P. HISTORY OF ' AATHEMATICAL - School of Mathematics

TRIGONOMETRY 177
for the calculus, inasmuch as, analogous to the usual forms of algebra,
1 1
there follows from T y = x immediately y = Tgx; from C z= u,
g ~
immediately z = Cos u; etc. "I
535. Persistence of rival notations for inverse funclions.--The
eighteenth-century notations, involving the word "arc" or its equivalent,
were only mild contractions of the words which defined the meanings
to be conveyed to the reader, and they maintained their place in
most works of Continental Europe during the nineteenth century. C.
Gudermannt in 1829 used the notation "arc (tang=z)," etc. In France,
J. Houel employed the easier form "arcsin." The only marked inroad of
Herschel's notation upon the Continent is found with G. Peano who in
1893 3 adopts the Eulerian cos x, for arc cos x, and later' writes sin-I,
COS-I, tang-I.
In Mexico the continental and English notations clashed. F. D.
Covarrubias- writes "arco (sec=x)" and lets cos-I x stand for _1_.
cos x
On the other hand, Manuel Torres Torija" uses "x=tan- I y" and
says that it means the same as "x=arc (tan=y)."
In England, Herschel's notation gained ground rapidly, but not
instantaneously. We find, for instance, in 1838, "arc (tan. ~)" in a
treatise by John. West.'
In the United States the British influence greatly predominated
during the nineteenth century. A rare exception is the occurrence of
the continental symbols "arc tan," "arc esc," in a book of E. P.
Seaver." The use of Herschel's notation underwent a slight change in
I M. Ohm, System der Mathematik, Vol. II (Berlin, 1829), p. 372.
'C. Gudermann, Journal fur r. u. a. Mathematik, Vol. IV (1829), p.
287.
a G. Peano, Lezioni di analisi injinite8'lmale, Vol. I (Turin, 1893), p. 43.
~ G. Peano, Formulaire matMmatique, Edition de l'an 1902-'03 (Turin, 1903),
p. 228, 229.
& F. D. Covarrubias, Elementos de analisis trascendente 6 cd.Iculo injinitessimal
(2d ed.; Mexico, 1890), p. 48, 49.
• Manuel Torres Torija, Neeiones de Algebra Superior y elementosfundamentalea
de cdlculo dijflJT'encial eintegral (Mexico, 1894), p. 181.
7 Rev. John West, Mathematical Treatises (ed. John Leslie; Edinburgh, 1838),
p.237.
• Edwin P. Seaver, Formulas of Plane and Spherical Trigonometry (Boston
and Cambridge, 1871), p. 43.