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SCIENCE 



[Vol. XX. No. 517 



SCIENCE; 



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NON-EUCLIDEAN GEOMETRY. 



BY G. A. MILLEE, PH.D., EUREKA COLLEGE, EUREKA, ILL. 



Euclid's elementary georaeti-y was written about three cen- 

 turies before the Christian era. We must conclude that it was 

 much superior to all preceding works on this subject. Proclus, 

 who wrote a commentary on Euclid's Elements in the fifth cen- 

 tury of our era, represents it such, and his statements are corrob- 

 orated by the facts that all similar works of Euclid's predecessors 

 have ceased to exist, and, if any elementary geometry was written 

 by a Greek after Euclid, there is no mention made of this any- 

 where.' 



The facts that Euclid's Elements are still used as a textbook — 

 especially in England —and that the works used in Its place are 

 generally based upon it, are perhaps still stronger evidences of its 

 excellence. 



No geometry can be written without making some assumptions 

 with respect to tbe space with which it deals. These are generally 

 of such a nature as to commend themselves to our full confidence 

 by their mere mention, and are commonly called axioms. It is 

 the duty of the geometer to demonstrate properties and relations 

 of magnitudes by non-contradictory statements which lest ulti- 

 mately upon these axioms. It is evident that the axioms should 

 be as few and as clear as possible. Upon essentially different 

 axioms essentially different geometries may be established. 



Among the axioms of Euclid there is at least one which is not 

 axiomatic.'' This is the axiom of parallels, which reads as fol- 

 lows : — 



" If a straight line meet two straight lines so as to make the two 

 interior angles on the same side of it taken together less than 

 two right-angles, these straight lines, being continually produced, 

 shall at length meet on that side on which are the angles which 

 are less than two right-angles." 



All the popular text-books on elementary geometry employ this 

 axiom either in this form or in some shorter form, such as, 

 "Through a point without a line only one line can be drawn 

 parallel to the given line." 



Many efforts have been made to demonstrate this axiom. Since 

 it does not depend upon more elementary axioms, such attempts 

 must be futile. If we assume it to be true, it follows directly 

 that the sum of the three angles of a plane triangle is two right- 

 angles; and, conversely, if we should assume that the sum of the 

 internal angles of a plane triangle is two right-angles, this axiom 

 would follow.^ 



As the geometers who do not adopt all the axioms of Euclid 

 deny this, non-Euclidean geometry is sometimes defined as the 

 geometry which does not assume that the sum of the three angles 

 of a plane triangle is two right-angles. A more satisfactory defi- 



» Cantor's Vorlesungen uber Geschlchte der Mathematik, Vol. I., p. 224. 

 - EncyclopEedla Britannlca, Vol. VIII., p. 657. 

 3 Frischauf's Absolute Geonietrie, pp. 14, 15. 



nition is, non-Euclidean geometry is a geometry which assumes 

 other properties of space in place of the following properties of 

 Euclidean space : — 



The sum of the three angles of a plane triangle is two right- 

 angles, space is an infinite continuity of three dimensions, and 

 rigid bodies may be moved in every way in space without change 

 of form. 



Just one hundred years ago (1792) the famous mathematician 

 Gauss began the study of a geometry free from the first of these 

 assumptions. He did not publish tbe results of his study. We 

 may infer something in regard to them from his letters '' It was 

 not until 1840 that a geometry was published in which Euclid's 

 axiom of parallels was replaced by another, and the sum of the 

 angles of a plane finite triangle was thus shown to be less than 

 two right angles. The work was written by a Russian mathe- 

 matician nanftd Lobatschewsky. It contains only sixty-one pages 

 and bears the title " Geometrische Untersucbungen znr Theorie 

 der Parallellinien." He began his treatment of parallels by ob- 

 servations, in substance, as follows : — 



Given a fixed line (L) and a fixed point (A) not on this line. 

 The lines through A lying in the plane determined by A and L 

 may be divided with respect to L into two classes — (1) those in- 

 tersecting i, and (2) those not intersecting L. The assumption 

 that the second class consists of the simple line which is at right- 

 angles with the perpendicular from .4 to L is the foundation of a 

 great part of the ordinary geometry and plane trigonometry. 

 While the assumption that the second class consists of more than 

 one line leads to a newer geometry, whose results are also free 

 from contradictions.'* This newer geometry was called non- 

 Euclidean geometry by Gauss, imaginary geometry by Lobat- 

 schewsky, and absolute geometry by Johann Bolyai." 



It is certainly of interest to learn what some of the foremost 

 mathematicians have said with respect to this geometry. Pro- 

 fessor Sylvester said in regard to Lobatschewsky 's work : — 



"In quaternions the example has been given of algebra released 

 from the yoke of the commutative principle of multiplication — 

 an emancipation somewhat akin to Lobatschewsky 's of geometry 

 from Euclid's noted empirical axiom." 

 Professor Cayley said : — 



"It is well known that Euclid's twelfth axiom, even in Play- 

 fair's form of it, has been considered as needing demonstration; 

 and that Lobatschewsky constructed a perfectly consistent theory 

 wherein this axiom was not assumed to hold good, or, say, a sys- 

 tem of non-Euclidean plane geometry." 



Another very eminent mathematician, Professor Clifford, in 

 speaking about the same work, said : — 



"What Vesalius was to Galen, what Copernicus was to Ptolemy, 

 that was Lobatschewsky to Euclid." 



Something of the nature of-this geometry may be inferred from 

 a few of its theorems which differ from the corresponding theo- 

 rems of the ordinary geometry. In addition to the important 

 theorem that the sum of the internal angles of a plane finite tri- 

 angle is less than two right-angles, it is proved that if we have 

 given a line (L) and a perpendicular (B) to L, the parallels to L 



through points on B will make angles with B varying from - toO; 



so that we can draw through B a parallel to L making any given 

 angle with B.' 



The locus of a point at a constant distince from a straight line 

 is a curved line.* 



The areas of two plane triangles are to each other in the ratio 

 of the excesses of two right-angles over the sums of their angles.' 



We proceed now to some observations on the second property 

 of Euclidean space mentioned above, viz., that space is an infinite 

 continuity of three dimensions. We shall not take up the ques- 

 tion of the infinitude of space nor Riemann's distinction between 



> Brietwechsel zwlschen Qausa und Schumacher,— especially Vol. [I., pp. 

 268-271. 



= LobatschewsSy's Theorle der Parallellinien, Art. 22. 



« Frischauf's Absolute Geometrle, Art. 13. 



' Lobatschewsky'a Theorle der Parallellinien, Art. 23. 



8 Frischauf's Absolute Ge metric, p. 18. 



^ Frischauf's Absolute Geometrle, p. 50. 



