PROPOSITION V.

263. If four quantities be in proportion, they will be in proportion by inversion.

264. If four quantities be in proportion, they will be in proportion by composition.

PROPOSITION VII.

265. If four quantities be in proportion, they will be in

266. In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent.

Let a b = c : d = e : ƒ=g: h.

or, a+c+e+g: b + d + f + h : a b.

Q. E. D.

PROPOSITION IX.

267. The products of the corresponding terms of two or more proportions are in proportion.

268. Like powers, or like roots, of the terms of a proportion are in proportion.

269. DEF. Equimultiples of two quantities are the products obtained by multiplying each of them by the same number. Thus ma and mb are equimultiples of a and b.

PROPOSITION XI.

270. Equimultiples of two quantities are in the same ratio as the quantities themselves.

Let a and b be any two quantities.

271. If two quantities be increased or diminished by like parts of each, the results will be in the same ratio as the quantities themselves.

Let a and b be any two quantities.

Then (1) a: (1) a : 6,

b,

b

272. DEF. Euclid's test of a proportion is as follows:

"The first of four magnitudes is said to have the same ratio to the second which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth ;

"If the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth; or,

"If the multiple of the first be equal to that of the second, the multiple of the third is also equal to that of the fourth; or, "If the multiple of the first be greater than that of the second, the multiple of the third is also greater than that of the fourth."

PROPOSITION XIII.

273. If four quantities be proportional according to the algebraical definition, they will also be proportional according to Euclid's definition.

Let a, b, c, d be proportional according to the alge

We are to prove a, b, c, d, proportional according to Euclid's definition.

if ma be less than n b, mc will also be less than nd;

if ma be equal to n b, mc will also be equal to n d;

if ma be greater than nb, mc will also be greater than n d.

.. a, b, c, d are proportionals according to Euclid's definition,

Q. E. D.