Chapter 1 All doctrine, and all intellectual discipline, arise from pre-existent knowledge. Now this is evident, if we survey them all, for both mathematical sciences are obtained in this manner, and also each of the other arts. It is the same also with arguments, as well those which result through syllogisms, as those which are formed through induction, for both teach through things previously known, the one assuming as if from those who understood them, the other demonstrating the universal by that which is evident as to the singular. Likewise also do rhetoricians persuade, for they do so either through examples, which is induction, or through enthymems, which is syllogism. It is necessary however to possess previous knowledge in a twofold respect; for with some things we must pre-suppose that they are, but with others we must understand what that is which is spoken of; and with others both must be known, as for instance, (we must pre-assume,) that of every thing it is true to affirm or deny that it is, but of a triangle, that it signifies so and so, and of the monad (we must know) both, viz. what it signifies and that it is, for each of these is not manifest to us in a similar manner. It is possible how ever to know from knowing some things previously, and receiving the knowledge of others at the same time, as of things which are contained under universals, and of which a man possesses knowledge. For he knew before that every triangle has angles equal to two right angles, but that this which is in a semi-circle is a triangle, he knew by induction at the same time. For of some things knowledge is acquired after this manner, nor is the extreme known through the middle, as such things as are singulars, and are not predicated of any subject. Perhaps however we must confess that we possess knowledge after a certain manner before induction or the assumption of a syllogism, but in another manner not. For what a man is ignorant about its existence at all, how could he know at all that it has two right angles? But it is evident that he thus knows because he knows the universal, but singly he does not know it. Still if this be not admitted, the doubt which is mentioned in the Meno will occur, either he will learn nothing, or those things which he knows, for he must not say, as some endeavour to solve the doubt, "Do you know that every duad is an even number or not?" for since if some one says that he does, they would bring forward a certain duad which he did not think existed, as therefore not even; and they solve the ambiguity, not by saying that he knew every duad to be even, but that he was ignorant as to what they know is a duad. Nevertheless they know that of which they possess and have received the demonstration, but they have received it not of every thing which they know to be a triangle or a number, but of every number and triangle singly, for no proposition is assumed of such a kind as the number which you know, or the rectilinear figure which you know, but universally. Still there is nothing (I think) to prevent a man who learns, in a certain respect knowing and in a certain respect being ignorant, for it is absurd, not that he should in some way know what he learns, but that he should thus know it, as he does when he learns it, and in the same manner.
Chapter 2 WE think that we know each thing singly, (and not in a sophistical manner, according to accident,) when we think that we know the cause on account of which a thing is, that it is the cause of that thing, and that the latter cannot subsist otherwise; wherefore it is evident that knowledge is a thing of this kind, for both those who do not, and those who do know, fancy, the former, that they in this manner possess knowledge, but those who know, possess it in reality, so that it is impossible that a thing of which there is knowledge simply should subsist in any other way. Whether therefore there is any other mode of knowing we shall tell hereafter, but we say also that we obtain knowledge through demonstration, but I call demonstration a scientific syllogism, and I mean by scientific that according to which, from our possessing it, we know. If then to know is what we have laid down, it is necessary that demonstrative science should be from things true, first, immediate, more known than, prior to, and the causes of the conclusion, for thus there will be the appropriate first principles of whatever is demonstrated. Now syllogism will subsist even without these, but demonstration will not, since it will not produce knowledge. It is necessary then that they should be true, since we cannot know that which does not subsist, for instance, that the diameter of a square is commensurate with its side. But it must be from things first and indemonstrable, or otherwise a man will not know them, because he does not possess the demonstration of them, for to know those things of which there is demonstration not accidentally is to possess demonstration. But they must be causes, and more known, and prior; causes indeed, because we then know scientifically when we know the cause; and prior, since they are causes; previously known also, not only according to the other mode by understanding (what they signify), but by knowing that they are. Moreover they are prior and more known in two ways, for what is prior in nature, is not the same as that which is prior in regard to us, nor what is more known (simply) the same as what is more known to us. Now I call things prior and more known to us, those which are nearer to sense, and things prior and more known simply, those which are more remote from sense; and those things are most remote which are especially universal, and those nearest which are singular, and these are mutually opposed. That again is from things first, which is from peculiar principles, and I mean by first, the same thing as the principle, but the principle of demonstration is an immediate proposition, and that is immediate to which there is no other prior. Now a proposition is one part of enunciation, one of one, dialectic indeed, which similarly assumes either (part of contradiction), but demonstrative which definitely (assumes) that one (part) is true. Enunciation is either part of contradiction, and contradiction is an opposition which has no medium in respect to itself. But that part of contradiction (which declares) something, of somewhat, is affirmation, and that (which signifies) something from somewhat is negation. Of an immediate syllogistic principle, I call that the thesis, which it is not possible to demonstrate, nor is it necessary that he should possess it, who intends to learn any thing; but what he who intends to learn any thing must necessarily possess, that I call an axiom, for there are certain things of this kind, and in denominating these, we are accustomed generally to use this name. But of thesis, that which receives either part of contradiction, as for instance, I mean that a certain thing is, or that it is not, is hypothesis, but that which is without this, is definition. For definition is a thesis, since the arithmetician lays down unity to be that which is indivisible, according to quantity, yet it is not hypothesis, since what unity is, and that unity is, are not the same thing.
Notwithstanding, since we must believe in and know a thing from possessing such a syllogism as we call demonstration, and this is, because these are so, of which syllogism consists—it is necessary not only to have a previous knowledge of the first, or all, or some things, but that they should be more known, for that on account of which any thing exists, always exists itself in a greater degree; for example, that on account of which we love is itself more beloved. Hence if we know and believe on account of things first, we also know and believe those first things in a greater degree, because through them (we know and believe) things posterior. A man however cannot believe more than what he knows, those things which he does not know, nor with respect to which he is better disposed than if he knew. This however will happen, unless some one should previously know of those who give credence through demonstration, since it is more necessary to believe either in all or in certain first principles, than in the conclusion. It is not only however requisite that he who is to possess knowledge through demonstration, should know in a greater degree first principles, and believe rather in them than in the thing demonstrated, but also that nothing else should be more credible or more known to him than the opposites of the principles, from which a syllogism of contra-deception may consist, since it behoves him who possesses knowledge singly to be unchangeable.
Chapter 3 To some, because it is necessary that first things should be known, science does not appear to exist, but to others to exist indeed, yet (they think) there are demonstrations of all things, neither of which opinions is true or necessary. For those who suppose that knowledge does not subsist at all, these think that we are to proceed to infinity as if we may not know things subsequent by things prior, of which there are no first, reasoning rightly, since it is impossible to penetrate infinites. And if (they say) we are to stop, and there are principles, these are unknown, since there is no demonstration of them, which alone they say is to know scientifically; but if it is not possible to know first things, neither can we know either simply or properly things which result from these, but by hypothesis, if these exist. Others however assent with respect to knowledge, for (they assert) that it is only through demonstration, but that nothing prevents there being a demonstration of all things, for demonstration may be effected in a circle, and (things be proved) from each other. We on the contrary assert, that neither is all science demonstrative, but that the science of things immediate is indemonstrable. And this is evidently necessary, for if it is requisite to know things prior, and from which demonstration subsists, but some time or other there is a stand made at things immediate, these must of necessity be indemonstrable. This therefore we thus assert, and we say that there is not only science, but also a certain principle of science, by which we know terms. But that it is impossible to demonstrate in a circle simply is evident, since demonstration must consist of things prior and more known, as it is impossible that the same should be prior and posterior to the same, unless in a different way, as for instance, some things with reference to us, but others simply in the manner in which induction makes known. If however this be so, to know simply will not be well defined, but it is two-fold, or the other demonstration is not simply so which is produced from things more known to us. Still there happens to those who assert there is demonstration in a circle, not only what has now been declared, but that they say nothing else than this is if it is, and in this manner we may easily demonstrate all things. Nevertheless it is evident that this occurs, when three terms are laid down, for to assert that demonstration recurs through many or through few terms, or whether through few or through two, makes no difference. For when A existing, B necessarily is, and from this last C, if A exists C will exist, if then, when A is, it is necessary that B should be, but this existing, A exists, (for this were to demonstrate in a circle,) let A be laid down in the place of C. To say therefore that because B is A is, is equivalent to saying that C is, and this is to say that A existing C is, but C is the same as A, so that it happens that they who assert there is demonstration in a circle, say nothing else than that A is because A is, and thus we may easily demonstrate all things. Neither however is this possible, except in those things which follow each other as properties: from one thing however being laid down, it has been proved that there will never necessarily result something else, (I mean by one thing, neither one term, nor one thesis being laid down,) but from two first and least theses, it is possible (to infer necessarily something else), since we may syllogize. If then A is consequent to B and to C, and these to each other, and to A, thus indeed it is possible to demonstrate all those things which are required from each other in the first figure, as we have shown in the books on Syllogism. It has also been shown that in the other figures there is either not a syllogism, or not one concerning the subjects assumed; but it is by no means possible to demonstrate in a circle those which do not reciprocate. Hence, since there are but few such in demonstrations, it is evidently vain and impossible to say, that there is demonstration of things from each other, and that on this account universal demonstration is possible.
Chapter 4 Since it is impossible that a thing, of which there is simply science, should have a various subsistence, it will be also necessary that what we know should pertain to demonstrative science, and demonstrative science is that which we possess from possessing demonstration, hence a syllogism is a demonstration from necessary (propositions). We must comprehend then of what, and what kind (of propositions), demonstrations consist; but first let us define what we mean by "of every," and "per se" and "universal."
I call that "of every," which is not in a certain thing, and in another certain thing is not, nor which is at one time, and not at another; as if animal is predicated of every man, if it is truly said that this is a man, it is true also that he is an animal, and if now the one is true, so also is the other; and in like manner, if a point is in every line. Here is a proof, for when we are questioned as it were of every, we thus object, either if a thing is not present with a certain individual, or if it is not sometimes. But I call those "per se" which are inherent in (the definition of) what a thing is, as line is in triangle, and point in line, (for the essence of them is from these, and they are in the definition explaining what it is:) also those things which are inherent in their attributes in the definition declaring what a thing is, as the straight and the curved are inherent in a line, and the odd and even in number, and the primary and composite, the equilateral and the oblong: and they are inherent in all these, in the definition declaring what a thing is, there indeed line, but here number. In a similar manner, in other things, I say that such are per se inherent in each, but what are in neither way inherent (I call) accidents, as the being musical, or white in an animal. Moreover, that which is not predicated of any other subject, as that which walks being something else, is that which walks, and is white, but essence and whatever things signify this particular thing, not being any thing else, are that which they are. Now those which are not predicated of a subject, I call "per se," but those which are so predicated, I call accidents. Again, after another manner, that which on account of itself is present with each thing is "per se," but that which is not on account of itself is an accident; thus it is an accident if while any body was walking it should lighten, for it did not lighten on account of his walking, but we say that it accidentally happened. If, however, a thing is present on account of itself, it is per se, as if any one having his throat cut should die, and through the wound, because he will die in consequence of his throat being cut, but it did not accidentally happen that he whose throat was cut died. Those therefore which are predicated in things which are simply objects of science per se, so as to be inherent in the things predicated, or which are themselves inherent in subjects, are on account of themselves, and from necessity, for it does not happen that they are not inherent either simply or as opposites, as the straight and the curved in a line, and the even or odd in number. For a contrary is either privation or contradiction in the same genus, as that is even which is not odd in numbers, so far as it follows: hence if it is requisite to affirm or deny, it is also necessary that those which are per se should be inherent.
Let then the expressions "of every" and "per se" be thus defined: I call that universal, however, which is both predicated "of every" and "per se," and so far as the thing is. Now it is evident that whatever are universal are inherent in things necessarily, but the expressions "per se," "and so far as it is," are the same; as a point and straightness are per se present in a line, for they are in it, in as far as it is a line, and two right angles in a triangle, so far as it is a triangle, for a triangle is per se equal to two right angles. But universal is then present, when it is demonstrated of any casual and primary thing, as to possess two right angles is not universally inherent in figure, yet it is possible to demonstrate of a figure that it has two right angles, but not of any casual figure, nor does a demonstrator use any casual figure, for a square is indeed a figure, yet it has not angles equal to two right. But any isosceles has angles equal to two right, yet not primarily, for triangle is prior. Whatever therefore is casually first demonstrated to possess two right angles, or any thing else, in this first is the universal inherent, and the demonstration per se of this is universal, but of other things after a certain manner not per se, neither is it universally present in an isosceles, but extends farther.
Chapter 5 We ought not to be ignorant that frequently error arises, and that what is demonstrated is not primarily universal, in so far as the primarily universal appears to be demonstrated. Now we are deceived by this mistake, when either nothing higher can be assumed, except the singular or singulars, or when something else can be assumed, but it wants a name in things differing in species, or when it happens to be as a whole in a part, of which the demonstration is made, for demonstration will happen to particulars, and will be of every individual, yet nevertheless it will not be the demonstration of this first universal. Still I say the demonstration of this first, so far as it is this, when it is of the first universal. If then any one should show that right lines do not meet, it may appear to be (a proper) demonstration of this, because it is in all right lines, yet this is not so, since this does not arise from the lines being thus equal, but so far as they are in some way or other equal. Also if a triangle should be no other than isosceles, so far as isosceles it may appear to be inherent: alternate proportion also, so far as regards numbers and lines and solids and times (as was once shown separately) it is possible at least to be demonstrated of all by one demonstration, but inasmuch as all these, numbers, length, time, are not one denominated thing, and differ from each other in species, they were assumed separately. But now the demonstration is universal, for it is not in so far as they are lines or numbers, that it is inherent, but in so far as this thing which they suppose to be universally inherent. For this reason neither if one should demonstrate each several triangle by one or another demonstration, that each has two right angles, equilateral, the scalene, and the isosceles separately, would he yet know that the triangle (itself) has angles equal to two right, except in a sophistical manner, nor triangle universally, though there should be no other triangle besides these. For he does not know it so far as it is triangle, nor does he know every triangle, except according to number, but not every, according to species, even if there be no one that he does not know. When then does he not know universally, and when knows he simply? It is clear that if there is the same essence of a triangle, and of an equilateral either of each or of all, he knows, but if there is not the same, but different, and it is inherent so far as it is triangle, he does not know. Whether however is it inherent, so far as it is triangle, or so far as it is isosceles? And when, according to this, is it primary? And of what is the demonstration universally? It is evident that it then is, when, other things being taken away, it is inherent in the primary, thus two right angles will be inherent in a brazen isosceles triangle, when the being brazen and the being isosceles are taken away, but not if the figure or boundary is taken away, nor if the primary are. But what primary? if indeed triangle (is taken away); according to this it is inherent in others, and of this universally is the demonstration.
Chapter 6 If then demonstrative science is from necessary principles, (for what is scientifically known cannot subsist otherwise,) and those which are per se inherent are necessarily so in things, (for some are inherent in the definition of what a thing is, but others are they in the very nature of which the subjects are inherent, of which they are so predicated, that one of opposites is necessarily present,) it is evident that the demonstrative syllogism will consist of certain things of this kind, for every thing is either thus inherent, or according to accident, but accidents are not necessary.
Either therefore we must say this, or that demonstration is a necessary thing, if we lay down this principle, and that if demonstration is given that a thing cannot subsist otherwise, wherefore the syllogism must be from necessary (matter). For it is possible without demonstration to syllogize from what are true, but we cannot do so from things necessary, except by demonstration, for this is now (the essence) of demonstration. An indication also that demonstration is from things necessary is, that we thus object to those who think they demonstrate that (the conclusion) is not necessary, whether we think that the matter may altogether be otherwise possible, or on account of the argument. Hence too the folly of those appears, who think they assume principles rightly, if the proposition be probable and true, as the Sophists (assume) that to know is to possess knowledge. For it is not the probable or improbable, which is the principle, but that which is primary of the genus about which the demonstration is made, nor is every thing true appropriate. But that it is necessary that the syllogism should consist of necessary things appears also from these; for if he who cannot assign a reason why a thing is, when there is a demonstration, does not possess knowledge, let A be necessarily predicated of C, but B the medium through which it is demonstrated not of necessity, (in this case) he does not know the cause. For this is not on account of the medium, for the latter may not exist, yet the conclusion is necessary. Besides, if some one does not know, though he now possesses a reason, and is safe, the thing also being preserved, he not having forgotten it, neither did he before know it. But the medium may perish if it is not necessary, so that he, being safe, will have a reason, the thing being preserved, and yet not know it, wherefore neither did he know it before. But if the medium is not destroyed, yet may possibly perish, that which happens will be possible and contingent, it is impossible however that one so circumstanced should know.
When therefore the conclusion is from necessity, there is nothing to prevent the medium through which the demonstration was made from being not necessary, since it is possible to syllogize the necessary even from things not necessary, just as we may the true from things not true. Still when the medium is from necessity the conclusion is also from necessity, as the true (results) from the true always: for let A be of necessity predicated of B, and this of C, then it is necessary that A should be with C. But when the conclusion is not necessary, neither possibly can the medium be necessary: for let A be present with C, not of necessity, but let it be with B, and this with C of necessity; A then will also be of necessity present with C, yet it was not supposed so. Since therefore what one knows demonstratively must be inherent of necessity, we must evidently obtain the demonstration through a necessary medium also, for otherwise, he will neither know why a thing exists, nor that it is necessary for it to exist, but he will either imagine not knowing, if he assumes what is not necessary as if it were necessary, or in like manner he will not imagine if he knows that it is through media, and why it is through the immediate.
Of accidents however which are not per se after the manner in which things per se have been defined, there is no demonstrative science, since it is not possible to demonstrate the conclusion of necessity, because accident may possibly not be present, for I speak of accident of this kind. Still some one may perhaps doubt why we must make such investigations about these things, if it is not necessary that the conclusion should be, for it makes no difference if any one interrogating casual things should afterwards give the conclusion: nevertheless we must interrogate not as if (the conclusion) were necessary on account of things interrogated, but because it is necessary for him who asserts these should assert this, and that he should speak truly if the things are truly inherent.
Since, however, whatever are inherent per se are necessarily inherent in every genus, and so far as each is, it is clear that scientific demonstrations are of things "per se" inherent, and consist of such as these. For accidents are not necessary: wherefore it is not necessary to know the conclusion why it is, nor if it always is, but not "per se," as, for instance, syllogisms formed from signs. For what is "per se" will not be known "per se," nor why it is, and to know why a thing is, is to know through cause, wherefore the middle must "per se" be inherent in the third, and the first in the middle.
Chapter 7 It is not therefore possible to demonstrate passing from one genus to another, as, for instance, (to demonstrate) a geometrical (problem) by arithmetic, for there are three things in demonstrations, one the demonstrated conclusion, and this is that which is per se inherent in a certain genus. Another are axioms, but axioms are they from which (demonstration is made), the third is the subject genus, whose properties and essential accidents demonstration makes manifest. Now it is possible that the things from which demonstration consists may be the same, but with those whose genus is different, as arithmetic and geometry, we cannot adapt an arithmetical demonstration to the accidents of magnitudes, except magnitudes are numbers, and how this is possible to some shall be told hereafter. But arithmetical demonstration always has the genus about which the demonstration (is conversant), and others in like manner, so that it is either simply necessary that there should be the same genus, or in a certain respect, if demonstration is about to be transferred; but that it is otherwise impossible is evident, for the extremes and the middles must necessarily be of the same genus, since if they are not per se, they will be accidents. On this account we cannot by geometry demonstrate that there is one science of contraries, nor that two cubes make one cube, neither can any science (demonstrate) what belongs to any science, but such as are so related to each other as to be the one under the other, for instance, optics to geometry, and harmonics to arithmetic. Nor if any thing is inherent in lines not so far as they are lines, nor as they are from proper principles, as if a straight line is the most beautiful of lines, or if it is contrary to circumference, for these things are inherent not by reason of their proper genus, but in so far as they have something common.
Chapter 8 It is also evident that if the propositions of which a syllogism consists are universal, the conclusion of such a demonstration, and in short of the demonstration of itself, must necessarily be perpetual. There is not then either demonstration, nor in short science of corruptible natures, but so as by accident, because there is not universal belonging to it, but sometimes, and after a certain manner. But when there is such, it is necessary that one proposition should not be universal, and that it should be corruptible, corruptible indeed, because the conclusion will be so if the proposition is so, and not universal, because one of those things of which it is predicated will be, and another will not be, hence it is not possible to conclude universally, but that it is now. It is the same in the case of definitions, since definition is either the principle of demonstration, or demonstration, differing in the position (of the terms), or a certain conclusion of demonstration. The demonstrations and sciences however of things frequently occurrent, as of the eclipse of the moon, evidently always exist, so far as they are such, but so far as they are not always, they are particular, and as in an eclipse, so also is it in other things.
Chapter 9