to lines, or of numbers to numbers, may be expressed by lines and numbers, and therefore by algebraical quantities. Hence these mathematical notations may be considered as the meafures of such physical quantities; they may be reasoned upon according to the principles of algebra, and from such reasonings, new relations of the quantities which they represent, may be discovered. In those branches of natural philofophy, therefore, in which the circumstances of the phenomena can be properly expressed by numbers, or geometrical magnitudes, algebra may be employed, both in promoting the investigation of physical laws by experience, and also in deducing the necessary consequences of laws investigated, and presumed to be juft. It is to be observed likewise, that if various hypotheses be assumed, concerning physical quantities, without regard to what takes place in nature, their consequences may be demonstratively deduced; and thus a science may be established, which may be properly called mathematical. The use of algebra 70 algebra in this science, which is sometimes called Theoretical Mechenics, is obvious from the principles already laid down. In conducting these inquiries, it is to be observed, that, for the sake of brevity, the language of algebraical operations is often used, with regard to physical quantities themselves ; though it is always to be understood, that, in strict propriety, it can be applied only to the mathematical notations of these quantities. Before illustrating this application of algebra by examples, it may be proper to exa plain a method of stating the proportion of variable quantities, and reasoning with regard to it; which is of general use in natural philosophy 1. Of the Proportion of Variable Quantitiesa Mathematical quantities are often so connected, that when the magnitude of one is varied, the magnitudes of the others are varied, according to a determined rule. Thus, if two straight lines, given in pofi tion, intersect each other; and, if a straight line, cutting both, moves parallel to itself, the two segments of the given lines between their intersection, and the moving line, however varied, will always have the same proportion. Thus also, if an ordinate to the diameter of a parabola move parallel to itself, the absciss will be increased or diininished, in proportion as the square of the ordinate is increased or diminished. In like manner may algebraical quantities be connected. If x, y, %, &c. represent any variable quantities, while a, b,c, represent such as are constant or invariable, then an equation containing two or more variable quantities, with any number of conftant quantities, will exhibit a relation of variable quantities, fimilar to those already mentioned. Thus, if ax=by, then x:y::b:a, that is, x has a constant proportion to y, in whatever way these two quantities may be varied. Likewise, if xy’=a2b, then yż: a ::6:x, or ya : : a*:, that is, yż has a constant proportion to the reciprocal of x, or g2 is increased in the same proportion as U x is diminished, and conversely, It is necessary to premise the following definitions. DEFINITION S. Let there be any number of variable quantities X, Y, Z,V, &c. connected in such a manner, that, when X becomes X, Y, Z, V, &c. become respectively y, %, v, &c. And let a, b, c, &c. represent any constant quantities, whether given or unknown. Then, 1. If two variable quantities X and Y are so connected, that, whatever be the values of x and y, X:x:: Y:y, this proportion is expressed thus, X=Y, and X is said to be directly as Y, or shortly, X is said to be as r. 2. If two variable quantities X and Y are so connected, that X: x::y:Y, or X : x :: their relation is thus expressed, r у X=1; and X is said to be inversely, or reciprocally as r. 3. If X, Y, Z, are three variable quantities, fo connected that X:*::YZ:y%, their relation I I ΥΖ y relation is fo expressed, X=rz, and X is said to be direetly as Y and Z, jointly; or X is said to be as Y and Z. 4. If any number of variable quantities as X, Y, Z, V, &c. are so connected, that rz XY:xy :: then XY= and XY V V is said to be directly as YZ, and inversely as V, or more explicitly, X and Y jointly, are direčily as Y and Z jointly, and inversely as V. In like manner are other combinations of variable qualities denoted and expressed. It is to be observed also, the fame definitions take place, when the variable quantities are multiplied or divided by any constant quantities. Thus, if aX : ax :: b b b then ax = &c. rij r 5. Let the preceding notation of proportion be called a proportional equation *, the equa* These terms are used only with a view to give more precision to the ideas of beginners. In order to avoid the ambiguity in the meaning of the sign =, son.e writers employ the character a to denote constant proportion; but this is seldom necessary, as the quantities compared are generally of different kinds, and the relation expressed is sufficiently obvious. See Emerson's Mathematics, Vol I. |