Brown. — Phenomena of Variation. 535 



is positive or negative. The usage being as it is, we often in 

 physical problems need to go back to the old arithmetical 

 notion of negative quantities being impossible or imaginary, 

 and consider the graph accordingly. For instance, take 

 Taylor's theorem. It is usually expressed in one formula 

 for the introduction of positive or negative increments ; but 

 Taylor himself (De Morgan, P. Cyc, p. 126) gave two 

 formulae, one for increments (or additions to x) and another 

 for decrements (or subtractions from x). If now we take 

 Maclaurin's form, we readily see that the second formula is 

 impossible if we cannot reduce the magnitude or quantity to 

 less than nothing. 



Thus, to take a typical case, the magnetisation, or B-H, 

 curve of iron, we cannot properly regard B and H as positive 

 and negative quantities, but as direct and reverse positive 

 magnitudes. A Taylor's series increment curve may then, 

 perhaps, hold for magnitudes in either direction. If, how- 

 ever, we adhere to the algebraic usage, we shall be unable 

 to express both of the symmetrical halves of the curve unless 

 we employ only odd-power terms in our formula. This is 

 obviously a very great disadvantage from a graphical point of 

 view. As an indication of the contrary advantage it may 

 be mentioned that a complete half of the sine curve can be 

 built up of added proportions of the standard parabolic and 

 quartic curves, with an extreme error of about 1 in 1,000 

 units, 7r radians forming the unit range of x. 



Further, unless we adhere to the arithmetical notions, we 

 are led to alternative values and imaginary quantities when, 

 as in the example of the B-H curve may be desirable, we em- 

 ploy formulae of fractional powers. Here the only alternative 

 is to drop the fractions which have even denominators, which 

 we can easily foresee may make formulae of this class imprac- 

 ticable for arbitrary approximation to a curve. 



To make clear what is meant, consider the expression ,/-l. 

 Arithmetically — 1 directs 1 to be subtracted from something 

 which appears in the context. To take the square root of 

 that which directs 1 to be subtracted from something else is 

 evidently meaningless arithmetically. So also with ( — l) 2 , 

 and so on. Algebraically we here take the symbol — to in- 

 dicate that the number to which it is attached is negative in 

 quality or impossible arithmetically. This quality is also in- 

 dicated by using a different symbol, ^ 2n , instead of + and — , 

 where i is an imaginary unit powers of which when com- 

 bined with arithmetical symbols make quantities impossible 

 in arithmetic. It is conventions as to the effect of powers 

 of i upon the directions to add or subtract which enable us 

 to perform calculations upon arithmetical quantities by alge- 

 braical methods with only occasional ambiguities. 



