282 Prof. Guthrie on */ — 1. 



cipal section inclined at an angle of 45° to those of the Nicols. 

 The needle instantly fell to zero, and went up to 90° on the other 

 side. 



And, for circular polarization : — The Nicols being crossed and 

 the needle pointing to 80° in favour of the platinum spiral, a 

 plate of rock-crystal cut perpendicular to the axis was placed 

 across the dark beam. The needle fell to zero, and went to 90° 

 on the other side. 



The penetrative power of the heat here employed may be in- 

 ferred from the fact that it traversed about 12 inches of Iceland 

 spar, and about 1J inch of the cell containing the solution of 

 iodine. 



Royal Institution, March 17, 1870. 



XXXVII. On V — 1. By Francis Guthrie, LL.B,, Professor 

 of Mathematics, Graaff Reinet College, South Africa*. 



"TV/TATHEMATICIANS will probably agree that, whereas 

 ■*-*-■■ V — 1, or some equivalent for it, can hardly be dispensed 

 with in practice, the justifications hitherto given for the use of 

 that symbol are extremely unsatisfactory. Double algebra 

 indeed supplies a complete and most ingenious solution of the 

 difficulty, but one which cannot but be considered as being too 

 laborious and (involving, as it does, conceptions of space and 

 direction) as somewhat inappropriate in a question of pure 

 algebra. The endeavour to put the laws of abstract magnitude 

 on a more simple elementary footing has led me to the following 

 way of considering this subject, which I am desirous of submit- 

 ting to those who take an interest in these matters." 



The first notion of a power is arrived at in arithmetic as the 

 abbreviation of the product of a number of equal numerical fac- 

 tors, the root being defined as the inverse of the power — a defi- 

 nition which, it may be observed, is preserved here and elsewhere, 

 whatever extensions are afterwards given to the word power. 



The first of these extensions results from the larger meaning 

 which it is found convenient to give to the word multiplication, 

 with a view to include negative factors. According to this defi- 

 nition, as x x (-f-m) means x added m times, soa?x (— m) means 

 x subtracted m times, and xx( — m) 2 or xx(—m)x(—m) 

 means xx (— m) subtracted m times, and so on. It is on ac- 

 count of this extension of the words multiplication and power that 

 the inverse function (the root) gives rise to ambiguity of sign, 

 so that Vx 2 — +x. 



* Communicated by Frederick Guthrie, Professor of Physics, Royal 

 School of Mines. 



