The Origin of the Logic of Symbolic Mathematics by Burt C. Hopkins

Author:Burt C. Hopkins
Language: eng
Format: epub, pdf
Publisher: Indiana University Press
Published: 2011-06-15T00:00:00+00:00

§ 104. The Auxiliary Status of Vieta’s Employment of the “General Analytic”

Klein writes, “Vieta explicitly notes that properly speaking these alone, ‘zetetic’ and ‘poristic,’ are intended in Theon’s definition of analysis.” Vieta defines ‘zetetic’ “as the procedure ‘through which the equation or the proportion is found which is to be constructed with the aid of the given magnitudes with a view to the magnitude sought,’ ” and ‘poristic’ “as the procedure ‘through which by means of the equation or proportion the truth of the theorem {!} set up {in them} is investigated’ ” (170/167).136 However, to these two types of analysis, which belong to the analytical art, “Vieta adds still a third” (170/166), which he defines “as the procedure ‘through which the magnitude sought is itself produced out of the equation or proportion set up {in canonical form}’ ” (170/167).137 Depending on whether the magnitude to which it leads is arithmetical or geometrical, this third type of analysis is called, respectively, “rhetic (ητικ) with respect to the definite amounts to which it leads and which can be expressed by the ordinary numeral names of our language” (172/167) and “exegetic (ξηγητικ) in respect to the geometric magnitudes which it makes directly available to sight.”

This third and final stage “in the solution of an equation, which, as we have seen . . . is actually already a part of the synthesis,” is “nevertheless understood by Vieta as an analytical procedure” (172–73/167). Indeed, according to Klein, “ ‘Synthesis’ in Vieta generally takes second place to ‘analysis,’ although in geometric problems he frequently makes use of it and recognizes its traditional priority” (172 n. 105/268 n. 235). Hence, Vieta “says expressly that the results of the analysis have to brought ‘under the order of the art’ (in artis ordinationem) according to the ‘laws’ (leges) κατ παντς, καθ’ ατ, καθλου πρτον (i.e., in school language: predicated ‘of every instance of its subject,’ ‘essentially,’ ‘commensurately with the universal’).” Klein reports that for Vieta the “ ‘law of every instance’ ” is something that “ ‘essentially’ demands that every ‘rule of the art’ (artis decretum) be ‘of the same genus and a member of the same body as it were’. . . .” Thus, “such results ‘as are demonstrated and firmly established by zetetic’ . . . must be subjected to ‘synthesis,’ ‘which is commonly considered the logically tighter way of demonstration’ . . . ; this means that ‘the tracks of analysis are thus repeated’. . . .” Vieta, however, “significantly adds, ‘this is itself also analytical’ (quod et ipsum analyticum est),” as well as “ ‘not troublesome, on account of the species calculation introduced’ . . . by him.”138

Klein reports that Vieta considers “rhetic” and “exegetic” to be “the most important part of the ‘analytic’ ” art (175/168), since—unlike the other two parts, “zetic” and “poristic,” which “consist essentially of ‘examples’ (exempla)”—each of these “comprises a series of ‘rules’ (praecepta).” Because of “[t]his organization of the ‘analytic’ ” (175/168), Klein maintains that the “ ‘general

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.