The mind that could conceive the world as a sphere and that
interested itself in weights and measures was, obviously, a mind
of the visualizing type. It is characteristic of this type of
mind to be interested in the tangibilities of geometry, hence it
is not surprising to be told that Pythagoras "carried that
science to perfection." The most famous discovery of Pythagoras
in this field was that the square of the hypotenuse of a
right-angled triangle is equal to the squares of the other sides
of the triangle. We have already noted the fable that his
enthusiasm over this discovery led him to sacrifice a hecatomb.
Doubtless the story is apocryphal, but doubtless, also, it
expresses the truth as to the fervid joy with which the
philosopher must have contemplated the results of his creative
imagination.

No line alleged to have been written by Pythagoras has come down
to us. We are told that he refrained from publishing his
doctrines, except by word of mouth. "The Lucanians and the
Peucetians, and the Messapians and the Romans," we are assured,
"flocked around him, coming with eagerness to hear his
discourses; no fewer than six hundred came to him every night;
and if any one of them had ever been permitted to see the master,
they wrote of it to their friends as if they had gained some
great advantage." Nevertheless, we are assured that until the
time of Philolaus no doctrines of Pythagoras were ever published,
to which statement it is added that "when the three celebrated
books were published, Plato wrote to have them purchased for him
for a hundred minas."[2] But if such books existed, they are lost
to the modern world, and we are obliged to accept the assertions
of relatively late writers as to the theories of the great
Crotonian.

Perhaps we cannot do better than quote at length from an
important summary of the remaining doctrines of Pythagoras, which
Diogenes himself quoted from the work of a predecessor.[3]
Despite its somewhat inchoate character, this summary is a most
remarkable one, as a brief analysis of its contents will show. It
should be explained that Alexander (whose work is now lost) is
said to have found these dogmas set down in the commentaries of
Pythagoras. If this assertion be accepted, we are brought one
step nearer the philosopher himself. The summary is as follows:


"That the monad was the beginning of everything. From the monad
proceeds an indefinite duad, which is subordinate to the monad as
to its cause. That from the monad and the indefinite duad proceed
numbers. And from numbers signs. And from these last, lines of
which plane figures consist. And from plane figures are derived
solid bodies. And from solid bodies sensible bodies, of which
last there are four elements--fire, water, earth, and air. And
that the world, which is indued with life and intellect, and
which is of a spherical figure, having the earth, which is also
spherical, and inhabited all over in its centre,[4] results from
a combination of these elements, and derives its motion from
them; and also that there are antipodes, and that what is below,
as respects us, is above in respect of them.

"He also taught that light and darkness, and cold and heat, and
dryness and moisture, were equally divided in the world; and that
while heat was predominant it was summer; while cold had the
mastery, it was winter; when dryness prevailed, it was spring;
and when moisture preponderated, winter. And while all these
qualities were on a level, then was the loveliest season of the
year; of which the flourishing spring was the wholesome period,
and the season of autumn the most pernicious one. Of the day, he
said that the flourishing period was the morning, and the fading
one the evening; on which account that also was the least healthy
time.

"Another of his theories was that the air around the earth was
immovable and pregnant with disease, and that everything in it
was mortal; but that the upper air was in perpetual motion, and
pure and salubrious, and that everything in that was immortal,
and on that account divine. And that the sun and the moon and the
stars were all gods; for in them the warm principle predominates
which is the cause of life. And that the moon derives its light
from the sun. And that there is a relationship between men and
the gods, because men partake of the divine principle; on which
account, also, God exercises his providence for our advantage.
Also, that Fate is the cause of the arrangement of the world both
generally and particularly. Moreover, that a ray from the sun
penetrated both the cold aether and the dense aether; and they
call the air the cold aether, and the sea and moisture they call
the dense aether. And this ray descends into the depths, and in
this way vivifies everything. And everything which partakes of
the principle of heat lives, on which account, also, plants are
animated beings; but that all living things have not necessarily
souls. And that the soul is a something tom off from the aether,
both warm and cold, from its partaking of the cold aether. And
that the soul is something different from life. Also, that it is
immortal, because that from which it has been detached is
immortal.

"Also, that animals are born from one another by seeds, and that
it is impossible for there to be any spontaneous production by
the earth. And that seed is a drop from the brain which contains
in itself a warm vapor; and that when this is applied to the womb
it transmits virtue and moisture and blood from the brain, from
which flesh and sinews and bones and hair and the whole body are
produced. And from the vapor is produced the soul, and also


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