=

ray of light coming from the sun, and beginning to be refracted at m; let it describe the curve mBn touching the earth at B, and at n let it be reflected into the curve nA, touching the earth at A, the place of the spectator; in this position therefore of the sun, the twilight just appears; draw the tangents Avz, mz, Bʊ, and join vnC. Then AO (the radius of curvature to the arc An) =7AC (193), considering An as a circle, from which it will differ but very little. Now suppose twilight to begin when the sun is 18° below the horizon, that being about the quantity found by computing the sun's depression from the observed time at which the twilight begins; it varies however in different seasons; hence, the angle 2162°; but the difference between the angle and the angle BoA is the refraction through mB, or 33'; therefore the angle AuB== 162°. 33', and AvC-81°. 16, consequently ACv=8°. 43, and hence nCO = 171°. 16; also, On: Oc:: 7 : 6; hence, On=7: Oc=6:: sin. nCO 171°. 16: sin. CnO=7°. 281, therefore nOC=1°. 15; hence, sin. nCO: sin. nOC:: On=7: Cn=1,01, from which take Ca=1, and we have nx=0,01= 39,64 miles. But (189) the ray begins to be refracted at the altitude of 77,25 miles; hence the reflection takes place at about half the altitude at which the refraction begins. This is upon supposition that the rays come to the spectator after one reflection. If we suppose them to come after 2, 3 or 4 reflections, the altitudes na will be about 12, 5,4 and 3 miles respectively, and the densities of the air 10,75, 2,9 and 1,8 less than at the earth's surface. Which of these is most probable, may admit of some doubt. That air at the altitude of 39,64 miles, where it is 2700 less dense than at the earth's surface, should have the power of reflecting rays so copiously, is almost incredible. And why should that particular density reflect, when it is not the boundary of the atmosphere, it having been shown that light is refracted at twice that altitude? It appears more probable that the reflection arises from the vapours and exhalations of various kinds with which the lower parts of the atmosphere are charged; for the twilight lasts till the sun is further below the horizon in the evening, than it is in the morning when it begins; and it is longer in summer than in winter. Now in the former case, the heat of the day has raised the vapours and exhalations; and in the latter, they will be more elevated from the heat of the season; therefore, upon supposition that the reflection is made by them, the twilight ought to be longer in the evening than in the morning, and longer in summer than in winter.

207. Another effect of refraction is that of giving the sun and moon an oval appearance, by the refraction of the lower limb being greater than that of the upper, whereby the vertical diameter is diminished. For suppose the diameter of the sun to be 32', and the lower limb to touch the horizon, then the mean refraction at that limb is 33', but the altitude of the upper limb being then 32', its refraction is only 28. 6", the difference of which is 4'. 54", the quantity by

which the vertical diameter appears shorter than that parallel to the horizon. When the body is not very near the horizon, the refraction diminishing nearly uniformly, the figure of the body is very nearly that of an ellipse. Now it is proved in that article where the diminution of weight of a body upon the surface of a spheroid is investigated, that the diameter (D) of an ellipse, which is nearly a circle, is diminished, in going from the major to the minor axis, as the square of the sine (s) of the angle which it makes with the major axis; hence, if d the diminution of the vertical diameter, rad.: s::d: the diminution of the diameter D. Thus we may find the diameter in any position; and in cases where extreme accuracy is required, such as measuring with a micrometer the distance of Venus or Mercury on the sun's disc from its limb, this circumstance may be considered.

CHAP. VIII.

ON THE SYSTEM OF THE WORLD.

Art. 208. WHEN any effect or phænomenon is discovered by experiment or observation, it is the business of Philosophy to investigate its cause. But there are very few, if any, enquiries of this kind, where we can be led from the effect to the cause by a train of mathematical reasoning, so as to pronounce with certainty upon the cause. Sir I. NEWTON therefore, in his PRINCIPIA, before he treats on the System of the World, has laid down the following Rules to direct us in our researches into the constitution of the universe.

RULE I. No more causes are to be admitted than what are sufficient to explain the phænomenon.

RULE II. Of effects of the same kind, the same causes are to be assigned, as far as it can be done.

RULE III. Those qualities which are found in all bodies upon which experiments can be made, and which can neither be increased nor diminished, may be looked upon as belonging to all bodies.

RULE IV. In Experimental Philosophy, propositions collected from phænomena by induction, are to be admitted as accurately or nearly true, until some reason appears to the contrary.

The principles, upon which the application of these Rules is admitted, are, the supposition that the operations of nature are performed in the most simple manner, and regulated by general laws. And although their application may, in many cases, be very unsatisfactory, yet in the instances to which we shall here want to apply them, their force is little inferior to that of direct demonstration, and the mind rests equally satisfied as if the matter could be strictly proved. 209. The diurnal motion of all the heavenly bodies may be accounted for, either by supposing the earth to be at rest, and all the bodies daily to perform their revolutions in circles parallel to each other; or by supposing the earth to revolve about one of its diameters as an axis, and the bodies themselves to be fixed, in which case their apparent diurnal motions would be the same. If we suppose the earth to be at rest, all the fixed stars must make a complete revolution, in parallel circles, every day. But it will be shown in a future part of this Work, that the nearest of the fixed stars cannot be less than 400000 times further from us than the sun is, and that the sun's distance from the earth is not less than 93 millions of miles. Also from the discoveries which are every day

making by the improvement of telescopes, it appears that the heavens are filled with an almost infinite number of stars, to which the number visible to the naked eye bears no proportion, and whose distances are, probably, incomparably greater than what we have stated above. But that an almost infinite number of bodies, most of them invisible except by the best telescopes, at almost infinite distances from us and from each other, should have their motions so exactly adjusted, as to revolve in the same time, and in parallel circles, and all this without their having any central body, which is a physical impossibility, is an hypothesis, which, by the Rules we have here laid down, is not to be admitted, when we consider, that all the phænomena may be solved simply by the rotation of the earth about one of its diameters. If therefore we had no other reason, we might rest satisfied that the apparent diurnal motions of the heavenly bodies are produced by the earth's rotation. But we have other reasons for this supposition. Experiments prove that all the parts of the earth have a gravitation towards each other. Such a body therefore, the greatest part of whose surface is a fluid, must, from the equal gravitation of its parts, form itself into a perfect sphere. But it appears from mensuration, that the earth is not a perfect sphere, but a spheroid, having the equatorial longer than its polar diameter. Now if we suppose the earth to revolve, the parts most distant from the axis must, from their greater velocity, have a greater tendency to fly off, and therefore that diameter which is perpendicular to the axis must be increased. That this must be the consequence appears from taking an iron hoop and making it revolve swiftly about one of its diameters, and that diameter will be diminished and the diameter perpendicular to it increased. The figure of the earth must therefore have arisen from its rotation, which is further confirmed from the following consideration. There can be but one diameter about which the earth can revolve, which can solve all the phænomena of the apparent revolution of the heavenly bodies; for if the diameter about which the earth is supposed to revolve were changed, it would change the situation of all the bodies in respect to the horizon and zenith; now that diameter about which the earth must revolve, in order to satisfy all the phænomena, is the diameter which, from mensuration, is found to be the shortest. Another reason for the earth's rotation is from analogy. The planets are opaque and spherical bodies like to our earth; now all the planets, on which sufficient observations have been made to determine the matter, are found to revolve about an axis, and the equatorial diameters of some of them are visibly greater than their polar. When these reasons, all upon different principles, are considered, they amount to a proof of the earth's rotation about its axis, which is as satisfactory to the mind as the most direct demonstration could be. These however are not all the proofs which might be offered; the situations and motions of the bodies in our system necessarily require this motion of the earth.

210. Besides this apparent diurnal motion, the sun, moon, and planets have another motion; for they are observed to make a complete revolution amongst the fixed stars, in different periods. But whilst they are performing these motions in respect to the fixed stars, they do not always appear to move in the same direction, or in that direction in which their complete revolutions are made, but sometimes appear stationary, and sometimes to move in a contrary direction. We will here briefly describe and consider the different systems which have been invented, in order to solve these appearances. PTOLEMY supposed the earth to be perfectly at rest, and all the other bodies, that is, the sun, moon, planets, comets and fixed stars, to revolve about it every day; but that, besides this diurnal motion, the sun, moon, planets and comets had a motion in respect to the fixed stars, and were situated, in respect to the earth, in the following order; the Moon, Mercury, Venus, the Sun, Mars, Jupiter, Saturn. These revolutions he first supposed to be made in circles about the earth placed a little out of the center, in order to account for some irregularities of their motions; but as their retrograde motions and stationary appearances could not thus be solved, he supposed them to revolve in epicycloids, in the following manner. Let ABC be a circle, S the center, E the earth, abcd another circle whose center v is in the circumference of the circle ABC. Conceive the circumference of the circle ABC to be carried round the earth every 24 hours according to the order of the letters, and at the same time let the center of the circle abcd have a slow motion in the opposite direction, and let a body revolve in this circle in the direction abcd; then it is manifest, that by the mo. tion of the body in this circle and the motion of the circle itself, the body may describe such a curve as is represented by klmnop; and if we draw the tangents El, Em, the body would appear stationary at the points 7 and m, and its motion would be retrograde through Im, and then direct again. Now to make Venus and Mercury always accompany the Sun, the center v of the circle abcd was supposed to be always very nearly in a right line between the earth and sun, but more nearly so for Venus than for Mercury, in order to give each its proper elongation. This system, although it will account for all the apparent motions of the bodies, yet it will not solve the phases of Venus and Mercury; for in this case, in both conjunctions with the sun they ought to appear dark bodies, and to lose their light both ways from their greatest elongations; whereas it appears from observation, that in one of their conjunctions they shine with a full face. This system therefore cannot be true.

211. The system received by the Egyptians was this: The Earth is immoveable in the center, about which revolve, in order, the Moon, Sun, Mars, Jupiter and Saturn; and about the Sun revolve Mercury and Venus. This dispo sition will account for the phases of Mercury and Venus, but not for the apparent motions of Mars, Jupiter and Saturn.

FIG.

40.