in the subject. The study begins with an explanation of certain ratios which are used in it continually, and most of the numbers that appear in the solution of its problems are incommensurable.

If one quantity is half as great as another quantity in magnitude, it is said that the ratio of the first quantity to the second is as one to two, or one-half. This ratio is sometimes indicated thus, 1:2; but more usually it is written in the fractional form,

In this example the magnitude of the second quantity is twice that of the first, and the ratio of the second quantity to the first is 2: 1, or, adopting the more usual style, , i.e. 2. The ratio of two quantities is simply the number which expresses the magnitude of the one when compared with the magnitude of the other. This ratio is obtained by finding how many times the one quantity contains the other, or by finding what fraction the one is of the other. It follows that a ratio is merely a pure number, and that it can be obtained only by comparing quantities of the same kind. Thus the ratio of the length 3 feet to the length 2 inches is 36, i.e. 18; the ratio of the weight 2 pounds to the weight 3 pounds is. But one cannot speak of the ratio of 3 weeks to 10 yards, for there is no sense in the questions: How many times does 3 weeks contain 10 yards? What fraction of 10 yards is 3 weeks?

When it is said that a line is ten inches long, this statement means that a line one inch long has been chosen for the unit of length, and that the first line contains ten of these units. Thus the number used in telling the length of a line is the ratio of the length of this line to the length of another line which has been chosen for the unit of length. The measure of any quantity, such as a length, a weight, a time, an angle, etc., is

the number of times the quantity contains

or, the fraction that the quantity is of

}

a certain quantity of the same kind which has been adopted as the unit of measurement. In other words, the measure of a quantity is the ratio of the quantity to the unit of measurement. For example, if half an inch is the unit of length, then the measure of a line 8 inches long is 16; if a foot is the unit of length, then the measure of the same line is ; if a second is the unit of time, then the measure of an hour is 3600; if an hour is the unit of time, then the measure of a second is 3600

If two quantities have a common unit of measurement, then their ratio is the ratio of their measures. For example, 1 pound being taken as the unit of weight, the ratio of a weight 3 pounds to a weight 7 pounds is, which is also the ratio of the measures 3 and 7. In general, if a quantity P contains m units, and a quantity Q contains n units of the same kind as is used in the case of P, then the ratio

m

The last fraction

n

I is the ratio of the numbers m and n, which

are the measures of the quantities P and Q respectively.

EXAMPLES.

1. What is the ratio of each of the following lengths to an inch, viz., 8 in., 2 ft., 3 ft. 6 in., 1.5 yd., 20 yd., a yd., b ft., c in.?

2. What is the ratio of each of the following lengths to a yard, viz., 6 yd., 3.75 yd., 8 ft., 2 ft. 6 in., 10 in., 5 in., a yd., b ft., c in. ?

3. What is the measure of each of the following lengths, when a foot is the unit of length, viz., 1.5 mi., 17 yd., 3 yd. 2 ft., 8.5 ft., 2 ft. 6 in., 9 in., 2 in., a yd., b ft., c in. ?

4. What is the measure of each of the following lengths, when 3 in. is the unit of length, viz., 2.5 yd., 1.5 ft., 8 in., a yd., b ft., c in. ?

5. Express the ratio of 2.5 mi. to 10 yd.; and the ratio of 2 in. to 31 yd.

6. Compare the ratio of a foot to a yard with the ratio of a square foot to a square yard.

7. What is the unit of measurement in each of the following cases: when the measure of 2 ft. is 4, of 1 yd. is 72, of .5 in. is 4, of 2.5 ft. is .25?

N.B. The following examples will be used again for purposes of illustration. The student is advised to draw figures neatly and accurately and to preserve the results carefully.

8. In a right-angled triangle the base is 6 ft. and the hypotenuse 10 ft. What is the perpendicular? Calculate the following ratios, viz.:

What are these ratios in a triangle whose base is 6 in., and hypotenuse 10 in.? What are they when the base is 6 yd., and the hypotenuse 10 yd.? When the base is 6 mi., and the hypotenuse 10 mi. ? When the base is 12 ft., and the hypotenuse 20 ft. ? When the base is 3 in., and the hypotenuse 5 in. ? Compare, if possible, the angles in these triangles.

9. In a right-angled triangle whose base is 35 ft. and perpendicular 12 ft., what is the hypotenuse? For this triangle calculate the ratios specified in Ex. 8. Calculate these ratios for a triangle whose base is 70 yd., and perpendicular 24 yd. Compare, if possible, the angles in these triangles.

10. Calculate these ratios for the triangle whose hypotenuse is 29 ft., and perpendicular 21 ft.; for the triangle whose hypotenuse is 2.9 in., and perpendicular 2.1 in. Compare, if possible, the angles in these triangles.

9. Incommensurable quantities. Approximations. If the side of a square is one foot in length, then the length of a diagonal of the square is √2 feet. Thus the ratio of the diagonal to the side is √2, a number which cannot be expressed as the ratio of two whole numbers. Two quantities whose ratio can be expressed by means of two integers are said to be commensurable the one with the other; when their ratio cannot be so expressed, the one quantity is said to be incommensurable with the other. For example, the diagonal of a square is incommensurable with the side, and the length of a circle with its diameter.* The quantities in the examples, Art. 8, are commensurable. Numbers such as √2, 4, √10 are incommensurable with unity, and their values cannot be found exactly. Their values, however, can be found to two, to three, to four, in fact, to as many places of decimals as one please. The greater the number of places of decimals, the more nearly will the calculated values represent the true values of the numbers. In other words, the values of incommensurable numbers can be found approximately; and the degree of approximation (that is, the nearness to the exact values) will depend only on the carefulness and patience of the calculator. In practical problems there frequently is occasion for the exercise of judgment as to the degree of approximation that is necessary and sufficient. For example, in calculating a length in inches in ordinary engineer

* See Appendix, Note C.

ing work there is no need to go beyond the third place of decimals, for engineers are satisfied when a measurement is correct to within of an inch. As a rule the results obtained in practical problems in mathematics are only approximate and not exact. There are two reasons for this: first, the data obtained by actual measurement can only be approximate, however excellent the instruments used in measuring may be, and however skilled and careful is the person who does the measuring; second, most of the numbers used in the subsequent computations are incommensurable.

The examples at the end of this article are intended to bring out more clearly the idea of an approximate result. The answers are to be calculated to three places of decimals. It is advisable to compare the values calculated to three places of decimals with the values calculated to two places of decimals, and to note the difference between them. The following facts are supposed to be known and will be taken for granted.

(a) In a right-angled triangle the square of the measure of the hypotenuse is equal to the sum of the squares of the measures of the other two sides.

(b) The ratio of the length of any circle to its diameter is a number which is the same for all circles.* The exact value of this ratio is incommensurable and is always denoted by the symbol π (read pi).† The approximate values commonly used for are 3.1416, 3.14159, (i.e. 3.1415929 ...), 27 (i.e. 3.142857); of these values the last is the least accurate, but it is accurate enough for many practical purposes.

(c) The length of a circle of radius r is 2 πr [by (b)]; and the enclosed area is #r2.

NOTE 1. If a number be calculated to three or more places of decimals, then the closest approximation to, say, two places of decimals is obtained by leaving the number in the second place of decimals unchanged when the number in the third place is less than 5, and by increasing the number in the second place by unity when the number in the third place is greater than 5 or 5 followed by numbers; thus, e.g., 3.72 for 3.724, 3.73 for 3.7261 and

* This ratio and facts (c) are considered in Note C, Appendix. The reading only requires a knowledge of elementary geometry.

† This symbol is the initial letter of periphereia, the Greek word for circumference. Its earliest appearances to denote this ratio are in Jones's Synopsis Palmariorum Mathesos, London, 1706, and in the Introductio in analysin infinitorum, published in 1748 by Leonhard Euler (1707–1783), a native of Switzerland, who was one of the greatest mathematicians of his time.

3.7257. When the number in the third place is 5 and this is followed by zeros only, the number in the second place is unchanged if it is even, and is increased by unity if it is odd; thus, e.g., 3.78 for 3.775, 3.78 for 3.785. In a series of calculations the errors made by following this rule tend to balance one another.

NOTE 2. A quantity measured to two places of decimals is correct to the hundredth part of the unit employed, and a quantity measured to three places is correct to the thousandth part of the unit. For example, the length of a circle of 10 feet diameter is 31.4159. . . feet. For this length 31.416 or 31.42 may be taken; the former result differs from the true result by less than one-thousandth of a foot, the latter by less than one-hundredth.

EXAMPLES.

1. A finds the square root of 3 correctly to two places of decimals, and B to three. How much closer than A does B come to the exact value of the

square root of 3 ?

2. A circle is 50 ft. in diameter. In calculating its length A takes 3.1416 as the ratio of the length of a circle to the diameter, B takes 3.14159, and C takes 22:7. What are the differences (in inches) between their results?

3. The radius of a circle is 49.95 ft. How nearly will a person come to the length of the circle if he assumes the radius to be 50 ft. ? [In this and the following example take = 22:7.]

4. It is known that the diameter of a certain circle does not differ from 100 ft. by more than 2 in. What will be the outside limits of the error made in calculating the area when the diameter is taken as 100 ft.?

5. Find the difference between the calculations of the numbers of revolutions per mile made by a 50-in. bicycle, for π = 22:7 and π = 3.1416.

6. A lot is 75 ft. by 200 ft. Find the diagonal distance across the lot correctly to within a tenth of an inch.

7. Find the height of an equilateral triangle whose side is 20 yd.

8. The side of an isosceles triangle is 40 ft. and the base is 30 ft.; find the height.

9. What is the length of the diagonal of a square whose side is 20 ft. ? 10. What is the length of the side of a square whose diagonal is 20 ft. ?

N. B. The following examples will be used again for purposes of illustration. The student is advised to draw figures and to preserve the results with those of Exs. 8, 9, 10, Art. 8.

11. (a) In a right-angled triangle the hypotenuse is 12 ft. and the base is 6 ft.; calculate the ratios specified in Ex. 8, Art. 8.

(b) What are these ratios when the lengths in (a) are taken twice, three times, one-half as great? Compare, if possible, the angles in these triangles.