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Again, 12: 4 :: 9: 3, and 12:9:: 8: 6,

Then 12±8:4:: 9±6:3. Hence, If to or from two homologous terms, &c.

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If four magnitudes be proportional, they will also be proportional by "composition;" that is, the sum of the first and second is to the second, as the sum of the third and fourth is to the fourth.

If A B C D, then, by composition, A+B B:: C+D: D;

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Again, 12 38: 2; then by composition 12+3: 38+22. Hence, If four magnitudes, &c.

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If four magnitudes be proportional, they will also be proportional by "division;" that is, the difference of the first and second is to the second, as the difference of the third and fourth to the fourth.

If A: B::C:: D, then by division, A-B: B:: C-D: D;

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Again, 12 : 3 - : 8:2; then, by division, 12-3:3::8—2:2. Hence, If four magnitudes, &c.

Cor. The sum of the first and second is to their difference, as the sum of the third and fourth is to their difference.

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Therefore, (8.3,) A+B : A-B:: C+D: C-D.

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If the corresponding terms of two or more ranks of proportionals be multiplied together, the products will be proportional.

If

A:B::C: D

And E: F::G: H, then AE: BF :: CG: DH.

A C

For, since by hypothesis, BD' and F=H'

E G

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Again, 123: 8:2

And

10:56: 3,

then 12×10:35::8×6:2×3.

In like manner it may be shown that the products of the corresponding terms of three, or any number of ranks of proportionals, are also proportional. Hence, If the corresponding terms, &c.

Cor. In compounding proportions equal factors in two analogous, or two homologous terms, may be rejected.

If A : B :: C: D

And B: H:

Hence,

Scholium.

DM, then A: H:: C : M.

For, AB BH :: CD: DM; (13. 3. Cor.,) A : H :

: C: M.

Compounding proportions should be distinguished from composition, which is an addition of the terms of a ratio. (16. 3.)

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If the terms of one rank of proportionals be divided by the corresponding terms of another rank, the quotients will be pro

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Scholium. Dividing the terms of one rank of proportionals by those of another, is called the “resolution of ratios." It must not be confounded with what is called division, which is a subtraction of one of the terms of a ratio from the other. (17. 3.)

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If four magnitudes are proportional, their like powers, or like roots, are also proportional.

If A B C D, then A2 : B2 :: C2 : D2, and A" : : C: D".

B":

:

Since A B C : D, we have, (5. 3,) AD=BC. Hence, (Ax. 6,) AD × AD=BC × BC; that is, A3D2=B'C'; Consequently, (6. 3,) A2 : B2 : : C2 : D2.

In a similar manner it may be proved that

A: B": : C" : D".

Again, if A : B :: C: D, then √A: √B:: √C: √D;

n

n

also VA: VB:: C: VD..

For, since A: B:: C: D, therefore AxD=B×C; hence, taking like roots of each,

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Consequently, (6. 3,) √A : √B:: √C: √D;

n

n

n

And VA : B :: VC: VD.

Likewise, 3: 46: 8, then 32 : 42: : 6a : 82.
For, 9x64-16x36.

And, 9 4 36: 16, then √9: √4:: √36: √16.
That is, 3 2 :: 6 : 4.

Scholium. It must not be inferred from this proposition that like powers, or roots of two couplets have the same ratio as the magnitudes have before they are involved, or their root extracted. (Def. 7.)

BOOK IV.

THE PROPORTIONS OF FIGURES, AND THE MEAS

UREMENT OF SURFACES.

DEFINITIONS.

1. Equal figures are those, which being applied to each other, coincide in all their parts; (Ax. 13;) as two circles, which have equal radii; two triangles, which have all their sides respectively equal, &c.

2. Equivalent figures are those which have equal surfaces. Note. Two figures may be equivalent though very dissimilar. A circle, e. g. may be equivalent to a square, a triangle to a rectangle, &c.

3. Similar figures are those which have their angles respectively equal each to each, and their homologous sides proportional.

Note. Homologous sides are those which have a corresponding position in the two figures, or which are adjacent to equal angles. Those angles are called homologous angles.

4. In two different circles, similar arcs, sectors, or segments, are those which correspond to equal angles at the centre.

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