P. HISTORY OF ' AATHEMATICAL - School of Mathematics

112 A HISTORY OF MATHEMATICAL NOTATIONS
-3.9030900 +). Certainly these can be placed more conveniently
one below the other.
"Other mathematicians understand by hypothetical logarithms
the complement of the negative logarithms with respect
to 1 O. 0 0 0 0 0 O. For example the hypothetical logarithm of
- 2.7482944 would be 7. 25 1 7056. But computation with
this sort of logarithms becomes in the first place too intricate, and
in the second place too mechanical."
478. Marking the last digit.-Gauss 1 states that von Prasse in
his logarithmic Tafeln of 1810 placed in different type the last figure
in the mantissa, whenever that figure represented an increase in the
value of the mantissa. Babbage denoted the increase by a point subscript
which the reader scarcely notices, but Lud. Schron (1860) used
a bar subscript which catches the eye at once "and is confusing."!
Many writers, for instance Chrystal," place a bar above the last
digit, to show that the digit has been increased by a unit. F. G.
Gausz! explains that 5 in the last digit means that it was raised from
4, that 5means that the remaining decimal has been simply dropped,
that a star (*) prefixed to the last three digits of the mantissa means
that the first "difference" given on the line next below should be taken.
479. Sporadic notations.-A new algorithm for logarithms was
worked out in 1778 by Abel Burja", who writes ~=m where b is the
base, a the power (dignite), and m the logarithm, so that, expressed
b
b
t
. he ordi . b H fi d hac a cd a:o
In t e ordinary notation, a= m. ens t at 'b-'d='b-'b = b
He calls proportion logarithmique four quantities of which the
second is the same power of the first that the fourth is of the third;
he designates it by the sign 8 as in 2, 8 ~ 4,64. Burja proceeds:
"a'! ou a" sera la n-ieme puissance de a
a:, " la n-ieme bipuissance de a .., .
1 K. F. Gauss, Werke, Vol. III (1866), p, 242.
2 J. W. L. Glaisher's "Report on Mathematical Tables" in Report of British
Association (1873), p. 58.
3 G. Chrystal, Algebra, Part II (Edinburgh, 1889), p. 218.
, F. G. Gausz, Logarithm. 'Und trigonom. Tafeln (Halle, a. S., 1906), "Erlauter­
.ungen."
6 Abel Buria in NO'UIJeauz mhnoires de l'acadbnie r, d. Bcience8 el des bel/eN
leUres, annee 1778 et 1779 (Berlin, 1793), p. 301, 321.