few symbols the immutable laws of the universe. This mighty instrument of human power itself originates in the primitive constitution of the human mind, and rests upon a few fundamental axioms, which have eternally existed in Him who implanted them in the breast of man when He created him after His own image. NOTES. 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, NS, 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. Fig. 1. m NOTE 7, p. 4. A hollow sphere. A hollow ball, like a tomb-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, Qq or NS is a diameter, and CQ, CN are radii. A great circle of the sphere has the same centre with the sphere as the circles Q E qd and QNq S. The circle A B is a lesser circle of the sphere. 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 con Fig. 2. trary, 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 Cq, Ca, CN, &c., are generally unequal; whereas in the sphere they are all equal. 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. S NOTE 11, pp. 4, 6. Poles and equator. Let fig. 1 or 3 represent the earth, C its centre, NCS 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, Fig. 3. B G G P the equatorial diameter, q Q, exceeding the polar diameter, NS, by about 26 miles. Lesser circles, A BG, 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 m 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 NQS, NGES, 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 GES be the meridian of Greenwich, the position of any place, B, is determined, when its latitude, QC B, and its longitude, ECQ, are known. 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 NS 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 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 PQ AD, fig. 6, be the orbit or path of the moon or of a planet; then PA is the major axis, C the centre, and CS is equal to CF. Now, since the earth or the sun is supposed to be in the point S according as PDAQ is regarded as the orbit of the moon or that of a planet, SA, SP are the greatest and least distances. But half the sum of SA and SP 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 SQ, SD, are each equal to CP, 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. 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. R Fig. 4. d V K C S rq p h T 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 mdg 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 mn during the time that she would fly off in the tangent m T by the action of the centrifugal force alone, or fall through mp by the earth's attraction alone. Tn, the deflection from the tangent, is parallel and equal to mp, 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. Tn, or mp, 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 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. NOTE 22, pp. 5, 11. Conic sections. Lines formed by any plane cutting A cone is a solid figure, like a sugar-loaf, fig. 5, of which A is a cone. the apex, AD the axis, and the plane B ECF the base. The axis may or may not be perpendicular to the base, and the base may be a circle, or |