P. HISTORY OF ' AATHEMATICAL - School of Mathematics

INTEGRAL CALCULUS
247
In Newton's Principia (1687), Book II, Lemma II, fluents are
represented simply by capital letters and their fluxions by the corresponding
small letters. Newton says: "If the moments of any quantities
A, B, C, etc., increasing or decreasing, by a perpetual flux, or the
velocities of the mutations which are proportional to them, be called
a, b, C, etc., the moment or mutation of the generated rectangle AB
will be aB+bA." Here a velocity or fluxion'is indicated by the same
symbol as a moment. With Newton a "fluxion" was always a velocity,
not an infinitely small quantity; a "moment" was usually, if not always,
an infinitely small quantity. Evidently, this notation was intended
only as provisional. Maclaurin does not use any regular sign
of integration. He says' simply: "yz+iy, the fluent of which is yz."
Nor have we been able to find any symbol of integration in Thomas
Simpson's Treatise of Fluxions (London 1737, 1750), in Edmund
Stone's Integral Calculus,2 in William Hale's Analysis jluxionum
(1804), in John Rowe's Doctrine of Fluxions (4th ed.; London, 1809),
in S. Vince's Principles of Fluxions (Philadelphia, 1812). In John
Clarke's edition of Humphry Ditton's text,3 the letter F. is used for
"fluent." Dominated by Wallis' concept of i~finity, the authors state,
n-l
"F.x x, will be Finite, Infinite, or more than Infinite, according as
x'"
n is >, =, or < than m." We have seen (§ 582) that the letter F
was used also for "fluxion."
623. Ch. Reyneau and others.-Perhaps no mathematical symbol
has encountered so little competition with other symbols as has f j
.the sign ~ can hardly be called a competitor, it being simply another
form of the same letter. The ~ was used in France by Reyneau' in
1708 and by L'Abbe Sauri in 1774; in Italy by Frisi," by Gherli," who,
in the case of multiple integrals, takes pains to indicate by vinculums
1 C. Maclaurin, Treatise of Fluxio1l8, Book II (1742), p, 600.
2 We have examined the French translation by Rondet, under the title AnalY86
de« injiniment petit8 comprenant le Galetd Integral, par M. Stone (Paris, 1735).
a An I1I8titution of Fluxion8 .... by Humphry Ditton (2d ed., John Clarke;
London, 1726), p, 159, 160.
• Ch. Reyneau, Usaae deL'Analy8e, Tome II (Paris, 1708), p, 734.
B Paulli FriBii Operum Tomus primus (Milan, 1782), p. 303.
B O. Gherli, Gli elementi teorico-praiici delle matematiche pure. Tomo V(
(Modena. 1775), D. L 334.