CHAPTER XIII.

RATIO.

The Relation of Number and Geometrical Magnitude.

Use of Numbers in comparing Magnitudes.-Up to this point only the absolute equality and inequality of geometrical magnitudes have been considered; we have now to lay down principles for a more definite comparison of magnitudes than that of simple equality and inequality.

The comparison required is exemplified in the case of numbers. For instance, the relation of the numbers 63 and 36 admits of a much more definite expression than simply that the first is larger than the second; for the first is exactly seven times the fourth part of the second. Again, the number 9 measures both these numbers; that is to say, 9 is their common measure, and the numbers are most definitely compared when it is ascertained that the first is seven times and the second four times their common measure.

Ratio. The relation of one number to another in respect of magnitude is called Ratio; and the ratio of one number to another is expressed by the fraction which the first is of the second. Any line which will measure exactly two given lines is also termed their common measure, and the lines are compared by stating how many times this common measure is contained in each.

Commensurable Magnitudes.—Any two magnitudes (as two lines, two angles, two surfaces, two solids) are said to be commensurable when a third magnitude of the same kind can be found which will measure both of them; and this third magnitude is called their greatest common measure when it is the greatest magnitude which will measure both of them. When two lines are commensurable, the ratio of the first to the second is the ratio of the numbers which show how many times they contain the G. C. M.

The greatest common measure of two numbers is found by dividing the greater by the smaller, and then taking the remainder and the divisor as the numbers whose common measure is required, these two numbers having the same G. C. M. as the original numbers. The

first divisor is divided by the first remainder, and a second remainder found, which is treated in the same way as the first, the process being continued until there is no remainder. The last divisor is the G. C. M.

This process, which will hereafter be referred to as that for finding the G. C. M., may be applied not only to numbers but to any two magnitudes of the same kind. It is applicable, for instance, to two straight lines, AB, C D.

To find their greatest common measure, mark off along AB, the greater, as many times CD as are contained in A B, and find the remainder, EB; set off E B along CD, and find the second remainder, FD; set off F D along E B to find the third remainder; and so on. That remainder which is contained any number of times exactly in the preceding remainder is the G. C. M. of A B and C D.

When applied to numbers, the process must terminate sooner or later; for, even when there is no other G. C. M., unity will always measure any two whole numbers.

Hence all whole numbers are commensurable, their common measure being unity; also all finite fractions, proper or improper, are commensurable, inasmuch as all such fractions may be reduced to equivalent ones having a common denominator, all of which will be measured at least by the fraction whose denominator is the common one, and whose numerator is unity.

Any two magnitudes having a common measure can be represented by the two numbers which express how many times the two magnitudes contain the common measure.

Incommensurable Magnitudes.-But number and magnitude do not correspond in all their relations. As stated above, all rational numbers are measured by unity; but with regard to magnitudes generally, though any quantity may be assumed as the unit, it does not follow that the same unit will measure every magnitude of like kind. Two such magnitudes may indeed have no common measure; that is to say, they may be incommensurable.

When the G. C. M. process is applied to two incommensurable magnitudes, it does not terminate; and whenever it can be shown that the process applied to two magnitudes is interminable, it may be concluded that the magnitudes are incommensurable. The side and

diagonal of a square are instances of incommensurable magnitudes. (Theorem LVIII.)

Two incommensurable magnitudes cannot be exactly represented by any two whole numbers or fractions whatever: thus it may be shown that if the side of a square contain one unit of length, the diagonal contains more than one, but less than two. If the side be divided into 10 units, the diagonal contains more than 14 and less than 15. If the side contain 100 units, the diagonal has more than 141 but less than 142 of such units. It is obvious that as the side is successively divided into a greater number of equal parts, or units, the error in the magnitude of the diagonal expressed in such units will continually diminish, and it is equally obvious that it can never be exhausted.

It will, however, be also evident that we can always express such incommensurable lines in terms of some unit small enough to be suffi ciently exact for all practical purposes.

The conclusions of geometry, being based upon principles applicable equally to incommensurable as to commensurable magnitudes, are more general than those of the corresponding theorems of arithmetic and algebra, whlch have reference to commensurable magnitudes only. If all lines were commensurable, the algebraical theorem

(a + b)2 = a2 + b2 + 2ab

would involve the geometrical Theorem XLVIII. of Chapter XII., inasmuch as a and b may stand for the number of units of length in any two lines. But two incommensurable lines cannot be exactly represented by two numbers, a and b. The demonstration of the geometrical Theorem XLVIII., on the other hand, applies to all lines, whether they have a common measure or not; hence that proposition includes the corresponding algebraical theorem. It is the same with all subsequent propositions on Ratio and Proportion.

Division of a Straight Line into Equal Parts.-Any straight line may be divided into any number of equal parts. If a series of parallels cutting two straight lines divide one of them into equal parts, they also divide the other into equal parts. Thus, if we take a sheet of paper ruled with equidistant lines, and draw a line across them, the intercepted parts of the newlydrawn line will be equal to one another. In order, therefore, to divide a given straight line, A B, into a number of equal parts-say sevenwe can draw the line so that A is on the first of these equidistant parallels, and B on the seventh beyond the first. (Fig. 152.) The following method is, however, more generally applicable :

A

1 2 3 4 5 6 7 8 9

FIG. 152.

B

Let it be required to divide the straight line A B into five equal parts. From the point A draw any straight line, A G, and upon it, starting from the point A, cut off any part (Fig. 153), and measure this length off five times upon A G, thus obtaining the points C, D, E, F, G. Join the last division, G, to the other extremity, B, of the Ggiven line, and through the points of division draw parallels to GB; the line A B will be divided into five equal parts. It is convenient in practice to take the length A C nearly equal to the fifth of A B.

E

FIG. 153

From this construction it will be evident that any given finite straight line can be divided into as many equal parts as we please; for we can draw from A an unlimited straight line, A G, along which we can set off any number of equal parts commencing at the point A, and by

B

FIG. 154.

joining the last point of division, X, to the point B, and drawing lines through the other points parallel to BX, AB will be divided into the same number of equal parts as A X.

It will be further evident that any finite straight line, A B, can be divided into equal parts, each of which shall be less than any given line; for it is only requisite to set off the given line along the indefinite straight line, A X, as many times as will make A X greater than A B, then each division of A B will be less than each division of AX; that is to say, less than the given straight line.

THEOREMS ON CHAPTER XIII.

DEFINITIONS.

51. When one magnitude is the sum of a number of parts each equal to another magnitude, the first is said. to be a multiple of the second, and the second is said to be an aliquot part or measure of the first.

52. When each of two magnitudes is a multiple of a third, the first two are said to be commensurable, and the third is termed their common measure.

53. When two magnitudes have no common measure, they are said to be incommensurable.

54. To investigate the commensurability of two given straight lines, A B and CD:—

Apply CD, the smaller of the two straight lines, to A B, the greater, and measure off as many equal parts as possible.

A

E

F

-D

B

If there be a remainder, E B, set it off in like manner upon the smaller line, C D.

Should there be a second remainder, F D, set it off in like manner upon the first, and so on.

The process will terminate only when a remainder is obtained which is an aliquot part of the preceding one.

THEOREM LVI.

When the G. C. M. process, applied to two given straight lines, terminates, the two lines will be commensurable, and have the last remainder for a common

measure.

Suppose M N to be the last remainder but one, and H K the last but two; then the last remainder, L K, will