Subject: [HM] (a) Heron's formula. (b) Negative numbers.
From: Abe Shenitzer (shenitze@pascal.math.yorku.ca)
Date: Thu Feb 10 2000 - 22:56:18 EST
I missed most of the discussions devoted to factorization and to negative
numbers, so that the remarks that follow may repeat what other list
members have said.
1. ad Heron's formula. On p.36 of "The Beginnings and Evolution of
Algebra" (MAA, vol. 23 in the Dolciani series) we read: "We encounter the
return to numerical algebra ... in the works of ... Heron ... . In
particular, they contain the famous `Heron formula' ... . Here the
expression under the square root sign is a product of four segments, and
thus an expression totally inadmissible in geometric algebra. It is clear
that Heron thought of segments as numbers whose products are likewise
numbers."
2. ad negative numbers. In all her discussions of the work of Diophantus,
Bashmakova credits him with the discovery of what we call today the field
of rational numbers (see, for example, pp.5-8 of her "Diophantus and
Diophantine Equations" MAA, vol. 20 in the Dolciani series). Her position
is championed by Prof. Klaus Barner of U. of Kassel, who was kind enough
to send me recently a copy of a preprint of his paper "Diophant und die
negativen Zahlen. Zu zwei Bemerkungen Norbert Schappachers".
Abe Shenitzer
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