the other quantity, 5547, is found by adding 9, the square of the second figure in the root, to the two preceding middle lines, 369 5169* We now add two ciphers, and repeat the whole process described in this paragraph. The remarks made above with respect to pointing the last figure of the integer, and also with respect to surd square roots, apply also to cube roots. Thus .01, 24.1 would be pointed for the cube root of .010, 24.100. Obtain the cube roots of (1.) 16194277. Ex. 142. (2.) 138188413. (3.) 437.245479. (6.) 2911954752. (8.) 308915776. (4.) 2024284.625. (5.) 363994314. (9.) Extract, to four figures, the cube roots of 20 and .002. (10.) Extract, to four figures, the cube roots of 300 and .3. (11.) What must be the length of a cubical box, which shall have room for 10 cubic feet? (12.) What must be the side of a cubical cistern, which shall contain exactly 1000 gallons of water, if a gallon contains 277.274 cubic inches? 151. In Decimal Arithmetic, the value of any figure is increased ten-fold by its being moved one place to the left, or diminished ten-fold by its being moved one place to the right. In Duodecimal Arithmetic, the value of any figure is increased or diminished twelve-fold by its being moved one place to the left, or right, respectively. So that 12 is, to numbers which are written in duodecimal notation, just exactly what 10 is to common numbers. Ex. 1. In common numbers, 37.5 denotes 3 tens +7 ones+5 tenths; but, in duodecimal notation, 37.5 would denote 3 twelves +7 ones+5 twelfths. Ex. 2. In common numbers, 8765 would denote 8 x 1000+7×100+6 x10+5; or, since 100 is the square of 10, 1000 the cube of 10, &c., if we denote the square (or second power) of 10 by 102, the cube (or third power) of 10 by 103, and so on, we may say that, in common numbers, 8765 denotes 8 x 103+7x102+6x10+5: but, in duodecimal notation, 8765 would be used to denote 8×123+7 x 122+6×12+5, which, in common numbers, is equivalent to 8 x 1728 +7 x 144 +6 × 12+5=14909. 152. From the last example it is seen, that it is very easy to convert a duodecimal whole number to a common number. But one remark must here be made. In order to write down any number in the decimal notation, we require the use of the nine digits and zero, that is, we require symbols for the nine numbers less than 10, and for zero, and for these we use the figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. So, likewise, in the duodecimal notation, we shall require symbols for the eleven numbers less than 12 and for zero: and, as there is no symbol in common use for the numbers 10 and 11, it is usual to represent them by the letters t and e. Ex. The duodecimal number te10 means 10 × 123 +11 x 122+1 × 12+0=10×1728 +11 × 144+1 × 12+0=18876. 153. So, too, a common number may be easily converted to a duodecimal, as follows: Ex. Reduce 67890 to a duodecimal. 12) 67890 12) 5657 ... 6 12) 471 ...5 12) 39... 3 3 ... 3 Here, dividing by 12, we find there are 5657 twelves in the given number, and 6 units over. Then, dividing this number of twelves by 12, we find that there are 471 hundred-and-fortyfours in the number, and 5 twelves over and so on. Ans. 33356 duodecimal; which, in common numbers, =3×12+3 × 123+3 × 122 +5 × 12+6 = 3 × 20736+3x1728 +3 × 144 +5 × 12+6 =62208+5184+432+60+6=67890, as it should be. 154. Our examples have hitherto been only of whole numbers; but fractions are also used in duodecimals, as well as in common numbers. 3 5 9 Ex. In common numbers, .359 would be used to denote + + 10 100 1000 5 9 or + + ; and so, in duodecimals, .359 would be used to denote 5 102 12+12+123. 103 Fractions may also be converted from one scale to another, without any real difficulty but, as the process is sometimes a little more complicated, and not much required in practice, we shall only illustrate it here by an example of each kind. Ex. 1. .359 (duodecimal)=3+1+1728 (decimal), = 3 x 144+5 x 12+9 = 502 251 1728 864 We have here added the three fractions together, by the usual process of finding their L. C. D., 1728; but it would be simpler to reason thus: 3 x 144+5 x 1249 (dec.)=1728 .359 (duod.)= 359 (duod.)= 1000 1728 502 Here we have only converted the numerator and denominator of the duodecimal fraction to common numbers. 257 Ex. 2. .379 (dec.)= 379 (dec.) = by the process of 153, 1000 6e4' =.4315 &c., by Ex. 4 of 155. 155. It will be very useful to observe that, whereas in decimals, is expressed by .5, in duodecimals or pressed by .6; so ==.3, 2=22.9, 122.2, &c. 12 is ex 156. Duodecimals are to be added, subtracted, &c., just like other numbers, only cbserving to borrow and carry 12, instead of 10, when necessary. Ex. 1. 7314 206e 1321 te89 26541 Ex. 2. 660452 271e43 3tt 50e The first column=9+1+e+4=25 (dec.), and 12 is in 25 twice, and 1 over; we set down 1, and carry 2; then 2+8+2+6+1=28 (dec.), and 12 is in 28 twice, and 4 over; we set down 4, and carry 2; and so on. Here 3 from 2, I cannot: I borrow 1, that is, 1 x 12, or 12; then 3 from 12 leaves 9, and 2 (of the upper line) makes 11; I set down e, and carry 1; and so on. Ex. 4. 157. In consequence of the foot containing 12 inches, duodecimals are often convenient for workmen in measuring their work. But, besides the foot, the inch also is supposed, for this purpose, to be divided into 12 equal parts, called seconds, and marked', the second into 12 thirds, marked", &c.; and thus, inches, seconds, thirds, &c., may be written down at once as duodecimals of a foot. Ex. 1. Multiply 6 ft. 3 in. 7' by 2ft. 4 in. 9'. 6.37 2.49 4883 2124 1072 13.1103 Here, in the Answer, we have the integer 13 (duod.) 15 (dec.), which will be 15 sq. ft.; then, since 144 sq. in. make 1 sq. ft., and 144 (dec.)=100 (duod.), if we multiply the fractional part of the Answer, .1103, by 100, or move the point two places to the right, we get 11.03 sq. in.=(dec.) 1324 sq. in.=134 sq. in. Since (duod.) .35t6 sq. ft.=35.16 sq. in.=(dec.) 41128 sq. in.=417 sq. in. Ex. 144. Work the following by duodecimals: (2.) 10 ft. 4 in. x 7 ft. 9 in. in. (1.) 5 ft. 6 in. x 3 ft. 9 in. (3.) 9 ft. 3 in. x 11 ft. 5 in. (5.) 32 ft. 10 in. × 22 ft. 1 in. (7.) 8 yds. 2 ft. 5 in. x 1 ft. 4 (8.) 17 yds. x 3 yds. 2 ft. (9.) 70 ft. 7 in. x 221 ft. (11.) 2 yds. 2 ft. 2 in. x 2 ft. 2 in. (12.) 47 ft. 0 in. x 1 ft. 15 in. 11 in. (10.) 15 ft. x 15 in. 158. Other problems in square and cubic measure may also be treated by duodecimals, as in the following examples. Ex. 1. Divide 376 sq. ft. 100 in. by 10 ft. 9 in. Here 376 sq. ft. 100 in. 274.84 sq. ft. (duod.) 10 ft. 9 in.= t.9 ft. (duod.) t.9) 274.8400 (2e.05e ft. 35 ft. nearly, omitting the small Ex. 2. Find the solid content of a cistern, that is 7 ft. 5 in. long, 5 ft. 8 in. broad, and 4 ft. 2 in. deep. 7.5 5.3 1t3 311 32.e3 4.2 Here, since 1728 cubic inches make 1 cubic foot, and 1728 (dec.)=1000 (duod.), if we multiply the fractional part of the final result, namely, .276 c. ft., by 1000, we shall obtain .2t6 c. in. (duod.)=414 c. in. (dec.). 6516 10e90 116.2t6 162 c. ft. 414 c. in. Ex. 3. What must be the depth of a cistern, whose base is 3 ft. square, that it may contain exactly 30 cubic feet? Herc 3 ft.-3.6 ft. (duòd.), and 30 c. ft.=26 c. ft. (duod.) Ex. 4. Find the cost of paving a cellar, 16 ft. 8 in. long and 14 ft. 4 in. broad, at 3s. 94d. per square yard. 16 ft. 8 in.-14.8 ft. (duod.) N.B. If pounds are given in the price, reduce them to shillings, and proceed as above. Ex. 145. (1.) Find the surface of a plank, 9 ft. 3 in. long by 2 ft. 4 in. broad. (2.) What is the size of a table, 8 ft. 7 in. long by 3 ft. 6in. broad? (3.) Find the cost of a marble chimney-piece, 4 ft. 4 in. long, and 1 ft. broad, at 7s. 6d. per square foot. |