192 REASONING. 



What is commonly called mathematical certainty, therefore, which com- 

 prises the twofold conception of unconditional truth and perfect accuracy, 

 is not an attribute of all mathematical truths, but of those only -which re- 

 late to pure Number, as distinguished from Quantity in the more enlarged 

 sense; and only so long as we abstain from supposing that the numbei's 

 are a precise index to actual quantities. The certainty usually asciibed to 

 the conclusions of geometry, and even to those of mechanics, is nothing 

 whatever but certainty of inference. We can have full assurance of par- 

 ticular results under particular suppositions, but we can not have the same 

 assurance tliat these suppositions are accurately true, nor that they include 

 all the data which may exercise an influence over the result in any given 

 instance. 



§ 4. It appears, therefore, that the method of all Deductive Sciences is 

 hypothetical. They proceed by tracing the consequences of certain as- 

 sumptions ; leaving for separate consideration whether the assumptions 

 are true or not, and if not exactly true, whether they are a sufficiently near 

 approximation 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 

 assumptions 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 consti- 

 tutes Demonstrative Science. 



It is of course quite as practicable to arrive at new conclusions from 

 facts assumed, as from facts observed ; from fictitious, as from real, induc- 

 tions. Deduction, as we have seen, consists of a series of inferences in this 

 form — a is a mark of h, 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 h, h of c, and c oi 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 attenipts to explain syn- 

 thetically 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. Some- 

 times the same thing is knowingly done, for the pm-pose of showing the 

 falsity of the assumption ; which is called a reductio ad absurdum. In 

 such cases, the reasoning is as follows : a is a mark of b, and b oi c; now 

 if c were also a mark of d, a would be a mark of d; but d is known to be 

 a mark of the absence of a; consequently a would be a mark of its own 

 absence, which is a contradiction ; therefore c is not a mark of d. 



