Chapter 9 Since however it is evident that we cannot demonstrate each thing except from its own principles, if what is to be demonstrated is inherent in a subject so far as the subject is that (which it is) to have a scientific knowledge of that thing is not this, it it should be demonstrated from true, indemonstrable, and immediate (propositions). For we may so demonstrate possibly, as Bryso did, the quadrature of the circle, since such reasonings prove through something common, that which is inherent in another thing, hence these arguments are adapted to other things not of the same genus. Wherefore that thing would not be scientifically known, as far as it is such, but from accident, for otherwise the demonstration would not be adapted also to another genus.
We know however each thing not accidentally when we know it according to that, after which it is inherent from principles which are those of that thing, so far as it is that thing; as that a thing has angles equal to two right angles, in which the thing spoken of is essentially inherent from the principles of this thing. Hence if that is essentially inherent in what it is inherent, it is necessary that the middle should be in the same affinity, but if not, yet it will be as harmonics are proved through an arithmetical principle. Such things however are demonstrated after a similar manner, yet they differ, for that they are, is part of another science, (for the subject genus is another,) but why they are, is a province of a superior science, of which they are the essential qualities. Hence from these things also it is apparent that we cannot demonstrate each thing simply, but from its proper principles, and the principles of these have something common.
If then this is evident, it is also clear that it is impossible to demonstrate the proper principles of each thing, for they will be the principles of all things, and the science of them the mistress of all (sciences): for the man has more scientific knowledge who knows from superior causes, since he knows from prior things when he knows not from effects, but from causes. So that if he knows more, he knows also most, and if that be science, it is also more, and most of all such. Demonstration however is not suitable to another genus, except as we have said, geometrical to mechanical or optical, and arithmetical to harmonical demonstrations.
Nevertheless it is difficult to know whether a man possesses knowledge or not, since it is hard to ascertain if we know from the principles of each thing or not, which indeed constitutes knowledge. We think however that we know, if we have got a syllogism from certain primary truths, but it is not so, since it is necessary that they should be of a kindred nature with the primary.
Chapter 10 I call those principles in each genus, the existence of which it is impossible to demonstrate. What then first things, and such as result from these signify, is assumed, but as to principles, we must assume that they are, but demonstrate the rest, as what unity is, or what the straight and a triangle are; it is necessary however to assume that unity and magnitude exist, but to demonstrate the other things.
Of those which are employed in demonstrative sciences, some are peculiar to each science, but others are common, and common according to analogy, since each is useful, so far as it is in the genus under science. The peculiar indeed are such as, that a line is a thing of this kind, and that the straight is, but the common are, as that if equals be taken from equals the remainders are equal. Now each of these is sufficient, so far as it is in the genus, for (a geometrician) will effect the same, though he should not assume of all, but in magnitudes alone, and the arithmetician in respect of numbers (alone).
Proper principles, again, are those which are assumed to be, and about which science considers whatever are inherent per se, as arithmetic assumes unities, and geometry points and lines, for they assume that these are, and that they are this particular thing. But the essential properties of these, what each signifies, they assume, as arithmetic, what the odd is, or the even, or a square, or a cube; and geometry, what is not proportionate, or what is to be broken, or to incline; but that they are, they demonstrate through things common, and from those which have been demonstrated. So also astronomy, for all demonstrative science is conversant with three things, those which are laid down as existing, and these are the genus, (the essential properties of which the science considers,) and common things called axioms, from which as primaries they demonstrate; and thirdly, the affections, the signification of each of which the demonstrator assumes. There is nothing however to prevent certain sciences overlooking some of these, as if the genus is not supposed to be, if it be manifest that it exists, (for it is not similarly manifest that number is, as that the cold and hot are,) and if (the science) does not assume what the affections signify, if they are evident, as neither does it assume what things common signify, (as what it is) to take away equals from equals, because it is known; nevertheless these things are naturally three, viz. that about which demonstration is employed, the things demonstrated, and the principles from which they are.
Neither however hypothesis nor postulate is that which it is necessary should exist per se, and be necessarily seen, for demonstration does not belong to external speech, but to what is in the soul, since neither does syllogism. For it is always possible to object to external discourse, but not always to internal. Whatever things then, being demonstrable, a man assumes without demonstration, these, if he assumes what appear probable to the learner, he supposes, and this is not an hypothesis simply, but with reference to the learner alone; but if, there being no inherent opinion, or when a contrary is inherent, the demonstrator assumes, he requires the same thing to be granted to him. And in this hypothesis and postulate differ, for postulate is any thing sub-contrary to the opinion of the learner, which though demonstrable a man assumes, and uses without demonstration.
Definitions then are not hypotheses, (for they are not asserted to be or not to be,) but hypotheses are in propositions. Now it is only necessary that definitions should be understood, but this is not hypothesis, except some one should say that the verb to hear is hypothesis. But they are hypotheses, from the existence of which, in that they are, the conclusion is produced. Neither does the geometrician suppose falsities, as some say, who assert, that it is not right to use a false (principle), but that the geometrician does so, when he calls a line a foot long when it is not so, or the line which he describes a straight line when it is not straight. The geometrician indeed concludes nothing from the lines being so and so, as he has said, but concludes those, which are manifested through these (symbols). Moreover postulate and every hypothesis are either as a whole or as in a part, but definitions are neither of these.
Chapter 11 That there should then be forms, or one certain thing besides the many, is not necessary, to the existence of demonstration, but it is necessary truly to predicate one thing of the many, for there will not be the universal unless this be so, and if there be not an universal, there will not be a medium, so that neither will there be a demonstration. It is essential then that there should be one and the same thing, which is not equivocal in respect of many: no demonstration however assumes that it is impossible to affirm and deny the same thing at one and the same time, unless it is requisite also thus to demonstrate the conclusion. It is demonstrated however by assuming the first to be true of the middle, and that it is not true to deny it, but it makes no difference whether we assume the middle to be or not to be, and in a similar manner also in respect of the third. For if that be granted in respect of which it is true to predicate man, even if (some one should think that man is) not man, (the conclusion) will be true, if only it is said that man is an animal, and not that he is not an animal, for it will be true to say that Callias, even if he be not Callias, yet is still an animal, but not that which is not an animal. The cause however is, that the first is not only predicated of the middle, but also of something else, in consequence of its being common to many, so that neither if the middle be that thing itself, or not that thing, does it make any difference in respect to the conclusion. But the demonstration which leads to the impossible, assumes that of every thing affirmation or negation is true, and these it does not always (assume) universally, but so far as is sufficient, and it is sufficient (which is assumed) in respect of the genus. I mean by the genus, as the genus about which a person introduces demonstrations, as I have observed before.
All sciences communicate with each other according to common (principles), and I mean by common those which men use as demonstrating from these, but not those about which they demonstrate, nor that which they demonstrate, and dialectic is (common) to all (sciences). If also any one endeavours to demonstrate universally common (principles), as that of every thing it is true to affirm or deny, or that equals remain from equals, or others of this kind. Dialectic however does not belong to certain things thus definite, nor to one particular genus; for it would not interrogate, since it is impossible for the demonstrator to interrogate, because the same thing is not proved from opposites: this however has been shown in the treatment of syllogism.
Chapter 12 Chapter 12 If syllogistic interrogation is the same as a proposition of contradiction, but there are propositions in each science, from which the syllogism which belongs to each consists, there will be a certain scientific interrogation, from which the syllogism, which is appropriate to each science, is drawn. It is clear, then, that not every interrogation would be geometrical, or medical, and so of the rest, but from what any thing is demonstrated about which geometry is conversant, or which are demonstrated from the same principles as geometry, as optics, and in like manner with other sciences. These also must be discussed from geometrical principles and conclusions, but the discussion of principles is not to be carried on by the geometrician so far as he is such; likewise with other sciences. Neither is every one who possesses science to be interrogated with every question, nor is every question about each to be answered, but those which are defined about the science. It is evident then that he does well, who disputes with a geometrician thus, so far as he is such, if he demonstrate any thing from these principles, but if not, he will not do well. Again, it is clear that neither does he confute the geometrician except by accident, so that there cannot be a discussion of geometry by those who are ignorant of geometry, since the bad reasoner will escape detection, and it is the same with other sciences.
Since there are geometrical interrogations, are there also those which are ungeometrical? and in each science are those ignorant questions which are of a certain quality geometrical? whether also is a syllogism, from ignorance, a syllogism composed from opposites or a paralogism, but according to geometry, or from another art, as a musical interrogation is ungeometrical, about geometry but to imagine that parallel lines meet is in a certain respect geometrical, and after another manner ungeometrical? For this is two-fold, in the same way as what is without rhythm; and the one is ungeometrical because it possesses not (what is geometrical), as what is without rhythm; but the other because it possesses it wrongly—and this ignorance which is from such principles, is contrary. In mathematics however there is not in like manner a paralogism, because the middle is always two-fold, for (one thing) is predicated of every individual of this, and this again of another every, but the predicate is not called universal; those, nevertheless, it is possible, we may see by common perception, but in argument they escape us. Is then every circle a figure? If any one should delineate it, it is clear. But what, are verses a circle? They are evidently not so.
Still it is improper to object to it, if it be an inductive proposition; for as neither is that a proposition which is not in respect of many things, (since it will not be in all, but syllogism is from universals,) neither, it appears clear, is that an objection, for propositions and objections are the same, as the objection which one adduces, may become either a demonstrative or a dialectic proposition.
It occurs that some argue contrary to syllogism, from assuming the consequences of both (extremes), as Cæneus does, that fire is in a multiple proportion, because, as he says, both fire and this proportion are rapidly generated. But thus there is no syllogism, though there will be, if the multiple is consequent to the most rapid proportion, and the most rapid proportion to fire in motion. Sometimes it does not happen that a conclusion is made from the assumptions, and sometimes it happens, but is not perceived: if however it were impossible to demonstrate the true from the false, it would be easy to resolve, for (the terms) would be necessarily converted. Thus let A exist, and this existing, these things also exist[,] the existence of which I know, as B, from these[,] then I will demonstrate that that exists. What pertain however to mathematics, are rather converted, because they take nothing accidental, (and in this they differ from dialectical subjects,) but definitions.
Yet they are increased, not through media, but through additional assumption, as A of B, this of C, this again of D, and so on to infinity. Also transversely, as A both of C and of E, as there is a number so great or even infinite, which is A, an odd number so great B, and an odd number C. A then is (true) of C, and the even is a number so great D, the even number is E, wherefore A is (true) of E.
Chapter 13 Now there is a difference between knowing that a thing is, and why it is, first in the same science, and in this in two ways, the one, if the syllogism is not formed through things immediate, (since the primary cause is not assumed, but the science of the why has respect to the first cause,) but the other if it is through things immediate indeed, yet not through the cause, but through that which is more known of the things, which reciprocate. Now nothing prevents that which is not a cause being sometimes more known amongst things which are mutually predicated, so that demonstration shall accrue through this, as that the planets are near, because they do not twinkle. Let C be the planets, B not to twinkle, A to be near, B therefore is truly predicated of C, since the planets do not twinkle, A also of B, for what does not twinkle is near, but this may be assumed by induction or by sense. It is necessary then that A should be present with C, so that it is demonstrated that the planets are near. This syllogism then is not of the "why," but of the "that" (a thing is), for the planets are not near because they do not twinkle, but they do not twinkle because they are near. It happens indeed that the one may be proved through the other, and the demonstration will be of the "why," as let C be the planets, B to be near, A not to twinkle, B then is present with C, so that A "not to twinkle" will be with C. It is also a syllogism of the "why," for the first cause was assumed. Again, as they show the moon to be spherical through increments (of light), for if what is thus increased be spherical, and the moon is increased, it is evident that the moon is spherical, thus then a syllogism of the "that" is produced, but if the middle is placed contrarily, there is a syllogism of the "why," for it is not spherical on account of the increments, but from being spherical she receives such increments: let the moon be C, spherical B, increase A. Where again the media do not reciprocate, and what is not the cause is more known, the "that" is indeed demonstrated, but not the "why;" further, where the middle is placed externally, for in these the demonstration is of the "that", and not of the "why," as the cause is not assigned. For example, why does not a wall breathe? because it is not an animal, for if this was the cause of its not breathing, it would be necessary that animal should be the cause of its breathing, since if negation is the cause of a thing not being, affirmation is the cause of its being, thus if the disproportion of hot and cold is the cause of not being well, the proportion of these is the cause of being well. Likewise if affirmation is the cause of being, negation is the cause of not being, but in things which have been thus explained, what has been stated does not occur, for not every animal respires. A syllogism of such a cause is nevertheless produced in the middle figure, for example, let A be animal, B to respire, C a wall, A then is present with every B, (for whatever respires is animal,) but with no C, so that neither is B present with any C, wherefore a wall does not respire. Such causes however resemble things spoken hyperbolically, and this is, when we turn aside to speak of the middle, which is more widely extended, as for instance, that saying of Anacharsis, that amongst the Scythians there are no pipers, since neither are there any vines.
As to the same science then, and the position of the media, these are the differences between a syllogism of, that a thing is, and of why it is, but in another respect the why differs from the that, because each is beheld in a different science. Now such are those things which so subsist with reference to each other, as that the one is under the other, such as optics with reference to geometry, mechanics to the measurement of solids, harmonics to arithmetic, and celestial phenomena to astronomy. Some of these sciences are almost synonymous, as astronomy is both the mathematical and the nautical; and harmony is both mathematical and that which belongs to the ear. For here to know that a thing is, is the province of those who exercise the sense, but to know why it is, belongs to mathematicians, since these possess the demonstrations of causes, and often are ignorant of the that, as they who contemplating universals, frequently are ignorant of singulars from want of observation. But these are such as being essentially something else use forms, for mathematics are conversant with forms, since they do not regard one certain subject, for though the geometrical are of a certain subject, yet not so far as they are geometrical are they in a subject. As optics also to geometry, so is some other science related to optics, as for example, the science about the rainbow, for to know that it is, appertains to the natural philosopher, but why it is, to the optician either simply or mathematically. Many sciences also which are not arranged under each other subsist thus, for example, medicine with regard to geometry, for to know that circular wounds heal more slowly is the province of the physician, but why (they do so) of the geometrician.
Chapter 14 Of the figures, the first is especially adapted to science, for both the mathematical sciences carry out their demonstrations by this, as arithmetic, geometry, optics, and nearly, so to speak, whatsoever sciences investigate the "why," since either entirely or for the most part, and in most sciences, the syllogism of the why is through this figure. Wherefore also, on this account, it will be especially adapted to science, for it is the highest property of knowledge to contemplate the "why;" in the next place, it is possible through this figure alone to investigate the science of what a thing is; for in the middle figure, there is no affirmative syllogism, but the science of what a thing is belongs to affirmation, and in the last figure, there is an affirmative, but not an universal; but the what a thing is belongs to universals, for man is not a biped animal in a certain respect. Moreover this has no need of those, but they are condensed and enlarged through this, till we arrive at things immediate: it is evident, then, that the first figure is in the highest degree adapted to scientific knowledge.
Chapter 15 Chapter 15 As it happened that A was present with B individually, so also it may happen not to be present, and I mean by being present with, or not, individually, that there is no medium between them, for thus the being present with or not, will not be according to something else. When then either A or B is in a certain whole, or when both are, it is impossible that A should not be primarily present with B. For let A be in the whole of C, if then B is not in the whole of C, (for it is possible that A may be in a certain whole, but that B may not be in this,) there will be a syllogism that A is not present with B, for if C is present with every A, but with no B[,] A will be present with no B. In like manner also, if B is in a certain whole, as for instance, in D, for D is with every B, but A with no D, so that A will be present with no B by a syllogism. In the same way it can be shown if both also are in a certain whole, but that it is possible that B may not be in the whole in which A is, or again A in which B is, is evident from those co-ordinations which do not interchange. For if none of those, which are in the class A C D, is predicated of any of those in B E F, but A is in the whole of H, which is co-arranged with it, it is evident that B will not be in H, for otherwise the co-ordinates would intermingle.
Likewise also if B is in a certain whole, but if neither is in any whole, and A is not present with B, it is necessary that it should not be present individually, for if there shall be a certain middle, one of them must necessarily be in a certain whole, for there will be a syllogism either in the first, or in the middle figure. If then it is in the first, B will be in a certain whole, (for it is necessary that the proposition in regard to this should be affirmative,) but if in the middle figure either of them may be (in the whole), for the negative being joined to both, there is a syllogism, but there will not be when both the propositions are negative.
It is manifestly possible then, that one thing may not be individually present with another, also when, and how this may happen, we have shown.
Chapter 16 The ignorance which is denominated not according to negation, but according to disposition, is a deception produced through syllogism, and this happens in two ways, in those things which are primarily present, or not present; for it happens either when one simply apprehends the being present, or not being present, or when he obtains this opinion through syllogism: of simple opinion, then, the deception is simple, but of that which is through syllogism, it is manifold. For let A not be present with any B individually, if then A is concluded to be present with B, assuming C as the middle, a person will be deceived through syllogism. Hence it is possible that both propositions may be false, but it is also possible that only one may be so, for if neither A is present with any C, nor C with any B, but each proposition is taken contrary, both will be false. But it may be that C so subsists with reference to A and B, as neither to be under A nor universally (present) with B, for it is impossible that B should be in a certain whole, since it was said that A is not primarily present with it; but A need not be universally present with all beings, so that both propositions are false. Nevertheless, we may assume one proposition as true, not either of them casually, but the proposition A C, for the proposition C B will be always false, because B is in none; but A C may be (true), for instance, if A is present individually, both with C and B, for when the same thing is primarily predicated of many things, neither will be predicated of neither; it makes no difference however if it (A) be not individually present with it (C).
The deception then of being present, is by these and in this way only, (for there was not a syllogism of being present in another figure,) but the deception of not being present with, is in the first and middle figure. Let us first then declare in how many ways it occurs in the first, and under what propositional circumstances. It may then happen when both propositions are false, e. g. if A is present individually with C and B, for if A should be assumed present with no C, but C with every B, the propositions will be false. But (deception) is possible, when one proposition is false, and either of them casually; for it is possible that A C may be true, but C B false; A C true, because A is not present with all beings, but C B false, because it is impossible that C should be with B, with nothing of which A is present; for otherwise the proposition A C will be no longer true, at the same time, if both are true, the conclusion also will be true. But it is also possible that C B may be true, when the other proposition is false, as if B is in C and in A, for one if must necessarily be under the other, so that if A should be assumed present with no C, the proposition will be false. It is clear then, that when one proposition is false, and also when both are, the syllogism will be false.
In the middle figure, however, it is not possible that both propositions should be wholly false, for when A is present with every B, it will be impossible to assume any thing, which is present with every individual of the one, but with no individual of the other; but we must so assume the propositions that the (middle) may be present with one (extreme), and not be present with the other, if indeed there is to be a syllogism. If then, when they are thus assumed, they are false, it is clear that, when taken contrarily, they will subsist vice versâ, but this is impossible. Still there is nothing to prevent each being partly false, as if C is with A, and with a certain B; for if it should be assumed present with every A, but with no B, both propositions indeed would be false, yet not wholly, but partially. The same will occur when the negative is placed vice versâ. But it is possible that one proposition, and either of them, may be false, for what is present with every A, will be also with B, if then C is assumed present with the whole of A, but not present with the whole of B, C A will be true, but the proposition C B false. Again, what is present with no B, will not be present with every A; for if with A, it would also be with B, but it was not present; if then C should be assumed present with the whole of A, but with no B, the proposition C B will be true, but the other false. The same will happen if the negative is transposed, for what is in no A, will neither be in any B; if then C is assumed not present with the whole of A, but present with the whole of B, the proposition A C will be true, but the other false. Again, also, it is false to assume that what is present with every B, is with no A; for it is necessary, if it is with every B, that it should be also with a certain A; if then C is assumed present with every B, but with no A, the proposition C B will be indeed true, but C A false. Hence, it is evident that when both propositions are false, and when one only is so, there will be a syllogism deceptive in individuals.
Chapter 17 In those which are not individually present, or which are not present, when a syllogism of the the false is produced through an appropriate medium, both propositions cannot be false, but only the major. But I mean by an appropriate medium, that through which there is a syllogism of contradiction. For let A be with B through the medium of C, since then we must take C B as affirmative, if there is to be a syllogism, it is clear that this will be always true, for it is not converted. A C, on the other hand, will be false, for when this is converted, a contrary syllogism arises. So also if the middle is assumed from another affinity, as for instance, if D is in the whole of A, and is predicated of every B, for the proposition D B must necessarily remain, but the other proposition must be converted, so that the one (the minor) will be always true, but the other (the major) always false. Deception also of this kind is almost the same as that which is through an appropriate medium, but if the syllogism should not be through an appropriate medium, when indeed the middle is under A, but is present with no B, it is necessary that both propositions should be false. For the propositions must be assumed contrary to the way in which they subsist, if a syllogism is to be formed, for when they are thus assumed both are false, as if A is with the whole of D, but D present with no B, for when these are converted, there will be a syllogism, and both propositions will be false. When however the