V58 THE PRINCIPLES OF SCIENCE. [CHAP 



man has ever said or even whispered. There, in their 

 mutable but unerring characters, mixed with the earliest 

 as well as the latest sighs of mortality, stand for ever 

 recorded, vows unredeemed, promises unfulfilled, perpe- 

 tuating in the united movements of each particle the 

 testimony of man's changeful will." 



When we read reflections such as these, we may con- 

 gratulate ourselves that we have been endowed with minds 

 which, rightly employed, can form some estimate of their 

 incapacity to trace out and account for all that proceeds 

 in the simpler actions of material nature. It ought to be 

 added that, wonderful as is the extent of physical pheno- 

 mena open to our investigation, intellectual phenomena are 

 yet vastly more extensive. Of this I might present one 

 satisfactory proof were space available by pointing out that 

 the mathematical functions employed in the calculations 

 of physical science form an infinitely small fraction of the 

 functions which might be invented. Common trigonometry 

 consists of a great series of useful formulae, all of which arise 

 out of the relation of the sine and cosine expressed in one 

 equation, sin z x + cos z x = I. But this is not the only 

 trigonometry which may exist ; mathematicians also recog- 

 nise hyperbolic trigonometry, of which the fundamental 

 aquation is cos z x sin z x = I. De Morgan has pointed 

 out that the symbols of ordinary algebra form but three 

 of an interminable series of conceivable systems. 1 As the 

 logarithmic operation is to addition or addition to multi- 

 plication, so is the latter to a higher operation, and so on 

 without limit. 



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

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

 the absence of any catastrophe to the species or the globe. 

 Within historical periods we can trace the rise of mathe- 

 matical science from its simplest germs. We can prove 

 our descent from ancestors who counted only on their 

 fingers. How 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 propo- 

 sition of Euclid, and the occasion was worthy of the sacrifice. 

 Archimedes was beside himself when he first perceived 



Trigonometry and Double Algebra chap, ix 



