Commentary on the Timaeus of Plato

1 Mr; is oinitud in the original. in.] TIM.EHS OF PLATO. ,jr, deriving ils subsistence from Itoih. For thus Theodonis flic Asimran, saya, who found this opinion in Porphyry us derived from (he Persian*. These thin-'stherefore, Antoninus relates \vlio was the disciple of Ammonius. To the first of these, however, i, miM he said, that since Plato dors not make the soul to \tc number, it is ahsurd to investigate the principles of number of which the soul consists. Jiut to the second, that Plato says the soul is incorporeal when com pared with every body, and that it has a self-motive essence : but that nothin" which possesses interval is a tiling of this kind. To the third it must he said ^ that Plato is not of opinion that the irrational is more ancient than the rational part. For divinity, as he says, did not think fit that the more ancient should lx> governed l»y the junior nature. To the fourth, that the discussion is not concern- " ing the psychical knowledge, hut the psychical essence. And hence it is not pro per to say, that the soul is a medium hetween the two gnostic powers, the intellectual and the sensitive. And to the fifth, that every intellect is unbegotten, and separate from bodies, liut Plato calls the soul a partible and generated essence, giving it a division opposite to the essence? which subsists with invariable sameness, and to that which is partible about bodies, and separating it from tlio essence which is external to bodies, and always is. As these particulars, however, have Ix'en suiicicntly discussed, lei us consider what follows.

" And again after the same manner, with respect to the nature of same and the nature of different, he constituted the soul in the middle of the impartiality of these, and of the nature which is divisible about bodies." Kssence, as we have said, has the first order in the genera, because it is as it were, the Vesta of being. Sameness, therefore, has the second ; and difference^ the third order. For some consider difference as having a dignity superior to sameness. But Plato, in what he before said, has clearly evinced that the similar is better than the dissimilar; and now assuming sameness after essence, directly gives it the preference to difference. And as we have said that the middle essence is inferior to the intellectual essence, but transcends that which is divisible about body, thus also we say that the sameness of the soul is inferior to the impartible, but is more united than the partible sameness, and in a similar manner with re.sjK'ct to difference. Hence in the essence of intellect, the sameness, being one. collects itself, and also the essential difference, to the essence which is there, and is one ; just as the difference. lx?ing one, separates itself, nnd the essence and same ness from each other. But m the soul sameness collects into one the differences \

4ft. PROCLUS ON THE [BOOK in. which are many in the many parts, and difference separates the samenesses. I know, therefore, that some Platonists, arrange sameness in impartible, but difference in partible natures, and thus make (he soul to consist of both, as a medium between sameness and difference. These, however, do not1 attend to what Plato here says, that the soul is a medium l>et\veen the impartiality of sameness and difference, and the partibility of the nature which is divisible about bodies. It is likewise requisite to know that these are the genera of being, and that it is necessary these should be every where appropriately, in impartible, and partible natures, and in the media between these, and again in intellectual es sences, in souls, in natures, and in corporeal masses. For I should be ashamed to divide the genera as they do, placing one here, but another there. For if they had said, that sameness predominates in intelligible ami impartible natures, but difference in sensibles, and partible natures, they would have spoken rightly. But if they assert that impartible* are separate from difference, neither will they be able to give to them sameness. For the one differs from iht same.* And if they say that partible natures are separate from sameness, they subvert the essence of them.

Following therefore things themselves, we must admit that the genus of same ness subsists with a demiurgic peculiarity, but that the species of it, is the imparti ble and the partible, and the medium between both. And again that the genus of difference is demiurgic, but the species of it, the impartible and the partible, and that which is intermediate. Admitting this Iikewi>e, we must assign media to the soul, ami complicate them with the middle form of essence, in order that we may constitute th»- existence of the soul. For thus I think we shall be able to adapt the words of Plato to things. For he says, that as in essence, so likewise in the nature of same and the nature of different, the Demiurgus mingled a third thing from both, and after the same manner. And as there, that which was mingled from both was a species or form of essence, so here, the medium between same and different, is a species or form. For it is possible to IK- a medium not as form, but as a whole composed of the extremes, a.s an animal which consists of soul and body. In order, therefore, that you may not ignorantly conceive this to be the rase, he adds, " and after the same manner;" that here also, that which consists of both may be a form and not a whole.

" And taking them as three beings, he mingled all of them into one idea." 1 Oi/i \"> Itfn- omitted in the ordinal. 1 For if imjiailiMt-) have no difference, llicv will be llic same with each other, and will be Iftt one. Fur in the ow (bete is no difference. So that thr onr will not differ from thr tame. BOOK in.] TIM/F.US OF PLATO. 47 That the three demiurgic genera, are the causes of hypostasis to second and third forms, Plato I think sufficiently manifests in saying, " and taking them as three things" For where do they subsist? Evidently in the Demiurgtis. For hecontains that which he takes. But he receives them from superior causes. For since sameness having in a greater degree the form of bound, consists of bound and infinity, and also difference which has more the form of infinity consists of these just as essence similarly proceeds according to both, it is evident, that \re must rather place in bound [than in infinity] the sameness and difference of the impartible, but those of the media similarly in both, and those of partible natures rather in the infinite than in bound ; just as the first difference lias the infinite, in a small degree only declining from bound, and the first sameness lias bound in a small degree declining from the infinite.1 Hence Plato says that the Demiurgus received them as three, l>eing separated from each other.

And since the forms that are in him,* hasten to the generation of other things, on this account also, he constitutes other things from these. If therefore we understand by hcings things prior to the generation of the media, we must say that these subsist in him according to cause. For these wore in him prior to the things generated by him. But if by bcingx we understand the media, we must understand them as things constituted. For they are now lyings, because they were produced by him prior to the mixture, each apart from the other; and essence, sameness, and difference were now generated each by itself. May not however the word she in terpreted more simply, vi/. the Demiurgus " taking them being three" for so many things he effected from the three, the extremes Ix'ing now now constituted by him, according to the preexistent causes which he contains ? And these genera indeed he produced according to demiurgic being ; but he added idea to all the three ac cording to the union in himself, and the deity in him, which is the cause of the union of multitude. And you see, that each of the three was a form, and that which was produced from the mixture of the three, was one idea. Hence it is" necessary to say, that the soul is a form of forms ; and, in short, to conceive nothing in it as a composite and corporeal.

The triad, therefore, is adapted to the essence itself of the soul ; since it was before shown that the soul is triadic. For we divided the whole of it into es sence, power and energy ; essence, into hyparxis, harmony, and form ; and hyparxis, into what is properly called essence, same, and different. Nor is it proper to wonder, if we make a part of essence lobe essence. For the one genus of being is called essence, and that also which is generated, as it were, from all the ele- 1 In the original there is in omissiou here of rov arcipov. * For cr rnirji in this place, it ia neccjtary to read <v ovry. PROCLUS ON TIIH [BOOK m. ments of being, is denominated essence. If, how ever, we should again inquire, what it is that makes this one idea not to be any casual soul, but the mundane soul, but elsewhere a different soul ; we reply, that it is the total nature of the genera that are assumed. For the mundane suul is a medium, not of casual extremes, but of a total intellect, and total corporeal nature, according to which the world is an animal ; just as it is endued with intellect according to its impartible nature, and animated according to its middle nature. The predominance likewise of essence, causes it to be the mundane soul ; for this makes it to be divine ; just as the prevalence of sameness alone, produces a da-moniacal soul, and of difference alone, a partial soul. A different habitude therefore to the extremes, produces a difference in the media. And the mixture of the media, defined according to the prevalence of one thing, evidently changes the whole.

" Co-adapting by force the nature of different which it was difficult to mingle, to the nature of same." How is the nature of different difficult Jo l>e mingled » because it has a sepa rating and dividing power, and is the cause of progressions and multiplications. \ But every divine In-ing begins his energy from himself. Hence also the nature of different separates itself from other things and from itself. For it generates multitude in itself. On this account, it is said in the Sophista, that it makes both other things and itself to be non-beings, separating them from other beings. Hence it is dillieult to be mingled, possessing this difficulty, not from accident or any deviation, but having an essence which is the cause of olluniess, of the nnconfused hyposlasis of forms, and of nnmingled simplicity. Possessing likewise such a power as this, it is a certain contrary both to sameness and to essence.1 And it is contrary indeed to sameness, because sameness is the cause ol union, communion and connexion ; but difference, of separation, of an inability to be mingled, and of otherness. But it is a certain contrary to essence because essence is being, but ditlereiice is non Ix-ing, as is demonstrated in the Sophista. For the nature of different being divided into minute parts, Iweomcs the principal of nonl>eing. That we may not, therefore, be involved in ambiguity by perceiving an at-once-collected mixture of the genera, he in tli<- tir>t place mingles same \\illi different' and >a\s that divinity co-adapted the nature of difference to that of sameness, as hajinoiiicjjly conjoining it with middle sameness. Afterwards, he

1 Tor troi TUV ain«» hero, it !•> uecossarv to r*ail jryxu rt\v uvitav. 1 In»lr;i<l of ro rui'roi- ro fliirffci* /jiyiu<m in tins place, il is «>lniuu>ly requisite to read ro TO.VTOV ry HOOK m.] TIM/EUS OF PLATO. 4., mingled both the>e with essence. For Pluto having said, that divinity co-ailapted the nature of difference to that of sameness adds, " that he mingled them with essence, and made OIK tiling from the three." For because essence is c< nnectivc of the two genera xamc and different, but tliese are co-ordinate to each other, it is necessary that tliese should in the first place be mingled with each other, and in the second place, that both should be mingled with essence. And thus much concorning the order in the mixture. The force, however, employed in the mixture, is not adventitious, nor such as the force which is preternatural, but indicates a trantccndcncy find abundance of power ; for such is demiurgic power. So that it is able to unite difference, to divide sameness, and to produce one harmony from both. " But having mingled these two with essence, and made one thing from the three, he again divided this whole, into appropriate parts."

As the equal and the unequal are conjoined with quantity,1 and as all quan tity is either equal or unequal, or rather is equal and at the same time unequal, for every quantity at once participates of both ; and as the similar and the dissimilar are conjoined with quality, and every quality is both similar and dissimilar ; thus, also, same and different are co-existent with essence ; and all essence participates of sameness and difference. For these are essentially, or according to existence itself, inherent in things, and not according to quantity or quality. Hence they are essential, being the media between the divine genera, and those things which are inherent in quantities and qualities. For sameness, indeed, is suspended from 1 hound, but difference from infinity ; just as similitude and equality, are suspended ] from sameness, but dissimilitude and inequality from difference. Hence, also, Plato in Hie Philelms, produces bound and infinity from [the highest] God : for they are divine genera. But in the Sophista, he denominates same and different the genera* of bcinsr. And the former subsist about the one, but the latter about essence. Again, also, you sec how much more venerable Plato is than all other physiologists, and even than Aristotle himself. For they making their principles to \w, contraries, introduce organic, material, and partible contrarieties. And even the most venerable of them, refer contraries to excess and defect; badly assuming the privation of measure in the principles of things. For measure is more divine than the privation of measure. But Plato refers contraries to

* I Of taeirrp rornv aif hcrr, read uoarp ry TOTJI. * For ycni rev m-roi Itrro, rend ycrij rov orroi. Tim. Plat. VOL. II. sameness and difference, through which he comprehends all the contrarieties in soul, in nature, and in body ; and he places these in the Demiurgus, in order that lie may give to them generative, cosmiirgir,1 immaterial, and exempt powers. Having likewise placed them there, he constitutes the soul from them, producing from these, that which is as it were the subject of it, and the l»eing of it, so far as being, and not so far as it is a being with a certain quality, in order that it may imitate primary beirn;. For lie afterwards assigns to the soul harmony, form, powers, and energies, thus rcnderin'j: the discussion of it perfect. For we shall be able to survey its dignity, and its order in the world, which is of a ruling nature, if we perceive how by all the genera that are in it, it contains mundane essences ; bow by its own harmony, it co-harmoni/.t-s the \\ hole world ; how by its own figure, it comprehends all mundane forms and fi-uivs ; after what manner, by the powers tlr.it are in it, it is able to perfeet all physical, and all rational, or artificial powers; and how by its own proper energies, it excites mundane productions.

Why, therefore, did not Plato say, that the soul is a medium between intellect and sense ? Pecaiise lie says, that intellect and sense, are certain gnostic powers. Hut his intention was to di liver the middle nature of it in essences, and not in po\\ci>. >Vhy, however, did he not say, that the soul is a medium between idea and things which are invested with form ? Heeau<e it was not now proposed by him to teach us \\liat the quality is of its form, hut what its essence is. Put it is not the same tiling to speak of the essence, ns it is to speak of the form of a cer- ^ tain thing. Fur form exhibits an essence of a certain quality. Why then, did he not place the soul as a medium between intelligible and sensibl«? numbers? Hecause in the following part of the discussion, he a->igns to it an harmony, accord ing to which it ranks as a medium between separate numbers, and separate harmony, and sensible numbers and the harmony which is inseparable from subjects. For according to tin.1 conception of Plato, the soul is neither harmony it>elf, nor the harmony which is in things harmoni/ed. Harmony itself, indeed, is uniform and separate, and is exempt from all harmonized natures, of whatever kind they may be; being that alone which is called harmony itself. Hut the harmony which is in things harmoni/ed, pertains to other things, and is naturally adapted to be moved by others. And the middle of both, is the harmony of the >oul. For this is that u'/iit.h is Jir&t /uinfiotiizcJ. Hence also it imparts harmony to other things. For that which is able Jo impart something to another thing, is either the form itself [which is participated], or prinaiily participates of it. This harmony therefore of the soul, is inferior to intelligible harmony, and to intelligible/

' i. c. Powers ffltclive of llie world. BOOK in.] TIM. TXTS OI' PLATO. ;t\ numbers, but transcending sensible harmony, it likewise transcends sensible numbers. And if it be requisite to speak concisely about each of these, each of them is fourfold. For with respect to number, the first is divine, the second essential, the third psychical, and the last physical. And the first, indeed, has the form of unity, the second is immoveable, the third is self-motive, and the fourth is alter-motive. With respect to harmony, also, the first is in the Gods, the second is in truly-existing beings, the third is in souls, and the last is in the natures which are harmonized by other things. If, therefore, it had been now proposed by Plato to speak concerning the psychical harmony, he would have said that it is a me dium IxHween impartible and partible harmony. But since th° present discussion is concerning the essence of the sotd, he says that it is a medium between same and different. Here, likewise, it is necessary to observe, that Plato in what is now paid, makes as it were a conversion of the progressions of the psychical essence to their prin ciples. Tor since the form of the mixtures is twofold, the one subsisting accord ing to the mixture of the extremes, which we have in the mixture of essence, same ness, and difference; but the other, according to again collecting the media into one whole ;

— this being the case, in the former mixture he began from essence, and rnded in the latter; but tire versa in the latter. For he first co-harmoni/ed the latter with sameness, and thus afterwards both with essence, and ended in essence, from which according to progression he bc^an. Every where, however, that which is a whole' is subordinate to the two principles. For when he pro duces the soul from impartibles and partibles, he places the impartible analogous to bound, but the partible to infinity. For the infinite is the cause of multitude, but bound of union. And when he co-harmonizes difference with sameness, he assumes difference as in the genera of being, belonging to the co-ordination of tin: infinite; but sameness, as belonging to the co-ordination of bound. When like wise he mingles the two with essence, lie assumes essence as having the form of unity ; but sameness and difference as dyadic, and opposed to each other. And he does not cease collecting the multitude together, till he evinces the whole to be one. For the one is more excellent than essence itself, and the biformed princi ples [bound and infinity]. The mixture, however, being as we have said, twofold, the one constituting the elements themselves, the other tains: that which consists of the elements, Porphyry rightly inquires whether the Demiurgus made both these in the Crater [i. e. Juno], or one of them out of, but the other in it ; and he

1 Instead of wayra\ov £« ni>, r\rrai r*iv ivo TO >v aft-^uf in this place, it is fff] .i'ilc from the TCf'.OO cf Leonicus Thomsras to read Trnrro\ev I* aw vwt\tTm tttv Ivo TO eXo* ap^vv. 54 I'ROCLUS ON THE [BOOK in. universe, since it is not lawful to refer the deteriorations' of partial souls to such as are divine, Plato adds, tjiat divinity divided the soul into appropriate part*. For here that which divides, divides according to an intellectual cause, and that vliicli is divided, is divided essentially. This mode of division, therefore, is adapted both to the divider, and the tiling divided ; since it is beneficent, and perfective of the essence of the sou!, introducing it to intellectual variety,' and making it all-perfect, by inserting in it all the reasons of beings. If then we assert these things rightly, it is not proper to separate the soul from union in the division of it, nor to consume the wholeness of it, into the genera tion of the parts, as the words of Tinntus seem to indicate when he says, that Un tiling mingled was consumed into these parts. Nor must we fancy that this \va> «s it were the division of a certain rule. J;or all these modes are corporeal, and by no means adapted to immaterial essences. For every thing which is generated bv the Deiniurgus, must necessarily remain the same, since he always produces after the same manner, being immoveuble and eternal in his energies.

It is neces sary, therefore, that the whole should always remain a whole ; that the generation of the parts should be effected, the wholeness remaining; and that this should not be consumed in the division of the parts. Hence, we must conceive, that the essence of the soul is one, and at the same time many, the whole remaining and being distributed into parts, and possessing continuity, and being, at the same time, divided. Nor must we imagine that the continuity of it is accompanied with interval; fur it is continued and without magnitude, like the continuity of time; nor that the division of it is according to monadic numbers. For a thing of this kind is quantity which does not accord with continuity.1 But we must collect these into one, as it is fit with incorporeal natures, anil must survey in the soul, the whole in conjunction with 4 the parts. Plato also manifests, that it is not proper to depart from union in the division, through the following words :

" At the same time minglingcach part from same, different, and csscnco. But he began to divide as follows." If the genera which constitute the soul, are in all the parts ' of it, and the whole 1 For ra tXarrw fitra here, read cAarrw/jara. ' lloittXtar is omitted in llm |>lace in tLe original, but ought from the version of Thomiru«, to be in serted. 1 Vi/. a lUiug of lliis kind is discrete quantity. * for fitf in this place, road ft<ra. 1 MrfMtfif is omitted in the original. consists of pnrls wmilar to itself, it will in no respect be separated from continuity and union. For if in bodies similars cohere with eaeli other without a medium, how much more in an ineorporeal nature must all the things that are as it were parts, be united, and the whole be vanquished by unity, neither the. parts being confused, through the position of the whole, nor the wholeness taken away through the separation of the parts? You may also assume from these things, that the soul according to all the parts of itself, is both impartible and partible. For if e\cry part of it participates of all the middle genera, nothing in it can be assumed w hich does not consist of these. The ancients also, looking to these things, concluded eseiy where concerning it, that all of it is beiri-^, life, and intellect, and that whiehe\er you may assume of the three, you may infer that it contains the remaining two ;

since all things in it proceed through all, the whole is one, the one of it is all-perfect, and a part in it corresponds to the whole. But if each of the parts of it being many, is a certain essence, as numerous as are the parts, so multitudinous also is the essence. In a similar manner also with respect to same ness and difference, each of these in intellect indeed is one, and on this account it is impartible. For in intellect one thins; is not a part different from another. But in the soul both these are divided according to essential mim)>er, and the parts of it are adapted to each other, causing it to l>e one tiling from many, and a whole from parts.

Moreover, this also deserves (o be considered, that according to the generation of the elements, he l>e«ran as we have said, from essence ; but according to the composition of the whole, from difference; co-adapting the nature of different, ^hich is difficult to be mingled wiih smite, and mingling both with essence, lint r.ccordintr to the division of the whole into harmonic ratios, he bewail from same ness. For he says' thai he divided each part mingled from same, different, and essence. For a commencement from essence is entirely * adapted to the genera tion of simple natures ; since essence is more simple than other things. To the composition, however, of the whole, a commencement from difference is adapted. For the generation of the whole from parts begins from things subordinate to the whole. And a commencement from sameness is adapted to the hypostasis of harmony. For the Demiurgus was willing that this should terminate in the sumene-s and communion of the things that were divided. And, in short, he was willing that harmony should l>e efVective of the sameness of the things harmonized. Those particulars, however, having been discussed by ns as far as we are able, I For •• r-r»i here it is obviously ncce^ary to read $1*1. * Instead of wurrvv here, read irarrwi.

rtCt PROCLUS ON THE [BOOK in. it is necessary in the next place to premise those things which ought to l>e readily known by us concerning numbers, and the harmonic ratios of the soul, in order that we may not attempt in vain the interpretation of what follows It is neces sary therefore to premise such things as are usually mentioned in harmonic dis cussions, viz. what sound, interval, and system are, and that the Pythagoreans did not assume the symphonies in harmony from any thing else than numbers, and not from all these, hut from multiples and super-particulars. For they said that the diatessaron is in a sesquitertian ratio ; hut thediapet.te in a scsquialter ; and the diapason in a duple ratio. And again they said, that the diapa>on and at the same time diapente is in a triple, but the disdiapason in a quadruple ratio. For the diapason and at the same time diatessaron, did not appear to them to be syrnphonious, because it consists i" a multiple super-partienl ratio, vi/. in the ratio of K to 3. For (j is a medium between the two, producing with the le^s numlKT a duple, but with the greater, a subsesquitertiau ratio. These things therefore, must be premised, and also that the sesquioctave is in the ratio of a tone; that the sesquitertian ratio consists of two tones and a leimma; and the sesquialter of three tones and a leimma.1 Hut we shall afterwards learn what the ratio of the leimma is.

Moreover, the Pythagoreans said, that there arcthree genera of harmonics, the diatonic, the enharmonic', and the chromatic. Likewise, that the diatonic consists of a .semitone (but this which I now call a semitone is not properly so, but a leimma), and of a tone, ami another tone. But the enharmonic consists of a diesis, another diesis, and a ditone. And the chro matic of a semitone, another .semitone, and a trisemitone. Hut diesis is as it were the fourth part of a tone, not being in reality a fourth, as neither is a leimma accurately a semitone. These tilings, however, we shall demonstrate in what follows.

Hut a> there are three genera, each of which is a certain division of the tetrachord, Plato appears to have used the diatonic genus alone. For he thinks tit to divide the sesquitertian ratios, into sesquiortaves and leimmas, but not into enharmonic dieses ; since some of the ancients called a semitone diesis. Plato, likewise, seems to have assumed this genus, 1 mean the diatonic, as more grand, simple, and generous, than the other genera; though tlie enharmonic appears to be more adapted to erudition. And if it be requisite to declare my own predic tion on this subject, the enharmonic genus, presides over all the life which is ' After X(Wiaro« in the original, it is necessary to su|>|ilv from the version of Tliom.Tus, the words TO ft ;||iitiX»-;»- ti r,,iw» ronjy »cu \f,/iftar»t. I relt-r the reader who it desirous of thoroughly undrrsIdDding \vliat it here, Jiid failher on, said, to mv Theorclk- Arithmetic. BOOK in.] TJM/ELS OF I'LATO. 57 divisible nliout bodies," just as the diatonic presides over the rational In.. Hence the enharmonic genus, is adapted to instruct and discipline the dixisiblc life. But the chromatic genus presides over the corporeal ide.i itself. Hence, it is effeminate and ignoble. The enharmonic genus therefore is deservedly dis;-iplinative. - Hence, Socrates in the Republic thinks fit to mention it particularly, in what lie says about harmony. And Tiuurus knowing this, and having heard Socrates asserting these things on the preceding day, at the same time constitutes the essence of the soul through the diatonic, and not through the enharmonic genus;

the latter, as we have said, being adapted to erudition. For on this account, the ancients called the leaders [or preceptors] of these disciplines Harmonic! [or skilled in music]. Ari.stoxemis therefore, in the first book of his Harmonic Elements, says, it happened that those were truly called Harmonici, who formerly employed themselves in what pertains to music. For being solely ••ugaged in harmony they neglected every other pursuit. In which Aristoxeuus also asserts what is wonderful, vi/. that the ancients had no knowledge of the diatonic diagram

For he thus writes: " As an indication of the truth of this, their diagram alone exhibits enharmonic systems, but no one ever saw a diatonic or chromatic dia gram delineated by them." It is worthy of admiration, however, that he should assert these things, since Plato exhibits a diagram according to the diatonic genus, and also Tima-us himself. Perhaps therefore what Adrastus says is true, who derides Aristoxenus as a man of not very elegant manners,' but studious of appearing to say something new. Plato, therefore, in the diatonic genus, makes a division of tetrachords, and proceeds not only as far as to the diapason, but also as far as to a quadruple diapason and diapente, adding likewise a tone. Or according to Severus, Plato did not produce the tetrachords without a tone, but ended in a leimma, and not in a tone. If, however, some one should doubt, how Plato produced the diagram to such an extent, let him attend to the words of Adrastus. For he says that Aristoxenus, extended the magnitude of his multiform diagram, as far as to the diapason and dialessaron, and the symphony of these, in consequence of prefer ring the information of the ears to the decision of intellect. But the more modern musicians extended the diagram as far as to the fifteenth mode, vi/. to the thrice diapason and tone, in so doing looking solely to our utility, and thinking that those ivhocontend in singing could not exceed this, nor their auditors judge clearly beyond

1 Initcad of ran wrpc ran vw/jairi fitpierait ill tliii place, it is necessary to read rijt n/n rttf ev^n*t * For ro tibot hf re, it u requisite from Uic version of Thomarus, to re«d ro qOoi. Tim. Plat. VOL. II. II it. Plato, however, looking to nature, constitutes the soul from all these, in order that it might proceed as far as to solid numbers, as it ought to preside over bodies. For the progression as far as to the quadruple diapason and diapente, necessarily follows the seven terms [or bounding numbers]. But this is evident from the greatest term being twenty seven. And thus much in answer to the doubt. in short, then- are these three things into which tin' consideration of harmony may be divided. One of these is the exposition of the se\en parts. The second is the insertion of the two media. The third is the division of the sesqnitertiaii and sesqiiialtcr ratios, into sesquioctaves and leimmas. Hence some, as Adrastus, are accustomed to make three triangles, and in one of them, which is the least, to describe the seven parts, making the summit of the triangle to be one of the parts, and distributing the other six about this. In one of the sides also, they describe the whole duple, order, but in the other, the whole- triple order. Moreover, in tinother triangle which is greater, and contain* the former, they increased the numbers, and again in a similar manner inserted two media, arranging the dupleseparate from the triple nmnlM-rs ;

and placing OIK; of the. parts at the summit. But in the third triangle, which compreliends both the others, they described after the same manner the whole digram. Others again, adopting a description in the form of the letter X, arrange the numbers successively, as in the section of a rule, according to three centers, assuming the first, second, and third numbers, as we also shall do. This method likewise is adopted by Porphyry and Severns. And such are the particulars which ought to be premised, and also tint Plato divides this Iv-ad into three parts, in the lir>t of the three, discussing the seven parts, in which there are three duple, and three triple interval.-, according to the geometrical middle, i. e. according to the same ratios. But in the second part, he discusses the insertion of the other two media, vi/. the harmonic and arith metic, into each interval of the duple and triple numbers. And in the third p;:rt, be considers the division of the se>quitei tian and sosquialter ratios, into sesquioetaves and leimmas, and as far as to these extends the discussion of the parts of the soul.

It is necessary however to be well acquainted with such things as are said about the three media, and to know their dilK-rences, and what the methods are through which they are discovered. The arithmetical medium, therefore, is that in which the middle term exceeds and is exceeded by an equal ' quantity, as may U- seen in all the numbers that are in a consequent order, conformably to the definition of Tima-us himself. But the harmonic medium is that in which the ' \aut> is omitted iu the original. BOOK MI.] TIM.l'US OF PLATO. :,-, middle term is exceeded l>y the greater, l>y Hie same part of the greater, by which it exceeds the less term, as in the numbers fj, 1 and 3. For here 4 is exceeded hy (J by '2, which is tin; third part of <i, and it exceeds 3 the less term by 1, which is the third part of 3.' And the geometrical medium is that, in which there is the K'une ratio of (he greater to the middle term, as there is of the middle to the less term. The methods however of discovering these, must in the next place be unfolded l>y us. Let two terms, therefore, be given, between which it is proposed to find an harmonic, and also an arithmetic medium; and let the terms have a duple ratio, as for instance 12 and 0. I take, therefore, the excess of the greater number above the less, which is evidently (J, and dividing it into two equal parts, I add the half to the less numl>er, and make this the middle term.

Hence 0 is the arith metical medium between 12 and 0. For the excess is three, both of the greater :<bove the middle, and of the middle above the least term. Again, taking the difference of the extremes, which is (J, I multiply this by the less term, and the product is 30, and dividing this by the sum of the extremes, i. e. by Itt, the quo tient 2 is produced, which is the breadth of the comparison.1 To this also, I add 0, and I have the harmonic middle 0. / Tor by that part of the greater term 12 by which 8 is exceeded by it, by this part of the less term <>, 0 exceeds the less. For it is exceeded by the third part of 12, and by a third part of (t it exceeds 6. Again, let there be a triple intenal. as lor instance I» and (>,' adding these together I make 24, of which taking the half, 1 have the arithmetical middle 12. Again, taking the e\c« ss of 1H above 0, i. c. 12, I multiply it by the less term 6, and the product is 72. This I divide by 24 the sum of the extremes, and 3 the breadth of the comparison is produced. Afterwards, 1 add this to (J, and 1 ha\e ,'Mor the harmonic medium, which exceeds and is exceeded by the same part of the extremes.

Thus also, if I and 2 were the extremes, by adding them together, and taking the half of both, I shall have 1 and the half of I, for the middle term of the arithmetical middle. Hut taking the excess of the greater term above unity, and multiplying it by the less term, viz. unity by unity, I have 1 from both. Afterwards dividing* this by 3, the sum of the extremes, I shall

1 Ihiuiiiiiiu- proportion may also hr <l< linni to lie that, in which the difference between the greatest and miiUlic It-mi, i> lo (lie (iilicmitr l»< tw<ni llir middle aud least term, as (lie greatest term is to thr I' ;ist. Tims in tlic niimlirrs 6, 4, 3, as 0 — t: i — 3 : : C> : 3 ; vi*. as '2 is •<> 1, so is 6 lo 3. 1 By Ihc bnatlth cf the com]iari"on, Fruclus means the ratio of the terms first proposed to each other, which in this in-t HUT i> duple. 1 or is omittctl in the urii;nul. * For TapaAa/jwr here, it i> obviously necessary to read /jr/»<£««'. <W PHOCI.US ON THE [BOOK in. have the breadth of the third part of 1 ; in order that 3 being compared to one may make the third part of I. Adding therefore, this i to 1, I shall have *, whichis the harmonic medium between 1 and ', in the same manner as before. Hence, by employing these methods, we shall ia a l>ecomin£ manner fill all the double and triple intervals, with arithmetical and harmoniacal middles ; which Timaeus has comprehended in the geometric middle, and which he increases by the insertion of the other middles.

In short, since Plato makes mention of the three middles, which are compre hended in the geometric middle, let the following theorem be added [as a corol lary] to what has been said. If the analogy consists in four terms, and one of the intermediate numl>eis produces an arithmetical middle, the other will produce an harmonic middle, and vice versa. For let then- be four terms, </, b, c, </, so that the first a,1 is to b, as c is to d, and let b be an arithmetical middle, [so that a, If, d, are in arithmetical proportion,] I say that c is an harmonic middle. For because the product of a by d is equal to the product of b by r, but b is an arithmetic middle and the product of c by a added to the product of c by d is the double of the product of b by c , as in the arithmetic middle; this l»einij the case, it follows that the product off by a added to the product off by </, is the double of the pro duct of a by d.1 But this was the property of the harmonic middle, viz. that the product of the middle by the extremes, is the double of the product of the ex tremes. Apiin, let c IK- an harmonic middle, I say that b is an arithmetic middle. For since the product of r by a added to the product of c by </, is the double of the product of b by c, the sum of a added to d is the double of /».' But this is an arithmetical middle, when the sum of the extremes is the double of the middle term.

Ajrain of these four terms, let b be an arithmetic, but c an harmonic mean, I say that as a is to b, so is c to d. For because the product of c by a, added to the product of c by d, is the double of the product of a by d, on account of the harmonic middle, but the sum of a added to d, is the double of b on account of the arithmetic middle, hence the product of a by d will l>c equal to the product of l> by c. As a then-fore is to bt so is c to d.* But this was the peculiarity of the geometric middle. Hence those two middles are contained in the geometric

1 a is omitted in thii place in the original. * A* a is to b: '.c: d by hypothesis and therefore ad=bc. Bui ca+cd:='jbc ; and because bc=ad, therefore 2bc=Cad. 1 Since ca-t-nt-bc, it follows since c multiplies all tbe three terms ca, cd, 2bf, thut a-t-d— '.?r>. • 4+dxc~.'ad. But 2bxc=Cad, and therefore bxc=axd. Hence a: b::c:d. The truth of tL.s may be steu iu numbers, by puttini; 6. 1C. i'. 18. for a. b. c.d. BOOK MI.] TIM/fcUS OF PLATO. 61 middle,1 and reciprocate with oach otlier. Since however we have premised thus much, let us proceed to the text of Plato. " In the first place, he took one part from the whole. After this, he separated a second part double of the first: and again, a third part, sesquinlterof the second, hut triple of the first." The mathematical theory i.s neither to be entirely despised [in the present discussion] nor to be alone embraced itself by itself. For the latter will not exhibit to ns the things which Plato intended to represent to us in images, and the former will cause the whole exposition to be unproductive' of advantage. For it is necessary to consider the essence of the things which are the subject of discussion, as on a secur • foundation. As we observed therefore before, we shall proceed in a middle way, first mathematically, in a manner adapted to the sub jects, and after this we shall unfold the division presented to our view in the text.

The Pythagoreans then conceive magnificently, respecting the division or section of the rule in this place, vi/.. that Plato unfolds in it tin: essential causes, and the reasons which are generative of mathematical theorems. I ,et us, therefore, as I have said, first mathematically .exercise the reasoning power of the reader, by contractedly explaining what is asserted by many, at the same time abstaining from controversy, and investigating the truth by itself. Our discourse, however, w ill be in short, concerning these five particulars ; viz. concerning multiple ratios ; the media that subsist U'lween these; the sesquitertian and sesquialter ratios, which present themselves in the middles ; the sesquioctaves which fill these inter vals; and the leimma. For it is necessary that the diagram should be comprehen sive of all these, and be condensed with all these ratios.

That we may proceed therefore in order, we shall assume the ratios which are first mentioned by Plato, in the numbers from unity. Let unity then be posited, and the double of this 2; afterwards .3, which is sesquialter indeed of 2, but triple of 1 ; then 4 which is the double* of 2 ; afterwards 0, the triple of 3 ; afterwards 0, the octuple of 1 ; and after all, the seventh term, which is twenty-seven times 1. Some, therefore, as we have said, arrange these numl>ersin the form of the letter X, making the monad the summit, and arranging the double numl>ers here, but the triple there. lint others more conformably to Plato, arrange them iu one order only. For he does not say, that the triple were apart from the duple numbers, 1 After the word ibiov in the original, it is necessary to nupply from the vcr»* n of Lcooirui Thoma'UJ the WonN, »fpi»^i,»ra« apn at tt*> [traoTrjTti ir ry ytvfitTpttr) ^firrrijn. * For rj)i7r\a<ria hrre, it is obviously necessary to reid liw\n<rin. Q-2 1'KOCLUS ON THE [BOOK in. hut he alternately mixes them, as proceeding in a right line. If, however, Plato had stopped here, there would have been nothing further for us to discuss. Cut since he himself exhorts us to hind tin- double and triple intervals with harmonic and aritlmit tic middles, and it is not possible t<> discover these middles between 1 and '2, some first number must be assumed, which Ix'iug the least, may have a half and a third part.

I -"or every number may have a double, and this must there fore IK- investigated. Let then fi be assumed, and the double of it 12, the former having the same ratio to the latter as 1 to '2. Between these therefore, vi/. 1 and "2 nmltiplied by 0, placing as media 8 and !>, we shall ha\e the above-mentioned middles. Tor 8 exceeds and is exceeded by the same part of the extremes ; but 0 exceeds and is exceeded according to an equal number. Hence by multiply ing 1 and 2 six times, \\e shall find numbers receiving tin; before-mentioned

middles. In .a similar manner by multiplying by 0 the remaining double and triple numbers in the before-mentioned order, we shall find the terms which we may be able to condense with arithmetic and harmonic middles. For sextuple numbers will l>e produced from all the before-mentioned orders, by arranging other numl>ers, only opening that 48 ought to be placed before f> I ; in this respect departing from the arrangement of I'lato, who places !) before 8, in order that he ininht alternately change the duple and triple ratios. We, however, make this alteration, as consentaneous to the multitude of the monads, and the nature of increasing number. Hence Hand 9 come between 0 and 12; but between 12 and the double of it 2-1, the harmonic mean is 1<>, and the arithmetic 18. And U-tween the third double1 21 and 48, the harmonic mean is 32, but tin; arithmetic 3f}. Hut in triple numlx-is, between fi and 18 which are th« first triple, the har monic middle is P, but the arithmetic 12. Between the second triple 18, and .Vf, the harmonic middle is 27, but the arithmetic 3(5. And between the third triple 54 and 16'2, the harmonic middle is 81, but the arithmetic 108. The double and triple intervals therefore, are divided by these two middles. So that these terms will be successive to each other, viz. <?. 8. 9. 12. 10. 10. 24. 27. 32. 30. 48. 54. 8i. 108. 16'2.

If, however, it was possible in the terms described by us, to divide the sesquitertian ratios, into sesquioctaves and leimmas, we should have no occasion to pro ceed any further. But now, as this is not possible, we are in want of another method. Since, therefore, it was proposed at first, to condense the duple ratio, with the before-mentioned middles, and with scsquioctavcs, it is necessary that the subduple term, should have the sesquitcrtian together with the two sesquiuctaves. 1 For rpir.Wiov lierc, it is obviously necessary to read f t*\a<nou. n.),,K in.] T1M/KUS OI; PLATO. (i:; I/et tliere be taken then in the first place, the third number from unity, according to an octuple ratio, viz. 04. From this it is possible to form two scsquioctavcs. For every multiple number is the leader of as many multiple ratios denominated iVni-i itself, as it is itself distant from unity. But it has not a sesquitertian.1 By tripling therefore 04, we shall have 192, the se.squi tertian of which is 2-10, but the sesquioctave 210. and of this the scsquioctavc is '243.* But the ratio of the leimma is that which remains after the ablation of the two sesquioctaves -243 and '210'. Tor from every sesquitertian two sesquioctaves being taken, the ratio of the leimma is left. But of 2-'j<» the sesquioctave is 208, which preserves an arithme tical mean In-tween 192 and 381, which has a duple ratio to 1!)2, and a sesqiiitertian to 208.

If, therefore, it were possible to form two sesquioctaves from '288, we might also condense this scsquitcrtian with sesquioctaves and a leimma. IS'ovv, however, this is not possible. For the sesquioctave of it, 324, has not an eighth part. Hence if we wish to preserve unity always undivided it is impos sible then; should be a sesquioctave ratio to it. For the eighth part of it is 40 and \. By doubling this, therefore, in order that we may make the half a whole, we shall be able to assume the eighth part of it. On this account, however, we .shall be compelled to double all the nnmlx.Ts prior to it, and also those posterior to it. Hence instead of 102, we shall have 384 ; instead of '2 1C, 432 ; instead of '213, 48r>; instead of 2-10, .11 2; and instead of 288, .070. And of this the se<quioctave, is 018, and of this 72.9.' Afterwards 708, which is the double of 381, has the ratio of a leimma to 729. After this manner, therefore, the double interval is filled with sesqm'alter, sesquiterlian, and scsquioctave ratios, in the numbers, 384. 132. -1HO, .112. .170. 048. 729. 7G8. Hence if we wish to till the whole diagram, and to describe all the numbers in a consequent order, instead of the first part we must assume 381; instead of the double of the first, 7G8 ; instead of the triple of the first, but the sesqnialter. of the second, 1152; instead of the quadruple of the first, 1530; instead of the fifth part which is triple of the third, 34-10; instead of the sixth part which is octuple of the first, 3072 ; and instead of the seventh part, which is the twenty-seventh pait of the first, 103G8.

If, therefore, we also wish to condense; these terms with harmonic and arithme tic middles, which bein^ inserted, make sesquialter and sesquilertian intervals, the intermediate numbers will l>e 384, and 708, the double of 381; 512, which produces an harmonic, and 570, which makes an arithmetic middle. But if we 1 The number fit lias not a srsquitertian in whole numbcrK. For a« 3 is to 4 »o is 01 to SjJ. 1 For as f» in to 8, so is 2*3 to 'Jl6. 1 For as 8 is to £) to is 57610 6 18 ; ami also M> is 648 to 7?f). g4 PROCLUS ON THE [BOOK in. wish to assume the above-mentioned middles of the triple interval, viz. of 384 and 1152, then 57(5 will preserve the harmonic ' middle, which filled for us the arithmetic* middle in the double1 interval; and 708 will be the arithmetic middle, which was the greater extreme of the double interval. Again, if we wish to assume the same middles of the duple and quadruple, i. e. of the middles between the terms 708 and 1530, the former of which is the double of 381, and the latter the quadruple, the harmonic middle \\ill be 1024, and 1152 the arithme tic middle. If also we wish to condense the second triple, the terms of which are 1 152, and 34.0(5 [the former being the double of 57(5, and the latter the triple of 1152] then 17*28 will give us the harmonic, and 2301 the arithmetic middle.

And if we wi>h to condense the third double, which consists in the terms 1536 and 3072, then 2')4» will be the harmonic, and '2301 the arithmetic middle. But if we wish to condense the third triple, with similar middles, but I mean the fifth and seventh part, the extremes will be for us 3450, and 10308 ; but the harmonic middle will be 5184, and the arithmetic 0912. If again, we should condense each of the sesquitertians which present themselves from those mid dles, audsesquialters, with sesqnioetaves and a leiramn, this will be manifest to us after the whole exposition, when we exhibit the whole diagram with all the terms in a consequent order, which has indeed 24 sesquioctaves but 9 leimmas.

These things therefore, having been elucidated by us, we shall observe thus much concerning the leiinma, that as it is not possible to divide any superparticular into equal ratios, a semitone cannot be assumed in numbers; but taking the ratios which are contiguous to each other, vi/. the seventeenth and the sixteenth part, and demonstrating that the seventeenth part is greaterthan that which is called the h imnva, and vtlrch is less than an accurate semi tone, it is inferred that the leiiuma and also the seventeenth part are le-,s than a semitone. But that it is less than a semitone, is demonstrated as follows : Let there be given the term 10 and the sesquioctave of it 10. Between these placintr 17 it will divide the sesquioctave into unequal ratios, which will be near to the semitonic interval, since 17 ililli-rs from tlu- extremes by unity alone. And it is evident that it will make a greater ratio with the l«-ss term; because in all arithmetical proportion, the ratio is greater which is in the less terms ; so that the seventeenth part is less than a semitone. Moreover, the leiinma is less than the seventeenth part, as is evident from the terms exhibited by Plato. For since 250 has to 243 the ratio of the leimma, as we shall demonstrate in what

i For apiO^^l' 1|p|*. '" tl* ""{-F'"al '• '* ne 1 For op/i«<i'(kij» also here, we mint read apiO 1 And for rpirXawiy, it is rt<iui»ite to read J no"* '"•] TIM/EUS OF PLATO. t^ follows, whore we shall show that the radical ratio of the leimma is in those numbers ; and since 250 exceeds 243 by less than the seventeenth part of it ; for it exceeds it by 13 unities, but the seventeenth part of 243 is more than 13; this bring the case, much more is the ratio of the leimma less than the semitonic. interval. Hence the ratio which remains to the completion of a tone, and which is called the ratio of an apotome, is necessarily greater than a semitone. Farther still, this may also be demonstrated after another manner, a* follows: Let the numbers 250 and 243 l>e given, and let there Iw assumed three numlxTS in a consequent order, in a ratio of this kind; from 256 indeed, 05536, but from 243 5.004.0, and from both O2208. These three numln-rs, therefore, arc analogous in the ratio of the leimma, which, if it is a semitone, will be the tonic ratio of the extremes. Hut if it is greater than a semitone, that also will be greater than a tone; but if less, that also will be less. The sesqiiioctaye however of 5909, is GG43 10 i. But this is greater than the greater term.

After another and a third way the same thing may also l>c demonstrated, vi/. that a tone cannot be divided into two equal parts, having the same ratio as that of 250 to 243. For if we take the eighth part of 243, which is 30 », and add this part toil, we shall make 75 j|, which has a sesquioctave ratio to 243. You see therefore that 250 has to 243 a less ratio than 2731 to 250. For 250 has to 243 a snpcrparticnt ratio, exceeding it by -Vj ; but 273 J exceeds 25r>. by j'oV.-JlJ. B"t the ratio is greater which exceeds by seven teen and more, than that which alone exceeds by thirteen, according to the ratio of excess. A tone, therefore, cannot be divided into equal parts, bu°t //„> js the leimma, as Plato also calls it, and that which has the greater ratio \sa/>o/ome, as musicians are accustomed to denominate it. For let 273 j have to 213 a sesquioctave ratio, but 250, to the same 213, the ratio of the leimma, which has a less - ratio than that of the seventeenth part, it is evident that 273,1, which hasthe ratio of a tone to 213, will have to 250 the ratio of the apotome, which is the remainder of the leimma, l>eing greater than the seventeenth part, which we have demonstrated to lie less than the ratio of the leimma. If therefore we multiply these eight times, we shall find the first numbers which in perfect unities have the ratio of the apotome.

For the octuple of 243 is 1914, of 250 is 2048, of 27.3j, 2187. Hence the ratio of the apotomc in radical numbers (n TvVff" ') is that of 2187 to 2048. And we shall be in want of these three terms which are in a consequent order, in the diagram. Let then these terms be, 243, 256, 273 J. But on account of [the fraction] ?,, let the octuple of these be the

' rivfyijr if a primary ratio, being a» it WOT* a bottom or root, from which other ratio* ari»*. Tim. Plat. VOL. II. I gfl 1'KOCLUS ON THK [BOOK in. numbers 1944, 2048, 2107, in order that the terms may be in perfect unities, and not in the parts of unity. Because however, it is necessary that the ratio of the leimma should !*• that of 250 to 243, we may demonstrate it to be so as follows: If from the sesquitertian interval, two sesquioctaves are taken away, the terms which comprehend the remaining interval, will have to each other the ratio of 25(5 to 243. For let a b l>e sesquitcrtian off, and let c be taken away, which is subses-. quioctave of the sesquioctave a b. And in a similar manner let d be taken from C. I say that d will have to c the proposed ratio. For from a b let c be taken \vhich is equal to ; b, and d which is equal <• b. Since therefore, as a b is to c, so in c to d ; for they are sesquioctaves ; it will also be as a z is to b :, so is b z to b e. Hence the n-mainder a z will be to the remainder z c, as whole to whole, i. e. asrt/'to b z. But a b is sesquioetave of b :. Hence I :

is sesquioctave afez. lx-t z h l>e placed equal to z e. 1 Fence z h is octuple of // a. But * e is equal to z //. Hence?// is eighteen times // a. Again, since z b is sesquioctave of be, fore is ses quioctave of d, hence b e is octuple of e =. Of Mich numbers therefore as - e is 8, of such cb is (Jl, and zb 72. For 72 is sesquioctave of oM. But the whole <i b is HI ; for this is the sesquioctave of 72. The numbers, then-fore, are quadruple. Hence of such numbers as « b is 324, of such e b, i. e. ,/, is 250. For 32 1 is qua druple of 01, and '25<! of 04. But numbers which are equally multiplied, have the same ratio as their parts. Since then-foil- « b is sesquitertian off, of such num bers as a b is 32 I, of such e will be 2 13. For 32 1 contains 243, and a third part of it, vi/. HI. But it has appeared, that of such numbers as a b is 324, of such d is 250. Hence of such numbers as d is 25'!, of such c is 2 13.

It is manifest, however, that this ratio of the leimma is in the least terms. For they are first terms with relation to each other. Arid this is evident from sub traction. For they end in unity, the less being always taken from the greater. But if they are first terms, it is evident they are the least of those that have the s;un,- ratio' with tlu-in. If, therefore, two sesquioctaves are taken from the sesquiti-i-lian interval, the remaining terms will have the ratio of 250 to 213. This therefore being demonstrated, let then- be taken in a consequent order a b for the tonic ratio, b c for the ratio of the h-imrna, a d for the ratio of that which is called a semitone, and d to c for the ratio of the comma. For the ratio of the c.xce-s of the apotome, above that which is truly a semitone, and which cannot be obtained in numbers, is thus called. This then is demonstrated. To what h:i» he.:n said however, it must be added, that we have called the ratio of d b a semi tone, not that a *e.squioctave is divided into two equal ratios ; for no superpartirular ratio is capable of being so divided; but because the followers of Aristoxenus assume a semitone after two sesquioctaves, the ratio of a semitone is as sumed, as we have said, according to their position, in order to discover what

HOOK in.] TIM7FXS OF PLATO. <i7 the ratio is of the comma and apotome to the ratio of the leimma. This therefore is assailed through the cause which has been mentioned by us. For that every superpartieular ratio is incapable of being divided into t\vo c<|iial ratios, is one among the things that are demonstrated. Thus much, however, must be added, for the sake of elegant erudition, that as the Pythagoreans neither admit that there is a semi tone from which together with t\vo sesquioctaves a sesquitertian ratio is produced, nor tin; symphony diapason and diatessaron, as the followers of Aristoxenus admit; — this being the case, the musicians posterior to him, the disciples of Ptolemy, grant with the Pythagoreans, that what is called a semitone, is not truly so, but reject the opinion, that tho diapason and diatessaron are not sym phonies. AVc, however, necessarily demonstrate the former, on account of the opinion of Plato ; but not l»eing compelled to demonstrate the latter, tacansv Plato says nothing about it, we shall at present omit it. Since then we have shown in what numbers the ratio of the leimma, and the ratio of the apotome are first found, we must likewise show, in what numtars the ratio of the comma, by which the apotome exceeds the leimma, is first disco vered. This ratio therefore is in perfect [i. e. in undivided] unities, as the ancients say, that of 531441 to 524*280.' But if to divide unity makes no difference, let the ratio of the leimma be taken in that of the numbers 256 to 243.

But the ses- <juioctave of 243 is 273 i1,, and of 250 200.* Another leimma is that of 209 to 243 -A. For tli is is the ratio of the leimma. For 2GD contains 250' and thirteen units, and 250 also contains 243 and thirteen units. Because therefore 250 con sists of 243, and besides this of thirteen units, which arc the numerator of 243 ; hence the 13 by which 250 exceeds 243, contains in it.-elf -,V. parts of 243. Each likewise of the thirteen units by which 250 exceeds 2 13 contains in itself. J- of 243. Hence 200 + /;, will have the same ratio to 250, as 250 to 243,' being in a superpartient ratio to it, and having ,V, parts of it, and 243 units. Hence that which remains, vi/. 273 .', has the ratio of the comma to 209 and -,'*\. So that it is shown in what numbers of the monad when divided, and in what two leimnias taken from the sesquioctave, the ratio of the comma is first found. It is evi dent therefore, from what has been said, that we have effected what we promised to do. The terms likewise, and all the intervals, arc condensed with harmonic and arithmetic middles, and the divisions of the sesquialter and M.-squitertian ratios, into sesquioctaves and leimmas, have bceu effected. For as there is a

* Lconicus Thomxus has in his versioo 524298. * 288 U omuied in the Grerk, and also in the version of TUomnrus. . ftS»HO^ «™ 2S« ISr!:.21H JT90« 68 PKOCLUS ON THE [BOOK in. duple interval between 384 and 700, the term 432 which is sesquioctave to 384, and 400 which is sesquioctave to X32, fall between them, and also 512 which makes a leimma with 400. And thus far the sesquitertian ratio consists of two tones and the leimma. Airain 570 is sesquioctave to 512, 0-10 to 570, 729 to 618, and 700 has the ratio of the leimma to 729. And from these the sesquialter is filled, having three sesquioctavcs, and one leimma.1 But the whole is duple, consisting of five sesquioctaves, and two leimmas. Again, according to the above described terms 304, and 700, the term 512 produces an harmonic, but 570 an arithmetical medium. Farther still, 00 1 is placed as sesquioctave to 700, but 972 is sesquioctave to 804, and 1024 has the ratio of the leimma to 972. To 1024 also 1152-is sesquioctave. And now after the duple the sesquialter ratio is produced, which makes a triple ratio, viz. the ratio of 1 162 to 301. But between this triple interval, 570 is the harmonic middle to the extremes, but 700 the arithmetic middle. For a theorem of the following kind is universally demonstrated, that if of the same term, one number is double, but another triple, and a certain mean of the double is assumed according to arith metical proportion, this mi an will be to the triple number an harmonic middle.

But the greater term in the duple ratio, -cill become the arithmetical mtan in the triple. Thus for instance, in the above terms, 708 is the double of 304, but 1152 is the triple. Between also the duple terms 700 and 301, an arithmetical mean 670 is assumed ; and the same mean between the triple terms 304 and 1152 is seen to !>e an har monic mean. And 700 which was duple, becomes between the triple terms an arithmetical mean. Afterwards, 1290 is sesquioctave to 1152, and of this MSBis the sesquioctave, to \\hirh 1630 has the ratio of the l.-imma. And as far as to this, the second duple is tilled, being composed of the sesquialter and sesqoitertian ratios, the extremes of which an- 700 and 1530, and are di\ided into five sesquiuctaves and two leimmas. It likewise has for the harmonic mean 1021, and for the arithmetical mean 1 152.

Again, 1720 is sesquioctavr to 1530, of this 1914 is sesquioctave, of this 2107 is sesquioclave, and to this 2301 has the ratio of the leimma. But the sesquioc tave of 230-1, is 2592 ; of this 2910 is the sesquioctave, and to this 3072 has the ratio of the leimma, which is octuple of the first part, tilling the third double. And farther still, the sesquioctave of 3072 is 3150. And as far as to this the second triple extends, having for its extremes 1 152 and 3150, and for its harmo nic mran 1720, but lor its arithmetical mean 23O1. In addition to this also, the ' l\>r 70S i» »e><iutaltcr to M-', and bclweiti these l*o terms, there are the above three »o<ju .mil one leLuima. BOOK in.] TIM/EUS OF PLATO. 09 sesqniortave of 3450 is 3800, but of this the sesquioctave is 4374, to which 4G08 lias the ratio of the leimma. The sesquioctave also of 4008 is 5184, and of this again, the sesquioctavc i* 5832, to which 0141 has the ratio of the leimma, the sesquioctave of which is 0!M2. And this again is another duple1 [vi/. 3450 and 0912] after the before-mentioned three duplcs [and afterwards another sesquioctave : for 7770 is sesquioctavo to (i.912]:1 for in the third triple, there is also a certain duple. And again, the sesquioctavc of 7770 is 8748, to which 0210 has the ratio of the leimma, and of i)210 the sesquioctave is 10308. And as far as to this, the third triple is extended, taring comprehended in the terms 3450 and 10308, and having two means, the harmonic and the arithmetical, the former of which is 518 J, hut the latter 01)12.

The double and triple intervals therefore are filled with middles, and with M-squioctaves and leimmas. The whole likewise of this diagram has nine leimmas, and twenty-four sesquioctaves. J;or the intervals are less in number than the terms by one. It also proceeds as far as to a quadruple diapason, and a diapente and tone. Adrastus however, who was a lover of the arts, makes the figure, as we have said, in the form of the letter x; and places the terms in certain triangles. And in the interior triangle, indeed, he places the ratios that are in monadic nmnl>ers [i. c. that consist in the numl>ers within ten] ; but in the trian gle next to this, the sextuple of these numbers, which have two middles according to each duple or triple interval. And in the outermost triangle he places the terms \\hich make the whole of the U-fore-mentioned diagram. What we have said, however, will become manifest from the delineation. But between the double and triple intervals, he inscribes all the above-mentioned numbers, which we have not thought fit to add. being unwilling to introduce a [needless] multitude of terms. For such a disposition of terms, and the insertion of the same numbers twice, is unmethodical. For many of the same media are found between the duple and triple intervals ; since the triple intervals themselves consist of duple terms and sesquialters. What is said by Plato, therefore, has Ixvu elucidated by us. For two media have been di-eov» red between all th«; duple and triple inter vals. And from these media scsquialter and sesquitcrtian ratios having been produced, these an? divided by the sesquioctave; a portion being left in both, which has the ratio of the leimma. From these likewise, assumed in an orderly

* Fur tiaratrvr licro, it i» nrcrssary lo read i(T.\n<rio>-. 1 The words within the brackets arc supplied from the >crsion of Thomsuus, where Luwerrr it ii to read sesquiocUvuin instead of sfsijuultcrura. manner, the terras which comprehend the whole diagram will be found to be thirty-four only.' Since, however, the Pythagoric Timarus says that the terms of the diagram are thirty-six, and yet assumes the same extremes as Plato, viz. 304 and 103G8, in order that these philosophers may not appear to be in any respect discordant with each other, let us show how the other t\vo terms are inserted. These mm there fore [i. e. the Pythagoreans] were willing that there should not only be the ratio of the leimrna in the diagram, but also that of the apotome, which they twice dis covered, both in radical numbers, and in those alone which are the triple of these. Adding likewise one term to each, they introduced this into the diagram. But Plato makes no mention of the apotome; whence also we being satisfied with the leimma, have alone employed the above-enumerated terms. For how, since he assumes the diatonic genus, could he make use of the apotome, the sesquioctave not being di\ided in this genus; the apotome being produced when the sesquioctave is divided ? For the part of the sesquioctave which remains after the leimma, is the apotome. Hence, since Plato does not mention the apotome, and it is not possible for it to occur in the diatonic genus, it would IM; ridiculous in MS to endeavour to insert other terms, in order that we may ha\e the apotome, the thirty-four terms being sufficient to the completion of the sesquioctLvea and leimmas. Jt seems also, thai the number 31 is adapted to the diatonic genus, in

' Thii will he evident from the following diagram, which also will be found to contain a quadruple diapason, together with the tliape rite and tone. Se§q. Sesq. Leimma. Sesq. Sesq. Sesq. Leimma. 384. 4.3-2. 486. 512. 576. fits. 7?9- 7^8. The first duple interval. Sesq. Sesq. Leimma. Srsq. Sesq. Sesq. Leimma. The second duple interval. Sesq. Sesq. L*iinuia. Stsq. Sesq. Leiuima. The third duple inter* al. • 2187 is the octuple of 273 j. Sesq. Sesq. Leimma. Sesq. Sesq. I^eimma. 3456. 3S8S. 4374. 400S. 6184. .')«32. 6lU. The fourth duple interval. Se«q. Sesq. Leimma. Re»q. <>9I2. 7770. 8748. QllCi. 103CS. The third triple interval. In this diagram it must be observed, that the last term of each interval forms • ie»quioct*ve with the 6r»t term of the interval that i» next in order. The first triple interval likewise begins with the term 384, and ends at the term 1152. The second triple interval begins at 1 152 and ends at 345C. And the third triple interval begins at 34 JO, and ends at J0308. nooKin.] TIM.EUS OF PLATO. which alone the sesquioetave ratio is fount]. For it consists of the terms 18 and \0, which are to each other in a sesquioctave ratio. For the sesquialter and sesquitertian ratios, and leimmas, are also in the other genera; hut the sesquioctaves are found in (his alone of the three genera. Hence this ratio of the scsquioctave, very properly produces liy composition the number of the parts; and this heinu the second, is adapted to the second progression of the soul from the first intelligible principles.

If therefore we assume the |es*i term of the third donhle. vi/. 15.T5, and airain the se>quiocta\v of this 1728. and afterwards the sesquioctave of this 11)44, and again the tritone of this 2107, there will he one interval of the extremes. Because however 20 10 has a sesquitertian ratio to l.WG, hut 1044 has to it the ratio of the leimnm.it is necessary that '2107 should makeanapotome to 2040. For an apotome is, as we have before said, that which remains to a tone, after the leimma. In a simi lar manner also, by assuming in the third triple, 4008, which contains the tritone ' C>56\,* and also ass-umin^ G144, which makes a sesquitertian ratio to 4608, but to 6832 has the ratio of the leimma, we shall necessarily have the apotome in the ratio of 0"><>1 to (i 1 44, which are triple of the radical terms that were l;>efore discovered by us in the third double.. For it is evident that the ratio of the apotome is radically in those terms. For 2107, and 2040, are demonstrated to J>e first terms to each other by the theorem of subtraction; first terms bein^ necessarily such as are least. The multitude indeed of the terms descril>ed by Tim;rus, is demonstrated by Philolaus ; but the diagram of Plato proceeds without the ratio of the apotome. And thus much concerning these particulars.

Since however we have l»efore observed, that if of one term two numl>ers are assumed, one of which is the double, but the other tin1 triple of it, the mean which between the duple terms is arithmetical, is between the triple terms harmonic, but the duple term is between the triple terms, an arithmetical mean, we will now concisely elucidate and at the same time demonstrate this theorem. Let then b l»e the double of a, but c the; triple of it, and between a and /;, let the arithmetical mean be d. I say that will happen which is enunciated in the proposition. For since b is the double, but c the triple of a, of such numl>ers as a is two, b will l>e four, and c will be six. Hence, of such as b is four, c will be six. By so much, ' For rptrof here, it is necessary lo rc;i<l rpirorov. 1 According to tin moderns, a tritone i* u dissonant interval, otherwise called a superfluous fourth. It is also a kind of redund:nit third, eon>isting of two tones, and two Jeruitones, one grciter and one less. And the ratio of the tritoni- i.i ,i- 13 to 32. This however does not accord with the ratio of the tritone given by Froclui, both iit this place, and above. For U>~OS is not to 0501 u32 lo 45, but u 32 to 4i ;££*. Nor in the other instance above, u iyu U» 2lt»7 «» 32 to 45.

74 PROCLUS ON THE [BOOK in. therefore, does c exceed b as b exceeds a. Hence b is an arithmetical mean between a and c. Again, because of such as a is two, of such b is four, but the arithmetical mean between them is d; hence d will be three of such numbers, as a is two, and b four.1 But of such as b is four, of such c is six. Of Mich there fore, as a is two, of such d is three, and c is six. I lei.ce d compared to a and to r, will produce an harmonic middle.1 Tor by the same part of the greater it is ex ceeded by the greater, and by the same part of the less exceeds the less. And thus much concerning this particular. Severus, however, thinks that this diagram should not end in a tone, but in the leiinma, because Plato terminates in this all the discussion concerning the division of the soul. In order, therefore, that it may terminate in the leimma, Severus transfers some of the terms, and makes all of them to be thirty-four. Hut as in the thirty-fourth term, the half of unity occurs, he doubles the terms, anil makes the first part to l>e 708, which is the double of 384. Of this, therefore, he places the sesqnioctave 801, and of this again the sesquioetave 972. To this also he adapts according to the leimma, 1021.

But of this he takes the sesquioctave 1152; of this the sesquioctave 1290; and of this again the sesquioctave 1458. But to this he adapts according to the leimma 1Mb', and places the sesquioctave of this, 1728, and of this again the sesquioetave 11)11. To this likewise he adapts according to the ratio of the leimma, '2107. And of this he assumes the sesqui octave 2301; of this the sesquioctave 2GU2 ; and of this again the sesquioctave 2U10. To this also he assumes 3702, which has the ratio of the leimma to it; to this the sesquioctave 3450; a'id to this in a similar manner 3888. To this likewise he adapts as the leiinma 4371; of this lie assumes the sesquioetave 4008; of this the sesquioctave 5181 ; and of this again the sesquiocta\e 5832. To this also he adapts according to the ratio of the leiinma, 0144 ; and of this he assumes the ses quioctave 0912 ; of this the sesquioctave 7770; and of this again the sesqiroetave 8748. To this likewise he adapts as a leimma 9210. But of this he makes the sesquioctave 10308; of this also 11001; and of this, again, he makes the sesqui octave 13122. To this he adapts as a leimma 13824 ; of this also he assumes the sesquioctave 15552; of this the sesquioctave 1741)0; and of this, again, the sesqui octave 19783. And to this he adapts 20030, having the ratio of the leimma. As far as to this, therefore, he gives completion to the diagram, making the leimma to be the end; except that in these terms, "there is first the sesquitei tian, after wards the sesquialter, then the sesquitertian, and afterwards the sesquialter ratio.

1 Hence a» 2, 3, 4 are in arithmetical proportion, so likewise *ill their equimultiple, C». :U. tx. * For 2, 3, and 6 are in harmonic proportion, and therefore thtir equimultiples also are in the same proportion. For 6 e\i eeds 3 bj the half of 6, aud 3 eiceedi C l>v the hulf of J. »<><>* in.] TIM/EUS OF PLATO. Ami again the sesqnitertian, afterwards the sesquialter. and then three s in a following order, as is evident from (lie above description. It happens, therefore, in this diagram, that there is a quadruple diapason, that the diapente occurs once, and tliat the tone is redundant. For three sesrnii;dters, make one diapason and tone. The diagram, however, does not end in atone. but in the leinnna. But this was the tiling proposed to be effected. Severus, thereIons does not take away the tone, but does not end in it. So that the whole diagram according to all the terms, consists of a quadruple diapason, and the diapentes, and one tone. If also we wish in monadic numbers to assume these intervals, we must survey the progression extended as far as to twenty-seven, For ~2 is double of unity, 4 is the double of '2, 8 of 4, and 1C, of 8. And as far as to this the qua druple diapason extends. But of 10, 24 is the sesquialter, and this is the diaprnte, and to this 27 has a tsesquioctave ratio.

80 that the before mentioned symphonies are j>erferted from one part, as far as to twenty-seven. This, therefore, as I lune said, is common to all the diagrams. But they differ from each other in this, that some are in the form of the letter x, but others are in a right line. For of the ancients, Adrastus employs figures in the form of ?., but Severus ri-ht lines, which in my opinion is better. For in the figures which are in the form of the letter x, the same numbers are found twice in different places. But this is discor dant with the things themselves. For there are not two parts of the soul which are the same. But all these numbers are parts of the soul. They difier also in this, that some of the diagrams end in a tone, but others in the leimma. Some also are more perspicuous, though the same numbers are assumed twice, as is the case with those diagrams which distribute the duple and triple ' ratios in the sides of triangles. But others place each nunnVronce in all the intervals, though this causes a more diflicult division of the duple terms, into super-particular and super-partieut ratios. We have thefore premised such things as may contribute to the theory of the psychogonic diagram, to those who survey it mathematically. As we are entering however on the more important explanation of the words of Plato, we think it requisite to speak in the first place, concerning the division itself according to which the soul is divided in these ratios, and to take away by argu ments such things as are an impediment to our apprehending the truth respecting it. Let no one, therefore, think that this division is corporeal. For it has been before demonstrated, that the middle nature of the soul is exempt from bodies, and at the

Both Ihc sense of what i» here *uid, and the version of Thomxuj. require the insertion in this place Tim. Plat. VOL. II. K PROCLUS ON THE [BOOE in. same time from every partible essence which is distributed about them. Nor must it be supposed that the soul is U-tter indeed than bodies, but that it is divided after the same manner as the terms or boundaries, and intervals by which bodies ' are measured. For things that have interval, are not wholly and through the whole present with themselves, and when divided, are not able to preserve an unconfused union. Hut the soul participating of an impartible allotment, is united to itself, and exhibits all the >ame rh -ments subsisting the same in all its parts. >*or again must this section of the soul, be considered as a division of nunilx-r. For" the soul is indeed number, yet not that which is according to quantity, but that which is essential, sclf-ln-gotten, ha\ing the form of unity, and converted to itself. Nor let the presence of these ratios in all the parts be com pared to spermatic reasons. For these are imperfect, corporeal-formed, and mat.-rial, and entirely fall short of the immaterial and pure essence of the psychi cal reasons. Nor let any one assimilate the before-mentioned parts to the theon-ms of M it-lice, because each theorem pus-esses the whole [science of which it is a theorem). For we do not now consider the knowledge, but the ess.

nee of the soul. Nor is it proper to think that the dilli-rcncrs «>f essences, are similar to the distinctions of habit*. |-or the latter are. entirely \aricd in (he natures tliat possess llum, but the form, r are establish* d vtith invariable .sameness in demiur gic boundaries. Ilcisce it i* necessary to biisp. ml the primordial principle of the psychogonie division from the demiurgic cause, and from perfect measures, which eternally pre-exist in real beings ; to \\liich also the Demiurgus looking, divides the soul. For as he duided this universe by intelligible paradigms, thus also he separates the essence of the soul by the most In-antil'til boundaries, assimilating it to the more ancient and primordial causes. The mode therefore of division, J.v nnj.iutenul, intellectual, utidflileil, perfect ire of the cwmc of the suitl, grmrarice of the multitude in it, collective through linnnony into one order, and connective of diviilal parts; at one and the same time being the cause of the iiiiiningled purity of the multitude in the soul, and proilm-ing a comimmion of reasons converging to the

And the Deiniurnus, indeed, appears to consume the whole by dinding it into parts. For thus also in a certain resjM^ct Tima-us says, '"•that he conxumcd the n-hule from uiiic/i lie cut o//' these parts" This, howe\er, is not the case, but it is necessary to preserve both impartible,1 and that the wholeness remaining imparti- 1 For aiTtjv lien1, it is nrerssary l« rtail aura. 1 For oXXu ImptptoTor lit-re, it ii necessary to read a\X' a^ttpiarov. ble, the division into multitude should l>c effected. For if we assume one of the>e only, I mean the section into parts, we shall make the soul to be partible alone. The whole therefore is <UxlrihuU<l into parts, and at the xmne time the whole rtmams. Again, (he impartible of the .soul equally participates of tin; impartible and the partible. For it is well said, as we have before observed, by the da-rnoniacal Aristotle, that there is something impartible in partible natures, which is connec tive of them ; so that it is much more necessary, that something impartible should remain, in things which have not only a partible, but also an impartible nature. For if this did not remain, that which consists of both would be alone partible. It is evident however, that it is necessary the whole should remain in the gene ration of the parts, if the Demiurgus is an eternally producing cause. But he constituted the soul one whole, prior to the division of it.

For he did not obli terate it in producing something else. But he always produces every ihinir, and eternally, and causes that which is produced to remain what it is. Hence the wholeness is not destroyed, when the parts are constituted, but it remains, and precedes the parts. For lie did not produce the parts prior to the whole, and afterwards generate the whole from these, but vice versa. The. essence therefore of the soul is at one and the same time a whole, and has parts, and is one and multitude. And such is the division of the soul which Tima-us assumes.

The mode, however, of unfolding it, should accord with the ss».ence of the soul, being liberated from visible, but elevating itself to essential and immaterial har mony, and transferring from images to paradigms. For the symphony which flows into the ears, and which consists in .sounds and pulsations, is terv different from that which is \ital and intellectual. No one, therefore, should stop at the mathematical theory, but should excite himself to a mode ' of survey adapted to the essence of the soul ; nor should he think that we ought to direct our attention to interval, or the differences of motions'. For these are assumed remotely, and are by no means adapted to the proposed subjects of investigation. But he should survey the assertions by themselves, and consider how they a/Ibrd an indication of the psychical middle, and look to the demiurgic providence as their end. In the first place therefore, if yon are willing thus to survey, since wholeness js triple, one l>eing prior * to parts, another consisting of parts and another l>eing in each of the parts, as we have frequently elsewhere demonstrated ; — this being the case, Plato lias already delivered the wholeness of the soul which is prior to parts. For

1 For roTov here, n ad rpowor. * InMcvd of wpoi THV fttfttf in this place, it it neceuarj lo read wpo r*v /..,.«.. 7fi PROCLUS ON THE [BOOK in. he made it to be one whole prior' to all division into parts, and which as we have said, remains what it is, without being consumed in the production of the parts. For 'to IK> willinjr to dissolve that which is w.-ll harmoni/ed is tin; province of an evil nature. But the dissolution is effected by consuming the whole into the parts. In what is now delivered however, he constitutes it a whole from parts, consum ing the whole mixture into the division of its essence, and through the harmony ofthe parts, rendering it a whole de ntivo, and canning it to l>e complete from all appropriate parts. lint he shortly afier teaches us the wholeness which is in each Of tin: parts, dividing the whole soul into certain circles, and in eaeh ofthe circles inserting all the realms, which he had already made manifest to us in what he had before said. For he had said, that in each of the parts there are three [i. c. sanus diflereiU, ami essence) in the same manner a* in the whole. Every part therefore, as NX ell as the whole, is in a certain respect a triadic whole. Hence it is necessary that the. soul should have three wholenesses, because it animates the turners"*-, which is a whole of v> holes, eaeh of which is a whole according to the wholeness which is in a part.

So that the soul animating the universe in a twofold respect, both as it is a whole, and as consisting of total parts, it requires two wholeness, and transcends the things that are animated, having something external to them, so as circularly to cover the universe, us Tima-ns says, as with a veil. 15v the wholeness, therefore, which is prior to parts, the soul entirely rims above the universe, but by the remaining two connects the universe and the parts it contain!*, these also being wholes.

In the next place, it must be observed, that Plato proceeding from the beginning to the end, preserves the monadic and at the same time dyadic nature of the soul. For he reduces the livparxis of it to essence, same, and dillerent, and distributes the number of it according to a twofold division, beginning from out part, into duple and tiiple numbers, lie also surveys the media or middles, in one of them comprehends the other two,1 and according to each of these unfolds twofold ses<|iiialter and sestjuitertiau ratios, and again cuts these into sesquioctaves and leimmas. In what follows likev ise, he divides the one leiiL'th into two, and the one figure of the soul into two periods. And, in short, he no where omits the monadic and at the same time dyadic, and this with the greatest • And I'ur rput xat-vi ^r^a^tou here, aUo, re.id *po k. X. 1 i.e. In tin' "eiiiuelric miiKlli- «>r |>ii.|iorli.)ii( « Lii ii coenpnlKUils arilliuielical ;tnd liarmotiical iirt>p»rl!oii. !'«>! it" to ;ni) llnce iiiiinl.er> in arilliiiittii.il |>rii|iorli»u, a fourth UumluT I* addtil, so a, t,, pr.Mliu-e i;i.)u.ctiii.il proporlioii, llu" t!.i> proporlion «ill i..iii|.reheu.l bulli lh.it *»liiili i> arithmfliiul. an.l that wliirli i» hur.m.mcal. TLu» if to the U-rni.- 1. 2. a a lourtl. ti-nii i- acMt-il MZ. 0, so ihat it may be 1 : 2 : : 3 : (i, then 1, '2 and J are ui arithmetical, and-', 3 and 6 in harruouic.

HOOK in.] TIMJEUS OF PLATO. propriety. For the monadic alone pertains to intellect, on which account also intellect is impartible. But the dyadic pertains to body, whence in the genera tion of the corporeal-formed nature, Plato began from the dnad, fire and earth, and arranged two other genera of elements between these. The soul, however, heinir a medium In-tween intellect and body, is a monad and at the same time a duad. But the cause1 of this is, that in a certain respect it equally participates of hound and infinity ; jn*t a* intellect indeed, is allied to hound, hut body rather pertains to infinity, on account of its subject matter, and divisibility ad infmitum. And if after this manner, some refer the impartible and the partible to the monad and indefinite duad, they speak conformably to things themselves ; but if as making the soul to IK- number, in no respect differing from monadic nunruVrs, they are very far from asserting that which happens to the essence of the soul. The soul, therefore, is a monad and at the same time a duad, adumbrating by the monadic, intellectual bound, but by the dyadic, infinity ; or by the former, lieing the image, indeed, of the impartible, but by the latter being the paradigm of partible natures. In addition to these things also, it is requisite to suney, how a two-fold work of the Deminrgns is here deli\cred. For he divides the soul into parts, harmoni/es the di\ided parts, and renders them concordant with each other.

But in eflecting thc<e things, he energi/es at one and the same time Dionysiacally [i. e. Bacchicallyjand Apolloniarally. For to divide, and produce wholes into parts, and to preside over the distribution of forms, is Dionysiacal ; but to perfect all things harmonically, is Apolloniacal. As the Demiurirus, therefore, comprehends in himself the cause of both these (iods, he both div ides and harmonizes the soul, lor the hebdomad is a number common to both these divinities, since theologists al.so say that Bacchus w;is divided into seven parts :

Into seven parts the Titans cut the boy. And they refer the heptad to A|>ollo, as containing all symphonies. For the duple diapason first subsists m the monad, duad, and tetrad, of which numbers the hebdomad consist.-.. Hence they rail the (iod Hebdomagetes, or born on the seventh day, and assert that this day is sacred to him : For on ihis Hay l.atona bore ' the God \\lio Mi-ars a pnMcn sword. Just as tin; sixth day is sacred to Diana. This number, indeed, in the same manner as the triad, is imparted to the soul from superior causes ; the latter from intelligible, but the former from intellectual natures. And it is also imparted from these very divinities [Apollo and Bacchus], in order that by a division into lor aittor *c here, it is obviously necessary to read air tor it. tor jy>v<ra«.pa yfiparo ilfrr, read g/wiraiopa yeoaro. 7tt 1'KOCLUS ON THE [BOOK in. seven parts, the soul may have a signature of the Dionysiacal series, and of the fabulous laceration • of Bacchus. For it is necessary that it should participate of the Pionysiacal intellect ; and as Orpheus says, that !>earing the. (lod on its head, it should IK' divided conformably to him. Hut it possesses harmony in thes-e parts, as a symbol of the Apolloniacal order. For in the lacerations of Bacchus, it is Apollo who collects and unites the distributed parts of Bacehus, according to the will of the father [JupiterJ.

In these nuiul>ers also, the three middles are com prehended.* The**1 therefore being three, adumbrate not only in the soul but every where, the three daughters of Themis. And the geometric middle, indeed, is the image of liiinomia.1 Hence IMato in the Laws nays, that she adonis polities, and dispose^ them in mi orderly manner, and he likewise celebrate* her as the judg ment of Jupiter, adorning the. univ« ise, and comprehending tin true political science. But the harmonic middle i.s an image o! Dice or Justice, distributing a <>reater ratio to greater, but a less to lesser terms. This liowe\er is the work ofjustice. And the arithmetical middle is an linage of IVaee. ' For i( is ihis, us it it also said in the Laws, which imparts to all things the <<|ii,il aicuidm- to quantity, and makes people at peace with people. For the solid analogy | i. e. tintriplicate proportion] prior to these, is sacred to their mother Themis, who com prehend.-, the powers of all of them. 4 And thus much universally concerning these thn-e middles.

These three middles ' however, may \n> said in a way adapted to whathns been before observed, to be the sources of union and connexion to the soul, or in other words, to l>e unions, analogies, and bonds. Hence al>o Tinni-us denominates them bonds. For prior to this, he had said, that the. geometric middle is the most beautiful of bonds, ami that the other middles are contained in this. * Hut fiery bond is a certain union. If therefore the middles are bunds, and bonds are the unions of the tilings that are bound, that which follows is evident. Hence these |>er\ade through all the essence of the soul, 7 and render it one from many 1 I'lir •»il)iii>/«'inr» III lllll (tl.lir. I., Ill rtit.1,,,1 , /.nl , ' I'u. 111 Ilir iiuinltrii I. '.'. II. 4. 0. lli«- lielnloiii.nl, »• l'nnlu» ul»M-rvr«, mnnlkliii)- of !.'.', urid I. lor I. ,'. unit J. jit in antlmirtu nl |iro|)orlioii. The tminlnr-. .'. .;. und (>. and aK<> 3. I. uiiil (>. art1 in barroouic.il proportion. And (lit numtirrs 1 . '2. .). and (i. .in iu gtouu Irical |iii>|iorlioii. 1 tnitr.nl uf rvfoiui here, it it nccra^arv lo rrjd I'.in.^m.

• For triplicate, conni.its of ^coinclrical proporlion, md ^romelrii.il ci>iiUiu> in il»«-lf. n« *t line beforr •liiivtn, llir uritlimrlu al, and ilie li.uini)nic.il jirnporiion ; and tlicnToic Iriplicutf, orsoliil <inulogj, cuiuprrlirnds in itvlf tlif (lircr nuddlfi. 1 Mffforrjrfi \\ omitted in tli«- original. * For tv Tovruii \trie, it is ob\iously necessary to read tv rovry. ' Tbe words ^-»\';i oi/«iai are omitted in iLe original, but evidently ought to b« inserted. wholes, as tlioy are allotted a power of binding together things of a various nature. A-J however they are three, the geometric middle hinds every tiling that is essential in souls. For essence is one. reason, proceeding through all things, and connect ing lirst, middle, and last natures, just as in the geometric middle, one and the same ratio, pervades perfectly through the three; terms [ofvvhieh the proportion consists]. But the harmonic middle connects all the divided sameness of souls, imparting to the extremes a communion of reasons, and a kindred conjunction. And sameness, indeed, is seen in a greater decree in more total, hut in a less degree in more partial natures. And the arithmetical mid die hinds the all-various diversity of the progression of the soul, and is less inherent in things which are greater, hut more in such as are less, according to order. For difference has dominion in more partial, just as same ness has in more total and more excellent natures. And these two middles have something l>y which they communicate with each other, in the same manner as sameness and difference.

As essence also is the monad of the latter, so the geo metrical middle is the monad of the former. The geometric middle, therefore, is the union of the essences in all the 34 terms ; the harmonic of the equally numerous samenesses; and the arithmetical, of the differences.1 Hence all these extend through all (he terms, or how could a certain whole he produced from them, unless they were as much as possible united to each other? Kssentiallv indeed, hy the geometric middle, hut in another and another way by the remaining two. On this account also the arithmetical and harmonic middles become the consum mation of the geometric middle, in the same manner as sameness and difference, contribute to the perfection of essence. For because the arithmetical and har monic middles subsist oppositely \>ith reference to each other, the geometric middle connects and, as it were, weaves' together their dissention. For the harmonic middle indeed, distributes as we have said, greater ratios to greater terms, and less ratios to less terms ; since it evinces that things which are essen tially greater and more total, are also more comprehensive in power than such a.s are of an inferior nature. lint vice versa, the arithmetical middle, distributes less ratios to greater terms, but greater ratios to less terms. For difference prevails more in inferior natures, just as sameness on the contrary, has greater authority in superior than in inferior natures. And the geometric middle extends the sarae ratio to all the terms ; imparting by illumination union to first, middle, and last natures, through the presence of essence- to all things.

The Demiurgus therefore imparts three connective unions to the nonl, which Plato denominates middles, as binding together the middle order of wholes. 1 It is here necessary In Mij>|>ly llie words w< t> noifyi/fmij rut- trrpoTifrvr. * For irvfi$<inrt here, it is necessary to read n>i ration. gO PKOCU7S ON THE [BOOK in. And of these, the geometric middle collects the multitude of essences, and causes essential progressions to lie one : for one ratio is the image of union. But the harmonic middle, hinds total samenesses, and the hyparxes of them into one communion. And the arithmetical middle, conjoins first, middle, and last dif ferences. For, in short, difl'erenre is the mother of numbers, as we learn in the Parmenides. These tlmv, however, viz. essence, sameness, and difference, are in each part of the soul, and it is requisite to conjoin all of them to each other through a medium and colligative reasons. In the next place, we say that the soul is a plenitude of reasons, she being more simple than sensibl.-s, hut more composite than intelligible*. Hence Tima-us assumes seven ratios in the soul, vi/.. the ratios of equality, multiplicity, suhmultiplicity, the superpurtieular and superpartient ratios, and the opposites of these, the sutisiiperparticular, and subsuperpartient, but not the ratios which are com pounded from these. For the*e an: adapted to corporeal reasons, since they are composite and partible. The reasons in the soul, however, proceed indeed into multitude and partibility, yet together with multitude, they exhibit simplicity, and the uniform in conjunction with a distribution into parts.

Hence they are not allotted an hypostasis in the monad, and the impartible, in the same manner as intellect. For intellect is alone monadic and impartible. Nor does the mul titude of them proceed into composite reasons. And multiple ratio indeed is in oneway only partible, viz. according to the pn>U.<u,s ' or greater term : for the hifpolo%os, or less term, is without dixision, and is not pre\ented from being unity. But the Miperparticular, is divisible in a twofold respect, uz. according to the prologos and hypologos; but is impartible according to difference.1 And the wiperpartieiit is partible, both according to the prologos and hypolo-os, and ac cording to difference.1 So that the first of these, is divisible in one \\ayonly, the second hilariously, and the third trifuriously. But equality is hnpartibh. The soul therefore constitutes the universe by these ratios ; the corporeal-formed nature indeed, by that which is trilariously partible; the nature of sujH-riicies by

' Thus for m,tance, 32 to 4 is a multiple ratio, which in in lowest terms, is lhatof S to I. But S i> divisible, and 1 accurately i])i-akii.B is not. Prothu, then-fore, .11 what he now SUNS, speaks of multiple ratio when reduced to its lowest terms. 1 Thu, the supcrparlicular ratio of 6 to 4, n in its lowest terms the rati" «.f 3 to:, «nd both 3 and 2 dm,ible. i5ut this ratio according to the dilfcrt-nce of the term,, i, indiv.s.ble; for this differ- .Iff nice !•> 1 • Thus for ii.staiicf.lhe su|M-rparlieut ratio 3 to 5, which is superl.iparlieni. heing in il» lowest ttrms. for this is the case with all superpartieut ratioi, is evidently partible, both according to the prologot and hypvlogoi; and it is also partible according to the difference of the terms. For this dif ference is -'. BOOK in.] T1M/EUS Ol- PLATO. 81 that which is bifariously partible ; every linear nature, by that which is partible in one way only ; and by the impartible the impartiality which comprehends all things. For there is something impartible in partible natures. These things, therefore, are truly asserted. It is necessary however, to survey these after another manner; premising, that numbers which are more simple, and nearer to the monad, ought to be conceived as more primary than those which are more composite. For I'lato aNo, having arranged one part prior to all the rest, refers all of them to this, and < nds in terms which are especially composite and solid.

Having therefore premised this, I say that equality and the ratio of equality, has the relation of a monad to all ratios. And what the monad is in quantity per se, that the equal is in relative quantity. Hence conformably to this, the soul introduces a common measure to all things which subsist according to the same reasons; which measure likewise, brings with it one idea the image of sameness. But according to the submultiple, and multiple ratio, it governs all the whole series of things, connectedly comprehend ing them, and exhibiting each total form of mundane natures frequently producer! by itself in all mundane beings. Thus, for instance, it produces the solar, and also the lunar form, in divine souls, in demoniacal and human souls, in irrational ani mals, in plants, and in stones themselves, and adonis the most universal genera by the more partial series. And according to the stiperparticular and subsuperparticular ratios, it adonis such things as are wholes in their participants, and which are participated according to one certain thing contained in them. But according to the buperpartient and subsuperpartient ' ratios, it adorns such things as are wholly participated by secondary natures, in conjunction with a division into multitude. For of animal indeed, man participates, anrl the whole of this form is in him, yet not alone, but the whole is in him according to one thing, viz. the human form; so that it is present to its participant with the whole, anrl one certain tiling which is a part of it.

But what are called common genera, par ticipate of one genus, yet not of this alone, but together with this of many other genera also, which are parts, and not a part of that one genus. Thus, for instance, a mule participates of the species from which it has a co-mingled generation. Each species, therefore, either participates of one genus according to one, and thus imitates the superparticular ratio which contains the whole, and one part of the whole ; or it participates that which is common and many things besides, and thus imitates the superpartient ratio, which together with the whole possesses also many parts of the whole. And besides these there is no other participation of

rpii is omitted in the original, Tim. Plat. VOL. II. 8-2 PKOCLUS ON THE [BOOK in. species or forms. Looking also to these things, we may be able to assign the s[>ecific censes of those natures which subsist according to one form, as, for in stance, of the sun, the moon, and man"; as likewise of those that subsist according to many forms, together with that which is common. For there are many tilings of this kind in the earth, and in the sea; such as animals with a human face and the extremities resembling those of a fish, and animals in the form of dragons, but with a h-onine face; these haxing an essence mingled from many things. All these ratios therefore, are very properly antecedently comprehended in the soul, as they define all the participations of forms in the unherse. Nor can there be any other ratios of communion besides these, since all things receive a specific distinction according to these. Acain, therefore, the hebdomad of ratios corresponds to the hebdomad of ports. And the soul is wholly through the whole of itself hchdomatic, in its parts, in its ratios, and in its circles. For if the demiurgic intellect is n monad, but the soul primarily proceeds from intellect, it will have the ratio of the hebdomad to it. For the hebdomad is paternal and motherless. And |>erhaps equality imparts to all the psychical ratios, a communion of the equal, in order that all may commu nicate with all.

But 1 1 ie multiple ratio a (lords an indication of the manner in which the ratios that are more single, measure those that are multitudinous, the former wholly proceeding through the whole of the latter ; those that are impartible, mea suring those that are more distributed into parts. The superparticular how ever, and siibsuperparticular ratios, indicate the difference according to which whole ratios do not communicate with whole, but have indeed a partial habi tude, yet are conjoined according to one certain most principal part of themsehes. And the superpurtieiit and siibMiperpartient ratios ailord an indica tion of the last nature, according to which there is a certain partible and mul tiplied communion of the psychical ratios, oil account of diminution and infe riority. For the more elevated of these reasons are united wholly to the whole of each other. Hut those of the middle rank are conjoined, not through the whole of themseUes, but according to the highest part. And those of the third rank, partibly coalesce according to multitudes. 1 say, for instance, essence communi cates with all the ratios, measuring all their progressions: for nothing in them is unessential. But1 sameness, being itself a ^cnus, especially collect*, the summils of them into one communion. And difference particularly measures their divisions and progressions. The communion, therefore, of the psychical ratios, is every where* exhibited. For it is either all-perfect, or alone subsists according to the summits, or according to extensions into multitude.

' For oi^t here, it is nercsiary to read »j It. 1 ln>Uail of wX'fK in this |>l.ue, il is requisite to read xfura^ou. »""K in.] TIM.EUS OK PLATO. a;{ Farther still, in the next place let us survey, how the seven parts ' are allotted their hypostasis. The first part then, is most intellectual and the summit of the soul, conjoining it to the one itself, and to the hyparxis of the first essence. Hence also it is culled one, as having the form of unity, and the number of it proreeding ' into multitude, is detained by union. It is likewise analogous to the cause and centre of the soul. For the soul abides according to this, and does not depart from wholes. And the tetrad indeed, is in the first monads, on account of its stability, and rejoicing in equality and sameness. Hut the onload is in the monads of the second order, on account of diminution, and the pro\idence of the soul which extends as far as to the last of things, and that which is most material.' And the triad is in the monads of the third order, on account of the circumduction to the all-perfect of the multitude which it contains. And at the same time, it is evident from these things as from images, that the summit of the soul, though it has the form of unity, yet is not purely one, but this also is an united multitude.

Just as the monad, is not indeed without multitude, yet at the same time is unity. But the one of the Gods is one alone. And the one of intellect is more one than multitude, though this also is multiplied. Hut the one of the soul is similarly one and multitude ; just as the unity of the natures posterior to it, which are divisible about body, is more multitude than one. And the one of bodies, ix not simply one, but the phantasm and image of unify. Hence the Elean guest, or stranger, says that every thin- corporeal is broken in pieces, as having an adventitious unity, and never ceasing to be divided. Hut the- second part multiplies the part prior to it, by generative progressions, \\hich the duad indicates, and unfolds all the progressions of essence. Hence it is said to Iw double of the first, as imitating the indefinite duad, and the intelligible infmitv. And the third part again converts all the soul to its principle ; and it is the third of it which is convolved to the principles. This, therefore, is measured by (he first part, as being filled with union from it ; but is conjoined more partially to the t-econd part. And on this account it is said to be triple of the first, but sesquialter of the second ; being half contained indeed by the second, as not having an equal power with it, but perfectly by the first.

Hut again, the fourth part, ~»nd besides this the fifth, evince that the soul ]>eculiarly presides over secondary natures. For these parLs are the intellectual causes of the incorporeals « hid, are divided about bodies, a* they are planes and squares; the former being the square of the second, and the latter of the third part. And the fourth part," indeed, is

Tliew seven parts arc the nuiulnr» 1. 2. 3. 4. «). 8. 27. *F*p\o(ttvot it omitted in the original. ' For atporarov litre, I rrarl t rcXoraro*. 84 PROCLUS ON THE [BOOK in. the cause of progression and generation, but the fifth, of conversion and per fection. For both are planes ; but one is from the second part, subsisting twice from it, and the other from the third part, thrice proceeding from it.1 And it seems, that the former of these planes, is imitative* of the generative natures which are divi sible about bodies; but that the latter is imitative of intellectual conversions.' Forall knowledgeconvertsthatwhich knowsto tin-thing known; just asevery nature wishes to "•enenite and to make a progression to that which is inferior. And the sixth and seventh parts, contain in themselves, the primordial causes of bodies and Kohd masses: for these numbers arc solids. And the former of them, indeed, is from the second, but the latter from the third part. Hut IMato in what he says convertin"- the last to the first parts, and the terminations of the soul to its t> ummit, places one part as octuple, but the other as twenty seven times the first. And thus the essence of the soul consists of seven parts as abiding, proceeding, and returning, and as the cause of the progression and regression of the essences divisible about bodies, and of bodies themselves. If, also, you are willing so to speak, Ix'cause the soul is allotted an hypostasis between impai tibles and partibles, she imitates the former through the triad of the terms, but antecedently assumes the latter from the tetrad.

Hut the whole of her consists of all the terms, l>ecause the whole of her is the centre of wholes. It is possible also to di\ide these parts according to the duple order, if you assume the summit of the soul, and consider the permanency, progression, and regression4 of it, and also the conversion to it of things proximately posterior to it, and the last subjection of solids, or rather the diminution of the cause of them, according to the duple ratio. For you will find that the whole of this co-ordination pertains to the prolific duad. Hut again the regression of itself to itself, and of the natures proximately posterior to it, and of those that rank in the third degree from it, to the uniform and collective essence of wholes, subsist according to the triple order. The arithmetical there fore and harmonic middles, give completion to these intervals, vt hich are essen tial, and surveyed according to existence itself; some of them as we have said, binding their samenesses, but others their differences.

Farther still \ou may also say, in a way more proximate to the things them selves, that the soul according to one part is united to the natures prior to it, and 1 The Iliinl part is 3, and }) u equal to 3 + 3 + 3. 1 rory<iM/ru>; IUTC, it i> ui'icss-ir) to read /ii/jijruij. > Instead i.f -,<>'•'/'•"• tt opui «.£«» rr,k wp-mv»* fupovfAttuv rr,i ^i»X»?« *»«<Tr,>«»ruoi> ill this pllCf.lt II ntcosary to n-u.l i ..nforni.il.U to tin- version ofTlioiuxiii, •; It rw* to<p*v twHrrpo.p** yt»7""», txcept that herealso for yo i ijru-ij, it is requisite to read ^t^rjnci}. * There is an omission here in tue original, of it t*i<n>x>f»i». BOOK in.] TIM/EL'S OF PLATO. 05 tliis part is the summit of the soul ; hut according to the duple and triple part, it proceeds from intellect, and returns to it;' and according to the douhle of the douhle, and the triple of the triple, it proceeds from itself, and again returns to itself, and through it>elf as a medium, to the principles of itself. For through being filled from these principles it is prolific * of secondary natures. And -.is indeed, the progression from itself, is suspended from the natures prior to itself, thus also the conversion or regression to itself1 is suspended from the regression of the beings that are prior to it. Hut the last parts, according to which it consti tutes the natures posterior to itself, are referred to the first part;

in order that a circle without a beginning may be unfolded to the view, the end In-inij conjoined to the beginning, and that the universe may become animated, and at the same time endued with intellect, the solid numbers being co-arranged with the first part. Moreover, he says that from these middles, sesquialter and sesquitertian ra tios, and also sesquioctaves, become apparent. What else, therefore, does he intend to indicate by these things, than the more partial dill'erence of the psychical ratios? And the sesquialter ratios indeed present us with an image of partible communion, but according to the first of the parts. The sesquitertian ratios, of partible communion according to the middle terms. And the sesquioctaves of this communion, according to the last terms. Hence also, the middles or pro portions are conjoined to each other, according to the sesquioctave ratio. For as being surveyed according to opposite genera, they have the smallest communion; but they are appropriately conjoined to both extremes.

Tiimrus also adds, that all the sesquitertiau ratios are filled by the interval of the sesquioctave, in conjunction with the leimma ; indicating that the terminations of all these ratios, end in more partial hypostases, in consequence of the soul com prehending the causes of the last and perfectly partible essences in the world, and pre-establishing in itself, the principles of the order and harmony of them, according to the demiurgic will. Tine soul therefore, possesses the principles of harmonious progression and regression, and of the division into things first, middle and last; and is one intellectual reason or ratio, receiving its completion from all ratios. And again, that all the harmony of the soul, consists of a quadruple diapason, diapentc and tone, is consonant to these things. For since According to the version of Thomacus, there is an omission Jicre in the original, of the words fara ic ri)v lir\aaiav tat Tfnx\affiav, OKO rnv rev rpotiai, KOI jrpot avror tiriorptt^t. iiidrrd the sens* require! this addition. * For powfAot in this place, it is necessary to read yon/jot. * The word* ovrwi ij m raw-i)* tvitrrpofti, which arc omitted io the original, ought to be inserted, conformably to the venion of Tbomcus.

80 PKOCLUS ON THK [BOOK nr. there is harmony in tlie world, and also in intellect anrl in soul, on which account Tiriiiviis says, that the soul participates of harmony, and is harmony, — hence the world participates of harmony decadieally, lint the soul tetradically, and liannonv pre-exists in intellect monadic-ally. And as the monad is the cause of the tetrad, but the tetrad of the decad, thus also, the intellectual harmony, is the supplier of the psychical, and the psychical of sensible, harmony. Hence Tim u.1 us conceived that the quadruple diapason, is adapted to the harmony of the soul. For the soul is the proximate paradigm of the harmony in the seusihle world. Since however, Jive ' figures and live centres in the universe, give completion to the whole, the harmony diapente also, impart* to the world the symphony which is in its parts. Since, likewise, the universe is divided into nine parts, the sescjuioctave produces the communion of the soul with the world. And here you mav see that the soul comprehends tin* world, and makes it to IK- a whole, according to cause, » as one, as consisting of four, and also of five parts, and as divided into nine parts, harmoni/ing and causally comprehending the whole of it. Tor the monad, the tetrad, the pentad, and the cnncad, ' procure for us the whole number, according to which all the parts of the world are divided.

Hence the ancients assert that the .Muses and Apollo Musagetes, preside over the universe; the lat'er supplying the one union of all its harmony ; but the former connecting the divided progression of this harmony, and rendering their number concordant with the eight Syrens mentioned in the Republic. Thus therefore, in the middle of the monad and the ennead, the universe is adorned tetradically and pentadically ; tetradically indeed, according to the four ideas of animals, which the para digm comprehends; but pentadicallv, according to the live figures, through which the Demiurgus distributed all things, himself, as Tima-us says, introducing the filth idea, and arranging this harmonically in the universe.

Again therefore from the Ix-ginnm;;, we mav sav, that the Demiurgus bavin" twofold powers, the one being eflective of sameness, as we learn in the 1'armenides, but the other, of dilli-reiice ; he both divides and binds the .soul. He is also the final cause of these, in order that the soul may become the middle of wholes, being similarly united and divided, two things existing prior to it, divine natures as unities, and beings as things united ; and two also being posterior to ' For niiru IHTC n-atl *trrt . Tht-M- livi- fi^uri-, arc the (i»e regular Ladies, \i/. llif dodecahedron, the pvr.iini I, I lit- <>< l.iht clion, llu- i< ovilii-ilrmi, .in. I I In- c ulic. Itut llic |i\o i-riilre* arc tlic nortlirrii, M. mil. in, i ,i-ii in, .ni.| « i >i, MI tfiilici, iiml (In- < i-iiin- nl (lit- miiM-or. rnnii I he \if.i«ui of 'riniui.cn-., ll it iu-(i^>.ir\ lo iiiM-H in llii> plji-c thr «or(l:i, 17 4V\1 r<>v ma^mv wff>it\(t, »ui t,\<,»' k.ir' diritu- »»<€i, which ure Mauling in ihc cii.uijl. 1 1'or <tai hrre, read oxui. nooiciii.J TLNf/KUS OF PLATO. X7 it, vi/. those which are dnided in conjunction with otliers.' and those which are in every respect partible, or if you are willing prior to tlie former, the one itself, hut posterior to the latter, matter itself. [Hut the efficient causes of these divisions and bonds, are mine and <//'// ire/;/, which are the i>eculiarities of the demiurgic

order. And the paradigmatic causes are the sections and bonds of the father. For he* lir>t cuts, and hinds with ine/lable honds. Tlif.sc things also, tire obscurely indicate,! In/ theorists when they speak of the Salunrian sections and bonds icith icliich the timber of the nnirerse is said to surround hi in.se If, and which arc mentioned by Socrates in tlie Cialylus. Hut the formal causes of the divisions indeed, are the numbers. For according to these the parts are distinguished. But of the bonds, the middles, and the ratios which ^i\e completion to these, are the formal causes. For it is impossible to survey concauses which have the relation of matter, in souls which are incorporeal.]' These things l>eing premised, it is evident that the Demiurirus ener^i/ini; with twofold powers, viz. with such as are of a dividing, and such as are of a binding nature, he divides the Informed essence of the soul, and the threefold mixture, the whole remaining that which it is, by the primordial causes of all di\ ision, and makes the w hole of it to consist of seven parts, and seven members, comprehended in intellectual boundaries. For since tjie Demiurgus constituted the soul as the medium between an impartible essence, and that essence which is divisible about bodies; but the impartible essence is triple, abiding, pro ceeding, and returning; — ho pre-established the similitude of this in three parts. And he adumbrated indeed, the permanency* of this essence- by the first part; but the progression of it by the second ; on which account, perhaps, it is said to be double of the first part. For every thin;; which proceeds, has permanency pro-existent to its progression. And he adumbrated its regression by the third part. Hence this part is triple of the first. For every thing which returns, has proceeded and been permanent.

Since, however, the soid produces the essence posterior to itself, it contains the total4 essences of it in itself; the whole of the incorporeal essence indeed, but which is inseparable from bodies, according to the fourth and fifth part ; but the ' The words firr' a\\v» are omitted in the original, but ought to be inserted conformably to the version ofThomirun. 1 For tturuv hrrc, rend tttuot. The Chaldean Oracles also say that the Demiurgus glitters with intellectual sections. J All this part within the brackets, is omitted in the version of Thoiuacus, with whom such omissions ar<; not unfrnjucnt. 4 For yovifinv here, it is necessary to read fionfior. 1 Instead of oX»n here, rcid oAoi. 8ft PROCLUS ON THE [BOOK ui. whole of the corporeal essence, according to the solid numt>ers, viz. the sixth and seventh part. Or [it may IM- said] that the soul being M-lf-stihsistent, and selfenergetic, produces itself and converts itself to its principle, according to the square numbers ; but all the partible, essence posterior to itself according to the cubes. These seven parts therefore, being divided as we have said, into three and four, the one ratio of geometric analogy, binds them essentially; but the harmonic middle, binds them according to sameness; and the arithmetical according to difference. These parts however are inserted between the geometric middle, and are said to give completion to the double and triple intervals ;' because all sameness and all difference, are.

uniformly comprehended by essence, and the harmony which subsists according to it. Hut from these middles, the multitude of ses(|iiialter and sesquitertian ratios, and sesqmoctaves, becomes apparent ; this multitude indeed, being of a binding and connecting nature, in the same manner as the middles, but more partially. For these indeed, are certain ratios; but each of the middles consists of many ratios, which are either the same or different. As analogy therefore, or proportion, is more comprehensive than ratio, thus also the above-mentioned middles, afford a greater cause to the soul, of conned, dly con taining the multitude which is in it, since they intellectually pervade1 through the •whole of it. Hence the sesquialter, and sesquiteitian ratios, nr» eeit.tiu more partial bonds, and are comprehended in the middles; not accord ug to different habitudes of them with reference to the extremes; for this is mathematical ; but according to causal comprehension, and a more total hypostasis.

Again, these bonds, contain the second and third progressions of the ratios ; the sesquialter indeed, constringing the harmony of the ratios, through the five centres; but the sesquitertian exhibiting their power, through the four elements which are every where ; and rendering all things intelligible and allied to each other; and the sesquioctaves co-hannonizing the division into fl and H. Hence, as the world consists of eight parts, and also of nine, the ancients, at one time establish eight Syrens, and at another nine Mus.-s, as presiding over the universe, since the harmony of the whole of things proceeds from these. The sesquiterlian there fore, anil sesquialter ratios, are more total than the sesquioctaves, becarse they are the suppliers of a more perfect symphonv, and antecedently comprehend in less numbers, the harmonious section of the world. Here therefore, the divisions are separated from each other in their participants; but in the incorporeal reasons 1 Tin- u-von term* I. '2. .1. 1. <). 8. -'7, are laid l»y IWIun I" l«e inserted between the geometric mid dle, l*cau« 1 . 2.4. «, and also 3. <). ?7, are iu geometric proportion. * For kiotovaa* faere, read tttifovva*. in.] TIMdiUS OF PLATO. y<,

of the soul, the more total contain the more partial. Since, however, thesesquiortaves are the causes of the most partial symphony, that which i* posterior to them is not undeservedly said to be impelled into the last order of the universe. Not that even this is dissonant to the whole itself, since it compels the partial rfllu.vions from each of the elements, into the place under the earth. For as the elements exist in many places, in the heavens, and in the sublunary region, this leimma collects the last dregs of them there, and conjoins them to wholes; so that both together may give completion to the w hole harmony of the universe. In short, therefore, the soul is the cause ' of all the harmony in the centres, and in the. elements of the universe. Hence also we say, that the harmony of it is entirely intellectual and essential, causally preceding sensible harmony. And Tim;rus wishing to indicate this through images,* employs harmonic ratios, and pre-supposes in the soul some causes more comprehensive than others, and which subsist prior to every form of the soul and to all its knowledge. Hence I think it is not proper to discuss any thing of this kind, so as to explain [in a merely mathematical way] either the parts, or the ratios, or the analogies; but it is requisite to survey all these essentially, according to the first division itself, and harmony of the soul.