Dr. 1013. The figure of the earth is so near that of a sphere, that in estimating its magnitude, we may consider it as a sphere whose radius is the mean radius of the earth. Now this mean radius is 3964 miles; hence, 3964 × 6,28318= 24907 miles the circumference; also, 3964 × 3,14159265 x 4= 197459101 the number of square miles upon the earth's surface; lastly, 39643 x 4,188790204785 -260909292265 the number of cubic miles contained in the earth. LONG estimated the proportion of the land and water upon the surface of the carth, so far as discoveries had then been made, in the following manner. He took the paper off a terrestrial globe, and then cut out the land from the sea, and weighed the two parts; by this means he found the proportion of the water to the land as 349: 124. The conclusion would be more accurate, if the land were cut out from the sea before the paper was put upon the globe. After all the modern discoveries, this method would probably give the proportion of land to water to a considerable degree of accuracy. ON THE PRECESSION OF THE EQUINOXES, AND THE NUTATION OF THE Art. 1014. IT has already been observed (148), that the equinoctial points have à retrograde motion of about 501" in a year. Sir I. NEWTON was the first who accounted for this motion. Having proved that, from the centrifugal force of the parts of the earth arising from its rotation, the equatorial diameter must be greater than the polar, he proceeded to show, that if we conceive a sphere to be inscribed in the earth, the attraction of the sun and moon upon the excess of the quantity of matter in the earth above that of the sphere will cause a motion in the plane of the equator, and make the points where it intersects the ecliptic go backwards upon it. But although he assigned the true cause of the precession, it is acknowledged that he fell into an error in his investigation of the effect. Without, however, any inquiry relative to the circumstances in which he erred, we shall show how to obtain a true solution from the common principles of motion. 1 1015. Let S be the sun, ABDC the earth, T its center, EQ the equator, P, p the poles; draw CTB perpendicular to SAD, and join SE, which produce to meet CB in K. Call the radius TE unity, and let the force of the sun on a particle at T be then the force on a particle at E = ; hence, if we resolve this latter force into two others, one in the direction ET and the other in a direction parallel to TS, we have SE: ST::- : the force in the ST ST 1 1 SE SE2 1 SEK + omitting the other Hence, the force with direction parallel to TS = which a particle at E is drawn from CB= 3EK ST3 consequently the effect of this force in a direction perpendicular to ET will be ; hence, this FIG. 224. KT: ST. Now if P= the periodic time of the earth, p=the periodic time of a body revolving at the earth's surface, then (858) the force of the earth to the sun force of the body to the earth, or the force of gravity,:: VOL. II. ST 1 FIG. 225. the force on a particle at E perpendicular to ET: force of gravity :: ŠEK×KT× p2: 1. 1016. Let v be the center of gyration, and put M= the quantity of matter in the earth; then the effect of the inertia of M placed at v to oppose the communication of motion is the same as the effect of the inertia of the earth; and hence, TE: T2 (= } TE') :: M: M, which is the quantity of matter to be } placed at E to have the same effect. 1017. Let PEPQ represent the earth, P, p its poles, EQ the equator, Pepq a sphere whose diameter is the axis PTP, TA a radius directed to the sun, and CTB a plane perpendicular to it, PXp a great circle perpendicular to PEAPQ, and let IR represent a small circle parallel to the equator; take the arc XL= Xl, and draw LM, lm, XY perpendicular to the plane CTB. Now (847) the disturbing force of the sun at L, I, in the directions ML, ml are as ML, ml, consequently these forces are the same as they would be if the corpuscles at L, 1 were orthographically projected upon the plane CABD; let us therefore conceive the whole matter in the earth to be thus projected. Draw XN, In parallel to BC; put p=3,14159, &c. a=the mean radius of the earth, Ee=m, r=IX on the projection, TX=v, s=sin. ATE, c=cos. ATE, the arc XI, or Xl=z, and y = sin. XL; then (in the projection) In=NX=sy, Xn=NL =cy, XY so and TY=co; therefore LM sv+cy, lm=sv-cy, TM = cv -sy, and Tm=co+sy. Now the forces at L and in the directions ML, ml, being as ML, ml, their effects to turn the earth about T'in the direction BAC are as MLX MT and mlxmT, or the whole effects are as su+cy x cv-sy+sv-cy × cv + sy=2cs xv-y; therefore the fluxion of the force of all the matter in the circumference IR is as 2cs xv-y'; hence, the fluxion of the force to ry'y turn the earth in the direction CAB is as 2cs xy-v2 = 2cs.x - 2csv', √r2—y2 whose fluent, when y=r, is pr× cs × r* - 2ʊ the force upon the semicircumference IR; hence, the force on that whole circumference = prx cs × r2 — 2v2 =(as r2 = a2—v2) pr × cs x a3-30. Now a ; r::m: Ii= ; hence, the flux × ion of the force of the annulus IieE is as pr× = mr mr a pmcs × csv × a2 — 3v2 4pmcs xat, which is as the whole force on the matter exterior to the sphere 15 Pepq, on one side of EQ; hence, 8pmcs x at is as the whole force of the sun upon the earth to turn it about in the direction CAB. Now a+m=TE, a÷\m= Te; hence, the solid content of the spheroid 8 2 p× a-m × a + § m2 =† pœ + § 8 8 pma' very nearly; and the content of the sphere p × a— m3 = pa3 = {} 1 $ 2pma very nearly; the difference of these is pma the content of the part exterior to the sphere; place one fifth of this matter, that is, pma, at E; then as EK=ca, and KT-sa, the effect of the sun to turn the matter is pma2 at E about T = 1 pmcsa, which is equal to the effect of the sun upon the whole earth to turn it about its center. Hence, the effect of the sun upon the matter of the earth exterior to the sphere to turn it about its center, is equal to the effect which would be produced if one fifth part of that matter were placed at E. 1018. Put q=the quantity of matter in the earth above that of its inscribed sphere; now (1017) the attraction upon the matter exterior to the sphere would generate an angular velocity about an axis perpendicular to CABD, equal to the angular velocity which would be generated in a quantity of matter = q placed at E. Let us therefore suppose the sun's attraction perpendicular to ET to be exerted upon a quantity of matter at E-to q, and at the same. time to have a quantity of matter to move = M, and then (1016, 1017) it appears, that the effect will be the same as the accelerative force of the sun to turn about the earth. Hence, that accelerative force is (1015) equal to 3EK × KT × p2 × 3q _ 3 EK × KT × p2 × q 3EK × KT × p3×9. Now if TE: TP::1:1—r, 2Mx P = M× P* quently the accelerative force = the earth being unity. then =r; conse 2M 3EK × KT x pxr, the force of gravity on p2 X 1019. Let the arc described by a point of the equator about its axis in an indefinitely small given time, which may therefore represent its velocity; and let aż represent the arc described in the same time by a body revolving about the earth at its surface; body in the same time, and consequently a the velocity generated by gravity whilst a point of the equator describes . 3 EK × KT × p2 × ˆ:: q31⁄23 r then a222 = the sagitta of the arc described by the a22 point E perpendicular to ET, generated by the action of the sun whilst the equator describes about its axis; consequently the ratio of these velocities is зEK × KT × p2 × r × a3 ż P FIG. 226. 1020. Let y be an arc described by the sun in the ecliptic to a radius equal to unity, whilst a point of the equator describes about its axis, then (as ap the time of the earth's rotation, and the arcs described in equal times to equal radii are inversely as the periodic times,) 1 1 P: ap :j:= Py, hence, ap if v and w be put for the sine and cosine of the sun's declination, the ratio of the velocities in the last Article becomes 3aprvwy : 1. P: P 10214 Hence, if SAL be the ecliptic to the radius unity, P the place of the sun, SBL the equator, PE the sun's declination, and we take Ec: dc (dc being perpendicular to Ec) :: 1: Saprowy, and through d, E, describe the great circle TEM, then will ST be the precession of the equinox, during the time the sun describes y in the ecliptic (861); hence, Ed or Ec, or 1, dc, or saprow x sin. SEx, therefore the sin. STV, or Saprowy P :: sin. SE: SV= Р 1022. Now sin. SP, and w= cos. ESP cos. SP ; hence, sin. ESP cos. ES cos. ES x tan. ES × cot. SP X sin. ES consequently ST= 3apr × sin. SP3 × cos. ESP xy=(if x = sine of SP) Sapr x cos. ESP × xaà (y being now=m a quadrant) the arc of precession whilst the sun describes 90° from the equinox; and to find the degrees, say 4m: 360° 3apr x cos. ESP ×m 3apr x cos. ESP 8P sion of the equinox arising from the attraction of the sun, if the earth were solid of an uniform density, and the ratio of the diameters as 229: 230; but, from what follows, if the greatest nutation of the earth's axis be rightly ascertained, the precession is only about 144"; which difference between the theory and what is deduced from observation, must arise, either from the fluidity of the earth's surface, an increase of density towards the center, or the ratio of |