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It is to be observed that the product of two quantities implies that at least one of the quantities is an abstract number. We cannot multiply dollars by dollars, or lines by lines. Still we may speak of the product of two lines, if we understand by the expression the product of the numbers which represent the lines when they are measured by a common unit.

Thus in the rectangle of the lines A B and A C, suppose A L to be a common measure of A B and A C, and to be contained in A B seven times, and in A C four times.

At the several points of division on AC draw lines perpendicular to AC,

and at the several points of division on A B draw lines perpendicular to A B.

Then the rectangle will be divided into squares, all equal to each other. There will be seven in a row, and four rows; that is four times seven squares.

PROPOSITION II. 260. A mean proportional between two quantities is equal to the square root of their product. In the proportion a : 6::6:c, 62 = ac,

$ 259 (the product of the extremes is equal to the product of the means). Whence, extracting the square root, b=Vac.

Q. E. D.

PROPOSITION III.

261. If the product of two quantities be equal to the product of two others, either two may be made the extremes of a proportion in which the other two are made the means.

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Divide both members of the given equation by bd.

Then

or,

a : 6 :: 0 :d.

Q. E. D.

PROPOSITION IV.

262. If four quantities of the same kind be in proportion, they will be in proportion by alternation.

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ad= b c,

§ 259 (the product of the extremes is equal to the product of the means).

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PROPOSITION V. 263. If four quantities be in proportion, they will be in proportion by inversion.

. Let a : 6::8: d.
We are to prove 6:a::d : c.

bc=ad,
(the product of the extremes is equal to the product of the means).
Divide by a c.

Then

or,

b: a ::d :c.

Q. E. D.

PROPOSITION VI.

264. If four quantities be in proportion, they will be in proportion by composition.

Let a : 6 ::C:d
We are to prove a + b : 6 ::c+d: do

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or,

a+b :b::c+d: d.

Q. E. D.

PROPOSITION VII. 265. If four quantities be in proportion, they will be in proportion by division.

Let a :b :: 0 :d.
We are to prove a b : 6 ::c-d : d.
Now

Subtract 1 from each member of the equation.
Then

6-1=: -1,

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PROPOSITION VIII. 266. In a series of equal ratios, of which all the terms are of the same kind, the sum of the antecedents is to the sum .. of the consequents as any antecedent is to its consequent.

Let a : b=c:d=e: f=g : h..
We are to prove a+c+e+g:6+d+f+h::a:b.
Denote each ratio by r.
Then r= = * = =
Whence, a=br, c=dr, e=fr, g=hr.
Add these equations.
Then a +c+e+ g = (b + d + f + h) r.
Divide by (b + d + f + h).
Then

a totetg a

6 + d + f + h *
or, atctetg : b + d + f + h :: a : b.

Q. E, D,

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PROPOSITION IX. 267. The products of the corresponding terms of two or more numerical proportions are in proportion.

Let a : 6 :: 0 :d,

e:f::9 :h,

k :1::m:n,
We are to prove aek : bfl :: cgm : dh n.

k m
Now
Whence by multiplication,

aek com

bfl - dhn'
or,
a ek : bfl :: cgm : dh n.

Q. E, D. PROPOSITION X. 268. Like powers, or like roots, of the terms of a proportion are in proportion.

Let a : 6 ::: d.
We are to prove an : 61 :: cn : d",

and

Now

By raising to the nth power,

an

in = in; or an : bn : :cn : d». By extracting the nth root,

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Q. E. D. 269. DEF. Equimultiples of two quantities are the products obtained by multiplying each of them by the same number. Thus ma and mb are equimultiples of a and b.

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