264 Mr DAVIES on the Equations of Loci 



pressed. The arc or angle never appeared in the notation before 

 the time of EULER : all the functions of it were expressed as 

 functions of some one selected function of the arc, rather than 

 directly as functions of the arc itself. Symbols of operation, 

 though the most natural symbols, and perhaps more easily ap- 

 prehended in the infancy of mathematical study, were then used 

 very sparingly, and were scarcely ever themselves rendered sub- 

 ject to subsequent operations ; except, indeed, when they assumed 

 a numerical form in a given position, as, for instance, the expo- 

 nent of a power. It was also undecided then, and is scarcely 

 agreed upon even now, in what way angles become subject to cal- 

 culation in conjunction with linear quantities. The result of tri- 

 gonometrical operations, thus shackled, became so extremely com- 

 plex, and the transformation which they required so difficult, 

 that any geometer who attempted it quickly abandoned it in 

 utter hopelessness. The trigonometrical notation of EULER, by 

 attaching the name of the line as a functional characteristic to 

 the symbol of the arc, opened a new and fertile field for disco- 

 very, and trigonometry instantly assumed a renovated aspect 

 under the talisman of his magic touch. Had there been then 

 any disposition to pursue the investigation of spherical loci, the 

 chief difficulty was removed ; and, no doubt, it would soon have 

 assumed, in his hands, the simplicity and symmetry which 

 MONGE was at the same time labouring to confer upon the geo- 

 metry of rectilinear co-ordinates. There was, indeed, still want- 

 ing the notation for inverse trigonometrical functions, which has 

 been more recently and elegantly supplied by a living mathema- 

 tician, whose varied intellectual powers approach more closely to 

 those of EULER than any one who has yet appeared since the 

 world was deprived of that singularly gifted mind. The want 

 of this notation was not so strongly felt ; but it would have been 

 sooner felt, and sooner discovered, had the cultivation of spheri- 

 cal loci, by means of trigonometrical functions, been earlier cul- 



