OF ARTS AND SCIENCES. 895 



XV. 



ox THE USES AND TRANSFORMATIONS OF LINEAR 



ALGEBRA. 



By Benjamin Peirce. 

 Presented, May 11, 1875. 



Some definite interpretation of a linear algebra would, at first sight, 

 appear indispensable to its successful application. Whereas it is a 

 singular fact, and one quite consonant with the principles of sound 

 logic, that its first and general use is mostly to be expected from its 

 want of significance. The interpretation is a trammel to the use. 

 Symbols are essential to comprehensive argument. The familiar 

 proposition that all A is B, and all B is C, and therefore all A is C, is 

 contracted in its domain by the substitution of significant words for the 

 symbolic letters. The A, B, and C, are subject to no limitation for 

 the purposes and validity of the proposition ; they may represent not 

 merely the actual, but also the ideal, the impossible as well as the pos- 

 sible. In Algebra, likewise, the letters are symbols which, passed 

 through a machinery of argument in accordance with given laws, are 

 developed into symbolic results under the name of formulas. When 

 the formulas admit of intelligible interpretation, they are accessions to 

 knowledge ; but independently of their interpretation they are invalu- 

 able as symbolical expressions of thought. But the most noted 

 instance is the symbol, called ,the impossible or imaginary, known also 

 as the square root of minus "one, and which, from a shadow of meanino- 

 attached to it, may be more definitely distinguished as the symbol of 

 semi-inversion. This symbol is restricted to a precise signification as 

 the representative of perpendicularity in quaternions, and this wonderful 

 algebra of space is intimately dependent upon the special use of the 

 symbol for its symmetry, elegance, and power. The immortal author 

 of quaternions has shown that there are other significations which may 

 attach to the svmbol in other cases. But the stronfft^st use of the 

 symbol is to be found in its magical power of doubling the actual 



