BOOK III.

THE

PHILOSOPHICAL and MATHEMATICAL

COMMENTARIES OF PROCLUS,

ON

THE FIRST BOOK OF EUCLID’S ELEMENTS.

TO WHICH ARE ADDED,

A History of the Restoration of PLATONIC THEOLOGY,

BY THE LATTER PLATONISTS:

And a Translation from the Greek of

PROCLUS’S THEOLOGICAL ELEMENTS.

IN TWO VOLUMES.

[Illustration]

VOL II.

LONDON, PRINTED FOR THE AUTHOR:

And Sold by T. PAYNE and SON; B. WHITE and SON; J. ROBSON; T. CADELL;

LEIGH and CO.; G. NICOL; R. FAULDER; and T. and J. EGERTON. 1792.

[ Price Two Guineas in Boards. ]

PREFACE.

I Now present the reader with the remaining part of the Commentaries

of Proclus on Euclid: with the addition of his Theological Elements,

and a History of the Restoration of the Platonic Theology, by the

latter disciples of Plato. Should my design be enquired into, in

combining works which the superficial observer will consider as

opposite and heterogeneous, I answer that it is no less novel than

certain, no less important than apposite and connected. Its novelty

will be evident by assuring the reader, that a vindication of

philosophical Polytheism, as embraced by the wisdom, and supported

by the general voice of antiquity, is the ultimate tendency of its

execution. Its connection too with geometry will be manifest to every

Tyro in Platonism, and has been so copiously proved in the former

volume, that it would be superfluous to repeat the demonstration in the

present. I am well aware that nothing has been so much the subject of

ridicule and declamation, of ignorant aspersion and impotent contempt,

as the theology of the ancients. It has supplied the harangues of the

pulpit with an endless variety of popular argument, and an exhaustless

source of _priestly_ elocution. It has been equally derided by the

man of learning and the merchant, by the noble and the vulgar, by

the peasant and the priest. But it still lives in the works of the

ancients, it is still capable of being supported by sound reasoning,

and sublime philosophy; and its intrinsic excellence and truth will

extend its existence beyond the wreck of modern systems, and the

desolation of ages. Like a strong and capacious ship it sails with

majestic security through the ocean of time; and sustains with careless

dignity the storms of opposition that roar round its well-compacted

sides. The blasts of calumny may indeed impede its progress, but are

unable to shatter its indissoluble fabric; and the prosperous gales of

philosophy will always succeed the tempests of folly, and waft it with

rapidity to the _enlightned_ regions of mankind. The time perhaps is

not far distant, when this fortunate change may commence. Above twelve

hundred years have elapsed since the vessel of ancient wisdom visited

the civilized parts of the world, and the nations were blest with its

invaluable contents: and during this dreadful interval, ignorance

and delusion, _jargon_ and _reverie_, have held an undisturbed and

universal reign. The depravity of the times is the subject of general

complaint: genius no longer soars; learning has evaporated into words;

and philosophy is but a name. Yet, though the restoration of ancient

theology is the object of my most ardent desires, I much fear that

a period still more barbarous, with respect to philosophy; that an

age still darker and more debased must precede its establishment on

the earth. Prodigies and destruction attended, as we shall observe

in the ensuing history, its departure from mankind; and desolation

will doubtless be the harbinger of its future appearance. The orb of

vicissitude produces renovation and decay in regular succession; and

marks, as it revolves, the dormant events of future periods with _the

ruinous_ characters of the past. Let us, therefore, patiently wait for,

and joyfully expect the happy moment when the breezes of philosophy

shall arise with abundance and vigour; and impel the vessel of theology

laden with the riches of wisdom, on our natal coast. The revolution is

certain, however remote: and the prospect is of itself sufficient to

increase the vigour of exertion, and animate the expectations of hope;

to enable us to brave the storms of ecclesiastical persecution, and

vanquish the resistance of folly.

COMMENTARIES

OF

PROCLUS.

BOOK III.

_Concerning Petitions and Axioms._

[1]Since the principles of geometry are triply divided into

Hypotheses, Petitions, and Axioms, the difference between these we

have explained in the preceding books. But we now intend to discourse

more accurately of petition and axiom, as especially necessary to our

present design. For hypotheses, which are also called definitions, we

have already explained. It is common, therefore, as well to axioms as

to petitions, to require no demonstration, and no geometrical faith:

but to be received as manifest, and to become the principles of the

rest. But they differ mutually from each other, in the same manner in

which we have distinguished theorems from problems. For as in theorems

we propose to perceive and know that which follows a subject; but

in problems we are ordered to compare and do something: in the same

manner also in axioms, we must receive whatever is manifest of itself,

and easily apprehended by our untaught conceptions; but in petitions

we must receive whatever is easy to be done and compared, (since in

admitting these, thought is not fatigued) and whatever requires no

variety, and no kind of construction. Hence evident and indemonstrable

cognition, and unconstructed assumption, distinguish petitions from

axioms. Just as demonstrative cognition, and an assumption of things

sought, together with construction, separates theorems from problems.

For it is every where requisite, that principles in simplicity,

indemonstrability, and self-evidence, should excel things posterior

to principles. For universally (says Speusippus) of the things which

cogitation pursues, some of its energies it produces without a various

progression, prepares them for future enquiry, and has a more evident

apprehension of these than of visible objects: but others which it

is not able immediately to follow, by a transition proceeding from

their nature, these it endeavours by _consequence_ to pursue. Thus

for example, _to draw a right line from one point to another_, it

receives as evident, and easy to be done. For since in this case the

line is composed from the indeclinable flux of a point, and at the

same time advances in an orderly progression, because it no where more

or less declines, it necessarily falls in another point. Again, if

one extremity of a right line abiding, the other is moved about it,

it will describe a circle without any labour. But if any one wishes

to describe a helix of one revolution, it requires a more various

operation. For it is generated by various motions. Likewise if any one

wishes to construct an equilateral triangle, he will require a certain

method for its construction. For the geometrical intellect says,

when I understand a right line, which abides according to one of its

extremities, but is moved about it according to the other, and at the

same time conceive a point, which is moved in the line from the abiding

extreme, I have described a helix of one revolution. For when at the

same time both the extremity of the right line, which describes the

circle, and the point which is moved in the right line, arrive at the

same point, and coincide, they produce for me such a helix. And again,

when I describe equal circles, and draw right lines from the common

section to the centre of the circles, and a right line from one centre

to the other, I shall have an equilateral triangle. The production

of these, therefore, is very remote from a simple apprehension, and

primary notion. For we are content to pursue the progressions of their

origin. Hence it happens that these are compared with greater ease or

difficulty, and are exhibited with many or fewer mediums, according to

the habit of those who enter on this undertaking: but that they require

demonstration and construction, on account of the property of the

things sought, which wants the evidence of petitions and axioms.

Petition, therefore, and axiom, are simple and easy to be apprehended.

But petition, indeed, commands us to fabricate, and provide a certain

matter, in order to the assignation of the _symptom_, which possesses

an easy and simple apprehension: but axiom pronounces a certain

essential accident, of itself known to the hearers. As that _fire is

hot_, or any other of those manifest truths, he who doubts of which,

we consider as either wanting sense or punishment. Hence, petition and

axiom are of the same genus; but they differ in the above-mentioned

manner. For each is an indemonstrable principle, but this after one

mode, and that after another, as we have already observed. But some

think that all these should be called petitions, in the same manner

as all problems, _things sought_. For Archimedes beginning his book

of _Equiponderants_, _we desire it may be granted_ (says he) _that

things equally heavy, from equal lengths, will equally ponderate_;

though some would rather chuse to call this an axiom. But others call

all these axioms, in the same manner as they denominate every thing

a theorem, which requires demonstration. For, according to the same

proportion, as it seems they pass from proper names to such as are

common. Nevertheless, as a problem differs from a theorem, so petition

from axiom: though both these last are indemonstrable, and the former

require demonstration. And the one, indeed, is assumed as easy to

be done, but the other is granted as easy to be known by the common

consent of all men. After this manner, therefore, Geminus distinguishes

petitions from axioms.

But others will perhaps say, that petitions are indeed proper to the

geometrical matter: but that axioms are common to the universal theory,

which is conversant about the _how-much_, and _the how-many_. For the

geometrician knows _that_ which requires _that all right angles are

equal_, and _that every finite right line may be produced straight

forwards_: but _that_ which says, _things equal to one and the same

are equal to each other_, is a common conception, which not only the

arithmetician employs, but every one endued with science, accommodating

that which is common to his own particular matter. But Aristotle (as we

have before observed[2]) says, that petition, since it is demonstrable,

is not granted by the hearer, yet is received as a principle: but that

axiom is of itself indemonstrable, and that this is confessed by all,

according to habit, though some, for the sake of disputation, have

doubted its evidence. Since then, there are these three differences,

according to the first, which by operating, and knowledge only

distinguishes petition from axiom, it is manifest that which says

_all right angles are mutually equal, is not a petition_. Nor the

fifth, which says, _if a right line falling on two right lines makes

the internal angles towards the same parts less than two right, those

right lines infinitely produced, shall coincide towards the parts in

which the angles less than two right subsist_. For these are neither

assumed in construction, nor do they command any thing to be done: but

they exhibit a certain _symptom_, inherent in right angles, and in

right lines, departing from angles less than two right. But, according

to the second difference, that will not be an axiom which says, that

_two right lines cannot comprehend space_, which some at present

consider as an axiom. For this is proper to the geometric matter, as

likewise that which affirms _that all right angles are equal_. But

according to the third difference, which is Aristotelic, all those

which produce their own credibility by a certain demonstration, are

petitions; but whatever are indemonstrable, are axioms. Apollonius,

therefore, in vain endeavours to deliver the demonstrations of

axioms: for Geminus very properly observes, that some have attempted

demonstrations of indemonstrables, and have endeavoured from more

unknown mediums, to prove things manifest to all, into which error

Apollonius has fallen, who wishes to prove the axiom true, which says,

_that things equal to one, and the same, are equal to each other_: but

that others assume in the place of indemonstrables, things requiring

demonstration. As is the case with Euclid himself, in the fourth and

fifth petition. For some say, that this last, as ambiguous, requires

demonstration. Indeed, is it not ridiculous, that theorems should be

assigned as indemonstrable, the converse of which are demonstrable?

For that the internal angles of coincident right lines are less than

two right, Euclid himself shews in the theorem, which says, _that two

angles of every triangle, however taken, are less than two right_:

besides, it may be perspicuously shewn, that not every thing equal

to a right angle is a right angle. Hence, says Geminus, the converse

of these are not to be granted indemonstrable. It seems therefore,

according to the ordination of this man, that there are, indeed, three

petitions: but that the other two, and the converse of these, require

demonstrating science: and that in the axioms, the one which says,

that two right lines cannot comprehend space, is superfluously added,

since its credibility must be derived from demonstration. And thus much

concerning the difference of petitions and axioms. Again, of axioms,

some are proper to arithmetic, but others to geometry; and others are

common to both: for that which says, _every number is measured by

unity_, is an arithmetical axiom. But that which says _equal right

lines agree amongst themselves_, as also this which affirms _that every

magnitude is divisible in infinitum_, are geometrical axioms: but the

one which says _that things equal to the same, are mutually equal_, and

all of this kind are common to both. However, it must be observed, that

each science uses such as the last, according to its proper subject;

as geometry in magnitudes, but arithmetic in numbers. In like manner

of petitions, some are peculiar to particular sciences, but others

are common to all. For you must call the petition which requires to

be granted, _that a number may be divided into the least parts_,

peculiar to arithmetic: but this, _that every finite straight line

may be produced straight forwards_, peculiar to geometry; and the one

which desires us to grant, _that quantity may be infinitely increased_,

common to both; for this passion is equally found to reside in number

and magnitude.

PETITIONS or POSTULATES.

I.

Let it be granted that a straight line may be drawn from any one

point to any other point.

II.

That a terminated straight line may be produced to any length in

a straight line.

III.

And that a circle may be described from any centre, at any

distance from that centre.

According to the opinion of Geminus, these three are necessarily

placed among petitions, as well on account of their facility, as

because they command us to do something. For this, _to draw a right

line from every point, to every point_, follows the definition, which

says, _that a line is the flux of a point_, and a right line _an

indeclinable and inflexible flow_. If then we conceive a point to

be moved with an uninclined, and the shortest motion, we shall fall

upon another point, and the first petition will be produced, and we

shall understand nothing various or difficult. But if when the right

line itself is terminated by a point, we conceive its extremity moved

with the shortest indeclinable motion, the second petition will arise

from an easy and simple apprehension. But if we again imagine that

the terminated right line abides according to its other extreme, but

that it moves about that which abides according to the rest, the

third petition will be produced; for the centre is the point which

abides, but the interval the right line. Since the distance of the

centre, from all parts of the circumference, is always equal to the

quantity of this interval. But if any one should doubt how we apply

motion in geometrical concerns, which have an immoveable existence;

and how we can move impartibles, (since this is impossible) we request

him to call to mind what we have demonstrated in the beginning of

these Commentaries. I mean that the reasons of things subsisting in

the phantasy, describe there all the images of cogitation, of which

cogitation itself possesses the reason: for an intellect of this kind

is an unwritten, ultimate, and passive tablet. Hence, it receives forms

from another, accompanied with motion; but we must not understand a

corporeal but imaginative motion, and must by no means admit that

impartibles are moved with corporeal motions, but that they suffer

imaginative progressions. For intellect, though impartible, is moved,

yet not according to place, and the phantasy has a proper motion

according to the impartible which it contains: but we only regarding

corporeal motions, neglect those which are made in things destitute

of interval. Impartibles, therefore, are pure from corporeal place,

and external motions: but another species of motion, and another place

congenial to such motions, is considered in their progressions. For,

indeed, we should say, that a point also has position in the phantasy,

and should not enquire how an impartible can abide, which is at the

same time moved elsewhere, and comprehended by place. Since the place

of things, with dimension, possesses itself dimension; but the place

of impartibles is destitute of all dimension. The proper species

therefore of geometrical concerns, are different from the things they

produce; and the motion of bodies is different from that of the forms

in the phantasy; and the place of partible is different from that of

impartible natures; and it is requisite, by distinguishing these,

neither to confound nor disturb the essences of things. But it appears

that the first of these three petitions declares to us in images, how

_the things which are_, are contained in their own impartible causes,

and are terminated by their immaterial bound; and that previous to

their constitution, they are on all sides comprehended in their

indivisible embrace: for the points existing, a right line is drawn

from the one to the other, is terminated by, and received between them.

But the second indicates how _the things which are_ by possessing

proper causes proceed to all things, preserving in them a continuation

not derived from the natures into which they proceed; but that through

a cause of infinite power, they endeavour to permeate every where,

with a never-failing progression. And the third petition shadows forth

the manner in which these progressions return again to their proper

principles: for the convolution of a point producing a circle, by

moving about an abiding point, imitates a circular regression. But it

is requisite to know, that every line cannot be infinitely produced,

for the circle and cissoid, and all such as describe figure, are

incapable of this property; as likewise some which produce no figure.

For the helix of one revolution cannot be infinitely produced, since

it is constituted between two points; nor any other lines similarly

formed. But neither is it possible to extend every line from every

point, to every point; for every line cannot subsist between all

points: and thus much for the three first petitions; let us now proceed

to the rest.

IV.

All right angles are equal to each other.

If the present petition is considered by us as manifest, and as

requiring no demonstration, it is not a petition according to the

opinion of Geminus, but an axiom; for it affirms a certain essential

accident of right angles, not commanding us to perform any thing

according to a simple conception. But neither is it a petition

according to the division of Aristotle: for petition, according to

his opinion, requires some demonstration. But if we should say it is

demonstrable, and enquire after its demonstration, yet according to

the opinion of Geminus, it ought not to be placed among petitions.

The equality, therefore, of right angles, appears from our common

conceptions; for since a right angle has the relation of unity or bound

to the infinite increase and decrease of the angles on each side, it is

equal with respect to every right angle, since we constitute the first

right angle after this manner, by a right line making angles on each

side of the right line on which it stands equal to each other; but if

it be requisite to produce a linear demonstration of this, let there be

two right angles, one _a b c_, the other _d e f_.

[Illustration]

I say that they are equal; for if they are not equal, one of them must

be greater, suppose the angle at _b_. If then the line _d e_ be adapted

to the line _a b_, the line _e f_ shall fall within. Let it fall as _b

g_, and let the line _b c_ be produced to _h_; because, then _a b c_ is

a right angle, _a b h_ also shall be a right angle, and they shall be

mutually equal to each other, from the tenth Definition: the angle _a b

h_ therefore, is greater than the angle _a b g_. Let again the line _g

b_ be produced to _k_, because, therefore _a b g_ is a right angle, the

successive angle _a b k_ shall be a right one, and consequently equal

to _a b g_. Hence, the angle _a b h_, shall be less than the angle _a

b g_; but it was also greater, which is impossible: but this has been

shewn by other expositors, and requires no great consideration. But

Pappus very properly admonishes us, that the converse of this Petition

is not true; I mean, that every thing equal to a right angle, is a

right angle; though if it be rectilinear, it is without doubt a right

angle. But a curvilinear angle may also be exhibited equal to one that

is right: for let there be conceived two equal right lines, _a b_, and

_b c_, making the angle at the point _b_, right;

[Illustration]

and on them let the semicircles _a e b_, _b f c_, with a proper centre

and interval be described; because, therefore, the semicircles are

equal, they shall have a mutual congruence, and the angle _e b a_, is

equal to the angle _f b c_, and _a b f_ is common: the whole right

angle, therefore, is equal to the lunular, i.e. to _e b f_, and yet

the lunular is not a right angle. In the same manner, if the angle _a

b c_ should be obtuse or acute, a lunular angle may be shewn equal

to it (for this is that genus of curvilinear angles which agrees

with such as are rectilinear), only this is to be observed, that in

a right and obtuse angle, it is requisite to add the middle angle,

which is contained by the line _a b_, and the circumference _b f_;

but in an acute angle to take this away: for the right line _c b_, in

these cases, cuts the circumference _b e_. The truth of which, will be

evident from the following figures:

[Illustration]

[Illustration]

And hence, it appears, that all right angles are mutually equal to

each other, and that not every thing equal to a right angle, is

consequently a right angle: for if it be not rectilinear, how can it be

called right. But it is also manifest from this Petition, that angular

rectitude is allied to equality, in the same manner as acuteness and

obtuseness are related to inequality. For rectitude and equality, as

also similitude, are of the same co-ordination, (for each exists under

bound): but acuteness and obtuseness, as also dissimilitude, are of the

same series with inequality. For they are all produced from _bound_ and

_infinite_. Hence some, regarding the quantity of angles, say, that a

right angle is equal to a right: but others, considering their quality,

affirm that one is similar to another. For similitude in qualities is

the same as equality in quantities.

V.

If a right line falling upon two right lines, makes the internal

angles towards the same part less than two right, those right

lines, if infinitely produced, shall coincide in that part, in

which the angles less than two right, are placed.

This ought to be entirely blotted out from the number of Petitions,

for it is a theorem including many doubts, which Ptolemy in one of his

books proposes to solve; but it requires in its demonstration both

many definitions and theorems; and Euclid also exhibits its converse

as a theorem. But perhaps some, from an erroneous conception, may

think that this should be placed among the petitions, as that which

produces credibility of itself, respecting the inclination of right

lines, on account of the diminution of two right angles. To such as

these, Geminus rightly answers, that from the authors of this science,

we learn not entirely to give credit to imaginative probabilities, for

the purpose of accomplishing geometrical reasons: for it is similar

(says Aristotle) to require demonstrations from a rhetorician, and

patiently listen to a geometrician, disputing from probability. And

Simmeas in the Phædo of Plato, says, “I know that those who demonstrate

from appearances, are vain.” Hence, in the present instance, it is

true and necessary that right lines should incline, while right

angles are diminished: but this, that the inclining lines, while they

are more produced, should at length coincide, is probable, but

not necessary, unless some reason demonstrates that this is true in

right lines: for there are certain lines infinitely inclining, and

never coinciding, and though this appears incredible and admirable,

yet it is true, and has been observed in other forms of a line. Is

it therefore possible that this can be accomplished in right lines

which takes place in others? For before we procure conviction of this,

from demonstration, the properties exhibited in other lines molest

the phantasy by the contrary images they produce. But if the reasons

doubting against the coincidence of lines are very strong, ought we

not much more to expel this improbable and irrational supposition from

our doctrine? And thus it appears that a demonstration is to be sought

for of the present theorem, and that it is foreign from the property

of Petitions: but how it is to be demonstrated, and by what reasons

the objections urged against it are to be removed, we shall shew in

our comment on the proposition, where it is used by the institutor of

the elements as manifest. For then it will be necessary to exhibit

its evidence, since it does not present itself to our view with

indemonstrable clearness, but becomes manifest through the medium of

demonstration alone.

AXIOMS.

I.

Things which are equal to the same, are equal to one another.

II.

If equals be added to equals, the wholes are equal.

III.

If equals be taken from equals, the remainders are equal.

IV.

If equals are added to unequals, the wholes are unequal.

V.

If equals be taken from unequals, the remainders are unequal.

VI.

Things which are double of the same, are equal to one another.

VII.

Things which are halves of the same, are equal to one another.

VIII.

Things which coincide with each other, are mutually equal.

IX.

The whole is greater than its part.

X.

Two right lines cannot comprehend space.

These are the things which, according to the opinion of all men, are

called indemonstrable axioms, so far as their certainty is admitted

by all, and no one disputes their evidence. For propositions also are

often simply called axioms, of whatever kind they may be, whether

they are immediately proper, or require some declaration; and the

Stoics, indeed, are accustomed to call every simple enunciative speech

an axiom: and when they write on dialectic arts, they say that they

discourse on axioms. But some, distinguishing more accurately axioms

from other propositions, give this appellation to a proposition

immediate, and producing credibility of itself, on account of its

evidence: as also Aristotle and geometricians themselves affirm.

For, according to the opinion of these, an axiom is the same as a

common conception. By no means, therefore, must we praise Apollonius

the geometrician, who writ (as it appears) demonstrations of axioms,

because he performs the very opposite to Euclid: for he, indeed,

enumerates that which is demonstrable among Petitions; but Apollonius

endeavours to find out demonstrations of indemonstrables. But these

naturally differ from each other, and the genus of the sciences is

different: I mean of the things which take place about immediate

propositions, which are entirely subject to our knowledge, on account

of their evidence; and of things which use demonstrations, which

receive principles from them; and which, when received, they orderly

employ in their proper conclusions. But that the demonstration of

the first axiom, which Apollonius persuades himself he has invented,

possesses a medium, not more known, but more dubious than the

conclusion may be known by any one from a slight inspection. For let

(says he) _a_ be equal to _b_, and _b_ to _c_, I say that _a_ also is

equal to _c_.

[Illustration]

For since _a_ is equal to _b_, it occupies the same place as _b_.

And because _b_ is equal to _c_, it occupies the same place as _c_; and

so _a_ occupies the same place as _c_, they are therefore equal. Now in

this demonstration it is requisite that two things must be previously

assumed; one, that things occupying the same place, are mutually equal;

but the other, that things occupying the same place, with the same

thing, mutually occupy the same place: but these are evidently more

obscure than the present axiom. For it is proper to enquire how are

things, which fill the same place equal, according to the whole, or

according to a part; or according to a figure of speech: hence we must

by no means admit a transition to place,[3] which is more unknown than

the natures it contains; for the invention of its essence is difficult

and ambiguous. That we may avoid prolixity, therefore, all axioms are

to be delivered as things immediate and self-manifest, since they are

of themselves known and credible; for he who brings demonstration to

things the most manifest, does not confirm their truth, but diminishes

the evidence we possess in the untaught and innate conceptions of the

soul: but this is to be received concerning axioms, as a judgment of

their peculiarity; and that all of them are of the common kind of the

mathematical sciences; and that each of them is said to be verified,

not only in magnitudes, but also in numbers, and motions, and times:

and this indeed is necessary. For equal and unequal, the whole and

part, and the more and the less, are common to discreet and continued

quantities. The contemplation, therefore, which is conversant with

times and motions, numbers and magnitudes, requires all these, as

things evident by their own intrinsic light; and in all of them both

that is true, which says, _things equal to the same, are equal to

one another_; as likewise each of the axioms we have assumed: but as

they exist in common, each science uses them according to its proper

matter, and one indeed, as in magnitudes; but another, as in numbers;

and another, as in times; and after this manner in each science, the

conclusions become peculiar and apposite, though the axioms are common.

Besides, it is likewise requisite not to contract the number of these

to the least, as is done by Heron, who only establishes three axioms;

for this also is an axiom, _the whole is greater than its part_, and

the geometrician every where assumes this in his demonstrations; as

also, that _things which mutually coincide, are equal_; for this is

employed with advantage in the solution of the fourth Proposition.

Nor is it proper to join some with others, of which some are proper

to the geometric matter, as _that two right lines cannot comprehend

space_, (since axioms are, as we have said, of a common kind); but

others are consequent to things established, as that which says,

_things double of the same, are equal_. For this is consequent to the

axiom, affirming, that _if to equals you add equals, the wholes are

equal_, since things equal to the half, because they assume the half,

become double to the same, and mutually equal, on account of an equal

addition: and according to this reason, not only the doubles, but also

the triples, and all multiples of the same quantity will appear equal.

But with these axioms, Pappus says, that certain others are to be

classed, as _if unequals are added to equals, the excess of the whole,

will be equal to the excess of the adjuncts_. And on the contrary, _if

equals are added to unequals, the excess of the wholes is equal to the

excess or difference of the unequals themselves_. And these also are

manifest from themselves, yet they may be made manifest as follows.

Let _a_ be equal to _b_, and add to each the unequals _c_ _d_, but let

_c_ be greater than _d_ by _e_, and the remainder be _f_; because,

therefore, _a_ is equal to _b_, and also _f_ to _d_; _a f_ will be

equal to _b d_. For if equals are added to equals, the wholes are

equals: _a c_, therefore, exceeds _b d_, by _e_ only, by which alone

_c_ exceeds _d_. Again, _c_ and _d_ are unequals, to which, let the

equals _a_ and _b_ be added, and let _e_ be the excess of _c_, above

_d_, and the remainder be _f_; because, therefore, _a_ is equal to _b_,

and _f_ to _d_ _a f_ will be equal to _b d_; the whole, therefore, _a

c_, will exceed _b d_, by _e_ only, by which _c_ also exceeds _d_.

These, therefore, are consequent to the aforesaid axioms, and are, not

undeservedly, in many copies, omitted. But whatever others he adds to

these, have been previously assumed by definitions, to which they are

consequent. As for example: _that all the parts of a plane and a right

line mutually agree_; for things placed in their extremities, possess

a nature of this kind; and _that a point divides a line, but a line a

superficies, and a superficies a solid_. For all things are divided by

the natures by which they are proximately bounded; and _that infinite

subsists in magnitudes, by addition and diminution, but according to

capacity only, in both these respects_: for every thing continuous may

be infinitely divided and increased. But, as we have summarily spoken

concerning these, it remains that we consider things consequent to

principles; for thus far principles extend themselves. But of those

who oppose geometry, some very much doubt concerning principles,

endeavouring to shew that the terms have no subsistence, whose

arguments, indeed, are known in common, who endeavour to take away all

science, and, like hostile foes from a foreign region, demolish the

fruits and fecundity of philosophy, as is the case with the Pyrrhonian

philosophers; but others only propose to themselves the subversion of

geometrical principles, as the Epicureans. Others, again, admitting

the principles, affirm, that things consequent to the principles

cannot be demonstrated, unless something else is granted, which was

not previously assumed in the principles. Zeno exercised this mode of

contradiction, who was a Sidonian by birth, but of the Epicurean sect,

against whom Possidonius wrote an entire book, exhibiting the whole

of his imbecile opinion; and thus much may suffice for the difference

of opinions concerning principles. We shall shortly consider the

troublesome objection of Zeno: but now, after we have briefly resumed

the consideration of theorems and problems, their difference, and the

divisions they receive, we shall proceed, to an exposition of the

things exhibited by the institutor of the elements, gathering the more

beautiful observations upon the propositions found in the writings

of the antients, and contracting the infinite prolixity of their

discourses; but delivering such things as are more artificial, and full

of methods producing science, dwelling more on an accurate treatise of

things than on the variety of cases and assumptions, to which young

men, for the most part, eagerly incline.

[Illustration]

PROPOSITION I. PROBLEM.

Upon a given terminated right line to describe an equilateral

triangle.

Since all science is two-fold, and one is conversant about immediate

propositions, but another about things, which are exhibited and

provided from the propositions, and universally about the consequents

to principles; this, again, divides itself in geometrical discourses,

into the solution of problems, and the invention of theorems. And

problems, indeed, geometry denominates things in which it proposes to

procure, manifest, and fabricate that, which, in a certain respect,

has no existence; but it calls theorems, things in which it appoints

to perceive, know, and demonstrate that which either exists, or

does not exist. For problems command us to undertake the origin,

positions, applications, descriptions, inscriptions, circumscriptions,

coaptations, and contacts of figures, and every thing of this kind:

but theorems endeavour to procure our assent to symptoms, and things

essentially inherent in the subjects of geometry, and to convince by

demonstrations. For geometry discourses concerning every object of

enquiry, which is possible to be effected, referring some things to

problems, but others to theorems; since it enquires concerning the

_what_, in a two-fold respect: for it either seeks for the reason

and intelligence of the thing; or for intelligence, and the essence

of the subject. I say, for example, as when it requires what a line

of similar parts may be: for in an enquiry of this kind, it either

desires to find the definition of such a line, as, _that a line of

similar parts is that which has all its parts agreeing with all_; or

to receive the species of lines of similar parts, as that it is either

_right_, or _circular_, or a _cylindric helix_. Besides, prior to this,

it enquires, by itself, concerning the _if_, and this especially in

its determinations, agitating, whether the object of its enquiry is

possible or impossible, what place it possesses, and in how many ways.

It likewise seeks concerning the _what kind_; for when it considers

the essential accidents of a triangle, circle, and parallels, it is

manifest, that in such cases it seeks after the _what kind_; but many

have thought that geometry very little contemplated the _cause_, and

_the why_. And of this opinion is Amphinomus, led by the decisions

of Aristotle: but (says Geminus) an enquiry into these may be found

in geometry. For does it not belong to geometry to enquire for _what

cause_ infinite equilateral multangles may be inscribed in circles,

but to describe solid equilateral and equiangular multangles, and

constructed from similar planes, in spheres, is impossible? To whom

does an investigation of this kind belong, except to a geometrician?

When, therefore, to geometricians the syllogism is by an impossibility,

they alone desire to find the symptom; but when by a principal

demonstration, then again if the demonstrations are in that which is

particular or partial, the cause is not yet manifest; but if in that

which is universal, and in all similars, the _why_ becomes immediately

manifest: and thus much concerning objects of enquiry.

But every problem and theorem which receives its completion from

its own perfect parts, ought to possess in itself all the following

parts: _proposition_, _exposition_, _determination_, _construction_,

_demonstration_, and _conclusion_. But of these, _proposition_ informs

us what the object of enquiry is from a given datum; for a perfect

proposition is composed from both; but _exposition_ receiving the

datum essentially, prepares for the question. Again, _determination_

separately explains the thing sought for according to the _what_; but

_construction_ adds to the datum what is wanting to the investigation

of the thing sought; and _demonstration_ skilfully collects the

proposition from the concessions. But the _epilogue_, or conclusion,

is again converted to the proposition, by confirming that which is

exhibited. And so many, indeed, are all the parts of problems and

theorems; but _proposition_, _demonstration_, and _conclusion_, are

especially necessary, and exist in all; for it is requisite that the

thing sought for should be previously known; and that this should

be shewn by proper mediums, and that what is exhibited should be

concluded; and it is not possible that any one of these three can be

wanting; but the rest are, indeed, received in many places; but in

many, because they produce no utility, are omitted. For _determination_

and _exposition_ are not found in the problem, which says, _to

construct an isosceles triangle, which will have each of the angles at

the base double of the other_; but _construction_ has frequently no

subsistence in many theorems, the demonstration being sufficient to

exhibit the thing proposed from the data, without any addition. When,

therefore, shall we say that _exposition_ fails, when no datum is given

in a proposition? Because, though _proposition_, for the most part,

is divided into _datum_, and _the thing sought for_, yet this is not

always the case; but sometimes _the thing sought for_, alone affirms

that which it is requisite to know or effect, as in the aforesaid

problem; for it does not previously say from what datum it is requisite

to construct an isosceles triangle, which shall have each of the angles

at the base, double of the remaining one; but that it is required to

effect this. And here, indeed, the admission of the proposition takes

place from things previously known; for we must know the meaning of

the terms _isosceles_, _equal_ and _double_ (since this, as Aristotle

observes, is the property of all ratiocinative discipline[4]), yet

nothing is subjected to us as in other problems, as in that which

says, _to bisect a given terminated right line_. For here the right

line is given, but we are ordered to divide it into two parts; and

the datum is separately determined from the object of enquiry. When,

therefore, a proposition has both of these, then also _determination_

and _exposition_ are found; but when the datum is deficient, these

also fail, since _exposition_ and _determination_ belong to the datum:

for this will be the same with the proposition. Indeed, what else

do we say, when determining in the aforesaid problem, unless that

it is requisite to find an isosceles of this kind? But such was the

proposition: if then the proposition has neither this _datum_, nor

_thing sought_, _exposition_ will, indeed, be silent, because there

is no datum; but _determination_ will be neglected, lest it should

become the same with the _proposition_: but you may find many other

problems of this kind, especially in arithmetic, and in the tenth book

of these Elements, as, _to find a medium comprehending two right lines

commensurable in power, and every thing of this kind_.

But every datum may be given in these four modes, either in

_position_, or _proportion_, in _magnitude_ or _form_; for a point,

indeed, is given in _position_ only, but a line and the rest in all

the four. Thus, when we say, _to bisect a given rectilineal angle_,

we declare the species of the angle given, as that it is right lined,

lest we should also seek to bisect a curvilinear angle by the same

methods. But when we say, _from the greater of two unequal right lines,

to cut off a part equal to the less_, the lines are given in magnitude;

for the less and the more, finite and infinite, are the proper

predications of magnitude. But when we say, that _if four magnitudes

are proportional, they shall be also alternately proportional_, the

same proportion is given in the four magnitudes: but when it is

requisite, _from a given point to place a right line equal to a given

right line_, then the point is given in position. From whence, since

position may be various, construction also receives variety; for the

point is given either without the right line, or in the right line,

and in the extremity, or without the extremity of the right line.

Since, therefore, a datum has a four-fold acceptation, it is manifest,

that exposition also is four-fold; but sometimes it connects two or

three modes. Again, we find that demonstration sometimes possesses

things proper to demonstration, exhibiting the thing sought for from

mediate definitions; for this is the perfection of demonstration, but

that sometimes it argues from certain signs. And it ought not to be

concealed, that geometrical discourses have every where that which is

necessary, on account of the subject matter, but are not every where

perfected by demonstrative methods. For when, because _the external

angle of a triangle is equal to the two internal and opposite ones_,

it is shewn, that _the three internal angles of the triangle are equal

to two right_, how is this demonstration from the cause? And is not a

sign the medium in this case? For the external angle not yet existing,

since the internal angles exist, they are equal to two right, since

it is a triangle, though the side is not produced; but when, by a

description of circles, the triangle, which is constituted, is shewn

to be equilateral, the apprehension takes place from the cause. For we

say, that the similitude and equality of the circles is the cause of

the triangle’s equality with respect to its sides.

But geometrical discourses are likewise accustomed to make the

conclusion, in a certain respect, two-fold. And this, when they exhibit

things agreeable to the data, and reason universally, recurring from

a particular conclusion to that which is universal; for when they do

not use the property of the subjects, but placing the data before

our eyes, describe an angle or right line, they think that which is

concluded in this, is to be concluded in every thing similar: they

pass on therefore to _universal_, lest we should think that the

conclusion is particular. But their transition is effected in the best

manner, since they employ, in demonstration, _the things placed_, not

considered as such, but considered as similar to others: for it is

not because such a particular angle is proposed that they effect a

bipartite section, but because it is rectilineal only. But quantity,

is indeed, proper to the proposed angle; but rectilineal is common to

all right lines: let then the given angle be a right one. If therefore,

we receive rectitude in the demonstration, we cannot pass to every

species of right lines; but if we do not subjoin its rectitude, or

being right angled, but alone consider its being rectilineal, the

discourse may be adapted to all right lined angles; and all that we

have previously observed we may contemplate in this first problem. For

that it is a problem, is evident, since it commands us to construct

an equilateral triangle: but _proposition_ in this, consists from a

_datum_ and _thing sought_. For a terminated right line is given, but

it is _enquired_ how an equilateral triangle may be constructed upon

it, and the datum indeed precedes, but the thing sought follows; so

that we may say, by conjoining the two, _if there be a terminated

right line, it is possible to construct upon it an equilateral

triangle_; for a triangle cannot be constructed without the existence

of a right line, since it is comprehended by right lines; nor upon

an unlimited line, for an angle cannot be constructed unless it is

made on one point, but in an infinite line there can be no extremity

or bounding point. But after proposition, _exposition_ follows, as,

_let there be given a terminated right line_. And here we may see that

_exposition_ alone pronounces the _datum_, but by no means subjoins

_the thing sought_; but after this we shall find _determination_:

_it is required upon the given terminated right line to construct an

equilateral triangle_; and here we may observe that _determination_

is in a certain respect, the cause of attention, for it makes us more

attentive to the demonstration, by pronouncing the thing sought, as

_exposition_ causes us to be more docile, by placing the datum before

our eyes. Again, after determination, _construction_ follows, _from

one extremity of the right line, as a centre, but with the remainder

as an interval, let a circle be described_. And again, _with the other

extremity, as a centre, and with the same interval, let a circle be

described; and from the common point of the sections of the circles, to

the extremities of the right line, let right lines be continued_. And

here we may observe, that Petitions are used in the construction, this

for one, _from every point to every point, to draw a right line_; and

also this, _with every centre and interval to describe a circle_; for

universally Petitions are the sources of utility to _constructions_,

but Axioms to _demonstrations_; _demonstration_ therefore follows,

_because, then each extremity of the given right line is the centre of

the circle surrounding it, the right line which reaches to the common

section is equal to the given right line; hence, because the other

extremity of the right line is the centre of its containing circle,

the right line reaching to the common section of the circles, is also

equal to the given line_. And the admonition of these, is derived from

the definition of the circle, which says, that _all lines from the

centre to the circumference are equal_. Each of these lines, therefore,

is equal to the same; but _things equal to the same, are equal among

themselves_, by the first axiom. The three right lines, therefore,

are mutually equal; hence, _upon this given right line an equilateral

triangle is constructed_; and this, indeed, is the first _conclusion_

which follows the exposition. But after this, that universal one; _upon

a given right line, therefore an equilateral triangle is constructed_:

for whether you make the line double of the one now proposed, or

triple; or receive any one greater or less, the same constructions and

demonstrations will accord. But to these he adds the particle _which

was required to be done_, shewing from hence, that the conclusion

is problematical; for in theorems, he adds the particle _which was

required to be shewn_; the former announcing the production of

something, but this the ostension and invention of a thing required. He

therefore subjoins this to the conclusions, for the purpose of shewing

that every part of the proposition is accomplished by this means,

uniting the end with the beginning, and imitating intellect convolved,

and again returning to its principle. But he does not always add the

same, but sometimes the particle _which was required to be done_, and

sometimes the particle _which was required to be shewn_, on account of

the difference between problems and theorems: and thus, in this one

problem, we have exercised and made perspicuous all this variety of

considerations. But the reader ought to make a similar enquiry in the

rest; investigating what propositions receive these leading properties,

and in what they are omitted. Likewise in how many ways a _datum_ is

given, and from what principles we receive either constructions or

demonstrations; for a perspicacious contemplation of these affords no

small exercise and meditation of geometrical discourses.

But here it is necessary that we should briefly determine the

nature of _assumption_, _case_, _corollary_, _instance_, (ενϛασις)

and _induction_. They say therefore that _assumption_ is often

predicated of every proposition assumed in the construction of another

proposition, affirming at the same time that the demonstration of such

a proposition is composed from so many _assumptions_. But _assumption_,

properly considered by those who are conversant in geometry, is a

proposition indigent of credibility; for when either in construction

or demonstration we assume any thing which has not been exhibited, but

requires a reason for its admission, then that which is assumed, as

of itself ambiguous, being considered as worthy of enquiry, we call

an _assumption_; and this differs from Petition and Axiom, because

it is demonstrable, but they are assumed without demonstration, for

the purpose of giving credibility to others. But the best aid in the

invention of _assumptions_, is an aptitude of cogitation; for we may

see many naturally acute in solutions, and discovering them without

any method, as was the case with our Cratistus, who was adapted to

the investigation of a thing sought from the first and shortest

methods possible; and had a natural promptitude for invention; but

there are nevertheless certain most excellent methods delivered,

one which reduces the thing sought, by resolution to its explored

principle, which, as they say, Plato delivered to Leodamas, and

from which he is reported to have been the inventor of many things

in geometry: but the second is that which has a power of division;

because it distributes the proposed genus into articles, but affords

an occasion of demonstration, by an ablation of other things from

the proposed construction. And this likewise is praised by Plato, as

that which affords assistance to all sciences; but the third is that

which by a deduction to an impossibility, does not of itself shew the

thing sought, but confutes its opposite, and discovers the truth by

accident; and thus far is the contemplation of _assumption_ extended.

But _case_ enunciates different modes of construction, and the

mutation of position, points, or lines, superficies, or solids being

transposed; and in fine, all its variety is beheld about description:

hence, it is also called case, because it is the transposition of

construction. Again, _Corollary_ is affirmed, indeed, of certain

problems, as the Corollaries which are ascribed to Euclid; but

Corollary is properly predicated, when, from the things demonstrated,

a certain unexpected theorem appears, which on this account they have

denominated Corollary, as a certain gain, exceeding the intention of

demonstrative science; but _instance_ impedes the whole passage of

the discourse, either opposing the construction or the demonstration:

and here it is not necessary, that as he who proposes a case, ought

to shew the proposition true; so he who proposes an _instance_: but

it is requisite to destroy the _instance_, and convict its employer

of falsehood. Lastly, _induction_ is a transition from one problem or

theorem to another, which being known or compared, the thing proposed

is also perspicuous. For example: when the duplication of the cube is

investigated, geometricians transfer the question into another to which

this is consequent, i.e. the invention of two mean proportionals, and

afterwards they enquire how between two given right lines two means

may be found. But Hippocrates Chius is reported to have been the first

inventor of geometrical induction; who also made a quadrangle equal to

a lunula, and invented many other things in geometry, and excelled all

in his ingenuity respecting appellations; and thus much for these.

But let us return to the proposed problem: that an equilateral

triangle, therefore, is the best among triangles, and is particularly

allied to a circle, having all lines from the centre to the

circumference equal, and one simple line for its external bound, is

manifest to every one; but the partial comprehension of two circles

in this problem, seems to exhibit in images how things which depart

from principles, receive from them perfection, identity, and equality.

For after this manner, things moving in a right line, roll round in

a circle, on account of continual generation; and souls themselves,

since they are indued with transitive intellections, resemble by

restitutions and circumvolutions, the stable energy of intellect. The

zoogonic or vivific fountain of souls too, is said to be contained by

two intellects. If, therefore, a circle is an image of the essence

of intellect, but a triangle of the first soul, on account of the

equality and similitude of angles and sides; this is very properly

exhibited by circles, since an equilateral triangle is included in

their comprehension. But if also every soul proceeds from intellect,

and to this finally returns and participates intellect in a two-fold

respect; on this account also it will be proper that a triangle, since

it is the symbol of the triple essence of souls, should receive its

origin comprehended by two circles. But speculations of this kind, as

from bright images in the mirror of phantasy, recall into our memory

the nature of things. And here, because some object to the constitution

of an equilateral triangle, thinking by this means to overthrow the

whole of geometry, let us briefly answer and confute them. Zeno then,

whom we have mentioned before, says, that if any one admits the

principles of geometry, yet he will not obtain from common consent,

things consequent to the principles, while this is not admitted, that

there are not the same segments of two right lines: for unless this is

given an equilateral triangle cannot be constructed. For let there be

(says he) a right line _a b_, upon which an equilateral triangle is to

be constructed.

[Illustration]

[Illustration]

But let circles be described, and from their common section let the

right lines _c e a_, _c e b_, be extended, having the common segment _c

e_. It will therefore happen, that the lines extended from the common

section, will be equal to the given line _a b_, and yet the sides of

the triangle will not be also equal, but two will be less than the

remainder, that is, than _a b_. And so this not being constituted,

neither can the rest be constructed. Can then (says Zeno) the rest

follow, though the principles are given, unless this also is previously

received, that there are no common segments either of circles or of

right lines? Against this objection then, we must affirm in the first

place, that it was in a certain respect previously understood, that two

right lines have no common segment. For the definition of a right line

comprehends this property, since _that is a right line which is equally

situated between its bounding points_; and the equality of the interval

between the points to the right line, causes that which joins the

points to be one, and the shortest line; so that if any one adapts it

to another line, according to one of its parts, it must also agree with

the line according to its remaining part; for since it is constituted

in its extremities, because it is the shortest line, it is necessary

that the whole should fall on the whole. But again, this was manifestly

received in the Petitions: for the Petition which says, _that a

terminated right line may be produced straight forwards_, perspicuously

shews that the produced line ought to be one, and produced by one

motion; but if any one is desirous to receive a demonstration of this

assumption, let, if possible, _a b_ be the common segment of _a c_ and

_a d_, and with the centre _b_, and interval _b d_, let the circle

_a c d_ be described; because therefore the right line _a b c_, is

drawn through the centre, _a f c_ is a semicircle; and because the

right line _a b d_ likewise is drawn through the centre, _a e d_ is a

semicircle. The semicircles, therefore, _a f c_, _a e d_, are equal to

each other, which is impossible. But against this demonstration Zeno

will perhaps say, that it is likewise requisite to demonstrate that

the diameter bisects the circle, because we previously assume that

there is not a common segment of two circumferences. Thus too we take

for granted, that one circumference coincides with another, or if it

does not coincide, that it either falls externally or internally. But

nothing hinders (he will say) that the whole may not coincide with the

whole, but according to some part. But to this Possidonius rightly

answers, who laughs at the acute Epicurean, as if conscious that

though the circumferences do not coincide according to a part, yet the

demonstration will succeed; for according to that part in which they

do not coincide, the one will fall within, and the other without, and

the same absurdities will follow when right lines are extended from

the centre to the external circumference; for those from the centre

will be equal, as well the greater which is drawn to the external, as

the less which is extended to the internal circle: either therefore

the whole will coincide with the whole, and they will be equal; or

coinciding according to a part, it will alternately vary according

to the remainder, or no part will coincide with no part; and in this

case it either falls within or without: but of this, enough. But Zeno

also condemns the following demonstration of this particular: Let _a

b_ be the common segment of two right lines _a c_, _a d_, and let be

_b e_ erected at right angles to _a c_, the angle _e b c_, therefore,

is a right one. Hence, if the angle _e b d_ is also right, they shall

be equal, which is impossible; but if not, let _b f_ be erected at

right angles to _a d_. The angle _f b a_, therefore, is right; but the

angle _e b a_ was also right; and they are therefore mutually equal,

which is impossible. This is the demonstration which Zeno opposes,

as assuming that which is to be exhibited afterwards; I mean from a

given point to raise a right line, at right angles, to a given right

line. However, Possidonius observes, that indeed, a demonstration of

this kind is never to be introduced into elementary institutions; but

that Zeno calumniates Geometricians using their own as a flagitious

demonstration; though there is some reason in their conduct. For there

are right lines existing at right angles; since any two right lines are

capable of forming a right angle; and this is previously assumed in

our definition of a right angle. For we alone constitute a right angle

from such an inclination; and it may perhaps be this which we have

erected. Indeed, Epicurus himself, and all other philosophers admit,

that not only many things possible may be supposed, but likewise many

of an impossible matter, for the purpose of contemplating something

consequent; and thus much concerning an equilateral triangle.

[Illustration]

[Illustration]

But it is requisite to construct other triangles, and in the first

place an isosceles. Let _a b_, therefore, be a right line, upon which

it is required to construct an isosceles triangle. Describe circles as

in the construction of an equilateral triangle, and produce the line

_a b_ on each side to the points _c d_; the line _c b_, therefore, is

equal to _a d_. Again, with the centre _b_, and interval _c b_, let the

circle _c e_ be described; and with the centre _a_, and the interval

_d a_, the circle _d e_; and from the point _e_, in which the circles

intersect each other, to the points _a_ and _b_, let the lines _e a_,

_e b_, be extended. Because therefore, _e a_ is equal to _a d_; but

_e b_ to _b c_, and _a d_ is equal to _b c_, _e a_ will also be equal

to _e b_; but they are also greater than _a b_. The triangle _a b e_,

therefore, is isosceles, which it was required to constitute. But let

it be ordered to construct a scalene triangle upon the given right line

_a b_. Describe circles with centres and intervals, as before, and let

there be taken in the circumference of the circle, whose centre is _a_,

the point _f_, and let the right line _a f_ be extended and produced

to the point _g_; and likewise let the right line _g b_ be extended.

Because, therefore, _a_ is a centre, _a f_ is equal to _a d_; and

hence, _a g_ is greater than _a d_, that is, than _g b_. But _b_ also

is a centre, _g b_, therefore, is equal to _c b_; and hence, _g b_ is

greater than _b a_: but _g a_ is greater than _g b_; the three lines

therefore _g b_, _b a_, _a g_, are unequal; and hence, the triangle

_a b g_ is scalene. Hence too, three triangles are constructed; but

these things are commonly known: however, this is beautiful in these

triangles, that the equilateral existing on all sides equal, is

constructed by one mode alone; but the isosceles, endued with equality

in two sides only, has a two-fold construction: for the given right

line is either less than both the equal ones (according to our present

construction), or it is greater than both; but the scalene being

unequal in all its sides, receives a triple construction; for the

given right line is either the greatest of the three, or the least;

or greater than the one, and less than the other; and indeed, it is

proper to be exercised in each supposition, either by enlarging or

contracting; but to us, what is already delivered, is sufficient. Let

us now contemplate problems universally, some of which are produced

simply, but others manifoldly, and others according to infinite modes.

But (as Amphinomus observes) those which are simply constructed are

_ordinate_: but those which receive a manifold composition, and are

constructed according to number, are _middle_; and those which are

varied in infinite ways, are _inordinate_. The manner, therefore, in

which problems are constructed, simply or manifoldly, becomes manifest

in the preceding triangles; for the equilateral is constituted simply;

but of the other two, the one receives a two-fold, and the other a

triple construction. But problems of the following kind, may take

place in infinite modes; I mean _to divide a given right line in three

proportional parts_; for if it be divided in a duple ratio, and the

deficient quadrangular form, resulting from the less, be applied to the

greater, it will be divided into three equal parts; but if the greater

segment be more than double of the less, as for instance, triple, and a

deficient quadrangular form, equal to that which results from the less,

be applied to the greater, the line will be divided into three unequal

parts. Because, therefore it may be divided into two parts, in infinite

ways, the greater of which is either double or triple, (for multiplex

proportion proceeds in infinitum), hence, it may be divided into three

parts, according to infinite variations.

But it is requisite to know, that problem also is manifoldly

predicated; for whatever is proposed may be called a problem,

whether it is proposed for the sake of learning or operating. But in

mathematical disciplines, that is properly called a problem, which is

proposed for the purpose of contemplative energy. Since that which is

performed in these, has contemplation for its end; and often, indeed,

certain things, impossible to be executed, are called problems: but

more properly that which is possible to be done, and neither exceeds,

nor is deficient, is allotted an appellation of this kind; and the

problem _exceeds_, which says, _to construct an equilateral triangle,

having its vertical angle two thirds of one right_; for this is

superfluous, and is added in vain: since it is a property inherent

in every equilateral triangle. But of those which exceed, whatever

are redundant with incongruous and non-existent symptoms, are called

_impossibles_; but whatever are redundant with accidents, are called

_greater problems_. But a _defective_ problem (which is also called a

_less problem_) is that which requires some addition, that it may be

reduced from inordination into order and scientific bound, as if any

one should say, _to constitute an isosceles triangle_: for this is

mutilated and indeterminate, and requires some one who may subjoin,

what kind of an isosceles triangle, whether that which has its base

greater than either of the equal sides; or that which has it less.

Likewise, whether that which has the vertical angle double of each at

the base, as a semiquadrangle; or that which has each of the angles at

the base double of the vertical angle; or that which possesses these

angles according to some other proportion, as triple or quadruple: for

it is possible that it may be varied in infinite modes. From hence,

therefore, it is manifest, that such things as are properly denominated

problems, ought to avoid indetermination, and not to be of the number

of things capable of infinite variation; though such as these are also

called problems, through an equivocation of the word problem. The first

problem, therefore, of these elements, excels the rest in the manner

we have explained; for it neither _exceeds_, nor is _deficient_; it is

neither constructed in a variety, nor according to infinite modes; and

such ought to be the conditions of that which is to be the element of

the rest.

PROPOSITION II. PROBLEM II.

To a given point to place a right line equal to a given

right line.

Of problems, as well as of theorems, some are without case, but

others possess a multitude of cases. Whatever, therefore, have the

same power acceding to many descriptions, and when their positions are

changed, preserve the same mode of demonstration, these are said to

have _case_; but such as proceed according to one position only, and

one construction, are without _case_; for simply, _case_, appears about

the construction both of theorems and problems. The second problem,

therefore, has many cases; but a point is given in it _in position_,

since it can only be given in this manner; but a right line, both in

form and position, (for it is not simply _line_, but of such a kind.)

For it is here enquired, _how to a given point to place a right line

equal to a given right line_. But it is manifest that the point is

entirely in the subject plane, in which the right line exists, and

not in one more elevated. For in all problems and theorems respecting

planes, we must conceive that one plane is subjected. But if any

one should doubt how a line is to be placed equal to a given right

line, for what if the given line be infinite? Since the present datum

pertains both to finite and infinite: for every datum signifies that

which is proposed and supposed by us for the sake of investigation.

But this Euclid himself declares sometimes, saying, _upon a given

terminated right line to construct an equilateral triangle_; but

at other times, _upon a given infinite right line to let fall a

perpendicular_. In answer then to this doubt, we must say, that when

he orders us to place the line equal to a given right line, at a given

point, he sufficiently evinces that the given line is finite; for

every thing placed at a point, is terminated according to that point.

Hence, the line equal to that which is given, must have a much prior

termination. At the same time, therefore, in which he says _to a given

point_, he terminates both the given right line, and its equal which is

investigated.

[Illustration]

[Illustration]

But that the cases of the present problem are formed from the

various position of a point, is manifest. For the given point is either

placed external to, or in the given right line; and if in it, it will

either be one of its extremities, or it will be situated within the

extremes; and if external, it will either have a lateral position, so

that a line drawn from it to the extremity of the given line will form

an angle, or a direct position; so that if the line were produced, it

would coincide with the external point. But the geometrician, indeed,

considers the point as external, and receives it according to a lateral

position; however, for the sake of exercise, all the positions are

to be assumed, the more difficult of which we shall exhibit. For let

there be given a right line _a b_, and a given point _c_, which lies

between its extremes, and let there be constituted according to the

doctrine of the elements, an equilateral triangle upon the right line

_a c_, and let _d c_, _d a_, be produced; then, with the centre _a_,

and the interval _a b_, let the circle _b e_ be described. And again,

with the centre _d_, but with the interval _d e_, let the circle _d

f_ be designed. Because, therefore, _a_ is the centre, _b a_ is equal

to _a e_; and hence, _d e_ is equal to _d f_, the parts of which, _d

a_, _d c_, are equal: for the triangle _d a c_ was established as

equilateral. The remainder, therefore, _a e_, is equal to _c f_; but _a

e_, as it was shewn, is equal to _a b_, and hence, _c f_ is equal to

_a b_. To a given point, therefore, _c_, a right line _c f_ is placed

equal to _a b_. With respect to the position of the point then, so many

cases arise. But there are many more with respect to the constitution

of the equilateral triangle, the extension of its sides, and the

description of circles. For let there be assumed, as in this element,

a point _a_, and a right line _b c_, but let _b a_ be extended. The

equilateral triangle, therefore, will not be constituted on _b a_,

with its vertex above (because there is no place for it), but beneath;

let it, therefore, be _a d b_; _a d_, therefore, is either equal to _b

c_, or greater or less. If then it be equal that which was required is

performed. But if less with the centre _b_, and the interval _b c_,

let a circle be described, and let _a d_, _d b_ be produced to the

points _e_ and _g_; and with the centre _d_, but the interval _d g_,

let a circle _g a_ be designed. Because, therefore, _d g_ is equal to

_d e_, for they are drawn from the centre; and likewise because _a d_

is equal to _d b_, for the triangle is equilateral, the remainder _a

e_ is equal to the remainder _b g_. But _b g_ is also equal to _b c_,

for they proceed from the centre; and hence, _a e_ is equal to _b c_,

which was required to be done. But if _a d_ is greater than _b c_ (for

this is the last case), then with the centre _b_, and the interval _b

c_, let a circle _e c_ be described. The line _d b_, therefore, shall

cut the circle _e c_. Again, with the centre _d_, and interval _d e_,

let the circle _e g_ be described. Because therefore, _d_ is the centre

of the circle _g e_, _g d_ is equal to _d e_. But _d a_ was also equal

to _d b_; the remainder, therefore, _a g_ is equal to the remainder

_b e_. But _b e_ is equal to _b c_, for both proceed from the centre.

Hence, _a g_ is equal to _b c_; and it is placed at the point _a_, as

was required to be done. And though there are many other cases, the

description of the above is sufficient for our present purpose. For

from these it is possible for the more curious to exercise themselves

in the rest. But formerly some destroying the construction and variety

of this problem, reasoned thus. Let _a_ be a given point, but _b e_ a

given right line, and with the centre _a_, but with an interval equal

to _b e_, let a circle _d e_ be described. Then let a certain right

line _a d_ be extended from the point _a_ to the circumference; and

this shall be equal to _b e_: for the magnitude of the line from the

centre, was equal to that of _b e_: and so that is done which was

required. But he who thus reasons, _begs_, in the very beginning.

For when he says with the centre _a_, but interval _b e_ describe a

circle _e d_, he receives, in a certain manner, a line equal to _b e_,

placed at the extremity _a_; and preserving the Petition, he makes

one extremity of the interval a centre, but with the other describes

a circle: however, in this case, the centre is in one place, but the

interval in another. We by no means, therefore, approve this method of

demonstration.

[Illustration]

[Illustration]

PROPOSITION III. PROBLEM III.

Two unequal right lines being given, from the greater

to cut off a part equal to the less.

[Illustration]

This third problem, likewise, has a variety of cases. For the given

unequal right lines are either mutually distant from each other, as

with the institutor of the elements, or they are united according

to one extreme; or the one cuts the other according to one of its

extremities, and this in a two-fold manner. For either the greater cuts

the less, or the less the greater. But if they are united according to

one extreme, the demonstration is manifest. For employing the common

extremity as a centre, and the lesser of the lines for an interval, you

will describe a circle, and cut off from the greater, a part equal to

the less; since as much as the circle intercepts within itself, will

be equal to the less. But if the one cuts the other according to its

extreme, either the greater will cut the greater, or the contrary.

And if they mutually cut each other, they will either be mutually cut

into equal parts, or into unequal; or the one will be cut into equal,

and the other into unequal parts, and this in a two-fold respect. For

all these present us with an admirable variety of exercise, some of

which, out of a many, we shall exhibit. Let there be given the unequal

right lines _a b_, _c d_, the greater of which is _c d_, and let it

cut _a b_ in one of its extremities _c_; then with the centre _a_, but

interval _a b_, let a circle _b f_ be described, and let an equilateral

triangle _a e c_ be constructed upon _a c_, and produce _e a_, _e c_.

Again, with the centre _e_, but interval _e f_, let the circle _g f_ be

described; and with the centre _c_, and interval _c g_, the circle _g

l_. Because therefore, _e f_ is equal to _e g_ (for the centre is _e_)

of which _e a_ is equal to _e c_, the remainder _a f_, shall be equal

to the remainder _c g_. But _a f_ is likewise equal to _a b_; for the

centre is _a_. Hence, _c g_ will be equal to _a b_, and this is equal

to _c l_, for the centre is the point _c_: _a b_, therefore, is equal

to _c l_, which was required to be done.

[Illustration]

[Illustration]

[Illustration]

[Illustration]

But let _c d_ be less than _a b_, and let it cut _a b_ according to

its extremity _c_; either, therefore, it will cut it in the middle,

or not in the middle. Let it in the first place cut it in the middle;

_c d_, therefore, is either the half of _a b_, and _a c_ is equal to

_c d_, or it is less than half. And in this case with the centre _c_,

and interval _c d_, describe a circle, and you will cut off from _a

b_ a part equal to _c d_: Or it is greater than half; and then at the

point _a_, placing _a f_, equal to _c d_, and describing a circle with

the centre _a_, and interval _a f_, you will cut off from _a b_ a part

equal to _a f_, that is to _c d_. But if _c d_ does not cut _a b_ in

the middle, _c d_ shall either be its half, or greater than the half,

or less. If therefore _c d_ is the half, or less than the half of _a

b_, employing _c_ as a centre, and _c d_ as an interval, you will cut

off from _a b_, a part equal to _e d_, as was required to be done. But

if _c d_ is greater than the half, again at the point _a_[5] placing _a

f_ equal to _c d_, you will accomplish the same. For with the centre

_a_, but interval _a f_, you will describe a circle, cutting off from

_a b_ a line equal to _a f_, that is, to _c d_. But if they mutually

intersect, as _c d_, _a b_, then with the centre _b_, but interval _b

a_, describe the circle _a f_, and let _b c_ be extended to the point

_f_. Because therefore, _b f_, _c d_, are the two unequal right lines,

and _c d_ cuts _b f_, according to its extremity, it is possible from

_c d_ to make a line equal to _b f_; for this has been shewn in the

first case of this problem. It is therefore possible, that a line equal

to _a b_ may be cut off from _c d_; for _a b_ and _b f_ are mutually

equal. Having, therefore, received these cases from division, we have

endeavoured to exhibit their variety. But the demonstration of the

elementary institutor is admirable, since it accords with all the

preceding constructions. And it is possible, in every position, at the

extremity of the greater, to place a line equal to the less, and using

the same extreme as a centre, and placing the interval to describe a

circle, which shall cut off from the greater, a line equal to the less,

whether they mutually intersect, or one cuts the other, or they are

constituted in a still different position.

PROPOSITION IV. THEOREM I.

If two triangles have two sides equal each to each; and have

likewise the angles equal; which are comprehended by the equal

sides; then they shall have their bases equal; and the two

triangles shall be equal; and the remaining angles opposite to

the equal sides shall be equal.

This is the first theorem in the institution of the elements, for

all those which preceded were problems. The first, indeed, treating

concerning the origin of triangles: but the second and third proposing

to procure one right line equal to another. And of these the one

produced an equal from an unequal line, but the other discovered an

equal line by an ablation from one unequal. Since, therefore, equality,

which is the first symptom in quantity, is to be constructed by us in

a triangle and right line, it is delivered in the following theorem.

For how can he who has not previously constructed triangles, and

procured their origin, be learned in their essential accidents, and

in the equality of angles and sides which they contain? How can he

receive sides equal to sides, and right lines to other right lines,

who has neither problematically investigated these, nor fabricated the

invention of equal right lines? For if he should say it may happen

before they are fabricated, that if two triangles have _this_ for a

_symptom_, they shall likewise have _this particular symptom_; would

it not, in this case, be easy to object to him, that we by no means

know whether a triangle can be constructed? And should it be afterwards

inferred, that if there are two triangles, they may have two sides

equal to two sides, may we not also doubt this, whether it is possible

that right lines may be mutually equal? And this particularly in

geometrical forms, in which inequality not entirely existing, equality

is likewise inherent. For we must learn that the cornicular is always

unequal to an acute angle, and the same is true of the semicircular

angle, and the transition from the greater to the less does not

entirely take place through that which is equal. The institutor of

the elements, therefore, first of all removing these objections,

delivers also the construction of triangles (for it is common to three

forms) and the origin of equal right lines, in a two-fold order. For

he produces the one, not yet existing: but he acquires the other by

an ablation from an unequal line. But after these he very properly

subjoins the theorem, by which it is shewn how triangles having two

sides equal to two, each to each, and the angles comprehended by the

equal sides equal, have also the base equal to the base, the area equal

to the area, and the remaining angles to the remaining angles. For

there are three particulars exhibited in these triangles: but two data.

Hence, the equality of the two sides is given, or two equal sides (and

it is manifestly given in proportion) and the equality of the angle

contained by the equal sides: but three particulars are investigated,

the equality of _base_ to _base_, of _triangle_ to _triangle_, and

of the _remaining angles_. But because it is possible that triangles

may have two sides equal to two, and yet the theorem not be true,

because the one is not equal to the other, but both together, on this

account he adds in the data, that the sides are equal not simply, but

one to the other. For if one of the triangles should have one of its

sides of three units, but the other of four; and again, if the sides

of the other triangle are respectively two, and five units, the angle

comprehended by these being right, the two sides of the one triangle,

will, indeed, taken together, be equal to the two sides of the other,

or to seven units, yet the two triangles will not be equal. For the

area of the one is six units[6], but of the other five. And the reason

of this is, because the sides are not equal each to each. Hence, many,

not observing this in the division of land, when they have received a

greater, have thought just the same as if they had received an equal

field; and this because both the sides containing one field, have been

together equal to both the sides containing the other field. It is

requisite, therefore, to receive the one equal to the other, and to

mark wherever the institutor of the elements subjoins this, because he

does not add it without occasion. For discoursing on the equality of

equal angles, he adds the particle _comprehended by equal sides_, lest

by speaking indeterminately we should assume some one of the angles at

the bases. Besides, when in triangles no side is previously named, we

must conceive the base to be the side opposite to our sight; but when

two are previously received, the remaining side is necessarily the

base. Hence, here too, the institutor of the elements having previously

assumed two sides equal to two, calls the remainder the bases of the

triangles. But a triangle is then said to be equal to a triangle, when

their areas are equal. For it is possible, that though the ambits are

equal, yet the areas may be unequal, on account of the inequality of

angles. But I call the area, the space intercepted by the sides of

the triangle: as also I denominate the ambit, the line composed from

the three triangular sides. Each, therefore, is different, and it is

requisite, indeed, that besides the equality of the ambits, according

to each side, the angles should also be equal, if also area ought to

be equal to area. But it happens in certain triangles, that though

the areas are equal, yet the ambits are unequal; and that the ambits

being equal, the areas are unequal. For if there be two isosceles

triangles, each of whose equal sides contains five units, but the base

of the one is eight, and of the other six units; he who is ignorant

of geometry, will say that the greater triangle is that whose base

contains eight units. For the whole ambit will be eighteen. But the

geometrician will say, that the area of each triangle contains twelve

units, and this he will demonstrate, by drawing in each triangle a

perpendicular from the vertex, and multiplying this with either part

of the segments of the base[7]. But it happens (as I have said) that

though the ambits are equal, the spaces are unequal. Hence, certain

persons formerly fraudulently deceived their partners in the division

of fields, on account of the equality according to ambit, receiving

a larger field. But one base is said to be equal to another, and one

right line to another, when their extremes conjoined make the whole

coincide with the whole. For every right line, indeed, agrees with

every right line; but equal right lines mutually coincide according

to their extremes. Again, one right-lined angle is said to be equal

to another, when one of the comprehending sides of one angle being

placed upon one of the other, the remaining side also coincides with

the remainder: but when one of the remaining sides falls external to

the other, the greater angle is that whose side falls externally; and

the less whose side falls within. For there, indeed, the one contains,

but in this case it is contained. But we must assume the equality of

angles according to the convenience of sides in right lines, and in all

of the same species, as in lunulars and systroides[8], and figures on

both sides convex; because, it is possible that they may be equal, and

yet the sides not mutually coincide. For a right angle is equal to a

certain lunular angle, and yet it is not possible that right lines can

coincide with circumferences. Besides, this also must be previously

understood, that the angles are said to subtend the opposite sides. For

every triangular angle is contained by two sides of the triangle, but

is subtended by the remaining side. Hence, the geometrician, when he

says that the angles are equal, adds, _which are opposite to the equal

sides_, lest we should conceive it of no consequence whatever angle is

received, and should think that he denominated any other two angles of

the triangles equal, but we must call those equal which subtend equal

sides. For equal sides mutually subtend equal angles. And such are the

considerations necessary to the declaration of the present theorem.

[Illustration]

But against the objection of our adversary[9], this must be previously

assumed, that two right lines cannot comprehend space. For this the

geometrician receives as evident. For if (says he) the extremes of

the bases mutually coincide, the bases also shall coincide: but if

not two right lines, will comprehend space. From whence, therefore,

is the impossibility of this derived? Let there then be two right

lines comprehending space _a c b_, _a d b_, and let them be infinitely

produced. Then with the centre _b_, and interval _a b_, let a circle

_a e f_ be described. Because, therefore, the line _a c b f_ is a

diameter, _a c f_ is the half of the circumference. Again, because

the line _a d b e_ is a diameter, _a e_, likewise, is one half

of the circumference. Hence, _a e_, and _a c f_ are equal to the

circumference, which is impossible. Two right lines therefore, cannot

comprehend space; which the institutor of the elements knowing said,

in the first Petition, _from every point, to every point, to draw a

right line_, because one right line is always capable of uniting two

points, but this is impossible for two right lines to effect. Many

circumferences, indeed, may conjoin two points, both in the same, and

in contrary parts: for by this means the extremities of a diameter

conjoin two circumferences, but only one right line. But it is possible

that both within and without semicircles, infinite circumferences

conjoining given points may be described. And the reason of this is,

because a right line is the least of lines, having the same extremes.

But there is every where one _minimum_, and this always becomes the

measure of the infinity of others. As therefore a right line; since

it is one, becomes the measure of the infinity of right-lined angles

(for by this we discover their quantity) so likewise a right line

procures us the greatest utility in the mensuration of such as are

non-rectilineal. And thus much may suffice concerning these.

But that the whole demonstration of the present theorem depends on

common conceptions, rising as it were spontaneously, and emerging

from the evidence of hypotheses, is manifest to every one. For since

two sides are equal to two sides, each to each, they will mutually

coincide. But since the angles contained by the equal sides are equal,

they also shall mutually coincide. And when angle is placed on angle,

and sides on sides, so as to touch, in every part, the extremities of

the sides beneath shall also coincide. But if these, then _base_, shall

agree with _base_. And if three with three, the whole triangle shall

accord with the whole triangle, and all shall be equal to all. Hence,

therefore, equality considered in things of the same species, appears

to be the cause of the whole demonstration. For here are two axioms

endued with a power of containing the whole method of the proposed

theorem. One, indeed, affirming, that _things which mutually coincide,

are equal_; and this is simply true, requiring no limitation, and is

employed by the institutor of the elements both in the base, and in the

space, and in the other angles. For these, says he, are equal, because

they mutually coincide. But the other affirming that _things which are

equal mutually coincide_. This, however, is not true in all, but in

those of a similar species. But I call things similar in species, such

as a right line when compared with a right line, one circumference with

another of the same circle, and the angles comprehended by similar

lines endued with a similar position. But of these, I say, that such as

are equal, mutually coincide: so that in short, the whole demonstration

is of this kind. These equals, therefore, are given, viz. two sides

equal to two sides, and the angles which they comprehend, and these

accord among themselves. But if these mutually coincide, the base also

shall agree with the base, and all coincide with all. And if these

accord, they are also equal. If then these are equal, it may at the

same time be shewn that all are equal to all. And this appears to be

the first mode of knowing triangles on all sides equal. And thus much

concerning the whole demonstration.

But Carpus, the mechanist, who, in an astrological treatise,

discourses of problems and theorems, says, “that they must not be

passed over in silence, since they opportunely present themselves

for investigation;” and lastly, entering on their distinction, he

observes, “that the problematical genus precedes theorems in order.

For in problems (says he) the invention of subjects is investigated

prior to symptoms. Likewise, a problematical proposition is simple, and

requires no artificial intelligence. For this commands us to accomplish

something evident, as _to construct an equilateral triangle_, or _from

two given unequal right lines, to cut off from the greater a part

equal to the less_. For what is there in these difficult and obscure.

But he affirms that the proposition of a theorem is difficult, and

requires the most accurate power, and a judgment productive of science,

that it may appear neither to exceed, nor to be deficient from truth;

such, indeed, as the present, which is the first of theorems. Add too,

that in problems, there is one common way invented by resolution, by

proceeding according to which, we can happily accomplish our purpose.

For after this manner the more easy kind of problems are investigated.

But the treatise of theorems is so very difficult, that even to our

time (says he) no one has been able to deliver any common method of

their invention. Hence, on account of facility also, the problematical

genus is more simple. But these being distinguished, it is on this

account (says he) that in the elementary institution problems precede

theorems, and from these the institution of the elements begins; and

the first theorem is in order the fourth, not because the fourth is

exhibited from the preceding, but because it is necessary they should

precede as being problems, and this a theorem, though it should

require none of the antecedent propositions for its demonstration.

For the present theorem entirely employs common conceptions; and in a

certain respect receives the same triangle in a different position.

Since coincidence, and its consequent equality possess a sensible and

manifest apprehension. But such being the demonstration of the first

theorem, problems with great propriety precede, because they are

universally allotted the primary place.” And perhaps, indeed, problems

antecede theorems in order; and particularly among those who ascend

to contemplation from the arts, which are conversant with sensible

particulars: but theorems excel problems in dignity of nature. And

it appears, that all geometry, so far as it conjoins itself with a

variety of arts, energizes problematically: but so far as it coheres

to the first science, it proceeds theorematically from problems to

theorems, from things secondary to such as are first, and from things

which more regard the arts, to such as are endued with a greater power

of producing science. It is, therefore, vain to accuse Geminus, for

affirming that theorems are prior to problems. For Carpus assigns a

precedency to problems, according to order: but Geminus to theorems,

according to a more perfect dignity. But of this fourth theorem, we

have already observed, that in a certain respect it is indigent of the

preceding problems, in which we learn the origin of triangles, and the

invention of equality. But we now add, that since it is the most simple

and principle of theorems (for it is naturally, as I may say, exhibited

from primary conceptions alone), but demonstrates a certain symptom

appearing about triangles, having two sides equal to two, each to each,

and the two angles equal contained by the equal sides, it is with great

propriety placed the first after problems, in which things subject to

this symptom, and the data themselves are constructed.

PROPOSITION V. THEOREM II.

The angles at the base of an isosceles triangle are mutually equal; and

the equal right lines being produced, the angles under the base shall

be mutually equal.

Of theorems some are _simple_, but others _composite_. I call those

_simple_, which, both according to hypotheses and conclusions, are

indivisible, possessing one _datum_, and one object of investigation.

Thus for example, if the institutor of the elements had said, _every

isosceles triangle has the angles at the base equal_, it would

have been a simple theorem. But theorems are composite, which are

composed from many particulars, either having composite hypotheses,

or conclusions from a simple hypothesis, or both. And of these, some

are _complex_, but others _incomplex_. The _incomplex_ are such

composites as cannot be divided into simple theorems, as the fourth

proposition. For in this, both the _datum_ is a composite, and its

consequent, yet it is impossible that the _datum_ can be divided into

things simple, and become theorems. For if a triangle has its sides

alone equal, or the angle at the vertex, the same consequences will

not ensue. But the _complex_ are such as may be divided into things

simple, as the theorem which says, _triangles and parallelograms of

the same altitude, have the same proportion as their bases_. For it

is possible to say by division, that _triangles of the same altitude,

have the same proportion as their bases_, and in parallelograms after

a similar manner. But of all composites, some are composed according

to the conclusion, being excited from the same hypothesis: but others

have their conclusion according to hypotheses, and infer the same

conclusion in all: and others, lastly, are composed both according to

the conclusion, and according to hypotheses. _Composition_, therefore,

in the present case, is according to the _conclusion_, for there are

three particulars concluded in this theorem, _that the bases are

equal, that the triangles are equal, and that the remaining angles,

under the base, are equal to the remaining angles_. But composition,

according to _hypothesis_, is found in the common theorem _of triangles

and parallelograms of the same altitude_. And _according to both_,

in the theorem that _the diameters both of circles and ellipses,

bisect as well the spaces as the lines containing the spaces_. But

of _complex_ theorems, some are universal: but others conclude that

which is universal from particulars. For if we should say that a

diameter divides a circle, ellipsis, and parallelograms, we receive,

indeed, every part of the complex, not universally, but we make that

universal which is composed from all. But if we should say, that _in

a circle, all lines passing through the centre, mutually bisect each

other, and make equal angles of all the segments_, we should affirm

a universal. For in an ellipsis all the angles of the segments are

not equal, but those only which are formed by the diameter. But these

compositions are entirely fabricated, for the sake of geometrical

brevity and resolutions. For many things incomposite are not resolved,

but composites alone afford convenience to a resolution tending to

principles.

In consequence of these previous considerations then, we must call

the fifth theorem a _composite_, and a composite, both with respect to

the _datum_, and the _object of investigation_; and this the institutor

of the elements exhibiting, divides this theorem, being one, and gives

a separate position to the _data_, and the _things to be investigated_,

for he says that _the angles at the base of an isosceles triangle are

equal_; and again, that _the equal sides being produced, the angles

under the base are equal_. For we must not think that there are two

theorems, but one; and that this is a composite, both according to

the _data_, and _thing sought_: and that each of these composites is

perfect and true. Hence, conversion also is true in each. For if the

angles at the base are equal, the triangle is isosceles: but if those

under the base are equal, the equal right lines are produced, and the

triangle is isosceles. But the institutor of the elements _converts_

the equality of the angles at the base; but not the equality of those

under the base, though this is likewise true; the reason of which we

shall shortly explain. But we shall now, in the first place, enquire on

what account he demonstrates that the angles under the base are equal.

For he never employs this in the construction or demonstration of other

problems or theorems. It may be doubted, therefore, why, since it is

useless, it was requisite to insert it in the present theorem? To this

we must reply, that though it is never employed in the elements, yet

it is most useful for the destruction of objections, and the solution

of oppositions to theorems[10]. But it is artificial, and belongs to

science to prepare solutions of things resisting its propositions, and

to provide subsidies of answers; that not only true demonstrations may

be fabricated from things previously demonstrated, but that from hence

confutations of error may be produced. And from this geometrical order,

you will likewise receive a rhetorical emolument. For he who can effect

this in the discourses of rhetoric, who can foresee the oppositions to

his following heads, and previous to their delivery, can first of all

prepare solutions of them to others, he, indeed, will fabricate in a

wonderful manner, a most excellent mode of disputation. The institutor

of the elements, therefore, teaching us this in reality, previous to

the theorems by which we solve opposing objections, employing such

as are now exhibited, at the same time demonstrates, that the angles

under the base of an isosceles triangle, are equal, and thus prepares

a confutation of the falsehood such objections contain. But that from

the present theorem we may solve the objections urged in the seventh

and ninth propositions, will be perspicuous as we proceed. Hence, it

appears, why Euclid does not convert the latter part of this theorem in

the sixth, because it does not produce a principal utility, but confers

to our advantage, accidentally, with respect to the whole of science.

But if any one should desire us without producing the equal right

lines, to prove the angles at the base of an isosceles triangle equal,

(for it is not requisite to demonstrate the equality of these, by

those under the base) by transposing, in a manner, the construction,

and fabricating those constructions within, which are made without the

isosceles triangle, we may exhibit the thing proposed. Thus let _a b c_

be an isosceles triangle, and in the side _a b_, take any point _d_,

and from _a c_, take _a e_, equal to _a d_, and draw the lines _b e_,

_d c_, _d e_. Because, therefore, _a b_ is equal to _a c_, and _a d_

to _a e_, and the angle _a_ is common, _b e_ also shall be equal to _c

d_, and the remaining angles to the remaining angles. Hence, the angle

_a b e_, is equal to the angle _a c d_. Again, because _d b_ is equal

to _e c_, and _b e_ to _d c_, and the angle _d b e_ to _e c d_; hence,

the base, since it is common to both, is equal to itself, and all are

equal to all. The angle, _e d b_, therefore, is equal to the angle _d

e c_: and the angle _d e b_, is equal to the angle _e d c_. Hence,

since the angle _e d b_, is equal to the angle _d e c_, from which the

equal angles _d e b_, _e d c_, are taken, the remaining angles _b d

c_, _c e b_ are equal. But the sides also _b d_, _d c_, are equal to

the sides _c e_, _e b_, each to each, and the base _b c_ is common.

All, therefore, are equal to all. Hence, the remaining angles also,

subtending equal sides, are equal. The angle, therefore, _d b c_, is

equal to the angle _e c b_. For the angle _d b c_, subtends the line _d

c_: but the angle _e c b_, the line _e b_. The angles, therefore, at

the base of an isosceles triangle, are equal, the equal right lines not

being produced.

[Illustration]

[Illustration]

But Pappus demonstrates this yet shorter, without any addition in

the following manner. Let _a b c_ be an isosceles triangle, having _a

b_, equal to _a c_. We must conceive, therefore, this one triangle as

if it was two, and reason thus. Because _a b_ is equal to _a c_, and

_a c_ to _a b_, the two sides _a b_, _a c_, are equal to the two _a

c_, _a b_, and the angle _b a c_, is equal to the angle _c a b_, (for

it is the same.) All, therefore, are equal to all. The base _b c_, to

the base _c b_. But the triangle _a b c_, to the triangle _a c b_;

and the angle _a b c_, to the angle _a c b_, and the angle _a c b_,

to the angle _a b c_. For they subtend equal sides, i.e. _a b_, _a

c_. The angles, therefore, at the base of an isosceles triangle, are

equal. And it seems that Pappus invented this mode of demonstration,

when he considered that the institutor of the elements also, in the

fourth theorem, when he had united two triangles, and had made them

mutually coincide, thus forming one of two, by this means observed

their equality throughout. In like manner it is possible, that we also,

by an assumption contemplating two triangles in one, may demonstrate

the equality of the angles at the base. Thanks, therefore are to be

given to the ancient Thales for the invention of this theorem, as well

as a multitude of others. For he, first, is said to have perceived

and affirmed, that the angles at the base of every isosceles triangle

are equal: and after the manner of the ancients, to have called them

similar. But still more deserving of praise are those moderns, who have

yet more universally demonstrated (among which number is Geminus) that

equal right lines falling from one point, on a line of similar parts,

form equal angles. For Geminus using this theorem, shews, that there

are only three lines, and not more of similar parts, the _right_, the

_circular_, and the _cylindric helix_; and this is properly universal,

to which this symptom first agrees, just as the possession of two

sides greater than the third, is shewn to be essentially inherent in

every triangle. It is not, therefore, the property universally of

every isosceles, though it belongs to every one, to possess angles at

the base equal: but of equal right lines falling on a line of similar

parts. For to subtend equal angles, is in these primarily inherent.

PROPOSITION VI. THEOREM III.

If two angles of a triangle be equal to each other, the sides

also which subtend the equal angles, shall be equal to one

another.

The present theorem exhibits these two properties of theorems,

_conversion_, and _a deduction to an impossibility_. For it is

converted, indeed, in the preceding theorem, but its certainty is

evinced by a deduction to an impossibility. It is requisite, therefore,

to speak of each, whatever belongs to the present treatise. One kind

of _conversion_ then, among geometricians, is denominated principally

and properly, when the conclusions and hypotheses alternately receive

theorems; so that the conclusion of the former becomes hypothesis in

the latter; and hypothesis is inferred as the conclusion. As _that

the angles at the base of an isosceles triangle are equal_. For here

the isosceles triangle is the _hypothesis_: but the _conclusion_, the

equality of the angles at the base. And _that where the angles at

the base are equal, the triangles are isosceles_, which the present

6th theorem affirms. For here the equality of the angles at the

base is the _hypothesis_; but the _conclusion_, the equality of the

sides subtending the equal angles. But another kind of conversion,

is alone according to a certain mutation of composites. For if the

theorem be composite, beginning from many hypotheses, and ending in

one conclusion, by receiving the conclusion, and one or more of the

hypotheses, we infer some one of the other hypotheses as a conclusion.

And after this manner the eighth theorem is the converse of the

fourth. For the one says, that _equal bases subtend equal sides and

angles_: but the other, that _equal sides being placed on equal bases,

contain equal angles_. Of which the predication concerning _equal

bases_ in the latter proposition, is the _conclusion_ of the former:

but the predication concerning the position of _equal sides_, is one

of the previously assumed hypotheses in the former theorem; and the

_comprehension of equal angles_ is another hypothesis which this

fourth proposition contains. In consequence therefore of these two

_conversions_, the one which is called the principle, is uniform and

determinate: but the other is various, advancing into a great number

of theorems, and not converting in one, but in many, on account of the

multitude of hypotheses, in composite theorems. But oftentimes in that

which begins from two hypotheses, there is one which is converted, when

the hypotheses are not all determinate, but some of them indeterminate.

It is here, however, requisite to observe, that many false and

improper conversions take place. As that _every sexangular is a

triangular number_[11]. For the converse is not also true, that _every

triangular number is sexangular_. But the reason of this is, because

the one is more common, but the other more particular. And one is alone

predicated _totally_[12] of the other. But things in which, _that which

is primary_, is inherent, and according to which it is received, in

these, conversion also follows. And these observations, indeed, were

not unknown to those mathematicians, the familiars of Menæchmus, and

Amphinomus. But of theorems receiving conversion, some are usually

called _precedents_, but others _converse_. For when supposing a

certain genus, they demonstrate some symptom of its nature, they call

this a _precedent_ theorem. But when on the contrary, they make the

hypothesis a symptom, and the conclusion a genus, they denominate

the theorem to which this happens _converse_. As for instance, the

theorem which says, _every isosceles triangle has the angles at the

base equal_, is a _precedent_. For that is subjoined which precedes by

nature. I mean the genus itself, or the isosceles triangle. But that

which says, _every triangle possessing two equal angles, has likewise

the sides subtending those equal angles equal, and is isosceles_, is

a _converse_ theorem. For it changes the subject, and its passion,

supposing the latter, and from this exhibiting the former. And thus

much concerning geometrical conversions.

But deductions to an impossibility, entirely end in an evident

impossible, the contrary of which is confessed by all. It happens,

however, that some of them end in such things as are opposed to Axioms,

or Petitions, or Hypotheses; but others in things contradicting prior

demonstrations. For the present sixth theorem shews that which happens

to be impossible, because it destroys the common conception, affirming

that _the whole is greater than its part_. But the eighth theorem

falls, indeed, on an impossible, yet not on that endued with a power

of destroying a common conception, but that exhibited by the seventh

theorem. For what the seventh denies, this affirming exhibits to such

as do not admit the object of investigation. But every deduction to

an impossibility, which being received, opposes _the thing sought_,

and on this hypothesis advances, until it falls upon the explored

absurdity, and by this means destroys the hypothesis, corroborates

that which was investigated from the first. But it is requisite to

know, that all mathematical proofs are either _from principles_, or

_to principles_, as Porphyry in a certain place affirms. And the

proofs _from principles_, are two-fold. For they either emanate from

common conceptions, and things self-evident: or from things previously

exhibited. But proofs _to principles_ are endued with a power of

either _establishing_ or _destroying principles_. And those, endued

with a power of _establishing principles_, are called _resolutions_;

and to these compositions are opposed. For it is possible that we may

proceed in an orderly method from those principles to the object of

investigation; and this is nothing else than composition. But those

possessing a power of _destroying principles_, are called _deductions

to an impossibility_. For it is the business of this mode to destroy

some of the concessions, and objects of investigation. And in this,

also, there is a certain ratiocination, though not the same as in

resolution. For in deductions to an impossibility, _complexion_ is

according to the second mode of hypothetical reasonings. As if _in

triangles possessing equal angles, the sides subtending the equal

angles are unequal; and the whole is equal to its part_: but this is

impossible. _In triangles, therefore, possessing two equal angles, the

sides subtending the equal angles are equal._ And thus much concerning

what is called by geometricians, deduction to an impossibility.

[Illustration]

But the institutor of the elements uses _conversion_ in the present

proposition, for he receives the conclusion of the fifth as a datum,

and adds its hypothesis as an object of enquiry: but he employs

_deduction to an impossibility_, in the construction and demonstration.

But if any should rise up, and assert that it is not necessary by

taking a part from _a c_ equal to _a b_, to make the ablation at the

point _c_, but at the point _a_, upon this hypothesis, we shall fall

into the same impossibility. For let _a b_ be equal to _a d_, and

having produced _b a_, let _a e_ be placed equal to _d c_. The whole _b

e_, therefore, is equal to the whole _a c_.

[Illustration]

Let _e c_ be connected. Because, therefore, _a c_ is equal to _b e_,

but _b c_ is common, the two are equal to the two, and the angle at the

point _b_, is equal to the angle _a c b_. For so it was established

in the hypothesis. All, therefore, are equal to all, by the fourth

theorem. Hence, the triangle _e b c_, is equal to the triangle _a b

c_, the whole to the part, which is impossible. But because this also

is manifest, it remains that we exhibit the rest of the conversion.

For the institutor of the Elements converts the whole sixth theorem

from a part of the fifth. But it is requisite to adjoin the remaining

conversion. This, then, he receives as an hypothesis, _that the angles

at the base of a certain triangle are equal_: but he shews that the

triangle is _isosceles_. Let _a c b_, therefore, be a triangle, and

let _a b_, _a c_, be produced to the points _d g_, and let the angles

under the base be equal. I say that the triangle _a b c_, is isosceles.

For let there be assumed in the line _a d_, the point _e_, and let _b

e_ be taken equal to _c f_; and connect the lines _e c_, _b f_, _e f_.

Because, therefore _b e_ is equal to _c f_, but _b c_ is common, the

two will be equal to the two. And the angle _e b c_, is equal to the

angle _f c b_; for they are under the base. All, therefore, are equal

to all, by the fourth theorem. Hence the base _e c_, is equal to the

base _f b_, and the angle _b e c_, to the angle _c f b_; and the angle

_c b f_, to the angle _b c e_: for they subtend equal sides. But the

whole angle _e b c_, was equal to the whole _f c b_, of which the angle

_f b c_, is equal to the angle _e c b_. The remainder, therefore, _e

b f_, is equal to the remainder _f c e_. But _b e_ is equal to _c f_,

and _b f_ to _c e_, and they contain equal angles. All, therefore,

are equal to all. Hence, also, the angle _b e f_, is equal to the

angle _c f e_. Wherefore, the side _a e_, is equal to the side _a f_

(for it is shewn by the sixth) of which _b e_, is equal to _c f_. The

remainder, therefore, _a b_, is equal to the remainder _a c_. And

hence, the triangle _a b c_, is isosceles. It is, therefore, as well

isosceles, if it possesses angles at the base equal: as if the sides

being produced it has the angles under the base equal. Why then did not

the institutor of the Elements convert the remaining part? Shall we say

it was because the equality of the angles under the base in the fifth

theorem, was exhibited for the sake of solving other doubts. But that

proving the triangle to be isosceles, from the equality of the angles

under the base, neither confers to a principal demonstration, nor to

the solution of things investigated, the truth of which is confirmed in

the following theorems, and that from the equality of the angles under

the base, he is enabled to demonstrate that the triangle is isosceles?

For if every right line, standing upon a right line, and forming two

angles, makes them equal to two right; when the angles under the

base are equal, those upon the base will be equal. And these being

equal, the sides subtending them shall be equal. Euclid, therefore,

having used this in the whole elementary institution, was enabled to

conclude, that when the angles under the base are equal, the triangle

is isosceles. Indeed he requires this also, for the demonstration of

certain theorems: For shortly a theorem will appear, evincing, that if

a right line standing on a right line, forms angles, it will either

make two right, or angles equal to two right. And the theorems, indeed,

preceding this, require no such conversion; but those which follow,

are indigent of this, and establish their credibility from the present

theorem.

PROPOSITION VII. THEOREM IV.

[13]Upon the same right line, two right lines cannot be

constituted equal to two other right lines each to each,

_drawn_ to different points, to the same parts, and having

the same extremes with the two right lines first drawn.

The present theorem possesses a rare property, which is not

frequently found in propositions producing science. For to be formed

by negation, and not by affirmation, is not their sufficiently

distinguishing property. Indeed, the propositions, as well of

geometrical as of arithmetical theorems, are for the most part

affirmations. But the reason of this is, (as Aristotle says) because,

an affirmative universal, especially agrees with sciences, as more

proper, and not indigent of negation: but a universal negative requires

affirmation, in order to produce evidence; for from negatives alone,

there is neither demonstration nor reasoning. Hence, demonstrative

sciences exhibit a multitude of affirmations, but rarely employ

negative conclusions. However, the proposition of this theorem is full

of admirable diligence, and is bound with every addition, by which it

is rendered so certain and indubitable, that it cannot be confuted and

overturned by the efforts of opposing calumniators. For in the first

place, the particle _upon the same right line_, is assumed, lest we

should exhibit upon _another_, two right lines equal each to each, and

employ the proposition for the purpose of circumvention. In the second

place, he does not say upon what right line, to constitute two right

lines simply equal to two (for this is possible) but _each to each_.

For what wonderful thing is it, that he should take both equal to both,

who extends one of the constituted lines, and contracts the other? But

each to each, (says he) is impossible. In the third place, he adds the

particle, _to different points_. For what, if some one, when he has

formed two lines equal to the first two, each to each, should connect

these with those in the same point, which joins the subject right

lines in the vertex; and should constitute these? For the extremes of

equal right lines perfectly coincide. In the fourth place, he adds the

particle _to the same parts_[14]. For what if one subject right line

being given, we should place two of the right lines on one side, and

the other two on the opposite side, so that this common right line

should be the basis of the two triangles with opposite vertexes? Lest,

therefore, we should form an erroneous figure, and charge our deception

on the institutor of the Elements, he adds the particle _to the same

parts_. In the fifth place, he subjoins, _having the same extremes with

the two right lines first drawn_. For it is possible to constitute

_upon the same right line, two right lines equal to two, each to each,

drawn to different points, and to the same parts_, by employing the

whole right line, and constructing upon it, these two right lines; but

then the lines last drawn, will not have the same extremes with those

constituted at first. For if we conceive in a quadrangle two diagonals

drawn on one of its sides, two lines shall be equal to two; a side and

diameter to its parallel side, and the other diameter. But in this case

the equal right lines will not have the _same_ extremes. For neither

the parallel sides, nor the diameters, will mutually possess the same

extremes; and yet they will be equal. These distinctions, therefore,

being preserved, the truth of the proposition, and the certainty of the

reasoning, is evinced.

[Illustration]

But perhaps, some, notwithstanding all these terms producing science,

will dare to object, that these hypotheses being admitted, it is

possible to effect what the geometrician affirms to be impossible. For

let there be a right line _a b_, and upon this two lines _a d_, _d

b_, equal to two _a c_, _c b_, and let the former be external to the

latter, being drawn to different points _d c_, and terminated in the

same extremes _a_ and _b_. Let _a c_ too, be equal to _a d_: but _b c_

to _b d_. This objection, then, we shall confute, by connecting the

line _d c_, and producing the lines _a c_, and _a d_, to the points _e

f_. For these being constructed, it is manifest that the triangle _a

c d_ is isosceles, _a d_, being equal to _a c_, from hypothesis; and

the angles under the base _e c d_, _f d c_ are equal. The angle _f d

c_, therefore, is greater than the angle _b d c_. Much more then is

the angle _b c d_ greater than the angle _b d c_. But again, because

the line _d b_, is equal to the line _b c_, the angles also at the

base are equal, i.e. the angle _b c d_, to the angle _b d c_. The same

angle, therefore, is both greater and equal, which is impossible. And

this is what we said in our exposition of the fifth theorem, that

though the equality of the angles under the base, was not useful to the

demonstrations of the following theorems, yet it procured the greatest

utility in the solution of objections. For in the present instance we

have confuted the objection, by inferring that, because _a c_, and _a

d_, are equal, the angles _e c d_, and _f d c_, are also equal. In

a similar manner in other theorems, it will appear to be peculiarly

useful for the solution of doubts[15].

But if any one should say that there may be constituted upon the right

line _a b_, right lines _b d_, _b c_, equal to the right lines _a c_,

_a d_, of which _b c_ may be equal to _a c_, but _b d_ to _a d_; and

that in this case they will be drawn to different points _a_ and _b_,

to the same parts, and will have the same extremes with _a c_, and _a

d_, viz. _c_, and _d_, what shall we reply to this assertion? Shall

we say that it is requisite to constitute the first lines, upon the

right line _a b_, and their equals upon the same right line? For this

is what the institutor of the Elements affirms in the proposition. But

here, _a c_, and _a d_, are not constituted upon the right line _a

b_, but only on one of its points. Hence, the lines _a c_, _c b_, and

_a d_, _d b_, which stand on the right line _a b_, are different from

the right lines, which were placed in the beginning, and to which they

ought to be constituted equal. Though at the same time it is necessary

that the right lines constituted upon _a b_, should be equal to those

constituted upon _a b_. And thus much may suffice for objections

against the present question, But that the present theorem is exhibited

by the institutor of the elements, by a deduction to an impossibility,

and that this impossible opposes the common conception, affirming that

_the whole is greater than its part_; and that _the same thing cannot

be both greater and equal_, is sufficiently manifest. But this theorem

seems to have been assumed for the sake of the eighth theorem. For it

confers to its demonstration, and is neither simply an element, nor

elementary: since it does not extend its utility to a multitude. And

hence, we find it very rarely employed by the geometrician.

PROPOSITION VIII. THEOREM V.

If two triangles have two sides equal to two, each to each, and

have the base equal to the base: then the angles contained by

the equal right lines, shall be equal to each other.