372 INDUCTION. 



There are, moreover, two or three of the principal laws of space or 

 extension which are unusually fitted for rendering one position or 

 magnitude a mark of another, and thereby contributing to render the 

 science largely deductive. First; the magnitudes of inclosed spaces, 

 whether superficial or solid, are completely determined by the magni- 

 tudes of the lines and angles which bound them. Secondly, the length 

 of any line, whether straight or curve, is measured (certain other things 

 beino- given,) by the angle which it subtends, and vice versd. Lastly, 

 the ano^le which any two straight lines make with each other at an inac- 

 cessible 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 

 accomplished (to borrow an observation from M. Comte), by measuring 

 a single straight line and a sufficient number of angles ; which is, indeed, 

 the plan actually pursued in the trigonometrical survey of a country; 

 and fortunate It is that this is practicable, the exact measurement of 

 straight lines being difficult, but that of angles very easy. Three such 

 generalizations as the foregoing affiard such facilities for the indirect 

 measurement of magnitudes, (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 conceivable 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 sub- 

 ject of the sciences of number and extension. The immense part which 

 those laws take in giving a deductive character to the other depart- 

 ments 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 upon, or, in mathematical language, is 

 a function of, the quantity of the agent; and generally of its position 

 also. We cannot, therefore, reason respecting causation, without intro- 

 ducing considerations 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 instruments 

 for calculating forward to an effect, or backward to a cause. That in 

 all other sciences,, as well as in geometry, questions of quality are 

 scarcely ever independent of questions of quantity, may be seen from 

 the most familiar phenomena. Even when several colors are mixed on 

 a painter's pallet, the comparative quantity of each entirely deter- 

 mines the color of the mixture. 



With this mere suggestion of the general causes which render math- 

 ematical principles and processes so predominant in those deductive 

 sciences which afford precise numerical data, I must, on the present 

 occasion, content myself; refen'ing the reader who desires a thorough 

 acquaintance with this great subject, to the fii"st two volumes of M. 

 Comte's systematic work. 



In the same work, and more particularly in the third volume, are 

 also ftilly discussed the necessary limits of the applicability of mathe- 

 matical principles to the improvement of other sciences. Such prin- 

 ciples are manifestly inapplicable, where tlie causes on which any class 



