Mathematical recreations and essays . axiom or a self-evident statement. This misplacement mayhave been due in the first instance to Theon of Alexandriawho, about 370 , lectured on Euclids Geometry. Ourmodern texts of Euclid are mainly based on Theons lectures,and it is only comparatively recently that the commentaries onEuclids teaching have been subjected to critical discussion. At any rate Euclid, either at the commencement of hiswork or more likely in the course of his demonstration, boldlyassumed that if a straight line meets two other straight linesso as to make the sum of the two in
Mathematical recreations and essays . axiom or a self-evident statement. This misplacement mayhave been due in the first instance to Theon of Alexandriawho, about 370 , lectured on Euclids Geometry. Ourmodern texts of Euclid are mainly based on Theons lectures,and it is only comparatively recently that the commentaries onEuclids teaching have been subjected to critical discussion. At any rate Euclid, either at the commencement of hiswork or more likely in the course of his demonstration, boldlyassumed that if a straight line meets two other straight linesso as to make the sum of the two interior angles on one sideof it less than two right angles, then these straight lines ifcontinually produced will meet upon that side on which these * Euclids Elements, book i, prop. 32. + Eudemus is our authority for this: see Prochis, ed. G. Fricdlein, Leipzig,1873, p. 379. CH. XV] THE PARALLEL POSTULATE 311 angles are situated. Accepting this or some similar assumption,the demonstration is rigorous, and was given by him as Take any triangle ABC, Produce the side BA to any distanceAH, and through A draw a line AK parallel to BC. On theassumption that his postulate is true, Euclid showed (Euc. i. 29)that the angle ABC must be equal to the angle HAK, and theangle AGB to the angle KAG. Hence the sum of the threeangles of the triangle ABG must be equal to the sum of theangles HAK, KAG, and GAB, that is, to two right angles. Euclids postulate and this theorem mutually involve theone the other: if we can prove his postulate this theoremis true, if otherwise we can prove this theorem, then hispostulate is true*. Hence the question with which I com-menced the chapter (namely whether the sum of the anglesof a triangle is, and can be shown to be, equal to two rightangles) comes in effect to asking whether Euclids postulate istrue and can be proved to be true. Features of the Problem, The postulate, as enunciatedby Euclid, has the semblance of a proposition. For manycenturies m
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