ill a Letter toJi.T. Graves, Esq. 491 



respecting the moduli is .*. fulfilled, if we suppress the term 

 involving /^ altogether, and what is more, a^ — 6^ — c^, 2rt^, 

 9,ac are precisely the coordinates of the square-point, so to 

 speak, deduced from the point a, i, c, in space, b}' a slight 

 extension of Mr. Warren's rule for points in a plane. [It is 

 a long time since I read his book, but the grand features of 

 his view cannot be foi'gotten.] In fact, if we double, in its 

 own plane, the rotation from the positive semiaxis o^ x to the 

 radius vector of the point «, h, c, we attain the direction of 

 the radius vector drawn to a^ — b'^—c'^, 2ab, 2ac. 



Behold me therefore tempted for a moment to fancy that 

 ij = 0. But this seemed odd and uncomfortable, and I per- 

 ceived that the same suppression of the term which was de 

 trop might be attained by assuming what seemed to me less 

 harsh, namely that ji=: —ij. I made therefore ij=:k,ji=--/c, 

 reserving to myself to inquire whether k was =0 or not. For 

 this purpose I next multiplied a + ib+jc, as a multiplier, into 

 x + ib+jCf as a multiplicand, keeping still, as you see, the two 

 factor lines in one common plane with the unit line. The re- 

 sult was ax—b' — c"^ + i{a + .v)b+j{a + s)c-\-/c{bc — bc), in which 

 the coefficient of Z: still vanishes; and ax— b'^—c% (« + j:)^, 

 {a -{-x)c are easily found to be the correct coordinates of the 

 product-point, in the sense that the rotation from the unit line 

 to the radius vector of a, b, c, being added in its own plane to 

 the rotation from the same unit-line to the radius vector of 

 the other factor-point x, b, c, conducts to the radius vector of 

 the lately mentioned product-point; and that this latter radius 

 vector is in length the product of the two former. Confirma- 

 tion of ij= —Ji ; but no information yet of the value of k. 

 Try boldly then the general product of two triplets, and seek 

 whether the law of the moduli is satisfied when we suppress 

 the k altogether. Is (a^ + b'^ + c^) {x- + j/^^ + ss^)=:{ax—bj/—czf 

 + {at/ + bx)'^ + [az + cxY? No, the first member exceeds the 

 second by (bz — cjjY. But this is just the square of the coeffi- 

 cient of /", in the development of the product {a + ib+Jc) 

 {x-\-iij-\-J2)i if we grant that ij-=k,ji=^ —k, as before. And 

 here there dawned on me the notion that we must admit, in 

 some sense, a fourth dimension^ of space for the purpose of 

 calculating with triplets ; or transferring the paradox to alge- 

 bra, must admit a third distinct imaginary symbol k, not to 

 be confounded with either / or J, but equal to the product of 



* The writer has this moment been inforinecl (in a letter from a friend) 

 that in the Cambridge Matliematical Journal for May last a pajier on Ana- 

 lytical Geometry of w dimensions has been published by Mr. Cayley, but 

 regrets that he docs not j'ct know how far Mr, Cayley's views and his own 

 may resemble or differ from each other. 



2 K2 



