Lectio 5 Lecture 5 Definitiones eorum quae pertinent ad hoc quod est tangere, et ad hoc quod est esse consequenter et ad continuum He defines “contact,” “consecutiveness,” “continuity,” and other related things Post haec autem dicamus quid est simul, et quid separatim, et quid est tangere, et quid est medium, et quid consequenter, et quid habitum, et quid continuum, et in qualibet re unumquodque horum inesse aptum natum sit. Let us now proceed to define the terms “together” and “apart,” “in contact,” “between,” “consecutive,” “contiguous,” and “continuous,” and to show in what circumstances each of these terms is naturally applicable. Simul igitur dicuntur haec esse secundum locum, quaecumque in uno loco sunt primo: separatim autem, quaecumque sunt in altero: tangere autem, quorum ultima simul. Things are said to be together in place when they are in one place (in the strictest sense of the word “place”), and to be apart when they are in different places. Things are said to be in contact when their extremities are together. Medium vero, in quod aptum natum est primum pertingere quod mutatur, quam in quod ultimum mutatur, secundum naturam continue mutationem patiens. In minus autem est medium in tribus: ultimum quidem enim mutationis contrarium est. Continue autem movetur, quod nihil aut paucissimum deficit rei, non temporis. That is “between” which a changing thing, if it changes continuously in a natural manner, naturally reaches before it reaches that to which it changes last. Thus, “between” implies the presence of at least three things, for in a process of change, it is the contrary that is “last” and a thing is moved continuously if it leaves no gap or only the smallest possible gap in the thing—not in the time Nihil enim prohibet deficientiam rei non temporis; et statim etiam post hypaten sonare ultimam; sed rei in qua movetur. (for a gap in the time does not prevent things having a “between,” while, on the other hand, there is nothing to prevent the highest note sounding immediately after the lowest), but in the material in which the motion takes place. Hoc autem et in his quae sunt secundum locum, et in aliis mutationibus manifestum est. This is manifestly true not only in local changes, but in every other kind as well. Contrarium autem secundum locum est, secundum rectitudinem distans plurimum. Minima enim finita: mensura autem finitum est. The contrary in respect of place is what is most distant in a straight line, for the shortest line is definitely limited, and that which is definitely limited constitutes a measure. Consequenter autem est, quod cum post principium solum sit, aut positione aut specie aut alio aliquo sic determinato, nullum medium est eorum quae sunt in eodem genere, et cuius consequenter est: A thing is “consecutive” when it is after the beginning in position or in form or in some other respect in which it is definitely so regarded, and when further there is nothing of the same kind as itself between it and that to which it is consecutive dico autem ut linea lineae secundum quod sunt lineae, aut unitati unitas secundum quod sunt unitates, aut domui domus: aliud autem nihil prohibet esse medium. —for instance, a line or lines if it is a line, a unit or units if it is a unit, a house if it is a house (there is nothing to prevent something of a different kind being between). Consequenter enim alicui consequenter, ac posterius aliquid. Non enim unum consequenter est duobus, neque nova luna secundae consequenter: sed haec illis. Habitum autem est, quod cum consequenter est, tangit. For that which is consecutive is consecutive to a particular thing and is something posterior: one is not consecutive to two, nor is the first day of the month to be second; in each case the latter is consecutive to the former. A thing that is consecutive and touches is contiguous. Quoniam autem omnis mutatio in oppositis est, opposita autem sunt et contraria et quae secundum contradictionem sunt, contradictionis autem nihil est medium; manifestum est quod in contrariis erit medium. Since all change is between opposites, while opposites include both the contrary and the contradictory, and there is no mean between contradictories, it is clear that there is a mean between contraries. Continuum autem est quidem quod habitum aliquid est. Dico autem esse continuum, cum idem fiat et unus utriusque terminus eorum quae tanguntur, et sicut significat nomen, contineatur: hoc autem esse non potest cum duo sint ultima. Sed hoc determinato, manifestum est quod in his est continuum, ex quibus unum aliquod aptum est fieri secundum contactum. Et sicut aliquando fit continuum unum, sic et totum erit unum; ut aut conclavatione aut colla aut tactu aut adnascentia. The “continuous” is a subdivision of the contiguous. Things are called continuous when the touching limits of each become one and the same and are, as the word implies, contained in each other: continuity is impossible if these extremities are two. This definition makes it plain that continuity belongs to things that naturally form a unity in virtue of their mutual contact. And in whatever way that which holds them together is one, so too will the whole be one, such as by a rivet or glue or contact or organic union. Manifestum autem et quod primum est id quod consequenter est. Contactum quidem enim necesse est consequenter esse: quod consequenter autem non omne tangere. Unde in prioribus ratione consequenter est, ut in numeris: tactus autem non est. It is obvious that, of these terms, “consecutive” is first in order of analysis, for that which touches is necessarily consecutive but not everything that is consecutive touches, and so consecutiveness is a property of things prior in definition, such as numbers, while contact is not. Et si continuum est, necesse est etiam tangere: si vero tangit, nondum continuum est. Non enim necesse est unum esse ipsorum ultima, si simul sint: sed si unum sunt, necesse est et simul esse. And, if there is continuity, there is necessarily contact, but if there is contact, that alone does not imply continuity, for the extremities of things may be together without necessarily being one, but they cannot be one without being necessarily together. Quare insertus ultimus est secundum generationem: necesse est enim tangere si adnata erunt ultima; contacta autem non omnia adnata sunt. In quibus autem non est tactus, manifestum est quod non est consertus in his. So, natural junction is last in generation, for the extremities must necessarily come into contact if they are to be naturally joined: things that are in contact are not all naturally joined, but where there is no contact, clearly there is no natural junction either. Quare si est unitas et punctum, qualia dicunt separata, non possibile est unitatem et punctum esse idem. His quidem inest tangere; unitatibus autem consequenter. Hence, if, as some say, “point” and “unit” have an independent existence of their own, it is impossible for the two to be identical: points can touch, but units can only be consecutive. Et horum quidem contingit esse medium; omnis enim linea medium punctorum est: harum autem non est necesse; nullum enim medium est dualitatis et unitatis. Moreover, there can always be something between points (for all lines are intermediate between points), but it is not necessary that there should possibly be anything between units, for there can be nothing between the numbers one and two. Quid quidem igitur est simul et separatim, et quid tangere, et quid medium, et quid consequenter, et quid habitum, et quid continuum, et in quolibet unumquodque eorum inest, dictum est. We have now defined what is meant by “together,” “apart,” “contact,” “between,” “consecutive,” “contiguous,” and “continuous”; and we have shown in what circumstances each of these terms is applicable. 684. Postquam Philosophus divisit mutationem et motum in suas species, hic procedit ad determinandum de unitate et contrarietate motus in suas species. 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. Et circa hoc duo facit: About this, he does two things: primo praemittit quaedam necessaria ad sequentia; first, he establishes a background of preliminary notions that will be of use; secundo prosequitur principale propositum, ibi: unus autem motus etc. second, he pursues his main objective, at there are many senses (227b3; [695]). Circa primum tria facit: About the first, he does three things: primo dicit de quo est intentio; first, he states his intention; secundo exequitur propositum, ibi: simul igitur etc., second, he pursues it, at things are said to be together in place (226a21; [685]); tertio recapitulat, ibi: quid quidem igitur etc. third, he makes a summary, at we have now defined (227a32; [694]). 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. He says first that we must now define the terms “together,” “extraneous” or “separate,” “touching,” “intermediate,” “consecutive to,” “contiguous,” and “continuous,” and in what circumstances each of these terms is naturally applicable. 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. 685. Deinde cum dicit: simul igitur dicuntur haec etc., exequitur propositum. 685. Then, at things are said to be together in place (226a21), he carries out his plan. Ea primo definit quae praemissa sunt; First, he defines the terms he mentioned; secundo comparat ea ad invicem, ibi: manifestum autem et quod primum etc. second, he compares one to the other, at it is obvious (227a17; [692]). Circa primum tria facit: About the first he does three things: primo definit ea quae pertinent ad tangere; first, he defines those that pertain to contact, that is, touching; secundo ea quae pertinent ad hoc quod est consequenter, ibi: medium vero etc.; second, those that pertain to consecutiveness, at that is “between” (226b23; [686]); tertio ea quae pertinent ad continuum, ibi: continuum autem etc. third, those that pertain to continuum, at the “continuous” is a subdivision (227a10; [691]). 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. Since “together” occurs in the definition of “in contact,” the Philosopher defines it first (227a7) and says that those things are said to be together in place that are in one first place, meaning the first place of each one, that is, its own proper 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.