terior angle AMB is therefore double the interior angle O; and the angle MAB being equal to the angle AMB, it follows that, in the isosceles triangle OAB, each of the angles A, B at the base, is double the angle O at the vertex, and as all three amount to two right angles, the angle O must be the fifth part of two right angles, or the tenth part of four right angles; consequently ten of these angles may be ranged round the point 0; and as they are subtended by equal chords and equal arcs, the chord of the arc AB may be applied exactly ten times round the circumference, forming the inscribed decagon required.

If the alternate corners of the decagon be joined, an inscribed regular pentagon will obviously be formed."

In a given circle, to inscribe a regular pentedecagon, or polygon of fifteen sides.

Let AB be a side of the inscribed regular decagon, found by last proposition; and let AC be a side of a regular inscribed hexagon; that is, apply the radius from A to Č (Prop. VI.). Join BC, then BC will be a side of the polygon required. For AB cuts off an arc equal to a tenth part of the circumference; and AC subtends an arc equal to a sixth of the circumference; the difference of these arcs, therefore, is a fifteenth part of the circumference; and as equal arcs are sub- B tended by equal chords, it follows that the chord BC may be applied exactly fifteen times round the circumference, thus forming a regular pentedecagon.

Scholium.

Since the perpendicular from the centre to a chord bisects the subtended arc, it is very easy, from having an inscribed polygon given, to insert another of double the number of sides. Thus, from having an inscribed square (Prop. XXIV. B. 4), we may inscribe in succession polygons of 8, 16, 32, 64, &c. sides; from the hexagon may be formed polygons of 12, 24, 48, 96, &c. sides; from the decagon polygons of 20, 40, 80,

&c. sides, and from the pentedecagon we may inscribe polygons of 30, 60, 120, &c. sides; and it is plain that each polygon will exceed the preceding in surface.

It is obvious that any regular polygon whatever might be inscribed in a circle, provided that its circumference could be divided into any proposed number of equal parts; but such division of the circumference, like the trisection of an angle, which, indeed, depends on it, is a problem which has not yet been effected. There are no means of inscribing in a circle a regular heptagon, or which is the same thing, the circumference of a circle cannot be divided into seven equal parts, by any method hitherto discovered. Indeed the polygons above noticed were, till about a quarter of a century ago, supposed to include all that could admit of inscription in a circle; but in 1801 a work was published by M. Gauss of Göttingen, (and afterward translated into French by M. Delisle, under the title of Recherches Arithmetiques), containing the curious discovery that the circumference of a circle could be divided into any number of equal parts capable of being expressed by the formula 2+1, provided it be a prime number, that is, a number that cannot be resolved into factors. The number 3 is the simplest of this kind, it being the value of the above formula when n=1; the next prime number is 5, and this also is contained in the formula. But polygons of 3 and of 5 sides have already been inscribed. The next prime number expressed by the formula is 17, so that it is possible to inscribe seventeen sided polygon in a circle. The investigation of Gauss's theorem, although it establishes the above geometrical fact, depends upon the theory of algebraical equations, and involves other considerations of a nature that do not enter into elements of geometry; we must, therefore, content ourselves with merely alluding to it *.

PROPOSITION IX. PROBLEM.

a

An inscribed regular polygon being given to circumscribe a similar polygon about the circle; and, conversely, from having a circumscribed regular polygon to form the similar inscribed one.

Let abcd, &c. be a regular inscribed polygon; it is required to describe a similar polygon about the circle.

On this subject the student may consult Barlow's Theory of Numbers.

At each of the points a, b, c, d, &c. draw tangents to the circle, and they will form the polygon ABCD, &c. similar to the polygon abcd, &c.

For, in the first place, there are as many tangents as the inscribed polygon has sides, and those drawn through the extremities of the same chord meet, otherwise the chord would be a diameter (Prop. IX. Cor. 3. B. III.). Next, the angles formed by these tangents and

chords are all equal to each other, for their sides include equal arcs (Prop. XV. B. III.). Hence the triangles fAa, aBb, bCc, &c. are isosceles, and they have equal bases fa, ab, bc, &c.; therefore these triangles are equal, and, consequently, the angles A, B, C, D, &c. are equal, and so are their including sides therefore the polygon A B C D &c. is regular; and it has the same number of sides as the polygon a b c d &c., it is, therefore, similar to it (Prop. I.),

Conversely. Let the circumscribed polygon A B C D &c. be given, then if the successive points of contact a, b, c, d, &c. be joined, a similar polygon will be inscribed in the circle:For the angles A, B, C, &c. are equal, as also the sides aB, Bb, bC, Cc, &c., each being half a side of the polygon; con-, sequently the sides ab, bc, cd, &c. are equal, and equal chords include equal angles (Prop. XV. Cor. IV. B. 3.); they, therefore, form a regular polygon, and as the sides are the same in number as those of the circumscribed polygon, it is similar to it.

Scholium.

1. It was remarked in the scholium to the preceding proposition that, from having an inscribed regular polygon, we might easily form another of double the number of sides. It may be, in like manner, here observed that, from having a circumscribed regular polygon, we may readily derive another of double the number of sides, nothing more being necessary than to draw tangents to the points of bisection of the arcs intercepted by the sides of the proposed polygon, limiting these tangents by those sides; and it is plain that each of the polygons so formed will be less in surface than the preceding, being entirely comprehended within it.

Let ab be a side of an inscribed polygon, and if aA, ¿A be tangents to the circle at the points a, b, cach will be one half

of the side of the similar circumscribed polygon, or which is the same thing, they will together be equal to the side of a circumscribed polygon, similar to the inscribed one whose' side is ab. Let M be the middle of the intercepted arc, and draw Ma, Mb, and the tangent BMC, then aM, Mb will be two consecutive sides of an inscribed

polygon, having double the number of sides that the polygon has whose side is ab; and, consequently, BC being a tangent at M, meeting the tangents at a and b must, by the proposition, be the side of a polygon, having double the number of sides that the polygon has, whose side is ab.

2. If polygons be thus successively circumscribed about the circle, their perimeters will decrease as the number of sides increase. For BC is less than AB+AC, and, consequently, aB+BC+Cb<aA+Ab; now, a B+Cb=BC; and aA+Ab is equal to a side of the first circumscribed polygon: hence two sides of the second circumscribed polygon are together less than one side of the first; and, therefore, the whole perimeter of the second is less than that of the first. It is obvious that with respect to the inscribed polygons, the perimeters increase in the same circumstance, thus :-The two sides aM, Mb being together longer than ab, it follows that the perimeter of the second inscribed polygon exceeds that of the first.

The successive circumscribed polygons that we have been considering continually approach nearer and nearer towards coincidence with the circle; for OB is nearer an equality to the radius Oa of the circle than OA, because the distance aB is less than the distance aA is; and in every succeeding polygon the difference between the radius of the circle and the distance of the centre from the remotest points in the perimeter will, in like manner, perpetually diminish; so that the perimeters continually approach towards coincidence with the circumference, and we have already seen that these perimeters continually diminish.

Now it is plain that if a series of magnitudes continually approach nearer and nearer towards coincidence with any proposed magnitude, and at the same time continually diminish; the magnitude to which they approach must be smaller than either of the approaching terms; we are, therefore, warranted in asserting that the circumference of a circle is a shorter line than the perimeter of any circumscribed polygon.

In a similar manner, by considering the successive inscribed polygons, it appears that they also continually approach towards coincidence with the circle; for Od is nearer an equality to the radius than OD, since the chord aM is shorter than ab (Prop. VII. B. III.); so that in each succeeding polygon the perimeter approaches nearer to coincidence with the circumference, and it has been shown that these perimeters successively increase hence we may infer that the circumference of a circle is a longer line than the perimeter of any inscribed polygon.

PROPOSITION X. THEOREM.

Two polygons may be formed, the one within, and the other about a circle, that shall differ from each other by less than any assigned magnitude however small. Let M represent any assigned surface, it is to be shown that two polygons may be described, the one within, and the other about the circle, whose centre is O, which will differ from each other by a magnitude less than M.

A

H

m

E

Let N be the side of a square, whose surface is less than the surface M, and inscribe in the circle a chord an equal to the line N. Then, by the methods already explained, inscribe in the circle a square, a hexagon, or indeed any regular polygon; let the arcs which its sides subtend be bisected, the chords of the half arcs will be the sides of a regular polygon, hav. ing double the number of sides: let, now, the arcs subtended by the sides of this second polygon be in like manner bisected, the chords will form a third polygon, having double the number of sides that the second has. Continue these successive bisections till the arcs become so small as to be each less than

B

C