P40: Whatever ideas follow in the Mind from ideas that are adequate in the Mind are also adequate.

Dem.: This is evident. For when we say that an idea in the human Mind follows from ideas that are adequate in it, we are saying nothing but that (by P11 C) in the Divine/Universal intellect there is an idea of which God/Nature is the cause, not insofar as he is infinite, nor insofar as he is affected with the ideas of a great many singular things, but insofar as he constitutes only the essence of the human Mind {NS: and therefore, it must be adequate}.

Schol. I: With this I have explained the cause of those notions which are called common, and which are the foundations of our reasoning. But some axioms, or notions, result from other causes which it would be helpful to explain by this method of ours. For from these {explanations} it would be established which notions are more useful than the others, and which are of hardly any use; and then, which are common, which are clear and distinct only to those who have no prejudices, and finally, which are ill-founded. Moreover, we would establish what is the origin of those notions they call Second, and consequently of the axioms founded on them, and other things I have thought about, from time to time, concerning these matters. But since I have set these aside for another Treatise, and do not wish to give rise to disgust by too long a discussion, I have decided to pass over them here.

But not to omit anything it is necessary to know, I shall briefly add something about the causes from which the terms called Transcendental have had their origin—I mean terms like Being, Thing and something. These terms arise from the fact that the human Body, being limited, is capable of forming distinctly only a certain number of images at the same time (I have explained what an image is in P17 S). If that number is exceeded, the images will begin to be confused, and if the number of images the Body is capable of forming distinctly in itself at once is greatly exceeded, they will all be completely confused with one another.

Since this is so, it is evident from P17 C and P18, that the human Mind will be able to imagine distinctly, at the same time, as many bodies as there can be images formed at the same time in its body. But when the images in the body are completely confused, the Mind also will imagine all the bodies confusedly, without any distinction, and comprehend them as if under one attribute, viz. under the attribute of Being, Thing, etc. This can also be deduced from the fact that images are not always equally vigorous and from other causes like these, which it is not necessary to explain here. For our purpose it is sufficient to consider only one.

For they all reduce to this: these terms signify ideas that are confused in the highest degree. Those notions they call Universal, like Man, Horse, Dog, etc., have arisen from similar causes, viz. because so many images (e.g., of men) are formed at one time in the human Body that they surpass the power of imagining—not entirely, of course, but still to the point where the Mind can imagine neither slight differences of the singular {men} (such as the color and size of each one, etc.) nor their determinate number, and imagines distinctly only what they all agree in, insofar as they affect the body. For the body has been affected most {NS: forcefully} by {what is common}, since each singular has affected it {by this property}. And {NS: the mind} expresses this by the word man, and predicates it of infinitely many singulars.

For as we have said, it cannot imagine a determinate number of singulars. But it should be noted that these notions are not formed by all {NS: men} in the same way, but vary from one to another, in accordance with what the body has more often been affected by, and what the Mind imagines or recollects more easily. For example, those who have more often regarded men’s stature with wonder will understand by the word man an animal of erect stature. But those who have been accustomed to consider something else, will form another common image of men—e.g., that man is an animal capable of laughter, or a featherless biped, or a rational animal.

And similarly concerning the others—each will form universal images of things according to the disposition of his body. Hence it is not surprising that so many controversies have arisen among the philosophers, who have wished to explain natural things by mere images of things.

Schol. 2: From what has been said above, it is clear that we perceive many things and form universal notions:

I. from singular things which have been represented to us through the senses in a way that is mutilated, confused, and without order for the intellect (see P29 C); for that reason I have been accustomed to call such perceptions knowledge from random experience;

II. from signs, e.g., from the fact that, having heard or read certain words, we recollect things, and form certain ideas of them, which are like them, and through which we imagine the things (P18 S). These two ways of regarding things I shall henceforth call knowledge of the first kind, opinion or imagination.

III. Finally, from the fact that we have common notions and adequate ideas of the properties of things (see P38 C, P39, P39 C, and P40). This I shall call reason and the second kind of knowledge.

IV. In addition to these two kinds of knowledge, there is (as I shall show in what follows) another, third kind, which we shall call intuitive knowledge. And this kind of knowing proceeds from an adequate idea of the formal essence of certain attributes of God/Nature to the adequate knowledge of the {NS: formal} essence of things.

I shall explain all these with one example. Suppose there are three numbers, and the problem is to find a fourth which is to the third as the second is to the first. Merchants do not hesitate to multiply the second by the third, and divide the product by the first, because they have not yet forgotten what they heard from their teacher without any demonstration, or because they have often found this in the simplest numbers, or from the force of the Demonstration of P7 in Bk. VII of Euclid, viz. from the common property of proportionals. But in the simplest numbers none of this is necessary. Given the numbers 1, 2, and 3, no one fails to see that the fourth proportional number is 6—and we see this much more clearly because we infer the fourth number from the ratio which, in one glance, we see the first number to have the second.