assumed according to a principal conversion. For it does not make the
whole of its hypothesis a conclusion; and the whole conclusion an
hypothesis. But connecting together some part of the hypothesis of the
fourth theorem, and some part of the objects of enquiry, it exhibits
one of the data which it contains. For the equality of two sides to
two, is in each an hypothesis; but the equality of base to base, is,
in the fourth, an object of investigation, but in the present a datum;
and the equality of angle to angle, is, in the former, a datum, but
in the latter, an object of enquiry. Hence, a change alone of data,
and objects of investigation, produces conversion. But if any one
desires to learn the cause why this theorem is placed in the order of
the eighth proposition, and not immediately after the fourth, as its
converse, in the same manner as the sixth after the fifth, of which
it is the converse, since many converted propositions follow their
precedents, and are exhibited after them without any intervening
medium, to this we must reply, that the eighth, indeed, is indigent
of the seventh proposition. For its truth is evinced by a deduction
to an impossibility, but the nature of an impossible becomes known
from the seventh. And, this again, in its demonstration, is indigent
of the fifth. Hence, the seventh and fifth theorems were necessarily
assumed, previous to the present. But because the _converse_ to the
fifth obtained a demonstration easy, and from _things first_, it was
very properly placed after the fifth, on account of its alliance
with that theorem; and because, since it is shewn by a deduction
to an impossibility, it confutes that which is impossible from
common conceptions, and not as the eighth from another theorem. For
things opposing common conceptions, are more evident for the purpose
of confutation than such as contradict theorems: since these are
assumed by demonstration, but the knowledge of axioms is better than
demonstration. But the institutor of the elements exhibits what is now
proposed from the previously demonstrated seventh theorem.
[Illustration]
But the familiars of Philo assert, that they can demonstrate this
theorem, without being indigent of any other. For let there be
conceived (say they) two triangles, _a b c_, _d e f_, having two sides
equal to two, and the base _b c_ equal to the base _e f_. Likewise let
the bases coincide with each other; and let the two triangles _a b
c_, _d e f_, be so placed in the same plane, that their vertices may
be opposite, and so that _e f g_ may be the equal substitute of _a b
c_. And let _e g_ be equal to _d e_, but _f g_ to _d f_. Hence, _f g_
will either be placed in a right line with _d f_, or not in a right
line. And if not in a right line, it will either make with it an angle
according to the internal part, or according to the external. Let it
first be placed in a right line. Because, therefore, _d e_ is equal to
_e g_, and _d f g_ is one line, the triangle _d e g_, is isosceles, and
the angle at the point _d_, is equal to the angle at the point _g_. But
if it does not lie in a right line, it will make an angle inward; and
in this case let _d g_ be connected.
[Illustration]
Because, therefore _e d_, _e g_, are equal, and the base is _d g_, the
angle _e d g_ also, is equal to the angle _e g d_. Again, because _d f_
is equal to _f g_, and the base is _d g_, the angle, also, _f d g_, is
equal to the angle _f g d_. But the angle _e d g_ was also equal to the
angle _e g d_. Hence, the whole _e d f_, is equal to the whole _f g e_,
which was required to be demonstrated. But in the third place, let _f
g_ make an angle with _d f_, externally, and let the right line _d g_
be connected. Because, therefore _d e_, _e g_, are equal, and the base
is _d g_, the angles _e d g_, _d g e_, are equal. Again, because _d f_,
_f g_, are equal, and the base is _d g_, the angle _f d g_, is equal to
the angle _f g d_. But the whole angles _e d g_, _d g e_, were mutually
equal. Hence, the remaining angles _e d f_, _f g e_, will be equal to
each other. And thus the thing proposed is invented according to any
position of the right line _f g_, and we may demonstrate the theorem,
without employing the seventh proposition.
[Illustration]
Is, then (say they), the seventh proposition introduced in vain by
the institutor of the elements? For if we only assume it on account
of the eighth, but the eighth may be exhibited without it, does not
the seventh appear entirely useless? To these enquiries we must reply
in the words of our predecessors, that the seventh theorem, being
demonstrated, is of the greatest utility to such as are skilled in
astronomical concerns, when they discourse concerning the eclipses of
the sun and moon. For, employing this theorem, they shew that three
consequent eclipses, distant from each other by an equal space, cannot
subsist. I say, in such a manner, that the second may be distant from
the first by as great a space of time as the third from the second. For
example, if the second is produced after the first, when six months
and twenty days are elapsed; the third, will by no means be produced
after the second, by the same, but by either a greater or less interval
of time. But that this is the case may be demonstrated by the seventh
theorem. And the institutor of the elements has not only exhibited
the present as conferring to astronomy, but a multitude of other
theorems and problems. For to what other end shall we say that the
last problem of the fourth book was proposed, by which we are taught
how to inscribe the side of a figure of fifteen angles in a circle,
than for its relation to astronomy? For those who describe in a circle
a quindecangle passing through the poles, will, by this means, obtain
the distance of the poles of the equator from the poles of the zodiac.
Since they are distant from each other by the side of a quindecangle.
The institutor of the elements, therefore, appears by regarding
astronomy, to have previously exhibited many things preparative to our
advancement in that science. But when, at the same time, he saw that
this seventh theorem is exhibited from the fifth, and proves the eighth
without any variety, he assigned it the present place. The addition of
Philo is, indeed, beautiful, but is not sufficiently adapted by its
variety of cases to an elementary institution. And thus much in reply
to the present question.
But if any one should doubt why he does not add so much in the eighth
as in the fourth theorem, I mean, _that the triangles and the remaining
angles are equal_; we must say, that because the equality of the
vertical angle is demonstrated, it follows, that all are equal to all,
by the fourth theorem. It was therefore alone necessary to demonstrate
this by itself, but to assume all the rest as consequents. But it seems
that the equality of the vertical angles causes the equality of the
bases, and of the sides comprehending those angles. For when the bases
are unequal, the same angles will not remain, though the containing
equal sides are supposed, but while the base becomes less, the angle is
at the same time diminished, and while that increases, the angle also
receives a correspondent increase. Nor while the same bases remain,
but the sides become unequal, will the angle remain; but while they
are diminished, it will be increased; and while they are increased, it
will be diminished: for angles, and their containing sides, suffer a
contrary passion. Thus, if upon the same base, you conceive the sides
descending to the lower part, you will diminish the sides, but increase
the angle which they comprehend, and enlarge their distance from each
other. But if you conceive the sides to be elevated, and to receive an
addition as they rise, you will diminish the angle which they contain:
for they will coincide the longer, when their vertex is more remote
from the base. We may therefore certainly affirm that the identity
of the basis and equality of the sides, in a triangle, determine the
equality of its angle.
PROPOSITION IX. PROBLEM IV.
To bisect a given rectilineal angle.
Our author mingles theorems with problems, and connects problems
with theorems, and through both completes the whole of his elementary
institution, comparing as well subjects as the _symptoms_ subsisting
about subjects themselves. Since, therefore, he had shewn in the
preceding propositions, both in one triangle, from the equality of
the sides, the consequent equality of the angles, and the contrary:
and in a similar manner in two triangles, with this exception, that
the mode of conversion in one and two triangles is different, he now
passes to problems, and orders us to bisect a rectilineal angle. And
it is manifest, that the angle here is given according to form: for it
is called right-lined, and not of any kind whatever. Indeed, we cannot
bisect every angle by the elementary institution; since it is doubtful
whether every triangle can be bisected. For, perhaps, you may doubt
whether it is possible to bisect a cornicular angle. But the ratio of
the section is also distinguished in this problem, and this again not
in vain. For to divide an angle in any given ratio, transcends the
present construction: as, for example, into three, four, or five equal
parts. Indeed, to trisect a right angle is possible, by employing a
few of the propositions which are afterwards delivered[16]: but this
cannot be effected in an acute angle, without passing on to other lines
of a mixt species.[17] And this is manifested by the geometricians who
propose to trisect a given rectilineal angle. For _Nichomedes_, indeed,
from conchoidal lines, the origin, order, and symptoms of which, he
delivers, as he was the inventor of their properties, trisects every
right-lined angle. But others effect this from the quadrantal lines
of _Hippias_ and _Nichomedes_, by employing mixt quadrantal lines.
Others, again, being incited from the Helices of _Archimedes_, divide
a given rectilineal angle, in a given ratio. But the consideration
of these, because difficult to learners, we shall for the present
omit; as it will, perhaps, be more convenient to examine this in the
third book[18], where the institutor of the Elements bisects a given
circumference. For there the same mode of enquiry presents itself with
respect not only to bisection, but also trisection; and the ancients
endeavoured, by employing the same lines, to divide every circumference
into three equal parts. With great propriety, therefore, he who only
mentions a right line and a circumference, alone bisects a right angle
and a circumference. But conceiving that the species composed from
these, through mixture, are difficult to explain and enumerate, without
a curious examination, he omits all such enquiries as involve mixt
lines in their consideration, and proposes to investigate in first and
simple forms alone, such things as can either be produced or considered
from these. And such, indeed, is the proposition of the present
problem, _to bisect a given right lined angle_. For in the construction
of this he uses one petition, and the first and third problem: but in
the demonstration he employs the eighth theorem alone. Since problems
entirely require demonstration (as we have already observed[19]) and
through this they obtain a power of producing science. But perhaps,
some may oppose the geometrician, by asserting that an equilateral
triangle may be constituted by him, not having its vertex within the
two right lines, but either upon, or external to each; and that this
may be manifested by the elements. For let there be an angle _b a c_,
which it is required to bisect. Then let _b a_ be taken equal to _a c_,
and let _b c_ be connected, and upon it, let an equilateral triangle _b
c d_ be constructed. This point _d_, therefore, is either within the
right lines _a b_, _a c_, or upon _a b_, or _a c_, or external to both.
Now the institutor of the Elements assumes them within; and hence,
those who oppose the demonstration, will say the point is either placed
on one of the right lines, or external to both. Let the point _d_
then be placed on the line _a b_, so that the triangle _b c d_ may be
equilateral: _d b_, therefore, is equal to _d c_, and the angles at the
base _c b d_, _b c d_, are equal. Hence, the whole, _b c e_, is greater
than the angle _c b d_. Again, because _a b_, _c a_, are equal, the
triangle _a b c_, is isosceles, and the angles under the base _b c_,
will be equal. The angle, therefore, _b c e_, is equal to the angle _c
b d_. But it was also greater, which is impossible. Hence, the vertex
of the equilateral triangle cannot be in the right line _a b d_. In
like manner we may shew that it cannot be in the right line _a c e_.
Let it therefore, if possible be placed externally. Because, then _b d_
is equal to _c d_, the angles at the base are equal, viz. _b c d_, and
_c b d_. Hence, the angle _b c d_, is greater than the angle _c b f_.
Much more, therefore, is the angle _b c e_, greater than _c b f_: but
it is also equal, because these angles are under the base _b c_, of an
isosceles triangle _a b c_, and this is impossible. Hence, the point
cannot fall in these parts external to the two right lines; and it may
be similarly shewn that this is impossible in other parts. Here too
you may again observe, that we destroy objections by using the second
part of the fifth proposition, _that the angles under the base of an
isosceles triangle are equal_. And this is what we have previously
observed, that many things opposing science, are shewn to be debile,
and easy of confutation, by the assistance of this theorem; and that
such is the utility it affords to geometry.
[Illustration]
[Illustration]
But if any one should say that there is no place under the base, and
yet that it is requisite to constitute the equilateral triangle at the
same parts, in which the lines _b a_, _a c_, are situated; it will be
necessary that the lines which are constituted should either coincide
with _b a_, _a c_, if they also are equal to the base _c b_: or that
they should fall external to them, if they are less than the base _b
c_: or within, if _b a_, _a c_, are greater than _b c_. Let them, in
the first place, coincide, and let _b a c_ be an equilateral triangle,
and let there be taken in the side _a b_, the point _d_, and make _a
e_ in the side _a c_, equal to _a d_, and connect the lines _d e_, _b
e_, _c d_, _a f_. Because, therefore, _a b_ is equal to _a c_, and
_a d_ to _a e_, the two _b a_, _a e_, are equal to the two _c a_, _a
d_, and they comprehend the same angle. Hence, they are all equal to
all, and the angle _d b e_, is equal to the angle _e c d_. But _d b_
is also equal to _e c_, and _b e_ to _c d_. All, therefore, are equal
to all. Hence, the angle _d e b_, is equal to the angle _e d c_: for
they subtend equal sides. And _d f_ is equal to _e f_, (by the sixth.)
Because, therefore, _a e_ is equal to _a d_, and _a f_ is common,
and the base _d f_, is equal to the base _e f_, the angle _d a e_ is
bisected, which was required to be done.
[Illustration]
But if the sides of the equilateral triangle fall external to the right
lines _b a_, _a c_, let them be _b d_, _d c_, and having connected _d
a_, let it be produced to the point _e_. Because, therefore _b d_, _d
c_, are equal, but _d a_ is common, and the bases _b a_, _a c_, are
equal, the angle, also, _b d a_, (by the eighth) is equal to the angle
_c d a_. Again, _b d_, _d c_, are equal, and _d e_ is common, and they
contain equal angles as we have shewn, the base also _b e_, is equal
(by the fourth) to the base _e c_. Because, therefore, _a b_ is equal
to _a c_, and _a e_ is common, the angle, also _b a e_, is equal to the
angle _c a e_, which was to be shewn.
But if the sides of the equilateral triangle fall within the right
lines _a b_, _a c_, as _b d_, _d c_, let again _a d_ be connected.
Because, therefore, _b a_, is equal to _a c_, and _a d_ is common,
but the base _b d_, is equal to the base _c d_, hence, the angle _b a
d_ (by the eighth) is equal to _c a d_. The angle, therefore, at the
point _a_, is bisected, in whatever manner the equilateral triangle
may be constituted. And having thus summarily spoken concerning these,
we shall now proceed to the following theorems, only adding, that the
given angle may be given in a four-fold respect. _In position_, as when
we say _to this right line, and to this point to place an angle_: for
after this manner it is given. But _in form_, as when we call the angle
right, or acute, obtuse, right-lined, or mixed. And _in proportion_, as
when we call it double, or triple, greater, or less. And lastly, _in
magnitude_, as when we call it the third part of a right angle. But the
present angle is only given in form.
[Illustration]
PROPOSITION X. PROBLEM V.
To bisect a given finite right line.
This, also, is a problem which supposes a finite right line, since we
cannot terminate a line on both sides infinite. But the section of a
line infinite on one side only, wherever the point is assumed, is made
in unequal parts. For that part of the section which takes place on
the infinite side, is necessarily greater than the remainder, because
finite. Hence, the line required to be bisected, must be necessarily
both ways finite. But perhaps, some excited by this problem, may think,
that the doctrine of a line, not being composed from impartibles, is
only previously received by geometricians as an hypothesis. For if it
consists from impartibles, it either becomes finite, and receives its
completion from _odd_, or from _even_ parts. But if from such as are
_odd_, it will appear that an impartible also may be cut, while a right
line is bisected. And if from such as are even, the section will be
unequal, because, one part, as composed from more impartibles, will
be greater than the remainder. It is therefore impossible to bisect a
given right line, if magnitude consists from impartibles. But if it
be not composed from impartibles, it may be divided in infinitum. It
appears, therefore, (say they) to be received by common consent, and
to be a geometrical principle, that magnitude is among the number of
things infinitely divisible. Against these we reply in the words of
Geminus, that geometricians previously receive according to a common
conception, that continued quantity is divisible. For we call that
continuous, which is composed from conjoined parts, and this it is in
every respect possible to divide. But that continued quantity may be
infinitely divided, they do not previously assume, but demonstrate from
proper principles. For when they shew that incommensurability is found
in magnitudes, and that all are not commensurable with each other,
what else can we say they evince by this means, except this, that
every magnitude may be divided into parts always divisible, and that
we can never arrive at an impartible, by the most unwearied analysis,
since this minimum would be the common measure of all magnitudes? This
then is demonstrable, but that which says, _every thing continuous
is divisible_, is an axiom. Hence, since a finite line also is
continuous, it is divisible. And from this conception the institutor
of the Elements cuts a finite right line into equal parts, but not
as pre-assuming, that it is divisible in infinitum. For to be merely
divisible, and to be infinitely divisible is not the same.
[Illustration]
But the discourse of Xenocrates inferring indivisible lines, is
confuted by this problem. For if it be a line, it is either right, and
may be bisected; or circular, and it is greater than a certain right
line; (since every circular has a certain right line less than itself);
or it is mixt, and on this account is the more divisible, since
composed from simple divisible lines. But this must be deferred to some
posterior speculation. However, the geometrician bisects a finite right
line, employing in the construction the first and ninth propositions;
but using in the demonstration the fourth alone; for by the angles he
shews the equality of the bases. But Apollonius Pergæus bisects a given
finite right line after the following manner. Let there be (says he)
a finite right line _a b_, which we are required to bisect, and with
the centre _a_, but interval _a b_, let a circle be described. And
again, with the centre _b_, but interval _b a_, let another circle be
described, and let the right line _c d_, connect the common sections of
the circles; this shall bisect the right line _a b_. For let the equal
lines _d a_, _d b_, _c a_, _c b_, be connected; these being equal,
because each is equal to _a b_. But _c d_ is common, and _d a_ is
equal to _d b_ on the same account. Hence the angle _a c d_, is equal
to the angle _b c d_; and so (by the fourth) _a b_ is bisected. Such
then, according to Apollonius, is the demonstration of this problem,
assumed, also, from an equilateral triangle; but instead of exhibiting
the bisection of the line, from the bisection of the angle at the point
_c_, it shews this from the equality of the bases. The demonstration,
therefore, of the institutor of the Elements, is much better, since it
is both more simple, and emanates from principles.
PROPOSITION XI. PROBLEM VI.
To raise a right line at right angles, to a given right line, from a
given point in that line.
Whether we receive a right line on both sides finite, or on both
sides infinite, or on one side infinite but on the other finite, and a
point in it, the construction of the present problem will conveniently
succeed to the geometrician. For though the given point should be on
the extremity of the right line, by producing it we can accomplish
our purpose. But it is manifest that the point in the present problem
is given in _position_, since it can only be placed in position in a
right line. But the right line is given according to _form_; since
its magnitude is not distinguished either by proportion or position.
Hence, the institutor of the Elements, employing the first and
third problem, together with the eighth proposition, and the tenth
definition, exhibits the thing proposed. But if any placing the point
on the extremity of the right line, should ask us without producing the
line, to erect upon this a right line at right angles, we can likewise
shew that this is possible to be effected. For let there be a right
line _a b_, and a given point in it _a_, and let there be assumed in
the line _a b_, any point _c_, and from this (as the present element
teaches us) let a right line _c e_ be erected at right angles to _a
b_. Then from _c e_, let _c d_ be taken equal to _a c_, and let the
angle at the point _c_ be bisected by the line _c f_, and at the point
_d_ let a right line be erected at right angles, coinciding with _f c_
in _f_; and lastly from the point _f_, to the point _a_, let _f a_ be
connected. I say that the angle at the point _a_ is right. For since _d
c_ is equal to _c a_, but _c f_ is common, and contains equal angles,
(for the angle at the point _c_ was bisected) hence, _d f_ is equal to
_f a_, and all in like manner (by the fourth) are equal to all. The
angle, therefore, at the point _a_, is equal to the angle at _d_. But
the angle at the point _d_ is right; and so consequently is the angle
at _a_. And thus the thing required is effected. But the institutor of
the Elements was not indigent of any such artifice: for he commands us
to raise a line at right angles, but not at one right. It is requisite,
therefore, not to receive the point in the extremity of the right line,
because the perpendicular line forms angles with its subject right
line, but not one angle alone.
[Illustration]
[Illustration]
But Apollonius raises a perpendicular as follows. Let the given right
line, says he, be _a b_, and a given point in it _c_, but let there be
assumed in _a c_ any point _d_, and from _c b_, take away _c e_, equal
to _c d_. Then with the centre _d_, but interval _d e_, let a circle
be described; and again with the centre _e_, but interval _e d_, let
another circle be described, and let a right line be drawn from _f_
to _c_. I say that _f c_ is a perpendicular. For if _f d_, _f e_, are
connected, they shall be equal. But _d c_, _c e_, are equal, and _f c_
is common. Hence, also, the angles at the point _c_ (by the eighth) are
equal. They are therefore right. And here, is it not again obvious,
that this demonstration is more various than that of Euclid, and
requires the description of circles, that by this means an equilateral
triangle may be described upon _d e_, and the problem exhibited? For
all the rest are common to the demonstrations. But the demonstration
by a semicircle is not worthy to be remembered, since it supposes many
things which are afterwards exhibited, and entirely falls from the
order of an elementary institution.
PROPOSITION XII. PROBLEM VII.
Upon a given infinite[20] right line, and from a given point which is
not in that line, to let fall a perpendicular.
Oenopides first investigated this problem, believing it useful
for astrological purposes. But he calls a perpendicular, after the
manner of the ancients, a gnomon, because a gnomon, also, is at right
angles to the horizon, but the same line is at right angles with a
perpendicular, from which it differs only in habitude, since, as he
observes a gnomon has the same subject with a perpendicular. But again,
a perpendicular is two-fold, that is, it is either plane or solid.
Hence, when the point from which the perpendicular right line is drawn,
is in the same plane, the perpendicular is called plane; but when the
point is on high, and external to the subject plane, it is called
solid. And the plane perpendicular, indeed, is drawn to a right line:
but the solid to a plane. Hence, it is necessary, that this last should
not only form right angles, with one right line, but with all right
lines in the same plane. For the perpendicular is let fall on a plane.
In the present problem, therefore, the institutor of the Elements
proposes to let fall a plane perpendicular. For the deduction is
proposed to a right line, and the discourse proceeds, so far as all are
supposed to be in the same plane. Hence, in the line at right angles we
do not require infinity, because the point is supposed to be in that
right line. But in the present problem, respecting a perpendicular, he
supposes the given right line infinite, because the point from which
the perpendicular is to be drawn is placed external to the right line.
For if it was not infinite, the point might be received externally, and
yet in a direct position, so that the protracted right line would fall
upon it, and the problem not succeed. Hence, he places the right line
infinite, so that the point may be received at either of its parts; and
that no place may be left, in which it can be in the same direction
with the given right line, unless it is in the line, and has not an
external position. And on this account the right line to which the
perpendicular is to be drawn is considered as infinite.
But in what manner infinite can subsist, is a matter well worthy our
contemplation. For it is manifest that a right line existing infinite,
a plane also will be infinite, and this in energy, if the thing
proposed by Euclid be true. That among sensible particulars, therefore,
there can be no magnitude infinite, according to any distance, both
the dæmoniacal Aristotle, and those who received their philosophy from
him, have abundantly shewn. For neither that which is moved circularly,
nor any other simple body can be infinite; since the place of each is
limited. But neither in separate and impartible reasons is an infinite
of this kind possible. For if they neither contain dimension, nor
magnitude, much less can they contain infinite magnitude. It remains,
therefore, that infinite can alone subsist in the phantasy, which at
the same time the phantasy does not comprehend. For as soon as it
understands, it induces form and bound to that which is understood,
stops the transit of the phantasm by its intellection, pursues its
progress, and infolds it in its shadowy embrace. The phantasy,
therefore, is not infinite by intellection, but rather by advancing
infinitely about that which is understood; and calling whatever it
leaves innumerable, and incomprehensible by intelligence, infinite. For
as the sight by not seeing understands darkness; so the phantasy by
not understanding perceives infinite. Hence it pursues the progress of
the infinite, because it is endued with an impartible power, capable
of perpetually advancing: but it understands as if stopping in its
progression, because infinite surpasses its comprehension. For it
calls that infinite, which it leaves as unable to pass over in its
pursuit. On this account when we place a given infinite line in the
phantasy, in the same manner as we establish all other geometrical
species, viz. triangles, circles, angles, lines, and all of this kind,
we must not wonder how a line is infinite in energy, and how advancing
infinitely, it applies itself to finite intellections. But cogitation,
in which reasons and demonstrations reside, does not use infinite for
the purpose of science, since infinite is by no means perceptible by
science, but receiving it from hypothesis, it employs finite alone in
its demonstrations, and assumes infinite not for the sake of infinite,
but of that which is bounded and finite. For if we should grant to
cogitation, that the given point, neither lies in a right line with the
given finite right line, nor yet is so distant from it, that no part of
the right line is subjected to the point, we shall no longer require
an infinite line. That cogitation, therefore, when employing a right
line, may use it without controversy and reproof, she supposes it to be
infinite; and employs the infinity of the phantasy, as the foundation
of infinite generation. And thus much may suffice for the present
concerning the nature of infinite.
[Illustration]
But it is now requisite that we should consider the objections
which are urged against the construction of this problem. Let
there be received, say they, an infinite right line _a b_, and let
the given point be _c_, from which it is required to let fall a
perpendicular, and let _d_ be a point on the other side, according
to the geometrician. But the circle which cuts the right line _a b_,
in the points _a_ and _b_, will cut it also in _f_, and will have a
situation according to the figure. In answer to this, we must say, that
it affirms an impossible case. For let the right line _a b_ be bisected
in _h_, and let _c h_ be connected, and produced to the circumference,
to the point _d_, and let _c a_, _c b_, _c f_, be connected. Because,
therefore, these lines are from the centre, and _a h_, is equal to _h
b_, but _c h_ is common, all are equal to all. Hence _c h_ forms right
angles at the point _h_. Again, because _c a_, _c b_, are equal, they
form equal angles at the points _a_ and _b_. But _c a_ also, is equal
to _c f_, on which account the angle _c a f_, is equal to the angle _c
f a_. In like manner the angle _c b f_ is equal to the angle _c f b_.
Because, therefore, the angles at the points _a_ and _b_, are equal,
the angle, also, _c f a_, is equal to the angle _c f b_, and they are
successive, and consequently right. But each of the angles at the point
_h_ is right. Hence, _c h_ is equal to _c f_. But _c f_ is also equal
to _c d_, since they are from the centre. Therefore _c h_ is equal to
_c d_, which is impossible. Hence, the circle does not cut the right
line in any other points than _a_ and _b_.
[Illustration]
But if any one should say, that he who describes a circle will bisect
_a b_ in _f_, we can again shew that this is impossible. For let all
be described as before, and let the right line _f b_, be bisected in
the point _h_. Because, therefore, _a f_, _f b_, are equal, but _c
f_ common, and the base _c a_, is equal to the base _c b_, all are
equal to all. Hence, the angles at the point _f_ are right. Again,
because _f h_ is equal to _h b_, and _c h_ being connected, is common,
and the base _c f_ is equal to the base _c b_, for they are from the
centre, the angles at the point _h_, are right; for they are equal and
successive. Because, therefore, each of the angles _c f h_, _c h f_, is
right, _c f_ is equal to _c h_. But _c f_ is equal to _c e_, for they
are from the centre, and hence _c h_ is not unequal to _c e_, which is
impossible.
[Illustration]
It now remains that we run over the third objection. For the circle
which is described (say they) will cut the right line in the points
_a_, _b_, and in the points _f_, _h_. We therefore bisecting the right
line _a b_ in the point _k_, and connecting the lines _c a_, _c f_, _c
k_, _c b_, can shew that this is impossible. For since _k a_, _k b_,
are equal, and _c k_ is common, and the bases _c a_, _c b_, are equal,
hence the angles at the points _a_ and _b_ are equal, and those at the
point _k_ right. But each of the lines is equal to _c f_; and hence,
the angles at the point _f_, are right; for they are equal, because
successive. Therefore, _c f_ is equal to _c k_: for they subtend right
angles. But _c f_ is equal to _c d_, since they are from the centre;
and hence, _c d_ is equal to _c k_, which is impossible. Hence then,
it is impossible that the circle which is described should cut the
line _a b_ in one, two, or in more points than _a b_. And such are the
objections against the present problem.
[Illustration]
But there are also cases of the construction of this problem, which
are to be distinguished from the objections. For case is not the same
with objection; since the former shews the same thing differently,
but the latter leads the objection to an inconvenience. But other
expositors, not distinguishing these from one another, bring all into
the same, so that it is uncertain, whether they enunciate to us in
their writings, cases, or objections. We therefore distinguishing
these, having enumerated the objections, shall now describe the cases
of the problem. Let there be then an infinite right line _a b_, and
a given point _c_. Now it may be said that there is no farther place
in the other part of the perpendicular right line, but in that only
where the point _c_ lies. Taking, therefore, in the right line _a b_
a point _d_, with the centre _c_, and interval _c d_, let us describe
the circumference of a circle _d e f_, and bisecting _d f_ in _h_, let
us connect the lines _c d_, _c h_, _c f_. Because, therefore, _d h_
is equal to _h f_, but _c h_ is common, and _c d_ is equal to _c f_,
(for they are from the centre,) hence, the successive angles at the
point _h_, are equal. They are, therefore, right. And hence, _c h_
is a perpendicular to _d f_. But if any one should also say that the
described circle does not cut the right line _a b_, but touch it as the
circle _d e_, by taking the point e externally, and using the centre
_c_, and interval _c e_, as in the preceding, we shall obtain the
object of our enquiry. And thus much we have said concerning the cases
of the problem, for the sake of exercising the attention of the reader.
[Illustration]
But if we are desirous of adding contemplation likewise to these
two problems, a right line erected at right angles, seems to imitate
a life tending on high from inferior concerns, ascending purely,
and without contamination, and abiding inflexibly with regard to
natures subordinate to its own. But a perpendicular is the image of
a life perpendicularly descending, and the least of all replete with
generative infinity. For a right angle is the symbol of an energy
inflexible, and restrained in the comprehension of equality, bound, and
finite. From whence, indeed, Timæus also calls the _other circle_
in the divine soul, possessing the reasons of sensible natures,
_right_; for in our souls it is bent with flexions of every kind,
and suffers various contortions and perturbations from the unceasing
whirls of generation: but among _wholes_ it resides immaculate,
uncontaminated, firm, and indeclinable, prior to sensible forms. But
if likewise an infinite right line is the symbol of the whole of
generation, which is moved infinitely and indeterminately, and besides
this, of matter itself, which is deprived of bound and form: and if a
point placed externally bears an image of an essence impartible, and
separate from material natures, doubtless the deduced perpendicular
will imitate that life which proceeds into generation with an undefiled
progress from unity, and an impartible essence. But if a perpendicular
cannot be shewn without circles, this also will be the symbol of an
inflexibility inherent in life, through the medium of intellect. For
life, indeed, since it subsists by itself as motion, is indeterminate:
but it becomes terminated, and is filled with a pure and immaculate
power, by participating and adhering to the circulations of intellect.
PROPOSITION XIII. THEOREM VI.
When a right line standing upon a right line forms angles, it either
forms two right, or angles equal to two right.
Euclid again passes on to theorems, consequent to things exhibited by
problems. For after a perpendicular had been drawn to a right line,
and a right line erected at right angles, it remained, to enquire if
it should not be a perpendicular, what angles it would form, and how
it would be affected to the line upon which it stands. This then he
proves universally, _that every right line standing upon a certain
line, and forming angles, either forms two right, if its state be
indeclinable, firm, and never verging: or angles equal to two right,
if it declines in one part, but is more distant from its subject line,
in the other part_. For as much as it takes away from a right angle
by its declination in one part, so much it adds by its distance in the
other. But it is requisite to take notice, that in this proposition
also, the Geometrician employs diligent care. For he does not simply
say that every right line, standing upon a right line, forms either two
right angles, or angles equal to two right, but he adds, _if it forms
angles_. For what if standing on the extremity of a right line, it
should form one angle, will it happen that this may be equal to two
right? This certainly is impossible. Since every rectilineal angle
is less than two right, as also every solid angle is less than four
right. Hence, though you should receive that which appears to be the
greatest of all obtuse angles, this also will increase, as that which
does not yet receive the measure of two right angles. It is requisite,
therefore, that the right line should stand in such a manner, that it
may form angles. And these observations regard the productive diligence
of science.
But what does he mean by adding the particle, _either two right, or
equal to two right_? For when he has constituted two right, he forms
angles equal to two right; since right angles are equal to each other.
Shall we say that one of the equal angles is also common, but that the
other of the equals is only proper? But we are accustomed when both
_proper_ and _common_ is verified, to express every particular from
that which is proper, but when we cannot effect this, we are content
with that which is common for the explication of the subject concerns.
This then, the equality of the successive angles, is common to right
angles, but is not predicated of these alone: but this, that they are
right, is peculiar to their equality. Hence, the assertion, _equal to
two right_, alone signifies the inequality of the angles. For in these
it is alone verified, but by no means in such as are equal. And this
also the institutor of the Elements divides in opposition to two right.
For since it is predicated by itself, it has a power of signifying that
the angles on each side are unequal. But through these observations
we may also perceive, that equality is the measure and bound of
inequality. For though the increase and decrease of an obtuse and
acute angle is indeterminate and infinite, yet it is said to receive
limitation, and bound from a right angle; and each of them, indeed,
separately, recedes from a similitude to the right; but both, according
to one harmonizing union, are reduced to its bound. But as they can by
no means perfectly equal the simplicity of a right angle, they receive
an equality to it when doubled, the duad being the exemplar of their
infinity, as of itself endued with an infinite nature. And this seems
to procure a manifest image of the progression of primary causes; and
of their abiding according to one boundary, in a manner perpetually
the same, about the infinity of generation. For how could otherwise
generation, which participates of the more and the less, and is carried
in indefinite whirls, agree with intelligibles, and be equalled with
them in a certain respect, unless by participating their natures,
whilst they advance with prolific powers, and only multiply themselves
in their progressions? For things which abide in their own simplicity
and impartibility, are entirely separated from generable natures. And
thus much is assumed from the present theorem, and applied to the
knowledge of universals.
PROPOSITION XIV. THEOREM VI.
If to any right line, and at a point in it, two right lines
being placed in a consequent order, and not towards the same
parts, make the successive angles equal to two right, those
right lines shall be in a direct position to each other.
The present theorem is the converse of the foregoing: for such as
are converse are always consequent to preceding theorems. Since,
therefore, the former had constituted a right line upon a right line,
and had shewn that it made the successive angles either two right, or
equal to two right; in the present theorem he receives the equality of
the angles to two right, which are formed at some right line, but he
shews that it is one right line which produces their equality. Hence,
that which was a _datum_ in the former, is in the present theorem an
object of enquiry; and is shewn by a deduction to an impossibility. For
after this manner the converse of theorems ought to be exhibited; but
in problems they should receive principal demonstrations. But in this
theorem we may also perceive the greatest and most admirable diligence
of this proposition producing science. For in the first place, after he
had said, _if to any right line_, he adds, _and at a point in it_; for
what if the two extremes of the right line existing, one of the right
lines should be drawn from the one extreme, but the other from the
remaining one, and should form angles at the right line, equal to two
right, would they on this account have a direct position? And how can
this take place in lines drawn from different points of the right line?
It is on this account also, that he adds, _and at a point in it_, since
he is willing that both should be in the same point. But in the second
place, because it is possible that the right lines which are drawn,
may be at the same point, and not consequent (since we may receive
infinite right lines placed at the same point) he adds the particle,
_in a consequent order_. And in the third place, because the word
_consequent_ may be considered as well at the same parts as on both
sides: but because it is impossible that lines which are consequent at
the same parts should be mutually in a direct position, this indeed he
explains, but affords us an opportunity of considering that consequent
right lines are to be received in position on both sides; since these
also can be shewn to be in a right line. Let there be placed at the
right line _a b_, and at a point in it _b_, towards the same parts,
two right lines _b c_, _b d_, these, therefore, shall be consequent
to each other. For no other right line is situated between them. But
those things are _successive_, between which there is nothing similar.
Thus we call the columns _consequent_, between which there is no other
column: for though the air intervenes, yet nothing of the same kind is
situated in the middle. Because, therefore, they lie towards the same
parts, they are by no means in a direct position, although they form
two angles equal to two right; I mean the angles at the point _b_. For
nothing hinders but that the angle _a b d_, may contain in itself,
one right, and a third part of a right angle: and that the angle _a b
c_, may be two thirds of a right angle. And thus much concerning the
proposition.
[Illustration]
But one petition is employed in the construction, viz. the second,
which begs _to produce a right line straight forwards_, as in the
demonstration he uses the preceding theorem, and two axioms; i.e. the
one which says, _things equal to the same, are equal to one another_;
and also the one which affirms, _that if from equal things equals are
taken away, the remainders shall be equal_. But at the collection of
the impossibility, he employs the axiom, which says, _the whole is
greater than its part_. For it is equal one common angle being taken
away, which is impossible. But that it is possible to the same right
line, and at a point in it, two right lines in a consequent position,
and yet, towards the same parts, may form angles belonging to that one
right line, equal to two right, we may shew with Porphyry, as follows.
Let there be a certain right line _a b_, and any point in it _c_, and
let _c d_ be raised at right angles to _a b_, and let the angle _d c b_
be bisected, by the line _c e_. Then from the point _e_, to the line
_a b_, let there be drawn the perpendicular _e b_, and let _e b_ be
produced, and place _f b_ equal to _e b_, and connect _c f_. Because,
therefore, _e b_ is equal to _b f_, but _b c_ is _common_, and they
contain equal angles (for they are right), hence, the base _e c_, is
equal to the base _c f_. All, therefore, are equal to all. Hence, the
angle _e c b_, is equal to the angle _f c b_. But the angle _e c b_ is
the half of a right angle: because the right angle _d c b_ was bisected
by the line _e c_. Hence, also the angle _f c b_, is the half of one
right. The angle, therefore, _d c f_, is equal to one right, and the
half of a right angle. But the angle _d c e_, also, is the half of a
right angle. Hence to the right line _c d_, and to a point in it _c_,
two right lines are consequently placed towards the same parts, viz. _c
e_ and _c f_, forming angles equal to two right, _c e_ causing the half
of a right angle, and _c f_ one and a half. Lest, therefore, we should
enquire after things impossible to be effected, viz. how the right
lines _c e_, _c f_, forming angles at the right line _d c_, equal to
two right, can be in a direct position to one another, the Geometrician
adds the particle _not towards the same parts_. It is requisite,
therefore, that the right lines which form angles equal to two right,
should be placed on both sides of the right line, being raised, indeed,
from one point, but drawn to different parts of the right line.
[Illustration]
PROPOSITION XV. THEOREM VIII.
If two right lines cut one another, they will form the angles at the
vertex equal.
[Illustration]
We must call _successive_ angles different from such as are
_vertical_. For these last originate from the section of two right
lines: but the former from the mere dissection of the one by the
other. Thus, if a right line remaining itself without section, but
cutting another in its extremity, forms two angles, we denominate
these _successive_ angles. But if the two right lines mutually cut
each other, they form _vertical_ angles. And they are so called,
because they have their vertices conjoined in the same point. But
their vertices are the points, at which the planes, while they are
contracted, form angles. This, therefore, is what the present theorem
evinces, that when two right lines mutually cut each other, the
vertical angles are equal. And it was first invented (according to
Eudemus) by Thales: but was thought worthy of a demonstration producing
science by the institutor of the Elements. But it is not exhibited
from all the particulars requisite to a perfect proposition. For
construction is wanting in the present theorem: but demonstration,
which must be necessarily inherent, depends on the thirteenth theorem.
But he uses two axioms, one of which is, _that things equal to the
same, are equal among themselves_: and the other, _if from equal things
equals are taken away, the remainders will be equal_. The theorem,
indeed, of Euclid, is manifest, but another such is converted to the
present theorem. _If to any right line, and at a point in it, two right
lines, not assumed towards the same parts, make the vertical angles
equal, those right lines shall be in a direct position to each other_.
For let there be a certain right line _a b_, and any point in it _c_,
and at the point _c_, let two right lines _c d_, _c e_, not towards
the same parts be assumed, forming equal angles _a c d_, _b c e_. I
say that _c d_, _c e_, are in a right line. For since the right line
_c d_, insists upon the right line _a b_, it forms angles equal to two
right, i.e. _d c a_, _d c b_. But the angle _d c a_, is equal to the
angle _b c e_. Therefore, the angles _d c b_, _b c e_, are equal to
two right. Because, therefore, to a certain right line _b c_, and at a
point in it _c_, two consequent right lines _c d_, _c e_, not placed
towards the same parts, form the successive angles equal to two right,
those right lines _c d_, _c e_, are in a direct position to each other.
The converse, therefore, to the present theorem, is exhibited. But
the Geometrician seems to have neglected this, because it is easy to
evince its truth, by the same method of deduction to an impossibility
as we employed in exhibiting the fourteenth proposition. For the same
things being supposed, I say that the right line _c d_, is in a direct
position to _c e_. For if it be not, let _c f_ be taken in a right line
with _c d_. Because, therefore, two right lines _a b_, _d f_, intersect
each other, they will form the angles at the vertex equal. Hence, the
angles _a c d_, _b c f_, are equal. But _a c d_, _b c e_, were also
equal. The angle, therefore, _b c e_, is equal to the angle _b c f_,
the greater to the less, which is impossible. Hence, no other right
line, besides _c d_, is in a direct position to _c e_. The right lines,
therefore, _c d_, _c e_, are in a direct position to each other, the
angles at the vertex being supposed equal. Since then, there is the
same demonstration which was pre-assumed in the fourteenth theorem,
would it not have been superfluous to have produced this conversion?
But for the sake of exercise, we have proved it as well by a deduction
to an impossible, as by an ostensive method. However, this fifteenth
theorem seems to rest upon the similitude of the parts of right lines,
and their situation in their extremities. Because lines with these
conditions, and mutually cutting each other, must necessarily possess
similar inclinations on both sides to each other. Since circumferences,
and universally non-right lines cutting one another, do not necessarily
form the vertical angles equal, but sometimes equal, and sometimes
unequal. For if two equal circles cut each other through the centres,
or even not through the centres, they will form the lunular angles at
the vertex equal: but not likewise the remaining angles, viz. those
on both sides concave, and on both sides convex, but the one will
be greater than the other. But in right lines, the situation in the
extremities, causes the distance of one segment, to be equal to the
distance of another.
COROLLARY.
From hence it is manifest that if two right lines cut each other, they
will make four angles equal to four right.
Corollary is one of the geometrical appellations, but it has a
two-fold signification. For they denominate corollaries, whatever
theorems are proved together with the demonstrations of others,
becoming as it were the unexpected gain and emolument of the
investigator: and likewise, whatever is the object of enquiry, but is
indigent of invention, and is neither investigated for the sake of
generation alone, nor of simple contemplation. For that the angles
at the bases of isosceles triangles are equal, it is requisite to
contemplate, and the knowledge of things in existence is of this
kind. But to bisect an angle, or constitute a triangle, to cut off,
or place an equal right line, all these demand that something may be
performed. And again, to find the centre of a given circle, or two
commensurable magnitudes being given to find their greatest common
measure, with every thing of this kind, are, after a manner, situated
between problems and theorems. For neither is the origin of _objects
of enquiry_ inherent in these, nor contemplation alone, but invention.
Since it is requisite to place the object of enquiry conspicuously and
before our eyes. Such then are whatever corollaries Euclid wrote, for
he constructed a book of corollaries. But we must now omit to speak
of corollaries of this kind. However, such as occur in the elementary
institution, appear at the same time with the demonstrations of other
things, but they themselves are not principally investigated, as is
evident in that which is proposed at present. For the design of the
proposition is to enquire whether if two right lines mutually cutting
each other, the angles at the vertex are equal. But whilst this is
evinced, it is at the same time demonstrated, that the four angles
which are formed, are equal to four right. For when we say let there
be two right lines, _a b_, _c d_, cutting each other in the point _e_:
because _a e_ stands upon _c d_, it makes the successive angles equal
to two right. And again, because _b e_ stands upon _c d_, it also makes
the successive angles equal to two right; then together with the object
of enquiry we demonstrate, that the angles about the point _e_, are
equal to four right. A corollary, therefore, is a theorem, unexpectedly
emerging from the demonstration of another problem, or theorem. For
we seem to fall upon corollaries, as it were, by a certain chance;
and they offer themselves to our inspection, without being proposed,
or investigated by us. Hence, we assimilate these also to gains. And
perhaps those skilled in mathematical concerns, have imposed on them
this appellation, shewing the vulgar, who rejoice in apparent gain,
that these are the true gifts of divinity, and true gains, and not
the objects of their sordid estimation. For this indeed produces that
faculty resident in our nature, and adds the prolific power of science,
to principal enquiries, manifesting the copious riches of theorems. And
such is the property of corollaries.
[Illustration]
But they are to be divided in the first place, according to sciences.
For of corollaries, some are geometrical, but others arithmetical.
Thus the present corollary is geometrical: but that which is added at
the end of the second theorem of the seventh book of the arithmetical
elements, is arithmetical. But afterwards they must be divided
according to the principal objects of enquiry. For some things are
consequent to problems, but others to theorems. Thus, the present is
consequent to a theorem: but that which is placed in the second of the
seventh book, is consequent to a problem. But in the third place, they
must be divided according to their ostensions. For some are exhibited,
together with ostensive methods, but others together with deductions
to an impossible. Thus the present is shewn by a direct ostension:
but that which is at the same time exhibited in the first of the
third book, appears, together with a deduction to an impossible. But
corollaries may also be divided in many other modes, but these may
suffice our present purpose. The present corollary, however, teaching
us that the place about one point is distributed into angles equal to
four right, is subservient to that admirable theorem, which shews that
the following three multangles about one point, can alone fill place,
viz. the equilateral triangle, the quadrangle, and an equilateral, and
equiangular sexangle. But the equilateral triangle must be six times
assumed; since six two-thirds, form four right angles. But the sexangle
must be three times formed; for every sexangular angle is equal to one
right, and a third part of a right. And a quadrangle must be four times
assumed: for every quadrangular angle is right. Hence, six equilateral
triangles conjoined according to their angles, fill four right
angles, as also three sexangles, and four quadrangles. But all other
multangles, however composed, according to angles, are either deficient
from four right, or exceed four right angles[21]; while these alone,
according to the aforesaid numbers, are equal to four right. And this
theorem is Pythagoric. But by the present corollary, if even more than
two right lines should cut each other in one point, as for instance,
three or four, or any other number, all the angles which they form,
may be shewn to be equal to four right. For they will vindicate to
themselves the place of four right angles. But it is manifest that the
angles always become double to the number of right lines. And thus two
right lines intersecting each other, there will be four angles equal
to four right: but from the intersection of three lines, there will
be six angles; and from four, eight, and so on, in infinitum. For the
multitude of the right lines is always doubled: but the angles increase
according to multitude, and are diminished according to magnitude,
because it is the same four right angles, which is perpetually divided.
PROPOSITION XVI. THEOREM IX.
In every triangle having one side produced, the external angle
is greater than either of the internal and opposite angles.
Those who enunciated this proposition, and at the same time omitted
the particle, _having one side produced_, perhaps afforded an occasion
of objection to many others, as well as to Philip, (according to
the narration of the mechanist Heron.) But such as were desirous of
entirely removing this calumny, enunciated the theorem, with the
proposed addition, corresponding with the general manner of the
geometrician. For in the fifth theorem, being desirous to shew, that
the angles under the base of an isosceles triangle are equal, he adds,
that when the equal right lines are produced, the angles under the
base are equal. Hence we infer, that though this proposition might
be defective and imperfect it various copies, yet it was perfect and
written entire, by the institutor of the Elements. What then does the
proposition assert? _That in every triangle, if you produce one of its
sides, you will find the angle constituted external to the triangle,
greater than either of the internal and opposite angles._ For a little
after, this angle will be shewn equal to both, but it is proved to be
greater than either in the present; and he necessarily compares it
with the opposite angles, and not with the _successive_ angle. For
to this last it may be both equal and less: but it is greater than
either of the former. Thus, if this triangle should be right angled,
and you conceive one of the sides comprehending the right angle to be
produced, the external will be equal to the successive angle. But if it
should happen to be obtuse-angled, the internal angle may be greater
than the external; and it is on this account that he does not compare
the external with the successive angle, but with the opposite angles.
For of the angles within a triangle, the successive angle borders on
the external, but the two others are opposite. Hence, the external
angle is greater than either of the successive, but may not exceed the
successive angle to which it is proximate. But some conjoining these
two theorems, I mean the present, and the following, enunciate the
proposition thus. _In every triangle having one side produced, the
external angle is greater than either of the internal and opposite
angles; and any two of the internal angles, are less than two right._
But there is occasion for the connection of these theorems, because
the geometrician himself, a little after, enunciates the proposition
after this manner, in equal angles, for he says: _In any triangle
having one of its sides produced, the external angle is equal to the
two interior, and opposite angles; and the three internal angles of a
triangle, are equal to two right_. Hence, they think it proper in the
present similar case, to connect the objects of investigation, and
to make the proposition a composite. But if the datum be enunciated
with this addition, it also will be a composite, (since it is
requisite to understand two things, viz. the subject triangle, and
one side produced:) and if the datum be given without this, it will
be a _composite_ in capacity, but _simple_ in energy; for this must
be received at the same time as a datum; since while we suppose an
external angle, we must pre-suppose the side as produced.
[Illustration]
[Illustration]
But we may assume from the present theorem, that it is impossible
from the same point, for three equal right lines to fall on the same
right line. Thus let there be drawn from one point _a_, three equal
right lines, _a b_, _a c_, _a d_, to the right line _b d_. Because,
therefore, _a b_ is equal to _a c_, the angles at the base are equal.
Hence, the angle _a b c_, is equal to the angle _a c b_. Again, because
_a b_ is equal to _a d_, the angle _a b d_, is equal to the angle _a
d b_. But the angle _a b c_, was equal to the angle _a c b_. Hence,
the angle _a c b_, is equal to the angle _a d b_, the external, to
the internal and opposite, which is impossible. From the same point
therefore, to the same right line, three equal right lines cannot be
drawn. But by the present theorem, we can also demonstrate, that if a
right line falling on two right lines, makes the external angle equal
to the internal and opposite, those right lines will by no means make
a triangle, nor coincide, because the same thing would be both greater
and equal, which is impossible. Thus for example, let _a b_, _c d_,
be right lines, and let the right line _e b_ falling on them make the
equal angles _a b d_, _c d e_, the right lines _a b_, _c d_, will not
coincide. For if they coincide, the equal angles remaining, the angle
_c d e_, will be equal to the angle _a b d_. And since it is external,
it will be greater than the internal and opposite angle. Hence, it is
necessary, if they coincide, that the angles remain no longer equal,
but that the angle at the point _d_, be augmented. For whether _a b_
remaining immoveable, we conceive that _c d_ is moved towards it, so
as to coincide in the point _c_, we shall produce a greater distance
in the angle _c d e_; since _c d_ approaches to _a b_, in the same
proportion as it recedes from _d e_. Or whether _c d_, abiding, we
conceive that _a b_ is moved towards it, in a similar manner, we shall
by this means diminish the angle _a b d_; for it is at the same time
carried towards _c d_, and to _b d_. Or whether we conceive both of
them tending to each other, we shall find that _a b_ by tending to _c
d_, contracts the angle _a b e_; and that _c d_, by receding from _d
e_, on account of the motion to the line _a b_, increases the angle _c
d e_. Hence, it is necessary, if it be a triangle, and if the right
lines _a b_, _c d_, coincide, that the external angle must be also
greater than the internal and opposite angle. For either the internal
angle remaining, the external is increased, or the external abiding,
the internal is diminished, or the internal is contracted, and the
external is more dilated. But the cause of these consequences is the
motion of the right lines, the one tending to those parts, where it
diminishes the internal angle, but the other to the parts where it
increases the external. And from this the reader should consider, how
the origin of things produces the true causes of enquiries, which we
have previously surveyed.
PROPOSITION XVII. THEOREM X.
The two angles of every triangle, taken all possible ways, are less
than two right.
[Illustration]
In the present theorem he shews indeterminately, that any two angles
of a triangle, are less than two right, but in the following theorems
he determines how much they are less, and that they are deficient by
the remaining angle of the triangle: for its three angles are equal to
two right; and on this account the two remaining angles are less than
two right. And, indeed, the demonstration of the elementary institutor
proceeds in a manifest order; for it uses the preceding theorem. But
if is necessary, as in the last proposition, by regarding the origin
of triangles, to find the cause of the present symptom. Let then the
right lines _a b_, _c d_, be at right angles to _b d_. If these lines
then are to form a triangle, it is requisite they should incline to
each other. But their inclination diminishes the internal angles, on
which account they become less than two right: for they were right
before their inclination. In like manner, if we conceive right lines
standing at right angles, on the side _a b_, the same consequences will
ensue respecting the inclination of the right lines; and the angles at
the points _a_, _b_, will be less than two right; and so of the other
side. This then is the cause of the proposition, and not the external
angle being greater than either of the internal, and opposite angles:
since it is not necessary that the side should be produced, nor that
any angle should be constituted external to the triangle; but it is
necessary that any two of the internal angles should be less than two
right. Hence, it is necessary, as I have said, that the cause of this
theorem should be the inclination of the right lines diminishing the
angles at the base. But as the institutor of the elements exhibits
the object of enquiry, by the external angle, we may accomplish this,
without producing any one of the sides. Thus let there be a triangle
_a b c_, and let there be taken in the side _b c_, any point _d_, and
let _a d_ be connected. Because, therefore, one side of the triangle _a
b d_, is produced, viz. _b d_, the external angle _a d c_, is greater
than the internal _a b d_. Again, because one side of the triangle _a
d c_ is produced, viz. _c d_, the external angle _a d b_, is greater
than the internal _a c d_. But the angles about the right line _a d_,
are equal to two right, by the thirteenth of this. Hence, the angles _a
b c_, _a c b_, are less than two right. In like manner, we may shew,
that the angles _b a c_, and _b c a_, are less than two right, by
taking a point in the side _a c_, and by connecting the point _b_ with
the assumed point. And again, we may affirm, that the angles _c a b_,
_a b c_, are less than two right, by taking a point in the side _a b_,
and by connecting a right line, from the point _c_, and the received
point. And thus the thing proposed, is exhibited by the same theorem,
without producing any side of the triangle. Hence, it is possible, that
by this, the theorem may be proved, which asserts, _that from the same
point, two perpendiculars cannot be drawn, to one right line_. For let
there be drawn, if possible, from the point _a_, two perpendiculars _a
b_, _a c_, to the right line _b c_. Then the angles _a b c_, _a c b_,
are right. But because _a b c_ is a triangle, two of its angles are
less than two right. The angles, therefore, _a b c_, _a c b_, are less
than two right. But they are also equal to two right, because they are
perpendiculars, which is impossible. Hence, from the same point, to the
same right line, two perpendiculars cannot be drawn.
[Illustration]
PROPOSITION XVIII. THEOREM XI.
The greater side of every triangle, subtends the greater angle.
That the equality of the sides in every triangle, forms the equality
of the angles which they subtend, and that in like manner the equality
of the angles shews the equality of their subtending sides, we learn
from the fifth and sixth theorems. But that the equality of those
angles, which are subtended by the sides, follows the inequality of
the sides, and the contrary we now learn by the present eighteenth
and nineteenth theorems. For the one shews that the greater angle is
contained under the greater side, but the other, that the greater side
subtends the greater angle; because these are mutually converted, but
the same symptoms are contemplated in things contrary, as in the fifth
and sixth theorems. But it is manifest, that we proportionally assume
the greater and less side, in scalene triangles, that we distinguish
the greatest, middle, and least, and the angles in a similar manner:
but in isosceles triangles, the greater and less, simply assumed, are
sufficient; for there is one side which is unequal to two, because
it is either greater or less, as these theorems cannot take place in
equilateral triangles. And here you may observe, that the theorems
which exhibit the equality of angles or sides, agree with equilateral
and isosceles triangles: but these which exhibit inequality to such
as are isosceles and scalene. But the cause of this is, because
of triangles, some are produced from equality alone, others from
inequality alone, and others from the conjunction of both, which are
partly constituted from equality, and partly through inequality. And
some are allied to _bound_, others to _infinity_, and others are
generated from the mixture of both. Hence the ternary permeates through
all geometrical forms, as through lines, angles, and figures; and among
figures, through such as are trilateral, quadrilateral, and all the
rest in a consequent order. But _bound_, likewise, must be considered
as inherent in geometrical forms, as well through similitude, as
equality; and _infinite_, both by dissimilitude, and inequality;
and that which is mixt, sometimes from the junction of similitudes,
and dissimilitudes, and sometimes from the union of equalities, and
inequalities. But the reason of this also, is because geometrical forms
regard both quantity and quality. And we have assigned these, because,
when we have determined these two, it will be manifest to us, that
when the institutor of the Elements says, _of every triangle_, he does
not also speak of the equilateral, but of that which has a greater and
less side: for it is necessary to consider the object of enquiry, as
consequent to the preceding datum; and that the triangle which has a
greater and less side, contains a greater angle, under the greater
side.
[Illustration]
But because the geometrician, when in the construction he receives
the triangle _a b c_, and the side _a c_, greater than the side _a b_,
in order that he may shew, that the angle at the point _b_ is greater
than at the point _c_, from the side _a c_, he cuts off a right line
_a d_, equal to the side _a b_: on this account it may be said that it
is necessary to make the ablation at the point _c_, let us therefore
exhibit the thing proposed upon this hypothesis, according to Porphyry,
as follows. Let _d c_ be equal to _a b_, and produce _a b_ to the point
_e_, and place _b e_ equal to _d a_. The whole, therefore, _a e_, is
equal to the whole _a c_. Connect _e c_. Because, therefore, _a e_ is
equal to _a c_, the angle, also, _a e c_, is equal to the angle _a c
e_, (by the fifth). Hence, the angle _a e c_ is greater than the angle
_a c b_. But the angle, also, _a b c_ is greater than the angle _a e
c_; because one side of the triangle _c b e_ is produced, viz. _b e_,
and so the angle _a b c_, since it is external, is greater than the
internal and opposite angle. Much more, therefore, is the angle _a b
c_, greater than the angle _a c b_, which was to be shewn. And such are
the geometrical exhibitions of the present theorem.
[Illustration]
[Illustration]
[Illustration]
But it is manifest that the cause of this symptom is the
amplification, or diminution according to magnitude, of the side
subtending the angle. For when it is greater, it more amplifies the
angle; but when less, at the same time it diminishes, and gives a
greater contraction to the angle. And this takes place on account of
the right line being situated in its extremities: for through its
being placed in its extremities, it changes likewise the magnitudes of
the angles, according to the increase and decrease which it receives.
And this we affirm in one triangle, since it is possible that the
same angle may be subtended by a greater or less right line; and that
the same right line may subtend a greater and less angle. For let the
triangle happen to be an isosceles one, _a b c_, and let there be
taken in the side _a b_, a point _a_, and let _a e_ be taken equal to
_a d_, and connect _d e_. The right lines therefore, _d e_, _b c_,
subtend the angle at the point _a_, of which the one is greater, but
the other less. And in the same manner infinite right lines, greater
and less subtending the angle _a_. Again, let the triangle _a b c_, be
isosceles, and let _b c_ be less than _b a_, _a c_, and construct upon
_b c_, an equilateral triangle _b d c_, and connect _a d_, and produce
it to _e_. Because, therefore, the angle _b d e_, of the triangle
_a b d_, is external, it is greater than the angle _b a d_. In like
manner the angle _c d e_, is greater than the angle _c a d_. The whole,
therefore, _b d c_, is greater than the whole _b a c_, and the same
right line subtends both, viz. the greater and the less angle. But it
is shewn, that likewise greater and less right lines subtend the same
angle. But in one and the same triangle, one right line subtends one
angle, and the greater always the greater, and the less always the
less, the cause of which we have contemplated.
PROPOSITION XIX. THEOREM XII.
The greater side of every triangle subtends the greater angle.
This is the converse of the preceding theorem; and the datum, as well
as the object of enquiry, is simple in each. Add too, that what was
conclusion there, is hypothesis here: and what was hypothesis there
is conclusion in this. But the former precedes, because it has the
inequality of the sides given; and this follows, because it supposes
unequal angles. For _sides_, indeed, seem to contain right-lined
figures, but the _angles_ appear to be contained; and the mode of
demonstration in the former is ostensive, but in this it concludes the
thing proposed by a deduction to an impossibility. The geometrician,
therefore, by division, reasons concerning that which is impossible:
for the angles being unequal, _I say_, (says he) _that the sides also
subtending the unequal angles are unequal; and the greater subtends
the greater given angle_. For if that which subtends the greater angle
is not greater, it is either equal, or less. But if it be equal, the
angles also which they subtend, are equal by the fifth. But if less,
the angle also which it subtends, is less by the preceding: for it was
shewn that the greater side subtends the greater angle, and the less
the lesser. But the angles have a contrary position; and hence, the one
side is greater than the other.
[Illustration]
But it is possible that we may exhibit the thing proposed, without
this division. For if the angle of a triangle be bisected, and the
right line drawn to the base, cutting the angle, divides it into
unequal parts, the sides containing that angle will be unequal, and the
greater will be that which coincides with the greater segment of the
base, but the less that which coincides with the lesser. Let there be
a triangle _a b c_, and let the angle at _a_ be bisected, by the right
line _a d_, and let _a d_ cut the base _b c_, into unequal parts, and
let _c d_ be greater than _b d_. I say that the side _a c_ is greater
than the side _a b_. Produce _a d_ to the point _e_, and place _d e_
equal to _a d_. And because _d c_ is greater than _d b_, place _d f_
equal to _b d_, and connect _e f_, and produce it to the point _g_.
Because, therefore, _a d_ is equal to _d e_, and _b d_ to _d f_, the
two are equal to the two, and they comprehend equal angles at the
vertex. Hence, the base _b a_, is equal to the base _e f_, and all,
therefore, are equal to all. On this account also the angle _d e f_, is
equal to the angle _d a b_. But this is not unequal to _d a g_. Hence,
the side _a g_, is equal to the side _e g_, by the sixth. The side,
therefore, _a c_, is greater than the side _e f_. But the side _f e_,
is equal to the side _a b_; and hence, the side _a c_, is greater than
the side _a b_, which was to be demonstrated.
[Illustration]
This being pre-assumed, we can shew that the greater side subtends
the greater angle. Let there be a triangle _a b c_, having the angle
at the point _b_, greater than the angle at the point _c_. I say that
the side _a c_, is greater than the side _a b_. Let _b c_ be bisected
in the point _d_, and connect _a d_, and draw _d e_, equal to _a d_,
and connect _b e_. Because, therefore, _b d_, is equal to _d c_, and _a
d_, to _d e_, the two are equal to the two, and they comprehend equal
angles at the vertex. Hence, the base _b e_, is equal to the base _a
c_, and all are equal to all. Hence too, the angle _d b e_, is equal to
the angle at the point _c_, but less than the angle _a b d_. The angle,
therefore _a b e_, is bisected by the right line _b f_. Hence, _e f_,
is greater than _f a_. Because, therefore, the angle at the point _b_,
of the triangle _a b e_, is bisected by the right line _b f_, and _e f_
is greater than _f a_, it follows from what has been previously shewn,
that the side _b e_, is greater than the side _b a_. But _b e_ has been
shewn to be equal to _a c_. The side, therefore, _a c_, is greater
than the side _a b_; and the object of enquiry is exhibited. And it is
manifest that the institutor of the Elements, avoiding a variety of
demonstration, refrains from this mode of demonstrating, and employs a
method of proof, which leads from division to an impossibility, because
he was willing to fabricate the converse to the preceding, without
any intervening medium. For the eighth theorem, indeed, which is the
converse of the fourth, brings great disturbance, because it makes
conversion difficult to be known. For it is more excellent to exhibit
converse theorems, by preserving the continuity through an impossible,
than to destroy the continuity by a principal demonstration. And
hence, Euclid shews almost all converse theorems by a deduction to an
impossibility.
PROPOSITION XX. THEOREM XIII.
Two sides of every triangle, however taken, are greater than the
remaining one.
The Epicureans oppose the present theorem, asserting that it is
manifest even to an ass; and that it requires no demonstration: and
besides this, that it is alike the employment of the ignorant, to
consider things manifest as worthy of proof, and to assent to such
as are of themselves manifest and unknown; for he who confounds
these, seems to be ignorant of the difference between demonstrable
and indemonstrable. But that the present theorem is known even to an
ass, they evince from hence, that grass being placed in one extremity
of the sides, the ass seeking his food, wanders over one side, and
not over two. Against these we reply, that the present theorem is
indeed manifest to sense, but not to reason producing science: for
this is the case in a variety of concerns. Thus for example, we are
indubitably certain from sense, that fire warms, but it is the business
of science to convince us how it warms; whether by an incorporeal
power, or by corporeal sections; whether by spherical, or pyramidal
particles. Again, that we are moved is evident to sense, but it is
difficult to assign a rational cause how we are moved; whether over an
impartible, or over an interval: but how can we run through infinite,
since every magnitude is divisible in infinitum? Let, therefore, the
present theorem, that the two sides of a triangle are greater than the
remainder, be manifest to sense, yet it belongs to science to inform us
how this is effected. And thus much may suffice against the Epicureans.
[Illustration]
[Illustration]
But it is requisite to relate the other demonstrations of the present
theorem, such as Heron, and the familiars of Porphyry have fabricated,
without producing the right line, after the manner of Euclid. Let there
be a triangle _a b c_, it is requisite, therefore, to shew, that
the sides _a b_, _a c_, are greater than the side _b c_. Bisect the
angle at _a_, by the right line _a e_. Because, therefore, the angle
_a e c_, is external to the triangle _a b e_, it is greater than the
angle _b a e_. But the angle _b a e_, was placed equal to the angle
_e a c_. The angle, therefore, _a e c_, is greater than the angle _e
a c_. Hence, the side also _a c_, is greater than the side _c e_. And
for the same reason the side _a b_, is greater than the side _b e_. For
the angle _a e b_ is external to the triangle _a e c_, and is greater
than the angle _c a e_; that is than the angle _e a b_. And on this
account the side _a b_, is greater than the side _b e_. The sides,
therefore, _a b_, _a c_, are greater than the whole side _b c_. And the
like may be shewn of the other sides. Let there again be a triangle
_a b c_. If therefore the triangle _a b c_, be equilateral, two sides
will be doubtless greater than the remaining one: for when there are
three equal quantities, any two are double of the remainder. But if it
be isosceles, it will have a base either less, or greater than each of
the equal sides. If therefore the base be less, the two sides are given
greater than the remainder. But if the base be greater, let it be _b
c_, and cut off from it a part equal to either of the sides, which let
be _b e_, and connect _a e_. Because, therefore, the angle _a e c_, is
external to the triangle _a e b_, it is greater than the angle _b a e_.
On the same account the angle _a e b_, is greater than the angle _c a
e_. Hence, the angles about the point _e_, are greater than the whole
angle about the point _a_, of which _b e a_ is equal to _b a e_, since
_a b_ is equal to _b e_. The remainder, therefore, _a e c_, is greater
than the remainder _c a e_. Hence, the side _a c_, is greater than the
side _c e_. But the side _a b_, was also equal to the side _b e_. The
sides, therefore, _a b_, _a c_, are greater than the side _b c_.
[Illustration]
But if the triangle _a b c_, be scalene, let the greatest side be
_a b_, the middle _a c_, and the least _c b_. The greatest side,
therefore, assumed with either of the others, exceeds the remainder:
for by itself it is greater than either. But if we are desirous of
shewing that the sides _a c_, _c b_, are greater than the greatest
side _a b_, we must employ the same construction as in the isosceles
triangle, cutting off from the greater side, a part equal to one of
the other sides, and connecting the line _c e_, and using the external
angles of the triangles.
[Illustration]
Let there be again any triangle _a b c_. I say that the sides _a b_,
_a c_, are greater than the side _b c_. For if they are not greater,
they are either equal or less. Let them be equal, and cut off _b e_,
equal to _a b_. The remainder, therefore, _e c_, is equal to _a c_.
Because then, _a b_, _b e_, are equal, they subtend equal angles; and
this is likewise true of _a c_, _e c_, because they are equal. Hence,
the angles at the point _e_, are equal to the angles at the point _a_,
which is impossible. Again let the sides _a b_, _a c_, be less than _b
c_, and cut off _b d_, equal to _a b_, and _e c_ to _a c_. Because,
therefore, _a b_ is equal to _b d_, the angle _b d a_, is not unequal
to the angle _b a d_. And because _a c_ is equal to _c e_, the angle
_c e a_, is equal to the angle _e a c_. Hence, the two angles _b d a_,
_c e a_, are equal to the two _b a d_, and _e a c_. Again, because
the angle _b d a_ is external to the triangle _a d c_, it is greater
than the angle _e a c_: for it is greater than _c a d_. By a similar
reason also, because the angle _c e a_, is external to the triangle
_a b e_, it is greater than the angle _b a d_: for it is greater than
the angle _b a e_. Hence, the angles _b d a_, _c e a_, are greater
than the two _b a d_, _e a c_. But they were also equal to them, which
is impossible. The sides, therefore, _a b_, _a c_, are neither equal
to, nor less than the side _b c_, but greater. And the like may be
exhibited in others.
[Illustration]
PROPOSITION XXIV. THEOREM XIV.
If upon one side of a triangle, two right lines beginning from
the extremities, are internally constituted, the constituted
right lines will be less than the other sides of the triangle,
but they will contain a greater angle.
[Illustration]
That which is expressed by the proposition, is, indeed, manifest; and
the demonstration adopted by the elementary institutor, is evident; and
the theorem is consequent to the first principles, since it depends on
two theorems, the one previously exhibited, and the sixteenth. For in
order to shew, that the lines internally constituted, are less than the
external, the theorem is required, which says, _the two sides of every
triangle, are greater than the remaining one_: but for the purpose of
confirming that the angle comprehended by them is greater than that
comprehended by the external sides, that theorem procures the greatest
utility, which says, _the external angle of every triangle, is greater
than the internal and opposite angle_. But you will receive at the
same time, conviction of geometrical diligence, and a commemoration of
things admirable in the mathematical disciplines, if we shall shew that
_it is possible within a certain triangle, upon one of its sides, not
upon the whole, but upon some one of its parts, to constitute two right
lines greater than the external right lines[22]; and again, others
comprehending a less angle, and comprehended in the angle made by the
external lines_. For this being exhibited, it will at the same time be
manifest, that the institutor of the Elements necessarily adds, that
the internally constituted lines must begin from the extremities of
the common basis; and must be constituted upon one whole side, and not
upon any one of its parts: but likewise, as I have said, one of the
admirable things which geometry contains, will be manifest. For is it
not, indeed, admirable, that the lines constituted upon the whole side,
should be less than the external sides: but that those constituted upon
a part should be greater? Let there be then a right angled triangle _a
b c_, having the angle at the point _b_ right, and take in the side _b
c_, any point _d_, and connect _a d_. Hence, _a d_ is greater than _a
b_. Take from _a d_, a part equal to _a b_, which let be _d e_, and
bisect _e a_, in the point _f_, and connect _f c_. Because, therefore,
_a f c_ is a triangle, the lines _a f_, _f c_, are greater than _a c_.
But _a f_ is equal to _f e_. The right lines therefore, _f e_, _f c_,
are greater than _a c_. But _d e_ is equal to _a b_. Hence, the right
lines _f c_, _f d_, are greater than the right lines _a b_, _a c_, and
they are internally constituted.
[Illustration]
Let there be again an isosceles triangle _a b c_, having the base
_b c_, greater than either of the equal sides. Then from _b c_, cut
off _b d_, equal to _a b_, and connect _a d_, and take in _a d_, any
point _e_, and connect _e c_. Because, therefore, _a b_, is equal to
_b d_, the angle _b a d_, is also equal to the angle _b d a_. And
because the angle _b d a_ is external to the triangle _e d c_, it
is greater than the internal and opposite _d e c_. Hence, the angle
_b a d_, is greater than the angle _d e c_. Much more, therefore,
is the angle _b a c_, greater than the angle _d e c_; and _b a c_
is contained by the external lines, but _d e c_ by internal lines.
Within a triangle, therefore, right lines _d e_, _e c_, comprehending
a lesser angle, are constituted within the angle comprehended by the
external lines; and the thing proposed is shewn without employing
the parallel lines of expositors. Hence, it is necessary that the
constituted right lines should begin from the extremities of the basis:
for those which are constituted upon any one of its parts, are shewn
to be sometimes greater than the external lines, and to comprehend a
lesser angle. But when they are constituted in this manner, beginning
from the extremities, the species of triangles, called (ἀκιδοειδῆ) or,
similar to the point of a spear[23], presents itself to our view; and
is one of the admirable things contained in geometry, viz. to find a
quadrilateral triangle. As for example, the triangle A B C. For it
is contained by the four sides B A, A C, B D, D C; but it has three
angles, one at B, the other at A, and the other at C. And hence, the
present figure is a quadrilateral triangle.
[Illustration]
PROPOSITION XXII. PROBLEM VIII.
To construct a triangle from three right lines, which are equal
to three given right lines. But it is requisite that two of the
lines must be greater than the remaining one, in whatever manner
they may be taken.
[Illustration]
We again pass to problems, and Euclid commands us to construct a
triangle from three proposed right lines, two of which are greater than
the remaining one, equal to given right lines. Because he knew this in
the first place, that it was impossible to construct a triangle from
those same lines, which had already received the declared position: but
that this was possible to be effected from their equals. In the next
place, he knew it was necessary that two of the right lines about to
complete the triangle, should be greater than the remaining one: for
the two sides of every triangle are greater than the remaining one,
however assumed, as we have shewn. On this account he adds, _that it
is necessary the first right lines remaining, to construct a triangle
from three equal to them: but that it is requisite, any two, however
taken, should be greater than the remainder, or there will not be a
triangle from three lines equal to the given right lines_. But by this
means he also destroys all the objections which are urged against the
construction, and which may be perfectly dissolved by this addition.
Hence, the present problem ranks among things determined, and not among
such as are indetermined: For of problems as well as of theorems, some
are indeterminate, but others without termination. Thus if we should
simply say, _from three right lines which are equal to three given
right lines, to construct a triangle_, the problem is indeterminate and
impossible. But if we add, _two of which, however assumed, are greater
than the remainder_, the problem is determined and possible. For as
the division of theorems takes place, according to true and false, so
that of problems according to a possible and impossible enunciation.
But that the objections which are urged against the construction, may
be from hence dissolved, we shall learn from a little inspection: for
we shall follow the words of the geometrician. Let there be three right
lines, _a b c_, of which any two, however taken, are greater than the
third, and let it be required to accomplish the thing proposed. Let
there be placed a certain right line _d e_, on one part finite, as
at the point _d_: and on the other part infinite. Then place _d f_
equal to _a_, but _f g_ to _b_: and _g h_ to _c_. And from the centre
_f_, but interval _f d_, let a circle _k_ be described. Again, with
the centre _g_, but interval _g h_, let the circle _l_ be designed;
and the circles will intersect each other. For this is assumed by the
institutor of the Elements. But it may be asked how this takes place?
For perhaps they either only touch each other, or they do not even
touch. Since it is necessary that they should suffer some one of three
cases, I mean that they should either intersect or touch, or be distant
from each other. I say, therefore, that they necessarily intersect each
other. For let them in the first place, touch each other. Because,
therefore, the point _f_ is the centre of the circle _k_, _d f_ is
equal to _f n_. And because the point _g_ is the centre of the circle
_l_, _h g_ is equal to _g m_. The two, therefore, _d f_, _g h_, are
equal to one, viz. to _f g_. But they were placed greater than one, as
also _a_, together with _c_, is greater than _b_. They are therefore
equal to it, and at the same time greater, which is impossible. Again,
if it be possible, let the circles be distant from each other, as _k_
and _l_. Because, therefore, the point _f_, is the centre of the circle
_k_, _d f_, is equal to _f n_. And because the point _g_, is the centre
of the circle _l_, _h g_ is equal to _g m_. The whole, therefore, _f
g_, is greater than the two, _d f_, _h g_: for _f g_, exceeds _d f_, _g
h_, by _n m_. But it was supposed that _d f_, _h g_, were greater than
_f g_, in the same manner as _a_ and _c_ are greater than _b_. For _d
f_ was placed equal to _a_, but _f g_, to _b_, and _h g_ to _c_. It is
necessary, therefore, that the circles _k_ _l_, should intersect each
other. Hence, the institutor of the Elements very properly receives
them cutting one another: since of the three right lines, he supposes
two greater than the third, however, they may be assumed, but neither
equal to, nor less than one. But it is necessary that when the circles
touch, two of the lines should be equal to the third; and that when
they are distant from each other, two should be greater than the
remainder.
[Illustration]
[Illustration]
PROPOSITION XXIII. PROBLEM IX.
On a given right line, and at a given point in it, to constitute an
angle equal to a given right lined angle.
[Illustration]
This also is a problem, whose invention according to Eudemus is
rather the gain of Oenopides than of Euclid: but it requires the
construction of an angle, on a given right line; and at a given
point in it equal to another right lined angle. This, then, Euclid
necessarily adds, that the given angle must be rectilineal; because
it is impossible that an angle can be construed on a right line equal
to every angle. For it has been shewn[24] that there are only two
curve-lined angles equal to right-lined angles, viz. the angle of a
lunular figure, which we have proved equal to every right-lined angle;
and the angle of that figure similar to an axe[25], which is equal
to two thirds of a right angle. But a lunular figure of this kind,
which is called (πελεκοειδὲς) Pelecoides, is formed from two circles
cutting each other through their centres. However, the construction
of an angle on a certain right line, causes the constituted angle to
become determinate, and not indifferent in species, but forms it either
right-lined, or mixt. But since no mixt can be equal to a right-lined
angle, it is manifest that this must be perfectly rectilineal. The
institutor of the Elements, therefore, simply using the present
problem, and constructing a triangle from three right lines, equal to
three given lines, accomplishes the thing proposed. But you may receive
a more exquisite construction of the triangle, by the following method.
Let there be a given right line _a b_, and a given point in it _a_,
and a given right-lined angle _c d e_. It is required, therefore, to
accomplish the problem. Connect _c e_, and produce _a b_ on both sides
to the points _f_, _g_. Then place _f a_, equal to _c d_, and _d e_
to _a b_, and _b g_ to _e c_. And with the centre _a_, but interval
_a f_, describe the circle _k_. And again, as in the preceding, with
the centre _b_, but interval _b g_, describe the circle _l_. The
circles, therefore, will cut each other, as we have shewn in the last
proposition. Let them cut each other in the points _m_, _n_, and from
these points draw right lines to the centres as in the figure. Because,
therefore, _f a_, is equal to _a m_, and _a n_, but _c d_, is equal to
_f a_; _a m_, and _a n_, will be each equal to _c d_. Again, because _b
g_ is equal to _b m_, and _b n_, but _g b_ is not unequal to _c e_; _b
m_, and _b n_, will be also equal to _c e_. But _a b_ is equal to _d
e_. The two therefore, _a b_, _a m_, are not unequal to the two _d e_,
_d c_, and the base _b m_, is equal to the base _c e_. Hence, the angle
_m a b_, is equal to the angle at the point _d_. And again, the two _n
a_, _a b_, are equal to the two _c d_, _d e_, and the base _n b_, is
equal to the base _c e_. The angle, therefore, _n a b_, is equal to
the angle _c d e_, and the thing proposed is doubly accomplished: for
we have not only constituted one, but two angles, equal to the given
angle, on each side of the right line _a b_; so that in whatever part
we may desire the construction to be made, it will be indubitable, and
without contradiction. And this we have added to the construction of
the elementary institutor.
[Illustration]
But we cannot praise the method of Apollonius, because it requires
the assistance of the third book. For he receives any angle _c d e_,
and a right line _a b_, and with the centre _d_, but interval _c d_,
he describes a circumference _c e_. In like manner with the centre
_a_, but interval _a b_, he describes a circumference _b f_; and
intercepting a circumference _c e_, equal to _b f_, he connects the
right line _a f_, and affirms that the angles _a_ and _d_, insisting
on equal circumferences, are equal. But it is necessary to pre-assume
that _a b_ is equal to _c d_, in order that the circles may be also
equal. We therefore think that a demonstration of this kind requiring
posterior propositions, is foreign from an elementary institution; and
we give the preference to that of the geometrician, as consequent to
principles.
PROPOSITION XXIV. THEOREM XV.
If two triangles have two sides equal to two, each to each, but
the one angle contained by the equal right lines greater than
the other: they shall also have the base of the one greater than
the base of the other.
Euclid again passes on to theorems, and speaks concerning inequality
in two triangles, in a manner similar to his discourse concerning
equality. For supposing two triangles, having two sides equal to two,
each to each, he sometimes places the vertical angle equal in each, and
sometimes unequal; and he proceeds in a similar manner with respect
to the base. Besides this, he demonstrates that the equality of the
bases is consequent to the equality of the vertical angle, and that the
equality of the vertical angles, is consequent to the equality of the
bases: but he now shews that the inequality of the one, follows the
inequality of the other. The present theorem, therefore, is opposite
to the fourth: for that, indeed, supposes the vertical angles of the
triangles equal, but this supposes them unequal. And that demonstrates
the equality of their bases; but this proves them unequal, in the
same manner as their angles. It precedes, however, the following
theorem: for that deduces its proof of inequality from the bases to
the angles subtending the bases: but this, on the contrary, reasons
from the angles to the bases, which are under the angles. Hence it
is, after this manner, the converse of its consequent proposition,
but opposite to the eighth theorem. For the one from the equality
of the bases, demonstrates the equality of the vertical angles, but
the other from the inequality of the bases, shews that the vertical
angles are unequal. It is, however, common to these four (two of which
are conversant with equality, I mean the fourth, and the eighth, but
two about inequality, the present and the following; and two begin
from angles, viz. the fourth, and the object of investigation in
the present, but two from bases, viz. the eighth, and the following
proposition); it is common, I say, to all these four, as well to the
fourth and the eighth, as to the twenty-fourth and twenty-fifth, to
have two sides equal to two, each to each. For these being unequal, all
enquiry is superfluous, and subject to deception. And thus much for a
universal speculation concerning the present theorem.
[Illustration]
[Illustration]
But let us now consider the construction of the elementary
institutor, and add to it where deficient. For Euclid receiving two
triangles, _a b c_, _d e f_, having the sides _a b_, _a c_, equal to
the sides _d e_, _d f_, each to each, and the angle at the point _a_,
being greater than the angle at the point _d_, and willing to shew that
the base _b c_, is greater than the base _e f_, on the right line _e
d_, and at a point in it _d_, constitutes an angle _e d h_, equal to
the angle at the point _a_. For the angle at the point _a_, is greater
than the angle at the point _d_, and he connects _d h_, equal to _a c_.
The right line, therefore, _e h_, produced to the point _h_, either
falls above, or upon, or beneath the line _e f_. The institutor of the
Elements, indeed, considers it as lying above the line. But let it be
upon the right line. Again, therefore, we may exhibit the same. For the
two _a b_, _a c_, are equal to the two _d e_, _d h_, and they contain
equal angles. Hence, the base _b c_, is equal to the base _e h_. But
_e h_ is greater than _e f_; and on this account _b c_ is greater than
_e f_. Again, let it be placed beneath _e f_. Connecting, therefore,
_e h_, we must say, that since _a b_, _a c_, are equal to _d e_, _d
h_, and they comprehend equal angles, _b c_ is also equal to _e h_.
Because, therefore, within the triangle _d e h_, two right lines _d
f_, _f e_, are constructed on the side _d e_, they are less than the
external sides. But _d h_, is equal to _d f_: for it is equal to _a
c_. Hence _h e_ is greater than _e f_. But _h e_ is equal to _b c_.
And therefore, _b c_ is greater than _e f_. The theorem, therefore,
is exhibited according to every position. Why then, as is the fourth
theorem, he at the same time demonstrated that the areas of triangles
are equal, does he not add in the present, that besides the inequality
of the bases, the areas also are unequal? Against this doubt we must
say, that there is not the same proportion in equal, as in unequal
angles and bases. For when the angles and bases are equal, the equality
also of the triangles follows: but when they are unequal, it is not
necessary that the inequality of the areas should be consequent;
since the triangles may as well be equal, as unequal; and that may
be greater, and likewise less, which contains the greater angle, and
the greater base. On this account, therefore, the institutor of the
Elements leaves the comparison of the triangles; to which we may add,
that the contemplation of these, requires the doctrine of parallels.
[Illustration]
[Illustration]
But if it be requisite, that anticipating things which are afterwards
exhibited, we at present make a comparison of areas, we must say,
that if the angles _a_, _d_, are equal to two right, the triangles
may be shewn to be equal: but when they are greater than two right,
the lesser triangle will be that which contains the greater angle;
and when they are less than two right, this will be the case with the
greater triangle. For let the construction in the element be given,
and produce _e d_, _f d_, to the points _k_, _h_; and let us suppose
the angles _b a c_, _e d f_, equal to two right. Because, therefore,
the angle _b a c_, is equal to the angle _e d g_, the angles _e d g_,
_e d f_, are equal to two right. But the angles _e d g_, _k d g_, are
also equal to two right. Let the common angle _e d g_ be taken away,
and the remainder _e d f_, will be equal to the remainder _k d g_. But
_e d f_ is equal to _h d k_; for they are vertical angles. Hence, the
angle _k d g_, is equal to the angle _h d k_. And because the angle
_g d h_, is external to the triangle _g d f_, it is equal to the two
internal and opposite angles at the points _g_ and _f_. But these
angles are equal to each other, because _d g_ is equal to _d f_. Hence,
the angle _g d h_, is double of the angle at the point _g_, and of the
angle at the point _f_. The angle, therefore, at the point _g_, is
equal to the angle _g d k_, and they are alternate; and consequently _d
e_ is parallel to _f g_. The triangles, therefore, _g d e_, _f d e_,
are upon the same base _d e_, and between the same parallels _d e_,
_g f_; and are consequently equal. But the triangle _g d e_, is equal
to the triangle _a b c_; and so the triangle _d e f_, is not unequal
to the triangle _a b c_. And here you may observe, that we require
three theorems belonging to the doctrine of parallels; one, indeed,
affirming, _that the external angle of every triangle is equal to the
two internal and opposite angles_: but the other, _that if a right line
falling upon two right lines, makes the alternate angles equal, the
right lines are parallel_; and the third, _that triangles constituted
upon the same base, and between the same parallels, are equal_, which
the institutor of the Elements also knowing, omits the comparison of
triangles.
[Illustration]
But let the angles _b a c_, _e d f_, be greater than two right, and
let the same things be constructed. Because, therefore, the angles _b
a c_, _e d f_, i.e. the angles _e d g_, _e d f_, are greater than two
right; but the angles _e d g_, _g d k_, are equal to two right, by
taking away the common angle _e d g_, the angle _e d f_, is greater
than the angle _g d k_. Hence, the angle _g d h_, is more than double
of the angle _g d k_; and so the angle _g d k_, is less than the angle
at the point _g_. Let _g d k_ be placed equal to _d g l_, and let _e
l_, and _d l_, be connected: _g l_, therefore, is parallel to _d e_;
and hence, the triangles _g d e_, _l d e_, are equal. But the triangle
_l d e_, is less than the triangle _f d e_. The triangle, therefore,
_g d e_, is less than the triangle _f d e_. But the triangle _g d e_,
is equal to the triangle _a b c_; and hence, the triangle _a b c_, is
less than the triangle _f d e_, viz. is less than the triangle which
contains the greater angle.
[Illustration]
In the third place, let the unequal angles be less than two right, and
let the same things be constructed. Because, therefore, the angles _e
d g_, _g d k_, are equal to two right, by taking away the common angle
_e d g_, the whole _g d h_, is less than double of _g d k_. But it is
double also of the angle at the point _g_. Hence, the angle _g d k_,
is greater than the angle at the point _g_. Let the angle _d g l_, be
placed equal to the angle _g d k_, and let _g l_ coincide with _e l_,
in the point _l_, and connect _d l_. Hence, _g l_ is parallel to _d
e_; and consequently the triangles _g d e_, _l d e_, are equal to each
other. But the triangle _l d e_, is greater than the triangle _f d e_;
and the triangle _g d e_, is equal to the triangle _a b c_. Hence, the
triangle _a b c_, is greater than the triangle _d f e_. It is shewn,
therefore, that the triangle _a b c_, is both equal to, and is also
greater and less than the triangle _d e f_, the angles at the points
_a_ and _d_, being either equal to, or greater or less than two right.
And thus, all the hypotheses may be accomplished. For what if the angle
at the point _a_, should be one right, and the half of a right angle,
but the angle at the point _b_, the half of one right, would not those
two angles be equal to two right? But what if the angle at the point
_a_, should be one right, and the half of a right, but the angle at the
point _b_, two thirds of one right, would they not be greater than two
right angles? And lastly, if the angle at the point _a_, should be one
right, and the half of a right angle, but the angle at the point _b_,
a third part of a right angle, would they not be less than two right,
and the angle _a_ be greater than the angle _d_? All these comparisons,
therefore, are produced by the assistance of parallels; and hence, they
are necessarily not found in the present elementary institution.
PROPOSITION XXV. THEOREM XVI.
If two triangles have two sides equal to two, each to each, but
have the base of the one greater than the base of the other;
they shall likewise have the angle contained by the equal sides
in the one, greater than the angle contained by the equal sides
in the other.
The present theorem is the opposite to the eighth, but the converse
of the preceding. For the institutor of the Elements produces theorems
concerning the equality and inequality of angles and bases, according
to _conjunction_; in each of the _conjunctions_, receiving some as
precedents, but others as converse. And in such as are precedent
indeed, he employs direct ostensions: but in such as are converse, he
uses deductions to an impossibility. After this manner he proceeds in
some particular triangle, sometimes from the equality of the sides
which it contains, shewing the consequent equality of the angles which
they subtend: but sometimes from their inequality evincing inequality.
And again, on the contrary, affirming that equality of sides is
consequent to equality of angles, but inequality to inequality.
However, that we may proceed to the thing proposed, we refer those
who are desirous of learning how the geometrician shews when this is
manifest, to his books on this subject. But we shall briefly relate the
demonstrations which others produce of this proposition; and in the
first place, that which Menelaus Alexandrinus invented and delivered.
Let there be two triangles _a b c_, _d e f_, having the two sides _a
b_, _a c_, equal to the two _d e_, _d f_, each to each, and the base _b
c_, greater than the base _e f_, I say that the angle at the point _a_,
is greater than the angle at the point _d_. For let there be cut from
the base _b c_, a line _b g_, equal to the base _e f_, and construct at
the point _b_, an angle _g b h_, equal to the angle _d e f_, and place
_b h_ equal to _d e_. Lastly, connect _h g_, and produce it to the
point _k_, and connect _a h_. Because, therefore, _b g_ is equal to _e
f_, but _b h_ to _e d_, the two are equal to the two, and they contain
equal angles. Hence, _g h_ is equal to _d f_, and the angle _b h g_,
is not unequal to the angle _e d f_. And because _g h_ is equal to _d
f_, but _d f_ to _a c_, _g h_, also, is equal to _a c_. Hence _h k_ is
greater than _a c_, and consequently is much greater than _a k_. The
angle, therefore, _k a h_, is greater than the angle _k h a_. Again,
because _b h_, is equal to _a b_, for it is equal to _d e_, the angle
_b h a_, is equal to the angle _b a h_. Hence, the whole angle _b h k_,
is less than the whole, _b a c_, but is shewn to be equal to the angle
at the point _d_. The angle, therefore, _b a c_, is greater than the
angle at the point _d_. And such is the demonstration of Menelaus.
[Illustration]
[Illustration]
But Heron, the mechanist, shews the same thing, in the following
manner, without leading to an impossibility, as is the case with the
demonstration of Euclid. Let there be two triangles _a b c_, _d e f_,
with the same hypotheses as above. And because _b c_ is greater than
_e f_, let _e f_ be produced, and place _e g_ equal to _b c_; and in
like manner extend _d e_, and place _d h_ equal to _d f_. The circle,
therefore, which is described with the centre _d_, and interval _d f_,
will pass also through the point _h_. Let it be described as _f k h_.
And because _a c_, _a b_, are together greater than _b c_, but these
are equal to _e h_, and _b c_ is equal to _g e_, hence the circle which
is described with the centre _e_, but interval _e g_, will cut _e h_.
Let it cut _e h_, as the circle _g k_, and connect from the common
section of the circles to the centres, the right lines _k d_, _k e_.
Because, therefore, the point _d_, is the centre of the circle _h k
f_, _d k_ is equal to _d h_, i.e. to _a c_. Again, because the point
_e_, is the centre of the circle _g k_, the line _e k_, is equal to _e
g_, i.e. to _b c_. Hence, since the two _a b_, _a c_, are equal to the
two _d e_, _d k_, and the base _b c_, is equal to the base _e k_, the
angle, also, _b a c_, is equal to the angle _e d k_. And thus the angle
_b a c_, is greater than the angle _f d e_.
PROPOSITION XV. THEOREM XVII.
If two triangles have two angles equal to two, each to each,
and one side equal to one side, either that which is adjacent
to the equal angles, or that which subtends one of the equal
angles: then they shall have the remaining sides equal to the
remaining sides, each to each, and the remaining angle equal to
the remaining angle.
[Illustration]
It is necessary, that he who wishes to compare triangles with each
other, according to sides, angles, and areas, should either, by
receiving the sides alone equal, enquire after the equality of angles;
or by assuming the angles alone equal, investigate the equality of the
sides; or by mingling the angles and sides, scrutinize the equality of
angles and sides. Since, therefore, Euclid alone receives the angles
equal, he could not likewise shew that the sides of the triangles are
equal. For the least triangles are equiangular with the greatest,
though at the same time they are excelled by them, both according
to sides and comprehended space: but the angles of the former are
separately equal to the angles of the latter. However, as he supposes
the sides alone to be equal, he demonstrates that all are equal, by
the eighth theorem, in which there are two triangles having two sides
equal to two, each to each, and the base to the base, and these are
shewn to be equiangular, and to possess a power of comprehending equal
spaces. And the institutor of the Elements omits this addition, as
necessarily following from the fourth, and requiring no demonstration.
But when receiving sides and angles, he ought to receive either one
side equal to one, and one angle equal to one; or one side, and two
angles of the triangles, equal to two; or on the contrary, one angle
and two sides; or one angle and three sides; or one side and three
angles; or more than one side, and more than one angle. But when he
had received one angle, and one side, he could by no means shew the
thing proposed. I mean, the equality of the rest. For it is possible
that two triangles which are equal, according to one side only, and
one angle, may be entirely unequal as to the rest. Thus let there be a
right line _a b_, perpendicularly erected upon the right line _c d_,
but let _b d_ be greater than _b c_, and connect _a c_, _a d_. In these
triangles, therefore, there is one common side, and one angle equal
to one, but all the rest are unequal. But it is lawful to receive one
side, and two angles, and to prove the rest equal, and this he performs
by the present theorem: though again, to suppose one side, and three
equal angles, is superfluous; since from the equality of two alone,
the equality of the rest is exhibited. Again, receiving one angle,
and two equal sides, he demonstrates that the rest are equal in the
fourth theorem. But it is superfluous to receive one angle, and three
equal sides: for two equals being alone assumed, conclude the equality
of the rest. Besides, it is superfluous to assume two sides, and two
equal angles; or two sides, and three equal angles; or two angles and
three sides; or three angles and three sides. For the consequents
to fewer hypotheses attend likewise a greater multitude, while the
hypotheses are received with proper conditions. Hence, three hypotheses
requiring demonstration, present themselves to our view, one, which
alone receives three sides; and another which assumes one side, and
two angles, which the geometrician now proposes; and a third, the
opposite to this. On this account, we have only these three theorems,
concerning the equality of triangles, which are conversant in sides
and angles; since all the other hypotheses are either invalid for the
purpose of shewing the object of enquiry; or they are valid indeed, but
superfluous, because the same things may be readily procured by fewer
hypotheses. As, therefore, when he assumed two sides equal to two, and
one angle equal to one, he did not, indeed, assume every angle, but (as
it was proposed by him) that contained by equal right lines, in the
same manner when he assumes two angles equal to two, and one side to
one, he does not assume any side, but either that which is adjacent to
the equal angles, or that which subtends one of the equal angles. For
neither is it possible in the fourth theorem, by assuming any equal
angle, nor in the present by assuming any side, to shew the equality of
the rest.
[Illustration]
[Illustration]
Thus for example, an equilateral triangle _a b c_, being given,
let the side _b c_ be divided into unequal parts, by the line _a d_.
Hence, there will be formed two triangles, having two sides _a b_, _a
d_, equal to the two _a c_, _a d_, and one angle at the point _b_,
equal to one angle at the point _c_, but the remaining sides will not
also be equal, as for instance, the side _b d_, to the side _d c_:
for they are unequal. But neither are the remaining angles equal: the
reason of which is, because we receive an angle equal to an angle, but
not the angle which is contained by equal sides. After the same manner,
indeed, the present theorem also will appear dubious, unless we assume,
according to the aforesaid condition, an equal side subtending one of
the equal angles, or adjacent to the equal angles. For let there be a
right angled triangle _a b c_, having the angle at the point _b_ right,
and the side _b c_, greater than the side _b a_, and let there be
constructed on the right line _b c_, and at a point in it _c_, an angle
_b c d_, equal to the angle _b a c_, and let _b d_, _c d_, produced,
coincide in the point _d_. There are two triangles, therefore _a b c_,
_b c d_, having one side _b c_ common, and two angles equal to two,
viz. _a b c_, to _c b d_ (for they are right), and _b a c_ to _b c
d_, according to construction. Hence, as it appears the triangles are
equal, and yet it may be shewn that the triangle _b d c_, is greater
than the triangle _a b c_. But the reason of this is, because in the
triangle _a b c_, we assume the common side _b c_, subtending one of
the equal angles, viz. the angle at the point _a_: but in the triangle
_b c d_, we assume the equal side, adjacent to the equal angles. It
was requisite, therefore, in each, either to subtend one of the equal
angles, or to be adjacent to the equal angles. But not observing this,
we affirmed that triangle to be equal, which is necessarily greater:
for is not the triangle _b c d_, greater than the triangle _a b c_? To
be convinced of this, let there be constructed on the right line _b c_,
and at a given point in it _c_, an angle _f c b_, equal to the angle
_a c b_: for the angle _b c d_, as well as the angle at the point _a_,
is greater than the angle _a c b_. Because, therefore, there are two
triangles, _a b c_, _b c f_, having two angles _a b c_, _b c a_, equal
to two _c b f_, _b c f_, each to each, and one side common, adjacent to
the equal angles, viz. _b c_, the triangles are equal. But the triangle
_b c d_ is greater than the triangle _b c f_, and consequently it is
also greater than the triangle _a b c_. But it was formerly shewn to
be equal, on account of the assumption of any side: And thus much the
diligence of Porphyry has supplied us on the present occasion. But
Eudemus, in his Geometrical Narrations, refers the present theorem to
Thales. For he says it is necessary to use this theorem in determining
the distance of ships at sea, according to the method employed by
Thales in this investigation. But from the preceding division we
may briefly assume all the contemplation concerning the equality of
triangles, and are enabled to relate the causes of things omitted,
confuting those hypotheses, as either false, or superfluous. And thus
far we determine the limits of the first section of the elementary
institutor, because he forms the constructions and comparisons of
triangles, according to equal and unequal. And by construction,
indeed, he delivers their essence: but by comparison, their identity
and diversity. For there are three things which are conversant about
being, _essence_, _same_, and _different_[26], as well in quantities,
as in qualities, according to the propriety of subjects. From these,
therefore, as images it may be shewn, that every thing is the _same_
with itself, and _differs_ from itself, on account of the multitude
which it contains; and that all things are the _same_ with one another,
and _different_ from themselves. For both, in every triangle, and in
more triangles than one, equality and inequality has been found to
reside.