Cartesian Method

The proposed difficulty is to be directly traversed, abstracting from the fact that some of its terms are known, others unknown, and the mutual dependence of each upon others is to be observed through true discourses.

The preceding four rules have taught how determined difficulties and those perfectly understood by each subject are to be abstracted from, and reduced to such a state that nothing else is sought afterwards, except certain magnitudes known from the fact that through this or that relation, they are referred to certain data.

But now in the following five rules, we will explain how these same difficulties are to be so subdued that however many unknown magnitudes will be in one proposition, all of them are subordinate to each other, and how the first will be to unity, so the second to the first, the third to the second, the fourth to the third, and so forth, if so many, they make a sum equal to some known magnitude; and that method so certain, that in this way, we may safely assert, that those difficulties were unable to be reduced by any skill to simpler terms.

As for the present, it must be noted in every question resolved through deduction that there is a plain and direct path by which we may most easily pass from one term to another, and all others are more difficult and indirect. For understanding this, it must be remembered what was said in the eleventh rule, where we explained what the chain of propositions is, each if compared with neighboring ones, we easily perceive how even the first and last look at each other, although we deduce the intermediates with difficulty from the extremes. Therefore, if we observe the dependence of each on each, in unbroken order, so that we infer how the last depends on the first, we will directly traverse the difficulty. But on the contrary, if from the fact that we know that the first and last are connected in a certain way, we want to deduce what are the means that connect them, we would follow an entirely indirect and disorderly order. However, because we are only concerned here with involved questions, in which some intermediates must be known from known extremes, the whole art of this place consists in this, that by supposing unknowns for knowns, we can propose an easy and direct way of seeking, even in however intricate difficulties; and nothing prevents it from always happening, since we have supposed from the beginning of this part, that we recognize those which are unknown in the question, as having such dependence on the known, that they are determined by them, so that if we reflect on those things themselves which first occur, while we recognize that determination, and even count the same unknown among the known, as we gradually deduce all the rest, as if they were unknown, by true discourses, we will execute everything that this rule prescribes: of which things, examples, and also many of those things which we will speak of next, we reserve to the twenty-fourth rule, since they will be more conveniently explained there.