P. HISTORY OF ' AATHEMATICAL - School of Mathematics

280 A HISTORY OF MATHEMATICAL NOTATIONS
writers using these two symbols are Senillosa,' Izquierdo,' LiBvano,J
and Porfirio da Motta Pegado." In German publications i for arithmetical
progression and for geometric progression occur less fre-
quently than among the French. In the 1710 publication in the Miscellanea
Berolinensia6 + is mentioned in a discourse on symbols
(5 198). The G was used in 1716 by Emanuel Swedenborg.6
Emerson7 designated harmonic progression by the symbol - and
harmonic proportion by :. .
249. Arithmetical proportion finds crude symbolic representation
in the Arithmetic of Boethius as printed at Augsburg in 1488 (see
Figure 103). Being, in importance, subordinate to geometrical proportion,
the need of a symbolism was less apparent. But in the seventeenth
century definite notations came into vogue. William Oughtred
appears to have designed a symbolism. Oughtred's language (Clawis
[1652], p. 21) is "Ut 7.4: 12.9 vel 7.7-3: 12.12-3. Arithmetic&
proportionales sunt." As later in his work he does not use arithmetical
proportion in symbolic analysis, it is not easy to decide whether the
symbols just quoted were intended by Oughtred as part of his algebraic
symbolism or merely as punctuation marks in ordinary writing.
John Newtons says: "As 8,5:6,3. Here 8 exceeds 5, as much as 6
exceeds 3."
Wallisg says: "Et pariter 5,3; 11,9; 17,15; 19,17. sunt in eadem
progressione arithmetica." In P. Chelucci's'o Inslituliones analyticae,
arithmetical proportion is indicated thus : 6.8 ': 10.12. Oughtred's
notation is followed in the article "Caract6re" of the Encyclopddie
Felipe Senillosa, Tratado elemental de Arismetica (Neuva ed.; Buenos Aires,
1844), p. 46.
Gabriel Izquierdo, Tratado de Aritdica (Santiago [Chile], 1859), p. 167.
Indalecio LiBvano, Tratado de Aritmetica (2. Bd.; Bogota, 1872), p. 147.
Luiz Porlirio da Motta Pegado, Tratade elementar de arithmetica (2. Bd.;
Lisboa, 1875), p. 253.
j Miscellanea Berolineda (Berolini, 1710), p. 159.
6Emanuel Swedberg, Daedalus hyperboreus (Upsala, 1716), p. 126. Facsimile
reproduction in Kungliga Veten-skaps Soeieteten-s i Upsala Tviihundraiirsminne
(Upsala, 1910).
' W. Emerson, Doctrine of Proportion (London, 1763), p. 2.
John Newton, Znstitutw d h i c a or mathematical Institution (London,
1654), p. 125.
0 John Wallis, op. cit. (Oxford, 1657), p. 229.
lo Paolino Chelucci, Znstitutiunes analyticoe (editio post tertiam Romanam
prima in Germania; Vienna, 1761), p. 3. See also the first edition (Rome, 1738),
p. 1-15.