Plato establishing two most simple and principal species of lines,

the right and the circular, composes all the rest from the mixture of

these; I mean such as are called curve lines, some of which are formed

from planes; but others subsist about solids; and whatever species

of curve lines are produced by the sections of solids. And it seems,

indeed, that a point (if it be lawful so to speak) bears an image of

the one itself, according to Plato: for unity has no part, as he also

shews in the Parmenides. But, because after unity itself there are

three hypostases, or substances, bound, infinite, and that which is

mixed from these, the species of lines, angles, and figures, which

subsist in the nature of things originate from thence. And, indeed, a

circumference and a circular angle, and a circle among plane figures,

and a sphere among solids, are analogous to _bound_. But a right

line corresponds to _infinity_, according to all these; for it

properly belongs to all, if it is beheld as existing in each. But that

which is mixed in all these, is analogous to the mixt which subsists

among intelligibles. For lines are mixed, as those which are called

spirals. And angles, as the semi-circular and cornicular[139]. And

plane figures, as segments and apsides; but solids, as cones and

cylinders, and others of that kind. Bound, therefore, infinite, and

that which is mixed, are participated by all these. But Aristotle[140]

likewise assents to Plato; for every species of lines, says he, is

either right or circular, or mixed from these two. From whence also

there are three motions, one according to a right line; the other

circular; and the third mixed. But some oppose this division, and

say that there are not two simple lines alone, but that there is a

certain third line given, i. e. a helix or spiral, which is described

about a cylinder[141], when, whilst a right line is moved round the

superficies 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 itself to all, according to

a similitude of parts, as Apollonius shews in his book concerning the

Cochlea; which passion, among all spirals, agrees to this alone. For

the parts of a plane helix are dissimilar among themselves; as also of

those which are described about a cone and sphere. But the cylindric

spiral alone, consists of similar parts in the same manner as a right

and circular line. Are there, then, three simple lines, and not two

only? To which doubt we reply, that a helix of this kind is, indeed, of

similar parts, as Apollonius teaches, but is by no means simple; since

among natural productions, gold and silver are composed of similar

parts, but are not simple bodies. But the generation of the cylindric

helix evinces that its mixture is from things simple; for it originates

while a right line is circularly moved round the axis of the cylinder,

a point at the same time flowing along in the right line. Two simple

motions, therefore, compose its nature; and, on this account, it is

among the number of mixt lines, and not among such as are simple: for

that which is composed from dissimilars is not simple, but mixt. Hence,

Geminus, with great propriety, when he admits that some simple lines

may be produced from many motions, does not grant that every such line

is mixt; but that alone, which arises from dissimilar motions. For if

you conceive a square, 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 simple motion, as we asserted of the cylindric

helix. Nor yet, if you suppose a right line, moving in a right angle,

and by a bisection to describe a circle[142], is the circular line, on

this account, produced with mixture: for the extremities of that which

is moved after this manner, since they are equally moved, will describe

a right line; and the bisection, since it is unequally devolved, will

delineate a circle; but the other points will describe an ellipsis. On

which account, the generation of a circular line is the consequence

of that inequality of lation arising from the bisection; because a

right line was supposed 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 simple lines, the right and the circular,

the right line is the more simple; for in this, dissimilitude cannot

be conceived, even in opinion. But in the circular line, the concave

and the convex, indicate dissimilitude. 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 said that a

circumference requires a right line to its construction! For if either

extreme of a right line remains fixt, but the other is moved, it will

doubtless describe a circle, whose 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

composes 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 same proportion to other lines, as bound to the

universality of things. For it is finite, and is alone among simple

lines perfective of figure. But a right line is proximate to infinity;

for its capacity of infinite extension never fails: and as all the

rest are produced from bound and infinite, in the same manner from the

circular and right line, every mixt genus of lines is composed, as

well of planes as of those which consist in solid bodies. And on this

account, the soul also[143] previously assumed into herself the right

and circular according to her essence, that she might moderate all

the co-ordination of infinite, and all the nature of bound, which the

world contains. By a right line, indeed, constituting the progression

of these principles into the universe; but by a circular line, their

return to their original source: and by the one, producing all things

into multitude; but by the other, collecting them into one. And not

only the soul, but he also who produced the soul, and endued her

with these powers, contains in himself both these primary causes.

For when he previously assumed the beginning, middle, and end of all

things, he terminated right lines (says Plato[144]), by a circular

progression according to nature. And proceeding to all things by

provident energies, and returning to himself, he established himself,

says Timæus, after his own peculiar manner. But a right line is the

mark or symbol of a providence, indeclinable, incapable of perversion,

immaculate, never-failing, omnipotent, and present to all beings, and

to every part of the universe. But a circumference, and that which

environs, is the symbol of an energy retiring into union with itself,

and which rules over all things according to one intellectual bound.

When, therefore, the demiurgus of the universe had established in

himself these two principles, the right and the circular line, and had

given them dominion, he produced from himself two unities; the one,

indeed, energizing according to the circular line, and being effective

of intellectual essences; but the other according to the right line,

and affording an origin to sensible natures. But because the soul is

allotted a middle situation between intellectuals and sensibles, so

far, indeed, as she adheres to an intellectual nature, she energizes

according to the circle; but so far as she presides over sensibles, she

provides for their welfare according to the right line: and thus much

concerning the similitude of these forms to the universality of things.

But Euclid, indeed, has properly delivered the present definition of

a line; by which he shews that a right line alone occupies a space

equal to that which is situated between its points: for as much as is

the distance of one point from another, so great is the magnitude of

the lines terminated by the points. And this is the meaning of being

equally situated between its extremes. For if you take two points in a

circumference, or in any other certain line, the space of line which

is included between these, exceeds their distance from each other;

and every line, besides a right one, appears to suffer this property.

Hence, according to a common conception, the vulgar also say, that he

who walks by a right line, performs only a necessary journey: but that

they necessarily wander much, who do not proceed in a right line. But

Plato thus defines it; a right line is that whose middle parts darken

its extremes. For this passion necessarily attends things which have

a direct position; but it is not necessary that things situated in the

circumference of a circle, or in another interval, should be endued

with this property. Hence, the astrologers also say, that the sun then

suffers an eclipse when that luminary, the moon, and our eye are in one

right line; for it is then darkened through the middle position of the

moon between us and its orb. And perhaps, the passion of a right line

will evince, that in the things which are, according to processions

emanating from causes, the mediums are endued with a power of dividing

the distance of the extremes, and their mutual communication with each

other. As also, according to regressions, such things as are distant

from the extremes, are converted by mediums to their primary causes.

But Archimedes defines a right line the least of things having the

same bounds. For since, according to Euclid, a right line is equally

situated between its points, it is on this account, the least of things

having the same bounds: for if a less line could be given, it would

not lie equally between its extremes: but all the other definitions of

a right line, fall into the same conclusions; as for instance, that

it is constituted in its extremities, and that one part of it is not

in its subject plane, but another, in one more sublime: and that all

its parts similarly agree to all: and that its extremes abiding, it

also abides. Lastly, that it does not perfect figure, with one line

similar in species to itself: for all these definitions express the

property of a right line, which it possesses from the simplicity of its

essence, and from its having one progression the shortest of all from

one extremity to another. And thus much concerning the definitions of a

right line. But again, Geminus divides a line first into an incomposite

and composite; calling a composite, that which is refracted, and forms

an angle; but all the rest of them, he denominates incomposites.

Afterwards, he divides a composite line into that which produces

figure, and that which may be infinitely extended. And he calls that

which produces figure, a circular line, and the line of a shield[145],

and that which is similar to an ivy leaf[146]; but that which is not

effective of figure, the section of a rectangular and obtuse angular

cone, the line similar to a shell[147], the right line, and all of

that kind. And again, after another manner, of the incomposite line,

one sort is simple, but the other mixt. And of the simple, one produces

figure, as the circular; but the other is indefinite, as the right

line. But of the mixt, one subsists in planes, but the other in solids.

And of that which is in planes, one coincides in itself, as the figure

of the ivy leaf, which is called the cissoid; but the other may be

produced in infinitum, as the helix. But of that which is in solids,

one may be considered in the sections of solids; but the other as

consisting about the solids themselves. For the helix, indeed, which

is described about a sphere or a cone, consists about solids; but

conic, or spirical sections are generated from a particular section of

solids. But, with respect to these sections, the conic were invented by

Mænechmus, which also Eratosthenes relating, says,

“Nor in a cone Mænechmian ternaries divide.”

But the spiric by Perseus, who composed an epigram on their invention,

to this purpose, “When Perseus had invented three spiral lines in five

sections, he sacrificed to the gods on the occasion.” And the three

sections of a cone, are the parabola, hyperbola, and ellipsis: but of

spiral sections, one kind is twisted and involved, like the fetlock of

a horse; but another is dilated in the middle, and deficient in each

extremity: and another which is oblong, has less space in the middle,

but is dilated on each side. But the multitude of the other mixt lines

is infinite. For there is an innumerable multitude of solid figures,

from which there are constituted multiform sections. For a right line,

while it is circularly moved[148], does not make a certain determinate

superficies, nor yet conical, nor conchoidal lines, nor circumferences

themselves. Hence, if these solids are multifariously cut, they will

exhibit various species of lines. Lastly, of those lines which consist

about solids, some are of similar parts, as the helixes about a

cylinder; but others of dissimilar parts, as all the rest. From these

divisions, therefore, we may collect, that there are only three lines

of similar parts, the right, the circular, and the cylindric helix. The

two simple ones, indeed, existing in a plane, but the one mixt, about

a solid. And this Geminus evidently demonstrates, when he shews, that

if two right lines are extended from one point, to a line of similar

parts, so as to make equal angles upon that line, they shall be equal

to each other. And the demonstrations of this may be received by the

studious, from his volumes; since in these he delivers the origin of

spiral, conchoidal, and cissoidal lines. But we have barely related

the names and divisions of these lines, for the purpose of exciting

the ingenious to their investigation; as we think, that an accurate

enquiry after the method of detecting the properties of each, would be

superfluous in the present undertaking: since the geometrician only

unfolds to us in this work, simple and primary lines, i.e. the right

line, in the present definition; but a circular line, in the tradition

of a circle. For he then says, that the line terminating the circle,

is the circumference. But he makes no mention of mixt lines, though

he was well acquainted with mixt angles, I mean, the semi-circular

and cornicular: as also with plane mixt figures, i.e. segments and

sectors; and with solids, viz. cones and cylinders. Of each of the

rest, therefore, he delivers three species; but of lines only two, i.

e. the right and circular: for he thought it requisite in discourses

concerning things simple, to assume simple species; and all the rest

are more composite than lines. Hence, in imitation of the geometrician,

we also shall terminate their explanation with simple lines.

DEFINITION V.

A SUPERFICIES is that which has only Length and Breadth.

After a point and a line, a superficies is placed, which is distant

by a twofold interval, length and breadth. But this also remaining

destitute of thickness or bulk, possesses a nature more simple than

body, which is distant by a triple dimension. On which account the

geometrician adds to the two intervals the particle _only_,

because the third interval does not exist in superficies. And this is

equivalent to a negation of bulk, as here also he shews the excellency

of superficies compared to a solid with respect to simplicity, by

negation, or by an addition equivalent to negation: but the diminution

which it possesses, if compared with the preceding terms, by the

affirmations themselves. But others define a superficies to be the

boundary of body, which is almost affirming the same as the definition

of Euclid; since that which terminates is exceeded in one dimension,

by that which is terminated. And others, a magnitude different by

two intervals. Lastly, others declaring the same affection, form its

assignation in a somewhat different manner. But they say we have a

knowledge of superficies when we measure fields, and distinguish their

extremities according to length and breadth; but that we receive a

certain sensation of it, when we behold shadows. For as they are

without bulk, because they cannot penetrate into the interior part of

the earth, they have only length and breadth. But the Pythagoreans

say, that it is assimilated to the triad; because the ternary is by

far the first cause to all the figures; which a superficies contains.

For a circle, which is the principle of orbicular figures, occultly

possesses the ternary, by its centre, interval, and circumference. But

a triangle, which ranks as the first among all right-lined figures, on

every side evinces that it is enclosed by the triad, and receives its

form from its perfect nature.

DEFINITION VI.

The Extremities of a SUPERFICIES are Lines.

From these also, as images, we may understand, that things more simple

procure bound and an end to every one of their proximate natures: for

soul perfects and determines the operations of nature; and nature

the motion of bodies. And prior to these, intellect measures the

convolutions of soul; and unity the life of intellect; for that is

the measure of all. Just as in these also, a solid is terminated by a

superficies; but a superficies by a line; and a line by a point; for

that is the boundary of them all. Hence, the line existing uniformly

in immaterial forms and impartible reasons, terminates and restrains

the various motion of a superficies in its progression, and proximately

unites its infinity. But in the images of these, when that which

bounds supervenes that which is bounded, it causes, by this means,

its limitation and bound. But if it should be enquired how lines are

the extremities of every superficies, since they are not the extremes

of every finite figure; for the superficies of a sphere is terminated

indeed, yet not by lines, but by itself? In answer to this, we must

say, that by receiving a superficies so far as it is distant by a

twofold interval, we shall find it terminated by lines according to

length and breadth. But if we behold a spherical superficies, we

must receive it as that which is endued with figure; which possesses

another quality, and conjoins the end with the beginning; and loses

its two extremities in the comprehensive embraces of one: and this one

extremity subsists in capacity only, and not in energy.

DEFINITION VII.

A Plane SUPERFICIES is that which is equally situated between

its bounding Lines.

It was not agreeable to the ancient philosophers to establish a plane

species of superficies; but they considered superficies in general,

as the representative of magnitude, which is distant by a twofold

interval. For thus the divine Plato[149] says, that geometry is

contemplative of planes, opposing it in division to stereometry, as

if a plane and a superficies were the same. And this was likewise the

opinion of the demoniacal Aristotle[150]. But Euclid and his followers

consider superficies as a genus, but a plane as its species, in the

same manner as rectitude of a line. And on this account he defines a

plane separate from a superficies, after the similitude of a right

line. For he defines this last as equal to the space, placed between

its points. And in like manner, he says, that two right lines being

given, a plane superficies occupies a place equal to the space situated

between those two lines. For this is equally situated between its

lines; and others also explaining the same boundary, assert that it

is constituted in its extremities. But others define it as that to

all the parts of which a right line may be adapted[151]. But perhaps

others will say, that it is the shortest of superficies, having the

same boundaries; and that its middle parts darken its extremities; and

that all the definitions of a right line may be transferred into a

plane superficies, by only changing the genus: since a right, circular,

and mixt line, commencing from lines, arrive even at solids, as we

have asserted above; for they are proportionally, both in superficies

and solids. Hence also, Parmenides says, that every figure is either

right, or circular, or mixt. But if you wish to consider the right in

superficies, take a plane, to which a right line agrees in various

ways; but if a circular receive a spherical superficies; and if a

mixt, a conic or cylindric, or some one of that genus. But it is

requisite (says Geminus) since a line, and also a superficies is

called mixt, to know the measure of mixture, because it is various.

For mixture in lines, is neither by composition, nor by temperament

only: since, indeed, a helix is mixed, yet one part of it is not

straight, and another part circular, like those things which are

mixed by composition: nor if a helix is cut after any manner, does

it exhibit an image of things simple, such as those which are mixed

through temperament; but in these the extremes are, at the same time,

corrupted and confused. Hence, Theodorus the mathematician, does not

rightly perceive, in thinking that this mixture is in lines. But

mixture in superficies, is neither by composition, nor by confusion;

but subsists rather by a certain temperament. For conceiving a circle

in a subject plane, and a point on high, and producing a right line

from the point to the circumference of the circle, the revolution of

this line will produce a conical superficies which is mixt. And we

again resolve it into its simple elements, by a parallel section: for

by drawing a section between the vertex and the base, which shall

cut the plane of the generative right line, we effect a circular

line. But the idea of lines, shews that the mode of mixture is not by

temperament; for neither does it send us back to the simple nature of

elements: on the contrary, when superficies are cut, they immediately

exhibit to us their producing lines. The mode of mixture, therefore, is

not the same in lines and superficies. But as among lines there were

some simple, that is, the right and circular, of which the vulgar also

possess an anticipated knowledge without any previous instruction;

but the species of mixt lines require a more artificial apprehension:

so among superficies, we possess an innate notion of those which are

especially elementary, the plane and spherical; but science and its

reason investigates the variety of those which are composed through

mixture. But this is an admirable property of superficies, that their

mixture in generation is oftentimes produced from a circular line;

and this also happens to a spiral superficies. For this is understood

by the revolution of a circle remaining erect, and turning itself

about the same point which is not its centre. And on this account, a

spiral also is threefold; for its centre is either in a circumference,

or within, or external to a circumference. If the centre is in

the circumference, a continued spiral is produced: if within the

circumference, an intangled one; if without, a divided one. And there

are three spiral sections corresponding to these three differences.

But every spiral line is mixt, although the motion from which it is

produced is one and circular. And mixt superficies are produced as

well from simple lines, (as we have said,) while they are moved with a

motion of this kind, as from mixt lines. Since, therefore, there are

three conic lines, they produce four mixt superficies, which they call

conoids. For a rectangular conoid, is produced from the revolution of

the parabola about its axis: but that which is formed by the ellipsis,

is called a spheroid; and is the revolution is made about the greater

axis, it is an oblong; but if about the lesser a broad spheroid.

Lastly, an obtuse-angled conoid is generated from the revolution of the

hyperbola. But it is requisite to know, that sometimes we arrive at the

knowledge of superficies from lines, and sometimes the contrary; for

from conical and spiral superficies, we apprehend conical and spiral

lines. Besides, this also must be previously received concerning the

difference of lines and superficies, that there are three lines of

similar parts (as we have already observed), but only two superficies,

the plane and the spherical. For this is not true of the cylindric,

since all parts of the cylindric superficies cannot agree to all. And

thus much concerning the differences of superficies, one of which

the geometrician having chosen (I mean the plane), this also he has

defined; and in this, as a subject, he contemplates figures, and their

attendant passions: for his discourse is more copious in this than in

other superficies: since, indeed, we may understand right lines, and

circles, and helixes in a plane; also the sections of circles and right

lines, contacts, and applications, and the constructions of angles

of every kind. But in other superficies, all these cannot be beheld.

For how in one that is spherical, can we apprehend a right line, or a

right-lined angle? How, lastly, in a conic or cylindric superficies,

can we behold sections of circles or right lines? Not undeservedly,

therefore, does he both define this superficies, and discuss his

geometrical concerns, by exhibiting every thing in this as in a

subject; for from hence he calls the present treatise plane. And, after

this manner, it is requisite to understand that which is plane, as

projected and constituted before the eyes: but cogitation as describing

all things in this, the phantasy corresponding to a plane mirror, and

the reasons resident in cogitation as dropping their images[152] into

its shadowy receptacle.

DEFINITION VIII.

[153]A PLANE ANGLE, is the inclination of two Lines to each

other in a Plane, which meet together, but are not in the same

direction.

Some of the ancient philosophers, placing an angle in the predicament

of relation, have said, that it is the mutual inclination of lines or

planes to each other. But others, including this in quality, as well

as rectitude and obliquity, say, that it is a certain passion of a

superficies or a solid. And others, referring it to quantity, confess

that it is superficies or a solid. For the angle which subsists in

superficies is divided by a line; but that which is in solids, by a

superficies. But (say they) that which is divided by these, is no other

than magnitude, and this is not linear, since a line is divided by a

point; and therefore it follows that it must be either a superficies

or a solid. But if it is magnitude, and all finite magnitudes of the

same kind have a mutual proportion; all angles of the same kind, i.

e. which subsist in superficies, will have a mutual proportion. And

hence, the cornicular will be proportionable to a right-lined angle.

But things which have a mutual proportion, may, by multiplication,

exceed each other; and therefore it may be possible for the cornicular

to exceed a right-lined angle, which, it is well known, is impossible,

since it is shewn to be less than every right-lined angle. But if

it is quality alone, like heat and cold, how is it divisible into

equal parts? For equality, inequality, and divisibility, are not less

resident in angles than in magnitudes; but they are, in like manner,

essential. But if the things in which these are essentially inherent,

are quantities, and not qualities, it is manifest that angles also are

not qualities. Since the more and the less are the proper passions

of quality[154], but not equal and unequal. On this hypothesis,

therefore, angles ought not to be called unequal, and this greater,

but the other less; but they ought to be denominated dissimilars, and

one more an angle, but the other less. But that these appellations

are foreign from the essence of mathematical concerns, is obvious to

every one: for every angle receives the same definition, nor is this

more an angle, but that less. Thirdly, if an angle is inclination, and

belongs to the category of relation, it must follow, that from the

existence of one inclination, there will also be one angle, and not

more than one. For if it is nothing else than the relation of lines

or planes, how is it possible there can be one relation of lines or

planes, but many angles? If, therefore, we conceive a cone cut by a

triangle from the vertex to the base, we shall behold one inclination

of the triangular lines in the semicone to the vertex; but two distinct

angles: one of which is plane, I mean that of the triangle; but the

other subsists in the mixt superficies of the cone, and both are

comprehended by the two triangular lines. The relation, therefore,

of these, do not make the angle. Again, if is necessary to call an

angle either quality or quantity, or relation; for figures, indeed,

are qualities, but their mutual proportions belong to relation. It is

necessary, therefore, that an angle should be reduced under one of

these three genera. Such doubts, then, arising concerning an angle,

and Euclid calling it inclination, but Apollonius the collection of

a superficies, or a solid in one point, under a refracted line or

superficies (for he seems to define every angle universally), we

shall affirm, agreeable to the sentiments of our preceptor Syrianus,

that an angle is of itself none of the aforesaid; but is constituted

from the concurrence of them all. And that, on this account, a doubt

arises among those who regard one category alone. But this is not

peculiar to an angle, but is likewise the property of a triangle. For

this, too, participates of quantity, and is called equal and unequal;

because it has to quantity the proportion of matter. But quality also,

is present with this, in consequence of its figure (since triangles

are called as well similar as equal); but it possesses this from one

category, and that from another. Hence, an angle is perfectly indigent

of quantity, the subject of magnitude. But it is also indigent of

quality, by which it possesses, as it were, its proper form and figure,

Lastly, it is indigent of the relation of lines terminating, or of

superficies comprehending its form. So that an angle consists from

all these, yet is not any one of them in particular. And it is indeed

divisible, and capable of receiving equality and inequality, according

to the quantity which it contains. But it is not compelled to admit

the proportion of magnitudes of the same kind, since it has also a

peculiar quantity, by which angles are also incapable of a comparison

with each other. Nor can one inclination perfect one angle: since the

quantity also, which is placed between the inclined lines, completes

its essence. If then we regard these distinctions, we shall dissolve

all absurdities, and discover that the property of an angle is not the

collection of a superficies or solid, according to Apollonius (since

these also complete its essence,) but that it is nothing else than

a superficies itself, collected into one point, and comprehended by

inclined lines, or by one line inclined to itself: and that a solid

angle is the collection of superficies mutually inclined to each

other. Hence, we shall find that a formed quantum, constituted in a

certain relation, supplies its perfect definition. And thus much we

have thought requisite to assert concerning the substance of angles,

previously contemplating the common essence of every triangle, before

we divide it into species. But since there are three opinions of an

angle, Eudemus the Peripatetic, who composed a book concerning an

angle, affirms that it is quality. For, considering the origin of an

angle, he says that it is nothing else than the fraction of lines:

because, if rectitude is quality, fraction also will be quality. And

hence, since its generation is in quality, an angle will be entirely

quality. But Euclid, and those who call it inclination, place it in the

category of relation. But they call it quantity, who say that it is the

first interval under a point, that is immediately subsisting after a

point. In the number of which is Plutarch, who constrains Apollonius

also into the same opinion. For it is requisite (says he) there should

be some first interval, under the inclination of containing lines

or superficies. But since the interval, which is under a point, is

continuous, it is not possible that a first interval can be assumed;

since every interval is divisible in infinitum. Besides, if we any

how distinguish a first interval, and through it draw a right line, a

triangle is produced, and not one angle. But Carpus Antiochenus says,

that an angle is quantity, and is the distance of its comprehending

lines, or superficies; and that this is distant by one interval, and

yet an angle is not on that account a line: since it is not true that

every thing which is distant by only one interval, is a line. But this

surely is the most absurd of all, that there should be any magnitude

except a line, which is distant only by one interval. And thus much

concerning the nature of an angle. But with respect to the division

of angles, some consist in superficies, but others in solids. And of

those which are in superficies, some are in simple ones, but others in

such as are mixt. For an angle may be produced in a cylindric, conic,

spherical, and plane superficies. But of those which consist in simple

superficies, some are constituted in the spherical; but others in the

plane. For the zodiac itself forms angles, dividing the equinoctial

in two parts, at the vertex of the cutting superficies. And angles

of this kind subsist in a spherical superficies. But of those which

are in planes, some are comprehended by simple lines, others by mixt

ones; and others, again, by both. For in the shield-like figure[155],

an angle is comprehended by the axis, and the line of the shield: but

one of these lines is mixt, and the other simple. But if a circle

cuts the shield, the angle will be comprehended by the circumference,

and the ellipsis. And when cissoids, or lines similar to an ivy leaf,

closing in one point like the leaves of ivy (from whence they derive

their appellation) make an angle, such an angle is comprehended by

mixt lines. Also, when the hippopede, or line familiar to the foot

of a mare, which is one of the spirals, inclining to another line,

forms an angle, it is comprehended by mixt lines. Lastly, the angles

contained by a circumference and a right line, are comprehended by

simple lines. But of these again, some are contained by such as are

similar in species, but others by such as are dissimilar. For two

circumferences, mutually cutting, or touching each other, produce

angles: and these triple, for they are either on both sides convex,

when the convexities of the circumferences are external: or on both

sides concave, when both the concavities are external; which they call

sistroides; or mixt from convex and concave lines, as the lines called

lunulas. But besides this, angles are contained in a twofold manner,

by a right line and a circumference: for they are either contained by

a right line, and a concave circumference, as the semi-circular angle;

or by a right line and a convex circumference, as the cornicular angle.

But all those which are comprehended by two right lines, are called

rectilinear angles, which have likewise a triple difference[156]. The

geometrician, therefore, in the present hypothesis, defines all those

angles which are constituted in plane superficies, and gives them the

common name of a plane angle. And the genus of these he denominates

inclination: but the place, the plane itself, for angles have position:

but their origin such, that it is requisite there should be two lines

at least, and not three as in a solid. And that these should touch

each other, and by touching, must not lie in a right line, as an angle

is the inclination and comprehension of lines: but is not distance

only, according to one interval. But if we examine this definition,

in the first place it appears that it does not admit, an angle can be

perfected by one line; though a cissoid, which is but one, perfects

an angle. And, in like manner, the hippopede. For we call the whole

a cissoid, and not its portions (lest any one should say, that the

conjunction of these forms an angle) and the whole a spiral, but not

its parts. Each, therefore, since it is one, forms an angle to itself,

and not to another. But after this, he is faulty, in defining an angle

to be inclination. For how, on this hypothesis, will there be two

angles, from one inclination? How can we call angles equal and unequal?

And whatever else is usually objected against this opinion. Thirdly,

and lastly, that part of the definition, which says, _and not placed

in a right line_, is superfluous in certain angles, as in those

which are formed from orbicular lines. For without the assistance of

this part, the definition is perfect; since the inclination of one of

the lines to the other, forms the angle. And it is not possible that

orbicular angles should be placed in a right line. And thus much we

have thought proper to say concerning the definition of Euclid; partly,

indeed, interpreting, and partly doubting its truth.

DEFINITION IX.

But when the Lines containing the Angle, are right, the

Angle is called RECTILINEAR.

An angle is the symbol and image of the connection and compression,

which subsists in the divine genera, and of that order which collects

divisibles into one, particles into an impartible nature, and the many

into conciliating community. For it is the bond of a multitude of lines

and superficies, the collector of magnitude into the impartibility of

points, and the comprehender of every figure which is composed by its

confining nature. On which account, the oracles[157] call the angular

junctions of figures, knots, so far as they bring with them an image of

connecting union, and divine conjunctions, by which discrete natures

mutually cohere with each other. The angles, therefore, subsisting in

superficies, express the more immaterial, simple, and perfect unions

which superficies contain: but those which are in solids, represent

the unions, which proceed even to inferiors, and supply a community

to things disjunct, and a construction of the same nature, to things

which on every side receive a perfect partition. But of the angles in

superficies, some shadow forth primary and unmixt unions; but others,

such as comprehend in themselves, an infinity of progressions. And

some, indeed, are the sources of union to intellectual forms; but

others, to sensible reasons; and others, again, are copulative of those

forms which obtain between these, a middle situation. Hence, the angles

which are made from circumferences, imitate those causes which envelop

intellectual variety in coercive union; for circumferences, hastening

to coalesce with each other, are images of intellect, and intellectual

forms. On the contrary, rectilineal angles, are the symbols of those

unions which preside over sensibles, and afford a conjunction of the

reasons subsisting in these: but mixt angles represent the preservers

of the communion, as well of sensible, as of intellectual forms,

according to one immoveable union. It is requisite, therefore, by

regarding these paradigms, or exemplars, to render the causes of each.

For among the Pythagoreans we shall find various angles dedicated to

various gods. Thus, Philolaus, consecrates to some a triangular, but to

others a quadrangular angle; and to others, again, different angles.

Likewise, he permits the same to many gods, and many to the same god,

according to the different powers which they contain. And with a view

to this, and to the demiurgic triangle, which is the primary cause of

all the ornament of the elements, it appears to me, that Theodorus

Asinæus the philosopher, constitutes some of the gods, according to

sides; but others, according to angles. The first, indeed, supplying

progression and power; but the second, the conjunction of the universe,

and the collection of progressive natures again into one. But these,

indeed, direct us to the knowledge of the things which are. And we must

not wonder that lines are here said to contain an angle. For the one

and impartible nature which is found in these, is adventitious: but

in the gods themselves, and in true beings, the whole, and impartible

good, precedes things many, and divided.

DEFINITION X.

When a Right Line standing on a Right Line, makes the successive

Angles on each side equal to one another, each of the equal

Angles, is a Right Angle; and the insisting Right Line, is called

a PERPENDICULAR to that upon which it stands.

DEFINITION XI.

An OBTUSE ANGLE is that which is greater than a RIGHT

ANGLE.

DEFINITION XII.

But an ACUTE ANGLE, is that which is less than a

RIGHT ANGLE.

These are the triple species of angles, which Socrates speaks of in

the Republic, and which are received by geometricians from hypothesis;

a right-line constituting these angles, according to a division into

species; I mean, the right, the obtuse, and the acute. The first of

these being defined by equality, identity and similitude; but the

others being composed through the nature of the greater and the lesser;

and lastly, through inequality and diversity, and through the more and

the less, indeterminately assumed. But many geometricians, are unable

to render a reason of this division, and use the assertion, that there

are three angles, as an hypothesis[158]. So that, when we interrogate

them concerning its cause, they answer, this is not to be required

of them as geometricians. However, the Pythagoreans, referring the

solution of this triple distribution to principles, are not wanting in

rendering the causes of this difference of right-lined angles. For,

since one of the principles subsists according to bound, and is the

cause of limitation, identity, and equality, and lastly, of the whole

of a better co-ordination: but the other is of an infinite nature,

and confers on its progeny, a progression to infinity, increase,

and decrease, inequality, and diversity of every kind, and entirely

presides over the worse series; hence, with great propriety, since

the principles of a right-lined angle are constituted by these, the

reason proceeding from bound, produces a right angle, one, with respect

to the equality of every right angle, endued with similitude, always

finite and determinate, ever abiding the same, and neither receiving

increment nor decrease. But the reason proceeding from infinity,

since it is the second in order, and of a dyadic nature, produces

twofold angles about the right angle, distinguished by inequality,

according to the nature of the greater and the lesser, and possessing

an infinite motion, according to the more and the less, since the one

becomes more or less obtuse; but the other more or less acute. Hence,

in consequence of this reason, they ascribe right angles to the pure

and immaculate gods of the divine ornaments, and divine powers which

proceed into the universe, as the authors of the invariable providence

of inferiors; for rectitude, and an inflexibility and immutability to

subordinate natures accords with these gods: but they affirm, that the

obtuse and acute angles should be ascribed to the gods, who afford

progression, and motion, and a variety of powers. Since obtuseness is

the image of an expanded progression of forms; but acuteness possesses

a similitude to the cause dividing and moving the universe. But

likewise, among the things which are, rectitude is, indeed, similar to

essence, preserving the same bound of its being; but the obtuse and

acute, shadow forth the nature of accidents. For these receive the more

and the less, and are indefinitely changed without ceasing. Hence,

with great propriety, they exhort the soul to make her descent into

generation, according to this invariable species of the right angle,

by not verging to this part more than to that; and by not affecting

some things more, and others less. For the distribution of a certain

convenience and sympathy of nature, draws it down into material error,

and indefinite variety[159]. A perpendicular line is, therefore, the

symbol of inflexibility, purity, immaculate, and invariable power,

and every thing of this kind. But it is likewise the symbol of divine

and intellectual measure: since we measure the altitudes of figures

by a perpendicular, and define other rectilineal angles by their

relation to a right angle, as by themselves they are indefinite and

indeterminate. For they are beheld subsisting in excess and defect,

each of which is, by itself, indefinite. Hence they say, that virtue

also stands according to rectitude; but that vice subsists according

to the infinity of the obtuse and acute, that it produces excesses and

defects, and that the more and the less exhibit its immoderation, and

inordinate nature. Of rectilineal angles, therefore, we must establish

the right angle, as the image of perfection, and invariable energy,

of limitation, intellectual bound, and the like; but the obtuse and

acute, as shadowing forth infinite motion, unceasing progression,

division, partition and infinity. And thus much for the theological

speculation of angles. But here we must take notice, that the genus

is to be added to the definitions of an obtuse and acute angle; for

each is right-lined, and the one is greater, but the other less than a

right-angle. But it is not absolutely true, that every angle which is

less than a right one, is acute. For the cornicular is less than every

right-angle, because less than an acute one, yet is not on this account

an acute angle. Also, a semi-circular is less than any right-angle, yet

is not acute. And the cause of this property is because they are mixt,

and not rectilineal angles. Besides, many curve-lined angles appear

greater than right-lined angles, yet are not on this account obtuse;

because it is requisite that an obtuse should be a right-lined angle.

Secondly, as it was the intention of Euclid, to define a right-angle,

he considers a right-line standing upon another right-line, and making

the angles on each side equal. But he defines an obtuse and acute

angle, not from the inclination of a right line to either part, but

from their relation to a right-angle. For this is the measure of

angles deviating from the right, in the same manner as equality of

things unequal. But lines inclined to either part, are innumerable,

and not one alone, like a perpendicular. But after this, when he says,

(_the angles equal to one another_) he exhibits to us a specimen

of the greatest geometrical diligence; since it is possible that angles

may be equal to others, without being right. But when they are equal

to one another, it is necessary they should be right. Besides, the

word _successive_ appears to me not to be added superfluously, as

some have improperly considered it; since it exhibits the reason of

rectitude. For it is on this account that each of the angles is right;

because, when they are _successive_, they are equal. And, indeed,

the insisting right-line, on account of its inflexibility to either

part, is the cause of equality to both, and of rectitude to each.

The cause, therefore, of the rectitude of angles, is not absolutely

mutual equality, but position in a consequent order, together with

equality. But, besides all this, I think it here necessary to call to

mind, the purpose of our author; I mean, that he discourses in this

place, concerning the angles consisting in one plane. And hence, this

definition is not of every perpendicular; but of that which is in one

and the same plane. For it is not his present design to define a solid

angle. As, therefore, he defines, in this place, a plane angle, so

likewise a perpendicular of this kind. Because a solid perpendicular

ought not to make right angles to one right-line only; but to all which

touch it, and are contained in its subject plane: for this is its

necessary peculiarity.

DEFINITION XIII.

A BOUND is that which is the Extremity of any thing[160].

A Bound, in this place, is not to be referred to all magnitudes, for

there is a bound and extremity of a line; but to the spaces which are

contained in superficies, and to solid bodies. For he now calls a

bound, the ambit which terminates and distinguishes every space. And

a bound of this kind, he defines to be an extremity: but not after

the manner in which a point is called the extremity of a line, but

according to its property of including and excluding from circumjacent

figures. But this name is proper to geometry in its infant state, by

which they measured fields, and preserved their boundaries distinct and

without confusion, and from which they arrived at the knowledge of the

present science. Since, therefore, Euclid calls the external ambit,

a bound, it is not without propriety that he, by this means, defines

the extremity of spaces. For by this, every thing comprehended is

circumscribed. I say, for example, in a circle, its bound and extremity

is the circumference; but itself, a certain plane space: and so of the

rest.

DEFINITION XIV.

A FIGURE is that which is comprehended by one or more

Boundaries.

Because figure is predicated in various ways, and is divided into

different species, it is requisite, in the first place, to behold

its differences; and afterwards to discourse concerning that figure

which is proposed in this Definition. There is, then a certain figure

which is constituted by mutation, and is produced from passion,

while the recipients of the figure are disturbed, divided, or taken

away; while they receive additions, or are altered, or suffer other

various affections. There is also a figure, which is produced by the

potter’s, or statuary’s art, according to the pre-existent reason,

which art itself contains: art, indeed, producing the form, but matter

receiving from thence, form, and beauty, and elegance. But there are

still more noble and more illustrious figures than these, the skilful

operations of nature. Some, indeed, existing in the elements under

the moon[161], and having a power of comprehending the reasons those

elements contain: but others are situated in the celestial regions,

distinguishing their powers, and endless revolutions. For the heavenly

bodies, both when considered by themselves, and with relation to each

other, exhibit an abundant and admirable variety of figures; and at

different times they present to our view different forms, bringing with

them a splendid image of intellectual species; and, by their elegant

and harmonious revolutions, describing the incorporeal and immaterial

powers of figures. But there are, again, besides all these, most pure

and perfect beauties, the figures of souls, which, because they are

full of life, and self-motive, have an existence prior to things moved

by another; and which, because they subsist immaterially, and without

any dimension, excel the forms which are endued with dimension and

matter. In the nature of which we are instructed by Timæus, who has

explained to us the demiurgic, and essential figure of souls[162].

But again, the figures of intellects are by far more divine than the

figures of souls; for these, on every side, excel partible essences;

are every where resplendent with impartible and intellectual light;

are prolific, effective, and perfective of the universe; are equally

present, and firmly abide in all things; and procure union to the

figures of souls; but recall the mutation of sensible figures to the

limitation of their proper bound. Lastly, there are, separate from

all these, those perfect, uniform, unknown, and ineffable figures of

the gods, which are resident, indeed, in the figures of intellects;

but jointly terminate all figures, and comprehend all things in their

unifying boundaries. The properties of which the theurgic art, also

expressing, surrounds various resemblances of the gods, with various

figures. And some, indeed, it fashions by characters, in an ineffable

manner; for characters of this kind, manifest the unknown powers of the

gods: but others it imitates by forms and images; fashioning some of

them erect, and others fitting; and some similar to a heart, but others

spherical, and others expressed by different figures. And again, some

it fabricates of a simple form; but others it composes from a multitude

of forms; and some are sacred and venerable; but others are domestic,

exhibiting the peculiar gentleness of the gods. And some it constructs

of a severe aspect; and lastly, attributes to others, different

symbols, according to the similitude and sympathy pertaining to the

gods[163]. Since, therefore, figure derives its origin from the gods

themselves, it arrives, by a gradual progression, even to inferiors,

in these also appearing from primary causes. Since it is requisite to

suppose the perfect before the imperfect, and things situated in the

stability of their own essence, prior to those which subsist in others,

and previous to things full of their own privation, such as preserve

their proper nature sincere. Such figures, therefore, as are material,

participate of material inelegance, and do not possess a purity

convenient to their nature. But the celestial figures are divisible,

and subsist in others. And the figures of souls are endued with

division, and variety, and involution of every kind; but the figures

of intellects, together with immaterial union, possess a progression

into multitude. And lastly, the figures of the gods are free, uniform,

simple, and generative; they subsist before all things, containing all

perfection in themselves, and extending from themselves to all things,

the completion of forms. We must not, therefore, listen to, and endure

the opinions of many, who affirm, that certain additions, ablations,

and alterations, produce sensible figures, (for motions, since they are

imperfect, cannot possess the principle and primary cause of effects;

nor could the same figures often be produced from contrary motions; for

the same form is sometimes generated from addition and detraction,)

but we must consider operations of this kind as subservient to other

purposes in generation, and derive the perfection of figure from other

primogenial causes. Nor must we subscribe to their opinion, who assert

that figures destitute of matter can have no subsistence; but those

only which appear in matter. Nor to theirs, who acknowledge, indeed,

that they are external to matter, but consider them as subsisting

alone, according to thought and abstraction. For where shall we

preserve in safety, the certainty, beauty, and order of figures, among

things which subsist by abstraction? For, since they are of the same

kind with sensibles, they are far distant from indubitable and pure

certainty. But from whence do they derive the certainty, order, and

perfection which they receive? For they either derive it from sensibles

(but they have no subsistence in these), or from intelligibles (but in

these they are more perfect), since, to say from that which is not, is

the most absurd of all. For nature does not produce imperfect figures,

and leave the perfect without any subsistence. Nor is it lawful, that

our soul should fabricate more certain, perfect, and orderly figures,

than intellect and the gods themselves. There are, therefore, prior

to sensible figures, self-moving, intellectual, and divine reasons of

figures. And we are excited, indeed, from the obscurity of sensible

forms, but we produce internal reasons, which are the lucid images

of others. And we possess a knowledge of sensible figures, by their

exemplars resident in soul (παραδειγματικῶς), but we comprehend by

images (εἰκονικῶς) such as are intellectual and divine. For the reasons

we contain, emerging from the dark night of oblivion, and propagating

themselves in sciential variety, exhibit the forms of the gods, and

the uniform bounds of the universe, by which they ineffably convert

all things into themselves. In the gods, therefore, there is both an

egregious knowledge of universal figures, and a power of generating and

constituting all inferiors. But in natures, figures are endued with

a power generative of apparent forms; but are destitute of cognition

and intellectual perception. And, in particular souls, there is,

indeed, an immaterial intellection, and a self-energizing knowledge;

but there is wanting a prolific, and efficacious cause. As, therefore,

nature, by her forming power presides over sensible figures, in the

same manner, soul, by her gnostic energy, drops in the phantasy as in

a mirror, the reasons of figures. But the phantasy receiving these in

her shadowy forms, and possessing images of the inherent reasons of

the soul, affords by these the means of inward conversion to the soul,

and of an energy directed to herself, from the spectres of imagination.

Just as if any one beholding his image in a mirror, and admiring the

power of nature, and his own beauty, should desire to see himself in

perfection, and should receive a power of becoming, at the same time,

the perceiver, and the thing perceived. For the soul, after this

manner, looking abroad into the bright mirror of the phantasy, and

surveying the shadowy figures it contains, and admiring their beauty

and order, pursues, in consequence of her admiration, the reasons from

which these images proceed; and being wonderfully delighted, dismisses

their beauty, as conversant about spectres alone; but afterwards

seeks her own purer beauty, and desires to pass into her own profound

retreats, and there to perceive the circle and the triangle, and all

things subsisting together, in an impartible manner, and to insert

herself in the objects, to contract her multitude into one; and lastly,

to behold the occult and ineffable figures of the gods, seated in the

most sacred and divine recesses of her nature. She is likewise desirous

of bringing into light, from its awful concealment, the solitary

beauty, of the gods, and of perceiving the circle, subsisting in its

true perfection, more impartible than any centre, and the triangle

without interval; and lastly, by ascending into an union with herself,

of surveying every object which is subject to the power of cognition.

The figure, therefore, which is self-motive, precedes that which is

moved by another; and the impartible that which is self-motive: but

that which is the same with _one_, precedes the impartible itself.

For all things are bounded, when they return to the unities of their

nature; since all things pass through these as a divine entrance into

being. And thus much for this long digression, which we have delivered

according to the sentiments of the Pythagoreans. But the geometrician,

contemplating that figure which is seated in the phantasy, and defining

this, in the first place, (since this definition agrees with sensibles,

in the second place) says, that figure is that which is comprehended by

one or more boundaries. For, since he receives it together with matter,

and conceives of it as distant with intervals, he does not improperly

call it finite and terminated[164]. [Since every thing which contains

either intelligible or feasible matter, is allotted an adventitious

bound; and is not itself bound, but that which is bounded.] Nor is it

the bound of itself; but one of its powers is terminating, and the

other terminated. Nor does it subsist in bound itself, but is contained

by bound. For figure is joined to quantity, and subsists together

with it; and, at the same time, quantity is subjected to figure; but

the reason and aspect of that quantity is nothing else than figure

and form. Since, indeed, reason terminates quantity, and adds to it

a particular character and bound, either simple or composite. For,

since this also exhibits the twofold progression of _bound_ and

_infinite_ in its proper forms, (in the same manner as the reason

of an angle,) it invests the objects of its comprehension with one

boundary and simple form, according to _bound_, but with many,

according to infinity[165]. Hence, every thing figured, vindicates to

itself either one boundary, or a many. Euclid, therefore, denominating

that which is figured and material, and annexed to quantity figure,

does not improperly say, that it is contained by one or more terms.

But Possidonius defines figure to be concluding bound, separating

the reason of figure from quantity; and considering it as the cause

of terminating, defining, and comprehending quantity. For that which

encloses, is different from that which is enclosed; and bound from that

which is bounded. And Possidonius, indeed, seems to regard the external

surrounding bound; but Euclid, the whole subject. Hence, the one calls

a circle a figure, with relation to its whole plane, and exterior

ambit; but the other with relation to its circumference only. And the

one defines that which is figured, and which is beheld together with

its subject: but the other desires to define the reason of the circle;

I mean that which terminates and concludes its quantity. But if any

logician, and captious person, should blame the definition of Euclid,

because he defines genus from species (for things contained by one or

more terms, are the species of figure,) we shall assert, in opposition

to such an objection, that genera also pre-occupy in themselves the

powers of species. And when men of ancient authority, were willing to

manifest genera themselves, from those powers which genera contain,

they appeared, indeed, to enter on their design from species, but, in

reality, they explained genera from themselves, and from the powers

which they contain. The reason of figure, therefore, since it is one,

comprehends the differences of many figures, according to the bound

and infinity residing in its nature. And he who defined this reason,

was not void of understanding, whilst he comprehended in a definition,

the differences of the powers it contained. But you will ask, From

whence does the reason of figure originate, and by what causes is

it perfected? I answer, that it first arises from _bound_ and

_infinite_, and that which is mixed from these. Hence it produces

some species from _bound_, others from _infinite_, and others

from the _mixt_. And this it accomplishes by bringing the form of

bound to circles; but that of infinite, to right lines: and that of

the mixt to figures composed from right and circular lines. But, in

the second place, this reason is perfected from that totality, which

is separated into dissimilar parts. From whence, indeed, it occasions

a whole to every form, and each figure is cut into different species.

For a circle, and every right-lined figure may be divided, by reason or

proportion, into dissimilar figures; which is the business of Euclid

in his book of divisions, where he divides one figure into figures

similar to each as are given; but another into such as are dissimilar.

In the third place, it is invigorated from accumulated multitude, and,

on account of this, extends forms of every kind, and produces the

multiform reasons of figures. Hence, in propagating itself, it does

not cease till it arrives at something last, and has unfolded all the

variety of forms. And, as in the intelligible world, _one_ is

shewn to abide in that which _is_; and, at the same time, that

which _is_ in _one_, so likewise, reason exhibits circular in

right-lined figures; and on the contrary, rectilinear comprehended in

circular figures. And it peculiarly manifests its whole nature in each,

and all these in all. Since the whole subsists in all collectively, and

in each separate and apart. From that order, therefore, it is endued

with this power. In the fourth place, it receives from the first of

numbers[166], the measures of the progression of forms. From whence it

constitutes all figures according to numbers; some, indeed, according

to the more simple, but others according to the more composite. For

triangles, quadrangles, quinquangles, and all multangles, proceed in

infinitum, together with the mutations of numbers. But the cause of

this is, indeed, unknown to the vulgar, though, to those who understand

where number and figure subsist, the reason is manifest. Fifthly,

it is replete with that division of forms, which divides forms into

other similar forms, from another second totality, which is also

distributed into similar parts. And by this, a triangular reason is

divided into triangles, and a quadrangular reason into quadrangles.

And hence, exercising our inward powers, we effect what I have said

in images, since it pre-existed by far the first in its principles.

But by regarding these distributions, we may render many causes of

figures, reducing them to their first principles. And the more common,

or geometrical figure, is allotted an order of this kind, and from so

many causes, receives the perfection of its nature. But, from hence

it advances to the genera of the gods, and is variously attributed

according to its various forms, and energizes differently in different

gods. To some, indeed, affording more simple figures; but to others,

such as are more composite. And to some, again, assigning primary

figures, and those which are produced in superficies; but to others

(entering the tumor of solid bodies) such figures, as in solids are

convenient to themselves. For all figures, indeed, subsist in all,

since the forms of the gods are accumulated, and full of universal

powers: but, by their peculiarity, they produce one thing according

to another. For one possesses all things circularly, another in a

triangular manner, but another according to a quadrangular reason. And

in a similar manner in solids.

DEFINITION XV.

A CIRCLE is a Plane Figure, comprehended by one Line, which is

called the Circumference, to which all Right Lines falling from

a certain Point within the Figure, are equal to each other.

DEFINITION XVI.

And that Point is called the CENTRE of the CIRCLE.

A Circle is the first, most simple, and most perfect of figures. For

it excels all solids, because it exists in a more simple place; but

it is superior to the figures subsisting in planes, on account of

its similitude and identity. And it has a corresponding proportion

to bound, and unity, and a better co-ordination of being. Hence, in

a distribution of mundane and super-mundane figures, you will always

find that the circle is of a diviner nature. For if you make a division

into the heavens, and the universal regions of generation, you must

assign to the heavens a circular form; but to generation, that of a

right line, For whatever among generable natures is circular, descends

from the heavens; since generation revolves into itself, through

their circumvolutions, and reduces its unstable mutation to a regular

and orderly continuance. But if you distribute incorporeal natures

into soul and intellect, you will say, that the circle belongs to

intellect, and the right line to the soul. And on this account, the

soul, by its conversion to intellect, is said to be circularly moved;

and it possesses the same proportion to intellect, as generation to

the heavens. For it is circularly moved, (says Socrates[167],) because

it imitates intellect. But the generation and progression of soul is

made according to a right-line. For it is the property, of the soul

to apply herself at different times to different forms. But if you

wish to divide into body and soul, you must constitute every thing

corporeal, according to the right line; but you must assign to every

animal a participation of the identity and similitude of the circle.

For body is a composite, and is endued with various powers, similar to

right-lined figures: but soul is simple and intelligent; self-motive,

and self-operative; converted into, and energizing in herself. From

whence, indeed, Timæus also, when he had composed the elements of

the universe from right-lined figures, assigned to them a circular

motion and formation, from that divine soul which is seated in the

bosom of the world. And thus, that the circle every where holds the

first rank, in respect of other figures, is sufficiently evident from

the preceding observations. But it is requisite to survey its whole

series, beginning supernally, ending in inferiors, and perfecting all

things, according to the aptitude of the natures which receive its

alliance. To the gods, therefore, it affords a conversion to their

causes, and ineffable union: it occasions their abiding in themselves,

prevents their departing from their own beatitude, strengthens their

highest unions, as centres desirable to inferior natures; and stably

places about these the multitude of the powers which the gods possess,

containing them in the simplicity of their essences. But the circle

affords to intellectual natures, a perpetual energy in themselves, is

the cause of their being filled with knowledge from themselves, and

of possessing in their essences, intelligibles contractedly; and of

perfecting intellections in themselves. For every intellect, proposes

to itself that which is intelligible; and this is as a centre to

intellect, about which it continually revolves: for intellect folds

itself, and operates about this, and is united within itself on all

sides, by universal intellectual energies. But it extends to souls by

illumination, a self-vital, and self-motive power, and an ability of

turning, and leaping round intellect, and of returning according to

proper convolutions, unfolding the impartibility of intellect. Again,

the intellectual orders excel souls after the manner of centres, but

souls energize circularly about their nature. For every soul, according

to its intellectual part, and the supreme _one_, which is the

very flower of its essence, receives a centre: but, according to its

multitude, it has a circular revolution, desiring, by this means, to

embrace the intellect which it participates. But, to the celestial

bodies, the circle affords an assimilation to intellect, equality, a

comprehension of the universe, in proper limits, revolutions which

take place in determinate measures, a perpetual subsistence, a nature

without beginning and end, and every thing of this kind. And to the

elements under the concave of the moon’s orb, it is the cause of a

period, conversant with mutations; an assimilation to the heavens; that

which is without generation, in generated natures; that which abides in

things which are moved; and whatever is bounded in partible essences.

For all things are perpetual, through the circle of generation; and

equability is every where preserved on account of the reciprocation

of corruption. Since, if generation did not return, in a circular

revolution, in a short space of time, the order, and all the ornament

of the elements would vanish. But again, the circle procures to animals

and plants, that similitude which is found in generations; for these

are produced from seeds, and seeds from these. Hence, generation here,

and a circumvolution, alternately takes place, from the imperfect to

the perfect, and the contrary; so that corruption subsists together

with generation. But, besides this, to unnatural productions it

imposes order, and reduces their indeterminate variety to the

limitation of bound; and, through this, nature herself is gracefully

ornamented in the last vestiges of her powers. Hence, things contrary

to nature have a revolution according to determinate numbers, and not

only fertility, but also sterility, subsists according to the alternate

convolutions of circles (as the discourse of the Muses evinces), and

all evils though they are dismissed from the presence of the gods,

into the place of mortals, yet these roll round, says Socrates, and to

these there is present a circular revolution, and a circular order;

so that nothing immoderate and evil is deserted by the gods; but that

providence, which is perfective of the universe, reduces also the

infinite variety of evils, to bound, and an order convenient to their

nature. The circle, therefore, is the cause of ornament to all things,

even to the last participations, and leaves nothing destitute of

itself, since it supplies beauty, similitude, formation, and perfection

to the universe. Hence too, in numbers it contains the middle centres

of the whole progression of numbers, which revolves from unity to the

decad (or ten). For five and six exhibit a circular power, because, in

the progressions from themselves, they return again into themselves,

as is evident in the multiplication of these numbers. Multiplication,

therefore, is an image of progression, since it is extended into

multitude; but an ending in the same species, is an image of regression

into themselves. But a circular power affords each of these, exciting,

indeed, as from an abiding centre, those causes which are productive

of multitude; but converting multitude after the productions to their

causes. Two numbers, therefore, having the properties of a circle,

possess the middle place between all numbers: of which one, indeed,

precedes every convertible genus of males and an odd nature; but the

other, recalls every thing feminine and even, and all prolific series,

to their proper principles, according to a circular power. And thus

much concerning the perfection of the circle. Let us now contemplate

the mathematical definition of the circle, which is every way perfect.

In the first place, therefore, he defines it a figure, because, indeed,

it is finite, and every where comprehended by one limit, and is not of

an infinite nature, but associated to bound. Likewise plane, because,

since figures are either beheld in superficies, or in solid bodies, a

circle is the first of plane figures, excelling solids in simplicity,

but possessing the proportion of unity to planes. But comprehended by

one line, because it is similar to one, by which it is defined, and

because it does not extrinsically receive a variety of surrounding

terms. And again, that this line makes all the lines drawn to it from

a certain point within equal, because of the figures which are bounded

by one line, some have all the lines proceeding from the middle equal;

but others not at all. For the ellipsis is comprehended by one line,

yet all the lines issuing from the centre, and bounded by is curvature,

are not equal, but only two. Also the plane, which is included by the

line called a cissoid, has one containing line, yet it does not contain

a centre, from which all the lines are equal. But, because the centre

in a circle is entirely one point (for there are not many centres of

one circle), on this account, the geometrician adds, that lines falling

from one point to the bound of the circle, are equal. For there are

infinite points within it, but of all these, one only has the power of

a centre. And because this one point, from which all the lines drawn to

the circumference of the circle are equal, is either within the circle,

or without (for every circle has a pole, from which all the lines drawn

to its circumference are equal), on this account he adds, _of the

points within the figure_, because, here he receives the centre

alone, and not the pole. For he wishes to behold all its properties in

one plane, but the pole is more elevated than the subject plane. Hence,

he necessarily adds, in the end of the definition, that this point,

which is placed within the circle, and to which all right lines drawn

from it to the circumference, are equal, is the centre of the circle.

For there are only two points of this kind, the pole and the centre.

But the former is without, and the other within the plane. Thus, for

instance, if you conceive a perpendicular standing on the centre of a

circle, its superior extremity is the pole: for all lines drawn from

it to the circumference of the circle, are demonstrated to be equal.

And, in like manner, in a cone, the vertex of the whole cone, is the

pole of the circle at the base. And thus far we have determined what a

circle is, and its centre, and what the nature is of its circumference,

and the whole circular figure. Again, therefore, from these, let us

return to the speculation of their exemplars, contemplating in them

the centre, according to one impartible and stable excellence. But the

distances from the centre, according to the progressions which are made

from one, to multitude infinite in capacity. And the circumference of

the circle, according to the regression of the progressions to the

centre, by means of which the multitude of powers are rolled round

their union, and all of them hasten to its comprehension, and desire

to energize about its indivisible embrace. And, as in the circle

itself, all things subsist together, the centre, intervals, and

external circumference; so in these which are its image, one thing

has not an essence pre-existent, and another consequent in time;

but all things are, indeed, together, permanency, progression, and

regression. But these differ from those, because the former subsist

indivisibly, and without any dimension; but the latter with dimension;

and in a divisible manner; the centre existing in one place, the lines

emanating from the centre, in another; and the external circumference

terminating the circle, having a still different situation. But there

all things abide in one: for if you regard that which performs the

office of a centre, you will find it the receptacle of all things. If

the progression distant from the centre, in this, likewise, you will

find all things contained. And, in a similar manner, if you regard

its regression. When, therefore, you are able to perceive all things

subsisting together, and have taken away the defect proceeding from

dimension, and have removed from your inward vision, the position about

which partition subsists, you will find the true circle, advancing to

itself, bounding, and energizing in itself, existing both one and many,

and abiding, proceeding, and returning; likewise firmly establishing

that part of its essence which is most impartible, and especially

singular; but advancing from this according to rectitude, and the

infinity which it contains; and rolling itself from itself to one, and

exciting itself by similitude and identity to the impartible centre

of its nature, and to the occult power of the one which it contains.

But this one, which the circle contains, and environs in its bosom, it

emulates according to the multitude of its own nature. For that which

is convolved, imitates that which abides, and the periphery is as a

centre which is distant with interval, and nods to itself, hastening

to receive, and to become one with the centre, and to terminate its

regress where it received the principle of its progression. For the

centre is every where in the place of that which is lovely, and the

object of desire, presiding over all things which subsist about its

nature, and existing as the beginning and author of all progressions.

And this the mathematical centre also expresses, by terminating all the

lines falling from itself to the circumference, and by affording to

them equality, as an image of proper union. But the oracles likewise

define the centre, after this manner: _The centre is that from which

and[168] to which all the lines to the circumference are equal_.

Indicating the beginning of the distance of the lines, by the particle

_from which_; but the middle of the circumference by the particle

_to which_: for this, in every part, is joined with the centre.

But if it be necessary to declare the first cause, through which a

circular figure appears and receives its perfection, I affirm, that

it is the supreme order of intelligibles. For the centre, indeed, is

assimilated to the cause of bound; but the lines emanating from this,

and which are infinite, with respect to themselves, both in multitude

and magnitude, represent infinity; and the line which terminates their

extension, and conjoins the circular figure with the centre, is similar

to that occult ornament, consisting from the intelligible orders;

which Orpheus also says, is circularly borne, in the following words,

_But it is carried with an unwearied energy, according to an infinite

circle_. For, since it is moved intelligibly, about that which is

intelligible, having it for the centre of its motion, it is, with great

propriety, said to energize in a circular manner. Hence, from these

also, the triadic god[169] proceeds, who contains in himself the cause

of the progression of right-lined figures. For on this account, wise

men, and the most mystic of theologists, have fabricated his name.

[[170]Hence too, it is manifest, that a circle is the first of all

figures:] but a triangle is the first of such as are right-lined.

Figures, therefore, appear first in the regular ornaments of the gods;

but they have a latent subsistence, according to pre-existent causes,

in intelligible essences.

DEFINITION XVII.

A DIAMETER of a Circle is a certain straight Line, drawn through

the Centre, which is terminated both ways by the Circumference

of the Circle, and, divides the Circle into two equal Parts.

Euclid here perspicuously shews, that he does not define every

diameter, but that which belongs to a circle only. Because there

is a diameter of quadrangles and all parallelograms, and likewise

of a sphere among solid figures. But in the first of these, it is

denominated a diagonal: but in a sphere, the axis; and in circles

the diameter only. Indeed, we are accustomed to speak of the axis of

an ellipsis, cylinder, and cone; but of a circle, with propriety,

the diameter. This, therefore, in its genus, is a right-line; but as

there are many right-lines in a circle, as likewise infinite points,

one of which is a centre, so this only is called a diameter, which

passes through the centre, and neither falls within the circumference,

nor transcends its boundary; but is both ways terminated by its

comprehensive bound. And these observations exhibit its origin. But

that which is added in the end, that it also divides the circle into

two equal parts, indicates its proper energy in the circle, exclusive

of all other lines drawn through the centre, which are not terminated

both ways by the circumference. But they report, that Thales first

demonstrated, that the circle was bisected by the diameter. And the

cause of this bisection, is the indeclineable transit of the right

line, through the centre. For, since it is drawn through the middle,

and always preserves the same inflexible motion, according to all its

parts, it cuts off equal portions on both sides to the circumference

of the circle. But if you desire to exhibit the same mathematically,

conceive the diameter drawn, and one part of the circle placed on

the other[171]. Then, if it is not equal, it either falls within, or

without; but the consequence either of these ways must be, that a less

right-line will be equal to a greater. Since all lines from the centre

to the circumference are equal. The line, therefore, which tends to

the exterior circumference, will be equal to that which tends to the

interior. But this is impossible. These parts of the circle, then,

agree, and are on this account equal. But here a doubt arises, if two

semi-circles are produced by one diameter, and infinite diameters may

be drawn through the centre, a double of infinities will take place,

according to number. For this is objected[172] by some against the

section of magnitudes to infinity. But this we may solve by affirming,

that magnitude may, indeed, be divided infinitely, but not into

infinites. For this latter mode produces infinites in energy, but the

former in capacity only. And the one affords essence to infinite,

but the other is the source of its origin alone. Two semi-circles,

therefore, subsist together with one diameter, yet there will never be

infinite diameters, although they may be infinitely assumed. Hence,

there can never be doubles of infinites; but the doubles which are

continually produced, are the doubles of finites; for the diameters

which are always assumed, are finite in number. And what reason can be

assigned why every magnitude should not have finite divisions, since

number is prior to magnitudes, defines all their sections, pre-occupies

infinity, and always determines the parts which rise into energy, from

dormant capacity?

DEFINITION XVIII.

A SEMI-CIRCLE is the Figure contained by the Diameter, and that

Part of the Circumference which is cut off by the Diameter.

DEFINITION XIX.[173]

But the CENTRE of the Semi-circle, is the same with that of

the Circle.

From the definition of a circle Euclid finds out the nature of the

centre, differing from all the other points which the circle contains.

But from the centre he defines the diameter, and separates it from the

other right lines, which are described within the circle. And from

the diameter, he teaches the nature of the semi-circle; and informs

us, that it is contained by two terms, always differing from each

other, viz. a right-line and a circumference: and that this right-line

is not any one indifferently, but the diameter of the circle. For

both a less and a greater segment of a circle, are contained by a

right-line and circumference; yet these are not semi-circles, because

the division of the circle is not made through the centre. All these

figures, therefore, are biformed, as a circle was monadic, and are

composed from dissimilars. For every figure which is comprehended by

two terms, is either contained by two circumferences, as the lunular:

or by a right-line and circumference, as the above mentioned figures;

or by two mixt lines, as if two ellipses intersect each other (since

they enclose a figure, which is intercepted between them), or by a

mixt line and circumference, as when a circle cuts an ellipsis; or by

a mixt and right-line, as the half of an ellipsis. But a semi-circle

is composed from dissimilar lines, yet such as are, at the same time,

simple, and touching each other by apposition. Hence, before he defines

triadic figures, he, with great propriety, passes from the circle to

a biformed figure. For two right-lines can, indeed, never comprehend

space. But this may be effected by a right-line and circumference.

Likewise by two circumferences, either making angles, as in the

lunular figure; or forming a figure without angles, as that which is

comprehended by concentric circles. For the middle space intercepted

between both, is comprehended by two circumferences; one interior,

but the other exterior, and no angle is produced. For they do not

mutually intersect, as in the lunular figure, and that which is on

both sides convex. But that the centre of the semi-circle is the same

with that of the circle, is manifest. For the diameter, containing in

itself the centre, completes the semi-circle, and from this all lines

drawn to the semi-circumference are equal. For this is a part of the

circumference of the circle. But equal right lines proceed from the

centre to all parts of the circumference. The centre, therefore, of the

circle and semi-circle is one and the same. And it must be observed,

that among all figures, this alone contains the centre in its own

perimeter, I say, among all plane figures. Hence you may collect, that

the centre has three places. For it is either within a figure, as in

the circle; or in its perimeter, as in the semi-circle; or without

the figure, as in certain conic lines[174]. What then is indicated by

the semi-circles, having the same centre with the circle, or of what

things does it bear an image, unless that all figures which do not

entirely depart from such as are first, but participate them after

a manner, may be concentric with them, and participate of the same

causes? For the semi-circle communicates with the circle doubly, as

well according to the diameter, as according to the circumference.

On this account, they possess a centre also in common. And perhaps,

after the most simple principles, the semi-circle is assimilated to

the second co-ordinations, which participate those principles; and by

their relation to them, although imperfectly, and by halves, they are,

nevertheless, reduced to that which is, and to their first original

cause.

DEFINITION XX.

RECTILINEAR FIGURES are those which are comprehended by

Straight Lines.

DEFINITION XXI.

TRILATERAL FIGURES, or TRIANGLES, by three Straight

Lines.

DEFINITION XXII.

QUADRILATERAL, by four Straight Lines.

DEFINITION XXIII.

MULTILATERAL FIGURES, or POLYGONS, by more than four

Straight Lines.

After the monadic figure having the relation of a principle to all

figures, and the biformed semi-circle, the progression of right-lined

figures in infinitum, according to numbers, is delivered. For on this

account also, mention was made of the semi-circle, as communicating

according to terms or boundaries; partly, indeed, with the circle,

but partly with right-lines: just as the duad is the medium between

unity and number. For unity, by composition, produces more than by

multiplication; but number, on the contrary, is more increased by

multiplication than composition: and the duad, whether multiplied into,

or compounded with itself, produces an equal quantity. As, therefore,

the duad is the middle of unity and number, so likewise, a semi-circle

communicates, according to its base, with right-lines; but according

to its circumference, with the circle. But right-lined figures proceed

orderly to infinity, attended by number and its bounding power,

which begins from the triad. On this account, Euclid also begins

from hence[175]. For he says, trilateral and quadrilateral, and the

following figures, called by the common name of multilateral: since

trilateral figures are also multilaterals; but they have likewise

a proper, besides a common denomination. But, as we are but little

able to pursue the rest, on account of the infinite progression of

numbers, we must be content with a common denomination. But he only

makes mention of trilaterals and quadrilaterals, because the triad and

tetrad are the first in the order of numbers; the former being a pure

odd among the odd; but the latter, an entire even among even numbers.

Euclid, therefore, assumes both in the origin of right-lined figures,

for the purpose of exhibiting their subsistence, according to all even

and odd numbers. Besides, since he is about to teach concerning these

in the first book, as especially elementary (I mean triangles and

parallelograms) he does not undeservedly, as far as to these, establish

a proper enumeration: but he embraces all other right-lined figures by

a common name, calling them multilaterals: but of these enough. Again,

assuming a more elevated exordium, we must say, that of plane figures,

some are contained by simple lines, others by such as are mixt, but

others again by both. And of those which are comprehended by simple

lines, some are contained by similars in species, as right-lines; but

others by dissimilars in species, as semi-circles, and segments, and

apsides, which are less than semi-circles. Likewise of those which

are contained by similars in species, some are comprehended by a

circular line; but others by a right-line. And of those comprehended

by a circular line, some are contained by one, others by two, but

others by more than two. By one, indeed, the circle itself. But by

two, some without angles, as the crowns[176] terminated by concentric

circles; but others angular (γεγωνιωμένα) as the lunula. And of those

comprehended by more than two, there is an infinite procession. For

there are certain figures contained by three and four and succeeding

circumferences. Thus, if three circles touch each other, they will

intercept a certain trilateral space; but if four, one terminated by

four circumferences, and in like manner, by a successive progression.

But of those contained by right lines, some are comprehended by three,

others by four, and others by a multitude of lines. For neither is

space comprehended by two right-lines, nor much more by one right-line.

Hence, every space comprehended by one boundary, or by two, is either

mixt or circular. And it is mixt in a twofold manner, either because

the mixt lines comprehend it, as the space intercepted by the cissoidal

line; or because it is contained by lines dissimilar in species,

as the apsis: since mingling is twofold, either by apposition or

confusion. Every right-lined figure, therefore, is either trilateral,

or quadrilateral, or gradually multilateral; but every trilateral, or

quadrilateral, or multilateral figure, is not right-lined; since so

great a number of sides is also produced from circumferences. And thus

much concerning the division of plane figures. But we have already

asserted[177], that rectitude of progression is both a symbol of motion

and infinity, and that it is peculiar to the generative co-ordinations

of the gods, and to the producers of difference, and to the authors

of mutation and motion. Right-lined figures, therefore, are peculiar

to these gods, who are the principles of the prolific energy of the

whole progression of forms. On which account, generation also, was

principally adorned by these figures, and is allotted its essence from

these, so far as it subsists in continual motion and mutation without

end.

DEFINITION XXIV.

Of three-sided FIGURES: an EQUILATERAL TRIANGLE is

that which has three equal Sides.

DEFINITION XXV.

An ISOSCELES TRIANGLE, is that which has only two Sides

equal.

DEFINITION XXVI.

A SCALENE TRIANGLE, is that which has three unequal

Sides.

DEFINITION XXVII.

A RIGHT-ANGLED TRIANGLE is that which has a Right

Angle.

DEFINITION XXVIII.

An OBTUSE-ANGLED TRIANGLE is that which has an

Obtuse Angle.

DEFINITION XXIX.

An ACUTE-ANGLED TRIANGLE is that which has three

Acute Angles.

The division of triangles sometimes commences from angles, but

sometimes from sides. And that, indeed, which originates from

sides, precedes as known; but that from angles follows as a proper

distribution. For these three angles alone belong to right-lined

figures, viz. the right, the obtuse, and the acute: but the equality

and inequality of sides subsist also in non-rectilinear figures.

Euclid says, therefore, that of triangles, some are equilateral,

others isosceles, and others scalene: for they have either all their

sides equal, or all unequal, or only two equal. And again, that of

triangles some are right-angled, others obtuse-angled, and others

acute-angled. And he defines a right-angled triangle, that which has

one right angle, as likewise an obtuse-angled triangle, that which has

one obtuse angle: for it is impossible that a triangle can have more

than one right, or obtuse angle[178]. But he defines an acute-angled

triangle, that which has all its angles acute. For here it is not

sufficient that it should have only one acute; since, in this case,

all triangles would be acute-angled, as every triangle has necessarily

two acute angles[179]. But, to possess three acute angles, is the

property of an acute-angled triangle alone. But Euclid appears to me to

have made a separate division into angles and sides, from considering

this alone, that every triangle is not also trilateral. For there are

quadrilateral triangles, which are called by mathematicians themselves

(ἀκιδοειδῆ) that is, similar to the point of a spear[180]: but by

Zenodorus (κοιλογώνια) that is, having an hollow angle. For on one of

the sides of a trilateral figure, constitute two right-lines inwardly;

by this means a certain space will be enclosed, which is comprehended

by external and internal right-lines, and which has three angles;

one, indeed, contained by the external lines; but two comprehended by

these and the internal lines, at the extremities in which these lines

are conjoined. A figure of this kind, therefore, is a quadrilateral

triangle. And hence, it does not immediately follow, that because a

figure has three angles (whether they are all acute, or one right,

or one obtuse), we shall find it trilateral; for it may be, perhaps,

quadrilateral. In like manner, you may also find quadrangles having

more than four sides. And therefore, we must not rashly determine the

number of sides from the multitude of angles. But of this enough. But

the Pythagoreans affirm that the triangle is simply the principle

of generation, and of the formation of generable natures. On which

account, Timæus says, that natural reasons, as well as those of the

construction of the elements, are triangular. For they are distant by a

triple interval, are on all sides collective of partible, and variously

mutable natures, are replete with material infinity, and bear before

themselves the conjunctions of material bodies, loosened and free: as,

indeed, triangles also are comprehended by three right-lines, but they

possess angles which collect the multitude of lines, and afford to them

an adventitious angle and conjunction. With great propriety, therefore,

Philolaus has consecrated the angle of a triangle to four gods,

Saturn, Pluto, Mars, and Bacchus, comprehending in these the whole

quadripartite ornament of the elements descending from the heavens,

or from the four segments of the zodiac. For Saturn constitutes an

essence wholly humid and frigid; but Mars a nature totally fiery; and

Pluto contains the whole terrestrial life; but Bacchus governs a humid

and hot generation; of which wine also is a symbol, for this is humid

and hot. Hence, all these gods differ according to their operations

in inferior concerns: but they are mutually united according to their

proper natures. And on this account, Philolaus collects their union

according to one angle. But if the differences of triangles contribute

to generation, we shall very properly confess that a triangle is

the principle and author of the constitution of sublunary natures.

For a right angle, indeed, affords them essence, and determines the

measure of being; and the reason of a right-angled triangle produces

the essence of the elements of generable natures; but an obtuse angle

assigns to them universal distance; and the reason of an obtuse-angled

triangle increases material forms in magnitude, and in mutation of

every kind. But an acute angle effects their divisible nature; and the

reason of an acute-angled triangle prepares them to receive infinite

division. But simply, a triangular reason constitutes the essence of

material bodies distant with interval, and on all sides divisible.

And thus much should we speculate concerning the nature of triangles.

But from these divisions you may understand, that all the species of

triangles are neither more nor less than seven. For the equilateral

triangle is one, since it is acute-angled only; but each of the rest

is triple. For the isosceles is either right-angled, or obtuse-angled,

or acute-angled; and, in like manner, the scalene triangle possesses

this triple difference. If then, these have a triple distinction, but

the equilateral has but one mode of existence, all the species of

triangles will be seven. But again, you will understand the proportion

of triangles to the things which _are_, according to the division

of sides; for the equilateral, entirely excelling in equality and

simplicity, is allied to divine souls; since it is the measure and

equality of things unequal, in the same manner as divinity of all

inferior concerns. But the isosceles triangle is allied to the better

genera, which govern a material nature, the greater part of which

genera is held by the limitation of measure; but their extremes extend

to inequality and material immoderation; for the two sides of an

isosceles triangle are equal, but the base is unequal. But a scalene

triangle symbolizes with partible lives, which are on all sides lame

and defective, which prepare themselves for generation, and are replete

with matter and material imperfection.

DEFINITION XXX.

Of Quadrilateral Figures, a QUADRANGLE or SQUARE is that which

has all its Sides equal, and all its Angles Right Angles.

DEFINITION XXXI.

An OBLONG is that which has all its Angles right Angles, but

has not all its Sides equal.

DEFINITION XXXII.

A RHOMBUS, is that which has all its Sides equal, but its

Angles are not right Angles.

DEFINITION XXXIII.

A RHOMBOID is that which has its opposite Sides equal to one

another, but all its Sides are not equal, nor its Angles Right

Angles.

DEFINITION XXXIV.

All other Quadrilateral Figures besides these, are called

TRAPEZIUMS.

It is requisite that the first division of quadrilateral figures should

take place in two numbers; and that some of them should be called

parallelograms, but others non-parallelograms. But of parallelograms

some are rectangular and equilateral, as quadrangles; but others

neither of these, as rhomboids: others again, are rectangular, but not

equilateral, as oblongs: but others, on the contrary, are equilateral,

but not rectangular, as the rhombuses. For it is requisite either

to possess both, viz. equality of sides and rectitude of angles, or

neither; or one of these, and this in a twofold respect. Hence a

parallelogram has a quadruple subsistence. But of non-parallelograms,

some have only two parallel sides, and not the rest; but others have

none of their sides parallel. And those are called Trapeziums, but

these Trapezoids. But of Trapeziums, some, indeed, have the sides

equal, by which the parallel sides of this kind are conjoined;

but others unequal; and the former of these are called isosceles

trapeziums; but the latter scalene trapeziums. A quadrilateral figure,

therefore, is constituted by us according to a seven-fold distribution.

For one is a quadrangle; but the other an oblong; the third a rhombus;

the fourth a rhomboides; the fifth an isosceles trapezium; the sixth

a scalene trapezium; the seventh a trapezoid. But Possidonius makes

a perfect division of right-lined quadrilateral figures into so many

members; for he establishes seven species of these; as likewise

of triangles. But Euclid could not divide into parallelograms and

non parallelograms, because he neither mentions parallels, nor

teaches us concerning the parallelogram itself. But trapeziums, and

all trapezoids, he calls by a common name, describing trapeziums

themselves, according to the difference of those four figures[181], in

which the property of parallelograms is verified. And this is to have

the opposite sides and angles equal. For a quadrangle and an oblong,

and a rhombus, have their opposite sides and angles equal. But in a

rhomboides he only adds this, _that its opposite sides are equal_,

lest he should define it by negations alone, since he neither calls it

equilateral, nor rectangular. For where we want proper appellations,

it is necessary to use such as are common. But we should hear Euclid

shewing that this is common to all parallelograms. But a rhombus

appears to be a quadrangle having its sides moved, and a rhomboides

a moved oblong. Hence, according to sides, these do not differ from

those; but they vary only according to the obtuseness and acuteness

of angles; since the quadrangle and the oblong are rectangular. For

if you conceive a quadrangle or an oblong, having its sides drawn in

such a manner, that while two of its opposite angles are dilated, the

other two are contracted; then the dilated angles will appear obtuse,

and the contracted, acute. And the appellation of rhombus[182] seems

to have been imposed from motion. For if you conceive a quadrangle

moving after the manner of a rhombus, it will appear to you changed in

order, according to its angles: just as if a circle is moved after the

manner of a sling, it will immediately exhibit the appearance of an

ellipsis. But here you may perhaps enquire concerning the quadrangle,

why it has this denomination? and why the appellation of quadrangle

may not be applied to other quadrilateral figures, as the name of

triangle is common to all those which are neither equiangular nor

equilateral, and in like manner of quinquangles or pentagons; for

the geometrician, in these, adds only the particle _an equilateral

triangle_, or a _quinquangle_, _which is equilateral and

equiangular_, as if these could not be otherwise than such as they

are? But when he mentions a quadrangle, he immediately indicates that

it must be equilateral and rectangular. But the reason of this is as

follows: a quadrangle alone has the best space, both according to

its sides and angles. For each of the latter is right, intercepting

a measure of angles, which neither receives intention nor remission.

As it excels, therefore, in both respects, it deservedly obtains a

common appellation. But a triangle, though it may have equal sides,

yet will in this case have all its angles acute, and a quinquangle all

its angles obtuse. Since, therefore, of all quadrilateral figures,

a quadrangle alone is replete with equality of sides, and rectitude

of angles, it was not undeservedly allotted this appellation: for,

to excellent forms, we often dedicate the name of the whole. But it

appeared also to the Pythagoreans, that this property of quadrilateral

figures, principally conveyed an image of a divine essence. For they

particularly signified by this, a pure and immaculate order. Since

rectitude imitates inflexibility, but equality a firm and permanent

power: for motion emanates from inequality, but quiet from equality

itself. The gods, therefore, who are the authors to all things

of stable disposition, of pure and uncontaminated order, and of

indeclinable power, are deservedly manifested as from an image, by a

quadrangular figure. But, besides these, Philolaus also, according to

another apprehension, calls a quadrangular angle, the angle of Rhea,

Ceres and Vesta. For, since a quadrangle constitutes the earth, and is

its proximate element, as we learn from Timæus, but the earth herself

receives from all these divinities, genital seeds, and prolific powers,

he does not unjustly consecrate the angle of a quadrangle to these

goddesses, the bestowers of life. For some call both the earth and

Ceres, Vesta[183], and they say that Rhea totally participates her

nature, and that all generative causes are contained in her essence.

Philolaus, therefore, says that a quadrangular angle comprehends, by

a certain terrestrial power, one union of the divine genera. But some

assimilate a quadrangle to universal virtue, so far as every quadrangle

from its perfection has four right angles. Just as we say that each of

the virtues is perfect, content with itself, the measure and bound of

life, and the middle of every thing which, in morals, corresponds to

the obtuse and acute. But it is by no means proper to conceal, that

Philolaus attributes a triangular angle to four, but a quadrangular

angle to three gods, exhibiting their alternate transition, and the

community of all things in all, of odd natures in the even, and of even

in the odd. Hence, the tetradic ternary, and the triadic quaternary,

participating of prolific and efficacious goods, contain the whole

ornament of generable natures, and preserve them in their proper

state. From which the duodenary, or the number twelve, is excited

to a singular unity, viz. the government of Jupiter. For Philolaus

says, that the angle of a dodecagon (or twelve-sided figure) belongs

to Jove, so far as Jupiter contains and preserves, by his singular

union, the whole number of the duodenary. For also, according to Plato,

Jupiter presides over the duodenary[184], and governs and moderates

the universe with absolute sway. And thus much we have thought proper

to discourse concerning quadrilateral figures, as well declaring the

sense of our author, as likewise affording an occasion of more profound

inspections to such as desire the knowledge of intelligible and occult

essences.

DEFINITION XXXV.

PARALLEL RIGHT LINES are such as being in the same Plane, and

produced both ways infinitely, will in no part mutually coincide.

What the elements of parallels are, and by what accidents in these

they may be known, we shall afterwards learn: but what parallel right

lines are, he defines in these words: “It is requisite, therefore (says

he), that they should be in one plane, and while they are produced

both ways have no coincidence, but be extended in infinitum.” For

non-parallel lines also, if they are produced to a certain distance,

will not coincide. But to be produced infinitely, without coincidence,

expresses the property of parallels. Nor yet this absolutely, but to be

extended both ways infinitely, and not coincide. For it is possible

that non-parallel lines may also be produced one way infinitely, but

not the other; since, verging in this part, they are far distant from

mutual coincidence in the other. But the reason of this is, because two

right-lines cannot comprehend space; for if they verge to each other

both ways, this cannot happen. Besides this, he very properly considers

the right-lines as subsisting in the same plane. For if the one should

be in a subject plane, but the other in one elevated, they will not

mutually coincide according to every position, yet they are not on this

account parallel. The plane, therefore, should be one, and they should

be produced both ways infinitely, and not coincide in either part. For

with these conditions, the right-lines will be parallel. And agreeable

to this, Euclid defines parallel right-lines. But Posidonius says,

parallel lines are such as neither incline nor diverge in one plane;

but have all the perpendiculars equal which are drawn from the points

of the one to the other. But such lines as make their perpendiculars

always greater and less, will some time or other coincide, because

they mutually verge to each other. For a perpendicular is capable of

bounding the altitudes of spaces, and the distances of lines. On which

account, when the perpendiculars are equal, the distances of the right

lines are also equal; but when they are greater and less, the distance

also becomes greater and less, and they mutually verge in those parts,

in which the lesser perpendiculars are found. But it is requisite to

know, that non-coincidence does not entirely form parallel lines. For

the circumferences of concentric circles do not coincide: but it is

likewise requisite that they should be infinitely produced. But this

property is not only inherent in right, but also in other lines: for

it is possible to conceive spirals described in order about right

lines, which if produced infinitely together with the right lines,

will never coincide[185]. Geminus, therefore, makes a very proper

division in this place, affirming from the beginning, that of lines

some are bounded, and contain figure, as the circle and ellipsis,

likewise the cissoid, and many others; but others are indeterminate,

which may be produced infinitely, as the right-line, and the section

of a right-angled, and obtuse angled cone; likewise the conchoid

itself. But again, of those which may be produced in infinitum, some

comprehend no figure, as the right-line and the conic sections; but

others, returning into themselves, and forming figure, may afterwards

be infinitely produced. And of these some will not hereafter coincide,

which resist coincidence, how far soever they may be produced; but

others are coincident, which will some time or other coincide. But

of non-coincident lines, some are mutually in one plane; and others

not. And of non-coincidents subsisting in one plane, some are always

mutually distant by an equal interval; but others always diminish the

interval, as an hyperbola in its inclination to a right-line, and

likewise the conchoid[186]. For these, though they always diminish

the interval, never coincide. And they mutually converge, indeed,

but never perfectly nod to each other; which is indeed a theorem in

geometry especially admirable, exhibiting certain lines endued with a

non-assenting nod. But the right-lines, which are always distant by an

equal interval, and which never diminish the space placed between them

in one plane, are parallel lines. And thus much we have extracted from

the studies of the elegant Geminus, for the purpose of explaining the

present definition.

END OF THE FIRST VOLUME.

FOOTNOTES:

[1] The Grecian literature of this writer will now prove of real

utility; and the graces and the sublimities of PLATO will

soon be familiarised to the English reader, by a hand that I am

persuaded will not appear inferior to his great original. Let me also

be permitted to recommend his version of PLOTINUS on THE

BEAUTIFUL.

[2] i.e. Capable of parts.

[3] i.e. Not capable of parts.