Abhandlungen zur Geschichte der Mathematik.

38 Florian Cajori
SPEIDELL did not advance a new theory. He simply aimed to make
all the logarithms in his table positive. To achieve this he subtracted
NAPIER'S numbers from 10 and then discarded the last two digits. NAPIER
gave sin. 30'= 87265 (the radius being taken 107 units) and the
log. sin. 30' = 47413852. Subtracting this logarithm from 108 leaves
52586148. SPEIDELL gives in his table log. sin. 30'= 525861. In both
NAPIER's and SPEIDELL'S tables the logarithms appear as integral numbers.
The same is true of the values for the sines in NAPIFRm' tables: Sin. 30'
is really equal to 0.0087265. The natural logarithm of this fraction is
5.25861. Hence it follows that SPEIDELL'S numiber is the natural logarithmi
of 0.0087265 with 10 added to the characteristi, the decimal point being
omitted. That SPEIDELL'S process of subtracting NAPIER'S logarithms from
10s will, in all cases, yield natural logarithms of x' with the charakteristic
in excess by 10 and with the decimal point left out, follows at once from
the examination of Equation I. For, subtracting 107 log - from 108
giVeS 107 xb 1 log, - from 1o~
gives 107 (10 +- log x' ), where x' -= l Strictly speaking SPEIDELL S
logarithms, as they stand, are not logarithms to a constant quantity taken
as a base. But, if the decimal point is inserted after the characteristic,
then we have natural logarithms to the base e 2.718..., with the
characteristic (in all cases except for the secants and the latter half of
the tangents) increased by 10.
The later editions of SPEIDELL's tables include also logarithms (to
six places) of numbers 1 (1) 1000. As before, the decimal point is left
out. In this table, as none of the characteristics are negative, he did not
add 10. For 770 he gives 6646388, the natural logarithm being 6.646388.
Editions of SPEIDELL'S tables appear to have been issued in
1620, 1621, 1622, 1623, 1624, 1627, 1628. Baron MASERES, in his
Scriptores Logarithinici, 1791-1807, reprinted the logarithms of numbers
from the "tenth inpression", dated 1628. DE MORGAN says that SPEIDELL,
in his Briefe Treatise of Sphaericall Triiangles, mentions and complains of
those who had printed his work without an atom of alteration, and yet
dispraised it in their prefaces for want of alterations. To themn he says:
"If thou canst amend it,
So shall the Arte increase:
If thou canst not: commend it,
Else, preethee, hould thy peace".
This unfair treatment of himself SPEIDELL attributes to his not having
been at Oxford or Cambridge - "not hauing seene one of the Vniuersities."