CHAP. XIV.

_How Dialectic is the Top of the Mathematical Sciences, and what

their Conjunction is, according to Plato._

Let us again consider after what manner Plato, in his Republic, calls

dialectic the top of the mathematical disciplines; and what their

conjunction is, according to the tradition of the author of the

Epinomis[90]. And in order to this we must assert, that as intellect

is superior to cogitation, supplying it with supernal principles,

and from itself giving perfection to cogitation; in the same manner

dialectic also, being the purest part of philosophy, excels in

simplicity the mathematical disciplines, to which it is proximate,

and with which it is conjoined. Indeed it embraces the complete

circle of these sciences, to which it elevates from itself various

energies, endued with a power of causing perfection, judgment, and

intelligence. And these energies consist in resolving, dividing,

defining, and demonstrating; by which mathematics itself, receiving

assistance and perfection, invents some things by resolution, but

others by composition: and some things it explains by division, others

by definition: but collects other subjects of its investigation by

demonstration; accommodating, indeed, these ways to its subjects, but

using each of them for the purpose of beholding its middle enquiries.

From whence indeed, both the resolutions, definitions, divisions, and

demonstrations which it contains, are peculiar, and adapted to its

nature, and revolve according to the mode of mathematical cognition.

Not undeservedly, therefore, is dialectic the vertex as it were,

and summit of mathematics. Since it perfects all which mathematics

contains of intelligence; renders its certainty free from reprehension,

preserves the stability of its immovable essence, and refers what it

contains destitute of matter and pure to the simplicity of intellect,

and a nature separated from material connections. Besides, it

distinguishes the first principles of these sciences, by definitions:

exhibits the separations of genera and forms contained under the

genera themselves: and besides this, teaches the compositions, which,

from principles, produce things consequent to principles: and the

resolutions which rise and mount up to things first, and to principles

themselves. But with respect to what remains, proportion itself is not

to be considered (as Eratosthenes thought it was) as the conjunction

of the mathematical disciplines. Since proportion is said to be, and

indeed is one of those things common to the mathematics. But in short,

many other things besides proportion regard all the mathematical

disciplines, which are essentially inherent in the common nature of

the mathematics. But as it appears to me, we should say, that there

is one proximate conjunction of these, and of the whole mathematical

science, which especially embraces in itself, in a more simple manner,

the principles of all sciences; which considers their community and

difference; teaches whatever is found in these the same; together

with what things are inherent in a many, and what in a few. So

that to those who aptly learn there is a reversion from many other

sciences to this alone[91]. But, dialectic is a conjunction of the

mathematical disciplines superior to the preceding; which Plato, as I

have already observed, calls in his Republic their vertex: for, indeed,

it perfects the whole of mathematics, brings it back to intellect by

its powers, shews it to be a true science, and causes it to be certain

and obnoxious to no reproof. But, intellect obtains the third order

between these conjunctions, which comprehends in itself uniformly

all the dialectic powers, contracts their variety by its simplicity,

their partition by its indivisible knowledge, and their multitude

by its occult union. Hence, intellect itself congregates indeed the

involutions and deviations of the dialectic paths, into an intelligible

essence, but it collects supernally all the progression of mathematical

discourses: and it is the best end both of the elevating power of the

soul, and of the energy consisting in cognition. And such are the

sentiments declared by me on the present enquiry.

CHAP. XV.

_From whence the Name Mathematics originated._

Again, from whence shall we say this name of mathematics, and

mathematical disciplines, was assigned by the ancients, and what

apt reason can we render of its position? Indeed, it appears to me,

that such an appellation of a science which respects cogitative

reasons, was not, like most names, invented by indifferent persons,

but (as the truth of the case is, and according to report) by the

Pythagoreans alone. And this, when they perceived, that whatever is

called mathesis or discipline, is nothing more than reminiscence;

which does not approach the soul extrinsically, like the images which

rising from sensible objects are formed in the phantasy: nor is it

adventitious and foreign, like the knowledge consisting in opinion,

but it is excited, indeed, from apparent objects, and is perfected

within, by thought intimately converted to itself. And when they

likewise perceived that though reminiscence might be shewn from many

particulars, yet it was evinced in a more eminent manner (as Plato

also says[92]) from the mathematical disciplines. For if any one, says

he, is led into the descriptions, he will there easily prove that

discipline is reminiscence. From whence Socrates also, in the Meno,

shews by this method of arguing, that learning is nothing else than the

soul’s recollection of her inherent reasons. And this, because that

which recollects, is alone the cogitative part of the soul; but this

perfects her essence in the reasons of the mathematical disciplines,

the sciences of which she previously received into herself, though she

does not always energize on their fair variety. Indeed, she contains

them all essentially and occultly; but she produces each of them when

she is freed from the impediments originating from sense. For sense

unites her with divisible objects: the phantasy fills her with forming

motions, and appetite bends her to an indulgent and luxurious life.

But every thing divisible is an obstacle to our self-conversion. And

whatever invests with form, disturbs and offends that knowledge which

is destitute of form. And whatever is obnoxious to perturbations is

an impediment to that energy which is unimpaired by affections. When,

therefore, we have moved all these from the cogitative power, then

shall we be able to understand by thought itself, the reasons which

thought contains: then shall we become scientific in energy; and

unfold our essential knowledge. But whilst we are captive and bound,

and winking with the eye of the soul, we cannot by any means attain

to a perfection convenient to our nature. Such then is mathesis or

discipline: a reminiscence of the eternal reasons contained in the

soul. And the mathematical or disciplinative science is on this account

particularly denominated that knowledge which especially confers to our

reminiscence of these essential reasons. Hence, the business and office

of this science[93], is apparent from its name. For its duty is to

move the inherent knowledge of the soul; to awaken its intelligence;

to purify its cogitation; to call forth its essential forms from their

dormant retreats; to remove that oblivion and ignorance, which are

congenial with our birth; and to dissolve the bonds arising from our

union with an irrational nature. It plainly leads us to a similitude

of that divinity who presides over this science, who manifests

intellectual gifts, and fills the universe with divine reasons; who

elevates souls to intellect, wakens them as from a profound sleep,

converts them by enquiry to themselves; and by a certain obstetric art,

and invention of pure intellect, brings them to a blessed life. To whom

indeed, dedicating the present work, we here conclude our contemplation

of the mathematical science.

BOOK II.

CHAP. I.

_What Part Geometry is of Mathematics, and what the Matter is of

which it consists._

In the preceding discourses we have considered those common properties

which respect the whole of the mathematical science; and this we have

done agreeable to the doctrine of Plato; at the same time collecting

such particulars as pertain to our present design. But consequent to

this it is requisite that we should discourse on geometry itself, and

on the proposed institution of the elements, for the sake of which we

have undertaken the whole of the present work. That geometry then, is

a part of the whole of mathematics, and that it obtains the second

place after arithmetic, since it is perfected and bounded by this,

(for whatever in geometry may be expressed and known, is determined

by arithmetical reasons) has been asserted by the ancients, and

requires no long discussion in the present enquiry. But we also may

be able to relate our opinion on this particular, if we consider what

place, and what essence its subject matter[94] is allotted among the

universality of things. For from a proper survey of this, the power

of the science which knows this subject matter, the utility arising

from it, and the good acquired by its learners, will immediately

appear. Indeed, some one may doubt in what genus of things he ought

to place geometrical matter, so as not to deviate from the truth it

contains. For if the figures concerning which geometry discourses,

exist in sensible natures, and cannot be separated from the dark

receptacle of matter; how can we assert that geometry frees us from

sensible objects, that it brings us to an incorporeal essence, that it

accustoms us to an inspection of intelligibles, and prepares us for

intellectual energy? Where shall we ever survey among sensible objects

a point without parts, or a line destitute of breadth, or a superficies

without profundity, or the equality of lines from the centre to the

circumference; or the multangles, and all the figures of many bases,

concerning which geometry informs us? Lastly, after what manner can the

reasons of such a science remain free from all possible confutation;

since, indeed, sensible forms and figures are susceptive of the more

and the less, are all moveable and mutable, and are full of material

variety; among which equality subsists mixt and confused with its

contrary inequality, and into which things without parts have proceeded

into partition, and interval, darkened with the shades of matter, and

lost in its infinite folds? But if the subjects of geometry are removed

from matter, are pure forms, and are separated from sensible objects:

they will be all of them, without doubt, void of parts, incorporeal,

and destitute of magnitude. For extension, tumor, and interval,

approach to forms, on account of the material receptacle in which they

are involved, and which receives things destitute of parts, distributed

into parts; things void of dimension, extended into dimension; and

immoveable natures accompanied with motion. How then, if this is the

case, shall we cut a right line, triangle, and circle? How can we speak

of the diversities of angles, and the increments and decrements of

triangular and quadrangular figures? Or how exhibit the contacts of

circles or right lines? For all these evince that the geometric matter

consists of parts, and does not reside among indivisible reasons.

Such then are the doubts concerning the matter of geometry, to which

we may add, that Plato considers the forms of geometry as placed in

cogitation; and grants, that we advance from sensibles to forms of

this kind, and that we rise from sensibles to intellect, though (as

we have previously observed) the reasons subsisting in cogitation

are indivisible, are separated by no interval, and subsist according

to the peculiarity of the soul. But if reasons are to be rendered

agreeable to things themselves, and to the doctrine of Plato, the

following division must be adopted. [95]Every universal, and one

thing containing many, is either naturally disposed to be thought of

in particulars, or to appear such, because it possesses its existence

in these; is inseparable from them; is disposed and distributed in

them; and together with these is either moved, or firmly and immoveably

abides. Or it is adapted to subsist prior to many, and to possess a

power of generating multitude, affording to many things images from

itself, being furnished with a nature destitute of parts, from the

essences which it participates, and raising various participations to

secondary natures: or it is disposed to be formed by thought, from the

many, to possess a generating existence, and to reside in the last

place in the many. For, according to these three modes of subsistence,

we shall find, I think, that some subsist before the many, others in

the many, and others from the relation and predication which they

possess to these. But, that I may absolve all in one word, universal

forms being threefold, we shall consider the differences of that form

which many participate, which exists in many, and fills particular

natures according to its subject matter. Besides this, establishing

a twofold order of participants, one subsisting in sensible objects,

but the other in the phantasy, (since matter is twofold; one indeed,

of things united with sense, but the other of such as fall under the

inspection of phantasy, as Aristotle asserts, in a certain place[96])

we must allow that the universal, which is distributed in the many,

is likewise twofold. The one, indeed, sensible, as being that which

sensible objects participate; but the other imaginative, as that

which subsists in the _many_ of the phantasy. For the phantasy,

on account of its forming motion, and because it subsists with, and

in body, always receives impressions which are both divided and

figured. So that whatever is known by it, is allotted a correspondent

existence: on which account, Aristotle[97] does not hesitate to call

it passive intellect. But if it is intellect, why is it not impassive,

and destitute of matter? And if it operates with passion, how can it

with propriety be called intellect? For impassivity, indeed, properly

belongs to intellect and an intelligent nature: but passivity is very

remote from such an essence. But (unless I am deceived) Aristotle

being willing to explain its middle nature between cognitions the most

primary, and such as are the last, calls it at the same time intellect,

because similar to primary cognitions, and passive from that alliance

which it possesses with such as are posterior. For first cognitions are

indeed destitute of figures and forms; comprehending in themselves,

intelligible natures, energizing about themselves, united with the

objects of knowledge, and free from all extrinsical impression and

passion. But last cognitions exercise themselves through the medium

of instruments, are rather passions than energies, admit extrinsical

knowledge, and move themselves together with their various subjects.

For such (says Plato) are the sensations which arise from violent

passions. But the phantasy, obtaining a middle centre in the order of

cognitions, is excited, indeed, by itself, and produces that which

falls under cogitation: but because it is not separate from body,

it deduces into partition, interval, and figure, the objects of its

knowledge, from the indivisibility of an intellectual life. Hence,

whatever it knows, is a certain impression and form of intelligence.

For it understands the circle, together with its interval, void,

indeed, of external matter, but possessing intelligible matter. On

this account, like sensible matter, it does not contain one circle

only: for we behold in its receptacle, distance, together with the

more and the less, and a multitude of circles and triangles. If then

an universal nature is distributed in sensible circles, since each of

these completes a circular figure, and they are all mutually similar,

subsisting in one reason, but differing in magnitudes or subjects:

in like manner, there is a common something in the circles, which

subsist in the receptacle of the phantasy, of which all its circles

participate, and according to which they all possess the same form;

but in the phantasy they possess but one difference only, that of

magnitude. For when you imagine many circles about the same centre,

they all of them exist in one immaterial subject and life, which is

inseparable from a simple body, which, by the possession of interval,

exceeds an essence destitute of parts; but they differ in magnitude

and parvitude, and because they are contained and contain. Hence,

that universal is twofold, which is understood as subsisting in the

many: one, indeed, in sensible forms; but the other in such as are

imaginative. And the reason of a circular and triangular figure, and of

figure universal, is twofold. The one subsisting in intelligible, but

the other in sensible matter. But prior to these is the reason which

resides in cogitation, and that which is seated in nature herself.

The former being the author of imaginative circles, and of the one

form which they contain; but the other, of such as are sensible.

For there are circles existing in the heavens, and universally

those produced by nature, the reason of which does not fall under a

cogitative distribution. For in incorporeal causes, things possessing

interval, are distinguished by no intervals: such as are endued with

parts, subsist without parts: and magnitudes without the diffusion of

magnitude, as on the contrary in corporeal causes, things without parts

subsist divisibly, and such as are void of magnitude with the extension

of magnitude. Hence, the circle resident in cogitation, is one, simple

and free from interval: and magnitude itself is there destitute of

magnitude; and figure expressed by no figure: for such are reasons

separate from matter. But the circle subsisting in the phantasy, is

divisible, figured, endued with interval, not one only, but one and

many, nor form alone, but distributed with form. And the circle, in

sensible objects, is composite, distant with magnitude, diminished by

a certain reason, full of ineptitude, and very remote from the purity

of immaterial natures. We must therefore say, that geometry, when it

asserts any thing of circle and diameter, and of the passions and

affections which regard the circle; as of contacts, divisions, and the

like: neither teaches nor discourses concerning sensible forms, (since

it endeavours to separate us from these), nor yet concerning the form

resident in cogitation, (for here the circle is one, but geometry

discourses of many, proposing something of each, and contemplating the

same of all: and here it is indivisible, but the geometric circle is

divisible); but we must confess, that it considers universal itself;

yet as distributed in imaginative circles. And that it beholds, indeed,

one circle[98]: and by the medium of another, contemplates the circle

resident in the depths of cogitation: but by another, different from

the preceding, fabricates the fair variety of its demonstrations.

For since cogitation is endued with reasons, but cannot behold them

contractedly, separated from material figure; it distributes and

removes them, and draws them forth seated in the shadowy bosom of the

phantasy, and placed in the vestibules of primary forms; revolving

in it, or together with it, the knowledge of these: loving, indeed,

a separation from sensibles, but finding imaginative matter proper

for the reception of its universal forms. Hence, its intellection

does not subsist without the phantasy. And the compositions and

divisions of figures are imaginative; and their knowledge is the way

which leads us to that essence pursued by cogitation: but cogitation

itself, does not yet arrive at this stable essence, while it looks

abroad to externals, contemplates its internal forms according to

these, uses the impressions of reasons, and is moved from itself to

external and material forms. But if it should ever be able to return

to itself, when it has contracted intervals and impressions, and

beholds multitude without impression, and subsisting uniformly; then

it will excellently perceive geometrical reasons, void of division and

interval, essential and vital, of which there is a copious variety.

And this energy will be the best end of the geometric study; and truly

the employment of a Mercurial gift, bringing it back as from a certain

Calypso, and her detaining charms, to a more intellectual knowledge;

and freeing it from those _forming_ apprehensions with which the

mirror of the phantasy is replete. Indeed, it is requisite that a

true geometrician should be employed in this meditation, and should

establish, as his proper end, the excitation and transition from the

phantasy to cogitation alone; and that he should accomplish this by

separating himself from intervals, and the passive intellect to that

energy which cogitation contains. For by this means he will perceive

all things without an interval, the circle and diameter without a part,

the polygons in the circle, all in all, and yet every one separate and

apart. Since, on this account, we exhibit also in the phantasy, both

circles inscribed in polygons, and polygons in circles; imitating the

alternate exhibition of reasons destitute of parts. Hence, therefore,

we describe the constitutions, the origin, divisions, positions, and

applications of figures: because we use the phantasy, and distances

of this kind proceeding from its material nature; since form itself

is immoveable, without generation, indivisible, and free from every

subject. But whatever form contains occultly, and in an indistant

manner, is produced into the phantasy subsisting with intervals,

divisibly and expanded. And that which, indeed, produces the forms of

geometric speculation, is cogitation: but that from which they are

produced, is the form resident in cogitation: and that in which the

produced figure resides is what is called the passive intellect. Which

folds itself about the impartibility of true intellect, separates from

itself the power of pure intelligence free from interval; conforms

itself according to all formless species, and becomes perfectly

every thing from which cogitation itself, and our indivisible reason

consists. And thus much concerning the geometric matter, as we are

not ignorant of whatever Porphyry the Philosopher has observed in his

miscellanies, and whatever many of the Platonists describe. But we

think that the present discussions are more agreeable to geometric

dissertations, and to Plato himself, who subjects to geometry the

objects of cogitation. For these mutually agree among themselves;

because the causes, indeed, of geometrical forms, by which cogitation

produces demonstrations, pre-exist in demonstration itself: but the

particular figures which are divided and compounded, are situated in

the receptacle of the phantasy.

CHAP. II.

_What kind of Science Geometry is._

But let us now speak of that science which possesses a power of

contemplating the universal forms participated by imaginative matter.

Geometry, therefore, is endued with the knowledge of magnitudes and

figures, and of the terms and reasons subsisting in these; together

with the passions, various positions and motions which are contingent

about these. For it proceeds, indeed, from an impartible point, but

descends even to solids, and finds out their multiform diversities.

And again, runs back from things more composite, to things more

simple, and to the principles of these: since it uses compositions

and resolutions, always beginning from suppositions, and assuming its

principles from a previous science; but employing all the dialectic

ways. In principles, by the divisions of forms from their genera, and

by defining its orations. But in things posterior to principles, by

demonstrations and resolutions. As likewise, it exhibits things more

various, proceeding from such as are more simple, and returning to

them again. Besides this, it separately discourses of its subjects;

separately of its axioms; from which it rises to demonstrations;

and separately of essential accidents, which it shews likewise are

resident in its subjects. For every science has, indeed, a genus, about

which it is conversant, and whose passions it proposes to consider:

and besides this, principles, which it uses in demonstrations; and

essential accidents. Axioms, indeed, are common to all sciences

(though each employs them in its peculiar subject matter), but genus

and essential accident vary according to the sciential variety. The

subjects of geometry are therefore, indeed, triangles, quadrangles,

circles, and universally figures and magnitudes, and the boundaries of

these. But its essential accidents are divisions, ratios, contacts,

equalities, applications, excesses, defects, and the like. But its

petitions and axioms, by which it demonstrates every particular are,

this, to draw a right line from any point to any point; and that,

if from equals you take away equals, the remainders will be equal;

together with the petitions and axioms consequent to these. Hence,

not every problem nor thing sought is geometrical, but such only as

flow from geometric principles. And he who is reproved and convicted

from these, is convinced as a geometrician. But whoever is convinced

from principles different from these, is not a geometrician, but is

foreign from the geometric contemplation. But the objects of the

non-geometric investigation, are of two kinds. For the thing sought

for, is either from entirely different principles, as we say that a

musical enquiry is foreign from geometry, because it emanates from

other suppositions, and not from the principles of geometry: or it is

such as uses, indeed, geometrical principles, but at the same time

perversely, as if any one should say, that parallels coincide. And on

this account, geometry also exhibits to us instruments of judging, by

which we may know what things are consequent to its principles, and

what those are which fall from the truth of its principles: for some

things attend geometrical, but others arithmetical principles. And why

should we speak of others, since they are far distant from these? For

one science is more certain than another (as Aristotle says[99]) that,

indeed, which emanates from more simple suppositions, than that which

uses more various principles; and that which tells the _why_,

than that which knows only the simple existence of a thing; and that

which is conversant about intelligibles, than that which touches and

is employed about sensibles. And according to these definitions of

certainty, arithmetic is, indeed, more certain than geometry, since its

principles excel by their simplicity. For unity is void of position,

with which a point is endued. And a point, indeed, when it receives

position, is the principle of geometry: but unity, of arithmetic. But

geometry is more certain than spherics; and arithmetic, than music.

For these render universally the causes of those theorems, which are

contained under them. Again, geometry is more certain than mechanics,

optics, and catoptrics. Because these discourse only on sensible

objects. The principles, therefore, of geometry and arithmetic, differ,

indeed, from the principles of other sciences; but the hypotheses of

these two, alternately differ and agree according to the difference we

have already described. Hence, also, with respect to the theorems which

are demonstrated in these sciences, some are, indeed, common to them,

but others peculiar. For the theorem which says, _every proportion

may be expressed_, alone belongs to arithmetic; but by no means

to geometry: since this last science contains things which cannot be

expressed[100]. That theorem also, which affirms, that _the gnomons

of quadrangles are terminated according to the least_[101], is the

property of arithmetic: for in geometry, a minimum cannot be given.

But those things are peculiar to geometry, which are conversant about

positions; for numbers have no position: which respect contacts; for

contact is found in continued quantities: and which are conversant

about ineffable proportions; for where division proceeds to infinity,

there also that which is ineffable is found[102]. But things common to

both these sciences, are such as respect divisions, which Euclid treats

of in the second book; except that proposition which divides a right

line into extreme and mean proportion[103]. Again, of these common

theorems, some, indeed, are transferred from geometry into arithmetic;

but others, on the contrary, from arithmetic into geometry: and others

similarly accord with both, which are derived into them from the whole

mathematical science. For the permutation, indeed, conversions,

compositions, and divisions of ratios are, after this manner, common

to both. But such things as are commensurable, arithmetic first

beholds; but afterwards geometry, imitating arithmetic. From whence,

also, it determines such things to be commensurables of this kind,

which have the same mutual ratio to one another, as number to number;

because commensurability principally subsists in numbers. For where

number is, there also that which is commensurable is found; and where

commensurable is, there also number. Lastly, geometry first inspects

triangles and quadrangles: but, arithmetic, receiving these from

geometry, considers them according to proportion. For in numbers,

figures reside in a causal manner. Being excited, therefore, from

effects, we pass to their causes, which are contained in numbers. And

at one time, we indifferently behold the same accidents, as when every

polygon is resolved by us into triangles[104]: but, at another time,

we are content with what is nearest to the truth, as when we find in

geometry one quadrangle the double of another, but not finding this

in numbers, we say that one square is double of another, except by a

deficience of unity. As for instance, the square from 7, is double the

square from 5, wanting one. But we have produced our discussion to this

length, for the purpose of evincing the communion and difference in the

principles of these two sciences. Since it belongs to a geometrician

to survey from what common principles common theorems are divided;

and from what principles such as are peculiar proceed; and thus to

distinguish between the geometrical, and non-geometrical, referring

each of them to different sciences.

CHAP. III.

_From whence the whole of Geometry originated, how far it proceeds,

and in what its Utility consists._

But, beginning still higher, let us contemplate the whole of geometry,

from whence it originated, and how far it proceeds in its energies:

for thus we shall properly perceive the ornament which it contains.

Indeed, it is necessary to understand that it is extended through the

universality of things: that it accommodates its animadversions[105]

to all beings; and contains in itself the forms of all things: that,

according to its supreme part, and which is endued with the highest

power of intelligence, it surveys true beings; and teaches by images

the properties of divine ornaments, and the powers of intellectual

forms: for it contains the reasons of these also in its peculiar

contemplations. And it exhibits what figures are convenient to the

god, to primary essences, and to the natures of souls. But, according

to its middle cognitions, it evolves cogitative reasons; explains and

beholds the variety which they contain; exhibits their existence, and

inherent passions; as also, their communities and diversities. From

which, indeed, it comprehends, in terminated bounds, the imaginative

formations of figures, and reduces them to the essential substance

of reasons. But, according to the third propagations of cogitative

intelligence, it considers nature, and delivers the manner in which

the forms of sensible elements, and the powers which they contain, are

previously received according to cause, in the reasons themselves. For

it possesses, indeed, the images of universal intelligible genera;

but the exemplars of such as are sensible: and completes its own

essence, according to such things as are subject to cogitation. And

through these, as through proper mediums, it ascends and descends to

those universals which truly are, and to sensible forms which are in a

state of perpetual formation. But always geometrically philosophising

concerning the things which are, it comprehends in all the proportions

of virtues, the images of intellectual, animal, and natural concerns.

And it delivers, in an orderly manner, all the ornaments of republics:

and exhibits in itself their various mutations. Such then are its

energies arising from a certain immaterial power of cognition: but

when it touches upon matter, it produces from itself a multitude of

sciences; such as geodæsia, mechanics, and perspective: by which it

procures the greatest benefit to the life of mortals. For it constructs

by these sciences, war-instruments, and the bulwarks of cities; and

makes known the circuits of mountains, and the situations of places.

Lastly, it instructs us in measures: at one time of the diversified

ways of the earth; and at another, of the restless paths of the deep.

Add too, that it constructs balances and scales, by which it renders

to cities a sure equality according to the invariable standard of

number. Likewise, it clearly expresses, by images, the order of the

whole orb of the earth; and by these, manifests many things incredible

to mankind, and renders them credible to all. Such, indeed, as Hiero

of Syracuse is reported to have said of Archimedes[106], when he had

fabricated a ship furnished with three sails, which he had prepared to

send to Ptolemy king of Egypt. For when all the Syracusians together,

were unable to draw this ship, Archimedes enabled Hiero to draw it

himself, without any assistance from others. But he, being astonished,

said, From this day, Archimedes shall be believed in whatever he shall

affirm. They also report, that Gelo said the same, when Archimedes

discovered the weight of the several materials from which his crown was

composed, without dissolving their union. And such are the narrations

which many of the ancients have delivered to our memory, who were

willing to speak in praise of the mathematics: and, on this account, we

have placed before the reader, for the present, a few out of the many,

as not foreign from our design of exhibiting the knowledge and utility

of geometry.

CHAP. IV.

_On the Origin of Geometry, and its Inventors._

But let us now explain the origin of geometry, as existing in the

present age of the world. For the demoniacal Aristotle[107] observes,

that the same opinions often subsist among men, according to certain

orderly revolutions of the world: and that sciences did not receive

their first constitution in our times, nor in those periods which are

known to us from historical tradition, but have appeared and vanished

again in other revolutions of the universe; nor is it possible to say

how often this has happened in past ages, and will again take place

in the future circulations of time. But, because the origin of arts

and sciences is to be considered according to the present revolution

of the universe, we must affirm, in conformity with the most general

tradition, that geometry was first invented by the Egyptians, deriving

its origin from the mensuration of their fields: since this, indeed,

was necessary to them, on account of the inundation of the Nile washing

away the boundaries of land belonging to each. Nor ought it to seem

wonderful that the invention of this as well as of other sciences,

should receive its commencement from convenience and opportunity.

Since whatever is carried in the circle of generation, proceeds

from the imperfect to the perfect. A transition, therefore, is not

undeservedly made from sense to consideration, and from this to the

nobler energies of intellect[108]. Hence, as the certain knowledge

of numbers received its origin among the Phœnicians, on account of

merchandise and commerce, so geometry was found out among the Egyptians

from the distribution of land. When Thales, therefore, first went into

Egypt, he transferred this knowledge from thence into Greece: and

he invented many things himself, and communicated to his successors

the principles of many. Some of which were, indeed, more universal,

but others extended to sensibles. After him Ameristus, the brother

of Stesichorus the poet, is celebrated as one who touched upon, and

tasted the study of geometry, and who is mentioned by Hippias the

Elean, as restoring the glory of geometry. But after these, Pythagoras

changed that philosophy, which is conversant about geometry itself,

into the form of a liberal doctrine, considering its principles in a

more exalted manner; and investigating its theorems immaterially and

intellectually; who likewise invented a treatise of such things as

cannot be explained[109] in geometry, and discovered the constitution

of the mundane figures. After him, Anaxagoras the Clazomenian

succeeded, who undertook many things pertaining to geometry. And

Oenopides the Chian, was somewhat junior to Anaxagoras, and whom Plato

mentions in his Rivals, as one who obtained mathematical glory. To

these, succeeded Hippocrates, the Chian, who invented the quadrature

of the lunula[110], and Theodorus the Cyrenean, both of them eminent

in geometrical knowledge. For the first of these, Hippocrates composed

geometrical elements: but Plato, who was posterior to these, caused

as well geometry itself, as the other mathematical disciplines, to

receive a remarkable addition, on account of the great study he

bestowed in their investigation. This he himself manifests, and his

books, replete with mathematical discourses, evince: to which we

may add, that he every where excites whatever in them is wonderful,

and extends to philosophy. But in his time also lived Leodamas the

Thasian, Archytas the Tarantine, and Theætetus the Athenian; by whom

theorems were increased, and advanced to a more skilful constitution.

But Neoclides was junior to Leodamas, and his disciple was Leon; who

added many things to those thought of by former geometricians. So that

Leon also constructed elements more accurate, both on account of their

multitude, and on account of the use which they exhibit: and besides

this, he discovered a method of determining when a problem, whose

investigation is sought for, is possible, and when it is impossible.

But Eudoxus the Cnidian, who was somewhat junior to Leon, and the

companion of Plato, first of all rendered the multitude of those

theorems which are called universals more abundant; and to three

proportions added three others; and things relative to a section,

which received their commencement from Plato, he diffused into a

richer multitude, employing also resolutions in the prosecution of

these. Again, Amyclas the Heracleotean, one of Plato’s familiars, and

Menæchmus, the disciple, indeed, of Eudoxus, but conversant with Plato,

and his brother Dinostratus, rendered the whole of geometry as yet

more perfect. But Theudius, the Magnian, appears to have excelled, as

well in mathematical disciplines, as in the rest of philosophy. For he

constructed elements egregiously, and rendered many particulars more

universal. Besides, Cyzicinus the Athenian, flourished at the same

period, and became illustrious in other mathematical disciplines, but

especially in geometry. These, therefore, resorted by turns to the

Academy, and employed themselves in proposing common questions. But

Hermotimus, the Colophonian, rendered more abundant what was formerly

published by Eudoxus and Theætetus, and invented a multitude of

elements, and wrote concerning some geometrical places. But Philippus

the Mendæan[111], a disciple of Plato, and by him inflamed in the

mathematical disciplines, both composed questions, according to the

institutions of Plato, and proposed as the object of his enquiry

whatever he thought conduced to the Platonic philosophy. And thus far

historians produce the perfection of this science. But Euclid was not

much junior to these, who collected elements, and constructed many

of those things which were invented by Eudoxus; and perfected many

which were discovered by Theætetus. Besides, he reduced to invincible

demonstrations, such things as were exhibited by others with a weaker

arm. But he lived in the times of the first Ptolemy: for Archimedes

mentions Euclid, in his first book, and also in others. Besides, they

relate that Euclid was asked by Ptolemy, whether there was any shorter

way to the attainment of geometry than by his elementary institution,

and that he answered, there was no other royal path which led to

geometry. Euclid, therefore, was junior to the familiars of Plato, but

more ancient than Eratosthenes and Archimedes (for these lived at one

and the same time, according to the tradition of Eratosthenes) but he

was of the Platonic sect, and familiar with its philosophy: and from

hence he appointed the constitution of those figures which are called

Platonic[112], as the end of his elementary institutions.

CHAP. V.

_What Mathematical Volumes Euclid composed._

There are, therefore, many other mathematical volumes of this man,

full of admirable diligence, and skilful consideration: for such are

his Optics[113], and Catoptrics: and such also, are his elementary

institutions, which conduce to the attainment of music[114]; and his

book concerning divisions[115]. But his geometrical institution of the

Elements is especially admirable, on account of the order and election

of those theorems and problems, which are distributed through the

Elements. For he does not assume all which might be said, but that

only which could be delivered in an elementary order. Besides this, he

exhibits modes of syllogisms of every kind; some, indeed, receiving

credibility from causes, but others proceeding from certain signs; but

all of them invincible and sure, and accommodated to science. But,

besides these, he employs all the dialectic ways, dividing, indeed, in

the inventions of forms; but defining in essential reasons: and again,

demonstrating in the progressions from principles to things sought,

but resolving in the reversions from things sought to principles.

Besides this, we may view in his geometrical elements, the various

species of conversions, as well of such as are simple as of such as

are more composite. And again, what wholes may be converted with

wholes: what wholes with parts; and on the other hand, what as parts

with parts[116]. Besides this, we must say, that in the continuation

of inventions, the dispositions and order of things preceding and

following, and in the power with which he treats every particular,

he is not deceived, as if falling from science, and carried to its

contrary, falsehood and ignorance. But because we may imagine many

things as adhering to truth, and which are consequent to principles

producing science, which nevertheless tend to that error which flows

from the principles, and which deceives ruder minds, he has also

delivered methods of the perspicacious prudence belonging to these.

From the possession of which, we may exercise those in the invention of

fallacies, who undertake this inspection, and may preserve ourselves

from all deception. And this book, by which he procures us this

preparation, is inscribed ψευδαρίος, or, concerning fallacies[117].

Because he enumerates in order their various modes, and in each

exercises our cogitation with various theorems. And he compares truth

with falsehood, and adapts the confutation of deception to experience

itself. This book, therefore, contains a purgative and exercising

power. But the institution of his elementary, skilful contemplation of

geometrical concerns, possesses an invincible and perfect narration.

CHAP. VI.

_Concerning the Purport of Geometry._

But, perhaps, some one may enquire in what the design of this treatise

consists? To this I answer, that its design is to be distinguished as

well according to the objects of enquiry, as according to the learner.

And, indeed, regarding the subject, we must affirm, that all the

discourse of geometry is concerning the mundane figures. Because it

begins from such things as are simple, but ends in the variety of their

constitution. And, indeed, it constitutes each of them separately,

but at the same time delivers their inscriptions in a sphere, and the

proportions which they contain. On which account some have thought,

that the design of each of the books is to be referred to the world;

and they have delivered to our memory, the utility which they afford

us in the contemplation of the universe. But distinguishing the

design with respect to the learner, we must affirm, that its purpose

is the institution of elements; and the perfection of the learners

cogitative powers in universal geometry. For beginning from these,

we are enabled to understand the other parts of this science, and to

comprehend the variety which they contain. And, indeed, without these,

the discipline of the rest, is to us impossible and incomprehensible.

For such theorems as are most principal and simple, and are most allied

to first suppositions, are here collected in a becoming order. And the

demonstrations of other mathematicians, use these as most known, and

advance from these in their most complicated progressions. For thus

Archimedes, in what he has writ concerning the sphere and cylinder, and

Apollonius, and the rest of mathematicians, use, as evident principles,

the things exhibited in this treatise. Its purpose, therefore, is the

institution of learners in the whole geometric science, and to deliver

the determinate constitutions of the mundane figures.

CHAP. VII.

_From whence the Name of Elementary Institution originated, and why

Euclid is called the Institutor of Elements._

But what gave rise to the name of elementary institution, and of

element itself, from which elementary institution was derived?

To this we shall reply, by observing, that of theorems some are

usually called elements, but others elementary, and others again are

determined beyond the power of these. Hence, an element is that whose

consideration passes to the science of other things, and from which we

derive a solution of the doubts incident to the particular science we

investigate. For as there are certain first principles of speech, most

simple and indivisible, which we denominate elements, and from which

all discourse is composed; so there are certain principal theorems of

the whole of geometry, denominated elements, which have the respect of

principles to the following theorems; which regard all the subsequent

propositions, and afford the demonstrations of many accidents essential

to the subjects of geometric speculation. But things elementary

are such as extend themselves to a multitude of propositions, and

possess a certain simplicity and sweetness, yet are not of the same

dignity with elements; because their contemplation is not common to

all the science to which they belong, as is the case in the following

theorem, that in triangles, perpendiculars, drawn from their angles

to their sides, coincide in one point[118]. Lastly, whatever neither

possesses a knowledge extended into multitude, nor exhibits any thing

skilful and elegant, falls beyond the elementary power. Again, an

element, as Menæchmus says, may have a twofold definition. For that

which confirms, is an element of that which is confirmed; as the first

proposition of Euclid with respect to the second, and the fourth with

regard to the fifth. And thus, indeed, many things may be mutually

called elements one of another; for they are mutually confirmed.

Thus, because the external angles of right-lined figures, are equal

to four right angles, the multitude of internal ones equal to right

angles; and, on the contrary, that from this is exhibited[119].

Besides, an element is otherwise called that into which, because it

is more simple, a composite is dissolved. But it must be observed,

that every element cannot be called the element of every thing: but

such as are more principal are the elements of such as are constituted

in the reason of the thing effected; as petitions are the elements

of theorems. And, according to this signification of an element,

Euclid’s elements are constructed. Some, indeed, of that geometry

which is conversant about planes; but others of stereometry. In the

same manner, likewise, in arithmetic and astronomy, many have composed

elementary institutions. But it is difficult, in each science, to chuse

and conveniently ordain elements, from which all the peculiarities of

that science originate, and into which they may be resolved. And among

those who have undertaken this employment, some have been able to

collect more, but others fewer elements. And some, indeed, have used

shorter demonstrations; but others have extended their treatise to an

infinite length. And some have omitted the method by an impossibility;

but others that by proportion; and others, again, have attempted

preparations against arguments destroying principles. So that many

methods of elementary institution have been invented by particular

writers on this subject. But it is requisite that this treatise should

entirely remove every thing superfluous, because it is an impediment to

science. But every thing should be chosen, which contains and concludes

the thing proposed; for this is most convenient and useful in science.

The greatest care, likewise, should be paid to clearness and brevity;

for the contraries to these, disturb our cogitation. Lastly, it should

vindicate to itself, the universal comprehension of theorems, in their

proper bounds: for such things as divide learning into particular

fragments, produce an incomprehensible knowledge. But in all these

modes, any one may easily find, that the elementary institution of

Euclid excels the institutions of others. For its utility, indeed,

especially confers to the contemplation of primary figures: but the

transition from things more simple to such as are more various, and

also that perception, which from axioms possesses the beginning of

knowledge, produces clearness, and an orderly tradition: and the

migration from first and principal theorems to the objects of enquiry,

effects the universality of demonstration. For whatever he seems to

omit, may either be known by the same ways, as the construction of a

scalene and isosceles triangle[120]: or because they are difficult,

and capable of infinite variety, they are far remote from the election

of elements, such as the doctrine of perturbate proportions, which

Apollonius has copiously handled: or, lastly, because they may be

easily constructed from the things delivered, as from causes, such as

many species of angles and lines. For these, indeed, were omitted by

Euclid, and are largely discoursed of by others, and are known from

simple propositions. And thus much concerning the universal elementary

institution of geometry.

CHAP. VIII.

_Concerning the Order of Geometrical Discourses._

But let us now explain the universal order of the discourses contained

in geometry. Because then, we assert that this science consists from

hypothesis[121], and demonstrates its consequent propositions from

definite principles (for one science only, I mean the first philosophy,

is without supposition, but all the rest assume their principles

from this) it is necessary that he who constructs the geometrical

institution of elements, should separately deliver the principles of

the science, and separately the conclusions which flow from those

principles; and that he should render no reason concerning the nature

or truth of the principles, but should confirm by reasons, the things

consequent to these geometric principles. For no science demonstrates

its own principles, nor discourses concerning them; but procures to

itself a belief of their reality, and they become more evident to the

particular science to which they belong than the things derived from

them as their source. And these, indeed, science knows by themselves;

but their consequents, through the medium of these. For thus, also, the

natural philosopher propagates his reasons from a definite principle,

supposing the existence of motion. Thus too, the physician, and he

who is skilled in any of the other sciences and arts. For if any one

mingles principles, and things flowing from principles into one and

the same, he disturbs the whole order of knowledge, and conglutinates

things which can never mutually agree; since a principle, and its

emanating consequent, are naturally distinct from each other. In the

first place, therefore (as I have said), principles in the geometric

institution are to be distinguished from their consequents, which is

performed by Euclid in each of his books; who, before every treatise,

exhibits the common principles of this science; and afterwards divides

these common principles into hypotheses, petitions, and axioms. For all

these mutually differ; nor is an axiom, petition, and hypothesis the

same, according to the demoniacal Aristotle; but when that which is

assumed in the order of a principle, is indeed known to the learner,

and credible by itself, it is an axiom: such as, that things equal

to the same, are mutually equal to each other. But when any one,

hearing another speak concerning that of which he has no self-evident

knowledge, gives this assent to its assumption, this is hypothesis. For

that a circle is a figure of such a particular kind, we presume (not

according to any common conception) without any preceding doctrine. But

when, again, that which is asserted was neither known, nor admitted by

the learner, yet is assumed, then (says he) we call it petition; as

the assumption that all right angles are equal. But the truth of this

is evinced by those who study to treat of some petition, as of that

which cannot by itself be admitted by any one. And thus, according to

the doctrine of Aristotle[122], are axiom, petition, and supposition

distinguished. But oftentimes, some denominate all these hypotheses,

in the same manner as the Stoics call every simple enunciation an

axiom. So that, according to their opinion, hypotheses also will

be axioms; but, according to the opinion of others, axioms will be

called suppositions. Again, such things as flow from principles are

divided into problems and theorems. The first, indeed, containing the

origin, sections, ablations, or additions of figures, and all the

affections with which they are conversant; but the other exhibiting

the accidents essential to each figure. For, as things effective of

science, participate of contemplation, in the same manner things

contemplative previously assume problems in the place of operations.

But formerly some of the ancient mathematicians thought that all

geometrical propositions should be called theorems, as the followers of

Speusippus and Amphinomus, believing, that to contemplative sciences,

the appellation of theorems is more proper than that of problems;

especially since they discourse concerning eternal and immutable

objects. For origin does not subsist among things eternal: on which

account, problems cannot have any place in these sciences; since they

enunciate origin, and the production of that which formerly had no

existence, as the construction of an equilateral triangle, or the

description of a square on a given right line, or the position of a

right line at a given point. It is better, therefore (say they), to

assert that all propositions are of the speculative kind; but that we

perceive their origin, not by production, but by knowledge, receiving

things eternal as if they were generated; and on this account we ought

to conceive all those theorematically, but not problematically. But

others, on the contrary, think that all should be called problems;

as those mathematicians who have followed Menæchmus. But that the

office of problems is twofold, sometimes, indeed, to procure the thing

sought; but at other times when they have received the determinate

object of enquiry, to see, either what it is, or of what kind it is,

or what affection it possesses, or what its relation is to another.

And, indeed, the assertions of each are right; for the followers of

Speusippus well perceive. Since the problems of geometry are not of

the same kind, with such as are mechanical. For these are sensibles,

and are endued with origin, and mutation of every kind. And, on the

other hand, those who follow Menæchmus do not dissent from truth: since

the inventions of theorems cannot by any means take place without an

approach into matter; I mean intelligible matter. Reasons, therefore,

proceeding into this, and giving form to its formless nature, are not

undeservedly said to be assimilated to generations. For we say that the

motion of our cogitation, and the production of its inherent reasons,

is the origin of the figures situated in the phantasy, and of the

affections with which they are conversant: for there constructions and

sections, positions and applications, additions and ablations, exist:

but every thing resident in cogitation, subsists without origin and

mutation. There are, therefore, both geometrical problems and theorems.

But, because contemplation abounds in geometry, as production in

mechanics, all problems participate of contemplation; but every thing

contemplative is not problematical. For demonstrations are entirely

the work of contemplation; but every thing in geometry posterior to

the principles, is assumed by demonstration. Hence, a theorem is more

common: but all theorems do not require problems; for there are some

which possess from themselves the demonstration of the thing sought.

But others, distinguishing a theorem from a problem, say, that indeed

every problem receives whatever is predicated of its matter, together

with its own opposite: but that every theorem receives, indeed, its

symptom predicate, but not its opposite. But I call the matter of

these, that genus which is the subject of enquiry; as for instance,

a triangle, quadrangle, or a circle: but the symptom predicate, that

which is denominated an essential accident, as equality, or section,

or position, or some other affection of this kind. When, therefore,

any one proposes to inscribe an equilateral triangle in a circle, he

proposes a problem: for it is possible to inscribe one that is not

equilateral. But when any one asserts that the angles at the base

of an isosceles triangle are equal, we must affirm that he proposes

a theorem; for it is not possible that the angles at the base of an

isosceles triangle should be unequal to each other. On which account,

if any one forming problematically, should say that he wishes to

inscribe a right angle in a semi-circle, he must be considered as

ignorant of geometry; since every angle in a semi-circle is necessarily

a right one. Hence, propositions which have an universal symptom,

attending the whole matter, must be called theorems; but those in

which the symptom is not universal, and does not attend its subject,

must be considered as problems. As to bisect a given terminated right

line, or to cut it into equal parts: for it is possible to cut it

into unequal parts. To bisect every rectilinear angle, or divide it

into equal parts; for a division may be given into unequal parts. On

a given right line to describe a quadrangle; for a figure that is not

quadrangular may be described. And, in short, all of this kind belong

to the problematical order. But the followers of Zenodotus, who was

familiar with the doctrine of Oenopides, but the disciple of Andron,

distinguish a theorem from a problem, so far as a theorem enquires what

the symptom is which is predicated of the matter it contains; but a

problem enquires what that is, the existence of which is granted. From

whence the followers of Possidonius define a theorem a proposition,

by which it is enquired whether a thing exists or not; but a problem,

a proposition, in which it is enquired what a thing is, or the manner

of its existence. And they say that we ought to form the contemplating

proposition by enunciating, as that every triangle has two sides

greater than the remaining one, and that the angles at the base of

every isosceles triangle are equal: but we must form the problematical

proposition, as if enquiring whether a triangle is to be constructed

upon this right line. For there is a difference, say they, absolutely

and indefinitely, to enquire whether the thing proposed is from a given

point to erect a right line at right angles to a given line, and to

behold what the perpendicular is. And thus, from what has been said, it

is manifest there is some difference between a problem and a theorem.

But that the elementary institution of Euclid, also, consists partly of

problems, and partly of theorems, will be manifest from considering the

several propositions. Since, in the conclusion of his demonstrations,

he sometimes adds (which was to be shewn) sometimes (which was to be

done) the latter sentence being the mark or symbol of problems, and

the former of theorems. For although, as we have said, demonstration

takes place in problems, yet it is often for the sake of generation;

for we assume demonstration in order to shew, that what was commanded

is accomplished: but sometimes it is worthy by itself, since the nature

of the thing sought after may be brought into the midst. But you will

find Euclid sometimes combining theorems with problems, and using them

alternately, as in the first book; but sometimes abounding with the one

and not the other. For the fourth book is wholly problematical; but the

fifth is entirely composed from theorems. And thus much concerning the

order of geometrical propositions.

CHAP. IX.

_Concerning the Design of the first Book,--its Division,--and a

previous Admonition to the Reader._

But, after these considerations, when we have determined the design

of the first book, and have exhibited its division, we shall enter

upon the treatise of the definitions. The design, then, of this book,

is to deliver the principles of the contemplation of right lines. For

though a circle, and its consideration, is more excellent than the

essence and knowledge of right lines, yet the doctrine concerning these

is more adapted to us, who are hastening to transfer our cogitation

from more imperfect and sensible natures, to such as are intelligible.

For, indeed, right lined figures are proper to sensibles, but a

circle to intelligibles. Because that which is simple, uniform, and

definite, is proper to the nature of the things which are: but that

which is various, and which increases indefinitely from the number

of its containing sides, regards the fluctuating essence of sensible

particulars. Hence, in this book, the first and most principal of right

lined figures are delivered; I mean the triangle and parallelogram.

For in these, as under their proper genus, the causes of the elements

are contained: viz. the isosceles and scalene, and those which are

formed from these, the equilateral triangle, and the quadrangle,

from which the four figures of the elements are composed. We shall

find, therefore, as well the origin of the equilateral triangle as

of the quadrangle; of the last, indeed, upon, but of the first from

a given right line. [[123]An equilateral triangle, therefore, is the

proximate cause of the three elements, fire, air, and water: but a

quadrangle is annexed to earth.] And lastly, the design of the first

book is adapted to the whole treatise, and confers to the universal

knowledge of the mundane elements. Besides, it instructs learners in

the science concerning right-lined figures; since it rightly invents,

and accurately collects, the first principles of these.

But this book is divided into three greatest parts, of which the first

declares the origin and properties of triangles, as well according

to angles, as also according to sides. Besides, it makes mutual

comparisons of these, and beholds every one by itself. For receiving

one triangle, sometimes it considers the angles from the sides; but

sometimes the sides from the angles: and this according to equality

and inequality. And supposing two triangles, it discovers the same

property again, by various methods. But the second part combines the

contemplation of parallelograms, describing their properties and

generations. And the third part shews the communication of triangles

and parallelograms, both in symptoms and mutual comparisons. For it

shews that triangles and parallelograms constituted on the same and on

equal bases, are affected with the same passions; and by complication,

when both stand upon one base: and again, after what manner a

parallelogram may be made equal to a triangle; and lastly, concerning

the proportion which in right angled triangles, the square made from

the side subtending, has to the squares containing the right angle. And

such is the division of the first Book.

But, previous to our enquiry into each of these parts, we think it

requisite to admonish the reader, that he must not require of us,

those small assumptions, and cases, and whatever else there may be of

that kind, which has been divulged by our predecessors. For we are

satiated with these, and shall, therefore, but rarely adopt them in our

discourse. But whatever has a more difficult contemplation, and regards

universal philosophy, of this we shall make a particular relation:

imitating the Pythagoreans, with whom this ænigma was common, “a[124]

figure and a step: but not a figure and three oboli,” shewing by this,

that it is requisite to pursue that philosophy which ascends every

theorem by a step, and raises the soul on high; but does not suffer it

to remain among sensibles, to fill up the use attendant on mortals,

and, consulting for this, to neglect the elevation which rises from

hence to an intelligible essence.

DEFINITIONS.

DEFINITION I.

A POINT is that which has no PARTS.

That geometry, according to the transition which takes place from

things more composite to such as are more simple, runs from body, which

is diffused into distance by three dimensions, to a superficies by

which it is bounded; but from superficies to a line, the boundary of

superficies; and from a line to a point destitute of all dimension,

has been often said, and is perfectly manifest. But because these

terms, in many places, on account of their simplicity, appear to be

more excellent than the nature of composites; but in many, as when they

subsist in things which they terminate, they are similar to accidents,

it is necessary to determine in what genera of beings each of these

may be beheld[125]. I say then, that such things as are destitute of

matter, which subsist in separate reasons, and in those forms which

are placed under themselves, are always allotted a subsistence of

more simple essences, superior to the subsistence of such as are more

composite. On this account, both in intellect, and in the ornaments,

as well of the middle kind as among those peculiar to the soul, and

in natures themselves, the terms which proximately vivify bodies,

excel according to essence the things which are terminated; and are

more impartible, more uniform, and more primary than these. For in

immaterial forms, unity is more perfect than multitude; that which is

impartible, than that which is endued with unbounded progression; and

that which terminates, than that which receives bound from another.

But such things as are indigent of matter, and abide in others, and

degenerate from the perfection of their essence, which are scattered

about subjects, and have an unnatural union, are allotted more

composite reasons, prior to such as are more simple. Hence, things

which appear in the phantasy invested with form, and the matter of

the figures which the phantasy contains, and whatever in sensibles is

generated by nature, have, in a preceding order, the reasons of the

things terminated; but the reasons which terminate, in a following

and adventitious rank[126].] For lest that which is distributed into

three dimensions, should be extended into infinite magnitude, either

according to intelligence or sense, it was every way terminated by

superficies. And lest a plane superficies should conceal itself in an

infinite progression, a line approaching opposed its diffusion, and

gave bound to its indefinite extension. And, in like manner, a point

limited the progressions of a line; composite natures deriving their

subsistence from such as are simple. For this also is again manifest,

that in separate forms the reasons of terms subsist in themselves,

but not in those which are terminated; and abiding such as they are

in reality, possess a power of constituting secondary natures. But,

in inseparable forms they give themselves up to things which are

terminated, reside in them, become, as it were, their parts, and

are replenished with baser natures. On which account, that which is

impartible is there endued with a partible essence, and that which is

void of latitude is diffused into breadth. And terms are no longer

able to preserve their simplicity and purity. For since they abide in

another, they necessarily change their own nature into the matter of

their containing subject. Matter, indeed, disturbs the perfection of

these, and causes the reason of a plane to become a profound plane;

but obscuring the one dimension of a line, causes it to be every way

partible; and gives corporeity to the indivisibility of a point, and

separates it together with the natures which it terminates. For all

these reasons falling into matter, the one kind from cogitation into

intelligible matter, but the other from nature into that which is

sensible, are replenished with their containing subjects; and depart

from their own simplicity, into foreign compositions and intervals. But

here a doubt arises how all these, existing in intellect and soul in

an impartible manner, and without any dimension, are distributed into

matter, some indeed, principally, but others on account of its nature?

Shall we say that there is a certain order in immaterial forms, so

that some are allotted the first, some the middle, and others the last

place; and that of forms some are more uniform, but that others are

more multiplied; and that some have their powers collected together,

but others tending into interval; and that some, again, border upon

bound, but that others are proximate to infinity? For though all

participate of these two principles, yet some originate from bound,

but others from infinity, of which they more largely participate.

Hence, a point is entirely impartible, since it subsists according

to bound, yet it occultly contains an infinite power, by which it

produces every interval, and the progression of all intervals, unfolds

its infinite power. But body, and the reason of body, participates

more of an infinite nature; on which account it is among the number of

things terminated by another, and divisible in infinitum, according

to all dimensions. But the mediums between these, according to the

distance of the extremes, are either among the number of things which

have an abundance of bound; or among such as have an affluence of

infinity: on which account they both terminate and are terminated.

For, indeed, so far as they consist from bound, they are able to

terminate others; but so far as they participate of infinity, they are

indigent of termination from others, Hence, since a point is also a

bound, it preserves its proper power in participation: but since it

likewise contains infinity occultly, and is compelled to be every where

present with the natures which it terminates, it resides with them

infinitely. And, because among immaterial forms there was a certain

infinite power capable of producing things distant from each other by

intervals, a point is present with its participants in capacity. For

infinity in intelligibles is the primary cause and prolific power of

the universe; but in material natures it is imperfect, and is alone

all things in dormant capacity. And in short, those forms which, on

account of their simplicity and impartibility, hold a superior rank

among principles, preserve, indeed, (in conformity to their nature,)

their own property in their participations, but become worse than

more composite reasons. For matter is able to participate these more

clearly, and to be prepared for their reception, rather than that of

the most simple causes of beings. On which account, the vestigies of

separate principles descend into matter; but the participations of

those in a second and third order, become more conspicuous. Hence,

matter participates more of the cause of body, than of a plane; and of

this more than the form of a line; and of this still more than that of

a point, which contains all these, and is the boundary of them all.

For the reason of a point presides over this whole series, unites

and contains all partible natures, terminates their progressions,

produces them all by its infinite power, and comprehends them in its

indivisible bound. On which account also, in the images of immaterial

forms, some are the boundaries of others; but a point is the limit

of them all. But that we must not think with the Stoics, that these

boundaries of bodies alone subsist from cogitation; but that there are

certain natures of this kind among beings, which previously contain

the demiurgical reasons of things, we shall be enabled to remember, if

we regard the whole world, the convolutions of its parts, the centres

of those convolutions, and the axes which penetrate through the whole

of these revolving circles. For the centres subsist in energy, since

they contain the spheres, preserve them in their proper state, unite

their intervals, and bind and establish to themselves the powers which

they possess. But the axes themselves being in an immoveable position,

evolve the spheres, give them a circular motion, and a revolution round

their own abiding nature. And the poles of the spheres, which both

terminate the axes, and bind in themselves the other convolutions, do

they not perspicuously evince, that points are endued with demiurgical

and capacious powers, that they are perfective of every thing distant

by intervals, and are the sources of union, and an unceasing motion?

From whence, indeed, Plato[127] also says, that they have an adamantine

subsistence; shewing by this, the immutable, eternal, and stable

power of their essence, ever preserving itself in the same uniform

mode of existence. He adds too, that the whole spindle of the Fates,

is turned about these, and leaps round their coercive union. But other

more recondite and abstruse discourses affirm, that the demiurgus

presides over the world, seated in the poles, and, by his divine love,

converting the universe to himself. But the Pythagoreans thought

that the pole should be called the Seal of Rhea[128]; because the

zoogonic, or vivific goddess, pours through these into the universe,

an inexplicable and efficacious power. And the centre they called the

prison of Jupiter; because, since Jupiter has placed a demiurgical

guard in the bosom of the world, he has firmly established it in the

midst. For, indeed, the centre abiding, the universe possesses its

immoveable ornament, and unceasing convolution: and the gods who

preside over the poles, obtain a power collective of divisible natures,

and unific of such as are multiplied: and those who are allotted the

government of the axes, restrain and eternally evolve their perpetual

convolutions. And, if it is lawful to offer our own opinion on this

subject, the centres and poles of all the spheres are the symbols of

the conciliating gods, shadowing forth their imperceptible and unifying

composition. But the axes express the coherencies of the universal

ornaments; and are endued with a power of comprehending the mundane

integrities and periods, in the same manner as their presiding deities,

of such as are intellectual. But the spheres themselves are images

of the gods, called perfectors of works, copulating the principle

with the end, and excelling all figures in simplicity, similitude,

and perfection. But we have been thus prolix, that we might evince

the power of impartibles, and of the terms which the world contains,

and that so far as they bear an image of primary and most principal

causes, they are allotted the most excellent order in the universe.

For centres and poles are not of the same kind with things which

are terminated; but they subsist in energy, and possess an essence,

and perfect power, which pervades through all partible natures. But

many beholding those terms which imperfectly subsist in terminated

essences, consider them as endued with a slender subsistence; and

some indeed say, that they are alone separated from sensibles by

thought; but others, that they have an essence no where but in our

thoughts. However, since the forms of all these are found both in

the nature of intellect, in the ornaments of soul, in the nature of

things, and in inferior bodies, let us consider how, according to the

order they contain, they subsist in the genera of beings. And indeed,

all of them pre-exist in intellect, but in an impartible and uniform

manner: so that they all subsist according to one form, the reason of

a point, which exists occultly and impartibly. But they all subsist

in soul according to the form of a line: on which account Timæus also

composes the soul from right and circular lines: for every circle is

a line alone[129]. But they all subsist in natures, according to the

reason of a plane; and on this account, Plato commands us to manifest

those natural reasons, which are endued with a power of constituting

bodies by a plane. And the resolution of bodies into planes leads us

to the proximate cause of appearances. Lastly, they all subsist in

bodies, but in a corporeal manner; since all forms have their being

in these, according to the partible nature of bodies. Hence, all of

them appear every where, and each according to its proper order; and

diversity arises from pre-dominating power. The point, indeed, is every

where impartible, and when that which is divisible into parts, excels

according to the diminution of beings, it vindicates to itself, an

illustrious subsistence of partible natures. And sometimes the point is

entirely superior, according to the excellence of cause; but sometimes

it is connected with divisibles, and sometimes it is allotted in them

an adventitious existence; and, as if swallowed up by the partition of

the lowest natures, loses its own proper impartibility. As, therefore,

with respect to the monad, one[130] is the mother of number, but the

other is as matter spread under, and the receptacle of numbers; and

each of them a principle, (yet neither of them is number), but in a

different respect: in the same manner a point also, is partly the

parent and author of magnitudes; but is partly a principle in another

respect, and not according to a generative cause. But is a point, then,

the only impartible? Or may we affirm this of the now in time, and of

unity in numbers? Shall we not say, that to the philosopher, indeed,

discoursing concerning the universality of things, it is proper to

behold every thing, however falling under distribution; but that to

him who is endued with the science of particulars, who produces his

contemplation from certain definite principles, and runs back even to

these, but very little scrutinizes the progressions of beings, it is

requisite to attempt, consider, and treat concerning that impartible

nature alone, which regards his first principles; and to behold that

simplicity which presides over all the particular subjects of his

knowledge? In consequence of this reasoning, therefore, a point alone,

according to the geometric matter, is destitute of partition; but unity

according to that which is arithmetical. And the reason of a point,

however in some other respects it may be imperfect, yet is perfect in

the present science. For, indeed, the physician also says, that the

elements of bodies are fire and water, and things similar to these; and

as far as to these the resolution of bodies proceeds. But the natural

philosopher passes on to more simple elements; and the one defines an

element simple as to sense, but the other simple as to reason; and both

of them properly as to their peculiar science. We must not, therefore,

think that the definition of a point is faulty, nor determine it as

imperfect; for so far as pertains to the geometric matter, and its

principles, it is sufficiently delivered. This alone, indeed, is

wanting to its completion, that the definition does not clearly say,

_that which is impartible with me is a point; and my principle, and

that which I contain as most simple, is nothing else than this_.

And after this manner it is proper to hear the geometrician addressing

us. Euclid, therefore, from a negation of parts, declares to us a

principle, leading to the theory of its whole subject nature. For

negative discourses are proper to principles, as Parmenides teaches

us, who delivers the doctrine concerning the first and last cause,

by negations alone. Since every principle consists of an essence

different from its flowing consequents; and the negations of these

exhibit to us the property of their source. For that it is, indeed, the

cause of these, yet at the same time has nothing in common with these,

becomes perspicuous from a doctrine of this kind. But here a doubt may

arise, how, since the phantasy receives all things invested with forms,

and in a partible manner, the geometrician beholds in it the point

destitute of parts? For it is not because they are reasons existing in

cogitation, but the phantasy receives the resemblances of intellectual

and divine forms according to its own proper nature, exhibiting in its

shadowy bosom the forms of formless natures, and clothing with figure

things entirely free from the affections of figure. To this ambiguity

we must say, that the species of imaginative motion is neither alone

partible, nor impartible; but that it proceeds from the impartible to

the partible, and from the formless nature to that which is expressed

by form. For if it was partible alone, it could not preserve in itself

many impressions of forms, since the subsequent would obscure the

pre-existent figures: for no body can contain at once, and according

to the same situation, a multitude of figures; but the former will

be blotted out by the succession of the latter. But if it was alone

impartible, it would not be inferior to cogitation, and to soul, which

surveys all things in an impartible manner. Hence, it is necessary that

it should indeed begin from an impartible according to its motion, and

from thence draw forth the folded and scattered form of every thing

falling under cogitation, and penetrating to its shadowy receptacle:

but, that it should at length end in form, figure, and interval. And

if it be allotted a nature of this kind, it will, after a certain

manner, contain an impartible essence: and a point, according to this,

must be said to have its principal subsistence: for the form of a

line is contracted in the phantasy according to this. Hence, because

it possesses a twofold power, impartible and partible, it will indeed

contain a point in an impartible, and intervals in a partible manner.

But as the Pythagoreans define a point to be unity having position, let

us consider what they mean. That numbers, indeed, are more immaterial

and more pure than magnitudes, and that the principle of numbers is

more simple than the principle of magnitudes, is manifest to every one:

but when they say that a point is unity endued with position, they

appear to me to evince that unity and number subsist in opinion: I mean

monadic number[131]. On which account, every number, as the pentad and

the heptad, is one in every soul, and not many; and they are destitute

of figure and adventitious form. But a point openly presents itself in

the phantasy, subsists, as it were, in place, and is material according

to intelligible matter. Unity, therefore, has no position, so far as it

is immaterial, and free from all interval and place: but a point has

position, so far as it appears seated in the bosom of the phantasy,

and has a material subsistence. But unity is still more simple than a

point, on account of the community of principles. Since a point exceeds

unity according to position; but appositions in incorporeals produce

diminutions of those natures, by which the appositions are received.

DEFINITION II.

A Line is a Length without Breadth.

A Line obtains the second place in the Definitions, as it is by far

the first and most simple interval, which the geometrician calls a

length, adding also without breadth; since a line, in respect of a

superficies, ranks as a principle. For he defines a point, as it is the

principle of all magnitudes, by negation alone; but a line, as well by

affirmation as by negation. Hence it is a length, and by this exceeds

the impartibility of a point; but it is without breadth, because it

is separated from other dimensions. For, indeed, every thing which

is void of breadth, is also destitute of bulk, but the contrary is

not true, that every thing void of bulk is also destitute of breadth.

Since, therefore, he has removed breadth from a line, he has also

removed at the same time bulk. On which account he does not add, that

a line also has no thickness, because this property is consequent

to the notion of being without breadth. But it is defined by others

in various ways: for some call it the flux of a point, but others a

magnitude contained by one interval. And this definition, indeed; is

perfect, and sufficiently explains the essence of a line; but that

which calls it the flux of a point, appears to manifest its nature from

its producing cause; and does not express every line, but alone that

which is immaterial. For this is produced by a point, which though

impartible itself, is the cause of being to partible natures. But the

flux of a point, shews its progression and prolific power, approaching

to every interval, receiving no detriment, perpetually abiding the

same, and affording essence to all partible magnitudes. However, these

observations are known, and manifest to every one. But we shall recall

into our memory, discourses more Pythagorical, which determine a point

as analogous to unity, a line to the duad, a superficies to the triad,

and body to the tetrad. [[132]Yet when we compare those which receive

interval together, we shall find a line monadic; but a superficies

dyadic, and a solid body triadic.] From whence also, Aristotle[133]

says; that body is perfected by the ternary number. And, indeed, this

is not wonderful, that a point, on account of its impartibility,

should be assimilated to unity; but that things subsequent to a point,

should subsist according to numbers proceeding from unity, and should

preserve the same proportion to a point, as numbers to unity; and that

every one should participate of its proximate superior, and have the

same proportion to its kindred, and following degree, as the superior

to this, which is the immediate consequent. [[134]For example, that

a line has the order of the duad with respect to the point, but of

unity to a superficies; and that this last has the relation of a triad

to the point, but of the duad to a solid.] And on this account, body

is tetradic, with respect to a point, but triadic as to a line. Each

order, therefore, has its proportion; but the order of the Pythagoreans

is the more principal, which receives its commencement from an exalted

source, and follows the nature of beings. For a point is indeed

twofold; since it either subsists by itself, or in a line; in which

last respect also, since as a boundary it is alone and one, neither

having a whole nor parts, it imitates the supreme nature of beings.

On which account too, it was placed in a correspondent proportion to

unity. [135]For as the oracle says, _Unity is there first, where the

paternal unity abides_. But a line is the first endued with parts

and a whole, and it is monadic because it is distant by one interval

only; and dyadic on account of its progression: for if it be infinite,

it participates of the indefinite duad; but if finite, it requires two

terms, from whence and to what place; since, on account of these it

imitates totality, and is allotted an order among totals. For unity,

according to the oracle, is extended[136], and generates two; and

this produces a progression into longitude, together with that which

is distant extendedly, and with one interval, and the matter of the

duad. But superficies, since it is both a triad and duad, as also the

receptacle of the primary figures, and that which receives the first

form and species, is in a certain respect similar to the triadic

nature, which first terminates beings; and to the duad, by which they

are divided and dispersed. But a solid, since it has a triple distance,

and is distinguished by the tetrad, which is endued with a power of

comprehending all reasons, is reduced to that order in which the

distinction of corporeal ornaments appears; as also the division of the

universe into three parts, together with the tetradic property, which

is generative and female. And these observations, indeed, might be more

largely discussed, but for the present, must be omitted. Again, the

discourse of the Pythagoreans, not undeservedly, calls a line, which is

the second in order, and is constituted according to the first motion

from an impartible nature, dyadic. And that a point is posterior to

unity, a line to the duad, and a superficies to the triad, Parmenides

himself shews, by first of all taking away multitude from one by

negation, and afterwards the whole. Because, if multitude is before

that which is a whole, number also will be prior to that which is

continuous, and the duad to the line, and unity to the point: since the

epithet _not many_, belongs to unity which generates multitude,

but to the point, the term _not a whole_, is proper, because it

produces a whole; for this is said to have no part. And these things

are affirmed of a line, while we more accurately contemplate its

nature. But we should also admit the followers of Apollonius, who say,

that we obtain a notion of a line, when we are ordered to measure the

lengths alone, either of ways or walls; for we do not then subjoin

either breadth or bulk, but only make one distance the object of our

consideration. In the same manner we perceive superficies, when we

measure fields; and a solid, when we take the dimensions of wells. For

then, collecting all the distances together, we say, that the space of

the well is so much, according to length, breadth, and depth. But a

line may become the object of our sensation, if we behold the divisions

of lucid places from those which are dark, and survey the moon when

dichotomized: for this medium has no distance with respect to latitude;

but is endued with longitude, which is extended together with the light

and shadow.

DEFINITION III.

But the Extremities of a Line are Points.

Every composite receives its bound from that which is simple, and every

thing partible from that which is impartible; and the images of these

openly present themselves in mathematical principles. For when it is

said that a line is terminated by points, it seems manifestly to make

it of itself infinite, because, on account of its proper progression,

it has no extremity. As, therefore, the duad is terminated by unity,

and reduces its own intolerable boldness under bound, when it is

restrained in its comprehensive embrace: so a line also is limited by

the points which it contains. For, since it is similar to the duad,

it participates of a point having the relation of unity, according to

the nature of the duad. Indeed, in imaginative, as well as in sensible

forms, the points themselves terminate the lines in which they reside.

But in immaterial forms, the reason of the impartible point pre-exists

separate and apart; but when proceeding from thence by far the first of

all, by determining itself with interval, moving itself, and flowing

in infinite progression, and imitating the indefinite duad, it is

restrained indeed, by its proper principle, is united by its power,

and on every side seized by its coercive bound. Hence it is, at the

same time, both infinite and finite: infinite, indeed, according to its

progression; but finite according to its participation of a terminating

cause. So that, when it approaches to this cause, it is detained in its

comprehension, and is terminated according to its union. Hence too, in

the images of incorporeal forms, a point is said to terminate a line,

by occupying its beginning and end. Bound, therefore, in immaterials,

is separated from that which is bounded: but here it is twofold; for

it subsists in that which is terminated. And this affords a wonderful

symptom, that forms; indeed, abiding in themselves, precede their

participants according to cause; but when giving themselves up to their

subordinate natures, subsist according to their diversified properties:

since they are multiplied and distributed together with these, and

receive the division of their subjects. Besides, this also must be

previously received concerning a line, that our geometrician uses it in

a threefold acceptation. As terminated on both sides, and finite; as

in the problem[137] which says, Upon a given terminated right line to

construct an equilateral triangle. And as partly infinite and partly

finite; as in the problem which commands us from three right lines,

which are equal to three given right lines, to construct a triangle;

for in the construction of the problem, he says, Let there be placed a

certain right line, on one part finite, but on the other part infinite.

And again, a line is received by Euclid as on both sides infinite; as

in the problem which says, Upon a given infinite right line, from a

given point, which is not in that line, to let fall a perpendicular.

But, besides this, the following doubts, since they are worthy of

solution, must not be omitted. How are points called the extremities

of a line? and of what line, since they can neither be the bounds of

one that is infinite, nor of every finite? For there is a certain line,

which is both finite, and has not points for its extremities. And such

is a circular line, which returns into itself, and is not bounded by

points, like a right line. And such also is the ellipsis, or line like

a shield. Is it therefore requisite to behold a line, considered as a

line? for we must receive a certain circumference, which is terminated

by points, and a part of the elliptic line; having, in like manner,

its extremities bounded by points. But every circular and elliptic

line, assumes to itself another certain property, by which it is not

line alone, but is also endued with a power of perfecting figure[138].

Lines, themselves, therefore, have their extremities terminated by

points; but those which are effective of such like figures, return into

themselves. And, indeed, if you conceive them to be described, you

will also find how they are bounded by points; but if you receive them

already described, and connect the end with the beginning, you can no

longer behold their extremes.

DEFINITION IV.

A Right Line, is that which is equally situated between its

_bounding_ Points.