LOGARITHMS. 
880 
5 • 
i 
7 . 2’ 
i 
9 . 2 9 
i 
- 
“ 
■5 
19.2 19 
I 
- *0000001003 
- *0000000227 
•OOOOOOOO5O 
- 
- 
•0416666666 
21.2 21 
I 
• 
- 
•00625 
23.2 23 
•549306142a 
- 
- 
•00111607x4 
whence -- ■■■*■■> or 4342944S X 5493061442 
2.30258509 
•4771212, which is the logarithm of 3 required. 
I I . 2* 
I 
15 ' 
.15 
•0002170138 
•OOOO443892 
•OOOOO939OO 
•0000020345 
•OOOOOO44S7 
This Tories, we have already obferved, will only anfwsr 
for the computation of the logarithms of fmall numbers; 
in other cafes different feries muff be employed, according - 
to the particular number under confideration. The limits 
of this article will not admit of an inveftigation of the 
feparate cafes. But for the fake of reference it will be 
ufeful to fubjoin a few of the molt ufeful formulae, for 
which purpofe we avail ourlelves of the fele&ion made 
by Mr. Bor.nycaftle, in his valuable treatife of Trigo¬ 
nometry. 
j 7.2 1 
1. Log. a = ^ X 
*• L °S- “ = H + 4 (“S - )* + 4 ~ &c '£ 
3. Log. a = i X + i (t+t)’ + i 
- ^ c-^y + 4 c-^y - -1 
* ^ r= r x \(r^) + 4 C-^y + 4 C-^> - H 
i =* >< j (£0+ 4 (if iy + * (ifi>+ 
7. Log.-. = log. (. - o + JJ X f f f + *'■ I 
8. Log. <2 = log. (a — 1) + x - , n ', 3 — &c. 
M C<2— I 2(0— l) 2 
9 . Log. a = log. (a — a) + ^ x 5 __L_ + ^ + See. ? 
IVJ la — 1 3 (a — j) 3 5 (a —1)? 5 
1°. Log. = X I (.-.-I) _ i («»_„-«) + 1 - &c. ? 
. ... Log. (. + .) = log. . + £ X - i £ + » £ - * £+*<■} 
... Log. (.-.) = log. . - i X f; + i £ + » *| + i "J +*'■} 
L °s- <« ± *) = <°g-«± ra x +»(rqb) a +t (rqb) 5 + ic.j 
14 . Log. « = - x J(V«-i) - £ (V«-0 9 + * (V«-0 3 - &c. J 
Thefe formulae might have been extended to a much greater length, but thofe that are given will be found to 
embrace the generality of cafes, and will be found ufeful on various occaiions. 
Kepler's Method of ConfruSlion. 
This was founded upon principles nearly fimilar to that 
of Napier. He nrft of all erefes a fyftem of proportions, 
and the meafures of proportion, founded upon principles 
purely mathematical; after which he applies thefe princi¬ 
ples to the conftrudtion of his table, containing only the 
logarithms of sooo numbers. The proportions on which 
his method is founded are in fubftance the following. 
1. All equal proportions equal among themfelv.es are 
exprefled by the fame quantity, be the terms many or 
Vol. XII. No. 880. 
few; as the proportion of 2, 4, 8, Sec. in geometrical pro- 
greffion is exprefled by 2; and of 2, 6, 18, 54, Sec. by 3. 
2. Hence the proportion of the extremes is compoTed 
of all the proportions of the intermediate terms; thus the 
proportion of 2 to 8 is compounded of that of 2 to 4, and 
of 4 to 8. 
3. The mean proportional betwixt two terms divides 
that proportion into two equal ones. Thus the propor¬ 
tion between 2 and 32 is divided by the mea.f propor¬ 
tional 8 into two equal proportions of 45 for 2 is to 8 as 
8 is to 32. 
is R 
4 * 
