gregory, elements of astronomy, physical and geometrical (english, 1715)

8/2/2019 Gregory, Elements of Astronomy, Physical and Geometrical (English, 1715)

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^f^^ .,;:'^C^'^^:^^^i'/THE

ELEMENTSO FAftronomy,Physical and Geometrical.By David Gregory M. D. Savilian

ProfefTor of Astronomy at Ox-ford^ and Fellow of the Royal-Society,

Done.into EngUjh^with Additions and Corre


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t"H E

AUTHOR'SPREFACE.MY defign in publifliing this Book, was,that the Celeftial Phyiics, which the

moft fagacious Kepler had got thefcent of, but the Prince of Geometers Sir IfaacNewton^thxou^t to fuch a pitch as furprifes allthe World, might, by my care and pains in il-luftrating, become eafier to fuch as are defi-reus of being acquainted with Philofophy andAftronomy, The Title informs you fuffici-ently that the Arithmetical orCalculatorypartof Aftronomy is here omitted, tho' that, per-haps, may be publiftied hereafter in its properplace. As for thePhyfics, it is all taken out ofthe abovemention'd Authors ^ but is here inter-mixed with Aftronomy, in fuch places asfeem'dproperand convenient ^ethe Geometry to be metwith in it, I have either borrowed eifewhere,


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ii The Authors P r f a c Eor delivered it Lemmatrically. Whateveris done in each Seftion, you have it exprefs deither in the Title or Preface thereof, in fucha manner, as that thofe who are lefs vers'din the more abftrufe parts of Geometry, orlefs concerned about the Phyfical parts, maypafs over, and only read the Aftronomy fepa-rately and diftind: from them.TheCeleftialPhyfics, or Phyfical Aftrono-my, is not only the firft in dignity of all inqui-ries into Nature whatever, but the firft in or-der, becaufe it is the eafieft* For the Suri andPlanets are feparated from one another by foimmenfe a diftance, as renders them incapableof exerting moft of thofe forces whereby allBodies aft upon one another 5 fo that theyhave no other force left them whereby theycan afFed one another, but the fingle force ofuniverfal Gravity: Whereas in theproduftioiiof feveral Phasnomena, that are obferv'd uponpur Earth,innumerable other forces ate exerted^fuch as are very hard to be diftinguifli'd fromone another 5 which notwithftanding, if notaccurately done, in vain do we attempt Nature,and make any inquiry into it. Upon this ac-count it is, that every Problem in the Terre-flrial Phyfics is very operofe and perplex'd, onthe contrary, in the Celeftial Phyfics, muchinore eafy^and fimple, tho* even the latter hasits diffieulties, arifing from the different di-ftances and magnitudes of the Celeftial Bodies,For the Fix d Stars are fo vaftiy diftant afun*


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Tke Author's Preface: mrfach other, obfervaKe by us who are the In-habitants of the Earth, The Primary Pla-nets are remov'd fo far from each other, that,tho' they have fome fmall power and effeftupon one another, yet we cannot be fenfiblepf it, till after many Years obfervation. TheSecondary Planets are not at fo great a di-ilance from their Primaries, or from the Sun,but that they may be confiderably affefted bythe powers pf both, (if regard be had to theiQiiantity of Matter that is in thefe latter,) andthis is the fpring of tbpfe manifpld inequali^ties found in them, fuch as, for inftance, mani-feftly fhews itfelf in our Moon ^ which yet isnothing at all, if compared with the inequali-ties found among Terreftrial Bodies, whichare afted upon by an innumerable variety ofother forces, prefGng everyway upon them*So that thofe perfons feem to apply theirtl;ioughts but to a very indifferent purpofe inthe ftudy pf Nature, that overlook this partpf Aftrononiy, from whence the principal an3fupft fimple Laws of Nature are to be learn d.That none may think the Phyfics delivered in

the followingWork intirely new and unknownin Aftronomy, I fliall take the liberty to (hewthat it W3S both known and diligently culti-vated by the moft ancient Philpfophers. AndIfliall dwella little longer upon this argument,becaufe there is no need of fpending a Preface,either upon the order of the parts of thisWork, which may be feen in the Index, ox


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iv The Author's Preface*grefs of Aftronomy, or even of the true Sy-5em of the World, approv d of hyFythagorasând others among the Ancients ^ thefe thingsbeing all of them treated of at largeJ^ thecommon Writers of Aftronomy. What I fhailnow therefore malê olit is, that we do ftilltread in the fteps of the Ancients in this Phy-fical Aftronomy 3 inafmuch as they ktifew thatthe Geleftial B6dies gravitated towards eachother, and were retain'd in their Orbits by theforce of Gravity^ and were alfo apprized ofthe Law of this Gravity,

For if we look back to the firft Rife ofA-flronomy, and take a view of it in its Infancy,as it were, we fliali find nothing better ap-prov'd of, nothing more nniverfally entertain-ed among the feveral Seds of Philofophers,than this notion of the Gravity of the Gele-ftial Bodies. That faying is well known, fooften ufed by ^ Anaxttgoras^ and his Scholars,Âchelaiis and "" Euripides, Namely, ** That** the Sun and Stars v/erc fiery or red-hot Stones" and Golden Clods!' Of the fame mind alfowere ^Democritus^ Metrodorus^ and ^Diogenes_^ By

a He affirmed the Sun to be a Mafs of Red-hot Iron. Diog*Laert.- in Atiaxag. That the fubflance of the Sun was Stonothat of the Moon, Earth. -Plat, in Apol. Socr.b That the Stars were Plates of Red hot Iron. Scob. Ed,Phyf. cap, 25.

c They fay that Euripides ufed to call the Sun a Clod of Goldor Qolden Mafs. Diog. Laer. in Anaxag.d Anaxagoras, Democricus and Tsletrodorus affirmed, thatthe Sun was a Mafs of Iron or Stone red- hot, Pltit. de Placit.


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The Author's Pri fa c e . vBy thefecxprdOSons they meant no more, thanthat they were heavy, denfe and fix d Bodies,(fuch as Stones are) fo as to bear a confidera-ble degree of heat: And that^this was reallyitheir meaning will evidently appear, if we dobut enquire more narrowly into the firft Au-thors of this Opinion. For, as we are toldby ^ Democritusy thefe notions about the Sunand Moon are not to be afcrib'd to Anaxago-ras as their original, for he had really bor-rowed them of the Ancients. Nor is it a diffi-cult matter to find out, who they were that heborrowed them of, or from whom they werehanded down to him. He had them from hisM2&QX ^ Anaximenes^ whofe Opinion we knowwas, that the Stars were of a fiery nature andfubftance, that there were alfo mingled withthem certain Earthly Bodies, which were car-ped round about them, tho'not vifible to us;By which words he plainly means. Planets ofa terreftrial nature, performing their revolu-tions in theSyftem of every Fix d Star, Thefepotions Anaximenes received from Anaximan-der^ A^iaximander from ^^ Thales himfelf, whowas the Head and Founder of the Jonk Philo-A 4 fophy 5

I II ! II I II I ! ' ',

' ''

f Favorinus, in his various f/iftory, relates, that Dcmocritusufed to fay of Anaxagoras, that the Opimtis which he taughtconcerning the Sun and Moon, were not his o?/?, but far more and'entthan AnaxagorasV time-y and that he hadftolien them. Laerc*in Democrir.

g Anaximenes /^/V, the nature of the Stars were fiery ^ andthat there were certain terreftrial Bodies that are inviftble, carriedtogether about them. Stob. Ecl.Phyf.c. 2$.


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vi The Authors Pr e fa c e*fophy^ and fpread this opinion of the Gravi-ty of the Fix'd Stars among his Seft. Nordid this Dodrine concerning the Stars flophere, but afterwards it diffused it felf thro'the Italic Philofophy, the ^ followers of whichtaught, that each Star was a World in thein-iinite Êthereal Space, containing Earth,Air and jEther^ and that the ^^Moon, not on-ly was like our Earth, but inhabited by Ani-mals of a larger fize, and furnifti'd with Plantsof a more beautiful appearance.Nor were they fo abfurd in their concep-tions about Gravity, as to think that it wasdone by the virtue of any point within theEarth, orofaCenteri to which all heavy Bo-dies placed any where tended 5 but they thoughtit was done by the ^ power of the wholeMatter in the Terreftrial Globe attrading allthings to it felf : And as the power of theLoadftone is composed of the powers of the

feveral1 The Pythagoreans affmd^Jhdt every Star is a W,orld in the

hifimie Ethereal Space, wherein are contain d Earthy Jir, andj.ther, Plut. (le Pkc. Philof. Lib. 2. c. ig.

k The Pythagoreans afferted, that the Moonfeeitid to be of afim'ilar nature with the Earth, it is inhabited as our Earth is^by Animals^ tho' of a larger fiiê than ours^ and fiird with thej(ime PIants y tho^ much more beautiful than ours. Pluc. de Plac.Phil. Lib. 2. c. 50.

1 Andyst if every heavy Body inclines towards the Jame placeând.does with every one of its parts tend to its middle or centerthe Earth lertaihly will not approfriiite'toitfelf thefe heavy Bodies:^which are its parts ^ becaufe it is the Center of the Vniverfe, butrather becaufe it is the whole^ of which they are the parts, Pluude facie in Orbe Luiia?'.

As for that which is incorporeal^ "'tis net probable^ nor will they


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The Author^s Preface. viifevcral parts combin'd together, fp they be-lieved that the Gravity towards the wholeEarth, refulted from the Gravity towardseach fingle part of it. Befides, they believ'dthere was a "^Gravity towards the Moon andSun, afting in the fame mann^ir as it dpes to-wards the Earthy and that each ^ Planet, likea Stone, whirl'd in a fling, was kept in itsiOrbit by the fame principle, and for the famereafon revolving always about us. Fromforae things mention d by Diogenes JuaertmT^concerning VlatOy which alfo are obfcurelyhinted at in his p Titn^us^ I am apt to believewith ^ GaUleOy that the divine Philofopherfuppos'd the Mundane Bodies, when they wereiirft formed, were moved with a Reftilinearmo-lion (by ik^ means Qf Gravity,) but after that

they

m And this meeting together of Bodies here, and th{ir poaVttjmli^'ewtfe with the Earth's Body, jherv us the manner how it is pro-pable that the parts, which are affembled at the Man's Body, con-tinue alfo there, i

n But the Moon is helped, and prefervd from falling down, byher very Motion and that impetmflty of her Revolu tion -, as Stonesand other weighty Bodies put in Slings and fxpung round, are keptfrom dropping out by tk-e fwiftnefs of their Motion, and their beingfnov'd circularly. Wherefore the Moon does not movedownwards, as her own weight n>ou'd naturally cap^ hqr, hef ten-dency thai w.ay hfing ftopt bj the violence of her circular revolu-tion. Ibid. p Thefe at firji were mov^d in a confus'd and irregular manner,but when they were duely adjufted and rightly fettled, then theW.oxld was eftabjifhld by God in juji order and proportion, .. Dio^t^QH. in Plat.

p He gavel t a Motion altogether agreeable to its nature as ^^


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viii The Author^s Preface;they had arrived to fome determined places,theybegan to revolve by degrees in a Curve, theReftilinear Motion being chang'd into >a Cur-vilinear one. Tis from this Doftrine of Gra-vity, that all Bodies gravitate mutually to oneanother, 'tis by this that ^ LucretiuSy taughtby Epicurus and Bemocritusy labours to prove,that theUnivcrfe hasno Center or loweft Place,but that there is an infinity of Worlds likeours in the immenfe Space. His Argumentruns thus:^ If the nature of things were bound-ed any where, th^n the outmoft Bodies, fincethey have no other beyond them, towardstvhich they may be made to tend by the forcecf Gravity, wou d not ftand in an Equilibrio,but make towards the inner and lower Bodies,being neceflarily inclin'd that way by theirGravity, and therefore having m^ide towardsone another, during an infinite fpace of time,would have long agp met, and lye in the mid-dle of the whole, as in the loweft place. Tisevident therefore from hence that Lucretius^and thofe whom he fbllowed, believ'd thatall Bodies did Gravitate towards the Mattel:placed around them, and that every fingle Bo-dy was carried by the more prevailing Graviry,towards that region where there was moftMatter.

As

r S :ppofe tbey all had Bounds^ Jttppofe an End y


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The Author''s P r e f a c e. ixAs it is manifeft that the Ancients were

apprized of, and had difcover'd the Gravity ofall Bodies towards one another, fp alfo theywere not unacquainted with the Law and Pro-portion which the adion of Gravity obferv'daccording to the different MalTes and Diftan-ces. For that Gravity is proportional to theQuantity of Mattei: in the heavy Body, ^Lm-tretius does fuAd'ettly declare, /^ as alfo thatwhat we call light Bodies, don't afcend oftheir own accord, but by the aftion of a forceunderneath them, impelling them upwards,juft as a piece of Wood is in Water 5 " an4further, that all Bodies, as well the heavy asthe light, do defcend i?i vacito^^ with an equalcderity. It will be plain likewife, from what IIhail prefently obferve,that the famousTheoremabout the proportion whereby Gravity decrea-fes in receding from the Sun, was not un-known at leaft to Vythagoras, This indeedfeems to be that which he and his followerswould fignify to us by the Harmony of theSpheres: That is, they feign d -^^^/^i? playingupon an Harp of feven Strings, by whichSymbol, as it is abundantly evident from^Pltny^ Macrohtus 2XiA.Cenforinus^ they meantthe Sun in Conjundion with the feven Planets,

for

1 Befides , why have not Bodies equal WeightWith thofe, whofe Figure is but juft as great/ Lucr, I. '1.^.41$.c And this I thivk a proper place to prove.

That nothing of it felfcan upwards move. Lucr. 1.2. v. 178.


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jsi The Mthor's V^ i F a c kfor they made him the Iqader of tl^at SeptenaryChorus, and Moderator of Nature 5 andthought that by his Attraftive force he aftediipon the Planets (and called it Jupiter's prilTon, becaufe it is by this Force that he retain?and keeps them in their Orbits, from flying offin Right Lines) in the Harmonical ratio oiFtheir Diftances. For tl^e fojrces, whereby crqual T^nfions ad upon Strings of differenflengths (being equal in other refpefts) are re-ciprocally as the Squarespf the lengths of the

y For Pythagoras as he was paffing by ^Smith's Shpp, took occafion to obferve, thatthe Sounds the Hammers made, were mpre ac-cute or grave in proportion to the weights ofthe Hammers 5 afterwards ftretching SheepsGuts, and faftning various Weights to them,he learn'd that here likewife the Sounds wereproportional to the Weights. Having fatisfy 'dhimfelf of this, he inveftigated the Numbers,according to which Confonant Sounds weregenerated. Whether the whole of this Storybe true, or but a Fable, 'tis certain Tj/thagQ-ras found out the true ratio between thefound of Strings and the Weights fatten d tothem. The fame Tenfipn a3:s upon a Stringas fhort again, four times more powerfully:For it produces an Odave, and an Odave 15founded by a force that is four times greater^for if a String, put upon the ftretch by a givei^

Weighty


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The Author^s P r e f a c'e; xi-Weight, generates a given Tone, the fameString ftretch*d by a Weight four times greater,will found an Odave. Thus likewife the*fame Tcnfion upon a fubfefquialteran Chord,afts ill a double fefquiquattan ratio: For itgenerates a Fifth or Diapente, and a Stringthat founds a given Note, \x^ith st given Weightought to be ftretch*d by a Weight that is adouble fefquiquartan to found a Fifth. Anduniverfally, the Weights which generate allTones in Strings, are reciprocally as theSquares of the lengths of Strings of equalTenfion, producing the fame found in anyMufical Inftrument. Pythagoras afterwardsapplied the proportion he had thus foundby experiments, to the Heavens, and fromthence learn'd the Harmony of the Spheres.And, by comparing thefe Weights withthe Weights of the Planets, and the inter-vals of the Tones, produced by the Weights,with the interval of the Spheres^ and laftly,the lengths of Strings with the Diftances ofthe Planets from the Center of the Orbs ^ heunderftood, a^ it were by the Harmony of theHeavens, that the Gravity of the Planets to-wards the Sun (according to whofe meafuresthe Planets move) were reciprocally as theSquares of their Diftances from the Sun.We have thus far been fliewing what wasthe Opinion of the Ancients concerning Gra-vity 5 and it is evident they were perfwadedthat Gravity was not an affedion of Terre-


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xii The Author^s Preface.'that all B odies gravitate towards one another;and that the Planets are retained in their Or-bits by the force of Gravity, and laftly, thatthe Gravity of the Planets towards the Sunare reciprocally as the Squares of their Di-ftances from it. What the induftry and (killof the Moderns have added to thefe invent!-ons of the Ancients, the following Pages dadeclare at large.

T H E

M I


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THEPUBLISHERT O T H E

READER.THE Science of Astko'N ouYywhich is as much efleenid and ad^mir^d for its great and manifold ufes forthe Service of Mankind^ as it is delightfuland entertaining to the more curious andcontemplative^ has in all ages been cul-tivated and improv^d^ by Men the mofieminent for their parts and learnings, andis now brought^ as it were^ to the ut*mofi degree of perfelion^ and that chieflyby the fuperior Genius and Indufiry ofthofe of our own Nation. But fince no-thing confiderable therein^ has been as yetwrit in our own Language^ I thought Icould not oblige my Country-Men morethan in publijlnng an Englifti Edition ofthe mofi valuable and finifhH piece of


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The Publiflicr to the Reader;reckoned to he a Book that contains not onlyall the Discoveries and Philofophical Senti^mcnts of the gf'eat Kepler, and the varibusHypothefes of the moft noted Jfironomersbefore and Jtnce his Time j but is chieflyvalued by the befl Judges^ for the largeand inflruSlive Comments delivered in it,on the Writings of the illujiriom Sir IfaacNewton, as well as on the feveral Aflro^nomical Differtations of the fagaciomDr.Halley, which the Reader will find hereevery where interfpers d.

And in order to render this work aseompleat as pojflble^J /hall^ in d verylittle time, prefent you with another Vo4lume, containing correSi Aflronomical Ta^^bjesy for the ready computing of the Pla^fiets PlaceSy Eclipfes^ &c. all done by aPerfon of known ability^ from the trueTheory of Gravity, deliver'*din this Book:For it was by no means judged proper thatI /Jjould annex to fo intire a piece as thisi>, any imperfeSt Tables, drawn from adifferent Principle from what is here efla^blifhed, fuch it feems allthofe asyetpub'-HfJjed are.


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ELEMENTSo PAftronomy,

Phyfical and Geometrical.' " - I

The First Book.Ofthe Syftem of the World.

- - -- l u llSection I*Concerning the Order, Diftances, and Peri-

ods of the Primary Planets revolving aboutthe Sun, and the principal Phenomenathence arifing.

Proposition LTO give a general accourJ of the- Ordery ^ndTeriods of The Vrimary Tlanets revolvingabout the Sun^ and their TXiftances from it}

us alfo what we are to think of the Comets and FioitStars.The Sun is to be look'd upon as immovable,


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i Tife^ Elements Bookl.in which the Planets perform their Revolutions.And there are fix opake Spherical Bodies thatrevolve about it, as their Center, from Weft toEaft, [Fig. I.] from^, along 5, C, andD, in thefollowing Order : Mercury neareft the Sun, corn-pleating its Revolution in about three Monthsfiext to Mercury, Venus in about {kvQn Monthsand an half; then the Earth in a Year ; Mars inabout two Years; Jupiter in twelve; and, laftofall, Saturn, which is outermoft, in thirty. TheirDiftances from the Sun are nearly the fame asthey are reprefented in the Scheme : namely,fuppofing the diftance of the Earth from the Sunto be divided into ten equal Parts, of thefe thediftance of Mercury will be about four, of Ve-nus feven, of Mars fifteen, ofJupiter fifty two,and that of Saturn ninety five.

'Tis to be obferv'd, thatall their Orbits arenot in the fame Plane, but vatioufiy inclin'd toone another; fo that fuppofing the Plane of theEarth's Orbit to coincide with the Plane of thisScheme, one half of the Plane of any otherPlanet's Orbit will be above, and the other halfBelow in ; fo that the Planes interfe


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Tlatc 1 3ook


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Book I. of AsTRqi^QU jijie Orbits ofthe Planets. Their Periods arenot as yet known from ObferVation, nor in-deed is it fully certain^ that tl^ey move in Linesthat return into themfelves. But ail thefe thingis^^perhaps^ will be determined by proper Obfer-vations made in future Ages.The reft of the Mundane Space is to be con-ceived as divided into Spaces juft like that wehave been defcribing, each having one of tholeStars in its Center^ which are called Fixt Stars,performing the office of a Sun to it, and having,it may be. Planets and Comets of its own re-volving about it.

It is enough at firft to have a general con-ception of thefe things as they are here defcribed:For tho* the Periods of the Planets, and theirDiftances from the Sun, as here laid down, arenot exadly true^ yet they are neareft theTruthin round Numbers. Again, tho* the Paths theydefcribe are not perfed Circles, concentric tothe Sun, and the Motion of the fame Planet notperfectly equable; yet the difference is fo fmall,that they need not, at prefent, be taken other-wife ; till by Obfervations and Methods ofufingthem, hereafter to be fliewn in their properplaces, all thefe things ftiall be precifely andexa^ly fettled, and thefe niceties examined ^ or,at leaft, till the fame are purpofely handled, inorder to inquire into their Phyfical Caufes.

Proposition ILTO defcrihe the 'Phenomena that arife fiom thtfituation of the Sun ^nd motion of the Earthds related above,Firft, If the Obferver be fuppos'd to be placedin the Sun, 'tis evident the Earth will feem to


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4 r^^ Elements BookLreally it does. Then fince, befides the Earth, ourObferver fees the Fixt Stars plac d round him^as it were (the nature of the Eye requiring it)in a concave Sphere, that has the Eye for itsCenter ; 'tis likewife evident, that he will ob-ferve the Earth moving, as it were, among theFix:t Stars, and approaching nearer and nearerthe more Eaftern ones3 till, in a Year's fpace,having compleated its Revolution, it returns tothe fame place among them again. And be-Caufe the Earth always goes the fame trackover again, the Obferver will take efpecial no-tice of the Stars the Earth paffes over ; alfo thePlane of the Earth's Orbit, and the Circle inthe Sphere of the Fixt Stars, called the Ecllpticymade by that Plane ; and this will be a greatCircle, becaufe it paffes thro' the Sun, or Eye,which is the Center of that Concave Spherethat terminates the Sight. But if che Obferver,for the advantage of making Obfervations,imagines this Ecliptic divided into twelve equalParts, or Signs, calling them by the name ofany neighbouring Conftellation, or Figurethofe Stars feem to make in this cafe, I fay,fFi^. 2.] the Earth will feem to move fromT to ^, and from thence to 3r, and fo on ,from Weft to Eaft, thro* all the Signs, till itreturn to T again.Secondly, If you imagine the Obferver to beremoved from the Sun to our Earth, if theEarth be at ^, where it is feen from the Sunamong the Fixt Stars, at Y; the Sun will ap-pear, when feen from the Earth, in the oppofiteSign zChy among the Fixt Stars the Earth beingthen the Center of the Sphere of the Fixe StarsFor the place of the Eye is the Centre of a


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Book I. 0/ Astronomy. 5ceiv'd to be plae'd. If the Earth be movedfrom A thro* By to C, in its Orbit^ or if look'dupon from the Sun, from Y thro'^, to 3i,in confequentiay or according to the order of theSigns ; the Sun, to an Obferver on our Earth,that thinks the place he (lands upon immovable,will appear to move among the Fixt Stars, ac-cording to the order of the Signs alfo, from:^,thro' m , to / , &c. in the fame Plane, duringthe fame time, and tovi^ards the fame Region ofthe Heavens, as the Earth feen from the Suridoes, but in the oppofite Points of the Ecliptic.SCHOLIUM.The like Phaenomena happen in refped ofthe Sun and any other Planet; nay indeed, thevery fame, excepting that the time of that Pla-nets revolution about the Sun, or the Sun s ap-parent revolution about the Planet, when view'dfrom that Planet, is various, according to thedifferent Period of each, mention'd in the fore-going Propofition * and that the Plane of theOrbit of that Planet produced, will cut otherStars than thofe which the Plane of the Earth'sOrbit does, when produced ; and confequently,that the Path of the Sun among the Fixt Stars,feen from any other Planet, is different from itsPath, when feen from the Earth, that is, fromthe Ecliptic.

Proposition III.TO defcrlhe the Vbanomena of the Vlanets fernfrom the SuHy arifing from their Motion inOrbitSy Tvhofe Vlanes ^re inclind to the Flane^ of theEcliptic,

Since the Orbits of the Earth and Planets arefo fituated, as that their Planes are inclin d toeach other, and interfed each other, (as was


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6 r^^ Elements Bookl.fhown in general in Trop. i.) in right Lines paf-fing thro' the Sun ; the Inclination of the Planeof the Orbit of each, to the Plane of the Eclip-tic, or Earth's Orbit, is to be taken into confi-deration, in explaining the Phaenomena of thePlanets viewed from the Sun. For the Plane ofthe Ecliptic is taken by Aftronomers as theStandard to which the Planes of the other Or-bits are judged to iftcline ; and that with verygood reafon, fince it is that in which the Earth(the habitation of tHe Aftronomer) moves roundthe Sun, or in which the Sun feems to moveround the Earth : And an Obferver placed inany other Planet, would make the Plane of thatPlanet's Orbit, the Standard of all the reft, andconfider them as inclin'd to it.The right Line which paffes thro' the Sun,and is the common Sed:ion of the Plane of theOrbit, with the Plane of the Ecliptic, is call'dthe Line ofthe Nodes of that Planet, and the Pointsfthemfelves, wherein the Orbit of the Planet cutsthe Ecliptic, are call'd theAW^j. Thus, [%.;.] letiOr f be the Plane of the Orbit of the Earthproduced indefinitely, N?n the Orbit of anyPlanet, interfering the Plane of the former Or-bit or Ecliptic in N^and n, which are the Nodesof that Planet^ fo as that one part N?n of thatOrbit be fappos'd above the Plane of this Scheme,and the other, npNy below it, (which makes itlook like an Ellipfe:) The right lAntNn joiningthe Nodes, being the common Sedion of thSPlane of the Orbit of the Planet with the Planef the Ecliptic, is the Line of the Nodes.

'Tis evident then, that if a Planet be feerifrom the Sun, when it is in one of the Nodes, ajST^it will appear to be in the Plane of theEclip-


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Book I. ^/Astronomy. 7feem to deviate from the Ecliptic: and (by Def*y. Elem.%1.) the Inclination of the right Lin,0P, to the Ecliptic, and confequently of thePlanet at P feen from the Sun, caii'd the PlanetsHelio'cetftrie Latitudey is meafur'd by the AngleTOE, whereP is fup'pos'd to be a perpendicularlet fai from P to the Plane of the Ecliptic. ThisHeliocentric Latitude is continually upon theiflcreafe, till the Planet is got as far as L, itsLimit ; where it is equal to the Inclination ofthe Plane of the Planet's Orbit, to the Plane ofthe Ecliptic. But during the Planet's paffagefrom thence to the other Node w, it is decre^a-fmgy till at laft it vaniflies at theNode. Havingpa&d the Node , it begins again, chang-ing its name, becaufe 'tis towards the con*trary parts of the Ecliptic, and grows biggerand bigger, till the Planet has arrived at theother Limit /, from whence again it grows leftand lefs, till it vaniflies at the other Node N".The Orbs of the Planets (that is, the Planes oftheir Orbits) are inclind to the Ecliptic in thefollowing manner : The Orb of Saturn makesan Angle of 2 ^ Degrees ; Jupiter, i f Degree ;Mars, a little lefs than 2 Degrees; Venus, fome-ihing above 5 ~ Deg. Mercury almoft 7 Degrees,An account of the Pofition of the Line of theNodes of each Planet fhall be given in amore proper place. We have, in thefe two laftProportions, confider'd the Planets as they ap-pear to one placed in the Sun: becaufe itwas neceffary to the underftanding theirMotiosiwhen view'd from the Earth.


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8 .The Elements Book I.Proposition IV.TO defcrihe the Vhanomenii arifing from the Mo'^tion of the Earthy and inferior Vianets^ Venusand Mercury^ when ^iew'd from the Earth.

Since Venus and Mercury revolve about theSun, in lefler Orbits than the Earth, as you feein the Scheme, \_Fig, 4.] where Treprefents tfieEarth, carried in its Orbit T $ . from Weft tpEall I zn&ACEG, the Orbit defcrib'd by Venu?in a lefs fpace of Time the fame way ; 'tis evi-dent, that when Venus is in D E F, that part ofits Orbit that is fartheft off the Earth, it willappear to us on the Earth to move in confequtntia,or according to the order of the Signs, and is't][XQn faid to be DlreB, When it is in G, mo-ving from thence to H, it will appear to moveas fwift as the Sun, becaufe then its Motiontends dired:ly towards the Earth, and it doesnot feem to move at all, but as its Orbit is car-ried along by the Sun, whofe Motion is toward?.the Eaft : Venus moves now therefore flower..than before, but is ftill DirecfV. When it is got.beyond Hin its Motion, thro' ^^ to B, it pafles; between the Earth and Sun, begaufe it is-nearer to us than the Sun, and moves fvyifter/rthan the^Earth, (the caufe of which we fhallhereafter affign,'*) confequently will feem to usto change its Place am.ong the Fixt Stars, andmove in antecedentia^ or contrary to the orderof the Signs, and then it is faid to ht Retro-gfade^ tho' neally Dired ftill, if view'd from

: fhe Sun; Between Direifi: and Retrograde, forinftance, about H, ie will app;af Stationarythe right Lines that join the Earth and Venus,continuing for fome fenfible' time parallel.Thus likewife, after its Retrogradation, before\x becomes Dirsffi.again^ it will appear Stati-


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I^LlU 2, .Booh J. ,


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a.ikits %^.appealor accothQn faid toving from thas fwift as u.tends dired:!'not feem toried along ^;the Eaftttiar be^'he !^


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Book I. ^/ASTRONO MY. ^onary afecond timCj apd is Stationary in orderto its being Dired, as it was before in prderto its being Retrograde at H. In all this ^ftair,regard is to be had to the Motion of the Earth,for Venus is Dired, Stationary or Retrogr^dje,Recording as it is pofited in fuch parts of itsiOrbit as have the fame relation to the Earth ijiits Motion as the Points aforementioned have.From what has been faid^ 'xk evident that Venus,when Retrograde, as at^^, is nearer to the Earthyand confequently appears bigger than at othertimes ; or the contrary when Dired, as at E, itis farther off, and confequently appears left.And becaufe Venus moves round the Sun at alefs diftance than the Earth j 'tis evident thatit will feem always to attend the Sun, fome-times to go to the Weft of it, fomerimes to theEaft ; all the Heavenly Bodies feeming to be atan equal diftance from the Eye of the Spedator.This digreffion to the Eaftward or Weftward ofthe Sun is call'd the Elongation, and is meafuredby theAngJe contain'd under a right Line drawnfrom the Eye to the Sun and Venus, which isnever greater than the Angle TC or TG,if the Lines TC pr TG, when drawn, are Tan-gents to the Orbit A D F. Confequently theElongation of Venus will never be above halfa Quadrant from the Sun, as is evident fromthe Semidiameters laid down in Frop, i. Andwhen it has arrived to its fartheft Elongationit will return to the Sun, and pafs as far be-yond on the other fide as if its Motion werePfcillatory.

Mercury has all the fame Phsenomena^ but itsDiredions, Stations, and Retrogradations hap-pen oftner, becaufe it finifhes its Courfe in a]iorter time^ and


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lo r/f^ Elements Book I,Earth oftner than Venus. And fince Mercury'$Orbit is lefs than that of Venus, its greateftElongations muft be alfo lefs, and it muftbe a nearer and more conflant attendant of theSun, being, by Frop, i. never a whole Sign di-ftant from it, and confequently feldom to befeen by us.

Proposition V.TO defcrihe the Vhanomena of the Latitude ofthe inferior Tlanets feen from the Earth.Let Ti t be the Earth's Orbit, whofe Plane is

the fame with that of the Ecliptic ; [Fig. f.]and letN 9 be the Orbit of an inferior Planet,for inftance Venus, whofe Plane is inclined tothat of the Ecliptic, and therefore will looklike an Ellipfe, whofe greater Axe is the inter-feiftion of the Planes, or Line of the Nodes Nn.And while Venus is in $ , let the Earth be in T\m which fuppofition Venus will be neareft theEarth, and Retrograde, by Trop. iv.

*Tis evident from Def. f. Elem. xi. that th^Inclination of the right Line ? 7^ to the Planeof the Ecliptic, or the Latitude of Venus in 9feen from the Earth (which is hereupon calledthe Geocentric Latitude) is meafured by the AngleS T'E, the right Line S , being made perpen-dicular to the Plane of the Elliptic. If Venus hefuppos'd to continue in 9 , and the Earth to beat f, in which fuppofition Venus is Direct, andfurtheft off the Earth ^ the Geocentric Latitudeof Venus will be the Angle ? t E, lefs than 9 TE,almoft in the ratio of 3" 9 to / 9 , the' the Helio-centric Latitude of Venus be in both cafes thefame.And all that has been now faid is true of


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BookL. (?/ Astronomy. iipthcr eircumftances being aUkc, the Latitudeq{ the inferior Planets is greater, when they areRetrograde and neareft the Earth ; lefs, whenDirect and fartheft off.Moreover if any inferior Pianette nriofl Retro-

grade and neareft the Earth, and at the fame timein or near a Node, it will be found diredly be-tween the Obferver and the Sun : If it be at a


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12 r^^ Elements BookLbeing fituated dire


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Book I. (/Astronomy. 13becaufe it appears fo, when it is at B, in theformer Scheme, and look'd upon from the Earthat Ty and fo on in the other cafes of C, D, &c.The HkePhaenomena will appear in Mercury,regard being had to its Orbit and Revolution.SCHOLIUM.

As the Phasnomena defcribed in the three laftPropofitions^manifeftly follow upon theSituationand Motion of the Earth, Venus and Mercury,laid down in Vrop. i. So it follows converfely,that the Obfervation of thefe Phsenomena inthem^ eftablifiies and confirms that Order andSituation, namely, that Venus and Mercury re-volve about the Sun, in Orbits, that are includedwithin the Earth's Orbit.

Proposition VII.TO dtfcrihe the Thanomena arijing from theMotion of the Earth, and of the fuperior Fla-nets. Mars, Jupiter, and Saturn. [^Fig. 8.]

LetM S he the Orbit of any one of the fupe-riorPlanets; for inftance Mars, ^CJGtheOr-bit of the Earth, nearer to the Sun, 'Tis evident,firft, that this Planet will not always attend theSun, but fometimes be diametrically oppofiteto it : For the Earth finifhing its Revolutionfooner than any of the fuperior Planets, willfometimes be exadly between the Sun andthat Planet- thus, when Mars is in M, theEarth may be at //; and, to fpeak univerfally,the Angle at the Earth, made by lines drawsthence to the Sun and that Planet, may be equalto any given one.

Let us fuppofeMars to be in M, and the Earthat the fame time in^- Mars in this cafe willappear Stationary, in order to its being Dired,


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1 4 Th^ E L EM E yi T s Book tPlanet at that moment^ will continue parallelfor fome fenfible time during the whole ofwhich, notwithftapdipg, M[^rs will feem to goforwards, as ufual, if view-ed from the Sun.

While the Earth moves along ti^ro' B, C, D, JE,Pand G, Mars likewifc will feem to go forwardsamong the Fixed Stars, upon a double account jfirft, becaufe it really does move about the Sun,in confequentia *j and then again, becaufe theEarth in the oppofite Semicircle is carried thefame way, and about the fame Center : And,confequently. Mars in this cafe being moft re-mote from the Earth, and in conjundion withthe Sun, view'd from the Earth, will feem tomove fafter than ordinary, in confequennay ancjbecome Dired. But when the Earth is arrivedto the Point G, in refped of Mars at M (whichfome time or other will happen, tho* Mars becarried in the mean while about the Sun, name-ly, when the Earth has almoft overtaken Mars)Mars will again become Stationary, in order toits being Retrograde, as it will be foon after.For when the Earth in its Motion from G,thro' H to Ay has pafs'd Mars, and that Planetis feen in oppofition to the Sun, and biggeft,becaufe neareft the Earth, which is lower andfwifter, willmake Mars appear to move in ante-cedentia from S thro' ^ to P y whereas in themean while viev/d from the Sun, it feem'd tomove as always before, in confeque7ttia.The like Phenomena will happen to Jupiterand Saturn, excepting that Saturn s Retrograda-tions are more frequent than Jupiter's, and Ju-piter's than Mars's: becaufe the Earth oftnerovertakes Saturn thanJupiter, andJupiter oftnerthan Mars, and paffes between them and the


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BookL i^/ Astronomy; 15Proposition VIILTO deferibe the Vhanomena ofthe Latitude ofthsSuperior Vianets feenfrom the Earth,

Let the Earth's Orbit be r f S ^ \_Fig. 9.] thatof any fuperior Planet as Mars 6 MyVJhok Planeis inclined to that of the Ecliptic, and cuts it inthe Line of the Nodes NQ)ni Let the fituationof Mars and the Earth, to the Sun, be fuch, as thatMars being in S , the Earth may be in T, almoftbetween Mars and the Sun^ in which cafe Marsis both neareft the Earth, and confequently big-geft and moft Retrograde, as was fhown in theforegoing Trofof Its Geocentric Latitude willbe meafured by the Angle S TE^ (S E being fup-pbs'd perpendicular to the Plana^ofthe Ecliptic.But if Mars continuing in the fame fituation inits own Orbit, and confequently having the fameHeliocentric Latitude, the Earth being fuppofedbe in ?, the Sun being between it and Mars in which Cafe, by the foregoing. Mars will befarcheft off, and confequently leaft, and moft

Dired in its Motion; Its Geocentric Latitudemeafured by the Angle StEy is always lefs thanthe Angle c? TE, in the ratio of the Diftances ofhe Earth from Mars, that is, of the right LinesTS ^tS , Thus inwhatever fituation Mars andthe Earth be placed, in refped of the Sun, itsGeocentric Latitude will be changed, fo as, ca-terls farihus, it will be lels as Mars is nearer toa Conjundion with the Sun, and its fwifteft di-red Motion ; and greater, as it is nearer its Re-trogradation and Oppofition to the Sun.

'Tis evident from what has been faid, thatnone of the Superior Planets can ever be feenfrom the Earth to cover the Sun; tho' any ofthem may be covered by the Sun, when it is


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i6 ' T'fo Elements BookL'Proposition IX.

TO defcrlhe the Vhdnomena of the Superior Tla-ncts^ arifing from their being Opake Bodies^and enlightned by the Sun,Saturn and Jupiter being Opkke Bodies, abd

illuminated by the Sun^ that h^lf of each Pla-net which is turn'd towards the' Sun, (that is,the illuminated half) is likewife turn d to-wards the Earth, which is never far off from theSun, or Center of Saturn and Jupiter's OrbitFor, by Prop. i. Jupiter's diftance front fheSun is above five times, and Saturn's almof?ten times greater than the Earth's diftance froiji'it.

In Mars indeed it is fomething dijBFerent : Forthe diftance of Mars from the Sun being buthalfas much more asthe'Earth'sDiftancefrom it^Its inlighten d Hemifphere, towards the Sun,is not always, as to Senfe, turn d towards theEarth. iFig. lo.] Let T be the Earth's place inits Orbit T^ , 'tis evident that Mars being at A *or B, in Conjunilion or Oppofition to the Sun,has the fame Face towards the Earth, as ithas towards the Sun, that is, its enlightenedone, and confequently appears Full j but in theiituation of the Points jD or C (when the AngleOct; or GDT" is greateft, or when rc, orTD is almoft a right Angle) neither is thewhole enlighten'd Face feen, nor is thatFace thatis feen entirely illuminated, but it appears Gib-bous, the light being deficient a little towardsthofe parts that are turn'd from the Sun.

SCHOLIUM.As the Pha^nomena defcribed in thefe threelaft Propofitions follow from the Order and Mo-


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Bookl. t?/ Astronomy. 17laid down in Trop. r. So by the converfe ofthem, the Obfervation of thefe Phenomenafettles and eftabliflies that Order.

Proposition X.TO defcrihe the Vhanomena of the Motion ofCo-*metSy feenfrom the Earth,Becaufe different Comets have different Or-

bits, which can't be determin'd tillfomc Obfer-vations have been made about them : 'Tis evi-dent that during their defcent, near their Peri-helium, or while they are in that part of theirOrbit, which is within the Region of the Pla-nets, they have the fame Ph^nom^na as thePlanet next them has, regard being had to theVelocity and Inclination of the Orbit of theComet.


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i8 ri^^ Elements Book I.Earth and the Comet, the Comet's Motion willappear fwifrer than really it is. On the con-trary, the Comets that move in antecedentia arefwifrer than they fhould be, when the Earth isbetween them and the Sun ; and flower, or per-haps moving according to the natural Order ofthe Signs in appearance, when the Earth is fitu-ated on the contrary fide. All this happensfrom the Motion of the Earth, and its variousPofjtion, as it docs in the Planets ; that are, ac-cording as the Motion of theEarth falls in with,or is contrary co their Motion, fomctimes Re-trograde, foraetimes Slower, fometimes fwifrerthan they fhould be, as has been ftiewn in Vrop,iv and vii. Thofe Phaenomena will be moft fen-fible, and eafily taken notice of, a little beforethe difappearance of the Comet its apparentMotion being at that time flower, and theEarth's Motion proportionably greater, withwhich the Spedator is carried.The Latitude of the Comet likewife, ceterisparibus^ is varied by the diiFerent Situation of theEarth, it being greater in Oppofition ; but inConjundlion greater or kfs, according as theComet is between the Earth and Sun, or theSun between the Comet and the Earth, as was4smonftrated of the Planets in Trop, v and viii.

Seo


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TUt^ ?, Maok


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Bookl. o/AstronoMy, 19Section II.

Of the Direftion of the Forces, which re-tain the Primary Planets in their Orbits.

Proposition XLIF a Body he moved according to the DireElion ofany given right line as A Zj [Fig. il.] and atthe fame time be urged by a Centripetal Force tend"ing towards a certain given immoveable Toint as S,fituated 'without the aforefaid right line ; the Linedefcrih*d by the Body "will be a Curve^ and Concave to^wards S, lying alt of it in the fame immoveableTlane faffing thro' the right line A Z and the Foint S:jind the Areas contain d under any portions of theCurve and Right lines draivn to the Center S, are toone anothery as the Times wherein thofe fortio?is of theCurve were dejcrlbed.

Let the Time be imagined to be divided intoequal parts ; in the firfl; of which let the Bodydefcribe the line AB, 9. part of A Z, by its Fisinjita alone^ by which it tends to move accord-ing to the Dire(5lion of the Line A Z ; in thefecond part of Time it will defcribe the LineB c, equal to the formerJ 5, if nothing hinderit: For by the firfl: and chief Law of Motion, allBodies once put into Motion, and not meetingwith any Impediment, will continue to moveuniformly on, in the fame right line they werefirfl: moved in. But when the Body is arrived atthe Point B^ let us fuppofe a Centripetal Forcetending towards the Point 5, to ad upon it bya fingle impulfe, fo as that, had it been impell'donly by that Impulfe, it would in the fecondpart of Time defcribe the line B G. 'Tis evident


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20 The Elements Book I,that drawing the Line c C parallel to B G, thro'the Point Cy and GC to Bc^ thro' G, the Bodyac9:ed upon by both Impulfes at once^ will ar-rive, in the fecond part of time, at the Point C,defcribing the right line B C. For as it is wellknown in Mechanics, a Body defcribes the Di-agonal of a Parallelogram, both the Forces corn-bin d, in the fame time as it would do the Sides,with the Forces feparated, 'Tis certain likewifethat the right line B C is in the Plane of the Pa-rallelogram BGCc, both of whofe Sides BGand B c is in the Plane of the Triangle ASB^that paffes thro' the Center 5, of the Centripe-tal Forces, and the immoveable right line A Z,Befides, the Triangles SC 5, ScB are equal, be-cause they are upon the fame Bafe BSy and be-tween the Parallels S B^ C c -^ hvit ScBy S B Aare equal, becaufe their Bafes are equal, andheight the fame: Confequently 5B^, SC B alfoare equal. By the fame method of reafoning^if in the third particle of Time, a Body defcribesany other right line as C D ^ it may be provedthat the Triangle SCD Is equal to the TriangleSBC^ and that the right line CD is in the fameplane with the right lines SB, BC ; that is, inthe fame with that which is drawn through theright line A 5, and the Point S. And fo weipay go on, as long as the Motion is continued,and in equal times the Area defcribed by Radiidrawn to the immoveable Center of the Forces,will be equally increafed ; and by compound-ing Ratios, any fums of Areas, are to one ano-ther, as the Times wherein they are defcrib'd.The Line defcrib'd by the Body will be in theimmoveable Plane, and it will be congave to-v/ards 5", becaufe every part of it tbat is aright line^ as^C^ declines from the foregoing


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BookL 0/ Astronomy. ziAB, towards the Center. If you fuppofe thenumber of the Triangles, S A B^ S BCy SCDto be augmented, and their breadth to be di-minifhed in infinitum i their Bafes, AB^ BCyCD,will form a Curve Line, concave towards thefame pares, and lying in the fame Plane ; andthe Centripetal Force, which before aded, asit were by ftarts, and 'at equal intervals of Time,whereby the Body is continually drawn off fromthe Tangent of that Curve ( which the bafes ofthe l^iangles meafure) now ad:s conftantly :And any Areas thus defcrib'd , S A B C S yS ABC DE Sy will be, as before, proporti-onal to the Times of defcription. ^ . D.

Propositio]5t XII.A Body mo'ved in a Curve Line ABC D,[Fig; 12.] defcriyd upon aVlane^ and con-cave to7i>ards the jnmefarts^ and by a Radius draivnto S, an immoveable Vant fituated in the fame Blanttowards the Concavity of the Curve, defcribing Areasfrofortional to the Timeiy is urged by a CentripetalForce tending to t^e Foint S.

Let fiich a Curve be imagined to be dividedinto the parts yi/ 5, BC^CD^ &c, differing as littleas can be from right lines, fo as to be defcribedih equa,l Particles of Time : Let the Centripe-tal Force likewife be conceiv'd to ad: only inthe points Bfi^D^&c. by ftarts, as in the forego-ing Vrof,. Let A B be produced to c, fo as thatBe ht equal to AB i in like manner BC to d^till C ^ be equal to B C , and fo on. The Tri-angle SAB, will be equal to S B C, becaufe, byHypothefis, the Areas defcribed are proportionnal to the Times ; and S AB is equal to S Bc^becaufe A B h ^qual to B c : Wherefore SBCh equal to SBc, and confequently (by


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21 The Elements Book I.Ekm. I.) Cc is parallel to S B, Moreover, theEody moved along A By in the firfl: particle ofTime, by its Vis infita alone would defcribe B r,in the fecond , but it really does defcribe B C,in that fecond Particle of Time : Whereforethe Force atîng in the Point By in conjundioi>with the Vis infita to carry the Body along BCyis directed according to a Right line parallel toCc'y that is, according to the Right line B S,After the fame manner the Force ading inthe point C, and in conjundHon with the Viînfitay ( by which alone the Body would defcribqthe Line Cd in the third Particle of Time) car-rying it in the fame time along C D, is di-reded according to a Right line parallel to dD^that is, according to CS : But the Right linesBS, CSy &c. tend towards the Point S. There-fore the Centripetal Force, that draws the Bo-dy oiF from the Tangent of the Curve, adsalong right lines tending to the immoveablePoint 5. ^ . P.

Proposition XIII.TH E Forces whereby the Vrimary TlanetSy Mer^curyy Venusy Marsy JufiteYy and Saturriy aredrawn off from their Reîlineal Motionsy and keptin their Orbitsy do not tend towards the Earthy bnttowards the Sun.

For every Body that is moved in a CurveLine, is turn'd off from its Rectilineal courfe,thsLt it naturally affeds, by fome Force or o-ther. And it is evident that the Planets movein Curve Lines, becaufe their Orbits return in-to themfelves. Now this Force exercifed uponthem, does not tend towards the Earth, becaufethe Orbits of two of them, namely. Mercuryand evi-


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Book L


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24 The Elements Book Llineal Motion, and retained in their Orbits,tend towards the Sun.

Proposition XIV.THE Forces, hy which the Comets are kept in theirTraje^ories (if they he Curvilineal) do not tendtowards the Earth, hut towards the Sun.

If the Trajedories of the Comets were Rightlines, they would not be urged by any Forces atall tending to a point fituated without thofeRight lines ^ for if they were, they would bedrawn off from a. Rediiineal Motion, and madeto defcribe Curvilineal Orbits, by Vrof. xi. Andif the Trajedories of Comets were Curve lines,yet the Force by which any fuch Comet is re-tained in that Curve, is not direded towards theEarth, becaufe the Earth is generally found tobe without the Plane of that Trajedory ^ be-Udes, in refped of the Earth, the Comet isfometinfes Dired and fometimes Retrograde,and confequently does not defcribe Areas pro-portional to the Times : Yet in refped to theSun, which is placed in the Planes of all theirTrajedories, a Comet always moves the fameway, and the ^nearer the Sun, the fwifter ^ ioas to encreafe its Area, which is equally defcri-bed by a Radius drawn from the Sun : Where-fore what was aflerted, is evident from Vro^, xii.

Section


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BookL (>/ Astronomy. 25

Section III.Of the Order, Diftances, and Periods of theSecondary Planets revolving about theirPrimary ones, andtheir Phaenomena^ to-gether with the Direftion of the Forces,whereby they are kept in their Orbits.

Proposition XV.TO defcrih'e the Order^ and Feriods of the Secon-dary VlanetSy or Satellites^ about their TrimaryOnesy and Difiances from them*Of the Six Primary Planets^ that revolve a-bout the Sun, there are but three, as wc are fureof by Obfervation, that have Satellites, or o-thers revolving about them, hence called. Se-condary Planets. The Earth has one, 'viz., theMoon, compleating its Revolution in ayfDays, and diftant about 60 Semidiameters ofthe Earth from it.

Jupiter has four; the inmoft of which revolvesi^ of a Day, at the diftance of jf Semidiame-ters of Jupiter from his Center; the fecond re-volves in 31 Days, at the diftance of 9 Semidi-ameters ; the third in 7-J- Days, at the diftanceof141 Semidiameters; the fourth and outermoftrevolves about Jupiter in the fpace of 16 J Days,being diftant from his Center 2 j f Semidiame-ters of Jupiter.

Saturn has five; of which the innermoft re-volves in i| of a Day, at the diftance of 4I. Se-midiameters of Saturn, from the Center of Sa-


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26 The Elements BookLthe diftance of yf Semidiameters of Saturn ;the third, in 4-f Days almoft, at the diftance of8 Semidiameters; the fourth, in almoft 16 Days,at the diftance of 18 Semidiameters; the fifthand outermoft of all, that have been difcovered,revolves about Saturn in 794- Days, at the di-ftance of ^"4 Semidiameters from the Center ofSaturn. And the Solar Syftem, as far as it hasbeen hitherto difcover^d is of a Figure muchlike that in the Scheme [^Fig 15,] But betweenthe two laft Satellites of Saturn, Hugenius fufpedsthere revolves a Sixth, the fpace being largerthanitfhould be, in proportion to the diftancesof the others, or that there are others beyondthe fifth revolving about Saturn, but not feenas yet, by reafon of their Obfcurity. ThePlanes of the Orbits of the Satellites of thefame primary Planet, do not coincide, but arevarioufly inclin d to one another, and to thePlane of the Orbit of the Primary one.

Saturn likewife is encompafs'd with a thinplane Ring^ that does not touch the Body of Sa-turn at all, but is like an Orbicular Arch, builtround about it. The Plane of this Ring is atthis time nearly parallel to the Plane of Earth'sEquator. And from the various Pofition of it^in refped: of the Sun illuminating it, it beingOpake like a Planet, and in refped of the Eyeof the Spedator^ the various Phafes of the Anfaof Saturn arife, which have fo long baffled theattempts of Aftronomers, with their variousShapes. This was firft difcover'd by Hngms^ inhis Syfiema Saturnm?n, Printed i6^


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Bookt


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28 The Elements BookLNode a or "O I all the reft of the time it iswide of it j and the Latitude of the Moon's de-viation is meafur'd by the Angle in which theRight line conneding the center of the Moonand Eye is inclin'd to the Plane of the Ecliptic,{by Dtf. ^. Elem.xi.) agreeable to what wasfaid above concerning the primary Planets,Trop, V. and viii.

Eefides^ theMoon^ like the other primary Pla-nets^ being an Opake^ Rough, Spherical Body,reflecting the Sun's Rays falling upon it- 'tisevident^ that half of it being turnd towards theSun is illuminated and bright^ while the otherhalf, that is turned from the Sun, continues ob-fcure and dark. Now only that Hemifphere ofthe Moon which is towards the Earth, can befeen by an Inhabitant of the Earth viewing it:Confequently the Phafes of the Moon will bevarious, according to the various habitude of theinlighten d Hemifphere to that which is turn'dtowards the Earth ^ as was ftiewn in the likecafe concerning the inferior Planets, in Frop.vuIf the Moon be at A in the Scheme [_Fig, 14.]the point juft oppofite to the Sun, which we willfuppofe to be at 0, and the Earth in T; 'tisevident that the whole enlighten'd face of theMoon is turn d towards the Earth ; this Phafisor appearance of the Moon is call'd the FullMoon. If the Moon be remov'd to B, the Sunand Moon continuing as they were, part ofthe bright or illuminated half will be turn'dfrom the Earth, and part of the obfcure ordarken'd half turn'd towards it hereupon theMoon will not appear full, but Gibbous, or alittle deficient or obfcure in oppofite to the Sun.The Moon arriving to the fituation C, where


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Book J. of Asr r o n o m y. 29of its ^adratures, half the Hemifphere whichis turn'd to the Earth is illuminated, and halfdark i on which account it will appear a halfMoon, and in the dqcreafe. The Moon havinggot farther, for inftance, to D, but a little of itsFace that is turn'd to the Earth is inHghten'd,much the greater part remaining dark: andConfequently the inlighten'd part will appearHorned, by reafon of the K^oon's really fpheri-cal, but apparently plane Figure; and the Hornswill appear turned away from the Sun, as fliallte demonftrated in the next Propofition, or inthe prefent cafe, turned towards the Weft. Atlength when the Moon is come to the point /


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JO r/?^ Elements Book I.Moon departing from the point K, where theNew Moon now happens, returns to it againthe Earth itfelf, together with the Moon its At-tendant^ is carried an intire Sign in confequentiafo that the faid point of the Orbit of the MoonKy is more towards the weft of the Sun; andconfequently the Moon is not yet arrived to aconjundion with the Sun ; but ftill lacks 2 daysand f hours to it, or to be become New ; thatis, to finifh an intire Lunation, and all the fe-veral Changes of its Appearances; which fpaceof time is call'd a Synodic Month, and confifts of29 days, 1 2 1: hours.A little before and after a New Moon, viz.wfien the Moon is in D or F, the Rays of theSun refledcd from the Earth, meeting with theMoon, are the caufe of that faint Light bywhich the reft of Moons Difc befides its Horns,is then render'd vifible. But when the Mooji isgot out of the way of the reflexion of theEarth's light, that faint light vanifiies, beingfalfly fuppos'd native.

Proposition XVILTO dra^i^ the Thajis or Appearance of the Moon atany given Time*Let a Plane, the fame with that of the Scheme,

for inftance, pafEng thro' the Centers of theSun, Earth and Moon, cut the Globe of theMoon, and let its Section be the Circle ADBC,iFig, 15.] Let CD be a Diameter of the Circle,and S L, a right line connecting the Centers ofthe Sun and Moon, perpendicular to it ; andA B another Diameter to which the Right lineTLy connecting the Centers of the Earth andMoon, is perpendicular. From the Point D, letP fall a perpendicular to AB^ meeting it in E,


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Book I. 0/ A S T R O N O M Y. 3 ISince a Plane perpendicular to that of thisScheme erected upon the Diameter CD, fepa-rates the enlighten'd Hemifphere from the ob-fcurc; and a Plane ereded after the fame man-ner upon A By feparates the vifible from the in-vifible Hemifphere : That part of the Globe ofthe Moon which is common to both Hemifpheres,^iz,, that which lies between the Planes erededupon ^L, D Ly will be the enlightened part ofthe vifible Hemifphere ; and that which lies be-tween the Planes erected upon L, DL^ willbe its obfcure or dark Portion. Both thefe Por-tions of the vifible Hemifphere, as well the in*lightened as the obfcure, are terminated in theoppofite Points of the Globe of the Moon;namely, thofe in which a Right line raifed from, perpendicular to the Plane of the CircleACBD3 and produced both ways, meets theMoon's Surface ; that line being the commonSedion of the two Planes ereded upon AB^ andCD: The greateft breadth of thefe parts is inthe Circumference ADB; the breadth of ob-fcure being B D, of the enlightened AD, TheArcBD is the meafure of the Angle J5 L I>, whichis equal to the Angle SLT, contained under theRight lines drawn from the Center of the Moonto the Centers of the Sun and Earth. For if tothe equal Angles 5 L D, TL 5, (becaufe rightones) you add the common Angle D LT, theAngles mention d will be formed, which areconfequently equal. But at any given time, theAngle of the Moon's Diftance from the Sun^'vlz* LTS is known; confequently S LT, itsComplement to two right : For the Diitance ofthe Moon from the Earth is fo fmall, in com-parifon with the diftance of the Earth from the|un^ that the Right lines T.^^ LS^ drawn from


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3 2 The Elements Book I.the Earth and Sun^ may be taken for parallels.Make therefore, that as twice the Radius is to theverfed line of the Angle TL S^ fo is B A toB E, And fince the Hemifphere of the Moonfeen from the Earth, appears like a Circle, as ithappens to any Globe feen at a great Diftance,according to the Principles of Optic : Let^M5iV[F/g^.i6.]be a Circle reprefentingtheDifcof the Moon, the fame with the former, made byth'* Sedion of the Moon with a Plane erededupon A By and confequently defcribed upon thefame Center L, and with the fame Diameter.Becaufe the Moon is at a confiderable diftance,each point of its Difc will be feen by Rightlines parallel to the Right line LT. And there-fore BEy and AEy will be the greateft Latitudesof the inlightened and obfcure parts in ths Difcof the Moon. Draw the Diameter MN per-peadicular to AB; and defcribe the Semi-EllipfeMENy whofe greater Axe is MNy and leffer'equal to twice LE, MA NEM will be the in-lightend Part, and MBNEM the obfcurePart of the Moon s Difc AMBN,For the boundary of illumination on the Sphe-rical Surface of the Moon, by the Sun, whichis likewife Spherical, is a Circle; and this Cir-cle ktn at a diftance obliquely, appears like anEllipfe, whofe Semiaxes are LMy LEy as is e-vident from Optics : 'Tis feen obliquely by TLybecaufe the Right line S L perpendicular to itsPlanes, is inclined to LT; excepting when theMoon is Full, where they coincide.

In the Conftrudion of this Problem, we fup-pos'd half the Globe of the Moon to be inligh-tenM, and half likewife to be feen by a Specta-tor, tho' neither of the Suppofitions are rigo-iouily true : For the Sun being bigger than the


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Book I. (>/ Astronomy. 33Moon, more than half the Moon is inlighten'd and the Spedator not being at an infinite Di-fiance from theMoon^, its intire Hemifpherewill not be vifible. But the difference in boththefe cafes is fo fmall, that it may very well benegleded : And the Phafis or Appearance ofthe Moon delineated as above, fufiiciently a-*grees with Obfervations. That the Scheme may-agree with the Heavens, the Line B Ay to whichMN connecting the Horns is perpendicular^muft be plac'd in fueh a Situation, as that it may-tend direcStly to the Sun : For the Right line L A^feen from T, coincides with the line L S tend-ing to the Sun.The Phafes of the inferior Planets and Mars^defcribed in Trop.YU and ix. are to be delineatedafter the fame manner j But then if L reprefenca Primary Planet, the Right lines TS, LS arefenfibly inclin'd to one another, and form aTriangle.

I^ROPOSItlON XVIIL' i 'O explain the Vhanomena of the Moo7t arlfingfrom^ the Opacity of the Earth cafii?^g a Shadowy thatisy an Eelipfe of the Moon.The Earth is an Opake Body, confequentlywhen it is inlighten'd by the Sun, it cafts a Sha-dow towards thofe part's which are turn'd fromthe Sun : and this Shadow muft always be inthe Plane of the Ecliptic ^ fince both Sun andEarth are always there. [F/g-. 17.] If at anytime, the Moon when Full, fiiould happen tobe in the Plane of the Ecliptic, or pretty nearit, (which it will be, when the Full Moon hap-pens in or near a Node,J 'tis evident the Moonwill be immers'd in the Shadow; and confe-quently deprived of the Sun's Light, by which


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34 T/;^ Elements BookLk fhines ; that is, it will fuffer an EcUpfe whichwill be Total or Partial^ according as the wholeor only a part of the Moon's Body enters theShadow.The Earth's Shadow terminates at lafl- in aPoint, and does not reach fo far as Mars: ForMars, tho' in the Plane of the Ecliptic, and op-polite to the Sun, is not Eclipfed; which it ne-ceffarily would be, if it were immers'd in theEarth's Shadow ; For Mars is an opake Body,as is fhewn in Vrop/ix. 'Tis evident at firfl: light,from this Figure of the Earth, that the Sun isbigger than the Earth: For if the lucid Bodybei^igger than the opake one, the Shadow willbe equally thick and cylindrical ; But if thelucid Body be lefs than the opake, the Shadowindeed will be conical, but growing bigger andbigger, and the farther off it is from the opakeBody, the thicker it is and in both cafes willbe extended infinitely. As the Sun is biggerthan the Earth, fo the Earth is bigger than theMoon ; becaufe the Moon fometimes is totallyEclipfed by entering into the Earth's Shadow but the Earth's Shadow is much fmaller wherethe Moon enters it, than nearer to the Earth^>as is evident from what has been already fhewn.

Let the Circle FM [Fig-. i8.] reprefent thetranfverfe Sedion of the Earth's Shadow, wherek croifes the Moon's Orbit, Let L^F be theOrbit of the Moon, C E the Ecliptic. The du-ration of fome Eclipfes is found to be fo long( for inftance, four hours,) as to let the Moongo the length of three of its Diameters in theShadow totally Eclipfed. This happens whenthe Center of the Moon palfes thro' the Centerof the Earth's Shadow or Circle FM: Andfuch an Edipfe is call'd a Tot^l and OpitralEcVipte.


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JBookL of Astro i


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i6 T^^ Elements BookLnumbra is difFufed all about the Shadow. LetGbe the Sun, [Fi^. 21.] E the Earth, (and letRight lines be drawn Juft as in the Schetue: Allabout the Shadow TVMRy where no part of theSun's light can come, a Penumbra VTVMRNis fpread, fome part of the Sun s light nor beenftop'd therein : And this Penumbra is darker to-wards jVy RM which are the extremities ofthe perfeA Shade; becaufe fewer Rays can arrivethither, the portion M the Sun, from whichthey are emitted, being fmaller ; but lefs ob-fcure towards TV, R N^ where more Rays canreach : And beyond which limit all the Rays of^the Sun can reach without any hindrance at all,and inlighten according to the degree of theirvigour

.

The Red colour, that the Moon in the middleof the compleat Shadow is aiFeded withal, andmakes her look like a Brick (for in fome Eclipfesthe Moon intirely difappears,) feems to arifefrom the Sun s Rays, either refraded in theirpaffage thro' the Earth's Atmofphere, or re-fleded by fome particles of Matter flying aboutthe Earth's Shadow, to the Moon ; or from thelight of the Sfars, or all thefe taken together.Some kind of Eclipfe or other of the Moon,happens generally twice a Year at leaf! : Forthere being two Nodes in which the Moon'sOrbit crofles the Ecliptic, and they moving inantccedentla^ (by Trof. xvi.) and the Sun appear-ing to go thro' the Ecliptic in confcquentia^ (byTro^. ii.) the Sun muft meet one or other of thefeNodes twice every Year; and confequently theEarth's Shadow muft perforate, as it were, theMoon's Orb in the other Node. If therefore aFull Moon happens juft at that time, the Moonmult neceffarily be totally and centrally Eclips'd,


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Lxt^ 1^ iBook 1 .


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BoofcL 0/ A STRONG MY. 37as was demonftrated above. And tho' a FullMoon does not happen juft as the Sun and Nodemeet ; yet the inclination of the Orbit of theMoon to the Ecliptic, and the depth of theEarth's Shadow is fo great^that tho' theFullMoonis diftant above ten Days from the aforefaid time,before or after it, and it can be diftant but fifteenDays, yet the Moon will fcarce get clear of theShadow. But if the abovemention d meeting ofthe Sun, and Lunar Node, happen on the veryDay of the New Moon, or a Day or two beforeor after, which can happen but feldom, theMoon will be far enough off from the Earth'sShadow in the next Full Moon, whether pre-ceding or following, and fo will efcape an E-clipfe; confequently there will be noEclipfethat half Year.

Proposition XIX.TO explain the Vhtznomena of the Sun fcen fromthe Earthy and arifing from the Opacity of thaMoon^ or Ecliffes of the Sun accountedfor.As the Moon upon the interpofition of theEarth is deprived of the Suns light, and faid tobe Eclipfed; fo, in like manner, if the Earthfliould be robbed of the Sun s light, by the in-terpofition of the Moon, this Phsenomena oughtto be caird an Eclipfe of the Earth. But theObferver of it being on the Enrth, and allow-ing of no lofs of Light or Eclipfe, nor Motion^,nor any other thing that feems to argHe an im-perfedion in his place of abode, calls this Phe-nomenon an Eclipfe of the Sun ; for the fam^reafon as an Inhabitant of the Moon, would faythe Sun was eclipfed, when the Moon is reallyentering into the Earth's Shadow,


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5 The Elements Book I.'Tis evident an Eclipfe of the Sun will happen

in every fuch New Moon, as has the Moon ator near any of the Nodes. For then the fliadowof the Moon lying diredly betwixt the Sun andEarth, reaches to the Earth, and caufes a totalEclipfe to the Inhabitants of the trad C D[F/^.22.] that areimmers'd in the thickefl: fliade.But the Moon's Shadow not being large enoughto cover the whole Earth, the circular tra6lJBC, ED, which furrounds the former CD, iscover'd with the Penumbra, and its Inhabitantsfee only a partial Eclipfe of the Sun ^ which isgreater towards C and D, becaufe a greater porti-on of the Sun will be cover'd by the Moonbut lefs towards B and E, the extremities of thePenumbra, and the defed of light is fcarce fen^fible. At the fame time, in other places, asE F, the bignefs of the Earth is the reafon whythere is no Eclipfe at all ; the Sun inlightningic without any hindrance or impediment.

Ail the preceding account happens in Nature,juft as it has been related. But if we look up-


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Book I. C?/ A S T R O N O M Y. 39Moon are more than ordinarily infleded, andfo fliorten the Moon's fhadow. The bignefs ofa Solar Eclipfe is to be eftimated juft as in theMoon^ by the digits of the Sun's diameter^ thatare darkened by the Moon at the time given.

'Tis evident likewife that the Moon movingtowards the Eaft^ or from iCt, thro' m to 7[F/g-.22.] the weftern part A of the Earth will bein the fhadow firft, which will pals along theEarth's difc, like a Spot, thro' B,'C, D, E, to F,where it leaves the Earth. But if the Moon b^look'd upon from the Earth, the eailern limb ofthe Moon will firft cover the weftern of the Sun,and the weftern of the Moon will laft uncoverthe eaftern limb of the Sun : And the greateftdarknefs that happens in a total Eclipfe is foonat an end ^ fome part of the Sun's disk beingprefently uncover'd, almoft fo foon as the wholewas cover'd : and that part, tho' never fo little,will mightily inlighten the Air.Tho' the Moon muft be in a Node the very

moment of the New Moon, to caufe the biggeflEclipfe of the Sun that is poffible, and that thefhadow of the Moon may go along the middleof the Earth : Yet if fiie be not far off fromthence, the fiiadow of the Moon, or at ieaftpart of the Penumbra, will fall upon fome tradof the Earth, being fo big, and there caufe atotal, or at Ieaft a partial Eclipfe : And in thisfenfe there are more Eclipfes of the Moon thanof the Sun. But Eclipfes of the Sun, in anygiven place, are much fewer than the Eclipfes ofthe Moon ; becaufe the Moon s fiiadow is lefsthan the Earth's, and confequently it does norinvolve any given place upon the Earth, fooften as the Earth's fliadow does fome part of


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40 The Elements Bookl.SC HOLIUM.The fame Pha^nomena will appear in Jupiter

and Saturn : For their Satellites or Moons, willbe eclipfed, by being immers'd in the fhadowof their primary Planet ; and thofe Eclipfes ofthem are obferved by us^ juft as the Eclipfes ofour Moon may be obferved from them. In likemanner^ every Satelles coming between the Sunand its Primary, qafts a fliadow upon the Pri-mary, which feems to move along the Disk ofthe Primary from Eaft to Weft like a Spot.But the Duration, Phafes, Periods, &c. of thefePhaenomena are various, and differing from thelike feen by us, and arifing from our Moon,according to the diverfity of the Shadows, Mo-tions and Magnitudes, both of the Primary andSecondary Planet.

Proposition XX.EJch of the Secondary Tlancts mention d inTrop. XV, is urged by a Force comfounded of a

Centripetal Force^ tending to the center of the Vrimaryâbout which it re'uol'vesy a7id of all the AcceleratingForce ivith njhich the Trimary is urged. And there-rfore the Forces whereby the Satellites are retain d intheir Orbits about the Trimary oneSy tend tewards thecenters of their Trimary ones rejjêîively,

Beoaufe the Satellites of Jupiter and Saturnrevolve equably in circular Orbits, concentricwith Jupiter and Saturn, they defcribe Areasabout thefe Centers refpecftively, proportionalto the Times. In like manner, if the Moon'sOrbit differ from a Circle concentric with theEarth, the lefs the Moon appears, (that is, thefarther it is from the Earth) the flower it movesround the Earth, fo that ftill the Area, that the


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Book I. t?/ A S T R O N O M Y, 41fcribes, is equably augmented. So that unirer-fally, each of the abovemention'd Satellites de-scribes Areas^ by a Radius drawn to the centerof its Primary Planet, about that Center, pro-portional to the Times. If therefore the Syftetnof any Primary Planet, and its revolving Sa-tellites, be fuppos'd to be urged along parallelLines, by a Force equal and contrary to that,whereby the Primary Planet in that Point ofits Orbit, where it then is, is urged towards theSun, the Primary Planet will no longer defcendtowards the Sun ; and the Secondary will con-tinue to defcribe the fame Areas about the Pri-jnary as before ; that is, proportional to theTimes : (For if the Bodies move any how inrefped of one another, and are urged by equalaccelerating Forces along parallel lines ; or,v/hich is all one, if the Space in which they per-form their Motions be moved uniformly in aright line, they will move all after the famemanner, as they would do if thofe Forces v/ereabfent, or the Space were at reft, in which theyare included). So that each Satelles, only ur-ged by the difference of the Forces, will go onto defcribe, about the center of its Primary,Areas proportional to the Times. Therefore,by Frop. xii. the difference of thofe Forces tendsto the Primary Planet as a Center. But beforethat the whole Syftem was urged along parallellines, by a Force equal arid contrary to that,whereby the Primary Planet is urged towardsthe Sun, that is, in its natural State, each Satellesis urged by a Force compounded of the Centri-petal Force tending to the center of its Primary^and of all the accelerative Force that the Pri-inary is urged by. Confequently the Yorefaid


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42 W


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Book I. (/Astronomy. 45and outmoft, arerefpedively as 1 1, 5 }, 7^^ and16 1 almoft; and their Diftances as yf, 9, 141-and 2^ , the Square of the Periodic Time ofthe inmoft, namely ;, is to i; the Square ofthe Periodic Tiie of the fecond Satelles, as 170,the Cube of the diftance of the inmoft fronithe Center of Jupiter, to 7 5 6, the Cube of thediftance of the fecond from thence. In likemanner ; is to yi, the Square of the PeriodicTime of the third Satelles, as 170 to 2890, theCube of the diftance of the third from the Cen-ter of Jupiter. And 3 is to 280^ the Square ofthe Periodic Time of the outmoft Satelles, as170 to i)'8oo, the Cube of the diftance of theoutmoft from Jupiter's Center. And therefore,ex aquoy the fame ratio holds between any o-ther two of them compared together^ a^ the fe-cond with the third or the laft, or the third withthe laft. This Ratio comes put more exad, ifthe Diftances and Periodic Times are takenmore accurat?ely.The fame will be found to hold in the Satel-

lites pf Saturn, if you take the numbers laiddown in Vro^. XV, that their diftances from Sa-turn, and their Periods about him may be efti-mated. Biit the Moon being a folitary Satel-^les, this Propofition can't be applied to her.

PapppsiTiON XXII.^H^ Motion of the TrimaryTlamts ahout the Sm,is fuch^ as that the Squares of their Teriodic

Times are in the fame ratio as the Cubes of theirJ)ifiances from the Sun,

Thus, for inftance in round numbers, the Pe-riod of Saturn is 30 Years, Jupiter 12; theirSquares are 900 and 144. The diftance of Sa


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44 The Elements Book Ldiftance of Jupiter from the fame, as 9 to 5- al-moft^ their Cubes are 729 and 12 f. But 'tisevident thofe Squares are nearly as thefe Cubes.Alfo the Period of the Earth is a little more thanfour times greater than the Period of Mercuryconfequently their Squares are as 17 to i almoft:Suppofmg the Diftance of the Earth from theSun to be ten parts, the Diftance of Mercuryis, by Obfervation, about 4, or 5, 9 j and theirCubes as 1000 to y9 : But 17 is to i almoft as1000 to ^9 ; And lb on in. the other Planets.The more corred the Diftances and Periods aretaken, the nearer you'll come to this Proportion.What has been now faid, may fuffice at prefent.-The Propoiition being more exadly to be madeout in the third Book.

Proposition XXIILTHE Spaces that a Body deferihes, aBedupon lyany kind of finite Force^ \whether it he deter-minate and unchangeahky or the fame in continualincreafey or continual decreafe) are in the very he"ginning of the Motion^ in the duplicate ratio of theTimes.

Let the Times be exprefs'd by the Right linesAD, AE; [F/^. 24.] and the Velocities acquiredat the end of thofe Times, by the OrdinatesD By EC : And the Spaces defcribed with thefeVelocities will be as the Areas AB Dy ACE,made up, as it were, of thefe Ordinates ; name-ly, the Space defcrib'd in every fmalleft parti*ele of Time, being as the Velocity and thatParticle of Time conjundly. But, in the be-ginning of the Motion, the Ordinates DB^ EC^are very near the Point A ; and in that cafe,theTrilineal Figures ^D 5, ACE, are firhilar


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Book I. (?/ A STRONOMY. 45j^BCy which belongs to the infinitely fmalltrilineal Figures, not extending it felf beyondthe Right line AG, which is a Tangent to theCurve in J. Now thefe fimilar redilineal Tri-angles, are in the duplicate ratio of their homo-logous Sides AD^ AE, Confequently the Spacesdefcrib'd in the beginning of the Motion, arein the duplicate ratio of the Time. Which wasto be demonftrated.COROLLART.From hence we may gather, that the Errorsof Bodies defcribing fimilar parts of fimilar Fi-gures in proportional Times, generated byanyequal difturbing Forces fimilarly applied to theBodies, and meafur'd by the diftances of theBodies from thofe places of the fimilar Figures,to which thefe Bodies would arrive, in the fameproportional Times, without thofe Forces, arevery near as the Squares of the Times where*in they are generated. For thefe Errors are theSpaces which the Bodies aded upon by the di-fturbing Force defcribe. But if thefe difturbingForces are not equal, but in a given ratio ,% theErrors are as thofe Forces and the Squares of theTimes conjunctly; fuppofing the Forces fimi-larly applied.

Proposition XXIV.*^He nafcent or evanefcent fuhtenfe of the Angle of

ContaB^ in any Circle^ is in the duplicate rati?of the conterminoids Arc.Let ADC \_Fig.^


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4^ T2^^ Elements BookL%cumftances^ as the Tquare of the Arc AD tothe fquare of the Arc A d.Draw the Diameter A C, which will be per-pendicular toAB, join the Right lines DA, dA,DCy dC, and draw /), ^f parallel to ^^ 5.

Firfl, let the Subtenfe B D be parallel to AC.Now ( by Sand ly.jE/cw.vi.) ^Dq==:C^x^E,and Ad^ = CAy. Ac Wherefore AD'i : Adi ::(AE : Ae ::) BD : hd. But fince BD, hd areneareft to the Point Ay or in their naltent con-dition^ the Arcs AD, Ad^ and their Subtenfesdo not differ from one another; that is, they areequal. Confequencly, in this cafe, 5Distohdy as the Square of the Arc ADy to the Squareof the Arc Ad,

Secondly, if the Subtenfe 5 D of the Angleof ContaA be not parallel to ACy but to y^ G^,[i^^f* 26 ] the fame ftill holds: For, drawingDFy df parallel to ^ C 5- then, becaufe DFBdfh are fimilar Triangles, BD : kd :: DF : df;but it has been already iliewn that DF is to df,as AD'i to Ad% Wherefore BD istohdy asAD'l to A d^l.

Thirdly, if B D be fuppos'd to be drawn ac-cording to any other certain Law, (for inftanceconverging toward the Center^) fmce BDyhd areas near as may be to the Point A, the Angles B, hwill be equal ; and confequently, in that cafe,Bdihd:: (DF:df::)AD'i:Ad^. Whichwas to be demonftrated.SCHOLIUM,

This is true likewife in any other Curve, towhich a Circle equi-curve may be drawn ; fuchas all the Conic Sections are. For the PointsD, d (being the neareft that can be to A) muftbe in that other Curve, as well as in the Circle,


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BookL 0/ Astronomy, 47fequently the Properties of B Dy h d^ which a-gree to the Circle, agree likewife to this Curve,to which the Circle is equi-curve. And fincethe foregoing Propofition is true of all Cir-cles, it muft likewife be true of all Curves,that can have equi-curve Circles drawn tothem.

Proposition XXV.Bodies defcrihing different Circles jvith an ecjuahkMotion^ are aBed upon hy Centripetal Forcestending to the Centers of tbofe Circles. And the For-ces are to one another as the Squares of the Arcs^ dc-fcrihed in the fame time^ applied to the Radii of theCircles* This likewife is true of the Centrifugal For-ces of the Bodies thus moved,Becaufe the Bodies, by their Vis i^fita alone,would defcribe Tangents,* 'tis evident that theyaredrawn off from their Redilineal Motion, andretained in Circular Orbits by Forces tendingto Points within the Circles. But fince, byilippofition, they are carried in the Circumfe'--rences by an equable Motion ^ the Areas de-fcribed by a Radius drawn to the Centers arequally augmented ; and therefore are propor-tional to the time which flows equably ^ andconfequendy thefe Forces, by Vrop, xii. tend tothe faid Centers.

Let the Bodies il^and 7"[F/^. 27.] revolvingm the Circumferences of the Circles MA, TR^defcribe the infinitely fmall Arcs M F, TD;From their extremities F, D, draw the Rightlines J^E, D C as far as the Tangents, either pa-rallel to SM, S r, or diverging from S: (For itcomes to the fame, fince MF^TD are only na--fcent Arcs.) Thefe Right lines are the efFedlsof the


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48 jTy^^ Elements BookL'confequently proportional to them, as their a-dequate Caufes; that is the Centripetal Force ofthe Body M is to the Centripetal Force of theBodyr, as EF to CD. Make the Fig. MGU fimilarto the Fig. TC D ; wherefore G H will be a na-fcent Right line i And (o C D is to G H, as theArc TDy to the Arcil^H. And, by the pra^ceed/GHisto E Fas MHI to Mfii And therefore,ex ie^uo, CD is to ET; as (TDyiMH'i toMHxMFl that is, zsTD X MH to M F%eras) TDvi MH^ ST to MF'i'^sr: Andfince jTCD, A/ G H are fimilar Figures, the ArcsTD, ikf H are fimilar ; that is, STi SMiiTD:MH. WhQrdoTQMFIyiST^^TD^SM. Thenby Subftitution CD : EF :: CrDq>c5Mj

r^q MF^MF^y^ST::) "^ : ;j^^But the Centripetal Forces of the Bodies T

and M were fliewn to be proportional to C D,E F. Therefore the Centripetal Force of theBody revolving in TR, is to the CentripetalTD^Force of the Body revolving in M J^ as -rj J.

fr. . And. (becaufe the Motion in bothis equable) TD and M F, b^ng defcrib'd at thefame time, have the fame ratio with any othersdefcrib'd at the fame time. Wherefore the Cen-tripetal Force of a Body revolving in TR, is tothe Centripetal Force of a Body revolving inM A, as the Square of an Arc defcribed in anytime in TR applied to the Radius S T^ is to theSqu

gregory, elements of astronomy, physical and geometrical (english, 1715)

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