591. Deinde cum dicit: sequitur autem et hoc etc., assignat eiusdem rationem ex parte lineae et puncti. Et dicit quod hoc quod dictum est de tempore et nunc, consequitur quodammodo ad id quod invenitur in linea et puncto: quia punctum continuat lineam, et distinguit ipsam inquantum est principium unius partis et finis alterius. 591. Then, at here, too, there is (220a9), he explains a case of the same in the matter of line and point. And he says that the conclusion drawn about time and the now in the preceding section follows in a way from what is found in a line and a point: the point continues the line and distinguishes its parts, inasmuch as it is the beginning of one part and the end of another. Sed tamen differenter se habet in linea et puncto, et tempore et nunc. Quia punctum est quoddam stans, et linea similiter: unde potest homo accipere idem punctum bis, et uti eo ut duobus, ut scilicet principio et ut fine. But there is a difference between the case of line and point and the case of time and the now. For both the point and the line are something stationary; hence, a person can consider the same point twice and use it as two, namely, as both a beginning and an end. Et cum sic utimur puncto ut duobus, accidit quies; sicut patet in motu reflexo, in quo id quod erat finis primi motus est principium secundi motus reflexi. Et propter hoc probatur infra in octavo, quod motus reflexus non est continuus, sed intercidit quies media. When we thus use the point as two, rest occurs, as is evident in a reflected motion, where that which was the end of the first motion is the beginning of the second and reflected motion. It is on this basis that we shall prove in book 8 that a reflected motion is not continuous, but that an intermediate pause occurs. Sed ipsum nunc non est stans, propter id quod respondet mobili, quod semper fertur durante motu; et propter hoc oportet nunc esse semper alterum et alterum secundum rationem, ut supra dictum est. But the now is not stationary, for it corresponds to the mobile that is always being carried along during the motion—which also accounts for the now having to be always different in account, as was said above. Et ideo, cum tempus sit numerus motus, non hoc modo numerat motum, quod aliquid idem temporis accipiatur ut principium unius et finis alterius; sed magis numerat motum accipiendo duo ultima temporis, scilicet duo nunc, quae tamen non sunt partes eius. Et quare competat iste modus numerandi in tempore magis quam alius, quo per punctum numerantur partes lineae, inquantum est principium et finis, ratio est quae dicta est, quia secundum hunc modum utitur aliquis puncto ut duobus; et sic accidit quies media, quae non potest esse in tempore et in motu. Non tamen intelligendum est per id quod dicitur, quod idem nunc non sit principium futuri et finis praeteriti, sed quod non percipimus tempus numerando motum per unum nunc, sed magis per duo, ut dictum est: quia sequeretur quod in numeratione motus idem nunc sumeretur bis. Therefore, since time is the number of motion, it does not number motion in the sense that some same time is taken as the beginning of one and the end of another, but rather it numbers motion by taking two boundaries of time, namely, two nows that are nevertheless not parts of time. The reason that this method of counting is used in numbering time, rather than the method used when a point numbers the parts of a line (where the same point is considered both a beginning and an end), is that, as was stated, in the latter method, we use the point as two things and this brings about an intermediate pause, which cannot exist in time or in motion. Now, this does not mean that the same now is not the beginning of the future and the end of the past, but that we do not perceive time by counting motion in terms of one now, but in terms of two, as was said; otherwise, in our counting of motion, the same now would be employed twice. 592. Deinde cum dicit: et adhuc manifestum quod nulla pars etc., assignat rationem eius quod dicitur, quod nunc non est pars temporis. Et dicit manifestum esse quod nunc non est pars temporis, sicut neque id per quod distinguitur motus, est pars motus, scilicet aliqua dispositio signata in mobili; sicut etiam nec puncta sunt partes lineae. Duae enim lineae sunt partes unius lineae. Manifestat autem proprietates ipsius temporis ex motu et linea: quia, sicut dictum est supra, motus est continuus propter magnitudinem, et tempus propter motum. 592. Then, at and further because (220a18), he explains why it is said that the now is not a part of time. And he says it is plain that the now is not a part of time, just as what distinguishes a motion is not a part of the motion—namely, some disposition in the mobile itself—just as points are not parts of a line. For two lines are the parts of a line. Now, he manifests the properties of time from the properties of motion and of line because, as was said above, motion is continuous on account of the magnitude, and time on account of the motion. Concludit ergo finaliter, quod ipsum nunc secundum quod est terminus quidam, non est tempus, sed accidit tempori, ut terminus terminato: sed secundum quod tempus vel nunc numerat alia, sic etiam nunc est numerus aliorum quam temporis. Et huius ratio est, quia terminus non est nisi eius cuius est terminus; sed numerus potest esse diversorum, sicut numerus decem equorum numerus est et aliarum rerum. Sic igitur nunc est terminus solius temporis, sed est numerus omnium mobilium quae moventur in tempore. He concludes, therefore, finally, that the now, insofar as it is a certain boundary, is not time, but it is an accident of time, as a boundary does to that which is bounded; but, insofar as time or the now numbers other things, the now is the number of things other than time. The reason is that a boundary can be only of that of which it is the boundary, but a number can be applied to various things, as the number of ten horses is also that of other things. Therefore, in this way, the now is the boundary only of time, but it is the number of all mobiles that are being moved in time. Lectio 19 Lecture 19 Manifestantur quaedam quae de tempore dici solent Clarification of certain things said about time Quod quidem igitur tempus numerus motus secundum prius et posterius sit, et continuum (continui namque), manifestum est. Minimus autem numerus, qui simpliciter quidem est, dualitas est: quidam autem numerus, est qui est quidem sic, est autem tanquam non sic; ut lineae minimum multitudine quidem est duae aut una, magnitudine autem non est minimum; semper enim dividitur omnis linea. Quare similiter et tempus: minimum enim quidem est secundum numerum, unum aut duo; secundum vero magnitudinem non est. It is clear, then, that time is “the number of motion in respect of the before and after” and is continuous, since it is an attribute of what is continuous. The smallest number, in the strict sense of the word “number,” is two. But of one certain number, sometimes there is a minimum, sometimes not: for instance, of a line, the smallest in respect of multiplicity is two (or, if you like, one), but in respect of size, there is no minimum; for every line is divided to infinity. Hence, it is so with time. In respect of number, the minimum is one (or two); in point of extent, there is no minimum. Manifestum est autem propter quid tardum et velox non dicitur: multum autem et paucum, et breve et longum. It is clear, too, that time is not described as fast or slow, but as many or few and as long or short. Secundum enim quod continuum est, longum et breve dicitur; secundum autem quod numerus, multum et paucum. Velox autem et tardum non est: neque enim numerus quo numeramus, velox et tardus ullus est. For as continuous, it is long or short, and as a number, many or few, but it is not fast or slow—any more than any number with which we number is fast or slow. Et idem autem ubique simul. Prius autem et posterius non idem: quia et mutatio praesens quidem una est; facta autem et futura altera est. Tempus autem numerus est, non quo numeramus, sed quod numeratur. Huic autem accidit prius et posterius semper esse alterum: ipsa enim nunc semper altera. Est autem numerus unus quidem et idem qui est centum equorum, et qui est centum hominum: quorum autem numerus est, altera sunt; ut equi ab hominibus. Further, time is the same everywhere at once, but not the same time before and after, for while the present change is one, the change that has happened and that which will happen are different. Time is not number with which we count, but the number of things that are counted, and this is always different according as it occurs before or after, for the “nows” are different. And the number of a hundred horses and a hundred men is the same, but the things numbered are different—the horses from the men. Amplius, sicut contingit motum esse eundem et unum iterum et iterum, sic et tempus contingit, ut hiemem aut ver aut autumnum. Further, as a motion can be one and the same again and again, so too can time—for instance, a year or a spring or an autumn. Non solum autem motum tempore metimur, sed motu tempus, propterea quod definiuntur ad invicem. Tempus quidem enim determinat motum, cum sit numerus ipsius: motus autem tempus. Et dicimus multum aut paucum esse tempus, motu mensurantes; sicut et numerabilibus numerum. Numero quidem equorum multitudinem cognoscimus; iterum autem uno equo equorum numerum ipsum. Similiter autem et in tempore et motu est. Tempore quidem enim motum, motu autem tempus mensuramus. Not only do we measure the motion by the time, but also the time by the motion, because they define each other. The time marks the motion, since it is its number, and the motion the time. We describe the time as much or little, measuring it by the motion, just as we know the number by what is numbered—for instance, the number of the horses by one horse as the unit. For we know how many horses there are by the use of the number; and again, by using the one horse as unit, we know the number of the horses itself. So it is with the time and the motion, for we measure the motion by the time and vice versa. Et hoc rationabiliter accidit. Imitatur enim magnitudinem quidem motus, hunc autem tempus, eo quod quanta et continua sint et divisibilia: propter magnitudinem enim esse huiusmodi, motus haec sustinet, propter autem motum tempus. It is natural that this should happen, for the motion goes with the distance and the time with the motion, because they are quanta and continuous and divisible. The motion has these attributes because the distance is of this nature, and the time has them because of the motion. Et mensuramus magnitudinem motu, et motum magnitudine: multam enim dicimus esse viam, si processus multus; et hunc multum, si via multa. Sic igitur et tempus si motus, et motum si tempus. And we measure both the distance by the motion and the motion by the distance, for we say that the road is long if the journey is long and that this is long if the road is long—the time, too, if the motion, and the motion, if the time. 593. Postquam Philosophus definivit tempus, hic ex definitione data reddit rationem eorum quae dicuntur de tempore. 593. Having defined time, the Philosopher now, in the light of that definition, gives an explanation of those things that are said about time. Et circa hoc quatuor facit: About this, he does four things: primo ostendit quomodo in tempore invenitur minimum, et quomodo non; first, he shows in what sense a smallest part is to be found in time, and in what sense it is not; secundo quare tempus dicitur multum et paucum, breve et longum, non autem velox et tardum, ibi: manifestum est autem propter quid etc.; second, he shows why time is said to be much and little, short and long, but not fast and slow, at it is clear, too, that time (220a32; [595]); tertio quomodo tempus sit idem, et quomodo non, ibi: et idem autem ubique etc.; third, he shows in what sense time is the same, and in what sense it is not, at further, there is the same (220b3; [596]); quarto quomodo tempus cognoscitur motu et e converso, ibi: non solum autem motum etc. fourth, he shows how time is known through motion and vice-versa, at not only do we (220b14; [598]). 594. Dicit ergo primo quod manifestum est ex definitione temporis prius data, quod tempus est numerus motus secundum prius et posterius, ut supra expositum est; et iterum manifestum est ex praemissis, quod tempus est quoddam continuum. Licet enim non habeat continuitatem ex eo quod est numerus, habet tamen continuitatem ex eo cuius est numerus: quia est numerus continui, scilicet motus, ut etiam supra dictum est. Non enim est tempus numerus simpliciter, sed numerus numeratus. In numero autem simpliciter est omnino invenire aliquem minimum numerum, scilicet dualitatem. 594. He says first (220a24) that the previously given definition of time makes clear that time is the number of motion according to before and after, as was expounded above, and that time is a type of continuum, as is likewise manifest from what has gone before. For although it does not have continuity insofar as it is a number, yet it has continuity by reason of that of which it is the number, for it is the number of a continuum, namely, of motion, as was said above. For time is not a number absolutely, but a number of something numbered. Among absolute numbers, there is unequivocally a minimum to be found, namely, two. Sed si accipiamus numerum quendam, scilicet numerum alicuius rei continuae, quodammodo est invenire minimum, et quodammodo non; quia secundum multitudinem est invenire minimum, non autem secundum magnitudinem. Sicut in multis lineis secundum multitudinem quidem est minimum, ut una linea vel duae lineae; una quidem si accipiatur id quod est minimum simpliciter in numero; duae autem si accipiatur id quod est minimum in genere numeri, habens rationem numeri. But, if we consider one certain number, namely, the number of something that is continuous, then there is in one sense a minimum and in one sense no minimum, because in the order of multitude there is a minimum, but not in the order of magnitude. For example, in a plurality of lines, there is a minimum according to plurality, namely, one line or two lines (one if you consider what is the minimum in number absolutely; two if you mean that which is least in the genus of number, having the account of number). Sed in lineis non est invenire minimum secundum magnitudinem, ut sit scilicet aliqua linea minima; quia semper est dividere quamcumque lineam. Et similiter dicendum est de tempore: quia est invenire in eo minimum secundum multitudinem, scilicet unum vel duo, ut puta aut unum annum aut duos annos, aut duos dies aut horas. Sed minimum secundum magnitudinem non est invenire in tempore; quia cuiuslibet temporis dati est accipere partes in quas dividitur. But, in respect of magnitude, there is no minimum in lines such that there would be some smallest lines, because it is always possible to divide any line whatsoever. A parallel situation is found in time, for there is a minimum according to multitude, namely, one or two—for example, one year or two years or two days or two hours. But, in the order of magnitude, there is no minimum, for of any given time, there are parts into which it may be divided. 595. Deinde cum dicit: manifestum est autem etc., assignat rationem quare tempus non dicitur tardum aut velox, sed dicitur multum et paucum, breve et longum. Iam enim ostensum est quod tempus et numerus est, et continuum est. Inquantum ergo est continuum, dicitur tempus et longum et breve, sicut et linea; inquantum autem numerus est, dicitur et multum et paucum. Esse autem velox et tardum, nullo modo competit numero: neque numero simpliciter, ut manifestum est; neque etiam potest convenire numero alicuius rei. Nam esse velox vel tardum, dicitur de aliquo secundum quod est numeratum: dicitur enim velox motus, eo quod parvo tempore numeratur; tardum autem e converso. Unde manifestum est quod tempus nullo modo potest dici velox vel tardum. 595. Then, at it is clear, too (220a32), he gives a reason that time is not said to be slow or fast, but great and small or short and long. For it has already been shown that time is both a number and a continuum. Insofar, therefore, as it is the latter, time is said to be long and short; insofar as it is a number, it is said to be great and small. But to be fast and slow in no way belongs to number, neither to number absolutely (as is plain) nor to the number of some things. For to be fast or slow is said of something insofar as it is numbered: for a motion is called fast insofar as it is counted off in a short time, and slow conversely. Hence, it is clear that, in no sense can time be called fast or slow. 596. Deinde cum dicit: et idem autem etc., ostendit quomodo tempus sit idem, et quomodo non idem. 596. Then, at further, there is the same (220b3), he shows how time is the same and how not the same: Et primo quomodo sit idem vel non idem simpliciter; first, how it is the same or not the same absolutely; secundo quomodo sit idem secundum quid, ibi: amplius sicut contingit etc. second, how it is the same in a certain respect, at further, as a motion (220b12; [597]). Dicit ergo primo quod tempus simul existens, est idem ubique, idest respectu omnium quae moventur ubicumque. Non enim diversificatur secundum diversa mobilia; sed diversificatur secundum diversas partes eiusdem motus. Et ideo tempus prius et tempus posterius non est idem. Et hoc ideo, quia prima mutatio praesens, cuius primo et principaliter numerus tempus est, una est; sed huius mutationis altera pars est, quae iam facta est et pertransiit, et altera, quae futura est. Unde et tempus alterum est quod prius fuit, et alterum quod futurum est. He therefore says first that the time existing at a given moment is the same everywhere; that is, it is the same in respect to everything that is being moved anywhere. For it is not diversified by reason of the diverse mobiles, but by reason of the diverse parts of the same motion. For this reason, a prior time and a later time are not the same. Why? Because the first and present motion, of which time is primarily and principally the number, is one; but one part of this motion has already taken place and is past, and another will be in the future. Hence, there is one time that is past and another time that is future. Et hoc ideo, quia tempus non est numerus simpliciter, sed numerus alicuius rei numeratae, scilicet prioris et posterioris in motu; et huic numero semper accidit esse alterum, et prius et posterius, propter hoc quod ipsa nunc, secundum quod se habent prius et posterius, semper sunt altera. Si autem esset numerus simpliciter, tunc esset idem tempus et mutationis quae praeteriit, et eius quae futura est; quia numerus simpliciter est unus et idem diversorum numeratorum, ut centum equorum et centum hominum. Sed numerus numeratus est alius diversorum: centum enim equi sunt aliud quid a centum hominibus. Et quia tempus est numerus prioris et posterioris in motu; quia alia sunt quae in motu se habent prius et posterius secundum id quod praeteriit de motu, et alia secundum id quod sequitur; propter hoc est aliud tempus praeteritum, et aliud futurum. This is so because time is not number absolutely, but the number of something numbered, namely, of the before and after in motion. And this number always varies and is before and after, because the nows, like before and after, are always other. But, if time were number absolutely, then the time corresponding to the change that is past and the time corresponding to the change that is to come would be the same, for number absolutely is one and the same of different things counted, such as in the case of one hundred horses and one hundred men. But number numbered varies with different things. For one hundred horses are not the same as one hundred men. Since time is the number of before and after in motion, and since the before and after of a past motion are not the same as those of that which follow, therefore the past time and the future time are different. 597. Deinde cum dicit: amplius sicut contingit etc., ostendit quomodo tempus reiteratur idem secundum quid. Et dicit quod sicut reiterari unum et eundem motum contingit, sic contingit reiterari unum et idem tempus. 597. Then, at further, as a motion (220b12), he shows how the same time returns in a certain respect. And he says that, in the same way that one and the same motion may be repeated, so may one and the same time. Reiteratur enim unus et idem motus specie, sed non numero: quia ab eodem signo arietis, a quo primo movebatur sol, et postea movebitur; et ideo sicut fuit hiems aut ver aut aestas aut autumnus, ita erit, non quidem unum numero, sed specie. For one and the same motion can be duplicated specifically, but not numerically; for it is from the same sign of the ram that the sun first moves and later will move the following year. Therefore, just as there has been winter or spring or summer or fall, so also there will be not, indeed, the same one in number, but in species. 598. Deinde cum dicit: non solum autem motum tempore etc., ostendit quod sicut motum cognoscimus tempore, ita et tempus motu: 598. Then, at not only do we (220b14), he shows that, just as we know motion from time, so also time from motion: et hoc primo ex ratione numeri et numerati; first, by reason of number and the thing numbered;