gents to the circle through the vertices of the inscribed polygon. If we draw the radii CA, CB, CD, &c. (fig. 80) to the vertices of the inscribed polygon ABDEFG, and draw through these vertices, perpendicular to radius, the straight lines tob, bol, &c, of, fg, ga, we shall have the regular circumscribed polygon a b d e fg, of the same number of sides. Remark. The perpendicular CA, CB, &c. drawn from the centre of the regular polygon to one of its sides, (which always bisects that side) is called the apothegm of the polygon. C in is the apothegm of the inscribed polygon. #43. It is manifest that by increasing the number of sides in the inscribed polygon, we increase its perimeter, which can never be greater than the circumference ; and by increasing the number of sides of the circumscribed pologon, we diminish its perimeter, which can never be less than the circumference. The circumference of the circle being then the limit of this approximation, or the value to which these two magnitudes approach ; we say that—The circumference of the circle is equivalent to the perimeter of the regular inscribed (or circumscribed) polygon of an infinite number of sides. Two circles may therefore be considered two similar polygons; whence (139)—The circumferences of two circles are to each other in the ratio of their radii, or of their diameters. Remark. The apothegm of the inscribed or circumscribed polygon of an infinite number of sides must be infinitely near to radius. The consideration of the ratio of the circumference to radius, or to the diameter, we shall defer for the present. PART FIRST. SECTION II.-Of the Measure and Comparison of Plane - . Surfaces. 149. In discussing the properties of plane figures thus far, we have considered only the lines and the angles, and their several relations to each other, without regard. ing the quantity of surface embraced by the outline or perimeter of the figure. The whole amount of surface or superficies in any geometrical figure is called its area. The word surface, in general, we use to signify superficial extent without regard to quantity. 150. In speaking of figures as equal, we have said that it is an indispensable condition of geometrical equality that the figures compared should coincide by superposition. But it is evident that two figures may have the same amount of surface, and still be very different in form. Such figures we call equivalent. A field of a circular form may have the same superficial extent with another field whose form is quadrangular ; we should say that the two fields are equivalent or equal in area. In comparing parallelograms we consider one side as the base of the figure, and the perpendicular distance of this side from the opposite, the height of the figure. Sometimes we call one side the inferior base, and the opposite, the superior base. In triangles one side is taken for the base, and the perpendicular distance of this side from the vertex of the opposite angle, is the height of the triangle. 151. To compare two parallelograms ABCD, ABEF, (fig. 81) whose bases are equal, and heights equal, place their bases together as in figure 82. On account of their equal heights and the parallelism of their opposite sides, their superior bases CD, EF, will be in the same straight line CF. We shall have two triangles CAE, DBF, of which the sides CA and DB are equal and parallel, being opposite sides of the same parallelogram (59); AE and BF are equal and parallel for the same reason. The two angles CAE, DBF, having their sides parallel and directed the same way, are equal; and the two triangles CAE, DBF, are equal by the second case of equal trian gles. If from the whole quadrilateral CABF, we take 5 Fig. 84. Fig. 85. the triangle DBF, there will remain the parallelogram . gle ABC (fig. 83) is then one half of the parallelo gram ABCD ; and therefore (152)—Every triangle is that is, the two rectangles are whose base is Ab by m ; as A b and AE are commen . ABCD — m surable, we shall have the proportion TAEGF = AB — d – **** If we apply the measure of AE one time more than it is contained in AB, it will reach beyond B to b'; and suppose this excess B b', which we denote by d', to be less than any assignable magnitude, (since we can divide AE into parts so small that this excess shall be as little as we please), if we draw through b' a line parallel to BD, meeting CD produced, we shall have a rectangle which, exceeds the rectangle ABCD by an indefinitely small excess which we designate by m'. According to what is proved in the preceding part of this article, we have >{ { f - W/ *...* = **. We see, therefore, that in comparing these two rectangles and their bases, if any magnitude however small be added to the rectangle ABCD, a corresponding magnitude must be added to its base AB to preserve the proportion ; and if any magnitude however small be subtracted from the rectangle ABCD, a corresponding magnitude must be subtracted from its base to preserve the proportion ; it therefore follows necessarily, that if nothing be added to, or subtracted from the rectangle, nothing must be added to, or subtracted from its base, to preserve the proportion ; and we ABCD AB . AECF T A F * gles are in the ratio of their bases though the bases are incommensurable. We therefore say—Two rectangles of the same height, are to cach other in the ratio of their bases. * 156. As the side AC might have been taken as the common base of the two rectangles whose heights are AB and AE; we have from the same proportion ABCD AB AECF T AF' of equal bases are to each other in the ratio of their heights. 157. As a triangle is equivalent to half the rectangle of the same base and height, it follows from the last two articles, that—Two triangles, of equal heights, are to consequently have that is, the rectan this general rule also—Two rectangles Fig. 86. Fig. 87. each other in the ratio of their bases ; and two triangles §§§ but in the first ratio of the proportion, the If we multiply will give |