P. HISTORY OF ' AATHEMATICAL - School of Mathematics

288 A HISTORY OF MATHEMATICAL NOTATIONS
nelle;l there the symbol H stands for complete logical disjunction,
X for logical product, I for "identity," C for "contains," and 0 for
"is contained in."
675. Signs of Bolyai.-Wolfgang Bolyai used logical symbols in
his mathematical treatise, the Tentamen,2 which were new in mathematical
works of that time. Thus A =.B signified A absolutely equal
to B; A~ signified A equal to B with respect to content; A(=B or
B=)A signified that each value of A is equal to some value of B;
A (=)B signified that each value of A is equal to some value of B,
and vice versa.
676. Signs of Bentham.-The earlier studies of logic in England
hardly belong to symbolic logic. George Bentham! in 1827 introduces
a few symbols. He lets = stand for identity, II for diversity, t for
in toto (i.e., universality), p for "partiality." Accordingly, "tX = tY"
means X in toto=Y in toto; "tX=pY" means X in toto=Y ex parte;
rin toto
"tX II ~Y" means X in toto II Y i or •
l ex parte
677. Signs of A. de Morgan.-The earliest important research in
symbolic logic in Great Britain is that of De Morqom:" He had not
seen the publications ofLambert, nor the paper of J. D. Gergonne.!
when his paper of 1846 was published. In 1831 De Morgan" used
squares, circles, and triangles to represent terms. He does this also
in his Formal Logic 7 but in 1831 he did not knows that Euler" had
1 J. D. Gergonne in Annales de matMmatiques pures et appliquees, Vol. VII
(Nismes, 1816-17), p. 189-228. Our information is drawn from G. Vacca's
article in Revue de mathematiques, Vol. VI (Turin, 1896-99), p, 184.
2 Wolfgangi Bolyai de Bolya, Tentamen • . . • in elementa matheseos (2d ed.),
Vol. I (Budapest, 1897), p. xi.
S George Bentham, Outline of a New System of Logic (London, 1827), p. 133.
• Augustus de Morgan, Formal Logic (London, 1847); five papers in the
Transactions of the Cambridge Philosophical Society, Vol. VIII (1846), p. 379-408;
Vol. IX (1850), p. 79-127; Vol. X (1858), p. 173-230; Vol. X (1860), p, 331-58;
Vol. X (1863), p. 428-87.
a J. D. Gergonne, "Essai de dialectique rationelle," Annales de matMmatique.
(Nismes, 1816, 1817).
e De Morgan, "The Study and Difficulties of Mathematics," Library of UsefUl
KnowWge (1831), p. 71-73.
7 De Morgan, Formal Logic (1847), p. 8, 9.
8 Ibid., p, 323, 324.
8 L. Euler, Leure« d une Prince88e d' AUemag716 sur quelques sujetsde Physique
et de Philosophie (Petersburg, 1768-72), Lettre CV.