? self; nothing like this postulate is mentioned however by any previous writer. Regarding the first question we can speak with more confidence. The researches of Peyrard show that in the earliest manuscripts now accessible, this postulate does not appear among the other postulates and common notions at the beginning of the text, but is found in the demonstration of Proposition 29 where it is introduced to support the proof of the equality of the alternate angles of parallel lines. Euclid himself then not only employed this postulate but the position in which he placed it seems to indicate clearly that he appreciated the difficulties which its use involves. Surely one who had formulated a system so rigorous, who was master of a logic so keen and true that the most critical efforts of modern thought have not destroyed but, on the contrary, have only strengthened his claims to rigor, could not have passed over such a manifest begging of the question as appears upon the very face of this postulate as he himself phrased it, without having first made a desperate effort to prove it. Euclid makes no attempt as did later writers to conceal the difficulty under the cloak of a subtle phraseology. He states it frankly as a petitio principii of the baldest type. It must then have appeared to him not as an "axiomatic" truth, but as a theorem calling for demonstration. Euclid proves propositions more obvious by far. He even demonstrates that two a sides of a triangle are greater than the third side, 66 11 The position of the postulate seems to indicate that Euclid struggled on as far as possible without it and postulated it finally only because he could neither prove it nor proceed any further without it. Moreover, the astronomical system of Eudoxus and the writings of Antolycus make it also apparent that Euclid must have had some knowledge of surface spherics and was therefore familiar with triangles whose angle sum contradicts the truth of this postulate. II. Attempts to Dispense with the Postulate.— The intersection of two slowly converging straight lines lies of course beyond the province of observation or construction. Hence it is obvious why the successors of Euclid, habituated by him to strict logical rigor, should have found fault with the parallel postulate and put forth their utmost endeavors to dispose of it in one way or another. In 1621 Sir Henry Saville 12 wrote: "In pulcherrimo Geometric corpore duo sunt nævi, duæ labes nec quod sciam plures in quibus elucendis et emaculendis cum veterum tum recentiorum vigilavit industria." One of these "blemishes" was the par 11 Proclus, op. cit., also Cajori, History of Elementary Mathematics, p. 74. 12 Lectures on Euclid, published at Oxford, 1621. allel postulate, the other Euclid's theory of proportion. Under the title "Parallel" in the "Encyclopädie der Wissenschaften und Kunste," published at Leipzig in 1838, Sohncke says that “in Mathematics there is nothing over which so much has been spoken, written and striven, and all so far without reaching a definite result and decision." Appended to this article there is a carefully prepared list of ninety-two authors who had dealt with the problem. These quotations show the extent to which these earlier efforts were carried. Indeed it appears that almost every writer on geometry of any note from Euclid to Sohncke had given more or less attention to this difficult subject. These earlier endeavors struck out in various directions which we shall now briefly state and consider. Some attempted to avoid the difficulty through a new definition of parallel lines; by others new assumptions which were considered less faulty were substituted for Euclid's. These in reality only concealed the difficulty; they did not remove it. A third class attempted to deduce the theory of parallels from Euclid's other postulates, by reasoning upon the nature of the straight line and the plane angle. These were by far the most desperate attempts. Finally there were those who decided that if this postulate is dependent upon the other assumptions which constitute the foundations of Euclid we shall by denying it and maintaining them, It become ultimately involved in contradiction. was this method of procedure which resulted in the first establishment of a non-Euclidean geometry. We shall consider these attempts in the order. named. (1) The Substitution of Different Definitions. Euclid's own definition was, that parallel lines are straight lines which lie in the same plane and will not meet however far produced. This definition is perhaps still best for elementary geometry. In 1525 Albrecht Dürer,13 a German painter, proposed the familiar definition that parallel lines are straight lines which are everywhere equally distant. Clavius 14 substituted for this the assumption that a line which is everywhere equidistant from a given straight line in the same plane is itself straight. Another definition which is often preferred because of its apparent simplicity is, that parallel lines are straight lines which have the same direction. This definition possesses the peculiar advantage that those who adopt it have no further difficulty; for they find no necessity to assume the parallel postulate or anything equivalent to it. This is a great advantage, certainly; but, as a matter of fact, any one of these definitions, though apparently more advantageous than Euclid's, is in reality more complex and less satisfactory. The first two make use 13 Cajori, p. 266. 14 Edition of Euclid, 1574. of the conception of distance. This of course involves measurement, which in turn embraces the whole theory of incommensurable quantities with its entire outfit of necessary presuppositions and attendant difficulties. What is more, these definitions only hold for Euclidean geometry; they are not true for pseudo-spherical space where parallel lines are still possible and where Euclid's definition is still valid. The objection to the third definition is its use of the term "direction," a word which because of its apparent simplicity, but real obscurity and vagueness, is exceedingly is exceedingly misleading and troublesome. For example, the straight line is often defined as one which does not change its direction at any point, and yet this same line is said to have opposite directions. Again, the angle is sometimes defined as a difference of direction. tion in the circumference of a circle is said to be in a clockwise or counter-clockwise direction, and in this sense a point may move all round the circumference without changing its direction, and yet we speak of this same circumference as a line which changes its direction at every point. Killing has shown that the word direction can only be defined when the theory of parallels is already presupposed.15 Mo Many other definitions have been proposed, but 15 Einfuehrung in die Grundlagen der Geometrie, Paderborn, 1898. |