finally, by means of the pentedecagon, those of 50, 60, 120, etc., sides.

Until the beginning of the present century, it was supposed that these were the only polygons that could be constructed by elementary geometry, that is, by the use of the straight line and circle only. GAUSS, however, in his Disquisitiones Arithmetica, Lipsia, 1801, proved that it is possible, by the use of the straight line and circle only, to construct regular polygons of 17 sides, of 257 sides, and in general of any number of sides which can be expressed by 2" + 1, n being an integer, provided that 2" + 1 is a prime number.

PROPOSITION VIII.-THEOREM.

22. The area of a regular polygon is equal to half the product of its perimeter and apothem.

For, straight lines drawn from the centre to the vertices of the polygon divide it into equal triangles whose bases are the sides of the polygon and whose common altitude is the apothem. The area of one of these triangles is equal to half the product of its base and altitude (IV. 13); therefore, the sum of their areas, or the area of the polygon, is half the product of the sum of the bases by the common altitude, that is, half the product of the perimeter and apothem.

PROPOSITION IX.-THEOREM.

23. The area of a regular inscribed dodecagon is equal to three times the square of the radius.

Let AB, BC, CD, DE, be four consecutive sides of a regular inscribed dodecagon, and draw the radii OA, OE; then, the figure OABCDE is one-third of the dodecagon, and we have only to prove that the area of this figure is equal to the square of the radius.

Draw the radius OD; at A and D draw the tangents AF and GDF meeting in F;

G

E

D

H

C

join AC and CE, and let AC and OE be produced to meet the tan

gent GF in H and G. The arc AD, containing three of the sides of the dodecagon, is one fourth of the circumference; therefore the angle AOD is a right angle, and OF is a square described on the radius.

Since AC and CE are sides of the regular inscribed hexagon, each is equal to the radius; therefore OA CE is a parallelogram. Hence also GOAHand GECH are parallelograms.

G

D H

F

C

E

B

The triangles DEC and BCA are equal (I. 80). The area of the triangle DEC is one-half that of the parallelogram EH (IV. 14); therefore the two triangles DEC and BCA are together equivalent to the parallelogram EH. Adding the parallelogram OC to these equals, we have the figure OABCDE equivalent to the parallelogram OH. But the parallelogram OH is equivalent to the square OF (IV. 11); therefore the figure OABCDE, or one-third the dodecagon, is equivalent to the square OF, that is, to the square of the radius. Therefore, the area of the whole dodecagon is equal to three times the square of the radius.

24. Scholium. The area of the circumscribed square is evidently equal to four times the square of the radius. The area of the circle is greater than that of the inscribed regular dodecagon, and less than that of the circumscribed square; therefore, if the square of the radius is taken as the unit of surface, the area of a circle is greater than 3 and less than 4.

PROPOSITION X.-PROBLEM.

25. Given the perimeters of a regular inscribed and a similar circumscribed polygon, to compute the perimeters of the regular inscribed and circumscribed polygons of double the number of sides.

Let AB be a side of the given inscribed polygon, CD a side of the similar circumscribed polygon, tangent to the arc AB at its middle point E. Join AE, and at A and B draw the tangents AF and BG; then AE is a side of the regular inscribed polygon of double

Since OC is the radius of the circle circumscribed about the polygon whose perimeter is P, we have (10),

and since OF bisects the angle COE, we have (III. 21),

Now FG is a side of the polygon whose perimeter is P', and is contained as many times in P' as CE is contained in P, hence (III. 9),

Again, the right triangles AEH and EFN are similar, since their

and since EN and EF are contained the same number of times in p' and P', respectively, we have

Therefore, from the given perimeters p and P, we compute P' by the equation [1], and then with p and P' we compute p' by the equation [2].

26. Definition. Two polygons are isoperimetric when their perimeters are equal.

PROPOSITION XI.-PROBLEM.

27. Given the radius and apothem of a regular polygon, to compute the radius and apothem of the isoperimetric polygon of double the number of sides.

Let AB be a side of the given regular polygon, O the centre of this polygon, OA its radius, OD its apothem. Produce DO to meet the circumference of the circumscribed circle in O'; join O'A, O'B; let fall OA' perpendicular to O'A, and through A' draw A'B' parallel to AB.

Since the new polygon is to have twice as many sides as the given polygon, the angle at its centre must be one-half the angle AOB;

therefore the angle AO'B, which is equal to one-half of AOB (II. 57), is equal to the angle at the centre of the new polygon.

Since the perimeter of the new polygon is to be equal to that of the given polygon, but is to be divided into twice as many sides, each of its sides must be equal to one-half of AB; therefore A'B', which is equal to one-half of AB (I. 121), is a side of the new polygon; O'A' is its radius, and O'D' its apothem.

If, then, we denote the given radius OA by R, and the given apothem OD by r, the required radius O'A' by R', and the apothem O'D' by r', we have

or

or

In the right triangle OA'O', we have (III. 44),

[1]

[2]

therefore, r' is an arithmetic mean between R and r, and R' is a geometric mean between R and r'.

MEASUREMENT OF THE CIRCLE.

The principle which we employed in the comparison of incommensurable ratios (II. 49) is fundamentally the same as that which we are about to apply to the measurement of the circle, but we shall now state it in a much more general form, better adapted for subsequent application.

28. Definitions. I. A variable quantity, or simply, a variable, is a quantity which has different successive values.

II. When the successive values of a variable, under the conditions imposed upon it, approach more and more nearly to the value of some fixed or constant quantity, so that the difference between the variable and the constant may become less than any assigned quantity, without becoming zero, the variable is said to approach indefi