BBV, T. P. iilBKMAN ON LINEAR CONSTEUCTIONS. 287 



K is a length found by the txihr only on the axis of y, and 

 positive or negative with the sum of the solids of fifteen di- 

 mensions, which is thus proved equivalent to one single 

 solid of the same dimensions; an equivalence rigorously 

 geometrical, and of the truth of which the reader can judge 

 for himself, although he has my word only for the existence 

 of such solids. 

 The term 

 C z:z:i^ .'i^-z^-xy.yz.zx.«.y.z.\ zzix.x.x.x.x-yz.yz-yz.yz.yz 



618 3341A50789 «66IS>2iSt«3&78 



can be reduced by drawing thirteen pairs of parallels, to 

 the equivalent C n: ^^S* a*o Yj which is a solid equal in 

 geometrical value to C. The whole of the terms contain- 

 ing iCo can thus be collected into one term, X^^r^ %, Xq. 



Let us suppose that we are in quest of all the points 

 (^oy© 2^0) in any line, passing through one of the given nine, 

 whose co-ordinates x^ y^ and z^ satisfy the geometrical con- 

 dition, 



2 + 4 . yf . 2I . 3:3^3 . y^Zi . 2-5X5 . ar, . ^7 . sTg . I9 = 0. 



Let the line pass through the point 9; let this be taken 

 fcMT the origin, and the hue for the axis of ar. We shall have 

 ajg ^i ^9 n: 2^9 zz n yo =^ ^jg in our paradigm, which will 

 thus contain no terms free from X(^^ The proposition before 

 us is then equivalent to 



a:oer^*{Kr„ — ^K,)=0, 

 which is satisfied by all the points which we are seeking 

 on the axis of a?, K and Ki being lengths on that of y from 

 O, found with the ruler only, and ^ being the a:-iinit, arbi- 

 trarily assumed 'on the axis of a?. There is plainly only one 

 such point, which is found by joining the extremities of § 

 and K, and then firom the extremity of K, drawing a 

 parallel to the joining line: the parallel cuts the axis of » 

 in the point required. If K, zz o, the axis of a? is a tangent 

 of the surface at the origin. We have thus proved that 

 our paradigm represents a surface, for it is the locus of a 

 point which can be assigned more than once upon any line 



