284. SCH. By the product of the extremes or of the means of a proportion is meant the product of the numerical measures, of those quantities. Hence, the product of two lines will be often used for brevity, meaning the product of the numbers which represent those lines.

Proposition 3.

285. If four quantities are in proportion, they are in proportion by inversion; that is, the second term is to the first as the fourth term is to the third.

286. If four quantities are in proportion, they are in proportion by alternation; that is, the first term is to the third as the second term is to the fourth.

Proposition 5.

287. If four quantities are in proportion, they are in proportion 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.

288. If four quantities are in proportion, they are in proportion by division; that is, the difference of the first and second is to the second as the difference of the third and fourth is to the fourth.

Proposition 7.

289. If four quantities are in proportion, they are in proportion by composition and division; that is, the sum of the first and second is to their difference as the sum of the third and fourth is to their difference.

290. The products of the corresponding terms of two or more proportions are proportional.

291. A greater quantity is said to be a multiple of a less, when the greater contains the less an exact number of times.

Equimultiples of two quantities are quantities which contain them the same number of times. Thus, ma and mb are equimultiples of a and b.

Proposition 9.

292. When four quantities are in proportion, if the first and second be multiplied, or divided, by any quantity, as also the third and fourth, the resulting quantities will be in proportion.

Multiply both terms of the first fraction by m, and both terms of the second by n.

293. SCH. Either m or n may be unity.

In a similar manner it may be shown that if the first and third terms be multiplied, or divided, by any quantity, and also the second and fourth, the resulting quantities will be in proportion.

That is, equimultiples of two quantities are in the same ratio as the quantities themselves.

Proposition 10.

295. If four quantities are in proportion, their like powers, or roots, are in proportion.

abc: d.

Hyp. Let

296. If any number of quantities are in proportion, any antecedent is to its consequent, as the sum of any number of the antecedents is to the sum of the corresponding consequents.

Hyp. Let a b c d e f etc.

To prove

=ef=

abcd etc. = a+c+e+ etc.: b+d+f+etc. Proof. abba,

ad be, and af= be, etc., etc.

Adding, a(b+d+ƒ + etc.) = b(a + c + e + etc.)

=

(281)

..a b c d etc. = a+c+e+etc.: b+d+f+etc. (283)

:

a+c+e+etc.:b+d+f+etc.

PROPORTIONAL LINES.

Q.E. D.

297. DEF. Two straight lines are said to be cut proportionally when the parts of one line are in the same ratio as the corresponding parts of the other line.

Thus, AB and CD are cut pro

portionally at P and Q if

AP: PB CQ: QD.

=

P

A

Q