to be moved.PROPOSITION XIII. THEOREM XIII.The planets move in ellipses tvhicli have their common focus in the centreof the sini ; and, by radii drawn, to tJtat centre, they describe areas proportional to the times of description.We have discoursed above of these motions from the Phenomena. Nowthat we know the principles on which they depend, from those principleswe deduce the motions of the heavens a priori. Because the weights ofthe planets towards the sun are reciprocally as the squares of their distances from the sun s centre, if the sun was at rest, and the other planets didnot mutually act one upon another, their orbits would be ellipses, havingthe sun in their common focus; and they would describe areas proportionalto the times of description, by Prop. I and XI, and Cor. 1, Prop. XIII,Book I. But the mutual actions of the planets one upon another are sovery small, that they may be neglected ; and by Prop. LXVI, Book I, theyless disturb the motions of the planets around the sun in motion than ifthose motions were performed about the sun at rest.It is true, that the action of Jupiter upon Saturn is not to be neglected;for the force of gravity towards Jupiter is to the force of gravity towardsthe sun (at equal distances, Cor. 2, Prop. VIII) as 1 to 1067; and thereforein the conjunction of Jupiter and Saturn, because the distance of Saturnfrom Jupiter is to the distance of Saturn from the sun almost as 4 to 9, thegravity of Saturn towards Jupiter will be to the gravity of Saturn towardsthe sun as 81 to 16 X 1067; or, as 1 to about 21 1. And hence arises aperturbation of the orb of Saturn in every conjunction of this planet withTupiter, so sensible, that astronomers are puzzled with it. As the planetBOOK III.] OF NATURAL PHILOSOPHY. 403is differently situated in these conjunctions, its eccentricity is sometimesaugmented, sometimes diminished; its aphelion is sometimes carried forward, sometimes backward, and its mean motion is by turns accelerated andretarded ; yet the whole error in its motion about the sun, though arisingfrom so great a force, may be almost avoided (except in the mean motion)by placing the lower focus of its orbit in the common centre of gravity ofJupiter and the sun (according to Prop. LXVII, Book I), and therefore thaterror, when it is greatest, scarcely exceeds two minutes ; and the greatesterror in the mean motion scarcely exceeds two minutes yearly. But in theconjunction of Jupiter and Saturn, the accelerative forces of gravity of thesun towards Saturn, of Jupiter towards Saturn, and of Jupiter towards thesun, are almost as 16, 81, and -~o^~~ >or 156609: and thereforethe difference of the forces of gravity of the sun towards Saturn, and ofJupiter towards Saturn, is to the force of gravity of Jupiter towards thesun as 65 to 156609, or as 1 to 2409. But the greatest power of Saturnto disturb the motion of Jupiter is proportional to this difference; andtherefore the perturbation of the orbit of Jupiter is much less than that ofSaturn s. The perturbations of the other orbits are yet far less, except thatthe orbit of the earth is sensibly disturbed by the moon. The commoncentre of gravity of the earth and moon moves in an ellipsis about the sunin the focus thereof, and, by a radius drawn to the sun, describes areas proportional to the times of description. But the earth in the mean time bya menstrual motion is revolved about this common centre.PROPOSITION XIV. THEOREM XIV.The aphelions and nodes of the orbits of the planets are fixed.The aphelions are immovable by Prop. XI, Book I; and so are theplanes of the orbits, by Prop. I of the same Book. And if the planes arefixed, the nodes must be so too. It is true, that some inequalities mayarise from the mutual actions of the planets and comets in their revolutions; but these will be so small, that they may be here passed by.COR. 1. The fixed stars are immovable, seeing they keep the same position to the aphelions and nodes of the planets.COR. 2. And since these stars are liable to no sensible parallax from theannual motion of the earth, they can have no force, because of their immense distance, to produce any sensible effect in our system. Not tomention that the fixed stars, every where promiscuously dispersed in theheavens, by their contrary attractions destroy their mutual actions, byProp. LXX, Book I.SCHOLIUM.Since the planets near the sun (viz. Mercury, Venus, the Earth, and404 THE MATHEMATICAL PRINCIPLES [B .-OK IILMars) are so small that they can act with but little force upon each other,therefore their aphelions and nodes must be fixed, excepting in so far asthey are disturbed by the actions of Jupiter and Saturn, and other higherbodies. And hence we may find, by the theory of gravity, that their aphelions move a little in consequentw, in respect of the fixed stars, and thatin the sesqui plicate proportion of their several distances from the sun. Sothat if the aphelion of Mars, in the space of a hundred years, is carried33 20" in consequent-la, in respect of the fixed stars, the aphelions of theEarth, of Venus, and of Mercury, will in a hundred years be carried forwards 17 40", 10 53 , and 4 16", respectively. But these motions areso inconsiderable, that we have neglected them in this Proposition,PROPOSITION XV. PROBLEM I.To find the principal diameters<>fthe orbits of the planets.They are to be taken in the sub-sesquiplicate proportion of the periodictimes, by Prop. XV, Book I, and then to be severally augmented in theproportion of the sum of the masses of matter in the sun and each planetto the first of two mean proportionals betwixt that sum and the quantity ofmatter in the sun, by Prop. LX, Book I.PROPOSITION XVI. PROBLEM II.To find the eccentricities and aphelions of the planets.This Problem is resolved by Prop. XVIII, Book I.PROPOSITION XVII. THEOREM XV.That the diurnal motions of the planets are uniform, and that thelibration of the moon arises from its diurnal motion.The Proposition is proved from the first Law of Motion, and Cor. 22,Prop. LXVI, Book I. Jupiter, with respect to the fixed stars, revolves in9 1. 5(5; Mars in 24h. 39 ; Venus in about 23h.; the Earth in 23 1. 56 ; theSun in 25 1days, and the moon in 27 days, 7 hours, 43 . These thingsappear by the Phasnomena. The spots in the sun s body return to thesame situation on the sun s disk, with respect to the earth, in 27 days ; andtherefore with respect to the fixed stars the sun revolves in about 25|days.But because the lunar day, arising from its uniform revolution about itsaxis, is menstrual, that is, equal to the time of its periodic revolution inits orb, therefore the same face of the moon wr ill be always nearly turned tothe upper focus of its orb ; but, as the situation of that focus requires, willdeviate a little to one side and to the other from the earth in the lowerfocus j and this is the libration in longitude ; for the libration in latitudearises from the moon s latitude, and the inclination of its axis to the planeof the ecliptic. This theory of the libration of the moon, Mr. N. Mercato*BOOK III.] OF NATURAL PHILOSOPHY. 4()in his Astronomy, published at the beginning of the year 1676. explainedmore fully out of the letters I sent him. The utmost satellite of Saturneeems to revolve about its axis with a motion like this of the moon, respecting Saturn continually with the same face; for in its revolution roundSaturn, as often as it comes to the eastern part of its orbit, it is scarcel)visible, and generally quite disappears ; which is like to be occasioned bysome spots in that part of its body, which is then turned towards the earth,as M. Cassini has observed. So also the utmost satellite of Jupiter seemato revolve about its axis with a like motion, because in that part of its bodywhich is turned from Jupiter it has a spot, which always appears as if itwere in Jupiter s own body, whenever the satellite passes between Jupiterand our eye.PROPOSITION XVIII. THEOREM XVI.That the axes of the planets are less than the diameters drawn perpendicular to the axes.The equal gravitation of the parts on all sides would give a sphericalfigure to the planets, if it was not for their diurnal revolution in a circle.By that circular motion it comes to pass that the parts receding from theaxis endeavour to ascend about the equator ; and therefore if the matter isin a fluid state, by its ascent towards the equator it will enlarge the diameters there, and by its descent to wards the poles it will shorten the axis.So the diameter of Jupiter (by the concurring observations of astronomers)is found shorter betwixt pole and pole than from east to west. And, bythe same argument, if our earth was not higher about the equator than atthe poles, the seas would subside about the poles, and, rising toward* Ikfequator, would lay all things there under water.PROPOSITION XIX. PROBLEM IIITofind the proportion of the axis of a planet to the dia meter j j*,rpendici/lar thereto.Our countryman, Mr. Norwood, measuring a distance of 005751 feet ofLondon measure between London and YorA:, in 1635, and obs,-rvino- thedifference of latitudes to be 2 28 , determined the measure of one degreeto be 3671 96 feet of London measure, that is 57300 Paris toises. MPicart, measuring an arc of one degree, and 22 55" of the meridian between Amiens and Malvoisine, found an arc of one degree to be 57060Paris toises. M. Cassini, the father, measured the distance upon the meridian from the town of Collionre in Roussillon to the Observatory ofPari; and his son added the distance from the Observatory to the Citadel of Dunkirk. The whole distance was 486156^ toises and the difference of the latitudes of Collionre and Dunkirk was 8 degrees, and 31106 THE MATHEMATICAL PRINCIPLES [BOOK 1IJ.llf". Hence an arc of one degree appears to be 57061 Paris toises.And from these measures we conclude that the circumference of the earthis 123249600, and its semi-diameter 19615800 Paris feet, upon the supposition that the earth is of a spherical figure.In the latitude of Paris a heavy body falling in a second of time describes 15 Paris feet, 1 inch, 1 J line, as above, that is, 2173 lines J. Theweight of the body is diminished by the weight of the ambient air. Letus suppose the weight lost thereby to be TT ^o-o- Par^ ^ ^he whole weight ;then that heavy body falling in, vacua will describe a height of 2174 linesin one second of time.A body in every sidereal day of 23 1. 56 4" uniformly revolving in acircle at the distance of 19615SOO feet from the centre, in one second oitime describes an arc of 1433,46 feet;the versed sine of which is 0,0523656 1feet, or 7,54064 lines. And therefore the force with which bodies descendin the latitude of Paris is to the centrifugal force of bodies in the equatorarising from the diurnal motion of the earth as 2174 to 7,54064.The centrifugal force of bodies in the equator is to the centrifugal forcewith which bodies recede directly from the earth in the latitude of Parin48 50 10" in the duplicate proportion of the radius to the cosine of thelatitude, that is, as 7,54064 to 3,267. Add this force to the force withwhich bodies descend by their weight in the latitude of Paris, and a body,in the latitude of Paris, falling by its whole undiminished force of gravity,in the time of one second, will describe 2177,267 lines, or 15 Paris feet,1 inch, and 5,267 lines. And the total force of gravity in that latitudewill be to the centrifugal force of bodies in the equator of the earth as2177,267 to 7,54064, or as 289 to 1.Wherefore if APBQ, represent the figure of theearth, now no longer spherical, but generated by therotation of an ellipsis about its lesser axis PQ, ; andACQqca a canal full of water, reaching from the poleQq to the centre Cc, and thence rising to the equatorArt ; the weight of the water in the leg of the canalACca will be to the weight of water in the other legQCcq as 289 to 288, because the centrifugal force arising from the circular motion sustains and takes off one of the 289 parts of the weight (in theone leg), and the weight of 288 in the other sustains the rest. But bycomputation (from Cor. 2, Prop. XCI, Book I) I find, that, if the matterof the earth was all uniform, and without any motion, and its axis PQ,were to the diameter AB as 100 to 101, the force of gravity in theplace Q towards the earth would be to the force of gravity in the sameplace Q towards a sphere described about the centre C with the radiusPC, or QC, as 126 to 125. And, by the same argument, the force ofgravity in the place A towards the spheroid generated by the rotation ofBOOK III.] OF NATURAL PHILOSOPHY. 407the ellipsis APBQ, about the axis AI3 is to the force of gravity in thesame place A, towards the sphere described about the centre C with theradius AC, as 125 to 126. But the force of gravity in the place A towards the earth is a mean proportional betwixt the forces of gravity towards the spheroid and this sphere; because the sphere, by having its diameter PQ, diminished in the proportion of 101 to 100, is transformed intothe figure of the earth ; and this figure, by having a third diameter perpendicular to the two diameters AB and PQ, diminished in the same proportion, is converted into the said spheroid ; and the force of gravity in A,in either case, is diminished nearly in the same proportion. Therefore theforce of gravity in A towards the sphere described about the centre C withthe radius AC, is to the force of gravity in A towards the earth as 126 to1251. And the force of gravity in the place Q towards the sphere described about the centre C with the radius QC, is to the force of gravityin the place A towards the sphere described about the centre C, with theradius AC, in the proportion of the diameters (by Prop. LXXII, Book I),that is, as 100 to 101. If, therefore, we compound those three proportions126 to 125, 126 to 125|. and 100 to 101, into one, the force of gravity inthe place Q towards the earth will be to the force of gravity in the placeA towards the earth as 126 X 126 X 100 to 125 X 125| X 101 ; or as:>01 to 500.Now since (by Cor. 3, Prop. XCI, Book I) the force of gravity in eitherleg of the canal ACca, or QCcy, is as the distance of the places from thecentre of the earth, if those legs are conceived to be divided by transverse.,parallel, and equidistant surfaces, into parts proportional to the wholes,the weights of any number of parts in the one leg ACca will be to theweights of the same number of parts in the other leg as their magnitudesand the accelerative forces of their gravity conjunctly, that is, as 10 J to100, and 500 to 501. or as 505 to 501. And therefore if the centrifugalforce of every part in the leg ACca, arising from the diurnal motion, wasto the weight of the same part as 4 to 505, so that from the weight ofevery part, conceived to be divided into 505 parts, the centrifugal forcemight take off four of those parts, the weights would remain equal in eachleg, and therefore the fluid would rest in an equilibrium. But the centrifugal force of every part is to the weight of the same part as 1 to 289 ;that is, the centrifugal force, which should be T y parts of the weight, isonly |g part thereof. And, therefore, I say, by the rule of proportion,that if the centrifugal force j ^ make the height of the water in the legACca to exceed the height of the water in the leg QCcq by one T | partof its whole height, the centrifugal force -^jj will make the excess of theheight in the leg ACca only ^{^ part of the height of the water in theother leg QCcq ; and therefore the diameter of the earth at the equator, isto its diameter from pole to pole as 230 to 229. And since the mean semi108 THE MATHEMATICAL PRINCIPLES [BooK III.diameter of the earth, according to PicarVs mensuration, is 19615800Paris feet, or 3923,16 miles (reckoning 5000 feet to a mile), the earthwill be higher at the equator than at the poles by 85472 feet, or 17^-miles. And its height at the equator will be about 19658600 feet, and atthe poles 19573000 feet.If, the density and periodic time of the diurnal revolution remaining thesame, the planet was greater or less than the earth, the proportion of thecentrifugal force to that of gravity, and therefore also of the diameter betwixt the poles to the diameter at the equator, would likewise remain thegame. But if the diurnal motion was accelerated or retarded in any proportion, the centrifugal force would be augmented or diminished nearly inthe same duplicate proportion ; and therefore the difference of the diameters will be increased or diminished in the same duplicate ratio very nearly.And if the density of the planet was augmented or diminished in any proportion, the force of gravity tending towards it would also be augmentedor diminished in the same proportion : and the difference of the diameterscontrariwise would be diminished in proportion as the force of gravity isaugmented, and augmented in proportion as the force of gravity is diminished. Wherefore, since the earth, in respect of the fixed stars, revolves in23h. 56 , but Jupiter in 9h. 56 , and the squares of their periodic times areas 29 to 5, and their densities as 400 to 94 , the difference of the diameters29 400 1of Jupiter will be to its lesser diameter as X ^^ X ^Tmto 1; or as 1 to9 f, nearly. Therefore the diameter of Jupiter from east to west is to itsdiameter from pole to pole nearly as 10 to 9|-. Therefore since itsgreatest diameter is 37", its lesser diameter lying between the poles willbe 33" 25" . Add thereto about 3 for the irregular refraction of light,and the apparent diameters of this planet will become 40 and 36" 25";which are to each other as 11 -jto 10^, very nearly. These things are soupon the supposition that the body of Jupiter is uniformly dense. Butnow if its body be denser towards the plane of the equator than towardsthe poles, its diameters may be to each other as 12 to 11, or 13 to 12, orperhaps as 14 to 13.And Cassini observed in the year 1691, that the diameter of Jupiterreaching from east to west is greater by about a fifteenth part than theother diameter. Mr. Pound with his 123 feet telescope, and an excellentmicrometer, measured the diameters of Jupiter in the year 1719, and foundthem as follow.K HI. OF NATURAL PHILOSOPHY. 409So thut the theory agrees with the phenomena ;for the planets are moreheated by the sun s rays towards their equators, and therefore are a lit fiemore condensed by that heat than towards their poles.Moreover, that there is a diminution of gravity occasioned by the diurnal rotation of the earth, and therefore the earth rises higher there than itdoes at the poles (supposing that its matter is uniformly dense), will appear by the experiments of pendulums related under the following Proposition.PROPOSITION XX. PROBLEM IV.Tofind and compare together the weights of bodies in the different regions of our earth.Because the weights of the unequal legs of the canalof water ACQqca are equal ; and the weights of theparts proportional to the whole legs, and alike situatedin them, are one to another as the weights of the P|wholes, and therefore equal betwixt themselves ; theweights of equal parts, and alike situated in the legs,will be reciprocally as the legs, that is, reciprocally as230 to 229. And the case is the same in all homogeneous equal bodies alikesituated in the legs of the canal. Their weights are reciprocally as the legs,that is, reciprocally as the distances of the bodies from the centre of the earth.Therefore if the bodies are situated in the uppermost parts of the canals, or onthe surface of the earth, their weights will be one to another reciprocally astheir distances from the centre. And. by the same argument, the weights inall other places round the whole surface of the earth are reciprocally as thedistances of the places from the centre ; and, therefore, in the hypothesisof the earth s being a spheroid are given in proportion.Whence arises this Theorem, that the increase of weight in passing fromtne equator to the poles is nearly as the versed sine of double the latitude ;or, which comes to the same thinir, as the square of the right sine of thelatitude ; and the arcs of the degrees of latitude in the meridian increasenearly in the same proportion. And, therefore, since the latitude of Parisis 48 50 , that of places under the equator 00 00 , and that of placesunder the poles 90 ; and the versed sines of double those arcs are11334,00000 and 20000, the radius being 10000 ; and the force of gravityat the pole is to the force of gravity at the equator as 230 to 229 ; andthe excess of the force of gravity at the pole to the force of gravity at theequator as 1 to 229 ; the excess of the force of gravity in the latitude ofParis will be to the force of gravity at the equator as 1 X Htll to 229,or as 5667 to 2290000. And therefore the whole forces of gravity inthose places will be one to the other as 2295667 to 2290000. Whereforesince the lengths of pendulums vibrating in equal times are as the forces of410 THE MATHEMATICAL PRINCIPLES [BOOK III.gravity, and in the latitude of Paris, the length of a pendulum vibratingseconds is 3 Paris feet, and S lines, or rather because of the weight ofthe air, 8f lines, the length of a pendulum vibrating in the same timearider the equator will be shorter by 1,087 lines. And by a like calculusthe following table is made.By this table, therefore, it appears that the inequality of degrees is scsmall, that the figure of the earth, in geographical matters, may be considered as spherical ; especially if the earth be a little denser towards theplane of the equator than towards the poles.Now several astronomers, sent into remote countries to make astronomicalobservations, have found that pendulum clocks do accordingly move slowernear the equator than in our climates. And, first of all, in the year I 72,M. Richer took notice of it in the island of Cayenne ; for when, in themonth of August, he was observing the transits of the fixed stars over themeridian, he found his clock to go slower than it ought in respect of themean motion of the sun at the rate of 2 29" a day. Therefore, fitting upa simple pendulum to vibrate in seconds, which were measured by an excellent clock, he observed the length of that simple pendulum ; and this hedid over and over every week for ten months together. And upon his return to France, comparing the length of that pendulum with the lengthiiJ.j OF NATURAL PHILOSOPHY. 411of the pendulum at Paris (which was 3 Paris feet and 8f lines), he foundit shorter by 1 j line.Afterwards, our friend Dr. Halley, about the year 1677, arriving at theisland of St. Helena, found his pendulum clock to go slower there than atIsondon without marking the difference. But he shortened the rod ofhis clock by more than the 。 of an inch, or l line; and to effect this, because the length of the screw at the lower end of the rod was riot sufficient,he interposed a wooden ring betwixt the nut and the ball.Then, in the year 1682, M. Varin and M. des Hayes found the lengthof a simple pendulum vibrating in seconds at the Royal Observatory ofParis to be 3 feet and S| lines. And by the same method in the islandof Goree, they found the length of an isochronal pendulum to be 3 feet and6 1 lines, differing from the former by two lines. And in the same year,going to the islands of Guadeloupe and Martinico, they found that thelength of an isochronal pendulum in those islands was 3 feet and 6^ lines.After this, M. Couplet, the son, in the month of July 1697, at the RoyalObservatory of Paris, so fitted his pendulum clock to the mean motion ofthe sun, that for a considerable time together the clock agreed with themotion of the sun. In November following, upon his arrival at Lisbon, hefound his clock to go slower than before at the rate of 2 13" in 24 hours.And next March coming to Paraiba, he found his clock to go slower thanat Paris, and at the rate 4 12" in 24 hours ; and he affirms, that the pendulum vibrating in seconds was shorter at Lisbon by 2 lines, and at Paraiba, by 3 1 lines, than at Paris. He had done better to have reckonedthose differences 。。 and 2f : for these differences correspond to the differences of the times 2 13" and 4 12". But this gentleman s observationsare so gross, that we cannot confide in them.In the following years, 1699, and 1700, M. des Hayes, making another