10. othe? Ob. Further, the existence beside the two monadfl fcctions drawn of the duad as a certain nature, and of the triad tureoftheduad beside the three monads, how, may I ask, is such and the triad, admissible 1 for one will either partake of the other, as a white man beside white and man — for he partakes of these — or will do so when the one amounts to a certain differ* ence of the other, as man beside animal and biped. Further^ some things are one in contact, and others by mixture, and others by position; not one of which is it admissible should be inherent in the monads from which the duad and the triad are compounded; but just as two men are not one certain thing beside both, so it is necessary, also, that the case should stand with the monads. And they will not be said to differ because they are indivisible, for on this account, also, are points indivisible; but, nevertheless, the duad of them will not be anything different from the two. But, imdoubtedly, neither should this escape our notice, that it happens that there will exist prior and subsequent duads; and in like manner doth the case stand with the rest of the numbers. For, indeed, even allowing the duads to rank in the tetrad cue along with another, yet these are antecedent to those in the octade: and they themselves have produced — as the duad has these — the tetrads that are contained in the octade itself; so that if, also, the first duad be an idea, these likewise will constitute certain ideas.
11. Confirmed ^^^ *^®^® ^ *^® ^™® reasoning applicable to from the case of the case of the monads also, for the monads in monads. ^j^^ ^^^ duad produce the four monads that are in the tetrad. Wherefore, all the monads become ideas, and an idea will be compounded of ideas. Wherefore, it is evident that those things of which the ideas themselves happen to be compounded will be composite natures, just as if one were to say that animals are compounded of animals; if there are ideas of these, ideas will be compounded of animals.* 12. To make ^"^» ^^ general, to make monads to involve a monads to in- mutual difference of any kind whatspever would dlfflrence is"*^ be an absurd and fictitious supposition — now, I absurd. mean by fictitious a thing that is forcibly con- trived so as to suit a particular hypothesis. For neither 1 iK C(i(»v I9lat tffovrai, Bekker has these words, and I have foUowod bioQu The French edition omits them.
Oh. VII.] ho MONADS MUTUALLY DIFFER? 87S according to quantity, nor according to quality, do we see a monad differing from a monad; and it is requisite that every number ^ould be either equal or unequal: but parti- cularly that which is monadic. Wherefore, if it be neither greater nor less it will be equal. But things that are equal, and, in short, devoid of mutual difference, we consider to be the same in numbers.
And, if this be not admitted, neither will there u.showninthe be in this decade duads that are without a dif- case* of a de- ference, seeing that they are equal; for what ^ ^' cause will one be able to bring forward who makes the assertion that they are devoid of this mutual difference 1 Moreover, if every monad and another monad make two, a monad which is taken from the duad itself, andofaduad and the duad which is taken from the triad «idt"ad. itself, will be derived from monads that are different; and the question may be put as to whether this duad will be antece- dent to the triad, or subsequent to it? But there appeai-s to exist a greater necessity for its being antecedent; for the one subsists along with a triad, and the other along with a duad of monads.
And we, indeed, in general, are inclined to j^ practical adopt the supposition that one and one are two, contradiction even whether they may be equal or unequal; ^*^**'*®8™*' as, for instance, what is good and what is evil, and man and horse. They who make assertions in this way do not make these assertions of the monads however.
But, if the number belonging to the triad itself „.-.„ be not a greater Bomber tUhat belonging to IVfi^^, the duad, it is astonishing: or, on the supposition dlfferences^'ihe of its being greater, it is evident that there is an ideas win bo equal number, also, in the duad. Wherefore, "*^ this will be without a difference from the duad itself. This, however, does not admit of taking place if there is a certain first number and a second number; neither will the ideas be numbers. For this very assertion do they correctly make who think that the monads should involve mutual differences, since they will constitute ideas, as has been previously stated;^ for the subject of both will be one form, ' Compare the beginning of chapter viL 374 THE METAFHTSICS OF ARI8T0TLB. [BOOK Xlt* 16 Theconte- But, if the monads do Dot involve this difference^ quences of both the duads and the triads will be indifferent SdlTo*^' likewise. Wherefore, to the authors of this devoid of mu- assertion it is necessary to say that in counting is previously existing, make any additional assumption of anything. For neither will there subsist generation from the indefinite duad, nor is it possible that an idea can exist, for there will be one idea inherent in another, and all forms will be parts of one. Wherefore, consistently, I admit, with their hypothesis do they make their assertions; yet, upon the whole, they do not make their assertions even consistentlY with thei; hy^thens. For they overturn ma»y things; sinw they are likely to say that this itself, at least, involves a certain doubt — namely, whether when we count and say one, two, three, we additionally assume anything in coimting^ or whether we carry on our reckoning according to parts 1 We do so, however, in both cases. Wherefore, it would be ridi- culous to reduce this into so great a difference of substance.
CHAPTER VII1.1 CHAPTER VII1.1 I If a number ^^ *^® ^™* plaoc, howcver, abovc all, it is well differs from a that we should comc to somo final distinctions S'*2c"?ordiS'''* as to what the difference is between a number to quality or and a monad, if there is any difference at alL quantity. j^qw, it is uccessary that this difference exist either according to quantity or according to quality; yet neither of these appears to be admissible. But, so &r forth as number is concerned, the difference subsists according to quantity.
2. Can monads -^^^j therefore, if monads likewise differ in differ in quan- quantity, ouc number also would differ from another number, though it may be equal in the multitude of the monads. Further may we ask whether the first monads are greater or less, and whether thej may subse* ' Some make this chapter ix.
CiE. Yin.J KUMBEBS DO KOT EXCLUDE IDEAS. 375 quently increase,^ or the contrary? for all these statements are irrational But, undoubtedly, neither is it ad- ^^ missible that they should differ according to '" " ^' quality, for it is not possible that there should reside subse- quently in them any passive condition; for also they say that there inheres in numbers quality subsequently to quantity. Further, neither would it happen unto them that this should be derived from unity, nor from the duad; for the one is not quality, whereas the other partakes of the nature of a con- stituent of quantity, for of the existence of many entities is the actual nature of them a cause.
-But if, then, this subsists after a certain man- niBr differently, we must declare that this is the monad^differ case likewise, in the most eminent degree, with neither in a first principle; and we must come to some quaiityfwhat final distinction respecting the difference of the?i?i7"°®/^^ '^ o., they involve?
monad — namely, that it is especially a necessary one, and why there exists a necessity that this should be the case. If monads, however, do not differ in quantity, nor yet in quality, what difference can speculators assume as existing in themf ^ That, indeed, therefore, on the supposition that ideas are numbers, it is admissible that all the monads neither should be capable of comparison, nor should be incapable of comparison one with another in either of these ways, this point is evident.
But, assuredly,^ after the manner in which 4 attack on certain other philosophers make statements re- those who ig- specting numbers neither are such assertions JnTeond"^?' made correctly. And these are such as do not Jor**^e^\®"?v consider that there are ideas in existence, neither of mathema- fiimply considered, nor as being certain numbers, ^^^^ entities. but lay down the existence of mathematical entities, and con- tend that numbers are most original amongst entities, and that actual unity constitutes a first principle of them. For it would be absurd to go on the supposition that unity should be something primary amongst the units, as those persona assert it is; but that a duad should not be something primary ^ hrt^i^aaiv. I have followed the Latin version, ''crescant;'* and find that it bears this sense in Herodotus, Euterpe, XIII., Reizii, edit Ozon, vol. I. p. 129. * Vide book IV. chap. iz.
' ' Some make chapter z to begii with these words.
S7S fflCIS liETAPHTSICS OF ABI8T0TLB. [BOOK-XIt,^^ amongst duads, nor the triad amongst triads; for all suob- points rest on the same reasoning.
5 ifmathe- ^^f indeed, therefore, the assertions in regard maticai number of number may be viewed after this manner, and unUyTa^pt'k i^ oue wiU Seek to establish that mathematical fi"t principle number exists solely, unity, in such a case, does o num rs. ^^^ constitute a first principle of numbers. For it is requisite that unity — such as this is —should differ from the rest of the monads; and, if this be admitted, there will necessarily exist a certain first duad that is different from the other duads, and in like manner, also, will it be so with the rest of the numbers — I mean, such as are consecutive. If, however, unity constitute a first principle, there subsistij the greater necessity that the case should stand just as Plato used to say the points regarding number were disposed, aud that there should exist a certain first duad and triad, and that numbers should be not capable of comparison with one another. But, on the other hand, if any one, again, should maintain these assertions, it has been declared that many impossibilities ensue.
But, certainly, it is, at any rate, necessary erJr^exposld^ t^^at the casc bo either in that way or this of confound- ^ay. Whercforc, on the supposition that it Ideal and ma- be in neither way, it would not be admissible number?*^ that number should involve a separate subsist- ence. It is evident, however, from these state- ments, that the third mode ^ is expressed even in the worst manner — I mean, that one which makes out that the number which belongs to forms, as well as mathematical number, are the same; for it is necessary that two errors at the same time should concur with one opinion. For neither is it possible that mathematical number should subsist in this manner; but, as regards a person indulging in peculiar hypo- theses, it is necessary that he should be prolix; and that he should enumerate the consequences also, whatsoever they are, which ensue unto those who denominate numbers aa forms, this is requisite likewise.
7. The Pytha- l^^t the plan of the Pythagorics partly, no fforean system doubt, iuvolves fcwcr difficulties than the state- * Tha three modes, I take it, are those severally adopted by Plato^ Pytha,^oras, and Xenocratea CH. vm.] BimcuLTna peculiar to the ptthaoobics. 37 1 ^ents that have been previously niade; but partly about numbers •t involves certain different difficulties peculiar wiShdifficuitiet to itself. For the constituting number as that that aw pecu- which possesses a subsistence not separable from ^ ^ "* °^* sensibles removes many of the impossibilities; but the asser* tion that bodies are compounded out of numbers, and that this number is mathematical, is impossible. For neither is it correct to say that it constitutes individual magnitudes; and, in the next place, because in the most eminent degree they are disposed after this jnode, »he monads, at any rate, do not involve magnitude: and how is it possible that magnitudes should be composed of things indivisible? But, assuredly, mathematical ^ number, at least, in its nature is monadic; yet those persons say that entities constitute number: at any rate, their speculations do they try and harmonize with bodies, as if numbers were derived from those. I^ therefore, it ia requisite, on the supposition of number being something essentially belonging to entities, that some one of those modes that have been mentioned should exist, but it is not admissible that any one of these should exist, it is evident, then, that there doth not subsist any such nature of numbera as those furnish who constitute number as that which pos-. Besses a separate subsistence.
Further, might the question be asked whether ^ ^^^^ ^^^ does each monad consist from the great and the each monad • small equalised; or whether is the one monad *'°"***'°^* from the little and another from the great 1 If, indeed, there- fore, the case stands thus, neither will each number consist from all the elements, nor will the monads be devoid of mutual difference; ^ for in this monad will be inherent the great, and in that the small — ^being what is in its own nature contrary. Further, how are those resident in the triad itself? for one of them is uneven. But, perhaps, on this account they make actual unity iu what is uneven a mean. But if each of the monads arises from both the elements equalised, how will the duad constitute one certain nature compounded from the great and small) or what difference will there be in this from the monad 1 Farther, the monad ^ Perhaps the better readmg is that found in Bekker and Didot< namely, dpi6firrrtK6s. * aSid<t>opoi is the word used by Ariftatl«.
378 THE METAPHYSICS OP ARISTOTLE. [BOOK Xll» is antecedent to the duad; for when it is taken away the duad is taken away. Therefore, it is necessary that this be an idea of an idea, being, at any rate, antecedent to an idea, and that it has been produced prior to such. Of what,^ then, will it be? for the indefinite duad would be formative of duality.
9. Number Further, it is necessary that, certainly, number roust be either be infinite or finite; for speculators make finite."' * number to be that which involves a separate subsistence, so that it is not possible that the other of these should not subsist.
10. It cannot be That, therefore, it is not possible that it should infinite. y^ infinite is evident, for neither is infinite number odd, nor is it even; but the generation of numbers is invariably either of an odd number or of an even: when unity, in one instance, fells upon an even number, an odd number is produced; and when the duad, in another case, £alls upon the even, that which is from unity is rendered two- fold; and when it falls, in a third way, upon the odd numbers, another even nimiber is produced. Further, if every idea belongs to some particular thing — ^but numbers are ideas — infinite number, also, will be the idea of something, either of sensibles or of something else; although neither does this admit of taking place according to position, nor according to reason; but philosophers arrange the ideas after this manner. «« T»-^« « «« On the supposition, however, that number how far does IS finite, how &r, m point of quantity, does it It extend! extend] for it is requisite that this should be declared — ^not only that the fact is so, but also why it is so. Undoubtedly, however, if number extends up to the decade, as certain say, in the first place, of course, will forms faXi quickly; as, for instance, if the triad constitute ideal man, what number will ideal horse be? for every ideal number reaches up to the decade.^ Therefore, it is necessary that certain numbers exist of those residing in these, for these are substances and ideas; notwithstanding, however, they will fail, for the species of animal already will be superabundant.^ At ' I have followed Didot. Bekker reads the sentence thus: trportpo* 4k rlvotrovv, *H yap dSpurros fivas, K.r.X. * Vide book L chaps, v. and viil ' TbiB is the reading in the French edition. Bekker has fhra^ltu OB. Vin.] 18 NUMBER FINITE OR INFINITE t 379 the same time it is, however, evident that, if the triad in this way be ideal man, the rest of the triads likewise will be 60, for similar are those that are inherent in the some numbers. Wherefore, will there exist infinite men; if, indeed, every triad constitutes an idea, each man will be an ideal man; but if not, yet, at any rate, men will be so.
And if the smaller ^ belong, as a portion, to the,2. The diffi- greater — namely, that which is of the monads cuity of Axing that are capable of comparison as a portion of ®°*"y ^^ those that are in the same number — and if the tetrad itself be an idea of anything, as of a horse or of what is white, man will be a part of horse, if man constitutes a duad. But absurd, also, is the supposition of there being an idea of the decade, but not of the endecade, nor of the numbers consecutive to this. Further, however, there both exist and are gene- rated certain things of which there are not forms. Wherefore, the question comes to this, on what account are there not forms of those also? In such a case the forms do not consti- tute causes. Moreover, it would be absurd to imagine that number, as far as the decade, should be a certain entity in a greater degree, and a form of the decade itself, although there is no generation of this, as of an unit^ but of that there is, - , Philosophers attempt, however, to alter their ^^ Theat- opinions, as if the supposition were true that tempted rel number up to the decade were a perfect one. SiSSlitJ,^*' They generate, at any rate, the things thereon fol- lowing: as, take the case of vacuity, proportion, the odd, and other things of this kind, within the decade; for some things they ascribe to first principles, — for example, motion, restj good, evil, — but other things to numbers. Wherefore, unity amounts to what is odd; for if it is resident in the triad, how will the pentad constitute what is odd ) Further, how far do magnitudes, and as many,4 canthe such bodies as there are, partake of quantity; Pythagoreans for instance, the first indivisible line, next a difficuhte* to duad, and next those numbers up to a decade? "f"J*®'^' Further, on the supposition that number in- prioruy^of ^ ▼olves a separate subsistence, one might feel "°**y' a doubt as to whether unity were antecedent, or the triad .And the duad. As far forth, therefore, as number is 00m ^ I have followed the punctuation of this passage adopted by Didot» 380 TUB UBTAPHTStOS OF ABI8T0TLB. [bOOK St pounded unity is antecedent, but, as &r forth as what il universal and is form are prior, number involves an ante- cedent subsistence; for each of the monads constitutes a portion of number as matter, but the other as form.
15 Illustrated '^^^^ ^^ doubt, in ouc way is the right prior to in'thecaseof the acutc angle, because it has been limited by •"r^ht^Mgie. ^^ definition, and in another way is the acute prior to the right, because it is a part of it, and the right angle is divided into the acute. Undoubtedly^ indeed, as matter, the acute angle and the element and the monad are prior; and, again, as in reference to form and substance—* such as subsists according to definition — is the right angla prior, and so with the entire, which is compounded of matter and form; for both are more proximate to form and to that which definition belongs unto, but in generation are they subsequent.
16. How, then. ^^"^y *^e°> ^^7 ^ ^^y ^ ^^^^7 «: firs* Prin- i« unity 8 first ciple? ^ because it is not, they say, divisible, but p ncipief jg indivisible, both that which is universal, and that which is particular, and that which is an element; but ux another manner is unity partly that which subsists accord<* ing to definition, and partly that according to duration. In what way, then, does unity constitute a first principle? for, as has been declared, both the right angle seems to be ante^ cedent to the acute, and the acute to ^he right, and each is one. Therefore, in both ways do speculators constitute unity as a first principle.
But, further, is this impossible; for the ond catesofthis subsists as form and substance, and the other aa M&wish it.*** * P*''* ^°^ ^ matter. For in a manner each one in reality subsists in capacity, if, at least, number is one certain thing and not as an aggregate heap; but different number subsists from different monads, as they say, and each monad does not subsist in actuality. 18 This fail "^ causc, howcver, of the error which ensues is accounted for this, that they are accustomed at the same time inqufr™pu?-°^ ^ pursuc their investigations from the mathe- tued by the matical scienccs and from universal definitions, Pythagoreans. ■^j^i^ej.gfQre, from thosc, uo doubt, as a point, havQ ' The student will remember how this question has been asked iM book II., and how Aristotle notices the theory itself in book L CH. IX.| DOI'BTS ARISING FROM THE FOREGOING. 381 they established unity, and the first principle; for the monad b a point without position. As, therefore, certain others, also, have compounded entities out of what is least, so do these .persons likewise. Wherefore, the monad becomes the matter of numbers, and at the same time is prior to the duad; and, again, is it subsequent to the duad existing as a certain whole, and as an unit, and as species. On account, however, of their being engaged in investigating that which has been predicated universally as an unit, they in this way, also, have spoken of it as a part. But it is impossible that these should reside in the same subject at the same time. But, on the supposi- tion of its being necessary that unity itself should subsist merely without position — for in no respect is there a differ^ ence, save that it constitutes a first principle, and that the duad is divisible, whereas that the monad is not so— -if this be the case, the monad would be more similar to unity itself; but, if the monad alone be without position, unity will be more similar to the monad than to the duad: so that, in either case, each monad would be prior to the duad. These specu- lators do not say so, however, at least they generate the duad^ first. Further, on the supposition that the duad itself is a certain unit, and the triad itself, both constitute a duad, from what, then, may I ask, does the duad itself consist ) CHAPTER IX.2 But one might also feel perplexed — since con- j nig^hatis tact, likewise, has not an existence in numbers, consecutive to but that which is consecutive has — in regard of Snityr^and whatsoever monads there is not to be found a other ques- medium, as those that are in the duad or the **°°'*. triad, whether what is consecutive is to be found in unity itself or not; and whether the duad be antecedent to those things that are consecutive, or anything whatsoever to the monads ) ^ Some copies read rhv hfKdda.
* These curious question* that follow in this chapter are <iuit« characteristic of the old Pnilosuphy. This chapter, which Bekkei reckons as ninth, some consider to be the eleventh. Vide nota^ p. 206* 382 THE METAPHTBIOS OF ARISTOTLE. [bOOK XtL 2 These diffl- "^^^ ^ ^^^^ manner, also, concerning the 8ul> .coities extend Sequent genera of number do these difficulties toe'Sfhlri^ ensue, both in the case of a line, and surface neraofnum- and bodj. Fcr some inquirers make lengtU ***"' from the species of the great and the small — ^for instance, the lengths, as it were, from the long as well as from the short — but surfaces from wide and narrow, and bulks from what is profound and low; and these are species of the great and the small In respect, however, of the principle that subsists according fo unity have different persons in different ways sought to establish their opinions upon points of this description: and in these, also, appear innumerable statements that are both impossible and fictitious, and which are contrary to all suppositions that are rational. For also it happens that they are severed in their connexion one with another, unless likewise the first principles are concomitant, so that there should exist what is broad and narrow, and long and short And if this be admitted, the surface will constitute a line, smd that which is solid a sur&ce. Further, however, angles, and figures, and such like, how will they be assigned? and the same consequence ensues unto the points respecting numbers; for these are passive states belonging to magnitude: but magnitude is not a passive condition belonging unto these; as neither is length of straightness and what is curved, nor solids of what is smooth and rough. 8. Common Common, however, to all these assumptions is