308 
ALGEBRA. 
On Logarithms. 
If there be a feries of magnitudes a”, a', a?, a 3 * ,, .. a v ; 
a a 2 , a 3 , .... a~y, the indices, o, i, 2, 3, . .. x ; 
— 1,—2,—3, .... — y, are called the meafures of the 
ratios of thofe magnitudes to 1, or the logarithms of the 
magnitudes. Thus x, the logarithm of any number c, is 
fucli a quantity that a x ~c. Here a may be affumed at 
pleafure ; and, for every different value fo affumed, a dif¬ 
ferent fyftem of logarithms will be formed. In the com¬ 
mon tabular logarithms a is 10, and confequently o, 1, 2, 3^ 
.... x, are the logarithms of 1, 10, 100, 1000, .... 
Cor. 1. Since the tabular logarithm of 10 is 1, the lo¬ 
garithm of a numbenbetween 1 and ioislefs than 1 ; and, 
in the fame manner, the logarithm of a number between 
ioand 100 is between 1 and 2; of a number between 100 
and 1000, is between 2 and 3, &c. If * be the logarithm 
r -N* 2 2 
of 5, then 10) =5; let - be fubftituted for x y and io^ 
3 
2 
is found to be lefs titan 5, therefore - is lefs than the lo- 
3 
3 3 
garithmof5; butio^ is greater titan 3, or-is greater than 
4 
the logarithm of 5 ; the logarithm found in the tables is 
.69897, and nearly. • 
Cor. 2. If quantities be in geometrical progreftion, 
a x , a zx , a 3X , See. their logarithms, x, ix, 3*, &c. are in 
arithmetical progreflion. 
The method of finding the logarithms of the natural 
numbers, or forming a table, will be explained in the 
Doftrineof Fluxions. 
The fum of the logarithms of two numbers is the lo¬ 
garithm of their produdt; and the difference of the loga¬ 
rithms is the logarithm of their quotient. Let x— log. of 
c, and yz^L. log. of d; then a x —c, and a^—d ; hence a x ^y 
c 
~dc, and a x 5 or x-\-y is the log. of dc , and x—y 
the log. of 
Ex. 1. Log. of 3X7= of 3+log. of 7. 
Ex. 2. Log. of pqr— log. of pqAr log. of r=z log. of 
p-\- log. of log. of r. s 
Ex. 3. Log. of - = log. of 5—log. of 7. 
7 
If the logarithm of a number be multiplied by n, the 
product is the logarithm of that number raifed to the nth 
power. Let d be the number whofe log. is x , or a x —d; 
then a’ ,x —d TI , that is, nx is the logarithm of d n . 
Exs. Log. of ITT’— 5 X log. 13. Log. o z —zX log. b. 
If the logarithm of a number be divided by n, the quo¬ 
tient is the logarithm of the nth root of that number. 
V 1 i 
i — x . *— 
lLcta x z=zdy then a n =d n , or - is the log. of d n . 
n 
3 . ”? 
Ex. Log. of 54— -x log. of 5. 
4 
The utility of a table of logarithms in arithmetical calr 
eolations will from hence be manifeft; the multiplication 
and divifion of numbers being performed by the addition 
and fubtradfion of thefe artificial reprefentatives ; and the 
involution or evolution of numbers, by multiplying or di¬ 
viding their logarithms by the indices of the powers or 
roots required. 
Let the value of 5 4/ qy/zX~\/3 be required. 
Log. of 7 ™ .845098 
- log. of 2 = .150515 
2 
~ log- of 3 — .1590404 
value required. 
5 ) 1 ■ *346534 fum 
.2309306 log. of 1. 70188, See. the 
On Intereji and Annuities. 
Intereft is the confideration paid for the ufe or forbear¬ 
ance of money. The rateoi intereft is the confideration 
paid for the ufe of a certain fum for a certain time, as of 
ll. for one year. When the intereft of the principal 
alone, or fum lent, is taken, it is called fimple interefl; 
but, if the intereft, as foon as it becomes due, be confider- 
edas principal, and intereft charged upon it, it is called 
compound intereft. The amount is the whole fum due at 
the end of any time, intereft and principal together. 
Difcounl is the abatement made for the payment of money 
before it becomes due. 
To find the amount of a given fum in any time at fimple 
intereft. 
Let P be the principal, 
r, the intereft of one pound for one year, 
n, the time for which the intereft is to be calculated, 
M, the amount. 
Then fimee the intereft of a given fum mud be proportional 
to the time, 1 (year) : n (years) :: r : nr the intereft of il. 
for n years, and the intereft of PI. muft be P times as great, 
or nrP ; therefore the amount M=;P-J-arP. 
In this fimple equation, any three of the quantities P, 
n, r, M, being given, the fourth may be found; thus P=s 
M 
i-j -nr 
Ex. What fum muft be paid down to receive 600I. at 
the end of nine months, allowing 5 per cent, difeount ? 
Or, which is the fame thing, what principal P will in 
nine months be equivalent, or amount, to 600I. allowing 
5 per cent, intereft ? 
In this cafe M=6oo, n—-, or .75, r~— —=.05; hence 
4 100 
P = - 
M 
600 
2578.313 &c. 1. 
1 +nr I+-75X-05 
To find the amount of an annuity or penfion left unpaid 
any number of years, allowing fimple intereft upon each 
fum or penfion from the time it becomes due. Let A be 
the annuity; then at the end of the firft year A becomes 
due, and, at the end of the fecond year, the intereft of the 
firft annuity is rA ; at the end of this year the principal be¬ 
comes 2 A, therefore the intereft due at the end of the third 
year is 2 rA ; in the fame manner, the intereft due at the end 
of the fourth year is 3 rA, Sec. hence the whole intereft is 
rA-^zrA^-^rA . -\-n — l.rA—n. —-— rA; and the 
fum of the annuities is nA ; therefore the whole amount 
71 - 1 
M-nA-\-n .- rA. 
2 
Required the prefent value of an annuity to continue a 
certain number of years, allowing fimple intereft for the 
money. Let P be the prefent value; then if P and the 
annuity, at the fame rate of intereft, amount to the fame 
fum, they are upon the whole of equal value. The 
amount of P in n-years is P-J-nrP, and the amount of the 
71 — 1 
annuity in the fame time is nA\n. ----- rA ; therefore, 
. . n—i . 
nAA-n. - rA 
n — 1 2 
PA-nrPz=n.AA-n, - rA, andP=---- 
2 i-ynr 
In 
