Organon (Owen)/The Posterior Analytics/Book 1

by Aristotle

Layout 2

Chap. 1. Upon the Nature of Demonstration.

1.1. All dianœtic discipline is produced from previous knowledge, possessed in a two-fold respect. (Cf. Mag. Moral. lib. i. 18, and Eth. Eud. lib. v. c. 1, 2, 3)

1.2. What we know universally and generally we may not know singly, although not in the same manner.

Chap. 2. Of Knowledge, and Demonstration, and its Elements.

2.1. Scientific knowledge is possessed, when we know the necessary connexion between a thing and its cause. Definition of Demonstration. (Vide Ethics, vi. 3, 4.)

2.2. Specified elements of true demonstrative science.

2.2.1. True.

2.2.2. First and indemonstrable.

2.2.3. Causes of the conclusion.

2.2.4. Prior and more known, in a two-fold respect.

2.2.5. Immediate.

2.3. Distinction of proposition.

2.4. Definition of thesis, considered by Pacius and Waits as synonymous with πτῶσις.

2.5. Of axiom.

2.6. Of hypothesis.

2.7. The necessity of knowing principles and their opposites, in order to possess science by demonstration.

Chap. 3. Refutation of certain opinions as to Science and Demonstration.

3.1. Refutation of those who deny the existence of science.

3.2. Also of those who declare all things capable of demonstration.

3.3. We cannot demonstrate in a circle things which do not reciprocate.

3.4. Example.

Chap. 4. Upon the terms "every," "per se," and "universal."

4.1. Definition of demonstration.

4.2. Of predication "de omni." τὸ κατὰ παντὸς.

4.3. Of "τὸ καθ’αὑτὸ," "per se."

4.4. Of accidents. συμβεβηκότα. (Cf. Phys. lib. ii, et Metap. lib. v.)

4.5. "Per se," recapitulation.

4.6. What is a contrary.

4.7. τὸ ἧ αὐτό, "quatenus ipsum," and το καθολου, explained.

Chap. 5. Of Errors about the primary Universal.

5.1. Sources of error in effecting universal demonstration. Example.

Chap. 6. Demonstration consists of Principles per se; and of a necessary Medium.

6.1. Recapitulation; true demonstration only from necessary propositions.

6.2. Proof of this.

6.2.1. Reply to objection.

6.3. If the conclusion be necessary, the premises need not be so, but when the latter are so the conclusion must be necessary.

6.4. The non-necessary, not to be neglected in disputation.

6.5. Necessity of the minor and major propositions being "per se."

Chap. 7. That we may not demonstrate by passing from one Genus to another.

7.1. Three things in demonstration, viz. a demonstrated conclusion, axioms, and the subject genus.

7.2. That the extremes and media must be of the same genus.

Chap. 8. Things which are subject to Change are incapable of Demonstration per se.

8.1. That there is no demonstration nor definition "per se" of mutable natures, because of the universal being non-existent.

Chap. 9. That the Demonstration of a thing ought to proceed from its own appropriate Principles: these last indemonstrable.

9.1. That true demonstration only results from principles appropriate to the subject of demonstration: the terms must either be homogeneous, or from two genera, of which one is contained in the other.

9.2. That the appropriate principles of each thing are themselves incapable of demonstration. What is the especial science.

9.3. Difficulty of deciding whether a thing is really known.

Chap. 10. Of the Definition and Division of Principles.

10.1. Definition of principles, (ἀρχαὶ,) their existence to be assumed. Example.

10.2. What are peculiar to each science, and what common.

10.3. All demonstration conversant with three things, of which we sometimes may neglect two.

10.4. Of the difference between ὰξιώμα, ὑπόθεσις, and αἴτημα.

10.5. That definition is not hypothesis.

Chap. 11. Of certain Common Principles of all Sciences.

11.1. Demonstration may exist without ειδη, but not without an universal conception.

11.2. Of the use of what is called the principle of contradiction in demonstration.

11.3. Of the common principles of the several sciences.

Chap. 12. Of Syllogistic Interrogation.

12.1. Method of deciding what proposition belongs to each science.

12.2. Of discovering the science to which each false syllogism appertains.

12.3. When an objection is not to be made.

12.4. Instance of a syllogistic argument, by employing a syllogism with both prop. affirm. in the 2nd figure.

12.5. Mathematical demonstrations rarely prove the same, by many media.

Chap. 13. The difference between Science, "that" a thing is, and "why" it is.

13.1. A two-fold difference if the syllogism be not through things immediate: next, if it be, but not through cause, in the same science.

13.2. Where the media do not reciprocate the ὅτι is demonstrated, also where the middle is externally placed.

13.3. Another difference between a syllogism of the ὅτι and the διότι, in respect of each belonging to a different science.

13.4. The knowledge of the ὅτι belongs to the perceptive, of the διότι to the mathematical arguer.

Chap. 14. The first Figure most suitable to Science.

14.1. Mathematical demonstrations effected through this figure.

14.2. Also the syllogism of the διότι. Cf book 2nd.

14.3. Also the science of τοῦ τί ἐστιν.

14.4. The other figures condensed by this one.

Chap. 15. Of immediate negative Propositions.

15.1. That one thing may possibly not be individually present with another. Examples.

Chap. 16. Of Ignorance, according to corrupt position of the Terms, where there are no Media.

16.1. Definition of ἄγνοια ἡ κατὰ διάθεσιν, and its kinds.

16.2. Examples of affirmative deception.

16.3. Negative deception instanced in the first and middle figures.

[16.3.1. First fig.]

16.3.2. Middle fig.

Chap. 17. Continuation of the same with Media.

17.1. Syllogism of the false produced in mediates, when the major is false.

17.2. Case of both propositions being false.

17.3. Both prop. cannot be wholly false in the middle figure, when deception is produced.

17.4. Affirmative deception.

Chap. 18. Of the Dependence of Universals upon Induction, and of the latter upon Sense.

18.1. Universals from which demonstration proceeds, depend upon induction, the latter upon sense. (Cf. Eth, b. vi. ch. 3; Rhet. b. i. ch. 2, and b. ii. ch. 23.

Chap. 19. Of the Principles of Demonstration, whether they are Finite or Infinite.

19.1. By those who syllogize κατὰ δόξαν it is to be considered whether the syllogisms arise from propositions especially probable.

19.2. An inquiry whether a stated series of terms proceeds to infinity.

19.3. The same as to negatives.

19.4. The doubt does not exist in the case of reciprocals.

Chap. 20. Of Finite Media.

20.1. Media not infinite where the predications stop—Explanation and example.

Chap. 21. It is shown that there are no Infinite Media in Negative Demonstration.

21.1. That there is not an infinity of media in negative demonstration, proved in the several figures.

Chap. 22. That there are no Infinite Media in Affirmative Demonstration.

22.1. Of predications, as to what a thing is, there cannot be infinity—a difference of predication pointed out.

22.2. True predications either define what the subject is, or are accidents.

22.3. In either case there cannot be an infinite series shown from the nature of category.

22.4. Hypothesis that a mediate proposition may be proved.

22.5.1. If there is an infinity of predication, demonstration cannot exist.

22.5.2. The same proved analytically from the nature of those things which are predicated καθ’αὑτά.

22.6. That there is not infinity of media.

Chap. 23. Certain Corollaries

23.1. Case where no common ground of inherency subsists.

23.2. Cases of propositional demonstration, when a certain medium is granted.

23.3. What position the connecting term should occupy in an affirmative and negative proposition.

Chap. 24. The superiority of Universal to Particular Demonstration proved.

24.1. The question stated.

24.2. Reasons why particular demonstration may appear eligible.

24.3. Reply to the above.

23.4.3. Universal alone is cognizant of cause.

24.4. It is true "non per aliud," but "per se."

24.5. Universals tend to the simple and finite hence are more scientific.

24.6. They come closer in demonstration to the principle.

24.7. The universal is above all superior, in that it comprehends the particular, and is more intellectual.

Chap. 25. The Superiority of Affirmative to Negative Demonstration proved.

25.1. That the demonstration which is through fewer postulates, etc., is, "cæteris paribus," the better—proved by example, and applied to affirmatives.

25.2. The negative requires the affirmative, but the latter does not need the former.

25.3. Affirmative comes nearer than negative to the nature of a principle.

Chap. 26. The Superiority of the same to Demonstration ad impossible proved.

26.1. The difference proved by example, between direct demonstration and that which leads "ad absurdum."

26.2. Scale of demonstrative superiority.

Chap. 27. Upon the Nature of more Accurate Science.

27.1. That one science is more subtle and accurate than another.

Chap. 28. What constitutes one, and what different Sciences.

28.1. Whatever things are demonstrated from principles of a common genus, these constitute one science. Nature of diverse sciences.

Chap. 29. That there may be several Demonstrations of the same thing.

29.1. The same thing demonstrable in many modes, both when the middles are taken from the same, or from a different genus.

Chap. 30. That there is no Science of the Fortuitous.

30.1. This class does not come under the proper subjects of demonstration.

Chap. 31. That we do not possess Scientific Knowledge through Sensation.

31.1. The perception of the senses is not science.

31.2. Though there are certain things unknown, from the deficiency of sensible perception.

Chap. 32. On the Difference of Priniciples according to the Diversity of Syllogisms.

32.1. The impossibility of principles of all syllogisms being identical, proved.

32.2. Reply to objection founded upon mistaken identity.

32.3. Principles (ἀρχαὶ) two-fold, ἐξ ὧν and περὶ ὃ.

Chap. 33. Upon the Difference between Science and Opinion.

33.1. Science is universal, and subsists through things necessary: intellect the principle of science.

33.2. Opinion conversant with the non-necessary.

33.3. Solution of an inquiry why in certain cases opinion may not be science.

33.4. We cannot, at one and the same time, know, and opine.

Chap. 34. Of Sagacity.

34.1. Definition of sagacity, ἐυστοχία τις ἐν ἀσκεπτῷ χρόνῶ τοῦ μέσοῦ: instances.

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

Since however it is evident that we cannot demonstrate each thing except from its own principles, if what is to be demonstrated is inherent in a subject so far as the subject is that (which it is) to have a scientific knowledge of that thing is not this, it it should be demonstrated from true, indemonstrable, and immediate (propositions). For we may so demonstrate possibly, as Bryso did, the quadrature of the circle, since such reasonings prove through something common, that which is inherent in another thing, hence these arguments are adapted to other things not of the same genus. Wherefore that thing would not be scientifically known, as far as it is such, but from accident, for otherwise the demonstration would not be adapted also to another genus.

We know however each thing not accidentally when we know it according to that, after which it is inherent from principles which are those of that thing, so far as it is that thing; as that a thing has angles equal to two right angles, in which the thing spoken of is essentially inherent from the principles of this thing. Hence if that is essentially inherent in what it is inherent, it is necessary that the middle should be in the same affinity, but if not, yet it will be as harmonics are proved through an arithmetical principle. Such things however are demonstrated after a similar manner, yet they differ, for that they are, is part of another science, (for the subject genus is another,) but why they are, is a province of a superior science, of which they are the essential qualities. Hence from these things also it is apparent that we cannot demonstrate each thing simply, but from its proper principles, and the principles of these have something common.

If then this is evident, it is also clear that it is impossible to demonstrate the proper principles of each thing, for they will be the principles of all things, and the science of them the mistress of all (sciences): for the man has more scientific knowledge who knows from superior causes, since he knows from prior things when he knows not from effects, but from causes. So that if he knows more, he knows also most, and if that be science, it is also more, and most of all such. Demonstration however is not suitable to another genus, except as we have said, geometrical to mechanical or optical, and arithmetical to harmonical demonstrations.

Nevertheless it is difficult to know whether a man possesses knowledge or not, since it is hard to ascertain if we know from the principles of each thing or not, which indeed constitutes knowledge. We think however that we know, if we have got a syllogism from certain primary truths, but it is not so, since it is necessary that they should be of a kindred nature with the primary.

Chapter 10

I call those principles in each genus, the existence of which it is impossible to demonstrate. What then first things, and such as result from these signify, is assumed, but as to principles, we must assume that they are, but demonstrate the rest, as what unity is, or what the straight and a triangle are; it is necessary however to assume that unity and magnitude exist, but to demonstrate the other things.

Of those which are employed in demonstrative sciences, some are peculiar to each science, but others are common, and common according to analogy, since each is useful, so far as it is in the genus under science. The peculiar indeed are such as, that a line is a thing of this kind, and that the straight is, but the common are, as that if equals be taken from equals the remainders are equal. Now each of these is sufficient, so far as it is in the genus, for (a geometrician) will effect the same, though he should not assume of all, but in magnitudes alone, and the arithmetician in respect of numbers (alone).

Proper principles, again, are those which are assumed to be, and about which science considers whatever are inherent per se, as arithmetic assumes unities, and geometry points and lines, for they assume that these are, and that they are this particular thing. But the essential properties of these, what each signifies, they assume, as arithmetic, what the odd is, or the even, or a square, or a cube; and geometry, what is not proportionate, or what is to be broken, or to incline; but that they are, they demonstrate through things common, and from those which have been demonstrated. So also astronomy, for all demonstrative science is conversant with three things, those which are laid down as existing, and these are the genus, (the essential properties of which the science considers,) and common things called axioms, from which as primaries they demonstrate; and thirdly, the affections, the signification of each of which the demonstrator assumes. There is nothing however to prevent certain sciences overlooking some of these, as if the genus is not supposed to be, if it be manifest that it exists, (for it is not similarly manifest that number is, as that the cold and hot are,) and if (the science) does not assume what the affections signify, if they are evident, as neither does it assume what things common signify, (as what it is) to take away equals from equals, because it is known; nevertheless these things are naturally three, viz. that about which demonstration is employed, the things demonstrated, and the principles from which they are.

Neither however hypothesis nor postulate is that which it is necessary should exist per se, and be necessarily seen, for demonstration does not belong to external speech, but to what is in the soul, since neither does syllogism. For it is always possible to object to external discourse, but not always to internal. Whatever things then, being demonstrable, a man assumes without demonstration, these, if he assumes what appear probable to the learner, he supposes, and this is not an hypothesis simply, but with reference to the learner alone; but if, there being no inherent opinion, or when a contrary is inherent, the demonstrator assumes, he requires the same thing to be granted to him. And in this hypothesis and postulate differ, for postulate is any thing sub-contrary to the opinion of the learner, which though demonstrable a man assumes, and uses without demonstration.

Definitions then are not hypotheses, (for they are not asserted to be or not to be,) but hypotheses are in propositions. Now it is only necessary that definitions should be understood, but this is not hypothesis, except some one should say that the verb to hear is hypothesis. But they are hypotheses, from the existence of which, in that they are, the conclusion is produced. Neither does the geometrician suppose falsities, as some say, who assert, that it is not right to use a false (principle), but that the geometrician does so, when he calls a line a foot long when it is not so, or the line which he describes a straight line when it is not straight. The geometrician indeed concludes nothing from the lines being so and so, as he has said, but concludes those, which are manifested through these (symbols). Moreover postulate and every hypothesis are either as a whole or as in a part, but definitions are neither of these.

Chapter 11

That there should then be forms, or one certain thing besides the many, is not necessary, to the existence of demonstration, but it is necessary truly to predicate one thing of the many, for there will not be the universal unless this be so, and if there be not an universal, there will not be a medium, so that neither will there be a demonstration. It is essential then that there should be one and the same thing, which is not equivocal in respect of many: no demonstration however assumes that it is impossible to affirm and deny the same thing at one and the same time, unless it is requisite also thus to demonstrate the conclusion. It is demonstrated however by assuming the first to be true of the middle, and that it is not true to deny it, but it makes no difference whether we assume the middle to be or not to be, and in a similar manner also in respect of the third. For if that be granted in respect of which it is true to predicate man, even if (some one should think that man is) not man, (the conclusion) will be true, if only it is said that man is an animal, and not that he is not an animal, for it will be true to say that Callias, even if he be not Callias, yet is still an animal, but not that which is not an animal. The cause however is, that the first is not only predicated of the middle, but also of something else, in consequence of its being common to many, so that neither if the middle be that thing itself, or not that thing, does it make any difference in respect to the conclusion. But the demonstration which leads to the impossible, assumes that of every thing affirmation or negation is true, and these it does not always (assume) universally, but so far as is sufficient, and it is sufficient (which is assumed) in respect of the genus. I mean by the genus, as the genus about which a person introduces demonstrations, as I have observed before.

All sciences communicate with each other according to common (principles), and I mean by common those which men use as demonstrating from these, but not those about which they demonstrate, nor that which they demonstrate, and dialectic is (common) to all (sciences). If also any one endeavours to demonstrate universally common (principles), as that of every thing it is true to affirm or deny, or that equals remain from equals, or others of this kind. Dialectic however does not belong to certain things thus definite, nor to one particular genus; for it would not interrogate, since it is impossible for the demonstrator to interrogate, because the same thing is not proved from opposites: this however has been shown in the treatment of syllogism.

Chapter 12

If syllogistic interrogation is the same as a proposition of contradiction, but there are propositions in each science, from which the syllogism which belongs to each consists, there will be a certain scientific interrogation, from which the syllogism, which is appropriate to each science, is drawn. It is clear, then, that not every interrogation would be geometrical, or medical, and so of the rest, but from what any thing is demonstrated about which geometry is conversant, or which are demonstrated from the same principles as geometry, as optics, and in like manner with other sciences. These also must be discussed from geometrical principles and conclusions, but the discussion of principles is not to be carried on by the geometrician so far as he is such; likewise with other sciences. Neither is every one who possesses science to be interrogated with every question, nor is every question about each to be answered, but those which are defined about the science. It is evident then that he does well, who disputes with a geometrician thus, so far as he is such, if he demonstrate any thing from these principles, but if not, he will not do well. Again, it is clear that neither does he confute the geometrician except by accident, so that there cannot be a discussion of geometry by those who are ignorant of geometry, since the bad reasoner will escape detection, and it is the same with other sciences.

Since there are geometrical interrogations, are there also those which are ungeometrical? and in each science are those ignorant questions which are of a certain quality geometrical? whether also is a syllogism, from ignorance, a syllogism composed from opposites or a paralogism, but according to geometry, or from another art, as a musical interrogation is ungeometrical, about geometry but to imagine that parallel lines meet is in a certain respect geometrical, and after another manner ungeometrical? For this is two-fold, in the same way as what is without rhythm; and the one is ungeometrical because it possesses not (what is geometrical), as what is without rhythm; but the other because it possesses it wrongly—and this ignorance which is from such principles, is contrary. In mathematics however there is not in like manner a paralogism, because the middle is always two-fold, for (one thing) is predicated of every individual of this, and this again of another every, but the predicate is not called universal; those, nevertheless, it is possible, we may see by common perception, but in argument they escape us. Is then every circle a figure? If any one should delineate it, it is clear. But what, are verses a circle? They are evidently not so.

Still it is improper to object to it, if it be an inductive proposition; for as neither is that a proposition which is not in respect of many things, (since it will not be in all, but syllogism is from universals,) neither, it appears clear, is that an objection, for propositions and objections are the same, as the objection which one adduces, may become either a demonstrative or a dialectic proposition.

It occurs that some argue contrary to syllogism, from assuming the consequences of both (extremes), as Cæneus does, that fire is in a multiple proportion, because, as he says, both fire and this proportion are rapidly generated. But thus there is no syllogism, though there will be, if the multiple is consequent to the most rapid proportion, and the most rapid proportion to fire in motion. Sometimes it does not happen that a conclusion is made from the assumptions, and sometimes it happens, but is not perceived: if however it were impossible to demonstrate the true from the false, it would be easy to resolve, for (the terms) would be necessarily converted. Thus let A exist, and this existing, these things also exist[,] the existence of which I know, as B, from these[,] then I will demonstrate that that exists. What pertain however to mathematics, are rather converted, because they take nothing accidental, (and in this they differ from dialectical subjects,) but definitions.

Yet they are increased, not through media, but through additional assumption, as A of B, this of C, this again of D, and so on to infinity. Also transversely, as A both of C and of E, as there is a number so great or even infinite, which is A, an odd number so great B, and an odd number C. A then is (true) of C, and the even is a number so great D, the even number is E, wherefore A is (true) of E.

Chapter 13

Now there is a difference between knowing that a thing is, and why it is, first in the same science, and in this in two ways, the one, if the syllogism is not formed through things immediate, (since the primary cause is not assumed, but the science of the why has respect to the first cause,) but the other if it is through things immediate indeed, yet not through the cause, but through that which is more known of the things, which reciprocate. Now nothing prevents that which is not a cause being sometimes more known amongst things which are mutually predicated, so that demonstration shall accrue through this, as that the planets are near, because they do not twinkle. Let C be the planets, B not to twinkle, A to be near, B therefore is truly predicated of C, since the planets do not twinkle, A also of B, for what does not twinkle is near, but this may be assumed by induction or by sense. It is necessary then that A should be present with C, so that it is demonstrated that the planets are near. This syllogism then is not of the "why," but of the "that" (a thing is), for the planets are not near because they do not twinkle, but they do not twinkle because they are near. It happens indeed that the one may be proved through the other, and the demonstration will be of the "why," as let C be the planets, B to be near, A not to twinkle, B then is present with C, so that A "not to twinkle" will be with C. It is also a syllogism of the "why," for the first cause was assumed. Again, as they show the moon to be spherical through increments (of light), for if what is thus increased be spherical, and the moon is increased, it is evident that the moon is spherical, thus then a syllogism of the "that" is produced, but if the middle is placed contrarily, there is a syllogism of the "why," for it is not spherical on account of the increments, but from being spherical she receives such increments: let the moon be C, spherical B, increase A. Where again the media do not reciprocate, and what is not the cause is more known, the "that" is indeed demonstrated, but not the "why;" further, where the middle is placed externally, for in these the demonstration is of the "that", and not of the "why," as the cause is not assigned. For example, why does not a wall breathe? because it is not an animal, for if this was the cause of its not breathing, it would be necessary that animal should be the cause of its breathing, since if negation is the cause of a thing not being, affirmation is the cause of its being, thus if the disproportion of hot and cold is the cause of not being well, the proportion of these is the cause of being well. Likewise if affirmation is the cause of being, negation is the cause of not being, but in things which have been thus explained, what has been stated does not occur, for not every animal respires. A syllogism of such a cause is nevertheless produced in the middle figure, for example, let A be animal, B to respire, C a wall, A then is present with every B, (for whatever respires is animal,) but with no C, so that neither is B present with any C, wherefore a wall does not respire. Such causes however resemble things spoken hyperbolically, and this is, when we turn aside to speak of the middle, which is more widely extended, as for instance, that saying of Anacharsis, that amongst the Scythians there are no pipers, since neither are there any vines.

As to the same science then, and the position of the media, these are the differences between a syllogism of, that a thing is, and of why it is, but in another respect the why differs from the that, because each is beheld in a different science. Now such are those things which so subsist with reference to each other, as that the one is under the other, such as optics with reference to geometry, mechanics to the measurement of solids, harmonics to arithmetic, and celestial phenomena to astronomy. Some of these sciences are almost synonymous, as astronomy is both the mathematical and the nautical; and harmony is both mathematical and that which belongs to the ear. For here to know that a thing is, is the province of those who exercise the sense, but to know why it is, belongs to mathematicians, since these possess the demonstrations of causes, and often are ignorant of the that, as they who contemplating universals, frequently are ignorant of singulars from want of observation. But these are such as being essentially something else use forms, for mathematics are conversant with forms, since they do not regard one certain subject, for though the geometrical are of a certain subject, yet not so far as they are geometrical are they in a subject. As optics also to geometry, so is some other science related to optics, as for example, the science about the rainbow, for to know that it is, appertains to the natural philosopher, but why it is, to the optician either simply or mathematically. Many sciences also which are not arranged under each other subsist thus, for example, medicine with regard to geometry, for to know that circular wounds heal more slowly is the province of the physician, but why (they do so) of the geometrician.

Chapter 14

Of the figures, the first is especially adapted to science, for both the mathematical sciences carry out their demonstrations by this, as arithmetic, geometry, optics, and nearly, so to speak, whatsoever sciences investigate the "why," since either entirely or for the most part, and in most sciences, the syllogism of the why is through this figure. Wherefore also, on this account, it will be especially adapted to science, for it is the highest property of knowledge to contemplate the "why;" in the next place, it is possible through this figure alone to investigate the science of what a thing is; for in the middle figure, there is no affirmative syllogism, but the science of what a thing is belongs to affirmation, and in the last figure, there is an affirmative, but not an universal; but the what a thing is belongs to universals, for man is not a biped animal in a certain respect. Moreover this has no need of those, but they are condensed and enlarged through this, till we arrive at things immediate: it is evident, then, that the first figure is in the highest degree adapted to scientific knowledge.

Chapter 15

As it happened that A was present with B individually, so also it may happen not to be present, and I mean by being present with, or not, individually, that there is no medium between them, for thus the being present with or not, will not be according to something else. When then either A or B is in a certain whole, or when both are, it is impossible that A should not be primarily present with B. For let A be in the whole of C, if then B is not in the whole of C, (for it is possible that A may be in a certain whole, but that B may not be in this,) there will be a syllogism that A is not present with B, for if C is present with every A, but with no B[,] A will be present with no B. In like manner also, if B is in a certain whole, as for instance, in D, for D is with every B, but A with no D, so that A will be present with no B by a syllogism. In the same way it can be shown if both also are in a certain whole, but that it is possible that B may not be in the whole in which A is, or again A in which B is, is evident from those co-ordinations which do not interchange. For if none of those, which are in the class A C D, is predicated of any of those in B E F, but A is in the whole of H, which is co-arranged with it, it is evident that B will not be in H, for otherwise the co-ordinates would intermingle.

Likewise also if B is in a certain whole, but if neither is in any whole, and A is not present with B, it is necessary that it should not be present individually, for if there shall be a certain middle, one of them must necessarily be in a certain whole, for there will be a syllogism either in the first, or in the middle figure. If then it is in the first, B will be in a certain whole, (for it is necessary that the proposition in regard to this should be affirmative,) but if in the middle figure either of them may be (in the whole), for the negative being joined to both, there is a syllogism, but there will not be when both the propositions are negative.

It is manifestly possible then, that one thing may not be individually present with another, also when, and how this may happen, we have shown.

Chapter 16

The ignorance which is denominated not according to negation, but according to disposition, is a deception produced through syllogism, and this happens in two ways, in those things which are primarily present, or not present; for it happens either when one simply apprehends the being present, or not being present, or when he obtains this opinion through syllogism: of simple opinion, then, the deception is simple, but of that which is through syllogism, it is manifold. For let A not be present with any B individually, if then A is concluded to be present with B, assuming C as the middle, a person will be deceived through syllogism. Hence it is possible that both propositions may be false, but it is also possible that only one may be so, for if neither A is present with any C, nor C with any B, but each proposition is taken contrary, both will be false. But it may be that C so subsists with reference to A and B, as neither to be under A nor universally (present) with B, for it is impossible that B should be in a certain whole, since it was said that A is not primarily present with it; but A need not be universally present with all beings, so that both propositions are false. Nevertheless, we may assume one proposition as true, not either of them casually, but the proposition A C, for the proposition C B will be always false, because B is in none; but A C may be (true), for instance, if A is present individually, both with C and B, for when the same thing is primarily predicated of many things, neither will be predicated of neither; it makes no difference however if it (A) be not individually present with it (C).

The deception then of being present, is by these and in this way only, (for there was not a syllogism of being present in another figure,) but the deception of not being present with, is in the first and middle figure. Let us first then declare in how many ways it occurs in the first, and under what propositional circumstances. It may then happen when both propositions are false, e. g. if A is present individually with C and B, for if A should be assumed present with no C, but C with every B, the propositions will be false. But (deception) is possible, when one proposition is false, and either of them casually; for it is possible that A C may be true, but C B false; A C true, because A is not present with all beings, but C B false, because it is impossible that C should be with B, with nothing of which A is present; for otherwise the proposition A C will be no longer true, at the same time, if both are true, the conclusion also will be true. But it is also possible that C B may be true, when the other proposition is false, as if B is in C and in A, for one if must necessarily be under the other, so that if A should be assumed present with no C, the proposition will be false. It is clear then, that when one proposition is false, and also when both are, the syllogism will be false.

In the middle figure, however, it is not possible that both propositions should be wholly false, for when A is present with every B, it will be impossible to assume any thing, which is present with every individual of the one, but with no individual of the other; but we must so assume the propositions that the (middle) may be present with one (extreme), and not be present with the other, if indeed there is to be a syllogism. If then, when they are thus assumed, they are false, it is clear that, when taken contrarily, they will subsist vice versâ, but this is impossible. Still there is nothing to prevent each being partly false, as if C is with A, and with a certain B; for if it should be assumed present with every A, but with no B, both propositions indeed would be false, yet not wholly, but partially. The same will occur when the negative is placed vice versâ. But it is possible that one proposition, and either of them, may be false, for what is present with every A, will be also with B, if then C is assumed present with the whole of A, but not present with the whole of B, C A will be true, but the proposition C B false. Again, what is present with no B, will not be present with every A; for if with A, it would also be with B, but it was not present; if then C should be assumed present with the whole of A, but with no B, the proposition C B will be true, but the other false. The same will happen if the negative is transposed, for what is in no A, will neither be in any B; if then C is assumed not present with the whole of A, but present with the whole of B, the proposition A C will be true, but the other false. Again, also, it is false to assume that what is present with every B, is with no A; for it is necessary, if it is with every B, that it should be also with a certain A; if then C is assumed present with every B, but with no A, the proposition C B will be indeed true, but C A false. Hence, it is evident that when both propositions are false, and when one only is so, there will be a syllogism deceptive in individuals.

Chapter 17

In those which are not individually present, or which are not present, when a syllogism of the the false is produced through an appropriate medium, both propositions cannot be false, but only the major. But I mean by an appropriate medium, that through which there is a syllogism of contradiction. For let A be with B through the medium of C, since then we must take C B as affirmative, if there is to be a syllogism, it is clear that this will be always true, for it is not converted. A C, on the other hand, will be false, for when this is converted, a contrary syllogism arises. So also if the middle is assumed from another affinity, as for instance, if D is in the whole of A, and is predicated of every B, for the proposition D B must necessarily remain, but the other proposition must be converted, so that the one (the minor) will be always true, but the other (the major) always false. Deception also of this kind is almost the same as that which is through an appropriate medium, but if the syllogism should not be through an appropriate medium, when indeed the middle is under A, but is present with no B, it is necessary that both propositions should be false. For the propositions must be assumed contrary to the way in which they subsist, if a syllogism is to be formed, for when they are thus assumed both are false, as if A is with the whole of D, but D present with no B, for when these are converted, there will be a syllogism, and both propositions will be false. When however the medium is not under A, for instance, D, A D will be true, but D B false, for A D is true, because D was not in A, but D B false, because if it were true the conclusion also would be true, but it was false.

Through the middle figure however, when deception is produced, it is impossible that both propositions should be wholly false, (for when B is under A, it is possible for nothing to be present with the whole of the one, but with nothing of the other, as has been observed before,) but one proposition may be false whichever may happen. For if C is with A and with B, if it be assumed present with A, but not present with B, the proposition A C will be true, but the other false; again, if C be assumed present with B, but with no A, the proposition C B will be true, but the other false.

If then the syllogism of deception be negative, it has been shown when and through what the deception will occur, but if it be affirmative, when it is through an appropriate medium, it is impossible that both should be false, for C B must necessarily remain, if there is to be a syllogism, as was also observed before. Wherefore C A will be always false, for it is this which is converted. Likewise also, if the middle be taken from another class, as was observed in negative deception, for the proposition D B must of necessity remain, but A D be converted, and the deception is the same as the former. But when it is not through an appropriate medium, if D be under A, this indeed will be true, but the other false, for A may possibly be present with many things which are not under each other. If however D is not under A, this will evidently be always false, (for it is assumed affirmative,) for D B may be as well true as false, since nothing prevents A being present with no D, but D with every B, as animal with (no) science, but science with (all) music. Again, (nothing prevents) A from being present with no D, and D with no B: it is clear then that when the medium is not under A, both propositions, and either of them, as it may happen, may be false.

In how many ways then, and through what, syllogistic deceptions are possible, both in things immediate, and in those which are demonstrated, has been shown.

Chapter 18

It is clear, also, that if any sense be deficient, a certain science must be also deficient, which we cannot possess, since we learn either by induction or by demonstration. Now demonstration is from universals, but induction from particulars, it is impossible however to investigate universals, except through induction, since things which are said to be from abstraction, will be known through induction; if any one desires to make it parent that some things are present with each genus, although they are not separable, so far as each is such a thing. Nevertheless, it is impossible for those who have not sense to make an induction, for sense is conversant with singulars, as the science of them cannot be received, since neither (can it be obtained) from universals without induction, nor through induction without sense.

Chapter 19

Every syllogism consists of three terms, and one indeed is able to demonstrate that A is with C from its being present with B, and this last with C, but the other is negative, having one proposition (to the effect) that one certain thing is in another, but the other proposition (to the effect) that it is not with it. Now it is clear, that the same are principles, and what are called hypotheses, since it is necessary to demonstrate by thus assuming these, e. g. that A is present with C through B, and again, that A is with B through another medium, and that B is with C in like manner. By those then who syllogize according to opinion only, and dialectically, this alone it is clear must be considered, viz. whether the syllogism is produced from propositions as probable as possible, so that if there is in reality a medium between A and B, but it does not appear, he who syllogizes through this, will have syllogized dialectically. But as to truth, it, behoves us to make our observations from things inherent: it happens thus. Since there is that, which is itself predicated of something else, not according to accident, but I mean by according to accident, as we say sometimes, that that white thing is a man, not similarly saying, that a man is a white thing, for man not being any thing else is white, but it is a white thing, because it happens to a man to be white: there are then some such things as are predicated per se. Let C be a thing of this kind which is not itself present with any thing else, but let B be primarily present with this, without any thing else between. Again, also let E be present in like manner with F, and this with B, is it then necessary that this should stop, or is it possible to proceed to infinity? Once more, if nothing is predicated of A per se, but A is primarily present with H, nothing prior intervening, and H with G, and this with B, is it necessary also that this should stop, or can this likewise go on to infinity? Now this so much differs from the former, that the one is, whether it is possible by beginning from a thing of that kind, which is present with nothing else, but something else present with it, to proceed upward to infinity; but the other is, beginning from that which is itself predicated of another, but nothing predicated of it, whether it is possible to proceed to infinity downward. Besides, when the extremes are finite, is it possible that the media may be infinite? I mean, for instance, if A is present with C, but the medium of them is B, and of B and A there are other media, and of these again others, whether it is possible or impossible for these also to proceed to infinity? To consider this however is the same as to consider whether demonstrations proceed to infinity, and whether there is demonstration of every thing, or whether there is a termination (of the extremes) relatively to each other.

I say also the same in respect of negative syllogisms and propositions, for instance, whether A is primarily present with no B, or there will be a certain medium with which it was not before present, as if G (is a medium), which is present with every B; and again, with something else prior to this, as whether (the medium is) H, which is present with every G; for in these also, either those are infinite with which first they are present, or the progression stops.

The same thing however does not occur in things which are convertible, since in those which are mutually predicated of each other, there is nothing of which first or last a thing is predicated; for in this respect all things subsist similarly with respect to all, whether those are infinite, which are predicated of the same, or whether both subjects of doubt are infinite, except that the conversion cannot be similarly made; but the one is as accident, but the other as predication.

Chapter 20

That media cannot be infinite, if the predications, both downward and upward, stop, is evident: I call indeed the predication upward, which tends to the more universal, but the downward that which proceeds to the particular. For if when A is predicated of F, the media are infinite, that is B, it evidently may be possible that from A in a descending series, one thing may be predicated of another to infinity, (for before we arrive at F, there are infinite media,) and from F in an ascending series, there are infinite (attributes) before we arrive at A. Hence, if these things are impossible, it is also impossible that there should be infinite media between A and F; for it does not signify if a man should say that some things of A B F so mutually adhere, as that there is nothing intermediate, but that others cannot be assumed. For whatever I may assume of B, the media with reference to A or to F, will either be infinite or not, and it is of no consequence from what the infinites first begin, whether directly or not directly, for those which are posterior to them are infinite.

Chapter 21

It is apparent also, that in negative demonstration the progression will stop, if indeed in affirmative it is stopped in both (series), for let it be impossible to proceed to infinity upward from the last, (I call the last that which is itself not present with any thing else, but something else with it, for instance, F,) or from the first to the last, (I call the first that which is indeed itself predicated of something else, but nothing else of it). If then these things are so, the progression must stop in negation, for the not being present is demonstrated triply, since either B is present with every individual with which C is, but A is present with none with which B is. In B C therefore, and always in the other proposition, it is necessary to proceed to immediates, for this proposition is affirmative. With regard to the other however it is clear, that if it is not present with something else prior, for instance, with D, it will be requisite that this (D) should be present with every B. Also if again it is not present with something else prior to D, it will require that to be present with every D, so that since the upward progression stops, the downward progression will also stop, and there will be something first with which it is not present. Moreover if B is with every A, but with no C, A will be with no C; again, if it is required to show this, it is evident, that it may be demonstrated either through the superior mode, or through this, or through the third, now the first has been spoken of, but the second shall be shown. Thus indeed it may demonstrate it, as, for instance, that D is present with every B, but with no C, if it is necessary that any thing should be with B, and, again, if this is not present with C, something else is present with D, which is not present with C, wherefore since the perpetually being present with something superior stops, the not being present will also stop. But the third mode was if A indeed is present with every B, but C is not present, C will not be present with every A; again, this will be demonstrated either through the above-mentioned modes, or in a similar manner, in those modes the progression stops, but if thus, it will again be assumed that B is present with E, with every individual of which C is not present. This again, also, will be similarly demonstrated, but since it is supposed that the downward progression stops, C also, which is not present with, will evidently stop.

Nevertheless, it appears plain, that if it should not be demonstrated in one way, but in all, at one time from the first figure, at another from the second or the third, that thus also the progression will stop, for the ways are finite, but it is necessary that finite things being finitely assumed should be all of them finite.

That in negation then the progression stops, if it does so in affirmation, is clear, but that it must stop in them is thus manifest to those who consider logically.

Chapter 22

In things predicated therefore as to what a thing is, this is clear, for if it is possible to define, or if the very nature of a thing may be known, but infinites cannot be passed through, it is necessary that those things should be finite which are predicated with respect to what a thing is. We must however speak universally thus: a white thing we may truly say walks, also that that great thing is wood; moreover, that the wood is great, and that the man walks, yet there is a difference between speaking in this way and in that. For when I say that that white thing is wood, then I say that what happens to be white is wood, but what is white is not, as it were, a subject to wood, since neither being white, nor what is a certain white thing, became wood, so that it is not (wood) except from accident. But when I say that the wood is white, I do not say that something else is white, but it happens to that to be wood, (as when I say that a musician is white, for then I mean that the man is white, to whom it happens to be a musician,) but wood is the subject which became (white), not being any thing else than what is wood, or a certain piece of wood. If indeed it is necessary to assign names, let speaking in this way be to predicate, but in that way be either by no means to predicate, or to predicate indeed, not simply, but according to accident. That which is predicated is as white, but that of which it is predicated as wood; now let it be supposed that the predicate is always spoken of what it is predicated of simply, and not according to accident, for thus demonstrations demonstrate. Therefore when one thing is predicated of one, it will be predicated either in respect of what a thing is, or that it is a quality, or a quantity, or a relative, or an agent, or a patient, or that it is some where, or at some time.

Moreover, those which signify substance, signify that the thing of which they are predicated, is that which it is, or something belonging to it, but whatever do not signify substance, but are predicated of another subject, which is neither the thing itself, nor something belonging to it, are accidents, as white is predicated of man, since man is neither white, nor any thing which belongs to white, but is perhaps animal, for man is that which is a certain animal. Such as do not signify substance it is necessary should be predicated of a certain subject, and not be something white, which is white, not being any thing else. For, farewell to ideas, for they are mere prattlings, and if they exist, are nothing to the subject, since demonstrations are not about such things.

Again, if this is not a quality of this, and that of this, neither a quality of a quality, it is impossible that they should be thus mutually predicated of each other, still they may possibly be truly said, but cannot truly be mutually predicated. For will they be predicated as substance, as being either the genus or the difference of what is predicated? It has been shown that these will not be infinite, neither in a descending nor in an ascending progression, as for instance, man is a biped, this an animal, this something else; neither can animal be predicated of man, this of Callias, this of something else, in respect to what a thing is. For we may define the whole of this to be substance, but we cannot penetrate infinites by perception, wherefore neither are there infinites upwards or downwards, for we cannot define that of which infinites are predicated. They will not indeed be mutually predicated of each other as genera, for genus would be a part itself, neither will quality nor any of the other categories be (mutually) predicated, except by accident, for all these are accidents, and are predicated of substances. But neither will there be infinites in ascending series, for of each thing, that is predicated, which signifies either a certain quality, or a certain quantity, or something of this kind, or those which are in the substance, but these are finite, and the genera of the categories are finite, since (a category) is either quality, or quantity, or relation, or action, or passion, or where, or when. One thing is however supposed to be predicated of one, but those not to be mutually predicated which do not signify what a thing is, since all these are accidents, but some are per se, others after a different manner, and we say all these are predicated of a certain subject, but that accident is not a certain subject, for we do not assume any thing of this kind to be, which not being any thing else, is said to be what it is said to be, but we say that it is predicated of something else, and certain other things of another thing. Neither then can one thing be predicated of one (infinitely) upwards, nor downwards, for those of which accidents are predicated, are such as are contained in the substance of each thing, but these are not infinite. Both these indeed and accidents are ascending, and both are not infinite, wherefore it is necessary that there should be something of which primarily something is predicated, and something else of this, also that this should stop, and that there should be something which is neither predicated of another prior thing, nor another prior thing of it.

This then is said to be one mode of demonstration, but there is another besides, if there is a demonstration of those of which certain things are previously predicated, but of what there is demonstration, it is not possible to be better affected towards them than to know them, nor can we know without demonstration. Still if this becomes known through these, but these we do not know, nor are better affected towards them than if we knew them, neither shall we obtain scientific knowledge of that which becomes known through these. If then it is possible to know any thing simply through demonstration, and not from certain things, nor from hypothesis, it is necessary that the intermediate predications should stop; for if they do not stop, but there is always something above what is assumed, there will be a demonstration of all things, so that if we cannot pass through infinites, we shall not know by demonstration those things of which there is demonstration. If then we are not better affected towards them than if we knew them, it will be impossible to know any thing by demonstration simply, but by hypothesis.

Logically then from these things a person may believe about what has been said, but analytically it is more concisely manifest thus, that there cannot be infinite predicates in demonstrative sciences, the subject of the present treatise, either in an ascending or descending series. For demonstration is of such things as are essentially present with things, essentially in two ways, both such as are in them in respect of what a thing is, and those in which the things themselves are inherent in respect of what a thing is, thus the odd in number which indeed is inherent in number, but number itself is inherent in the definition of it, again also, multitude or the divisible is inherent in the definition of number. Still neither of these can be infinites, nor as the odd is predicated of number, for again there will be something else in the odd, in which being inherent, (the odd) would be inherent, and if this be so, number will be first inherent in those things which are inherent in it. If then such infinites cannot be inherent in the one, neither will there be infinites in ascending series. Still it is necessary that all should be inherent in the first, for example, in number, and number in them, so that they will reciprocate, but not be more widely extensive. Neither are those infinite which are inherent in the definition of a thing, for if they were, we could not define, so that if all predicates are predicated per se, and these are not infinite, things in an upward progression will stop, wherefore also those which descend.

If then this be so, those also which are between the two terms will be always finite, but if this be the case, it is clear now that there must necessarily be principles of demonstrations, and that there is not demonstration of all things, as we observed in the beginning, certain persons assert. For if there be principles, neither are all things demonstrable, nor can we progress to infinity, since that either of these should be, is nothing else than that there is no proposition immediate and indivisible, but that all things are divisible, since what is demonstrated is demonstrated from the term being inwardly introduced, and not from its being (outwardly) assumed. Wherefore if this may possibly proceed to infinity, the media between two terms might also possibly be infinite, but this is impossible, if predications upwards and downwards stop, and that they do stop, has been logically shown before, and analytically now.

Chapter 23

From what has been shown it appears plain that if one and the same thing is inherent in two, for instance, A in C and in D, when one is not predicated of the other, either not at all or not universally, then it is not always inherent according to something common. Thus to the isosceles and to the scalene triangle, the possession of angles equal to two right, is inherent according to something common, for it is inherent so far as each is a certain figure, and not so far as it is something else. This however is not always the case, for let B be that according to which A is inherent in C D, then it is evident that B is also inherent in C, and in D, according to something else common, and that also according to something else, so that between two terms, infinite terms may be inserted, but this is impossible. It is not then necessary that the same thing should always be inherent in many, according to something common, since indeed there will be immediate propositions; it is moreover requisite that the terms should be in the same genus, and from the same individuals, since that which is common will be of those which are essentially inherent, for it is impossible to transfer things which are demonstrated from one genus to another.

But it is also manifest that when A is with B, if there is a certain middle, we may show that B is with A, and the elements of this are these and whatever are media, for immediate propositions, either all of them, or those which are universal, are elements. Yet if there is not (a medium) there is no longer demonstration, but this is the way to principles. In like manner, if A is not with B, if there is either a middle, or something prior to which it is not present, there is a demonstration, but if not, there is no demonstration, but a principle, and there are as many elements as terms, for the propositions of these are the principles of demonstration. As also there are certain indemonstrable principles, that this is that, and that this is present with that, so there are also that this is not that, and that this is not present with that, so that there will be some principles that a thing is, but others that it is not. Still when it is required to demonstrate, that which is first predicated of B must be assumed; let this be C, and let A, in like manner, (be predicated) of this; by always proceeding thus, there is never a proposition externally, nor is that which is present with A assumed in the demonstration, but the middle is always condensed till they become indivisible and one. They are one indeed when the immediate is produced, and one proposition simply, an immediate one, and as in other things the principle is simple, but this is not the same every where, but in weight it is a minor, in melody a demi-semi-quaver, and something else in another thing, thus in syllogism, "the one" is an immediate proposition, but in demonstration and science it is intuition. In syllogisms then, which demonstrate the being inherent, nothing falls beyond (the middle), but in negatives here, nothing falls external of that which ought to be inherent, as if A is not present with B through C. For if C is present with every B, but A with no C, and if, again, it should be requisite to show that A is with no C, we must assume the medium of A and C, and thus we must always proceed. If however it should be required to show that D is not with E, because C is with every D, but with no, or not with every, the medium will never fall external to E, and this is with what it need not be present. As to the third mode, it will never proceed external to that from which, nor which it is necessary to deny.

Chapter 24

As one demonstration is universal, but another particular, one also affirmative, but the other negative, it is questioned which is preferable, likewise also about what is called direct demonstration, and that which leads to the impossible. Let us first then consider the universal and the particular, and having explained this, speak of what is called direct demonstration, and that to the impossible.

Perhaps then to some considering the matter in this way, the particular may appear the better, for it that demonstration is preferable, by which we obtain better knowledge, for this is the excellence of demonstration, but we know each thing better when we know it per se, than when through something else, (as we know Coriscus is a musician, when we know that Coriscus is a musician rather than when we know that a man is a musician, and likewise in other things,) but the universal demonstrates because a thing is something else, not because it is that which it is, as that an isosceles triangle (has two right angles), not because it is isosceles, but because it is a triangle,) but the particular demonstrates because a thing is what it is, if then the demonstration per se is preferable, and the particular is such rather than the universal, particular demonstration would be the better. Besides, if the universal is nothing else than particulars, but demonstration produces opinion that this thing is something according to which it demonstrates, and that a certain nature of this kind is in things which subsist, (as of triangle besides particular (triangles), and of figure besides particular (figures), and of number besides particular (numbers)[)], but the demonstration about being is better than that about non-being, and that through which there is no deception than that through which there is, but universal demonstration is of this sort, (since men proceeding demonstrate as about the analogous, as that a thing which is of such a kind as to be neither line nor number, nor solid nor superficies, but something besides these, is analogous,) if then this is more universal, but is less conversant with being than particular, and produces false opinion, universal will be inferior to particular demonstration.

First then may we not remark that one of these arguments does not apply more to universal than to particular demonstration? For if the possession of angles equal to two right angles is inherent, not in respect of isosceles, but of triangle, whoever knows that it is isosceles knows less essentially than he who knows that it is triangle. In short, if not so far as it is triangle, he then shows it, there will not be demonstration, but if it is, whoever knows a thing so far as it is what it is, knows that thing more. If then triangle is of wider extension (than isosceles), and there is the same definition, and triangle is not equivocal, and the possession of two angles equal to two right angles is inherent in every triangle, triangle will have such angles, not so far as it is isosceles, but the isosceles will have them, so far as it is triangle. Hence he who knows the universal, knows more in regard to the being inherent than he who knows particularly, hence too the universal is better than the particular demonstration. Moreover if there is one certain definition, and no equivocation, the universal will not subsist less, but rather more than certain particulars, inasmuch as in the former there are things incorruptible, but particulars are more corruptible. Besides, there is no necessity that we should apprehend this (universal) to be something besides these (particulars), because it shows one thing, no more than in others which do not signify substance, but quality, or relation, or action, but if a person thinks thus, it is the hearer, and not demonstration, which is to blame.

Again, if demonstration is a syllogism, showing the cause and the why, the universal indeed is rather causal, for that with which any thing is essentially present, is itself a cause to itself, but the universal is the first, therefore the universal is cause. Wherefore the (universal) demonstration is better, since it rather partakes of the cause and the why, besides up to this we investigate the why, and we think that then we know it, when this is becoming, or is, not because something else (is), for thus there is the end and the last boundary. For example, on what account did he come? that he might receive money, but this that he might pay his debts, this that he might not act unjustly, and thus proceeding, when it is no longer on account of something else, nor for the sake of another thing, then we say that he came, and that it is, and that it becomes on account of this as the end, and that then we especially know why he came. If then the same occurs, as to all causes and inquiries into the why, but as to things which are so causes as that for the sake of which, we thus especially know, in other things also we then chiefly know, when this no longer subsists because another thing does. When therefore we know that the external angles are equal to four right angles, because it is isosceles, the inquiry yet remains, why because isosceles, because it is a triangle, and this because it is a rectilinear figure. But if it is this no longer on account of something else, then we pre-eminently know, then also universally, wherefore the universal is better. Again, by how much more things are according to the particular, do they fall into infinites, but the universal tends to the simple and the finite, so far indeed as they are infinite, they are not subjects of science, but so far as they are finite they may be known, wherefore so far as they are universal, are they more objects of scientific knowledge, than so far as they are particular. Universals however are more demonstrable, and of things more demonstrable is there pre-eminent demonstration, for relatives are at one and the same time more, whence the universal is better, since it is demonstration pre-eminently. Besides, that demonstration is preferable, according to which this and something else are known, to that, by which this alone is known, now he who has the universal knows also the particular, but the latter does not know the universal, wherefore even thus the universal will be more eligible. Again, as follows: it is possible rather to demonstrate the universal, because a person demonstrates through a medium which is nearer to the principle, but what is immediate is the nearest and this is the principle; if then that demonstration which is from the principle is more accurate than that which is not from the principle, the demonstration which is in a greater degree from the principle, is more accurate than that which is from it in a less degree. Now the more universal is of this kind, wherefore the universal will be the better, as if it were required to demonstrate A of D, and the media should be B C, but B the higher, wherefore the demonstration through this is more universal.

Some of the above arguments are logical, it is chiefly clear however that the universal is more excellent, because when of two propositions we have that which is the prior, we also in a certain degree know and possess in capacity that which is posterior; thus if a man knows that every triangle has angles equal to two right, he also in a certain respect knows in capacity that an isosceles triangle has angles equal to two right, even if he does not know that the isosceles is a triangle, but he who has this proposition by no means knows the universal, neither in capacity nor in energy. The universal proposition also is intuitively intelligible, but the particular ends in sense.

Chapter 25

That universal is better than particular demonstration, let so much be alleged, but that the affirmative is preferable to the negative, will be evident from this. Let that demonstration be better, cæteris paribus, which consists of fewer postulates, or hypotheses, or propositions. For if they are similarly known, quicker knowledge will be obtained through these, which is more eligible. The reason however of this proposition, that that which consists of fewer is better, universally is this; for if the media are similarly known, but things prior are more known, let the demonstration be through the media of B C D, that A is present with E, but through F G, that A is present with E. That A is present with D, and that A is present with E subsists similarly, but that A is with D, is prior and more known than that A is with E, for that is demonstrated through this, and that is more credible through which (a thing is demonstrated). Also the demonstration which is through fewer things is therefore better, cæteris paribus; both then are demonstrated through three terms, and two propositions, but the one assumes that something is, and the other, that something is and is not, hence through a greater number of things (the demonstration is made) so that it is the worse.

Moreover since it has been shown impossible for a syllogism to be produced with both propositions negative, but that one must of necessity be such (negative), and the other that a thing is present with, (that is affirmative,) we must in addition to this assume this, for it is necessary that affirmative (propositions) when the demonstration is increased, should become more, but it is impossible that the negatives should be more than one in every syllogism. For let A be present with nothing of those with which B is, but B be present with every C, if indeed, again, it should be necessary to increase both propositions, a middle must be introduced. Of A B then let the middle be D, but of B C let the middle be E, E then is evidently affirmative, but D is affirmative indeed of B, yet is placed negatively as regards A, since it is necessary that D should be present with every B, but A with no D; there is then one negative proposition, viz. A D. The same mode also subsists in other syllogisms, for the middle of affirmative terms is always affirmative in respect of both (extremes), but in the case of a negative (syllogism), the middle must be necessarily negative in respect to one of the two, so there is one proposition of this kind, but the others are affirmative. If then that is more known and credible through which a thing is demonstrated, but the negative is shown through the affirmative, and the latter not through the former, this, since it is prior, more known, and more credible, will be better. Again, since the principle of syllogism is an universal immediate proposition, but the universal proposition in an ostensive (demonstration) is affirmative, but in a negative is negative, and since the affirmative is prior to, and more known than, the negative, for negation is known through affirmation, and affirmation is prior, just as being is prior to not being, therefore the principle of affirmative is better than that of negative demonstration, but that which uses better principles is better. Moreover it partakes more of the nature of principle, since without affirmative there is no negative demonstration.

Chapter 26

Since affirmative is better than negative demonstration, it is evidently also better than that which leads to the impossible, it is necessary however to know what the difference between them is. Let A then be present with no B, but let B be with every C, wherefore it is necessary that A should be with no C, (the terms) then being thus assumed, the negative proposition proving that A is not present with C will be ostensive. The demonstration however to the impossible is as follows: if it is required to show that A is not present with B it must be assumed present, also that B is with C so that it will happen that A is with C. Let this however be known and acknowledged impossible, then it is impossible that A should be with B; if then B is acknowledged present with C, it is possible that A should be with B. The terms then indeed are similarly arranged, but it makes a difference which negative proposition is more known, viz. whether that A is not present with B, or that A is not present with C. When then the conclusion is more known that it is not, there is a demonstration to the impossible produced, but when that which is in the syllogism (is more known) the demonstration is ostensive. Naturally, however, that A is not present with B is prior to A is not present with C, for those things are prior to the conclusion, from which the conclusion (is collected), and that A is not with C is the conclusion, but that A is not with B is that from which the conclusion is derived. For neither if a certain thing happens to be subverted, is this the conclusion, but those (the premises) from which (the conclusion is derived). That indeed from which (it is inferred) is a syllogism, which may so subsist as either a whole to a part, or as a part to a whole, but the propositions A C and A B do not thus subsist with regard to each other. If then that demonstration which is from things more known and prior be superior, but both are credible from something not existing, yet the one from the prior, the other from what is posterior, negative demonstration will in short be better, than that to the impossible, so that as affirmative demonstration is better than this, it is also evidently better than that leading to the impossible.

Chapter 27

One science is more accurate than, and prior to, another, both the science that a thing is, and the same why it is, but not separately that it is, than the science of why it is, also that which is not of a subject than that which is of a subject, for instance, metic then harmonic science, and that which consists of fewer things than that which is from addition, as arithmetic than geometry. I mean by "from addition," as unity is a substance without position, but a point is substance with position, this is from addition.

Chapter 28

One science is that which is of one genus of those things which are composed of first (principles), and are the parts or affections of these per se; but a science is different from another, whose principles are neither from the same things, nor one from the other. A token of this is when any one arrives at things indemonstrable, for it is necessary that they should be in the same genus with those that are demonstrated; it is also a sign of this when things demonstrated through them are in the same genus and are cognate.

Chapter 29

There may possibly be many demonstrations of the same thing, not only when one assumes an un-continued medium from the same class, as if C D and F (were assumed) of A B, but also from another (series). Thus, let A be to be changed, D to be moved, B to be delighted, and again G to be tranquillized. It is true then to predicate D of B and A of D, for whoever is delighted is moved, and what is moved is changed: again, it is true to predicate A of G, and G of B, for every one who is delighted is tranquillized, and he who is tranquillized is changed. Wherefore there is a syllogism through different media, and not from the same class, yet not so that neither is predicated of neither medium, since it is necessary that both should be present with something which is the same. We must also consider in how many ways there may be a syllogism of the same thing through the other figures.

Chapter 30

There is no science through demonstration of that which is fortuitous, since the fortuitous is neither as necessary nor as for the most part, but that which is produced besides these, and demonstration is of one of these. For every syllogism is through premises, either necessary, or through those which are for the most part (true), and if indeed the propositions are necessary, the conclusion also is necessary; but if for the most part (true), the conclusion also is of the same character. Hence if the fortuitous is neither as for the most part nor necessary, there cannot be demonstration of it.

Chapter 31

Neither is it possible to have scientific knowledge through sensation, for although there is sensible perception of such a thing as this, and not of this particular thing, yet it is necessary to have a sensible perception of this particular thing, and some where and now. But it is impossible sensibly to perceive the universal and in all things, for it is not this particular thing, nor now, otherwise it would not be universal, since we call the universal that which is always and every where. Since then demonstrations are universal, but these cannot be perceived by sense, it is plain that neither can scientific be possessed through sense. In fact, it is clear, that even if we could perceive by sense that a triangle has angles equal to two right, we should require demonstration, and not, as some say, know this scientifically, for it is necessary sensibly to perceive the singular, but science is from the knowledge of the universal. Wherefore also if we were above the moon, and saw the earth opposite, we should not know the cause of an eclipse (of the moon). For we should perceive that it is eclipsed, but in short should not perceive why, since there would not be a sensible perception of the universal. Nevertheless, from observing this frequently to happen, by investigation of the universal, we should obtain demonstration, for the universal is manifest from many singulars, but is valuable, because it discloses the cause, wherefore the universal (knowledge) about such things, of which there is another cause, is more honourable than the senses and apprehension: about first principles however there is another reason.

It is clearly then impossible to possess scientific knowledge of any thing demonstrable by sensible perception, unless some one should affirm that sensible perception is this, to possess science through demonstration. There are indeed certain problems which are referred to the deficiency of our sensible perception, for some if we should see them we should not investigate, not as knowing from seeing, but as possessing the universal from seeing. For instance, if we saw glass perforated, and the light passed through it, it would be also manifest why it illuminates in consequence of our seeing separately in each, and at the same time perceiving that it is thus with all.

Chapter 32

That there should be the same principles of all syllogisms is impossible, first (this will be seen) by those who consider logically. For some syllogisms are true, others false, since it is possible to conclude the true from the false, yet this but rarely happens, for instance, if A is truly predicated of C, but the middle B is false, for neither is A present with B nor B with C. If however the media of these propositions are assumed, they will be false, because every false conclusion is from false principles, but the true from true principles, and the false and the true are different. Next, neither are the false (deduced) from the same (principles) with themselves, for they are false and contrary to each other, and cannot be simultaneous, for instance, it is impossible that justice should be injustice or timidity, that man should be a horse or an ox, or that the equal should be greater or less. From these positions indeed (we may prove it) thus, since neither are there the same principles of all the true (conclusions), for the principles of many are different in genus, and are not suitable, as units do not suit points, for the former have not position, but the latter have it. At least it is necessary to adapt (either) to media or from above or below, or to have some terms within but others without. Nor can there possibly be certain common principles from which all things may be demonstrated: I mean by common as to affirm or to deny every thing, for the genera of beings are different, and some are present with quantities, but others with qualities alone, with which there is demonstration through the common. Again, principles are not much fewer than conclusions, for the propositions are principles, but the propositions subsist when a term is either assumed or introduced. Moreover, conclusions are infinite, but terms finite; besides, some principles are from necessity, but others contingent.

To those therefore who thus consider, it will be impossible that there should be the same finite principles when the conclusions are infinite, but if any one should reason in some other way, for instance, that these are the principles of geometry, but these of reckoning, and these of medicine, what is this statement other than that there are principles of the sciences? but to say that there are the same principles because they are the same with themselves is ridiculous, for thus all things become the same. Still neither is to demonstrate any thing from all things to investigate whether there are the same principles of all, since this would be very silly. For neither does this happen in evident disciplines, nor is it possible in analysis, since immediate propositions are principles, and another conclusion arises, when an immediate proposition is assumed. If however any one should say that the first immediate propositions are the same principles, there is one in each genus, but if it is neither possible that any thing can be demonstrated as it ought to be from all (principles), nor that they should be so different, as that there should be different ones of each science, it remains that the principles of all are the same in genus, but that from different principles different sciences (are demonstrated). Now this is evidently impossible, for it has been shown that the principles are different in genus of those things which are generically different, for principles are two-fold, viz. from which and about which, those indeed from which are common, but those about which are peculiar, for instance, number and magnitude.

Chapter 33

The object of scientific knowledge and science (itself) differs from the object of opinion, and from opinion, because science is universal, and subsists through things necessary, and what is necessary cannot subsist otherwise than it does: some things however are true, and subsist, yet may possibly subsist otherwise. It is evident then that science is not conversant with these, (for else things which are capable of subsisting otherwise could not possibly subsist otherwise). Yet neither is intellect conversant with such, (for I call intellect the principle of science,) nor indemonstrable science, and this is the notion of an immediate proposition. But intellect, science, and opinion, and what is asserted through these, are true, wherefore it remains that opinion is conversant with the true or false, which yet may have a various subsistence, but this is the notion of an immediate and not necessary proposition. This also agrees with what appears, for both opinion is unstable, and its nature is of this kind, besides, no one thinks that he opines, but that he knows, when he thinks it impossible for a thing to subsist otherwise than it does, but when he thinks that it is indeed thus, yet that nothing hinders it being otherwise, then he thinks that he opines; opinion as it were being conversant with a thing of this kind, but science with what is necessary.

How then is it possible to opine and know the same thing, and why will opinion not be science, if a person admits that every thing which he knows he may opine? for both he who knows and he who opines will follow through media till they come to things immediate, so that if the former knows, he also who opines knows. For as it is possible to opine that a thing is, so likewise why it is, and this is the medium. Or if he so conceives things which cannot subsist otherwise, as if he had the definitions through which the demonstrations are framed, he will not opine, but know; but if that they are true, yet that these are not present with them essentially, and according to form, he will opine and not know truly both the that and the why, if indeed he should opine through things immediate; but if not through the immediate, he will only opine that they are. Still opinion and science are not altogether conversant with the same thing, but as both the true and the false opinion are in a manner about the same thing, thus also science and opinion are conversant with the same. For as some say that true and false opinion are of the same; absurd consequences follow both in other respects, and also that he who opines falsely does not opine. Now since the same thing is stated in several ways, in one way there may be, and in another there cannot be (a true and false opinion of the same). For to opine truly that the diameter of a square is commensurate with its side, is absurd, but because the diameter about which there are (contrary) opinions is the same thing, thus also they are of the same thing, but the essence of each according to the definition is not the same. In like manner also knowledge and opinion are conversant with the same thing, for the former is so conversant with animal as that it is impossible animal should not exist, but the latter so as that it may possibly not exist, as if the one should be conversant with that which is man essentially, but the other with man indeed, yet not with what is man essentially; for it is the same thing, that is, man, but not the same as to the manner.

From these then it is clearly impossible to opine and know the same thing at the same time, for otherwise at one and the same time a man might have a notion that the same thing could and could not subsist otherwise, which is impossible. In different (men) indeed each (of these) may be possible about the same thing, as we have said, but in the same (man) it is impossible even thus, since he would have a notion at the same time, for instance, that man is essentially animal, (for this it is to be impossible not to be an animal,) and is not essentially an animal, for this it is to be possible not to be an animal.

For the rest, how it is necessary to distinguish between discourse and intellect, and science and art, and prudence and wisdom, belongs rather partly to the physical, and partly to the ethical theory.

Chapter 34

Sagacity is a certain happy extempore conjecture of the middle term, as if a man perceiving that the moon always has that part lustrous which is towards the sun, should straightway understand why this occurs, viz. because it is illuminated by the sun, or seeing a man talking to a rich person, should know that it is in order to borrow money of him, or that persons are friends, because they are enemies of the same man; for he who perceives the extremes knows all the middle causes. Let to be lustrous in the part toward the sun be A, to be illuminated by the sun B, the moon C. Wherefore B to be illuminated by the sun is present with the moon C, but A to be lustrous in the part turned towards that by which it is illuminated is present with B, hence also A is present with C through B.