Ut, Hypate, i.e. Principalis. Re, Parhypate, — Postprincipalis. Mi, lLychanos, — Index.
Fa, Mese, — Media.
Sol, Paramese, — Postmedia.
Re, Trite, — Tertia.
Mi, Paranete, — Antepenultima.
Fa, Nete, — Ultima, vel suprema.
P. 80. I swear by him who the tetractys found.
The tetrad was called by the Pythagoreans every
number, because it comprehends in itself all the
numbers as far as to the decad, and the decad itself;
for the sum of I, 2, 3, and 4, is 10. Hence both the
decad and the tetrad were said by them to be every
number; the decad indeed in energy, but the tetrad
ncapacity. The sum likewise of these four numbers
was said by them to constitute the tetractys, in which
all harmonic ratios are included. For 4 to 1, which is
a quadruple ratio, forms the symphony bisdiapason;
the ratio of 3 to 2, which is sesquialter, forms the
symphony diapente; 4 to 3, which Is sesquitertian, the
symphony diatessaron; and 2 to 1, which is a duple
ratio, forms the diapason.
In consequence, however, of the great veneration paid to the tetractys by the Pythagoreans, it will be proper to give it a more ample discussion, and for this purpose to show from Theo of Smyrna,+ how many tetractys there are: “‘The tetractys,” says he, “‘was not only principally honored by the Pythagoreans, because all symphonies are found to exist within it, but also because it appears to contain the nature of all things.” Hence the dollowine was their oath: “Not by him who delivered to our soul the tetractys, which contains the fountain and root of everlasting nature.” But by him who delivered the tetractys they mean Pythagoras; for the doctrine concerning it appears to have been his invention. The above-mentioned tetractys, therefore, is seen in the composition of the first numbers I. 2. 3. 4. But the second tetractys arises from the increase by multiplication of even and odd numbers beginning from the monad.
Of these, the monad is assumed as the first, because,
1 In Mathemat. p. 147.
as we have before observed, it is the principle of all even, odd, and evenly-odd numbers, and the nature of it is simple. But the three successive numbers receive their composition according to the even and the odd; because every number is not alone even, nor alone odd. Hence the even and the odd receive two tetractys, according to multiplication; the even indeed, in a duple ration; for 2 is the first of even numbers, and increases from the monad by duplication. But the odd number is increased in a triple ratio; for 3 is the first of odd numbers, and is itself increased from the monad by triplication. Hence the monad is common to both these, being itself even and odd. The second number, however, in even and double numbers is 2; but in odd and triple numbers 3. The third among even numbers is 4; but among odd numbers is 9. And the fourth among even numbers is 8; but among odd numbers is 27.
{h 2. 4. oy Is. 3s -Q..- 27.
In these numbers the more perfect ratios of sym-
phonies are found; and in these also a tone is compre-
hended. ‘The monad, however, contains the produc-
tive principle of a point. But the second numbers 2
and 3 contain the principle of a side, since they are
incomposite, and first, are measured by the monad, and
naturally measure a right line. The third terms are
4 and 9g, which are in power a square superficies,
since they are equally equal. And the fourth terms
8 and 27 being equally equal, are in power a cube.
Hence from these numbers, and this tetractys, the
increase takes place from a point toa solid. Fora side
follows after a point, a superficies after a side, and a
solid after a superficies. In these numbers also, Plato
in the Timzus constitutes the soul. But the last of
these seven numbers, i.e. 27, is equal to all the numbers
that precede it; for 1+2+3+4+8+9=27. There
are, therefore, two tetractys of numbers, one of which
subsists by addition, but the other by multiplication,
and they comprehend musical, geometrical, and arith-
metical ratios, from which also the harmony of the
universe consists.
But the third tetractys is that which according to
the same analogy or proportion comprehends the nature
of all magnitude. For what the monad was in the
former tetractys, that a point is in this. What the
numbers 2 and 3, which are in power a side, were in
the former tetractys, that the extended species of a
line, the circular and the right, are in this; the right
line indeed subsisting in conformity to the even number,
since it is terminated ! by two points; but the circular
in conformity to the odd number, because it is compre-
hended by one line which has noend. But what in the
former tetractys the square numbers 4 and g were, that
the two-fold species of planes, the rectilinear and the
circular, are in this. And what the cube numbers 8
and 27 were in the former, the one being an even, but
the other an odd number, that the two solids, one of
which has a hollow superficies, as the sphere and the
cylinder, but the other a plane superficies, as the cube
and pyramid, are in this tetractys. Hence, this is
the curd tetractys, which gives completion to every
magnitude, from a point, a line, a superficies, and a
solid.
The fourth tetractys is of the simple bodies fire, air,
water, and earth, which have an analogy according to
numbers. For what the monad was in the first tetractys,
that fire is in this. But the duad is air, the triad is
water, and the tetrad is earth. For such is the nature
of the elements according to tenuity and density of
parts. Hence fire has to air the ratio of 1 to 2; but
to water, the ratio of I to 3; and to earth, the ratio of
1 Instead of wepirrovrat, it is necessary to read mwepardvrac; the
necessity of which emendation, I wonder the learned Bullialdus did not observe.
1 to 4. In other respects also they are analogous to each other.
The fifth tetractys is of the figures of the simple
bodies. For the pyramid, indeed, is the figure of fire;
the octaedron, of air; the icosaedron, of water; and
the cube, of earth.
The sixth tetractys is of things rising into existence
through the vegetative life. And the seed, indeed, is
analogous to the monad anda point. But if it increases
in length it is analogous to the duad and a line; if in
breadth, to the triad and a superficies; but if in thick-
ness, to the tetrad and a solid.
The seventh tetractys is of communities; of which
the principle indeed, and as it were monad, is man;
the duad is a house; the triad a street; and the tetrad
acity. Fora nation consists of these. And these indeed
are the material and sensible tetractys.
The eighth tetractys consists of the powers which
form a judgment of things material and sensible, and
which are of a certain intelligible nature. And these
are, intellect, science, opinion, and sense. And in-
tellect, indeed, corresponds in its essence to the monad;
but science to the duad; for science is the science of a
certain thing. Opinion subsists between science and
ignorance; but sense is as the tetrad. For the touch
which is common to all the senses being fourfold, all
the senses energize according to contact.
The ninth tetractys is that from which the animal
is composed, the soul and the body. For the parts of
the soul, indeed, are the rational, the irascible, and the
epithymetic, or that which desires external good; and
the fourth is the body in which the soul subsists.
The tenth tetractys is of the seasons of the year,
through which all things rise into existence, viz. the
spring, the summer, the autumn, and the winter.
And the eleventh is of the ages of man, viz. of the infant, the lad, the man, and the old man.
Hence there are eleven tetractys. The first is that
which subsists according to the composition of numbers.
The second, according to the multiplication of numbers.
The third subsists according to magnitude. The fourth
is of the simple bodies. ‘The fifth is of figures. The -
sixth is of things rising into existence through the
vegetative life. The seventh is of communities. The
eighth is the judicial power. The ninth is of the parts
of the animal. The tenth is of the seasons of the year.
And the eleventh is of the ages of man. All of them
however are proportional to each other. For what the
monad is in the first and second tetractys, that a point
is in the third; fire in the fourth; a pyramid in the
fifth; seed in the sixth; man in the seventh; intellect
in the eighth; and so of the rest. Thus, for instance,
the first tetractys 1s I. 2. 3. 4. The second is the
monad, a side, a square, and a cube. The third is a
point, a line, a superficies, and a solid. The fourth is
fire, air, water, earth. The fifth the pyramid, the
octaedron, the icosaedron, and the cube. The sixth,
seed, length, breadth and depth. The seventh, man,
a house, a street, acity. The eighth, intellect, science,
opinion, sense. The ninth, the rational, the irascible,
and the epithymetic parts, and the body. The tenth,
the spring, summer, autumn, winter. The eleventh,
the infant, the lad, the man, and the old man.
The world also, which is composed from these
tetractys, is perfect, being elegantly arranged in geo-
metrical, harmonical, and arithmetical proportion;
comprehending every power, all the nature of number,
every magnitude, and every simple and composite body.
But it is perfect, because all things are the parts of it,
but it is not itself the part of any thing. Hence, the
Pythagoreans are said to have first used the before-
mentioned oath, and also the assertion that “ all things
are assimilated to number.”
P. 81. This number ts the first that partakes of
every number, and when divided in every possible
way, receives the power of the numbers subtracted,
and of those that remain.
Because 6 consists of I, 2 and 3, the two first of
which are the principles of all number, and also because
2 and 3 are the first even and odd, which are the
sources of all the species of numbers; the number 6
may be said to partake of every number. In what
Iamblichus afterwards adds, I suppose he alludes to 6
being a perfect number and therefore equal to all its
parts.
P. 98. Not to step above the beam of the balance.
This is the 14th Symbol in the Protreptics of Iam-
blichus, whose explanation of it is as follows: “This
symbol exhorts us to the exercise of justice, to the
honoring equality and moderation in an admirable
degree, and to the knowledge of justice as the most
perfect virtue, to which the other virtues give com-
pletion, and without which none of the rest are of any
advantage. It also admonishes us, that it is proper
to know this virtue not in a careless manner, but through
theorems and scientific demonstrations. But this
knowledge is the business of no other art and science
than the Pythagoric philosophy alone, which in a tran-
scendent degree honors disciplines before every thing
else.”
The following extract also from my Theoretic Arith-
metic, (p. 194.), will in a still greater degree elucidate
thissymbol. The information contained in it is derived
from the anonymous author of a very valuable work
entitled Ocoroyovueva ’ApOuntucjs Theologumena Arith-
metice, and which has lately been reprinted at Leipsic.
“The Pythagoreans called the pentad providence and
justice, because it equalizes things unequal, justice being
a medium between excess and defect, just as 5 is the
middle of all the numbers that are equally distant from
it on boths sides as far as to the decad, some of which
it surpasses, and by others is surpassed, as may be seen
in the following arrangement:
ATS \o oon
If 2 is added to §, and likewise taken from it, 7 and 3
will be produced. And by adding 1 to 5, and sub-
tracting 1 from it, 6 and 4 will be the result; in
all which instances, the numbers produced are equi-
distant from 5, and the sum of each couple is equal
to 10.”
P. 116. Such as dig not fire with a sword.
This is the 9th Symbol in the Protreptics, and is
thus explained by Iamblichus. ‘This symbol exhorts
to prudence. For it excites in us an appropriate con-
ception with respect to the propriety of not opposing
sharp words to a man full of fire and wrath, nor con-
tending with him. For frequently by words you will
agitate and disturb an ignorant man, and will yourself
suffer things dreadful and unpleasant. Heraclitus also
testifies to the truth of this symbol. For he says, “It
is difficult to fight with anger: for whatever is necessary
to be done redeems the soul.” And this he says truly.
For many, by gratifying anger, have changed the con-
dition of their soul, and have made death preferable to
life. But by governing the tongue, and being quiet,
friendship is produced from strife, the fire of anger
being extinguished; and you yourself will not appear
to be destitute of intellect.”