Principia Mathematica (Motte Translation, 1848) — part 34 of 45

X AZ X TG X p^ to AT^ that is, as ^^,p X p^ to 2AT, that

is, as the sine of double the distance of the moon from the quadratures

multiplied into .^ to twice the radius, the sum of all the horary varia- PG

tions during the time that the moon, in this situation of the nodes, passes

from the quadrature to the syzygy (that is, in the space of 177| hours) will

be to the sum of as many angles 33" 10"' 33^ or 5878", as the sum of all

the sines of double the distance of the moon from the quadratures multi-

Fp

plied into ^ to tl\e sum of as many diameters ; that is, as the diameter PG

P»

multiplied into ;~ to the circumference ; that is, if the inclination be 5°

V, as 7 X tI Ho to 22, or as 27S to 10000. And, therefore, the whole variation, composed out of the sum of all the horary variations in the aforesaid time, is 163", or 2' 43".

Book III.] of natural philosophy. 443

PROPOSITION XXXV. PROBLEM XVI.

To a s^iven time to find the inclination of the mooji's orbit to the plane

of the ecliptic. Let AD be the sine of the greatest inclination, and AB the sine of the least. Bisect BD in C ; and round the centre C, with the interval BC, describe the circle BGD. In AC take CE in the same proportion to EB

A.

as EB to twice BA. And if to the time given we set off the angle AEG equal to double the distance of the nodes from the quadratures, and upon AD let fall the perpendicular GH, AH will be the sine of the inclination required.

For GE^ is equal to GH^ -}- HE^ = BHD + HE^ = HBD + HE^ — BH2 =- HBD + BE2 __ 2BH X BE =- BE^ + 2E0 X BH = 2EC X AB + 2EC X BH = 2EC X AH; wherefore since 2EC is given, GE^ will be as AH. Now let AEo- represent double the distance of the nodes from the quadratures, in a given moment of time after, and the arc Gg, on account of the given angle GE^-, will be as the distance GE. But HA is to G^ as GH to GO, and, therefore, HA is as the rectangle GH X Gg, or

GH GH

GH X GE, that is, as ^:;^^ X GE% or p^^T X AH; that is, as AH and

the sine of the angle AEG conjunctly. If, therefore, in any one case, AH be the sine of inclination, it will increase by the same increments as the sine of inclination doth, by Cor. 3 of the preceding Prop, and therefore wdll always continue equal to that sine. But when the point G falls upon either point B or D, AH is equal to this sine, and therefore remains always equal thereto. Q..E.D.

In this demonstration I have supposed that the angle BEG, representing double the distance of the nodes from the quadratures, increaseth uniform- ly ; for I cannot descend to every minute circumstance of inequality. Now suppose that BEG is a right angle, and that Gg is in this case the ho- rary increment of double the distance of the nodes from the sun ; then, by Cor. 3 of the last Prop, the horary variation of the inclination in the same case will be to 33" 10"' 33''\ as the rectangle of AH, the sine of the incli- nation, into the sine of the right angle BEG, double the distance of the nodes from the sun, to four times the square of the radius ; that iS; as AH,

444 THE MATHEMATICAL PRINCIPLES [BoOK III.

the sine of the mean inclination, to four times the radius ; that is, seeing the mean inclination is about 5° 8^, as its sine 896 to 40000, the quad- ruple of the radius, or as 224 to 10000. But the whole variation corres- ponding to BD, the diiference of the sines, is to this horary variation as the diameter BD to the arc Gg, that is, conjunctly as the diameter BD to the semi- circumference BGD, and as the time of 2079 y\ hours, in which the node proceeds from the quadratures to the syzygies, to one hour, that is, as 7 to 11, and 2079yV to 1. Wherefore, compounding alL these pro- portions, we shall have the whole variation BD to 33" 10'" 33^ as 224 X 7 X 20793-V to 110000, that is, as 29645 to 1000; and from chence that variation BD will come out 16' 23|".

And this is the greatest variation of the inclination, abstracting from the situation of the moon in its orbit ; for if the nodes are in the syzygies, the inclination suffers no change from the various positions of the moon. But if the nodes are in the quadratures, the inclination is less when the moon is in the syzygies than when it is in the quadratures by a difference of 2' 43", as we shewed in Cor. 4 of the preceding Prop. ; and the whole mean variation BD, diminished by 1' 2U", the half of this excess, becomes 15' 2", when the moon is in the quadratures; and increased by the same, becomes 17' 45" when the moon is in the syzygies. If, therefore, the moon be in the syzygies, the whole variation in the passage of the nodes from the quadratures to the syzygies will be 17' 45" ; and, therefore, if the inclination be 5° 17' 20", when the nodes are in the syzygies, it will be 4° 59' 35" when the nodes are in the quadratures and the moon in the syzy- gies. The truth of all which is confirmed by observations.

Now if the inclination of the orbit should be required when the moon is in tlie syzygies, and the nodes any where between them and the quadratures, let AB be to AD as the sine of 4° 59' 35" to the sine of 5° 17' 20", and take the angle AEG equal to double the distance of the nodes from the quadratures ; and AH will be the sine of the inclination desired. To this inclination of the orbit the inclination of the same is equal, when the moon is 90° distant from the nodes. In other situations of the moon, this men- strual inequality, to which the variation of the inclination is obnoxious in the calculus of the moon's latitude, is balanced, and in a manner took off, by the menstrual inequality of the motion of the nodes (as we said before), and therefore may be neglected in the computation of the said latitude.

SCHOLIUM.

By these computations of the lunar motions I was willing to shew that by the theory of gravity the motions of the moon could be calculated from their physical causes. By the same theory I moreover found that the an- nual equation of the mean motion of the moon arises from the various

Book III] of natural philosophy. 445

dilatation which the orbit of the moon suffers from the action of the sun, according to Cor. 6, Prop. LXVI, Book 1. The force of this action is greater in the perigeon sun, and dilates the moon's orbit ; in the apogeon sun it is less, and permits the orbit to be again contracted. The moon moves slower in the dilated and faster in the contracted orbit ; and the annual equation, by which this inequality is regulated, vanishes in the apogee and perigee of the sun. In the mean distance of the sun from the earth it arises to about 11' 50" ; in other distances of the sun it is pro- portional to the equation of the sun's centre, and is added to the mean motion of the moon, while the earth is passing from its aphelion to its perihelion, and subducted while the earth is in the opposite semi-circle. Taking for the radius of the orbis magnus 1000, and 16| for the earth's eccentricity, this equation, when of the greatest magnitude, by the theory of gravity comes out 11' 49". But the eccentricity of the earth seems to be something greater, and with the eccentricity this equation will be aug- mented in the same proportion. Suppose the eccentricity 16|i, and the greatest equation will be 11' 51".

Farther ; I found that the apogee and nodes of the moon move faster in the perihelion of the earth, where the force of the sun's action is greater, than in the aphelion thereof, and that in the reciprocal triplicate propor- tion of the earth's distance from the sun ; and hence arise annual equa- tions of those motions proportional to the equation of the sun's centre. Now the motion of the sun is in the reciprocal duplicate proportion of the earth s distance from the sun ; and the greatest equation of the centre which this inequality generates is 1° 56' 20", corresponding to the above- mentioned eccentricity of the sun, 16||-. But if the motion of the sun had been in the reciprocal triplicate proportion of the distance, this ine- quality would have generated the greatest equation 2° 54' 30" ; and there- fore the greatest equations which the inequalities of the motions of the moon's apogee and nodes do generate are to 2° 54' 30" as the mean diur- nal motion of the moon's apogee and the mean diurnal motion of its nodes are to the mean diurnal motion of the sun. Whence the greatest equation of the mean motion of the apogee comes out 19' 43", and the greatest equation of the mean motion of the nodes 9' 24". The former equation is added, and the latter subducted, while the earth is passing from its perihelion to its aphelion, and contrariwise when the earth is in the opposite semi-circle.

By the theory of gravity I likewise found that the action of the sun upon the moon is something greater when the transverse diameter of the moon's orbit passeth through the sun than when the same is perpendicu- lar upon the line which joins the earth and the sun ; and therefore the moon's orbit is something larger in the former than in the latter case. And hence arises another equation of the moon's mean motion, depending

446 THE MATHEMATICAL PRINCIPLES [BoOK 111.

upon the situation of the moon's apogee in respect of the sun, which is in it^/ greatest quantity when the moon's apogee is in the octants of the sun, and vanishes when the apogee arrives at the quadratures or syzygies ; and it is added to the mean motion while the moon's apogee is passing from the quadrature of the sun to the syzygy, and subducted while the apogee is passing from the syzygy to the quadrature. This equation, which I shall call the semi-annual, when greatest in the octants of the apogee, arises to about 3' Ad", so far as I could collect from the phsenomena : and this is its quantity in the mean distance of the sun from the earth. But it is increased and diminished in the reciprocal triplicate proportion of the sun's distance, and therefore is nearly 3' 34" when that distance is greatest^ and 3' 56" ^vhen least. But when the moon's apogee is without the octants, it becomes less, and is to its greatest quantity as the sine of double the distance of the moon's apogee from the nearest syzygy or quad- rature t(f the radius.

By the same theory of gravity, the action of the sun upon the moon is something greater when the line of the moon's nodes passes through the sun than when it is at right angles with the line which joins the sun and the earth ; and hence arises another equation of the moon's mean motion, which I shall call the second semi-annual ; and this is greatest when the nodes are in the octants of the sun, and vanishes when they are in the syzygies or quadratures : and in other positions of the nodes is propor- tional to the sine of double the distance of either node from the nearest syzygy or quadrature. And it is added to the mean motion of the moon, if the sun is in cmtecedejitia, to the node which is nearest to him, and subducted if in consequent ia ; and in the octants, where it is of the greatest magnitude, it arises to 4/" in the mean distance of the sun from the earth, as I find from the theory of gravity. In other distances of the sun, this equation, greatest in the octants of the nodes, is reciprocally as the cube of the sun's distance from the earth ; and therefore in the sun's perigee 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 the greatest rate when it is either in conjunction with or in opposition to the sun, but in its quadratures with the sun it goes backward : and the ec- centricity comes, in the former case, to its greatest quantity ; in the latter to its least, by Cor. 7, 8, and 9, Prop. LXYI, Book 1. And those ine- qualities, by the Corollaries we have named, are very great, and generate the principal which I call the semi-annual equation of the apogee ; and this semi-annual equation in its greatest quantity comes to about 12° 18', as nearly as I could collect from the phrenomena. Our countryman, Horrox, was the first who advanced the theory of the moon's moving in an ellipsis about the earth placed in its lower focus. Dr. Halley improved the notion, by putting the centre of the ellipsis in an epicycle whose cen-

Book III.] of natural philosophy. 447

tre is uniformly revolved about the earth; and from the motion in this epicycle the mentioned inequalities in the progress and regress of the apo- gee, and in the quantity of eccentricity, do arise. Suppose the mean dis- tance of the moon from the earth to be divided into 100000 parts, and let T represent the earth, and TC the moon's mean eccentricity of 5505 such parts. Produce TC to B, so as CB may be the sine of the greatest semi-annual equation 12'' 18' to the radius TC; and the circle BDA de- scribed about the centre C, with the interval CB, will be the epicycle spoken of, in which the centre of the -jB moon's orbit is placed, and revolved according to the order of the letters BDA. Set off the angle BCD equal to twice the annual argument, or twice the distance of the sun's true place from the place of the moon's apogee once equated, and CTD will be the semi-annual equation of the moon's apogee, and TD the eccentricity of its orbit, tending to the place of 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 of its 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 the methods 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 the aphelion, and that in the reciprocal triplicate proportion of the sun's dis- tance from the earth. But, because the equation of the sun's centre is included in the annual argument, the centre of the moon's orbit moves faster in its epicycle BDA, in the reciprocal duplicate proportion of the sun's distance from the earth. Therefore, that it may move yet faster in the reciprocal simple proportion of the distance, suppose that from D, the centre of the orbit, a right line DE is drawn, tending towards the moon's apogee once equated, that is, parallel to TC ; and set off the angle EDF equal to the excess of the aforesaid annual argument above the distance of the moon's apogee from the sun's perigee in consequentia ; or, which comes to the same thing, take the angle CDF equal to the complement of the sun's true anomaly to 360° ; and let DP be to DC as twice the eccen- tricity of the 07-bis magnus to the sun's mean distance from the earth, and the sun's mean diurnal motion from the moon's apogee to the sun's mean diurnal motion from its own apogee conjunctly, that is, as 33f to 1000, and 52' 27" W" to 59' 8" 10'" conjunctly, or as 3 to 100; and imagine the centre of the moon's orbit placed in the point F to be revolved in an epicycle whose centre is D, and radius DF, while the point D moves in the circumference of the circle DABD ; for by this means the centre of

448 THE I\rATHEMATICAL PRINCIPLES [BoOK III.

the moon's orbit comes to describe a certain curve line about the centre C, with a velocity which will be almost reciprocally as the cube of the sun's distance from the earth, as it ought to be.

The calculus of this motion is difficult, but may be rendered more easy by the following approximation. Assuming, as above, the moon's mean distance from the earth of 100000 parts, and the eccentricity TC of 5505 such parts, the line CB or CD will be found 1172|, and DF 35} of those parts ; 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 F generates in the motion of this centre: and double this line DF in a parallel position, at the distance of the upper focus of the moon's orbit from the earth, subtends at the earth the same angle as DF did before, which that removal generates in the motion of this upper focus ; but at the dis- tance 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 angle may be therefore called the second equation of the moon's centre ; and this equation, in the mean distance of the moon from the earth, is nearly as the sine of the angle which that line DF contains with the line drawn from the point F to the moon, and when in its greatest quantity amounts to 2' 25". But the angle which the line DF contains with the line drawn from the point F to the moon is found either by subtracting the angle EDF from the mean anomaly of the moon, or by adding the distance of the moon from the sun to the distance of the moon's apogee from the apogee of the sun ; and as the radius to the sine of the angle thus found, so is 2' 25" to the second equation of the centre : to be added, if the forementioned sum be less than a semi-circle ; to be subducted, if greater. And from the moon's place in its orbit thus corrected, its longitude may be found in the syzygies of the luminaries.

The atmosphere of the earth to the height of 35 or 40 miles refracts the sun's light. This refraction scatters and spreads the light over the earth's shadow ; and the dissipated light near the limits of the shadow dilates the shadow. Upon which account, to the diameter of the shadow, as it comes out 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 the phsenomena, first in the syzygies, then in the quadratures, and last of all in the octants; and whoever pleases to undertake the work will find it not amiss to assume the following mean motions of the sun and moon at the Royal Observatory of Greenwich, to the last day of Decejnber at noon, aimo 1700, O.S. viz. The mean motion of the sun V5' 20° 43' 40", and of its apogee 25 7° 44' 30"; the mean motion of the moon ox 15° 21' 00"; of its apogee, X 8° 20' 00" ; and of its ascending node U 27° 24' 20" ; and the difference of meridians betwixt the Observatory at Greemvich and

Book III.] of natural philosophy. 449

the Royal Observatory at Paris, 0^ 9' 20" : but the mean motion of the moon and of its apogee are not yet obtained with sufficient accuracy.

PROPOSITION XXXVI. PROBLEM XVII.

To find the force of' the sun to move the sea.

The sun's force ML or PT to disturb the motions of the moon, was (by Prop. XXY.) in the moon's quadratures; to the force of gravity with us, as 1 to 638092,6 ; and the force TM — LM or 2PK in the moon's syzygies is double that quantity. But, descending to the surface of the earth, these forces are diminished in proportion of the distances from the centre of the earth, that is, in the proportion of 60|- to 1 ; and therefore the former force on the earth's surface is to the force of gravity as 1 to 38604600 ; and by this force the sea is depressed in such places as are 90 degrees distant from the sun. But by the other force, which is twice as great, the sea is raised not only in the places directly under the sun, but in those also which are directly opposed to it ; and the sum of these forces is to the force of gravity as 1 to 12868200. And because the same force excites the same motion, whether it depresses the waters in those places which are 90 degrees distant from the sun, or raises them in the places which are directly under and di- rectly opposed to the sun, the aforesaid sum will be the total force of the sun to disturb the sea, and will have the same effect as if the whole was employed in raising the sea in the places directly under and directly op- posed to the sun, and did not act at all in the places which are 90 degrees removed 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 iii its mean distance from the earth. In other positions of the sun, its force to raise the sea is as the versed sine of double its altitude above the horizon of the place di- rectly, and the cube of the distance from the earth reciprocally.

Cor. Since the centrifugal force of the parts of the earth, arising from the 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 the poles by 85472 Paris feet, as above, in Prop. XIX., the force of the su?i^ 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 to 44527, will be able to raise the waters in the places directly under and di- rectly opposed to the sun to a height exceeding that in the places which are 90 degrees removed from the sun only by one Paris foot and 113 g'^- inches ; for this measure is to the measure of 85472 feet as 1 to 44527,

PROPOSITION XXXYII. PROBLEM XYIH,

To find the force of the moon to move the sea.

The force of the moon to move the sea is to be deduced from its propor--

29

450 TPIE MATHEMATICAL PRINCIPLES [BoOK IIT.

tion to the force of the sun, and this proportion is to be collected from the proportion 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 height of the ascent of the water in the vernal and autumnal syzygies of the lu- minaries (by the observations of Samuel Sturmy) amounts to about 45 feet, but in the quadratures to 25 only. The former of those heights ari- ses from the sum of the aforesaid forces, the latter from their difference. If, therefore, S and L are supposed to represent respectively the forces of the sun and moon while they are in the equator, as well as in their mean distances from the earth, we shall have L + S to L — S as 45 to 26, or as 9 to 5.

At Ply7nouth (by the observations of Samuel Colepress) the tide in its mean height rises to about 16 feet, and in the spring and autumn the height thereof in the syzygies may exceed that in the quadratures by more than 7 or 8 feet. Suppose the greatest difference of those heights to be 9 feet, and L + S will be to L — S as 20|- to 11|-, or as 41 to 23; a pro- portion that agrees well enough with the former. But because of the great tide at Bristol, we are rather to depend upon the observations of Sturmy ; and, therefore, till we procure something that is more certain, we shall use the proportion of 9 to 5.

But because of the reciprocal motions of the waters, the greatest tides do not happen at the times of the syzygies of the luminaries, but, as we have said before, are the third in order after the syzygies ; or (reckoning from the syzygies) follow next after the third appulse of the moon to the me- ridian of the place after the syzygies ; or rather (as Sturmy observes) are the third after the day of the new or full moon, or rather nearly after the twelfth hour from the new or full moon, and therefore fall nearly upon the forty-third hour after the new or full of the moon. But in this port they fall out about the seventh hour after the appulse of the moon to the me- ridian of the place ; and therefore follow next after the appulse of the moon to the meridian, when the moon is distant from the sun, or from op- position with the sun by about 18 or 19 degrees in consequentia. So the summer and winter seasons come not to their height in the solstices them- selves, but when the sun is advanced beyond the solstices by about a tenth part of its whole course, that is, by about 36 or 37 degrees. In like man- ner, the greatest tide is raised after the appulse of the moon to the meridian of 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 the next folloioing' greatest tide. Suppose that distance about 18|- degrees; and the sun's force in this distance of the moon from the syzygies and quadratures will be of less moment to augment and diminish that part of the motion of the sea which proceeds from the motion of the moon than in the '^ysjgies and nuarlratures themselyes ir, the proportion of the radius to

Book III] of natural philosophy. 451

the co-sine of double this distance, or of an angle of 37 degrees ; that is, in proportion of 10000000 to 79S6355 ; and, therefore, in the preceding an- alogy, in place of S we must put 0,79S6355S.

But farther ; the force of the moon in the quadratures must be dimin- ished, on account of its declination from the equat(p ; for the moon in those quadratures, or rather in 1S| degrees past the quadratures, declines from the equator by about 23° 13' ; and the force of either luminary to move the sea is diminished as it declines from the equator nearly in the duplicate proportion of the co-sine of the declination ; and therefore the force of the moon in those quadratures is only 0,8570327L ; whence we have 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 69 to 70 ; and therefore the moon's distance from the earth in the syzygies is to its distance in the quadratures, cceteris paribus^ as 69 to 70 ; and its distances, when 18|- degrees advanced beyond the syzygies, where the great- est tide was excited, and when 18|- degrees passed by the quadratures, where the least tide was produced, are to its mean distance as 69,098747 and 69,897345 to 69^. But the force of the moon to move the sea is in the reciprocal triplicate proportion of its distance; and therefore its forces, in the greatest and least of those distances, are to its force in i\s mean distance as 0;9830427 and 1,017522 to 1. From whence we have 1,017522L X 0,79863558 to 0,9830427 X 0,8570327L — 0,79863558 as 9 to 5; and 8 to L as 1 to 4,4815. Wherefore since the force of the sun is to the force of gravity as 1 to 12868200, the moon's force will be to the force of gravity as 1 to 2871400.

Cor. 1 . Since the. waters excited by the sun's force rise to the height of a foot and 11 3^-5- inches, the moon's force will raise the same to the height of 8 feet and 7-^^ inches ; and the joint forces of both will raise the same to the height of 10^ feet ; and when the moon is in its perigee to the height of 12i feet, and more, especially when the wind sets the same way as the tide. And a force of that quantity is abundantly sufficient to ex- cite all the motions of the sea, and agrees well with the proportion of those motions ; for in such seas as lie free and open from east to west, as in the Pacific sea, and in those tracts of the Atlantic and Ethiopic seas which lie without the tropics, the waters commonly rise to 6, 9, 12, or 15 feet ; but in the Pacfic sea, which is of a greater depth, as well as of a larger extent, the tides are said to be greater than in the Atlantic and Ethiopic seas ; for to have a full tide raised, an extent of sea from east to west is required of no less than 90 degrees. In the Ethiopic sea, the waters rise to a less height within the tropics than in the temperate zones, he- cause of the narrowness of the sea between Africa and the southern parts of America. In the middle of the open sea the waters cannot rise with-

452 THE MATHEMATICAL PRINCIPLES [BoOK 111.

out falling togetherj and at the same timCj upon both the eastern and west- ern shores, when, notwithstanding, in our narrow seas, they ought to fall on those shores by alternate turns ; upon which account there is com- monly but a small flood and ebb in such islands as lie far distant from the continent. On the contrary, in some ports, where to fill and empty the bays alternately the waters are with great violence forced in and out through shallow channels, the flood and ebb must be greater than ordinary ; as at Plymouth and Chepstoio Bridge in England, at the mountains of St. Michael, and the town of Auranches, in Normandy, and at Cambaia and Pegu in the East Indies. In these places the sea is hurried in and out with such violence, as sometimes to lay the shores under water, some- times to leave them dry for many miles. Nor is this force of the influx and 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 shal- low channels or straits, such as the Magellanic straits, and those chan- nels which environ England. The tide in such ports and straits, by the violence of the influx and efflux, is augmented above measure. But on such shores as lie towards the deep and open sea with a steep descent, where the waters may freely rise and fall without that precipitation of influx and efflux, the proportion of the tides agrees with the forces of the sun and moon.

Cor. 2. Since the moon's force to move the sea is to the force of gravity as 1 to 2871400, it is evident that this force is far less than to appear sensibly in statical or hydrostatical experiments, or even in those of pen- dulums. . It is in the tides only that this force shews itself by any sensi- ble effect.

Cor. 3. Because the force of the moon to move the sea is to the like force 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 the cubes of their apparent diameters conjunctly, the density of the moon will be to the density of the sun as 4,4815 to 1 directly, and the cube of the moon's diameter to the cube of the sun's diameter inversely ; that is (see- ing the mean apparent diameters of the moon and sun are 31' 16|-", and 32' 12"), as 4891 to 1000. But the density of the sun was to the den- sity of the earth as 1000 to 4000 ; and therefore the density of the moon is to the density of the earth as 4891 to 4000, or as 11 to 9. Therefore the body of the moon is more dense and more earthly than the earth itself.

Cor. 4. And since the true diameter of the moon (from the observations of astronomers) is to the true diameter of the earth as 100 to 365, the mass of matter in the moon will be to the mass of matter in the earth as 1 to 39,788.

Cor. 5. And the accelerative gravity on the surface of the moon will be

Book IIL] of natural philosophy. 453

about three times less than the accelerative gravity on the surface of the earth.

Cor. 6. And the distance of the moon's centre from the centre of the earth will be to the distance of the moon's centre from the common centre of gravity of the earth and moon as 40,788 to 39,788.

Cor. 7. And the mean distance of the centre of the moon from the centre of the earth will be (in the moon's octants) nearly 60 f of the great- est semi-diameters of the earth; for the greatest semi- diameter of the earth was 1 9658600 Paris feet, and the mean distance of the centres of the earth and moon, consisting of 60| such semi-diameters, is equal to 1187379440 feet. And this distance (by the preceding Cor.) is to the dis- tance of the moon's centre from the common centre of gravity of the earth and moon as 40,788 to 39,788 ; which latter distance, therefore, is 1158268534 feet. And since the moon, in respect of the fixed stars, per- forms its revolution in 27'^. 7'\ 43 f, the versed sine of that angle which the moon in a minute of time describes is 12752341 to the radius 1000,000000,000000 ; and as the radius is to this versed sine, so are 1158268534 feet to 14,7706353 feet. The moon, therefore, falling tow- ards the earth by that force which retains it in its orbit, would in one minute of time describe 14,7706353 feet ; and if we augment this force in the proportion of 178 ff to 177||, we shall have the total force of gravity at the orbit of the moon, by Cor. Prop. Ill ; and the moon falling by this force, in one minute of time would describe 14,8538067 feet. And at the 60th part of the distance of the moon from the earth's centre, that is, at the distance of 197896573 feet from the centre of the earth, a body falling by its weight, would, in one second of time, likewise describe 14,8538067 feet. And, therefore, at the distance of 19615800, which compose one mean semi-diameter of the earth, a heavy body would de- scribe in falling 15,11175, or 15 feet, 1 inch, and 4.j\ lines, in the same time. 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 in the latitude of Paris will be a little greater by an excess of about | parts of a line. Therefore, by this computation, heavy bodies in the latitude of Paris falling iji vacuo will describe 15 Paris feet, 1 inch, 4|f lines, very nearly, in one second of time. And if the gravity be diminished by tak- ing away a quantity equal to the centrifugal force arising in that latitude from the earth's diurnal motion, heavy bodies falling there will describe in one second of time 15 feet, 1 inch, and l-i line. And with this velo- city heavy bodies do really fall in the latitude of Paris, as we have shewn above in Prop. lY and XTX.

Cor. 8. The mean distance of the centres of the earth and moon in the syzygies of the moon is equal to 60 of the greatest semi-diameters of the earth, subducting only about one 30th part of a semi-diameter; and in the

454 THE MATHEMATICAL PRINCIPLES [BoOK III.

moon's quadratures the mean distance of the same centres is 60| such semi- diameters of the earth ; for these two distances are to the mean distance of the moon in the octants as 69 and 70 to 69|^; by Prop. XXVIII.