^ This is one of the passages in the Aristotelian writings where the famous distinction of the Stagyrite's works into acroatic and exoteric is recognised. Vide book I. the Less, chap. iii.
^ In this and two of the following chapters Aristotle discussei a question m regard of mathematical entities which had been alread| mooted in the enumeration of the doubts to be found in boo^ IX.
S50 TH£ MBTAPTTTST08 OF ARISTOTLE. [BOOK XII.
reside in sonsibles, and that no one of them should possess a separable subsistence. These things^ then, have been already declared.^ 2 Further ^^*» ^^ addition to these statements, it is proofs of this evident that it is impossible that any body what- SXwuty o"f" soever should be divided; for it wiU be divided mathematical according to a superficies, and this according to ^ ' a line, and a line according to a point. Wherefore, supposing that it is impossible to divide a point, it la also impossible to divide a line; and if it is impossible to divide a line, the case is the same with the other mathema- tical quantities likewise. What, therefore, is the difference in allowing either that natures of this desciiption should exist, or that these do not exist at all, but that such natures should be found in sensibles? For the same consequence will ensue; for, on the supposition of a division of the sensibles, they also will be divided, or they will not be of the nature of sensibles. But the fact is, neither is it possible noiusepua-' that such naturcs should be actually, at least, ^^'nsibies'^ Separated; for if independent of such as are cognisant by the senses, there should exist other solids that are actually in a condition of separation there- from, and which are antecedent to those that are cognisant by sense, it is evident that it is also necessary that beside sur- faces there shoidd exist other sur&ces that involve a separable subsistence; and in like manner other points and lines, for this deduction rests upon the same reasoning.
And if these points be admitted, again, in ad- preluTpt^e*^''' dition to the surfaces, and Hues, and points o' separate sur- a mathematical solid, there will be different ones thcwe inherent* subsistiug in a Separate condition. For incom- ucaiTour"**^ posite natures are antecedent to those that are composite. And if antecedent to sensibles there exist bodies which do not fall under the notice of the senses, by the same reasoning those very surfaces which subsist essentially will likewise be antecudent to those sur- faces that are to be found in immovable solids. Wherefore, those surfaceH and lines are different from those which at the same tim6 are inherent in separated solids; for the latter, indeed, are capable of consubiistence with mathematical ' For insUnoe, in book YIXL OELIL] mathematical entities XOT in SENSIBLEa 351 solids, but the former are antecedent to mathematical solids. Again, therefore, there will be lines belonging to these sur- faces prior to which there will needs be different lines and points for the same reason. And of those points con- tained in the lines that have an antecedent subsistence to those cognisant by sense there will be ether prior points to which there will no longer belong different ones that have this prior subsistence.
Wherefore, also, such an accumulation ^ as the foregoing would be absurd; for it- happens that pUcatlonof independent of such as fall under the notice of Jg^^JindrMc'e the senses there subsist single solids, no doubt, towards a ded- yet that there are three ranks of sur&ces beside qlSSSon?* those that are cognisant by the senses, and that one of these subsists beside those that are sensible, and that the second resides in mathematical solids, and that the third subsists beside those sensibles that are inherent in these, and that there exists a fourfold classification of lines, and that there are five ranks of points. Wherefore, let me ask, respecting which of these will the mathematical sciences be conversant) for, undoubtedly, they are not conversant re- specting the surfaces, and lines, and points that are resident in an immovable solid; for a science is always conversant about subjects that involve a priority of subsistence.
And the same reasoning holds good respecting numbers also; for beside each of the points reasoning holds will there exist other monads, and beside each of J^jJum&s*!"* the entities that &11 under the notice of sense; next in order will subsist those that are objects of perception for the mind: wherefore, there will exist infinite genera of mathematical numbers.
Further, how is it possible that we should decide the questions of controversy which we refuution have taken a review of in the doubts enumerated „JJJJ^ oTSSS above? For the objects about which Astronomy nomy as a is conversant will in like manner be different ^ science, from those that are cognisant by sense; and this will be the case, too, with those particulars about which Geometry is concerned. But let me ask the question how it is possibJa ^ (r(op€^ts is the word I have translated ''aocuxnulAtion.** ' Some copies I'ead mpd, others^ wtpi 352 THE METAPHYSIOS OF ARISTOTLE. [BOOK Xlt that Heaven, and the parts thereof, subsist, or any other thing whatsoever that involves motion t And the case stands the same in regard of those objects that pertain unto Optics and Harmonics; for there will exist both voice and a power of vision in addition to the things that fall beneath the notice of our senses, and to singulars. Wherefore, it is evident that there will be in existence both other senses and other objects of the senses; for why, may I ask, should these exist rather than those '< If, however, these do exist, there will also be in existence other animals, if the truth be that also there are other senses.
7. Another re- Further, are some things described by the unhrersais of* Diathematioiaus as universal in addition to these the mathema- substauccs. Therefore will this also constitute a ticians. certain other separated substance intermediate between both ideas and media, and which will be neither number, nor points, nor magnitude, nor duration. But if this is impossible, it is evident that it is impossible that those natures, also^ should be separated from sensibles.
Now, the short of the matter is this, that the coiSrning^™* vcry contrary takes place, both to what is in fact mathematical true and habitually supposed to be true, if one contrary to ^ will in this way seek to establish the existence of takes 'pface. mathematical entities as certain natures pos- sessed of a separated subsistence. For it is necessary, from the fact of the subsistence of these in this manner, that they should be antecedent to magnitudes that are cognisant by the senses, when yet in reality they are sub- sequent to them. For an imperfect magnitude is prior in generation, but subsequent in substance, in the same way as what is inanimate is prior to that which is animated. 9 Rovr wii Further, in what way also at all will these mathese raathe- thcmatical magnitudes be one,^ and when will this [Sde8*be"'o^e"/" bc the casc 1 for the things, of course, that are here reside in the soul, or a portion of the soul, or in something else that is endowed with reason. And if this be not the case, many things are exposed to dissolution. But now, what is the cause of those things which are divisible How this applies to the present question will be better understood by consulting, in book lY. chap, vi, what Aristotle consido|:s as the o^racteristics of unity* and pertaiDing to quantity being one, and remaining in c()u« junction with one another as such?
Further, do generations make this evident; for lo. This diffi. in the first place, no doubt, such make a transition by^'^^enwa-** into what pertains unto length, in the next place, 'i^n.^ into what pertains unto breadth, and lastly, into what relates to depth, and has reached an end. If, therefore, that wliich is subsequent in generation may be antecedent in substance, cor- poreity would be antecedent to a surface and a length, and will be both perfect and an entirety in this way in preference, because it is rendered a thing that is animated; but how, one may ask, would a line or a surface become animated? fot such an axiom as this would be above the grasp of our senses.
Further, it is true, corporeity constitutes a cer- ^^ ^^^ ^^ tain substance, for already doth it in a manner in- corporeUy can, volve that which is perfect; but how are lines said tfcauinef^a^n- to be substances?^ for neither are they substances not, be sub- in the same manner as species, and a certain ^^^^^^' form — for example, if in such a case we should admit that soul were a thing of this sort, — nor are they substances in the same way as matter — for instance, take the case of body as a thing of this description, — for nothing appears as endued with a capacity of consisting either from lines, or surfaces, or points. But supposing that it were a certain material substance, this woidd appear as one that is endued with ^ capacity of assuming passive states.
In definition, then, granting that mathema- tical natures will be antecedent to sense, yet it on'ty «.f^mathe does not follow that all things whatsoever that maticai entities , 1 ij 1.,. in delmition are prior m definition should be prior also m does not prove substance. For those things that are prior in i^^ubstMce^ substance, indeed, are whatsoever things which, involving a separate subsistence, are transcendent in their essence; but all those things are prior in definition of which there are definitions compounded of definitions. These, however, are not inherent at the same time. For if there are not in existence passive conditions, independent of the substances to which they belong — as, for example, a some- thing that has motion imparted to it, or which is white — whiteness will be prior to a white man, and will be prior ia ^ Vide book II. cbapa. i. and ii« 3fir4' THE MElAfHTSICS OF ARISTOTLE. [bOOK ZH^ accordance with the definition, but not in accordance witb the substance; for it does not admit of a separate subsistence, but it always subsists in conjunction with a thing in ita entirety — now, I mean by entirety a man, for instance, who ia white. Wherefore, it is evident that neither is that prior which subsists by abstraction, nor is that subsequent which subsists by addition, for by addition is a man styled white by reason of whiteness.
18. Recapituia- That, indeed, therefore, neither are mathe- tion as regards matical entities in a greater degree existences inatUiideiitn than bodies, and that they are not antecedent ties. in their essence to those objects that DbJI under the notice of the senses, but are so merely in point of definition, and that it is not possible that they should be made to involve a separate subsistence in any place, haa been declared with sufficient clearness. Since, however,^ neither in sensibles is it possible for these to subsist, it is evident that either, in short, they have no existence at all, or they subsist after some mode or other; and on this account not simply do they exist, for existence we predicate midtifiariously.
CHAPTER III.
CHAPTER III.
For in the same manner also as universals in . That there mathematics are not conversant about thingil tion* and "dc- *^^* ^^^® ^®®^ Separated, and in this condition monstration as of separation subsist independent of magnitudes weSagnitudes. and numbers, but are concerned about these— but not so far forth as they are things of such a kind as to involve magnitude, or to be divisible — ^it is evident that there is a possibility of there likewise being in existence both definitions and demonstrations respecting those magni- tude which fall under the notice of our senses; not, how- ever, so far forth as they are things cognisable by sense, but 80 far forth as thev are universals.
* Didot's e^iition begins chap. i:L with tl.ese words. I liave foUowtd IHLker.
C&. m.] RELATION BETWEEN THESE TWO. S5S For in like manner as, also, SO for forth as things „...j jRre in motion merely, there are many formal prin- tiated in thT ciples of them independent of the essence of each of S^motion!"** the things of this sort, and of their accidents, and since there is no necessity, on account of these things, either that there should exist anything that is being moved in a oondition of actual separation from seusibles, or that there shoidd be in things that are such as these any separated nature at all, so, therefore, likewise, in the case of things that are being moved, will there be rational principles and sciences; not, however, so &r forth as they are things that are in motion, but so far forth as they are bodies merely: and, again, so &r forth as they are sur&ces merely, and so far forth as they are lengths merely, and so i&r as they are divisible, and so &r as they are indivisible and things which involve position,^ and so far forth as they are indivisible merely.
Wherefore, since it is absolutely true to affirm, not only that things capable of a separate sub- ^tting°mathe. sistence exist, but also things that are not maticaS nature* capable of this separable subsistence— as, for they are said instance, that things in motion exist — so, as re- ^i5®no?prove gards mathematical entities, it is absolutely true their in-, to affirm that such mathematical entities exist, ginsfbiesf and that, at any rate, they are such as they are asserted to be. And, likewise, as it is absolutely true to affirm, in respect of the rest of the sciences, that there are sciences conversant with this particular thing, and not with that which is accidental to it — for instance, that there is one of what is white, if that which is salubrious should be what is white, but so far forth as it is salubrious — ^yet they are not conversant with that, I say, which is salubrious, but with that to which each science of it belongs, if it is salubrious, that is, in this case, with the salubrious,^ and if so &r forth as such is a man it is conversant with man, so also that this is the case with Geometry. It does not, however, follow, even though sensibles happen to belong to those objects about which Geometry is conversant, and though it may not be ^ I have followed the Paris edition. Bekker reads, tx^^^ tp^w, * There is a discordance in the MSS. as to the reading of this passage.
I have eudearoured to select the most intelligible one, and have followed Xaylor.
aa2 conversant with them so fiir forth as they are sensibleSy that the mathematical sciences will be concerned with objects that fall under the notice of the senses, ^nd thej wiU not, certainly, be conversant with these ^ while there are in existence other separate natures.
But many things are essentially accidental in Jirfeces" &c. things, as far forth as each peculiar quality of be mere acci- guch is inherent in each. Since both as &r as things separ- an animal IS female, and so far forth as it is not^maufema^ male, thcsc are its peculiar affections, although tics be conver- there is uot anything that is female, or anything arsuch?^ ^^^^ *^^* is male, which involves a subsistence separ- able from animals: wherefore, also, the case is the same so far forth as there are lengths merely, and so &t as there are surfaces.
And by so much the more as Geometry is of geometry employed about those things that are prior in nif^y! '^** definition, and which are more simple, by so much the more does it involve the considera- tion of what is accurate; but the accurate is what is simple. Wherefore, Geometry speculates into things that are without magnitude, rather than into those that are connected with magnitude, and especially are without motion. But if it contemplates motion, especially will it contemplate that motion which is primary or original, for this is most simple, and of this is that motion which is equable. ^ ^,. And there is the same mode of reasoning both mise coniinned in the case 01 the scienccs of Harmonics and fromTheobjects Gptics; for neither are the speculations of either of optics and carried on as far forth as the power of vision, or as far forth as voice is concerned, but as far forth as lines and numbei-s are the objects of inquiry; for these, of course, are the appropriate affections of those: and this is the case with mechanical science in like manner. 7. May not the Wherefore, if any one, admitting the existence matliemat^ar ^^ thosc things which iuvolvc a separate subsist- natures be eucc from accidcuts, makes any inquiry respecting pure y menta ^j^ggg qq fg^j. ft^fth as they are such, he will not for this reason utter any falsehood; just as neither does he do ^ Thi3 is a better reading which Didot gives, than the oue adopted by Bekker; the latter has *%pA instead of vtpi.
ea. ni.] analogy confirms the FORfiGonra. 351 BO when he describes anything on the earth, and says that that ia the measure of a foot which is not the measure of a foot; for not in the propositions^ doth the felsehood hirk. But thus would esich particular be investigated in the most excellent manner, if any one, having effected, as he thought, a separation, should regard as such that which does not in reality possess a separate subsistence, as is done by the arith- metician and geometrician.
For one, indeed,' and indivisible is man, so Arithmeti £ar forth as he is man; but the arithmetician and geometry bas established an indivisible one; and next ^v^^thafftu. he considers whether there is anything that is an accident in man so far forth as he is indivisible. The geometrician, on the other hand, carries on specu- lations relative to man neither as far forth as he is man, nor as &r forth as he is indivisible, but. as far forth as he is a solid. For what things, even though he were not indivisible anywhere, would be inherent in him is evident, because, even without these, that which is endued with capa- city admits of being inherent in this very man. Wherefore, on this account, geometricians, with correctness, make asser- tions, and discourse concerning entities, and entities have an existence, (for twofold is entity,) the one subsisting in actu- ality and the other materially.
Since, however, that which is good is different ^ j^ j^ ^^^ ^^ from that which is fair — for the one is always in say that mathe- conjunction with the method of doing a thing, « ™n^'^'„*^^°**^ but that which is fair also resides in things that about what is are immovable — those who assert that the ma- ^^ thematical sciences make no afiGj'mation about what is &ir or good make a false' assertion; for they do speak of these, ' This is a &Yourite principle with Aristotle, ov yitp iv rout wpardtrttn rd ^tvS^s. Vide Archbishop Whately's Elements of Logic, book IL chap, il, and Appendix of ambiguous terms — the word '' Truth."
^ iv irpd^€i — "is evidenced in the way of doing a thing;" this is the force of vpa^is compared with irpSryfui, which is the thing done. For example, Upd^€is ray k'roar6h^v means, not the acts, but the ways o' acting pursued by the Apostles. Archbishop What«^y uses the word in this sense in Appendix III. of his Logic, where ne gives us " A Praxis of Logical Analysis."
' Aristotle is here attacking Arislippus, and men of that class who ■ought to bring mathematical studies into disrepute. Vide book IL dup. ii« and frame demonstrations of them, in the most eminent sense of the word. For if they do not actually empl($y these names, they do not exhibit even the results and the reasons of these, and therefore they can hardly be said to make any assertion about them. Of what is feir, however, the most important species are order and symmetry, and that which is definite, which the mathematical sciences make manifest in a most eminent degree. And since, at least, these appear to be the causes of many things — ^now, I mean, for example, order, and that which is a definite thing — ^it i» evident that they would assert, also, the existence of a cause of this description, and its subsistence after the same manner as that which is fair subsists in. We will, however, declare our sentiments in regard of these points, in a more intel<* ligible form, elsewhere.^ CHAPTER IV.« CHAPTER IV.« Respecting, indeed, therefore, mathematical tiVe Ideal natures, that they are entities, and how &r they mine7 *^** ^^ entities, and how, in one respect, they are not antecedent to sense, and how, in another, they are antecedent, let thus much suffice to have been said on this subject. Concerning ideas, however, we must, in the first instance, examine into the actual opinion in regard of the idea which would not in any degree connect it with the nature of numbers, but in accordance with the hypothesis that has prevailed from the earliest age amongst those who originally were the first to affirm the existence of ideas.
The opinion, however, in regard of forms, hap- system a reac- pened to be adopted by those who make assertions o?Hora!ciitu8* ^^ *^^^ ^^y» ^^ account of their being persuaded, respecting the reality of this dogma, by the arguments adduced by Heraclitus, to show that all entities that &11 imder the notice of the senses are in a state of ^ Possibly Aristotle alludes to some of his mathematical writings, fragments of which havt "^nly come down to us; or, perhaps, this topie was investigated in his lost Treatise, Ufpl Ayadov.
' In this and the following chapter we have a most elaborate refuta»i tion of the Ideal Hypothesis. I have followed Didot's text. Bekkef V«gins chap. iv. with the words, vtpi Si rw IS^»k CH.IV.J BQOBATBS KO PATRON 07 IDJSALISM. 35S contiuual flux. Wherefore, if there are systems of science, and of practical wisdom, conversant about anything, wo affirm that some different natures, in a condition of permanence, must necessarily exist beside those that are cognisant by the senses, for it is plain that a science of those things that are in a state of flux has no existence.
Now, seeing that Socrates^ was engaged in forming systems in regard of the ethical or moral Jrov^CTti in virtues, and was the first to institute an investi- wience intro- gatiou in regard of the universal definition of socntes^aud these— for, to be sure, Democritus to a small ex- Jjjj.^ ^®* ^ tent merely busied himself iu physical inquiries, and defined afber what mode that which is hot, and that which is cold, subsisted, but the Pythagoreans, previously to his time, brought forward^ definitions in respect of some few things, the formal principles of which these philosophers con- nected with numbers, as, for example, take the instance what opportunity constitutes, or justice, or marriage — Socrates, notwithstanding, I say, from time to time investigated into quiddity or what a thing is, and this, too, on rational grounds. For his aim was to form syllogisms, and we know that quid- dity is a first principle of syllogisms. For dialectical strength not as yet had at that time any existence; so that they were able, even without the possession of quiddity or the substance of a thing, to institute inquiries into those things that are contraries, even though we should suppose that there would be the same science of contraries. For there are two improvements in science which one might justly ascribe to Socrates; now, I allude to his employment of inductive argu- ments, and his definition of the universal: for both of these belong to a science that is conversant about a first principle.
Socrates,^ however, did not, it is true, consti- 4. Yet not tute universals as thines involving a separable Socrates, but o o ST otiiers were tnv subsistence, nor did be regard the definitions as authors of the such; the other philoBophei*s, however, invested ^*®^ Theoo-. them with a separate subsistence, and, in addition, they denominated things of this sort as the ideas of entities.
* A repetition of this and other parts of iliese two chapters may b« found in book I. chap. iz.
' I haye followed the Paris edition. Bekker reads, iv^m-oy, ^ Aristotle will not allow the advocates of the Ideal Theory to claim Bocrates as a patroo of their system.
S60 THB XETAPHTSrCS CF ABI8T0TLE. [BOOK XII.
t tie Wherefore, it occurred to them, almost for the m-gnes against camo rcason, that there exist ideas of all things Theo?Mhatit ^^^ich are predicated tmiversally; and this proves too assumption is just as if one desirous of reckoning there are more ^ particular sum, whou, in &ct, the component forms than parts were fewer in number, should consider it an impossibility to do so, but when he had made them more numerous should succeed in counting them. For more numerous, so to say, are forms than singulars that fidl under the notice of sense: from an investigation into the causes of which did these speculators advance from sensibles to ideas; for a form is a thing that is of the same import with a sensible singular, and it subsists independent of sub- stances; ^ and forms are there in the case of many other things — namely, both in these particular things and in those that are eternal.
6. The hypo- Further, in the modes in which it is demon- thesis fain in strated that forms exist, according to none of eiiistence^of ^ these is it apparent that they really do exist; for these forms. f^Qjj^ some of them it is not necessary that a syllogism should arise, but from certain others: and in the case of things where they do not suppose that there are forms in existence, of these are there generated forms. For, according to the rational principles that may be adduced from the other sciences, there will subsist forms of all things of whatsoever there are sciences; and according to the notion of the unity that is involved in plurality will there subsist forms also of negations, and according to the perception of something belonging to what has been corrupted will there be forms of things subject to corruption, for of these is there a certain impression on the mind.
7 The best ^^^' further, with respect to the most accurate