By Rule II. Here the nearest root to the first period is 3, hence r=3000, and r2= 3000) 5, N=3076828211, &c.; n+1=6, and n−1=4, therefore 6×3076828211, &c.+4x300015 4×3076828211, &c.+6x3000) 5 -×3000=3144, the root nearly, and by taking r=3144, and repeating the operation, the root will be had. (2.) Extract the square root of 2. (3.) Required the cube, or third, root of 5. (9.) Required the 9th root of 1.551328215978515625. (10.) Required the 365th root of 1·05. (11.) Required the 40th root of 1·04. (12.) The amount of £1. for 40 years at compound interest is £48010206, what is the rate per cent.? Definition. DUODECIMALS. Duodecimals are so called because every superior place is 12 times its next inferior in that scale of notation. This way of conceiving an unit to be divided is chiefly in use among artificers who generally take the linear dimensions of their work in feet, inches, and parts. Note, 12 Inches' = 1 Foot, | 12 Thirds" 1 Second", 12 Seconds" = 1 Inch, 12 Fourths iv 1 Third", &c. Different works are computed by different measures, viz. glazing, &c. by the foot; painting, plastering, paving, &c. by the yard; flooring, roofing, tiling, &c. by the square of 100 feet; bricklayer's work, &c. by the rod of 16 feet, the square of which is 2724 feet. Bricklayers always value their work at the rate of 12 brick thick, therefore the content of the wall, &c. must be multiplied by the number of bricks it is in thickness, and then be divided by 3, before the value of the work is estimated. Several other observations, equally useful, might here be inserted, but this part rather belongs to mensuration than arithmetic. See Keith's Mensuration. A general rule for multiplying duodecimally, or squaring the dimensions of artificers' work. Under the multiplicand write the corresponding denominations of the multiplier. Multiply each term in the multiplicand, beginning at the lowest, by the feet in the multiplier; write each result under its respective term, observing to carry an unit for every 12, from each lower denomination to its next superior. In the same manner multiply all the multiplicand by the inches in the multiplier, and write the result of each term one place removed to the right-hand of those in the multiplicand. Work in a similar manner with the seconds in the multiplier, setting the result of each term removed two places to the right-hand of those in the multiplicand.Proceed in like manner with the rest of the denominations, and their sum will give the answer required. Note. This may be performed by the rule of practice; thus, after you have multiplied by the feet, take aliquot parts of the multiplicand, with the inches, &c. Or the inches, &c. may be reduced to the fraction of a foot, by Prop. 9, Vulgar Fractions, and then multiplied together. Or, turn the inches, &c. into the decimal of a foot by Prop. 2, Rule 2, in Reduction of Decimals, and then multiply them together by some of the rules in Multiplication of Decimals. By reducing the inches, &c. into decimals of a superior name, it will often happen, that these decimals will be infinite; and hence the scholar may have a good opportunity of examining the truth and ceertainty of the rules I have laid down for managing recurring, or infinite, decimals; for, though the multiplier and multiplicand may be infinite in a decimal scale, yet they will be finite in a fractional or duodecimal one. Examples. (1.) Multiply 4ft. 6in. 5 parts by 9ft. 4in. 7 parts. Ft. In. Pts. By Practice. Ft. In. Pts. -Note. The same answer may be exactly found either by fractions or decimals. (2.) Mult. 7ft. 5in. by 4ft. 7in. (10.) Mult. 10ft. 4in. 5pts. by 7ft. 8in. 9pts. CLASS II. (12.) If a window be 7ft. 3in. high, and 3ft. 5in. broad, how many square feet of glazing are contained therein? (13.) There is a house with three tiers of windows, 7 in a tier; the height of the first tier is 6ft. 11in., of the second 5ft. 4in., and of the third 4ft. 3in., the breadth of each window is 3ft. 6in. What will the glazing come to at 14 d. per foot? (14.) What will the paving a court-yard come to at 3s. 4d. per yard, the length being 24ft. 5in., and breadth 12ft. 7in.? (15.) What will be the expence of paving a rectangular court-yard, its length being 62ft. 7in., and breadth 44ft. 5in., and in which there is laid a foot path the whole length of it, and 5 feet broad, with broad stones at 3s. per yard, the rest being paved with pebbles at half a crown a yard? (16.) If the national debt be £500,000,000, how long a foot path, of a yard wide, would this sum pave if reduced to guineas ?-a guinea being one inch in diameter. (17.) What will be the expence of plastering a ceiling at 11 d. per yard, supposing the length 22ft. 7in., and breadth 13ft. 11in.? (18.) A gentleman had a room painted at 8d. per square yard, the measure whereof is as follows: the height 11ft. 7in., the compass 74ft. 10in., the door 7ft. 6in. by 3ft. 9in.; 5 window-shutters, each 6ft. 8in. by 3ft. 4in., the breaks in the windows 14in. deep and 8ft. high, the chimney 6ft. 9in. by 5ft., the shutters and doors being coloured on both sides; what will the whole come to ? U (19.) If a house measure 57ft. 7in. in length, and 31ft. 5in. in breadth, and if the roof be of a true pitch, what will it cost roofing at half a guinea per square? (20.) How many square rods are there in a wall 631⁄2 feet long, 14ft. 11in. high, and 24 bricks in thickness? (21.) Admit the end-wall of a house to be 28ft. 10in. in breadth, and the height of the roof from the ground 55ft. 8in., the gable (or triangular part above the side walls) to rise 42 courses of bricks, reckoning 4 courses to a foot; and that 20 feet high be 24 bricks thick, 20 feet more 2 bricks thick, and the remaining 15ft. 8in. 11 brick thick. What will the work come to at 51. 168. per rod, the gable being 1 brick in thickness? END OF THE FIRST PART. 444 THE COMPLETE PRACTICAL ARITHMETICIAN. PART II. ALLIGATION. DEFINITION. When different sorts of wine, corn, spices, metals, &c., or any number of simples, of different qualities are required to be mixed together, the method of proportioning such a mixture is called Alligation, from the quantities being generally linked, or joined, together by lines. Note 1. The first proposition and rule are usually called Alligation medial; the second Alligation alternate, and is the reverse of Alligation medial; the third Alligation partial; and the fourth Alligation total. Proposition 1. Given the particular quantities mixed, and their respective rates, or prices to find the mean rate, or price, of the compound. Rule. Multiply the quantities of the mixture by the respective rates, or prices, reduced to one denomination, and divide the sum of the products by the sum of the quantities, the quotient will be the mean rate, or price. The method of proof. Find the whole value of the mixture at the mean price, and if it be the same with the total value of the several ingredients, at their respective prices, the work is right. |