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Download The Works of Archimedes: Edited in Modern Notation with by Thomas L. Heath PDF

By Thomas L. Heath

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With the aid of such auxiliary lines or (what is the same thing) auxiliary fixed points in a figure, combined with the use of proportions, Archimedes is able to effect some remarkable eliminations. Thus in the proposition On the Sphere and Cylinder 11. 4 he obtains three relations connecting three as yet undetermined points, and xliv INTRODUCTION. proceeds at once to eliminate two of the points, so that the problem is then reduced to finding the remaining point by means of one equation. Expressed in an algebraical form, the three original relations amount to the three equations 3a — x y 2a - x ~ x a+x X % z 2a-x n and the result, after the elimination of y and z, is stated by Archimedes in a form equivalent to m+n a+x 4a2 n ' (2a —x)*' a Again the proposition On the equilibrium of Planes II.

To which Amthor added (ibid. p. ) a discussion of the problem itself. The general result of Krumbiegel's investigation is to show (1) that at the Museum which was found at Chaourse (Aisne) in France and is of a section sufficiently like the curve in the Salinon. The other explanations of a-dXivov which have been suggested are as follows. (1) Cantor connects it with

Equilib. II. 9) he actually gives as the Archimedean proof a paraphrase of Eutocius somewhat retouched and abridged, and in many other instances he has inserted corrections and interpolations from another Greek MS. which he once names. This MS. appears to have been a copy made from F, with interpolations due to some one not unskilled in the subject-matter; and this interpolated copy of F was apparently also the source of the Niirnberg MS. now to be mentioned. Na (= Codex Norimbergensis) was written in the 16th century and brought from Rome to Niirnberg by Wilibald Pirckheymer.

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