and, by substituting these values in the series for

Making x1, we have, since hyp. log. (1)=0,

and, therefore, the differential of the hyperbolic logarithm of any quantity, is equal to the differential of that quantity, divided by the quantity itself.

COR. 2. Hence, writing (− x) in series (B), we

and, therefore, the hyp. log. of a quantity less than

unity is negative.

COR. 3. Hence, subtracting series (C) from series (B), we have

hyp.log. (1 +x) – hyp.log. (1-x) or hyp.log. (1)

hyp. log. (*) or hyp. log. (1 +8) – hyp. log. (*)

1+%

=

we have

Z

COR. 6. If, in the series (E), we. make y = 2,

From whence hyp. log. (2) may be calculated by a series, which converges rapidly. But in the series (B), if we had made x=1, we should have had

a series, which converges, indeed, but so slowly, as to be never used in calculation.

Again, if in the series (E), we had made y=3,

whence, hyp. log. (3) might be calculated. But having before calculated hyp. log. (2), we might have used the series (F), and have made %=2, whence

hyp. log. (3) = hyp. log. (2) + 2 { +

1

3.53

a series which converges with greater rapidity than the one above.

Again, hyp.log. (4) = hyp. log. (2)=2.hyp.log. 2.

as appears by making =4, in the series (F).

Again, hyp.log. (6)=hyp. log. (2) + hyp. log. (3), both which hyp. logs. have been already calculated.

Again, hyp.log. (7) = hyp. log. (6)

hyp.log. (8) hyp.log. (2) = 3 hyp.log. (2),

=

=

hyp.log. (9) hyp.log. (3) 2 hyp. log. (3),

hyp. log. (10) = hyp.log. (5) + hyp. log. (2).

In a similar manner, the hyperbolic logarithms of higher numbers might have been calculated. As, however, it is not the object of academical education to send out into the world a multitude of expert logarithmic calculators, it would be useless to prosecute this subject any further. Those, who have undertaken the laborious task of constructing logarithmic tables, have discovered a great variety of curious artifices, by which they have materially diminished the labour of calculation. A knowledge of these artifices, ingenious as they may be, is, like the knowledge of verbal criticism, useful in its way, but, since it has no tendency either to enlarge the mind, or to strengthen the understanding, it should be left to the pursuit of those, who intend to devote their lives to the study of Mathematics.

PROP. XXXIV.

Having found the hyperbolic logarithms of numbers, to find their logarithms to any other base.

Let (a) be the given base,

(y) the number, whose logarithm is required, (x) the logarithm required;

then y = a*;

.. hyp. log. (y)=hyp. log. (a)=x. hyp. log. (a);

hyp. log. (y)

hyp. log. (a)

COR. 1. Hence, the common logarithm of (2), hyp. log. (2), and having calculated hyp. log. (10) '

or log. (2) =

the hyperbolic logarithms of (2) and of (10) by the former proposition, we may, by division, find the common logarithm of (2). In the same way we may find the common logarithm of all other numbers.

COR. 2. Hence, if (a) and (a') be the bases of two systems of logarithms, and (y) any number,

and log. (y) to base (a) = hyp. log. (y)

hyp. log. (a)

;

hence, by division, we derive the above equation.