After the very fame manner the hyperbolic logarithm of any other number may be determined; but, as the series converges, flower and flower, 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 fimple, 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 progreffion, whofe common difference is unity; then, a being and c=b+1, we shall have ac = b2-1, and con ac + I sequently = ас ac = b I, I Whence, by the nature of logarithms, we likewife have 2 log. b - log. 2x3 2x5 2x7 + + &c. (by what has been already 3 5 7 fhewn) which being denoted by S, we shall log.blog.a+log. c + S. have log. a 2 log. b log. c S. log. 2 log. b- log. a 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 fhall, in this x3 &c. Therefore S(++&c.)=,058891517 &c. and confequently hyp. log. 3. byp. log. 2+ byp. log. 4. (byp +S) = 1,098612288 &c. 2 2. Let the hyperbolic logarithm of 10 be required. The logarithms of 8 and 9 being given, from thofe of 2 and 3 (already found), a may, here, be 8,69 and c= 10; and then x (2+1) being we shall have S (x + + 3 5 &c.) = ,006211180 &c. +,000000079 &c. &c. = ,006211259 &c. -S) And therefore log. 10 (2 log. 9 — log. 8 — =2,302585092 &c. Hitherto we have had regard to logarithms of the hyperbolic kind: but thofe of any other kind may be derived from thefe, 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 tiplying the hyperbolic logarithm of the fame number by the fraction ,434294481 &c. which is the proper modulus of this form. For, fince the logarithms of all forms preferve the fame proportion with refpect to each other, it will be, as 2,302585092 &c. the hyperbolic log. of 10 (above found) is to (H) the hyperbolic logarithm of any other number, fo is 1, the common logarithm of 10, to to (253 H 2,302585092 &c. Hx,434294481 &c. the common logarithm of the fame number. But (to avoid a tedious multiplication, which will always be required when a great degree of accuracy is infifted on) the best way to find the logarithms of this form is from the feries 2x + 2** 2x5 3 + ·&c. x 0,434294481 &c. which expreffes 5 x the common logarithm of1+ (by what has been I X already fhewn), and which, by making R = ,868588963 &c. will stand more commodiously Rx3 Rx5 Rx7 thus, Rx+ + 3 5 + &c. 7 For an example hereof, let the common logarithm of 7 be required: in which cafe (the logarithms of 8 and 9 being known, from those of 2 and 3), we fhall have log. 72 log. 8 Rx3 log. 9 S (by the Theor.), S being = Rx +- + 3 164) and x(= 64-63 64+63 6129) we fhall have Rx 3 5 &c. Rx3 Rx5 ,006839424 &c. and 2 log. 8- - log. 9 S= ,845098040 &c. the common logarithm of 7, required. But the fame conclufion may be brought out by fewer terms of the feries, if the logarithms of the three first primes 2, 3 and 5 be fuppofed known; because those of 48 and 50 (which are compofed of them) will likewife be known; from whence the logarithm of 7 (= log. 49 = log. 48+ log. 50 + S 4 will come out =,845098040 &c. (as before) which value will be true to 11 places of figures by taking the first term of the feries, only. Again, let the common logarithm of the next prime number, which is 11, be required. Here a may be taken = 10, b = 11, and c = 12; but fewer terms of the feries will fuffice, if other three numbers, compofed of 11 and the inferior primes, be taken, whereof the common difference is an unit. Thus, because 98 2×7×7, 99= 3x 3 x 11 (9×11), and 100 = 2 × 2 × 5×5 (or 10 x 10), let there be taken a=98, b= 99, and c = 100, and then, by the firft 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 alfo known. But But notwithstanding all these artifices and compendiums, a method (fimilar to that in page 18.) for finding the logarithms of large numbers, one from another, by addition and fubtraction, only, ftill feems wanting in the calculation of tables; I fhall, therefore, here fubjoin fuch a method. 1. Let A, B and C denote any three numbers in arithmetical progreffion, not less than 10000 each, whereof the common difference is 100. 2. From twice the logarithm of B, fubtract the fum 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 part of the difference of the logarithms of A and B; then the fum will be the excess of the logarithm of A+ 1 above that of A. 4. From this excefs let the quotient (found by Rule 2.) be continually fubtracted, 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 faid excefs, and to the fum add the firft of the remainders; to the laft fum add the next remainder, &c. &c. then the several fums, thus arifing, will exhibit the logarithms of A + 1, A+ 2, A+ 3, &c. refpectively. Thus, let it be propofed to find the logarithms of all the whole numbers between 17900 and 18100; thofe of the two extremes 17900 and 18100, and that of the mean (18900) being given. Then E |