A chord, passing through the extremity of the diameter, is a mean proportional between the diameter and the segment formed by the perpendicular let fall from the other extremity of that chord.--To find a mean proportional between two given lines. To divide a line in extreme and mean ratio.--The length of the line being given numerically, to calculate the numerical value of each of the segments. _Of polygons inscribed and circumscribed to the circle._ To inscribe or circumscribe a circle to a given triangle. Every regular polygon can be inscribed and circumscribed to the circle. A regular polygon being inscribed in a circle, 1º to inscribe in the same circle a polygon of twice as many sides, and to find the length of one of the sides of the second polygon; 2º to circumscribe about the circle a regular polygon of the same number of sides, and to express the side of the circumscribed polygon by means of the side of the corresponding inscribed polygon. To inscribe in a circle polygons of 4, 8, 16, 32, sides. To inscribe in a circle polygons of 3, 6, 12, 24, sides. To inscribe in a circle polygons of 5, 10, 20, 40, sides. To inscribe in a circle polygons of 15, 30, 60, sides. Regular polygons of the same number of sides are similar, and their perimeters are to each other as the radii of the circles to which they are inscribed or circumscribed.--The circumferences of circles are to each other as their radii. To find the approximate ratio of the circumference to the diameter. _Of the area of polygons and of that of the circle._ Two parallelograms of the same base and of the same height are equivalent.--Two triangles of the same base and height are equivalent. The area of a rectangle and that of a parallelogram are equal to the product of the base by the height.--What must be understood by that enunciation.--The area of a triangle is measured by half of the product of the base by the height. To transform any polygon into an equivalent square.--Measure of the area of a polygon.--Measure of the area of a trapezoid. The square constructed on the hypothenuse of a right-angled triangle is equivalent to the sum of the squares constructed on the other two sides.--The squares constructed on the two sides of the right angle of a right-angled triangle and on the hypothenuse are to each other as the adjacent segments and entire hypothenuse. The areas of similar polygons are to each other as the squares of the homologous sides of the polygons. Notions on surveying for content (_arpentage_).--Method of decomposition into triangles.--Simpler method of decomposition into trapezoids.--Surveyor’s cross.--Practical solution, when the ground is bounded, in one or more parts, by a curved line.