are participated by all thefe. But Ariftotle* likewife affents to Plato; for every species of lines, fays he, is either right or circular, or mixed from these two. From whence alfo there are three motions, one according to a right line; the other circular; and the third mixed. But fome oppofe this divifion, and fay that there are not two fimple lines alone, but that there is a certain third line given, i. e. a helix or spiral, which is described about a cylinder †, when, whilst a right line is moved round the fuperficies of the cylinder, a point in the line is carried along with an equal celerity. For by this means, a helix, or circumvolute line, is produced, which adapts all the parts of itfelf to all, according to a fimilitude of parts, as Apollonius fhews in his book concerning the Cochlea; which paffion, among all spirals, agrees to this alone. For the parts of a plane helix are diffimilar among themselves; as alfo of those which are defcribed about a cone and sphere. But the cylindric spiral alone, confifts of fimilar parts in the fame manner as a right and circular line. Are there, then, three fimple lines, and not two only? To which doubt we reply, that a helix of this kind is, indeed, of fimilar parts, as Apollonius teaches, but is by no means fimple; fince among natural productions, gold and filver are compofed of fimilar parts, but are not fimple bodies. But the generation of the cylindric helix evinces that its mixture is from things fimple; for it originates while a right line is circularly moved round the axis of the cylinder, a point at the fame time flowing along in the right line. Two fimple motions, therefore, compose its nature; and, on this account, it is among the number of mixt lines, and not among fuch as are fimple: for that which is composed from diffimilars is not fimple, but mixt. Hence, Geminus, with great propriety, when he admits that fome fimple lines may be produced from many motions, does not grant that every such line is mixt; but that alone, which arifes from diffimilar motions. For if you conceive fopher, that the effence of an angle does not fubfift in either quantity, quality, or inclination, taken fingly, but in the aggregate of them all. For if we regard the inclination of a circular line to its tangent, we fhall find it poffefs the property, by which Euclid defines an angle: if we refpect its participation of quantity, we fhall find it capable of being augmented and diminished; and if we regard it as poffeffing a peculiar quality, we fhall account for its being incommenfurable with every right-lined angle. See the Comment on the 8th Definition.

In i. De Calo.

mixed from thefe, the fpecies of lines, angles, and figure
fubfift in the nature of things originate from thence. A
a circumference and a circular angle, and a circle amor
and a sphere among folids, are analogous to bound
correfponds to infinity, according to all these
longs to all, if it is beheld as exifting in
mixed in all thefe, is analogous to the
intelligibles. For lines are mixed, as

COMMENTARIES OF PROCLUS.

1:9

of the cylinder, a point in the line a certain third line given, i e. a helix cylinder †, when, whilft a right ivifion, and fay that there are not two fimple line; the other circular; and the third mixed cies of lines, fays he, is either right or circular, or mixed From whence alfo there are three motions, one ipated by all thefe. But Ariftotle likewife aflents to Plato,

And angles, as the femi-circular and
as fegments and apfides; but foli
of that kind. Bound, there

For
by
this
means,
a
helix,

adapts
all
the
parts
of
itfelf

Vallomus fhews in his
all Spirals, agrees
Bacilar among

and

e ε

be a right angle, and D E the line to be moved, which is bifected in G. Now, conceive it to be moved along the lines A B, B C, n fuch a manner, that the point D may always remain in A B, and the point E in B C. Then, when the line D E, is in the fituations de, d, the point G, fhall be in g, y, and these points G, g, y, fhall be in a circle. And any other point F in the line DE, will, at the fame time, defcribe an ellipfis; the greater axis being in the line A B, when the point F is between D and G; and in the line B C, when the point Fis between G and E,

generation

generation of a circular line is the confequence of that inequality of
lation arifing from the bifection; because a right line was fuppofed to
be moved in a right angle, but not in a natural manner. And thus
much concerning the generation of lines. But it feems, that of the
two fimple lines, the right and the circular, the right line is the more
fimple; for in this, diffimilitude cannot be conceived, even in opinion.
But in the circular line, the concave and the convex, indicate diffimi-
litude. And a right line, indeed, does not infer a circumference
according to thought; but a circumference brings with it a right line,
though not according to its generation, yet with respect to its centre.
But what if it should be faid that a circumference requires a right line
to its conftruction! For if either extreme of a right line remains fixt,
but the other is moved, it will doubtlefs defcribe a circle, whofe centre
vill be the abiding extreme of the right line. Shall we fay that the
enerator of the circle is the point which is carried about the abiding
int, but not the right line itself? For the line only determines the
lance, but the point compofes the circular line, while it is moved
circular manner: but of this enough. Again, a circumference
ears to be proximate to bound, and to have the fame proportion to
r lines, as bound to the univerfality of things. For it is finite, and
one among simple lines perfective of figure. But a right line is
nate to infinity; for its capacity of infinite extension never fails:
all the reft are produced from bound and infinite, in the fame
r from the circular and right line, every mixt genus of lines
pofed, as well of planes as of those which confift in folid
And on this account, the foul alfo * previously affumed into
herfelt ne right and circular according to her effence, that he might
moderate all the co-ordination of infinite, and all the nature of bound,
which the world contains. By a right line, indeed, conftituting the
progreffion of these principles into the univerfe; but by a circular line,
their return to their original fource: and by the one, producing all
things into multitude; but by the other, collecting them into one.
And not only the foul, but he alfo who produced the foul, and endued
her with thefe powers, contains in himself both thefe primary caufes.

That is, the foul of the world.
S 2

For

a fquare, and two motions which are performed with an equal celerity, one according to the length, but the other according to the breadth, a right line or the diameter will be produced; but the right line will not, on this account, be mixed: for no other line precedes it, formed by a fimple motion, as we afferted of the cylindric helix. Nor yet, if you fuppofe a right line, moving in a right angle, and by a bisection to defcribe a circle *, is the circular line, on this account, produced with mixture: for the extremities of that which is moved after this manner, fince they are equally moved, will defcribe a right line; and the bisection, fince it is unequally devolved, will delineate a circle; but the other points will defcribe an ellipfis. On which account, the * The present very obfcure paffage, may be explained by the following figure. Let ABC,

be a right angle, and D E the line to be moved, which is bifected in G. Now, conceive it to be moved along the lines A B, B C, n fuch a manner, that the point D may always remain in A B, and the point E in BC. Then, when the line D E, is in the fituations de, dt, the point G, fhall be in g, 7, and thefe points G, g, y, fhall be in a circle. And any other point F in the line DE, will, at the fame time, defcribe an ellipfis; the greater axis being in the line AB, when the point F is between D and G; and in the line B C, when the point F is between G and E.

generation

i

generation of a circular line is the confequence of that inequality of lation arising from the bifection; because a right line was fuppofed to be moved in a right angle, but not in a natural manner. And thus much concerning the generation of lines. But it seems, that of the two fimple lines, the right and the circular, the right line is the more fimple; for in this, diffimilitude cannot be conceived, even in opinion. But in the circular line, the concave and the convex, indicate diffimilitude. And a right line, indeed, does not infer a circumference according to thought; but a circumference brings with it a right line, though not according to its generation, yet with refpect to its centre. But what if it should be faid that a circumference requires a right line to its conftruction! For if either extreme of a right line remains fixt, but the other is moved, it will doubtless describe a circle, whofe centre will be the abiding extreme of the right line. Shall we say that the generator of the circle is the point which is carried about the abiding point, but not the right line itself? For the line only determines the distance, but the point compofes the circular line, while it is moved in a circular manner: but of this enough. Again, a circumference appears to be proximate to bound, and to have the fame proportion to other lines, as bound to the universality of things. For it is finite, and is alone among fimple lines perfective of figure. But a right line is proximate to infinity; for its capacity of infinite extension never fails: and as all the reft are produced from bound and infinite, in the fame manner from the circular and right line, every mixt genus of lines is compofed, as well of planes as of thofe which confift in folid bodies. And on this account, the foul alfo * previously affumed into herself the right and circular according to her effence, that he might moderate all the co-ordination of infinite, and all the nature of bound, which the world contains. By a right line, indeed, conftituting the progreffion of these principles into the universe; but by a circular line, their return to their original fource: and by the one, producing all things into multitude; but by the other, collecting them into one. And not only the foul, but he also who produced the foul, and endued her with these powers, contains in himself both these primary causes.

That is, the foul of the world.
S 2

For