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Published online by Cambridge University Press: 03 November 2016
This great work, generally attributed to Eudoxus, has attracted the attention of mathematicians of all time, but is now neglected both in the Schools and in the Universities.
The first person to discover a flaw in the logic of any of the propositions, as given in the text that has come down to us, was (I believe) Saccheri, who, in his Euclides ab omni naevo vindicatus, pointed out that in the demonstration of the 18th proposition it was assumed that if there be three magnitudes of which the first and second are of the same kind, then a fourth magnitude of the same kind as the third must exist such that the ratio of the first magnitude to the second was the same as that of the third to the fourth. This proposition he endeavoured unsuccessfully to prove, but it is now known that its validity depends on the Axiom of Continuity.
Page 215 of note * I leave out of account Euclid’s definition of ratio, because he makes no use of it in his argument.
Page 216 of note * See the writer’s Theory of Proportion, p. 88. (Constable & Co., 1914.)
Page 216 of note † Euclid assumes the possibility of such division in X. 6.
Page 217 of note * The rest of the work amounts to proving that if X > F + Z, then an integer t exists such that X > tz > Y.
Page 217 of note † As has been previously remarked, the fundamental assumption regarding unequal ratios was employed in constructing the test for equality of ratios, but the test for equality of ratios having once been found, propositions concerned with unequal ratios need not be employed to prove propositions about equal ratios.
Page 218 of note * Simson’s proof can be put much more simply (see the writer’s Contents of Euc. V. and VI. 2nd edition, p. 113).
Page 219 of note * If A :B=C:D, then A-C : B-D = A : B.
Page 220 of note * I am indebted to Sir T. L. Heath lor this reference.