Calendar: 1874-1875 Page 544

544 GENERAL LITERATURE AND SCIENCE than the sum of the sides of the triangle and greater than half their sum To describe parallelogram that shall be equal to given triangle and have one of its angles equal to given rectilineal angle Describe parallelogram equal to given triangle in perimeter as well as in area If BD be diagonal of parallelogram ABCD prove that the area of the triangle PBD will be jual to the sum or difference of the triangles Ρ AB and PBC as the point Ρ is taken without or within the parallelogram In any right-angled triangle the square described on tbe side subtending the right angle is equal to tbe sum of the squares described on the sides which contain the right angle If from the middle point of one of the sides of right-angled triangle perpendicular be drawn to the hypotenuse the differ- ence of the squares of the segments into which tbe hypotenuse is divided equals the square of the remaining side of the triangle If straight line be divided into two equal and also into two unequal parts the squares on the two unequal parts are together double of the square on half the line and of the square on the line between the points of section To describe square that shall be equal to given rec- tilineal figure If two circles touch externally the straight line which joins their centres shall pass through the point of contact If two circles touch externally and parallel diameters be drawn prove that the straight line which joins the ends of these diameters towards opposite parts passes through the point of contact The straight line drawn at right angles to the diameter of circle from the extremity of it falls without the circle and no straight line can be drawn from the extremity between that straight line and the circumference so as not to cut the circle 10 The opposite angles ef any quadrilateral figure inscribed in circle are together equal to two right angles If be point within circle BC tbe diameter and through there be drawn AD perpendicular to BC and BAB meeting the circum-