1. Since it is your opinion that it would be requisite for me to
collect together what I have discoursed and written dispersedly in
several treatises explaining, as we apprehended his sense and meaning,
what opinion Plato had concerning the soul, as requiring a particular
commentary by itself; therefore I have compiled this discourse,
which asks for your consideration and pardon not only because the
matter itself is by no means easy to be handled, but also because the
doctrines herein contained are somewhat contrary to those held by most
of the Platonic philosophers. And I will first rehearse the words as
they run originally in the text itself of Timaeus.[188]
“There being one substance not admitting of division, but continuing
still the same, and another liable to be divided among several bodies,
out of both these he produced for a middle mixture a third sort of
Substance, partaking of the nature of the Same and of the nature of the
Other, and placed it in the midst between that which was indivisible
and that which was subject to be corporeally divided. Then taking all
three, he blended them into one form, forcibly adapting to the Same
the nature of the Other, not readily condescending to a mixture. Now
when he had thus mixed them with the Substance, and reduced the three
into one, he again divided this whole matter into so many parts as were
thought to be necessary; every one of these parts being composed of the
Same, the Other, and the Substance. And thus he began his division.”
By the way, it would be an endless toil to recite the contentions and
disputes that have from hence arisen among his interpreters, and to you
indeed superfluous, who are not ignorant yourselves of the greatest
part.
But seeing that Xenocrates won to his opinions several of the most
eminent philosophers, while he defined the substance of the soul to be
number moved by itself; and that many adhered to Crantor the Solian,
who affirmed the soul to consist partly of an essence perceptible to
the mind, partly of a nature concerned with sensible things and subject
to opinions; I am apt to believe that the perspicuity of these matters
clearly dilucidated will afford you a fair entrance into the knowledge
of the rest.
2. Nor does either of the two conjectures require many words of
explanation. For the one side pretends that by the mixture of the
divisible and indivisible substance no other thing is meant than the
generation or original of number, seeing that the unit is undividable
but multitude is subject to division; however, that out of these is
begot number, unity terminating plurality and putting a period to
infinity, which they call the unlimited binary. This binary Zaratas,
the scholar of Pythagoras, named the mother, but the unit the father
of number; and therefore he believed those numbers were the best which
approached nearest in resemblance to the unit. Nevertheless, this
number cannot be said to be the soul; for it neither has the power
to move, neither can it be moved. But the Same and the Other being
blended together, of which one is the original of motion and mutation,
the other of rest and stability, from these two springs the soul, which
is no less active or passive itself to stay or to be stayed, than to
move or to be moved.
But the followers of Crantor, supposing the proper function of the
soul to consist in judging of those things which are discernible to
the understanding and those which are liable to sense, as also of the
differences and similitudes of these things, as well in themselves as
in reference one to another, allege the soul to be composed of all,
to the end she may have a true knowledge of the whole. Now the things
of which the All is composed are fourfold,—the intelligible nature,
always immutable and still the same, and the sensitive nature, which
is passive and subject to alteration; and also the nature of the Same,
and the nature of the Other, in regard the two former in some measure
participate also of diversity and identity.
3. All these philosophers likewise equally hold that the soul neither
derives its beginning from time nor is the product of generation, but
that it is endued with several faculties and virtues, into which Plato,
as it were, melting and dissolving its substance for contemplation’s
sake, supposes it in his discourse to have had its original from
procreation and mixture.
The same was his opinion concerning the world; for he knew it to be
uncreated and without end, but not perceiving it so easy to apprehend
how the structure was reared, or by what order and government
supported, unless by admitting its beginning and the causes thereto
concurring, he followed that method to instruct himself. These things
being thus generally by them laid down, Eudorus will allow to neither
side any share of probability; and indeed to me they both seem to have
wandered from the opinion of Plato, if we intend to make the most
likely rule our guide,—which is not to advance our own conceits,
but to come as close as we can to his sense and meaning. Now as to
this same mixture (as they call it) of the intelligible and sensitive
substance, no reason appears why it should be more the original of the
soul than of any other thing that ye can name. For the whole world
itself and every one of its parts pretend to no other composition
than of a sensitive and an intelligible substance, of which the one
affords matter and foundation, the other form and figure to the whole
mass. And then again, whate’er there is of material substance, framed
and structured by participation and assimilation of the intelligible
nature is not only to be felt but visible to the eye; whenas the soul
still soars above the reach of all natural apprehension. Neither did
Plato ever assert the soul to be number, but a perpetually self-moving
nature, the fountain and principle of motion. Only he embellished and
adorned the substance of it with number, proportion, and harmony;
as being a subject capable of receiving the most goodly form which
those ornaments could produce. So that I cannot believe it to be the
same thing to compose the soul according to number, and to affirm the
soul to be number itself. Nor can it be said to be harmony because
harmoniously composed, as he has clearly demonstrated in his Treatise
of the Soul. But plain it is, that those philosophers understood not
the meaning of the Same and the Other. For they tell us how the Same
contributes rest, the Other motion toward the generation of the soul.
Though Plato himself, in his treatise entitled the Sophist, disposes
and distinguishes Essence, the Same, the Other, together with Motion
and Rest, as being five things altogether differing one from another
and void of mutual affinity.
4. But these men are generally, as the most part of Plato’s readers,
timorous and vainly perplexed, using all their endeavors by wresting
and tormenting his sense to conceal and hide what he has written,
as if it were some terrible novelty not fit for public view, that
the world and the soul neither had their beginning and composition
from eternity, nor had their essence from a boundless immensity of
time,—of which we have particularly spoken already. So that now it
shall suffice to say no more than this, that these writers confound and
smother (if they do not rather utterly abolish) his eager contest and
dispute in behalf of the Gods, wherein Plato confesses himself to have
been transported with an ambitious zeal, even beyond the strength of
his years, against the atheists of his time. For if the world had no
beginning, Plato’s opinion vanishes,—that the soul, much elder than the
body, is the principle of all motion and alteration, or (to use his own
words) their chieftain and first efficient cause, whose mansion is in
Nature’s secret retirements. But what the soul is what the body, and
why the soul is said to have been elder than the body, shall be made
appear in the progress of this discourse. The ignorance of this seems
to have been the occasion of much doubt and incredulity in reference to
the true opinion.
5. First therefore, I shall propose my own sentiments concerning these
things, desiring to gain credit no otherwise than by the most probable
strength of arguments, explaining and reconciling to the utmost of
my ability truth and paradox together; after which I shall apply
both the explication and demonstration to the words of the text. In
my opinion then the business lies thus. The world, saith Heraclitus,
neither did any one of all the Gods nor any mortal man create,—as if
he had been afraid that, not being able to make out the creation by a
Deity, we should be constrained to acknowledge some man to have been
the architect of the universe. But certainly far better it is, in
submission to Plato’s judgment, to avow, both in discourse and in our
songs of praise, that the glory of the structure belongs to God,—for
the frame itself is the most beautiful of all masterpieces, and God the
most illustrious of all causes,—but that the substance and materials
were not created, but always ready at the ordering and disposal of the
Omnipotent Builder, to give it form and figure, as near as might be,
approaching to his own resemblance. For the creation was not out of
nothing, but out of matter wanting beauty and perfection, like the rude
materials of a house, a garment, or a statue, lying first in shapeless
confusion. For before the creation of the world there was nothing but
a confused heap; yet was that confused heap neither without a body,
without motion, nor without a soul. The corporeal part was without form
or consistence, and the moving part stupid and headlong; and this was
the disorder of a soul not guided by reason. God neither incorporated
that which is incorporeal, nor conveyed a soul into that which had
none before; like a person either musical or poetical, who does not
make either the voice or the movement, but only reduces the voice with
harmony, and graces the movement with proper measures. Thus God did not
make the tangible and resistant solidity of the corporeal substance,
nor the imaginative or moving faculties of the soul; but taking these
two principles as they lay ready at hand,—the one obscure and dark, the
other turbulent and senseless, both imperfect without the bounds of
order and decency,—he disposed, digested, and embellished the confused
mass, so that he brought to perfection a most absolute and glorious
creature. Therefore the substance of the body is no other than that
all-receiving Nature, the seat and nurse of all created beings.
6. But the substance of the soul, in Philebus, he called an infinite
being, the privation of number and proportion; having neither
period nor measure either of diminution or excess or distinction or
dissimilitude. But as to that order which he alleges in Timaeus to
be the mixture of nature with the indivisible substance, but which
being applied to bodies becomes liable to division,—he would not have
it thought to be a bulk made up by units or points, nor longitude
and breadth, which are qualities more consentaneous to bodies
than to the soul, but that disorderly unlimited principle, moving
both itself and other substances, that which he frequently calls
necessity, and which within his treatise of laws he openly styles
the disorderly, ill-acting, or harm-doing soul. For such was this
soul of herself; but at length she came to partake of understanding,
ratiocination, and harmony, that she might be the soul of the world.
Now that all-receiving principle of matter enjoyed both magnitude,
space, and distance; but beauty, form, and measure of proportion it
had none. However, all these it obtained, to the end that, when it
came to be thus embellished and adorned, it might assume the form
of all the various bodies and organs of the earth, the sea, the
heavens, the stars, and of all those infinite varieties of plants
and living creatures. Now as for those who attribute to this matter,
and not to the soul, that which in Timaeus is called necessity, in
Philebus vast disproportion and unlimited exorbitancy of diminution
and excess,—they can never maintain it to be the cause of disorder,
since Plato always alleges that same matter to be without any form or
figures, and altogether destitute of any quality or effectual virtue
properly belonging to it; comparing it to such oils as have no scent
at all, which the perfumers mix in their tinctures. For there is no
likelihood that Plato would suppose that to be the cause and principle
of evil which is altogether void of quality in itself, sluggish, and
never to be roused on to action, and yet at the same time brand this
immensity with the harsh epithets of base and mischievous, and call it
necessity repugnant and contumaciously rebellious against God. For
this same necessity, which renverses heaven (to use his own phrase
in his Politicus) and turns it the quite contrary way from decency
and symmetry, together with innate concupiscence, and that inbred
confusion of ancient nature, hurly-burly’d with all manner of disorder,
before they were wrought and kneaded into the graceful decorum of
the world,—whence came they to be conveyed into several varieties of
forms and beings, if the subject, which is the first matter, were void
of all quality whatsoever and deprived of all efficient cause; more
especially the Architect being so good of himself, and intending a
frame the nearest approaching to his own perfections? For besides these
there is no third principle. And indeed, we should stumble into the
perplexed intricacies of the Stoics, should we advance evil into the
world out of nonentity, without either any preceding cause or effect of
generation, in regard that among those principles that have a being,
it is not probable that either real good or that which is destitute of
all manner of quality should afford birth or substance to evil. But
Plato escaped those pitfalls into which they blundered who came after
him; who, neglecting what he carefully embraced, the third principle
and energetic virtue in the middle between God and the first matter,
maintain the most absurd of arguments, affirming the nature of evils
to have crept in spontaneously and adventitiously, I know not how nor
by what strange accidents. And yet they will not allow an atom of
Epicurus so much as a moment’s liberty to shift in its station, which,
as they say, would infer motion out of nonentity without any impulsive
cause; nevertheless themselves presuming all this while to affirm that
vice and wickedness, together with a thousand other incongruities and
vexations afflicting the body, of which no cause can be ascribed to any
of the principles, came into being (as it were) “by consequence.”
7. Plato however does not so; who, despoiling the first matter of
all manner of distinction, and separating from God, as far as it is
possible, the causes of evil, has thus delivered himself concerning
the world, in his Politicus. “The world,” saith he, “received from the
Illustrious Builder all things beautiful and lovely; but whatsoever
happens to be noxious and irregular in heaven, it derives from its
ancient habit and disposition, and conveys them into the several
creatures.” And a little farther in the same treatise he saith: “In
process of time, when oblivion had encroached upon the world, the
distemper of its ancient confusion more prevailed, and the hazard
is, lest being dissolved it should again be sunk and plunged into
the immense abyss of its former irregularity.” But there can be no
dissimilitude in the first matter, as being void of quality and
distinction.
Of which when Eudemus with several others was altogether ignorant, he
seems deridingly to cavil with Plato, and taxes him with asserting the
first matter to be the cause, the root, and principle of all evil,
which he had at other times so frequently dignified with the tender
appellations of mother and nurse. Whereas Plato gives to matter only
the titles of the mother and nurse; but the cause of evil he makes
to be the moving force residing within it, not governed by order and
reason, though not without a soul neither, which, in his treatise of
the Laws, he calls expressly the soul repugnant and in hostility with
that other propitiously and kindly acting. For though the soul be the
principle of motion, yet is it the understanding and intelligence which
measures that motion by order and harmony, and is the cause of both.
For God could not have brought to rest mere sleepy and sluggish matter,
but he brought it to rest when it had been troubled and disquieted by
a senseless and stupid cause. Neither did he infuse into nature the
principles of alteration and affections; but when it was under the
pressure of those unruly disorders and alterations, he discharged it
of its manifold enormities and irregularities, making use of symmetry,
proportion, and number. For these are the most proper instruments, not
by alteration and lawless motion to distract the several beings with
passions and distinctions, but rather to render them fixed and stable,
and nearest in their composition to those things that in themselves
continue still the same upon the equal poise of diuturnity. And this,
in my judgment, is the sense and meaning of Plato.
8. Of which the easy reconciliation of his seeming incongruities and
contradiction of himself may serve for the first proof. For indeed no
men of judgment would have objected to the most Bacchanalian sophister,
more especially to Plato, the guilt of so much inconvenience and
impudent rashness in a discourse by him so elaborately studied, as to
affirm the same nature in one place never to have been created, in
another to have been the effects of generation;—in Phaedrus to assert
the soul eternal, in Timaeus to subject it to procreation. The words
in Phaedrus need no repetition, as being familiar to nearly every one,
wherein he proves the soul to be incorruptible in regard it never had a
beginning, and to have never had a beginning because it moves itself.
But in Timaeus, “God,” saith he, “did not make the soul a junior to
the body, as now we labor to prove it to have been subsequent to the
body. For he would never have suffered the more ancient, because linked
and coupled with the younger, to have been governed by it; only we,
guided I know not how by chance and inconsiderate rashness, frame odd
kind of notions to ourselves. But God most certainly composed the soul
excelling the body both in seniority of origin and in power, to be
mistress and governess of her inferior servant.”[189] And then again
he adds, how that the soul, being turned upon herself, began the divine
beginning of an eternal and prudent life. “Now,” saith he, “the body
of heaven became visible; but the soul being invisible, nevertheless
participating of ratiocination and harmony, by the best of intelligible
and eternal beings she was made the best of things created.”[190] Here
then he determines God to be the best of sempiternal beings, the soul
to be the most excellent of temporal existences. By which apparent
distinction and antithesis he denies that the soul is eternal, and that
it never had a beginning.
9. And now what other or better reconciliation of these seeming
contrarieties than his own explanation, to those that are willing
to apprehend it? For he declares to have been without beginning the
never procreated soul, that moved all things confusedly and in an
irregular manner before the creation of the world. But as for that
which God composed out of this and that other permanent and choicest
substance, making it both prudent and orderly, and adding of his own,
as if it were for form and beauty’s sake, intellect to sense, and
order to motion, and which he constituted prince and chieftain of
the whole,—that he acknowledges to have had a beginning and to have
proceeded from generation. Thus he likewise pronounces the body of the
world in one respect to be eternal and without beginning, in another
sense to be the work of creation. To which purpose, where he says
that the visible structure, never in repose at first but restless in
a confused and tempestuous motion, was at length by the hand of God
disposed and ranged into majestic order,—where he says that the four
elements, fire and water, earth and air, before the stately pile was by
them embellished and adorned, caused a prodigious fever and shivering
ague in the whole mass of matter, that labored under the combats of
their unequal mixtures, by his urging these things, he gives those
bodies room in the vast abyss before the fabric of the universe.
Again, when he says that the body was younger than the soul, and that
the world was created, as being of a corporeal substance that may
be seen and felt,—which sort of substances must necessarily have a
beginning and be created,—it is evidently demonstrable from thence that
he ascribes original creation to the nature of bodies. But he is far
from being repugnant or contradictory to himself in these sublimest
mysteries. For he does not contend, that the same body was created
by God or after the same manner, and yet that it was before it had
a being,—which would have been to act the part of a juggler; but he
instructs us what we ought to understand by generations and creation.
Therefore, says he, at first all these things were void of measure and
proportion; but when God first began to beautify the whole, the fire
and water, earth and air, having perhaps some prints and footsteps of
their forms, lay in a huddle jumbled all together,—as probable it is
that all things are, where God is absent,—which then he reduced to a
comely perfection varied by number and order. Moreover, having told us
before that it was a work not of one but of a twofold proportion to
bind and fasten the bulky immensity of the whole, which was both solid
and of a prodigious profundity, he then comes to declare how God, after
he had placed the water and the earth in the midst between the fire and
the air, incontinently closed up the heavens into a circular form. Out
of these materials, saith he, being four in number, was the body of the
world created, agreeing in proportion, and so amicably corresponding
together, that being thus embodied and confined within their proper
bounds, it is impossible that any dissolution should happen from their
own contending force, unless he that riveted the whole frame should
go about again to rend it in pieces;—most apparently teaching us,
that God was not the parent and architect of the corporeal substance
only, or of the bulk and matter, but of the beauty and symmetry and
similitude that adorned and graced the whole. The same we are to
believe, he thought, concerning the soul; that there is one which
neither was created by God nor is the soul of the world, but a certain
self-moving and restless efficacy of a giddy and disorderly agitation
and impetuosity, irrational and subject to opinion; while the other is
that which God himself, having accoutred and adorned it with suitable
numbers and proportions, has made queen regent of the created world,
herself the product of creation also.
10. Now that Plato had this belief concerning these things, and did
not for contemplation’s sake lay down these suppositions concerning
the creation of the world and the soul,—this, among many others, seems
to be an evident signification that, as to the soul, he avers it to be
both created and not created, but as to the world, he always maintains
that it had a beginning and was created, never that it was uncreated
and eternal. What necessity therefore of bringing any testimonies out
of Timaeus? For the whole treatise, from the beginning to the end,
discourses of nothing else but of the creation of the world. As for
the rest, we find that Timaeus, in his Atlantic, addressing himself
in prayer to the Deity, calls God that being which of old existed
in his works, but now was apparent to reason. In his Politicus, his
Parmenidean guest acknowledges that the world, which was the handiwork
of God, is replenished with several good things, and that, if there be
any thing in it which is vicious and offensive, it comes by mixture
of its former incongruous and irrational habit. But Socrates, in the
Politics, beginning to discourse of number, which some call by the name
of wedlock, says: “The created Divinity has a circular period, which
is, as it were, enchased and involved in a certain perfect number;”
meaning in that place by created Divinity no other than the world
itself.
* * * * *
11. The first pair of these numbers consists of one and two, the second
of three and four, the third of five and six; neither of which pairs
make a tetragonal number, either by themselves or joined with any other
figures. The fourth consists of seven and eight, which, being added all
together, produce a tetragonal number of thirty-six. But the quaternary
of numbers set down by Plato have a more perfect generation, of even
numbers multiplied by even distances, and of odd by uneven distances.
This quaternary contains the unit, the common original of all even and
odd numbers. Subsequent to which are two and three, the first plane
numbers; then four and nine, the first squares; and next eight and
twenty-seven, the first cubical numbers (not counting the unit). Whence
it is apparent, that his intention was not that the numbers should be
placed in a direct line one above another, but apart and oppositely one
against the other, the even by themselves, and the odd by themselves,
according to the scheme here given. In this manner similar numbers will
be joined together, which will produce other remarkable numbers, as
well by addition as by multiplication.
1 2
3 4
5 6
7 8
[Illustration:
1
/\
/ \
2 / 5 \ 3
/ \
4 / 13 \ 9
/ \
8 / 35 \ 27
- - - - - - - -
]
12. By addition thus: two and three make five, four and nine make
thirteen, eight and twenty-seven make thirty-five. Of all which numbers
the Pythagoreans called five the nourisher, that is to say, the
breeding or fostering sound, believing a fifth to be the first of all
the intervals of tones which could be sounded. But as for thirteen,
they called it the remainder, despairing, as Plato himself did, of
being ever able to divide a tone into equal parts. Then five and
thirty they named harmony, as consisting of the two cubes eight and
twenty-seven, the first that rise from an odd and an even number, as
also of the four numbers, six, eight, nine, and twelve, comprehending
both harmonical and arithmetical proportion. Which nevertheless will be
more conspicuous, being made out in a scheme to the eye.
[Illustration]
Admit a right-angled parallelogram, A B C D, the lesser side of which
A B consists of five, the longer side A C contains seven squares. Let
the lesser division be unequally divided into two and three squares,
marked by E; and the larger division in two unequal divisions more of
three and four squares, marked by F. Thus A E F G comprehends six, E B
G I nine, F G C H eight, and G I H D twelve. By this means the whole
parallelogram, containing thirty-five little square areas, comprehends
all the proportions of the first concords of music in the number of
these little squares. For six is exceeded by eight in a sesquiterce
proportion (3: 4), wherein the diatessaron is comprehended. And six is
exceeded by nine in a sesquialter proportion (2: 3), wherein also is
included the fifth. Six is exceeded by twelve in duple proportion (1:
2), containing the octave; and then lastly, there is the sesquioctave
proportion of a tone in eight to nine. And therefore they call that
number which comprehends all these proportions harmony. This number
is 35, which being multiplied by 6, the product is 210, which is the
number of days, they say, which brings those infants to perfection that
are born at the seventh month’s end.
13. To proceed by way of multiplication,—twice 3 make 6, and 4
times 9 thirty-six, and 8 times 27 produce 216. Thus six appears to
be a perfect number, as being equal in its parts; and it is called
matrimony, by reason of the mixture of the first even and odd. Moreover
it is composed of the original number, which is one, of the first
even number, which is two, and the first odd number, which is three.
Then for 36, it is the first number which is as well quadrangular as
triangular, being quadrangular from 6, and triangular from 8.[191] The
same number arises from the multiplication of the first two square
numbers, 4 and 9; as also from the addition of the three cubical
numbers, 1, 8, and 27, which being put together make up 36. Lastly,
you have a parallelogram with unequal sides, by the multiplication of
12 by 3, or 9 by 4. Take then the numbers of the sides of all these
figures, the 6 of the square, the 8 of the triangle, the 9 for the one
parallelogram, and the 12 for the other; and there you will find the
proportions of all the concords. For 12 to 9 will be a fourth, as nete
to paramese. To eight it will prove a fifth, as nete to mese. To six it
will be an octave, as nete to hypate. And the two hundred and sixteen
is the cubical number proceeding from six which is its root, and so
equal to its own perimeter.
14. Now these numbers aforesaid being endued with all these properties,
the last of them, which is 27, has this peculiar to itself, that it
is equal to all those that precede together; besides, that it is the
periodical number of the days wherein the moon finishes her monthly
course; the Pythagoreans make it to be the tone of all the harmonical
intervals. On the other side, they call thirteen the remainder, in
regard it misses a unit to be half of twenty-seven. Now that these
numbers comprehend the proportions of harmonical concord, is easily
made apparent. For the proportion of 2 to 1 is duple, which contains
the diapason; as the proportion of 3 to 2 sesquialter, which embraces
the fifth; and the proportion of 4 to 3 sesquiterce, which comprehends
the diatessaron; the proportion of 9 to 3 triple, including the
diapason and diapente; and that of 8 to 2 quadruple, comprehending the
double diapason. Lastly, there is the sesquioctave in 8 to 9, which
makes the interval of a single tone. If then the unit, which is common,
be counted as well to the even as the odd numbers, the whole series
will be equal to the sum of the decade. For the even numbers[192] (1
+ 2 + 4 + 8) give 15, the triangular number of five. On the other
side, take the odd numbers, 1, 3, 9, and 27, and the sum is 40; by
which numbers the skilful measure all musical intervals, of which they
call one a diesis, and the other a tone. Which number of 40 proceeds
from the force of the quaternary number by multiplication. For every
one of the first four numbers being by itself multiplied by four, the
products will be 4, 8, 12, 16, which being added all together make 40,
comprehending all the proportions of harmony. For 16 is a sesquiterce
to 12, duple to 8, and quadruple to 4. Again, 12 holds a sesquialter
proportion to 8, and triple to 4. In these proportions are contained
the intervals of the diatessaron, diapente, diapason, and double
diapason. Moreover, the number 40 is equal to the two first tetragons
and the two first cubes being taken both together. For the first
tetragons are 1 and 4, the first cubes are 8 and 27, which being added
together make 40. Whence it appears that the Platonic quaternary is
much more perfect and fuller of variety than the Pythagoric.
15. But since the numbers proposed did not afford space sufficient for
the middle intervals, therefore there was a necessity to allow larger
bounds for the proportions. And now we are to tell you what those
bounds and middle spaces are. And first, concerning the medieties (or
mean terms); of which that which equally exceeds and is exceeded by
the same number is called arithmetical; the other, which exceeds and
is exceeded by the same proportional part of the extremes, is called
sub-contrary. Now the extremes and the middle of an arithmetical
mediety are 6, 9, 12. For 9 exceeds 6 as it is exceeded by 12, that
is to say, by the number three. The extremes and middle of the
sub-contrary are 6, 8, 12, where 8 exceeds 6 by 2, and 12 exceeds 8
by 4; yet 2 is equally the third of 6, as 4 is the third of 12. So
that in the arithmetical mediety the middle exceeds and is exceeded
by the same number; but in the sub contrary mediety, the middle term
wants of one of the extremes, and exceeds the other by the same part
of each extreme; for in the first 3 is the third part of the mean; but
in the latter 4 and 2 are third parts each of a different extreme.
Whence it is called sub-contrary. This they also call harmonic, as
being that whose middle and extremes afford the first concords; that is
to say, between the highest and lowermost lies the diapason, between
the highest and the middle lies the diapente, and between the middle
and lowermost lies the fourth or diatessaron. For suppose the highest
extreme to be placed at nete and the lowermost at hypate, the middle
will fall upon mese, making a fifth to the uppermost extreme, but a
fourth to the lowermost. So that nete answers to 12, mese to 8, and
hypate to 6.
16. Now the more readily to find out these means Eudorus hath taught
us an easy method. For after you have proposed the extremities, if you
take the half part of each and add them together, the product shall be
the middle, alike in both duple and triple proportions, in arithmetical
mediety. But as for sub-contrary mediety, in duple proportion, first
having fixed the extremes, take the third part of the lesser and the
half of the larger extreme, and the addition of both together shall
be the middle; in triple proportion, the half of the lesser and the
third part of the larger extreme shall be the mean. As for example, in
triple proportion, let 6 be the least extreme, and 18 the biggest; if
you take 3 which is the half of 6, and 6 which is the third part of
18, the product by addition will be 9, exceeding and exceeded by the
same proportional parts of the extremes. In this manner the mediums are
found out; and these are so to be disposed and placed as to fill up the
duple and triple intervals. Now of these proposed numbers, some have no
middle space, others have not sufficient. Being therefore so augmented
that the same proportions may remain, they will afford sufficient space
for the aforesaid mediums. To which purpose, instead of a unit they
choose the six, as being the first number including in itself a half
and third part, and so multiplying all the figures below it and above
it by 6, they make sufficient room to receive the mediums, both in
double and triple distances, as in the example below:—
12 2 | | 3 18
24 4 | 6 | 9 54
48 8 | | 27 162
Now Plato laid down this for a position, that the intervals of
sesquialters, sesquiterces, and sesquioctaves having once arisen from
these connections in the first spaces, the Deity filled up all the
sesquiterce intervals with sesquioctaves, leaving a part of each, so
that the interval left of the part should bear the numerical proportion
of 256 to 243.[193] From these words of Plato they were constrained
to enlarge their numbers and make them bigger. Now there must be two
numbers following in order in sesquioctave proportion. But the six
does not contain a sesquioctave; and if it should be cut up into parts
and the units bruised into fractions, this would strangely perplex
the study of these things. Therefore the occasion itself advised
multiplication; so that, as in changes in the musical scale, the whole
scheme was extended in agreement with the first (or base) number.
Eudorus therefore, imitating Crantor, made choice of 384 for his
first number, being the product of 64 multiplied by 6; which way of
proceeding the number 64 led them to, having for its sesquioctave 72.
But it is more agreeable to the words of Plato to introduce the half of
384. For the remainder of that will bear a sesquioctave proportion in
those numbers which Plato mentions, 256 and 243, if we make use of 192
for the first number. But if the same number be made choice of doubled,
the remainder (or leimma) will have the same proportion, but the
numbers will be doubled, i.e. 512 and 486. For 256 is in sesquiterce
proportion to 192, as 512 to 384. Neither was Crantor’s reduction of
the proportions to this number without reason, which made his followers
willing to pursue it; in regard that 64 is both the square of the
first cube, and the cube of the first square; and being multiplied by
3, the first odd and trigonal, and the first perfect and sesquialter
number, it produces 192, which also has its sesquioctave, as we shall
demonstrate.
17. But first of all, we shall better understand what this leimma or
remainder is and what was the opinion of Plato, if we do but call
to mind what was frequently bandied in the Pythagorean schools. For
interval in music is all that space which is comprehended by two sounds
varied in pitch. Of which intervals, that which is called a tone is
the full excess of diapente above diatessaron; and this being divided
into two parts, according to the opinion of the musicians, makes two
intervals, both which they call a semitone. But the Pythagoreans,
despairing to divide a tone into equal parts, and therefore perceiving
the two divisions to be unequal, called the lesser leimma (or defect),
as being lesser than the half. Therefore some there are who make the
diatessaron, which is one of the concords, to consist of two tones and
a half; others, of two tones and leimma. In which case sense seems
to govern the musicians, and demonstration the mathematicians. The
proof by demonstration is thus made out. For it is certain from the
observation of instruments that the diapason has double proportion,
the diapente a sesquialter, the diatessaron a sesquiterce, and the
tone a sesquioctave proportion. Now the truth of this will easily
appear upon examination, by hanging two weights double in proportion
to two strings, or by making two pipes of equal hollowness double in
length, the one to the other. For the bigger of the pipes will yield
the deep sound, as hypate to nete; and of the two strings, that which
is extended by the double weight will be acuter than the other, as nete
to hypate; and this is a diapason. In the same manner two longitudes
or ponderosities, being taken in the proportion of 3: 2, will produce
a diapente; and three to four will yield a diatessaron; of which the
latter carries a sesquiterce, the former a sesquialter proportion.
But if the same inequality of weight or length be so ordered as nine
to eight, it will produce a tonic interval, no perfect concord, but
harmonical enough; in regard the strings being struck one after another
will yield so many musical and pleasing sounds, but all together a
dull and ungrateful noise. But if they are touched in consort, either
single or together, thence a delightful melody will charm the ear. Nor
is all this less demonstrable by reason. For in music, the diapason
is composed of the diapente and diatessaron. But in numbers, the
duple is compounded of the sesquialter and sesquiterce. For 12 is a
sesquiterce to 9, but a sesquialter to 8, and a duple to 6. Therefore
is the duple proportion composed of the sesquialter and sesquiterce,
as the diapason of the diapente and diatessaron. For here the diapente
exceeds the diatessaron by a tone; there the sesquialter exceeds
the sesquiterce by a sesquioctave. Whence it is apparent that the
diapason carries a double proportion, the diapente a sesquialter, the
diatessaron a sesquiterce, and the tone a sesquioctave.
18. This being thus demonstrated, let us see whether the sesquioctave
will admit a division into two equal parts; which if it will not do,
neither will a tone. However, in regard that 9 and 8, which make the
first sesquioctave, have no middle interval, but both being doubled,
the space that falls between causes two intervals, thence it is
apparent that, if those distances were equal, the sesquioctave also
might be divided into equal parts. Now the double of 9 is 18, that
of 8 is 16, the intermedium 17; by which means one of the intervals
becomes larger, the other lesser; for the first is that of 18 to 17,
the second that of 17 to 16. Thus the sesquioctave proportion not being
to be otherwise than unequally divided, consequently neither will the
tone admit of an equal division. So that neither of these two sections
of a divided tone is to be called a semitone, but according as the
mathematicians name it, the remainder. And this is that which Plato
means, when he says, that God, having filled up the sesquiterces with
sesquioctaves, left a part of each; of which the proportion is the same
as of 256 to 243. For admit a diatessaron in two numbers comprehending
sesquiterce proportion, that is to say, in 256 and 192; of which two
numbers, let the lesser 192 be applied to the lowermost extreme, and
the bigger number 256 to the uppermost extreme of the tetrachord.
Whence we shall demonstrate that, this space being filled up by two
sesquioctaves, such an interval remains as lies between the numbers 256
and 243. For the lower string being forced a full tone upward, which is
a sesquioctave, it makes 216; and being screwed another tone upward it
makes 243. Which 243 exceeds 216 by 27, and 216 exceeds 192 by 24. And
then again of these two numbers, 27 is the eighth of 216, and 24 the
eighth of 192. So the biggest of these two numbers is a sesquioctave
to the middle, and the middle to the least; and the distance from the
least to the biggest, that is from 192 to 243, consists of two tones
filled up with two sesquioctaves. Which being subtracted, the remaining
interval of the whole between 243 and 256 is 13, for which reason they
called this number the remainder. And thus I am apt to believe the
meaning and opinion of Plato to be most exactly explained in these
numbers.
19. Others, placing the two extremes of the diatessaron, the acute part
in 288, and the lower sound in 216, in all the rest observe the same
proportions, only that they take the remainder between the two middle
intervals. For the base, being forced up a whole tone, makes 243; and
the upper note, screwed downward a full tone, begets 256. Moreover 243
carries a sesquioctave proportion to 216, and 288 to 256; so that each
of the intervals contains a full tone, and the residue is that which
remains between 243 and 256, which is not a semitone, but something
less. For 288 exceeds 256 by 32, and 243 exceeds 216 by 27; but 256
exceeds 243 by 13. Now this excess is less than half of the former. So
it is plain that the diatessaron consists of two tones and the residue,
not of two tones and a half. Let this suffice for the demonstration of
these things. Nor is it a difficult thing to believe, by what has been
already said, wherefore Plato, after he had asserted that the intervals
of sesquialter, sesquiterce, and sesquioctave had arisen, when he comes
to fill up the intervals of sesquiterces with sesquioctaves, makes
not the least mention of sesquialters; for that the sesquialter is
soon filled up, by adding the sesquiterce to the sesquioctave, or the
sesquioctave to the sesquiterce.
20. Having therefore shown the manner how to fill up the intervals,
and to place and dispose the medieties, had never any person taken the
same pains before, I should have recommended the further consideration
of it to the recreation of your fancies; but in regard that several
most excellent musicians have made it their business to unfold these
mysteries with a diligence more than usually exact,—more especially
Crantor, Clearchus, and Theodorus, all born in Soli,—it shall suffice
only to show how these men differed among themselves. For Theodorus,
varying from the other two, and not observing two distinct files or
rows of numbers, but placing the duples and triples in a direct line
one before another, grounds himself upon that division of the substance
which Plato calls the division in length, making two parts (as it were)
out of one, not four out of two. Then he says, that the interposition
of the mediums ought to take place in that manner, to avoid the trouble
and confusion which must arise from transferring out of the first
duple into the first triple the intervals which are ordained for the
supplement of both.... But as for those who take Crantor’s part, they
so dispose their numbers as to place planes with planes, tetragons with
tetragons, cubes with cubes, opposite to one another, not taking them
in file, but alternatively odd to even. [Here is some great defect in
the original.]
21. ... Which, being in themselves permanently the same, afford the
form and species; but being subject to corporeal division, they become
the matter and subject to receive the other’s impression, the common
mixture being completed out of both. Now the indivisible substance,
which is always one and the same, is not to be thought to be incapable
of division by reason of its smallness, like the most minute of bodies,
called atoms. But as it is unmixed, and not to be any way affected,
but pure and altogether of one sort, it is said not to consist of
parts, but to be indivisible. By means of which purity, when it comes
in any manner whatsoever to approach and gently touch compounded
divisible and differing substances, all their variety ceases and they
crowd together into one habit by sympathy and similitude. If now any
one will call that substance which admits corporeal separation matter,
as a nature subject to the former and partaking of it, the use of
that equivocal term will nothing disadvantage our discourse. But they
are under a mistake that believe the corporeal to be blended with the
indivisible matter. First, for that Plato does not here make use of any
one of its names; whereas in other places he calls it the receptacle
and nurse, capable of both receiving and fostering the vast infinity
of created beings; not divisible among bodies, but rather the body
itself parted and divided into single individuals. Then again, what
difference would there be between the creation of the world and that
of the soul, if the composition of each proceeded from both matter and
the intelligible essence? Certainly Plato, as endeavoring to separate
the generation of the body from that of the soul, tells us that the
corporeal part was by God seated and deposited within it, and that
it was outwardly covered and enveloped by it; and after he had thus
wrought the soul to its perfection out of proportion, he then proceeds
to this argument concerning matter, of which he had no occasion to make
mention before when he was producing the soul, as being that which had
not its existence from matter.
22. The same may be said against the followers of Posidonius. For they
seem not altogether to separate the soul from matter; but imagining
the essence of limitations to be divisible in reference to bodies,
and intermixing it with the intelligible essence, they defined the
soul to be an idea (or essential form) of that which has extension in
every direction, subsisting in an harmonical proportion of numbers.
For (they say) all mathematical objects are disposed between the first
intelligible and sensible beings; and since the soul contains the
sempiternal nature of things intelligible and the pathetic nature of
things subjected to sense, it seems but rational that it should consist
of a substance between both. But they were ignorant that God, when
the soul was already brought to perfection, afterwards making use of
the limitations of bodies to form and shape the matter, confined and
environed the dissipated and fleeting substance within the compass of
certain surfaces composed of triangles adapted together. And it is
even more absurd to make the soul an idea. For the soul is always in
motion; the idea is incapable of motion; the one never to be mixed with
that which is subjected to sense, the other wrought into the substance
of the body. Moreover, God could be said only to imitate an idea, as
his pattern; but he was the artificer of the soul, as of a work of
perfection. Now enough has been already said to show that Plato does
not assert number to be the substance of the soul, only that it is
ordered and proportioned by number.
23. However this is a common argument against both the former
opinions, that neither in corporeal limits nor in numbers there is
the least footstep or appearance of that power by which the soul
assumes to itself to judge of what is subject to sense. For it was
the participation of the intelligible principle that endued it with
understanding and the perceiving faculty. But as for opinion, belief,
imagination, and its being affected with qualities relating to the
body, no man could ever dream that they proceeded simply either from
units, or lines, or surfaces. For not only the souls of mortals have
a power to judge of what is subject to sense; but the soul of the
world also, says Plato, “when it revolves upon itself, and happens
once to touch upon any fluid and roving substance or upon any thing
indivisible, then being moved throughout its whole self, it gives
notice with what this or that thing is identical, to what heterogeneal,
and in what relations especially and in what manner it happens to be
and to be affected towards each created thing.”[194] Here he gives at
the same time an intimation of the ten Categories or Predicaments; but
afterwards he gives us a clearer manifestation of these things. “For
when true reason,” says he, “is fixed upon what is subject to sense,
and the circle of the Other, observing a just and equal motion, conveys
its intelligence to the whole soul, then both opinion and belief
become steadfast and certain; on the other side, when it is settled
upon ratiocination, and the circle of the Same, turning readily and
easily, furnishes its intimations, then of necessity knowledge arrives
to perfection. And indeed, whoever shall affirm that any thing in
which these two operations take place is any thing besides a soul, may
deservedly be thought to speak any thing rather than the truth.”
From whence then does the soul enjoy this motion whereby it comprehends
what is subject to sense, different from that other intelligible motion
which ends in knowledge? This is a difficult task to resolve, unless
we steadfastly assert that Plato here did not compose the soul, so
singly considered, but the soul of the world also, of the parts above
mentioned,—of the more worthy indivisible substance, and of the less
worthy divisible in reference to bodies. And this soul of the world is
no other than that motion which gives heat and vigor to thought and
fancy, and sympathizes with what is subject to sense, not created, but
existing from eternity, like the other soul. For Nature, which had
the power of understanding, had also the power of opining. But the
intelligible power is subject neither to motion nor affection, being
established upon a substance that is still the same. The other is
movable and fleeting, as being engaged to an unstable, fluctuating, and
disunited matter. In regard the sensible substance was so far from any
order, that it was without shape and boundless. So that the power which
is fixed in this was capable of producing no clear and well-grounded
notions and no certain or well-ordered movements, but only sleepy
dreams and deliriums, which amuse and trouble corporeal stupidity;
unless by accident they lighted upon the more worthy substance. For
it was in the middle between the sensible and discerning faculty, and
had a nature conformable and agreeable to both; from the sensible
apprehending substance, and borrowing from judgment its power of
discerning things intelligible.
24. And this the express words of Plato declare. “For this is my
opinion,” saith he, “in short, that being, place, and generation were
three distinct things even before the heavens were created.”[195] By
place he means matter, as being the seat and receptacle; by being or
existence, the intelligible nature; and by generation, the world not
being yet created, he designs only that substance which was subject to
change and motion, disposed between the forming cause and the thing
formed, transmitting hither those shapes and figures which were there
contrived and moulded. For which reason it was called divisible;
there being a necessity of distributing sense to the sensitive, and
imagination to the imaginative faculty. For the sensitive motion, being
proper to the soul, directs itself to that which is outwardly sensible.
As for the understanding, it was fixed and immovable of itself, but
being settled in the soul and becoming its lord and governor, it turns
upon itself, and accomplishes a circular motion about that which is
always permanent, chiefly laboring to apply itself to the eternally
durable substance. With great difficulty therefore did they admit
a conjunction, till the divisible at length intermixing with the
indivisible, and the restlessly hurried with the sleepy and motionless,
constrained the Other to meet and join with the Same. Yet the Other was
not motion, as neither was the Same stability, but the principle of
distinction and diversity. For both the one and the other proceed from
a different principle; the Same from the unit, the Other from the duad;
and these were first intermixed with the soul, being fastened and bound
together by number, proportion, and harmonical mediums; so that the
Other being riveted into the Same begets diversity and disagreement;
and the Same being fermented into the Other produces order. And this
is apparent from the first powers of the soul, which are judgment and
motion. Motion immediately shows itself in the heavens, giving us an
example of diversity in identity by the circumvolution of the fixed
stars, and of identity in diversity by the order of the planets. For
in them the Same bears the chiefest sway; in terrestrial bodies,
the contrary principle. Judgment has two principles,—understanding
from the Same, to judge of things in general, and sense from the
Other, to judge of things in particular. Reason is a mixture of both,
becoming intellect in reference to things intelligible, and opinion
in things subject to sense; making use of the interdisposed organs
of imagination and memory, of which these in the Same produce the
Other, and those in the Other make the Same. For understanding is the
motion of the considerative faculty about that which is permanent and
stable. Opinion is a continuance of the perceptive faculty upon that
which is continually in motion. But as for fancy or imagination, being
a connection of opinion with sense, the Same has placed it in the
memory; and the Other moves it again in the difference between past and
present, touching at the same time upon diversity and identity.
25. But now let us take a draught of the corresponding composition
of the soul from the structure of the body of the universe. There we
find fire and earth, whose nature is such as not to admit of mixture
one with another but with great difficulty, or rather is altogether
obstinately refractory to mixture and constancy. God therefore, placing
air and water in the middle between both,—the air next the fire, the
water next the earth,—first of all tempered the middlemost one with
another, and next, by the assistance of these two, he brought the
two extreme elements not only to mix with the middlemost, but also
to a mutual closure or conjunction between themselves. Then he drew
together those contrary powers and opposing extremes, the Same and the
Other, not immediately, the one adjoining to the other, but placing
other substances between; the indivisible next the Same, and the
divisible next the Other, disposing each to each in convenient order,
and mixing the extremes with the middlemost. After which manner he
interweaved and tissued the whole into the form and composition of the
soul, completing, as far as it was possible, similitude out of things
different and various, and one out of many. Therefore it is alleged by
some, that Plato erroneously affirmed the nature of the Other to be an
enemy to mixture, as being not only capable to receive it, but a friend
of change. Whereas that should have been rather said of the nature of
the Same; which, being stable and an utter adversary to mutability, is
so far from an easy and willing condescension to mixture, that it flies
and abhors it, to the end it may preserve itself pure and free from
alteration. But they who make these objections against Plato betray
their own ignorance, not understanding that the Same is the idea (or
essential form) of those things that always continue in the same state
and condition, and that the Other is the idea of those things which are
subject to be variously affected; and that it is the peculiar nature
of the one to disjoin and separate into many parts whatever it happens
to lay hold upon, and of the other to cement and assimilate scattered
substances, till they resume one particular form and efficacy.
26. And these are the powers and virtues of the soul of the universe.
And when they once enter into the organs of corruptible bodies, being
themselves incorruptible, there the form of the binary and boundless
principle shows itself most briskly, while that of the unmixed and
purer principle lies as it were dormant in obscurity. And thus
it happens, that a man shall rarely observe any human passion or
motion of the understanding, void of reason, where there shall not
something appear either of desire or emulation, joy or grief. Several
philosophers therefore will have the passions to be so many sorts of
reasonings, seeing that desire, grief, and anger are all the effects
of judgment. Others allege the virtues themselves to be derived from
passions; fortitude depending on fear, temperance on voluptuousness,
and justice on love of gain. Now the soul being both speculative and
practical, contemplating as well generals as particulars, and seeming
to comprehend the one by the assistance of the intellect and the
other by the aid of sense, common reason, which encounters the Same
in the Other and the Other in the Same, endeavors by certain limits
and distinctions to separate one from many and the divisible from the
indivisible; but she cannot accomplish her design nor be purely in one
or the other, in regard the principles are so oddly interwoven and
intermixed and confusedly huddled together.
For this reason did God constitute a receptacle for the Same and the
Other, out of the indivisible and divisible substance, to the end there
might be order in variety. Now this was generation. For without this
the Same could have no variety, and therefore no motion or generation;
and the Other could have no order, and therefore no consistence or
generation. For should we grant the Same to be different from the
Other, and the Other to be the Same with itself, such a commixture
would produce nothing generative, but would want a third something,
like matter, to receive both and be disposed of by both. And this is
that matter which God first composed, when he bounded the movable
nature of bodies by the steadfastness of things intelligible.
27. Now then, as voice, merely voice, is only an insignificant and
brutish noise, but speech is the expression of the mind by significant
utterance; as harmony consists of sounds and intervals,—a sound being
always one and the same, and an interval being the difference and
diversity of sounds, while both being mixed together produce air and
melody;—thus the passive nature of the soul was without limits and
unstable, but afterwards became determinate, when limits were set and a
certain form was given to the divisible and manifold variety of motion.
Thus having comprised the Same and the Other, by the similitudes and
dissimilitudes of numbers which produce concord out of disagreement, it
becomes the life of the world, sober and prudent, harmony itself, and
reason overruling necessity mixed with persuasion. This necessity is by
most men called fate or destiny, by Empedocles friendship and discord,
by Heraclitus the opposite straining harmony of the world, as of a
bow or harp, by Parmenides light and darkness, by Anaxagoras mind and
infinity, by Zoroaster God and Daemon, naming one Oromasdes, the other
Arimanius. Though as for Euripides, he makes use of the disjunctive
erroneously for the copulative, where he says,
Jove, whether he be
Necessity, that Nature’s force controls,
Or the intelligence of human souls
For, indeed, the powers which bear dominion over the universe are
necessity and wisdom. This is that therefore which the Egyptians
intimate in their fables, feigning that, when Horus was punished and
dismembered, he bequeathed his spirit and blood to his father, but his
flesh and his fat to his mother. There is no part of the soul which
remains pure and unmixed, or separate from the rest; for, according
to the opinion of Heraclitus, “harmony latent is of greater value
than that which is visible,” as being that wherein the blending Deity
concealed and sunk all varieties and dissimilitudes. Nevertheless,
there appears in the irrational part a turbulent and boisterous
temerity; in the rational part, an orderly and well-marshalled
prudence; in the sensitive part, the constraint of necessity; but in
the understanding, entire and perfect command of itself. The limiting
and bounding power sympathizes with the whole and the indivisible,
by reason of the nearness of their relations; on the other side,
the dividing power fixes itself upon particulars, by virtue of the
divisible substance; and the whole rejoices at the mutation of the Same
by means of the Other, as occasion requires. In the like manner, the
various inclinations of men to virtue and vice, to pleasure and toil,
as also the enthusiasms and raptures of lovers, the combats of honor
with lustful desires, plainly demonstrate the mixture of the divine and
impassible with the moral and corporeal part; of which Plato himself
calls the one concupiscence of pleasures, natural to ourselves; the
other an opinion introduced from without, aspiring to the chiefest
good. For passible qualities of the soul arise from herself; but she
participates of understanding, as being infused from without, by the
more worthy principle.
28. Nor is the celestial nature privileged from this double society
and communion. For sometimes it is seen to incline one way or the
other, but it is set right again by the more powerful revolution of
the Same, and governs the world. Nay, there shall come a time, as it
has happened already, when the world’s moving wisdom shall grow dull
and drowsy, drowned in oblivion of its own duty; while that which
is familiar and agreeable to the body from the beginning draws and
winds back the right-hand motion of the universe, causing the wheels
to go slow and heavy. Yet shall it not be able to dash in pieces the
whole movement, for that the better part, rousing and recollecting
herself and observing the pattern and exemplar of God, shall with
his aid reduce all things again into their former order. Thus it is
demonstrable by many proofs, that the soul was not altogether the
workmanship of the Deity, but that having in itself a certain portion
of innate evil, it was by him digested and beautified who limited
infinity by unity, to the end it might be a substance within the
compass of certain limits; intermixing order and mutation, variety and
resemblance, by the force of the Same and the Other; and lastly working
into all these, as far as it was possible, a mutual community and
friendship by the assistance of numbers and harmony.
29. Concerning which things, although you have heard frequent
discourses, and have likewise read several arguments and disputes
committed to writing upon the same subjects, it will not be amiss for
me also to give a short account, after a brief repetition of Plato’s
own words. “God,” said he, “in the first place withdrew one part from
the whole; which done, he took away the double of that; then a third
part, sesquialter in proportion to the second, and triple to the first;
then a fourth part, double to the second; next a fifth part, being the
triple of the third; then a sixth, eight times the first; and lastly a
seventh, being twenty-seven times the first. This done, he filled up
the duple and triple intervals, retrenching also from thence certain
other particles, and placing them in the midst of those intervals; so
that in every interval there might be two medieties, the one exceeding
and being exceeded by one and the same part of the extremes, the other
exceeding and being exceeded by the same number. Now in regard that
from these connections in the first spaces there arose the intervals
of sesquialters, sesquiterces, and sesquioctaves, he filled up all the
sesquiterce intervals with sesquioctaves, leaving a part of each, so
that the interval left of the part might bear the numerical proportion
of 256 to 243.”[196]
Here the question will be first concerning the quantity, next
concerning the order, and in the third place concerning the force and
virtue of the numbers. As to the quantity, we are to consider which he
takes in the double and triple intervals. As to the order, whether they
are to be placed in one row, according to the direction of Theodorus,
or (as Crantor will have them) in the form of a _Λ_, placing the
unit at the top, and the duples and triples apart by themselves in two
several files. Lastly, we are to examine of what use and virtue they
are in the structure and composition of the soul.
30. As to the first, we shall relinquish the opinion of those who
affirm that it is enough, in proportions, to consider the nature of
the intervals, and of the medieties which fill up their vacancies; and
that the demonstration can be made out for any numbers whatsoever that
have spaces sufficient to receive the aforesaid proportions. For this
being granted, it makes the demonstration obscure, without the help of
schemes, and drives us from another theory, which carries with it a
delight not unbecoming philosophy.
1
2 3
4 9
8 27
Beginning therefore from the unit, let us place the duples and triples
apart; and there will be on the one side, 2, 4, 8; on the other 3, 9,
27;—seven numbers in all, proceeding forward by multiplication four
steps from the unit, which is assumed as the common base.... For not
only here, but upon other occasions, the sympathy of the quaternary
number with the septenary is apparent. There is this peculiar to that
tetractys or quaternary number thirty six, so much celebrated by the
Pythagoreans, which is more particularly worthy admiration,—that it is
composed of the first four even numbers and the first four odd numbers;
and it is the fourth connection made of numbers put together in order.
The first connection is of one and two; the second of odd numbers....
For placing the unit, which is common to both, before, he first takes
eight and then twenty-seven, as it were pointing out with the finger
where to place each particular sort.
[These places are so depraved in the original, that the sense is lost.]
But it belongs to others to explain these things more accurately and
distinctly; while we content ourselves with only what remains, as
peculiarly proper to the subject in hand.
31. For it was not out of vain-glory, to boast his skill in the
mathematical sciences, that Plato inserted in a treatise of natural
philosophy this discourse of harmonical and arithmetical medieties, but
believing them both apt and convenient to demonstrate the structure and
composition of the soul. For some there are who seek these proportions
in the swift motions of the spheres of the planets; others rather in
the distances, others in the magnitude of the stars; others, more
accurate and nice in their enquiry, seek for the same proportions in
the diameters of the epicycles; as if the Supreme Architect, for the
sake of these, had adapted the soul, divided into seven parts, to
the celestial bodies. Many also there are, who hither transfer the
inventions of the Pythagoreans, tripling the distances of bodies from
the middle. This is done by placing the unit next the fire; three next
the Antichthon, or earth which is opposite to our earth; nine next the
Earth; 27 next the Moon; 81 next to Mercury; 243 upon Venus; and 729
upon the Sun. The last (729) is both a tetragonal and cubical number,
whence it is, that they also call the sun a tetragon and a cube. By
this way of tripling they also reduce the other stars to proportion.
But these people may be thought to dote and to wander very much from
reason, if there be any use of geometrical demonstration, since by
their mistakes we find that the most probable proofs proceed from
thence; and although geometers do not always make out their positions
exactly, yet they approach the nearest to truth when they say that the
diameter of the sun, compared with the diameter of the earth, bears
the proportion of 12 to 1; while the diameter of the earth to that of
the moon carries a triple proportion. And for that which appears to be
the least of the fixed stars, the diameter of it is no less than the
third part of the diameter of the earth, and the whole globe of the
earth to the whole globe of the moon is as twenty-seven to one. The
diameters of Venus and the earth bear a duple, the globes or spheres
of both an octave proportion. The width of the shadow which causes an
eclipse holds a triple proportion to the diameter of the moon; and the
deviation of the moon from the middle of the signs, either to the one
or the other side, is a twelfth part. Her positions as to the sun,
either in triangular or quadrangular distances, give her the form when
she appears as in the first quarter and gibbous; but when she comes
to be quite round, that is, when she has run through half the signs,
she then makes (as it were) a kind of diapason harmony with six notes.
But in regard the motions of the sun are slowest when he arrives at
the solstices, and swiftest when he comes to the equinoxes, by which
he takes from the day or adds to the night, the proportion holds thus.
For the first thirty days after the winter solstice, he adds to the
day a sixth part of the length whereby the longest night exceeds
the shortest; the next thirty days he adds a third part; to all the
rest till the equinox he adds a half; and so by sextuple and triple
distances he makes even the irregularity of time.
Moreover, the Chaldaeans make the spring to hold the proportion of a
diatessaron to autumn; of a diapente to the winter, and of a diapason
to the summer. But if Euripides rightly divides the year, where he says,
Four months the parching heats of summer reign,
And four of hoary winter’s cold complain;
Two months doth vernal pride the fields array,
And two months more to autumn tribute pay,
then the seasons shall be said to change in octave proportion.
Others there are, who fancy the earth to be in the lowest string of the
harp, called proslambanomenos; and so proceeding, they place the moon
in hypate, Mercury and Venus in the diatoni and lichani; the sun they
likewise place in mese, as in the midst of the diapason, a fifth above
the earth and a fourth from the sphere of the fixed stars.
32. But neither doth this pleasant conceit of the latter come near the
truth, neither do the former attain perfect accuracy. However, they
who will not allow the latter to depend upon Plato’s sentiments will
yet grant the former to partake of musical proportions; so that, there
being five tetrachords, called ὑπάτων, μέσων, συνημμένων, διεζευγμένων,
and ὑπερβολαίων, in these five distances they place all the planets;
making the first tetrachord from the Moon to the Sun and the planets
which move with the Sun, that is, Mercury and Venus; the next from the
Sun to the fiery planet of Mars; the third between this and Jupiter;
the fourth from thence to Saturn; and the fifth from Saturn to the
sphere of the fixed stars. So that the sounds and notes which bound
the five tetrachords bear the same proportion with the intervals
of the planets. Still further, we know that the ancient musicians
had two notes called hypate, three called nete, one mese, and one
paramese, thus confining their scale to seven standing notes, equal
in number to the number of the planets. But the moderns, adding the
proslambanomenos, which is a full tone in descent from hypate, have
multiplied the scheme into the double diapason, and thereby confounded
the natural order of the concords; for the diapente happens to be
before the diatessaron, with the addition of the whole tone in the
bass. Whereas Plato makes his addition in the upper part; for in his
Republic[197] he says, that every one of the eight spheres rolls about
a Siren which is fixed upon each of the tuneful globes, and that they
all sing one counterpoint without diversity of modulation, taking
every one their peculiar concords, which together complete a melodious
consort.
These Sirens sing for their pleasure divine and heavenly tunes, and
accompany their sacred circuit and dance with an harmonious song of
eight notes. Nor was there necessity of a fuller chorus, in regard that
within the confines of eight notes lay the first bounds and limits of
all duple and triple proportions; the unit being added to both the even
and odd numbers. And certainly from hence it was that the ancients
raised their invention of nine Muses; of which eight were employed in
celestial affairs, as Plato said; the ninth was to take care of things
terrestrial, and to reduce and reform the inequality and confusion of
error and jarring variance.
33. Now then consider whether the soul does not roll and turn and
manage the heavens and the celestial bodies by means of those
harmonious concords and equal motions that are wrought and fermented
within her, being herself most wise and most just. And such she became
by virtue of harmonical proportions, whose images representing things
incorporeal are imprinted into the discernible and visible parts and
bodies of the world. But the chief and most predominating power is
visibly mixed in the soul, which renders her harmonious and obedient
to herself, the other parts unanimously yielding to her as the most
supreme and the divinest part of all. For the Sovereign Artificer and
Creator finding a strange disorder and erroneous confusion in the
motions of the decomposed and unruly soul, which was still at variance
with herself, some things he divided and separated, others he brought
together and reconciled to a mutual sympathy, making use of harmony and
numbers. By virtue of which, the slightest and meanest of insensible
substances, even stocks and stones, the rinds of trees, and sometimes
even the rennets of beasts, by various mixtures, compositions, and
temperatures, may become the charming objects of the sight, or afford
most pleasing perfumes and wholesome medicaments for the relief of
mankind, or be wrought and hollowed to send forth pleasing musical
sounds. And for this reason it was that Zeno of Citium encouraged and
persuaded youth to frequent the theatres, there to observe the variety
of melodious sounds that proceeded from horns or cornets, wooden
hautboys, flutes and reeds, or any other musical instruments to which
the contrivance of art had rightly applied the reason of number and
proportion. Not that we will here maintain, with the Pythagoreans, that
all things resemble number, for that requires a long discourse to prove
it. But where mutual society and sympathy arise out of discord and
dissimilitude, that the cause of this is moderation and order, produced
by the power of harmony and number, was a thing not concealed even from
the poets. And these give to what is friendly and kind the epithet
“evenly fitted;” while, on the other side, men of rugged and malicious
dispositions they called “unevenly tempered,” as if enmity and discord
were nothing but a sort of a disproportion. For this reason, he who
writes Pindar’s elegy gives him this encomium,