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

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7. If 2 S, S3, 4.... Sn, be the sums to 2, 3, 4.... n terms of
any series A + B +3 C + D +.+....; shew that
log =S log S+log(1+) 40log ( ) + (l +-)logl+ + ) &c.
XXI.
1. TWhat multiple of x must z be, that az may be equal to e"?
2. If a, b, c be in geometrical progression, shew that logca, logic,
and logab are in arithmetical progression and their common difference
is:.
aoa2.
3. If log ablo = lobb logb3=&, here a, e,, &c., are in geometrical progression, show that b, b2, b3, &c., will also be in geometrical
progression.
4. If ax = b = c &c., and x, y, z, &c., be in harmonical progression, then shall a, b, c, &c., be in geometrical progression::and conversely, if a, b, c, &c., be in geometrical progression, then shall x, y,
z, &c., be in harmonical progression.
5. If loga, log.b, logxc.... be in arithmetical progression, shew
that logay, logy, logy.... are in harmonical progression.
6. Shew that if the bases of different systems of logarithms increase
in geometrical progression, their moduli will decrease in harmonical
progression.
XXII.
1. By means of the expansion of ex, show that
t~- t' ~- =V X2 4 X
e~';f-~ +6e-x/-l-= 2 { 1 + 1.2. 3.4 1.2.3.4.5.G
e 1 -.= 2 IT {- &C 6 }
1.2.3 1.2.3.4.5 f
2. Shew that whatever be the value of x, the series
(x-X-1) 1 -a3-x- 3) + I(X- a-5) + &c. =log,/-1.
3. Express loge (x + y\/- 1) in the form of a + 6 b/- 1.
4. The imaginary part of log,(a- + b - _)vi -- is / -- log,^V/a + b
5. Find the value of log,{(l + x) (1 + a x) (1 + /3 x)} in a series
Nwhere a, P, are the two impossible cube roots of unity.
Xa 293 X4 x5 2X6 x7
6. Shew that lo+ge,(1+ +a-+-)=x + ---+ --- --- + —&C.
In. 2pd 3 4 5 6 7
7. Expand loge ( 1 + + — x) in a series of ascending powers of x.
1 - x X2
XXIII.
1. Shew how far the error committed in obtaining the logaritlm
of a number of six digits from the logarithms of numbers of five digits
by means of proportional parts, can affect the first seven decimal
places of the required logarithm.
2. Explain the method of constructing the tables of proportional
parts for a table of logarithms. Given logo,22713=4 43562745,