tranfparent body fhall be lefs than the intervals of the fits, thofe intervals fhall fcarce be difturbed by fiich a body, but the light fliall pafs through without any re- £ea ion b 1 8. What the power in nature is, whereby this adion between light and bodies is caufed, our author has not difco- vered. But the effeds, which he has xiifcovered, of this power are very furprifing, and altogether wide from any con- jedures that had ever been framed concerning it ; and from thefe difcoveries of his no doubt this power is to be dedu- ced, if we ever can come to the knowledge of it. Sir I s a a c Newton has in general hinted at his opinion concerning it ; that probably it is owing to fome very fubtle and elaftic fub- ftance diffufed through the univcrfe, in which fuch vibrations may be excited by the rays of light, as they pafs through k, that fliall occafion it to operate fo differently upon the light in different places as to give rife to thefe alternate fits of refledion and tranfmilBon, of which we have now been fpeaking ^ He is of opinion, that fuch a fubftance may pro- duce this, and other effeds alfo in nature, though it be fo rare as not to give any fenfible refiftance to bodies in mo- * Ibid, p op.^iy. >> Ibid prop, 13. Opt. Qii. iS, £vc. tion; Chap. 4- PHILOSOPHY. 377 tion ^ ; and therefore not inconiiftent with what has been faid above, that the planets move in fpaces free from refiftance ^ 1 9. In order for the more full difco\'ery of this action between light and bodies, our author began another fet of experi- ments, wherein he found the light to be a£led on as it pafTes near the edges of folid bodies ; in particular all fmall bo- dies, fuch as the hairs of a man's head or the like, held in a very fmall beam of the fun's Hght, caft extremely broad fha- dows. And in one of thefe experiments the fhadow was 5 S times the breadth of the body ^ Thefe fhadows are alfo obferved to be bordered with colours'^. This our author calls the inflexion of light ; but as he informs us, that he was interrupt- ed from profecuting thefe experiments to any length, I need not detain my readers with a more particular account of them. Chap. IV. Of OPTIC GLASSES. O IR Isaac Newton having deduced from his dodrine ^^^ of light and colours a furpriling improvement of tele- fcopes, of which I intend here to give an account, I {hall hrfl premife fomething in general concerning thofe inflru- ments. « Sdb Cond. S. i: 1 = Opt. B. III. Obf. r. iB. II, Ch, I. 1 * Ibid. Obf. 2, C c c a. Ir 378 Sir Isaac Newton's Book III. a. It will be underftood from what has been faid above, that when light falls upon the furface of glafs obliquely, after its entrance into the glafs it is more inclined to the line drawn through the point of incidence perpendicular to that furface, than before. Suppofe a ray of light iffuing from the point A (in fig. I g 6) falls on a piece of glafs BCDE, whofe furface BC, whereon the ray falls, is of a fpherical or globu- lar figure, the center whereof is F. Let the ray proceed in the line A G filling on the furface B C in the point G, anddraw F G H. Here the ray after its entrance into the glafs will pafs on in fome 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 per- pendicular to the furface B C in the point G. By this means, if a number of rays proceeding from any one point fall on a convex fpherical furface of glafs, they fhall be infledled (as is reprefented in fig. i g 7,) fo as to be gathered pretty clofe together about the line drawn through the center of the glafs from the point, whence the rays proceed ; which line henceforward we fliall call the axis of the glafs : or the point from whence the rays proceed may be fo near the glafs* that the rays fhall after entring the glafs ftill go on to fpread themfelves, but not fo much as before ; fo that if the rays were to be continued backward (as in fig. i g 8,) they fhould gather together about the axis at a place more remote from the glafs, than the point is, whence they aftually proceed. In thefe and the following figures A denotes the point to which the rays are related before refradion, Bthe point to which they are direded afterwards, and C the center of the refrading fur- face. Here we may obferve, that it is poflible to form the glafs of fuch a figure, that all the rays which proceed from one point fhall Chap.4. philosophy. 379 fliall after refradion be reduced again exactly into one point on the axis of the glafs. But in glafles of a fpherical foim though this does not happen; yet tlie rays, which fall within a moderate di- ftance from the axis, will unite extremely near together. If the light fall on a concave fpherical furface, after refradion it fliall fpread quicker than before (as in fig. 1^9,) unlefs the rays proceed from a point between the center and the furface of the glafs .If we fuppofe the rays of light, which fall upon the glafs, not to proceed from any point, but to move fb as to tend all to fome point in the axis of the glafs beyond the furface ; if the glafs have a convex furflice, the rays fhall unite about the axis fooner, than otherwife they would do (as in fig. 140,) unlels the point to which they tended was between the furface and the center of that furface. But if the furface be concave, they fliall not meet fo foon : nay perhaps converge. (Sec fig. 141 and 14.2.) 5. Farther, becaufe the light in pallnig out of glafs in- to the air is turned by the refradion farther off from tlie line drawn through the point of incidence perpendicular to the refrading furface, than it was before ; the light which fpreads from a point fliall by pafling through a convex fur- face of glafs into the air be made either to fpread lefs than before (as in fig. 143,) or to gather about the axis beyond the glafs (as in fig. 144.) But if the rays of light were pro- ceeding to a point in the axis of the glafs, they fliould b}' the refradion be made to unite fooner about that avis (as in fig. 1 4 y.) If the furflice of the glafs be concave, rays which proceed from a point fhall be made to fpread fifber (as ia fig. 146,) but rays which are tending to a point in the axis of C c c 1 the 38 o Sir Isaac N e w t o n's Book III. the glafs, fliall be made to gather about the axis farther from the glafs (as in fig. 1 47) or even to diverge (as in fig. 148,) unlefs the point, to which the rays are directed, Hes between the furface of the glafs and its center. 4. The rays, which fpread themfelves from a point, are called diverging ; and fuch as move toward a point, are called converging rays. And the point in the axis of the glafs, about which the rays gather after refradion, is called the focus of thofe rays. 5". I F a glafs be formed of two convex fpherical furfaces (as in fig. 149,) where the glafs AB is formed of the furfaces A C B and A D B, the line drawn throvigh the centers of the two furfaces, as the line E F, is called the axis of the glafs ; and rays, which diverge from any point of this axis, by the refradion of the glafs will be caufed to converge toward fome part of the axis, or at leaft to diverge as from a point more remote from the glafs, than that from whence they proceed- ed ; for the two furfaces both confpire to produce this effecfl upon the rays. But converging rays will be caufed by fuch a glafs as this to converge fooner. If a glafs be formed of two concave furfaces, as the glafs A B (in fig. I)0,) the line CD drawn through the centers, to which the two furfaces are formed, is called the axis of the glafs. Such a glafs fhall caufe diverging rays, which proceed from any point in the axis of the glafs, to diverge much more, as if they came from fome place in the axis of the glafs nearer to it than the point, whence / c Chap. 4: PHILOSOPHY. 381 whence the rays adually proceed. But converging rays will be made either to converge lefs, or even to diverge. 6. In thefe glaffes rays, which proceed from any point near the axis, will be affefted as^ it were in the fame man- ner, as if they proceeded from the very axis it felf, and fuch as converge toward a point at a fmall diftance from the axis will fuffer much the fame effeds from the glafs,as if they converged to fbme point in the very axis. By this means any luminous body expofed to a convex glafs may have an image formed upon any white body held beyond the glafs. This may be ea- fily tried with a common fpedacle-glafs. For if fuch a glais be held between a candle and a piece of white paper, if the diftances of the candle, glafs, and paper be properly adjufted? the image of the candle will appear very diftindly upon the paper , but be feen inverted ; the reafon- whereof is this . Let AB (infig. i^i) be the glafs, CD an objedl placed crofs the axis ot the glafs. Let the rays of light, which if- fue from the point E, where the axis of the glafs crofTes the ob- jed:, be fo retraded by the glafs, as to meet again about the point F. The rays, which diverge from the point C of the objed, fliall meet again almoft at the fame diftance from the glafs, but on the other lide of the axis, as at G ; for the rays at the glafs crofs the axis. In like manner the rays, which proceed from the point D, will meet about H on the other fide of the axis. None of thefe rays, neither tliofe which proceed from the point H in the axis, nor thofe which ilfue from C or D, will meet again exactly in one point ; but yet in one place, as is here fuppofed at F, G, and H, they will 382 Sir Isaac Newton's Book III. . will be croiided (o clofe together, as to make a diftlnct image of the object upon any body proper to reflect it> which fliall be held there. 7. If the object be too near the glafs for the rays to converge nfter the refraction, the rays fliall iflTiie out of the glafs, as if they diverged from a point more diftant from the glafs, than that from whence they really proceed (as in fig. 1 yi,) where the rays coming from the point E of the object, which lies on the axis of the glafs A B, if- fue out of the glafs , as if they came from the point F more remote from the glafs than E ; and the rays proceed- ing from the point C ifliie out of the glafs, as if they pro- ceeded from the point G ; likewife the rays which ifliie from the point D emerge out of the glafs, as if they came from the point H. Here the point G is on the fame fide q( the axis, as the point C ; and the point H on the fame fide, as the point D. In this cafe to an eye placed beyond the glafs the object fliould appear, as if it were in the fltii- ation G F H. 8. 1 F the glafs A B had been concave (as in, fig. j yg,) to an eye beyond the glafs the objeft C D would appear in the fituation G H, nearer to the glafs than really it is. Here nlfo the objcd: will not be inverted; but the point G is on the fame fide the axe with the point C, and H on the £ime fide as D. 9. Hence Chap.4. philosophy. 383 9. Hence may be iinderflood, why fpedacles made with convex glaffes help the fight in old age : for the eye in that age becomes unfit to fee objeds diftindlly, except fucli as are remov'd to a very great diftance ; whence all men, when they firft ftand in need of fpedtacles, are ob- ferved to read at arm's length, and to hold the objedt at a greater diftance, than they ufed to do before. But when an objed: is removed at too great a diftance from the fight, it cannot be feen clearly, by reafon that a lefs quantity of light from the objed: will enter the eye, and the whole objed will alfo appear fmaller. Now by help of a con- vex glafs an objedt may be held near, and yet the rays of light ifluing from it will enter the eye, as if the objedt v\'ere farther removed. 10. After the fame manner concave glafi^es afiift fuch, as are fliort fighted. For thefc require the objedt to be brought inconveniently near to the eye, in order to their feeing it diftindly ; but by fuch a glafs the object may be removed to a proper diftance, and yet the rays of light enter the eye, as it they came from a place much nearer. 11. Whence thefe defects of the fight arifb, that iti old age objects cannot be [cqii diftinct within a moderate diftance, and in fliort-fightednefs not without being brought too near, will be eafily underftood, when the manner of vifion in general fhall be explained ; which I fhall now en- deavour to do, in order to be better underftood in what 3 fellows. 38+ Sir Is A AC Newton's Book III. follows. The eye is form'd, as is reprefented in fig. i f^. ■It is of a globular figure, the fore part whereof fcarce more protuberant than the reft is tranfparent. Underneath this tranlparent part is a fmall collection of an humour in appearance like water, and it has alfo the fame refractive power as common water ; this is called the aqueous hu- mour, and fills the fpace A B C D in the figure. Next beyond lies the body DEFG; this is folid but tranfparent, it is compofed with two convex furfaces, the hinder furface E F G being more convex, than the anterior E D G. Between the outer membrane ABC, and this body EDGE is placed that membrane, which exhibits the colours, that are feen round the fight of the eye ; and the black fpot, which is called the fight or pupil, is a hole in this membrane, through which the light enters, whereby we fee. This membrane is fixed on- ly by its outward circuit, and has a mufcular power, where- by it dilates the pupil in a weak light, and contracts it in a ftrong one. The body DEFG is called the cryftalline humour, and has a greater refracting povi^er than water. Behind this the bulk of the eye is filled up with what is called the vitreous humor, this has much the fame refra- ctive power with water. At the bottom of the eye toward the inner fide next the nofe the optic glafs enters, as at H, and fpreads it felf all over the infide of the eye , till within a fmall diftance from A and C. Now any object, as IK, being placed before the eye, the rays of light ifiiiing from each point of this object arc fo refracted by the coi> vex furface of the aqueous humour, as to be caufed to con- verge j after this being received by the convex furface E D G of Chap. 3. PHILOSOPHY. 385 of the cryftalline humour, which has a greater refradlive power than the aqueous, the rays, when they are entered into this furface, ftill more converge, and at going out of the furface E F G into a humour of a lefs refradive power than the cryftaUine they are made to converge yet farther. By all thefe fucceilive refradlions they are brought to converge at the bottom of the eye, fo that a diftind image of the ob- jed: as L M is imprefs'd on the nerve. And by this means the objed is (een. 11. It has been made a difficulty, that the image of the objedt imprelTed on the nerve is inverted, fo that the upper part of the image is impreiTed on the lower part of the eye. But this difficulty, I think, can no longer re- main, if we only confider, that upper and lo\\'er are terms merely relative to the ordinary pofition of our bodies : and our bodies, when view'd by the eye, have their ima, gh^ gly in the 'fame man- ner inclined to each other, as they were before their inci- dence upon the point o-, and will make the fame angles with the furface of the drop at the points hy k, I, as at the points h, dj /, after their entrance ; and therefore after their emer- gence out of the drop each ray will be inclined to the fur- face in the fame angle, as when it firft entered it ; whence the lines hm, knylo, in which the rays emerge, muft be parallel to each other, as well as the lines a by c dy efy in which they were incident. But thefe emerging rays being parallel will not fpread nor diverge from each other in their palTage from the drop, and therefore will enter the eye conveniently fituated in fufficient plenty to caufe a fenfation. Whereas all the other rays, whether thofe nearer the center of the drop, as ^ 5', r j-, or thofe farther oiF, as tu, -w X, will be refledled from other points in the hin- der furface of the drop ; namely, the ray £ q from the point jy r s from Zy tv from «, and 'w x from /3. And for this reafon by their reflection and fucceeding refraClion they will be fcattered after their emergence from the toremen- tioned rays and from each other, and therefore cannot en- ter the eye placed to receive them copious enough to excite any diftind fenfation. Eee X 3. The 39<5 Sir Isaac Newton's Book III. 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 hy g ky g /, at the points hy hy /, do not wholly pafs out of the drop, but are in part refledled back ; though the fecond refledlion of thefe particular rays does not form the outer bow. For this bow is made by thofe rays, which after their en- trance into the drop are by the refraction of it united, be- fore they arrive at the farther furface, at fuch a diftance from it, that when they fall upon that furface, they may be re- fleded in parallel lines, as is reprefented in fig. idl ; where the rays a hy c dy efy are colledled by the refraftion of the drop into the point gy and paffing on from thence ftrike upon the furface of the drop in the points h, A, /, and are thence reflected to m, «, (?,pafling from h to m, from k to Uy and from I to in parallel lines. For thefe rays after reflexion at nty n, a will meet again in the point p, at the fame diftance from thefe points of refleftion 2W, w, 0, as the point g is from the former points of reflection by ky I. Therefore thefe rays in pafling from ^ to the furface of the drop will fall upon that furface in the points q, y, J- in the fame angles, as thefe rays made with the furface in by dy f) after refraClion. Confequently, when thefe rays emerge out of the drop into the air, each ray will make with the furface of the drop the fame angle, as it made at its firft incidence ; fo that the lines qty r v, s isoy in which they come from the drop, will be parallel to each other, as well as the lines ahy c dj cfy in which they came to the drop. Chap. 5. PHILOSOPHY. 397 drop. By this means thefe rays to a fpedator commodioiifly fituated will become vifible. But all the other rays, as well thofe nearer the center of the drop x j, 2i «,, as thofe more remote from it /3 y, J\ «, will be refleded in lines not paral- lel to the lines bra, kn, lo\ namely, the ray xj, in the line ^ «, the ray 2: * in the line 9 x, the ray jS y in the line A|^, and the ray J\6 in the line »^. Whence thefe rays after their next reflediion and fiibfequent refradion will be fcattered from the foremcntioned rays, and from one ano- ther, and by that means become invifible. 4. I T is farther to be remarked, that if in the firft cafe the incident rays « ^, c of, e fy and their correfpondent e- mergent rays h nty kn, 1 0, are produced till they meet,, they will make with each other a greater angle, than any other incident ray will make with its correfponding emer- gent ray. And in the latter cafe, on the contrary, the e- mergent rays qt, r v, S'w make with the incident rays an acuter angle, than is made by any other of the emergent rays. .J-. Our author delivers a method of finding each of thefe extream angles from the degree of refradiion being given ; by which method it appears, that the firft of thefe angles is the lefs, and the latter the greater, by how much the refradive power of the drop, or the refrangibility of the rays is greater. And this laft confideration fully com- pleats the dodrine of the rainbow, and fhews, why the co- lours of each bow are ranged in the order wherein they are feen. 5 J, (J. SUF. ~N1 398 Sir Isaac Newton's Book III. 6. Suppose A (in fig. 1 6±.) to be the eye, B,C,D,E,F,drops of rain, M w, O^, Qr, StyV ^w parcels of rays of the fun, which entring the drops B, C, D, E, F after one reflection pafs out to the eye in A. Now let Mn he produced to m till it meets with the emergent ray likewife produced, let O^ produced meet its emergent ray produced in x, let Qj- meet its emergent ray in A, let S t meet its emergent ray in ^, and let V -iw meet its emergent ray produced in ». If the angle under M>iA be that, which is derived from the refradlion of the violet-making rays by the method we have here fpoken of, it follows that the violet light will only enter the eye from the drop B, all the other coloured rays pafTmg below it, that is, all thofe rays which are not fcattered, but go out parallel fo as to caufe a fenfation. For the angle, which thefe parallel em.ergent rays makes with the incident in the moft refrangible or violet-making rays, being lefs than this angle in any other fort of rays, none of the rays which emerge parallel, except the violet-making, will enter the eye under the angle M « A, but the reft mak- ing with the incident ray M n a greater angle than this will pafs below the eye. In like manner if the angle under O » A agrees to the blue-making rays, the blue rays only fhall en- ter the eye from the drop C, and all the other coloured rays will pafs by the eye, the violet-coloured rays pafling above,^ the other colours below. Farther, the angle QaA corre-' fponding to the green-making rays, thofe only fhall enter the eye from the drop D, the violet and blue-making rays pailing above, and the other colours, that is the yellow and red, Chap. 5- PHILOSOPHY. 399 red, below. And if the angle S f* A anfwers to the refra- ction of the yellow-making rays, they only fhall come to the eye from the drop E. And in the laft place, if the an- gle V y A belongs to the red-making and Icaft refrangible rays, they only fhall enter the eye from the drop F, all the other coloured rays palling above. 7. But now it is evident, that all the drops of water found in any of the lines A x,, A a, A ^, A v, whether farther from the eye, or nearer than the drops B, C, D, E, F, will give the fame colours as thefe do, all the drops upon each line giving the fame colour ; fo that the light refledted from a number of thefe drops will become copious enough to be vifible; whereas the reflection from one minute drop alone could not be perceived. But befldes, it is farther manifefl, that if the line A 3 be drawn from the fun through the eye, that is, parallel to the lines Mn, O ^, Qj^, S f , V -ijy, and if drops of water are placed all round this line, the fame colour w411 be exhibited by all the drops at the fame diftance from this line. Hence it follows, that when the fun is moderately elevated above the horizon, if it rains oppo- fite to it, and the fun fhines upon the drops as they fall, a fpedlator with his back turned to the fun muft obferve a co- loured circular arch reaching to the horizon, being red with- out, next to that yellow, then green, blue, and on the in- ner edge violet ; only this laft colour appears faint by being diluted with the white light of the clouds, and from another caufe to be mentioned hereafter \ »§ II. 8. Thus 400 Sir Isaac Newton's Book III. 8. Thus is caufed the interior or primary bow. The drops, of rain at fome diftance without this bow will caufe the exterior or fecondary bow by two refledlions of the fun's light. Let thefe drops be G, H, I, K, L; Xj, Z a, r|3, A «, © ^ denoting parcels of rays which enter each drop. Now it has been remarked, that thefe rays make with the vijQble refraded rays the greateft angle in thofe rays, which are moft refrangible. Suppofe therefore the vifible refracted rays, which pafs out from each drop after two reflections, and enter the eye in A, to interfedthe incident rays in w, p, Ibid. ' Opt B. II. part 4. "" Ch. 3. § 14. F £ £ .eafy ^o2 Sir I s A A c N E w T o n's Book III. cafy tranfmilTion and refledlion, fucceeding each other at equal intervals ; infomuch that if they reach the farther fu- perficies in one fort of thofe fits, they fhall be tranfmitted ; if in the other kind of them, they fhali rather be refledled back. Whence the rays that proceed from B to C, and emerge in the line C D, being in a fit of eafy tranfmiffion, the fcattered rays, that fall at a fmall diftance without thefe on either fide {fuppofe the rays that pafs in the lines BE, BG) fhall fall on the furface in a fit of eafy refledion, and fhall not emerge ; but the fcattered rays, that pafs at fome diftance without thefe laft, fhall arrive at the furface of the or lobule in a fit of eafy tranfmifiion, and break through that furface. Suppofe thefe rays to pafs in the lines BF, BH; the former of which rays fhall have had one fit more of eafy tranfmifiion, and the latter one fit lefs, than the rays that pafs from B to C. Now both thefe rays, when they go out of the globule, will proceed by the refraction of the water in the lines FI, H K, that will be inclined almoft equally to the rays incident on the globule, which come from the fun ; but the angles of their inclination will be lefs- than the angle, in which the rays emerging in the line CD are inclined to thofe incident rays. And after the fame manner rays fcatter- ed from the point B at a certain diftance without thefe will emerge out of the globule, while the intermediate rays are intercepted; and thefe emergent rays will be inclined to the lays incident on the globule in angles ftill lefs than the angles, in which the rays F I and H K are inclined to them ; and without thefe rays will emerge other rays, that fliall be inclined to tlie incident rays in angles yet lefs. . Now by -f^.40; - A - r-^. j/>'o . ■X^'..V.'v--^^^!s^... 1' - ^'■^::-;^:::^^. Z - ^~~-~-^r""~"~~-^^^^ \ '" -wii^;--- X- "'■•I'"". -~~~,^^^ ^^J^C^;^^ sA "'">... '■p- ^^^^/"^^C^^^^^ ^^^; H,f _/« c Chap. 5. PHILOSOPHY. 4.03 by this means may be formed of every kind of rays, bciides the principal arch, which goes to the formation of the rain- bow, other arches within every one of the principal of the fame colour, though much more faint; and this for divers fucceflions, as long as thefe weak lights, which in every arch grow more and more obfcure, lliall continue vifiblc. Now as the arches produced by each colour will be vari- ously mixed together, the diverfity of colours obferv'd in thefe fecondary arches may very poflibly arife from them. II. In the darker colours thefe arches may reach below the bow, and be feen diftindt. In the brighter colours thefe arches are loft in the inferior part of the principal light of the rainbow ; but in all probability they contribute to the red tin- cture, which the purple of the rainbow ufually has, and is moft. remarkable when thefe fecondary colours appear ftrongeft. However thefe fecondary arches in the brighteft colours may pofTibly extend with a very faint light below the bow, and tinge the purple of the fecondary arches with a reddifli hue. II. The precife diftances between the principal arch and thefe fainter arches depend on the magnitude of the drops, wherein they are formed. To make them any degree feparate it is necelfary the drop be exceeding fmall. It is moft likely, that they are formed in tlie \'apour 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 reafon, why thefe colours appear under the upper F f f a part ^04- Sir Isaac Newton's, >&€. Book III. part of the bow only, this vapour not defcending very low. As a farther confimiation of this-, thefe colours are feen flrongeft, when the rain falls from ver)^ black clouds, which caufe the iierce^ft rains, by the fall whereof tfie air will be moft agitated. 13. To the like alternate return of the fits of eafy tranf- mifiion and reflection in the pallagc of light through the globules of water, which compofe the clouds, Sir Isaac Newton afcribcs fome of thofe coloured chcles, which at times appear about the fiin and moon \ * Opt. B. II. part 4.. obf. i^. C O N- CONCLUSION. 4.0$ i' '/.'/*' .'iM^ . CONCLUSION. IR Isaac Newton having concluded each of his philofophical treatifes with fome general refledions, I fhall now take leave of my readers with a fhort account of what he has there delivered. At the end of his mathematical prin- Ij ciples of natural philofophy he has given us his thoughts concerning the Deity. Wherein he firft obferves, that the fimilitude found in all parts of the univerfe makes it undoubted, that the whole is governed by one fupreme being, to whom tlie original is owing of the frame of nature, which evidently is the cffe6l of choice and defign. He then proceeds briefly to ftate the beft me- taphyseal notions concerning God. In fliort, we cannot conceive either of fpace or time otherwife than as necef- farily 4-o(J CONCLUSION. farily exifting ; this Being therefore, on whom all others de- pend, muft certainly exift by the fame neceflity of nature. Confequently wherever fpace and time is found, there God muft alfo be. And as it appears impoflible to us, that fpace fhould be limited, or that time iliould have had a beginning, the Deity mufl: be both immenfe and eternal. 2. At the end of histreatife of optics he has propofed fome thoughts concerning other parts of nature, which he had not diftindly fearchcd into. He begins with fome farther refiedions concerning light, which he had not fully examined. In particular he declares his fentiments at large concerning the power, whereby bodies and light adt on each other. In fome parts of his book he had given fhort hints at his opinion concerning this', but here he exprefly de- clares his conjecture, which we have already mentioned ^ that this power is lodged in a very fubtle fpirit of a great elaftic force diffufed thro' the univ^erfe, producing not only this, but many other natural operations. He thinks it not impoffible, that the power of gravity it felf fhould be ov/ing to it. On this occafion he enumerates many natural appearances, the chief of which are produced by chymical experiments. From numerous obfervations of this kind he makes no doubt, that the fmallefl: parts of matter, when near contad:, ad flrongly on each other, fometimes being mutually attraded, at other times repelled. 7. The attradive power is more manifeft than the Other, Yor the patts of all bodies adhere by this principle. And the » Opt. pig, 2/^. "ch. 3. §is, name CONCLUSION. 407 name of attraction, wliich our author has given to it, has been very freely made ufe of by many writers, and as much obje(£led to by others. He has often complained to me of having been mifunderllood in this matter. What he fays upon this head was not intended by him as a phi~- lofophical explanation of any appearances, but only to point out a power in nature not hitherto diftinctly obferved, the caufe of which, and the manner of its ading, he thought was worthy oi a diligent enquiry. To acquiefce in the explanation of any appearance by afferting it to be a gene- ral power of attraction, is not to improve our knowledge in philofophy, but rather to put a ftop to our farther fearch. .^. urxirv ,?>/ /''/y/*' .^t'w^/' A ■ \ / UnlversWY of CalMornIa ^ IN REGIONAL LIBRARY FACILITY SOUTHERN REG'0^''";r,""' ca 90024-1388 from which it was borrowed. fi^ mil 'b^mc PRtON , ,_ "IP r=_" z-^-fo^ 0^r;Loo| QL JAN 1 6 2001 4JH1VERS1TY OF OALIFOHNIA AT LOS ANGELES UBRARX 'flffl D 000 001 090 jfliitjiliifliiti; ■iiiiifeini' :lliii)!!|illltli!i|il}iiil!!i!ii-: