158 INDUCTION. 



and angles which bound them. Secondly, the length of any 

 line, whether straight or curve, is measured (certain other 

 things [being given) by the angle which it subtends, and vice 

 versa. Lastly, the angle which any two straight lines make 

 with each other at an inaccessible point, is measured by the 

 angles they severally make with any third line we choose to 

 select. By means of these general laws, the measurement 

 of all lines, angles, and spaces whatsoever might be accom- 

 plished by measuring a single straight line and a sufficient 

 number of angles ; which is the plan actually pursued in the 

 trigonometrical survey of a country ; and fortunate it is that 

 this is practicable, the exact measurement of long straight 

 lines being always difficult, and often impossible, but that of 

 angles very easy. Three such generalizations as the foregoing 

 afford such facilities for the indirect measurement of magni- 

 tudes, (by supplying us with known lines or angles which are 

 marks of the magnitude of unknown ones, and thereby of the 

 spaces which they inclose,) that it is easily intelligible how 

 from a few data we can go on to ascertain the magnitude of 

 an indefinite multitude of lines, angles, and spaces, which we 

 could not easily, or could not at all, measure by any more 

 direct process. 



9. Such are the few remarks which it seemed necessary 

 to make in this place, respecting the laws of nature which are 

 the peculiar subject of the sciences of number and extension. 

 The immense part which those laws take in giving a deductive 

 character to the other departments of physical science, is well 

 known ; and is not surprising, when we consider that all 

 causes operate according to mathematical laws. The effect is 

 always dependent on, or is a function of, the quantity of the 

 agent ; and generally of its position also. We cannot, there- 

 fore, reason respecting causation, without introducing consi- 

 derations of quantity and extension at every step ; and if the 

 nature of the phenomena admits of our obtaining numerical 

 data of sufficient accuracy, the laws of quantity become the 

 grand instrument for calculating forward to an effect, or back- 

 ward to a cause. That in all other sciences, as well as in 



