Proportion is the equality of ratios: hence, there must, at least, be three numbers or quantities to form a Proportion. The width of the narrower figure No. 1, is 6; and has a ratio to 9, the width of the wider, but equal figure No. 2, of rds. But, since the areas of the two figures are equal, their lengths must bear to each other the same ratio, but invertedly, of their widths. Accordingly, 4, the length of the wider, has to 6, the length of the narrower figure, the same ratio, or rds. Hence, we have the proportion expressed thus 6:9: 4:6. s A Triangle, or Trigon, is a figure inclosed by three straight lines. Triangles are distinguished as under : The Scalene, or unequal-sided triangle. The Equilateral, or triangle of three equal sides. They are likewise named from their angles, Obtuse Angled. Right Angled. Acute Angled. 9 T All the three angles of any Triangle, amount exactly to two right angles or 180°. This may easily be verified, in any particular instance, by describing arcs from the angular points as centres, (as in figure Q,) and setting off these arcs on a circumference drawn with the same opening of the compasses, as in fig. q.: they will be found, invariably, to measure off half of that circumference. Hence, no triangle can have more than one of its angles a right angle; and, if one of them is a right angle, each of the remaining two is the Complement (N on page 2,) of the other. V Any one of the sides of a triangle may be considered as the Base, according to the position in which we view it. W If a triangle be Isosceles, or have two of its sides equal, the angles at its base will be equal. DEFINITIONS, ILLUSTRATIONS, ETC. 13 That is, each equal side is supported by the other equal side, at an equal inclination. As the two sloping portions of the roofing of a house, which, if equal, will necessarily incline equally over the ceiling of the upper story. Referring to T p. 12, if the angle made at the uppermost part or ridge of the roof A be 90° or a right angle (M p. 2); then, since all the three angles equal twice 90°, and the other two angles are equal, each of the latter must contain 45°. × When the sun or moon is elevated 45°, the shadow of any object will be exactly equal to the perpendicular height of that object; and may be measured instead of it. For, (as in the figure) the extremity of the object, A, is in the same plane, and also in the same straight line, with the sun and with the extremity of its shadow, B: and since the horizontal shadow, and this plane or the straight line A B, make an angle of 45°; and the upright object itself, forming the other side of a triangle, makes 90° with the ground or shadow, the angle at the top, A, must also be 45°; and thus the shadow and the height of the object correspond in length. 3 A Luminary of equal brightness all around, such as the sun, or a star, or a ball of fire in the higher regions of the atmosphere, will send forth a globe of rays of light or heat. The influence of these rays will be intense in the inverse ratio of the squares of the distances of any objects upon which they fall from that luminary. This will, perhaps, be more readily understood if we express it thus: The diffusiveness and consequent weakness of the influence of such rays of light or heat, must be considered by comparing the squares of the distances of any objects enlightened or warmed by them, from the source of such rays. C 1 Illustration. Let us suppose a taper could be so placed as that its flame should occupy the exact centre of several concentric and transparent globes, (as C in fig. below). It may be perceived that the influence (as brightness or warmth) of the rays we have drawn, and, consequently, of all lying within them, will be diffuse and weak in proportion to the area or extent of surface over which they fall or spread. The globes drawn have their radii as the numbers 1, 2, and 3; but it will be understood that the portions of the surfaces enlightened, will be as the squares of these distances from the centre or flame; viz., that on the smallest globe the rays comprised within the lines we have drawn from the candle to the surface, occupy one space; but on the next a fourfold space, and on the outermost a ninefold space. Now 1, 4, and 9, are, respectively, the squares of the numbers or distances 1, 2, and 3. If we could place an attracting body, as a magnetic ball, at this centre, instead of the taper flame; or, if we suppose a sounding body to be stationed there, as a buzzing bee; the weakness of the attracting force, or the weakness of the sound (impediments being removed) would follow the same law. This is, in fact, the great law of gravity, and of any influence proceeding from a point. D We may remember, from the foregoing illustration, that the Surfaces of Spheres, are in proportion to the squares of their radii, or of their diameters. Thus, our two feet wide globe has a surface four times that of a one foot wide globe, and our three feet wide globe, a ninefold surface and the mapping of these globes will, of course, follow these ratios; for each of the several portions we have drawn as enlightened by the taper, is contained the same number of times in its respective globe-surface. F The Solid Contents of Globes of equal densities must, however, be considered by comparing the cubes of their diameters, like the contents of all solid bodies that are similar" to one another. Thus, a two feet globe has a DEFINITIONS, ILLUSTRATIONS, ETC. 15 solidity eight times; a three-feet globe twenty-seven times; and a four-feet globe sixty-four times that of a one-foot globe. Circles (circular spaces) must, in like manner, be compared by considering the squares of their radii, or of their diameters. Illustration. The three following are Quadrants, (quarters of circular areas) described with the radii 1, 2, and 3, respectively, and included in squares constructed on those radii. We think the Pupil will allow, without difficulty, that each of the larger quadrants, with its including square, is only a magnified representation of that which precedes it; and, therefore, bears to it the ratio which the square of its radius bears to the square of the radius of the other. Thus, the third quadrant bears to the first quadrant, the ratio which 9 (the square of 3, its radius,) bears to 1, the square of 1; and to the second quadrant, the ratio which 9 bears to 4, the square of 2. Hence, the circles of these quadrants are large in the same ratio. H The quantity of light received from an object into each of two telescopes, will, therefore, be in the ratio of the squares of the diameters of the tubes, or of the glasses called the "object-glasses" at their ends; or, if the telescopes be of the reflecting construction, in the ratio of the squares of the diameters of the reflectors; and the internal arrangements being equally good, their powers of revealing the features of an object, will be in that ratio. B The earth reflects the sun's rays upon the moon (as she upon the earth); so that, with the unassisted sight, we can frequently perceive the darker portion of her disk. Dr. Herschel's ten feet reflector, which has, perhaps, an aperture of fifteen inches diameter, collected sufficient of this earth-light, to show him even the most obscure of her spots on that portion. C It was formerly usual to form deep well-holes in observatories, for the purpose of seeing the stars in the day time; because in the old uncorrected object-glass of telescopes, all the more oblique rays, that is, those admitted into the object-glass at a considerable distance from its centre, as those beyond a and p, (see fig. below,) were decomposed in passing through it, into their parts or colours, (see 'Refraction,") and, causing distortion and confusion, were necessarily "stopped out." The improvements introduced by Dollond and others,* in the construction of the object glasses of telescopes, now allow of their apertures being so enlarged, (see A P,) as to admit the light from a star in quantity sufficient (even when mixed with strong daylight,) to reveal the star to the eye in the focus of the instrument. A large star may be thus seen, even when near the sun; and Dr. Dick informs us, in his "Celestial Scenery," that he has frequently observed the planet Venus, even when at her superior conjunction; i. e., just northward or southward of the sun, and at her greatest distance from us. D The planet Jupiter, at its more than fivefold distance from the sun, receives, on any portion of its surface, less than one twenty-fifth of the light which a similar portion of our own planet's surface receives. (C on. p. 13.) If imagination can enable us to furnish its inhabitants with eyes, the pupils of which shall have a six-fold width, and consequently present a thirty-six fold surface for the reception of light, we may conceive of them as admitting to nerves, fitted to receive and enjoy it, a daily splendour far exceeding our own. A C E H B If we now drop a perpendicular, D E, from D, the extremity of the measuring arc, upon C B, (the radius with which it was formed), * Mr. Ross of Regent Street has very recently made material improvements. |