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more is DN greater than DE. Therefore, AC and its equal FG are farther from the center than AB.

193. Corollary-Conversely of these two theorems, when the radii are equal, chords which are equally distant from the center are equal; and of two chords which are unequally distant from the center, the one nearer to the center is longer than the other.

194. Problem in Drawing.—To bisect a given arc.

Draw the chord of the arc, and erect a perpendicular at its

center.

State the theorem and the problems in drawing here used.

195. "The most simple case of the division of an arc, after its bisection, is its trisection, or its division into three equal parts. This problem accordingly exercised, at an early epoch in the progress of geometrical science, the ingenuity of mathematicians, and has become memorable in the history of geometrical discovery, for having baffled the skill of the most illustrious geometers.

"Its object was to determine means of dividing any given arc into three equal parts, without any other instruments than the rule and compasses permitted by the postulates prefixed to Euclid's Elements. Simple as the problem appears to be, it never has been solved, and probably never will be, under the above conditions." -Lardner's Treatise.

ANGLES AT THE CENTER.

190. Angles which have their vertex at the center of a circle are called, for this reason, angles at the center. The arc between the sides of an angle is called the intercepted are of the angle.

197. Theorem.-The radii being equal, any two angles at the center have the same ratio as their intercepted arcs. This theorem presents the three following cases: 1st. If the arcs are equal, the angles are equal.

For the arcs may be placed one upon the other, and will coincide. Then BC will coincide with AO, and DC with EO. Thus the angles may coincide, and are equal. The converse is proved in the same manner.

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2d. If the arcs have the ratio of two whole numbers, the angles have the same ratio.

Suppose, for example, the arc BD: arc AE :: 13: 5.

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Then, if the arc BD be divided into thirteen equal parts, and the arc AE into five equal parts, these small arcs will all be equal. Let radii join to their respective centers all the points of division.

The small angles at the center thus formed are all equal, because their intercepted arcs are equal. But BCD is the sum of thirteen, and AOE of five of these equal angles. Therefore,

angle BCD angle AOE :: 13 : 5;

that is, the angles have the same ratio as the arcs. Geom.-6

3d. It remains to be proved, that, if the ratio of the arcs can not be expressed by two whole numbers, the angles have still the same ratio as the arcs; or, that the radius being the same, the

arc BD arc AE :: angle BCD: angle AOE.

If this proportion is not true, then the first, second,

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and third terms being unchanged, the fourth term is either too large or too small. We will prove that it is neither. If it were too large, then some smaller angle, as AOI, would verify the proportion, and

arc BD : arc AE :: angle BCD : angle AOI.

Let the arc BD be divided into equal parts, so small that each of them shall be less than EI. Let one of these parts be applied to the arc AE, beginning at A, and marking the points of division. One of those points must necessarily fall between I and E, say at the point U. Join OU.

Now, by this construction, the arcs BD and AU have the ratio of two whole numbers. Therefore,

arc BD are AU :: angle BCD : angle AOU,

These last two proportions may be written thus (19); arc BD angle BCD :: arc AE : angle AOI; arc BD angle BCD :: arc AU ; angle AOU.

Therefore (21),

arc AE angle AOI :: arc AU: angle AOU;

or (19),

arc AE arc AU :: angle AOI : angle AOU. But this last proportion is impossible, for the first antecedent is greater than its consequent, while the second antecedent is less than its consequent. Therefore, the supposition which led to this conclusion is false, and the fourth term of the proportion, first stated, is not too large. It may be shown, in the same way, that it is not too small.

Therefore, the angle AOE is the true fourth term of the proportion, and it is proved that the arc BD is to the arc AE as the angle BCD is to the angle AOE.

DEMONSTRATION BY LIMITS.

198. The third case of the above proposition may be demonstrated in a different manner, which requires some explanation.

We have this definition of a limit: Let a magnitude vary according to a certain law which causes it to approximate some determinate magnitude. Suppose the first magnitude can, by this law, approach the second indefinitely, but can never quite reach it. Then the second, or invariable magnitude, is said to be the limit of the first, or variable one.

The

199. Any curve may be treated as a limit. straight parts of a broken line, having all its vertices in the curve, may be diminished at will, and the broken line made to approximate the curve indefinitely. Hence, a curve is the limit of those broken lines which have all their vertices in the curve.

200. The arc BC, which is cut off by the secant AD, may be diminished by successive bisections, keeping the remainders next to B. Thus AD, re

volving on the point B, may
approach indefinitely the tan-
gent EF.
Hence, the tangent

at any point of a curve is the
limit of the secants which may
cut the curve at that point.

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201. The principle upon which all reasoning by the method of limits is governed, is that, whatever is true up to the limit is true at the limit. We admit this as an axiom of reasoning, because we can not conceive it to be otherwise.

Whatever is true of every broken line having its vertices in a curve, is true of that curve also. Whatever is true of every secant passing through a point of a curve, is true of the tangent at that point.

We do not say that the arc is a broken line, nor that the tangent is a secant, nor that an arc can be without extent; but that the curve and the tangent are limits toward which variable magnitudes may tend, and that whatever is true all the way to within the least possible distance of a certain point, is true at that point.

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202. Having proved (first and second parts, 197) that, when two arcs have the ratio of two whole numbers, the angles at the center have the same ratio, we may then suppose that the ratio of BD to BF can not be expressed by whole numbers.

Now, if we divide BF into two

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equal parts, the point of division will be at a certain

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