Volume 2 · pp. 85–87

Treatise III, Chapter VII Continuation. On The Difference Between Demonstration That And Why In Diverse Sciences

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Latin (Borgnet)English

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quod una scientia speculatur propter quid, et alia scientia speculatur quia. Hujusmodi autem scientiae sunt quaecumque sic se habent ad invicem, quod alterum (hoc est, subjectum quod in una earum determinatur) est sub altero quod in altera scientia determinatur, sicut subalternatum est sub subalternante, sicut se habent speculativae sive perspectivae scientiae ad geometriam. Et sicut se habet scientia machinativa de machinis quae proprio nomine vocatur scientia de ingeniis, ad stereometriam, hoc est, ad illam partem geometriae, quae est de stereis, hoc est, de mensura solidorum sive corporum, quae traditur in undecimo et duodecimo Euclidis: machinativa enim sive de ingeniis scientia sequitur mensuras solidorum in ponderibus et quantitate et erectione ingeniorum et proportione. Et sicut se habet harmonica in musicis cantibus et instrumentis, monochordo scilicet, vigella et lyra, et hujusmodi ad arithmeticam: quia proportiones numerorum considerat arithmeticus: quam proportionem in cantu et divisione chordarum ponit harmonica: sicut passiones lineae considerat geometria, quas in linea visuali (quae radius vocatur) ponit perspectiva. Et sicut scientia navalis (quae apparentia vocatur) se habet ad astrologiam quantum ad illam partem quae est de angulis et de diversitate aspectus: ad hanc enim navalis secundum apparentiam in stellis vias navium imaginatur et dirigit. Vel forte melius ad lunae quartas: quia ex illis elementum aqueum movetur ad effluxum et refluxum, et ex illis oriuntur venti secundi vel non secundi de fundo maris. Et hoc apparet in crispatione undarum. Ad quamcumque enim plagam moventur crispantes undulae, primo in aquae superficie, ad illam flabit ventus in puncto proximae quartae, maxime si ibi erit luna. Quartas autem voco puncta Orientis, medii caeli, Occidentis, et angulum mediae terrae.

Harum autem scientiarum subalternantium et subalternatarum quaedam fere sunt univocae, hoc est, de univoco subjecto et passionibus. Et dico fere propter id quod addit subalternata super subalternantem, sicut linea visualis addit supra lineam, et sicut angulus reflexionis ad speculum addit supra angulum simpliciter qui est contactus duarum linearum. Hoc autem modo fere univocae sunt astrologia mathematica quae considerat diversitates motuum et praeventionum et conjunctionum et aspectuum in stellis, et navalis astrologia quae fere idem considerat referendo hoc ad actum et vias navigantium. Etiam sic fere univocae sunt harmonica mathematica sive musica, et illa quae est secundum auditum in chordis et fidibus et instrumentis musicis: hoc enim in omnibus talibus ipsum quia est scire sensibilium, hoc est, subjectum sensibile et passiones considerantium, sed propter quid considerare habent mathematicae quae non considerant sensibile, sed abstractum a sensibili secundum rationem diffinitivam quae medium est in demonstratione.

Attendendum autem quod passiones subjecti possunt considerari dupliciter, scilicet in quantum sunt subjecti quod est per se et secundum ipsum, vel in quantum sunt hujus subjecti determinati ad hanc naturam vel materiam. Quantitatem autem per omnes differentias quantitatis (quae sunt linea, superficies, et corpus, quae sunt differentiae quantitatis continuae primae secundum ipsam naturam quantitatis exeuntes ab ea) habet considerare geometer: et quia passiones quantitatis causantur ab ipsa natura quantitatis secundum se et secundum seipsam, ideo geometer habet dicere propter quid in omnibus illis sive sint abstractae, sive sint ad materiam determinatae: tales enim passiones sunt quantitatis secundum quod est quantitas, et non istius vel illius secundum quod est hoc vel illud secundum ipsum, sed sunt quantitatis in isto. Si autem passiones aliquae sunt, quae causantur ab hoc vel illo secundum quod hoc vel illud, in illis non dicit propter quid scientia superior, sed inferior.

that one science considers the why, and another science considers the that. But sciences of this sort are whatever sciences are so related to one another that the one, that is, the subject which is determined in one of them, is under the other which is determined in the other science, as what is subalternated is under what subalternates, just as the speculative or perspective sciences are related to geometry. And as the machinative science concerning machines, which by its proper name is called the science concerning devices, is related to stereometry, that is, to that part of geometry which concerns solids, that is, the measure of solid things or bodies, which is handed down in the eleventh and twelfth books of Euclid. For machinative science, or the science concerning devices, follows the measures of solids in weights and quantity and the erection of devices and proportion. And as harmonics in musical songs and instruments, namely in the monochord, the viol, the lyre, and things of this kind, is related to arithmetic, because the arithmetician considers proportions of numbers; harmonics posits that proportion in song and in the division of strings. Just as geometry considers the properties of a line, which perspective posits in the visual line, which is called a ray. And as naval science, which is called appearance, is related to astrology with respect to that part which concerns angles and diversity of aspect. For naval science, according to appearance in the stars, imagines and directs the routes of ships with respect to this. Or perhaps it is better related to the quarters of the moon, because from them the aqueous element is moved to ebb and flow, and from them favorable or unfavorable winds arise from the bottom of the sea. And this appears in the ruffling of waves. For toward whatever region the rippling little waves are moved first on the surface of the water, toward that region the wind will blow at the point of the next quarter, especially if the moon will be there. But I call the quarters the points of the East, the mid-heaven, the West, and the angle of the middle of the earth.

But of these subalternating and subalternated sciences, some are almost univocal, that is, concerning a univocal subject and properties. And I say "almost" on account of that which the subalternated science adds above the subalternating science, just as the visual line adds above line, and as the angle of reflection toward a mirror adds above angle simply, which is the contact of two lines. In this way mathematical astrology is almost univocal, which considers diversities of motions and preventions and conjunctions and aspects in the stars, and naval astrology, which considers almost the same thing by referring it to the act and routes of those sailing. In this way also mathematical harmonics or music is almost univocal, and that which is according to hearing in strings and viols and musical instruments. For in all such sciences, knowing the that belongs to sensible things, that is, to those considering the sensible subject and properties; but the mathematical sciences, which do not consider the sensible but the abstracted from the sensible according to the definitive account which is the middle in demonstration, have the consideration of the why.

But it must be attended to that the properties of a subject can be considered in two ways, namely insofar as they are of the subject which is per se and according to itself, or insofar as they are of this subject determined to this nature or matter. But the geometer has to consider quantity through all the differences of quantity, which are line, surface, and body, which are the differences of first continuous quantity proceeding from it according to the very nature of quantity. And because the properties of quantity are caused by the very nature of quantity according to itself and according to itself alone, therefore the geometer has to state the why in all those cases, whether they are abstract or determined to matter. For such properties belong to quantity according as it is quantity, and not to this or that according as it is this or that according to itself, but they are properties of quantity in this thing. But if there are any properties which are caused by this or that according as it is this or that, in those the superior science does not state the why, but the inferior science does.

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Hujus exemplum est radius qui est linea: quia linea secundum quod est linea, habet rectum esse vel curvum: et per contactum alterius facere angulum et anguli quantitatem ex hoc quod perpendiculariter vel oblique contingit: quae omnia quia quantitatis sunt, secundum ipsum considerat secundum causam geometer. Sed in quantum radius habet reflecti ad tersum et politum, et in speculo concavo reflecti ad medium, et in rotundo pervio reflecti ad punctum oppositum: quae passiones quia causantur non a linea secundum quod quantitas est, ideo lineae secundum ipsam non conveniunt, nec possunt a geometra causari ex propriis principiis quantitatis: et ideo non dicit propter quid in illis: sed sunt passionis lineae visualis secundum ipsum quod visualis est: et ideo propter quid in talibus dicitur a perspectiva. Nec etiam in hoc, si quis subtiliter inspiciat, subalternatur perspectiva geometriae: quinimo scientia est per se considerativa passionum sui subjecti. Et hoc est quod dicit Aristoteles, quod mathematicorum, astrologorum et harmonicorum et non sensibilium est scire propter quid; eo quod illi in talibus habent demonstrationem facere ex quibus tales causae talium passionum accipi possunt.

Sunt autem quidam dicentes quod tria exiguntur ad hoc quod una scientia alteri subalternetur. Et dicunt primum esse quod subjectum subalternatae concipiat in se subjectum subalternantis cum additione differentiae contrahentis ipsum ad naturam vel materiam determinatam, ut linea visibilis se habet ad lineam, et pyramis visibilis se habet ad pyramidem, et circulus visibilis qui est basis pyramidis, se habet ad circulum, et diameter pyramidis se habet ad diametrum, et triangulus pyramidis ex radiis exeuntibus ad illuminatum, se habet ad triangulum cujus motu imaginativo generatur pyramis, et sic numerus modulorum se habet ad numerum, et sic de aliis. Secundum dicunt esse, quod id quod addit inferior et subalternata scientia, oportet quod contrahat ipsum ad rem alterius naturae et generis: quia aliter scientia de speciebus et differentiis divisivis subalternata esset scientiae de genere, quod falsum esse dicunt. Tertium autem dicunt esse, quod id quod addit supra subjectum superioris, non sit causatum vel procedens a substantia et natura subjecti superioris: aliter scientia paris et imparis subalternaretur scientiae de numeris, quod falsum est, cum sint una et eadem scientia, et sic etiam si hoc esset verum, prima philosophia sibi subalternaret omnes alias, eo quod ex suo subjecto stabilit subjecta scientiarum omnium aliarum.

Haec igitur sunt quae dicuntur communiter et satis subtiliter et bene: sed quantum ex natura demonstrationis est, et ex dictis Aristotelis perpendi potest, non exigit nisi quod subalternans et subalternata sint circa idem in substantia subjectum, et diversum secundum esse: quia aliud habet esse in suis principiis consideratum, et alias passiones fluentes ab ipso secundum esse quod habet in suis principiis substantialibus acceptum, quod secundum hoc ipsum causatur ab ipso: et aliud habet esse secundum hoc quod relatum est ad hoc vel illud, secundum quod in inferiori scientia consideratur: et quia accidit ei esse in hoc, et sic, ideo causas illarum passionum quae insunt ei secundum se, non potest scire scientia inferior, sed expectat eas superiori, sicut patet: quia sic medicina considerat corpus humanum, non secundum quod est magnum, sed secundum quod est ad complexionis aequalitatem commixtum: inveniens tamen impeditivum in ipso ex corruptione compositionis et continuitatis et figurae, causam illius medii non considerat, sed scientia superior, geometria, vel alia: sicut maxime scientia de ponderibus et distantia graduum exigitur in medicina, ut in libro de Gradibus dicit Alchindus. Et hoc modo in iride quam secundum causam considerat physicus, secundum causam dico suae generationis in nube aquosa hoc modo: secundum causam autem reflexionis causata ab opposito sole consideratur a perspectivis.

An example of this is a ray, which is a line, because line according as it is line has being straight or curved, and by the contact of another thing has to make an angle and the quantity of an angle from the fact that it touches perpendicularly or obliquely. Because all these are matters of quantity, the geometer considers them according to cause and according to the line itself. But insofar as a ray has to be reflected toward something smooth and polished, and in a concave mirror to be reflected to the middle, and in a round transparent body to be reflected to the opposite point, since these properties are caused not by line according as it is quantity, therefore they do not belong to line according to itself, nor can they be caused by the geometer from the proper principles of quantity. And therefore he does not state the why in them. But they belong to the property of a visual line according to the very thing that it is visual; and therefore in such matters the why is stated by perspective. Nor even in this, if someone inspects subtly, is perspective subalternated to geometry; rather, it is a science considering per se the properties of its own subject. And this is what Aristotle says: that knowing why belongs to mathematicians, astrologers, and harmonicians, and not to sensible things, because in such matters they have the demonstration made from things from which such causes of such properties can be received.

But there are certain persons saying that three things are required for one science to be subalternated to another. And they say the first to be that the subject of the subalternated science should conceive within itself the subject of the subalternating science, with the addition of a difference contracting it to a determinate nature or matter, as visible line is related to line, and visible pyramid is related to pyramid, and visible circle, which is the base of a pyramid, is related to circle, and the diameter of a pyramid is related to diameter, and the triangle of a pyramid from rays going out to the illuminated thing is related to the triangle by whose imaginative motion the pyramid is generated, and so the number of measures is related to number, and so concerning the others. They say the second to be that what the inferior and subalternated science adds must contract the subject to a thing of another nature and genus, because otherwise a science concerning species and divisive differences would be subalternated to the science concerning the genus, which they say is false. But they say the third to be that what is added above the subject of the superior science is not caused by or proceeding from the substance and nature of the subject of the superior science; otherwise the science of even and odd would be subalternated to the science concerning numbers, which is false, since they are one and the same science. And thus also, if this were true, first philosophy would subalternate all other sciences to itself, because from its own subject it establishes the subjects of all other sciences.

Therefore these are the things which are commonly said, and with sufficient subtlety and well. But so far as it can be weighed from the nature of demonstration and from the sayings of Aristotle, it requires only that the subalternating and the subalternated science be concerned with the same subject in substance, and with a diverse subject according to being. For one thing has being considered in its principles, and has other properties flowing from it according to the being which it has when accepted in its substantial principles, because according to this very being it is caused by itself. And it has another being according as it is related to this or that, according as it is considered in the inferior science. And because it happens to it to be in this thing, and in this way, therefore the inferior science cannot know the causes of those properties which are present in it according to itself, but awaits them from the superior science, as is clear. For thus medicine considers the human body, not according as it is large, but according as it is mixed to equality of complexion; nevertheless, when it finds in that body something impeditive from corruption of composition and continuity and figure, it does not consider the cause of that middle, but the superior science does, geometry or another. Thus especially the science concerning weights and distance of degrees is required in medicine, as Alchindus says in the book On Degrees. And in this way, concerning the rainbow which the physicist considers according to cause, I mean according to the cause of its generation in a watery cloud in this mode, according to the cause of reflection caused by the opposite sun it is considered by perspectivists.

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Ex hoc patet quare quaedam subalternantes et subalternatae sunt vere univocae, in quibus scilicet inferior recipit nomen superioris et rationem: numerus enim relatus ad morulas et modulos sonorum est numerus, et linea visualis est linea non additione contrahente. Sed quaedam sunt, in quibus subjectum superioris non nomine et ratione praedicatur de subjecto inferioris, sicut magnitudo continua non praedicatur de corpore humano: quia non consideratur a medico: et id quod addit, non est de quantitate nec de principiis quantitatis: sed omnino praeter ipsam, quamvis sit in ipsa secundum esse aliquod.

His igitur sic notatis, planum est quod dicitur: et sicut una considerat quia, et altera propter quid in talibus: ita contingit multoties, quod considerantes ipsum propter quid, nesciunt ipsum quia. Et hujus causam quidam dicunt esse, quia non petractant. Sed vera causa est, quia universale (ex quo accipitur causa propter quid) cognoscunt: et ex illo particulare (quod ex universali in propria natura cognosci non potest) pertractare ex propriis non possunt: et hoc est quod additur: quia superius sicut universale considerantes, multoties quaedam singularium nesciunt: quia in subjecto universali non habent unde hoc ex principiis universalis facere possint: et etiam propter id quod particulari et propriae naturae non intendunt ex principiis universalis, unde sunt considerantes. Haec autem sunt et talia quaecumque per additionem ad superioris scientiae subjectum alterum quiddam sunt: quia id quod additur subjecto superioris scientiae, trahitur in alterum genus, et non causatur illud a principiis subjecti universalis scientiae et superioris: illae enim scientiae utuntur speciebus et formis universalis secundum substantiam et materiam sui subjecti, hoc est, secundum quod relatae sunt ad hoc vel illud: mathematica autem quae simpliciter mathematica et abstracta sunt (sicut ea quae sunt scientiae altioris) sunt simpliciter circa species abstractas: tales enim non sunt de subjecto aliquo sensibili: quia quamvis geometrica aliqua de subjecto aliquo sensibili sint, non sunt tamen de illo secundum quod est subjectum aliquod sensibile, sicut patet per antedicta.

In talibus autem inferior etiam subalternat alterum ad minus quantum ad modum: et hoc est quod dicit, habet autem se ad speculativam sive perspectivam sicut se habet perspectiva ad geometriam: ita enim se habet alia quae sub ista (quoad aliquid) ad istam, sicut perspectiva se habet ad geometriam, quamvis non simpliciter subalternetur eidem: sicut est id quod est de iride, de quo (sicut dictum est) ipsum quia physici est scire, qui iridem quantum ad materiam secundum quod est in nube aquosa habet considerare: sed propter quid est iris quantum ad reflexionem radiorum solis, est considerare perspectivi: aut simpliciter, quia ex principiis suae scientiae dicit propter quid est iris: aut secundum hoc quod est doctor vel secundum doctrinam, quia docendis est scire causam propter quid.

Multae autem etiam earum quae non sunt sibi invicem scientiarum, ut subalternans et subalternata, sic se habent ad invicem, sicut medicina ad geometriam se habet in aliquo: quod enim vulnera circularia tardius sanantur, medici chirurgici est scire: sed propter hoc est sic scire geometrae. Hoc enim ex hoc est, quod circulus est capacissima figurarum: et ex hoc est capacissima, quia latera ipsius ad circulum undique distantia sunt et magis difficulter unibilia, et ex hoc sequitur quod tardius sanabilia.

From this it is clear why certain subalternating and subalternated sciences are truly univocal, namely those in which the inferior science receives the name and account of the superior. For number related to pauses and measures of sounds is number, and visual line is line without a contracting addition. But there are some sciences in which the subject of the superior science is not predicated in name and account of the subject of the inferior, as continuous magnitude is not predicated of the human body, because it is not considered by the physician; and that which it adds is not from quantity nor from the principles of quantity, but is altogether besides it, although it is in it according to some being.

Therefore, with these things noted in this way, what is said is plain: and as one science considers the that, and another the why in such matters, so it happens many times that those considering the why itself do not know the that itself. And certain persons say that the cause of this is because they do not treat it throughout. But the true cause is that they know the universal, from which the cause why is received, and from that universal they cannot treat the particular, which cannot be known from the universal in its proper nature, from proper principles. And this is what is added: because, considering above as the universal, they many times do not know certain singular things, because in the universal subject they do not have something from which they can do this from the principles of the universal, and also because they do not direct attention to the particular and proper nature from the principles of the universal from which they are considering. But these things are also whatever such things through addition to the subject of the superior science are something other, because that which is added to the subject of the superior science is drawn into another genus, and it is not caused by the principles of the universal and superior science's subject. For those sciences use the species and forms of the universal according to the substance and matter of their own subject, that is, according as they are related to this or that. But the mathematical things which are simply mathematical and abstract, such as those which belong to the higher science, are simply concerned with abstract species. For such things are not concerned with some sensible subject, because although some geometrical things are about some sensible subject, nevertheless they are not about it according as it is some sensible subject, as is clear through the things said before.

But in such matters the inferior science also subalternates another science at least with respect to mode. And this is what he says: it is related to the speculative or perspective science as perspective is related to geometry. For in this way another science which is under this one with respect to something is related to this one, as perspective is related to geometry, although it is not simply subalternated to the same science. Such is the matter concerning the rainbow, concerning which, as was said, it belongs to the physicist to know the that, who has to consider the rainbow as to matter according as it is in a watery cloud; but as to why the rainbow is, with respect to the reflection of the rays of the sun, it belongs to the perspectivist to consider it, either simply because from the principles of his own science he states why the rainbow is, or according to the fact that he is a teacher or according to teaching, because for those being taught it belongs to him to know the cause why.

But many also of those sciences which are not mutually related as subalternating and subalternated are so related to one another, as medicine is related to geometry in some respect. For that circular wounds are healed more slowly belongs to the surgical physician to know; but that on account of which this is so belongs to the geometer to know. For this comes from the fact that the circle is the most capacious of figures; and it is most capacious from this, that its sides are everywhere distant toward the circle and are more difficult to unite, and from this it follows that they are more slowly healable.