S 38. The fourth book contains only problems, all relating to the
construction of triangles and polygons inscribed in and circumscribed
about circles, and of circles inscribed in or circumscribed about
triangles and polygons. They are nearly all given for their own sake,
and not for future use in the construction of figures, as are most of
those in the former books. In seven definitions at the beginning of
the book it is explained what is understood by figures inscribed in or
described about other figures, with special reference to the case
where one figure is a circle. Instead, however, of saying that one
figure is described about another, it is now generally said that the
one figure is circumscribed about the other. We may then state the
definitions 3 or 4 thus:--
_Definition._--A polygon is said to be inscribed in a circle, and the
circle is said to be circumscribed about the polygon, if the vertices
of the polygon lie in the circumference of the circle.
And definitions 5 and 6 thus:--
_Definition._--A polygon is said to be circumscribed about a circle,
and a circle is said to be inscribed in a polygon, if the sides of the
polygon are tangents to the circle.
S 39. The first problem is merely constructive. It requires to draw in
a given circle a chord equal to a given straight line, which is not
greater than the diameter of the circle. The problem is not a
determinate one, inasmuch as the chord may be drawn from any point in
the circumference. This may be said of almost all problems in this
book, especially of the next two. They are:--
Prop. 2. _In a given circle to inscribe a triangle equiangular to a
given triangle;_
Prop. 3. _About a given circle to circumscribe a triangle equiangular
to a given triangle._
S 40. Of somewhat greater interest are the next problems, where the
triangles are given and the circles to be found.
Prop. 4. _To inscribe a circle in a given triangle._
The result is that the problem has always a solution, viz. the centre
of the circle is the point where the bisectors of two of the interior
angles of the triangle meet. The solution shows, though Euclid does
not state this, that the problem has but one solution; and also,
_The three bisectors of the interior angles of any triangle meet in a
point, and this is the centre of the circle inscribed in the
triangle._
The solutions of most of the other problems contain also theorems. Of
these we shall state those which are of special interest; Euclid does
not state any one of them.
S 41. Prop. 5. _To circumscribe a circle about a given triangle._
The one solution which always exists contains the following:--
_The three straight lines which bisect the sides of a triangle at
right angles meet in a point, and this point is the centre of the
circle circumscribed about the triangle._
Euclid adds in a corollary the following property:--
The centre of the circle circumscribed about a triangle lies within,
on a side of, or without the triangle, according as the triangle is
acute-angled, right-angled or obtuse-angled.
S 42. Whilst it is always possible to draw a circle which is inscribed
in or circumscribed about a given triangle, this is not the case with
quadrilaterals or polygons of more sides. Of those for which this is
possible the regular polygons, i.e. polygons which have all their
sides and angles equal, are the most interesting. In each of them a
circle may be inscribed, and another may be circumscribed about it.
Euclid does not use the word regular, but he describes the polygons in
question as _equiangular_ and _equilateral_. We shall use the name
regular polygon. The regular triangle is equilateral, the regular
quadrilateral is the square.
Euclid considers the regular polygons of 4, 5, 6 and 15 sides. For
each of the first three he solves the problems--(1) to inscribe such a
polygon in a given circle; (2) to circumscribe it about a given
circle; (3) to inscribe a circle in, and (4) to circumscribe a circle
about, such a polygon.
For the regular triangle the problems are not repeated, because more
general problems have been solved.
Props. 6, 7, 8 and 9 solve these problems for the square.
The general problem of inscribing in a given circle a regular polygon
of n sides depends upon the problem of dividing the circumference of a
circle into n equal parts, or what comes to the same thing, of drawing
from the centre of the circle n radii such that the angles between
consecutive radii are equal, that is, to divide the space about the
centre into n equal angles. Thus, if it is required to inscribe a
square in a circle, we have to draw four lines from the centre, making
the four angles equal. This is done by drawing two diameters at right
angles to one another. The ends of these diameters are the vertices of
the required square. If, on the other hand, tangents be drawn at these
ends, we obtain a square circumscribed about the circle.
S 43. To construct a _regular pentagon_, we find it convenient first
to construct a _regular decagon_. This requires to divide the space
about the centre into ten equal angles. Each will be 1/10th of a right
angle, or 1/5th of two right angles. If we suppose the decagon
constructed, and if we join the centre to the end of one side, we get
an isosceles triangle, where the angle at the centre equals 1/5th of
two right angles; hence each of the angles at the base will be 2/5ths
of two right angles, as all three angles together equal two right
angles. Thus we have to construct an isosceles triangle, having the
angle at the vertex equal to half an angle at the base. This is solved
in Prop. 10, by aid of the problem in Prop. 11 of the second book. If
we make the sides of this triangle equal to the radius of the given
circle, then the base will be the side of the regular decagon
inscribed in the circle. This side being known the decagon can be
constructed, and if the vertices are joined alternately, leaving out
half their number, we obtain the regular pentagon. (Prop. 11.)
Euclid does not proceed thus. He wants the pentagon before the
decagon. This, however, does not change the real nature of his
solution, nor does his solution become simpler by not mentioning the
decagon.
Once the regular pentagon is inscribed, it is easy to circumscribe
another by drawing tangents at the vertices of the inscribed pentagon.
This is shown in Prop. 12.
Props. 13 and 14 teach how a circle may be inscribed in or
circumscribed about any given regular pentagon.
S 44. The _regular hexagon_ is more easily constructed, as shown in
Prop. 15. The result is that the side of the regular hexagon inscribed
in a circle is equal to the radius of the circle.
For this polygon the other three problems mentioned are not solved.
S 45. The book closes with Prop. 16. To inscribe a regular quindecagon
in a given circle. If we inscribe a regular pentagon and a regular
hexagon in the circle, having one vertex in common, then the arc from
the common vertex to the next vertex of the pentagon is 1/5th of the
circumference, and to the next vertex of the hexagon is 1/6th of the
circumference. The difference between these arcs is, therefore, 1/5 -
1/6 = 1/30th of the circumference. The latter may, therefore, be
divided into thirty, and hence also in fifteen equal parts, and the
regular quindecagon be described.
S 46. We conclude with a few theorems about regular polygons which are
not given by Euclid.
_The straight lines perpendicular to and bisecting the sides of any
regular polygon meet in a point. The straight lines bisecting the
angles in the regular polygon meet in the same point. This point is
the centre of the circles circumscribed about and inscribed in the
regular polygon._
We can bisect any given arc (Prop. 30, III.). Hence we can divide a
circumference into 2n equal parts as soon as it has been divided into
n equal parts, or as soon as a regular polygon of n sides has been
constructed. Hence--
_If a regular polygon of n sides has been constructed, then a regular
polygon of 2n sides, of 4n, of 8n sides, &c., may also be
constructed._ Euclid shows how to construct regular polygons of 3, 4,
5 and 15 sides. It follows that we can construct regular polygons of
3, 6, 12, 24 sides
4, 8, 16, 32 "
5, 10, 20, 40 "
15, 30, 60, 120 "
The construction of any new regular polygon not included in one of
these series will give rise to a new series. Till the beginning of the
19th century nothing was added to the knowledge of regular polygons as
given by Euclid. Then Gauss, in his celebrated _Arithmetic_, proved
that every regular polygon of 2^n + 1 sides may be constructed if this
number 2^n + 1 be prime, and that no others except those with 2^m(2^n
+ 1) sides can be constructed by elementary methods. This shows that
regular polygons of 7, 9, 13 sides cannot thus be constructed, but
that a regular polygon of 17 sides is possible; for 17 = 2^4 + 1. The
next polygon is one of 257 sides. The construction becomes already
rather complicated for 17 sides.