5. Show that it is possible to trisect the central angle of a regular n-gon, when n is any one of the first or third series of Euclid's numbers (111).

6. In the same cases show that it is possible to trisect the interior or exterior angle of the regular n-gon.

7. Show that it is possible to divide a right angle into n equal parts, when n is any of Euclid's numbers.

8. Show that it is possible to divide the angle of an equilateral triangle into n equal parts, when n is any number belonging to the first three series of Euclid's numbers.

NOTE ON EXS. 5-8. The general problem to trisect a given arbitrary angle is one of the famous problems of antiquity, and has never been solved by methods permitted in elementary geometry. Modern mathematicians have demonstrated that this general problem cannot be analyzed into simpler ones that require only the drawing of straight lines and circles.* Thus the construction cannot be performed by means of only a pair of compasses and an unmarked straightedge. Several general solutions are known which overstep these limitations to a greater or less degree. One of the simplest employs the sliding motion of a straightedge on which two points are marked. There are, however, certain special angles that can be trisected by the methods of elementary geometry. (See exs. 5-8 above.)

The still more general problem of dividing a given arbitrary angle into n equal parts can be solved only when n is one of the first series of Euclid's numbers (111; I. 73); but in the case of certain special angles the problem can be solved for some other values of n (exs. 7, 8).

MAXIMA AND MINIMA †

141. Certain principles of maxima and minima relating to triangles were considered in Book II. 92-108. Similar principles can now be extended to polygons in general, subject to certain given conditions. The theorems here considered fall into five groups according to the nature of the assigned conditions.

* See Klein's "Vorträge über ausgewählte Fragen der Elementar Geometrie." (Translated by Professors Beman and Smith.)

This topic is discussed here on account of its intimate connection with the properties of the circle, and of inscribed and circumscribed polygons. It may, however, be postponed without inconvenience.

GIVEN SIDES

This group of two theorems with their corollaries will show how to make the surface of a polygon a maximum subject to various assigned conditions relating to the magnitude of the sides. In each case the additional condition is to be proved both necessary and sufficient for a maximum; and accordingly each theorem is accompanied by its converse (II. 93).

Greatest polygon with one arbitrary side.

142. THEOREM 40. Among the polygons that have all the sides but one equal respectively to given lines taken in order, any one that is a maximum is circumscribable by a semicircle having the undetermined side as diameter.

Let the polygon ABCDEF be a maximum subject to the condition that the sides AB, BC, CD, DE, EF are respectively equal to given lines taken in

D

E

the semicircle does not pass through C; and draw CA, CF. Then the angle ACF is not a right angle (55, 56).

F

Hence, by rotating the figures ABC and FEDC about the point C until ACF becomes a right angle, the triangle ACF could be increased (II. 94); and therefore the whole polygon ABCDEF could be increased without changing any of the given sides. This is contrary to the hypothesis that ABCDEF is a maximum under the given conditions.

Hence the semicircle described on AF passes through the point C. Similarly it passes through the other vertices.

143. Cor. Among the polygons that have all the sides but one equal respectively to given lines taken in order, any polygon that is circumscribable by a semicircle having the undetermined side as diameter, is a maximum.

For all polygons that satisfy the given conditions, and the further condition of being circumscribable by a semicircle having the undetermined side as diameter, are equal (140), and are therefore equal to any one that is a maximum (142). Ex. Show how to enunciate II. 94 so as to make it a particular case of 143.

Greatest polygon with all the sides given.

144. THEOREM 41. Among the polygons that have their sides equal respectively to given lines taken in order, any polygon that is circumscribable by a circle is a maximum.

Let ABCD and A'B'C'D' be two polygons that satisfy the conditions of having their sides equal respectively to given lines taken in order, and let the former be circumscribable and the latter not.

B

A

P

B'

First to prove that ABCD is greater than A'B'C'D'.

Draw the diameter AP; and join CP, DP. On C'D', which equals CD, construct a triangle C'D'P' equal to the triangle CDP; and draw A'P'.

The circle whose diameter is A'P' does not pass through all the points B', C', D' (hyp.). Use 143 and add; then subtract the equal triangles.

Next, to prove that ABCD is a maximum under the given conditions.

The polygon ABCD is superposable on any other polygon that satisfies the given conditions and the further condition of being circumscribable (139); and it has just been proved greater than any polygon that satisfies the given conditions without satisfying the further condition. Therefore the polygon ABCD is a maximum under the given conditions.

145. Cor. I. Among the polygons that have their sides respectively equal to the given lines taken in order, any polygon that is a maximum is circumscribable. (Indirect proof.)

Equilateral n-gon with given side.

146. Cor. 2. Of all equilateral polygons having a given side and a given number of sides, one that is regular is a maximum.

Among the polygons that satisfy the given conditions, one that is equilateral and equiangular is circumscribable; and one that is equilateral and not equiangular is not circumscribable (109, ex. 2); hence one that is equilateral and equiangular is a maximum (144).

147. Cor. 3. Of all equilateral polygons having a given side and a given number of sides, any polygon that is a maximum is regular. (Use 145.)

GIVEN PERIMETER

The following theorems show how to make the surface of a polygon a maximum, when the perimeter is given, and when another assigned condition is fulfilled.

The n-gon of greatest surface.

148. THEOREM 42. Among the polygons that have a given perimeter and a given number of sides, one that is a maximum is regular.

Let the polygon ABCD... be a maximum, subject to the conditions of having a given perimeter and a given number of sides.

Suppose, if possible, that the two adjacent sides AB and BC are not equal.

On AC as base construct an isosceles

triangle B'AC having the sum of the

sides B'A and B'C equal to the sum of BA and BC.

The isosceles triangle B'AC is greater than the isoperimetric triangle BAC (II. 101).

Therefore the polygon AB'CD... is isoperimetric with, and greater than, the polygon ABCD...; but this is impossible, since ABCD... is one of the greatest of the isoperimetric set, by hypothesis.

Hence the supposition fails, and the adjacent sides AB and BC are equal.

Similarly all the sides are equal. Therefore the polygon ABCD... is equilateral.

Next, to prove that the equilateral polygon ABCD... is a regular polygon.

Since ABCD... is an equilateral polygon having a given perimeter (that is to say, a given side) and a given number of sides, and since it is a maximum, hence it is a regular polygon (147).

149. Cor. Among the polygons that have a given perimeter and a given number of sides, one that is regular is a maxi

тит.

For all polygons that satisfy the given conditions and the additional condition of being regular are equal, and are hence equal to any one that is a maximum.