Volume 1 · pp. 203–204
End of Chapter V and Chapter VI. How certain quantities have position in their parts, and certain do not
autem locati est intra locatum, et est aliquid ejus, et est contenta non continens. Actu autem mensurandi: quia omnes diametri distantiæ quæ vocatur locus, sunt mensuræ mensurantes in longum, latum et profundum, et nulla est mensurata. Omnes autem diametri ejus quod locatum est, sunt diametri mensuratæ et contentæ, et sic mensuratum non mensurat distantiam loci, sed quantitatem corporis non ut locatam sed secundum quod ipsum est quantum.
Sic autem accipiendo locum, est locus mathematica quantitas, copulata punctis usque ad centrum locati ejus; quia diametri secundum esse sunt eadem cum diametris ad idem centrum copulatis: quia centrum loci et centrum locati idem sunt secundum esse, differunt autem secundum actum et rationem: et quia distantia locati est continua quantitas, necesse est quod locus sit continua quantitas in hoc differens ab aliis quod non ex se, sed ex distantia diametrorum corporis usque ad centrum suam habet copulationem. Aliæ autem distantiæ linearum et superficierum et corporum ex seipsis et inter seipsa suam habent copulationem et suas copulationes. Sunt enim diametri secundum aliquid differentes quæ diversa habent suæ protractionis principia: diameter enim distantiæ loci suæ productionis principium habet superficiem extra continentem: diameter autem distantiæ quæ est distantia locati, suæ protractionis principium habet superficiem locati intra locatum existentem: unum autem est centrum utrorumque secundum esse et non secundum actum, ut dictum est. Et sic locus erit diversa secundum modum istum quantitas ab omnibus aliis speciebus continuæ quantitatis: sic enim mathematica abstractione separatur loci quantitas a quantitate locati. Et sic est verum quod dicitur, quod singulæ particulæ loci continent singulas particulas corporis locati. Et ideo tam loci quam corporis locati particulæ ad unum terminum copulantur. Et sic loquendo de loco mathematicis dandus est locus, ut in prima philosophia dicit Aristoteles.
Est autem aliter consideratus locus quando est de consideratione naturalis Philosophi, secundum quod scilicet locus est, ad quem et ex quo est motus: sic enim est superficies continentis corporis habens in se virtutes omnium continentium tam cælorum quam elementorum circumdantium, per quas virtutes generat et salvat id quod locatum est in ipso: et de hoc loco hic Aristoteles non loquitur, quia per illas proprietates non est in genere quantitatis, quamvis in seipso sit quantus. Primo igitur modo considerando locum continua quantitas erit locus, eo quod particulæ ejus quæcumque ad unum communem terminum copulantur, et est speciale genus quantitatis propter causam in ante habitis assignatam.
CAPUT VI.
Qualiter quædam quantitates habent positionem in partibus, et quædam non.
Amplius autem aliqua de numero specierum quantitatis continuæ ex particulis quæ in eis sunt, constant ex talibus positionem habentibus in continuo cujus sunt particulæ. Alia vero quæ sunt de genere et de numero quantitatum, constant etiam ex particulis positionem non habentibus in continuo vel discreto, cujus sunt particulæ.
Positionem autem habere tria concernit, scilicet ut assignetur ubi in continuo sita sit particula, et ut una pars permaneat stans in eodem continuo cum altera, et ut teneatur cum altera copulatione. Et quodcumque horum trium deficiat, non habebit in partibus positionem. Positio enim dicit ordinem, et propter hoc oportet ut assignetur ubi in continuo posita sit. Positio autem in continuo dicit permanentiam, quia positum est fixum immobile secundum esse, et ideo oportet ut dicat permanentiam. Positio autem in continuo dicit copulationem unius cum
of the located thing is inside the located thing, and is something of it, and is contained, not containing. But by the act of measuring: because all the diameters of the distance which is called place are measures measuring in length, breadth, and depth, and none is measured. But all the diameters of that which is located are measured and contained diameters; and thus the measured thing does not measure the distance of place, but the quantity of the body, not as located but according as it itself is quantified.
But taking place in this way, place is a mathematical quantity, joined by points all the way to the center of the located thing; because the diameters according to being are the same as the diameters joined to the same center, because the center of the place and the center of the located thing are the same according to being, but they differ according to act and account. And because the distance of the located thing is continuous quantity, it is necessary that place be continuous quantity, differing from others in this, that it has its joining not from itself, but from the distance of the diameters of the body all the way to its center. But the other distances of lines and surfaces and bodies have their joining and their joinings from themselves and among themselves. For diameters are different according to something, which have diverse principles of their drawing-out: for the diameter of the distance of place has as the principle of its production the surface containing externally; but the diameter of the distance which is the distance of the located thing has as the principle of its drawing-out the surface of the located thing existing inside the located thing. But the center of both is one according to being and not according to act, as has been said. And thus place according to this mode will be a quantity different from all the other species of continuous quantity; for thus by mathematical abstraction the quantity of place is separated from the quantity of the located thing. And thus what is said is true, that the individual particles of place contain the individual particles of the located body. And therefore the particles both of place and of the located body are joined to one term. And speaking thus of place, a place must be given to mathematical things, as Aristotle says in First Philosophy.
But place is considered otherwise when it belongs to the consideration of the natural Philosopher, namely according as place is that to which and from which motion is. For thus it is the surface of the containing body, having in itself the powers of all the containing things, both of the heavens and of the surrounding elements, through which powers it generates and preserves that which is located in it; and about this place Aristotle does not speak here, because through those properties it is not in the genus of quantity, although in itself it is quantified. Therefore, considering place in the first mode, place will be continuous quantity, because all its particles whatsoever are joined to one common term, and it is a special genus of quantity on account of the cause assigned in the things held before.
CHAPTER VI.
How certain quantities have position in their parts, and certain do not.
Furthermore, certain things from the number of the species of continuous quantity, from the particles which are in them, consist of such particles having position in the continuous of which they are particles. But other things which are of the genus and of the number of quantities also consist of particles not having position in the continuous or discrete thing of which they are particles.
But to have position concerns three things, namely that it be assigned where in the continuous the particle is situated, and that one part remain standing in the same continuum with another, and that it be held together with another by joining. And whatever of these three is lacking will not have position in its parts. For position says order, and on account of this it must be assigned where it has been placed in the continuous. But position in the continuous says permanence, because what has been placed is fixed, immobile according to being, and therefore it must say permanence. But position in the continuous says the joining of one with
alio, et ideo oportet ut dicat copulatorum connexionem per communem terminum, qui aliquid est utriusque copulatorum.
Ex his patet quod lineæ quidem particulæ positionem habent in continuo quod est linea, et ad se invicem sunt habentes positionem. Singulum namque signatum in particulis lineæ situm est alicubi, vel ante in linea, vel post: et permanentibus partibus sic positis sic habes unde sumas partem permanentem, et assignes in continuo unumquodque quod est de numero particularum, ubi sive in qua parte ordine partium in loco situm est suppositum: et hoc planum est valde et facile ad intelligendum. Et habes unde assignes ad quam particulam, et per quem communem terminum cæteræ quæcumque particulæ sumptæ fuerint, copulantur: quod planissimum est ex prædictis. Similiter autem et particulæ plani sive superficiei quamdam secundum latitudinem habent positionem. Similiter namque sicut et in linea facile ostenditur, ubi jacent in superficie positæ, et quæ sunt quæ copulantur ad invicem, et quo copulante ut communi termino. In linea enim copulat punctum: in superficie autem copulat linea et communis terminus. Soliditatis quoque sive corporis particulæ eodem modo et eadem de causa habent positionem et permanentiam et copulationem. Et loci particulæ secundum quod mathematice sumitur locus, talem in partibus habent positionem et permanentiam et copulationem.
In numero autem in partibus positionem et permanentiam et copulationem non potest quisquam per intellectum perspicere: cujus causa est, quod natura partium numeri repugnat ad tria quæ dicta sunt: et ideo non invenitur quod particulæ numeri positionem habeant ad invicem. Primo enim deficit situs in continuo alicubi, quia unitas (cujus iteratio facit numerum) non est indivisibile habens in continuo positionem, sed potius a natura positionis penitus separatum: et ideo non potest dici quod unitas vel pars numeri sit sita alicubi in continuo. Nec potest dici quod aliquæ particulæ numeri ad se invicem in aliquo (quod aliquid utriusque sit) habeant connexionem sive copulationem, sicut in capitulo de discretione numeri est probatum: sed omnes particulæ manentes discretæ et separatæ ab invicem, ad unum ultimum habent ordinem, cujus additione perficitur species numeri.
Sed nec ea quæ sunt particulæ temporis habent in suis partibus positionem, quia quamvis habeant connexionem et terminum communem in quo connectuntur et continuantur, tamen nullam habent in esse permanentiam: quod autem in esse non permanet, de illo non potest dici ubi situm sit. Nusquam enim situm est, quod secundum esse non est.
Sed secundum hoc videtur, quod non habeat copulationem vel connexionem, quia quod secundum esse non est, nulli potest copulari. Sed ad hoc dicendum est, quod in successione est quod copulatur, et in successione esse, est secundum aliquid esse. Est et præteriti in hoc quod copulat usque ad quid sicut ad finem (quem intra se ut aliquid sui habet) ipsa successio et similiter futurum cui copulantur est secundum successionis principium: quod iterum ipsa successio futuri intra se habet ut aliquid sui, quia est successionis principium: et sic in instanti (quod est medium, scilicet finis præteriti et principium futuri) copulantur præteritum et futurum 1. Et aliquo modo est præteriti successio in præsenti ut in termino, et successio futuri est in eodem sicut in principio. Sic ergo copulationem habent partes temporis, et tamen nullam habent permanentiam neque positionem.
Sed in hoc conveniunt cum partibus numeri, quod ordinem quemdam dices
another, and therefore it must say the connection of the things joined through a common term, which is something of both of the joined things.
From these things it is clear that the particles of line indeed have position in the continuous thing which is line, and toward one another they have position. For each marked thing among the particles of line is situated somewhere, either before in the line or after; and when the parts remain thus positioned, thus you have from where you may take a permanent part, and may assign in the continuous each thing which is of the number of the particulars, where or in what part, in the order of parts, the supposed thing is situated in place; and this is very plain and easy to understand. And you have from where you may assign to what particle, and through what common term, the other particles, whatever have been taken, are joined; which is most plain from the aforesaid things. Similarly also the particles of a plane or surface have a certain position according to breadth. For similarly, just as also in line, it is easily shown where they lie when placed in the surface, and which are the things that are joined to one another, and by what joining thing as by a common term. For in line a point joins; but in surface a line joins and is the common term. The particles of solidity also, or of body, in the same mode and from the same cause have position and permanence and joining. And the particles of place, according as place is taken mathematically, have such position and permanence and joining in their parts.
But in number no one can discern by understanding position and permanence and joining in the parts; the cause of this is that the nature of the parts of number is repugnant to the three things that have been said; and therefore it is not found that the particles of number have position toward one another. For first, situation somewhere in a continuum is lacking, because unity, whose repetition makes number, is not an indivisible having position in a continuum, but rather something wholly separated from the nature of position; and therefore it cannot be said that unity or a part of number is situated somewhere in a continuum. Nor can it be said that some particles of number have connection or joining toward one another in something which is something of both, as was proved in the chapter on the discreteness of number; but all the particles, remaining discrete and separated from one another, have order toward one last thing, by whose addition the species of number is perfected.
But neither do those things which are particles of time have position in their parts, because although they have connection and a common term in which they are connected and continued, nevertheless they have no permanence in being; but concerning what does not remain in being, it cannot be said where it is situated. For nowhere is that situated which according to being is not.
But according to this it seems that it does not have joining or connection, because what according to being is not can be joined to nothing. But to this one must say that in succession there is what is joined, and to be in succession is to be according to something. And there is also of the past, in this that it joins up to something as to an end, which succession itself has within itself as something of itself, and similarly the future to which they are joined exists according to the principle of succession; and again the very succession of the future has this within itself as something of itself, because it is the principle of succession; and thus in the instant, which is the middle, namely the end of the past and the principle of the future, past and future are joined 1. And in some mode the succession of the past is in the present as in a term, and the succession of the future is in the same as in a principle. Thus therefore the parts of time have joining, and nevertheless they have no permanence nor position.
But in this they agree with the parts of number, that you would say a certain order
Notes
- Note nevertheless that the succession of the imperfect past is better in the present than the succession of the perfect past and of the future, and therefore ampliations are given with respect to the first, but not with respect to the others. (Nota tamen quod successio præteriti imperfecti melius est in præsenti quam successio præteriti perfecti et futuri, et ideo respectu primi dantur ampliationes, non autem respectu aliorum.) ↩
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