4o6 



INDUCTION. 



are completely determined by the 

 magnitudes of the lines 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 accomplished 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 measure- 

 ment of long straight lines being al- 

 ways difficult, and often impossible, 

 but that of angles very easy. Three 

 such generalisations as the foregoing 

 afford such facilities for the indirect 

 measurement of magnitudes, (by sup- 

 plying us with known lines or angles 

 which are marks of the magnitude of 

 unknown ones, and thereby of the 

 spaces which they enclose,) that it is 

 easily intelligible how from a few data 

 we can go on to ascertain the mag- 

 nitude 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 remarks which it 

 seems necessary to make in this place 

 respecting the laws of nature which 

 are the peculiar subject of the sciences 

 of number and extension. The im 

 mense 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 alv\ays depen- 

 dent on or is a function of the quantity 

 of the agent, and generally of its 

 position also. We cannot, therefore, 

 reason respecting causation without 



introducing 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 quan- 

 tity become the grand instrument 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 col- 

 ours are mixed on a painter's palette, 

 the comparative quantity of each en- 

 tirely determines the colour of the mix- 

 ture. 



With this mere suggestion of the 

 general causes which render mathe- 

 matical principles and processes so pre- 

 dominant in those deductive sciences 

 which afford precise numerical data, 

 I must, on the present occasion, con- 

 tent myself : referring the reader who 

 desires a more thorough acquaintance 

 with the subject to the first two 

 volumes of M. Comte's systematic 

 work. 



In the same work, and more parti- 

 cularly in the third volume, are also 

 fully discussed the limits of the ap- 

 plicability of matriematical principles 

 to the improvement of other sciences. 

 Such principles are manifestly inap- 

 plicable, where the causes on which 

 any class of phenomena depend are 

 so imperfectly accessible to our obser- 

 vation, that we cannot ascertain, by 

 a proper induction, their numerical 

 laws ; or where the causes are so 

 numerous, and intermixed in so com- 

 plex a manner with one another, that 

 even supposing their laws known, the 

 computation of the aggregate effect 

 transcends the powers of the calculus 

 as it is, or is likely to be ; or, lastly, 

 where the causes themselves are in a 

 state of perpetual fluctuation ; as in 

 physiology, and still more, if possible, 

 in the social science. The mathe- 

 matical solutions of physical ques- 

 tions become progressively more diffi- 

 cult and imperfect in proportion as 

 the questions divest themselves of 



