222. The number that expresses how many times a quantity contains the unit is called the numerical measure of that quantity. The numerical measure may be a number of any kind, integral or fractional, rational or irrational, or any letter denoting number.

E

F

C

D

B

Thus if AB is divided into 11 equal parts, of which CD is found to contain 83, and EF, a parts, then 11, 83, and x are the numerical measures of AB, CD, and EF, respectively.

If two quantities of the same kind, but expressed in different units, are to be compared as quantities, it is evident that both must be expressed as consisting of units of the same kind before the comparison can be effected.

Thus if AB were 8 rods and CD 15 yards, we cannot compare them until we have expressed them as 132 feet and 45 feet, for example. In every case, then, where numerical measures are compared, they are to be understood as referring to the same unit. It may also be observed that though abstract numbers are properly only the numerical measures of quantities, yet they are conveniently regarded as quantities whose unit is not expressed.

223. If a quantity is contained an exact number of times in each of two like quantities, it is called a common measure of these quantities, which are then said to be commensurable with each other.

Thus a rod and a yard are commensurable, since they have 2, 3, 6, 9 inches as common measures. 71⁄2 inches and 13 inches are commensurable, since they have of an inch as a common measure; and any two quantities are commensurable when, referred to the same unit, their numerical measures are rational numbers.

224. If two quantities have no common measure, they are said to be incommensurable with each other.

Thus every unit of our common system of measures is

incommensurable with every like unit of the metric system; and, as afterwards will be seen, the diagonal and side of a square are incommensurable with each other, as are also the diameter and circumference of a circle, etc. When such quantities are compared, their numerical measures are either approximate, as when we say 3 meters = 118.11237 + inches, or the numerical measures are such symbols as √2, √3, π, etc.

RATIO.

225. The ratio of two quantities is their relative greatness as expressed by the quotient of the one by the other. Thus the ratio of 15 inches to 7 inches is 15 in. 7 in. = 15; the ratio of 8 rods to 15 yards is (reducing to a common measure) 132 ft. 45 ft. 132; and, generally, if the numerical measures of two quantities are a and b, the ratio of these quantities is a and b being any numbers whatever.

a

b

=

226. The ratio of the quantity 4 to a like quantity B is denoted symbolically by either of the expressions

each of which is read, the ratio of A to B. In each, A is called the first term or antecedent; B, the second term or consequent.

227. When A and B are commensurable quantities, the value of their ratio is expressed exactly by the fraction denoting the quotient of the numerical measure of the antecedent by that of the consequent. But if the quantities are incommensurable, no rational fraction can express their ratio exactly, since then they would not be incommensurable. Yet, by taking the unit of measure sufficiently small, we can find a fraction that expresses the true value of the ratio to as near a degree of approximation as we please.

Geom.-8

B.

Thus suppose we have two lines A and B, whose numerical measures are √2 and 1, respectively. Now, carried to seven decimals, √21.4142135+; that

A

is, √2>1.414213 and <1.414214, so that the ratio of 4 to B or √2:1 lies between

and must differ from either by less than one millionth. As we can find the value of √2 to any number of decimals, we can find a fraction that differs from √2:1 by less than any assignable quantity.

To generalize, let A and B be any two incommensurable quantities. If we suppose B divided into any number of equal parts, so that BnP, P denoting one of the parts, then A must contain some number m of such parts, with a remainder less than P; or 4>m P and <(m+1)P. Thus

1

either of these fractions by less than a fraction that may

n

be made less than any assigned quantity by taking n sufficiently great. Hence

A rational fraction can be found that expresses the ratio of any two given incommensurable quantities within any required degree of precision.

PROPORTION. DEFINITIONS.

228. If two pairs of quantities have equal ratios, they are said to be proportionals or to be in proportion. Thus each of the equalities A: B = C : D,

=

A C
B

expresses a proportion that may be read, the ratio of A to B is equal to the ratio of C to D; or more briefly, A is to B as C is to D.

229. The first and fourth terms of a proportion are called its extremes; the second and third, its means. The fourth term is also called a fourth proportional to the other three.

230. Although ratio, from its very nature, can exist between like quantities only, yet as we may have, for example, 4B, and also arc Parc Q, the proportion

LA: ZB arc Parc Q,

may be stated between these pairs of unlike quantities, since the proportion simply states that the angle 4 is just as great compared with the angle B as is the arc P compared with the arc Q.

231. Of the following theorems concerning proportions and their transformations, some apply to pairs of quantities whether like or unlike, and the given proportion will be stated under the form

A:BP: Q.

Others apply only when the pairs of quantities are like, and the given proportions will be stated under the form

A: BC: D.

Others, again, that apply properly to numbers only, will have the given proportion stated under the form.

a: b = c: d.

232. It follows at once, from the definitions of quantity, ratio, and proportion, that (1) if four quantities are in proportion, their numerical measures are also in proportion; i.e.

and conversely; (2) if four numbers are in proportion, quantities of which these numbers are numerical measures are also in proportion; i.e.,

if

then

α p

a:b=p: q, or b

A. BP: Q.

It also follows from the same definitions, and Ax. 1, that (3) ratios equal to the same ratio are equal to each other.

EXERCISES.

QUESTIONS.

285. Taking the inch as unit, what is the ratio of 1 ft. to 7 in.? To 13 in. ? To 1 ft. ? To 21 ft. ? Toyd. ?

286. A train goes at the rate of at the rate of 105 miles in 23 hrs.

112 miles in 31 hrs.; a second train What is the ratio of the speed of

the first train to that of the second?

287. What is the ratio of 1 lb. to 9 oz. ? To 33 oz.? To 21 lbs. ? To 20ğ lbs. ?

288. That 216 grs. of silver may be worth 13 grs. of gold, what should be the ratio of the value of gold to that of silver?

290. The vertical angle of an isosceles triangle is 50°. ratio of that angle, (1) to a right angle? (2) to. each angles?

What is the

of the base

291. A base angle of an isosceles triangle is 75°. What is the ratio of that angle, (1) to a right angle? (2) to the vertical angle?

292. The ratio of a base angle of an isosceles triangle to the vertical angle is. What is that angle, and what is its ratio to a right angle? 293. An acute angle of a right triangle is 35°. What is the ratio of that angle to the other acute angle?

294. A certain angle has to an angle of an equilateral triangle the same ratio that the latter has to a right angle. How many degrees in the first angle?