divide the number into periods at the commencement of the process." The cube of 1, 10, 100, is formed of the unit followed by three times as many cyphers as there are in the number taken. A number, therefore, between 10 and 100 has its cube between 1000 and 1000000 ; consequently its cube is formed of 4, 5, or 6 figures, according to the proximity of its value to 10, or 100. Generally, the cube of any number will contain three times its number of figures, minus one or two, according to the value of the first figure in the root. Hence the reason for dividing a number, whose cube root is to be extracted into periods of three figures each, commencing from the units' place. Now for the cube root of '04; we must add seven cyphers in order to obtain three decimal places in the root; for the periods commence "from the units' place." The method here employed, is published in “Tate's Principles of Arithmetic," a copy of which every teacher The process for obtaining the trial divisor is original and elegant. It will be comprehended should possess. from the following explanation, in which it is assumed that the mode of deriving the rule for the extraction of the cube root is understood. By this method, squaring the root already found, and then multiplying by 3 for a new divisor, are superseded by the simple process of addition. The operation commences after the second figure in the root is obtained; when the two figures found form that part of the root in the formula, (a + b)s = a23 + 3 a2 b + 3 a b2 + b3, represented by a. Now in the preceding example (leaving out of consideration that the figures found are decimals, as the reasoning will not be affected thereby), the a is 34, which may be resolved into 30 + 4; squaring these by the binomial and multiplying by 3, we have, 3 (302 + 2 x 30 x 4 + 42) = 27002 × 360 + 3 × 16, the first of which numbers and the greater factor of the others are all previously obtained (marked (c), (d), and (e), respectively); for 27 stands two places to the left, and 36 one. Hence the sum of once (c), twice (d), and thrice (e) will give three times the square of 34. The addition is conveniently performed, as in the example, by placing the sum of (d) and (e) under the obtained sum of the three products, and then adding the three numbers which are directly under each other. The advantage of this artifice becomes immensely great, as the number of figures in the root increases. 11 MENSURATION. SECTION I. 1. Prove the rule of cross multiplication. 2. Prove a rule for determining the number of standard rods of brick-work in a wall. 3. Prove a rule for determining the area of a trapezoid. 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? 2. In a wall 10 ft. high, 15 ft. long, and 2 bricks thick, there is an arched door-way 4 ft. wide and 6 ft. high to the springing of the arch, which is semi-circular; how many standard rods of brickwork are there in the wall? 3. What is the weight of a circular iron ring whose inner diameter is 18 in., and whose section is a eircle 2 in. in diameter, the weight of a cubic foot of the iron being 450 lbs. ? SECTION III. 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 the cloth? 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 of its diameter ? N.B. An imperial gallon contains 277,274 cubical inches. 3. Show that less tin will be used in making a vessel of the dimensions given in the last example, than in making one of any other dimensions but of a cylindrical form and the same capacity. 4. Investigate Thomas Simpson's rule for determining the area of a plane surface bounded by an irregular line. 5. Investigate a formula for determining the quantity of earth to be taken out of a cutting for a road, the width of the road and the slope of the banks being given. SECTION IV. 1. Describe Gunter's chain. 2. Construct a field-book for a three-sided field of which one side has an irregular form, and the other two are straight lines; assuming any dimensions whatever. 3. Describe and explain the vernier. 4. Describe the spirit level and its adjustments. SECTION I. 1. "Prove the rule of cross multiplication." Let us find the area of the floor in ques. 1. sec. ii. to exemplify the rule. Here the product of the 6 by 15 is divided by 12, and the remainder put in the inches' place; the quotient is carried to the square feet. In multiplying by the 3 in., the first remainder from the division of 12 is put one place further to the right; the process is then continued, dividing each product by 12 and placing the remainder in order towards the left hand. Now the result which is generally termed 266 feet, 10 inches, 6 parts, is really 266 feet, 10 parts, 6 inches; the parts being rectangles of 1 by 12 in., which, of course, are each 12 square inches in area. The result expressed in feet and fractions of a foot would stand thus, Let us now prove that this is the true result, by working with the 17 ft. 6 in. and the 15 ft. 3 in. in the form of their equivalents, 17 ft.; thus, 17+12 15 +12 and 15 262 +12 4 +1 +144 266 + 1 + 14± The first product is obtained by multiplying by the 15 whole numbers, and the second by the fraction, care being taken to retain the duodecimal scale, or the decrease of the numbers from the left in a twelvefold ratio. 66 2. Prove the rule for determining the number of standard rods of brickwork in a wall." Rule:-Multiply the number of feet in the surface of the wall by the number of half bricks in the thickness of the wall, and divide the result by 3 × 272. The standard rod is 272 feet of surface of brickwork, |