OP ARTS AND SCIENCES. 397 



• QUADRATES. 



The best definition of quadrates is that proposed by Mr. Charles S. 

 Peirce. If the letters A, B, G, &,c., represent absolute quantities, 

 differing in quality, the vids may represent the relations of these quaa- 

 tities, and may be written in the form 



{A: A) {A:B){A:C)... {B:A) {B:B) . . . {C:A),&c. 

 subject to the equations 



(A:B) (B:C)=z(A:C) 

 {A:B) (C:B)=0. 



i. e. every product vanishes, in which the second letter of the multiplier 

 differs from the first letter of the multiplicand, and, Avhen these two 

 letters are identical, both are omitted, and the product is the vid which 

 is compounded of the remaining letters which retain their relative 

 position. 



Mr. Peirce has shown by a simple logical argument that the quadrate 

 is the legitimate form of a complete linear algebra, and that all the 

 forms of the algebras given by me must be imperfect quadrates, and 

 has confirmed this conclusion by actual investigation and reduction. 

 His investigations do not however dispense with the analysis, by which 

 the independent forms have been deduced in my treatise, but they 

 seem to throw much light upon their probable use. 



UNITY. 



The sum of the vids (A: A), {B:B), (C:C), &c., extended so as to 

 include all the letters which represent absolute quantities in a given 

 algebra, whether it be a complete or an incomplete quadrate, has tlie 

 peculiar character of being idempotent, and of leaving any factor 

 unchanged with which it is combined as multiplier or multiplicand. 

 Tins is the distinguishing property of unity, so that this combination 

 of the vids can be regarded as unity, and may be introduced as such, 

 and called the vid of unity. There is no other combination which 

 possesses this property. 



But either of the vids {A: A), (B:B), &c., or the sum of any of 

 these vids is idempotent. There are many other idempotent combina- 

 tions, such as 



^ {A:A)-{- ^ (A: B) -{- ^ {B: A) -^ ^ (B:B), 



which may deserve consideration in making transformations of an alge- 

 bra preparatory to its application. 



