10. In order that the mind may acquire sagacity, it is necessary to give it practice in investigating what has already been discovered by others; and it ought to traverse methodically even the most trifling inventions of men, but especially those which best explain or presuppose order.

11. After we have grasped by intuition a certain number of simple propositions, if we wish to infer some other proposition from them, it is useful to run over them in a continuous and uninterrupted movement of thought in order to reflect on their relations to one another, and as far as possible to conceive distinctly several at a time. For it is in this way that our knowledge becomes much more certain and the power of our mind is greatly increased.

12. Finally we ought to use all the aids of intellect, imagination, sense, and memory, partly in order to have a distinct intuition of simple propositions; partly to compare correctly what we seek with what we know so that we may recognize it; partly in order to discover those things which should be so compared with one another so that no human resources may be neglected.

13. If we understand a question perfectly, we must abstract it from every superfluous concept, simplify it as much as possible, and divide it by enumeration into the smallest possible parts. [ . . .]

14. The same question must be applied to the real extension of bodies and represented in its entirety to the imagination by means of bare figures; for in this way it will be much more distinctly perceived by the understanding.

15. It is also useful in many cases to describe these figures and to exhibit them to the external senses, in order that by this device our thought should more easily be kept attentive. [. . .]

16. As for the things which do not demand the immediate attention of the mind, although they are necessary for the conclusion it is better to designate them by very brief signs rather than by complete figures; for thus the memory cannot err, and meanwhile the thought will not be distracted for the purpose of retaining them, while it is applying itself to deducing other things. [. . .]

17. A given difficulty should be run through directly, in abstraction from the fact that some of its terms are known and others unknown, and with the intuition, obtained by taking the right road, of the mutual dependence of each term on the others. [. . .]

18. For this only four operations are required, addition, subtraction, multiplication, and division, among which the last two often do not need to be carried out here, as much to keep from complicating things needlessly as because they can be executed more easily later. [. . .]

19. By this method of ratiocination we should seek out as many magnitudes expressed in two different modes, as we suppose unknown terms directly bearing on the difficulty in place of known ones: for thus we shall have as many comparisons between two equals.

20. When the equations have been found, we must finish the operations which we have left aside, never making use of multiplication whenever there is room for division.

21. If there are several equations of this sort, we should reduce them all to a single one, that is to say, to the one whose terms will occupy the least number of degrees in the sequence of magnitudes in continuous proportion, according to which they are to be ordered.

16. As for the things which do not demand the immediate attention of the mind, although they are necessary for the conclusion it is better to designate them by very brief signs rather than by complete figures; for thus the memory cannot err, and meanwhile the thought will not be distracted for the purpose of retaining them, while it is applying itself to deducing other things. [. . .]

17. A given difficulty should be run through directly, in abstraction from the fact that some of its terms are known and others unknown, and with the intuition, obtained by taking the right road, of the mutual dependence of each term on the others. [. . .]

18. For this only four operations are required, addition, subtraction, multiplication, and division, among which the last two often do not need to be carried out here, as much to keep from complicating things needlessly as because they can be executed more easily later. [. . .]

19. By this method of ratiocination we should seek out as many magnitudes expressed in two different modes, as we suppose unknown terms directly bearing on the difficulty in place of known ones: for thus we shall have as many comparisons between two equals.

20. When the equations have been found, we must finish the operations which we have left aside, never making use of multiplication whenever there is room for division.

21. If there are several equations of this sort, we should reduce them all to a single one, that is to say, to the one whose terms will occupy the least number of degrees in the sequence of magnitudes in continuous proportion, according to which they are to be ordered.