Mary Somerville

Note 1, page 2. Diameter. A straight line passing through the centre, and terminated both ways by the sides or surface of a figure, such as of a circle or sphere. In fig. 1, q Q, N S, are diameters.

Note 2, p. 2. Mathematical and mechanical sciences. Mathematics teach the laws of number and quantity; mechanics treat of the equilibrium and motion of bodies.

Note 3, p. 2. Analysis is a series of reasoning conducted by signs or symbols of the quantities whose relations form the subject of inquiry.

Note 4, p. 3. Oscillations are movements to and fro, like the swinging of the pendulum of a clock, or waves in water. The tides are oscillations of the sea.

Note 5, p. 3. Gravitation. Gravity is the reciprocal attraction of matter on matter; gravitation is the difference between gravity and the centrifugal force induced by the velocity of rotation or revolution. Sensible gravity, or weight, is a particular instance of gravitation. It is the force which causes substances to fall to the surface of the earth, and which retains the celestial bodies in their orbits. Its intensity increases as the squares of the distance decrease.

Note 6, p. 4. Particles of matter are the indefinitely small or ultimate atoms into which matter is believed to be divisible. Their form is unknown; but, though too small to be visible, they must have magnitude.

Note 7, p. 4. A hollow sphere. A hollow ball, like a bomb-shell. A sphere is a ball or solid body, such, that all lines drawn from its centre to its surface are equal. They are called radii, and every line passing through the centre and terminated both ways by the surface is a diameter, which is consequently equal to twice the radius. In fig. 3, Q q or N S is a diameter, and C Q, C N are radii. A great circle of the sphere has the same centre with the sphere as the circles Q E q d and Q N q S. The circle A B is a lesser circle of the sphere.

Note 8, p. 4. Concentric hollow spheres. Shells, or hollow spheres, having the same centre, like the coats of an onion.

Fig. 1.

Note 9, p. 4. Spheroid. A solid body, which sometimes has the shape of an orange, as in fig. 1; it is then called an oblate spheroid, because it is flattened at the poles N and S. Such is the form of the earth and planets. When, on the contrary, it is drawn out at the poles like an egg, as in fig. 2, it is called a prolate spheroid. It is evident that in both these solids the radii C q, C a, C N, &c., are generally unequal; whereas in the sphere they are all equal.

Fig. 2.

Note 10, p. 4. Centre of gravity. A point in every body, which if supported, the body will remain at rest in whatever position it may be placed. About that point all the parts exactly balance one another. The celestial bodies attract each other as if each were condensed into a single particle situate in the centre of gravity, or the particle situate in the centre of gravity of each may be regarded as possessing the resultant power of the innumerable oblique forces which constitute the whole attraction of the body.

Note 11, pp. 4, 6. Poles and equator. Let fig. 1 or 3 represent the earth, C its centre, N C S the axis of rotation, or the imaginary line about which it performs its daily revolution. Then N and S are the north and south poles, and the great circle q E Q, which divides the earth into two equal parts, is the equator. The earth is flattened at the poles, fig. 1, the equatorial diameter, q Q, exceeding the polar diameter, N S, by about 26¹/₂ miles. Lesser circles, A B G, which are parallel to the equator, are circles or parallels of latitude, which is estimated in degrees, minutes, and seconds, north and south of the equator, every place in the same parallel having the same latitude. Greenwich is in the parallel of 51° 28' 40. Thus terrestrial latitude is the angular distance between the direction of a plumb-line at any place and the plane of the equator. Lines such as N Q S, N G E S, fig. 3, are called meridians; all the places in any one of these lines have noon at the same instant. The meridian of Greenwich has been chosen by the British as the origin of terrestrial longitude, which is estimated in degrees, minutes, and seconds, east and west of that line. If N G E S be the meridian of Greenwich, the position of any place, B, is determined, when its latitude, Q C B, and its longitude, E C Q, are known.

Fig. 3.

Note 12, p. 4. Mean quantities are such as are intermediate between others that are greater and less. The mean of any number of unequal quantities is equal to their sum divided by their number. For instance, the mean between two unequal quantities is equal to half their sum.

Note 13, p. 4. A certain mean latitude. The attraction of a sphere on an external body is the same as if its mass were collected into one heavy particle in its centre of gravity, and the intensity of its attraction diminishes as the square of its distance from the external body increases. But the attraction of a spheroid, fig. 1, on an external body at m in the plane of its equator, E Q, is greater, and its attraction on the same body when at m' in the axis N S less, than if it were a sphere. Therefore, in both cases, the force deviates from the exact law of gravity. This deviation arises from the protuberant matter at the equator; and, as it diminishes towards the poles, so does the attractive force of the spheroid. But there is one mean latitude, where the attraction of a spheroid is the same as if it were a sphere. It is a part of the spheroid intermediate between the equator and the pole. In that latitude the square of the sine is equal to ¹/₃ of the equatorial radius.

Note 14, p. 4. Mean distance. The mean distance of a planet from the centre of the sun, or of a satellite from the centre of its planet, is equal to half the sum of its greatest and least distances, and, consequently, is equal to half the major axis of its orbit. For example, let P Q A D, fig. 6, be the orbit or path of the moon or of a planet; then P A is the major axis, C the centre, and C S is equal to C F. Now, since the earth or the sun is supposed to be in the point S according as P D A Q is regarded as the orbit of the moon or that of a planet, S A, S P are the greatest and least distances. But half the sum of S A and S P is equal to half of A P, the major axis of the orbit. When the body is at Q or D, it is at its mean distance from S, for S Q, S D, are each equal to C P, half the major axis by the nature of the curve.

Note 15, p. 4. Mean radius of the earth. The distance from the centre to the surface of the earth, regarded as a sphere. It is intermediate between the distances of the centre of the earth from the pole and from the equator.

Note 16, p. 5. Ratio. The relation which one quantity bears to another.

Note 17, p. 5. Square of moon’s distance. In order to avoid large numbers, the mean radius of the earth is taken for unity: then the mean distance of the moon is expressed by 60; and the square of that number is 3600, or 60 times 60.

Fig. 4

Note 18, p. 5. Centrifugal force. The force with which a revolving body tends to fly from the centre of motion: a sling tends to fly from the hand in consequence of the centrifugal force. A tangent is a straight line touching a curved line in one point without cutting it, as m T, fig. 4. The direction of the centrifugal force is in the tangent to the curved line or path in which the body revolves, and its intensity increases with the angular swing of the body, and with its distance from the centre of motion. As the orbit of the moon does not differ much from a circle, let it be represented by m d g h, fig. 4, the earth being in C. The centrifugal force arising from the velocity of the moon in her orbit balances the attraction of the earth. By their joint action, the moon moves through the arc m n during the time that she would fly off in the tangent m T by the action of the centrifugal force alone, or fall through m p by the earth’s attraction alone. T n, the deflection from the tangent, is parallel and equal to m p, the versed sine of the arc m n, supposed to be moved over by the moon in a second, and therefore so very small that it may be regarded as a straight line. T n, or m p, is the space the moon would fall through in the first second of her descent to the earth, were she not retained in her orbit by her centrifugal force.

Note 19, p. 5. Action and reaction. When motion is communicated by collision or pressure, the action of the body which strikes is returned with equal force by the body which receives the blow. The pressure of a hand on a table is resisted with an equal and contrary force. This necessarily follows from the impenetrability of matter, a property by which no two particles of matter can occupy the same identical portion of space at the same time. When motion is communicated without apparent contact, as in gravitation, attraction, and repulsion, the quantity of motion gained by the one body is exactly equal to that lost by the other, but in a contrary direction; a circumstance known by experience only.

Note 20, p. 5. Projected. A body is projected when it is thrown: a ball fired from a gun is projected; it is therefore called a projectile. But the word has also another meaning. A line, surface, or solid body, is said to be projected upon a plane, when parallel straight lines are drawn from every point of it to the plane. The figure so traced upon a plane is a projection. The projection of a terrestrial object is therefore its daylight shadow, since the sun’s rays are sensibly parallel.

Note 21, p. 5. Space. The boundless region which contains all creation.

Fig. 5.

Fig. 6.

Note 22, pp. 5, 11. Conic sections. Lines formed by any plane cutting a cone. A cone is a solid figure, like a sugar-loaf, fig. 5, of which A is the apex, A D the axis, and the plane B E C F the base. The axis may or may not be perpendicular to the base, and the base may be a circle, or any other curved line. When the axis is perpendicular to the base, the solid is a right cone. If a right cone with a circular base be cut at right angles to the base by a plane passing through the apex, the section will be a triangle. If the cone be cut through both sides by a plane parallel to the base, the section will be a circle. If the cone be cut slanting quite through both sides, the section will be an ellipse, fig. 6. If the cone be cut parallel to one of the sloping sides as A B, the section will be a parabola, fig. 7. And if the plane cut only one side of the cone, and be not parallel to the other, the section will be a hyperbola, fig. 8. Thus there are five conic sections.

Fig. 7.

Fig. 8.

Note 23, p. 5. Inverse square of distance. The attraction of one body for another at the distance of two miles is four times less than at the distance of one mile; at three miles, it is nine times less than at one; at four miles, it is sixteen times less, and so on. That is, the gravitating force decreases in intensity as the squares of the distance increase.

Note 24, p. 5. Ellipse. One of the conic sections, fig. 6. An ellipse may be drawn by fixing the ends of a string to two points, S and F, in a sheet of paper, and then carrying the point of a pencil round in the loop of the string kept stretched, the length of the string being greater than the distance between the two points. The points S and F are called the foci, C the centre, S C or C F the excentricity, A P the major axis, Q D the minor axis, and P S the focal distance. It is evident that, the less the excentricity C S, the nearer does the ellipse approach to a circle; and from the construction it is clear that the length of the string S m F is equal to the major axis P A. If T t be a tangent to the ellipse at m, then the angle T m S is equal to the angle t m F; and, as this is true for every point in the ellipse, it follows that, in an elliptical reflecting surface, rays of light or sound coming from one focus S will be reflected by the surface to the other focus F, since the angle of incidence is equal to the angle of reflection by the theories of light and sound.

Note 25, p. 5. Periodic time. The time in which a planet or comet performs a revolution round the sun, or a satellite about its planet.

Note 26, p. 5. Kepler discovered three laws in the planetary motions by which the principle of gravitation is established:—1st law, That the radii vectores of the planets and comets describe areas proportional to the time.—Let fig. 9 be the orbit of a planet; then, supposing the spaces or areas P S p, p S a, a S b, &c., equal to one another, the radius vector S P, which is the line joining the centres of the sun and planet, passes over these equal spaces in equal times; that is, if the line S P passes to S p in one day, it will come to S a in two days, to S b in three days, and so on. 2nd law, That the orbits or paths of the planets and comets are conic sections, having the sun in one of their foci. The orbits of the planets and satellites are curves like fig. 6 or 9, called ellipses, having the sun in the focus S. Several comets are known to move in ellipses; but the greater part seem to move in parabolas, fig. 7, having the sun in S, though it is probable that they really move in very long flat ellipses; others appear to move in hyperbolas, like fig. 8. The third law is, that the squares of the periodic times of the planets are proportional to the cubes of their mean distances from the sun. The square of a number is that number multiplied by itself, and the cube of a number is that number twice multiplied by itself. For example, the squares of the numbers 2, 3, 4, &c., are 4, 9, 16, &c., but their cubes are 8, 27, 64, &c. Then the squares of the numbers representing the periodic times of two planets are to one another as the cubes of the numbers representing their mean distances from the sun. So that, three of these quantities being known, the other may be found by the rule of three. The mean distances are measured in miles or terrestrial radii, and the periodic times are estimated in years, days, and parts of a day. Kepler’s laws extend to the satellites.

Fig. 9.

Note 27, p. 5. Mass. The quantity of matter in a given bulk. It is proportional to the density and volume or bulk conjointly.

Note 28, p. 5. Gravitation proportional to mass. But for the resistance of the air, all bodies would fall to the ground in equal times. In fact, a hundred equal particles of matter at equal distances from the surface of the earth would fall to the ground in parallel straight lines with equal rapidity, and no change whatever would take place in the circumstances of their descent, if 99 of them were united in one solid mass; for the solid mass and the single particle would touch the ground at the same instant, were it not for the resistance of the air.

Note 29, p. 5. Primary signifies, in astronomy, the planet about which a satellite revolves. The earth is primary to the moon.

Note 30, p. 6. Rotation. Motion round an axis, real or imaginary.

Note 31, p. 7. Compression of a spheroid. The flattening at the poles. It is equal to the difference between the greatest and least diameters, divided by the greatest, these quantities being expressed in some standard measure, as miles.

Note 32, p. 7. Satellites. Small bodies revolving about some of the planets. The moon is a satellite to the earth.

Note 33, p. 7. Nutation. A nodding motion in the earth’s axis while in rotation, similar to that observed in the spinning of a top. It is produced by the attraction of the sun and moon on the protuberant matter at the terrestrial equator.

Note 34, p. 7. Axis of rotation. The line, real or imaginary, about which a body revolves. The axis of the earth’s rotation is that diameter, or imaginary line, passing through the centre and both poles. Fig. 1 being the earth, N S is the axis of rotation.

Note 35, p. 7. Nutation of lunar orbit. The action of the bulging matter at the earth’s equator on the moon occasions a variation in the inclination of the lunar orbit to the plane of the ecliptic. Suppose the plane N p n, fig. 13, to be the orbit of the moon, and N m n the plane of the ecliptic, the earth’s action on the moon causes the angle p N m to become less or greater than its mean state. The nutation in the lunar orbit is the reaction of the nutation in the earth’s axis.

Note 36, p. 7. Translated. Carried forward in space.

Note 37, p. 7. Force proportional to velocity. Since a force is measured by its effect, the motions of the bodies of the solar system among themselves would be the same whether the system be at rest or not. The real motion of a person walking the deck of a ship at sea is compounded of his own motion and that of the ship, yet each takes place independently of the other. We walk about as if the earth were at rest, though it has the double motion of rotation on its axis and revolution round the sun.

Note 38, p. 8. Tangent. A straight line which touches a curved line in one point without cutting it. In fig. 4, m T is tangent to the curve in the point m. In a circle the tangent is at right angles to the radius, C m.

Note 39, p. 8. Motion in an elliptical orbit. A planet m, fig. 6, moves round the sun at S in an ellipse P D A Q, in consequence of two forces, one urging it in the direction of the tangent m T, and another pulling it towards the sun in the direction m S. Its velocity, which is greatest at P, decreases throughout the arc to P D A to A, where it is least, and increases continually as it moves along the arc A Q P till it comes to P again. The whole force producing the elliptical motion varies inversely as the square of the distance. See note 23.

Note 40, p. 8. Radii vectores. Imaginary lines adjoining the centre of the sun and the centre of a planet or comet, or the centres of a planet and its satellite. In the circle, the radii are all equal; but in an ellipse, fig. 6, the radius vector S A is greater, and S P less than all the others. The radii vectores S Q, S D, are equal to C A or C P, half the major axis P A, and consequently equal to the mean distance. A planet is at its mean distance from the sun when in the points Q and D.

Note 41, p. 8. Equal areas in equal times. See Kepler’s 1st law, in note 26, p. 5.

Note 42, p. 8. Major axis. The line P A, fig. 6 or 10.

Note 43, p. 8. If the planet described a circle, &c. The motion of a planet about the sun, in a circle A B P, fig. 10, whose radius C A is equal to the planet’s mean distance from him, would be equable, that is, its velocity, or speed, would always be the same. Whereas, if it moved in the ellipse A Q P, its speed would be continually varying, by note 39; but its motion is such, that the time elapsing between its departure from P and its return to that point again would be the same whether it moved in the circle or in the ellipse; for these curves coincide in the points P and A.

Note 44, p. 8. True motion. The motion of a body in its real orbit P D A Q, fig. 10.

Fig. 10.

Note 45, p. 9. Mean motion. Equable motion in a circle P E A B, fig. 10, at the mean distance C P or C m, in the time that the body would accomplish a revolution in its elliptical orbit P D A Q.

Note 46, p. 9. The equinox. Fig. 11 represents the celestial sphere, and C its centre, where the earth is supposed to be. q ? Q ? is the equinoctial or great circle, traced in the starry heavens by an imaginary extension of the plane of the terrestrial equator, and E ? e ? is the ecliptic, or apparent path of the sun round the earth. ? ?, the intersection of these two planes, is the line of the equinoxes; ? is the vernal equinox, and ? the autumnal. When the sun is in these points, the days and nights are equal. They are distant from one another by a semicircle, or two right angles. The points E and e are the solstices, where the sun is at his greatest distance from the equinoctial. The equinoctial is everywhere ninety degrees distant from its poles N and S, which are two points diametrically opposite to one another, where the axis of the earth’s rotation, if prolonged, would meet the heavens. The northern celestial pole N is within 1° 24' of the pole star. As the latitude of any place on the surface of the earth is equal to the height of the pole above the horizon, it is easily determined by observation. The ecliptic E ? e ? is also everywhere ninety degrees distant from its poles P and p. The angle P C N, between the poles P and N of the equinoctial and ecliptic, is equal to the angle e C Q, called the obliquity of the ecliptic.

Fig. 11.

Note 47, p. 9. Longitude. The vernal equinox, ?, fig. 11, is the zero point in the heavens whence celestial longitudes, or the angular motions of the celestial bodies, are estimated from west to east, the direction in which they all revolve. The vernal equinox is generally called the first point of Aries, though these two points have not coincided since the early ages of astronomy, about 2233 years ago, on account of a motion in the equinoctial points, to be explained hereafter. If S ?, fig. 10, be the line of the equinoxes, and ? the vernal equinox, the true longitude of a planet p is the angle ? S p, and its mean longitude is the angle ? C m, the sun being in S. Celestial longitude is the angular distance of a heavenly body from the vernal equinox; whereas terrestrial longitude is the angular distance of a place on the surface of the earth from a meridian arbitrarily chosen, as that of Greenwich.

Note 48, pp. 9, 58. Equation of the centre. The difference between ? C m and ? S p, fig. 10; that is, the difference between the true and mean longitudes of a planet or satellite. The true and mean places only coincide in the points P and A; in every other point of the orbit, the true place is either before or behind the mean place. In moving from A through the arc A Q P, the true place p is behind the mean place m; and through the arc P D A the true place is before the mean place. At its maximum, the equation of the centre measures C S, the excentricity of the orbit, since it is the difference between the motion of a body in an ellipse and in a circle whose diameter A P is the major axis of the ellipse.

Note 49, p. 9. Apsides. The points P and A, fig. 10, at the extremities of the major axis of an orbit. P is commonly called the perihelion, a Greek term signifying round the sun; and the point A is called the aphelion, a Greek term signifying at a distance from the sun.

Note 50, p. 9. Ninety degrees. A circle is divided into 360 equal parts, or degrees; each degree into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds. It is usual to write these quantities thus, 15° 16' 10, which means fifteen degrees, sixteen minutes, and ten seconds. It is clear that an arc m n, fig. 4, measures the angle m C n; hence we may say, an arc of so many degrees, or an angle of so many degrees; for, if there be ten degrees in the angle m C n, there will be ten degrees in the arc m n. It is evident that there are 90° in a right angle, m C d, or quadrant, since it is the fourth part of 360°.

Note 51, p. 9. Quadratures. A celestial body is said to be in quadrature when it is 90 degrees distant from the sun. For example, in fig. 14, if d be the sun, S the earth, and p the moon, then the moon is said to be in quadrature when she is in either of the points Q or D, because the angles Q S d and D S d, which measure her apparent distance from the sun, are right angles.

Note 52, p. 9. Excentricity. Deviation from circular form. In fig. 6, C S is the excentricity of the orbit P Q A D. The less C S, the more nearly does the orbit or ellipse approach the circular form; and, when C S is zero, the ellipse becomes a circle.

Note 53, p. 9. Inclination of an orbit. Let S, fig. 12, be the centre of the sun, P N A n the orbit of a planet moving from west to east in the direction N p. Let E N m e n be the shadow or projection of the orbit on the plane of the ecliptic, then N S n is the intersection of these two planes, for the orbit rises above the plane of the ecliptic towards N p, and sinks below it at N P. The angle p N m, which these two planes make with one another, is the inclination of the orbit P N p A to the plane of the ecliptic.

Note 54, p. 9. Latitude of a planet. The angle p S m, fig. 12, or the height of the planet p above the ecliptic E N m. In this case the latitude is north. Thus, celestial latitude is the angular distance of a celestial body from the plane of the ecliptic, whereas terrestrial latitude is the angular distance of a place on the surface of the earth from the equator.

Fig. 12.

Note 55, p. 9. Nodes. The two points N and n, fig. 12, in which the orbit N A n P of a planet or comet intersects the plane of the ecliptic e N E n. The part N A n of the orbit lies above the plane of the ecliptic, and the part n P N below it. The ascending node N is the point through which the body passes in rising above the plane of the ecliptic, and the descending node n is the point in which the body sinks below it. The nodes of a satellite’s orbit are the points in which it intersects the plane of the orbit of the planet.

Note 56, p. 10. Distance from the sun. S p in fig. 12. If ? be the vernal equinox, then ? S p is the longitude of the planet p, m S p is its latitude, and S p its distance from the sun. When these three quantities are known, the place of the planet p is determined in space.

Note 57, pp. 10, 59. Elements of an orbit. Of these there are seven. Let P N A n, fig. 12, be the elliptical orbit of a planet, C its centre, S the sun in one of the foci, ? the point of Aries, and E N e n the plane of the ecliptic. The elements are—the major axis A P; the excentricity C S; the periodic time, that is, the time of a complete revolution of the body in its orbit; and the fourth is the longitude of the body at any given instant—for example, that at which it passes through the perihelion P, the point of its orbit nearest to the sun. That instant is assumed as the origin of time, whence all preceding and succeeding periods are estimated. These four quantities are sufficient to determine the form of the orbit, and the motion of the body in it. Three other elements are requisite for determining the position of the orbit in space. These are, the angle ? S P, the longitude of the perihelion; the angle A N e, which is the inclination of the orbit to the plane of the ecliptic; and, lastly, the angle ? S N, the longitude of N the ascending node.

Note 58, p. 10. Whose planes, &c. The planes of the orbits, as P N A n, fig. 12, in which the planets move, are inclined or make small angles e N A with the plane of the ecliptic E N e n, and cut it in straight lines, N S n passing through S, the centre of the sun.

Note 59, p. 11. Momentum. Force measured by the weight of a body and its speed, or simple velocity, conjointly. The primitive momentum of the planets is, therefore, the quantity of motion which was impressed upon them when they were first thrown into space.

Note 60, p. 11. Unstable equilibrium. A body is said to be in equilibrium when it is so balanced as to remain at rest. But there are two kinds of equilibrium, stable and unstable. If a body balanced in stable equilibrium be slightly disturbed, it will endeavour to return to rest by a number of movements to and fro, which will continually decrease till they cease altogether, and then the body will be restored to its original state of repose. But, if the equilibrium be unstable, these movements to and fro, or oscillations, will become greater and greater till the equilibrium is destroyed.

Note 61, p. 14. Retrograde. Going backwards, as from east to west, contrary to the motion of the planets.

Note 62, p. 14. Parallel directions. Such as never meet, though prolonged ever so far.

Fig. 13.

Fig. 14.

Note 63, pp. 14, 16. The whole force, &c. Let S, fig. 13, be the sun, N m n the plane of the ecliptic, p the disturbed planet moving in its orbit n p N, and d the disturbing planet. Now, d attracts the sun and the planet p with different intensities in the directions d S, d p: the difference only of these forces disturbs the motion of p; it is therefore called the disturbing force. But this whole disturbing force may be regarded as equivalent to three forces, acting in the directions p S, p T, and p m. The force acting in the radius vector p S, joining the centres of the sun and planet, is called the radial force. It sometimes draws the disturbed planet p from the sun, and sometimes brings it nearer to him. The force which acts in the direction of the tangent p T is called the tangential force. It disturbs the motion of p in longitude, that is, it accelerates its motion in some parts of its orbit and retards it in others, so that the radius vector S p does not move over equal areas in equal times. (See note 26.) For example, in the position of the bodies in fig. 14, it is evident that, in consequence of the attraction of d, the planet p will have its motion accelerated from Q to C, retarded from C to D, again accelerated from D to O, and lastly retarded from O to Q. The disturbing body is here supposed to be at rest, and the orbit circular; but, as both bodies are perpetually moving with different velocities in ellipses, the perturbations or changes in the motions of p are very numerous. Lastly, that part of the disturbing force which acts in the direction of a line p m, fig. 13, at right angles to the plane of the orbit N p n, may be called the perpendicular force. It sometimes causes the body to approach nearer, and sometimes to recede farther from, the plane of the ecliptic N m n, than it would otherwise do. The action of the disturbing forces is admirably explained in a work on gravitation, by Mr. Airy, the Astronomer Royal.

Note 64, pp. 16, 74. Perihelion. Fig. 10, P, the point of an orbit nearest the sun.

Note 65, p. 16. Aphelion. Fig. 10, A, the point of an orbit farthest from the sun.

Note 66, pp. 16, 17. In fig. 15 the central force is greater than the exact law of gravity; therefore the curvature P p a is greater than P p A the real ellipse; hence the planet p comes to the point a, called the aphelion, sooner than if it moved in the orbit P p A, which makes the line P S A advance to a. In fig. 16, on the contrary, the curvature P p a is less than in the true ellipse, so that the planet p must move through more than the arc P p A, or 180°, before it comes to the aphelion a, which causes the greater axis P S A to recede to a.

Fig. 15.

Fig. 16.

Note 67, pp. 16, 17. Motion of apsides. Let P S A, fig. 17, be the position of the elliptical orbit of a planet, at any time; then, by the action of the disturbing forces, it successively takes the position P' S A', P S A, &c., till by this direct motion it has accomplished a revolution, and then it begins again; so that the motion is perpetual.

Fig. 17.

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 p' A, in consequence of the decrease in the excentricity C S; then the forms P p A, P p''' 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.

Fig. 18.

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.

Fig. 19.

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 p'; then, instead of moving in the orbit p n, it will tend to move in the orbit p p' n', whose plane cuts the ecliptic in the straight line S n'. If the disturbing force acts again upon the body when at p', so as to draw it into the arc p' p, the planet will now tend to move in the orbit p' p n, whose plane cuts the ecliptic in the straight line S n. The action of the disturbing force on the planet when at p will bring the node to n''', and so on. In this manner the node goes backwards through the successive points n, n', n, n''', &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 n', n' p' n, &c., are much smaller than the corresponding angles n S n', S n, &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.

Fig. 20.

Fig. 21.

Note 81, p. 24. If each particle, &c. Let P, P', P, &c., fig. 21, be planets moving in their orbits about the centre of gravity of the system. Let P S M, P' S M', &c., be portions of these orbits moved over by the radii vectores S P, S P', &c., in a given time, and let p S m, p' S m', &c., be their shadows or projections on the invariable plane. Then, if the numbers which represent the masses of the planets P, P', &c., be respectively multiplied by the numbers representing the areas or spaces p S m, p' S m', &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.

Fig. 22.

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° 5' 30.

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.

Fig. 23.

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.

Fig. 24.

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.

Fig. 25.

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 J' C' 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.

Fig. 26.

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 S', the diagonal of the figure A B S' 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, S' 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 S' E A. The star appears to be in the direction A S', when it really is in the direction A S; hence the angle S A S' 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.

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 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 n' is the path of the moon through the shadow.

Fig. 27.

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.

Fig. 28.

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, P' 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 S' b is less than a S b.

Fig. 29.

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.

Fig. 30.

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 Z' 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 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 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 m' be measured, all the angles and the side B m of the triangle B m m' are known, whence the length of the line m m' may be computed, so that the portion A m' of the meridian is determined, and in the same manner it may be prolonged indefinitely.

Fig. 31.

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.

Fig. 32.

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 m', m, &c., the parallax H m' E, H m 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 m' Z; and as a person at H sees m' in the direction H m' A, when it really is in the direction E m' 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 v' a'.

Fig. 33.

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 n', another in n, and so on; all the points n, n', n, n''', &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 n', P n, &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 n', &c., when added together, is the least possible. Each line P n, P n', &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.

Fig. 34.

Note 139, p. 61. The true longitude of Uranus was in advance of the tables previous to 1795, and continued to advance till 1822, after which it diminished rapidly till 1830-1, when the observed and calculated longitudes agreed, but then the planet fell behind the calculated place so rapidly that it was clear the tables could no longer represent its motion.

Note 140, p. 65. An axis that, &c. Fig. 20 represents the earth revolving in its orbit about the sun in S, the axis of rotation P p being everywhere parallel to itself.

Note 141, p. 65. Angular velocities that are sensibly uniform. The earth and planets revolve about their axis with an equable motion, which is never either faster or slower. For example, the length of the day is never more nor less than twenty-four hours.

Note 142, p. 68. If fig. 1 be the moon, her polar diameter N S is the shortest; and of those in the plane of the equator, Q E q, that which points to the earth is greater than all the others.

Note 143, p. 73. Inversely proportional, &c. That is, the total amount of solar radiation becomes less as the minor axis C C', fig. 20, of the earth’s orbit becomes greater.

Note 144, p. 75. Fig. 35 represents the position of the apparent orbit of the sun as it is at present, the earth being in E. The sun is nearer to the earth in moving through ? P ? than in moving through ? A ?, but its motion through ? P ? is more rapid than its motion through ? A ?; and, as the swiftness of the motion and the quantity of heat received vary in the same proportion, a compensation takes place.

Note 145, p. 76. In an ellipsoid of revolution, fig. 1, the polar diameter N S, and every diameter in the equator q E Q e, are permanent axes of rotation, but the rotation would be unstable about any other. Were the earth to begin to rotate about C a, the angular distance from a to the equator at q would no longer be ninety degrees, which would be immediately detected by the change it would occasion in the latitudes.

Note 146, pp. 50, 80. Let q ? Q, and E ? e, fig. 11, be the planes of the equator and ecliptic. The angle e ? Q, which separates them, called the obliquity of the ecliptic, varies in consequence of the action of the sun and moon upon the protuberant matter at the earth’s equator. That action brings the point Q towards e, and tends to make the plane q ? Q coincide with the ecliptic E ? e, which causes the equinoctial points ? and ? to move slowly backwards on the plane e ? E, at the rate of 50·41 annually. This part of the motion, which depends upon the form of the earth, is called luni-solar precession. Another part, totally independent of the form of the earth, arises from the mutual action of the earth, planets, and sun, which, altering the position of the plane of the ecliptic e ? E, causes the equinoctial points ? and ? to advance at the rate of O·31 annually; but, as this motion is much less than the former, the equinoctial points recede on the plane of the ecliptic at the rate of 50·1 annually. This motion is called the precession of the equinoxes.

Note 147, p. 81. Let q ? Q, e ? E, fig. 36, be the planes of the equinoctial or celestial equator and ecliptic, and p, P, their poles. Then suppose p, the pole of the equator, to revolve with a tremulous or wavy motion in the little ellipse p c d b in about 19 years, both motions being very small, while the point a is carried round in the circle a A B in 25,868 years. The tremulous motion may represent the half-yearly variation, the motion in the ellipse gives an idea of the nutation discovered by Bradley, and the motion in the circle a A B arises from the precession of the equinoxes. The greater axis p d of the small ellipse is 18·5, its minor axis b c is 13·74. These motions are so small that they have very little effect on the parallelism of the axis of the earth’s rotation during its revolution round the sun, as represented in fig. 20. As the stars are fixed, this real motion in the pole of the earth must cause an apparent change in their places.

Note 148, p. 83. By means of a transit instrument, which is a telescope mounted so as to revolve only in the plane of the meridian, the instant of the transit or passage of a celestial body across the meridian can be determined. The transits of the principal stars are used to ascertain the time, or, which is the same thing, the amount of the error of clocks. A system of equidistant wires, as represented in the figure, is placed in the focus of the eye-piece, so that the middle wire is perpendicular and at right angles to the axis of the telescope. It consequently represents a portion of the celestial meridian; and when a star is seen to cross that wire it then crosses the celestial meridian of the place of observation. A clock beating seconds being close at hand, the duty of an observer is to note the exact second and part of a second at which a star crosses each wire successively in consequence of the rotation of the earth. Then the mean of all these observations will give the time at which the star crosses the celestial meridian of the place of observation to the tenth of a second, provided the observations are accurate. Now it happens that the simultaneous impression on the eye and ear is estimated differently by different observers, so that one person will note the transit of a star, for example, as happening the fraction of a second sooner or later than another person; and as that is the case in every observation he makes, it is called his personal equation, that is to say, it is a correction that must be applied to all the observations of the individual, and a curious instance of individuality it is. For instance, M. Otto Struve notes every observation O·11 too soon, M. Peters O·13 too late; M. Struve noted every observation one second later than M. Bessel, and M. Argelander estimated the transit of a star 1·2 later than M. Bessel. All these gentlemen were or are first-rate observers; and when the personal equation is known it is easy to correct the observations. However, to avoid that inconvenience Mr. Bond has introduced a method in the Observatory at Cambridge in the United States in which touch is combined with sight instead of hearing, which is now used also at Greenwich. The observer at the moment of the observation presses his fingers on a machine which by means of a galvanic battery conveys the impression to where time is measured and marked, so that the observation is at once recorded and the personal equation avoided.

Note 149, p. 84. Let N be the pole, fig. 11, e E the ecliptic, and Q q the equator. Then, N n m S being a meridian, and at right angles to the equator, the arc ? m is less than the arc ? n.

Note 150, p. 85. Heliacal rising of Sirius. When the star appears in the morning, in the horizon, a little before the rising of the sun.

Note 151, p. 87. Let P ? A ?, fig. 35, be the apparent orbit or path of the sun, the earth being in E. Its major axis, A P, is at present situate as in the figure, where the solar perigee P is between the solstice of winter and the equinox of spring. So that the time of the sun’s passage through the arc ? A ? is greater than the time he takes to go through the arc ? P ?. The major axis A P coincided with ? ?, the line of the equinoxes, 4000 years before the Christian era; at that time P was in the point ?. In 6468 of the Christian era the perigee P will coincide with ?. In 1234 A.D. the major axis was perpendicular to ? ?, and then P was in the winter solstice.

Note 152, p. 88. At the solstices, &c. Since the declination of a celestial object is its angular distance from the equinoctial, the declination of the sun at the solstice is equal to the arc Q e, fig. 11, which measures the obliquity of the ecliptic, or angular distance of the plane ? e ? from the plane ? Q ?.

Note 153, p. 88. Zenith distance is the angular distance of a celestial object from the point immediately over the head of an observer.

Note 154, p. 89. Reduced to the level of the sea. The force of gravitation decreases as the square of the height above the surface of the earth increases, so that a pendulum vibrates slower on high ground; and, in order to have a standard independent of local circumstances, it is necessary to reduce it to the length that would exactly make 86,400 vibrations in a mean solar day at the level of the sea.

Note 155, p. 90. A quadrant of the meridian is a fourth part of a meridian, or an arc of a meridian containing 90°, as N Q, fig. 11.

Note 156, p. 93. Moon’s southing. The time when the moon is on the meridian of any place, which happens about forty-eight minutes later every day.

Note 157, p. 96. The angular velocity of the earth’s rotation is at the rate of 180° in twelve hours, which is the time included between the passages of the moon at the upper and under meridian.

Note 158, p. 96. If S be the earth, fig. 14, d the sun, and C Q O D the orbit of the moon, then C and O are the syzygies. When the moon is new, she is at C, and when full she is at O; and, as both sun and moon are then on the same meridian, it occasions the spring-tides, it being high water at places under C and O, while it is low water at those under Q and D. The neap-tides happen when the moon is in quadrature at Q or D, for then she is distant from the sun by the angle d S Q, or d S D, each of which is 90°.

Note 159, p. 97. Declination. If the earth be in C, fig. 11, and if q ? Q be the equinoctial, and N m S a meridian, then m C n is the declination of a body at n. Therefore the cosine of that angle is the cosine of the declination.

Note 160, pp. 99, 131. Fig 37 shows the propagation of waves from two points C and C', where stones are supposed to have fallen. Those points in which the waves cross each other are the places where they counteract each other’s effects, so that the water is smooth there, while it is agitated in the intermediate spaces.

Note 161, p. 100. The centrifugal force may, &c. The centrifugal force acts in a direction at right angles to N S, the axis of rotation, fig. 30. Its effects are equivalent to two forces, one of which is in the direction b m perpendicular to the surface Q m n of the earth, and diminishes the force of gravity at m. The other acts in the direction of the tangent m T, which makes the fluid particles tend towards the equator.

Note 162, p. 106. Analytical formula or expression. A combination of symbols or signs expressing or representing a series of calculation, and including every particular case that can arise from a general law.

Note 163, p. 106. Fig. 38 is a perfect octahedron. Sometimes its angles, A, X, a, a, &c., are truncated, or cut off. Sometimes a slice is cut off its edges A a, X a, a a, &c. Occasionally both these modifications take place.

Note 164, p. 107. Prismatic crystals of sulphate of nickel are somewhat like fig. 62, only that they are thin, like a hair.

Note 165, p. 108. Zinc, a metal either found as an ore or mixed with other metals. It is used in making brass.

Note 166, p. 108. A cube is a solid contained by six plane square surfaces, as fig. 39.

Note 167, p. 108. A tetrahedron is a solid contained by four triangular surfaces, as fig. 40: of this solid there are many varieties.

Note 168, p. 108. There are many varieties of the octahedron. In that mentioned in the text, the base a a a a, fig. 38, is a square, but the base may be a rhomb; this solid may also be elongated in the direction of its axis A X, or it may be depressed.

Note 169, pp. 109, 192, 273. A rhombohedron is a solid contained by six plane surfaces, as in fig. 63, the opposite planes being equal and similar rhombs parallel to one another; but all the planes are not necessarily equal or similar, nor are its angles right angles. In carbonate of lime the angle C A B is 105°·55, and the angle B or C is 75°·05.

Note 170, p. 109. Sublimation. Bodies raised into vapour which is again condensed into a solid state.

Note 171, p. 112. Platinum. The heaviest of metals; its colour is between that of silver and lead.

Note 172, p. 113. The surface of a column of water, or spirit of wine, in a capillary tube, is hollow; and that of a column of quicksilver is convex, or rounded, as in fig. 41.

Note 173, p. 113. Inverse ratio, &c. The elevation of the liquid is greater in proportion as the internal diameter of the tube is less.

Note 174, p. 114. In fig. 41 the line c d shows the direction of the resulting force in the two cases.

Note 175, p. 115. When two plates of glass are brought near to one another in water, the liquid rises between them; and, if the plates touch each other at one of their upright edges, the outline of the water will become an hyperbola.

Note 176, p. 115. Let A A', fig. 42, be two plates, both of which are wet, and B B' two that are dry. When partly immersed in a liquid, its surface will be curved close to them, but will be of its usual level for the rest of the distance. At such a distance they will neither attract nor repel one another. But, as soon as they are brought near enough to have the whole of the liquid surface between them curved, as in a a', b b', they will rush together. If one be wet and another dry, as C C', they will repel one another at a certain distance; but, as soon as they are brought very near, they will rush together, as in the former cases.

Note 177, p. 123. In a paper on the atmospheric changes that produce rain and wind, by Thomas Hopkins, Esq., in the Geographical Journal, it is shown that, when vapour is condensed and falls in rain, a partial vacuum is formed, and that heavier air presses in as a current of wind. Thus the vacuum arising from the great precipitation at the tropics causes the polar winds to descend from the upper regions of the atmosphere and blow along the surface to the equator as trade winds to supply the place of the hot currents that are continually raising them into the higher regions. This circumstance removes the only difficulty in Lieutenant Maury’s theory of the winds.

Note 178, p. 134. Latent or absorbed heat. There is a certain quantity of heat in all bodies, which cannot be detected by the thermometer, but which may become sensible by compression.

Note 179, p. 137. Reflected waves. A series of waves of light, sound, or water, diverge in all directions from their origin I, fig. 43, as from a centre. When they meet with an obstacle S S, they strike against it, and are reflected or turned back by it in the same form as if they had proceeded from the centre C, at an equal distance on the other side of the surface S S.

Note 180, p. 138. Elliptical shell. If fig. 6 be a section of an elliptical shell, then all sounds coming from the focus S to different points on the surface, as m, are reflected back to F, because the angle T m S is equal to t m F. In a spherical hollow shell, a sound diverging from the centre is reflected back to the centre again.

Note 181, p. 142. Fig. 44 represents musical strings in vibration; the straight lines are the strings when at rest. The first figure of the four would give the fundamental note, as, for example, the low C. The second and third figures would give the first and second harmonics; that is, the octave and the 12th above C, n n n being the points at rest; the fourth figure shows the real motion when compounded of all three.

Note 182, p. 143. Fig. 45 represents sections of an open and of a shut pipe, and of a pipe open at one end. When sounded, the air spontaneously divides itself into segments. It remains at rest in the divisions or nodes n n', &c., but vibrates between them in the direction of the arrow-heads. The undulations of the whole column of air give the fundamental note, while the vibrations of the divisions give the harmonics.

Note 183, p. 144. Fig. 1, plate 1, shows the vibrating surface when the sand divides it into squares, and fig. 2 represents the same when the nodal lines divide it into triangles. The portions marked a a are in different states of vibration from those marked b b.

Note 184, p. 145. Plates 1 and 2 contain a few of Chladni’s figures. The white lines are the forms assumed by the sand, from different modes of vibration, corresponding to musical notes of different degrees of pitch. Plate 3 contains six of Chladni’s circular figures.

Note 185, p. 145. Mr. Wheatstone’s principle is, that when vibrations producing the forms of figs. 1 and 2, plate 3, are united in the same surface, they make the sand assume the form of fig. 3. In the same manner, the vibrations which would separately cause the sand to take the forms of figs. 4 and 5, would make it assume the form in fig. 6 when united. The figure 9 results from the modes of vibration of 7 and 8 combined. The parts marked a a are in different states of vibration from those marked b b. Figs. 1, 2, and 3, plate 4, represent forms which the sand takes in consequence of simple modes of vibration; 4 and 5 are those arising from two combined modes of vibration; and the last six figures arise from four superimposed simple modes of vibration. These complicated figures are determined by computation independent of experiment.

Note 186, p. 146. The long cross-lines of fig. 46 show the two systems of nodal lines given by M. Savart’s laminÆ.

Note 187, p. 146. The short lines on fig. 46 show the positions of the nodal lines on the other sides of the same laminÆ.

Note 188, p. 146. Fig. 47 gives the nodal lines on a cylinder, with the paper rings that mark the quiescent points.

Note 189, pp. 138, 153, 156. Reflection and Refraction. Let P C p, fig. 48, be perpendicular to a surface of glass or water A B. When a ray of light, passing through the air, falls on this surface in any direction I C, part of it is reflected in the direction C S, and the other part is bent at C, and passes through the glass or water in the direction C R. I C is called the incident ray, and I C P the angle of incidence; C S is the reflected ray, and P C S the angle of reflection; C R is the refracted ray, and p C R the angle of refraction. The plane passing through S C and I C is the plane of reflection, and the plane passing through I C and C R is the plane of refraction. In ordinary cases, C I, C S, C R, are all in the same plane. We see the surface by means of the reflected light, which would otherwise be invisible. Whatever the reflecting surface may be, and however obliquely the light may fall upon it, the angle of reflection is always equal to the angle of incidence. Thus I C, I' C, being rays incident on the surface at C, they will be reflected into C S, C S', so that the angle S C P will be equal to the angle I C P, and S' C P equal to I' C P. That is by no means the case with the refracted rays. The incident rays I C, I' C, are bent at C towards the perpendicular, in the direction C R, C R'; and the law of refraction is such, that the sine of the angle of incidence has a constant ratio to the sine of the angle of refraction; that is to say, the number expressing the length of I m, the sine of I C P, divided by the number expressing the length of R n, the sine of R C p, is the same for all the rays of light that can fall upon the surface of any one substance, and is called its index of refraction. Though the index of refraction be the same for any one substance, it is not the same for all substances. For water it is 1·336; for crown-glass it is 1·535; for flint-glass, 1·6; for diamond, 2·487; and for chromate of lead it is 3, which substance has a higher refractive power than any other known. Light falling perpendicularly on a surface passes through it without being refracted. If the light be now supposed to pass from a dense into a rare medium, as from glass or water into air, then R C, R' C, become the incident rays; and in this case the refracted rays, C I, C I', are bent from the perpendicular instead of towards it. When the incidence is very oblique, as r C, the light never passes into the air at all, but it is totally reflected in the direction C r', so that the angle p C r is equal to p C r'; that frequently happens at the second surface of glass. When a ray I C falls from air upon a piece of glass A B, it is in general refracted at each surface. At C it is bent towards the perpendicular, and at R from it, and the ray emerges parallel to I C; but, when the ray is very oblique to the second surface, it is totally reflected. An object seen by total reflection is nearly as vivid as when seen by direct vision, because no part of the light is refracted. When light falls upon a plate of crown-glass, at an angle of 4° 32' counted from the surface, the glass reflects 4 times more light than it transmits. At an angle of 7° 1' the reflected light is double of the transmitted; at an angle of 11° 8' the light reflected is equal to that transmitted; at 17° 17' the reflected is equal to ¹/₂ the transmitted light; at 26° 38' it is equal to ¹/₄, the variation, according to Arago, being as the square of the cosine.

Note 189, p. 154. Atmospheric refraction. Let a b, a b, &c., fig. 49, be strata, or extremely thin layers, of the atmosphere, which increase in density towards m n, the surface of the earth. A ray coming from a star meeting the surface of the atmosphere at S would be refracted at the surface of each layer, and would consequently move in the curved line S v v v A; and as an object is seen in the direction of the ray that meets the eye, the star, which really is in the direction A S, would seem to a person at A to be in s. So that refraction, which always acts in a vertical direction, raises objects above their true place. For that reason, a body at S', below the horizon H A O, would be raised, and would be seen in s'. The sun is frequently visible by refraction after he is set, or before he is risen. There is no refraction in the zenith at Z. It increases all the way to the horizon, where it is greatest, the variation being proportional to the tangent of the angles Z A S, Z A S', the distances of the bodies S S' from the zenith. The more obliquely the rays fall, the greater the refraction.

Note 190, p. 154. Bradley’s method of ascertaining the amount of refraction. Let Z, fig. 50, be the zenith or point immediately above an observer at A; let H O be his horizon, and P the pole of the equinoctial A Q. Hence P A Q is a right angle. A star as near to the pole as s would appear to revolve about it, in consequence of the rotation of the earth. At noon, for example, it would be at s above the pole, and at midnight it would be in s' below it. The sum of the true zenith distances, Z A s, Z A s', is equal to twice the angle Z A P. Again, S and S' being the sun at his greatest distances from the equinoctial A Q when in the solstices, the sum of his true zenith distances, Z A S, Z A S', is equal to twice the angle Z A Q. Consequently, the four true zenith distances, when added together, are equal to twice the right angle Q A P; that is, they are equal to 180°. But the observed or apparent zenith distances are less than the true on account of refraction; therefore the sum of the four apparent zenith distances is less than 180° by the whole amount of the four refractions.

Note 191, p. 155. Terrestrial refraction. Let C, fig. 51, be the centre of the earth, A an observer at its surface, A H his horizon, and B some distant point, as the top of a hill. Let the arc B A be the path of a ray coming from B to A; E B, E A, tangents to its extremities; and A G, B F, perpendicular to C B. However high the hill B may be, it is nothing when compared with C A, the radius of the earth; consequently, A B differs so little from A D that the angles A E B and A C B are supplementary to one another; that is, the two taken together are equal to 180°. A C B is called the horizontal angle. Now B A H is the real height of B, and E A H its apparent height; hence refraction raises the object B, by the angle E A B, above its real place. Again, the real depression of A, when viewed from B, is F B A, whereas its apparent depression is F B E, so E B A is due to refraction. The angle F B A is equal to the sum of the angles B A H and A C B; that is, the true elevation is equal to the true depression and the horizontal angle. But the true elevation is equal to the apparent elevation diminished by the refraction; and the true depression is equal to the apparent depression increased by refraction. Hence twice the refraction is equal to the horizontal angle augmented by the difference between the apparent elevation and the apparent depression.

Note 192, p. 155. Fig. 52 represents the phenomenon in question. S P is the real ship, with its inverted and direct images seen in the air. Were there no refraction, the rays would come from the ship S P to the eye E in the direction of the straight lines; but, on account of the variable density of the inferior strata of the atmosphere, the rays are bent in the curved lines P c E, P d E, S m E, S n E. Since an object is seen in the direction of the tangent to that point of the ray which meets the eye, the point P of the real ship is seen at p and p', and the point S seems to be in s and s'; and, as all the other points are transferred in the same manner, direct and inverted images of the ship are formed in the air above it.

Note 193, p. 156. Fig. 53 represents the section of a poker, with the refraction produced by the hot air surrounding it.

Note 194, p. 156. The solar spectrum. A ray from the sun at S, fig. 54, admitted into a dark room, through a small round hole H in a window-shutter, proceeds in a straight line to a screen D, on which it forms a bright circular spot of white light, of nearly the same diameter with the hole H. But when the refracting angle B A C of a glass prism is interposed, so that the sunbeam falls on A C the first surface of the prism, and emerges from the second surface A B at equal angles, it causes the rays to deviate from the straight path S D, and bends them to the screen M N, where they form a coloured image V R of the sun, of the same breadth with the diameter of the hole H, but much longer. The space V R consists of seven colours—violet, indigo, blue, green, yellow, orange, and red. The violet and red, being the most and least refrangible rays, are at the extremities, and the green occupy the middle part at G. The angle D g G is called the mean deviation, and the spreading of the coloured rays over the angle V g R the dispersion. The deviation and dispersion vary with the refracting angle B A C of the prism, and with the substance of which it is made.

Note 195, pp. 159, 164. Under the same circumstances, and where the refracting angles of the two prisms are equal, the angles D g G and V g R, fig. 54, are greater for flint-glass than for crown-glass. But, as they vary with the angle of the prism, it is only necessary to augment the refracting angle of the crown-glass prism by a certain quantity, to produce nearly the same deviation and dispersion with the flint-glass prism. Hence, when the two prisms are placed with their refracting angles in opposite directions, as in fig. 54, they nearly neutralize each other’s effects, and refract a beam of light without resolving it into its elementary coloured rays. Sir David Brewster has come to the conclusion that there may be refraction without colour by means of two prisms, or two lenses, when properly adjusted, even though they be made of the same kind of glass.

Note 196, p. 165. The object glass of the achromatic telescope consists of a convex lens A B, fig. 55, of crown-glass placed on the outside, towards the object, and of a concave-convex lens C D of flint-glass, placed towards the eye. The focal length of a lens is the distance of its centre from the point in which the rays converge, as F, fig. 60. If, then, the lenses A B and C D be so constructed that their focal lengths are in the same proportion as their dispersive powers, they will refract rays of light without colour.

Note 197, p. 165. If the mean refracting angle of the prism D g G, fig. 54, were the same for all substances, then the difference D g V - D g R would be the dispersion. But the angle of the prism being the same, all these angles are different in each substance, so that in order to obtain the dispersion of any substance the angle D g V - D g R must be divided by the angle D g G or its excess above unity, to which the mean refraction is always proportional. According to Mr. Fraunhofer the refraction of the extreme violet and red rays in crown-glass is 1·5466 and 1·5258; so D g V - D g R = 1·5466 - 1·5258 = ·0208, and half the sum of the excess of each above unity is = ·5362; consequently

so that the dispersive power of diamond is a little less than that of crown-glass; hence the splendid refracted colours which distinguish diamond from every other precious stone are not owing to its high dispersive power, but to its great mean refraction.—Sir David Brewster.

Note 198, p. 168. When a sunbeam, after having passed through a coloured glass V V', fig. 56, enters a dark room by two small slits O O' in a card, or piece of tin, they produce alternate bright and black bands on a screen S S' at a little distance. When either one or other of the slits O or O' is stopped, the dark bands vanish, and the screen is illuminated by a uniform light, proving that the dark bands are produced by the interference of the two sets of rays. Again, let H m, fig. 57, be a beam of white light passing through a hole at H, made with a fine needle in a piece of lead or a card, and received on a screen S S'. When a hair, or a small slip of card h h', about the 30th of an inch in breadth, is held in the beam, the rays bend round on each side of it, and, arriving at the screen in different states of vibration, interfere and form a series of coloured fringes on each side of a central white band m. When a piece of card is interposed at C, so as to intercept the light which passes on one side of the hair, the coloured fringes vanish. When homogeneous light is used, the fringes are broadest in red, and become narrower for each colour of the spectrum progressively to the violet, which gives the narrowest and most crowded fringes. These very elegant experiments are due to Dr. Thomas Young.

Note 199, pp. 171, 200. Fig. 58 shows Newton’s rings, of which there are seven, formed by screwing two lenses of glass together. Provided the incident light be white, they always succeed each other in the following order:—

1st ring, or 1st order of colours: Black, very faint blue, brilliant white, yellow, orange, red.

2nd ring: Dark purple, or rather violet, blue, a very imperfect yellow green, vivid yellow, crimson red.

3rd ring: Purple, blue, rich grass green, fine yellow, pink, crimson.

4th ring: Dull blueish green, pale yellowish pink, red.

5th ring: Pale blueish green, white, pink.

6th ring: Pale blue green, pale pink.

7th ring: Very pale blueish green, very pale pink.

After the seventh order the colours become too faint to be distinguished. The rings decrease in breadth, and the colours become more crowded together, as they recede from the centre. When the light is homogeneous, the rings are broadest in the red, and decrease in breadth with every successive colour of the spectrum to the violet.

Note 200, p. 172. The absolute thickness of the film of air between the glasses is found as follows:—Let A F B C, fig. 59, be the section of a lens lying on a plane surface or plate of glass P P', seen edgewise, and let E C be the diameter of the sphere of which the lens is a segment. If A B be the diameter of any one of Newton’s rings, and B D parallel to C E, then B D or C F is the thickness of the air producing it. E C is a known quantity; and when A B, the diameter, is measured with compasses, B D or F C can be computed. Newton found that the length of B D, corresponding to the darkest part of the first ring, is the 98,000th part of an inch when the rays fall perpendicularly on the lens, and from this he deduced the thickness corresponding to each colour in the system of rings. By passing each colour of the solar spectrum in succession over the lenses, Newton also determined the thickness of the film of air corresponding to each colour, from the breadth of the rings, which are always of the same colour with the homogeneous light.

Fig. 59.

Note 201, p. 174. The focal length or distance of a lens is the distance from its centre to the point F, fig. 60, in which the refracted rays meet. Let L L' be a lens of very short focal distance fixed in the window-shutter of a dark room. A sunbeam S L L' passing through the lens will be brought to a focus in F, whence it will diverge in lines F C, F D, and will form a circular image of light on the opposite wall. Suppose a sheet of lead, having a small pin-hole pierced through it, to be placed in this beam; when the pin-hole is viewed from behind with a lens at E, it is surrounded with a series of coloured rings, which vary in appearance with the relative positions of the pin-hole and eye with regard to the point F. When the hole is the 30th of an inch in diameter and at the distance of 6¹/₂ feet from F, when viewed at the distance of 24 inches, there are seven rings of the following colours:—

1st order: White, pale yellow, yellow, orange, dull red.

2nd order: Violet, blue, whitish, greenish yellow, fine yellow, orange red.

3rd order: Purple, indigo blue, greenish blue, brilliant green, yellow green, red.

4th order: Blueish green, blueish white, red.

5th order: Dull green, faint blueish white, faint red.

6th order: Very faint green, very faint red.

7th order: A trace of green and red.

Note 202, p. 175. Let L L', fig. 61, be the section of a lens placed in a window-shutter, through which a very small beam of light S L L' passes into a dark room, and comes to a focus in F. If the edge of a knife K N be held in the beam, the rays bend away from it in hyperbolic curves K r, K r', &c., instead of coming directly to the screen in the straight line K E, which is the boundary of the shadow. As these bending rays arrive at the screen in different states of undulation, they interfere, and form a series of coloured fringes, r r', &c., along the edge of the shadow K E S N of the knife. The fringes vary in breadth with the relative distances of the knife-edge and screen from F.

Note 203, p. 177. Fig. 43 represents the phenomena in question, where S S is the surface, and I the centre of incident waves. The reflected waves are the dark lines returning towards I, which are the same as if they had originated in C on the other side of the surface.

Note 204, p. 180. Fig. 62 represents a prismatic crystal of tourmaline, whose axis is A X. The slices that are used for polarising light are cut parallel to A X.

Note 205, p. 181. Double refraction. If a pencil of light R r, fig. 63, falls upon a rhombohedron of Iceland spar A B X C, it is separated into two equal pencils of light at r, which are refracted in the directions r O, r E: when these arrive at O and E they are again refracted, and pass into the air in the directions O o, E o, parallel to one another and to the incident ray R r. The ray r O is refracted according to the ordinary law, which is, that the sines of the angles of incidence and refraction bear a constant ratio to one another (see Note 184), and the rays R r, r O, O o, are all in the same plane. The pencil r E, on the contrary, is bent aside out of that plane, and its refraction does not follow the constant ratio of the sines; r E is therefore called the extraordinary ray, and r O the ordinary ray. In consequence of this bisection of the light, a spot of ink at O is seen double at O and E, when viewed from r I; and when the crystal is turned round, the image E revolves about O, which remains stationary.

Note 206, p. 182. Both of the parallel rays O o and E o, fig. 63, are polarised on leaving the doubly refracting crystal, and in both the particles of light make their vibrations at right angles to the lines O o, E o. In the one, however, these vibrations lie, for example, in the plane of the horizon, while the vibrations of the other lie in the vertical plane perpendicular to the horizon.

Note 207, p. 183. If light be made to fall in various directions on the natural faces of a crystal of Iceland spar, or on faces cut and polished artificially, one direction A X, fig. 63, will be found, along which the light passes without being separated into two pencils. A X is the optic axis. In some substances there are two optic axes forming an angle with each other. The optic axis is not a fixed line, it only has a fixed direction; for if a crystal of Iceland spar be divided into smaller crystals, each will have its optic axis; but if all these pieces be put together again, their optic axes will be parallel to A X. Every line, therefore, within the crystal parallel to A X is an optic axis; but as these lines have all the same direction, the crystal is still said to have but one optic axis.

Note 208, p. 184. If I C, fig. 48, be the incident and C S the reflected rays, then the particles of polarised light make their vibrations at right angles to the plane of the paper.

Note 209, p. 184. Let A A, fig. 48, be the surface of the reflector, I C the incident and C S the reflected rays; then, when the angle S C B is 57°, and consequently the angle P C S equal to 33°, the black spot will be seen at C by an eye at S.

Note 210, p. 185. Let A B, fig. 48, be a reflecting surface, I C the incident and C S the reflected rays; then, if the surface be plate-glass, the angle S C B must be 57°, in order that C S may be polarised. If the surface be crown-glass or water, the angle S C B must be 56° 55' for the first, and 53° 11' for the second, in order to give a polarised ray.

Note 211, p. 186. A polarising apparatus is represented in fig. 64, where R r is a ray of light falling on a piece of glass r at an angle of 57°: the reflected ray r s is then polarised, and may be viewed through a piece of tourmaline in s, or it may be received on another plate of glass, B, whose surface is at right angles to the surface of r. The ray r s is again reflected in s, and comes to the eye in the direction s E. The plate of mica, M I, or of any substance that is to be examined, is placed between the points r and s.

Note 212, p. 187. In order to see these figures, the polarised ray r s, fig. 64, must pass through the optic axis of the crystal, which must be held as near as possible to s on one side, and the eye placed as near as possible to s on the other. Fig. 65 shows the image formed by a crystal of Iceland spar which has one optic axis. The colours in the rings are exactly the same with those of Newton’s rings given in Note 199, and the cross is black. If the spar be turned round its axis, the rings suffer no change; but if the tourmaline through which it is viewed, or the plate of glass, B, be turned round, this figure will be seen at the angles 0°, 90°, 180°, and 270° of its revolution. But in the intermediate points, that is, at the angles 45°, 135°, 225°, and 315°, another system will appear, such as represented in fig. 66, where all the colours of the rings are complementary to those of fig. 65, and the cross is white. The two systems of rings, if superposed, would produce white light.

Note 213, p. 188. Saltpetre, or nitre, crystallises in six-sided prisms having two optic axes inclined to one another at an angle of 5°. A slice of this substance about the 6th or 8th of an inch thick, cut perpendicularly to the axis of the prism, and placed very near to s, fig. 64, so that the polarised ray r s may pass through it, exhibits the system of rings represented in fig. 67, where the points C and C mark the position of the optic axes. When the plate B, fig. 64, is turned round, the image changes successively to those given in figs. 68, 69, and 70. The colours of the rings are the same with those of thin plates, but they vary with the thickness of the nitre. Their breadth enlarges or diminishes also with the colour, when homogeneous light is used.

Note 214, p. 189. Fig. 71 represents the appearance produced by placing a slice of rock crystal in the polarised ray r s, fig. 64. The uniform colour in the interior of the image depends upon the thickness of the slice; but whatever that colour may be, it will alternately attain a maximum brightness and vanish with the revolution of the glass B. It may be observed, that the two kinds of quartz, or rock crystal, mentioned in the text, are combined in the amethyst, which consists of alternate layers of right-handed and left-handed quartz, whose planes are parallel to the axis of the crystal.

Note 215, p. 193. Suppose the major axis A P of an ellipse, fig. 18, to be invariable, but the excentricity C S continually to diminish, the ellipse would bulge more and more; and when C S vanished, it would become a circle whose diameter is A P. Again, if the excentricity were continually to increase, the ellipse would be more and more flattened till C S was equal to C P, when it would become a straight line A P. The circle and straight line are therefore the limits of the ellipse.

Note 216, p. 194. The coloured rings are produced by the interference of two polarised rays in different states of undulation, on the principle explained for common light.

Note 217, p. 225. According to Mr. Joule, that heat is produced by motion, and that it is equivalent to it, Mr. Thompson of Glasgow investigates from whence the sun derives his heat, since he shows that neither combustion nor his primitive heat could have supplied the waste during 6000 years. He concludes that the solar heat is maintained by myriads of minute bodies that are revolving at the edge of his dense nebulosity or atmosphere, some of which are often seen by us as falling stars. These, vaporized by his heat, and drawn by his attraction, meet with intense resistance on entering the solar atmosphere as a shower of meteoric rain; through it they descend in spiral lines to the sun’s surface, producing enormous heat by friction during their fall, and serving for fuel on their arrival.

Note 218, p. 252. The class Cryptogamia contains the ferns, mosses, funguses, and sea-weeds; in all of which the parts of the flowers are in general too minute to be evident.

Note 219, p. 254. Zoophytes are the animals which form madrepores, corals, sponges, &c.

Note 220, p. 254. The Saurian tribe are creatures of the crocodile and lizard kind.

Note 221, p. 266. If heat from a non-luminous source be polarised by reflection or refraction at r, fig. 64, the polarised ray r s will be stopped or transmitted by a plate of mica M I, under the same circumstances that it would stop or transmit light; and if heat were visible, images analogous to those of figs. 65, 67, &c., would be seen at the point s.

Note 222, pp. 275, 329, 357. The foot-pound, or unit of mechanical force established by Mr. Joule, is the force that would raise one pound weight of matter to the height of one foot; or it is the impetus or force generated by a body of one pound weight falling by its gravitation through the height of one foot.

Impetus, vis viva, or living force, is equal to the mass of a body multiplied by the square of the velocity with which it is moving, and is the true measure of work or labour. For if a weight be raised 10 feet, it will require four times the labour to raise an equal weight 40 feet. If both these weights be allowed to descend freely by their gravitation, at the end of their fall their velocities will be as 1 to 2; that is, as the square roots of their heights; but the effect produced will be as their masses multiplied by 1 and 4; but these are the squares of their velocities: hence the impetus or vis viva is as the mass into the square of the velocity.

Thus impetus is the true measure of the labour employed to raise the weights, and of the effect of their descent, and is entirely independent of time. Now heat is proportional to impetus, and impetus is the true measure of labour. In percussion the heat evolved is in proportion to the force of the impetus, and is thus measured by labour.

Travail is a word used in mechanics, to express that work done is equal to the labouring force employed. The work done may be resistance overcome or any other effect produced, while the labouring force may be a horse, a steam-engine, wind, falling water, &c.

Note 223, p. 313. When a stream of positive electricity descends from P to n, fig. 72, in a vertical wire at right angles to the plane of the horizontal circle A B, the negative electricity ascends from n to P, and the force exerted by the current makes the north pole of a magnet revolve about the wire in the direction of the arrow-heads in the circumference, and it makes the south pole revolve in the opposite direction. When the current of positive electricity flows upwards from n to P, these effects are reversed.

Note 224, p. 314. Fig. 73 represents a helix or coil of copper wire, terminated by two cups containing a little quicksilver. When the positive wire of a Voltaic battery is immersed in the cup p, and the negative wire in the cup n, the circuit is completed. The quicksilver ensures the connection between the battery and the helix, by conveying the electricity from the one to the other. While the electricity flows through the helix, the magnet S N remains suspended within it, but falls down the moment it ceases. The magnet always turns its south pole S towards P, the positive wire of the battery, and its north pole towards the negative wire.

Note 225, p. 316. A copper wire coiled in the form represented in fig. 73 was the first and most simple form of the electro-dynamic cylinder. When its extremities P and n are connected with the positive and negative poles of a Voltaic battery, it becomes a perfect magnet during the time that a current of electricity is flowing through it, P and n being its north and south poles.

Note 226, p. 344. It is to Halley we are indebted for the first declination chart and the theory of 4 poles of maximum magnetic intensity, since confirmed by observation, as well as the earliest authentic values of the magnetic elements in London and St. Helena, where he went on purpose to make observations on terrestrial magnetism. Since that time M. Gauss has formed charts of the magnetic lines, and published a theory which very nearly represents the magnetic state of the globe. The mass of observations daily making by our cruizers and our Government surveys in every part of the earth is enormous.

Note 227, p. 360. In fig. 74 the hyperbola H P Y, the parabola p P R, and the ellipse A E P L, have the focal distance S P, and coincide through a small space on each side of the perihelion P; and, as a comet is only visible when near P, it is difficult to ascertain which of the three curves it moves in.

Note 228, p. 363. In fig. 75, E A represents the orbit of Halley’s comet, E T the orbit of the earth, and S the sun. The proportions are very nearly exact.

Note 229, p. 382. Fig. 74 represents the curves in question. It is evident that, for the same focal distance S P, there can be but one circle and one parabola p P R, but that there may be an infinity of ellipses between the circle and the parabola, and an infinity of hyperbolas H P Y exterior to the parabola p P R.

Note 230, p. 387. Let A B, fig. 26, be the diameter of the earth’s orbit, and suppose a star to be seen in the direction A S' from the earth when at A. Six months afterwards, the earth, having moved through half of its orbit, would arrive at B, and then the star would appear in the direction B S', if the diameter A B, as seen from S', had any sensible magnitude. But A B, which is 190,000,000 of miles, does not appear to be greater than the thickness of a spider’s thread, as seen from 61 Cygni, supposed to be the nearest of the fixed stars.

Note 231, p. 389. Stars whose parallax and proper motions are known.

The space run through in one second by these stars is therefore—

There are three great discrepancies in the parallax of the star Argelander or 1830 Groombridge. M. Otto Struve makes it 0·034, which gives it a velocity of 251 leagues per second, while M. Faye finds the parallax to be between 0·03 and 0·01, which makes its velocity from 30 to 85 leagues per second.

These are all minimum velocities, because we can only determine on the celestial vault a projection perhaps much foreshortened of the real motions of the stars.

Note 232, pp. 398, 401. The following are the binary systems whose orbits have been accurately determined:—

Note 233, p. 403. The mass is found in the manner explained in the text; but the method of computing the distance of the star may be made more clear by what follows. Though the orbit of the satellite star is really and apparently elliptical, let it be represented by C D O, fig. 14, for the sake of illustration, the earth being in d. It is clear that, when the star moves through C D O, its light will take longer in coming to the earth from O than from C, by the whole time it employs in passing through O C, the breadth of its orbit. When that time is known by observation, reduced to seconds, and multiplied by 190,000, which is the number of miles light darts through in a second, the product will be the breadth of the orbit in miles. From this the dimensions of the ellipse will be obtained by the aid of observation; the length and position of any diameter as S p may be found; and as all the angles of the triangle d S p can be determined by observation, the distance of the star from the earth may be computed.

Note 234, p. 405. The mean results of MM. Argelander, Otto Struve, and Luhndahl for stars in the northern hemisphere and the epoch 1790, places the point to which the sun is tending in 259° 5' of right ascension and 55° 23' of north polar distance. Mr. Gallaway computed from stars in the southern hemisphere, at the same epoch, the point to have been in 260° 1' right ascension and 55° 37' north polar distance, results nearly identical, though from very different data.

Note 235, p. 414. One of the globular clusters mentioned in the text is represented in fig. 1, plate 8. The stars are gradually condensed towards the centre, where they run together in a blaze. The more condensed part is projected on a ground of irregularly scattered stars, which fills the whole field of the telescope. There are few stars near this cluster.

Note 236, p. 420. Plate 8 shows five nebulÆ as seen in Sir John Herschel’s 20-feet telescope.

1. An enormous ring seen obliquely with a dark centre and a small star at each extremity.

2. The ring in the constellation Lyra.

3. The dumb-bell nebula in Vulpicula.

4. The spiral nebula or brother system in the 20-feet telescope.

5. A spindle-shaped nebula.

Plate 9 represents some of the same objects as seen by Lord Rosse.

1. Nebula in the girdle of Andromeda.

2. The circular nebula of Lyra.

3. The dumb-bell nebula in Vulpicula.

The spiral nebulÆ of 51 Messier, as seen by Lord Rosse, 1 in plate 10, represents fig. 4 of plate 8; and fig. 2 in the same plate is part of the great nebula in Orion, for the whole has never been seen, on account of extreme remoteness.

Note 237, pp. 32, 427. The motion of the earth is visibly proved by M. Foucault’s experiments. If a pendulum be left to oscillate quite freely, the forces producing the oscillations being in the vertical plane, there is no cause that can produce an absolute change in its position with regard to space; but the motion of the earth changes the position of a spectator with respect to the vertical plane, and he refers his own motion to it, which seems gradually to turn away from its position, precisely as a person in a boat refers his own motion to that of the land, and thus the motion of the earth is truly and visibly proved.