OF ARTS AND SCIENCES. 115 



III. 



NOTE ON GRASSMANN'S CALCULUS OF EXTENSION. 

 By C. S. Peirce. 



Read Oct. 10, 1877. 



The last " Matheraatische Annalen " contains a paper by H. Grass- 

 mann, on the application of his calculus of extension to Mechanics. 



He adopts the quaternion addition of vectors. But he has two mul- 

 tiplications, internal and external, just as the principles of logic 

 require. 



The internal product of two vectors, v^ and v^^ is simply what is 

 written in quaternions as — S. v^ v.,. He writes it \y^ | v^. So 

 that 



[^1 1 «^2] = \y-i I ^i]» 



t;2= {Tvf. 



The external product of two vectors is the parallelogram they form, 

 account being taken of its plane and the direction of running round it, 

 which is equivalent to its aspect. We therefore have : — 



[y^v^ = ^1^2 si'i <ll' L 



[^1^2] = — [^2^'i]. v"" = 0, 



where I is a new unit. This reminds me strongly of what is written 

 in quaternions as — V{vy^). But it is not the same thing in fact, 

 because [t\vj]v.^ is a solid, and therefore a new kind of quantity. In 

 truth, Grassmau has got hold (though he did not say so) of an eight- 

 fold algebra, which may be written in my system as follows : — 



Three Rectangular Vectors. 



i = M\ A — B '. Z^ a I Y-^ X: N 



j=zM:B—C: X-^A : Z -\- T : N 



k = M: C—A: T-\-B \ X-^ Z '. N 



