366. Also the first is to the excess of the first above the second as the third is to the excess of the third above the

or a-bac-d: c; therefore a: a-b::c: c-d.

367. When four numbers are proportionals, the sum of the first and second is to their difference as the sum of the third and fourth is to their difference; that is, if abcd, then a+b: a−b :: c+d: c-d.

368. It is obvious from the preceding Articles that if four numbers are proportionals we can derive from them many other proportions; see also Art. 356.

EXAMPLES. XXXVI.

Find the value of x in each of the following propor

tions.

8.

x2+x+1 : 62(x+1) :: x2−x+1 : 63(x−1).

10. If pq=rs, and qt=su, then pr::t: u.

11. If a b c d, and a' : b' :: c' : d', then

12. If a b :: b : c, then (a2+b2) (b2 + c2) = (ab+bc)”.

13. There are three numbers in continued proportion; the middle number is 60, and the sum of the others is 125: find the numbers.

14. Find three numbers in continued proportion, such that their sum may be 19, and the sum of their squares

133.

If a b c d, shew that the following relations are

true.

XXXVII. Variation.

369. The present Chapter consists of a series of propositions connected with the definitions of ratio and proportion stated in a new phraseology which is convenient for some purposes.

370. One quantity is said to vary directly as another when the two quantities depend on each other, and in such a manner that if one be changed the other is changed in the same proportion.

Sometimes for shortness we omit the word directly, and say simply that one quantity varies as another.

371. Thus, for example, if the altitude of a triangle be invariable, the area varies as the base; for if the base be increased or diminished, we know from Euclid that the area is increased or diminished in the same proportion. We may express this result with Algebraical symbols thus ; let A and a be numbers which represent the areas of two triangles having a common altitude, and let B and b be numbers which represent the bases of these triangles reA B spectively; then And from this we deduce a b'

A α

b

=

=, by Art. 363. If there be a third triangle having the B same altitude as the two already considered, then the ratio of the number which represents its area to the number which represents its base will also be equal to . Put=m, b

a

a

A then =m, and A=mB. Here A may represent the B area of any one of a series of triangles which have a common altitude, and B the corresponding base, and m remains constant. Hence the statement that the area varies as the base may also be expressed thus, the area has a

constant ratio to the base; by which we mean that the number which represents the area bears a constant ratio to the number which represents the base.

These remarks are intended to explain the notation and phraseology which are used in the present Chapter. When we say that A varies as B, we mean that A represents the numerical value of any one of a certain series of quantities, and B the numerical value of the corresponding quantity in a certain other series, and that A=mB, where m is some number which remains constant for every corresponding pair of quantities.

It will be convenient to give a formal demonstration of the relation A= mB, deduced from the definition in Art. 370.

372. If A vary as B, then A is equal to B multiplied by some constant number.

Let a and b denote one pair of corresponding values of two quantities, and let A and B denote any other pair; A B

then -=

a

b

, by definition. Hence A=B=mB, where

m is equal to the constant

373. The symbol is used to express variation; thus A & B stands for A varies as B.

374. One quantity is said to vary inversely as another, when the first varies as the reciprocal of the second. See Art. 323.

when m is constant, A is said to vary

375. One quantity is said to vary as two others jointly, when, if the former is changed in any manner, the product of the other two is changed in the same proportion.

Or if AmBC, when m is constant, A is said to vary jointly as B and C.

376. One quantity is said to vary directly as a second and inversely as a third, when it varies jointly as the second and the reciprocal of the third.

377. If A & B, and B x C, then A ∞ C.

For let A= mB, and B=nC, where m and n are constants; then A=mnC; and, as mn is constant, A x C.

378. If A C, and B∞ C, then A±B∞ C, and √(AB) × C.

For let A=mC, and B=nC, where m and n are constants; then A+B=(m±n)C; therefore A±B C.

Also √(AB)=√(mnC3)=C√(mn); therefore √(AB) ∞ C.

380. If A & B, and C x D, then AC ∞ BD.

For let A=mB, and C=nD; then AC=mnBD; therefore AC & BD.

381. If A B, then A" B".

For let A=mB, then A"=m"E"; therefore A" ∞ B".

382. If A B, then AP BP, where P is any quantity variable or invariable.

For let A= mB, then AP = mBP; therefore AP × BP.