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equations formerly treated of being in this place, for the sake of distinction, called absolute.

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

Prop. I. If one side of a proportional equation be either multiplied'or divided by any constant quantity, it will continue to be true. Thus, if X=;, then X .

For, since

or
X =ş (Def. 3.) X:* ::
(X x

it follows, r

у

a2 a2 (Chap. 2.), that X : x ::

therefore

br.by' (Def. 4.) X=

br PROP. 2.

If the two sides of a proportional equation be both multiplied, or both divided by the same quantity, it will continue to be true.

1st, If the quantity be constant, it is manifest from Prop. I.

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I

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2d, If the quantity be variable, let XY, and Z a variable quantity, then XZ= YZ. For,since X=Y, (Def. 2.) X: x ::Y:y; multiply the antecedents by Z, and the consequents by x, then XZ : xz :: YZ : yx, therefore (Def. 5.) XZ=rz.

In like

X r manner, if X=r,

, ž=Z:

Z

Υ

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vr Xū, and also V–

Cor. Any variable quantity, which is a factor of one side of a proportional equation, may be made to stand alone. Thus, Z

z
if XY=ý, then X=ir; also, Z=XYV;

V
Z

Z and r=

= &c. Hence, XV

xr also, if one side of a proportional equation be divided by the other, the quotient is a constant quantity, viz. I. PROP. 3.

If two proportional equations have a common side, the remaining two fides will form a proportional equation. Also, that common side will be as the sum or difference of the other two.

Thus, if X=Y, and r=Z, then X=2.
For X : x :: Y: y, and Y:y:: Z : %,

r
Y

therefore multiplying these ratios, XY : xy :: YZ

: ya,

:

a

T

:yz, and by dividing antecedents and con; sequents, X : x :: Z :%, therefore, (Def. 1.) X=Z.

Likewise, if X=Y, and Y=28=X+2. For, since X:* :: Y:y:: 2:2. (Chap. 2.) Y:y :: XIZ: x£x, therefore (Def. 5.) N=X2.

Cor. Hence one side of a proportional equation will be as the lum, or as the difference of the two sides; and the sum of the two sides will be as their difference. Thus, if X=Y+Z, then X = x+7+Z and X=x-Y-Z, and also X+1 +2=X-Y

Z.

Prop. 4. If the two sides of a proportional equation be respectively multiplied or divided by the two sides of

any

other proportional equation, the products or quo- . tients will form a proportional equation.

Thus, if X=Y, and Z=V, then X2=YV. For, since X : « :: Y: y, and Z:z::V: v, by multiplying these proportions (Chap. 1. 2.

2.) XZ : XZ:: YV: yv, therefore (Def. 5.) XZ= Yv. In like manner in the case of division.

Cor. 1. The two sides of a proportional equation may be raised to any power, or,

any

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two.

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

Thus, if X=Y, then Xm_Ym, for, since X=Y, X: x :: Y : y, and therefore Xm: mm :: Ym : ym; therefore Xm_Ym.

And, if X=r, also xn--rn

COR. 2 If two proportional equations have a common fide, that side will be as the square root of the product of the other Thus, if X=Y, and Y=2, by this

=: Z, prop. Ya=XZ, and (Cor. 1.) ř=vXZ. Hence also, in this case, VXZ=XZ; for (prop. 3.) r=XZ.

If one side of a proportional equation be a factor of a side of another proportional equation, the remaining side of the former may be inserted in the latter, in

, place of that factor. Thus, if X=ZY, and

r
Z=j, then X=
V

V?

as appears by multiplying the two equations, and dividing by 2.

PROP. 5. Any proportional equation may be made absolute, by multiplying one side by a constant quantity.

Thus,

COR. 3.

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Thus, if X=Y, then let two particular corresponding values of thete variable

quantities be assumed as constant, and let them be a and b, then X:a:: Y:b, and Xb=ay, or X=rx , an absolute equation. b

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

2. If two variable physical quantities be so connected, that the one increases in the same proportion as the other is diminished, and conversely; or, if the magnitudes of the one,

in any two situations, be reciprocally proportioned to the magnitudes of the other,

in

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