But now, to determine what this value muft be, 2x3 2x5 + &c. will give the hyperbolic logarithm 3 5 of the respective number. Example. Let it be propofed to find the hyperbolic logarithm of the number 2. Which values being refpectively divided by the numbers, 1, 3, 5, 7, 9, &c. and the feveral quotients added together, (fee the general feries) we fhall have 346573590 &c. whofe double, being ,693147180 &c. is the hyperbolical logarithm of the number 2. After 1 After the very fame manner the hyperbolic logarithm of any other number may be determined; but, as the feries converges, flower and flower, the higher we go, it is ufual, 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, efpecially, 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 b-1 and c=b+1, we fhall have ac-b-1, and confequently ac ac + 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 fhall As an example hereof, let it be propofed 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++ (x++ &c.)=,058891517&c.and 3 5 confequently hyp. log. 3. (hyp. log. 2+hyp. log. 4• + 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 I =8,6=9 and c=10; and then *(2+1 (2) being I ,006211180 &c. +,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 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 multiplying 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 H common logarithm of 10, to 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 + 2x3 2x5 + &c. X0,434294481 &c. which expreffes 3 5 the common logarithm of 1+ (by what has been already fhewn), and which, by making R ,868568963 &c, will ftand more commodioufly Rx3 Rx5 Rx7 thus, Rx + + + 3 5 7 &c. 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 -S (by the Theor.), S being log. 9 Rx3 Rx + + 3 5 R* (=,8685 &c.) =,006839283 &c. 127 Confequently S (R≈ + + &c.) = 3 5 S= ,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 firft 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) will come out =,845098049 4 &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 1 and the inferior primes, be taken, whereof the common difference is an unit. Thus, because 982x7×7, 99-3X 3x11(9×11), and 100=2×2×5×5 (or 10x10), let there be taken a 98, 699, and c=100; and then, by the first term of the feries only, the log. of 99 will be found true to 14 places; whence that of ii (log. 99-log. 9.) is also known. But |