DEMONSTRATION, AND NECESSARY TRUTHS. 1G9 



comprises the two-fold conception of unconditional truth and perfect 

 accuracy, is not an attribute of all mathematical truths, but of those 

 only which relate to pure Number, as distinguished from Quantity, 

 in the more enlarged sense ; and only so long as we abstain fi-om sup- 

 posing that the numbers are a precise index to actual quantities. The 

 certainty usually ascribed to the conclusions of geometiy, and even to 

 those of mechanics, is nothing whatever but certainty -of inference. 

 We can have full assurance of particular results under particular sup- 

 positions, but we cannot have the same assurance that these suppositions 

 are accurately true, nor that they include all the data which may exer- 

 cise 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 

 assumptions ; leaving for separate consideration whether the assump- 

 tions are true or not, and if not pxactly true, whether they are a suffi- 

 ciently near approximation to the ti-uth. The reason is obvious. Since 

 it is only in questions of pure number that the assumptions are exactly 

 ti-ue, and even there, only so long as no conclusions except purely nu- 

 merical ones are to be founded upon them ; it must, in all other cases 

 of deductive investigation, form a part of the inquiry, to detennine 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 obsei-vation, may re- 

 quii-e, in every difterent case, different evidence, and present every de- 

 gi-ee of difficulty from the lowest to the highest. But the other part of the 

 process — viz., 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 dravni 

 fi-om assumptions, is what properly constitutes Demonstrative Science. 



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

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

 ductions. 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 tliat a were a mark of i, h of c, 

 and cofd,a would be a mark of ^Z, which last conclusion was not thought 

 of by those who laid down the premisses. A system of propositions as 

 complicated as geometry might be deduced fi-om assumptions which are 

 false ; as was done by Ptolemy, Des,cartes, and others, in their attempts 

 to explain synthetically the phenomena of the solar system, on the sup- 

 position that the apparent motions of the heavenly bodies were the real 

 motions, or were produced in some way more or less different fi-om the 

 true one. Sometimes the same thing is knowingly done, for the pur- 

 pose of showing the falsity of the assumption ; which is called a reduc- 

 tio ad absurdum. In such cases, the reasoning is as follows : a is a 

 mark ofb, and b of c ; now if c were also a mark of J, a would be a mark 

 of d ; but d is known to be a mark of the absence of a ; conso(iuently a 

 would be a mark of its own absence, which is a contradiction ; there- 

 fore c is not a mark of d. 

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