subducted if in consequential and in the octants, where it is of thegreatest magnitude, it arises to 47" in the mean distance of the sun fromthe earth, as I find from the theory of gravity. In other distances of thesun, this equation, greatest in the octants of the nodes, is reciprocally asthe cube of the sun s distance from the earth ; and therefore in the sun sperigee it comes to about 49", and in its apogee to about 45".By the same theory of gravity, the moon s apogee goes forward at thegreatest rate when it is either in conjunction with or in opposition to thesun, but in its quadratures with the sun it goes backward ; and the eccentricity comes, in the former case, to its greatest quantity ;in the latterto its least, by Cor. 7, 8, and 9, Prop. LXVI, Book 1. And those inequalities, by the Corollaries we have named, are very great, and generatethe principal which I call the semi-annual equation of the apogee ; andthis semi-annual equation in its greatest quantity comes to about 12 18 ,as nearly as I could collect from the phenomena. Our countryman,HorroXj was the first who advanced the theory of the moon s moving inan ellipsis about the earth placed in its lower focus. Dr. Halley improvedthe notion, by putting the centre of the ellipsis in an epicycle whose cenBOOK III.] OF NATURAL PHILOSOPHY. 447tre is uniformly revolved about the earth ; and from the motion in thisepicycle the mentioned inequalities in the progress and regress of the apogee, and in the quantity of eccentricity, do arise. Suppose the mean distance of the moon from the earth to be divided into 100000 parts, andlet T represent the earth, and TC the moon s mean eccentricity of 5505such parts. Produce TC to B, so as CB may be the sine of the greatestsemi-annual equation 12 18 to the radius TC; and the circle BOA described about the centre C, with the( interval CB, will be the epicyclespoken of, in which the centre of themoon s orbit is placed, and revolvedaccording to the order of the lettersBDA. Set off the angle BCD equalto twice the annual argument, ortwice the distance of the sun s true place from the place of the moon sapogee once equated, and CTD will be the semi-annual equation of themoon s apogee, and TO the eccentricity of its orbit, tending to the placeof the apogee now twice equated. But, having the moon s mean motion,the place of its apogee, and its eccentricity, as well as the longer axis ofits orbit 200000, from these data the true place of the moon in its orbit,together with its distance from the earth, may be determined by themethods commonly known.In the perihelion of the earth, where the force of the sun is greatest,the centre of the moon s orbit moves faster about the centre C than in theaphelion, and that in the reciprocal triplicate proportion of the sun s distance from the earth. But, because the equation of the sun s centre isincluded in the annual argument, the centre of the moon s orbit movesfaster in its epicycle BDA, in the reciprocal duplicate proportion of thesun s distance from the earth. Therefore, that it may move yet faster inthe reciprocal simple proportion of the distance, suppose that from D, thecentre of the orbit, a right line DE is drawn, tending towards the moon sapogee once equated, that is, parallel to TC ; and set off the angle EDFequal to the excess of the aforesaid annual argument above the distanceof the moon s apogee from the sun s perigee in conseqiientia ; or; whichcomes to the same thing, take the angle CDF equal to the compleiuent ofthe sun s true anomaly to 360 ; and let DF be to DC as twice the eccentricity of the orbis magnus to the sun s mean distance from the earth.and the sun s mean diurnal m:tion from the moon s apogee to the sun smean diurnal motion from its own apogee conjunctly, that is, as 33f to1000, and 52 27" 16 " to 59 8" 10 "conjunctly, or as 3 to 100; andimagine the centre of the moon s orbit placed in the point F to be revolvedin an epicycle whose centre is D, and radius DF, while the point D movesin the circumference of the circle DABD : for by this means the centre ofTHE MATHEMATICAL PRINCIPLES [BOOK IIIthe moon s orbit comes to describe a certain curve line about the centre Cwith a velocity which will be almost reciprocally as the cube of the sun sdistance from the earth, as it ought to be.The calculus of this motion is difficult, but may be rendered more easyby the following approximation. Assuming, as above, the moon s meandistance from the earth of 100000 parts, and the eccentricity TC of 5505Buch parts, the-line CB or CD will be found 1172f, and DF 35} of thoseparts : and this line DF at the distance TC subtends the angle at the earth,which the removal of the centre of the orbit from the place D to the place P generates in the motion of this centre; and double this line DF in aparallel position, at the distance of the upper focus of the moon s orbit fromthe earth, subtends at the earth the same angle as DF did before, whichthat removal generates in the motion of this upper focus; but at the distance of the moon from the earth this double line 2DF at the upper focus,in a parallel position to the first line DF, subtends an angle at the moon,which the said removal generates in the motion of the moon, which anglemay be therefore called the second equation of the moon s centre; and thisequation, in the mean distance of the moon from the earth, is nearly as thesine of the angle which that line DF contains with the line drawn fromthe point F to the moon, and when in its greatest quantity amounts to 225". But the angle which the line DF contains with the line drawn fromthe point F to the moon is found either by subtracting the angle EDFfrom the mean anomaly of the moon, or by adding the distance of the moonfrom the sun to the distance of the moon s apogee from the apogee of thesun ; and as the radius to the sine of the angle thus found, so is 2 25" tothe second equation of the centre: to be added, if the forementioned sumbe less than a semi-circle ;to be subducted, if greater. And from the moon splace in its orbit thus corrected, its longitude may be found in the syzygiesof the luminaries.The atmosphere of the earth to the height of 35 or 40 miles refracts thesun s light. This refraction^scattersand spreads the light over the earth sshadow ; and the dissipated^ light near the limits of the shadow dilates theshadow. Upon which account, to the diameter of the shadow, as it corneaout by the parallax, I add 1 or 1^ minute in lunar eclipses.But the theory of the moon ought to be examined and proved from thephenomena, first in the syzygies, then in the quadratures, and last of allin the octants: and whoever pleases to undertake the work will find itnot amiss to assume the following mean motions of the sun and moon atthe Royal Observatory of Greenwich, to the last day of December at noon,anno 1700, O.S. viz. The mean motion of the sun Y5> 20 43 40", and ofits apogee s 7 44 30"; the mean motion of the moon ^ 15 21 00";of its apogee, X 8 20 00"; and of its ascending node Si 27 24 20";and the difference of meridians betwixt the Observatory at Greenwich andBOOK III.] OF NATURAL PHILOSOPHY. 449the Royal Observatory at Paris, Oh. 9 20 : but the mean motion >f theinoon and of its apogee are not yet obtained with sufficient accuracy.PROPOSITION XXXVI. PROBLEM XVII.Tofind the force of the sun to move the sea.The sun s force Ml, or PT to disturb the motions of the moon, was (byProp. XXV.) in the moon s quadratures, to the force of gravity with us, as1 to 638092.6; and the force TM LM or 2PK in the moon s syzygiesis double that quantity. But, descending to the surface of the earth, theseforces are diminished in proportion of the distances from the centre of theearth, that is, in the proportion of 60| to 1; and therefore the former forceon the earth s surface is to the force of gravity as 1 to 38604600 ; and bythis force the sea is depressed in such places as are 90 degrees distant fromthe sun. But by the other force, which is twice as great, the sea is raisednot only in the places directly under the sun, but in those also which aredirectly opposed to it; and the sum of these forces is to the force of gravityas 1 to 12868200. And because the same force excites the same motion,whether it depresses the waters in those places which are 90 degrees distantfrom the sun, or raises them in the places which are directly under and directly opposed to the sun, the aforesaid sum will be the total force of thesun to disturb the sea, and will have the same effect as if the whole wasemployed in raising the sea in the places directly under and directly opposed to the sun, and did not act at all in the places which are 90 degreesremoved from the sun.And this is the force of the sun to disturb the sea in any given place,where the sun is at the same time both vertical, and in its mean distancefrom the earth. In other positions of the sun, its force to raise the sea isas the versel sine of double its altitude above the horizon of the place directly, and the of the distance from the earth reciprocally.COR. Since the centrifugal force of the parts of the earth, arising fromthe earth s diurnal motion, which is to the force of gravity as 1 to 289,raises the waters under the equator to a height exceeding that under thepoles by 85472 Paris feet, as above, in Prop. XIX., the force of the sun,which we have now shewed to be to the force of gravity as 1 to 12868200,and therefore is to that centrifugal force as 289 to 12868200, or as 1 to44527, will be able to raise the waters in the places directly under and directly opposed to the sun to a height exceeding that in the places which arc90 degrees removed from the sun only by one Paris foot and 113 V inches ;for this measure is to the measure of 85472 feet as 1 to 44527.PROPOSITION XXXVII. PROBLEM XVIILTofind the force of the moon to move the sea.The force of the moon to move the sea is to be deduced from its proper-29450 THE MATHEMATICAL PRINCIPLES [BoOH IIItion to the force of the sun, and this proportion is to he collected from theproportion of the motions of the sea, which are the effects of those forces.Before the mouth of the river Avon, three miles below Bristol, the heightof the ascent of the water in the vernal and autumnal syzygies of the luminaries (by the observations of Samuel Sturmy} amounts to about 45feet, but in the quadratures to 25 only. The former of those heights arises from the sum of the aforesaid forces, the latter from their difference.If, therefore, S and L are supposed to represent respectively the forces ofthe sun arid moon while they are in the equator, as well as in their meandistances from the earth, we shall have L + S to L S as 45 to 25, or as9 to 5.At Plymouth (by the observations of Samuel Colepress) the tide in itsmean height rises to about 16 feet, and in the spring and autumn tluheightthereof in the syzygies may exceed that in the quadratures by morethan 7 or 8 feet. Suppose the greatest difference of those heights to be 9feet, and L -f S will be to L S as 20 to ll|, or as 41 to 23; a proportion that agrees well enough with the former. But because of the greattide at Bristol, we are rather to depend upon the observations of Sturmy ;and, therefore, till we procure something that is more certain, we shall usethe proportion of 9 to 5.But because of the reciprocal motions of the waters, the greatest tides donot happen at the times of the syzygies of the luminaries, but, as we havesaid before, are the third in order after the syzygies ;or (reckoning fromthe syzygies) follow next after the third appulse of the moon to the meridian of the place after the syzygies ; or rather (as Sturmy observes) arethe third after the day of the new or full moon, or rather nearly after thetwelfth hour from the new or full moon, and therefore fall nearly upon theforty-third hour after the new or full of the moon. But in this port theyfall out about the seventh hour after the appulse of the moon to the meridian of the place ; and therefore follow next after the appulse of themoon to the meridian, when the moon is distant from the sun, or from opposition with the sun by about IS or 19 degrees in. consequent-la. So thesummer and winter seasons come not to their height in the solstices themselves, but when the sun is advanced beyuni the solstices by about a tenthpart of its whole course, that is, by about 36 or 37 degrees. In like manner, the greatest tide is raised after the appulse of the moon to the meridianof the place, when the moon has passed by the sun, or the opposition thereof.by .about a tenth part of the whole motion from one greatest tide to thenext following greatest tide. Suppose that distance about 18^ degrees:and the sun s force in this distance of the moon from the syzygies andquadratures will be of less moment to augment and diminish that part o1the motion of the sea which proceeds from the motion of the moon than inIhe syzygies and quadratures themselves in the proportion of the radius tuBOOK III.] OF NATURAL PHILOSOPHY 451the co-sine of double this distance, or of an angle of 37 degrees ;that is- inproportion of 10000000 to 798)355; and, therefore, in the preceding analogy, in place of S we must put 0,79863558.But farther ; the force of tne moon in the quadratures must be diminished, on account of its declination from the equator ;for the moon inthose quadratures, or rather in 18^ degrees past the quadratures, declinesfrom the equator by about 23 13 ; and the force of either luminary tomove the sea is diminished as it declines from the equator nearly in theduplicate proportion of the co-sine of the declination ; and therefore theforce of the moon in those quadratures is only 0.85703271. ; whence wehave L+0,7986355S to 0,8570327L 0,79863558 as 9 to 5.Farther yet ; the diameters of the orbit in which the moon should move,setting aside the consideration of eccentricity, are one to the other as 69to 70 ; and therefore the moon s distance from the earth in the syzygiesis to its distance in the quadratures, c&teris paribus, as 69 to 70 ; and itsdistances, when 18i degrees advanced beyond the syzygies, where the greatest tide was excited, and when 18^ degrees passed by the quadratures,where the least tide was produced, are to its mean distance as 69,098747and 69,97345 to 69 1. But the force of the moon to move the sea is inthe reciprocal triplicate proportion of its distance ; and therefore itsforces, in the greatest and least of those distances, are to its force in itsmean distance ;is 0.9830427 and 1,017522 to 1. From whence we have1,0175221, x 0,79863558. to 0,9830427 X 0,8570327L 0,79863558as 9 to 5 ; and 8 to L as 1 to 4,4815. Wherefore since the force of thesun is to the force of gravity as 1 to 12868200, the moon s force will beto the force of gravity as 1 to 2871400.COR. 1. Since the waters excited by the sun s force rise to the height ofa foot and ll^V inches, the moon s force will raise the same to the heightof 8 feet and 7/ inches ; and the joint forces of botli will raise the sameto the height of 10^ feet; and when the moon is in its perigee to theheight of 12 ifeet, and more, especially when the wind sets the same wayas the tide. And a force of that quantity is abundantly sufficient to excite all the motions of the sea, and agrees well with the proportion ofthose motions; for in such seas as lie free and open from east to west, asirithe Pacific sea. and in those tracts of the Atlantic and Ethiopia seaswhich lie without the tropics, the waters commonly rise to 6, 9,* 12, cr 15feet; but in the Pacific sea, which is of a greater depth, as well as- of alarger extent, the tides are said to be greater than in the Atlantic andiEthiopic seas ;for to have a full tide raised, an extent of sea from east 1 towest is required of no less than 90 degrees. In the Ethiopic sea, the watersriseto a less height within the tropics than in the temperate zones, because of the narrowness of the sea between Africa and the southern partsof America. In the middle of the open sea the waters cannot rise with*J52 THE MATHEMATICAL PRINCIPLES [BOOK 111,out falling together, and at the same time, upon both the eastern and western shores, when, notwithstanding, in our narrow seas, they ought to fallon those shores by alternate turns ; upon which account there is commonly but a small flood and ebb in such islands as lie far distant fromthe continent. On the contrary, in some ports, where to fill and emptythe bays alternately the waters are with great violence forced in and outthrough shallow channels, the flood and ebb must be greater than ordinary ;as at Plymouth and Chepstow Bridge in England, at the mountains ofSt. Michael, and the town of Auranches, in Normandy, and at Combaiaand Pegu in the East Indies. In these places the sea is hurried in andqjit with such violence, as sometimes to lay the shores under water, sometimes to leave them dry for many miles. Nor is this force of the influxand efflux to be broke till it has raised and depressed the waters to 30, 40,or 50 feet and above. And a like account is to be given of long and shallow channels or straits, such as the Mugellrniic straits, and those channels which environ England. The tide in such ports and straits, by theviolence of the influx and efflux, is augmented above measure. But onsuch shores as lie towards the deep and open sea with a steep descent,where the waters may freely rise and fall without that precipitation ofinflux and efflux, the proportion of the tides agrees with the forces of thesun and moon.COR. 2. Since the moon s force to move the sea is to the force of gravityas 1 to 2871400, it is evident that this force is far less than to appearsensibly in statical or hydrostatical experiments, or even in those of pendulums. It is in the tides only that this force shews itself by any sensible effect.COR. 3. Because the force of the moon to move the sea is to the likeforce of the sun as 4,4815 to 1, and those forces (by Cor. 14, Prop. LXVI,Book 1) are as the densities of the bodies of the sun and moon and thecubes of their apparent diameters conjunctly, the density of the moon willbe to the density of the sun as 4,4815 to 1 directly, and the cube of themoon s diameter to the cube of the sun s diameter inversely ;that is (seeing the mean apparent diameters of the moon and sun are 31 161", and32 12"), as 4891 to 1000. But the density of the sun was to the density of the earth as 1000 to 4000; and therefore the density of the moonis to the density of the earth as 4891 to 4000, or as 11 to 9. Thereforethe body of the moon is more dense and more earthly than the earthitself.COR. 4. And since the true diameter of the moon (from the observationsof astronomers) is to the true diameter of the earth as 100 to 365, themass of matter in the moon will be to the mass of matter in the earth as1 to 39,788.Cor. 5. And the accelerative gravity on the surface of the moon will be30OK HI.] OF NATURAL PHILOSOPHY. 453about three times less than the accelerative gravity on the surface of threarth.COR. 6. And the distance of the moon s centre from the centre of theearth will be to the distance of the moon s centre from the common centreof gravity of the earth and moon as 40,783 to 39,788.COR. 7. And the mean distance of the centre of the moon from thecentre of the earth will be (in the moon s octants) nearly 60f of the greatest semi-diameters of the earth; for the greatest semi- diameter of theearth was 1 9658600 Paris feet, and the mean distance of the centres ofthe earth and moon, consisting of 60| such semi-diameters, is equal to1187379440 feet. And this distance (by the preceding Cor.) is to the distance of the moon s centre from the common centre of gravity of theearth and moon as 40.788 to 39,788 : which latter distance, therefore, is1158268534 feet. And since the moon, in respect of the fixed stars, performs its revolution in 27d. 7h. 43f , the versed sine of that angle whichthe moon in a minute of time describes is 12752341 to the radius1000,000000,000000; and as the radius is to this versed sine, so are1158268534 feet to 147706353 feet. The moon, therefore, falling towards the earth by that force which retains it in its orbit, would in oneminute of time describe 147706353 feet; and if we augment this forcein the proportion of 17Sf to l?7-, we shall have the total force ofgravity at the orbit of the moon, by Cor. Prop. Ill; and the moon fallingby this force, in one minute of time would describe 14.8538067 feet. Andat the 60th part of the distance of the moon from the earth s centre, thatis, at the distance of 197896573 feet from the centre of the earth, a bodyfalling by its weight, would, in one second of time, likewise describe14,8538067 feet. And, therefore, at the distance of 19615800, whichcompose one mean serni -diameter of the earth, a heavy body would describe in falling 15,11175, or 15 feet, 1 inch, and 4^ lines, in the sametime. This will be the descent of bodies in the latitude of 45 degrees.And by the foregoing table, to be found under Prop. XX, the descent inthe latitude of Paris will be a little greater by an excess of about | partsof a line. Therefore, by this computation, heavy bodies in the latitude ofParis falling in vacno will describe 15 Paris feet, 1 inch, 4|f lines, verynearly, in one second of time. And if the gravity be diminished by taking away a quantity equal to the centrifugal force arising in that latitude."rom the earth s diurnal motion, heavy bodies falling there will describein one second of time 15 feet, 1 inch, and l line. And with this velocity heavy bodies do really fall in the latitude of Paris, as we have shewnabove in Prop. IV and XIX.COR. 8. The mean distance of the centres of the earth and moon in thesyzygies of the moon is equal to 60 of the greatest semi-diameters of theearth, subducting only about one 30th par1; of a semi- diameter : and in the45.4 THE MATHEMATICAL PRINCIPLES [BOOK III,moon s quadratures the mean distance of the same centres is 60f such semidiametersof the earth ;for these two distances are to the mean distance oithe moon in the octants as 69 and 70 to 69|, by Prop. XXVIII.COR. 9. The mean distance of the centres of the earth and moon in thesyzygies of the moon is 60 mean semi-diameters of the earth, and a 10thpart of one semi-diameter; and in the moon s quadratures the mean distance of the same centres is 61 mean semi- diameters of the earth, subducting one 30th part of one semi-diameter.COR. 10. In the moon s syzygies its mean horizontal parallax in the latitudes of 0. 30, 38, 45, 52, 60, 90 degrees is 57 20", 57 16", 57 14", 5712", 57 10", 57 8", 57 4", respectively.In these computations I do not consider the magnetic attraction of theearth, whose quantity is very small and unknown : if this quantity shouldever be found out, and the measures of degrees upon the meridian, thelengths of isochronous pendulums in different parallels, the laws of the motions of the sea, and the moon s parallax, with the apparent diameters ofthe sun and moon, should be more exactly determined from phenomena : woshould then be enabled to bring this calculation to a greater accuracy.PROPOSITION XXXVIII. PROBLEM XIX.To find the figure of the moon s body.If the moon s body were fluid like our sea, the force of the earth to raisethat fluid in the nearest and remotest parts would be to the force of themoon by which our sea is raised in the places under and opposite to themoon as the accelerative gravity of the moon towards the earth to the accelerativegravity of the earth towards the moon, and the diameter of themoon to the diameter of the earth conjunctly ; that is, as 39,788 to 1, and100 to 365 conjunctly, or as 1081 to 100. Wherefore, since our sea, bythe force of the moon, is raised to Sf feet, the lunar fluid would be raisedby the force of the earth to 93 feet; and upon this account the figure ofthe moon would be a spheroid, whose greatest diameter produced wouldpass through the centre of the earth, and exceed the diameters perpendicular thereto by 186 feet. Such a figure, therefore, the moon affects, andmust have put on from the beginning. Q.E.I.COR. Hence it is that the same face of the moon always respects theearth ;nor can the body of the moon possibly rest in any other position,but would return always by a libratory motion to this situation ; but thoselibrations, however, must be exceedingly slow, because of the weakness ofthe forces which excite them ; so that the face of the moon, which shouldbe always obverted to the earth, may, for the reason assigned in Prop. XVI I.be turned towards the other focus of the moon s orbit, without being im
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