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tween two series will be necessarily equal to it: it being assumed that the

others

which have inverse ratio of any two consecutive coefficients of those series

the same

limit.

is always finite.

Thus, if the three series representing the three quantities be

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where the inverse ratio of any two consecutive coefficients is always finite, then values of x exist which will make their arithmetical values differ from their first terms by quantities less than any which may be assigned: let such values of the series be a +d, b+d1, a + d2: and since it is assumed that a +d is greater than b + d1, and b+d, greater than a + d2, it will follow that the values of

a-b+d-d, and b-a +8,- d are arithmetical and positive: and, if possible, let us suppose b=a+d or a-d: in the first case, the preceding expressions become

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and inasmuch as these expressions are necessarily positive, it will follow that, in the first case, d-d, is greater than d, or d greater than d+d, which is contrary to the hypothesis, since d is supposed less than any quantity which may be assigned: and in the second case, d1-d, must be greater than d, or d, greater than do, which is also contrary to the hypothesis, since d has been supposed less than any quantity which may be assigned it follows, therefore, that b is necessarily equal to a, which is the proposition to be proved.

:

This is a proposition of very extensive application, inasmuch as it will very frequently happen that an expression, which is not capable of direct development into a series by the aid of assumed definitions or known theorems, may be shewn, by other considerations, to be included in value between expressions which admit of direct development, and which have therefore ascertainable limits; and it will follow that, if those limits be the same, the limit of the unknown expression or undeveloped series is necessarily the same likewise.

attending

a limit.

922. The correct notion of a limit is not easily formed, Difficulties inasmuch as it is necessarily connected with a conception of a the constate of existence of magnitude, either in itself, or in some ception of quantity upon which it is dependent, which is incapable of arithmetical or geometrical representation: and like all our notions, therefore, which ultimately involve considerations of zero or infinity (Chap. xxxvIII.), it is entirely negative: it is on this account that it becomes of the utmost importance that we should confine our attention exclusively to the definition (Art. 917), which we have given of it, and altogether disconnect it, like other definitions, from every consideration which is not essentially involved in it.

nity.

A limit, in conformity with its definition, may be zero, but A limit not infinity: for though we are incapable of conceiving zero may be zero, but as one of the successive states of existence of magnitude, we not infiare capable of conceiving its existence in a state in which it differs from zero by a quantity less than any which may be assigned, and therefore when zero becomes, as it were, the fixed limit of the definition: but we are utterly incapable of conceiving the existence of a quantity which is not infinite, but which at the same time differs from infinity by a quantity less than any which may be assigned: and therefore, under no circumstances can infinity answer the conditions of a limit, which the definition assigns to it.

CHAPTER XXXVII

The series

ON THE SERIES AND EXPONENTIAL EXPRESSIONS FOR THE
SINE AND COSINE OF AN ANGLE.

923. IT has been already shewn (Art. 808) that the value

for the sine of a in the exponential expressions

and cosine

of x.

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is indeterminate, as far as it is dependent upon the definition of the sine and cosine of an angle only: we shall proceed, in the Articles which follow, to shew that it ceases to be indeterminate when the measure of angles ceases to be so.

The exponential series deduced in Chapter xxxv give us

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If we further replace k by c√-1, these series become,

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The mea

924. It may be easily shewn that these series will satisfy

sure of an the equation (Art. 758)

angle

which is

assumed in

sinx + cosr=1

when substituted in it, whatever be the value of c: but if we

the value

assume, as we have already done (Art. 746), x or the measure Art. 746, of an angle, to be the ratio of the arc which subtends it to the determines radius of the circle in which it is taken, then it may be demon- of the symstrated that 1 is the only value of c, in the series for sin x and cos x, which will satisfy the conditions to which it leads. For, if we assume x or the measure of the angle BAC to be

BEC

T

bol c which

they in

volve.

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AB we get

AB'

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of the series which is equivalent to it), is required to be determined.

For this purpose, we observe that the ratio

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CP
chord BC
since the chord BC is less than

is

x

the arc BEC: but CP= AB sin x, and the chord BC=2 AB sin

-:

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and again, inasmuch as the tangential line BT is greater than the

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sin x
СР
we get = COS x =
tan x

BT

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This may be easily shewn from geometrical considerations: if from the extremity C of the arc BEC, we draw Ct a tangent meeting BT in t, then we have Bt Ct: but Tt is greater than Ct, being opposite to the greater angle TCt: therefore BT is greater than Bt + Ct: again, if from the middle point E of BEC, we draw the tangent cEb, meeting Ct in c and Bt in b, then we have bt+ct,

greater

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are arranged in the order of their magnitude; and since the first and third have a common limit, which is 1, the limit c of the second series is identical with it, and therefore equal to 1. (Art. 919).

greater than be, and therefore Bt + Ct, and therefore a fortiori BT, greater than Bb+be+cC: in a similar manner, if we bisect the arcs BE and CE, and from their middle points draw tangents meeting Bb, be and Cc, then the sum of the tangents thus formed will be less than Bb+be+ Cc, and therefore a fortiori than BT: by continuing this process, we should increase the number and diminish the magnitude of these small tangents, until their sum, which is necessarily less than BT, shall differ from the arc BEC by a quantity or line less than any that may be assigned: or in other words, the arc BEC is necessarily less than BT: by a similar course of reasoning, the arc BEC may be shewn to be necessarily greater than the chord BC.

Again, if we take r for the radius of the circle, the series

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will express BT, and the several sums of the successive circumscribing tangents: in a similar manner the series

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will express the chord BC and the several sums of the successive inscribed chords : the ratio of the nth term of the second series to the nth term of the first is

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and the limit of this value is 1: this is another mode of arriving at the conclusion in the text.

This relation of the chord, arc and circumscribing tangent, is one of fundamental importance in the application of Algebra to the theory of curves: it applies to the arcs of all curves of continuous curvature, as well as to arcs of the circle.

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