PRINCIPLES OF RATIO, ETC., TREATED ARITHMETICALLY. 13

5. A theorem, is the formal statement of a mathematical truth.

6. Theorem I. If the terms of a ratio are multiplied or divided by thesame number, the ratio is unaltered.

Take any ratio 12:20, and let its terms he multiplied by anynumber 3 ; then shall the ratio 36 : 60 be equal to the ratio 12 : 20.

36 , q

For the ratio 36 : 60 is measured by the fraction v

12

and the ratio 12 : 20 is measured by the fraction J § 3.

but — = —, by vulgar fractions :

60 20 J 6

the ratio 36 : 60 is equal to the ratio 12:20. § 4.

It can be proved in a similar manner that a ratio is unaltered if itsterms are divided by the same number. .

7. The generality of the reasoning in Theorem I., and all otherTheorems, should be carefully noted; the argument would be pre-cisely the same if any other ratio were taken instead of 12 : 20, orany other number instead of 3.

8. A theorem consists of two parts, tli e'Jiypothesis or part given, theconclusion or part to be proved.

In Theorem I., the hypothesis is that the terms of the ratio are multi-plied or divided by the same number, the conclusion is that the ratiois unaltered.

9. The bearing of Theorem I. will be seen more clearly if we ex-amine the effect of adding or subtracting the same quantity to or fromthe terms of a ratio. It will be found that by this operation theratio is always altered, except in the case in which its terms are equal.

10. Def. 2. If there are two or more ratios, the ratios formed bymultiplying together all the antecedents for a new antecedent, and allthe consequents for a new consequent, is called the ratio compoundedof the original ratios.

11. Theorem II. If there are two sets of ratios, and the ratios of thefirst set are respectively equal to those of the second set, then the ratio com-pounded of the first set is equal to the ratio compounded of the second set.

Take the two sets of ratios 3 : 5, 7 :11 ; and 6 : 10, 21 : 33 ; theratios of the first set being respectively equal to those of the second;then shall the ratio 3x7:5x11 be equal to the ratio 6x21 : 10x33.

For 3:5 = 6:10 by hypothesis,

and 7 :11 = 21 : 33 „

3 _ 6^ , 7__21

5 “ 10 11 — 33’

3 7 6 21 since if equals are multiplied by equals the

5 X IT” To X 33’ products are equal,

§4.

. 3x7 _ 6x21“ 5xll — 10x33’.-. 3x7 : 5x11 = 6

by vulgar fractionsX 21 : 10x23

§ 4 .