188 THIRD REPORT — 1833. 



speculative geometry, as well as between physical and specula- 

 tive mechanics ; and if in speculative geometry we regarded 

 the actual construction and mensuration of the figures and solids 

 in physical geometry alone, the transition from one science to 

 the otlier being made by interpretation, then speculative geo- 

 metry and speculative mechanics must be regarded as sciences 

 which were similar in their character, though different in 

 their objects : but we cultivate speculative geometry without 

 any such exclusive reference to physical geometry, as an in- 

 strument of investigation more or less applicable, by means 

 of interpretation, to all sciences which are reducible to mea- 

 sure, and whose abstract conclusions, in whatever manner 

 suggested or derived, possess a great practical value altogether 

 apart from their applications to practical geometry ; whilst the 

 conclusions in speculative mechanics are valuable from their 

 applications to physical mechanics only, and are not other- 

 wise separable from the conclusions of those abstract sciences 

 which are employed as instruments in their investigation. 



This separation of speculative and physical geometry was 

 perfectly understood by the ancients, though their views of its 

 application to the physical sciences were extremely limited; 

 and it is to the complete abstraction of the principles of specu- 

 lative geometry that we must in a great measure attribute the 

 vast discoveries which were made by its aid in the hands of 

 Newton and his predecessors, when a more enlarged and phi- 

 losophical knowledge of the laws of nature supplied those phy- 

 sical axioms or truths which were required as the medium of 

 its applications ; and though it was destined to be superseded, 

 at least in a great degree, by another abstract science of much 

 greater extent and applicability, yet it was enabled to maintain 

 its ground for a considerable time against its more powerful 

 rival, in consequence of the superior precision of its prin- 

 ciples and the superior evidence of its conclusions, when con- 

 sidered with reference to the form under which the principles 

 and conclusions of algebra were known or exhibited at that 

 period. 



Algebra was denominated in the time of Newton specious or 

 universal arithmetic, and the view of its principles which gave 

 rise to this synonym (if such a term may be used) has more or 

 less prevailed in almost every treatise upon this subject which 

 has appeared since his time. In a similar sense, algebra has 

 been said to be a science which arises from that generalization 

 of the processes of arithmetic which results from the use of 

 symbolical language : but though in the exposition of the prin- 

 ciples of algebra, arithmetic has always been taken for its foun- 



