Works › Posterior Analytics, Books I–II
Volume 2 · pp. 61–72
Treatise II, Chapter XVI Continuation. On Passing From Genus Into Genus
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et geometria: alia enim subjecta et aliae passiones sunt arithmeticae et geometriae: et ideo non est vel non contingit arithmeticam demonstrationem, quae perfectam numeri passionem concludit inesse numero, convenire in accidentia sive passiones magnitudinis continuae, quae considerat geometer per se et proprie, nisi magnitudines sint numeri. Et hoc dicimus ne aliquis calumniator calumniari possit hoc quod dictum est. Si enim magnitudines sunt numeri, cum subjectum quod est magnitudo, efficitur sub subjecto quod est numerus, et tunc ita efficiuntur unius generis sicut superius et inferius: et tunc ex causis et subjectis et passionibus superioris inferiori appropriatis per principia superioris probantur ea quae sunt in inferiori scientia sicut magnitudines numeri, prout in decimo Euclidis linea rationalis numeratur et mensuratur numero per digitum iteratum in ipsa, vel manum, vel palmam, vel pedem, vel cubitum, vel passum, et numeratur quantitate ipsius: cum enim ipsa linea numerus est numeratus, tunc per causas et principia numeri probantur multae passiones de tali linea. Et hoc modo arithmetica descendit in geometriam: quia subjectum geometriae quoad hoc efficitur sub subjecto arithmeticae. Qualiter autem hoc saepius contingit in quibusdam scientiis, posterius in proprio loco dicetur.
Sed arithmetica demonstratio secundum se sumpta semper habet genus proprium, circa quod fit sua ex propriis demonstratio. Et similiter de aliis: quia aliae scientiae similiter habent sua propria genera, ex quibus in ipsis scientiis propriae fiunt demonstrationes. Quare autem sive propter quid simpliciter in demonstratione necesse idem esse genus quod sit proprium principium generationis eorum quae sunt in scientia: aut necesse est sic esse genus, quod unius scientiae subjectum sit sub subjecto alterius, ut dictum est de magnitudine et numero, si debeat esse propria et vera demonstratio, ut patet per praedicta. Et si demonstratio unius scientiae secundum suas causas debeat descendere in genus alterius scientiae, aut scientia demonstrationis ferri de genere in genus aliter quam dictum est, manifestum est impossibile esse.
Et hujus causa est: quia in demonstratione potissima ultima, hoc est, extrema, necesse est esse ex eodem genere: quia aliter non primum medio et medium postremo per se inesset: si enim talia non sunt sive non insunt per se, sequitur quod accidentia erunt quoad hoc quod accidentaliter insunt. Et propter hoc quod non descendit demonstratio de genere in genus, nisi sicut dictum est, propter hoc geometriae quae est specialis scientia, non est monstrare, quod contrariorum eadem scientia sive disciplina: quae est scientiae communioris et universalioris, quia est philosophiae primae. Sed nec est geometriae ostendere id quod est alterius scientiae particularis, sicut quod duo cubi sunt unus cubus, hoc est, quod cubus numerus ductus in cubum numerum, facit duos cubos: quod demonstratio specialis scientiae est quae est arithmetica. Neque universaliter loquendo alteri scientiae convenit demonstrare quod alterius est scientiae: eo quod nec adhuc sunt subjecta propria, nec passiones, nec media, quae sunt propriae causae: sed aut, hoc est solum quaecumque habent sic se ad invicem, sicut dictum est, quod alterum, hoc est, alterius subjectum sit sub altero, hoc est, alterius subjecto, sicut specialibus sub generaliori, sicut perspectiva quae est de linea visibili, quae radius est, se habet ad geometriam quae est de linea simpliciter: et sicut se habet consonativa quae est de cantu consonante, quae dicitur harmonica sive musica, ad arithmeticam: arithmetica enim proportiones numerorum demonstrat: musica autem determinat proportiones numerorum in cantu. In talibus enim quando subjectum est sub subjecto, superior scientia circa inferiorem, per causas quae in ipsa communes sunt, appropriatas inferiori, multas passiones de inferiori subjecto demonstrat.
and geometry. For the subjects and the properties of arithmetic and geometry are different; and therefore there is not, or it does not happen, that an arithmetical demonstration, which concludes a perfect property of number to be present in number, belongs to the accidents or properties of continuous magnitude which the geometer considers per se and properly, unless magnitudes are numbers. And we say this lest some cavilling critic be able to calumniate what has been said. For if magnitudes are numbers, since the subject which is magnitude is brought under the subject which is number, then they are made of one genus in the way superior and inferior are; and then from the causes and subjects and properties of the superior, appropriated to the inferior through the principles of the superior, those things which are in the inferior science are proved, as magnitudes of number are proved when, in the tenth book of Euclid, a rational line is numbered and measured by number through a digit repeated in it, or a hand, or palm, or foot, or cubit, or pace, and is numbered by its quantity. For when the line itself is a numbered number, then through the causes and principles of number many properties are proved about such a line. And in this mode arithmetic descends into geometry, because the subject of geometry in this respect is brought under the subject of arithmetic. But how this happens more often in certain sciences will be said later in its proper place.
But arithmetic demonstration, taken according to itself, always has its proper genus, about which its demonstration is made from its proper principles. And similarly with the other sciences, because other sciences likewise have their proper genera, from which proper demonstrations are made in those sciences. But why, or for what reason, it is necessary in demonstration simply that the genus be the same which is the proper principle of generation of the things that are in the science, or that the genus be such that the subject of one science is under the subject of another, as was said about magnitude and number, if there must be a proper and true demonstration, is evident through the things already said. And if the demonstration of one science according to its causes ought to descend into the genus of another science, or the science of demonstration be carried from genus into genus otherwise than has been said, it is manifest that this is impossible.
And the cause of this is that in the most powerful demonstration the last things, that is, the extremes, must be from the same genus, because otherwise the first would not be present per se to the middle nor the middle to the last. For if such things are not, or are not present, per se, it follows that they will be accidents with respect to the fact that they are present accidentally. And because demonstration does not descend from genus into genus except as has been said, for this reason it does not belong to geometry, which is a special science, to show that there is one and the same science or discipline of contraries; this belongs to a more common and more universal science, because it belongs to first philosophy. Nor does it belong to geometry to show that which belongs to another particular science, as that two cubes are one cube, that is, that a cubical number multiplied by a cubical number makes two cubes; this belongs to the special demonstrative science which is arithmetic. Nor, speaking universally, does it belong to one science to demonstrate what belongs to another science, because as yet the subjects are not proper, nor the properties, nor the middles which are the proper causes. But only those things which are so related to one another as has been said, namely that one, that is, the subject of one, is under another, that is, under the subject of the other, as special things are under a more general thing: just as perspective, which is about the visible line, which is a ray, is related to geometry, which is about line simply; and as the consonative science, which is about consonant song, which is called harmonics or music, is related to arithmetic. For arithmetic demonstrates the proportions of numbers, but music determines the proportions of numbers in song. For in such cases, when a subject is under a subject, the superior science demonstrates many properties about the inferior subject through causes which are common in it and appropriated to the inferior.
Hujus autem probatio est: quia si aliquid inest lineis, hoc est, aliqua passio inest lineis, non secundum quod lineae sunt in genere vel specie consideratae, quod non inest eisdem lineis secundum quod lineae sunt, nec in quantum ex principiis propriis sunt linearum: hoc etiam non habet demonstrare geometria quae considerat de lineis: hoc enim accidit lineis ex alio quodam genere subjecti et scientiae: ut si debeat demonstrari si pulcherrima linearum sit recta linea: hoc enim non accidit lineae in quantum linea est, sed ex hoc quod recta linea naturalis vel physica est. Aut si demonstratur si linea recta contrario modo se habeat in positione partium ad lineam circularem: hoc enim non nisi ex physicis cognoscitur causis: talia enim non accidunt eis quae lineae sunt secundum quod est proprium genus ipsarum linearum, sed in quantum quoddam commune generalioris scientiae subjectum. Sic ergo tripliciter descendit demonstratio de genere in genus, scilicet vel quia subjectum est sub subjecto, sicut speciale sub generali, sicut perspectiva sub geometria: vel subjectum unius est sub subjecto alterius per accidens aliquod determinans subjectum unius ad formam subjecti alterius, ut linea rationalis et numerabilis ad arithmeticam se habet, et musica ad arithmeticam: aut siquidem eidem subjecto de quo probat una scientia passiones, accidat aliquid gratia materiae in qua est, cujus causa non sit in superiori scientia, ut quod linea recta sit pulcherrima vel contraria lineae circulari: hoc enim non accidit nisi per causam materiae recti vel circularis. Ex his igitur patet qualiter demonstratio unius generis scientiae de genere descendit in genus.
But the proof of this is that if something is present to lines, that is, if some property is present to lines, not according as lines are considered in genus or species, and is not present to the same lines according as they are lines, nor insofar as they are from the proper principles of lines, then geometry, which considers lines, does not have to demonstrate this either. For this happens to lines from another genus of subject and science, as if it has to be demonstrated whether the most beautiful of lines is a straight line; for this does not happen to line insofar as it is line, but from this, that a straight line is natural or physical. Or if it is demonstrated whether a straight line is related in a contrary mode in the position of its parts to a circular line, this is known only from physical causes. For such things do not happen to those things which are lines according to what is the proper genus of those lines, but insofar as a certain common subject of a more general science is involved. Thus therefore demonstration descends from genus into genus in three ways: either because a subject is under a subject, as the special under the general, as perspective is under geometry; or because the subject of one science is under the subject of another through some accident determining the subject of one to the form of the subject of the other, as rational and numerable line is related to arithmetic, and music to arithmetic; or because something happens, by reason of the matter in which it is, to the same subject about which one science proves properties, and its cause is not in the superior science, as that a straight line is most beautiful or contrary to a circular line. For this happens only through the cause of the matter of the straight or circular. Therefore from these things it is clear how the demonstration of one genus of science descends from genus into genus.
Caput XVII. Quod demonstratio est ex his quae sunt per se et secundum ipsum quantum ad conclusionem, et de diffinitione quod sit incorruptibilium et demonstratio et scientia similiter.
Manifestum est ex praeinductis, quod si sint propositiones ex quibus est syllogismus et necessariae et per se, quod necesse est conclusionem perpetuam esse: et hoc patet per universale quod est in demonstrativis quod de omni, et non de quodam et de quodam non, sed de omni simpliciter, et quod non est quandoque et quandoque non, sed semper: sic enim de omni secundum quod cadit in diffinitione universalis, in praehabitis determinatum est. Ex tali autem universali sequitur conclusionem perpetuam et incorruptibilem esse, quando principia sive praemissae sic sunt universales. Sic ergo manifestum est quod si sint propositiones universales ex quibus est syllogismus demonstrativus, necesse est et conclusionem perpetuam esse et de omni et semper, et hujus demonstrationis quae tales habet propositiones: et haec est demonstratio quae simpliciter et absolute dicitur demonstratio.
Ex quo ulterius concluditur, quod corruptibilium quae sunt singularia, non est demonstratio: sed, sicut dicit Alfarabius in commento, alia ratiocinatio est de talibus ex causis materialibus, quae non est demonstratio, quamvis in hoc demonstrationi in aliquo sit similis, et in hoc quod concludit. Neque de talibus, singularibus scilicet et corruptibilibus est scientia, quae simpliciter scientia est, secundum quod scire in ante habitis diffinitum est. Sed sic est de eis sicut secundum accidens. Et exponens Alfarabius dicit. Sic est de eis scientia sicut sunt: sunt autem per casum et fortunam quae sunt causae per accidens: sic ergo scientia est de ipsis: et haec non est scientia per se, sed habet aliquod accidens ad scientiae acceptionem, scilicet per materialem diffinitionem per quam per accidens scitur quod scitur 1.
Quamvis autem haec expositio sit Alfarabii et etiam Themistii et Alexandri: tamen forte melius dicitur, quod id quod est universale, hoc est, illa simplex natura cui, a relatione ad multa quae possunt eam participare, accidit, quod sit universale, habet duplex esse. Unum quidem in lumine intelligentiae, cujus ipsa natura est actus sive effectus sicut efficientis causae, quae ex forma sua facit: et hoc est esse principium et incorruptibile, et forte ubique et semper. Aliud autem esse habet secundum quod est natura istius vel illius, et secundum hoc esse est corruptibile. Et primo quidem modo per se intellectus est et scientia de ipso. Secundo autem modo non nisi secundum accidens. Quod autem est esse universaliter ipsius talis naturae, non est secundum quod est in isto, sed est aliquando: et sic prout est in isto, et non per se, scientia est de ipso secundum quod esse suum, quod est in isto, refertur ad esse simpliciter, et secundum quod est aliquando, non refertur ad esse semper: et hoc modo per accidens est de ipso scientia. Et haec expositio bona est et concordat cum his quae dicit Aristoteles in sexto primae philosophiae 2. Sed syllogizare universaliter de talibus non erit: quia deficit in tempore et aliquando non erit, sed aliquando erit: et sic prout scilicet hoc in particulari ens refertur ad universale.
Chapter XVII. That demonstration is from those things which are per se and according to itself with respect to the conclusion, and, concerning definition, that there is demonstration and science similarly of incorruptible things.
It is manifest from the things brought forward that if the propositions from which the syllogism proceeds are both necessary and per se, the conclusion must be perpetual. And this is clear through the universal which exists in demonstrative matters, because it is of every, and not of some and not of some, but of every simply, and because it is not at one time and at another time not, but always. For thus of every, according as it falls in the definition of the universal, has been determined in the foregoing. But from such a universal it follows that the conclusion is perpetual and incorruptible, when the principles or premises are universal in this way. Thus therefore it is manifest that if the propositions from which there is a demonstrative syllogism are universal, then it is necessary both that the conclusion be perpetual and that it be of every and always, and this belongs to the demonstration which has such propositions; and this is the demonstration which is called demonstration simply and absolutely.
From this it is further concluded that there is no demonstration of corruptible things which are singulars. But, as Alfarabi says in the commentary, there is another reasoning about such things from material causes, which is not demonstration, although in this respect it is in some way similar to demonstration, namely in the fact that it concludes. Nor is there science of such things, namely of singular and corruptible things, which is science simply, according as to know has been defined in the preceding matters. But it is about them as according to accident. And Alfarabi, explaining, says: there is science about them as they are; but those things which are by chance and fortune are from causes per accidens; thus therefore science is about them, and this is not science per se, but has some accident belonging to the reception of science, namely through a material definition through which what is known is known per accidens 1.
But although this exposition belongs to Alfarabi and also to Themistius and Alexander, nevertheless perhaps it is better said that that which is universal, that is, that simple nature to which, from a relation to many things which can participate it, it happens that it is universal, has a twofold being. One is in the light of intelligence, of which its very nature is the act or effect as of an efficient cause, which makes from its form; and this is to be a principle and incorruptible, and perhaps everywhere and always. But it has another being according as it is the nature of this or that, and according to this being it is corruptible. And in the first mode, intellect and science are per se about it; but in the second mode, only according to accident. But the being universal of such a nature is not according as it is in this thing, but is at some time; and thus, insofar as it is in this thing and not per se, science is about it according as its being, which is in this thing, is referred to being simply, and according as it is at some time is not referred to being always. And in this mode science is about it per accidens. And this exposition is good and agrees with what Aristotle says in the sixth book of the Metaphysics 2. But to syllogize universally about such things will not occur, because it fails with respect to time and at some time will not be, but at some time will be, and thus only insofar as this particular being is referred to the universal.
Cum autem sic accipitur: tunc ad minus necesse est alteram talis syllogismi propositionem non esse universalem, quam necesse est esse corruptibilem, quae aliquando erit et aliquando non erit: corruptibilis autem erit altera. Quod si sic, etiam conclusionem necesse est esse non universalem. Et in tali quidem propositione hoc quidem erit praedicatum, vel subjectum: aut non erit, scilicet praedicatum vel subjectum: et sic compositio non erit semper, vel altera quidem propositio erit semper, et altera non erit semper, sed aliquando. Ex quibus quidem propositionibus est talis conclusio non universalis in omnibus et semper: propter quod ex talibus non contingit syllogizare universaliter: sed si syllogizatur ex talibus, hoc est, quoniam nunc est, et non simpliciter quoniam semper est.
Similiter autem est et de diffinitione: quia ipsa corruptibilium et singularium est per accidens, in quantum scilicet participant sua universalia, et per se incorruptibilium et universalium. Hujus autem probatio: quoniam diffinitio tripliciter se habet ad demonstrationem: quoniam quidem diffinitio aut est principium demonstrationis, quando scilicet est medium. Aut est demonstratio, positione sola sive situ differens a demonstratione: quod secundum Alfarabium est, quia partim ponatur in majori propositione, partim in minori, et partim in conclusione: et sic positione differens est, non substantia. Aut etiam conclusio demonstrationis: quoniam sicut ostendit Aristoteles in principio secundi de Anima 3, concluditur diffinitio dicens quid per diffinitionem dicentem propter quid, sicut quod anima est endelechia physici corporis potentia vitam habentis, per hoc quod anima est principium et causa vitae talis corporis. Et haec est expositio Alfarabii et Commentatorum.
But when it is received thus, then at least one proposition of such a syllogism must not be universal, since it must be corruptible, and it will be at some time and will not be at another; for one of them will be corruptible. But if this is so, it is also necessary that the conclusion be non-universal. And in such a proposition this indeed will be the predicate, or the subject; or it will not be, namely the predicate or the subject; and thus the composition will not be always, or one proposition indeed will be always and the other will not be always, but at some time. From such propositions, therefore, there is a conclusion which is not universal in all things and always; on account of which it does not happen that one syllogizes universally from such things. But if one syllogizes from such things, this is that it is now, and not simply that it is always.
Similarly it is also with definition, because the definition itself of corruptible and singular things is per accidens, namely insofar as they participate their universals, and per se it is of incorruptible and universal things. The proof of this is that definition is related to demonstration in three ways. For definition either is a principle of demonstration, namely when it is the middle; or it is a demonstration differing from demonstration only in position or arrangement, which according to Alfarabi is because it is placed partly in the major proposition, partly in the minor, and partly in the conclusion, and thus it differs in position, not in substance. Or it is also the conclusion of demonstration, since, as Aristotle shows at the beginning of the second book On the Soul 3, a definition saying what is is concluded through a definition saying why, as that the soul is the entelechy of a physical body having life in potency, through this, that the soul is the principle and cause of the life of such a body. And this is the exposition of Alfarabi and of the Commentators.
Dicunt tamen satis subtiliter quidam modernorum quod est diffinitio formalis, et est diffinitio materialis, et forte diffinitio perfecta colligitur ex utraque istarum, ut dicit Isaac in libro Diffinitionum 4. Et in demonstratione in qua concluditur diffinitio materialis de subjecto per diffinitionem formalem, diffinitio formalis est principium, et materialis est conclusio: et tertia quae composita est, demonstratio positione differens est: quia continet in virtute quidquid habet demonstratio: quamvis secundum situm non ita ponatur in diffinitione sicut in demonstratione. Cujus exemplum est, quod formalis irae diffinitio est appetitus contrarii doloris: materialis autem, accensus circa cor sanguis ex evaporatione fellis: composita autem, appetitus contrarii doloris ortus ex accensione circa cor sanguinis ex evaporatione fellis. Et si fiat demonstratio sic, quicumque habet appetitum contrarii doloris ei inest accensio circa cor sanguinis ex evaporatione fellis: iratus habet appetitum contrarii doloris: ergo iratus habet accensionem caloris et sanguinis ex evaporatione fellis: ex evaporatione enim fellis, addunt Damascenus et Nyssenus, ut materia quae sola causa non est, cum habeat principium efficiens et motivum.
Alii autem dixerunt, quod quidquid est, aut est sicut hoc constans et subjectum: aut sicut hujus, sicut praedicatum et passio: utriusque autem horum diffinitio est in demonstratione: diffinitio ergo subjecti principium est demonstrationis; eo quod oportet quod demonstratio ipsam habeat sicut et subjectum. Diffinitio autem passionis duplex est: aut enim solam indicat essentiam passionis: aut indicat causam inhaerentiae subjecto. Si primo modo: tunc potest concludi de subjecto sicut passio: et tunc est sicut conclusio. Si secundo modo: tunc dicitur positione differens a demonstratione: quia in ipsa est quidquid de substantia demonstrationis: sed primo ponit passionem, secundo subjectum, et ultimo causam passionis, sicut patet sic dicendo, eclipsis est privatio luminis in luna per objectum terrae eveniens. Demonstratio autem primo locat subjectum, secundo medium, tertio passionem.
Alii dicunt, quod diffinitio dicitur sola positione differre a demonstratione: quia omnia tria praedicta, subjectum scilicet, passionem, et medium, sive causam sub unius incomplexi comprehendit intentione: in demonstratione autem considerantur ut plura et ut in majorem et in minorem et conclusionem posita: et hoc fere idem est cum eo quod primo positum est ex commento Alfarabii Arabico. Adhuc dicunt quod eadem quae in diffinitione per abstractionem dicuntur, ut cum dicitur, eclipsis est privatio luminis in luna objectu terrae inter solem et lunam in nodo capitis vel caudae proveniens, dicuntur in demonstratione sub concretione, sicut eclipsatur privatur lumine ex objectu umbrae terrae inter solem et lunam: talis enim diffinitio medium est demonstrans passionem de subjecto: dicit enim quid est et propter quid est passionis: illa vero quae concludi potest de subjecto, non dicit nisi quid est: illa vero diffinitio quae est subjecti, dicitur diffinitio secundum speciem quae non demonstratur.
Yet certain moderns say quite subtly that there is a formal definition, and there is a material definition, and perhaps a perfect definition is gathered from both of these, as Isaac says in the book of Definitions 4. And in the demonstration in which the material definition is concluded of the subject through the formal definition, the formal definition is the principle, and the material is the conclusion; and the third, which is composite, is the demonstration differing in position, because it contains virtually whatever demonstration has, although according to arrangement it is not placed in definition as in demonstration. An example of this is that the formal definition of anger is appetite for contrary pain; but the material definition is blood kindled around the heart from the evaporation of bile; and the composite is appetite for contrary pain arising from the kindling of blood around the heart from the evaporation of bile. And if demonstration is made thus: whoever has appetite for contrary pain has kindling around the heart of blood from the evaporation of bile; the angry person has appetite for contrary pain; therefore the angry person has a kindling of heat and blood from the evaporation of bile. For Damascene and Nyssen add "from the evaporation of bile" as matter, which alone is not a cause, since it has an efficient and motive principle.
But others have said that whatever exists is either as this constant thing and subject, or as of this, as predicate and property; and the definition of each of these is in demonstration. Therefore the definition of the subject is the principle of demonstration, because demonstration must have it as it must have the subject. But the definition of the property is twofold: for either it indicates only the essence of the property, or it indicates the cause of inherence in the subject. If in the first mode, then it can be concluded of the subject as a property, and then it is like a conclusion. If in the second mode, then it is said to differ in position from demonstration, because in it is whatever belongs to the substance of demonstration; but first it places the property, secondly the subject, and lastly the cause of the property, as is clear by saying thus: eclipse is the deprivation of light in the moon occurring through the interposition of the earth. But demonstration first locates the subject, secondly the middle, and thirdly the property.
Others say that definition is said to differ from demonstration only in position because it comprehends all three aforesaid things, namely subject, property, and middle or cause, under the intention of one incomplex thing; but in demonstration they are considered as many and as placed in the major and in the minor and in the conclusion. And this is almost the same as what was first posited from the Arabic commentary of Alfarabi. Moreover they say that the same things which are said in definition by abstraction, as when it is said, eclipse is the deprivation of light in the moon arising from the object of the earth between the sun and the moon in the node of the head or tail, are said in demonstration under concretion, as the eclipsed thing is deprived of light from the object of the earth's shadow between the sun and the moon. For such a definition is the middle demonstrating the property of the subject; for it says what the property is and why it is. But that definition which can be concluded of the subject says only what it is; but that definition which belongs to the subject is called the definition according to species, which is not demonstrated.
Prima magis esse videtur secundum intentionem Aristotelis ut sit sensus, quod formalis est principium, materialis vero conclusio, et composita differens, hoc est, diversa positione, quia partim ponitur in majori, partim autem in minori, et continet substantiam demonstrationis. Et omnis modus diffinitionis sic tactus est praeter illum qui est per nomina alia magis nota: qui modus magis est etymologia quam diffinitio.
Quamvis autem sit eorum quae sunt semper sic diffinitio, sic etiam demonstratio est eorum quae saepe fiunt et non semper, et scientiae sunt de ipsis, sicut est lunae defectus vel solis et hujusmodi. Sed scientiae et demonstrationes sunt de his: quia manifestum est quod haec secundum quod hujusmodi sunt, hoc est, relata ad causas proximas et necessarias, semper sunt. In quantum autem non semper sunt: tunc sunt secundum partem, hoc est, secundum singulare: haec autem causa decidens: et sic defectus est universalis quod est de omni et semper. Similiter autem est in aliis hujusmodi. Eclipsis enim vel alia passio potest comparari ad subjectum tantum, sicut ad lunam, et sic non semper est. Potest etiam comparari ad subjectum cum causa sicut ad lunam existentem in nodo capitis vel caudae latitudine determinata: et sic semper est eclipsis universalis, quae cum eveniat secundum ordinem motus solis et lunae et motum capitis et caudae, semper est necessaria. In se autem haec vel illa considerata, neque necessaria est, neque semper, sive aeterna.
The first exposition seems to be more according to Aristotle's intention, so that the sense is that the formal is the principle, the material is the conclusion, and the composite is different, that is, diverse in position, because it is placed partly in the major and partly in the minor, and contains the substance of demonstration. And every mode of definition thus touched on is besides that mode which is through other names more known; that mode is etymology more than definition.
But although in this way there is definition of those things which are always, there is also in this way demonstration of those things which occur often and not always, and there are sciences of them, as there is of an eclipse of the moon or of the sun and the like. But sciences and demonstrations are of these things because it is manifest that these things, according as they are such, that is, referred to proximate and necessary causes, are always. But insofar as they are not always, then they are according to a part, that is, according to the singular; but this is a falling-away cause, and thus an eclipse is universal, that is, of every and always. Similarly it is also in other things of this sort. For eclipse or another property can be compared to the subject only, as to the moon, and thus it is not always. It can also be compared to the subject with the cause, as to the moon existing in the node of the head or tail with determinate latitude; and thus there is always a universal eclipse which, since it occurs according to the order of the motion of the sun and moon and the motion of the head and tail, is always necessary. But considered in itself as this or that, it is neither necessary, nor always, or eternal.
Caput XVIII. Quod demonstratio non est ex communibus, sed ex propriis.
Quoniam autem ex praedeterminatis manifestum est, quod non est vel contingit unumquodque vel quodlibet demonstrare: sed aut, hoc est, solum illa demonstrare contingit, quae sunt ex unoquoque propriorum principiorum suorum per quae demonstrari habet: hoc enim necesse est dicere si concedatur quod ante probatum est, quod scilicet id quod demonstratur, hoc est, passio demonstrata de subjecto, et insit subjecto secundum ipsum, hoc est, appropriate secundum quod illud est sibi attributum, sicut paulo ante probatum est. Ex hoc autem accipitur quod demonstrative non est quidem scire quodcumque demonstrabile propter hoc, si ex veris et communissimis non appropriatis, quamvis tamen indemonstrabilibus monstratur et immediatis, quia communissima non sunt demonstrabilia, eo quod non habent alia priora quae sint media quibus possint demonstrari.
Si autem ex communioribus demonstretur et non appropriatis: tunc fieret demonstratio sicut Brisso tetragonismon, hoc est, quadrangulum seu figuram quadrangulam demonstravit aequalem esse circulo: rationes enim hujusmodi qualis est Brissonis, secundum commune medium quod alteri inest et convenit, tales sumunt rationes: propter quod tales rationes etiam in aliis demonstrationibus, quae non sunt ejusdem generis, conveniunt et non sunt ex proximis: et ex hoc sequitur, quod non conveniunt isti secundum ipsum secundum quod illud vel ipsum est: nec sic sciens per tales rationes, scit secundum quod illud est, sed secundum accidens, hoc est, secundum aliud, quia convenit isti secundum communius quod est in ipso a quo ipsum non est hoc quod ipsum est secundum hoc quod ipsum est: et sic sciens non scit: quia aliter oporteret quod demonstratio descenderet de genere in genus, quod superius improbatum est. Unumquodque autem quod per sui communius et non convertibile cognoscitur, cognoscitur secundum accidens: et quod sic cognoscitur, non cognoscitur secundum hoc quod ipsum est: non enim cognoscitur secundum quod est ex principiis propriis illius in quantum ipsum est, sicut in exemplo quod superius induximus, est videre: tres angulos enim aequos duobus rectis habere jam quidem dictum est cui subjecto per se inest: illi enim inest ex quo fluit sicut ex propriis principiis ipsius, et quod secundum seipsum tota et sufficiens causa est passionis: propter quod si per se inest, et illud inest quod est passio, et secundum ipsum, necesse est medium per quod probatur inesse, in eadem esse cum subjecto et passione proximitate: quia proprium per commune non probatur inesse, nisi sicut est in communi: hoc est autem in potentia tantum, et secundum scire in universali, quod est scire imperfectum.
Chapter XVIII. That demonstration is not from common things, but from proper things.
But since from the things predetermined it is manifest that it is not, or it does not happen, that one demonstrates each thing or anything whatever, but only those things happen to be demonstrated which are from the proper principles of each thing through which it has to be demonstrated, this must be said if what was previously proved is granted, namely that what is demonstrated, that is, the property demonstrated of the subject, is present to the subject according to itself, that is, appropriately according as that property is attributed to it, as was proved a little before. But from this it is received that it is not indeed to know demonstratively just any demonstrable thing for this reason, if it is shown from true and most common things that are not appropriated, although nevertheless indemonstrable and immediate, because the most common things are not demonstrable, since they do not have other prior things which are middles by which they could be demonstrated.
But if something is demonstrated from more common and non-appropriated things, then the demonstration would be like Brisso's quadrature, that is, he demonstrated a quadrangle or quadrangular figure to be equal to a circle. For arguments of this sort, such as Brisso's is, receive such reasons according to a common middle that is present to and belongs to another thing. On account of this such reasons also belong in other demonstrations which are not of the same genus and are not from proximate things; and from this it follows that they do not belong to this thing according to itself according as that or it itself is. Nor does one who knows through such reasons know according to what that thing is, but according to accident, that is, according to another thing, because it belongs to this thing according to something more common which is in it, from which it itself is not what it itself is according to what it itself is. And thus one who knows does not know, because otherwise demonstration would have to descend from genus into genus, which was disapproved above. But each thing which is known through something more common than itself and not convertible is known according to accident; and what is known thus is not known according to what it itself is. For it is not known according as it is from the proper principles of that thing insofar as it is itself, as can be seen in the example which we introduced above. For to have three angles equal to two right angles has already been said to be present per se to its subject, because it is present to that from which it flows as from its own proper principles, and which according to itself is the whole and sufficient cause of the property. On account of this, if it is present per se, and that which is the property is present according to itself, the middle through which it is proved to be present must be in the same proximity with the subject and the property, because the proper is not proved to be present through the common except as it is in the common; but this is only in potency and according to knowing in the universal, which is imperfect knowing.
Brissonis autem talis dicitur ex communioribus fuisse demonstratio, quod omne id quod est minus aliquo, et invenitur majus eodem, si minus debeat excrescere in aequalitatem majoris, aliquando attingit aequale quod est inter haec. Est autem quadrangulum minus circulo, sicut quadrangulum inscriptum circulo. Cum igitur quadrangulum circumpositum gnomone, possit crescere in infinitum, aliquando de minori ad majus crescens, aequale erit circulo: et sic quandoque quadrangulum vel quadratum aequale erit circulo. Et patet quod principia sunt valde communia non tantum in hac demonstratione de quadratura circuli, sed etiam in physicis et in multis aliis communia. Vel forte aliter Brisso voluit quadrare circulum in quo non est vis, nisi in hoc quod communiora accepit principia et non appropriata huic passioni et huic subjecto, ut scilicet primum medio et medium minori insit secundum ipsum et per se: et ad hoc tantum inducimus Brissonis rationes.
Si vero non sic fiat demonstratio per propria: tunc descendit demonstratio de genere in genus, sicut superius dictum est, scilicet a subalternante scientia in subalternatam: sicut harmonica passio aliquando demonstratur per arithmeticam: eo quod harmonica consistit ex proportionali numero notarum taliter se habentium: similiter demonstratur hoc ex eisdem principiis. Sed haec differunt in hoc quod quia demonstratum, alterius est scientiae quam ipsum propter quid: genus enim quod est subjectum, est alterum, quia subalternatum quod alterum est a subalternante: sed propter quid, sicut dictum est, superioris sive subalternantis est scientiae, cujus etiam illae quae probantur de inferiori subjecto, per se sunt passiones, sicut dictum est de perspectiva subalternata sub geometria, sicut si tres angulos habere aequales duobus rectis probetur de triangulo qui resultat ex radiis reflexis: hujusmodi enim passio, quia superioris scientiae est per se, oportet quod probetur per superioris scientiae principia. Propter quod manifestum est quod non est demonstrare in inferiori scientia unumquodque demonstrabilium simpliciter, hoc est, universaliter: sed si debeat demonstrari, oportet quod demonstretur appropriatum per demonstrationem superioris et contractionem secundum quod competit proprium demonstrari ex propriis uniuscujusque principiis, quod sicut contrahitur subjectum et sicut contrahitur passio et determinatur, sic etiam contrahatur medium quod est causa.
But Brisso's demonstration is said to have been such from more common things, because everything which is less than something and is found greater than the same, if the lesser must grow up to the equality of the greater, at some time reaches the equal which is between these. But a quadrangle is less than a circle, as a quadrangle inscribed in a circle. Therefore since a quadrangle circumscribed by a gnomon can grow to infinity, at some time, growing from the lesser to the greater, it will be equal to the circle; and so at some time a quadrangle or square will be equal to the circle. And it is clear that the principles are very common, not only in this demonstration about the quadrature of the circle, but also common in physics and in many other matters. Or perhaps Brisso wished to square the circle otherwise, in which there is force only in this, that he received more common principles and not principles appropriated to this property and to this subject, namely so that the first be present to the middle and the middle to the minor according to itself and per se; and only for this do we introduce Brisso's reasons.
But if demonstration is not made thus through proper things, then demonstration descends from genus into genus, as was said above, namely from a subalternating science into a subalternated one, as a harmonic property is sometimes demonstrated through arithmetic, because harmonics consists of a proportional number of notes related in such a way. Similarly this is demonstrated from the same principles. But these differ in this, that the demonstration that belongs to one science, while the reason why belongs to another. For the genus which is the subject is another, because it is subalternated, which is other than the subalternating; but the reason why, as has been said, belongs to the superior or subalternating science, to which also those things which are proved of the inferior subject are per se properties. This was said about perspective subalternated under geometry, as if having three angles equal to two right angles is proved of a triangle which results from reflected rays; for a property of this sort, because it belongs per se to the superior science, must be proved through the principles of the superior science. On account of this it is manifest that it is not possible in the inferior science to demonstrate each of the demonstrable things simply, that is, universally; but if it must be demonstrated, it must be demonstrated as appropriated through the demonstration of the superior and through contraction according as it belongs to the proper thing to be demonstrated from the proper principles of each, so that just as the subject is contracted and just as the property is contracted and determined, so also the middle which is the cause is contracted.
Sed horum sic appropriatorum principia propria habent commune, quo et ipsa demonstrentur. Si autem quod nunc dictum est, manifestum est esse verum, tunc etiam manifestum est cum hoc, quod non est uniuscujusque demonstrabilis propria principia demonstrare: quia communis scientiae propria principia sunt communia: illa enim communis scientia est prima philosophia, et illius principia erunt omnia principia communiter et communia: et scientia illorum communium principiorum est demonstrans omnibus propria principia. Quod autem hoc verum sit, per hoc patet quod ante habitum est, quod ille magis scivit ex superioribus causis: quia habitum est quod propter quod unumquodque tale, illud est magis: ex superioribus enim scivit propter quid, cum taliter sciens non sciat causam ex causatis demonstratione quia, sed potius proprias causas scivit ex superioribus causis: propter quod si magis scivit qui scivit ex superioribus causis, quam ille qui scivit ex inferioribus appropriatis, tunc sequitur quod magis, scivit qui scivit ex omnium primis: ergo et illa communissima et prima philosophia erit, et magis scientia quam particularis et propria, et simpliciter erit maxima scientia, quia est omnium prima.
Sed non sequitur propter hoc quod demonstrationi conveniat descendere de uno genere in aliud genus omnino diversum: sed si descendit, non descendit, nisi aut, hoc est, solum, sicut dictum est, a subalternante in subalternatam et communi in propriam, sicut geometricae descendunt in mechanicas quae sunt manuales, ut cosmetria, altimetria, et profundimetria: vel sicut geometricae descendunt in mechanicas quae dicuntur scientiae de ingeniis: aut sicut descendunt geometricae in speculativas quae dicuntur perspectivae scientiae: et sicut arithmeticae scientiae descendunt in harmonicas in quibus est proportio numeri notarum. Hoc autem ideo est, quia subalternatum habet in se actu et intellectu suum subalternans: et ita tunc subalternatum est ex primis in se contentis sicut subalternans est ex propriis simpliciter.
But principles proper to things appropriated in this way have something common by which they too are demonstrated. But if what has now been said is manifestly true, then with this it is also manifest that it is not possible to demonstrate the proper principles of each demonstrable thing, because the proper principles of the common science are common. For that common science is first philosophy, and its principles will be all principles commonly and common principles; and the science of those common principles demonstrates proper principles for all. But that this is true is clear through what was had before, namely that one knows more from superior causes, because it has been had that that on account of which each thing is such is more such. For from superior causes he knows the reason why, since one knowing in such a way does not know the cause from caused things by a demonstration that, but rather knows proper causes from superior causes. On account of this, if the one who knew from superior causes knew more than he who knew from appropriated inferior causes, then it follows that the one who knew from the first causes of all knew more. Therefore that most common and first philosophy also will be more a science than the particular and proper science, and simply will be the greatest science, because it is first of all.
Yet it does not follow on account of this that it belongs to demonstration to descend from one genus into another wholly diverse genus. But if it descends, it descends only, as has been said, from the subalternating into the subalternated and from the common into the proper, as geometrical sciences descend into the mechanical sciences which are manual, such as cosmetria, altimetry, and profundimetry; or as geometrical sciences descend into the mechanical sciences which are called sciences about instruments; or as geometrical sciences descend into the speculative sciences which are called perspective sciences; and as arithmetical sciences descend into harmonics, in which there is the proportion of the number of notes. But this is because the subalternated has in itself, in act and in intellect, its subalternating science; and thus then the subalternated is from first things contained in itself, just as the subalternating is from things proper simply.
Tractatus III. De usu principiorum in demonstrationibus.
Caput I. De usu principiorum incomplexorum.
Postquam autem jam conditiones principiorum determinatae sunt, consequens ratio postulat, ut de usu ipsorum in demonstrationibus determinetur, sive sint subjecta, sive passiones et dignitates: et primo ostendatur hoc in incomplexis, et deinde in complexis. Sed ante hoc dicimus quod ex quo oportet scire secundum ipsum et per se et proprium si scitur aliquid per demonstrationem, difficile est nosse si aliquis vere scivit vel non scivit illud. Cujus causa et probatio est, quia difficile est nosse si scimus aliquid ex uniuscujusque principiis propriis, aut non scimus ex propriis: quod tamen scire ex propriis solum est vere scire in particulari et in agere: quia in quantum scitur ex propriis, scitur in propria natura: et in quantum scitur ut ex principiis, est scire in agere. Opinamur autem communiter, quamvis decepti sumus, scire aliquid vere, si habeamus ex veris aliquibus principiis et propositionibus et primis, secundum quod primum dicitur ante quod nihil secundum quod communia indemonstrabilia dicuntur principia, syllogismum concludentem illud quod scire opinamur: sed tamen hoc non est, quod scilicet ex talibus veris et primis vera sciamus: sed oportet principia ex quibus verissime scimus esse proxima primis: et hoc cum hoc quod sunt principia oportet esse proxima: et hoc non potest esse nisi cum sunt prima per coarctantia approximata: quia tunc in proximis sunt propria virtute et actu et intellectu et causa, sicut superiora sunt his tribus modis in suis inferioribus subalternatis: et sicut primae causae sunt in proximis, quando essentiales sunt et primae et proximae.
Dico autem principia sive complexa sive incomplexa in unoquoque genere sive subjecto, quae cum sint in ipso subjecto, non contingit ea demonstrare per aliqua quae sunt illius generis vel scientiae cujus sunt principia. Et hujus causa est, quia ad illa demonstranda non habet aliqua priora: nam si haberet aliqua priora ex quibus illa demonstraret, tunc ipsa non essent principia. In qualibet ergo scientia demonstrativa accipitur sine demonstratione quid significent per nomen, diffinitionem, vel expositionem, et prima quae sunt subjecta, et quae sunt ex his, hoc est, passiones quae causantur a subjecto: et quid significent dignitates quas scimus in quantum terminos cognoscimus: et similiter cognoscitur quid significent per nomina quae sunt ex his ut passiones probatae de subjecto quae sunt conclusiones.
Adhuc autem si sciuntur quid significetur per nomina in incomplexis et in his quae sunt ex his, scilicet incomplexa. Omnia autem haec vera sunt, et a diversis hic locus sic exponitur. Alia vero contingit demonstrare, ut scilicet passionem esse: vel quia sit, quia passionis ut passionis esse est inesse subjecto: et haec est conclusio quae non scire contingit nisi demonstretur. Horum autem exempla in incomplexis sint: quia contingit accipere sine demonstratione quid unitatis, hoc est, quid significetur per nomen unitatis: quia est indivisibile primum in discretis non habens positionem. Aut quid rectum, hoc est, quid significetur per nomen recti: quia rectum est extensio inter duo puncta latitudine indivisibilis, cujus medium, ut dicit Plato, non exit ab extremis, vel cujus primum punctum recta positione tegit totum. Et quid triangulus: quia est figura tribus rectis lineis conclusa habens tres aequos duobus rectis. Est autem unitatem accipere quia est, et magnitudinem similiter accipere quia est: quia unitas subjectum est in arithmeticis, de qua multa demonstrantur: et magnitudinem similiter in geometria oportet accipere quia est, quia de ipsa in geometria multa demonstrantur. Altera autem quae sunt passiones et praedicata, non est accipere quia sunt: sed potius quia sunt, talia oportet in scientiis demonstrativis demonstrare. Cujus ratio jam dicta est: quia passionis esse est inesse, et sic cadit in intentionem conclusionis, in qua concluditur passio de subjecto.
Treatise III. On the use of principles in demonstrations.
Chapter I. On the use of incomplex principles.
But after the conditions of principles have now been determined, the consequent account requires that the use of these principles in demonstrations be determined, whether they are subjects, or properties and dignities; and first this should be shown in incomplex things, and then in complex things. But before this we say that, because one must know according to itself and per se and proper if something is known through demonstration, it is difficult to know whether someone has truly known that thing or has not known it. The cause and proof of this is that it is difficult to know whether we know something from the proper principles of each thing, or whether we do not know it from proper principles. Yet only to know from proper principles is truly to know in the particular and in acting, because insofar as it is known from proper principles, it is known in its proper nature; and insofar as it is known as from principles, it is knowing in act. But commonly, although we are deceived, we think that we truly know something if from some true principles and propositions and first things, according as first is said of that before which there is nothing, as common indemonstrables are called principles, we have a syllogism concluding that which we think we know. Yet this is not so, namely that from such true and first things we know true things; rather the principles from which we know most truly must be proximate to first things. And, together with this, because they are principles they must be proximate; and this cannot be unless they are first things approximated through restricting things, because then in proximate things proper things exist in power and act and intellect and cause, as superior things exist in these three modes in their subalternated inferiors, and as first causes exist in proximate causes when they are essential and first and proximate.
But I call principles, whether complex or incomplex, in each genus or subject, those things which, since they are in that subject, cannot be demonstrated through some things which belong to that genus or science whose principles they are. And the cause of this is that for demonstrating those things it does not have any prior things; for if it had some prior things from which it demonstrated them, then they themselves would not be principles. Therefore in any demonstrative science there is received without demonstration what the first things which are subjects signify by name, definition, or exposition, and also what things are from these, that is, properties caused by the subject, signify; and what the dignities signify, which we know insofar as we know the terms; and similarly it is known what those things which are from these signify by names, as properties proved of the subject which are conclusions.
Moreover, if what is signified by names is known in incomplex things and in those things which are from them, namely the incomplex things. But all these things are true, and this passage is explained in this way by diverse persons. Other things, however, happen to be demonstrated, namely that the property exists, or that it is, because the being of a property as property is to be present to a subject; and this is the conclusion which one cannot know unless it is demonstrated. Let examples of these things in incomplex things be these: because one can receive without demonstration what unity is, that is, what is signified by the name unity, because it is the first indivisible thing in discrete things, not having position. Or what the straight is, that is, what is signified by the name straight, because the straight is an extension between two points, indivisible in breadth, whose middle, as Plato says, does not go outside the extremes, or whose first point by straight position covers the whole. And what a triangle is, because it is a figure enclosed by three straight lines, having three angles equal to two right angles. But one can receive that unity is, and similarly receive that magnitude is, because unity is the subject in arithmetic, about which many things are demonstrated; and similarly one must receive that magnitude is in geometry, because many things are demonstrated about it in geometry. But the other things which are properties and predicates are not to be received because they are; rather, that they are must be demonstrated in demonstrative sciences. The reason for this has already been stated, because the being of a property is to be present, and thus it falls into the intention of the conclusion, in which the property is concluded of the subject.
Sed quantum ad dignitates hic considerandum est, quod ea quibus pro dignitatibus utimur in demonstrativis scientiis, alia quidem sunt propria unicuique scientiae, et illi scientiae appropriata, quibus in aliis scientiis demonstrativis uti non contingit ad proprias conclusiones. Alia vero sunt communia, prima scilicet non appropriata et approximata per coarctationem. Communia autem sunt in genere, quia non sunt plures scientiae de genere uno: sed communia pluribus scientiis secundum analogiam, hoc est, secundum proportionem ad eam cui convenit id ut primo, et aliis per relationem et respectum ad illud: sicut id quod utile est, in totum id quod in genere est ordinabile sub scientia illa quantumcumque illud est, hoc est, quantumcumque commune est secundum analogiam, quod sicut ad indivisibilem ordinatum est. Et propria principia quidem sunt ut geometriae lineam esse hujusmodi, scilicet extensionem inter duo puncta indivisibilem: et rectum in continuo esse hujusmodi, hoc est, extensionem brevissimam quae est inter duo puncta. Haec enim appropriata insunt huic subjecto quod proprium est geometriae.
Communia autem secundum analogiam sunt, ut aequalia si auferas ab aequalibus, reliqua, hoc est, relicta sunt aequalia. Aequalitas enim prima in numeris est, per analogiam est in quantis continuis numeratis. Si ergo contingat uti propriis, non oportet coarctari proprium. Si autem contingat uti communibus, sufficiens est ad usum si coarctantur quantumcumque conveniens est in genere subjecto illius scientiae, in quo quis utitur ipso communi. Idem enim faciet quantum ad demonstrationem in illo genere subjecto sic coarctatum principium, etsi, hoc est, quamvis non de omnibus generibus accipiat, de quibus fuit acceptum in communi sive in sua prima communitate, ut sit hoc principium, ut si ab aequalibus aequalia demas, quae relinquuntur, aequalia demas, acceptum in magnitudinibus solum in geometria idem facit quod faceret acceptum in communi quantum ad conclusionis sive probationis passionem: et idem faciet arithmetico acceptum in numeris proprie vel appropriatum quod faceret acceptum in communi: et eamdem virtutem demonstrandi habebit in hoc genere coarctatum, quam in communi acceptum habuit in omnibus. Et hujus causa est: quia semper priora actu et intellectu et essentia et virtute remanent in secundis, quae per additionem coarctantem exeunt a primis.
But with respect to dignities one must consider here that the things which we use as dignities in demonstrative sciences are, some, proper to each science and appropriated to that science, and it does not happen that they are used in other demonstrative sciences for proper conclusions. But others are common, namely first things not appropriated and approximated through restriction. But they are common in genus, because there are not several sciences about one genus; rather they are common to several sciences according to analogy, that is, according to proportion to that to which this belongs as first, and to the others by relation and respect to it, as that which is useful is ordered to the whole of what is in the genus under that science, however much that is, that is, however much it is common according to analogy, which is ordered as to the indivisible. But the proper principles are such as these: for geometry, that line is of this sort, namely an indivisible extension between two points; and that the straight in a continuum is of this sort, that is, the shortest extension between two points. For these appropriated things are present to this subject which is proper to geometry.
But common things according to analogy are such as: if equal things are taken away from equal things, the remaining things, that is, the things left, are equal. For equality is first in numbers and exists by analogy in numbered continuous quantities. Therefore if it happens that one uses proper things, the proper thing need not be restricted. But if it happens that one uses common things, it is sufficient for use if they are restricted to the degree that is fitting in the subject genus of that science in which someone uses that common thing. For the principle restricted in this way in that subject genus will do the same thing with respect to demonstration, even if, that is, although, it is not received about all genera about which it was received in common or in its first commonness, as this principle: if equal things are subtracted from equal things, equal things which are left are subtracted. Received only in magnitudes in geometry, it does the same thing that it would do received in common with respect to the conclusion or proof of the property; and received in numbers properly or as appropriated it will do the same for the arithmetician as it would do received in common. And restricted in this genus it will have the same power of demonstrating which, received in common, it had in all things. And the cause of this is that prior things always remain in act and intellect and essence and power in second things, which go out from first things through a restricting addition.
Si autem quaeritur, quae sunt propria? Dicimus quod illa sunt propria principia quidem et incomplexa, quae sine demonstratione accipiuntur esse, hoc est, quia sunt et quia vere sunt constantia in esse, sicut subjecta circa quae demonstrativa scientia per demonstrationem speculatur sive ratiocinatur ea quae sunt et insunt per se tali subjecto convenientia: sicut speculatur unitates arithmetica, de quibus multas speculatur passiones: similiter autem ut subjecta propria geometriae speculatur signa sive puncta et lineas, de quibus multas concludit passiones. Haec enim talia subjecta non recipiunt esse quo sunt in seipsis, sed recipiunt hoc esse quod habent ex hoc quod passiones insunt eis: eas autem quae sunt de numero passiones horum principiorum, scientiae particulares accipiunt sine demonstratione quid sit vel quid quidem significet unaquaeque, hoc est, quid significetur per nomen uniuscujusque passionis per se separatim. Quia aliter nec posset demonstrari de subjecto quid est et quid significet passio: quia diffinitio passionis aliquando medium est in demonstratione in qua concluditur passio de subjecto: et idem est concludere passionem de subjecto et concludere diffinitionem quae dicit quid est passio per essentiam, et non dicit propter quid est de subjecto: et ideo de passione oportet accipere quid per essentiam et quid significet per nomen.
Si enim arithmetica in demonstratione sua accipit quid est impar, aut quid est par, aut quid quadrangulus numerus, aut cubus: cum tamen omnia haec passiones sint absoluti numeri cujus passiones sunt par et impar, ut patet in primo Arithmeticae. Numeri autem figuralis et corporalis passiones sunt quadrangulus et cubus, sicut patet in secundo Arithmeticae Pythagorae. Geometria vero in demonstrationibus geometricis accipit et supponit quid est per diffinitionem dicentem quid, et quid significetur per nomen cum dicitur rationale sive rationalis: haec est enim, ut dicit Euclides in decimo Geometriae qua ratiocinamur, sicut est linea per aliquem numerum proportionalis: aut quid reflecti, quod etiam est passio lineae a geometria considerata et de linea probata, accipit et supponit quid sit per essentiam et quid significetur per nomen: aut quid sit curvare, quod est passio de linea probata, de qua etiam accipit quid curvum per diffinitionem dicentem quid, et quid significetur per nomen, quam etiam demonstrando considerat geometer.
Quod autem sint sive quia sint hae passiones, cum earum esse sit subjecto inesse, nec sit inesse vere nisi quia insint subjecto, demonstrant scientiae demonstrativae ex his principiis communibus, per quae omnia demonstrantur: et ex his, propriis scilicet sive communibus appropriatis, demonstrantur ea quae demonstrantur ut passiones de subjectis. Similiter autem his scientiis facit astrologia et circa subjectum et circa passiones et circa dignitates. Omnis enim scientia demonstrativa in demonstrando circa tria est, sicut ante diximus: et haec sunt quaecumque in demonstrativis esse ponuntur secundum quidditatem et nominis significationem. Haec autem tria sunt; genus quidem quod est in genere illo primum et per se subjectum et secundum ipsum, cujus generis subjecti illa scientia passionum quae per se illi subjecto insunt speculativa est. Et secundo illae quae communes dicuntur dignitates, quae verissimae et notissimae sunt: et ideo primae dignitates in scientia demonstrativa obtinent locum. Ex his enim ut primis et immediatis ut indemonstrabilibus demonstrant in potissima demonstratione quae est demonstratio ostensiva dicens propter quid et quid in medio per quod demonstrat. Tertium circa quod versantur scientiae demonstrativae, passiones, quarum, hoc est, de quibus accipit ipsa scientia quid significet per nomen et diffinitionem dicentem quid unaquaeque passionum.
But if it is asked what proper things are, we say that those are proper principles, indeed, and incomplex ones, which are received without demonstration to be, that is, because they are and because they truly are constant in being, as subjects about which demonstrative science through demonstration speculates or reasons concerning those things which are and are present per se as fitting such a subject. Thus arithmetic speculates about unities, about which it speculates many properties. Similarly, as proper subjects of geometry, it speculates signs or points and lines, about which it concludes many properties. For such subjects do not receive the being by which they are in themselves, but receive this being which they have from the fact that properties are present to them. But particular sciences receive without demonstration those properties of these principles which concern number, what each one is or what each one signifies, that is, what is signified by the name of each property per se separately. For otherwise it could not be demonstrated of the subject what the property is and what the property signifies, because the definition of the property is sometimes the middle in the demonstration in which the property is concluded of the subject; and to conclude the property of the subject is the same as to conclude the definition which says what the property is through essence and does not say why it is of the subject. And therefore concerning a property one must receive what it is through essence and what it signifies by name.
For if arithmetic in its demonstration receives what the odd is, or what the even is, or what a quadrangular number is, or what a cube is, although all these are properties of absolute number, whose properties are even and odd, as is clear in the first book of Arithmetic. But the properties of figural and corporeal number are quadrangle and cube, as is clear in the second book of the Arithmetic of Pythagoras. Geometry, however, in geometrical demonstrations receives and supposes what it is through a definition saying what, and what is signified by the name when rational or rational line is said. For this is, as Euclid says in the tenth book of Geometry, what we reason about, as a line proportional through some number. Or what to be reflected is, which is also a property of line considered by geometry and proved of line, it receives and supposes what it is through essence and what is signified by the name. Or what to be curved is, which is a property proved of line, concerning which also the geometer receives what the curved is through a definition saying what, and what is signified by the name, which he also considers by demonstrating.
But that these properties are, or because they are, since their being is to be present to a subject, and there is no true being-present unless they are present to a subject, demonstrative sciences demonstrate from these common principles through which all things are demonstrated; and from these, namely proper principles or common principles appropriated, those things are demonstrated which are demonstrated as properties of subjects. Astrology acts similarly to these sciences concerning subject, properties, and dignities. For every demonstrative science, in demonstrating, is concerned with three things, as we said before; and these are whatever things are posited in demonstrative matters according to quiddity and the signification of the name. These three are: first, the genus which in that genus is the first and per se subject according to itself, and the science of that subject genus is speculative of the properties which are present per se to that subject. Secondly, there are those things which are called common dignities, which are most true and most known; and therefore the first dignities hold the place in demonstrative science. For from these as first and immediate and indemonstrable things they demonstrate in the most powerful demonstration, which is ostensive demonstration stating the reason why and what exists in the middle through which it demonstrates. Third, the things about which demonstrative sciences are occupied are properties, of which, that is, concerning which, the science receives what each property signifies by name and by a definition saying what.
Quamvis autem omnes demonstrativae haec tria considerent et determinent, tamen quasdam demonstrativarum scientiarum nihil prohibet quaedam horum trium despicere, ita quod propter hoc quod sensui manifesta sunt, nec determinant quid sint et quid significent per nomen: eo quod non oportet studium ponere ad ea manifestanda, quia sensu manifesta sunt, et de quibus nemo est qui dubitet nisi sensu indigens: sicut nec oportet, quod petat a respondente genus illud supponere esse: si etiam secundum sensum sit manifestum quoniam est, et essentiam habet constantem: et hoc maxime est in physicis quibusdam, ut quod calidum sit aut frigidum: quia sensu manifestum est. In aliis autem in quibus secundum sensum non est manifestum, oportet cum respondente pertractare, quod tale subjectum esse supponatur, sicut in arithmetica quod numerus sit: non enim similiter sive aequaliter manifestum est in sensu, quod numerus sit et constans sit secundum esse et quod calidum et frigidum. Numerus enim fit ex iteratione unitatis secundum mentem acceptae: propter quod etiam numerus dicatur nutus mentis, vel memeris, hoc est, divisionis: quia, sicut dicit Aristoteles in tertio Physicorum 5, numerum cognoscimus divisione continui. Calidum autem et frigidum manifesta sunt in primo et communi animalis sensu, ita quod etiam apud bruta animalia haec nota sunt.
Similiter autem et passiones non est in pertractatione cum respondente recipere quid significent per nomen si sint apud sensum cujuslibet manifestae, sed etiam talia, sicut neque communia, hoc est, propositiones communes quae dignitates dicuntur, quoniam manifestae sunt, non recipit cum pertractatione significationis terminorum quid significent, sicut est illa dignitas de qua nemo dubitat, quod si aequalia, ab aequalibus demantur, quae relinquuntur, aequalia sunt. Et hoc ideo est, quia talia omnibus nota sunt. Sed tamen nihil minus, quamvis forte quaedam talium concernantur, cum natura, hoc est, de natura demonstrationis tria haec quae dicta sunt circa quae et demonstrant et demonstrando versantur demonstrationes, hoc est, subjectum circa quod demonstrat, et ea quae demonstrant de subjectis quae sunt passiones, et principia sive dignitates ex quibus demonstrant.
But although all demonstrative sciences consider and determine these three things, nevertheless nothing prevents some demonstrative sciences from passing over certain of these three, so that because they are manifest to sense they do not determine what they are and what they signify by name, because it is not necessary to place study upon manifesting those things, since they are manifest to sense, and there is no one who doubts about them unless he lacks sense. Nor, similarly, is it necessary that it ask the respondent to suppose that genus to exist, if also according to sense it is manifest that it is and has an essence constant in being; and this is especially so in certain physical matters, as that heat or cold exists, because it is manifest to sense. But in other things in which it is not manifest according to sense, one must treat with the respondent that such a subject is supposed to exist, as in arithmetic that number exists; for it is not similarly or equally manifest in sense that number exists and is constant according to being as it is that heat and cold exist. For number arises from the repetition of unity received according to the mind; on account of this number is also called a nod of the mind, or of memory, that is, of division, because, as Aristotle says in the third book of the Physics 5, we know number by the division of the continuum. But heat and cold are manifest in the first and common sense of an animal, so that these are known even among brute animals.
Likewise with properties, it is not necessary in discussion with the respondent to receive what they signify by name if they are manifest to the sense of anyone, but also such things, just as common things too, that is, the common propositions which are called dignities, because they are manifest, do not receive with a discussion of the signification of the terms what they signify, as is that dignity about which no one doubts, that if equal things are subtracted from equal things, the things which remain are equal. And this is because such things are known to all. Yet no less, although perhaps some of such things are concerned, with the nature, that is, concerning the nature of demonstration, are these three things which have been stated, about which demonstrations demonstrate and around which they are occupied in demonstrating: that is, the subject about which it demonstrates, the things which they demonstrate about subjects, which are properties, and the principles or dignities from which they demonstrate.
Caput II. De usu principiorum complexorum in demonstrationibus.
Sunt autem demonstrationis non tantum principia incomplexa, sed etiam complexa, quae sunt quinque, scilicet dignitas, suppositio, diffinitio, petitio, et quaestio. Et horum quidem quaedam sunt indemonstrabilia, sicut dignitas, et diffinitio: quaedam autem demonstrabilia sunt, sicut suppositio, petitio, et quaestio. Haec ergo demonstranda sunt in se quid sint, et comparanda sunt inter se per convenientiam et differentiam, et ostendenda sunt in relatione ad demonstrationem. Dicamus igitur primo quod dignitas est, ut dicit Boetius, propositio quam propter sui evidentiam quisque probat auditam. Et hoc dicit etiam Themistius: dicit enim quod dignitas est quam discens non habet a doctore, sed scit eam per naturam intellectus naturalem, quam habet apud seipsum: nec oportet quod habeat aliquid ad notitiam ejus nisi notitiam terminorum. Propter quod etiam communis animi conceptio vocatur: quia communiter ab omni habente rationem concipitur, et ei propter seipsam et non propter aliquid aliud demonstrans ipsam consentitur.
De hac igitur tali dignitate dicit Aristoteles, quod non est suppositio, neque est petitio: quia ipsa est tale aliquid in propositionibus, quod necesse est esse verum, non propter aliud, sed propter seipsum, cum sit primum, et non habeat aliquid ante se primum: et quod necesse est videri propter seipsum, cum ipsa notissima sit, et nihil habeat ante se notius ea, per quod possit notificari. Hujus autem causa est, quia dignitas non est vera vel nota ad rationem sive ratiocinationem quae exterius afferri possit ad probandum ipsam, sed est necessaria et nota ad rationem quae est in anima, hoc est, ad habitum naturalem intellectus qui est in anima. Quia sicut extra quaedam visibilia sunt non ad lucem alienam super se cadentem, sed sua propria luce manifesta sunt, sicut sol, sub cujus luce omnia alia videntur: sic est in intelligibilibus, quod quaedam propria luce videntur ab intellectu, quorum omnia alia manifestantur: et haec apud se habet intellectus, et sunt in ipso sicut prima instrumenta, per quae accipit omnium aliarum scientiarum, in omnibus scibilibus nihil accipit quod est contra illa, et per singularem intellectum accipit illa nullo alio indigens, nisi quod extendat intellectum ad illa, sicut visus accipit per se visibilia nullo indigens nisi quod convertat visum ad ea.
Quod autem ad talia non sit extrinsecus ratio dicens vel probans ea, per hoc patet quod nec syllogismus ad illa, qui potissimus est inter omnes extrinsecas rationes. Et ratio hujus est, quia priora non habent ex quibus doceri vel concludi possint. Alia autem ratio ad idem est: quia illa quae habent exterius rationem, talia sunt, quod semper ad ea contingit instare, secundum quod instantiam habere dicimus omne illud quod in aliquo accipitur et in aliquo non recipitur: nisi enim in aliquo non reciperetur a respondente et in aliquo reciperetur, non haberet extrinsecus rationem: sed universaliter et simpliciter in omnibus concederetur. Sed ad illa quae habent interius rationem acceptam, non semper contingit instare respondentem, ita quod illa in aliquo non accipiat, sed illa propter lucem rationis suae intrinsecam universaliter et specialiter accipit 6. Haec igitur et hac de causa sunt dignitates ut indemonstrabiles acceptae sine syllogismo et sine instantia.
Chapter II. On the use of complex principles in demonstrations.
But the principles of demonstration are not only incomplex but also complex, and these are five, namely dignity, supposition, definition, petition, and question. And of these, some are indeed indemonstrable, such as dignity and definition; but some are demonstrable, such as supposition, petition, and question. Therefore these things must be demonstrated, namely what they are in themselves, and they must be compared with one another according to agreement and according to difference, and they must be shown in their relation to demonstration. Therefore let us first say that a dignity is, as Boethius says, a proposition which, on account of its own evidentness, each person approves when it has been heard. And Themistius also says this; for he says that a dignity is what the learner does not have from a teacher, but knows through the nature of natural intellect, which he has within himself, and it is not necessary that he have anything for knowledge of it except the knowledge of its terms. On account of this it is also called a common conception of the mind, because it is commonly conceived by each and every one having reason, and consent is given to it on account of itself and not on account of some other thing demonstrating it.
Therefore concerning such a dignity Aristotle says that it is not a supposition, nor is it a petition, because it is something in propositions which must be true not on account of another thing but on account of itself, since it is first and does not have anything first before itself; and it must be seen on account of itself, since it itself is most known and has nothing before itself more known than itself, through which it could be made known. But the cause of this is that a dignity is not true or known by an account or reasoning which could be brought externally to prove it, but is necessary and known by the reason which is in the soul, that is, by the natural habit of intellect which is in the soul. For just as outside there are certain visible things which are not manifest by an alien light falling upon them, but by their own proper light, as the sun, under whose light all other things are seen, so it is in intelligible things, that certain things are seen by the intellect by their own proper light, by which all other things are manifested. And the intellect has these within itself, and they are in it as first instruments through which it receives all other sciences; among all knowable things it receives nothing which is contrary to them, and by singular understanding it receives them, needing nothing else except that it extend the intellect to them, as sight receives things visible per se needing nothing except that it turn sight toward them.
But that there is no extrinsic reason stating or proving such things is clear through this, that neither is there a syllogism directed to them, which is the most powerful among all extrinsic reasons. And the reason for this is that prior things do not have things from which they can be taught or from which they can be concluded. Another reason for the same thing is that those things which have a reason externally are such that it always happens that an instance can be raised against them, according as we say that everything which is received in one case and not received in another has an instance. For unless it were not received by the respondent in some case and were received in some other case, it would not have an extrinsic reason, but would be conceded universally and simply in all cases. But against those things which have a reason received internally, it does not always happen that the respondent can raise an instance so that he does not receive them in some case; rather he receives them universally and especially on account of the intrinsic light of his own reason 6. Therefore these things, and for this cause, are dignities, received as indemonstrable without syllogism and without instance.
Aliorum autem conceptuum sive acceptorum quaecumque quidem de numero talium ut demonstrabilia accipit intellectus, ita quod ipse opponens non demonstrat ea, cum tamen demonstrabilia sint: haec siquidem intellectus respondentis vel discentis accipit ut probabilia sibi, per hoc quod sibi sine syllogismo vera videntur, illa supponuntur et suppositiones vocantur. Et si quidem in hoc supponuntur, quod vera discenti videntur: tunc non suppositiones simpliciter, sed tantum ad illum qui utitur eis tanquam suppositis: et per hoc habetur, quod si utitur eis veris simpliciter, sunt suppositiones simpliciter: sicut saepe facit Euclides et ipse Aristoteles, qui probatis in uno, in aliis pro suppositionibus utuntur.
Si vero demonstrabilia quidem sint, sed talia quod discens vel audiens ea, nec opinetur de ipsis quod vera sunt, nec habeat opinionem contrarii de ipsis, et tamen opinans non demonstret ea: tunc illa talia opponentem oportet petere a discente, et dicuntur tales propositiones petitiones. Petitio enim est propositio non ex seipsa vel ex demonstratione supposita, vel ex opinione discentis, sed ab audiente petita. Et in hoc quidem differt suppositio et quaestio, quod quidem quaestio est quae est in contrarium discentis opinionis, et de qua dubitat audiens. Unde Boetius dicit quod quaestio est dubitabilis propositio. Aut quaestio est quaerens aliquid si hoc est: et si quis accipiat non demonstrans illud et utatur eo sine demonstratione qua demonstretur, cum tamen sit demonstrabile et dubitabile apud audientem. Haec igitur sunt suppositiones et petitiones et quaestiones.
Termini autem sive diffinitiones non sunt suppositiones, quia non sunt propositiones: nihil enim dicunt enuntiative esse per affirmationem, aut non esse per negationem: et ideo nec affirmativae nec negativae sunt propositiones: sed in propositionibus sicut in genere sunt suppositiones. Terminos autem sive diffinitiones.
But among other concepts or things received, whatever the intellect receives from the number of such things as demonstrable, in such a way that the objector himself does not demonstrate them although they are demonstrable: these indeed the intellect of the respondent or learner receives as probable to itself, through the fact that without a syllogism they seem true to it; those things are supposed and are called suppositions. And if indeed they are supposed in this, that they seem true to the learner, then they are not suppositions simply, but only in relation to the one who uses them as things supposed. And through this it is had that, if he uses them as true simply, they are suppositions simply, as Euclid often does and Aristotle himself, who use as suppositions in other matters things proved in one matter.
But if they are indeed demonstrable, yet such that the learner or hearer neither opines concerning them that they are true, nor has an opinion of the contrary concerning them, and nevertheless the one opining does not demonstrate them, then the objector must ask such things from the learner, and such propositions are called petitions. For a petition is a proposition supposed not from itself or from demonstration, or from the opinion of the learner, but requested from the hearer. And in this supposition and question differ, because a question is what is contrary to the opinion of the learner and about which the hearer doubts. Hence Boethius says that a question is a doubtful proposition. Or a question is asking something whether this is so; and if someone receives it without demonstrating it and uses it without a demonstration by which it may be demonstrated, although nevertheless it is demonstrable and doubtful for the hearer. Therefore these are suppositions and petitions and questions.
But terms or definitions are not suppositions, because they are not propositions: for they say nothing enunciatively to be through affirmation, or not to be through negation; and therefore they are neither affirmative nor negative propositions; but suppositions are in propositions as in a genus. But terms or definitions.
Notes
- The same is the opinion of Themistius and Alexander concerning science of singulars per accidens. (Eadem est sententia Themistii et Alexandri de scientia singularium per accidens.) ↩
- Aristotle, in the sixth book of the Metaphysics, text-commentary 4. (ARISTOTELES, In 6 primae philosophiae, tex. com. 4.) ↩
- The same author, in the second book On the Soul, text-commentary 12. (ID. In 2 de Anima, tex. com. 12.) ↩
- This is the opinion of the Lincolnian, and it is the second exposition. (Ista est sententia Linconiensis et est secunda expositio.) ↩
- Aristotle, in the third book of the Physics, text-commentary 68. (ARISTOTELES, In 3 Physicorum, tex. com. 68.) ↩
- Compare Themistius on these things, because he explains this part otherwise. P. J. (Cf. Themistium his, quia aliter exponit hanc partem. P. J.) ↩
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