Part IV, Chapter 3: Of the Method of Composition, and particularly of that which the Geometers observe
WHAT we have said in the preceding chapter has already given us some idea of the method of composition, which is the most important, inasmuch as it is that which is employed for the explanation of all the sciences.
This method consists principally in commencing with the most general and simple things, in order to pass to those which are less general and more complex. In this way we avoid repetitions, since, were we to treat of the species before the genus, as it is impossible to know well a species without knowing its genus, it would be necessary to explain the nature of the genus many times in the explanation of each species.
There are still many things to be observed in order to render this method perfect, and fully fitted to the end which it ought to propose, which is, that of giving us a clear and distinct knowledge of truth. But as general precepts are more difficult to comprehend when they are separated from all matter, we will consider the method which the geometers follow, that being always considered best adapted for proving the truth, and for fully convincing the mind of it. We shall first consider what is its excellence; and, in the second place, wherein it appears to be defective.
The geometers having for their aim the advancing only of that which is convincing, have believed that they could secure this by observing three things in general: —
The first is, to leave no ambiguity in the terms, which they have provided for by the definition of words, of which we have spoken in the First Part.
The second is, to establish their reasonings only on principles clear and evident, and which cannot be contested by any one; which leads them, first of all, to lay down axioms which they require to be granted to them, as being so clear that they would only be obscured by any attempt to prove them.
The third is, to prove demonstratively all the conclusions which they advance, by availing themselves only of the definitions which they have laid down of principles which have been accorded to them as being very evident, or of propositions which they have derived from these by the force of reasoning, which afterwards become to them the same as principles.
Thus we may reduce to these three heads all that the geometers have observed for convincing the mind, and include the whole in these five most important rules: —
For Definitions.
1. To admit no terms in the least obscure or equivocal without defining them.
2. To employ in the definitions only terms already known or perfectly explained.
For Axioms.
3. To demand as axioms only things perfectly evident.
For Demonstrations.
4. To prove all propositions which are at all obscure, by employing in their proof only the definitions which have preceded, or the axioms which have been accorded, or the propositions which have been already demonstrated, or the construction of the thing itself which is in dispute, when there may be any operation to perform.
5. Never to abuse the equivocation of terms by failing to substitute for them, mentally, the definitions which restrict and explain them.
This is what the geometers have judged necessary in order to render their proofs convincing and invincible. It must be confessed that attention to the observation of these rules is sufficient to enable us to avoid false reasoning in the treating of the sciences, which is, without doubt, the main thing, since all the rest may be called useful rather than necessary.