P. HISTORY OF ' AATHEMATICAL - School of Mathematics

ADDITION AND SUBTRACTION 239
of varying shape, found, for example, in James Gregory,' Corachan12
W~lff,~ and Hindenb~rg.~
Sometimes the ordinary Greek cross has the horizontal stroke
very much heavier or wider than the vertical, as is seen, for instance,
in Fort~natus.~ A form for plus -/- occurs in Johan Albert.6
208. Varieties of minus signs.-One of the curiosities in the history
of mathematical notations is the fact that notwithstanding the
extreme simplicity and convenience of the symbol - to indicate subtraction,
a more complicated symbol of subtraction i should have
been proposed and been able to maintain itself with a considerable
group of writers, during a period of four hundred years. As already
shown, the first appearance in print of the symbols + and - for
"plus" and "minus" is found in Widman's arithmetic. The sign - is
one of the very simplest conceivable; therefore it is surprising that a
modification of it should ever have been suggested.
Probably these printed signs have ancestors in handwritten documents,
but the line of descent is usually difficult to trace with certainty
(5 201). The following quotation suggests another clue: "In
the west-gothic writing before the ninth century one finds, as also
Paoli remarks, that a short line has a dot placed above it -, to indicate
m, in order to distinguish this mark frpm the simple line which
signifies a contraction or the letter N. But from the ninth century
down, this same west-gothic script always contains the dot over the
line even when it is intended as a general mark."'
In print the writer has found the sign -- for "minus" only once.
It occurs in the 1535 edition of the Rechenbiichlin of Grammateus
(Fig. 56). He says: "Vnd maii brauchet solche zeichen als + ist
mehr / vnd - / minder."8 Strange to say, this minus sign does not
occur in the first edition (1518) of that book. The corresponding passage
of the earlier edition reads: "Vnd man braucht solche zaichen
Geometriue pars universalis (Padua, 1668), p. 20, 71, 105, 108.
Juan Bautista Corachan, An'lhmelica demonslrada (Barcelona, 1719), p. 326.
Christian Wolff, Elementa mdheseos unioersae, Tomus I (Halle, 1713), p. 252.
' Op. cit.
P. F. Fortunatus, Elemenfa malheseos (Brixia, 1750), p. 7.
eJohan Albert, New Rechenhchlein auff der fedem (Witternberg, 1541);
taken from Glaisher, op. cit., p. 40, 61.
Adriano Cappelli, Lexicon abbrmialuram (Leipzig, 1901), p. xx.
Henricus Grammateus, Eyn new Kumtlich behend and gevriss Rechenbtichlin
(1535; lat ed., 1518). For a facsimile page of the 1535 edition, see D. E. Smith,
Rara arithmetica (1908), p. 125.