fkom number to quaternion. 83 



Reflections on Algebra. 

 As the whole algebra is logically deduced from the fundamental 

 properties of the operations which are verified for complex quanti- 

 ties generally, the process of algebraical calculation will not allow 

 us to distinguish between positive, negative or imaginary quantities. 



In other words, when we treat a question by Algebra — what- 

 ever be the real data — as soon as we translate our question into 

 algebraical formulae we transform it into a more general question 

 on complex quantities. Therefore it is easily understood that the 

 solution may be of a quite different nature from the data, for 

 instance, imaginary although the data be real. But the ordinary 

 way of dealing with natural questions is to consider only positive 

 or negative quantities, i.e., to consider quantities only in magni- 

 tude (within the sign) not in direction. In that case, the complex 

 quantity which might be the solution of the corresponding 

 algebraical question is not solution of the physical question. 



We may remark nevertheless, that, as the sum and product of 

 two scalar quantities does not give an imaginary quantity, as long 

 as in our calculations we use only addition and multiplication, 

 the nature of our quantities remains the same. Such is the case 

 for instance for the questions which can be solved by equations 

 of first degree. But if we have to extract roots, then we may 

 introduce by so doing an imaginary quantity and therefore a 

 solution which does not correspond to the physical problem. 



In many questions however, quantities having direction as well 

 as magnitude have to be considered and it might be possible to 

 treat them directly.* In that case the complex solution furnishes 

 the solution of the physical problem. But, as I will explain, it 

 is not always possible to do so. The condition, necessary and 



* Numerous examples of this nature will be found in the works of 

 Giusto Bellavitis. and specially in its " Metodo delle equipollenze," Ann. 

 Sci. Lomb. Veneto vn., 1837, pp. 243 . 261 ; vin., pp. 17 . 37—85 . 121, 

 translated in French, " Exposition de la methode des equipollences " 

 Nouv. Ann. Math, xn., 1873. See also C. A. Laisant, Theorie et applica- 

 tion des equipollences, Paris, Gauthier Villars, 1887. 



