This amounts to saying that if _x_ = _p_ + _q_ + _r_ + ..., then _ax_ = _ap_ + _aq_ + _ar_ + .... We also materially simplify Euclid's Book V. He, for example, proves that "If four magnitudes be proportional, they will also be proportional alternately." This he proves generally for any kind of magnitude, while we merely prove it for numbers having a common measure. We say that we may substitute for the older form of proportion, namely, _a_ : _b_ = _c_ : _d_, the fractional form _a_/_b_ = _c_/_d_. From this we have _ad_ = _bc_. Whence _a_/_c_ = _b_/_d_. In this work we assume that we may multiply equals by _b_ and _d_. But suppose _b_ and _d_ are cubes, of which, indeed, we do not even know the approximate numerical measure; what shall we do? To Euclid the multiplication by a cube or a polygon or a sphere would have been entirely meaningless, as it always is from the standpoint of pure geometry. Hence it is that our treatment of proportion has no serious standing in geometry as compared with Euclid's, and our only justification for it lies in the fact that it is easier. Euclid's treatment is much more rigorous than ours, but it is adapted to the comprehension of only advanced students, while ours is merely a confession, and it should be a frank confession, of the weakness of our pupils, and possibly, at times, of ourselves. If we should take Euclid's Books II and V for granted, or as sufficiently evident from our study of algebra, we should have remaining only one hundred thirty-four propositions, most of which may be designated as basal propositions of plane geometry. Revise Euclid as we will, we shall not be able to eliminate any large number of his fundamental truths, while we might do much worse than to adopt these one hundred thirty-four propositions _in toto_ as the bases, and indeed as the definition, of elementary plane geometry. =Bibliography.= Heath, The Thirteen Books of Euclid's Elements, 3 vols., Cambridge, 1908; Frankland, The First Book of Euclid, Cambridge, 1906; Smith, Dictionary of Greek and Roman Biography, article Eukleides; Simon, Euclid und die sechs planimetrischen Buecher, Leipzig, 1901; Gow, History of Greek Mathematics, Cambridge, 1884, and any of the standard histories of mathematics. Both Heath and Simon give extensive bibliographies. The latest standard Greek and Latin texts are Heiberg's, published by Teubner of Leipzig. FOOTNOTES: [23] Riccardi, Saggio di una bibliografia Euclidea, Part I, p. 3, Bologna, 1887. Riccardi lists well towards two thousand editions. [24] Hermotimus of Colophon and Philippus of Mende. [25] Literally, "Who closely followed the first," i.e. the first Ptolemy. [26] Menaechmus is said to have replied to a similar question of Alexander the Great: "O King, through the country there are royal roads and roads for common citizens, but in geometry there is one road for all."