Note 68, p. 17. Sidereal revolution. The consecutive return of an object to the same star.

Note 69, p. 17. Tropical revolution. The consecutive return of an object to the same tropic or equinox.

Note 70, p. 17. The orbit only bulges, &c. In fig. 18 the effect of the variation in the excentricity is shown where P p A is the elliptical orbit at any given instant; after a time it will take the form P pʹ A, in consequence of the decrease in the excentricity C S; then the forms P pʺ A, P pʹʹʹ A, &c., consecutively from the same cause; and, as the major axis P A always retains the same length, the orbit approaches more and more nearly to the circular form. But, after this has gone on for some thousands of years, the orbit contracts again, and becomes more and more elliptical.

Note 71, pp. 18, 19. The ecliptic is the apparent path of the sun in the heavens. See note 46.

Note 72, p. 18. This force tends to pull, &c. The force in question, acting in the direction p m, fig. 13, pulls the planet p towards the plane N m n, or pushes it farther above it, giving the planet a tendency to move in an orbit above or below its undisturbed orbit N p n, which alters the angle p N m, and makes the node N and the line of nodes N n change their positions.

Note 73, p. 18. Motion of the nodes. Let S, fig. 19, be the sun; S N n the plane of the ecliptic; P the disturbing body; and p a planet moving in its orbit p n, of which p n is so small a part that it is represented as a straight line. The plane S n p of this orbit cuts the plane of the ecliptic in the straight line S n. Suppose the disturbing force begins to act on p, so as to draw the planet into the arc p pʹ; then, instead of moving in the orbit p n, it will tend to move in the orbit p pʹ nʹ, whose plane cuts the ecliptic in the straight line S nʹ. If the disturbing force acts again upon the body when at pʹ, so as to draw it into the arc pʹ pʺ, the planet will now tend to move in the orbit pʹ pʺ nʺ, whose plane cuts the ecliptic in the straight line S nʺ. The action of the disturbing force on the planet when at pʺ will bring the node to nʹʹʹ, and so on. In this manner the node goes backwards through the successive points n, nʹ, nʺ, nʹʹʹ, &c., and the line of nodes S n has a perpetual retrograde motion about S, the centre of the sun. The disturbing force has been represented as acting at intervals for the sake of illustration: in nature it is continuous, so that the motion of the node is continuous also; though it is sometimes rapid and sometimes slow, now retrograde and now direct; but, on the whole, the motion is slowly retrograde.

Note 74, p. 18. When the disturbing planet is anywhere in the line S N, fig. 19, or in its prolongation, it is in the same plane with the disturbed planet; and, however much it may affect its motions in that plane, it can have no tendency to draw it out of it. But when the disturbing planet is in P, at right angles to the line S N, and not in the plane of the orbit, it has a powerful effect on the motion of the nodes: between these two positions there is great variety of action.

Note 75, p. 19. The changes in the inclination are extremely minute when compared with the motion of the node, as evidently appears from fig. 19, where the angles n p nʹ, nʹ pʹ nʺ, &c., are much smaller than the corresponding angles n S nʹ, S nʺ, &c.

Note 76, p. 20. Sines and cosines. Figure 4 is a circle; n p is the sine, and C p is the cosine of an arc m n. Suppose the radius C m to begin to revolve at m, in the direction m n a; then at the point m the sine is zero, and the cosine is equal to the radius C m. As the line C m revolves and takes the successive positions C n, C a, C b, &c., the sines n p, a q, b r, &c., of the arcs m n, m a, m h, &c., increase, while the corresponding cosines C p, C q, C r, &c., decrease; and when the revolving radius takes the position C d, at right angles to the diameter g m, the sine becomes equal to the radius C d, and the cosine is zero. After passing the point d, the contrary happens; for the sines e K, l V, &c., diminish, and the cosines C K, C V, &c., go on increasing, till at g the sine is zero, and the cosine is equal to the radius C g. The same alternation takes place through the remaining parts g h, h m, of the circle, so that a sine or cosine never can exceed the radius. As the rotation of the earth is invariable, each point of its surface passes through a complete circle, or 360 degrees, in twenty-four hours, at a rate of 15 degrees in an hour. Time, therefore, becomes a measure of angular motion, and vice versâ, the arcs of a circle a measure of time, since these two quantities vary simultaneously and equably; and, as the sines and cosines of the arcs are expressed in terms of the time, they vary with it. Therefore, however long the time may be, and how often soever the radius may revolve round the circle, the sines and cosines never can exceed the radius; and, as the radius is assumed to be equal to unity, their values oscillate between unity and zero.

Note 77, p. 20. The small excentricities and inclinations of the planetary orbits, and the revolutions of all the bodies in the same direction, were proved by Euler, La Grange, and La Place, to be conditions necessary for the stability of the solar system. Subsequently, however, the periodicity of the terms of the series expressing the perturbations was supposed to be sufficient alone, but M. Poisson has shown that to be a mistake; that these three conditions are requisite for the necessary convergence of the series, and that therefore the stability of the system depends on them conjointly with the periodicity of the sines and cosines of each term. The author is aware that this note can only be intelligible to the analyst, but she is desirous of correcting an error, and the more so as the conditions of stability afford one of the most striking instances of design in the original construction of our system, and of the foresight and supreme wisdom of the Divine Architect.

Note 78, p. 22. Resisting medium. A fluid which resists the motions of bodies, such as atmospheric air, or the highly elastic fluid called ether, with which space is filled.

Note 79, p. 23. Obliquity of the ecliptic. The angle e ♈ q, fig. 11, between the plane of the terrestrial equator q ♈ Q, and the plane of the ecliptic E ♈ e. The obliquity is variable.

Note 80, p. 23. Invariable plane. In the earth the equator is the invariable plane which nearly maintains a parallel position with regard to itself while revolving about the sun, as in fig. 20, where E Q represents it. The two hemispheres balance one another on each side of this plane, and would still do so if all the particles of which they consist were moveable among themselves, provided the earth were not disturbed by the action of the sun and moon, which alters the parallelism of the equator by the small variation called nutation, to be explained hereafter.

Note 81, p. 24. If each particle, &c. Let P, Pʹ, Pʺ, &c., fig. 21, be planets moving in their orbits about the centre of gravity of the system. Let P S M, Pʹ S Mʹ, &c., be portions of these orbits moved over by the radii vectores S P, S Pʹ, &c., in a given time, and let p S m, pʹ S mʹ, &c., be their shadows or projections on the invariable plane. Then, if the numbers which represent the masses of the planets P, Pʹ, &c., be respectively multiplied by the numbers representing the areas or spaces p S m, pʹ S mʹ, &c., the sum of the whole will be greater for the invariable plane than it would be for any plane that could pass through S, the centre of gravity of the system.

Note 82, p. 24. The centre of gravity of the solar system lies within the body of the sun, because his mass is much greater than the masses of all the planets and satellites added together.

Note 83, pp. 25, 36. Conjunction. A planet is said to be in conjunction when it has the same longitude with the sun, and in opposition when its longitude differs from that of the sun by 180 degrees. Thus two bodies are said to be in conjunction when they are seen exactly in the same part of the heavens, and in opposition when diametrically opposite to one another. Mercury and Venus, which are nearer to the sun than the earth, are called inferior planets; while all the others, being farther from the sun than the earth, are said to be superior planets. Suppose the earth to be at E, fig. 24; then a superior planet will be in conjunction with the sun at C, and in opposition to him when at O. Again, suppose the earth to be in O, then an inferior planet will be in conjunction when at E, and in opposition when at F.

Note 84, p. 26. The periodic inequalities are computed for a given time; and consequently for a given form and position of the orbits of the disturbed and disturbing bodies. Although the elements of the orbits vary so slowly that no sensible effect is produced on inequalities of a short period, yet, in the course of time, the secular variations of the elements change the forms and relative positions of the orbits so much, that Jupiter and Saturn, which would have come to the same relative positions with regard to the sun and to one another after 850 years, do not arrive at the same relative positions till after 918 years.

Note 85, p. 26. Configuration. The relative position of the planets with regard to one another, to the sun, and to the plane of the ecliptic.

Note 86, p. 27. In the same manner that the excentricity of an elliptical orbit may be increased or diminished by the action of the disturbing forces, so a circular orbit may acquire less or more ellipticity from the same cause. It is thus that the forms of the orbits of the first and second satellites of Jupiter oscillate between circles and ellipses differing very little from circles.

Note 87, p. 28. The plane of Jupiter’s equator is the imaginary plane passing through his centre at right angles to his axis of rotation, and corresponds to the plane q E Q e, in fig. 1. The satellites move very nearly in the plane of Jupiter’s equator; for, if J be Jupiter, fig. 22, P p his axis of rotation, e Q his equatorial diameter, which is 6000 miles longer than P p, and if J O and J E be the planes of his orbit and equator seen edgewise, then the orbits of his four satellites seen edgewise will have the positions J1, J2, J3, J4. These are extremely near to one another, for the angle E J O is only 3° 5ʹ 30ʺ.

Note 88, p. 28. In consequence of the satellites moving so nearly in the plane of Jupiter’s equator, when seen from the earth, they appear to be always very nearly in a straight line, however much they may change their positions with regard to one another and to their primary. For example, on the evenings of the 3rd, 4th, 5th, and 6th of January, 1835, the satellites had the configurations given in fig. 23, where O is Jupiter, and 1, 2, 3, 4, are the first, second, third, and fourth satellites. The satellite is supposed to be moving in a direction from the figure towards the point. On the sixth evening the second satellite was seen on the disc of the planet.

Note 89, p. 28. Angular motion or velocity is the swiftness with which a body revolves—a sling, for example; or the speed with which the surface of the earth performs its daily rotation about its axis.

Note 90, p. 29. Displacement of Jupiter’s orbit. The action of the planets occasions secular variations in the position of Jupiter’s orbit J O, fig. 22, without affecting the plane of his equator J E. Again, the sun and satellites themselves, by attracting the protuberant matter at Jupiter’s equator, change the position of the plane J E without affecting J O. Both of these cause perturbations in the motions of the satellites.

Note 91, p. 29. Precession, with regard to Jupiter, is a retrograde motion of the point where the lines J O, J E, intersect fig. 22.

Note 92, p. 30. Synodic motion of a satellite. Its motion during the interval between two of its consecutive eclipses.

Note 93, p. 30. Opposition. A body is said to be in opposition when its longitude differs from that of the sun by 180°. If S, fig. 24, be the sun, and E the earth, then Jupiter is in opposition when at O, and in conjunction when at C. In these positions the three bodies are in the same straight line.

Note 94, p. 30. Eclipses of the satellites. Let S, fig. 25, be the sun, J Jupiter, and a B b his shadow. Let the earth be moving in its orbit, in the direction E A R T H, and the third satellite in the direction a b m n. When the earth is at E, the satellite, in moving through the arc a b, will vanish at a, and reappear at b, on the same side of Jupiter. If the earth be in R, Jupiter will be in opposition; and then the satellite, in moving through the arc a b, will vanish close to the disc of the planet, and will reappear on the other side of it. But, if the satellite be moving through the arc m n, it will appear to pass over the disc, and eclipse the planet.

Note 95, pp. 30, 43. Meridian. A terrestrial meridian is a line passing round the earth and through both poles. In every part of it noon happens at the same instant. In figures 1 and 3, the lines N Q S and N G S are meridians, C being the centre of the earth, and N S its axis of rotation. The meridian passing through the Observatory at Greenwich is assumed by the British as a fixed origin from whence terrestrial longitudes are measured. And as each point on the surface of the earth passes through 360°, or a complete circle, in twenty-four hours, at the rate of 15° in an hour, time becomes a representative of angular motion. Hence, if the eclipse of a satellite happens at any place at eight o’clock in the evening, and the Nautical Almanac shows that the same phenomenon will take place at Greenwich at nine, the place of observation will be in the 15° of west longitude.

Note 96, p. 31. Conjunction. Let S be the sun, fig. 24, E the earth, and J O Jʹ Cʹ the orbit of Jupiter. Then the eclipses which happen when Jupiter is in O are seen 16^m 26^s sooner than those which take place when the planet is in C. Jupiter is in conjunction when at C, and in opposition when in O.

Note 97, p. 31. In the diagonal, &c. Were the line A S, fig. 26, 100,000 times longer than A B, Jupiter’s true place would be in the direction A Sʹ, the diagonal of the figure A B Sʹ S, which is, of course, out of proportion.

Note 98, p. 31. Aberration of light. The celestial bodies are so distant that the rays of light coming from them may be reckoned parallel. Therefore, let S A, Sʹ B, fig. 26, be two rays of light coming from the sun, or a planet, to the earth moving in its orbit in the direction A B. If a telescope be held in the direction A S, the ray S A, instead of going down the tube, will impinge on its side, and be lost in consequence of the telescope being carried with the earth in the direction A B. But, if the tube be held in the position A E, so that A B is to A S as the velocity of the earth to the velocity of light, the ray will pass through Sʹ E A. The star appears to be in the direction A Sʹ, when it really is in the direction A S; hence the angle S A Sʹ is the angle of aberration.

Note 99, p. 32. Density proportional to elasticity. The more a fluid, such as atmospheric air, is reduced in dimensions by pressure, the more it resists the pressure.

Note 100, p. 32. Oscillations of pendulum retarded. If a clock be carried from the pole to the equator, its rate will be gradually diminished, that is, it will go slower and slower: because the centrifugal force, which increases from the pole to the equator, diminishes the force of gravity.

Note 101, p. 34. Disturbing action. The disturbing force acts here in the very same manner as in note 63; only that the disturbing body d, fig. 14, is the sun, S the earth, and p the moon.

Note 102, pp. 35, 36, 86. Perigee. A Greek word, signifying round the earth. The perigee of the lunar orbit is the point P, fig. 6, where the moon is nearest to the earth. It corresponds to the perihelion of a planet. Sometimes the word is used to denote the point where the sun is nearest to the earth.

Note 103, p. 35. Evection. The evection is produced by the action of the radial force in the direction S p, fig. 14, which sometimes increases and sometimes diminishes the earth’s attraction to the moon. It produces a corresponding temporary change in the excentricity, which varies with the position of the major axis of the lunar orbit in respect of the line S d, joining the centres of the earth and sun.

Note 104, p. 35. Variation. The lunar perturbation called the variation is the alternate acceleration and retardation of the moon in longitude, from the action of the tangential force. She is accelerated in going from quadratures in Q and D, fig. 14, to the points C and O, called syzygies, and is retarded in going from the syzygies C and O to Q and D again.

Note 105, p. 36. Square of time. If the times increase at the rate of 1, 2, 3, 4, &c., years or hundreds of years, the squares of the times will be 1, 4, 9, 16, &c., years or hundreds of years.

Note 106, p. 37. In all investigations hitherto made with regard to the acceleration, it was tacitly assumed that the areas described by the radius vector of the moon were not permanently altered; that is to say, that the tangential disturbing force produced no permanent effect. But Mr. Adams has discovered that, in consequence of the constant decrease in the excentricity of the earth’s orbit, there is a gradual change in the central disturbing force which affects the aërial velocity, and consequently it alters the amount of the acceleration by a very small quantity, as well as the variation and other periodical inequalities of the moon. On the latter, however, it has no permanent effect, because it affects them in opposite directions in very moderate intervals of time, whereas a very small error in the amount of the acceleration goes on increasing as long as the excentricity of the earth’s orbit diminishes, so that it would ultimately vitiate calculations of the moon’s place for distant periods of time. This shows how complicated the moon’s motions are, and what rigorous accuracy is required in their determination.

Note 107, p. 37. Mean anomaly. The mean anomaly of a planet is its angular distance from the perihelion, supposing it to move in a circle. The true anomaly is its angular distance from the perihelion in its elliptical orbit. For example, in fig. 10, the mean anomaly is P C m, and the true anomaly is P S p.

Note 108, pp. 38, 68. Many circumferences. There are 360 degrees or 1,296,000 seconds in a circumference; and, as the acceleration of the moon only increases at the rate of eleven seconds in a century, it must be a prodigious number of ages before it accumulates to many circumferences.

Note 109, p. 39. Phases of the moon. The periodical changes in the enlightened part of her disc, from a crescent to a circle, depending upon her position with regard to the sun and earth.

Note 110, p. 39. Lunar eclipse. Let S, fig. 27, be the sun, E the earth, and m the moon. The space a A b is a section of the shadow, which has the form of a cone or sugar-loaf, and the spaces A a c, A b d, are the penumbra. The axis of the cone passes through A, and through E and S, the centres of the sun and earth, and n m nʹ is the path of the moon through the shadow.

Note 111, p. 39. Apparent diameter. The diameter of a celestial body as seen from the earth.

Note 112, p. 40. Penumbra. The shadow or imperfect darkness which precedes and follows an eclipse.

Note 113, p. 40. Synodic revolution of the moon. The time between two consecutive new or full moons.

Note 114, p. 40. Horizontal refraction. The light, in coming from a celestial object, is bent into a curve as soon as it enters our atmosphere; and that bending is greatest when the object is in the horizon.

Note 115, p. 40. Solar eclipse. Let S, fig. 28, be the sun, m the moon, and E the earth. Then a E b is the moon’s shadow, which sometimes eclipses a small portion of the earth’s surface at e, and sometimes falls short of it. To a person at e, in the centre of the shadow, the eclipse may be total or annular; to a person not in the centre of the shadow a part of the sun will be eclipsed; and to one at the edge of the shadow there will be no eclipse at all. The spaces P b E, Pʹ a E, are the penumbra.

Note 116, p. 43. From the extremities, &c. If the length of the line a b, fig. 29, be measured, in feet or fathoms, the angles S b a, S a b, can be measured, and then the angle a S b is known, whence the length of the line S C may be computed. a S b is the parallax of the object S; and it is clear that, the greater the distance of S, the less the base a b will appear, because the angle a Sʹ b is less than a S b.

Note 117, p. 44. Every particle will describe a circle, &c. If N S, fig. 3, be the axis about which the body revolves, then particles at B, Q, &c., will whirl in the circles B G A a, Q E q d, whose centres are in the axis N S, and their planes parallel to one another. They are, in fact, parallels of latitude, Q E q d being the equator.

Note 118, p. 44. The force of gravity, &c. Gravity at the equator acts in the direction Q C, fig. 30. Whereas the direction of the centrifugal force is exactly contrary, being in the direction C Q; hence the difference of the two is the force called gravitation, which makes bodies fall to the surface of the earth. At any point, m, not at the equator, the direction of gravity is m b, perpendicular to the surface, but the centrifugal force acts perpendicularly to N S, the axis of rotation. Now the effect of the centrifugal force is the same as if it were two forces, one of which acting in the direction b m, diminishes the force of gravity, and another which, acting in the direction m t, tangent to the surface at m, urges the particles towards Q, and tends to swell out the earth at the equator.

Note 119, p. 45. Homogeneous mass. A quantity of matter, everywhere of the same density.

Note 120, p. 45. Ellipsoid of revolution. A solid formed by the revolution of an ellipse about its axis. If the ellipse revolve about its minor axis Q D, fig. 6, the ellipsoid will be oblate, or flattened at the poles like an orange. If the revolution be about the greater axis A P, the ellipsoid will be prolate, like an egg.

Note 121, p. 45. Concentric elliptical strata. Strata, or layers, having an elliptical form and the same centre.

Note 122, p. 46. On the whole, &c. The line N Q S q, fig. 1, represents the ellipse in question, its major axis being Q q, its minor axis N S.

Note 123, p. 46. Increase in the length of the radii, &c. The radii gradually increase from the polar radius C N, fig. 30, which is least, to the equatorial radius C Q, which is greatest. There is also an increase in the lengths of the arcs corresponding to the same number of degrees from the equator to the poles; for, the angle N C r being equal to q C d, the elliptical arc N r is less than q d.

Note 124, p. 46. Cosine of latitude. The angles m C a, m C b, fig. 4, being the latitudes of the points a, b, &c., the cosines are C q, C r, &c.

Note 125, p. 47. An arc of the meridian. Let N Q S q, fig. 30, be the meridian, and m n the arc to be measured. Then, if Zʹ m, Z n, be verticals, or lines perpendicular to the surface of the earth, at the extremities of the arc m n they will meet in p. Q a n, Q b m, are the latitudes of the points m and n, and their difference is the angle m p n. Since the latitudes are equal to the height of the pole of the equinoctial above the horizon of the places m and n, the angle m p n may be found by observation. When the distance m n is measured in feet or fathoms, and divided by the number of degrees and parts of a degree contained in the angle m p n, the length of an arc of one degree is obtained.

Note 126, p. 47. A series of triangles. Let M Mʹ, fig. 31, be the meridian of any place. A line A B is measured with rods, on level ground, of any number of fathoms, C being some point seen from both ends of it. As two of the angles of the triangle A B C can be measured, the lengths of the sides A C, B C, can be computed; and if the angle m A B, which the base A B makes with the meridian, be measured, the length of the sides B m, A m, may be obtained by computation, so that A m, a small part of the meridian, is determined. Again, if D be a point visible from the extremities of the known line B C, two of the angles of the triangle B C D may be measured, and the length of the sides C D, B D, computed. Then, if the angle B m mʹ be measured, all the angles and the side B m of the triangle B m mʹ are known, whence the length of the line m mʹ may be computed, so that the portion A mʹ of the meridian is determined, and in the same manner it may be prolonged indefinitely.

Note 127, pp. 47, 49. The square of the sine of the latitude. Q b m, fig. 30, being the latitude of m, e m is the sine and b e the cosine. Then the number expressing the length of e m, multiplied by itself, is the square of the sine of the latitude; and the number expressing the length of b e, multiplied by itself, is the square of the cosine of the latitude.

Note 128, p. 48. The polar diameter of the earth determined by the survey of Great Britain is 7900 miles; the equatorial is 7926, which gives a compression of ¹⁄_299·33.

Note 129, p. 50. A pendulum is that part of a clock which swings to and fro.

Note 130, p. 52. Parallax. The angle a S b, fig. 29, under which we view an object a b: it therefore diminishes as the distance increases. The parallax of a celestial object is the angle which the radius of the earth would be seen under, if viewed from that object. Let E, fig. 32, be the centre of the earth, E H its radius, and m H O the horizon of an observer at H. Then H m E is the parallax of a body m, the moon for example. As m rises higher and higher in the heavens to the points mʹ, mʺ, &c., the parallax H mʹ E, H mʺ E, &c., decreases. At Z, the zenith, or point immediately above the head of the observer, it is zero; and at m, where the body is in the horizon, the angle H m E is the greatest possible, and is called the horizontal parallax. It is clear that with regard to celestial bodies the whole effect of parallax is in the vertical, or in the direction m mʹ Z; and as a person at H sees mʹ in the direction H mʹ A, when it really is in the direction E mʹ B, it makes celestial objects appear to be lower than they really are. The distance of the moon from the earth has been determined from her horizontal parallax. The angle E m H can be measured. E H m is a right angle, and E H, the radius of the earth, is known in miles; whence the distance of the moon E m is easily found. Annual parallax is the angle under which the diameter of the earth’s orbit would be seen if viewed from a star.

Note 131, p. 52. The radii n B, n G, &c., fig. 3, are equal in any one parallel of latitude, A a B G; therefore a change in the parallax observed in that parallel can only arise from a change in the moon’s distance from the earth; and when the moon is at her mean distance, which is a constant quantity equal to half the major axis of her orbit, a change in the parallax observed in different latitudes, G and E, must arise from the difference in the lengths of the radii n G and C E.

Note 132, p. 52. When Venus is in her nodes. She must be in the line N S n where her orbit P N A n cuts the plane of the ecliptic E N e n, fig. 12.

Note 133, p. 53. The line described, &c. Let E, fig. 33, be the earth, S the centre of the sun, and V the planet Venus. The real transit of the planet, seen from E the centre of the earth, would be in the direction A B. A person at W would see it pass over the sun in the line v a, and a person at O would see it move across him in the direction vʹ aʹ.

Note 134, p. 54. Kepler’s law. Suppose it were required to find the distance of Jupiter from the sun. The periodic times of Jupiter and Venus are given by observation, and the mean distance of Venus from the centre of the sun is known in miles or terrestrial radii; therefore, by the rule of three, the square root of the periodic time of Venus is to the square root of the periodic time of Jupiter as the cube root of the mean distance of Venus from the sun to the cube root of the mean distance of Jupiter from the sun, which is thus obtained in miles or terrestrial radii. The root of a number is that number which, once multiplied by itself, gives its square; twice multiplied by itself, gives its cube, &c. For example, twice 2 are 4, and twice 4 are 8; 2 is therefore the square root of 4, and the cube root of 8. In the same manner 3 times 3 are 9, and 3 times 9 are 27; 3 is therefore the square root of 9, and the cube root of 27.

Note 135, p. 55. Inversely, &c. The quantities of matter in any two primary planets are greater in proportion as the cubes of the numbers representing the mean distances of their satellites are greater, and also in proportion as the squares of their periodic times are less.

Note 136, p. 55. As hardly anything appears more impossible than that man should have been able to weigh the sun as it were in scales and the earth in a balance, the method of doing so may have some interest. The attraction of the sun is to the attraction of the earth as the quantity of matter in the sun to the quantity of matter in the earth; and, as the force of this reciprocal attraction is measured by its effects, the space the earth would fall through in a second by the sun’s attraction is to the space which the sun would fall through by the earth’s attraction as the mass of the sun to the mass of the earth. Hence, as many times as the fall of the earth to the sun in a second exceeds the fall of the sun to the earth in the same time, so many times does the mass of the sun exceed the mass of the earth. Thus the weight of the sun will be known if the length of these two spaces can be found in miles or parts of a mile. Nothing can be easier. A heavy body falls through 16·0697 feet in a second at the surface of the earth by the earth’s attraction; and, as the force of gravity is inversely as the square of the distance, it is clear that 16·0697 feet are to the space a body would fall through at the distance of the sun by the earth’s attraction, as the square of the distance of the sun from the earth to the square of the distance of the centre of the earth from its surface; that is, as the square of 95,000,000 miles to the square of 4000 miles. And thus, by a simple question in the rule of three, the space which the sun would fall through in a second by the attraction of the earth may be found in parts of a mile. The space the earth would fall through in a second, by the attraction of the sun, must now be found in miles also. Suppose m n, fig. 4, to be the arc which the earth describes round the sun in C, in a second of time, by the joint action of the sun and the centrifugal force. By the centrifugal force alone the earth would move from m to T in a second, and by the sun’s attraction alone it would fall through T n in the same time. Hence the length of T n, in miles, is the space the earth would fall through in a second by the sun’s attraction. Now, as the earth’s orbit is very nearly a circle, if 360 degrees be divided by the number of seconds in a sidereal year of 365¹⁄₄ days, it will give m n, the arc which the earth moves through in a second, and then the tables will give the length of the line C T in numbers corresponding to that angle; but, as the radius C n is assumed to be unity in the tables, if 1 be subtracted from the number representing C T, the length of T n will be obtained; and, when multiplied by 95,000,000, to reduce it to miles, the space which the earth falls through, by the sun’s attraction, will be obtained in miles. By this simple process it is found that, if the sun were placed in one scale of a balance, it would require 354,936 earths to form a counterpoise.

Note 137, p. 59. The sum of the greatest and least distances S P, S A, fig. 12, is equal to P A, the major axis; and their difference is equal to twice the excentricity C S. The longitude ♈ S P of the planet, when in the point P, at its least distance from the sun, is the longitude of the perihelion. The greatest height of the planet above the plane of the ecliptic E N e n, is equal to the inclination of the orbit P N A n to that plane. The longitude of the planet, when in the plane of the ecliptic, can only be the longitude of one of the points N or n; and, when one of these points is known, the other is given, being 180° distant from it. Lastly, the time included between two consecutive passages of the planet through the same node N or n, is its periodic time, allowance being made for the recess of the node in the interval.

Note 138, p. 60. Suppose that it were required to find the position of a point in space, as of a planet, and that one observation places it in n, fig. 34, another observation places it in nʹ, another in nʺ, and so on; all the points n, nʹ, nʺ, nʹʹʹ, &c., being very near to one another. The true place of the planet P will not differ much from any of these positions. It is evident, from this view of the subject, that P n, P nʹ, P nʺ, &c., are the errors of observation. The true position of the planet P is found by this property, that the squares of the numbers representing the lines P n, P nʹ, &c., when added together, is the least possible. Each line P n, P nʹ, &c., being the whole error in the place of the planet, is made up of the errors of all the elements; and, when compared with the errors obtained from theory, it affords the means of finding each. The principle of least squares is of very general application; its demonstration cannot find a place here; but the reader is referred to Biot’s Astronomy, vol. ii. p. 203.