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45 After the very same manner the hyperbolic lo
any other number may be determined but, as the series converges, flower and nower, the higher we go, it is usual, in computing of tables, to derive the logarithms we would find, by help of others already known; for which there are various methods, but the following is the most commodious and simple, that has occurred to me, especially, when a great degree of accuracy is required.
It is thus. Let a, b and c denote any three numbers in arithmetical progression, whose common difference is unity ; then, a being and c=6+1, we shall have ac =b2 - 1, and con
ac + I sequently
Whence, by the nature of logarithms, we likewise have 2 log. b- log. a - log. c = = log.
ac +I: but the logarithm
2% of by putting
= X, will be +
2ac + 1 2x3
&c. (by what has been already 3 5 7 Thewn): which being denoted by S, we shall
log. b = log. 'a + log. c +4S. have log. a
log. a = 2 log. b log. ( - S.
As an example hereof, let it be proposed to find the hyperbolic logarithm of 3.
Then, the hyperbolic logarithm of 2 being already found = ,693147180 &c. that of 4, which is the double thereof, will also be known. Therefore, taking a = 2, b = 3, and c = 4, we shall, in
Therefore S(x++&c.)=,058891517 &c. and
byp. log. 2 + byp. log. 4. consequently hyp. log. 3. +S) = 1,098612288 &c.
2. Let the hyperbolic logarithm of 10 be required.
The logarithms of 8 and 9 being given, from those of 2 and 3 (already found), a may, here, be =8,b=9 and c=10; and then x
5 ,006211180 &c. t ,000000079 &c. &c. = ,006211259 &c.
And therefore log. 10 (2 log. 9 - log. 8 S) = 2,302585092 &c.
Hitherto we have had regard to logarithms of the hyperbolic kind : but those of any other kind may be derived from these, by, barely, multiplying by the proper multiplicator, or modulus.
Thus, in the Brigean (or common) form, where an unit is affumed for the logarithm of 10, the logarithm of any number will be found, by mul
tiplying the hyperbolic logarithm of the same number by the fraction 2434294481 &c. which is the proper modulus of this form.
For, since the logarithms of all forms preserve the same proportion with respect to each other, it will be, as 2,302585092 &c. the hyperbolic log. of 10 (above found) is to (H) the hyperbolic logarithm of
H common logarithm of 10, to
2,302585092 &c. Hx ,434294481 &c. the common logarithm of the same number.
But (to avoid a tedious multiplication, which will always be required when a great degree of accuracy is insisted on) the best way to find the logarithms of this form is from the series 2x +
&c. x 0,434294481 &c. which expresses 3 5
1 + x the common logarithm of (by what has been already shewn), and which, by making R = ,868588963 &c. will stand more commodiously
Rx? R25 Rx7 thus, Rx + + +
5 7 For an example hereof, let the common logarithm of
be required : in which cafe (the logarithms of 8 and 9 being known, from those of 2 and 3), we shall have log. 7 = 2 log. 8
R33 R.x5 S (by the Theor.), S being = Rx tt
64-63 &c. (= the common log. of
and x 63
64 +63 : whence (x2 being
we shall have 127
- log. 9
Rx3 Rx5 Consequently S (Rx + +
3 5 ,006839424 &c. and 2 log. 8 - log. 9-S ,845098040 &c. = the common logarithm of 7, required. But the same conclufion may be brought out by fewer terms of the series, if the 'logarithins of the three first primes 2, 3 and 5 be fupposed known; because those of 48 and 50 (which are composed of them) will likewise be known; from whence the logarithm of 7(=_log. 49 =
S 4 &c. (as before) which value will be true to II places of figures by taking the first term of the series, only.
Again, let the common logarithm of the next prime number, which is it, be required. Here a may be taken = 10,b=ll, and c = 12 ; but fewer terms of the series will fuffice, if other three numbers, composed of 11 and the inferior primes, be taken, whereof the common difference is an unit. Thus, because 98 = 2x7x7, 99= 3* 3x11 (9*11), and 100 = 2 * 2 * 5x5 (or 10 x 10), let there be taken a=98,b=99, and c = 100; and then, by the first term of the series only, the log. of 99 will be found true to 14 places ; whence that of 11 (log. 99 - log. 9) is also known.
But notwithstanding all these artifices and compendiums, a method (similar to that in page 18.) for finding the logarithms of large numbers, one from another, by addition and subtraction, only, still seems wanting in the calculation of tables ; I hall, therefore, here subjoin such a method.
1. Let A, B and C denote any three numbers in arithmetical progression, not less than 10000 each, whereof the common difference is 100.
2. From twice the logarithm of B, subtract the sum of the logarithms of A and C, and let the remainder be divided by 10000.
3. Multiply the quotient by 49,5, and to the product add to part of the difference of the logarithms of A and B; then the sum will be the excess of the logarithm of A + 1 above that of A.
4. From this excess let the quotient (found by Rule 2.) be continually subtracted, that is, first from the excess itself, then from the remainder, then from the next remainder, &c. &c.
5. To the logarithm of A add the said excess, and to the sum add the first of the remainders ; to the last sum add the next remainder, &c. &c. then the several sums, thus arising, will exhibit the logarithms of A + 1, A + 2, A + 3, &c. respectively.
Thus, let it be proposed to find the logarithms of all the whole numbers between 17900 and 18100 ; those of the two extremes 17900 and 18100, and that of the mean (18000) being given.