NON-EUCLIDEAN GEOMETRY. 645 



In vain would one seek to treat analytically propositions of this species, 

 even as all the theory of parallels. One would never succeed, just as one 

 would not be able to do without synthesis for measuring plane rectilineal 

 figures, or solids terminated by plane surfaces. It is incontestable that in the 

 beginnings of geometry or mechanics, analysis can not serve as sole method. 



One may compress the circle of synthesis; but it is impossible completely 

 to suppress it. 



From this, however, it follows that all investigations, such as those 

 of Sophus Lie which start with the idea of number-manifold, involve 

 a petitio principii, if interpreted directly as researches on the founda- 

 tions of geometry. In the same way, the non-Euclidean geometry 

 stops the old wrangle as to whether the axioms of geometry are a 

 priori or empirical by showing that they are neither, but are conven- 

 tions, disguised definitions, or unprovable assumptions pre-created by 

 auto-active animal and human minds. 



As Lambert insisted, for the space problem the mathematical treat- 

 ment is in essence the treatment by logic. The start is from a system 

 of axioms, assumptions. We postulate that between the elements of 

 a system of entities certain relations shall hold, e. g., two points 

 determine a straight, three a plane. There is to be shown that these 

 axioms are independent and not contradictory, presupposing pure logic 

 and the applicability to the entities of an arithmetic founded by and 

 made of pure logic. That the assumptions considered should be axioms 

 of geometry, they must satisfy a further condition, which Hilbert 

 formulates thus: 



A system of assumptions is called a system of axioms of geometry if it 

 gives the necessary and sufficient and independent conditions to which a system 

 of things must be subjected in order that every property of these things should 

 correspond to a geometric fact, and inversely; so that therefore in Hertz's sense 

 these things should be a complete and simple picture of geometric reality. 



The physiologic-psychologic investigation of the space problem 

 must give the meaning of the words geometric fact, geometric reality. 



It is the set of assumptions which makes the geometry what it is, 

 which determines it. Thus, in my ' Rational Geometry,' one system 

 of assumptions about the elements, points and straights on a plane, 

 makes Euclidean planimetry. Another set makes Eiemannean planim- 

 etry, in which when we picture it as in Euclidean space, we may call 

 the straights straightests (great circles), and the plane sphere. 



In the light of all this we see how the importance of non-Euclidean 

 geometry for the teacher is still emphasized by the text-books of France, 

 which have never recovered from Clairaut and Legendre. Even the 

 latest and best French geometry, that of Hadamard, published under 

 the editorship of Gaston Darboux, never presents nor consciously con- 

 siders the question of its own foundations. It seems childishly uncon- 

 scious of what is now requisite for any geometry pretending to be scien- 

 tific or rigorous. This lack of foundation is allowable in a preliminary 



