I. Comparison of Rectangles contained by Straight Lines and their Segments. HYPOTHESES. If three straight lines be proportionals. If four straight lines be proportionals. If in three straight lines the rectangle under the extremes is equal in area to the square on the mean. If in four straight lines the rectangle under the extremes be equal in area to the rectangle under the means. K. Of Polygons. HYPOTHESES. If polygons are similar Idem. If rectilineal figures are similar to the same rectilineal figure. If three straight lines be proportionals. If four straight lines be proportionals. If the similar rectilineal figures similarly-described upon four straight lines be proportionals. CONSEQUENCES. (The rectangle under the extremes is equal in area to the square on the mean. (The rectangle under the extremes is equal in area to the rectangle under the means. -The lines are proportionals. The lines are proportionals. CONSEQUENCES. They may be divided into the same number of similar triangles, having the same ratio to one another that the polygons have. The polygons have to one another the duplicate ratio of that which their homologous sides have. Their perimeters are as their homologous sides. They are similar to one another. As the first is to the third, so is any rectilineal figure upon the first to a similar and similarly-described rectilineal figure upon the second. The similar rectilineal figures similarly described upon them shall also be proportionals. The lines shall also be proportionals. CONSEQUENCES. (It may have one circle circumscribed about it, and another inscribed in it. And the same point is the center of both circles. They are to one another as the squares on their diameters. It is equilateral. } It is equiangular. Tangents to the circle drawn through the angular points, will form an equilateral and equiangular figure of the same number of sides, circumscribed about the circle. The radius of the circle is equal to the side of the hexagon. |