Authors/Thomas Aquinas/physics/L5/lect5

Lecture 5 The definitions of “in contact,” “consecutive,” “continuous”

Latin	English
	Lecture 5 The definitions of “in contact,” “consecutive,” “continuous”
lib. 5 l. 5 n. 1 Postquam philosophus divisit mutationem et motum in suas species, hic procedit ad determinandum de unitate et contrarietate motus in suas species. Et circa hoc duo facit: primo praemittit quaedam necessaria ad sequentia; secundo prosequitur principale propositum, ibi: unus autem motus et cetera. Circa primum tria facit: primo dicit de quo est intentio; secundo exequitur propositum, ibi: simul igitur etc., tertio recapitulat, ibi: quid quidem igitur et cetera. Dicit ergo primo, quod dicendum est post praedicta, quid est simul, et quid est extraneum vel separatum, et quid est tangere, et quid est medium, et quid consequenter, et quid habitum, et quid continuum, et in quibus haec nata sunt esse. Praemittit autem haec, quia horum definitionibus utitur in demonstrationibus consequentibus per totum librum; sicut et in principio Euclidis ponuntur definitiones, quae sunt sequentium demonstrationum principia.	684. After dividing change and motion into its species, the Philosopher now begins to discuss the senses in which motion is said to be one, and the senses in which motions are said to be contrary. About this he does two things: First he establishes a background of preliminary notions that will be of use; Secondly, he pursues his main objective, at L. 6. About the first he does three things: First he states his intention; Secondly, he pursues it, here at 684; Thirdly, he makes a summary, at 694. He says therefore first that we must now define the terms together, extraneous or separate, touching [in contact], intermediate [or between], consecutive to. The reason for positing these definitions now is that they will be used in later demonstrations, just as in the beginning of Euclid are posited definitions that serve as principles of later demonstrations.
lib. 5 l. 5 n. 2 Deinde cum dicit: simul igitur dicuntur haec etc., exequitur propositum. Ea primo definit quae praemissa sunt; secundo comparat ea ad invicem, ibi: manifestum autem et quod primum et cetera. Circa primum tria facit: primo definit ea quae pertinent ad tangere; secundo ea quae pertinent ad hoc quod est consequenter, ibi: medium vero etc.; tertio ea quae pertinent ad continuum, ibi: continuum autem et cetera. Et quia in definitione eius quod est tangere, ponitur simul, ideo primo definit ipsum: et dicit quod illa dicuntur esse simul secundum locum, quae sunt in uno loco primo; et dicitur primus locus uniuscuiusque, qui est proprius locus eius. Ex hoc enim aliqua dicuntur esse simul, quod sunt in uno loco proprio: non autem ex hoc quod sunt in uno loco communi; quia secundum hoc posset dici quod omnia corpora essent simul, quia omnia continentur sub caelo. Dicit autem quod simul dicuntur haec esse secundum locum, ad differentiam eorum quae dicuntur esse simul tempore: hoc enim non est nunc ad propositum. Per oppositum autem dicuntur esse separatim vel seorsum, quaecumque sunt in alio et alio loco. Tangere autem se dicuntur, quorum sunt ultima simul. Ultima autem corporum sunt superficies, et ultima superficierum sunt lineae, et ultima linearum sunt puncta. Si ergo ponatur quod duae lineae se tangant in suis ultimis, duo puncta duarum linearum se tangentium continebuntur sub uno puncto loci continentis. Nec propter hoc sequitur quod locatum sit maius loco: quia punctum additum puncto nihil maius efficit. Et eadem ratione se habet in aliis.	685. Then at (506 226 b22) he carries out his plan. First he defines the terms he mentioned; Secondly, he compares one to the other, at 692. About the first he does three things: First he defines those that pertain to contact, i.e., touching; Secondly, those which pertain to consecutiveness, at 686; Thirdly, those that pertain to continuum, at 691. Since “together” occurs in the definition of in contact, the Philosopher defines it first (506 226 b22) and says that those things are said to be together in respect of place which are in one first place, where first place refers to proper rather than common place. For things are said to be together not because they are in one common place but in one proper place; otherwise, we should be able to say that all bodies are together, since they are all contained under the heavens. He speaks of such things that are together in respect of place, in distinction to those that are said to be together in time—a point we are not now discussing, Conversely, whatever things are one in one place, and another in another place, are said to exist separate or apart. But in contact is said of things whose termini are together. The termini of bodies are surfaces and of surfaces, lines, and of lines, points. Therefore, if two lines are in contact as to their termini, the two points of the two lines in contact will be contained under one point of the place containing them. From this, however, it does not follow that the thing in place is greater than the place, for point added to point does not make anything larger. And the same holds for the others.
lib. 5 l. 5 n. 3 Deinde cum dicit: medium vero in quod aptum etc., definit ea quae pertinent ad hoc quod consequenter se habet. Et circa hoc tria facit: primo definit medium, quod ponitur in definitione eius quod est consequenter; secundo definit hoc quod est consequenter, ibi: consequenter autem est etc.; tertio infert quoddam corollarium ex dictis, ibi: quoniam autem omnis mutatio et cetera. Dicit ergo primo, quod medium est, in quod primo aptum natum est pervenire id quod continue mutatur secundum naturam, quam in ultimum terminum motus, in quem mutatur: sicut si aliquid mutatur de a in c per b, dummodo sit continuus motus, primo pervenit ad b, quam ad c. Medium autem potest quidem esse in pluribus; quia inter duo extrema possunt esse multa media, sicut inter album et nigrum sunt multi colores medii: sed ad minus oportet quod sit in tribus, quorum duo sunt extrema, et unum medium. Sic igitur medium est per quod in mutatione pervenitur ad ultimum; sed ultimum mutationis est contrarium. Dictum est enim supra quod motus est de contrario in contrarium.	686. Then at (507 226 b24) he defines the things that pertain to consecutiveness, About this he does three things: First he defines between, which is placed in the definition of consecutive to; Secondly, he defines consecutive to, at 689; Thirdly, he draws a corollary, at 690. He says therefore first (507 226 b24) that the between is what a naturally and uninterruptedly changing thing is apt to arrive at before it reaches the ultimate terminus of the motion, into which terminus it is being changed; for example, if something is changing from A to C through B, then, provided it is a continuous motion, it reaches B before C. In some cases there are a number of “betweens” to be traversed as you pass from one extreme to the other, as from black to white there are many colors between; but there must be at least three things involved, two of which are extremes and one the between. Consequently, the between is what must be passed through before arriving at the terminus of a change: but the terminus of a change is a contrary; for it has already been stated that motion goes from contrary to contrary.
lib. 5 l. 5 n. 4 Et quia in definitione medii posuerat continuationem motus, consequenter ostendit quid dicatur continue moveri. Potest autem continuatio motus ex duobus attendi: et ex tempore in quo movetur, et ex re per quam transit, sicut est magnitudo in motu locali. Ad hoc igitur quod sit motus continuus, requiritur quod nulla interpolatio sit in tempore: quia quantumcumque modicum interpolaretur motus secundum tempus, non esset continuus. Sed ex parte magnitudinis per quam transit motus, potest esse aliqua modica interpolatio sine praeiudicio continuationis motus; sicut patet in transitibus viarum, in quibus ponuntur lapides modicum ab invicem distantes, per quos homo transit de una parte viae ad aliam, motu continuo. Hoc est ergo quod dicit, quod continue movetur illud quod nihil aut paucissimum deficit rei, idest quod non habet interpolationem ex parte rei per quam transit; aut si deficit, paucissimum deficit. Sed temporis non potest nec paucissimum deficere, si sit motus continuus. Quomodo autem res possit deficere in motu continuo, manifestat, subdens quod nihil prohibet aliquod moveri continue cum defectu rei, sed non temporis; sicut si aliquis citharizans, statim post hypaten, idest primam chordam gravem, sonet ultimam acutam, intermissis quibusdam chordis in medio. Sed iste defectus est rei in qua est motus, non autem temporis. Hoc autem quod dictum est de continuitate motus, intelligendum est tam in motu locali, quam in aliis motibus.	687. Because the definition of between made mention of continuity of motion, he now shows what continuous movement means. Now continuity of motion may be viewed from two aspects: first, from the time during which the movement occurs and, secondly, from the thing through which the motion takes place for example, the magnitude, in local motion. For a motion to be continuous it is required that there be no interruptions in time, because even the slightest interruption of the motion as to time prevents the motion from being continuous. But on the side of the magnitude through which the motion passes there can be slight variations without prejudice to the continuity of the motion. This is clear in crossings over streets, at which stones are placed slightly distant from each other, and over which a person passes from one side of the street to another without interrupting his motion. This, therefore, is what he says: that continuity of motion is present when there is no gap or only the slightest in the thing, i.e., when there is no interruption in the thing over which the motion passes or, if there is, it is very slight. But there cannot be the slightest interruption of time, if the motion is to be continuous. How there can be a gap in continuous motion he explains by adding that a motion will be continuous even if there is a gap in the material, as long as there is no time-gap; for example, if in playing the harp one strikes the highest note immediately after having sounded the lowest and none of the intermediate ones. But this is not a gap in time, but in the material in which the motion takes place. What has been said about the continuity of motion applies not only to local motion but to all the others as well.
lib. 5 l. 5 n. 5 Sed quia non est manifestum quomodo ultimum in motu locali sit contrarium, quia locus non videtur esse contrarius loco, ideo hoc manifestat. Et dicit quod contrarium secundum locum est, quod plurimum distat secundum lineam rectam. Et intelligendum est plurimam distantiam esse secundum comparationem ad motum et mobilia et moventia: sicut maxime distant secundum locum per comparationem ad motum gravium et levium, centrum et extremitas caeli quoad nos; secundum autem motum meum vel tuum, maxime distat id quo intendimus ire, ab eo a quo incepimus moveri. Quid autem sibi velit quod dixit secundum rectitudinem, exponit subdens, minima enim finita et cetera. Ad cuius intellectum considerandum est, quod minima distantia quae est inter quaecumque duo puncta signata, est linea recta, quam contingit esse unam tantum inter duo puncta. Sed lineas curvas contingit in infinitum multiplicari inter duo puncta, secundum quod duae lineae curvae accipiuntur ut arcus maiorum vel minorum circulorum. Et quia omnis mensura debet esse finita (alias non posset certificare quantitatem, quod est proprium mensurae), ideo distantia maxima quae est inter duo, non potest mensurari secundum lineam curvam, sed solum secundum lineam rectam, quae est finita et determinata.	688. But because it is not evident how the terminus of a local motion is a contrary, since one place does not seem to be contrary to another, he now gives an explanation. And he says that the contrary in respect of place is the greatest rectilinear distance, where greatest distance is taken in relation to the motion and the mobiles and the movers, for example, for the motion of heavy and light things the distance from the center of the earth to the extremity of the sky is the greatest distance, while in regard to my motion and your motion, the greatest distance is the interval between where we start and where we intend to arrive. What he means by the phrase “in a straight line” he explains by adding “that the shortest line is definitely limited”. To understand this, consider that the shortest distance between two points is a straight line, for between any two points there is only one straight line. But there are any number of curved lines between two points, where by curved lines we mean the arcs of major or minor circles, Now since every measure should be finite (otherwise there would be no way of knowing the quantity of a thing—for that is the purpose of measuring), the greatest distance between two objects is not measured by a curved line but by a straight line which is finite and determinate.
lib. 5 l. 5 n. 6 Deinde cum dicit: consequenter autem est etc., definit hoc quod est consequenter, et quandam speciem eius, scilicet habitum. Et dicit quod ad hoc quod aliquid dicatur esse consequenter ad alterum, duo requiruntur. Quorum unum est, quod sit post aliquod principium quodam ordine; vel secundum positionem, sicut in iis quae habent ordinem in loco; vel secundum speciem, sicut dualitas est post unitatem; vel quocumque alio modo aliqua determinate ordinentur, sicut secundum virtutem, secundum dignitatem, secundum cognitionem, et huiusmodi. Aliud quod requiritur est, quod inter id quod est consequenter, et id cui est consequenter, non sit aliquod medium de numero eorum quae sunt in eodem genere: sicut linea consequenter se habet ad lineam, si nulla linea sit in medio; et similiter est de unitate ad unitatem, et de domo ad domum. Sed nihil prohibet, ad hoc quod aliquid sit alteri consequenter, quin aliquid sit medium inter ea alterius generis; sicut si aliquod animal sit medium inter duas domus. Quare autem dixerit et cuius est consequenter, et quod est post principium, manifestat subdens, quod omne quod dicitur consequenter, est consequenter respectu alicuius, et non tanquam prius, sed tanquam posterius. Non enim dicitur quod unum sit consequenter duobus, neque nova luna secundae, sed e converso. Deinde definit quandam speciem eius quod est consequenter, quae dicitur habitum. Et dicit quod non omne quod est consequenter, est habitum; sed quando sic est consequenter, quod tangit; ita quod nihil sit medium, non solum eiusdem generis, sed nec alterius.	689. Then at (508 226 b34) he defines what is meant by consecutive to and a species of it, namely, contiguous. And he says that two things are required in order that something be called consecutive to another. One is that it be after the first and in a certain order: either according to position, as things that are in order in place; or according to species, as 2 comes after 11 or in any way in which things can be in order, as according to virtue, according to dignity, according to knowledge, and so on. The other requirement is that between that which is consecutive and that to which it is consecutive there not be anything of the same kind intervening; for example, one line is consecutive to another, if there is no line between—likewise from one unit to another and one house to another. However, this does not forbid something else intervening. For example, an animal could be found between two houses. Why he said “to which it is consecutive” and “that it is after the first” he explains by adding that whatever is said to be consecutive is so in respect to something else, not as being prior to it but as following it. For 1 is not said to be consecutive to 2, or a new moon to a second new moon; rather, it is just the opposite. Then he defines a certain species of consecutive called contiguous. And he says that not everything consecutive is also contiguous, but only when it is consecutive and in contact, so that there is nothing at all between, i.e., nothing of the same genus or of any other genus.
lib. 5 l. 5 n. 7 Deinde cum dicit: quoniam autem omnis mutatio etc., concludit ex praemissis quod, cum medium sit per quod aliquid mutatur in ultimum, et omnis mutatio sit inter opposita, quae vel sunt contraria vel contradictoria, in contradictoriis autem nihil est medium; relinquitur quod omne medium sit inter contraria aliquo modo.	690. Then at (509 226 b31) he concludes from the foregoing that since the between is that through which something is changed into what is final, and since every change is between opposites which are either contrary or contradictory, although there is no between in contradictories, it follows that it is between contraries that the between is found.
lib. 5 l. 5 n. 8 Deinde cum dicit: continuum autem est quidem etc., manifestat quid sit continuum: et dicit quod continuum est aliqua species habiti. Cum enim unus et idem fiat terminus duorum quae se tangunt, dicitur esse continuum. Et hoc etiam significat nomen. Nam continuum a continendo dicitur: quando igitur multae partes continentur in uno, et quasi simul se tenent, tunc est continuum. Sed hoc non potest esse cum sint duo ultima, sed solum cum est unum. Ex hoc autem ulterius concludit, quod continuatio esse non potest, nisi in illis ex quibus natum est unum fieri secundum contactum. Ex eadem enim ratione aliquod totum est secundum se unum et continuum, ex qua ex multis fit unum continuum, vel per aliquam conclavationem, vel per aliquam incollationem, vel per quemcumque modum contingendi, ita quod fiat unus terminus utriusque; vel etiam per hoc quod aliquid naturaliter nascitur iuxta aliud, sicut fructus adnascitur arbori et continuatur quodammodo ei.	Then at (510 227 a10) he shows what a continuum is and he says that it is a species of the contiguous. For when the terminus of two things in contact is one and the same, then something is continuous. And the very word “continuum” denotes this. For “continuum” is derived from “continere” (to hold together): when, therefore, many parts are held together in a unit and, as it were, keep themselves together, then there is a continuum. But this cannot be while the endings are two but only when they are one. From this he further concludes that continuity can occur only in things from which a unity through contact is naturally apt to come about. For in whatever way a whole is naturally one and continuous in the same way is a continuous unity formed from many things, whether by riveting, by gluing or by any form of contact that makes one terminus for two parts, or even by being born of another, as fruit is born of a tree and forms a sort of continuum with it.
lib. 5 l. 5 n. 9 Deinde cum dicit: manifestum autem et quod primum etc., comparat tria praemissorum ad invicem, de quibus principaliter intendit, scilicet consequenter se habens, contactum et continuum. Et circa hoc tria facit: primo comparat consequenter se habens ad contactum; secundo contactum ad continuum, ibi: et si continuum etc.; tertio infert quoddam corollarium ex dictis, ibi: quare si est unitas et cetera. Dicit ergo primo manifestum esse, quod consequenter se habens est primum inter tria praemissa ordine naturae, secundum quod dicitur esse prius, a quo non convertitur consequentia essendi; quia omne contactum necesse est esse consequenter: oportet enim inter ea quae se contingunt, esse aliquem ordinem, ad minus positione. Sed non oportet omne quod consequenter se habet, esse tangens: quia ordo potest esse in quibus non est tactus, sicut in separatis a materia. Unde hoc quod est consequenter, invenitur in iis quae sunt priora secundum rationem: invenitur enim in numeris, in quibus non invenitur tactus, qui invenitur solum in continuis. Numeri autem secundum rationem sunt priores continuis quantitatibus, sicut magis simplices et magis abstracti.	692. Then at (511 227 a17) he compares three of the foregoing with one another; namely, the consecutive to the continuous, and the continuum. About this he does three things: First he compares consecutiveness to contact; Secondly, contact with continuum, at 693; Thirdly, he draws a corollary, at 694. He says therefore first (511 227 a17) that it is clear why among these three, consecutiveness is naturally first in the order of nature, for in the cases of contact there is always consecutiveness, since there must be an order, at least of position, among things that are in contact. Bat not all cases of consecutiveness involve contact, for an order can exist among things in which there is no contact, as in substances separated from matter. Hence, consecutiveness is present in things that are prior in definition, for it is found in numbers, in which there is no contact, which is present only among continua. Numbers, however, are prior to continuous quantities in definition, for they are more simple and more abstract.
lib. 5 l. 5 n. 10 Deinde cum dicit: et si continuum est etc., comparat contactum ad continuum. Et dicit quod eadem ratione contactum est prius quam continuum: quia si aliquid est continuum, necesse est quod sit tangens; sed non est necessarium, si tangit quod sit continuum. Et hoc probat per rationem utriusque. Non enim necessarium est quod ultima aliquorum sint unum, quod est de ratione continui, si sunt simul, quod est de ratione contacti: sed necesse est e converso, si ultima sunt unum, quod sint simul, ea ratione qua potest dici, quod unum sit simul sibi ipsi. Si autem hoc quod dico simul, importat habitudinem distinctorum, non possunt esse unum quae sunt simul: et secundum hoc nec contacta esse possunt quae sunt continua, sed communiter accipiendo. Unde concludit quod insertus, idest continuatio secundum quam una pars inseritur alteri in uno termino, est ultimus in ordine generationis, prout specialia sunt posteriora communibus, sicut prius generatur animal quam homo. Et ideo dico esse ultimum insertum, quia necesse est aliqua se tangere ad invicem, si ultima eorum sunt adnata, idest naturaliter unita; sed non est necessarium quod omnia quae se tangunt, quod sint naturaliter adnata ad invicem. Sed in quibus non potest esse contactus, manifestum est quod in his non potest esse consertus, idest continuatio.	693. Then at (512 227 a21) he compares in contact with continuous and says that for the same reason in contact is prior to continuous, because if a thing is continuous it must be in contact, but it does not necessarily follow that if it is in contact it is continuous, And he proves this from the definitions of the two. For it is not necessary that the endings of things be one (which is implied in the notion of continuum), if they are together (which is implied in the notion of contact). But, on the other hand, if the endings are one, they must be together, for what is one is together unto itself. However, if “together” implies a relationship between distinct things, then things that are together are not one: and according to this, continua are not in contact. But they are, if we do not speak so precisely. Hence he concludes that natural junction, i.e., continuity, in which one part is joined to another at one terminus, is last in coming to be, in the sense that what is specific comes to be after what is general, as animal comes to be before man. And, therefore, I say that natural junction is last, because things must mutually touch if their extremities are naturally united; however, it is not necessary that all things that touch be naturally joined to one another. But in regard to things which cannot touch, it is clear that continuity is impossible.
lib. 5 l. 5 n. 11 Deinde cum dicit: quare si est unitas etc., concludit quoddam corollarium ex dictis; scilicet quod si unitas et punctum sunt separata, sicut quidam dicunt, ponentes mathematica separari secundum esse, sequitur quod unitas et punctum non sunt idem. Et hoc manifestum fit duabus rationibus. Primo quidem, quia puncta sunt in his quae nata sunt se tangere, et secundum puncta aliqua se tangunt ad invicem: in unitatibus autem non invenitur contactus, sed solum hoc quod est consequenter. Secundo vero, quia inter duo puncta contingit esse aliquid medium; omnis enim linea est media inter duo puncta: sed inter duas unitates non necesse est esse aliquod medium. Patet enim quod inter duas unitates, quae constituunt dualitatem, et ipsam primam unitatem, nihil est medium. Ultimo ibi: quid quidem igitur est simul etc., epilogat quae dicta sunt: et est planum in littera.	694. Then at (513 227 a27) he draws a corollary from the preceding: i.e., if point and unit have an independent existence of their own, as some say who suppose a separated existence for mathematical objects), it follows that unity and point are not the same. And this is clear for two reasons: first, because points are present in things that are capable of mutual contact and certain things touch at points; but in units contact is never found, but only consecutiveness. Secondly, because there must be something existing between two points, but between two unities there is not necessarily anything between. For it is evident that between the two unities that form 2 and the very first unity, which is 1, there is nothing intermediate. Finally, at (514 227 a32) he makes a summary and says that we have defined what is meant by together and apart, contact, between, consecutiveness, contiguous and continuous. Also we have shown in which circumstances each of these terms is applicable,