DEMONSTRATION, AND NECESSARY TRUTHS. 



exactly true, whether they are a sufficiently near approxima- 

 tion to the truth. The reason is obvious. Since it is only in 

 questions of pure number that the assumptions are exactly 

 true, and even there, only so long as no conclusions except 

 purely numerical ones are to be founded on them ; it must, in 

 all other cases of deductive investigation, form a part of the 

 inquiry, to determine how much the assumptions want of being 

 exactly true in the case in hand. This is generally a matter 

 of observation, to be repeated in every fresh case; or if it has 

 to be settled by argument instead of observation, may require 

 in every different case different evidence, and present every 

 degree of difficulty from the lowest to the highest. But the 

 other part of the process namely, to determine what else may 

 be concluded if we find, and in proportion as we find, the as- 

 sumptions to be true may be performed once for all, and the 

 results held ready to be employed as the occasions turn up for 

 use. We thus do all beforehand that can be so done, and leave 

 the least possible work to be performed when cases arise and 

 press for a decision. This inquiry into the inferences which 

 can be drawn from assumptions, is what properly constitutes 

 Demonstrative Science. 



It is of course quite as practicable to arrive at new conclu- 

 sions from facts assumed, as from facts observed; from fic- 

 titious, as from real, inductions. Deduction, as we have seen, 

 consists of a series of inferences in this form a is a mark of b, 

 b of c, c of d, therefore a is a mark of d, which last may be a 

 truth inaccessible to direct observation. In like manner it is 

 allowable to say, suppose that a were a mark of b, b of c, and 

 c of d, a would be a mark of d, which last conclusion was not 

 thought of by those who laid down the premises. A system of 

 propositions as complicated as geometry might be deduced 

 from assumptions which are false ; as was done by Ptolemy, 

 Descartes, and others, in their attempts to explain syntheti- 

 cally the phenomena of the solar system on the supposition 

 that the apparent motions of the heavenly bodies were the real 

 motions, or were produced in some way more or less different 

 from the true one. Sometimes the same thing is knowingly 



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