Abstract
Theory of demonstrative science: all teaching arises from pre-existing knowledge, acquired through syllogism or induction; it distinguishes knowledge of the particular through experience from universal knowledge through science, and grounds the indemonstrable first principles on which every demonstration rests.
Traduzione di Octavius Freire Owen · Wikisource, \"The Organon\" (Octavius Freire Owen translation, London: Henry G. Bohn, 1853), 1853 · pubblico dominio
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 apparent 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
The subjects of investigation are equal in number to the things which we scientifically know; but we investigate four things; that a thing is, why it is, if it is, what it is. For when we inquire whether it is this, or that, having reference to a number (as whether the sun is eclipsed or not) we investigate the that, and a sign of this is that when we have found that it is eclipsed we desist from our inquiries, and if we knew from the first that it is eclipsed, we do not inquire whether it is so. But when we know the that, we investigate the why, for instance, when we know that there is an eclipse, and there is an earthquake, we inquire why there is an eclipse, and an earthquake. These things indeed we investigate thus, but some after another manner, for instance, if there is, or is not, a centaur or a God. I say if there is or is not, simply, and not if it is white or not. When however we know that a thing is, we inquire what it is, for instance, what God, or what man is.
The things then which we investigate, and which having discovered we know, are such and so many, but when we inquire the that or if a thing is simply, then we inquire whether there is a medium of it or not, but when knowing, either that it is, or if is, either in part or simply, we again investigate why it is, or what it is, then we inquire what the middle is. But I mean by the that if it is in a part and simply, in a part indeed (as) is the moon eclipsed or increased? for in such things we inquire if a thing is or is not; but simply (as) if there is a moon or not, or if night is or not. In all these inquiries it occurs that we investigate either if there is a middle or what the middle is, for the cause is the middle, and this is investigated in all things. Is there then an eclipse? is there a certain cause or not? after this, when we know that there is, we inquire what this is. For the cause of a thing not being this or that, but simply substance, or not simply, but something of those which subsist per se, or accidentally, is the middle. I mean by what is simply (substance) the subject, as the moon, or the earth, or the sun, or a triangle, but by a certain thing, (as) an eclipse, equality, inequality if it is in the middle or not. For in all these it is evident that what a thing is and why it is are the same; what is an eclipse? a privation of light from the moon through the interposition of the earth. Why is there an eclipse, or why is the moon eclipsed? because its light fails through the interposition of the earth. What is symphony? a ratio of numbers in sharp and flat. Why does the sharp accord with the flat? because the sharp and flat have the ratio of numbers. Do then the sharp and flat accord? is there then a ratio of them in numbers? assuming that there is, what then is the ratio?
That the inquiry is of the middle those things prove whose middle falls within the cognizance of the senses, since we inquire when we have not a sensible perception, as of an eclipse, whether it is or not. But if we were above the moon we should not inquire neither if, nor why, but it would be immediately evident, as from sensible perception we should also obtain knowledge of the universal; for sense (would show us) that the earth is now opposed, for it would be evident that there is now an eclipse, and from this there would arise the universal.
As therefore we say, the knowledge of the what is the same as the knowledge of the why, and this is either simply, and not somewhat of things inherent, for it is of things inherent, as that there are two right angles or that it is greater or less.
That all investigations then are an inquiry of the middle is evident, but let us show how what a thing is, is demonstrated, and what is the method of training up a thing to its principles, also what a definition is, and of what subjects doubting first about these. But let the commencement of the future (doubts) be that which is most appropriate to the following discussion, since perhaps a man might doubt whether it is possible to know the same thing, and according to the same by definition and demonstration, or whether it is impossible? For definition seems to be of what a thing is, but every thing (which signifies) what a thing is, is universal and affirmative, but some syllogisms are negative, others not universal; for instance, all those in the second figure are negative, but those in the third not universal. Next, neither is there definition of all affirmatives in the first figure, as that every triangle has angles equal to two right angles; the reason of this is, because to know scientifically that which is demonstrable, is to possess demonstration, so that if there is demonstration in regard to things of this kind, there can evidently not be also definition of them, for a person might know by definition without demonstration, since nothing prevents the possession of it at one and the same time. A sufficient evidence of this is also derived from induction, for we have never known by definition, any of those which are inherent per se nor which are accidents; besides, if definition be a certain indication of substance, it is evident that such things are not substances.
Clearly then, there is not definition of every thing of which there is also demonstration, but what, is there then demonstration of every thing of which there is definition or not? there is one reason and the same also of this. For of one thing, so far as it is one, there is one science, so that if to know that which is demonstrable be to possess demonstration, an impossibility would happen, for he who possesses definition would know scientifically without demonstration. Besides, the principles of demonstration are definitions, of which it has been shown before, there will not be demonstrations, since either principles will be demonstrable, and principles of principles, and this would proceed to infinity, or the first (principles) will be indemonstrable definitions.
Yet if there are not of every thing and the same, may there not be definition and demonstration of a certain thing and the same? or is it impossible? since there is not demonstration of what there is definition. For definition is of what a thing is, and of substance, but all demonstrations appear to suppose and assume what a thing is, as mathematics, what is unity and what an odd number, and the rest in like manner. Moreover every demonstration shows something of somewhat, as that it is, or that it is not, but in definition one thing is not predicated of another, as neither animal of biped, nor this of animal, nor figure of superficies, for superficies is not figure, nor figure superficies. Again, it is one thing to show what a thing is, but another to show that it is, definition then shows what a thing is, but demonstration that this thing, either is or is not of this. Of a different thing indeed there is a different demonstration, unless it should be as a certain part of the whole. I say this because the isosceles has been shown (to have angles equal) to two right, if every triangle has been shown (to have them), for that is a part, but this a whole: these however, that a thing is, and what it is, do not thus subsist in reference to each other, since the one is not a part of the other.
Evidently then there is neither entirely demonstration of what there is definition, nor entirely definition of what there is demonstration; hence in short it is impossible to have both of the same thing, so that it is also evident that definition and demonstration will neither be the same, nor the one contained in the other, otherwise their subjects would subsist similarly.
Let then so far these things be matters of doubt, but as to what a thing is whether is there, or is there not, a syllogism and a demonstration of it, as the present discussion supposed? for a syllogism shows something in respect of somewhat through a medium, but the (definition) what a thing is, is both peculiar and is predicated in respect of what it is. Now it is necessary that these should reciprocate: for if A is the property of C, it is evidently also that of B, and that of C, so that all reciprocate with each other. Nevertheless, if A is present with every B in respect of what it is, and universally B is predicated of every C in respect of what it is, it is also necessary that A should be predicated of C in the question what it is. Still if some one should assume without this reduplication, it will not be necessary that A should be predicated of C in the question what a thing is, though A should be predicated of B in the same question, but not of those of which B is predicated in this question. Now both these will signify what a thing (C) is, wherefore B will also be the definition of C, hence if both signify what a thing is, and what the very nature of it is, there will be the very nature of a thing prior in the middle term. Universally also, if it is possible to show what man is, let C be man, but A what he is, whether biped animal, or any thing else; in order then that a conclusion should be drawn, A must necessarily be predicated of every B, and of this there will be another middle definition, so that this also will be a definition of a man, wherefore a person assumes what he ought to show, for B also is the definition of a man.
We must however consider it in two propositions, and in first and immediate (principles), for what is stated becomes thus especially evident: they therefore who show what the soul is, or what man or any thing else is, by conversion, beg the question, as if a man should assume the soul to be that which is the cause to itself of life, and that this is number moving itself, he must necessarily so assume as a postulate that the soul is number moving itself, as that it is the same thing. For it does not follow if A is consequent to B, and this to C, that A will therefore be the definition of the essence of C, but it will be only possible to say that this is true, nor if A is that which is predicated essentially of every B. For the very nature of animal is predicated of the very nature of man, since it is true that whatever exists as man, exists as animal, (just as every man is animal,) yet not so, as for both to be one thing. If then a person does not assume this, he will not conclude that A is the very nature and substance of C, but if he thus assume it, he will assume prior to the conclusion that B is the definition of the essence of C. Therefore there has been no demonstration, for he has made a “petitio principii.”
Nevertheless, neither does the method through divisions infer a conclusion, as we observed in the analysis about figures, since it is never necessary that when these things exist, that should exist, as neither does he demonstrate who forms an induction. For the conclusion ought not to inquire nor to exist from being granted, but it necessarily is, when they exist, although the respondent does not acknowledge it. Is man (for instance) animal or inanimate, if he has assumed him to be an animal, it has not been syllogistically concluded. Again, every animal is either pedestrian or aquatic, he assumes it pedestrian, and that man is that whole animal pedestrian, is not necessary from what is said, but he assumes also this. It signifies nothing however, whether he does this in respect of many things or few, since it is the same thing; to those therefore who thus proceed, and in what is capable of syllogistic conclusion, this use is unsyllogistic. For what prevents the whole of this being true of man, yet without enunciating what a thing is, or the very nature of it? Again, what prevents something being added to, or taken away from, or exceeding the essence?
Negligence then happens about these things, but we may avoid it by assuming all things (as granted) in respect of what a thing is, and the first being made a postulate by arranging the order in division, omitting nothing. This however is requisite for it is necessary that there should be an individual, yet nevertheless there is not a syllogism, but if so it indicates after another manner. And this is not at all absurd, since neither perhaps does he who makes an induction demonstrate, though at the same time he renders something manifest, but he who selects definition from division does not state a syllogism. For as in conclusions without media, if a man state that from such things being granted, this particular thing necessarily exists, it is possible to inquire why, thus also is it definitions by division. What is man? A mortal animal, pedestrian, biped, without wings. Why? according to each addition, for he will state and show by division as he thinks that every one is either mortal or immortal. The whole however of such a sentence is not definition, wherefore though it should be demonstrated by division, yet the definition does not become a syllogism.
Is it however possible to demonstrate what a thing is according to substance, but from hypothesis assuming that the very nature of a thing in the question what it is, is something of its peculiar principles, and that these alone indicate its substance, and that the whole is its peculiarity? for this is its essence. Or again, has a person assumed the very nature of a thing in this also? for we must necessarily demonstrate through a middle term. Moreover, as in a syllogism, we do not assume what is to have been syllogisticaliy concluded, (for the proposition is either a whole or a part, from which the syllogism consists,) thus neither ought the very nature of a thing to be in a syllogism, but this should be separate from the things which are laid down, and in reply to him who questions whether this has been syllogistically concluded or not, we must answer that it is, for this was the syllogism. And to him who asserts that the very nature of the thing was not concluded, we must reply that it was, for the very nature of the thing was laid down by us, so that it is necessary that without the definition of syllogism, or of the definition itself, something should be syllogistically inferred.
Also, if a person should demonstrate from hypothesis, for instance, if to be divisible is the essence of evil; but of a contrary, the essence is contrary of as many things as possess a contrary; but good is contrary to evil, and the indivisible to the divisible, then the essence of good is to be indivisible. For here he proves assuming the very nature of a thing, and he assumes it in order to demonstrate what is its very nature: let however something be different, since in demonstrations is assumed that this is predicated of that, yet not that very thing, nor that of which there is the same definition, and which reciprocates. To both however there is the same doubt against him who demonstrates by division, and against the syllogism thus formed, why man will be an animal biped pedestrian, but not an animal and pedestrian, for from the things assumed, there is no necessity that there should be one predicate, but just as the same man may be both a musician and a grammarian.
How then will he who defines show the essence of a thing, or what it is? for neither as demonstrating from things which are granted will he render it evident that when they exist, it is necessary that something else should be, for demonstration is this, nor as forming an induction by singulars which are manifest, that every thing thus subsists, from nothing subsisting otherwise; since he does not show what a thing is, but that it is, or is not. What remaining method is there? for he will not indicate by sense nor by the finger.
Moreover how will he show what it is? for it is necessary that he also who knows what man is, or any thing else, should also know that he is, for no one knows with respect to non-being that it is, but what the definition or the name signifies, as when I say “tragelaphos,” it is impossible to know what tragelaphos is. Moreover, if he should show what a thing is, and that it is, how will he show this in the same sentence? for both definition and also demonstration manifest one certain thing, but what man is is one thing, and the essence of man is another.
We next say that it is necessary to show by demonstration every thing, that it is, except it be substance, but to be, is not substance to any thing, for being is not the genus. There will then be demonstration that it is, and this the sciences now effect. For what a triangle means, the geometrician assumes, but that it is, he demonstrates. What then will he who defines what it is, prove? that it is a triangle? he then who knows what it is by definition, will not know if it is, but this is impossible.
Evidently then those who define according to the present methods of definition, do not demonstrate that a thing is, for although those lines be equal which are drawn from the middle, yet why is it the thing defined? and why is this a circle? for we might say that there is the same definition of brass. For neither do definitions demonstrate that it is possible for that to be which is asserted, nor that that thing is, of which they say there are definitions, but it is always possible to say why.
If then he who defines shows either what a thing is or what the name signifies, except there is, by no means (an explanation) of what a thing is, definition will be a sentence signifying the same thing as a name, but this is absurd. For in the first place there would be a definition of non-essences and of non-entities, since it is possible even for non-entities to have a signification. Again, all sentences will be definitions, for we might give a name to any sentence, so that we might all discuss in definitions, and the Iliad would be a definition. Besides, no science would demonstrate that this name signifies this thing, neither therefore do definitions manifest this.
From these things therefore it appears that neither definition nor syllogism are the same thing, nor are syllogism and definition of the same thing, moreover that definition neither demonstrates nor shows any thing, and that we can know what a thing is neither by definition nor by demonstration.
Moreover we must consider which of these things is well, and which is not well asserted, also what definition is, and whether there is in a certain way or by no means a demonstration and definition of what a thing is. Now since it is the same thing as we have said to know what a thing is, and to know the cause wherefore it is, and the reason of this is, that there is a certain cause, and this is either the same or another, and if it is another, it is either demonstrable or indemonstrable; if then it is another, and is capable of demonstration, it is necessary that the cause should be a medium, and should be demonstrated in the first figure, for that which is demonstrated is both universal and affirmative. Now one method will be that which has been now investigated, viz. to demonstrate what a thing is through something else, for of those things which are predicated in respect of what a thing is, it is necessary that the medium should be what it is, and a property in respect of properties, wherefore of two essential natures of the same thing, it will demonstrate the one, but not the other.
That this method then is not demonstration, has been shown before, but it is a logical syllogism of what a thing is, still let us show in what method this is possible, discussing it again from the beginning. For as we investigate why a thing is, when we know that it is, but sometimes those become evident at the same time, but it is not possible to know why it is, prior to knowing that it is, it is clear that in like manner the very nature of a thing or what it is, cannot be known, without knowing that it is, since it is impossible to know what a thing is, when ignorant if it is. We sometimes indeed know if it is, accidentally, knowing sometimes something belonging to the thing, as thunder we know, because it is a certain sound of the clouds, and an eclipse, because it is a certain privation of light, and a man, because it is a certain animal, and soul, because it moves itself. As regards then whatever we know accidentally that they are, it is by no means necessary that we should possess any thing by which to know what they are, for neither do we (really) know that they are, and to inquire what a thing is, when we do not know that it is, is to inquire about nothing. In those things however of which we know something, it is easy (to inquire) what they are; hence as we know that a thing is, so also are we disposed to know what it is, now of those things, of whose essential nature we know something, let this be first an example, an eclipse A, the moon C, the opposition of the earth B. To inquire then whether there is an eclipse or not, is to inquire whether B is or not, but this does not at all differ from the inquiry if there is a reason of it, and if this is, we say that that also is. Or we (inquire) of which contradiction there is a reason, whether of possessing, or of not possessing, two right angles, but when we have discovered, we know at the same time, that it is, and why it is, if it is inferred through media; but if it is not so inferred, we know the that, but not the why. Let C be the moon, A an eclipse, not to be able to produce a shadow when the moon is full and nothing is seen interposed between us, B, if then B, that is, not to be able to produce a shadow when there is nothing between us, be present with C, and A, to be eclipsed, present with this, that there is an eclipse, is indeed evident, but why is not yet so, and that there is an eclipse, we indeed know, but what it is we do not know. Yet as it is clear that A is with C, (to inquire) why it is, is to investigate what B is, whether it is the opposition (of the earth), or the turn of the moon, or the extinction of light, but this is the definition of the other extreme, as in those (examples) of A, since an eclipse is the interposition of the earth. What is thunder? the extinction of fire in a cloud: why does it thunder? because fire is extinguished in a cloud. Let C be a cloud, A thunder, B the extinction of fire, hence B is present with C, that is, with the cloud, for fire is extinguished in it, but A, sound, is present with this, and B is the definition of A, the first extreme; if there be again another medium of this it will be from the remaining definitions.
We have shown therefore thus, how what a thing is, is assumed, and becomes known, wherefore there is neither syllogism nor demonstration of what a thing is, still it will become evident through syllogism, and through demonstration; and hence without demonstration it is neither possible to know what a thing is, of which there is another cause, nor is there demonstration of it, as we have already observed in the doubts.
Of some things indeed there is a certain other cause, but of others there is not, so that it is plain that some of them are immediate, and principles, whose existence and what they are, we must suppose, or make manifest after another manner, which indeed the arithmetician does, for he both supposes what unity is, and that it is. Of those however which have a medium, and of whose essence there is another cause, it is possible, as we have said, to produce a manifestation through demonstration, yet not by demonstrating what they are.
Since definition is said to be a sentence (explanatory) of what a thing is, it is evident that one definition will be of what a name signifies, or another nominal sentence, as what a thing signifies, which is so far as it is a triangle, which when we know that it is, we inquire why it is. Still it is difficult thus to assume things, the existence of which we do not know, and the cause of this difficulty has been explained before, because neither do we know whether it is or is not, except accidentally. One sentence is indeed in two ways, the one by conjunction, as the Iliad, but the other from signifying one thing of one, not accidentally.
The above-named then is one definition of a definition, but the other definition is a sentence showing why a thing is, so that the former signifies, but does not demonstrate, but the latter will evidently be, as it were, a demonstration of what a thing is, differing from demonstration in the position (of the terms). For there is a difference between saying, why does it thunder? and what is thunder? for thus a person will answer, because fire is extinguished in the clouds; but what is thunder? the sound of fire extinguished in the clouds; hence there is the same sentence spoken in another manner, and in the one way there is a continued demonstration, but in the other there is a finition. Moreover the definition of thunder is, a sound in the clouds, but this is the conclusion of the demonstration of what it is; now the definition of things immediate is, the indemonstrable thesis of essence.
One definition then is, an indemonstrable sentence (significative) of essence, but another is a syllogism of essence, differing from demonstration in case, and a third is the conclusion of the demonstration of what a thing is. Wherefore, from what we have said, it is evident how there is, and how there is not, a demonstration of what a thing is, also of what things there is, and of what there is not; moreover in how many ways definition is enunciated, and how it demonstrates the essence of a thing, and how it does not; also of what things there is, and of what there is not, definition; yet more, how it subsists with respect to demonstration, and how it may, and how it may not be, of the same thing.
Since we think that we scientifically know, when we are cognizant of the cause, but causes are four, one indeed as to the essence of a thing, another that which from certain things existing, this necessarily exists, a third that which first moves something, and a fourth on account of which a thing (exists); all these are demonstrated through a medium. For the one that this existing it is necessary that that should be, is not from one proposition being assumed, but from two at the least, but this is, when they have one medium; this one therefore being assumed, there is necessarily a conclusion, which is evidently thus: Why is the angle a right one in a semicircle, or from the existence of what, is it right? Let then A be a right angle, B the half of two right angles, and the angle in the semicircle C. Hence B is the cause why A the right angle is inherent in C, i. e. in the angle of a semicircle; for this angle is equal to A, but C is equal to B, for it is the half of two right angles; B then being the half of two right angles, A is inherent in C, and this was for the angle in a semicircle to be a right angle. This however is the same as the explanation of the essence of a thing, because definition signifies this, but the cause of the essence of a thing has been shown to be the middle. Why was there a Median war with the Athenians? What was the cause of waging war with the Athenians? Because the latter with the Eretrians attacked Sardis; this was the first cause of the movement. Let war then be A, first made the attack B, the Athenians C, B then is present with C, i.e. to have first made the attack is present with the Athenians, but A is also with B, for they make war with the aggressors, A then is present with B, i. e. to wage war is present with the aggressors, but this, B, is present with the Athenians, for they were the aggressors. Wherefore the middle is the cause here, and that which first moves; but of those things, whose cause is for the sake of something, as, why does he walk? that he may be well; why is a house built? that furniture may be preserved; the one is for the sake of health, but the other for the sake of preservation. Still there is no difference between why is it necessary to walk after supper, and for the sake of what is it necessary? but let walking after supper be C, the food not to rise B, to be well A. Let then walking after supper be the cause why the food does not rise to the mouth of the stomach, and let this be healthy; for B, that is, for the food not to rise, appears to be present with walking, C, and with this A, salubrious. What then is the cause that A, which is that for the sake of which (the final cause), is present with C? B (is the cause), that is, the food not rising, this however is as it were, the definition of it, for A will be thus explained. Why is B present with C? because to be thus affected is to be well: we must nevertheless change the sentences, and thus the several points will be more clear. The generations here indeed, and in causes respecting motion, subsist vice versâ, for there it is necessary that the middle should be first generated, but here C, which is the last, and that for the sake of which is generated the last.
Possibly indeed the same thing; may be for the sake of something, and from necessity; for instance, why does light pass through a lantern? for necessarily that which consists of smaller particles passes through larger pores, if light is produced by transit, also (it does so) on account of something, that we may not fall. If then it possibly may be, is it also possible to be generated? as if it thunders, fire being extinguished, it is necessary that it should crash and rumble, and, as the Pythagoreans say, for the sake of threatening, that those in Tartarus may be terrified. Now there are many things of this kind, especially in those which are constituted and consist from nature, for nature produces one thing for the sake of something, and another from necessity; but necessity is two-fold, one according to nature and impulse, another with violence, contrary to impulse; thus a stone is borne from necessity both upward and downward, yet not from the same necessity. In things however which are from reason, some never subsist from chance, as a house, or a statue, nor from necessity, but for the sake of something, whilst others are also from fortune, as health and safety. Especially in those which are capable of a various subsistence, as when the generation of them is not from fortune, so that there is a good end, on account of which it takes place, and either by nature or by art: from fortune however nothing is produced for the sake of something.
The cause of things which are, is the same also as that of things which are generated, which have been generated, and which will be, for the middle is the cause, except that being is the cause to be, what is generated, to those which are generated, what has been, to those which have been, and what will be to those that will be. Thus why was there an eclipse? because the earth was interposed, but an eclipse is generated, because an interposition of the earth is generated, but there will be, because the earth will be, and there is, because it is interposed. What is ice? Let it be assumed to be congealed water; let water be C, congealed A, the middle cause B, a perfect defect of heat; B then is present with C, but with this A, viz. to be congealed, but ice is generated, when B is generated, it was so, when the latter was so, and it will be, when the latter will be.
Hence that which is thus a cause, and that of which it is the cause, are generated at one and the same time, when they are generated; are simultaneously when they are; and in like manner, in respect to the having been, and the will be, generated. In the case of things which are not simultaneous, are there in a continued time, as it seems to us, different causes of different things? for instance, is another thing having been generated the cause of this thing having been generated, and another thing which will be, the cause that this will be, and of this being, something which was generated before? the syllogism however is from what was afterwards generated. And the principle of these are those things which have been generated, wherefore the case is the same as to things which are generated. From the prior indeed there is no (syllogism), as that this thing was afterwards generated, because that thing was generated, it is the same also in regard to the future. For whether the time be indefinite or definite, it will not result that because that thing was truly said to have been generated, this which is posterior is truly said to have been generated, since in the interval it will be false to say this, when already another thing has been produced. The same reasoning also happens to what will be, nor because that was produced, will this be, as the middle must be generated at the same time; of things that have been that which has been, of the future the future, of what are produced that which is produced, of things which are that which is, but of what was generated, and of that which will be, the middle cannot possibly be produced at one and the same time. Moreover neither can the interval be indefinite, nor definite, since it will be false to assert it in the interval; but we must consider what is connected with it, so that after the having been generated, to be generated may exist in things. Or is it evident that what is generated is not connected with what was generated? for the past does not cohere with what was generated, since they are terms and individuals. As then neither points are mutually connected, those things which have been produced are not so, for both are indivisible; nor for the same reason does that which is, cohere with that which has been generated, for that which is generated is divisible, but that which has been is indivisible. As a line then is to a point, so is that which is to that which was generated, for infinite things which have been, are inherent in that which is; we must however enunciate these matters more clearly in the universal discussions about motion.
Concerning then the manner in which, when there is a successive generation, the middle cause subsists, let so much be assumed, for in these also it is necessary that the middle and the first should be immediate, thus A was generated because C was so, but C was after, A before. The principle indeed is C, because it is nearer to the now, which is the principle of time, but C was generated if D was, hence from D having been, it is necessary that A should have been. The cause however is C, for from D having been, it is necessary that C should have been generated, but C having been, A must of necessity have been produced before. When however we thus assume the middle, will (the process) at any time stop at the immediate, or on account of the infinity will a medium always intervene? for, as we have stated, what has been generated is not connected with what has been; nevertheless we must commence at least from the immediate and from the first now. Likewise with regard to the “will be,” for if it is true to say that D will be, it is necessary that, prior to this, it should be true to say that A will be, the cause however of this is C, for if D will be, prior to it C will be, but if C will be, prior to it A will be. Likewise also in these the division is infinite, for things which will be, are not mutually coherent, but an immediate principle must also be assumed in these. It is thus in the case of works, if a house has been built, stones must necessarily have been cut, and formed; and why this? because the foundation must of necessity have been laid, if the house was built, but if the foundation was laid, stones must necessarily have been prepared before. Again, if there shall be a house, in like manner there will be stones prior to this, still the demonstration is in like manner through a medium, for the foundation will have a prior subsistence.
Notwithstanding, since we see in things which are, that there is a certain generation in a circle, this happens when the middle and the extremes follow each other, for in these there is a reciprocation; this however was shown in the first treatise, viz. that the conclusions are converted; but the case of being in a circle is thus. In works it appears after this manner, when the earth has been moistened, vapour is necessarily produced, from the production of this, there is a cloud, from this last, water, and from the presence of this, the earth is necessarily moistened, this however was the (cause) at first, so that it has come round in a circle, for any one of these existing, another is, and if that is, another, and from this, the first.
There are some things which are generated universally, (for always, and in every thing, they either thus subsist, or are generated,) but others not always, but for the most part; thus not every vigorous man has a beard, but this is generally the case, now or such things it is necessary that the medium also should be for the most part; for if A is universally predicated of B, and this of C universally, it is necessary that A also should be predicated always, and of every C, (for the universal is that which is present with every individual and always,) but it was supposed to be for the most part, wherefore it is necessary that the medium also, B, should be for the most part: hence of those which are for the most part, the principles are immediate, as many as thus subsist for the most part, or are generated.
We have before shown how what a thing is, is attributed to definitions, and in what way there is or is not a demonstration or definition of it, how therefore it is necessary to investigate things which are predicated in respect to what a thing is, let us now discuss.
Of those then, which are always present with each individual, some have a wider extension, yet are not beyond the genus. I mean those have a wider extension, as many as are present with each individual universally, yet also with another thing, thus there is something which is present with every triad, and also with that which is not a triad, as being is present with a triad, but also to that which is not number. Nevertheless the odd is present with every triad, and is of wider extension, for it is with five, but it is not beyond the genus, for the five is number, and nothing out of number is odd. Now such things we must take so far until so many are first assumed, each of which is or wider extension, but all of them together are not of greater extent, for it is necessary that this should be the substance of a thing. For example, number, the odd is present with every triad, the first in both ways, both as not being measured by number and as not being composed of numbers. Now therefore the triad is this, viz. the first odd number, and the first in this way, for each of these is present, the one with all odd numbers, but the last also with the dual, yet all of them (together) with none (but the triad). Since however we have shown above, that those things which are predicated in respect of what a thing is are necessary, but universals are necessary, but what are thus assumed of a triangle, or any other thing, are assumed in respect to what a thing is, thus from necessity the triad will be these things. That this however is its essence appears from this, since it is necessary, unless the very nature of a triad were not this, that this should be a certain genus, either denominated or anonymous. It will be therefore of wider extension than to be with a triad alone, for let the genus be supposed of that kind as to be more widely extended according to power, if then it is present with nothing else than individual triads, this will be the essence of the triad. Let this also be supposed, that an ultimate predication like this of individuals is the essence of each thing, wherefore in like manner, when any thing is thus demonstrated, it will be the essence of that thing.
Nevertheless it is right when any one is conversant with a certain whole, to divide the genus into the individuals which are first in species, for instance, number into triad and dual, then to endeavour thus to assume the definitions of these, as of a straight line, of a circle, and of a right angle; afterwards assuming what the genus is, for instance, whether it is quantity or quality, he should investigate the peculiar passions through common first (principles.) For those which happen to the composites from individuals will be evident from the definitions, because definition and that which is simple are the principles of all things, and accidents are essentially present with simple things alone, but with others according to them. The divisions indeed by differences are useful for our progression in this way, but how indeed they demonstrate we have shown before, but they would thus be useful only for syllogizing what a thing is, and indeed they may appear to do nothing, but to assume every thing immediately, just as if any one assumed from the beginning without division. It makes some difference, however, whether what is predicated be so, prior or posterior, as for instance, whether we call animal, mild biped, or biped, animal mild, for if every thing consists of two, and one certain thing is animal mild, and again from this, and the difference, man or any thing else which is one, consists, we must necessarily make a postulate by division. Besides, thus only is it possible to leave out nothing in the definition, since when the first genus is assumed, if a person takes a certain inferior division, every thing will not fall into this; for instance, not every animal has entire or divided wings, but every animal which is winged, for this is the difference of it, but the first difference of animal is that into which every animal falls. Likewise in regard to each of the rest, both of those genera which are external to animal, and of those which are contained under it, as of bird, is that into which every bird falls, and of fish that into which every fish falls. Thus proceeding we may know that nothing is omitted, but otherwise we must omit something, and not know it. It is not at all necessary that he who defines and divides, should know all things that subsist, though some say it is impossible to know the differences of each thing without knowing each; but it is impossible to know each thing without differences, for that from which this does not differ, is the same with this, but that from which it differs is something else than this. In the first place then this is false, for it is not something else according to every difference, since there are many differences in things which are the same in species, yet not according to substance, nor per se. Next, when any one assumes opposites, and difference, and that every thing falls into this or that, and assumes also that the question is in one part of the two, and knows this, it is of no consequence whether he knows or does not those other things of which the differences are predicated. For it is evident that thus proceeding, if he should arrive at those of which there is no longer a difference, he will obtain the definition of the substance; but that every thing will fall into division, if there should be opposites of which there is no medium, is not a postulate, since every thing must necessarily be in one of them, if indeed it will be the difference of it.
In order to frame definition by divisions, we must attend to three things, viz. to assume the things predicated in respect of what a thing is; to arrange these, which shall be first or second; and that these are all. Now the first of these arises from our being able as syllogistically to collect accident, that it is inherent, so to construct through genus. There will however be a proper arrangement if what is first be assumed, and this will be if that be taken which is consequent to all, but all not consequent to it; for there must be something of this kind. This then being taken, there must now be the same method in the things inferior, since the second will be that which is first of the rest, and the third that which is first of the following, for what is superior being taken away, whatever succeeds will be the first of the others; there is also similar reasoning in the other cases. Still that all these should be, is clear from assuming what is first in the division, that every animal is either this or that, but this is inherent; and again the difference of this whole but that of the last there is no longer any difference, or immediately with the last difference this does not differ in species from the whole: for it is clear that neither more (than is necessary) is added, for every thing has been assumed in reference to what a thing is, nor is any thing deficient, for it would be either genus or difference. Both the first then is genus, and this assumed together with differences, but all the differences are contained, for there is no longer any posterior difference. Otherwise the last would differ in species, this however has been shown not to differ.
Still we must investigate, looking to those which are similar and do not differ, first (considering) what that is which is the same in all these, then again in other things which are in the same genus with them, and which are among themselves the same in species, but different from those. Yet when in these that is assumed which all have the same, and in others similarly, we must consider in the things assumed whether it is the same, until we arrive at one reason, for this will be the definition of the thing. Yet if we do not arrive at one, but at two or more, it is evident that the question will not be one, but many for instance, I mean if we should inquire what magnanimity is, we must consider in the cases of certain magnanimous persons, whom we know what one thing they all possess, so far as they are such. Thus if Alcibiades is magnanimous, or Achilles, or Ajax, what one thing have they all? intolerance of insult, for one of them fought, another sulked, another slew himself. Again, in other instances, as in that of Lysander or Socrates. If then (it is common to these) to behave in the same manner, in prosperity and adversity, taking these two, I consider what indifference with regard to fortune, and what impatience under insult possess in common; if they have nothing there will be two species of magnanimity.
Every definition is nevertheless universal, for the physician does not prescribe what is wholesome for a certain eye, but defines what is fit for every eye, or for the species. The singular however is easier to define than the universal, wherefore we must pass from singulars to universals, for equivocations lie more concealed in universals, than in things without a difference. But as in demonstrations the power of syllogizing must necessarily be inherent, so also perspicuity must be in definitions, and there will be this, if through things which are singularly enunciated, what is in each genus be separately defined; as with the similar, not every similar, but that which is in colours and in figures, and the sharp that which is in voice, and so to proceed to what is common, taking care that equivocation does not occur. But if it is not right to use metaphors in disputation, we must clearly not define by metaphors, nor by those things which are spoken by metaphor, otherwise it will be necessary to use metaphors in disputation.
Now that we may have problems, we must select sections and divisions, and thus select, the common genus of all being supposed, as for example, if animals were the subjects of consideration, (we must first consider,) what kind of things are present with every animal. When these have been taken, we must again see what kind of things are consequent to every first individual of the rest, thus if this is a bird, what things follow every bird, and so always that which is nearest, for we shall evidently now be able to say why things are present, which are consequent to those under what is common, as why they are present with man or horse. Let then animal be A, B things consequent to every animal, C D E certain animals, why then B is present with D is evident, for it is present through A: in a similar manner with the rest, and in others there is always the same reasoning.
Now then we speak according to presented common names, but we must not only consider in these, but also assume if any thing else should be seen to be common, afterwards consider to what things this is consequent, and the quality of the things consequent to this, as those consequent to having horns are the possession of a rough muscular lining to the stomach, and the not having teeth in both jaws. Moreover to what things the possession of horns is consequent, for it will be evident why what has been mentioned is present with them, for it will be so in consequence of their possessing horns.
There is yet another mode of selection by analogy, since it is impossible to assume one and the same thing, which it is necessary to call sepium, spine, and bone, there are also things consequent to these, as if there were one certain nature of this kind.
Some problems are the same from having the same medium, for instance, because all things are an antiperistasis, but of these some are the same in genus, which have differences from belonging to other things, or from subsisting differently, e. g. why is there an echo, or why is there a reflection, and why a rainbow? for all these are the same problem in genus, (for all are reflection,) but they differ in species. Other problems differ from the medium being contained under another medium, as why does the Nile have a greater flow during the fall of the month? because the fall of the month is more winterly: but why is the fall more winterly? because the moon fails, for thus do these subsist towards each other.
Some one may perhaps doubt concerning cause and that of which it is the cause, whether when the effect is inherent, the cause also is inherent, as if the leaves fall from a tree, or there is an eclipse, will there also be the cause of the eclipse, or of the fall of the leaves? As if the cause of this, is the having broad leaves, but of an eclipse the interposition of the earth, for if this be not so, something else will be the cause of these, and if the cause is present, at the same time the effect will be, thus if the earth be interposed, there is an eclipse, or if a tree have broad leaves, it sheds them. But if this be so, they would be simultaneous, and demonstrated through each other, for let the leaves to fall be A, the having broad leaves B, and a vine C, if then A is present with B, (for whatever has broad leaves sheds them,) but B is present with C, for every vine has broad leaves, A is present with C, and every vine sheds its leaves, but the cause is B, the middle. We may also show that the vine has broad leaves, from its shedding them, for if D be what has broad leaves, E to shed the leaf, F a vine, E then is present with F, (for every vine sheds its leaf,) but D with E, (for every thing which sheds its leaf, has broad leaves,) every vine then has broad leaves, the cause is, its shedding them. Nevertheless if they cannot be the cause of each other, (since cause is prior to that of which it is the cause,) the cause of an eclipse indeed is the interposition of the earth, but an eclipse is not the cause of the earth interposing. If then the demonstration by cause (shows) why a thing is, but that which is not through cause, that it is, one knows indeed that the earth is interposed, but why it is, he does not know. Yet that an eclipse is not the cause of the interposition, but this of an eclipse, is plain, since in the definition of an eclipse, the interposition of the earth is inherent, so that evidently that is known through this, but not this through that.
Or may there be many causes of one thing? for if the same thing may be predicated of many primary, let A be present with B a first, and with C another first, and these with D E, A then will be present with D E, but the cause why it is with D will be B, and C the cause why it is with E, hence from the existence of the cause there is necessarily the existence of the thing, but when the thing exists, it is not necessary that every cause should exist, still some cause indeed, yet not every cause. Or if the problem is always universal, is the cause also a certain whole, and that of which it is the cause universal? as to shed the leaf is present definitely with a certain whole, though there should be species of it, and with these universally, i. e. either with plants or with such plants. Hence in these, the medium and that of which it is the cause must be equal, and reciprocate, for instance, why do the trees shed their leaves? if indeed through the concretion of moisture, whether the tree casts its leaf, there must of necessity be concretion, or whether there is concretion not in any thing indiscriminately, but in a tree, the latter must necessarily shed its leaf.
Whether however may there not be possibly the same cause of the same thing in all things, but a different one, or is this impossible? or shall we say it cannot happen, if it is demonstrated per se and not by a sign or accident? for the middle is the definition of the extreme, but if it is not thus, (shall we say that) it is possible? We may however consider that of which and to which it is the cause by accident still they do not appear to be problems, but if not, the medium will subsist similarly, if indeed they are equivocal, the medium will be equivocal, if however as in genus the medium will be similar. For instance, why is there alternate proportion? for there is a different cause in lines, and in numbers, and the same (medium) so far as they are lines, is different, but so far as it has an increase of the same kind, it is the same, the like also occurs in all things. There is indeed a different cause in a different subject, why colour is similar to colour, and figure to figure, for the similar in these is equivocal, for here perhaps it is to have the sides analogous, and the angles equal, but in colours it consists in there being one sense (of their perception) or something else of the kind. Things however analogically the same, will have also the same medium by analogy, and this is so from cause, and that of which, and to which it is the cause following each other; but by assuming each singly, that of which it is the cause is more widely extended, as for the external angles to be equal to four, is or wider extension than triangle or square, but equal in all, for whatever have external angles equal to four right, will also have the medium similarly. The medium however is the definition of the first extreme, wherefore all sciences are produced by definition, thus to shed the leaf, is at the same time consequent to the vine, and exceeds, and to the fig tree, and exceeds, yet does not exceed all (plants), but is equal to them. If then you take the first middle it is the definition of shedding the leaf, for the first will be the middle of one of them, because all are such, next the middle of this is, that sap is congealed, or something else of the sort, but what is it to shed the leaf? it is for the sap to be congealed, at the junction of the seed.
In figures, to those who investigate the consequence of the cause, and of what it is the cause, we may explain the matter thus: let A be present with every B, and B with every D, but more extensively, B then will be universal to D, I call that universal which does not reciprocate, but that the first universal, with which each singular does not reciprocate, but all together reciprocate, and are of similar extension. B then is the cause why A is present with D, wherefore it is necessary that A should be more widely extended than B, for if not, why will this be rather the cause than that? If then A is present with all those of E, all those will be some one thing different from B, for if not, how will it be possible to say that A is present with every thing with which E is, but E not with every thing with which A is? for why will there not be a certain cause as there is why A is present with all D? wherefore will all those of E be one thing? We must consider this, and let there be C, hence there may be many causes of the same of the same thing, but not to the same in species, for instance, the cause why quadrupeds are long-lived, is their not having bile, but why birds live long, their being of a dry complexion, or something else: if however they do not arrive immediately at an individual, and there is not one medium only, but many, the causes also are many.
Which of the media is the cause to singulars, whether that which belongs to the first universal, or that to the singular? Evidently the nearest to the singular to which it is cause. For this is the cause why the first, under the universal, is inherent, C is the cause that B is inherent in D, hence C is the cause why A is inherent in D, but B is the cause why it is in C, yet to this itself is the cause.
Concerning syllogism then and demonstration, what either of them is, and how it is produced, is clear, and at the same time about demonstrative science, for it is the same: but about principles, how they become known, and what is the habit which recognises them, is manifest hence to those who have previously doubted it.
That it is then impossible to have scientific knowledge through demonstration, without a knowledge of first immediate principles, has been elucidated before, still some one may doubt the knowledge of immediate principles, both whether it is the same or not the same, also whether there is a science of each or not, or a science of one, but a different kind (of science) of another, and whether non-inherent habits are ingenerated, or when inherent are latent. If then, indeed, we possess them, it is absurd, for it happens that it (the principle) escapes those who have a more accurate knowledge than demonstration, but if, not having them before, we acquire them, how can we know and learn without pre-existent knowledge? for this is impossible, as we said also in the case of demonstration. It is evident then, that they can neither be possessed, nor ingenerated in the ignorant, and in those who have no habit, wherefore it is necessary to possess a certain power, yet not such an one as shall be more excellent according to accuracy than these. Now this appears inherent in all animals, for they have an innate power, which they call sensible perception, but sense being inherent in some animals, a permanency of the sensible object is engendered, but in others it is not engendered. Those, therefore, wherein the sensible object does not remain, either altogether or about those things which do not remain, such have no knowledge without sensible perception, but others when they perceive, retain one certain thing in the soul. Now since there are many of this kind, a certain difference exists, so that with some, reason is produced from the permanency of such things, but in others it is not. From sense, therefore, as we say, memory is produced, but from repeated remembrance of the same thing, we get experience, for many remembrances in number constitute one experience. From experience, however, or from every universal being at rest in the soul, that one besides the many, which in all of them is one and the same, the principle of art and science arises, if indeed it is conversant with generation, of art, but if with being, of science. Neither, therefore, are definite habits inherent, nor are they produced from other habits more known, but from sensible perception, as when a flight occurs in battle, if one soldier makes a stand, another stands, and then another, until the fight is restored. But the soul has such a state of being, as enables it to suffer this, what, however, we have before said, but not clearly, let us again explain. When one thing without difference abides, there is (then) first, universal in the soul, (for the singular indeed is perceived by sense, but sense is of the universal, as of man, but not of the man Callias,) again, in these it stops, till individuals and universals stop, as such a kind of animal, until animal, and in this again (it stops) after a similar manner. It is manifest then that primary things become necessarily known to us by induction, for thus sensible perception produces the universal. But since, of those habits which are about intellect, by which we ascertain truth, some are always true, but others admit the false, as opinion, and reasoning, but science, and intellect, are always true, and no other kind of knowledge, except intellect, is more accurate than science, but the principles of demonstrations are more known, and all science is connected with reason, there could not be a science of principles: but since nothing can be more true than science except intellect, intellect will belong to principles, and to those who consider from these it is evident also, that as demonstration is not the principle of demonstration, so neither is science the principle of science. If then we have no other true genus (of habit) besides science, intellect will be the principle of science: it will also be the principle (of the knowledge of the principle, but all this subsists similarly with respect to every thing.