BOOK IV.

Whatever can be said in an elementary institution, concerning the

origin and equality of triangles, we may learn from the preceding

discourse. But after this, the narration of Euclid is concerning

quadrilateral figures, and he particularly teaches us concerning

parallelograms, together with the contemplation of these delivering

the doctrine of trapeziums. For a quadrilateral figure, (as we

have formerly observed in our discourse on hypotheses,) is divided

into parallelogram and trapezium; and a parallelogram into other

certain species, and in like manner a trapezium. But because a

parallelogram, on account of its participation of equality, possesses

disposition and order, but a trapezium has neither the same, nor

a similar order; Euclid’s principal discourse, is with propriety,

concerning parallelograms, but he also contemplates together with

these a trapezium. For from the section of parallelograms, the origin

of trapeziums will appear, as will be manifest as we proceed. But

because again, it is not possible that any thing can be said of the

construction or equality of parallelograms, without the consideration

of parallels, (for as it is manifest from the very name, that is,

a parallelogram, which is circumscribed by parallel right lines in

an opposite position,) hence, he necessarily assumes from parallels

the beginning of his doctrine, but having advanced a little from

these, he enters on the doctrine of parallelograms, employing one

middle theorem, between the elementary institution of each, because

he appears to contemplate a certain symptom inherent in parallels:

but he delivers the first origin of a parallelogram. For such is the

proposition, which says, that _right lines which join equal and

parallel right lines towards the same parts, are themselves equal

and parallel_. For in this theorem, indeed, a certain accident to

equal and parallel right lines is considered: but from the connection

a parallelogram appears, having its sides opposite and parallel. And

from hence it is manifest that the discourse concerning parallels,

was necessarily pre-assumed. But three things are to be assumed,

essentially inherent in parallels, which they essentially express,

and are converted with them, not only the three together, but every

one separately assumed from the rest. Of these, one is, _that when

a right line cuts parallel lines, the alternate angles are equal_;

but the second, _that when a right line cuts parallel lines, the

internal angles are equal to two right_; and the third, _that in

consequence of a right line cutting parallel lines, the external is

equal to the internal and opposite angle_. For when any one of these

_symptoms_ is demonstrated, we have sufficient authority to affirm

that the right lines are parallel. But other mathematicians, also,

have been accustomed to discourse after this manner concerning lines,

delivering the symptoms of every species. For Apollonius, in each of

the conical lines, shews what a _symptom_ is, as also Nicomedes in

his Treatise on Conchoids, and Hippias in his Quadratics, and Perseus

in his Spirals. Since after their origin, that which is essentially

inherent in these lines, and according to what it is inherent, being

assumed, distinguishes a constructed form from all others. After the

same manner, therefore, the institutor of the Elements, first of all

investigates, the symptoms of parallels.

PROPOSITION XXVII. THEOREM XVIII.

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

alternate angles equal to each other, those right lines shall be

parallel to each other.

[Illustration]

In the present theorem it was not pre-assumed as evident that the

right lines are in one plane, but this ought rather to be previously

admitted in all theorems which are considered in a plane. This,

however, is added, because it does not universally follow, that when

the alternate angles are equal, the right lines will be parallel,

unless they are in the same plane. For nothing hinders, but that a

right line falling on right lines disposed in the shape of the letter

X, one of which is situated in one plane, but the other in a different

one, may make the alternate angles equal; and yet the right lines thus

disposed will not be parallel. It was pre-assumed[27], therefore,

that in a treatise on planes, we conceive every thing described in

one and the same plane: and on this account, he does not require this

addition in the present proposition. But it is requisite to know that

the geometrician considers the particle _alternate_, in a two-fold

respect, sometimes, indeed, according to a certain situation, but

sometimes according to a certain consequence of proportions. And

according to this last signification, the particle _alternate_ is used

in the fifth book, and in such as are arithmetical: but agreeable

to the former, both in this, and in all the other books concerning

parallel right lines, and that which falls upon these. For he calls

the angles alternate, which are not formed at the same parts, and are

not successive to each other, which are distinct, indeed, from the

incident line, but both of them exist within parallels, and differ in

this, that the one has an upward, but the other a downward position.

I say, for example, that when a right line _e f_, falls on the right

lines _a b_, and _c d_, he calls the angles _a e f_, _d f e_, and also

the angles _c f e_, _b e f_, _alternate_, or _altern_, because they

have an alternate, or changed order, according to their position. But

this too must be known, that from such a situation of right lines, all

the symptoms become by division, six; three of which the geometrician

alone receives; and three he omits. For we either assume the angles at

the same parts, or not at the same. And if at the same parts, either

both within the right lines, which shews them to be parallels; or both

without, or one without, and the other within. And if not at the same

parts, again, after the same manner, they are either both without the

right lines, cutting the lines it is necessary to receive; or within;

or one within, and the other without. But what we have said will become

manifest by the same description as above. For let there be certain

right lines _a b_, _c d_, and let a right line _e f_, fall upon them,

and let it be produced to the points _h_ and _g_. If then you assume

angles at the same parts, you will either place them both within, as _b

e f_, and _e f d_, or as _a e f_, and _e f c_; or both without, as _h

e b_, and _d f g_, or as _h e a_, and _c f g_; or one within, and the

other without, as _h e b_, and _e f d_, or as _g f d_, and _f e b_, or

as _h e a_, and _e f c_, or as _g f c_, and _a e f_: for these last are

received in a quadruple respect. But if you assume the angles not at

the same parts, you will either place both within, as _a e f_, and _e f

d_, or as _c f e_, and _f e b_; or both without, as _a e h_, and _d f

g_, or as _h e b_, and _c f g_; or one within, and the other without,

and this again in a quadruple respect. For they will either be the

angles _a e h_, and _e f d_; or _h e b_, and _e f c_; or _g f c_ and

_f e b_; or _g f d_, and _f e a_. And besides these, there is no other

assumption.

As, therefore, angles are assumed according to six modes, the

geometrician combines three assumptions alone; and these consequent

symptoms, are naturally adapted to express parallels. But of these

three assumptions, one belongs to those angles which are not at the

same parts, viz. to those which are only assumed within; and these he

calls alternate, so that those, which are both external, and those, one

of which is external, but the other internal, are omitted: but two of

these assumptions belong to angles at the same parts, to those, indeed,

which are both internal, which he says are equal to two right, and

to those, one of which is internal, but the other external, which he

says are equal, leaving indeed one assumption which supposes both the

angles to be external. We therefore affirm that the same things will

be consequent to the three omitted hypotheses. Thus, in the preceding

figure, let both the external angles _h e b_, _d f g_, be at the same

parts, I say that these are equal to two right angles. For the angle

_d f e_, is equal to the angle _h e b_, and the angle _b e f_, to the

angle _d f g_. But if the angles _b e f_, _e f d_, are equal to two

right, the angles _d f g_, _h e b_, are equal to two right. Let again

the angles _a e h_, _e f d_, not be towards the same parts, of which

the one is within, but the other external, I say that these also are

equal to two right angles. For if the angle _a e h_, is equal to the

angle _b e f_, but the angles _b e f_, and _e f d_, are equal to two

right, the angles, also, _a e h_, and _e f d_, are equal to two right.

Again, let them not be at the same parts, but both without the right

lines as _a e h_, _d f g_. I say that these are equal to one another.

For if the angles _a e h_, and _b e f_, are equal to each other, but

the angle _d f g_, is equal to the angle _b e f_, hence the angle _a

e h_, is not unequal to the angle _d f g_. If, therefore, the things

assumed by the geometrician, in three hypotheses are verified, all

the same follow in the remaining three as indisputably true. Besides

this too is to be observed, that in such as the geometrician receives

these, according to two assumptions, the angles are supposed equal to

each other, but when according to one assumption, equal to two right:

but in these last on the contrary, according to two assumptions, they

are supposed equal to two right angles, but according to one equal

to each other. For since all the assumptions are six, it happens,

indeed, from three, that the angles are equal to two right, but from

the other three, that they are equal to each other. Hence, those which

are omitted are not undeservedly contrary to the assumptions which

are reckoned worthy of relation. But the geometrician appears to have

chosen such hypotheses as either abound in affirmation, or are more

simple, and, on this account of those angles which are not at the same

parts, he assumed alone the internal, which he calls alternate: but of

those at the same parts, he assumes as well the internal, as well as

one internal and the other external, but he avoids the rest, either

because they are more declared by negation, or because they are more

various. However, whether this or some other be the cause, the number

of the consequents to those hypotheses is from hence sufficiently

manifest.

PROPOSITION XXVIII. THEOREM XIX.

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

equal to the internal angle, placed opposite, and at the same

parts, or makes the angles internally situated, and at the same

parts equal to two right, those right lines shall be parallel to

each other.

[Illustration]

The preceding theorem receiving the angles, not at the same parts,

but situated within right lines, shews that the right lines are

parallel among themselves: but the present theorem proposes the two

remaining hypotheses, of which one separates the angles according to

the particles _without_ and _within_, but the other supposes them both

within, and exhibits the same conclusion. But it may seem, perhaps,

that the institutor of the Elements has inconveniently distributed the

theorems. For it was necessary either to receive three hypotheses in

a divided manner, and to make three theorems; or to collect all into

one theorem, as Æneas Hierapolites does, who wrote a compendium of

the Elements; or willing to divide them into two, to make an orderly

division, and to assume the hypotheses separately, which contain equal

angles, and separately that in which the angles are equal to two

right. But in the present propositions, in one theorem he supposes the

alternate angles equal, but in the other, the external to the internal,

and the internal angles situated at the same parts equal to two right.

What then is the cause of this division? Does he regard the equality of

the angles to each other, or to two right, and on this account does not

separate the proposed theorems from each other; or does he respect the

angles being received at the same, or not at the same parts? For the

preceding theorem does not respect angles at the same parts, since such

as these are alternate: but the present regards such as are situated

at the same parts, as is perspicuous from the proposition. But how the

institutor of the Elements shews, that from the internal angles being

equal to two right, the right lines are parallel, appears from his

writings on this subject. Ptolemy, however, in the theorems in which he

proposes to demonstrate that right lines produced from angles less than

two right, coincide at the same parts, in which the angles less than

two right are situated, shewing before all his theorems, that from the

internal angles being equal to two right, the right lines are parallel,

proves it in the following manner. Let there be two right lines _a b_,

_c d_, and let a certain right line _e g f h_, so cut them, that it may

make the angles _b f g_, and _f g d_, equal to two right, I say that

those right lines are parallel, that is, will never coincide. For if

it be possible, let them coincide while the right lines _b f_, _g d_,

are produced in the point _k_. Because, therefore, the right line _e

f_, stands upon the right line _a b_, it makes the angles _a f e_, _b

f e_, equal to two right. In like manner because _f g_ stands upon _c

d_, it makes the angles _c g f_, _d g f_, equal to two right. Hence,

the four angles _b f e_, _a f e_, _c g f_, _d g f_, are equal to four

right, two of which _b f g_, _f g d_, are supposed equal to two right.

The remainders, therefore, _a f g_, _c g f_, are equal to two right.

If then the right lines _f b_, _g d_, when produced, coincide, the

internal angles being equal to two right, _f a_, and _g c_, also, shall

coincide when produced: for the angles _a f g_, _c g f_, are also equal

to two right. Either therefore the right lines shall coincide in both

parts, or in neither, since these, as well as the former, are equal to

two right. Let the right lines then _f a_, _g c_, coincide in the point

_l_. But if this be admitted two right lines _l a f k_, _l c g k_, will

comprehend space, which is impossible. It is not therefore possible,

that the internal angles being equal to two right, the right lines can

coincide. They are therefore parallel.

PROPOSITION XXIX. THEOREM XX.

A right line falling upon parallel right lines, makes the

alternate angles equal to each other; and the external equal to

the internal angle, oppositely situated, and at the same parts;

and the internal angles at the same parts equal to two right.

[Illustration]

The present theorem is converted in both the preceding. For that

which is _the object of investigation_, in each of them, forms the

hypothesis: but what are _data_ in the preceding, he proposes to shew

in the present. And this difference of converse theorems is not to be

passed over in silence. I mean that every thing which is converted, is

either converted as one to one, as the sixth proposition to the fifth;

or as one to a many, as the present to the preceding; or as many to

one, as will shortly be manifest[28]. But in the present theorem, the

institutor of the Elements first employs the petition, which says: _If

a right line falling upon two right lines, makes the angles situated

internally, and at the same parts less than two right, those right

lines whilst they are infinitely produced, will coincide at those

parts in which the angles less than two right are situated_. But in

our exposition of things prior to theorems[29], we have asserted, that

this petition is not allowed by all to be indemonstrably evident. For

how can this be the case when its converse is delivered among the

theorems as demonstrable? For the theorem which says that the two

internal angles of every triangle are less than two right, is the

converse of this petition. Besides, the perpetual inclination of right

lines, more and more, while they are produced, is not a certain sign

of coincidence, because other lines are found perpetually inclining,

and never coinciding, as we have already observed. Formerly, therefore,

some, when they had pre-ordained this as a theorem, considered that

which is assumed by the institutor of the Elements as a petition, to

be worthy of demonstration. But this seems to be shewn by Ptolemy

himself, in a book entitled: _That right lines which are produced

from less than two right angles, coincide_. And this he proves by

pre-assuming many things, which as far as to the present theorem, are

already demonstrated by the elementary institutor; and he supposes

that all are true (lest we should also superadd another confusion) and

that this, as a small assumption, may be exhibited from the preceding.

But this also is one of the things previously exhibited, which says,

_that the right lines produced from two angles equal to two right, will

never coincide_. I say, therefore, that the converse also is true,

which says, _that right lines being parallel, if they are cut by one

right line, the angles situated internally, and at the same parts,

shall be equal to two right angles_. For it is necessary that a line

cutting parallels, should either make the angles internally situated,

and towards the same parts, equal to two right, or less, or greater

than two right. Let the lines then, _a b_, _c d_, be parallel, and let

the right line, _g f_, fall upon them, I say that it will not make the

angles internal, and at the same parts greater than two right. For if

the angles _a f g_, _c g f_, are greater than two right, the remainders

_b f g_, _d g f_, are less than two right. But the same are also

greater than two right. For _a f_, and _c g_, are not more parallel

than _f b_, and _g d_. Hence, if the line which falls upon _a f_, _c

g_, makes the internal angles greater than two right, that also which

falls upon _f b_, _g d_, will make the internal greater than two right.

But they are also less than two right (since the four, _a f g_, _c g

f_, _b f g_, _d g f_, are equal to four right) which is impossible.

In like manner we may plainly shew, that the right line which falls

upon parallels, does not make the angles internal, and at the same

parts, less than two right. But if it makes them neither greater nor

less than two right, it remains that the incident line must make the

angles internal, and at the same parts equal to two right. This then

being previously shewn, the thing proposed, is doubtless demonstrated.

For I say, that if a right line falling upon two right lines, makes

the angles situated internally, and at the same parts, less than two

right, if those right lines are produced they will coincide at those

parts in which the angles less than two right are situated. For let

them not coincide. But if they are non-coincident at those parts in

which the angles less than two right are situated, much more will they

be non-coincident at the other parts, in which the angles greater than

two right are situated. Hence, the right lines will be non-coincident

at both parts; and if this be true, they will be parallel. But it was

shewn that the right line which falls on parallels, makes the angles

internal and at the same parts equal to two right. The same, therefore,

are both equal to, and less than two right, which is impossible.

Ptolemy having previously shewn this, and proceeding to the thing

proposed, wishes to add something more accurate, and to shew that if

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

and at the same parts, less than two right, the lines are not only

coincident as has been shewn, but likewise that their coincidence takes

place at those parts, in which the angles less than two right, and not

at those in which the angles greater than two right are situated. For

let there be two right lines _a b_, _c d_, and let a right line _e f g

h_, falling upon them make the angles _a f g_, and _c g f_, less than

two right. The remainders, therefore, are greater than two right; and

thus it is shewn that the right lines coincide. But if they coincide,

they will either coincide at the points _a_ and _c_, or at the points

_b_ and _d_. Let them coincide at the points _b_ and _d_ in the point

_k_. Because, therefore, the angles _a f g_, and _c g f_, are less than

two right, but the angles _a f g_, _b f g_, are equal to two right,

by taking away the common angle _a f g_, the angle _c g f_, will be

less than the angle _b f g_. The external angle, therefore, of the

triangle _g f k_, is less than the internal and opposite angle, which

is impossible. Hence then, they do not coincide at these parts. But

they do coincide; and consequently they will be coincident at the other

parts, in which the angles less than two right are situated. And thus

far Ptolemy.

[Illustration]

But it is necessary to scrutinize this demonstration, lest perhaps

there should be any perverse and captious reasoning in the assumed

hypotheses, in those, I say, in which he affirms, that a right line

cutting non-coincident right lines, by forming four internal angles,

forms the angles at the same parts on each side, either equal to two

right, or greater, or less than two right. For the division is not

perfect; since nothing hinders our calling those lines non-coincident,

which are produced from angles less than two right, denominating,

indeed, the two angles at the same parts, greater than two right, but

the two at the remaining parts less than two right and not admitting in

these, one and the same proportion. But the division being imperfect,

the thing proposed is by no means demonstrated. Besides this, also,

is not to be passed over in silence against his demonstration, that

he does not essentially shew that which is impossible. For it is not

because a certain right line cutting parallels, makes the angles at

the same parts on each side, greater or less than two right, that an

absurdity on this account follows these hypotheses. Nevertheless,

because the four angles within the lines which are cut, are equal to

four right, on this account each of these hypotheses is impossible;

since, if parallel right lines are not assumed, yet, when the same

hypotheses are assumed, the same consequences will be the result. And

such are our animadversions against the demonstration of Ptolemy: for

the imbecility of his demonstration appears from what has been said.

[Illustration]

Let us now consider those, who say it is impossible that lines

produced from angles less than two right, should coincide. For when

they have assumed two right lines _a b_, _c d_, and a right line _a c_,

falling upon them, and making the two internal angles less than two

right, they say it is possible that the right lines _a b_, _c d_, may

be shewn to be non-coincident. For let _a c_ be bisected in _e_, and

cut off from _a b_, a part _a f_, equal to _a e_: but from _c d_, a

part _c g_, equal to _e c_. It is manifest, therefore, that the right

lines _a f_, _c g_, will not coincide in the points _f_ and _g_. For

if they coincide, these two in the triangle will be equal to _a c_,

which is impossible. Let again _f g_ be connected, and bisected in _h_,

and cut off equal parts. These, therefore, will not coincide on the

same account, and this will be the case, in infinitum, by connecting

the non-coincident points; and bisecting the connecting line, and

by cutting from the right lines, lines equal to the halves of the

connecting lines; for by this means they say, that the right lines _a

b_, _c d_, will never coincide. To such as these we reply, that they

indeed affirm that which is true, but not so much as they imagine. For

it is not true that the point of coincidence is simply determined by

this means, nor is it true that the lines by no means coincide. Thus,

when the angles _b a c_, and _d c a_, are determined, the lines _a b_,

and _c d_, will not coincide in the points _f_ and _g_, yet nothing

hinders their coinciding in the points _k_ and _l_, though _f k_ and

_g l_ should be equal to _f h_, and _h g_. For when _a k_ and _c l_

coincide, the angles _k f h_, _l g h_, will not remain the same, and

a certain part of the right line _f g_, will be left external to the

right lines _a k_ and _c l_; and so again the two lines _f k_, and

_g l_, are so much greater than the base, as the interior parts of

the right line _f g_, which they intercept. Besides this also is to

be said to such as affirm the non-coincidence of lines extended from

angles less than two right, that they destroy what they are unwilling

to destroy. For let the same description be given. Whether, therefore,

is it possible, or impossible to connect a right line from the point

_a_, to the point _g_? For if it be impossible, besides destroying the

fifth petition, they also destroy that which says, _that a right line

may be drawn from every point to every point_: but if possible let it

be connected. Because, therefore, the angles _f a c_, _g c a_, are less

than two right, it is manifest that the angles also, _g a c_, _g c a_,

are much less than two right. The right lines, therefore, _a g_, _c g_,

will coincide in the point _g_, being produced from angles less than

two right. Hence, it is not possible to affirm indeterminately, that

lines produced from angles less than two right, will not coincide. It

is however manifest, that some right lines produced from angles less

than two right will coincide, though the present discourse seems to

investigate this in all. For it may be said, that when the diminution

of two right lines is indefinite, the lines will remain non-coincident

according to such a diminution: but will coincide according to another

less than this. But he who desires to behold a demonstration of this

affair, must be informed that it is requisite for this purpose to

pre-assume such an axiom as is employed by Aristotle[30] in proving the

world to be finite, viz. _If from one point two right lines forming an

angle are produced in infinitum, the distance of the lines infinitely

produced will exceed every finite magnitude._ For he shews that when

infinite right lines are produced from the centre to the circumference,

the interval also contained between them will be infinite: since, if

it be only finite, it is possible that the distance may be increased;

and on this account the right lines will not be infinite. Right lines,

therefore, infinitely produced, are distant from each other by an

interval greater than every finite magnitude.

[Illustration]

This being pre-supposed, I say that if any right line cuts the one of

parallel right lines, it will also cut the other. For let _a b_ and _c

d_ be parallels, and let the right line _e f g_ cut _a b_. I say that

it will also cut _c d_. For since there are two right lines, which are

produced infinitely from the point _f_, viz. _b f_, and _f g_, they

shall have a distance greater than every magnitude. Hence, they shall

exceed the quantity of the interval contained between the parallel

lines. Since, therefore, their distance from each other is greater

than that of the parallels, _f g_ shall cut _c d_. But this being

demonstrated, we can exhibit the thing proposed in a consequent order.

For let there be two right lines _a b_, _c d_, and let a right line

_e f_, fall upon them, making the angles _b e f_, _d f e_, less than

two right. I say that the right lines will coincide in those parts, in

which the angles less than two right are situated. For since the angles

_b e f_, _d f e_, are less than two right, let the angle _h e b_, be

equal to the excess of two right angles above these angles, and produce

_h e_, to the point _k_. Because, therefore, a right line _e f_, falls

upon the right lines _h k_, _c d_, and makes the internal angles equal

to two right, viz. the angles _h e f_, _d f e_, the right lines _h k_,

_c d_, are parallel; and _a b_ cuts _k h_. It will therefore also cut

_c d_, by the assumption previously exhibited. Hence, the right lines

_a b_, _c d_, will coincide in those parts, in which the angles less

than two right are situated. And on this account the thing proposed, is

evinced[31].

[Illustration]

PROPOSITION XXX. THEOREM XXI.

Right lines parallel to the same right line, are parallel to

each other.

The geometrician in these discourses which are conversant with

relation, is accustomed to shew identity permeating through all

quantities, having the same relation to the _same_. Thus among the

axioms also he says, _things equal to the same, are equal to each

other_: and in the following books he says, _things similar to

the same, are similar to each other_, and _things having the same

proportion to the same, have the same proportion to each other_. After

this manner, therefore, he now also demonstrates, _that right lines

parallel to the same, are parallel to each other_. But it happens

that this is not true in all respects. For quantities double of the

same, are not also double of each other: nor are those which are

sesquialter of the same, sesquialter likewise to each other, but it

appears to take place in those alone, which are univocally converted

in equality, similitude, identity, and parallel position. For that

which is parallel to a parallel, is itself also parallel. As that which

is equal to an equal, is itself equal; and that which is similar to

a similar, is itself similar. For the relation of parallels to each

other, is similitude of position. He affirms, therefore, and shews,

in the present theorem, that lines parallel to the same, are entirely

so related, that they are also parallel to each other. And he also

exhibits the parallels with an external position, and likewise a

medium, to which these have a similar relation, that what he asserts

may become manifest from a common conception. For if they coincide with

each other on either side, and coincide with that which is situated in

the middle, they will no longer be parallel to it.

[Illustration]

But it is possible that he who changes the position may shew the

same thing, and by the same methods which the geometrician employs in

exhibiting his proposition. For instance, he assumes both _c d_, and

_e f_, parallel to _a b_, both of them situated above, and _a b_ being

beneath, and not in the middle. For a right line _h k l_, falling upon

them, makes each of the angles _h k d_, _k l f_, equal to _a h k_,

because they are alternate; and on this account it makes the angles _h

k d_, _k l f_, equal to each other. The right lines, therefore, _c d_,

_e f_, are parallel. But if any one should say that _a h_, _h b_, are

parallel to _c d_, and are therefore parallel to each other, we reply

that _a h_, _h b_, are parts of one parallel, and are not two parallel

lines. For parallels are conceived to be infinitely produced, but _a

h_, when produced, falls upon _h b_. It is therefore the same with _h

b_, and not a different line. Hence, all the parts of a parallel, are

parallel both to the right line, to which the whole was parallel, and

to its parts. As for example, _a h_ is as well parallel to _k d_, as

_h b_, to _c k_. For if they are infinitely produced, they will never

coincide. And these remarks must be considered as not foreign from

the purpose, both on account of sophistical importunities, and the

juvenile habits of mathematical auditors. For the vulgar rejoice to

find captious reasonings of this kind, and to procure vain molestation

to the possessors of science. But it is not requisite to convert the

present theorem, and to shew that lines parallel to each other, are

also parallel to the same. For if we again suppose one line parallel to

some other, the remainder also of these shall be parallel to it, and

they will be parallel to the same, and we shall again return to the

same proposition.

PROPOSITION XXXI. PROBLEM X.

Through a given point to draw a right line parallel to a

given right line.

It is requisite that we should not only learn the essential accidents

of parallels, in the discourses of the elementary institutor, but

also that we should relate their origin, and know how one right line

becomes parallel to another: for origin every where renders the essence

of subjects more known to us. And this the institutor of the Elements

effects by the present problem. For having received a point and a

right line, he draws through that point a line parallel to a right

line. But we ought to pre-assume as necessary, that the point should

entirely be placed external to the right line: for we must not place it

in the right line, because it is said, _through a given point_; since

no other, besides the given line, can be that which is drawn parallel

through the point. Since, therefore, the point and the right line is

divided, it indicates that the point is to be received external to

the right line, which he manifests in a perpendicular by addition,

commanding, _upon a given infinite right line, and from a given point

which is not in it, to let fall a perpendicular_. One thing, therefore,

which is common to both these problems, is, the external position of

the point: but the other, that from the same point two perpendiculars

cannot be let fall to the same right line, and that through the same

point, two lines cannot be drawn parallel to the same right line.

Hence, the institutor of the Elements commands in the singular number

_to draw a right line_, in the former problem, _a perpendicular_,

but in the present _a parallel_. And, _that_ indeed, has been shewn,

but _this_ is manifest, from what is previously demonstrated. For if

through the same point two parallels are drawn to the same right line,

they would be parallel to each other, and coincide in the given point,

which is impossible. But it is requisite to observe the differences

of these two propositions, _from a given point_, and _through a given

point_. For sometimes the point is the beginning of the right line

which is drawn, and on this account the deduction is made from it: but

sometimes the point is in the drawn right line, and on this account the

drawing is made through the point. For the particle _through_, was not

asserted, because the right line cuts a given point, but because it

coincides with it, and terminates its own interval, in respect of that

right line, by the distance of the point and the right line. Since as

much as the given point is distant from the given right line, so much

also is the interval of the parallel between itself and the right line.

PROPOSITION XXXII. THEOREM XXII.

One side of every triangle being produced, the external angle of

the triangle is equal to the two internal and opposite angles;

and the three internal angles of a triangle are equal to two

right angles.

[Illustration]

As much as was deficient in the sixteenth and seventeenth theorem,

so much Euclid adds in the present. For we not only learn by this

theorem that the external angle of a triangle is greater than either

of the internal and opposite angles, but likewise how much it is

greater; since as it is equal to both, it is greater than either of the

remaining angles. Nor do we alone know from this theorem, that any two

angles of a triangle, are less than two right, but by how much they

are less: for they are deficient by the remaining third. The former,

therefore, were more indefinite theorems: but this brings with it, on

both sides, a boundary to science. We must not, however, call them on

his account superfluous: for they are of the greatest utility in many

demonstrations; and the present is proved by their assistance. And

besides this, it is necessary that our knowledge, proceeding from the

imperfect to the perfect, should pass from indeterminate apprehensions,

to determinate and certain propositions. But the institutor of the

Elements, by drawing a parallel externally, exhibits each of the

objects of investigation. It is, however, possible that the same thing

may be shewn without drawing the parallel externally; and this, by only

changing the order of the things exhibited. For Euclid first shews,

that the external angle is equal to the internal and opposite, and

from this he proves the remainder. But we shall demonstrate this by a

contrary mode of proceeding. Let there be then a triangle _a b c_, and

let the side _b c_ be produced to the point _e_. Then take a point _f_

in _b c_, and connect _a f_, and through the point _f_, let _f d_ be

drawn parallel to _a b_. Because, therefore, _f d_ is parallel to _a

b_, and a right line _a f_, falls upon these parallels, as also a right

line _b c_, hence, the alternate angles are equal, and the external is

equal to the internal angle. The whole, therefore, _a f c_, is equal

to _f a b_, added to _a b f_. In like manner we may shew by drawing a

parallel, that the angle _a f b_, is equal to the angles _f a c_, _a c

f_. The two, therefore, _a f b_, _a f c_, are equal to the three angles

of the triangle: and hence, the three angles of a triangle are equal to

two right, viz. to _a f b_, added to _a f c_. But _a c f_, _a c e_, are

also equal to two right angles. Let, therefore, the common angle _a c

f_, be taken away; and then the remaining external angle will be equal

to the internal and opposite angles. And after this manner may the

present theorem be exhibited.

[Illustration]

But Eudemus, the Peripatetic, ascribes the invention of this theorem

to the Pythagoreans, I mean that every triangle has its internal angles

equal to two right, and says that they demonstrate it in the following

manner. Let there be a triangle _a b c_, and let there be drawn through

the point _a_, a line _d e_, parallel to _b c_. Because, therefore, the

right lines _d e_, _b c_, are parallel, the alternate angles are equal.

Hence, the angle _d a b_, is equal to the angle _a b c_; and the angle

_e a c_, to the angle _a c b_. Let the common angle _b a c_, be added.

The angles, therefore, _d a b_, _b a c_, _c a e_, that is, the angles

_d a b_, _b a e_, and that is two right, are equal to the three angles

of the triangle. And such is the demonstration of the Pythagoreans.

[Illustration]

But it is here requisite to deliver such theorems as are converse

to the present theorem of the elementary institutor. For two are

converted to one, since this is a composite, both, according to _the

object of enquiry_, and the _datum_: for the _datum_ is two-fold, viz.

the triangle, and one of its sides produced; and in like manner _the

object of enquiry_. For one part says, that the external angle is equal

to the internal and opposite angles: but the other, that the three

internal angles are equal to two right. If therefore, we suppose that

the external is equal to the internal and opposite angles, we may shew

that one side is produced, and that the right line externally situated,

is in a direct position with one of the sides of the triangle: but if

the three internal angles are equal to two right, we may shew that

the given figure is a triangle. And so _the whole object of enquiry_,

is converse to _the whole datum_. Let there be then a triangle _a b

c_, and let the external angle _a c d_, be equal to the internal and

opposite angles, I say that the side _b c_, is produced to the point

_d_, and that _b c d_, is one right line. For since the angle _a c d_,

is equal to the internal and opposite angles, let the common angle _a

c b_ be added. The angles, therefore, _a c d_, _a c b_, are equal to

the three angles of the triangle _a b c_. But the three angles of the

triangle _a b c_, are equal to two right. And hence, the angles _a

c d_, _a c b_, are equal to two right. But if two right lines being

consequently placed, and not at the same parts to any right line, and

at a point in it, make the successive angles equal to two right, those

right lines shall be in a direct position to each other. The right

line, therefore, _b c_, is in a direct position to _c d_.

[Illustration]

Let there be again, a certain right lined figure _a b c_, having three

angles alone equal to two right, viz. _a_, _b_, and _c_, I say that the

figure is a triangle, and that _a c_, is one right line. For let the

right line _b d_ be connected. Because, therefore, the three angles

of each of the triangles _a b d_, _b d c_, are equal to two right,

of which the angles of the figure _a b c_, are equal to two right,

the remainders _a d b_, _c d b_, are equal to two right, and they are

placed about a right line _b d_. Hence _d c_, is in the same direction

with _d a_; and so the side _a c_, is one right line. In like manner we

may shew that the side _a b_, and the side _b c_, are each of them one

right line. And consequently the figure _a b c_, is a triangle. If then

a figure having internal angles equal to two right, is right-lined,

it is perfectly a triangle: but it does not follow that a figure is

a triangle merely because it has internal angles equal to two right.

For you will find a figure constructed from circumferences, having

its internal angles equal to two right. For let there be a quadrangle

_a b c d_, and upon the side _a b_, let a semicircle _a e b_, be

internally described: but upon the other sides, let the semicircles be

externally described, as _f_, _g_, _h_. The figure, therefore, which

is comprehended by the semicircles, has two angles _g a e_, _e b h_,

equal to two right, viz. to _c a b_, _d b a_. For this was shewn in

the petitions[32], and these angles alone are in this figure. There

is, therefore, a certain figure not a triangle, which has its internal

angles equal to two right. And thus much may suffice concerning

converse theorems.

[Illustration]

But as we have discovered that the three angles of every triangle are

equal to two right, we ought to determine a certain method, by which

we may find how many angles, of all other multangles, are equal to so

many right angles; as for instance, of a quadrangle, quinquangle, and

of all consequent multilateral figures. In the first place, therefore,

it must be known, that every right-lined figure may be resolved into

triangles, since a triangle is the principle of the constitution of

all things, which Plato also asserts in the Timæus, when he teaches us

that the rectitude of a plane basis is composed from triangles. But

every figure is resolved into triangles less in number, by the binary,

than its proper sides. If a quadrilateral figure, into two triangles:

if a figure of five sides, into three: if of six sides, into four.

For two triangles composed together, immediately form a quadrilateral

figure. But the number of composite triangles by which the first

constituted figure differs from its sides, is the measure of difference

to the rest. Hence, every multilateral figure possesses more sides,

by the binary, than the triangles into which it may be dissolved. But

every triangle has been shewn to contain angles equal to two right.

And hence, if the number of the angles be made double to that of the

composite triangles, it will afford a multitude of right angles, to

which the angles of every multangle will be equal. On this account

every quadrilateral figure has angles equal to four right, since it

is composed from two triangles: but every figure of five sides has

angles equal to six right; and after the same manner of the rest in a

consequent order. This one thing, therefore, is to be assumed from the

present theorem, concerning all multangular and right-lined figures.

But there is another consequent to this, which is summarily as

follows. In every right-lined figure, each of its sides being at the

same time produced, the angles externally constituted are equal to four

right. For it is requisite, indeed, that the successive right angles

should be double of the multitude of the sides; because, in each they

are constituted equal to two right. But the right angles equal to the

internal angles being taken away, the remaining external angles are

equal to four right. As for example, if the figure is a triangle,

while every one of its sides is produced, at the same time internal

and external angles are constructed equal to six right angles, of

which the internal angles are equal to two right, but the remaining

external angles to four right. But if the figure be quadrilateral,

the angles are in all eight, since they are double of the sides,

of which the internal are equal to four right, and the external to

the four remaining angles, and the consequences will be similar in

infinitum. But after these observations we may also collect, that by

this theorem every angle of an equilateral triangle is two thirds of a

right angle; but that an isosceles triangle, when the vertical angle

is right, has each of its remaining angles the half of one right, as

a semiquadrangle; and that a scalene triangle, when it is the half

of an equilateral triangle, formed by a perpendicular drawn from any

angle to its opposite side, has one angle right, but the other (which

likewise belonged to the equilateral triangle) two thirds of a right

angle, and the remainder by a necessary consequence, a third part of

a right angle. For it is requisite that the three should be equal to

two right. But I do not conceive that these remarks are foreign from

our purpose, since they prepare us for the doctrine of Timæus. This

also must be observed, that the possession of internal angles equal to

two right is inherent essentially, and answering to the predication

_according to what_, in a triangle. And on this account, Aristotle

in his Treatise on Demonstration[33], employs this as an example,

considering it _according to what_. As therefore _to be terminated_,

is essentially and primarily inherent in every figure, so likewise the

possession of internal angles equal to two right, is essentially and

primarily inherent in a triangle, though not in every figure. And the

truth of this theorem seems to present itself to us according to common

conceptions. For if we conceive a right line, and two right lines

standing on its extremities, and inclining to each other, so as to form

a triangle, we shall find that in proportion to their inclination they

diminish the right angles, which they form with the right line. Hence,

obtaining as much angular quantity, by their inclination at the vertex,

as they take away from the base, they necessarily form three angles

equal to two right.

PROPOSITION XXXIII. THEOREM XXIII.

The right lines which join equal and parallel right lines at the

same parts, are themselves also equal and parallel.

The present theorem is, as it were, the confine of the consideration

of parallels and parallelograms: for it seems to declare a certain

symptom of parallel right lines, and delivers the latent origin of

parallelograms. For a parallelogram is formed, as well from those equal

and parallel right lines, which are drawn in the beginning, as from

those which conjoin them, and which are in like manner shewn to be

equal and parallel. Hence, the proposition which immediately follows

the present, contemplates the properties essentially inherent in

these spaces, in a parallelogram as it were already constructed. And

these things are indeed manifest. But it is requisite to consider the

diligence which this preposition contains. In the first place, indeed,

that it is not sufficient, that the lines which are conjoined should

be equal: for the lines which connect equals, are not entirely equal,

unless they are also parallel. For a triangle being isosceles, and a

point being assumed in one of the equal sides, and through this a line

being drawn parallel to the basis, equal lines shall indeed conjoin

parallels to the basis, and the basis itself, yet these parallels shall

not also be equal; and the sides will not be parallel, because they

coincide at the vertex of the triangle.

[Illustration]

[Illustration]

In the second place, he considers that the subject right lines being

parallel, is not sufficient to constitute the equality of the lines

which conjoin them. For this is evident from the preceding construction

of the isosceles triangle; since the drawn right line, and the basis,

are parallel, and yet the lines which connect them are not parallel,

because they are parts of the sides of the isosceles triangle. The

parallel position, therefore, of the lines which are conjoined, is

requisite to the equality of the connecting lines: but the equality

of the latter is necessary to the parallel position of the former. On

this account the institutor of the Elements assumes each, in those

which are conjoined, for the purpose of exhibiting, that the connecting

lines are as well equal, as parallel to one another. But in the third

place, he intimates, that right lines being supposed both equal and

parallel, their connecting lines will not be universally equal and

parallel. For unless we make the conjunctions at the same parts as in

this case, the connecting lines cannot be parallel (since they will

cut each other), so they may be sometimes equal, and sometimes not.

For if you assume a quadrangle, or oblong, as _a b c d_, and connect

the right lines _a d_, _b c_, the diameters are indeed equal, but not

parallel, and they conjoin the equal and parallel opposite sides of the

aforesaid spaces. But if the figure be a Rhombus, or a Rhomboides, the

diameters of these, are not only non-parallels, but also unequal. For

since _a b_, is equal to _c d_, but _a c_ is common, and the angle _b

a c_, is unequal to the angle _a c d_, the bases also are unequal. The

institutor of the Elements, therefore, very properly considered, that

the lines which conjoin equal and parallel lines, ought to make the

conjunction at the same parts, lest _a c_, _b d_, being supposed equal

and parallel, we should assume _a d_, _b c_, as the connecting lines,

and not _a b_, and _c d_. For he shews that these latter are equal and

parallel: but that the former are, indeed, never parallel, but equal,

as we have observed in a quadrangle and oblong, but never in a rhombus

and rhomboides; as the opposite to this has been proved to be true,

because they are unequal, on account of the inequality of the angles

internal, and situated at the same parts.

PROPOSITION XXXIV. THEOREM XXIV.

The opposite sides and angles of parallelogrammic spaces are

equal to each other and they are bisected by the diameter.

[Illustration]

[Illustration]

As from the preceding theorem, he had assumed a parallelogram already

constructed, he now contemplates its primarily inherent properties, and

such things as express its peculiar constitution. But these are the

following: _that the sides and angles which are opposite, are equal,

and that the spaces themselves are bisected by the diameter_. For

that part of the proposition relates to the spaces, which says: _and

they are bisected by the diameter_. So that the area itself, is that

whole which is bisected, and not the angles through which the diameter

passes. These three properties then, are essentially inherent in

parallelograms, the equality of the opposite sides and angles, and the

bisection of the spaces by the diameter. And you may observe that the

properties of parallelograms are investigated from all these, viz. from

the sides, from the angles, and from the areas. But as there are four

kinds of parallelograms, which Euclid defines in the hypotheses[34],

viz. a quadrangle, oblong, rhombus, and rhomboides, it deserves

to be remarked, that if we divide these four into rectangles, and

non-rectangles, we shall find, that not only the diameters bisect these

spaces, but that the diameters themselves, are, indeed, in rectangles

equal, but in non-rectangles unequal, as was observed in the preceding

theorem. But if we divide them into equilateral, and non-equilateral,

we shall again find that in the equilateral figures, not only the

spaces are bisected by the diameters, but likewise the angles through

which they are drawn: but in non-equilaterals this is never the case.

For in a quadrangle, and a rhombus, the diameters bisect the angles,

and not the spaces only: but in an oblong, and a rhomboides, they alone

bisect the spaces. For let there be a quadrangle, or a rhombus, _g c

a b_, and a diameter _g b_. Because, therefore, the sides _g c_, _c

b_, are equal to the sides _g a_, _a b_ (for they are equilateral),

and the angles _g c b_, _g a b_, are equal (for they are opposite),

and the basis also is common, hence, all are equal to all; and on this

account the angles _c g a_, _a b c_, are bisected. Again, let there

be an oblong, or rhomboides given. If, therefore, the angle _b a c_,

and the angle _c d b_, is bisected by the diameter, but the angle _c

a d_, is equal to the angle _a d b_, the angle also, _b a d_, will be

equal to the angle _a d b_. Hence, the side also, _a b_, will be equal

to the side _b d_. But they are unequal; and consequently the angle

_b a c_, is not bisected by the diameter, nor its equal the angle _c

d b_. That I may therefore comprehend the whole in a few words, in a

quadrangle the diameters are equal, on account of the rectitude of the

angles, and the angles are bisected by the diameters, on account of the

equality of the sides, and the areas are bisected by the diagonal, on

account of the common property of parallelograms: but in an oblong, the

diameters are indeed equal, because it is a rectangle, but the angles

are not bisected by the diameters, because it is not equilateral,

though the division of spaces into equal parts, is also inherent in

this figure, so far as it is a parallelogram: but in a rhombus the

diameters are unequal, because it is not a rectangle, but the spaces

are not only bisected by these, because it is a parallelogram, but the

angles also, because it is equilateral; and in the remaining figure,

i.e. a rhomboides, the diameters are unequal, because it is not a

rectangle, and the angles are cut by these into unequal parts, because

it is not equilateral, and the spaces alone situated at each part of

the diagonals, are equal, because it is a parallelogram. And thus much

concerning observations of this kind, which exhibit the diversity found

in the four divisions of parallelograms.

But we must not pass over in silence, the artificial consequence

appearing in this theorem, that of theorems, some are universals, but

others non-universals. But we shall speak concerning each of these,

when we divide _the object of investigation_, which has, indeed, one

part universal, but the other non-universal. For though every theorem

may seem to be universal, and every thing exhibited by the elementary

institutor may appear to be of this kind (as in the present he may

not only seem to assert, that in all parallelograms universally, the

opposite sides and angles are equal, but likewise that each is bisected

by the diameter), yet we must say that some things are universally

exhibited, but others not universally. For it is customary to call the

_universal_ which affirms the truth concerning every thing of which it

is predicated, differently from _that universal_, comprehending all

things in which the same symptom is inherent. Thus it is universal,

that every isosceles triangle has three angles equal to two right,

because it is true of all isosceles triangles: and it is universal

that every triangle has three angles equal to two right, because it

comprehends all things, in which this is essentially inherent. On which

account we affirm that the possession of three angles equal to two

right, is to be primarily manifested of a triangle. According to this

signification, therefore, of theorems, calling some universal, but

others non-universal, we must affirm that the present theorem, has,

indeed, one of its objects of investigation universal, but the other

non-universal. For the possession of opposite sides and angles that are

equal, is a universal, since it is alone inherent in parallelograms:

but that the diameter bisects the space, is not universal, because it

does not comprehend all things in which this symptom is beheld; for

this is inherent in a circle and ellipsis. And it appears, indeed,

that primary conceptions of such like concerns, are more particular,

but that in their progress they comprehend the whole. For when the

ancients had contemplated that a diameter bisects an ellipsis, circle,

and parallelogram, they afterwards surveyed that which was common in

these. But we are deceived (says Aristotle[35]) when a non-universal

is exhibited as universal, because that common something in which the

symptom is primarily inherent, is nameless. For we cannot say what that

is, which is common to numbers and magnitudes, motions and sounds; and

it is likewise difficult to express what is common to an ellipsis,

circle, and parallelogram. For one of these figures is right-lined, but

the other circular, and the third mixt; and on this account we conceive

that he exhibits universally, who demonstrates that a diameter bisects

every parallelogram, because we do not at the same time perceive that

common something, on account of which, this is true. This then in

parallelograms, is not an universal of this kind, on account of the

aforesaid cause; but the proposition is universal, which asserts, that

every parallelogram has its opposite sides and angles equal. For if

any figure is supposed, having its opposite sides and angles equal, it

may be shewn to be a parallelogram. Thus let such a figure be _a b c

d_[36], and its diameter _a d_. Because, therefore, the sides _a b_, _b

d_, are equal to the sides _a c_, _c d_, and the angles comprehended

by them are equal, and the base common, all will be equal to all. The

angle, therefore, _b a d_, is equal to the angle _a d c_, and the

angle _a d b_, to the angle _c a d_. Hence, _a b_, is parallel to _c

d_, and _a c_ to _b d_. And on this account the figure _a b c d_, is a

parallelogram. And thus much may suffice for observations of this kind.

[Illustration]

But the institutor of the Elements seems to have composed the name

of parallelograms, by taking an occasion from the preceding theorem.

For when he had shewn that right lines, which conjoin equal and

parallel right lines at the same parts, were themselves also equal and

parallel, it is evident that he pronounces as well the opposite sides

which conjoin, as those which are conjoined, to be parallel: but that

he very properly calls the figure which is contained by parallels,

a parallelogram, in the same manner as he denominates that which is

comprehended by right lines rectilineal. And it is evident that the

institutor of the Elements places a parallelogram among quadrilateral

figures. But it is worthy our observation and enquiry, whether every

right-lined figure, which is composed from equal sides, since it is

equilateral and equiangular, is to be called a parallelogram. For a

figure of this kind also, has its opposite sides equal and parallel,

as likewise the opposite angles equal. As for example, a sexangle, and

an octangle, and a decangle. Thus, if you conceive a sexangle _a b c

d e f_, and connect a right line _a c_, you may shew that _a f_ is

parallel to _c d_. For the angle at the point _b_, is one right, and

the third part of a right angle; and this is true of every angle of a

sexangle, since it is equiangular. Besides the side _a b_, is equal

to the side _b c_, for it is placed equilateral. Each of the angles,

therefore, _b a c_, _b c a_, is a third part of a right angle. Hence,

the angles _f a c_, _a c d_, are right angles. And on this account

_a f_, is parallel to _c d_. In like manner we may shew that the

other opposite sides are parallel, and the same may be evinced in an

octangle, and in the remaining figures of this kind. If, therefore,

that is a parallelogram which is comprehended by parallels oppositely

situated, a parallelogram will likewise subsist among non-quadrilateral

figures. But it appears that with the institutor of the Elements a

parallelogram is quadrilateral. And this is particularly perspicuous

in that theorem, in which he says, _that a parallelogram which has the

same base with a triangle, and is between the same parallels, is double

of the triangle_: for this is alone true in quadrilateral figures.

PROPOSITION XXXV. THEOREM XXV.

Parallelograms which are upon the same base, and between

the same parallels, are equal to each other.

As we have said that of theorems, some are universal, but others

particular, and as dividing these we have subjoined, that some are

also simple, but others composite, and have shewn the nature of each,

so according to another distinction, we assert that some of these are

local, but others non-local. But I call those local, to which the same

symptom happens in a certain place; and I denominate the place of a

line or a superficies, that situation, which produces one and the same

symptom. For of local theorems some are constructed in lines, but

others in superficies. And because of lines, some are plane, but others

solid, the plane being those of which there is a simple conception in

a plane, as of a right line: but the solid those whose origin appears

from a certain section of a solid figure, as of cylindric, spiric,

and conic lines, I should say, that of the local theorems which are

constructed in lines, some have a plane, but others a solid place. The

present theorem, therefore, is both local, and local in lines, and a

plane. For the whole space which lies between the parallels, is the

place of the parallelograms constructed upon the same base; and which

the institutor of the Elements shews to be equal to each other. But

of those local theorems which are called solid, let the following be

an example[37]. _The parallelograms which are inscribed within the

asymptotes and the hyperbola, are equal_: for it is evident that the

hyperbola is a solid line.

But Chrysippus, as we are informed by Geminus, assimilates theorems of

this kind to ideas. For as ideas comprehend the origin of infinites,

in terminated limits, so in these also there is a comprehension of

infinites, in terminated places, and by this boundary equality appears,

since the altitude of the parallels remaining the same, if infinite

parallelograms are conceived upon the same base, they may all be

shewn to be equal to each other. The present, therefore, is with the

institutor of the Elements, the first local theorem. And he appears,

when, agreeable to an elementary mode, he had distinguished theorems by

a variety, according to all possible divisions, with great propriety

not to have omitted, considering their idea of this kind. Nevertheless,

as his discourse, for the present, is concerning right lines, he

delivers local plane theorems in right lines: but in the third book,

as he treats concerning things which may be contemplated of circles,

and their symptoms, he likewise teaches the particulars, which are

constructed in circumferences belonging to local, and at the same time,

plane theorems. And such, among these, is the theorem, which says,

_that angles in the same segment, are equal to one another_. Also

this which asserts, _that the angles in a semicircle are right_.

For if infinite angles are constructed in a circumference, the same

base remaining, they are all shewn to be equal; but if that which is

comprehended by the base and the circumference, is a semicircle, they

are all shewn to be right. And these, indeed, correspond in proportion

to triangles and parallelograms upon the same base, and between the

same parallels. And such is the species of theorems called local, by

the ancient mathematicians.

[Illustration]

But perhaps it may seem perfectly worthy of admiration, to such as

are unskilled in contemplations of this kind, that parallelograms

constructed upon the same base, and between the same parallels, should

be equal to each other. For it may be asked, how is this possible,

since the longitude of the spaces, constructed on the same base,

increases in infinitum? Since as much as we produce the parallels, by

so much we may also increase the longitudes of the parallelograms. But

some one may not improperly enquire how, while this takes place, the

equality of the spaces remains. For if the breadth is the same (since

the base is one), but the length is greater, will not the space also

be greater? The present theorem, therefore, and that which follows

concerning triangles, are among the number of mathematical theorems,

which are denominated admirable. For mathematicians in theorems, as

the Stoics in arguments, have established a _place_, which is called

admirable, and they place the present among theorems of this kind.

The vulgar, therefore, are immediately astonished, when they hear

that the multiplication of length does not destroy the equality of

spaces on the same base. We must nevertheless assert, that equality

and inequality possess the greatest power in increasing or diminishing

the spaces of angles. For in proportion as we make angles unequal,

in such proportion we diminish the space, if the length and breadth

remain the same. Hence, the increase of length is necessary, that we

may preserve equality. Thus, for example, let there be a parallelogram

_a b c d_, and let the side _a c_ be produced in infinitum, and let

it be a right-angled parallelogram; and lastly, on the base _b d_,

construct another parallelogram _b e f d_. That the length, therefore,

is increased is evident: for the side _b e_, is greater than the

side _a b_, since the angle at the point _a_, is right. But this

necessarily takes place, as the angles of the parallelogram _b e f

d_, are unequal, and some of them are acute, but others obtuse: and

this happens, because the side _b e_, approaches after a manner to

the side _b d_, and contracts the space. For let _b g_ be taken equal

to _a b_, and through _g_, draw _g h_ parallel to _b d_. The length,

therefore, of the parallelogram _b d g h_, is equal to the length of

the parallelogram _a b c d_, and the breadth is the same, and yet one

space is less than the other; for it is less than _b e f d_. Hence, the

inequality of angles diminishes the area, but the increment of length

adding as much as the inequality of angles takes away, preserves the

equality of the spaces. But the boundary of the increase of length,

is the place of the parallel lines. For when both the parallelograms

are rectangular, and have an equal ambit, the quadrangle is shewn to

be greater than the oblong[38]: but when they are both equilateral,

and have consequently an equal ambit, that which is rectangular, is

shewn to be greater than that which is non-rectangular[39]. For the

rectitude of angles, and the equality of sides, possesses universal

power in the augmentation of spaces. It is on this account that a

quadrangle is the greatest of all figures with an equal ambit, and a

rhomboides the least. And these observations we shall demonstrate in

another place[40]: for they more properly belong to the hypotheses of

the second book.

[Illustration]

[Illustration]

But with respect to the present theorem, it is requisite to know,

that when Euclid calls parallelograms equal, he means the spaces, and

not the sides: for he now discourses of areas. And we must likewise

observe, that he first mentions trapeziums in the demonstration of

this theorem: from whence also it is manifest, that he does not

improperly teach us concerning a trapezium, in the definitions, when

he informs us that it is indeed of a quadrilateral species, but is not

a parallelogram. For the figure which has not its opposite sides and

angles equal, falls from the order of parallelograms. The institutor

of the Elements, therefore, as he had chosen a more difficult case,

demonstrates the thing proposed. But if any one should say, let the

parallelograms _a b c d_, and _b d c e_, be upon the same base _d

b_, so that the side _c d_ may be the diameter of the parallelogram

_a b_, we can shew that according to this position they are equal.

For the triangle _b c d_, is the half of each parallelogram: because

_c d_ is the diameter of _a b_, but _c b_ of _d e_; and diameters

bisect parallelograms. Hence, _a b_ is equal to the parallelogram

_d e_. Again, if any one should suppose that the side _a c_, of

the parallelogram _a b_, is cut by the side _d c_, and that the

parallelograms are situated as _a d b e_, _b d c f_, we can shew that

these also are equal. For since the side _a e_, is equal to the side _c

f_ (each because opposite being equal to _d b_), let the common right

line _c e_ be taken away. Hence, _a c_ is equal to _e f_. But _a d_,

also, is equal to _e b_, and the angle _c a d_, to the angle _f e b_.

For _a d_ is parallel to _e b_; and hence, the base _c d_, is equal to

the base _f b_, and the whole triangle _a d c_, is equal to the whole

triangle _e b f_. Let the common trapezium _c b_, be added. The whole,

therefore, _a b_, is not unequal to the whole _d f_. And here you

may observe that these are the only three cases. For the side _d c_,

either cuts the side _e b_, according to the position of the elementary

institutor; or it falls on the point _c_, as in the penultimate

description: or it cuts the line _a e_, according to the present

supposition. And thus the theorem is shewn to be true according to all

its cases. Lastly, as there is a two-fold difference of trapeziums,

and one kind has neither of its opposite sides parallel, but the other

has one side parallel to one, this latter species of trapeziums is

alone employed by the geometrician throughout the elements, and in the

present description: for _c e_ is parallel to _d b_.

PROPOSITION XXXVI. THEOREM XXVI.

Parallelograms which are upon equal bases, and between

the same parallels, are equal to each other.

[Illustration]

The preceding theorem assumed, indeed, the same bases, but this

receives them equal, and different from each other. But it is common to

both, to suppose the parallelograms between the same parallels. It is

requisite, therefore, that they should neither fall within, nor without

their subject parallel lines. For parallelograms are said to be between

the same parallels, when their bases and opposite sides are adapted to

the same parallels. As to the rest, the institutor of the Elements,

as he had assumed the bases entirely separate, exhibits the theorem.

But nothing hinders our receiving them with this hypothesis, so that

they may have a common part. For let _a b_, _c d_, be parallelograms

upon equal bases _e b_, _f d_, having a common part, and constructed

between the same parallels, I say that they are equal. Let the lines

_e c_, _b g_, be connected. Because, therefore, _e f_, is equal to _b

d_ (for the base _e b_, was supposed equal to the base _f d_), but the

side _c f_, is equal to the side _d g_, and the angle _c f e_, is equal

to the angle _g d b_, and hence, _c e_ is equal to _b g_. But it is

also parallel to it. Hence, _c b_ is a parallelogram, and has the same

base with each of the parallelograms _a b_, _c d_, and is between the

same parallels. The parallelogram, therefore, _a b_, is equal to the

parallelogram _c d_.

[Illustration]

But if any one should suppose that the bases of the parallelograms

have neither a common part, nor are separate from each other, but

(which is the only remaining hypothesis) that they touch each other in

one point, as in the parallelograms _a e_, _e d_, we must say that the

base _b e_ is equal to the base _e f_, and to the side _c d_. Hence,

also, the right line _c b_, is equal to the right line _d e_, and is

parallel to it. For the lines which join equal and parallel lines, are

themselves also equal and parallel. Hence, _b d_ is a parallelogram,

and is upon the same base, and between the same parallels, with the

parallelograms _c b_, _d e_. The parallelograms, therefore, _c b_, _d

e_, are equal. But according to the first conception of a theorem, we

may divide the constructions by asserting that the bases have either a

common part, or touch each other, or are distant from each other. It is

however possible, that though they may touch each other, as _b e_, _e

f_, yet the whole parallelogram _d e_ may be supposed external to the

side _c e_; or one side of the parallelogram _c f_, may be the diameter

of the parallelogram _a e_; or the side _c e_, may cut the side _a c_;

or the side _a c_, being produced beyond _a_, the side _c e_, may fall

as the diameter of the parallelogram increased towards _a_, when the

side _d f_ becomes the same as a line drawn from _a_ to _f_, or the

side _c e_, may cut the side _a c_, produced beyond _a_; or the side _a

c_, may be still farther produced beyond _a_, so that the side _c e_

may fall beyond the point, to which _a c_ was extended in the preceding

case, and the side _d f_, may cut the line produced beyond _a_[41]. * *

*

PROPOSITION XXXVII. THEOREM XXVII.

Triangles which are upon the same base, and between

the same parallels, are equal to each other.

_The beginning of this Commentary is wanting._

* * * * *

* * for those being equal, the spaces are unequal; and when these

are unequal, those are shewn to be equal. And this is the case with

Chorographers, when they reason concerning the magnitudes of cities,

from their ambits. But formerly, certain persons deceived their

partners, in the distribution of their possessions, deluding them by

an excess of ambit, so as to make them believe that they received

a greater portion of land, when they received a greater ambit; and

that they were gainers, by changing spaces into areas of less ambit.

[42]Thus two isosceles triangles being proposed, one of which has

each of its equal sides, containing five parts, but the base six:

and the other has each of its equal sides five parts, but the base

eight; and let these parts be, for instance, cubits, or digits, these

triangles will very much deceive the ignorant in their choice. For

the ambit of the one is eighteen, and of the other sixteen measures.

But a geometrician is not ignorant that the spaces are equal, though

the ambits are unequal; since the area of each is twelve measures.

For if you draw a perpendicular from the vertex, you will bisect the

bases, and cause the half of the one to be three but of the other

four measures: but the perpendicular on the contrary, will be there

equal to four, but here equal to three; since it is requisite that

the square from the quinary, should be equal to the squares from

the perpendicular, and the half of the base. But if the base of the

one is equal to three, the perpendicular must be four; and if the

base of the other is equal to four, its perpendicular must be three.

When, therefore, you have multiplied the half of the base with the

perpendicular, you will have a space equal to the triangle: but this

is the same in each, whether you multiply the quaternary with the

ternary, or the ternary with the quaternary. And we have made these

observations for the purpose of shewing that the equality of spaces

is not to be entirely received from the ambits. Nor should we wonder,

that though triangles upon the same base, may be infinitely increased

between the same parallels, according to the remaining sides, yet the

equality of the spaces immutably remains. But those triangles are said

to be between the same parallels, which have their bases upon one of

the parallel lines, and fix their vertices on the remainder; and whose

vertices being connected, form one right line, parallel to the bases on

the same right line.

PROPOSITION XXXVIII. THEOREM XXVII.

Triangles which are upon equal bases, and between the

same parallels, are equal to each other.

The present theorem also is local, because it corresponds in

proportion with parallelograms, and supposes the situation of triangles

upon equal bases. But Euclid seems, to me, to have delivered one

demonstration by the first proposition of the sixth book of these

four theorems, two of which are exhibited in parallelograms, and two

in triangles: and two of which are on the same base, and the other

two on equal bases. But that Euclid has performed this is unknown to

the vulgar. For after he had shewn that triangles and parallelograms,

which are under the same altitude, have the same proportion to each

other as their bases, nothing demonstrates all these four theorems

more universally, from proportion, than this theorem: since to possess

the same altitude, is nothing else than being constituted between

the same parallels. For all figures between the same parallels, are

under the same altitude, and the contrary: since the altitude is the

perpendicular, which extends itself from one parallel to the rest. In

that proposition, therefore, it is shewn by proportion, that triangles

and parallelograms, under the same altitude, that is, situated between

the same parallels, are to each other as their bases, and so when the

bases are equal, the spaces are equal; and when those are double,

these will be double; and when the bases have any other proportion,

the spaces also will have to each other the same proportion. But for

the present, because it is not proper that he should use proportion,

who has not yet explained its nature, he is content with equality and

identity alone: for the identity of bases is collected from equality.

Hence, these four theorems are comprehended in that one; not only

because he shews by one demonstration, whatever are contained in

these four, but likewise, because he adds what was wanting to their

perfection, viz. identity of proportion, though the bases are unequal.

But that this theorem, also, has many cases, and that it is possible

that the bases of the triangles may be assumed, either having the same

part as in parallelograms; or possessing no common part, but touching

each other according to one point; or entirely separate, so that a line

may intervene between them, is manifest, even to such as are endued

with slender capacities. And this too is evident, that according to all

cases, however the bases or vertices may be situated, the same method

of proceeding must be adopted as in parallelograms; viz. parallels to

the sides must be drawn, and produced both ways, and the equality of

the triangles exhibited.

PROPOSITION XXXIX. THEOREM XXXIX.

Equal triangles, which are upon the same base, and at the

same parts, are between the same parallels.

When it was proposed to exhibit equality to us, then it was requisite

to make four theorems, receiving two in parallelograms, but the other

two in triangles, situated either upon the same, or upon equal bases.

But now by conversion, we neglect the theorems which are converse in

parallelograms, and esteem such as are converse in triangles worthy of

relation. And the reason of this is, because the mode of demonstration

in parallelograms, is the same indifferently, by a deduction to an

impossibility, and the construction is similar. But we are content when

we have exhibited the way in more simple figures, I mean triangles,

to leave to the more curious the same mode of reasoning in the rest:

since it is easy, at the same time, to perceive that there is the same

method in these. For when we assume equal parallelograms, upon the same

base, or upon equal bases, we must say that they are also between the

same parallels. For if they are not, either one of them falls within,

when the parallels which are in the other are produced; or without.

But which ever case is assumed, when we receive it and its parallels,

we may exhibit the same consequences as in triangles, I mean that the

whole will be equal to its part: but this is impossible. It is however

manifest, that the institutor of the Elements very properly adds the

particle, _and at the same parts_. For it is possible that equal

triangles, may be assumed upon the same base, one, indeed, at these

parts, but the other at different parts, and yet these will not be

entirely between the same parallels: for neither will they be contained

under the same altitude. And on this account he added the particle.

But since a parallel may be drawn in a two-fold respect, according

to an absurd hypothesis, i.e. either within or without, Euclid draws

it within: but we can exhibit the same consequences, by drawing it

without. For let the equal triangles _a b c_, _d b c_, be upon base,

and at the same parts, I say that they are between the same parallels,

and that the right line connected at their vertices, is parallel to

the base. Let the right line _a d_ be connected. But if this is not

parallel, let the line, external to this, i.e. _a e_ be parallel,

and let _b d_ be produced to the point _e_, and connect _e c_. The

triangle, therefore, _a b c_, is equal to the triangle _e b c_, the

whole to the part. But this is impossible; and hence, the parallel line

does not fall external to _a d_. But it is shewn by the institutor of

the Elements, that neither does it fall within: and hence _a d_, is

parallel to _b c_. Hence too, equal triangles, which are at the same

parts, and upon the same base are parallel to each other. And thus the

remaining part of the deduction to an impossibility is demonstrated.

But it is worthy of observation, that since the conversion of theorems

is triple (for either the whole is converted to the whole, as we have

noticed, in the eighteenth and nineteenth theorems; or the whole to

the part, as the sixth and fifth; or the part to the part, as the

eighth and the fourth: for the whole is not a _datum_, in the one,

and _an object of investigation_ in the other: nor is the _object of

investigation_, _a datum_, but a part) these triangular theorems appear

to be of this kind. For, that the triangles are equal, is an object of

investigation in the preceding; but this is not a datum alone in these,

because it assumes, besides this, a part of that which was hypothesis

in those. For to stand upon the same, or upon equal bases, is a datum

in these, as well as in those, except that in these hypotheses he adds

something which was neither an object of investigation, nor a datum

in these; since the particle _at the same parts_, is over and above

extrinsically assumed.

[Illustration]

PROPOSITION XL. THEOREM XXX.

Equal triangles which are upon equal bases, and at the

same parts, are between the same parallels.

There is the same mode of conversion too in the present theorem,

and a similar demonstration; and that part of the deduction to an

impossibility, which is omitted by the institutor of the Elements,

is demonstrated after the same manner, and there is no occasion

for repetition. But since these three conditions are in the

aforesaid propositions, _situation upon equal, or on the same bases;

position between the same parallels; and equality of triangles and

parallelograms_, it is manifest that we may variously convert, by

always connecting two, and leaving one. For we either supposed the

bases the same, or equal, and triangles and parallelograms between

the same parallels, and thus we form four theorems; or we consider

the triangles and parallelograms equal, and the bases the same, or

equal, and thus we produce another four, two of which the elementary

institutor omits, viz. those which respect parallelograms, but the

other two relative to triangles, he exhibits; or lastly, when we have

assumed them equal, and between the same parallels, we prove the

remainder, that they are either upon the same, or upon equal bases, and

produce another four, which the institutor of the Elements entirely

neglects. For there is the same demonstration in these, except that

two of these four are not essentially true. Thus, equal parallelograms

or triangles, between the same parallels, are not necessarily upon

the same base: but all this is true in these hypotheses, that they

are upon the same or equal bases; but the other does not entirely

follow the assumed hypotheses. Hence, as all these theorems are ten,

the geometrician speaks of six, and neglects four, lest he should

labour in vain, by repetition, since the demonstration is the same.

For it may be shewn in triangles, that if they are equal, and between

the same parallels, they will either be upon the same, or upon equal

bases. For let it be denied, and if possible, let the triangles _a b

c_, _d e f_, have these conditions, upon unequal bases _b c_, _e f_.

Let too, _b c_, be the greater, and cut off _b h_, equal to _e f_, and

connect _a h_. Because, therefore, the triangles _a b h_, _d e f_,

are upon equal bases, _b h_, _e f_, and between the same parallels,

they are equal. But the triangles also, _a b c_, _d e f_, are supposed

equal. Hence, the triangles _a b c_, _a b h_, are equal, which is

impossible. The bases, therefore, of the triangles _a b c_, _d e f_,

are not unequal. And the mode of demonstration will be the same in

parallelograms. Since, therefore, the ostensive method is the same,

and the impossibility the same, viz. that the whole is equal to its

parts, it is not improperly omitted by the elementary institutor. And

thus we have shewn, that there are necessarily ten theorems, and have

enumerated what are omitted, and shewn the reason of their omission.

But let us now pass to the following propositions.

[Illustration]

PROPOSITION XLI. THEOREM XXXI.

If a parallelogram has the same base with a triangle, and is

between the same parallels, the parallelogram shall be double

the triangle.

The present theorem also is local, but it mingles the constructions

of triangles and parallelograms, situated under the same altitude. As,

therefore, we have separately surveyed parallelograms and triangles,

so when we assume each of them in conjunction, and with the same

condition, we contemplate their proportion to each other. In the

former, therefore, an equality of proportion is apparent, since all

upon the same bases, and between the same parallels, have a mutual

equality, whether they are triangles, or parallelograms. But in

these latter, the first of unequal proportions, I mean the duple,

is exhibited: for he demonstrates that a parallelogram is double of

a triangle, on the same base, and possessing the same altitude. But

the elementary institutor shews the thing proposed, by supposing the

vertex of the triangle external to the parallelogram. We can, however,

demonstrate the consequence, by assuming the line which is parallel

to their common base, in the other side of the parallelogram: for

these are two cases of the theorem. Since in consequence of the two

having the same base, it is necessary that the vertex of the triangle

should either be within, or without the parallelogram. Let there be,

therefore, a parallelogram _a b e d_, and a triangle _e c d_, and let

a point _c_ be placed between the points _a_ and _b_, and connect the

right line _a d_. Because, therefore, the parallelogram is double

of the triangle _e c d_, but the triangle _a d c_, is equal to the

triangle _e d c_, hence, the parallelogram is double of the triangle

_e c d_. And hence it is evident that a parallelogram is double of a

triangle on the same base. But if the bases are equal, we can shew

the same by drawing the diameters of the parallelograms: for if the

triangles are equal, the parallelogram which is double of the one, will

also be double of the other. But triangles are equal, on account of

the equality of bases, and the identity of altitude. The geometrician,

therefore, very properly omits this, for the demonstration is the

same: since they will either have the same part, or they will be

conjoined in one point only, or they will be separate from each other.

But in whatever manner they may receive this variety, there is one

demonstration according to all the cases.

[Illustration]

We can likewise demonstrate the converse propositions to this theorem,

after the same manner. One of which is: _If a parallelogram is double

of a triangle, and they have the same or equal bases, and are at the

same parts, they shall be between the same parallels._ For if they are

not the whole shall be equal to the part, and the same proportion shall

prevail: since it is necessary that the vertex of the triangle should

either fall within, or external to the parallels. But in either case,

the same impossibility will be the result, by drawing a parallel to

the base, through the vertex of the triangle. But the second converse

theorem is: _If a parallelogram is double of a triangle, and both

are between the same parallels, they will either be situated upon

one base, or upon equal bases._ For if they are upon unequal bases,

since we have assumed the figures to be equal, we may shew that the

whole will be equal to its part. Hence, all these theorems end in this

common impossible: and on this account, the institutor of the Elements

leaves us to investigate the variety they contain, as he himself, has

contracted his speculation to such as are more simple, and of a more

primary nature. However, as we have recognized these observations, let

us see for the sake of exercise, by not assuming a parallelogram, but

a trapezium, two of whose sides only are parallel (because it has the

same base with the triangle, while it is situated between the same

parallels), let us, I say, consider what proportion it possesses to the

triangle. That it has not, therefore, a duple proportion is evident:

for if it had a duple ratio, it would be a parallelogram, since it

is a quadrilateral figure. But I say that it is either greater than

double or less; for since the two sides are parallel, one is greater,

but the other less; because if equal, the sides conjoining them will

be parallel. If, therefore, the triangle has its greater side for

the base, the quadrilateral figure will be less than double of the

triangle: but if the lesser side, it will be more than double. For let

_a b c d_, be a quadrilateral figure, and let the side _a b_, be less

than the side _c d_, and produce the side _a b_, in infinitum, and let

the triangle _e c d_ have the same base with the quadrilateral figure,

that is _c d_; and lastly, through _d_, draw _d f_, parallel to _a c_.

Hence, the parallelogram _a c d f_, is double of the triangle _e c d_;

and so the quadrilateral figure _a b c d_, is less than double of the

triangle.

[Illustration]

[Illustration]