erties of zero. He does not discuss a symbol, but he shows by his
treatment that in some way zero had acquired a special significance not
found in the Greek or other ancient arithmetics. A still more scientific
treatment is given by Bh[=a]skara,[192] although in one place he permits
himself an unallowed liberty in dividing by zero. The most recently
discovered work of ancient Indian mathematical lore, the
Ganita-S[=a]ra-Sa[.n]graha[193] of Mah[=a]v[=i]r[=a]c[=a]rya (c. 830 A.D.),
while it does not use the numerals with place value, has a similar
discussion of the calculation with zero.

What suggested the form for the zero is, of course, purely a matter of
conjecture. The dot, which the Hindus used to fill up lacunae in their
manuscripts, much as we indicate a break in a sentence,[194] would have
been a more natural symbol; and this is the one which the Hindus first
used[195] and which most Arabs use to-day. There was also used for this
purpose a cross, like our X, and this is occasionally found as a zero
symbol.[196] In the Bakh[s.][=a]l[=i] manuscript above mentioned, the word
_['s][=u]nya_, with the dot as its symbol, is used to denote the unknown
quantity, as well as to denote zero. An analogous use of the {54} zero, for
the unknown quantity in a proportion, appears in a Latin manuscript of some
lectures by Gottfried Wolack in the University of Erfurt in 1467 and
1468.[197] The usage was noted even as early as the eighteenth
century.[198]

The small circle was possibly suggested by the spurred circle which was
used for ten.[199] It has also been thought that the omicron used by
Ptolemy in his _Almagest_, to mark accidental blanks in the sexagesimal
system which he employed, may have influenced the Indian writers.[200] This
symbol was used quite generally in Europe and Asia, and the Arabic
astronomer Al-Batt[=a]n[=i][201] (died 929 A.D.) used a similar symbol in
connection with the alphabetic system of numerals. The occasional use by
Al-Batt[=a]n[=i] of the Arabic negative, _l[=a]_, to indicate the absence
of minutes {55} (or seconds), is noted by Nallino.[202] Noteworthy is also
the use of the [Circle] for unity in the ['S][=a]rad[=a] characters of the
Kashmirian Atharva-Veda, the writing being at least 400 years old.
Bh[=a]skara (c. 1150) used a small circle above a number to indicate
subtraction, and in the Tartar writing a redundant word is removed by
drawing an oval around it. It would be interesting to know whe