November 8, 1901.] 



SCIENCE. 



709 



of tbese. If the general proposition can 

 be demonstrated in terms general for these, 

 such should be preferred, even if, to attain 

 this, it be necessary to subject the ordinary 

 form to some modification. To cite only 

 one example, we take the convex quadri- 

 lateral inscribed in a circle. 



''In Euclidean geometry, the sum of two 

 opposite angles is constant and equal to two 

 right angles; in non-Euclidean geometry 

 this sum is variable. Notwithstanding this, 

 the two forms may be reconciled, since in 

 both cases the sum of two op^Josite angles equals 

 that of the other two, and this is sufiicient for 

 a convex quadrilateral to be inscriptible. 



" Confronting the proposition with that 

 which concerns the circumscribed quadri- 

 lateral, we put in full light a correlation 

 which, a priori, ought evidently to exist. 



" This correlation, which is the very heart 

 of general geometry, and which does not 

 always appear in the ordinary geometry 

 with the same clearness, can be utilized for 

 finding new properties of the figures. 



" Example : Every conic is the locus of the 

 points such that the suyn of the tangents from 

 these drawn to two circles is constant; every 

 conic then will also be the curve envelope of the 

 straights which cut tivo given circles under 

 angles of which the sum is constant. (Excellent 

 problem for investigating directly.) 



'' III. Is it expedient to associate the 

 non-Euclidean geometry with instruction, 

 and in what measure ? 



" If we treat of higher instruction, with 

 ardor we respond affirmatively. 



" In the courses of higher geometry of 

 the universities the names of Bolyai, Lo- 

 bachevski, Riemann have their assigned 

 place, and there are still divers unexplored 

 domains on the road which these scientists 

 have opened. 



" In so far as it refers ta secondary in- 

 struction, the question is more delicate. 

 The program? of preparatory courses at 

 the high schools contain all, or almost all, 



special mathematics and spherical geom- 

 etry. 



" It would not be then a great incon- 

 venience to there make opportunely a dis- 

 crete allusion to general geometry : on the 

 contrary, the attention of the students and 

 their critical spirit would be held awake by 

 the necessity of investigating if such propo- 

 sition which is expounded to them is of 

 order particular or general. 



"At least two indispensable conditions 

 should be satisfied ; it is requisite : 



"1°. That in all the books put in the hands of 

 the students, the hypothetical and wholly facti- 

 tious character of the Euclidean postidate be put 

 ivell into relief. 



" In my classes I have recourse with suc- 

 cess to the simple procedure which fol- 

 lows, and which I recommend. Take the 

 straight AB and the two equal perpen- 

 diculars AB, BD: the angles AGD, BDC 

 are equal, and may be right, acute or ob- 

 tuse. But whichever be the one among 

 these three hypotheses which we assume 

 for this particular quadrilateral, we must 

 conserve it for all the other like quadrilat- 

 erals. We choose the system of geometry 

 in which these are right angles, and which 

 corresponds to the Euclidean- hypothesis. 



" 2°. That the invertibility of the postu- 

 late of Euclid be completely given up in all 

 the demonstrations in which it can be done 

 without and where nevertheless it is wrongly 

 used. 



" See, for example, the theorem on the 

 face angles of a trihedral or polyhedral 

 angle. 



" We should recognize that great ad- 

 vances have been made in these latter 

 years in the sense indicated. 



' ' If the ideas of general geometry tend 

 to become popularized, the honor of it is 

 due above all to the periodicals which have 

 given their hospitality, and in special 

 manner to Mathesis, so ably edited by our 

 excellent confrere, P. Mansion of Ghent. 



