FROM NUMBER TO QUATERNION. 81 



Negative quantities. — Now, if we remark that the subtraction 

 of a is replaced by the addition of a^ we see that — incorporating 

 the sign with the quantity we consider — we have 



—a = a^ 

 so that a quantity of argument tt may be replaced by the same 

 quantity of argument zero but with sign minus. And we come 

 then to the notion of negative quantity which may be developed 

 at this stage of our study. As a natural result, tensor and argu- 

 ment can be considered as well positive as negative. 



Any positive or negative quantity is called a scalar quantity. 



Imaginary quantities. — Let us consider again a complex quantity 

 under fiorm x + y v 



we see that it can be written also 



and putting for shortness 2 



^77 — i 



2 

 we may write any complex quantity under the form 



x + i y 



From definition of quantity i 



i s = In. 1^ = — i i* = l 27r = 1 



2 



or more generally denoting by n any whole number 

 #* = 1 i 4ft + 2 = — 1 



i^n+l — I ^4n+3 — — { 



and keeping the same definition of the square root as that usedjin 

 arithmetic, i.e., calling the square root of a complex quantity a 

 second quantity such that squared it gives the first one, we see that 



as i 2 = — 1 i = VZTi 



a quantity with a perfectly real signification, the symbol of a 

 rotation by tt. 



2 



Trigonometrical Expressions. 

 All that we have hitherto done till now supposes only a know- 

 ledge of arithmetic and of Euclid.* Adding to these mathematical 



* We have only differed from Euclid in one respect, that is in consider- 

 ing a portion of a straight line as a symbol of translation and an angle 

 as a symbol of rotation. I may point out here that Euclid's elements 

 would gain a good deal by the introduction at once of these notions, which 

 are universally recognised in modern mathematics. 



F— Jvme 6, 1894. 



