June 29, 1883.] 



SCIENCE. 



593 



instance, the parameters are the homogeneous 

 co-ordinates of a point in a five-dimensional flat, 

 one gets, hy permuting them, seven hundred 

 and twent}' points, which correspond in twelves 

 to the sixt}' Pascal lines. The analog}' is pre- 

 cise ; for the two figures ha\'e the same alge- 

 braic base, namel}', the substitutions. In his 

 former paper, Veronese forms, for convenience, 

 out of the six fundamental points, fifteen tri- 

 angles, and, out of the sixty Pascal lines and 

 Kirkman points, six configurations 77, consist- 

 ing each of ten Pascal lines and ten Kirkman 

 points, poles and polars with respect to a 

 conic. He finds, that, in a five-fold flat, to 

 the triangles correspond fifteen surfaces of the 

 second order in four dimensions ; to the sixtj' 

 Pascal lines, sist}- surfaces of the fourth order 

 in three dimensions ; to the twenty Steiner 

 points, twenty surfaces of the sixth order in 

 two dimensions ; and, to the six figures 77, six 

 configurations 77, represented in the theorj^ of 

 groups hy the six remarkable six-valued func- 

 tions found bj' Serret {Lioxiville, 1850). As 

 a sample of the vast multitude of propositions 

 given concerning these figures and spaces, we 

 may take the following : the seven hundred 

 and twent}- points obtained b}- permuting the 

 six co-ordinates form a hundred and twenty' 

 cj'cles of six points on rational curves of the 

 fifth order. They lie in sixes on conies in 

 twentj'-four hundred planes, which i>ass bj' 

 hundred and twenties through the twentj' in- 

 tersections of the space unitj^ with the faces of 

 the fundamental pyramid. They are in twen- 

 tj'-fours in four hundred and fifty threefold 

 spaces, which go b^' thirties through the in- 

 tersections of the space unit3^ with the fifteen 

 threefold faces of the fundamental pyramid ; 

 and in hundred and twenties on thirtj'-six four- 

 fold spaces, which go b^' sixes through the 

 intersections of the space unit}' with the six 

 fourfold faces of the fundamental pyramid. 

 Such properties as these are simple and inter- 

 esting in space of high degrees ; but it is well 

 to utilize them also for space of two and three 

 dimensions, which Veronese does hy means of 

 his method of projection {Math, ann., xix.). 

 Thus for everj' complete tetrahedron, penta- 

 gon, and hexagon, in space of three dimen- 

 sions, he gets configurations of points, lines, 

 and curves, like those of the Pascal hexagram, 

 and so for everj- triangle, quadrilateral, penta- 

 gon, and hexagon of the plane ; and he remarks 

 that the same method might be applied to 

 configurations determined by an}' value of n 

 in a space of n — 1 or less dimensions. An- 

 other geometrical interpretation of the groups 

 of substitutions of six letters is given bj- six 



linear complexes of lines in involution two and 

 two (Klein, math, ann., ii.). The}' determine 

 fifteen surfaces of the second order, whose in- 

 tersections are sixty curves of the fourth order 

 corresponding to the sixtj' Pascal lines. There 

 is also a theorem analogous to the Pascal theo- 

 rem for a rational quartic in fourfold space. 



M. Folie, in his report on this paper, com- 

 plains that the contestant has refused to 

 understand the question in the plain sense in 

 which it was proposed ; that he should have 

 started out from the propositions which in 

 M. Folie's book, ' Sur les fondements d'une 

 g^om^trie supdrieure Cart^sienne,' are said to 

 be analogous to the Pascal properties, namel}^, 

 that in a plane cubic curve opposite sides of 

 two quadrilaterals cut in a line, and that in 

 a cubic surface opposite faces of two tetrahe- 

 drons cut in four lines in a plane ; that, after 

 having extended the question as far as possi- 

 ble in this direction, it was open to him to 

 take another point of view, and even that 

 which he has taken, though that is perhaps 

 least of all susceptible of generalization. 

 This work, he says, is remarkable and highly 

 original, and would have deserved the prize 

 had it been the aim of the academ}' simply to 

 call forth a work of that description ; but its 

 object was to engage j'oung geometers in the 

 wa}' alreadj' opened in his own memoirs, and 

 to provoke them to researches which should 

 complete those of the Belgian school of 

 geometers, according to the expression of 

 M. Chasles. This the author has not done : 

 the question, hence, remains unattacked, and 

 will continue to be retained upon the pro- 

 gramme of the academj'. Veronese, in reply, 

 very pertinentl}' inquires whj' it was not 

 equall}' incumbent upon the contestant to 

 follow in the waj' marked out by the Italian 

 and the French schools, b}' Cremona and hy 

 Serret, and maintains that the prize is wrong- 

 1}^ withheld on account of his having followed 

 a new and original waj' instead of that which 

 M. Folie professes to have pointed out to the 

 geometers of the future. He admits that his 

 results are not very susceptible of generaliza- 

 tion, for the reason that they are alreadV so 

 extremelj' general. He complains that M. 

 Folie has given no idea of the contents of his 

 paper, — the usual task of a rapporteur, — and 

 that, in each instance in which he refers to it, 

 he fails to understand it. M. Folie says, for 

 example, that Veronese has applied his method 

 to cubics in space because he could, but not to 

 plane curves or surfaces of order higher than 

 the second, because his method was not there 

 applicable ; while, in fact, Veronese obtains 



