P. HISTORY OF ' AATHEMATICAL - School of Mathematics

POWERS 339
294. Two general plans for marking powers.-In the early develop
ment of algebraic symbolism, no signs were used for the powers of given
numbers in an equation. As given numbers and coefficients were not
represented by letters in equations before the time of Vieta, but were
specifically given in numerals, their powers could be computed on the
spot and no symbolism for powers of such numbers was needed. It was
different with the unknown numbers, the determination of which constituted
the purpose of establishing an equation. In consequence,
NB. Cod ~tuh G so ~OL aae, sea' b d erru rpatw-&--g.en
FIG. 104.-Signs found in German manuscripts and early German books.
(Taken from J. Tropfke, op. cit., Vol. I1 [2d ed., 19211, p. 112.)
one finds the occurrence of symbolic representation of the unknown
and its powers during a period extending over a thousand years before
the introduction of the literal coefficient and its powers.
For the representation of the unknown there existed two general
plans. The first plan was to use some abbreviation of a name signifying
unknown quantity and to use also abbreviartions of the names
signifying the square and the cube of the unknown. Often special
symbols were used also for the fifth and higher powers whose orders
were prime numbers. Other powers of the unknown, such as the
fourth, sixth, eighth powers, were represented by combinations of
those symbols. A good illustration is a symbdism of Luca Paciola,
in which co. (cosa) represented x, ce. (censo) x2, cu. (cubo) 9, p.r.
(primo relato) 9; combinations of these yielded ce.ce. for d, ce.cu. for