[bakkr] the foundations OF GEOMETRY 125 



of the proposed magnitudes." In the demonstration of this Lemma, 

 Euclid says, " For C (the less of the magnitudes) may be multiplied 

 £0 as at length to become greater than AB (the greater of the mag- 

 nitudes)/' which is a statement of this principle of continuity, and 

 which Euclid evidently bases on Definition 4, Book V. Euclid uses 

 this Lemma in the demonstration of Prop. 2, Book XII, " Circles are 

 to one another as the squares on their diameters," and Archimedes 

 frequently assumes it. Indeed, it constitutes the foundation of the 

 Method of Exhaustions from which descended our Infinitesimal Calculus. 

 I have thought this historical reference worth making that the place 

 this assumption of Archimedes occupies in science may be thle more 

 manifest. 



Such then are the assumptions which Hilbert makes the foundations 

 of geomet^}^ It will be observed that they all refer to geometrical 

 magnitude, whereas, of the axioms of Euclid only three are really 

 geometrical, — the eighth (magnitudes wliich coincide are equal), the 

 eleventh, and the twelfth (two straight lines cannot enclose a finite 

 space), the tenth (all right angles are equal to one another) being 

 capable of proof. 



It is, of course, essential that these assumptions should be (1) con- 

 sistent with one another, and (2) independent of one another. First 

 with respect to the question of consistency: "As geometry is built 

 up by the indefinitely repeated application of the axioms, the possi- 

 bility is not excluded that a contradiction might appear only after an 

 unlimited repetition of such application" (Sommer, Bulletin, Am. 

 Math. Soc, Vol. VI, p. 291). To settle the question Hilbert trans- 

 lates his groups of assumptions into the domain of numbers, the number 

 foncept being presumably entirely abstract, and, therefore, independent 

 cX experience. Any inconsistency would then appear in the arith- 

 metical form of the assumptions, and the search may be a possible 

 performance. Xext, with respect to the independence of the assump- 

 tions of one another: The examination is here made by leaving out 

 each assumption in turn, and showing that without it a perfectly inde- 

 pendent and consistent system of geometry can be constructed in which 

 the omitted assumption does not hold. The method, of course, is 

 suggested by the way in which non-Euclidean geometry has been 

 built up. 



Professor Halsted's presentation of Hilbert's system of geometry, 

 a presentation which, having regard to the newness of the ideas, must 

 be spoken of as consummately able, has been undertaken with a view 

 to popularizing this rational geometry. I conjecture that most edu- 

 cators will think it impossible for school purposes. Let me, however. 



