equations formerly treated of being in this place, for the fake of diftinction, called abfolute.

COR. Every abfolute equation, containing more than one variable quantity, may be confidered as a proportional equation; and, in a proportional equation, if at any particular correfponding values of the variable quantities, the equation becomes absolute, it will be univerfally abfolute.

PROP. I. If one fide of a proportional equation be either multiplied or divided by any constant quantity, it will continue to be

PROP. 2. If the two fides of a proportional equation be both multiplied, or both divided by the fame quantity, it will continue to be true.

ist, If the quantity be constant, it is manifest from Prop. I.

2d,

2d, If the quantity be variable, let X`r, and Z a variable quantity, then XZ= YZ. For,fince X=Y, (Def. 2.) X :x :: Y: y; multiply the antecedents by Z, and the confequents by z, then XZ : xx :: YZ : yz, therefore (Def. 5.) XZ-YZ. In like

COR. Any variable quantity, which is a factor of one fide of a proportional equation, may be made to ftand alone. Thus,

Ꮓ

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if XY==, then X= ; alfo, Z=XYV;

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also, if one fide of a proportional equation be divided by the other, the quotient is a conftant quantity, viz. 1.

PROP. 3. If two proportional equations have a common fide, the remaining two fides will form a proportional equation. Alfo, that common fide will be as the fum or difference of the other two.

Thus, if X=Y, and Y=Z, then X=Z. For X:x::ry, and r: y :: Z: z, therefore multiplying these ratios, XY: xy:: YZ

:

yz, and by dividing antecedents and confequents, Xx: Z: x, therefore, (Def. 1.) X=Z.

r

Likewife, if X=Y, and Y-Z,Y=X±Z. For, fince Xx :: Y: y :: Z: z. (Chap. 2.) Y: y :: X±Z: xx, therefore (Def. 5.) Y=X±2.

COR. Hence one fide of a proportional equation will be as the fum, or as the difference of the two fides; and the fum of the two fides will be as their difference. Thus, if XYZ, then XX++Z and X=X-Y-Z, and alfo X+Y+Z=X−r Z.

PROP. 4. If the two fides of a proportional equation be refpectively multiplied or divided by the two fides of any other proportional equation, the products or quotients will form a proportional equation.

Thus, if X=Y, and Z=V,then XZ=YV. For, fince X:x:: Y: y, and Z: z :: V: v, by r multiplying these proportions (Chap. 1. 2. XZ : x≈ :: YV : yv, therefore (Def. 5.) XZ= YV. In like manner in the cafe of divifion. COR. I. The two fides of a proportional equation may be raised to any power, or,

any

any root may be extracted out of both, and the equation will continue to be true.

Thus, if XY, then XmYm, for, fince X=r, X: x :: ry, and therefore Xm: xm :: rm: : ym; therefore Xm=Ym. And, if X=Y, alfo Xm=rm.

COR. 2 If two proportional equations have a common fide, that fide will be as the square root of the product of the other Thus, if X=Y, and Y-Z, by this prop. YXZ, and (Cor. 1.) Y=√XZ.

two.

Hence alfo, in this cafe,

for (prop. 3.) Y=X÷Z.

XZ=X÷Z;

COR. 3. If one fide of a proportional equation be a factor of a fide of another proportional equation, the remaining fide of the former may be inferted in the latter, in place of that factor. Thus, if X=ZY, and

PROP. 5. Any proportional equation may be made abfolute, by multiplying one fide by a conftant quantity.

Thus,

Thus, if XY, then let two particular correfponding values of thefe variable

quantities be affumed as conftant, and let them be a and b, then X: a :: Y: b, and Xb=a¥, or X=Y×—-, an absolute equation.

SCHOLIU M.

1. If there be two variable phyfical quantities, either of the fame, or of different kinds, which are fo connected, that, when the one is increased or diminished, the other is increased or diminished in the fame proportion; or, if the magnitudes of the one, any two fituations, have the fame ratio to each other, as the magnitudes of the other in the correfponding fituations, the relation of the mathematical measures of these quantities may be expreffed by a proportional equation, according to Def. 1.

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2. If two variable physical quantities be fo connected, that the one increases in the fame proportion as the other is diminished, and conversely; or, if the magnitudes of the one, in any two fituations, be reciprocally proportioned to the magnitudes of the other,

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