INDUCTION IN GENERAL. 209 



sively brought; and finally, the legitimacy of the inductions themselves 

 must be decided by other rules, and these it is now our purpose to investi- 

 gate. If this third part of the operation be, in many of the questions of 

 practical life, not the most, but the least arduous portion of it, we have 

 seen that this is also the case in some great departments of the field of sci- 

 ence ; in all those which are principally deductive, and most of all in math- 

 ematics ; where the inductions themselves are few in number, and so obvi- 

 ous and elementary that they seem to stand in no need of the evidence of 

 experience, while to combine them so as to prove a given theorem or solve 

 a problem, may call for the utmost powers of invention and contrivance 

 with which our species is gifted. 



If the identity of the logical processes which prove particular facts and 

 those which establish general scientific truths, required any additional con- 

 firmation, it would be suflftcient to consider that in many branches of sci- 

 ence, single facts have to be proved, as well as principles ; facts as com- 

 pletely individual as any that are debated in a court of justice; but which 

 are proved in the same manner as the other truths of the science, and with- 

 out disturbing in any degree the homogeneity of its method. A remark- 

 able example of this is afforded by astronomy. The individual facts on 

 which that science grounds its most important deductions, such facts as 

 the magnitudes of the bodies of the solar system, their distances from one 

 another, the figure of the earth, and its rotation, are scarcely any of them 

 accessible to our means of direct observation : they are proved indirectly, 

 by the aid of inductions founded on other facts which we can more easily 

 reach. For example, the distance of the moon from the earth was deter- 

 mined by a very cii'cuitous process. The share which direct observation 

 had in the work consisted in ascertaining, at one and the same instant, the 

 zenith distances of the moon, as seen from two points very remote from 

 one another on the earth's surface. The ascertainment of these angular 

 distances ascertained their supplements ; and since the angle at the earth's 

 centre subtended by the distance between the two places of observation 

 was deducible by spherical trigonometry from the latitude and longi- 

 tude of those places, the angle at the moon subtended by the same line 

 became the fourth angle of a quadrilateral of which the other three 

 angles were known. The four angles being thus ascertained, and two 

 sides of the quadrilateral being radii of the earth ; the two remaining 

 sides and the diagonal, or, in other words, the moon's distance from the 

 two places of observation and from the centre of the earth, could be as- 

 certained, at least in terms of the earth's radius, from elementary theo- 

 rems of geometry. At each step in this demonstration a new induction 

 is taken in, represented in the aggregate of its results by a general propo- 

 sition. 



Not only is the process by which an individual astronomical fact was 

 thus ascertained, exactly similar to those by which the same science estab- 

 lishes its general truths, but also (as we have shown to be the case in all 

 legitimate reasoning) a general proposition might have been concluded in- 

 stead of a single fact. In strictness, indeed, the result of the reasoning is 

 a general proposition ; a theorem respecting the distance, not of the moon 

 in particular, but of any inaccessible object; showing in what relation that 

 distance stands to certain other quantities. And although the moon is al- 

 most the only heavenly body the distance of which from the earth can real- 

 ly be thus ascertained, this is merely owing to the accidental circumstances 

 of the other heavenly bodies, which render them incapable of affording such 



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