PROP. XX. PROB. In a given circle, to inscribe regular polygons of fifteen and of thirty sides. Let AB and BC be the sides of an inscribed decagon, and AD the side of a hexagon inscribed; the arc BD will be the fifteenth part of the circumference of the circle, and DC the thirtieth part. For, if the circumference were divided into thirty equal portions, the arc AB would be equal to three of these, and the C arc AD to five; consequently the B excess BD is equal to two of these portions, or it is the fifteenth part of the whole circumference. A gain, the double arc ABC being equal to six portions, and ABD to five, the defect DC is equal to one portion, or to the thirtieth part of the circumference. Scholium. From the inscription of the square, the pentagon, and the hexagon, may be derived that of a variety of other regular polygons: For, by continually bisecting the intercepted arcs and inserting new chords, the inscribed figure will, at each successive operation, have the number of its sides doubled. Hence polygons will arise of 6, 8, and 10 sides; then of 12, 16, and 20; next of 24, 32, and 40; again, of 48, 64, and 80; and so forth repeatedly. The excess of the arc of the hexagon above that of the decagon, gives the arc of a fifteen-sided figure; and the continued bisection of this arc will mark out polygons with 30, 60, or 120 equal sides, in perpetual succession. The same results might also be obtained from the differences of the preceding arcs. Of the regular polygons, three only are susceptible of perfect adaptation, and capable therefore of covering, by their repeated addition, a plane surface. These are the equilateral triangle, the square, and the hexagon. The angles of an equilateral triangle are each two-thirds of a right angle, those of a square are right angles, and the angles of a hexagon are each equal to four-third parts of a right angle. Hence there may be constituted about a point, six equilateral triangles, four squares, and three hexagons. But no other regular polygon can admit of a like disposition. The pentagon, for instance, having each of its angles equal to six-fifths of a right angle, would not fill up the whole space about a point, on being repeated three times; yet it would do more than cover that space, if added four times. On the other hand, since each angle of a polygon which has more than six sides must exceed four-third parts of a right angle, three such polygons cannot stand round a point. Nor can the space about a point ever be bisected by the application of any regular polygons, of whatever number of sides; for their angles are always necessarily each less than two right angles. ELEMENTS OF GEOMETRY. BOOK V. OF PROPORTION, THE preceding Books treat of magnitude as concrete, or having mere extension; and the simpler properties of lines, of angles, and of surfaces, were deduced, by a continuous process of reasoning, grounded on the principle of superposition. But this mode of investigation, how satisfactory soever to the mind, is by its nature very limited and laborious. By introducing the idea of Number into geometry, a new scene is opened, and a far wider prospect rises into view. Magnitude, being considered as discrete, or composed of integrant parts, becomes assimilated to multitude; and under this aspect, it presents a vast system of relations, which may be traced out with the utmost facility. Numbers were at first employed, to denote the aggregation of separate, though kindred, objects; but the subdivision of extent, whether actually effected or only conceived to exist, bestowing on each portion a sort of individuality, they came afterwards to acquire a more comprehensive application. In comparing together two quantities of the same kind, the one may contain the other, or be contained by it; that is, the one may result from the repeated addition of the other, or it may in its turn produce this other by a successive composition. The one quantity is, therefore, equal, either to so many times the other, or to a certain aliquot part of it. Such seems to be the simplest of the numerical relations. It is very confined, however, in its application, and is evidently, in this shape, insufficient altogether for the purpose of general comparison. But that object is attained, by adopting some intermediate term of reference. Though a quantity neither contain another exactly, nor be contained by it; there may yet exist a third and smaller quantity, which is at once capable of measuring them both. This measure corresponds to the arithmetical unit; and as number denotes the collection of units, so quantity may be viewed as the aggregate of its component measures. But mathematical quantities are not all susceр |