```Date: Dec 6, 2012 4:22 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Matheology § 172

Matheology § 172Wallis in 1684 [?] accepts, without any great enthusiasm, the use ofStevin's decimals. He still only considers finite decimal expansionsand realises that with these one can approximate numbers (which forhim are constructed from positive integers by addition, subtraction,multiplication, division and taking nth roots) as closely as onewishes. However, Wallis understood that there were proportions whichdid not fall within this definition of number, such as thoseassociated with the area and circumference of a circle:Real numbers became very much associated with magnitudes. Nodefinition was really thought necessary, and in fact the mathematicswas considered the science of magnitudes. Euler, in Completeintroduction to algebra (1771) wrote in the introduction:"Mathematics, in general, is the science of quantity; or, the sciencewhich investigates the means of measuring quantity." He also definedthe notion of quantity as that which can be continuously increased ordiminished and thought of length, area, volume, mass, velocity, time,etc. to be different examples of quantity. All could be measured byreal numbers.Cauchy, in Cours d'analyse (1821), did not worry too much about thedefinition of the real numbers. He does say that a real number is thelimit of a sequence of rational numbers but he is assuming here thatthe real numbers are known. Certainly this is not considered by Cauchyto be a definition of a real number, rather it is simply a statementof what he considers an "obvious" property. He says nothing about theneed for the sequence to be what we call today a Cauchy sequence andthis is necessary if one is to define convergence of a sequencewithout assuming the existence of its limit.[J.J. O'Connor and E.F. Robertson: "The real numbers: Stevin toHilbert"]http://www-history.mcs.st-and.ac.uk/HistTopics/Real_numbers_2.htmlRegards, WM
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