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And, since

A:B::E F.

A E we have

BF с E

A But and D being severally equal to must be equal to

B each other, and therefore

C:D::E:F. Cor. If the antecedents of one proportion are equal to the antecedents of another proportion, the consequents are proportional. If

A:B::C:D and

A:E::C:F; then will

B:D::E:F. For, by alternation (Prop. III.), the first proportion becomes

A:C::B:D, and the serond, A:C::E:F. Therefore, by the proposition,

B:D::E:F.

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A:

If four quantities are proportional, they are also proportional when taken inversely. Let

B::C:D; then will

B:A::D:C. For, since

A:B::C:D by Prop. I.,

AxD=BXC, or,

BxC=AXD; therefore, by Prop. II.,

B:A::D:C.

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If four quantities are proportional, they are also proportion. al by composition.

Let

A:B::C:D then will

A+B:A:: C+D.C. For, since

A:B::C:D, by Prop. I.,

BXC=AXD. To each of these equals add

AxC=A1C, then

A XC+BXC=AXC+AXD,

(A+B) XC=A x(C+D). Therefore, by Prop. II.,

A+B:A::C+D: C.

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If four quantities are proportional, they are also proportion al by division. Let

A:B::C:D; then will

A-B:A::C-D: C. For, since

A:B::C:D, by Prop. I.,

BxC=AXD. Subtract each of these equals from AXC; then

AXC-BxC=AXC-AxD, or,

(A — B) RC=AX(C—D). Therefore, by Prop. II.,

A —B:A::C-D:C. Cor. A+B:A-B::C+D:C-D.

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Equimultiples of two quantities have the same ratio as the quantities themselves.

Let A and B be any two quantities, and mA, mB their equimultiples; then will

A:B:: MA : mB. For

mx A x B=mX AXB, or,

AXmB=BXmA. Cherefore, by Prop. II.,

A:B::MA: mB.

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If any number of quantities are proportional, any one anle cedent is to its consequent, as the sum of all the antecedents, is to the sum of all the consequents. Let

A:B::C:D::E:F, &c.; then will A:B::A+C+E:B+D+F For, since

A:B::C:D, we have

AxD=BxC. And, since

A:B::E:F, we have

AXF=BXE. To these equals add

AXB=AXB

and we have

AXB+AD+AxF=AXB+B XC+BXE; or,

A (B+D+F)=BX(A+C+E). Therefore, by Prop. II.,

A:B :: A+C+E:B+D+F.

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If four quantities are proportional, their squares or cubes are also proportional. Let

A:B ::C:D; then will

A’: B’ ::C: D’, and

AS: B: ::C: D'. For, since

A:B::C:D, hy Prop. I.,

AXD=BxC; or, multiplying each of these equals by itself (Axiom 1), we have

A'xD'=B'XC; and multiplying these last equals by AxD=B*C, we have

A® XD'=B*XC%. Therefore, by Prop. II.,

A': B'::C: D', and

Al: B::C:: D'.

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If there are two sets of proportional quantities, the products of the corresponding terms are proportional. Let

A:B::C:D, and

E:F::G:H; then will AXE:BF::CXG:DXH. For, since

A:B::C:D by Prop. I.,

AXD=BxC. And, since

E:F::G:H, by Prop. I.,

EXH=FxG.
Multiplying together these equal quantities, we have

A XD XEXH=BxCxFXG; or,

(A xE) X (D x H)=(BF) × (CxG); therefore, by Prop. II.,

AXE:BXF::CxG: DxH. Cor. If

A:B::C:D, and

B:F::G:H; then

A:F::CxG:DXH.

For, by the proposition,

AXB:BXF::CxG:D xH Also, by Prop. VIII.,

AXB: BXF::A:F; hence, by Prop. IV.,

A:F::CxG:DXH.

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If three quantities are proportional, the first is to the third, as the square of the first to the square of the second.

Thus, if

A:B::B :C; then

A:C:: A’: B'. For, since

A:B::B:C, and

A:B::A:B; therefore, by Prop. XI.,

A': B' :: AXB:BxC. But, by Prop. VIII.,

AXB:BxC::A:C; hence, by Prop. IV., A:C::A': B'.

BOOK III.

THE CIRCLE, AND THE MEASURE OF ANGLES.

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Definitions. 1. A circle is a plane figure bounded by a line, every point of which is equally distant from a point within, called the center.

This bounding line is called the circumference of the circle.

2. A radius of a circle is a straight line drawn from the center to the circumference.

A diameter of a circle is a straight line passing through the center, and terminated both ways by the circumference.

Cor. All the radii of a circle are equal ; all the diameters are equal also, and each double of the radius.

3. An arc of a circle is any part of the circumference.

The chord of an arc is the straight line which joins its two extremities.

4. A segment of a circle is the figure included between an arc and its chord.

5. A sector of a circle is the figure included between an arc, and the two radii drawn to the extremities of the arc.

6. A straight line is said to be inscribed in a circle, when its extremities are on the circumference.

An inscribed angle is one whose sides are inscribed.

7. A polygon is said to be inscribed in a circle, when all its sides are inscribed. The circle is then said to be described about the polygon.

8. A secant a line which cuts the circumference, and lies partly within and partly without the circle.

9. A straight line is said to touch a circle, when it meets the circumference, and, being produced, does not cut it. Such a line is called a tangent, and the point in which it meets the circumference, is called the point of contact.

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