S 26. The third book of the _Elements_ relates exclusively to
properties of the circle. A circle and its circumference have been
defined in Book I., Def. 15. We restate it here in slightly different
words:--
_Definition_.--The circumference of a circle is a plane curve such
that all points in it have the same distance from a fixed point in the
plane. This point is called the "centre" of the circle.
Of the new definitions, of which eleven are given at the beginning of
the third book, a few only require special mention. The first, which
says that circles with equal radii are equal, is in part a theorem,
but easily proved by applying the one circle to the other. Or it may
be considered proved by aid of Prop. 24, equal circles not being used
till after this theorem.
In the second definition is explained what is meant by a line which
"touches" a circle. Such a line is now generally called a tangent to
the circle. The introduction of this name allows us to state many of
Euclid's propositions in a much shorter form.
For the same reason we shall call a straight line joining two points
on the circumference of a circle a "chord."
Definitions 4 and 5 may be replaced with a slight generalization by
the following:--
_Definition_.--By the distance of a point from a line is meant the
length of the perpendicular drawn from the point to the line.
S 27. From the definition of a circle it follows that every circle has
a centre. Prop. 1 requires to find it when the circle is given, i.e.
when its circumference is drawn.
To solve this problem a chord is drawn (that is, any two points in the
circumference are joined), and through the point where this is
bisected a perpendicular to it is erected. Euclid then proves, first,
that no point off this perpendicular can be the centre, hence that the
centre must lie in this line; and, secondly, that of the points on the
perpendicular one only can be the centre, viz. the one which bisects
the parts of the perpendicular bounded by the circle. In the second
part Euclid silently assumes that the perpendicular there used does
cut the circumference in two, and only in two points. The proof
therefore is incomplete. The proof of the first part, however, is
exact. By drawing two non-parallel chords, and the perpendiculars
which bisect them, the centre will be found as the point where these
perpendiculars intersect.
S 28. In Prop. 2 it is proved that a chord of a circle lies altogether
within the circle.
What we have called the first part of Euclid's solution of Prop. 1 may
be stated as a theorem:--
_Every straight line which bisects a chord, and is at right angles to
it, passes through the centre of the circle._
The converse to this gives Prop. 3, which may be stated thus:--
_If a straight line through the centre of a circle bisect a chord,
then it is perpendicular to the chord, and if it be perpendicular to
the chord it bisects it._
An easy consequence of this is the following theorem, which is
essentially the same as Prop. 4:--
_Two chords of a circle, of which neither passes through the centre,
cannot bisect each other._
These last three theorems are fundamental for the theory of the
circle. It is to be remarked that Euclid never proves that a straight
line cannot have more than two points in common with a circumference.
S 29. The next two propositions (5 and 6) might be replaced by a
single and a simpler theorem, viz:--
_Two circles which have a common centre, and whose circumferences have
one point in common, coincide._
Or, more in agreement with Euclid's form:--
_Two different circles, whose circumferences have a point in common,
cannot have the same centre._
That Euclid treats of two cases is characteristic of Greek
mathematics.
The next two propositions (7 and 8) again belong together. They may be
combined thus:--
_If from a point in a plane of a circle, which is not the centre,
straight lines be drawn to the different points of the circumference,
then of all these lines one is the shortest, and one the longest, and
these lie both in that straight line which joins the given point to
the centre. Of all the remaining lines each is equal to one and only
one other, and these equal lines lie on opposite sides of the shortest
or longest, and make equal angles with them._
Euclid distinguishes the two cases where the given point lies within
or without the circle, omitting the case where it lies in the
circumference.
From the last proposition it follows that if from a point more than
two equal straight lines can be drawn to the circumference, this point
must be the centre. This is Prop. 9.
As a consequence of this we get
_If the circumferences of the two circles have three points in common
they coincide._
For in this case the two circles have a common centre, because from
the centre of the one three equal lines can be drawn to points on the
circumference of the other. But two circles which have a common
centre, and whose circumferences have a point in common, coincide.
(Compare above statement of Props. 5 and 6.)
This theorem may also be stated thus:--
_Through three points only one circumference may be drawn; or, Three
points determine a circle._
Euclid does not give the theorem in this form. He proves, however,
_that the two circles cannot cut another in more than two points_
(Prop. 10), and _that two circles cannot touch one another in more
points than one_ (Prop. 13).
S 30. Propositions 11 and 12 assert that _if two circles touch, then
the point of contact lies on the line joining their centres_. This
gives two propositions, because the circles may touch either
internally or externally.
S 31. Propositions 14 and 15 relate to the length of chords. The first
says _that equal chords are equidistant from the centre, and that
chords which are equidistant from the centre are equal_;
Whilst Prop. 15 compares unequal chords, viz. _Of all chords the
diameter is the greatest, and of other chords that is the greater
which is nearer to the centre_; and conversely, _the greater chord is
nearer to the centre_.
S 32. In Prop. 16 the tangent to a circle is for the first time
introduced. The proposition is meant to show that the straight line at
the end point of the diameter and at right angles to it is a tangent.
The proposition itself does not state this. It runs thus:--
Prop. 16. _The straight line drawn at right angles to the diameter of
a circle, from the extremity of it, falls without the circle; and no
straight line can be drawn from the extremity, between that straight
line and the circumference, so as not to cut the circle._
_Corollary_.--The straight line at right angles to a diameter drawn
through the end point of it touches the circle.
The statement of the proposition and its whole treatment show the
difficulties which the tangents presented to Euclid.
Prop. 17 solves the problem _through a given point, either in the
circumference or without it, to draw a tangent to a given circle_.
Closely connected with Prop. 16 are Props. 18 and 19, which state
(Prop. 18), _that the line joining the centre of a circle to the point
of contact of a tangent is perpendicular to the tangent_; and
conversely (Prop. 19), _that the straight line through the point of
contact of, and perpendicular to, a tangent to a circle passes through
the centre of the circle_.
S 33. The rest of the book relates to angles connected with a circle,
viz. angles which have the vertex either at the centre or on the
circumference, and which are called respectively angles at the centre
and angles at the circumference. Between these two kinds of angles
exists the important relation expressed as follows:--
Prop. 20. _The angle at the centre of a circle is double of the angle
at the circumference on the same base, that is, on the same arc._
This is of great importance for its consequences, of which the two
following are the principal:--
Prop. 21. _The angles in the same segment of a circle are equal to one
another_;
Prop. 22. _The opposite angles of any quadrilateral figure inscribed
in a circle are together equal to two right angles._
Further consequences are:--
Prop. 23. _On the same straight line, and on the same side of it,
there cannot be two similar segments of circles, not coinciding with
one another_;
Prop. 24. _Similar segments of circles on equal straight lines are
equal to one another._
The problem Prop. 25. _A segment of a circle being given to describe
the circle of which it is a segment_, may be solved much more easily
by aid of the construction described in relation to Prop. 1, III., in
S 27.
S 34. There follow four theorems connecting the angles at the centre,
the arcs into which they divide the circumference, and the chords
subtending these arcs. They are expressed for angles, arcs and chords
in equal circles, but they hold also for angles, arcs and chords in
the same circle.
The theorems are:--
Prop. 26. _In equal circles equal angles stand on equal arcs, whether
they be at the centres or circumferences_;
Prop. 27. (converse to Prop. 26). _In equal circles the angles which
stand on equal arcs are equal to one another, whether they be at the
centres or the circumferences_;
Prop. 28. _In equal circles equal straight lines_ (equal chords) _cut
off equal arcs, the greater equal to the greater, and the less equal
to the less_;
Prop. 29 (converse to Prop. 28). _In equal circles equal arcs are
subtended by equal straight lines._
S 35. Other important consequences of Props. 20-22 are:--
Prop. 31. _In a circle the angle in a semicircle is a right angle; but
the angle in a segment greater than a semicircle is less than a right
angle; and the angle in a segment less than a semicircle is greater
than a right angle_;
Prop. 32. _If a straight line touch a circle, and from the point of
contact a straight line be drawn cutting the circle, the angles which
this line makes with the line touching the circle shall be equal to
the angles which are in the alternate segments of the circle._
S 36. Propositions 30, 33, 34, contain problems which are solved by
aid of the propositions preceding them:--
Prop. 30. _To bisect a given arc, that is, to divide it into two equal
parts_;
Prop. 33. _On a given straight line to describe a segment of a circle
containing an angle equal to a given rectilineal angle_;
Prop. 34. _From a given circle to cut off a segment containing an
angle equal to a given rectilineal angle_.
S 37. If we draw chords through a point A within a circle, they will
each be divided by A into two segments. Between these segments the law
holds that the rectangle contained by them has the same area on
whatever chord through A the segments are taken. The value of this
rectangle changes, of course, with the position of A.
A similar theorem holds if the point A be taken without the circle. On
every straight line through A, which cuts the circle in two points B
and C, we have two segments AB and AC, and the rectangles contained by
them are again equal to one another, and equal to the square on a
tangent drawn from A to the circle.
The first of these theorems gives Prop. 35, and the second Prop. 36,
with its corollary, whilst Prop. 37, the last of Book III., gives the
converse to Prop. 36. The first two theorems may be combined in one:--
_If through a point A in the plane of a circle a straight line be
drawn cutting the circle in B and C, then the rectangle AB.AC has a
constant value so long as the point A be fixed; and if from A a
tangent AD can be drawn to the circle, touching at D, then the above
rectangle equals the square on AD._
Prop. 37 may be stated thus:--
_If from a point A without a circle a line be drawn cutting the circle
in B and C, and another line to a point D on the circle, and AB.AC =
AD^2, then the line AD touches the circle at D._
It is not difficult to prove also the converse to the general
proposition as above stated. This proposition and its converse may be
expressed as follows:--
_If four points ABCD be taken on the circumference of a circle, and if
the lines AB, CD, produced if necessary, meet at E, then_
EA.EB = EC.ED;
_and conversely, if this relation holds then the four points lie on a
circle, that is, the circle drawn through three of them passes through
the fourth._
That a circle may always be drawn through three points, provided that
they do not lie in a straight line, is proved only later on in Book
IV.