Volume 1 · pp. 205–206
End of Chapter VI and Chapter VII. On doubts concerning those things which have been said about line
habere particulas temporis in eo quod quædam sunt priores et quædam posteriores tam in præterito quam in futuro, sed priores in præterito sunt a præsenti longinquiores, priores autem in futuro sunt quæ ad præsens sunt priores: et sic aliquid particularum temporis est prius, et aliquid suarum particularum est posterius. Sic etiam de numero similiter est, quod scilicet in particulis numeri non est positio, sed ordo enumerationis: quia prius enumeratur unus quam duo, et prius duo quam tres, et sic de aliis numeri partibus: et sic est in numero quod unitati quæ iteratur in numero est propinquius: et sic ordinem quemdam habebunt numeri partes, positionem vero non habebunt aliquo modo, nec etiam copulationem sive connexionem, nec communem terminum, qui aliquid sit utriusque partis, prioris scilicet et posterioris, et ideo manent partes divisæ discretæ. Quod etiam ostendit ratio nominis numeri. Numerus enim dicitur quasi numus verus, hoc est, divisionis: quia quæque unitates in partibus numeri inveniuntur divisæ et segregatæ. Patet igitur quod nullam hujusmodi partes perficiunt positionem.
Similiter et oratio se habet in partibus suis: hæc enim in discretione partium convenit cum numero, in successione autem convenit cum tempore quo pronuntiatur orationis pronuntiatio, et ideo partes ejus discretæ sunt et non permanentes. Oratio igitur est de numero eorum quæ dicta sunt, quæ scilicet in partibus non habent positionem. Nihil enim permanent particulæ orationis et minus stare possunt quam particulæ temporis, quia ad minus in tempore est accipere nunc vel præsens instans, in quo aliquo modo est præteritum, et aliquo modo est futurum. Sed in oratione nihil est accipere aliquo modo: cum omnia indivisibilia elementorum discreta sunt et in nullo connexa, sed quodlibet eorum dictum est et transit, et ideo non potest amplius sumi ut denominetur permanens: propter quod nulla erit positio particularum orationis: propter quod nihil re permanet particularum ejus. Et sic probatum est inducendo quod intendimus, quod scilicet alia de genere quantitatis constant ex particulis quæ in eis sunt essentialiter positionem habentibus ad se invicem, alia autem ejusdem generis constant ex particulis non habentibus ad se invicem positionem.
CAPUT VII.
De dubitationibus circa ea quæ dicta sunt de linea.
Quia autem quantitates de quibus dictum est sunt mensuræ substantiæ, antequam dicamus de proprietatibus, in se habent dubitationes quæ sunt circa ipsas determinatæ. Et primo de linea: hæc enim videtur extensio infinita secundum longitudinem, cum dicat Euclides quod licet lineam protrahere in infinitum et continuum et directum. Adhuc cursus puncti, qui mathematicam facit lineam, non sistit in continuo. Videtur ergo quod procedat in infinitum. Sed contra hoc videtur esse, quod natura et intellectus abhorrent infinitum. Ergo videtur quod linea sit de his quæ abhorret intellectus.
Sed quæstio penitus nulla est, quia linea non est infinita in actu, sed secundum potentiam, per hoc solum quod in continuo secundum quod continuum est, assignari non potest ei terminus ad quem stat protractio ipsius. Et propterea dicit Euclides quia licet lineam in infinitum producere in continuum et directum, tale autem infinitum aliquid potentia est, nec natura abhorret neque intellectus.
Adhuc autem dubitant quidam utrum linea sit mensura substantiæ corporalis? si enim non est mensura, cum per nullam aliam dispositionem possit reduci ad subjectum quod est substantia, videbitur quod non sit accidens: constat autem, quod nec est substantia: ergo nihil est quod stare non potest. Si autem est
to be had by the particles of time in this, that certain ones are prior and certain ones posterior, both in the past and in the future; but the prior ones in the past are farther from the present, whereas the prior ones in the future are those which are prior toward the present; and thus something of the particulars of time is prior, and something of its particulars is posterior. Thus also concerning number it is similarly the case, namely that in the particles of number there is not position, but the order of enumeration, because one is enumerated before two, and two before three, and so concerning the other parts of number; and thus in number what is nearer to the unity which is repeated in number is in this way, and thus the parts of number will have a certain order, but they will not have position in any mode, nor even joining or connection, nor a common term which is something of each part, namely of the prior and posterior; and therefore the parts remain divided and discrete. The account of the name of number also shows this. For number is said as though true numus, that is, of division, because each unit in the parts of number is found divided and segregated. Therefore it is clear that parts of this sort perfect no position.
Similarly also speech relates itself in its parts: for this agrees with number in discreteness of parts, but in succession it agrees with the time by which the pronunciation of speech is pronounced, and therefore its parts are discrete and not permanent. Therefore speech is of the number of those things which have been said, namely those which do not have position in their parts. For the particles of speech in no way remain and can stand less than the particles of time, because at least in time it is possible to take the now or present instant, in which in some mode there is the past and in some mode there is the future. But in speech there is nothing to take in some mode, since all the indivisibles of the elements are discrete and connected in nothing, but each of them has been spoken and passes away, and therefore it cannot any longer be taken so as to be denominated permanent; on account of which there will be no position of the particles of speech; on account of which nothing of its particulars remains in reality. And thus by inducing we have proved what we intend, namely that some things from the genus of quantity consist of particles which are in them essentially having position toward one another, but other things of the same genus consist of particles not having position toward one another.
CHAPTER VII.
On doubts concerning those things which have been said about line.
But because the quantities about which it has been spoken are measures of substance, before we speak about their properties, in themselves they have doubts which are determined concerning them. And first about line: for this seems to be an infinite extension according to length, since Euclid says that it is permitted to draw a line out into the infinite and continuous and straight. Further, the course of the point, which makes a mathematical line, does not stop in the continuous. Therefore it seems that it proceeds to infinity. But against this there seems to be that nature and understanding abhor the infinite. Therefore it seems that line is among those things which understanding abhors.
But the question is wholly nothing, because line is not infinite in act, but according to potency, through this alone, that in the continuous according as it is continuous, a term at which its drawing-out stands cannot be assigned to it. And for this reason Euclid says that it is permitted to produce a line into the infinite in a continuous and straight way; but such an infinite is something in potency, and neither nature nor understanding abhors it.
Further, certain people doubt whether line is the measure of corporeal substance. For if it is not a measure, since through no other disposition can it be reduced to the subject which is substance, it will seem that it is not an accident; but it is agreed that it is not substance either; therefore it is nothing, because it cannot stand. But if it is
mensura, tunc videtur quod sit imperfecta: quia corpus mensuratur longo, lato et profundo: linea autem est longitudo tantum. Ad hoc autem dicendum quod linea est prima et perfecta mensura substantiæ. Est enim linea mensurans in longum, et mensurans in latum, et mensurans in profundum. Dicit autem Avicenna in Sufficientia quod corpus mensuratur tribus diametris. Diametri autem lineæ sunt. Ergo corpus mensuratur lineis, et incrementum omnis corporis naturalis est secundum extensionem trium diametrorum et omnium natura constantium terminus est et ratio magnitudinis et augmenti. Quod autem superficies mensuratur, hoc est secundum totam corporis extensionem inter diametros mensurantes in altum, in latum, in profundum; et quod corporea quantitas mensurat corpoream substantiam, hoc est secundum repletionem quantitativam quæ secundum intellectum est inter omnes superficies: et sic dicendum est de istis mensuris substantiæ. Nec dicendum, ut quidam male dicunt, quod linea sit mensura non separata a superficie actu vel corporea quantitate, et ideo juncta cum aliis fit mensura perfecti corporis, quamvis in esse non sit separata a superficie et corpore, tamen in actu mensurandi secundum naturam et intellectum separata in eo modo quo diximus: et ideo specialis species est quantitatis continuæ.
Dubitant etiam quidam quid linea habeat pro materia et pro forma, dicentes quod in linea non sunt nisi duæ partes, scilicet lineæ quæ copulantur et communis terminus copulationis; aut ergo erunt ei partes pro materia aut terminus copulans. Dicunt autem quod terminus copulans non potest esse lineæ pro materia, eo quod terminus copulans est punctum: linea autem non est ex punctis, ut probat Aristoteles 1. Si autem partes copulatæ erunt pro materia, adhuc sequitur major abusio: partes enim lineæ sunt lineæ: ergo linea est materia lineæ: et illa linea iterum habet materiam: ergo illius lineæ erunt materia lineæ, et ibitur in infinitum, aut erit idem materia sui ipsius, quod absurdum est: ergo videtur quod nullo modo possunt esse lineæ. Adhuc autem si punctum dicatur forma lineæ, videtur inconveniens esse: quia ea quæ sunt ex materia et forma, non possunt intelligi id quod sunt sine forma: potest autem intelligi linea sine puncto si intelligatur longitudo infinita ex utraque parte.
Ulterius etiam quæritur, utrum linea possit esse ex punctis, et videtur per Aristotelem 2 quod non, quia non se tangentium non est continuatio, punctum autem non tangit punctum. Aut enim tangeret secundum partem aut secundum totum. Si secundum partem, sequitur quod punctum habet partem, quod falsum est: quia dicit Euclides quod punctum est cujus pars non est. Si autem secundum totum tangitur, hoc est iterum impossibile, quia contactus est eorum quorum ultima sunt simul, et sic puncta se tangentia haberent ultima: sed quidquid habet ultimum, aliud est ipsum, et aliud est ultimum ipsius: igitur punctum est aliud ab ultimo ipsius: et sic iterum sequitur quod punctum secundum quantitatem partes habet: ultimum scilicet et id quod habet ultimum: non ergo linea componitur ex punctis.
Si hoc autem conceditur, tunc quæritur quare numerus componitur ex indivisibilibus, et continuum ex indivisibilibus componi non potest?
Hoc autem facile est solvere si ad memoriam revocentur quæ de natura continuæ quantitatis dicta sunt. Dicendum enim pro certo, quod linea forma est et non habet materiam ex qua fit: et ideo quæstio nulla est. Si autem sic formaretur quæstio, quid habeat linea pro substantia secundum id quod est, et quid pro formali principio? tunc quæstio valet: et quod dicatur, quod partes lineæ sunt in
a measure, then it seems that it is imperfect, because body is measured by length, breadth, and depth, but line is length only. But to this one must say that line is the first and perfect measure of substance. For line is measuring in length, and measuring in breadth, and measuring in depth. But Avicenna says in the Sufficientia that body is measured by three diameters. But diameters are lines. Therefore body is measured by lines, and the increase of every natural body is according to the extension of three diameters, and of all things standing by nature it is the term and account of magnitude and increase. But that surface is measured, this is according to the whole extension of body among the diameters measuring in height, in breadth, and in depth; and that corporeal quantity measures corporeal substance, this is according to the quantitative filling which according to understanding is among all the surfaces; and thus one must speak about these measures of substance. Nor must it be said, as certain people badly say, that line is a measure not separated in act from surface or corporeal quantity, and therefore, joined with the others, it becomes the measure of a perfect body; although in being it is not separated from surface and body, nevertheless in the act of measuring according to nature and understanding it is separated in that mode which we have stated, and therefore it is a special species of continuous quantity.
Certain people also doubt what line has for matter and for form, saying that in line there are only two parts, namely lines which are joined and the common term of joining; therefore either parts will be for it as matter, or the joining term will be. But they say that the joining term cannot be for line as matter, because the joining term is a point; but line is not from points, as Aristotle proves 1. But if the joined parts will be for matter, a greater abuse still follows: for the parts of line are lines; therefore line is the matter of line; and that line in turn has matter; therefore the lines of that line will be the matter of line, and there will be a going into infinity, or the same thing will be the matter of itself, which is absurd; therefore it seems that they can in no mode be lines. Further, if a point is called the form of line, it seems to be unfitting, because those things which are from matter and form cannot be understood as that which they are without form; but line can be understood without a point, if infinite length from each side is understood.
Further also it is asked whether line can be from points, and through Aristotle it seems 2 that it cannot, because of things not touching one another there is no continuation, but point does not touch point. For either it would touch according to part or according to whole. If according to part, it follows that point has a part, which is false, because Euclid says that a point is that whose part does not exist. But if it is touched according to the whole, this again is impossible, because contact belongs to those things whose outermost parts are together, and thus points touching one another would have outermost parts; but whatever has an outermost part is itself one thing, and its outermost part another; therefore a point is other than its outermost part, and thus again it follows that point has parts according to quantity, namely an outermost part and that which has an outermost part. Therefore line is not composed from points.
But if this is conceded, then it is asked why number is composed from indivisibles, and the continuous cannot be composed from indivisibles?
But this is easy to solve if those things which have been said about the nature of continuous quantity are recalled to memory. For it must be said as certain that line is form and does not have matter from which it is made; and therefore the question is nothing. But if the question were formed in this way, what line has for substance according to that which it is, and what it has for a formal principle, then the question is valid; and as to the fact that it is said that the parts of line are in
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