Elementary arithmetic, with brief notices of its history... by Robert Potts.

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XVII.
1. + = <( + X) —1++ - See Art. 11. Also for the other four series.
XVIII.
1. Art. 12.
For the next five examples see Art. 14, note.
7. The logarithms given suffice to find the values of (1 44)-6 and (1 44)-3.
8. 10'=101, xlo0g10-loge(10-l)=loge. 102. (1+12)}
=2 log,10 f-logc((l- )
=2log.l0+ 1 1 -1 &c
1 — el + 102- 2-.+ &c. C
1 (11 1
x=2+lolo 10'10l 0- + &c.
XIX.
1. Since a, b, c are three consecutive numbers, a==b-l andi c=b+l, also b-=
ac + 1. See Art. 14.
2. In the product of the expansions of (1 + x)" and c, find the coefficient of a":
so for the other example.
3. Since log,{(1 +X-) } =(1- ) -.log(1 + ).
The difference of the coefficients of x2 — 1 and x~2 in the product of the two series
for (1 - x)- and loge(1 +x) will be found to be e.
4. Expand the logarithms in series and find their sum.
5. The third equation having been found by eliminating z between the two given
equations,
Next take the Napierian logarithms of each of the three equations, expand the
intembers on the right side, and then find the sum of the three equations.
6. See Art. 14.
XX.
1. See Section I., p. 5, and Art. 6, for the advantages of logarithms calculated to
base 10, the same number as the base of the system of numerical notation.
2. The following two series are here added for an exercise of the student's
ingenuity:1 1.3 1 1.3.5 1 1.3.5.7 1 1.3.5.7.9
2' + e 2.4 2 2.64 3 2.4.6.8 4 2.4.6.8.10
loge{2}=1 +3 1+3+3+2 &
^ ^.1 - ' 18 ^ ~ -&e.
4. Note that 1 — -(1-x)-1.
5.2 + 5^+6 =(+3)( +2) = (1 )( +3 )=X2(1+ ) (1+)
3 )xI 1-)-2
loge(x-+5a-6) 2 logx+loge(1+) +0log(1 +)
Expand by the formula Art. 14.
And log(x2 +5x+6)21o+g+ - -*-'+ 2-j.- + &c.
S1-A, S2-S1+A=-S (1+ ) 3s.+B-S2(1+ ) ),