SECOND OPERATION. In the second operation the work 6.8 is performed as in pure duodecimals (Art. 594). The point (.) separates 4:5 lower duodecimal orders from those be2 9 4 ginning with feet. As to the number 2 2 8 of feet in the multiplicand and multi25-5 4 = 29sq. ft. 5'4". plier, no change is required in that part of either of the given numbers in expressing them according to the duodecimal scale. In performing the multiplication, we make the several reductions required according to the radix 12; and have, after pointing off, 25.54 square feet, expressed according to the duodecimal scale. The units, or feet, at the left of the point, are readily changed to the decimal scale by multiplying the left-hand figure, 2, by 12, the number of units in the radix, and adding the right-hand figure, 5, and giving the figures to the right of the point their proper notation, we have then 29sq. ft. 5' 4" for the answer, as before. RULE. - Write the multiplier under the multiplicand, so that units of the same orders shall stand in the same column. Beginning at the right, multiply each term in the multiplicand by each term of the multiplier, and write the first term of each partial product under its multiplier, observing to carry a unit for every twelve from each lower order to the next higher. The sum of the partial products will be the product required. 2. How many square feet in a floor 48 feet 6 inches long, and 24 feet 3 inches broad? Ans. 1176 sq. ft. 1' 6':. 3. The length of a room being 20 feet, its breadth 14 feet 6 inches, and height 10 feet 4 inches, how many square yards of painting are in it, deducting a fireplace of 4 feet by 4 feet 4 inches, and 2 windows, each 6 feet by 3 feet 2 inches? Ans. 73. square yards. 4. Required the solid contents of a wall: 53 feet 6 inches long, 10 feet 3 inches high, and 2 feet thick. 5. There is a house with four tiers of windows, and four windows in a tier; the height of the first is 6 feet 8 inches; the second, 5 feet 9 inches; the third, 4 feet 6 inches; the fourth, 3 feet 10 inches ; and the breadth is 3 feet 5 inches; how many square feet do they contain in the whole ? Ans. 283sq. ft. 7. 6. How many cords in a pile of wood 97 feet 9 inches long, 4 feet wide, and 3 feet 6 inches high? Ans. 10137 cords. 7. Required the number of cords of wood in a pile 100 feet long, 4 feet wide, and 6 feet 11 inches high. Ans. 2136 DIVISION OF DUODECIMALS. FIRST OPERATION. SECOND OPERATION. 7e 601. To divide one duodecimal by another. Ex. 1. A board in the form of a rectangle, whose area is 27 sq. ft. 8' 6", is ift. 7in. wide; what is its length ? Ans. 17ft. 6in. We find how many times 27 1ft. 7') 27sq. ft. 8' 6" (17ft. 6' square feet contains the divisor, 26 11 and obtain 17 feet for the quo tient, we multiply the entire di9 6 visor by the 17ft., and subtract 9 6 the product, 26ft. 11', from the corresponding portion of the dividend, and obtain 9', to which remainder we bring dcwn the 6", and, dividing, obtain 6' for the quotient. Multiplying the entire divisor by 6”, we obtain Oft. 9'6", which, subtracted as before, leaves no remainder. Therefore 17 feet 6 inches is the length required. In the second operation we re1.7) 23.86 ( 15.6 = 17ft. 6' duce the feet of the given multi17 plicand and multiplier to the duo decimal scale, and thus obtain 88 23.86 and 1.7. We then conduct the division with reference to the 96 radix 12, as is ordinarily done with respect to 10 (Art. 594). The re96 sult obtained is 15.6 in the duo decimal, which, on changing the figures to the left of the point to the decimal scale, and giving the proper notation to the figure on the right of the point, becomes transformed to 17tt. 6' 17ft. 6in., the answer, as before. RULE. — Find how many times the highest term of the dividend will contain the divisor. By this quotient multiply the entire divisor, and subtract the product from the corresponding terms of the dividend. To the remainder annex the next denomination of the dividend, and divide as before, and so continue till the division is complete. 2. It required 834 sq. ft. 3' of board to cover the side of a certain building. The height was 17ft. Iin.; what was the length of the side? Ans. 47 feet. 3. How many feet wide is a plank of uniform width, whose length is 18ft. Iin., thickness 3 inches, and solid contents 84ft. 4' 6:1 ? 4. An alley has an area of 792ft. 6' 9" 21". Its width is 12ft. 7' 8". Required its length. Ans. 62ft. 8' 6". MISCELLANEOUS EXAMPLES. per month ? 1. A merchant engages a clerk at the rate of $ 20 for the first month, $ 25 for the second, $30 for the third, &c., thus in, creasing his salary by $ 5 per month. How long must the clerk retain his situation, so as to receive on the whole as much as he would have received had his salary been fixed at $ 52.50 Ans. 14 months. 2. A mason has plastered 3 rooms; the ceiling of each is 20 feet by 16 feet 6 inches, the walls of each are 9 feet 6 inches high, and 90 yards are to be deducted for doors, windows, &c. For how many yards must he be paid ? Ans. 25lyd. 1ft. 6'. 3. A man of wealth, dying, left his property to his ten sons, and the executor of his will, as follows: to his executor, $ 1024; to his youngest son, as much and half as much more; and increasing the share of each next elder in the ratio of 13. What was the share of the eldest ? 4. A butcher, wishing to buy some sheep, asked the owner how much he must give him for 20; on hearing his price, he said it was too much ; the owner replied, that he should have 10, provided he would give him a cent for each different choice of 10 in 20, to which he agreed. How much did he pay for the 10 sheep, according to the bargain? Ans. $ 1847.56. 5. If 340 square feet of carpeting are required to cover the floor of a room, how many yards will be required, provided the width of the carpeting is 3 feet 9 inches ? Ans. 30yd. Sin. 6. If a clergyman's salary of $ 700 per annum is 6 arrears, how much is due him, allowing compound interest at 6 Ans. $ 4882.72. 7. Suppose a clock to have an hour-hand, a minute-hand, and a second-hand, all turning on the same center. At 12 o'clock all the hands are together and point at 12. (1.) How long will it be before the second-hand will be between the other two hands, and at equal distances from each? Ans. 601482seconds. (2.) Also before the minute-hand will be equally distant between the other two hands? Ans. 61983 seconds. (3.) Also before the hour-band will be equally distant between the other two hands? Ans. 597 seconds. years in per cent. ? MENSURATION. DEFINITIONS. A B В D 602. A point is that which has neither length, breadth, nor thickness, but position only. 603. A line is length, without breadth or thickness. A straight line is one which has the same direction in its whole extent; as the line A B. A curved line is one which continually changes its direction ; as the line CD. 604. An angle is the inclination or opening of two lines, which meet in a point. A right angle is an angle formed by a straight line and a perpendicular to it; as the angle A B C. An acute angle is one less than a right angle; as the angle EBC. An obtuse angle is one greater than a right angle; " as the angle F B C. A B E B с B 605. A surface is that which has length and breadth, without thickness. A plane surface, or simply a plane, is that in which, if any two points whatever be taken, the straight line that joins them will lie wholly in it. Every surface, which is not a plane, or composed of planes, is a curved surface. 606. The area of a figure is its quantity of surface; and is estimated in the square of some unit of measure, as a square inch, a square foot, &c. 607. A solid, or body, is that which has length, breadth, and thickness. 608. The solidity, or volume of a solid, is estimated in the cube of some unit of measure; as a cubic inch, a cubic foot, &c. 609. MENSURATION is the process of determining the areas of surfaces, and the solidity or volume of solids. MENSURATION OF SURFACES. 610. A plane figure is an enclosed plane surface ; if bounded by straight lines only, it is called a rectilineal figure, or polygon. The perimeter of a figure is its boundary, or contour. 611. Three-sided polygons are called triangles; those of four sides, quadrilaierals; those of tive sides, pentagons, and so on. TRIANGLES. 612. An equilateral triangle is one whose sides are all equal; as CAD. Note. — The line A B, drawn from the angle A perpendicular to the base C D, is the altitude of the triangle CAD. C B An isosceles triangle is one which has two of its sides equal; as EFG. A scalene triangle is one which has its three sides unequal; as HIJ. H M A right-angled triangle is one which has a right angle; as K L M. area. 613. To find the area of a triangle. Multiply the base by half the altitude, and the product will be the Or, Add the three sides together, take half that sum, and from this subtract each side separately; then multiply the half of the sum and these remainders together, and the square root of this product will be the area. Ex. 1. What are the contents of a triangle whose perpendicular height is 12 feet, and whose base is 18 feet ? Ans. 108 feet. 2. There is a triangle, the longest side of which is 15.6 feet, the shortest side 9.2 feet, and the other side 10.4 feet. What are the contents ? Ans. 46.139+ feet. 3. The triangular gable of a certain building has a base of 40 feet and an altitude of 15 feet; how many square feet of boards will cover it ? Ans. 300 są: ft. 4. The perimeter of a certain field in the form of an equilateral triangle is 336 rods; what is the area of the field ? Ans. 33 acres 152 sq. rd. QUADRILATERALS. 611. A parallelogram is any quadrilateral whose opposite sides are parallel. 615. A rectangle is any right-angled parallelogram ; as ABCD. D A B |