254 Mr. DE MORGAN, ON TRIPLE ALGEBRA. 



ADDITION. 



In single Algebra, we use no angles, and, so far as geometrical interpretation is concerned, 

 only one dimension of space. In double Algebra, we use two dimensions of space, and the rec- 

 tilinear angle. It might be supposed that in triple Algebra we should use three dimensions of 

 space, and solid angles, considered as proportional to the areas of their subtending equi-radial 

 spherical triangles. I can make no use of these solid angles ; but others may be inclined to try 

 tliem : I accordingly give the following results, connecting the solid angles of a system of co-ordi- 

 nates with the plane ones. 



Let the positive sides of the rectangular axis of x, y, z, meet the sphere in X, Y, Z; let 

 P be any point on the sphere, and let the cosines of the angles PX, PY, PZ, be X, fj., v. Let 

 the spherical excesses of the triangles PYZ, PZX, PXY, be a, /3, 7 : their signs being taken 

 so that the equation a + /3 + 7 =47r, which obviously exists when P is inside the triangle XYZ, 

 may be permanent. We then easily obtain 



, X-' . \{l + ^ + v) , 



cos a = 1 - y- r-r- r , sm a = —: r , 81C. 



(1 + /a) (1 + v) (1 + ,1) (1 + •') 



1 + \ 1 + iU 1 + v (1 + \) (1 + ,u) (1 + >0 



1 + sin a 1 + sin /3 1 + sin 7 (1 + x) (1 + ^u) (1 + v) - X/uf ' 



2(1 + X) (1 + m)(1 + v) 



which, since X^ -1- fi' + v'^ = I 



Also, 1 - cos a + sin a 



(1 -(- X + M + ")' 

 (1+X)(l +,.)(! + .') 



2(1 + sin a) 

 (1 - coso + sin a) + (1 - cos j3 + sin /3) + (1 - C0S7 + sin 7) 



Having since I read this paper in proof, examined Sir W. Hamilton's system of quaternions, I 

 may state that, in my view of the subject, it is not quadruple, but triple, since every symbol is 

 explicable by a line drawn in space. His object has been, to secure interpretation, though it 

 sliould cost the loss of some of the symbolic forms of Algebra ; and his success has been of a 

 most remarkable character. My object has been to detect systems in which the symbolic forms 

 of common Algebra are true, without making any sacrifice to interpretation. The redundant 

 biquadratic system in J 4 of this paper has a resemblance to Sir W. Hamilton's quaternion 

 system in some of its formulse, and a still greater one in its redundant character. It yet remains 

 to be seen what systems exist in which the axes of y and x are not symmetrically related to 

 that of ,v. 



December 17, 1844. 



