timation to from the Obfervations of Sturmy concludes that the greatest Tide follows next after the Appulfe of the Moon to the Meridian when the Moon is diftant from the Sun about 184. the Sun's Force in this Distance of the Moon from Syfigies being to the Force [S] in the Syfigies, as the Cofine [7986355] of double this Distance, or of an Angle of 37 Degrees, to the Radius [10000000] in the Place of L+S in the preceding Analogy L+o, 7986355 S is to be Subftituted. In the Quadratures the conjoint Forces of the Sun and Moon being leaft, it was alfo fuppofed that the leaft Tide happens at that Time, but the Sea loofes its Motion by the Reduction fame Degrees that it acquired it, fo that the Tides are not at their leaft of this ef Height until fome Time after the Moon has paffed the Quadratures, and the mean Newton from the fame Obfervations of Sturmy concluded that the leaft ftate of the Tide follows next after the Appulfe of the Moon to the Meridian when variable cir the Moon is diftant from the Quadratures 184. Now the Sun's Force in this Distance of the Moon from the Quadratures being to the Force [S] in the Quadratures, as the Cofine (7986355) of double this Distance or of an Angle of 37 Degrees, to Radius (10000000) in the Place of L-S in the preceding Analogy, L-0, 7986355S is to be Substituted. It is to be observed 20 that the Orbit of the Moon was fuppofed to Coinfide with the Plane of the Equator, but the Moon in the Quadratures, or rather 180 paft the Quadratures, declines from the Equator by about 22d 13m, now the Force of the Moon in this diftance from the Equator being to its Force (L) in the Equator, as the Square of the Cofine (8570327) of its Declination 22d 13m, to Radius (10000000) in the Place of Lo, 7986355S in the preceding Analogy 0,8570327L0,7986355S is to be Substituted. It is to be obferved 30 that the Orbits of the Sun and Moon were fuppofed to be perfectly Circular, and confequently thofe Luminaries to be in their mean Diftances from the Earth. But Newton demonftrated that the lunar Orbit (abstracting from its Excentricity) ought to be an Elliptic Figure, having its Centre in the Centre of the Earth and the fhortest Diameter directed to the Sun; and determined (Prop. 28. B. 3.) the Ratio of this shortest Diameter to the longeft or the Distance of the Moon from the Earth in the Syfigies and Quadratures to be as 69 to 70. To find its Distance when 18 Degrees advanced beyond the Syfigies, and when 18 Degrees pafsed by the Quadratures, it is to be obferved that in an Ellipfis if the longest Semidiameter be expreffed by (a) its fhorteft by [b] and the Difference of the Squares of the longest and fhorteft Semidiameters by [cc] and the Sine of the Angle which any Diameter [y] makes with the longest Semidi: ameter by [s] yy wherefore fubftituting fucceffively in this Expreffion 69 for [a] 70 for [b] for [s] 3173047 and 9483236 the Sines of 18 Degrees and 71 Degrees : thofe Distances will be 69,098747 and 69,897345 and the mean Distance will be 69 as equal to half the Sum = aabb aa sscc cumstances. of the moon of the the longest and shortest femidiameters. 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 leaft of thofe Distances are to its Force in its mean Distance, as 0,9830427 and 1,017522 to 1. confequently The force in the preceding Analogy, in the Place of L+ 0, 7986355S, we must put is to that of 1,017522L+0, 7986355 S, and in the Place of 0,8570327 Lthe fun as 0,7986355S we must put 0,9830427X0,8570327 L-0,7986355S; from whence we have 1,017522L+0,79863555, to 0,9830427X0,8570327 L The force -0,79863558 as 9 to 5, confequently 1,017522 LX5 + 0,7986355S X5 of the fun =0,9830427X9X0,857032 L-0,7986355SX9, and by tranfpofition, S is mited raile, to L, as 0,9830427X0,8570327X9 0,17522X5 to 0,7986355×5+ the waters 0,7986355×9, that is, $ is to L as I to 4,4815 nearly. 4, 5 to 1. & moon u of the fea to the height of 10 feet 12 feet when the 30 XII, Newton having thus determined the Force of the Moon to raise the and even to Waters of the Sea, affigns the Quantity of this Elevation. The Force (1) of the Sun being to the Force (4,4815) of the Moon, as the Elevation (1 Foot moon is pe- II Inches) arifing from the Action of the Sun, to the Elevation (8 Feet 7 Inches) arifing from the Action of the Moon. So that the Sun and Moon together may produce an Elevation of about 10 Feet in their mean Distances from the Earth, and an Elevation of about 12 Feet when the Moon is nearest the Earth. rigee. How New The Influence of the Moon on the Tides has enabled Newton to Estimate ton invefti- her Denfity, her Quantity of Matter, and what Bodies weigh on her Surgated the face, compared with the Denfity and Quantity of Matter of the Earth, and denfity and the Weights of Bodies on its Surface. For fince the Force (v) of the Moon of matter of to move the Sea is to the like Force (V) of the Sun as 4, 4815 to 1,and v is to the moon & Vas abfolute Force of the Moon divided by the Cube of its Distance from what bodies g quantity of weigh on G her furface the Earth to abfolute Force of the Sun divided by the Cube of its Dif compaired with the density and quantity of and the tance from the Earth (Cor. 14 Prop. 66); 4, 4815 is to I as matter of the abfolute Force (g) of the Moon is to the abfolute Force (G) of the Sun, the earth, as the Denfity of the Moon and Cube of its Diameter conjointly (dxq') to the Density of the Sun and Cube of its Diameter conjointly (DXp'), and the apparent Diameter (31m. 16) of the Moon being to the apparent Diameter (32m. 125) of the Sun as to is to P1 weight of bodies on its furface. wherefore by the Composition of Ratios is to G R as dx,141583 to DX,154508, confequently 4, 4815 is to 1 as d X,141583 to DX,154508 that is, as the Denfities of the Moon and Sun and the Cubes of their apparent Diameters conjunctly, from whence it follows that the Denfity (d) of the Moon is to the Denfity (D) of the Sun, as- 4, 4815 to 141583 154508 or as 4891 to 1000, but the Density (D) of the Sun is to the Denfity (c) of the Earth, as 1000 to 4000, confequently Dxd is to DXc, or the Denfity (d) of the Moon is to the Denfity (c) of the Earth as 4891X1000 to 4000X1000 Deafity of or as II to 9, therefore the Body of the Moon is more Denfe and more the moon. Earthly than the Earth its self. And fince the true Diameter of the Moon [from the Obfervations of the Aftronomers] is to the true Diameter of the Earth as 100 to 365, the Quan- matter in tity of Matter in the Earth, is to the Quantity of Matter in the Moon as the moon. 1000000 X 11 to 48627125 X9, that is, as 1 to 39, 788. And fince the accelerative Gravity on the Surface of the Moon is to the accelerative Gravity on the Surface of the Earth as the Quantity of Weight of Matter in the Moon to the Quantity of Matter in the Earth, directly, and bodies on its as the Square of the Distances from the Center inversely, they will be surface. to each other as I X 13324 to 39,788X1000 that is as 1 to 3 nearly: consequently the accelerative Gravity on the Surface of the Moon will be about three Times lefs than the accelerative Gravity on the Surface of the Earth. XIII on & why. Daniel Bernoully, in his Piece on the Tides which carried the Prize of the Academy of Sciences in the Year 1738, obferves that the Method is of a difBernoully of eftimating the Proportion of the Force of the Sun to that of the Moon by ferent opinithe greatest and leaft Heights of the Tides as employ'd by Newton is very uncertain; because in the Ports of England and France the Tides are not immediately produced by the Actions of the two Luminaries, but are rather a Confequence of the great Tides of the Ocean, as the Tides of the Adriatic Seal are a Confequence of the Small Tides of the Mediterranean, and that the primitive Tides may differ very fenfibly in every Respect from the fecondary Tides which is confirmed by Obfervation; the Proportion of the Spring and Neap Tides being found to be very different in the different Ports. At St. Malo's, for Example, the greatest and leaft Height of the Waters are to one another as 10 to 3, and below Bristol according to Sturmy they are to each other as 9 to 5. He obferves further that the Motion of Rotation of the Earth being very rapid with Respect to the Motion of the Sun and Moon; The Sea cannot every Inftant affume its Figure of Equilibrium without any fenfible Motion, hence the Waters which were raised by the combined Actions of the Luminaries tending on one Hand to conferve as much as poffible by their Force. of inertia the Elevation they had acquired, and on the other tending as they recede from the Moon to loose a Part of their Elevation, they will be lefs Elevated than they would be if the Earth was at Reft, and confequently the Neap Tides are greater and the Spring Tides lefs than what refults from a Computation founded on the Laws of Equilibrium, wherefore the great Spring Tides and Neap Tides are in a greater Ratio according to the Laws of Equilibrium than that of 9 to 5. Bernoully fuppofes them to be to each other as 7 to 3, confequently that Force of the Force (L) of the Moon is to the Force (5) of the Sun as 5 to 2. A proaccording to portion which anfwers better to the Obferved Variations in the duratiBernoully on and interval of the Tides (Variations which receive no Alteration from the moon Singular moon. the above mentioned fecondary Caufes) and to the other Theories which depend on a Determination of the Force of the Moon. Hence the Density of the Moon is to the Denfity of the Earth as 7 to 10, the Quantity of Matter in the Moon is to the Quantity of Matter in the Earth as 1 to 70, and finally the accelerative Gravity at the Surface of the Moon is to the accelerative Gravity on the Surface of the Earth as 1 to 5. XIV. If the Moon's Body were Fluid like our Sea it would be elevated by the figure of the Action of the Earth in the Parts which are nearest to it and in the Parts oppofite to these, and Newton enquires into the Quantity of this Elevation. He obferves that the Elevation (8) of the Earth produced by the Action of the Moon would be to the Elevation (E) of the Moon (if it had the fame Diameter as the Earth) produced by the Action of the Earth as the Quantity of Matter in the Moon to the Quantity of Matter in the Earth, or as I to 39,788. and the Elevation (E) produced by the Action of the Earth in the Moon if it had the fame Diameter as the Earth, is to the real Elevation (x) produced in the Moon by the Action of the Earth, as the Diameter of the Earth to the Diameter of the Moon or as 365 to 100. wherefore by the Compofition of Ratios 8XE is to Exx or the Elevation of the Earth (83) produced by the Action of the Moon is to the real Elevation of the Moon produced by the Action of the Earth as 1 × 365 to 39,788 X 100 or as 1081 to 100 or x=93 Feet. confequently the Diameter of the Moon that paffes through the Centre of the Earth, must exceed the Diameter which is perpendicular to it by 186 Feet. Hence it is, that the Moon always turns the fame Side towards the Earth. Effect of the oblong figure of the fpheroidal moon. In Effect La Grange in his Piece which carried the Prize of the royal Academy of Sciences in the Year 1764, fuppofing with Newton that the Moon is a Spheroid having its longeft Diameter directed towards the Earth, has found that this Planet fhould have a libratory or ofcillatory Motion about its Axis, whereby its Velocity of Rotation is fometimes accelerated and other Times retarded, and that then the Moon fhould always turn the fame side nearly towards the Earth, though it did not receive in the Beginning a Motion of Rotation whofe Duration was equal to that of its Revolution. La Grange has demonftrated also that the Figure of the Moon might be fuch that the Preceflion of its equinoctial Points or the Retrogradation of the Nodes of the lunar Equator, would be equal to the retrograde Motion of the Nodes of the lunar Orbit; and in this Cafe he found that the lunar Axis would have no fenfible Nutation. The Action of the Sun in all thofe Inquiries, is almoft infenfible with refpect to that of the Earth; it is the Earth which produces the Motion of the Nodes of the lunar Equator, by acting more or lefs obliquely on the lunar Spheroid; hence the Preceffion of the lunar Equator, and the Law of the Motion produced in the lunar Spheroid, differ very much from that which is obferved in the Equator of the Earth. The Researches of this eminent Mathematician of Turin, fhall be explained hereafter. XV. Newton having fhewn that the Tides proceed from the combined Actions of the Sun and Moon, and determined the Quantity that each of those Luminaries contribute to their Production, enters into an Explanation of the Circumstances which attend the Phenomena of the Tides. ons have There has been observed in all Times, three Kinds of Motions in the Three kinds Sea, its diurnal Motion, whereby it ebbs and flows twice a Day, the of variati regular Alterations which this Motion receives every Month, and which been obfollow the Pofition of the Moon with refpect to the Sun, and those served in which arrive every Year and which depend on the Pofition of the Earth the motion with refpect to the Sun. of the fea. To deduce those Motions from their Caufe, we are to obferve that Diurnal the Sea yielding to the Force of the Sun and Moon impreffed on it in variations. Proportion to their Quantity, acquires its greatest Height by a Force compounded of those two Forces; hence this greatest Height (even abftracting from the Force of Inertia of the Waters) should not be immediately under the Moon, for immediately under the Sun, but in an intermediate Point, which correfponds more exactly to the Motion of the Moon than to that of the Sun, because the Force of the Moon on the Sea is greater than that of the Sun. To determine the Position of this Point, it is manifeft that at High-Water in any Place, ssS-4-ttL is a Maximum, and at Low Water a Minimum or Ssds+Ltdt=o. But the inftantaneous Increment (ds) of the Sine of the Altitude of the Sun, is to the correfponding Increment (dz) of the Sun's diurnal Arc, as the Cofine (VI-ss) of the Altitude of the Sun to Radius (1), or ds= VI-Xdz and the correfponding Decrement (dt) of the Sine of the Moon's Altitude, is to the correfponding Increment (dx) of the Moon's diurnal Arc, as the Cofine (VI-tt) of its Altitude to Radius (1), or -dt dxXVI-tt=32dzXv1-tt, dx being to dz as 29 to 30, on account of the Motion of the Moon. Subftituting thofe Values of ds and dt in the Expreffion Ssds+Ltdt-o, we will have Ssy1-ss-30 XL from whence it appears that at the Time 29 L Vitt 30 S |