and on the Theory of Tessarines. 409 



angled cone, whose axis is the axis of y, and whose vertex 

 is the origin, the axes being rectangular. Hence, taking 

 i'x+j'y-\-k'z to denote a point whose rectangular co-ordinates 

 are .r, ?/, 2, we see that if two points be taken in the same 

 generatrix of the cone (w.), their tessarine product, considering 

 the vertex as origin, will be the point* whose co-ordinates are 



(1.) (supi'h, ]). 133) A is the axe, B the perpe, and C the norme. Let me 

 add that the equations 



a—x—0, a2_j_y— 0, //«+ '/«=0, 

 (of which the respective solutions are 



x=:a, i/=i'a, z=j'a,) 



denote, when considered separately, three points, each at a distance a from 

 the origin, but in axes at right angles to each other. 



* My friend Professor Davies has {supra, p. 37, and vol. xxix. p. 171- 

 175) intimated or expressed an opinion adverse to the interpretability of 

 the symbol \/ — i in geometry. If eminence in geometric science can 

 confer a right, not only to express such an opinion, but to have that ex- 

 pression duly weighed, then I think that there are few, if any, English geo- 

 meters who possess those rights more unquestionably. I must however 

 confess that I do not see the force of the reasoning employed in Mr. Davies's 

 proposition {lb. p. 174). The inconsistency alluded to in paragraph 4 of the 

 proposition could never arise — at least 1 am unable to perceive how itcould. 

 In saying that a rectangle is equal to the product of its sides, we mean that 

 the numbers of linear units in the sides, when multiplied together, give a 

 number equal to the number of square units in the rectangle. But the 

 signs are not elements in the consideration when we multiply the sides. 

 Unless I am mistaken, the inconsistency in question must arise in some such 

 manner as the following : — Take A in BB'; and let it be required that the 

 rectangle B' A X AB shall equal half the square on BB'. We should then (as 

 to this vide supra, p. 43) find AC=4:a'V/- 1, and that would be a solution 

 of the problem. I consider, then, that my much-valued friend's argument 

 rests solely on the inductive ground previously assigned (vol. xxix. p. 172). 

 But there are strong inductive reasons on the other side of the question. 

 The symbol V' — 1 appears to indicate that move dimensions o? the sub- 

 ject-matter must be taken into consideration than are stated in the data of 

 a question. If we are dealing with a two-dimensioned subject, and meet 

 with the symbol s/ — 1, that symbol indicates impossibility or not according 

 to the fact of our having or not having two dimensions given in our data. 

 Thus, in that spherical geometry with which Mr. Davies's name must ever 

 be associated, and in which a point may be determined by its longitude (<p) 

 and its latitude (^^) (see Camb. Math. Journ., vol. i. p. 193; ii.p. 37, &c.), 



such an expression as 



cos (p -f cos;^V — 1 



is unmeaning; while in plane analytical geometry the expression x-\-yV — 1 

 indicates possibility in space. I may here observe, that if we regard space 

 under a purely graphic aspect, and consider all lines drawn through a point 

 as determined by their inclination to two fixed lines, then we have a strictly 

 two-dimensioned science. With reference to the subject of space, I would 

 add that " Symbolical Geometry " may be made to take an almost infinite 



