In the Protreptics this is the 37th Symbol; and Iamblichus has not
developed for us the more mystical signification of this symbol. For he
only says that “it admonishes us to beware of every thing which is
corruptive of our converse with the Gods and divine prophecy.” But
Aristotle appears to have assigned the true mystical reason why the
Pythagoreans abstained from beans. For he says, (apud Laert.) “that
Pythagoras considered beans as a symbol of generation [i. e. of the
whole of a visible and corporeal nature,] which subsists according to a
right line, and is without inflection; because a bean alone of almost
all spermatic plants, is perforated through the whole of it, and is not
obstructed by any intervening joints.” Hence he adds, “it resembles the
gates of Hades.” For these are perpetually open without any impediment
to souls descending into generation. The exhortation, therefore, to
abstain from beans, is equivalent to admonishing us to beware of a
continued and perpetual descent into the realms of generation. Hence the
true meaning of the following celebrated lines in Virgil;
——facilis descensus Averno.
Noctes atque dies patet atri janua Ditis:
Sed revocare gradum, superasque evadere ad auras,
Hoc opus, hic labor est.
i. e.
The gates of Hell are open night and day,
Smooth the descent, and easy is the way;
But to return, and view the cheerful skies,
In this, the mighty task and labor lies.
Dryden.
P. 98. _Such as infallible predictions of earthquakes, rapid expulsions
of pestilence, &c. &c._
Since Pythagoras, as Iamblichus informs us, p. 9. was initiated in all
the mysteries of Byblus and Tyre, in the sacred operations of the
Syrians, and in the mysteries of the Phœnicians, and also (p. 12.) that
he spent two and twenty years in the adyta of temples in Egypt,
associated with the Magi in Babylon, and was instructed by them in their
venerable knowledge;—it is not at all wonderful that he was skilled in
magic or theurgy, and was therefore able to perform things which surpass
_merely human power_, and which appear to be perfectly incredible to the
vulgar. For “magic,” (as we learn from Psellus in his MS. treatise on
Dæmons) “formed the last part of the sacerdotal science.” He farther
likewise informs us, “that magic investigates the nature, power, and
quality of every thing sublunary; viz. of the elements and their parts,
of animals, all-various plants, and their fruits, of stones, and herbs:
and in short, it explores the essence and power of every thing. From
hence, therefore, it produces its effects. And it forms statues which
procure health, makes all-various figures, and things which become the
instruments of disease. If asserts too, that eagles and dragons
contribute to health; but that cats, dogs, and crows, are symbols of
vigilance, to which therefore they contribute. But for the fashioning of
certain parts, wax and clay are used. Often, too, celestial fire is made
to appear through magic; and then statues laugh, and lamps are
spontaneously enkindled.” See the original in the Notes to my Pausanias,
p. 325. And that theurgy was employed by the ancients in their
mysteries, I have fully proved in my treatise On the Eleusinian and
Bacchic Mysteries.[100]
Conformably to this, Plato also in the First Alcibiades says, that the
magic of Zoroaster consisted in the worship of the Gods, on which
passage, I shall present the reader with what I have said, in the first
volume of my Plato, p. 63, as it will enable him to see that the theurgy
of the ancients is founded in a theory equally scientific and sublime.
“The following account of magic by Proclus, originally formed, as it
appears to me, a part of the Commentary written by him on the present
passage. For the MS. Commentary of Proclus, which is extant on this
dialogue, does not extend to more than a third part of it; and this
Dissertation on Magic, which is only extant in Latin, was published by
Ficinus the translator, immediately after his Excerpta from this
Commentary. So that it seems highly probable, that the manuscript from
which Ficinus translated his Excerpta, was much more perfect, than that
which has been preserved to us, in consequence of containing this
account of the magic of the ancients.
“In the same manner as lovers gradually advance from that beauty which
is apparent in sensible forms, to that which is divine; so the ancient
priests, when they considered that there is a certain alliance and
sympathy in natural things to each other, and of things manifest to
occult powers, and discovered that all things subsist in all, they
fabricated a sacred science from this mutual sympathy and similarity.
Thus they recognized things supreme in such as are subordinate, and the
subordinate in the supreme: in the celestial regions, terrene properties
subsisting in a causal and celestial manner; and in earth celestial
properties, but according to a terrene condition. For how shall we
account for those plants called heliotropes, that is, attendants on the
sun, moving in correspondence with the revolution of its orb, but
selenitropes, or attendants on the moon, turning in exact conformity to
her motion? It is because all things pray, and hymn the leaders of their
respective orders; but some intellectually, and others rationally; some
in a natural, and others after a sensible manner. Hence the sun-flower,
as far as it is able, moves in a circular dance towards the sun; so that
if any one could hear the pulsation made by its circuit in the air, he
would perceive something composed by a sound of this kind, in honor of
its king, such as a plant is capable of framing. Hence, too, we may
behold the sun and moon in the earth, but according to a terrene
quality; but in the celestial regions, all plants, and stones, and
animals, possessing an intellectual life according to a celestial
nature. Now the ancients, having contemplated this mutual sympathy of
things, applied for occult purposes, both celestial and terrene natures,
by means of which, through a certain similitude, they deduced divine
virtues into this inferior abode. For, indeed, similitude itself is a
sufficient cause of binding things together in union and consent. Thus,
if a piece of paper is heated, and afterwards placed near a lamp, though
it does not touch the fire, the paper will be suddenly inflamed, and the
flame will descend from the superior to the inferior parts. This heated
paper we may compare, to a certain relation of inferiors to superiors;
and its approximation to the lamp, to the opportune use of things
according to time, place, and matter. But the procession of fire into
the paper, aptly represents the presence of divine light, to that nature
which is capable of its reception. Lastly, the inflammation of the paper
may be compared to the deification of mortals, and to the illumination
of material natures, which are afterwards carried upwards like the
enkindled paper, from a certain participation of divine seed.
“Again, the lotus, before the rising of the sun, folds its leaves into
itself, but gradually expands them on its rising: unfolding them in
proportion to the sun’s ascent to the zenith; but as gradually
contracting them, as that luminary descends to the west. Hence this
plant, by the expansion and contraction of its leaves, appears no less
to honor the sun, than men by the gesture of their eye-lids, and the
motion of their lips. But this imitation and certain participation of
supernal light, is not only visible in plants, which possess nothing
more than a vestige of life, but likewise in particular stones. Thus the
sun-stone, by its golden rays, imitates those of the sun; but the stone
called the eye of heaven, or of the sun, has a figure similar to the
pupil of an eye, and a ray shines from the middle of the pupil. Thus too
the lunar stone, which has a figure similar to the moon when horned, by
a certain change of itself, follows the lunar motion. Lastly, the stone
called helioselenus, i. e. of the sun and moon, imitates, after a
manner, the congress of those luminaries, which it images by its color.
So that all things are full of divine natures; terrestrial natures
receiving the plenitude of such as are celestial, but celestial of
supercelestial essences;[101] while every order of things proceeds
gradually in a beautiful descent from the highest to the lowest. For
whatever particulars are collected into one above the order of things,
are afterwards dilated in descending, various souls being distributed
under their various ruling divinities.
“In the next place, there are many solar animals, such as lions and
cocks, which participate, according to their nature, of a certain solar
divinity; whence it is wonderful how much inferiors yield to superiors
in the same order, though they do not yield in magnitude and power.
Hence it is said, that a cock is very much feared, and as it were
reverenced, by a lion; the reason of which we cannot assign from matter
or sense, but from the contemplation alone of a supernal order. For thus
we shall find that the presence of the solar virtue accords more with a
cock than with a lion. This will be evident from considering that the
cock, as it were, with certain hymns, applauds and calls to the rising
sun, when he bends his course to us from the antipodes; and that solar
angels sometimes appear in forms of this kind, who though they are
without shape, yet present themselves to us who are connected with
shape, in some sensible form. Sometimes too there are dæmons with a
leonine front, who, when a cock is placed before them, unless they are
of a solar order, suddenly disappear; and this, because those natures
which have an inferior rank in the same order, always reverence their
superiors; just as many, on beholding the images of divine men, are
accustomed, from the very view, to be fearful of perpetrating any thing
base.
“In fine, some things turn round correspondent to the revolutions of the
sun, as the plants which we have mentioned, and others after a manner
imitate the solar rays, as the palm and the date; some the fiery nature
of the sun, as the laurel; and others a different property. For, indeed,
we may perceive that the properties which are collected in the sun, are
every where distributed to subsequent natures constituted in a solar
order; that is, to angels, dæmons, souls, animals, plants, and stones.
Hence the authors of the ancient priesthood discovered from things
apparent, the worship of superior powers, while they mingled some things
and purified others. They mingled many things indeed together, because
they saw that some simple substances possessed a divine property (though
not taken singly) sufficient to call down that particular power, of
which they were participants. Hence, by the mingling of many things
together, they attracted upon us a supernal influx; and by the
composition of one thing from many, they produced an assimilation to
that one which is above many; and composed statues from the mixture of
various substances conspiring in sympathy and consent. Besides this,
they collected composite odours, by a divine art, into one,
comprehending a multitude of powers, and symbolizing with the unity of a
divine essence; considering that division debilitates each of these, but
that mingling them together, restores them to the idea of their
exemplar.
“But sometimes one herb, or one stone, is sufficient to a divine
operation. Thus, a thistle is sufficient to procure the sudden
appearance of some superior power; but a laurel, raccinum, (or a thorny
kind of sprig) the land and sea onion, the coral, the diamond, and the
jasper, operate as a safeguard. The heart of a mole is subservient to
divination, but sulphur and marine water to purification. Hence, the
ancient priests, by the mutual relation and sympathy of things to each
other, collected their virtues into one, but expelled them by repugnancy
and antipathy; purifying when it was requisite with sulphur and bitumen,
and sprinkling with marine water. For sulphur purifies, from the
sharpness of its odour; but marine water, on account of its fiery
portion. Besides this, in the worship of the Gods, they offered animals,
and other substances congruous to their nature; and received, in the
first place, the powers of dæmons, as proximate to natural substances
and operations; and by these natural substances they convoked into their
presence those powers to which they approached. Afterwards, they
proceeded from dæmons to the powers and energies of the Gods; partly,
indeed, from dæmoniacal instruction, but partly by their own industry,
interpreting convenient symbols, and ascending to a proper intelligence
of the Gods. And lastly, laying aside natural substances and their
operations, they received themselves into the communion and fellowship
of the Gods.”
It will doubtless be objected by most of the present period, who believe
in nothing beyond the information of their senses, that plants, animals,
and stones, no longer possess those wonderful sympathetic powers, which
are mentioned by Proclus in the above extract. In answer to any such
objector, whose _little_ soul, (in the language of the Emperor Julian)
is indeed acute, but sees nothing with a vision healthy and sound, it
must be said, that this is not at all wonderful at a period, when, as
the author of the Asclepian dialogue justly observes, “there is a
lamentable departure of divinity from man, when nothing worthy of
heaven, or celestial concerns, is heard or believed, and when every
divine voice is by a _necessary_ silence dumb.”[102] But to the
philosophic reader, it must be observed, that as in the realms of
generation, or in other words, the sublunary region, wholes, viz. the
spheres of the different elements, remain perpetually according to
nature; but their parts are sometimes according, and sometimes contrary
to nature; this must also be true of the parts of the earth. When those
circulations therefore take place, during which the parts of the earth
subsist according to nature, and which are justly called, by Plato,
fertile periods, the powers of plants, animals, and stones, magically
sympathize with superior natures, in consequence of a more abundant
participation of them, through a greater degree of aptitude to receive,
and alliance to the participated powers. But during those circulations,
in which the parts of the earth subsist contrary to nature, as at
present, and which Plato calls barren periods, the powers of plants,
animals, and stones, no longer possess a magic sympathy, and
consequently are no longer capable of producing magical operations.
P. 106. _The eternal essence of number is the most providential
principle of the universe_, &c.
The following account of the manner in which the Pythagoreans
philosophized about numbers, is extracted from my Theoretic Arithmetic,
and the information contained in it is principally derived from the
great Syrianus.
“The Pythagoreans, turning from the vulgar paths, and delivering their
philosophy in secret to those alone who were worthy to receive it,
exhibited it to others through mathematical names. Hence, they called
forms, numbers, as things which are the first separated from impartible
union; for the natures which are above forms, are also above
separation.[103] The all-perfect multitude of forms, therefore, they
obscurely signified through the duad; but they indicated the first
formal principles by the monad and duad, as not being numbers; and also
by the first triad and tetrad, as being the first numbers, the one being
odd, and the other even, from which by addition the decad is generated;
for the sum of 1, 2, 3, and 4, is ten. But after numbers, in secondary
and multifarious lives, introducing geometrical prior to physical
magnitudes; these also they referred to numbers, as to formal causes and
the principles of these; referring the point indeed, as being
impartible, to the monad; but a line, as the first interval, to the
duad; and again, a superficies, as having a more abundant interval, to
the triad; and a solid to the tetrad. They also called, as is evident
from the testimony of Aristotle, the first length the duad; for it is
not simply length, but the _first_ length, in order that by this they
might signify _cause_. In a similar manner also, they denominated the
_first_ breadth, the triad; and the _first_ depth the tetrad. They also
referred to formal principles all psychical knowledge. And intellectual
knowledge indeed, as being contracted according to impartible union,
they referred to the monad; but scientific knowledge, as being evolved,
and as proceeding from cause to the thing caused, yet through the
inerratic, and always through the same things, they referred to the
duad; and opinion to the triad, because the power of it is not always
directed to the same thing, but at one time inclines to the true, and at
another to the false. And they referred sense to the tetrad, because it
has an apprehension of bodies; for in the duad, indeed, there is one
interval from one monad to the other; but in the triad there are two
intervals from any one monad to the rest; and in the tetrad there are
three. They referred, therefore, to principles every thing knowable,
viz. beings, and the gnostic powers of these. But they divided beings
not according to breadth, but according to depth; into intelligibles,
objects of science, objects of opinion, and sensibles. In a similar
manner, also, they divided knowledge into intellect, science, opinion,
and sense. The extremity, therefore, of the intelligible triad, or
animal itself, as it is called by Plato in the Timæus, is assumed from
the division of the objects of knowledge, manifesting the intelligible
order, in which forms themselves, viz. the first forms and the
principles of these, are contained, viz. the idea of the one itself, of
the first length, which is the duad itself, and also the ideas of the
first breadth and the first depth; (for in common the term _first_ is
adapted to all of them), viz. to the triad itself, and the tetrad
itself.
“Again, the Pythagoreans and Plato did not denominate idea from one
thing, and ideal number from another. But since the assertion is
eminently true, that all things are similar to number, it is evident
that number, and especially every ideal number, was denominated on
account of its paradigmatic peculiarity. If any one, however, wishes to
apprehend this from the appellation itself, it is easy to infer that
idea was so called, from rendering as it were its participants similar
to itself, and imparting to them _form_, _order_, _beauty_, and _unity_;
and this in consequence of always preserving the same form, expanding
its own power to the infinity of particulars, and investing with the
same species its eternal participants. _Number_ also, since it imparts
proportion and elegant arrangement to all things, was allotted this
appellation. For the ancients, says Syrianus,[104] call to _adapt_ or
_compose_ αρσαι _arsai_, whence is derived αριθμος _arithmos number_.
Hence αναρσιον anarsion among the Greeks signifies _incomposite_. Hence
too, those Grecian sayings, _you will adapt the balance_, _they placed
number together with them_, and also _number and friendship_. From all
which number was called by the Greeks _arithmos_, as that which measures
and orderly arranges all things, and unites them in amicable league.
“Farther still, some of the Pythagoreans discoursed about inseparable
numbers alone, i. e. numbers which are inseparable from mundane natures,
but others about such as have a subsistence separate from the universe,
in which as paradigms they saw those numbers are contained, which are
perfected by nature. But others, making a distinction between the two,
unfolded their doctrine in a more clear and perfect manner. If it be
requisite, however, to speak concerning the difference of these monads,
and their privation of difference, we must say that the monads which
subsist in quantity, are by no means to be extended to essential
numbers; but when we call essential numbers monads, we must assert that
all of them mutually differ from each other by _difference_ itself, and
that they possess a privation of difference from _sameness_. It is
evident also, that those which are in the same order, are contained
through mutual comparison, in _sameness_ rather than in difference, but
that those which are in different orders are conversant with much
diversity, through the dominion of _difference_.
“Again, the Pythagoreans asserted that nature produces sensibles by
numbers; but then these numbers were not mathematical but physical; and
as they spoke symbolically, it is not improbable that they demonstrated
every property of sensibles by mathematical names. However, says
Syrianus, to ascribe to them a knowledge of sensible numbers alone, is
not only ridiculous, but highly impious. For they received indeed, from
the theology of Orpheus, the principles of intelligible and intellectual
numbers, they assigned them an abundant progression, and extended their
dominion as far as to sensibles themselves.”
Again, their conceptions about mathematical and physical number, were as
follow:
“As in every thing, according to the doctrine of Aristotle, one thing
corresponds to matter, and another to form, in any number, as for
instance the pentad, its five monads, and in short its quantity, and the
number which is the subject of participation, are derived from the duad
itself; but its form, i. e. the pentad itself, is from the monad; for
every form is a monad, and unites its subject quantity. The pentad
itself, therefore, which is a monad, proceeds from the principal monad,
forms its subject quantity, which is itself formless, and connects it to
its own form. For there are two principles of mathematical numbers in
our souls: the monad, which comprehends in itself all the forms of
numbers, and corresponds to the monad in intellectual natures; and the
duad, which is a certain generative principle of infinite power, and
which on this account, as being the image of the never-failing and
intelligible duad, is called indefinite. While this proceeds to all
things, it is not deserted in its course by the monad, but that which
proceeds from the monad continually distinguishes and forms boundless
quantity, gives a specific distinction to all its orderly progressions,
and incessantly adorns them with forms. And as in mundane natures, there
is neither any thing formless, nor any vacuum among the species of
things, so likewise in mathematical number, neither is any quantity left
innumerable; for thus the forming power of the monad would be vanquished
by the indefinite duad, nor does any medium intervene between the
consequent numbers, and the well-disposed energy of the monad.
“Neither, therefore, does the pentad consist of substance and accident,
as a white man; nor of genus and difference, as man of animal and biped;
nor of five monads mutually touching each other, like a bundle of wood;
nor of things mingled, like a drink made from wine and honey; nor of
things sustaining position, as stones by their position complete the
house; nor lastly, as things numerable, for these are nothing else than
particulars. But it does not follow that numbers themselves, because
they consist of indivisible monads, have nothing else besides monads,
(for the multitude of points in continued quantity is an indivisible
multitude, yet it is not on this account that there is a completion of
something else from the points themselves); but this takes place because
there is something in them which corresponds to matter, and something
which corresponds to form. Lastly, when we unite the triad with the
tetrad, we say that we make seven. The assertion, however, is not true:
for monads conjoined with monads, produce indeed the subject of the
number 7, but nothing more. Who then imparts the heptadic form to these
monads? Who is it also that gives the form of a bed to a certain number
of pieces of wood? Shall we not say that the soul of the carpenter, from
the art which he possesses, fashions the wood, so as to receive the form
of a bed, and that the numerative soul, from possessing in herself a
monad which has the relation of a principle, gives form and subsistence
to all numbers? But in this only consists the difference, that the
carpenter’s art is not naturally inherent in us, and requires manual
operation, because it is conversant with sensible matter; but the
numerative art is naturally present with us, and is therefore possessed
by all men, and has an intellectual matter which it instantaneously
invests with form. And this is that which deceives the multitude, who
think that the heptad is nothing besides seven monads. For the
imagination of the vulgar, unless it first sees a thing unadorned,
afterwards the supervening energy of the adorner, and lastly, above all
the thing itself, perfect and formed, cannot be persuaded that it has
two natures, one formless, the other formal, and still further, that
which beyond these imparts form; but asserts, that the subject is one,
and without generation. Hence, perhaps, the ancient theologists and
Plato ascribed temporal generations to things without generation, and to
things which are perpetually adorned, and regularly disposed, privation
of order and ornament, the erroneous and the boundless, that they might
lead men to the knowledge of a formal and effective cause. It is,
therefore, by no means wonderful, that though seven sensible monads are
never without the heptad, these should be distinguished by science, and
that the former should have the relation of a subject, and be analogous
to matter, but the latter should correspond to species and form.
“Again, as when water is changed into air, the water does not become
air, or the subject of air, but that which was the subject of water
becomes the subject of air, so when one number unites itself with
another, as for instance the triad with the duad, the species or forms
of the two numbers are not mingled, except in their immaterial reasons
(or productive principles), in which at the same time that they are
separate, they are not impeded from being united, but the quantities of
the two numbers which are placed together, become the subject of the
pentad. The triad, therefore, is one, and also the tetrad, even in
mathematical numbers: for though in the ennead or number nine, you may
conceive a first, second, and third triad, yet you see one thing thrice
assumed; and in short, in the ennead there is nothing but the form of
the ennead in the quantity of nine monads. But if you mentally separate
its subject, (for form is impartible) you will immediately invest it
with forms corresponding to its division; for our soul cannot endure to
see that which is formless, unadorned, especially as she possesses the
power of investing it with ornament.
“Since also separate numbers possess a demiurgic or fabricative power,
which mathematical numbers imitate, the sensible world likewise contains
images of those numbers by which it is adorned; so that all things are
in all, but in an appropriate manner in each. The sensible world,
therefore, subsists from immaterial and energetic reasons, and from more
ancient causes. But those who do not admit that nature herself is full
of productive powers, lest they should be obliged to double things
themselves, these wonder how from things void of magnitude and gravity,
magnitude and gravity are composed; though they are never composed from
things of this kind which are void of gravity and magnitude, as from
parts. But magnitude is generated from essentially impartible elements;
since form and matter are the elements of bodies; and still much more is
it generated from those truer causes which are considered in demiurgic
reasons and forms. Is it not therefore necessary that all dimensions,
and all moving masses, must from these receive their generation? For
either bodies are unbegotten, like incorporeal natures; or of things
with interval, things without interval are the causes; of partibles
impartibles; and of sensibles and contraries, things insensible and void
of contact: and we must assent to those who assert that things
possessing magnitude are thus generated from impartibles. Hence the
Pythagorean Eurytus, and his followers, beholding the images of things
themselves in numbers, rightly attributed certain numbers to certain
things, according to their peculiarity. In consequence of this, he said
that a particular number is the boundary of this plant, and again,
another number of this animal; just as of a triangle 6 is the boundary,
of a square 9, and of a cube 8. As the musician, too, harmonizes his
lyre through mathematical numbers, so nature through her own natural
numbers, orderly arranges, and modulates her productions.
“Indeed, that numbers are participated by the heavens, and that there is
a solar number, and also a lunar number, is manifest according to the
adage, even to the blind. For the restitutions of the heavenly bodies to
their pristine state (αποκαταστασεις) would not always be effected
through the same things, and in the same manner, unless one and the same
number bad dominion in each. Yet all these contribute to the procession
of the celestial spheres, and are contained by their perfect number. But
there is also a certain natural number belonging to every animal. For
things of the same species would not be distinguished by organs after
the same manner, nor would they arrive at puberty and old age about the
same time, or generate, nor would the fœtus be nourished or increase,
according to regular periods, unless they were detained by the same
measure of nature. According to the best of the Pythagoreans also, Plato
himself, number is the cause of better and worse generations. Hence
though the Pythagoreans sometimes speak of the squares and cubes of
natural numbers, they do not make them to be monadic, such as the number
9, and the number 27; but they signify through these names, from
similitude, the progression of natural numbers into, and dominion about,
generations. In like manner, though they call them equal or double, they
exhibit the dominion and symphony of ideas in these numbers. Hence
different things do not use the same number, so far as they are
different, nor do the same things use a different number, so far as they
are the same.
“In short, physical numbers are material forms divided about the subject
which receives them. But material powers are the sources of connexion
and modification to bodies. For form is one thing, and the power
proceeding from it another. For form itself is indeed impartible and
essential; but being extended, and becoming bulky, it emits from itself,
as if it were a blast, material powers which are certain qualities.
Thus, for instance, in fire, the form and essence of it is impartible,
and is truly the image of the cause of fire: for in partible natures,
the impartible has a subsistence. But from form which is impartible in
fire, and which subsists in it as number, an extension of it accompanied
with interval takes place about matter, from which the powers of fire
are emitted, such as heat, or refrigeration, or moisture, or something
else of the like kind. And these qualities are indeed essential, but are
by no means the essence of fire. For essences do not proceed from
qualities, nor are essence and power the same thing. But the essential
every where precedes power. And from this being one the multitude of
powers proceeds, and the distributed from that which is undistributed;
just as many energies are the progeny of one power.”
P. 107. _For Pythagoras always proclaimed, that nothing admirable
pertaining to the Gods or divine dogmas, should be disbelieved._
This in the Protreptics forms the fourth symbol, and is thus explained
by Iamblichus:—“This dogma sufficiently venerates and unfolds the
transcendency of the Gods, affording us a viaticum, and recalling to our
memory that we ought not to estimate divine power from our judgment. But
it is likely that some things should appear difficult and impossible to
us, in consequence of our corporeal subsistence, and from our being
conversant with generation and corruption; from our having a momentary
existence; from being subject to a variety of diseases; from the
smallness of our habitation; from our gravitating tendency to the
middle; from our somnolency, indigence and repletion; from our want of
counsel and our imbecility; from the impediments of our soul, and a
variety of other circumstances, although our nature possesses many
illustrious prerogatives. At the same time however we perfectly fall
short of the Gods, and neither possess the same power with them, nor
equal virtue. This symbol therefore in a particular manner introduces
the knowledge of the Gods, as beings who are able to effect all things.
On this account it exhorts us to disbelieve nothing concerning the Gods.
It also adds, nor about divine dogmas; viz. those belonging to the
Pythagoric philosophy. For these being secured by disciplines and
scientific theory, are alone true and free from falsehood, being
corroborated by all-various demonstration, accompanied with necessity.
The same symbol, also, is capable of exhorting us to the science
concerning the Gods: for it urges us to acquire a science of that kind,
through which we shall be in no respect deficient in things asserted
about the Gods. It is also able to exhort the same things concerning
divine dogmas, and a disciplinative progression. For disciplines alone
give eyes to, and produce light about, all things, in him who intends to
consider and survey them. For from the participation of disciplines, one
thing before all others is effected, viz. a belief in the nature,
essence, and power of the Gods, and also in those Pythagoric dogmas,
which appear to be prodigious to such as have not been introduced to,
and are uninitiated in, disciplines; So that the precept _disbelieve
not_ is equivalent to _participate_ and _acquire_ those things through
which you will not disbelieve; that is to say, acquire disciplines and
scientific demonstrations.”
P. 88. _After this manner therefore it is said that music was discovered
by Pythagoras._
The following particulars relative to music are added for the purpose of
elucidating what is said about it in this chapter.
“Take two brazen chords, such as are used in harps; for those chords
which are made from the intestines of sheep are for the most part either
false or obnoxious to the change of the air,
[Illustration: A—————————B
C—————————D
|
E]
“Let these chords be perfectly equal, and equally stretched, so as to be
in unison, i. e. so that there may be only one sound, though there are
two strings. But it is requisite that they should be placed upon some
oblong and polished rule. The ancients called this rule an harmonic
rule, or also a monochord, by which instrument all consonances and
dissonances, and likewise musical intervals, were tried. Let now one of
these chords be bisected in E. Afterwards under the point E place what
is vulgarly called the _tactus_, but which was denominated by the
ancients, from its figure, a hemisphere. The tactus, therefore, being
placed under E, press there the chord, so that one half of it only, as
for instance ED, may be wholly struck and resound. Having therefore
struck each of the chords at the same time, viz. the whole of AB, and
the half ED, so that they may resound at one and the same time, you will
hear the sweetest of all consonances, composed from the sound of the
whole chord AB, and the sound of the half ED. This consonance the
ancients called diapason, i. e. _through all_ [the chords], because in
the musical instruments of the ancients, the two extreme chords, i. e.
the most grave, and the most acute of all the chords, contained this
consonance; so that, from the gravest chord having made a transition
through all the chords to the supreme and most acute of all, they would
hear this sweetest consonance. It was, likewise, said to be in a duple
ratio of the proportion of one sound to the other. For the sound of the
chord AB is doubly greater or more grave than the sound of the half ED.
For as sounding bodies are to each other, so are their sounds. But the
chord AB is the double of ED. This, however, is now commonly called the
octave, because from the first sound, and that the gravest, which is
called _ut_, as far as to that sound which corresponds to it in the
consonance diapason, there are these eight sounds, _ut_, _re_, _mi_,
_fa_, _sol_, _re_, _mi_, _fa_. And of these the first ut, and the last
_fa_, which is the eighth, produce the consonance diapason, or the
double, or the octave.
“Again, let the same chord CD be divided into three equal parts in the
points F, G.
[Illustration: A————————————B
C————————————D
| |
F G]
“FD, therefore, will be two-thirds as well of the whole CD as of the
whole AB. Let the tactus now be placed in F, and let AB and FD be struck
at the same time, and a consonance very sweet and perfect will indeed be
heard, yet not so sweet as the diapason. This the ancients called
diapente (i. e. through five chords), because the first and the fifth
chord produce this consonance. But according to proportion it is called
sesquialter, because the chord AB is sesquialter to FD, and consequently
the sounds of these chords also are in the same ratio. But sesquialter
ratio is when the greater quantity AB contains the less FD once, and the
half of it besides. It is, indeed, commonly called the fifth, because it
is composed from the first sound _ut_, and the fifth, _sol_.
“Again, let the same chord be cut into four equal parts in the points H,
E, I,
[Illustration: A——————————————————————————B
C——————————————————————————D
| | | | | | | | |
K L H F M N E G I]
“so that the chord HD, may be three-fourths of the whole CD. The tactus,
therefore, being placed in H, let AB and HD be struck at one and the
same time, and a consonance will be heard, indeed, yet more imperfect
than the preceding two. This was called by the ancients diatessaron, i.
e. through four chords or sounds, for a similar reason to that by which
the former were denominated. With reference, however, to the ratio of
the chords and sounds, it is called sesquitertian, because the greater
AB contains the less once, and a third part of it besides. But it is now
commonly called a fourth, because it is found between the first sound
_ut_, and the fourth _fa_. If now the point F be added in the preceding
figure, and at one and the same time two chords HD and FD are compared
in arithmetical ratios, we shall find that the greater HD will have to
the less FD a sesquioctave[105] ratio, and the sound of the greater HD
to the less FD will have the same ratio, i. e. in modern terms, that
between _fa_ and _sol_ there is a sesquioctave ratio. But if these two
sounds are heard together, they will be discordant to the ear. Again,
the distance between these sounds _fa_, _sol_, or between the chords HD
and FD, or between the two harmonic intervals HD and FD, the ratio of
which was sesquioctave, was called by the ancients a tone. Afterwards
they divided the whole of CD into nine equal parts, the first of which
is divided in K, so that the whole CD may have to the remainder KD,
which contains eight of those parts, a sesquioctave ratio. This, in like
manner, will be the interval of a tone, the first sound of which, i. e,
of the whole CD, is now called _ut_, but the second sound of the rest of
the chord KD is called _re_. Afterwards they in a similar manner divided
the remainder KD into nine parts, the first part of which is marked in
the point L. And for the same reason between the chord KD and the chord
KD, and their sounds, there will be a sesquioctave ratio. The sound of
the chord LD is now called _mi_; but the interval which remains between
the chord LD and the chord HD has not a sesquioctave ratio, but less
than it almost by half, and therefore an interval of this kind was
called a semitone, and also diesis or a division. But that interval
which remains between the points F and E they divided after the same
manner, as the space between C and H was divided, and they again found
the same sounds. Let those divisions be marked by the points M and N;
and here, also, between N and E, or between _mi_ and _fa_, there is in
like manner another semitone. These eight sounds, therefore, are _ut_,
_re_, _mi_, _fa_, _sol_, _re_, _mi_, _fa_, which compose the whole
diapason. For as we have before observed, between _ut_ and the last _fa_
is the consonance diapason, or between the chord CD or AB, and the chord
ED. But from the intervals which are between the sounds there are two
semitones, viz. one between _mi_ and _fa_, denoted by the letters L, N,
and the other between the last _mi_ and _fa_, denoted by the letters N,
E. The remaining five intervals are entire tones. It must, also, be
observed, that from _ut_ to the first _sol_ is the consonance diapente,
which contains three tonic intervals, and one semitone; nevertheless in
all there are five sounds, _ut_, _re_, _mi_, _fa_, _sol_.
“Again, from _sol_ to the last _fa_ there are four sounds, _sol_, _re_,
_mi_, _fa_, which are perfectly similar to the first four, _ut_, _re_,
_mi_, _fa_. Nevertheless these are more grave, but those are more acute.
And as from _ut_ to the first _fa_ is the diatessaron, so likewise from
_sol_ to the last _fa_ is another diatessaron, from which, in the last
place, it must be observed, it follows that the two consonances
diatessaron and diapente constitute the whole diapason; or that the
diapason is divided into one diatessaron, and one diapente. For from
_ut_ to _sol_ is the diapente, but from _sol_ to the last _fa_ is the
diatessaron. This will also be the case if we should say that from _ut_
to the first _fa_ is the diatessaron, as is evident from the division of
the chord; but from the first _fa_ to the last _fa_ is the diapente, as
is evident from the four intervals of the chord, three of which are
tones, and the remaining interval is a semitone, which also in the other
diapente were contained between _ut_ and _sol_.
“Now again, let the tactus be placed in I; but I is the fourth part of
the whole CD. Let, also, AB and ID be struck at one and the same time,
and the sweetest consonance, called bisdiapason, will be produced; which
is so denominated, because it is composed from two diapasons, of which
the first is between AB or CD, and ED, but the second is between ED and
ID; for the ratio of these is double as well as of those. The ratio,
also, of the bisdiapason is quadruple, as is evident from the division;
and is commonly called a fifteenth, because from the first _ut_ to this
sound, which is also denominated _fa_, there would be fifteen sounds, if
the interval EI were divided after the same manner as the first CE is
divided.
“Farther still, let GD be a third part of the whole CD, and let the
tactus be placed in G. Then at one and the same time let AB and GD be
struck, and a sweet consonance will be heard, which is called
diapasondiapente, because it is composed from one diapason contained by
the interval CE, or the two chords CD, ED, and one diapente, contained
by the interval EG, or the chords ED, GD. For the chord ED is
sesquialter to the chord GD; which ratio constitutes the nature of the
diapente. The proportion, also, of this consonance is triple. For the
chord AB or CD is triple of GD; and it is commonly called the twelfth,
because between _ut_ and _sol_, denoted by the letter G, there would be
twelve sounds, if the interval EG received its divisions. From all which
it is manifest by the experience of the ear, that there are altogether
five consonances, three simple, the diapason, the diapente, and the
diatessaron; but two composite, the bisdiapason, and the
diapasondiapente.”
In the last place, it is necessary to observe that those ancient Greeks
differently denominated these sounds, _ut_, _re_, &c. For the first,
i. e. the gravest sound or chord, which is now called _ut_, they,
denominated hypate, and the others in the following order:
Ut, Hypate, i. e. Principalis.
Re, Parhypate, — Postprincipalis.
Mi, Lychanos, — Index.
Fa, Mese, — Media.
Sol, Paramese, — Postmedia.
Re, Trite, — Tertia.
Mi, Paranete, — Antepenultima.
Fa, Nete, — Ultima, vel suprema.