215. For measuring angles the standard unit is the right angle, defined in Article 15, page 9.

This unit again is too large for convenient use, and a smaller unit is commonly employed in making actual measurements, namely, a degree.

A degree is defined to be of a right angle.

A minute is of a degree, and a second, of a minute.

EXERCISES

1. How many degrees in two right angles? In two-thirds of a right angle? In one-fifth of a right angle?

2. How many degrees in the sum of the angles of a triangle?

3. How many degrees in each of the angles of an equilateral triangle? 4. How many degrees in each of the angles of an isosceles right triangle ? 5. One angle of a right triangle equals 30°. How many degrees in each of the others?

6. The vertical angle of an isosceles triangle equals 18°. How many degrees in each of the base angles?

7. One of the base angles of an isosceles triangle equals 50°. How large is the vertical angle ?

8. One angle of a parallelogram equals 45°. What is the measure of each of the other angles ?

2. ON RATIO

216. When we speak of the ratio of one quantity to another, we have in mind their relative magnitude. By this we mean not how much the one is greater or less than the other, but how many times the one is as great as the other.

A ratio can be expressed only between two quantities of the same kind.

When dealing with numbers we say that the ratio of one to another is the quotient arising from dividing the first by the

second, since the quotient expresses how many times the one is as great as the other.

Thus the ratio of 8 to 4 is 2, the ratio of 7 to 3 is 21, the ratio of 5 to 9 is 5, etc.; but the division of one geometrical magnitude by another has no meaning unless the first is an exact multiple of the second, or until a meaning is assigned by definition. Consequently, the numerical value of the ratio of two such magnitudes must sometimes be got at in a roundabout way.

217. The same two quantities A and B have two different ratios, viz. the magnitude of A as compared with B, and the magnitude of B as compared with A.

The first is expressed A: B, and should be read ‘the ratio of A to B'; the second is expressed B: A and should be read 'the ratio of B to A.'

In any ratio the first term is called the antecedent, the second term, the consequent.

218. The following postulates will serve to add definiteness to the meaning of the term 'ratio.'

POSTULATE 6. If P and Q are any two equal magnitudes and R is a third magnitude of the same kind, then the ratio of P to R is equal to the ratio of Q to R, i.e. if P: Q, then P: R Q: R; and, conversely, if P and Q are such that P: R= Q: R, then P=Q.

=

=

POSTULATE 7. If P and Q are two unequal magnitudes, and R is a third magnitude of the same kind, then the ratio of P to R is greater or less than the ratio of Q to R, according as P is greater or less than Q.

219. In order to show how the ratio of one geometrical magnitude to another can be expressed numerically, it is necessary to consider two distinct cases.

(a) When the two given magnitudes have a measure, or are commensurable.

common

(b) When the two given magnitudes have no common measure, or are incommensurable.

(a) RATIO OF COMMENSURABLE MAGNITUDES

220. DEFINITION. The ratio of two commensurable magnitudes is the ratio of their numerical measures by a common unit.

Let A and B be any two magnitudes of the same kind which have a common measure S, and let S be contained in Am times, and in B n times, so that m is the measure of A, and n the measure of B, by the common unit S.

Then by definition

A: B: =m: n.

But the ratio of the number m to the number n is the quotient

m
n

221. If instead of A and B we take any equimultiples of these magnitudes, say pA and pB, their measures by the common unit S would be pm and pn.

Therefore, the ratio of any two commensurable magnitudes is equal to the ratio of any equimultiples of those magnitudes, taken in the same order.

The ratio of 5 ft. to 7 ft. is the same as the ratio of 10 ft. to 14 ft., or of 25 ft. to 35 ft.

222. Again, if instead of the common measure S we use a

becomes tm and the measure of B becomes tn, then

That is, the ratio of A to B is not altered by a change in the unit of measure.

The ratio of 5 ft. to 7 ft. is not altered if those lengths are expressed in inches, or in yards, instead of feet.

223. For convenience we frequently write A: B in the A form even when A and B represent geometrical magnitudes; B

but the symbol so used should always be read as 'the ratio of A to B,' and should not be confounded with an ordinary fraction, or symbol of division.

Thus,

ZABC

Z PQR

expresses the ratio of ABC to ≤ PQR, and may be written in that form or in the usual ratio form

If

ZABC:/PQR.

AOB is at the centre of a circle subtended by the arc AB, and ACB is at a point of the circle subtended by the

(Art. 178.)

being taken for unit of measure in each case.

224. Whenever the first of the two given magnitudes is an exact multiple of the second, the second may be taken as the common unit of measure; and then the ratio of the first to the second is equal to the measure of the first by the second.

The ratio of 8 to 4 is 2, of 20 to 5 is 4, of the diameter to a radius of a circle is 2, of the perimeter to a side of an equilateral triangle is 3, etc.

L

(b) RATIO OF INCOMMENSURABLE MAGNITUDES

225. Next suppose that A and B are two magnitudes such that no unit however small will measure them both integrally, that is, suppose that A and B are incommensurable.

In that case the ratio of A to B cannot be expressed either as a whole number (Art. 224) or as a fraction whose numerator and denominator are both whole numbers (Art. 220), since they have no common measure.

But while the ratio of two such magnitudes does not absolutely equal any integer or common fraction, it is always possible to find a common numerical fraction which will differ in value from their ratio by less than any assigned quantity however small, as we shall now proceed to show.

226. It is necessary before going further to introduce a new idea, which we shall do by means of an illustration.

A boy is to walk from P to Q, a distance of two miles. He goes half the distance the first hour, half the remaining distance the second hour,

half the remaining dis- P

tance the third hour,

and so on. Would he ever absolutely reach his distination? Could you fix a point between P and Q, as near as you like to Q, beyond which he would not pass in time?

ever get beyond Q?

Would he

In the first hour the boy would go 1 mile, in the second, mile; in the third, mile; in the fourth, mile; and so on. Take these numbers 1,,,, 16, 32,, and let S represent the sum of n of them.

Then when n = 1, S = 1;

when n=2, S=1+1=3;

when n=3, S=1+1+1=1;

when n = 4, S=1+} + {+} = 15;

and so on.