Naked shapes to the imagination

The same principles that are used for transferring real extension of bodies should also be proposed entirely through naked figures to the imagination; for in this way it will be perceived much more distinctly by the intellect.

Furthermore, in order to also use the aid of imagination, it must be noted whenever something unknown is deduced from something else already known, that not a new kind of being is found, but rather the entire knowledge is extended to perceive the sought object participating in the nature of those things which are given in the proposition.

For example, if someone is blind from birth, it is not to be hoped that we could ever convince them of perceiving true ideas of colors such as we receive from the senses; but if someone has seen primary colors at some point and never the intermediate or mixed ones, it is possible that they may form images of even those they have not seen, based on similarity to others.

Similarly, if there exists some kind of being in a magnet which our intellect has never perceived as similar, we should not hope to understand it through reasoning alone; rather, we would need to be equipped with a new sense or by divine intellect. Whatever can be accomplished by human ingenuity in this matter, we will believe we have achieved, if we distinctly perceive that the mixture of already known beings or natures produces effects similar to those apparent in the magnet.

Indeed, all these beings already known, such as extension, figure, motion, and similar ones which need not be listed here, are known through the same idea in different subjects, and we do not imagine the figure of a crown differently if it is silver than if it is gold; and this common idea is transferred from one subject to another only by simple comparison, through which we affirm that the sought thing is similar to, the same as, or equal to a given thing, so that in every reasoning we precisely know the truth only through comparison.

For example, in this way: all A is B, all B is C, therefore all A is C; the sought and the given are compared to each other, namely A and C, according to whether each is B, etc. But because, as we have often pointed out, the forms of syllogisms do not help in perceiving the truth of things, it will benefit the reader if these are plainly rejected, to conceive that all knowledge, which is not acquired through simple and pure intuition of a solitary thing, is acquired through the comparison of two or more things with each other. Indeed, nearly the whole industry of human reason lies in preparing for this operation; for when it is open and simple, art is needed for nothing, but only the light of nature is required to perceive the truth obtained through it.

It must be noted that comparisons are called simple and open only when the sought and the given share some nature equally; all others, however, need no preparation for another reason than that the common nature is not equally present in both, but according to other relations or proportions in which it is involved; and the chief part of human industry is to be placed not elsewhere than in reducing these proportions so that equality between the sought and something known is clearly seen.

Moreover, it must be noted that nothing can be reduced to this equality except what receives more and less, and all that is comprehended through the term “magnitude”; so that after the difficulties of the term have been abstracted from every subject, from now on we shall understand ourselves to be engaged only with magnitudes in general.

However, to imagine something even then and not to use pure intellect, but with species depicted in the imagination as an aid: it must finally be noted that nothing is said about magnitudes in general which cannot also be referred to any particular species.

From which it is easily concluded that it will be very helpful if we transfer what we understand to be said about magnitudes in general to that species of magnitude which is easiest and most distinctly depicted in our imagination: however, this extension of real body abstracted from everything else, except that it is figured, follows from what has been said in rule twelve, where we conceive the imagination itself with the ideas existing in it to be nothing else than a real extended and figured body. Which is itself also evident, since in no other subject are all the differences of proportion displayed so distinctly; for although one thing can be said to be more or less white than another, likewise one sound more or less acute, and so on, nevertheless we cannot define exactly whether such an excess consists in a double or triple proportion, etc., except by some analogy to a certain kind of extended body. Therefore, let it remain fixed and determined that perfectly defined questions contain hardly any difficulty, except that which consists in unfolding proportions into equalities; and that all that in which this difficulty is precisely found can easily and ought to be separated from every other subject, and then transferred to extension and figures, about which alone henceforth we shall treat, omitting all other considerations until rule twenty-five.

We would wish the reader here to acquire a liking for the study of arithmetic and geometry, even if they have not yet engaged in them, rather than to be educated in the common way; for the use of the rules which I shall present here in learning those subjects is far easier than in any other kind of questions; and their usefulness is so great for attaining higher wisdom that I do not hesitate to say that this part of our method was not invented for the sake of mathematical problems, but rather these arts should be learned almost solely for their cultivation. And I assume nothing from these disciplines, except perhaps what is known by itself and obvious to everyone; but the knowledge of them, as usual, though not corrupted by any open errors, is obscured by many oblique and ill-conceived principles, which I will attempt to correct throughout the following.

By extension, we understand everything that has length, breadth, and depth, without inquiring whether it is a true body or merely space; nor does it seem to need any further explanation, since nothing is easier for our imagination to perceive. However, since scholars often use distinctions so sharp that they dissipate natural light and find darkness even in things which are never unknown to the common people: it must be warned that here by extension we do not designate something distinct and separated from the subject itself, nor do we recognize in general such philosophical entities which truly do not fall under imagination. For even if someone may persuade themselves, for example, that whatever is extended in the nature of things is reduced to nothingness, it does not conflict in the meantime that extension alone exists per se, not as a body for this concept, but only by a judgment of the intellect. Which he himself will admit, if he reflects carefully on that image of extension itself which he will then try to fashion in his imagination: for he will notice that he does not perceive it stripped of every subject, but imagines it altogether differently than he judges; so that abstract entities (whatever the intellect may believe about the truth of the thing) are never formed in imagination separated from subjects.

But since we are not going to act on anything henceforth without the aid of imagination, it is worth distinguishing cautiously which single ideas of words should be proposed to our intellect. Therefore, we propose to consider these three forms of speech: extension occupies a place, body has extension, and extension is not a body.

The first of these shows how extension is taken for what is extended; for I conceive the same thing if I say extension occupies a place, as if I say what is extended occupies a place. Nevertheless, it is not better to use the word “extended” to avoid ambiguity; for it would not so distinctly signify what we conceive, namely that some subject occupies a place, because it is extended; and someone might interpret it only as that which is extended occupies a place, just as if I said an animated being occupies a place. This reasoning explains why we have said that we will act upon extension here rather than upon what is extended, even if we think it should not be conceived otherwise than as extended.

Now let us proceed to these words: body has extension, where we understand by extension something different from body; nevertheless, we do not form two distinct ideas in our imagination, one of body and the other of extension, but only one of an extended body; and it is nothing on the part of the thing, as if I said a body is extended, or rather, the extended is extended. This peculiarity is in those beings which are only in something else, and never can be conceived without a subject; and it happens differently in those which are really distinguished by subjects: for if I said, for example, Peter has riches, the idea of Peter is plainly different from that of wealth; similarly, if I said Paul is rich, I would imagine something entirely different than if I said wealth is rich. Many people mistakenly think without distinguishing these things, that extension contains something distinct from what is extended, just as wealth of Paul is something different than Paul.

Finally, if it is said extension is not a body, then the term extension is taken in a far different sense than before; and in this meaning no peculiar idea corresponds to it in our imagination, but this whole proposition is perfected by pure intellect, which alone has the ability to separate such abstract entities.

This is a common cause of error for those who do not notice that extension taken in this way cannot be comprehended by imagination, so they represent it to themselves as a true idea; such an idea necessarily involves the concept of a body, if they say extension conceived in this way is not a body, they are imprudently involved in the notion that the same thing is both a body and not a body. It is of great importance to distinguish statements in which such terms as extension, figure, number, surface, line, point, unity, etc., have such a strict meaning that they exclude something from which they are not really distinct, as when it is said: extension or figure is not a body; number is not a counted thing; surface is the boundary

of a body; line is the division of a surface, or a line of which the surface is a part; point is the boundary of a line; unity is one part of a number; etc.