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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.

To give an idea of the labour requisite merely to perfect or correct the lunar tables, the moon's place was determined by observation at the Greenwich Observatory in 6000 different points of her orbit, each of which was compared with the same points calculated from Baron Plana's formulæ, and to do that sixteen computers were constantly employed for eight years. Since the longitude is determined by the motions of the moon, the lunar tables are of the greatest importance.

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 PC m, and the true anomaly is P Sp.

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

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has the form of a cone or sugar-loaf, and the spaces A ac, Abd, 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 n' 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 Eb is the moon's shadow, which some

Fig. 28.


times 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 Pb E, P'a E, are the penumbra.

Fig. 29.

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

NOTE 117, p. 44. Every particle will describe a circle, &c. If NS, 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 qd 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 a to the surface, but the centrifugal force acts perpendi

cularly to NS, 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 bm, diminishes the force of gravity, and another which, acting in the direc


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tion mt, 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 QD, fig. 6, the ellipsoid will be oblate, or flattened at the poles like an

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. having an elliptical form and the same centre.

Strata, or layers,

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

NOTE 123, p. 46. Increase in the length of the radii, &c. The radii gradually increase from the polar radius CN, 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 Cr being equal to q Cd, the elliptical arc Nr is less than q d.

NOTE 124, p. 46. Cosine of latitude. The angles m Ca, m Cb, fig. 4, being the latitudes of the points a, b, &c., the cosines are Cq, Cr, &c.

NOTE 125, p. 47. An arc of the meridian. Let NQ Sq, fig. 30, be the meridian, and m n the arc to be measured. Then, if Z'm, Zn, be verticals, or lines perpendicular to the surface of the earth, at the extremities of the arc mn they will meet in p. Qan, Qbm, are the latitudes of the points m and n, and their difference is the angle mpn. 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 pn 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 mpn, the length of an arc of one degree is obtained.

NOTE 126, p. 47. A series of triangles. Let M M', 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 Bm, Am, may be obtained by computation, so that A m, a small part of

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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 CD, BD, computed. Then, if the angle Bm m' be measured, all the angles and the side Bm of the triangle B mm' are known, whence the length of the line m m' may be computed, so that the portion Am' 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. Qbm, fig. 30, being the latitude of m, em is the sine and be the cosine. 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 199.33.

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

Fig. 32.

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NOTE 130, p. 52. Parallax. The angle a Sb, fig. 29, under which we view an object ab: 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, EH its radius, and m H O the horizon of an observer at H. Then Hm E is the parallax of a body m, the moon for example. As m rises higher and higher in the heavens to the points m', m", &c., the parallax Hm' E, Hm" E, &c., decreases. At Z, the zenith,

or point immediately above

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the head of the observer, it is zero; and at m, where the body is in

the horizon, the angle Hm 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 m' Z ; and as a person at H sees m' in the direction Hm' A, when it really is in the direction Em' 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 Em H can be measured. EHm is a right angle, and E H, the radius of the earth, is known in miles; whence the distance of the moon Em 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 NS n where her orbit P N An cuts the plane of the ecliptic E N e n, fig. 12. The line described, &c. Let E, fig. 33, be the earth,

NOTE 133, p. 53.

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The real transit of the

S the centre of the sun, and V the planet Venus. 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 v' a'.

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

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