PROPOSITION VI. THEOREM

428. If four quantities are in proportion, they are in proportion by division.

Let the student derive the solution of Prop. VI. from the analysis.

430. DEFINITION. A proportion is arranged by composition and division, when the sum of antecedent and consequent is compared with the difference of antecedent and consequent.

PROPOSITION VII. THEOREM

431. If four quantities are in proportion, they are in proportion by composition and division.

433. If four quantities are in proportion, like powers of those quantities are proportional.

Proof.

bn

3815

an

dn

[Raise both members of (1) to the nth power.] Q.E.D.

434. COROLLARY. If four quantities are in proportion, like roots of those quantities are proportional.

PROPOSITION IX. THEOREM

437. If four quantities are in proportion, equimultiples of the antecedents are proportional to equimultiples of the consequents.

442. DEFINITION. A continued proportion is a proportion made up of several ratios that are successively equal to each

PROPOSITION X. THEOREM

443. In a continued proportion the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent.

Add (6), (7), (8), and (9), and factor.

f(a+c+e+g)=e(b+d+f+h).

Apply § 411 to (10).

(1)

(10)

Q.E.D.

PROPOSITION XI. THEOREM

446. If the terms of one proportion are multiplied by the corresponding terms of another proportion, the products are proportional.

Proof. [The proof is left to the student.]

Q.E.D.

447. EXERCISE. If the terms of one proportion are divided by the corresponding terms of another proportion, the quotients are proportional.

449. If a number of parallels intercept equal distances on one of two transversals, they will intercept equal distances on the other also.

Let AB, CD, EF, and GH be a number of parallels cut by the transversals xy and zr, making

Proof. [Proof similar to that of § 240.]

Q.E.D.