The concepts and theories of modern physics / By J.B. Stallo.

  • John Stallo
Date:
1900
Catalogue details

Licence: In copyright

Credit: The concepts and theories of modern physics / By J.B. Stallo. Source: Wellcome Collection.

    293/376 (page 243)
    form now generally given to it, viz., that through a given point but one straight line can be drawn parallel to another straight line, presupposes the detinition, not only of the straight line, but of parallelism which, in elementary geometry, presents the difficulty of involv- ing the concept of intinite extension, and has given rise to innumerable (piandaries (such as that of infinitely distant, and yet real, points of intersection), among which those of the pangeometncal sort arc not the least. The Euclidean list of definitions, postulates and axioms is vitiated, not, or, at least, not only, by the fact that his lines of distinction between these several pre- requisites of geometrical reasoning are not correctly drawn—that he confounds definitions with axioms and postulates with both,* and, besides, fails to discriminate between axioms of quantity in general and axioms of spatial quantity—but by his ignorance or disregard of the fact, to which I have referred, that every axiom, which is geometrically feifile, involves a definition. lines can not Inclose a space ”) are of the same nature with those of the 12th (usually called the 11th—the axiom of parallelism) has long been known. “ La definition ct les proprietes de la ligne droite,” says D’Alem- bert (ItHdmens de Philosophic, § 12—OCuvres, tome i, p. 280) “ainsi que des lignes parallelcs sont done I’ccucil et, pour ainsi dire, le scandale des ei6mens de geometric.” * Rankel (V'orlesimgcn ueber die complexen Zahlcn und ihre Func- tionen, p. 62) draws attention to the fact that the confusion above referred to is chargeable, not to Euclid, but to his editors and commen- tators. “In all the manuscripts,” says Ilankel, “which F. Peyrard has collected in preparing his excellent edition of Euclid (Q'iuvres d’Eucliile trad, en Latin et en Fran9ais, tome 1, p. 464) the famous 11th principle of the theory of parallels appears, not among the Koivat (vvoiai relating to equal and unequal quantities, but as the 5th postulate {ciTT^na). Simi- larly the 10th axiom in all these manuscripts appears as the 4th postulate, while the MSS. vary in respect to the 12th axiom, it being thus evident that the three axioms owe the place which unaccountably they still oc- cupy in the list*of axioms to a misunderstanding.”