These things are better designated by very brief notes

Those things which do not require the present attention of the mind, even though they are necessary for reaching a conclusion, are better designated by very brief notes than by complete figures.

For in this way, memory cannot fail, and yet the thought is not distracted by the need to retain these things while it is engaged in deducing others.

Moreover, because we have said that no more than two distinct dimensions, out of the countless ones that can be pictured in our imagination, are to be contemplated by the same visual or mental intuition, it is worthwhile to retain all others so that they easily come to mind whenever necessity demands;

Towards which end, memory seems to have been instituted by nature. But because memory is often unstable, and lest we be compelled to expend any part of our attention in renewing it while engaged in other thoughts, the art of writing has most aptly provided a solution.

Relying on its aid, we commit absolutely nothing to memory here, but leaving our imagination free and wholly occupied with present ideas, we will note down whatever needs to be retained on paper, and that through very brief notes, so that after inspecting each distinct thing clearly according to the ninth rule, we may, according to the eleventh, run through everything with the swiftest movement of thought and contemplate many things simultaneously.

Therefore, whatever is to be regarded as one for the solution of difficulty, we will designate by a single note, which can be assumed at will. But, for the sake of simplicity, we will use characters a, b, c, etc., to denote known magnitudes, and A, B, C, etc., to denote unknowns; to these we will often prefix the numbers 1, 2, 3, 4, etc., to explain their multitude, and again we will add the numbers to the relations which will be understood in the same way: so that if I write 2a3, it will be the same as if I said twice the magnitude noted by the letter a containing three relations.

By this industry we will not only make a compendium of many words, but, what is most important, we will exhibit the terms of difficulty so pure and naked that, although nothing useful is omitted, nothing will ever be found in them to be superfluous, and that which may occupy the power of the mind in vain, while more things will be comprehended in the mind at the same time.

In order to understand all these things more clearly, it must first be noted that Logicians are accustomed to denote single magnitudes by several units or by some number; but we in this place abstract no less from numbers than a little before from Geometric figures, or anything else.

We do this both to avoid the weariness of long and superfluous reckoning, and especially so that the parts of the subject which pertain to the nature of difficulty always remain distinct, and are not entangled with useless numbers; so that if the base of a right triangle is sought, whose sides are given as 9 and 12, the Logician will say that it is √225 or 15; but we will put down a and b for 9 and 12, find that the base is √a2 + b2, and those two parts, a2 and b2, which are confused in number, will remain distinct.

It must also be noted that by the number of relations are to be understood proportions, which succeed each other in continuous order; which others in common Algebra attempt to express through several dimensions and figures, and whose first they call root, second square, third cube, fourth biquadrate, etc. I confess that I myself have been deceived by these names for a long time; for nothing seemed to be able to be proposed more clearly to my imagination, after the line and the square, than the cube and other figures shaped after their likeness; and I solved not a few difficulties by their aid.

But finally, after many experiments, I discovered that I had never found anything through that manner of conceiving which I could not have recognized much more easily and distinctly without it; and that such names are to be entirely rejected, lest they disturb the concept, since the same magnitude, although it may be called cube or biquadrate, is nevertheless never to be proposed to the imagination other than as a line or surface according to the preceding rule.

Therefore, it must be especially noted that root, square, cube, etc., are nothing else than magnitudes continuously proportional, to which that assumed unit is always presupposed, of which we have spoken above; to which unit the first proportional is referred immediately and by a single relation; the second, through the first, and hence by two relations; the third, through the first and second, and by three relations, etc. We will therefore call the first proportional the magnitude which is called root in Algebra; the second proportional, that which is called square, and so on.

Finally, it must be noted that although here we abstract from numbers some of the terms of difficulty in examining its nature, it often happens that it can be resolved in a simpler way with given numbers, than if it had been abstracted from them; which is done by the double use of numbers which we touched on before, because they are expressed, namely, in order and measure; and therefore, after we have sought it expressed in general terms, we must return it to given numbers to see whether by chance they will supply us with any simpler solution: for example, after we have seen that the base of a right triangle from the sides a and b is √a2 + b2, if we put down 81 for a2 and 144 for b2, which, when added, are 225, whose root or mean proportional between unity and 225 is 15; whence we shall know that the base 15 is commensurable with the sides 9 and 12, not generally because it is the base of a right triangle, of which one side is to the other as 3 to 4. All these things we distinguish, we who seek the clear and distinct knowledge of things, not the Logicians, who are content if the sum sought occurs to them, even if they do not notice how the same depends on the data, in which however one science properly consists.

But in general, it must be observed that nothing is ever to be committed to memory from those things which do not require continuous attention, if we can deposit them on paper, lest some part of our intellect be occupied in vain with the superfluous remembrance of the present object; and a certain index must be made, in which the terms of the question, as they will be proposed for the first time, will be written; then how the same are abstracted, and by what notes they are designated, so that, after the solution has been found in those notes, we may easily apply the same, without any aid of memory, to the particular subject about which the question is. For nothing is ever abstracted except from something less general. Therefore, I will write in this way: The base AC in the right triangle ABC is sought, and I abstract the difficulty, so that the magnitude of the base is sought generally from the magnitudes of the sides; then for AB, which is 9, I put down a, for BC, which is 12, I put down b, and so on.

And it must be noted that we will still use these four rules in the third part of this Treatise, and take them a little more broadly than they have been explained here, as will be said in their proper place.