Suppose

a

с

> then it will be possible to find some fraction b d'

which lies between them, q and p being positive integers.

26. It should be noticed that the geometrical definition of proportion deals with concrete magnitudes, such as lines or areas, represented geometrically but not referred to any common unit of measurement. So that Euclid's definition is applicable to incommensurable as well as to commensurable quantities; whereas the algebraical definition, strictly speaking, applies only to commensurable quantities, since it tacitly assumes that a is the same determinate multiple, part, or parts, of b that c is of d. But the proofs which have been given for commensurable quantities will still be true for incommensurables, since the ratio of two incommensurables can always be made to differ from the ratio of two integers by less than any assignable quantity. This has been shewn in Art. 7; it may also be proved more generally as in the next article.

27. Suppose that a and b are incommensurable; divide b into m equal parts each equal to ß, so that b= mß, where m is a positive integer. Also suppose ẞ is contained in a more than n times and less than n + 1 times;

can choose ẞ (our unit of measurement) as small as we please, m can

1

be made as great as we please. Hence can be made as small

m

as we please, and two integers n and m can be found whose ratio will express that of a and b to any required degree of accuracy.

28. The propositions proved in Art. 23 are often useful in solving problems. In particular, the solution of certain equations is greatly facilitated by a skilful use of the operations componendo and dividendo.

7. √ a2 + c 2 : √ b2 + d2 = √ √ ac + √ bd + .

c3

a

:

d3

b

If a, b, c, d are in continued proportion, prove that

9. 2a+3d 3a-4d=2a3+3b3 : 3a3 - 4b3.

10. (a2+b2+c2) (b2+c2+d2)=(ab+be+cd)2.

11. If b is a mean proportional between a and c, prove that

12. If a b c d, and e: f=g: h, prove that

ae+bf: ae-bf=cg+dh: cg-dh.

17. If a, b, c, d, e are in continued proportion, prove that

(ab+be+cd+de)2= (a2+b2+c2+d2) (b2+c2+d2+e2).

18. If the work done by x-1 men in +1 days is to the work done by x+2 men in x-1 days in the ratio of 9: 10, find x.

19. Find four proportionals such that the sum of the extremes is 21, the sum of the means 19, and the sum of the squares of all four numbers is 442.

20. Two casks A and B were filled with two kinds of sherry, mixed in the cask A in the ratio of 2: 7, and in the cask B in the ratio of 15. What quantity must be taken from each to form a mixture which shall consist of 2 gallons of one kind and 9 gallons of the other?

21. Nine gallons are drawn from a cask full of wine; it is then filled with water, then nine gallons of the mixture are drawn, and the cask is again filled with water. If the quantity of wine now in the cask be to the quantity of water in it as 16 to 9, how much does the cask hold?

22. If four positive quantities are in continued proportion, shew that the difference between the first and last is at least three times as great as the difference between the other two.

23. In England the population increased 15.9 per cent. between 1871 and 1881; if the town population increased 18 per cent. and the country population 4 per cent., compare the town and country popula

tions in 1871.

24. In a certain country the consumption of tea is five times the consumption of coffee. If a per cent. more tea and b per cent. more coffee were consumed, the aggregate amount consumed would be 7c per cent. more; but if b per cent. more tea and a per cent. more coffee were consumed, the aggregate amount consumed would be 3c per cent. more compare a and b.

25. Brass is an alloy of copper and zinc; bronze is an alloy containing 80 per cent. of copper, 4 of zinc, and 16 of tin. A fused mass of brass and bronze is found to contain 74 per cent. of copper, 16 of zinc, and 10 of tin: find the ratio of copper to zinc in the composition of brass.

26. A crew can row a certain course up stream in 84 minutes; they can row the same course down stream in 9 minutes less than they could row it in still water: how long would they take to row down with the stream?

CHAPTER III.

VARIATION.

29. DEFINITION.

One quantity A is said to vary directly as another B, when the two quantities depend upon each other in such a manner that if B is changed, A is changed in the same ratio.

NOTE. The word directly is often omitted, and A is said to vary as B.

For instance: if a train moving at a uniform rate travels 40 miles in 60 minutes, it will travel 20 miles in 30 minutes, 80 miles in 120 minutes, and so on; the distance in each case being increased or diminished in the same ratio as the time. This is expressed by saying that when the velocity is uniform the distance is proportional to the time, or the distance varies as the time.

30. The symbol is used to denote variation; so that A a B is read "A varies as B."

31. If A varies as B, then A is equal to B multiplied by some constant quantity.