Similarly, also, all men will admit that demonstration is discourse, ^[3705] agreeable to reason, producing belief in points disputed, from points admitted.

Now, not only demonstration and belief and knowledge, but foreknowledge also, are used in a twofold manner. There is that which is scientific and certain, and that which is merely based on hope.

In strict propriety, then, that is called demonstration which produces in the souls of learners scientific belief. The other kind is that which merely leads to opinion. As also, both he that is really a man, possessing common judgment, and he that is savage and brutal,—each is a man. Thus also the Comic poet said that "man is graceful, so long as he is man." The same holds with ox, horse, and dog, according to the goodness or badness of the animal. For by looking to the perfection of the genus, we come to those meanings that are strictly proper. For instance, we conceive of a physician who is deficient in no element of the power of healing, and a Gnostic who is defective in no element of scientific knowledge.

Now demonstration differs from syllogism; inasmuch as the point demonstrated is indicative of one thing, being one and identical; as we say that to be with child is the proof of being no longer a virgin. But what is apprehended by syllogism, though one thing, follows from several; as, for example, not one but several proofs are adduced of Pytho having betrayed the Byzantines, if such was the fact. And to draw a conclusion from what is admitted is to syllogize; while to draw a conclusion from what is true is to demonstrate.

So that there is a compound advantage of demonstration: from its assuming, for the proof of points in question, true premisses, and from its drawing the conclusion that follows from them. If the first have no existence, but the second follow from the first, one has not demonstrated, but syllogized. For, to draw the proper conclusion from the premisses, is merely to syllogize. But to have also each of the premisses true, is not merely to have syllogized, but also to have demonstrated.

And to conclude, as is evident from the word, is to bring to the conclusion. And in every train of reasoning, the point sought to be determined is the end, which is also called the conclusion. But no simple and primary statement is termed a syllogism, although true; but it is compounded of three such, at the least,—of two as premisses, and one as conclusion.

Now, either all things require demonstration, or some of them are self-evident. But if the first, by demanding the demonstration of each demonstration we shall go on ad infinitum; and so demonstration is subverted. But if the second, those things which are self-evident will become the starting points [and fundamental grounds] of demonstration.

In point of fact, the philosophers admit that the first principles of all things are indemonstrable. So that if there is demonstration at all, there is an absolute necessity that there be something that is self-evident, which is called primary and indemonstrable.

Consequently all demonstration is traced up to indemonstrable faith. ^[3706]

It will also turn out that there are other starting points for demonstrations, after the source which takes its rise in faith,—the things which appear clearly to sensation and understanding. For the phenomena of sensation are simple, and incapable of being decompounded; but those of understanding are simple, rational, and primary. But those produced from them are compound, but no less clear and reliable, and having more to do with the reasoning faculty than the first. For therefore the peculiar native power of reason, which we all have by nature, deals with agreement and disagreement. If, then, any argument be found to be of such a kind, as from points already believed to be capable of producing belief in what is not yet believed, we shall aver that this is the very essence of demonstration.

Now it is affirmed that the nature of demonstration, as that of belief, is twofold: that which produces in the souls of the hearers persuasion merely, and that which produces knowledge.

If, then, one begins with the things which are evident to sensation and understanding, and then draw the proper conclusion, he truly demonstrates. But if [he begin] with things which are only probable and not primary, that is evident neither to sense nor understanding, and if he draw the right conclusion, he will syllogize indeed, but not produce a scientific demonstration; but if [he draw] not the right conclusion, he will not syllogize at all.

Now demonstration differs from analysis. For each one of the points demonstrated, is demonstrated by means of points that are demonstrated; those having been previously demonstrated by others; till we get back to those which are self-evident, or to those evident to sense and to understanding; which is called Analysis. But demonstration is, when the point in question reaches us through all the intermediate steps. The man, then, who practices demonstration, ought to give great attention to the truth, while he disregards the terms of the premisses, whether you call them axioms, or premisses, or assumptions. Similarly, also, special attention must be paid to what suppositions a conclusion is based on; while he may be quite careless as to whether one choose to term it a conclusive or syllogistic proposition.

For I assert that these two things must be attended to by the man who would demonstrate—to assume true premisses, and to draw from them the legitimate conclusion, which some also call "the inference," as being what is inferred from the premisses.

Now in each proposition respecting a question there must be different premisses, related, however, to the proposition laid down; and what is advanced must be reduced to definition. And this definition must be admitted by all. But when premisses irrelevant to the proposition to be established are assumed, it is impossible to arrive at any right result; the entire proposition—which is also called the question of its nature—being ignored.

In all questions, then, there is something which is previously known,—that which being self-evident is believed without demonstration; which must be made the starting point in their investigation, and the criterion of apparent results.