186 THIRD REPORT — 18o3. 



assumption and establishment of those principles involve meta- 

 physical difficulties of a very serious kind, which present them- 

 selves to a learner at a period of his studies when his mind has 

 not been subjected to such a system of mathematical discipline 

 as may enable it to cope with them : in tlie second place, we are 

 commonly taught to approach those difficulties under the cover 

 of a much more simple and much less general science, by steps 

 which are studiously smoothed down, in order to render the 

 transition from one science to the other as gentle and as little 

 startling as possible ; and lastly, from the peculiar relation 

 which the first principles of algebra, in common with those of 

 other sciences of strict demonstration, bear to the great mass 

 of facts and reasonings of which those sciences are composed. 



It is this last circumstance which constitutes a marked distinc- 

 tion between those sciences which, like algebra and geometry, 

 are founded upon assumed principles and definitions, and the 

 physical sciences : in one case we consider those principles and 

 definitions as ultimate facts, from which our investigations pro- 

 ceed in one direction only, giving rise to a series of conclusions 

 which have reference to those facts alone, and whose correct- 

 ness or truth involves no other condition than the existence of 

 a necessary connexion between them, in whatever manner the 

 evidence of that existence may be made manifest ; whilst in the 

 physical sciences there are no such ultimate facts which can be 

 considered as the natural or the assumable limits of our inves- 

 tigations. It is true, indeed, that in the application of algebra 

 or geometry to such sciences, we assume certain facts or prin- 

 ciples as possessing a necessary existence or truth, investing 

 them, as it were, with a strictly mathematical character, and 

 making them the foundation of a system of propositions, whose 

 connexion involves the same species of evidence with that of the 

 succession of propositions in the abstract sciences ; but in as- 

 signing to such propositions their proper interpretation in the 

 physical world, our conclusions are only true to an extent which 

 is commensurate with the truth and universality of application 

 of our fundamental assumptions, and of the various conditions 

 by which the investigation of those propositions has been sup- 

 posed to be limited ; in other words, such conclusions can be 

 considered as approximations only to physical truth ; for such 

 assumed first principles, however vast may be the superstruc- 

 ture which is raised upon them, form only one or more links in 

 the great chain of propositions, the tennination and foundation 

 of which must be for ever veiled in the mystery of the first 

 cause. 



It is not my intention to enter upon the examination of the 



