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to lines, or of numbers to numbers, may be expreffed by lines and numbers, and therefore by algebraical quantities. Hence thefe mathematical notations may be confidered as the meafures of fuch physical quantities; they may be reasoned upon according to the principles of algebra, and from fuch reasonings, new relations of the quantities which they reprefent, may be discovered.
In those branches of natural philosophy, therefore, in which the circumftances of the phenomena can be properly expreffed by numbers, or geometrical magnitudes, algebra may be employed, both in promoting the investigation of phyfical laws by expe rience, and alfo in deducing the neceffary confequences of laws investigated, and prefumed to be just.
It is to be obferved likewife, that if various hypothefes be affumed, concerning phyfical quantities, without regard to what takes place in nature, their confequences may be demonftratively deduced; and thus a fcience may be eftablished, which may be properly called mathematical. The use of
algebra in this science, which is fometimes called Theoretical Mechanics, is obvious from the principles already laid down.
In conducting these inquiries, it is to be obferved, that, for the fake of brevity, the language of algebraical operations is often used, with regard to phyfical 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 explain a method of ftating the proportion of variable quantities, and reasoning with regard to it; which is of general ufe in na tural philofophy.
1. Of the Proportion of Variable Quantities.
Mathematical quantities are often fo 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, interfect each other; and, if a straight line, cutting both, moves parallel to itself, the two fegments of the given lines between their interfection, and the moving line, however varied, will always have the fame proportion. Thus alfo, if an ordinate to the diameter of a parabola move parallel to itself, the abfcifs will be increased or diminished, in proportion as the fquare of the ordinate is increased or diminished.
In like manner may algebraical quantities be connected. If x, y, z, &c. reprefent any variable quantities, while a, b, c, reprefent fuch as are conftant 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 axby, then x:y::b:a, that is, x has a conftant proportion to y, whatever way thefe two quantities may be varied. Likewise, if xy2=a2b, then y2 : a2 :x, or y2: y2: 1 :: a2:—, that is, y2 has a
conftant proportion to the reciprocal of x, or y2 is increafed in the fame proportion as
x is diminished, and conversely. It is neceffary to premise the following definitions.
Let there be any number of variable quantities X, Y, Z, V, &c. connected in fuch a manner, that, when X becomes x, Y, Z, V, &c. become refpectively y, z, v, &c. And let a, b, c, &c. reprefent any conftant quantities, whether given or unknown. Then,
1. If two variable quantities X and Y are fo connected, that, whatever be the values of x and X:x:::y, this proportion is expreffed thus, X=Y, and X is faid to be directly as Y, or fhortly, X is said to be as Y.
2. If two variable quantities X and Y are fo connected, that X: xy: Y, or X:x::
their relation is thus expreffed,
X ; and X is faid to be inversely, or
reciprocally as Y.
3. If X, Y, Z, are three variable quantities, fo connected that X: x :: YZ : yz,
relation is fo expreffed, X=rZ, and X is faid to be directly as Y and Z, jointly; or X is faid to be as Y and Z.
4. If any number of variable quantities
as X, Y, Z, V, &c. are fo connected, that
is faid to be directly as YZ, and inversely as V, or more explicitly, X and Y jointly, are directly 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 obferved also, the fame definitions take place, when the variable quantities are multiplied or divided by any conftant quantities. Thus, if aX : ax ::
5. Let the preceding notation of proportion be called a proportional equation *,
* These terms are ufed only with a view to give more precision to the ideas of beginners. In order to avoid the ambiguity in the meaning of the fign=, fome writers employ the character x to denote constant proportion; but this is feldom neceffary, as the quan-tities compared are generally of different kinds, and the relation expreffed is fufficiently obvious. See Emer fon's Mathematics, Vol. I.