P. HISTORY OF ' AATHEMATICAL - School of Mathematics

POWERS 355
--- I I I etc., as having the "index -4." Our modern notation
/i' /2' /ii'
involving fractional and negative exponents was formally introduced
a dozen years later by Newton1 in a letter of June 13, 1676, to Oldenburg,
then secretary of the Royal Society of London, which explains
the use of negative and fractional exponents in the statement, "Since
algebraists write a2, a3, a4, etc., for aa, aaa, aaaa, etc., so I write a$,
1 1 1
a$, a!, for /a, /aa, /c a5; and I write a-l, a-S, etc., for ---
a' aa' aaa'
etc." He exhibits the general exponents in his binomial formula first
announced in that letter:
m m
p+p&)"= P"+? AQ+- m-n
m-2n m-3n
2n
BQ+n
3n CQ+-&- DQ+ , etc.,
m
m
m
where A = P", B = -PnQ, etc., and where - may represent any real
n
n
and rational number. It should be observed that Newton wrote here
literal .exponents such as had been used a few times by Walli~,~ in
1657, in expressions like ddRd= R, ARmXAR"= A2Rm+", which arose
in the treatment of geometric progression. Wallis gives also the
division ARm)ARm+"(R". Newton3 employs irrational exponents in his
letter to Oldenburg of the date October 24, 1676, where he writes
~Wlfl4=
y. Before Wallis and Newton, Vieta indicated general
exponents a few times in a manner almost ~hetorical;~ his
is our
E potestate-A potesta
A potestas+-- E gradui+A gradu
in A gradum
the two distinct general powers being indicated by the words potestas
and gradus. Johann Bernoulli5 in 1691-92 still wrote 3 0 vax+xx for
1 Isaaei Newtoni Opera (ed. S. Horsely), Tom. IV (London, 1782), p. 215.
2 J. Wallis, Mathesis universalis (Oxford, 1657), p. 292,293, 294.
% J. Collins, Commercium epistolicum (ed. J. B. Biot and F. Lefort; Paris,
1856), p. 145.
Vieta, Opera maihematica (ed. Fr. van Schooten, 1634), p. 197.
Iohannis I Bernoulli, Lectiones de calcvlo diffeentialium . . . . von Paul
Schafheitlin. Separatabdruck aus den Vethandlungen der Naiurfotachenden Gesellschqft
in Basel, Vol. XXXIV, 1922.