154 INDUCTION. 



Positive, on the philosophy of the higher branches of mathe 

 matics, are among the many valuable gifts for which philosophy 

 is indebted to that eminent thinker. 



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 somewhat difficult effort 

 of abstraction to conceive those laws as being in reality phy 

 sical 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 senses, and suggesting eminently distinct 

 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 circumstances. 

 One of these is the characteristic property, already noticed, of 

 the facts of geometry, that they may be collected from our 

 ideas or mental pictures of objects as effectually as from the 

 objects themselves. The other is, the demonstrative character 

 of geometrical truths; which was at one time supposed to 

 constitute a radical distinction between them and physical 

 truths, the latter, as resting on merely probable evidence, being 

 deemed essentially uncertain and unprecise. The advance of 

 knowledge has, however, made it manifest that physical 

 science, in its better understood branches, is quite as demon 

 strative as geometry. The task of deducing its details from 

 a few comparatively simple principles is found to be anything 

 but the impossibility it was once supposed to be; and the 

 notion of the superior certainty of geometry is an illusion, 

 arising from the ancient prejudice which, in that science, mis 

 takes the ideal data from which we reason, for a peculiar class 

 of realities, while the corresponding ideal data of any deduc 

 tive physical science are recognised as what they really are, 

 mere hypotheses. 



Every theorem in geometry is a law of external nature, 

 and might have been ascertained by generalizing from obser 

 vation and experiment, which in this case resolve themselves 

 into comparison and measurement. But it was found prac- 



