*Ex. 343. The two common exterior tangents of two circles intercept on a common interior tangent a segment, CD, equal to the exterior tangent, AB.

203. A ratio of two quantities of the same kind is the quotient obtained by dividing the first quantity by the second. Thus, the ratio of two quantities, a and b, is or a ÷ b; the

α

ratio of four yards and two yards is or 2. A ratio is used to compare the magnitude of two quantities.

204. To measure a quantity is to find how many times it contains another quantity of the same kind, called a unit of measurement. Thus a line is measured by finding that it contains a certain number of yards or inches.

205. A number which expresses how many times a quantity contains a unit is called the numerical measure of the quantity.

206. Two quantities are commensurable when there is a third quantity, called the common measure, which is contained an integral number of times in each.

207. Two quantities are incommensurable when there is no common measure. The ratio of such quantities is called an incommensurable ratio.

208. Such a ratio cannot be found exactly in figures, but an approximate value can be found that differs from the true value by less than any assigned value, however small. If AB and CD are two lines whose ratio is the square root of two, then

AB
CD

=

√2 = 1.41421 +·

Thus the true value lies between 1.41421 and 1.41422, and differs from either of the approximate values by less than 0.00001. It is evident that by continuing the decimal this difference may be diminished to one millionth or one billionth or any assigned value, however small.

If a quantity, A, be divided into m equal parts, and another quantity, B, is found to contain n of these parts, with a remainder less than one of the parts,

The error in taking one of these approximate values as a ratio is less than

1

m

As m may be made indefinitely large, the fraction may be

made indefinitely small, or the value of

B

may be found with

A

any assigned degree of accuracy.

LIMITS

209. A constant is a quantity that maintains the same value throughout the same discussion.

210. A variable is a quantity whose value changes during the same discussion.

211. A limit of a variable is a constant which the variable does approach indefinitely near, but which it can never reach.

212. For example, suppose a point P to move from A to B in such a way as to move in the first second over half of AB

to C, in the second second, over half of the remainder, CB, to D, in the third second over half of the new remainder, DB, to E, and so on indefinitely.

It is evident that the moving point P can never reach B, while it will approach nearer to B than any quantity which we can assign.

The distance from A to the moving point P is a variable whose value can be made to differ from AB by less than any assigned quantity, while it never can be made equal to AB.

AB is therefore the limit of the variable.

213. THEOREM. If two variables are always equal and each approaches a limit, the two limits are equal.

A

P

C B

Hyp. The equal variables AP and A'P', having the limits. AB and A'B' respectively.

off

Proof. Suppose AB is greater than A'B', and on AB lay

AC = A'B'.

Then the variable AP attains values greater than AC, while the variable A'P' is always less than AC, or the two variables are unequal, which is contrary to the hypothesis.

Whence AB cannot be greater than A'B', and in like manner it can be shown that A'B' cannot be greater than AB.

Ex. 344. What is the limit of the circulating decimal .999... ?

214. In the same circle, or in equal circles, two central angles have the same ratio as their intercepted arcs.

C

A

B

A

Hyp. In the equal circles ABC and A'B'C', two central angles AOB and A'O'B' intercept the arcs AB and A'B' respectively.

Proof. CASE I. The arcs are commensurable.

Let m be a common measure contained in AB five times and A'B' four times.

Connect the points of division with the center. Then ZAOB will have been divided into five parts and ▲ A'O'B' into four, all being equal.

(174)

CASE II. The arcs are incommensurable.

Divide AB into any number of equal parts, and apply one of those parts to A'B' as often as possible.

Since AB and A'B' are incommensurable, there must be a remainder C'B' less than one of the equal parts. Draw O'C'. Since the arcs AB and A'C' are commensurable,

A'C LA'O'C'

AB ZAOB

By increasing the number of parts into which AB is divided, we can diminish the length of each part, and, therefore, the length of C'B' indefinitely.

Hence the arc A'C' approaches A'B' as a limit, and ▲ A'O'C' approaches A'O'B' as a limit.

215. SCHOLIUM. The circumference is divided into 360 equal parts called degrees, and therefore a central angle of 1° will intercept an arc of 1°.

The numerical measure of any central angle is equal to the numerical measure of the intercepted arc, or more briefly :

216. A central angle is measured by the intercepted arc.