456 THE PRINCIPLES OF SCIENCE. 



invented. Common trigonometry, for instance, consists 

 of a great series of useful formulas, all of which arise 

 out of the simple fundamental relation of the sine and 

 cosine expressed in the one equation 



sin 2 x + cos 2 x = i. 



But this is not the only trigonometry which may exist ; 

 mathematicians also recognise the so-called hyperbolic 

 trigonometry of which the fundamental equation is 



cos 2 x sin 2 x = i . 



De Morgan has pointed out that the symbols of ordinary 

 algebra form but three of an interminable series of con- 

 ceivable systems 8 . As the logarithmic operation is to addi- 

 tion or addition to multiplication, so is the latter to a 

 higher operation, and so on without limit. 



We may rely upon it that indefinite, and to us incon- 

 ceivable, advances will be made by the human intellect, in 

 the absence of any unforeseen catastrophe to the species 

 or the globe. Almost within historical periods we can 

 trace the rise of mathematical science from its simplest 

 germs. We can prove our descent from ancestors who 

 counted only on their fingers, but how almost infinitely 

 is a Newton or a Laplace above those simple savages. 

 Pythagoras is said to have sacrificed a hecatomb when 

 he discovered the Forty-seventh Proposition of Euclid, and 

 the occasion was worthy of the sacrifice. Archimedes was 

 beside himself when he first perceived his beautiful mode 

 of determining specific gravities. Yet these great dis- 

 coveries are the simplest elements of our schoolboy-know- 

 ledge. Step by step we can trace upwards the acquirement 

 of new mental powers. What could be more wonderful 

 and unexpected than Napier's discovery of logarithms, a 

 wholly- new mode of calculation which has multiplied 

 perhaps a hundred-fold the working powers of every 

 computer, and indeed has rendered easy calculations which 



8 ' Trigonometry and Double Algebra,' chap. IX. 



