§ 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 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 constitutes 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, 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 synthetically 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 done, for the purpose 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_ of _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_.