172 INDUCTION. 



analysis of the truths and processes of algebra any 

 farther ; which is moreover the less needful, as the 

 task has been recently and thoroughly performed by 

 other writers. Professor Peacock's Algebra, and 

 Mr. Whewell's Doctrine of Limits, should be studied 

 by every one who desires to comprehend the evidence 

 of mathematical truths, and the meaning of the 

 obscurer processes of the calculus ; while, even after 

 mastering these treatises, the student will have much 

 to learn on the subject from M. Comte, of whose 

 admirable work one of the most admirable portions is 

 that in which he may truly be said to have created the 

 philosophy of the higher mathematics*. 



7. If the extreme generality and remoteness, not 

 so much from sense as from the visual and tactual 

 imagination,, of the laws of number, renders it a some- 

 what difficult effort of abstraction to conceive those 

 laws as being in reality physical truths obtained by 

 observation ; the same difficulty does not exist with 

 regard to the laws of extension. The facts of which 

 those laws are expressions, are of a kind peculiarly 

 accessible to the sense, and suggesting eminently dis- 

 tinct images to the fancy. That geometry is a strictly 

 physical science would doubtless have been recognised 

 in all ages, had it not been for the illusions produced 

 by two causes. One of these is the characteristic 

 property, already noticed, of the facts of geometry, 

 that they may be collected from our ideas or mental 



* In the concluding pages of his Cours de Philosophic Positive, of 

 which the final volume has but recently appeared, M. Comte 

 announces the intention of hereafter producing a special and syste- 

 matic work on the Philosophy of Mathematics. All competent 

 judges who are acquainted with what M. Comte has already accom- 

 plished in that great department of the philosophy of the sciences, 

 will look with the highest expectations to this promised treatise. 



