of a brick and a half thick. This rod is in surface nearly the same as that in land measure for the square of 5 yards, or 16.5 feet equals 272.25. .. a x b x c 3 x 272 the number of feet in the wall of the number of standard rods. 3. "Prove a rule for determining the area of a trapezoid." "The area of a trapezoid is equal to half the area of a rectangle, having the same altitude, and whose base equals the sum of the parallel sides of the trapezoid."Tate's Geometry. Art. 43. SECTION II. 1. "How many cubical feet of timber are there in the flooring of a room in. thick and 17 ft. 6 in. in length by 15 ft. 3 in. in breadth?" 66 2. In a wall 10 ft. high, 15 ft. long, and 2 bricks thick, there is an arched doorway 4 ft. wide and 87 16: 16128 6 ft. high to the springing of the arch, which is semicircular; how many standard rods of brick-work are there in the wall?" 15 X 10 X 5 3 x 272 doorway. 6 x 4 x 5 3 x 272 = number of rods including the = number of rods in the door-way to 66 number of rods in the arch of /6 X 4 X 5 6 × 4 3 x 272 3 x 272 8 x 7854)5 3. What is the weight of a circular iron ring whose inner diameter is 18 in., and whose section is a circle 2 in. in diameter, the weight of a cubic foot of iron being 450 lbs.?" 18+ 2 = 20 diameter of ring to the centre of circular rod. 20 × 3·1416 = 62.832 = the length of circular rod in inches. 22 × 7854 rod in inches. 3.1416 = area of section of circular 62.832 × 31416 197.393, solid content in in. 197.393 1728 X 450 = 51.4044 lbs. weight. SECTION III. 66 1. After measuring a piece of cloth, to contain 90 yards, I find that the yard measure that I have used is too short by 1-30th part; what is the true measure of of the cloth?" Every length measured contained 1 yard minus. 1-30th yard. 1 .. 90 lengths = 90 yards minus 90 x yards. 30 90 90 30 = 90 387 yards, the true measure. 2. "How many square inches of tin plate are required to make an open cylindrical vessel to contain a gallon whose height is equal to one half its diameter ?" "N.B.-An imperial gallon contains 277 274 cubical inches.' မိ But the vessel is to contain 277.274 cubical inches. Having found the diameter of the vessel we can now find the surface. 3. "Show that less tin will be used in making a vessel of the dimensions given in the last question, than in making one of any other dimensions but of a cylindrical form and the same capacity." Now let a (the base) be increased or diminished by any quantity n, such that a―n shall be positive. Then an diameter of base, The contents of the cylinder under the new conditions remain unaltered. the given case: and accordingly as this ratio is greater than, equal to, or less than unity, so will the surface in the assumed case be greater than, equal to, or less than : n2 (3 a + n) =1+ a quan 3 a2 (a±n)' tity greater than unity for if we take the positive sign, this is evidently the case; and if we take the negative, the fractional part of the ratio will be addative, since a > n. 4. "Investigate Thomas Simpson's rule for determining the area of a plane surface bounded by an irregular line." CD Let A B C be a plane surface bounded by an irregular line A a B b Cc. In order to find the area by Thomas Simpson's rule, let it be divided into three parts by the right lines A C and B D drawn (or in practice measured) perpendicularly to each other. We propose to investigate the rule in the portion A BD: the point B in our figure is left unlimited, so that there may be n ordinates drawn at equal distances |