When the number of points of division S1, S2, S3, etc., is indefinitely increased the polygon OAP,P,. . P approaches coincidence with the hyperbolic sector OAP,P2.. P, that is,

÷ area of sector OAP,.. P, when n∞;

54. Agenda. The Addition Theorem for Hyperbolic Ratios. From the foregoing definitions of the hyperbolic ratios deduce the following formula:

(I). sinh (u+v)= sinh a cosh + cosh a sinh v
(2). cosh(u+v)=cosha cosh a – sinh “ sinh .

(3). tanh (u+v)

=

tanh u + tanh v I +tanh u tanh v

(4). Deduce these formulæ also geometrically from the constructions of Arts. 48, 53, assuming for the definitions of sinh u and cosh u the ratios NP /a and ON a. [Burnside: Messenger of Mathematics, vol. xx, pp. 145-148.]

(5). In the figure of Art. 53 show that the trapezoids SAPS, SP,P,S,, etc., are equal in area to the corresponding triangles OAP, OPP, etc., and consequently to each other.

(6). Show that when the hyperbolic sector OAP (Art. 53) increases uniformly, the corresponding segment OS, laid off on the asymptote, increases proportionately to its own length.

(7). Assuming a=1 in the equilateral hyperbola of Art. 48, and that the area of any sector is u, prove that limit [(sinh u) /u]= 1. (Use the method of Art. 55.)

55. An Approximate Value of Natural Base. We may determine between what integers the numerical value of e must lie, by substituting their equivalents in x and y for the terms of the inequality:

S

Triangle OAP > sector OAVP > triangle ORS, as represented in Fig. 28. For our present purpose it will involve no loss of generality and it will simplify the computation to assume OA = 1, so that the equation of the hyperbola is x2 — y2 = 1.

RA

Fig. 28.

The sectorial area OAVP, as previously found in Art. 53, is then u, the area of OAP is

obviouslyy, and for that of ORS we may write

OR X (ordinate of S).

To determine OR and this ordinate, write the equations to the tangent RS and the line OP, and find the ordinate of their intersection, and the intercept of the former on the x-axis. The results are:

n

the equation to the tangent RS, έ and ʼn being the current co-ordinates of the line;

the required intercept on the x-axis; and

n = √ 2 (x − 1),

the ordinate of S, found by eliminating έ from the equations to RS and OP. Hence the area ORS is

*Had the assumption a = 1 not been made, this inequality would have been

The equations for RS and OP and the expressions for OR and would have been correspondingly changed, but the final results would have been the same as those given above.

A nearer approximation to the value of e is found by other methods. To nine decimal places it is 2.718281828.

56. Agenda. Logarithmic Forms of Inverse Hyperbolic Ratios. It is customary to represent by sinh-1y, cosh-ix, the arguments whose sinh, cosh,

y', x, etc. Let sinh-y; then

y= sinht= === (et —e-t),

whence, multiplying by e and re-arranging terms,

a quadratic equation in et, the solution of which gives

or

et =y±√ y2+ 1,
t=ln(y±1y2 + 1).

are

If y be real, the upper sign must be chosen; for √y2+1>y and et is positive for all real values of † (Art. 23). Hence (r). sinh-y=ln(+V1+i).

Prove by similar methods the following formulæ :

-I

(5). sech-1x=ln ( 1 / x + √ 1 / x2 — 1).

(6). csch-1y= ln (1 / y + √ I / y2 + 1).

(Cf. Art. 97.)

CHAPTER III.

THE ALGEBRA OF COMPLEX QUANTITIES.

XI.

57.

GEOMETRIC ADDITION AND MULTIPLICATION.

Classification of Magnitudes: Definitions. It was pointed out in Art. 2 that any one straight line suffices for the complete characterization of all so-called real quantities; in fact the real magnitudes of algebra were defined as lengths set off upon such a line. But because, in this representation, no distinction in direction was necessary, all line-segments were taken to be real magnitudes, and comparisons of direction were made, by means of the principles of geometrical similarity, for the sole purpose of determining lengths. Such comparisons will still be necessary whenever the product or quotient of two real magnitudes is called for, but into the real magnitudes themselves no element of direction enters; their sole characteristics are length and sense, that is, length and extension forwards or backwards.

If the attempt be made to apply the various algebraic processes to all real magnitudes, negative as well as positive, another kind of magnitude, not yet considered, is necessarily introduced. For example, if x be positive, no real magnitude can be made to take the place of either (-x)/2 or logm (-x); for the square of a real quantity is always positive (Art. 14), and the definition of an exponential given in Art. 23 precludes its ever assuming a negative value.

In order that forms like these may be admitted into