Translation © George MacDonald Ross, 1998

Part IV: On Method

One can call method in general the art of well organising a sequence of many thoughts in order to discover the truth when we don’t know it, or to prove it to others when we already know it.

So there are two sorts of method. One is for discovering the truth, which is called analysis or the method of resolution, and which can also be called the method of discovery [invention]. The other, for making it understood to others when it has been discovered, is called synthesis, or the method of composition, and which can also be called the method of exposition [doctrine].

It is rare for the whole body of a science to be dealt with by analysis, and it is used only for resolving certain questions. [Footnote: The bulk of what is said here is drawn from a manuscript of Descartes, which Mr. Clerselier has been so good as to lend us.]

But all questions are either about words or about things.

Here I call questions of words, not those where one is searching for words, but those where one is searching for things by means of words, for example, where it is a question of finding the meaning of a riddle, or of explaining what an author meant by obscure and ambiguous expressions.

Questions about things can be reduced to four principal kinds.

The first is when one is searching for causes by means of effects. For example, one knows the various effects of the magnet, and one searches for the cause; one knows the various effects which are usually attributed to the horror of the vacuum, and one searches whether it is the true cause of them (and the answer is that it isn’t); one knows of the ebb and flow of the tide, and one asks what can be the cause of such a large and regular motion.

The second is when one searches for effects by means of causes. For example, it has always been known that wind and water have sufficient force to move bodies; but the ancients hadn’t sufficiently studied what could be the effects of these causes. Consequently, they never applied them, as was done subsequently by means of windmills and waterwheels, to a large number of things of great use to human society, and which significantly reduce people’s labour. That ought to be the fruit of genuine physics, so that one could say that the first sort of question (that of searching for causes by means of effects) constitutes the whole of speculative physics, and that the second sort constitutes the whole of practical physics.

The third sort of question is when one searches for the whole by means of the parts, as when having some numbers, one searches for their sum by adding one to the other; or when having two numbers, one searches for their product by multiplying them together.

The fourth is when, having a whole and a part, one searches for another part; as when having a number and that which one wants to subtract from it, one searches for the remainder; or having a number, one searches for the so-manyeth (tantième) part of it. . .

So that’s what is called analysis or resolution, and it should be noted that (1) one must always (just as much as in the method of composition) move from what is better known to what is less known; for there is no true method which can dispense with this rule.

(2) But it differs from the method of composition in that these known truths are derived from the study of the particular thing which one sets out to know, and not from things more general, as happens in the method of exposition. So, in the example we have just given, we didn’t start by establishing general maxims such as that no substance perishes strictly speaking, but that what is called ‘destruction’ is only a dissolution of parts; and thus that what has no parts cannot be destroyed, etc. Instead, one gradually ascends to these general truths.

(3) In the analytic method, clear and evident maxims are put forward only when necessary; whereas in the synthetic method they are established at the beginning, as we shall say below.

(4) Finally, these two methods differ only as the route one takes when climbing from a valley up a mountain differs from the route one takes when descending from the mountain into the valley. . . . This is what is ordinarily done in the sciences, where, having used analysis to find a certain truth, one uses the other method to explain what one has discovered.

This helps us understand what analysis is in geometry, and here is what it consists in. Suppose a question is proposed, and it is unknown whether it is true or false (if it’s a theorem), or whether it is possible or impossible (if it’s a problem). It is then supposed that it is true/possible, and if, on examining what follows from it, the outcome is something which is clearly true and a necessary consequence of what was proposed, the conclusion is that what was proposed is true. Then returning from the finishing point, it is proved by the other method, called that of composition. But if the outcome is an absurdity or an impossibility which is a necessary consequence of what was proposed, the conclusion is that it was false and impossible.

This is all that can be said in general about analysis, which consists more in judgment and creative intelligence (adresse de l’esprit) than in detailed rules.

What we have said in the last chapter has already given us some idea of the method of composition. It is the more important of the two, since it is the one which is used for expounding all the sciences.

This method consists mainly in starting with the most general and simplest things, in order to proceed to the less general and more complex. It thus enables us to avoid repetition, since, if we were to consider the species along with the genus (since it is impossible to know a species properly without knowing its genus), we would have to explain the nature of the genus many times, along with the explanation of each species.

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