Elements of Geometry, with Trigonometry

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1872 edition. Excerpt: ...in the polygon. A regular polygon of 360 sides would, therefore, divide the circumference into arcs of 1; but 360 does not occur in any of the above series, nor is there any elementary method of trisecting the arc of 3 subtended by the side of a regular polygon of 120 sides. Hence, though many multiples of the unit arc are easily determined, the unit arc itself, so important in the construction of mathematical and astronomical instruments, must be determined by approximation. S IV. A regular polygon being inscribed in a given circle, tc circumscribe a similar regular polygon about the same circle. At the vertices of the angles of the regular inscribed polygon ABCDEF (Fig. 89), draw the tangents PH, HK, KL, &c, and the circumscribed polygon which they form is similar to the inscribed polygon. For since AF = AB = BC... and the angle AFP= FAP=BAH=ABH..., all being angles which are N( measured by the halves of equal arcs (39), therefore the triangles FPA, AH.B, BKC.... are isosceles and equal. Hence FP = AP = AH = HB..., and the angle P = H = K... Consequently the circumscribed polygon is regular, and, having the same number of sides, it is similar to the inscribed polygon. Tangents drawn at the middle points of the arcs AF, AB, &c, will be parallel to the chords, and will also form a regular circumscribed polygon similar to the inscribed one. The converse problem of inscribing a regular polygon similar to a given circumscribed polygon, can likewise, as is obvious, be solved, either by joining the points of contact, or by drawing chords parallel to the sides of the circumscribed polygon. V. To find the ratio of the circumference of a circle to its diameter. Of the numerous solutions given of this celebrated problem, one of the...