P. HISTORY OF ' AATHEMATICAL - School of Mathematics

AGGREGATION 389
a few times in John Wallis.' In the Ada eruditorum (1709)) page 327,
one finds ny6=3J[(z-nna)a], where the [ ] makes, we believe, its
first appearance in this journal, but does so as a redundant symbol.
348. Aggregation expressed by dots.-The denoting of aggregation
by placing a dot before the expression affected is first encountered in
Christoff Rudolff (Q 148). It is found next in the An'thmetica integra
of M. Stifel, who sometimes places a dot also at the end. He writes2
Jz. 12++ 6+ ./z. 12-/z 6 for our -+/12- &; also
Jz.144-6+/z.144-6 for /mi-d144-6. In 1605 C. Dibuadius3
writes d.2-/.2+/.2+/.2+/.2+1/2
as the side of
a regular polygon of 128 sides inscribed in a circle of unit radius, i.e.,
42- d2+42+42+/~ (see also g 332). ~t must be admitted
that this old notation is simpler than the modern. In Snell's
translation4 into Latin (1610) of Ludolph van Ceulen's work on the
circle is given the same notation, /. 2+/.2-
d.2- J.2+/2+-
J23. In Snell's 1615 translation5 into Latin of Ludolph's arithmetic
and geometry is given the number J. 2- /. 24 +dl$ which, when
divided by / .2+1/.24+1/1$, gives the quotient /5+1- /.5+
J20. The Swiss Joh. Ardiiser6 in 1646 writes /.2+/.2+/.2+
J.2+/2+/.2+/.2+/3, etc., as the side of an inscribed polygon
of 768 sides, where + means "minus."
The substitution of two dots (the colon) in the place of the single
dot was effected by Oughtred in the 1631 and later editions of his
Clavis mathematicae. With him this change became necessary when
he adopted the single dot as the sign of-ratio. He wrote ordinarily
Jq: BC,- BA,: for I/BC2- BA~, placing colons before and after
the terms to be aggregated (Q 181).'
John Wallis, Treatise of Algebra (London, 1685), p. 133.
M. Stifel, Arithmetica integra (Niirnberg, 1544), fol. 135v0. See J. Tropfke,
op. cit., Vol. I11 (Leipzig, 1922), p. 131.
C. Dibvadii in arithmeticam irrationulivrn Evclidis decimo elementmm libto
(Arnhem, 1605).
Willebrordus Snellius, Lvdolphi b Cevlen de Cirwlo et adscriptis liber ... d
ventaMllo Latina fecit ... (Leyden, 1610), p. 1, 5.
Fvndamenta arithmetica el gemetrica. ... Lvdolpho a Cevlen, ... in Latinum
tzalt9lata a Wil. Sn. (Leyden, 1615), p. 27.
Joh. Ardiiser, Geometriae theorim el pradim XI1 libri (Ziirich, 1646),
fol. 181b.
' W Oughtred, Clavis mnthematicae (1652), p. 104.