Page:Encyclopædia Britannica, Ninth Edition, v. 2.djvu/593

Rh ARITHMETIC 533 92100; the 6 is 60 ; and so on. This explains the principle of pointing in periods of two figures. In place of doubling the whole root for every trial divisor, it will be sufficient to add to the preceding com plete divisor its last figure; thus, 140 + 6 = 152. The 7 has been already doubled in 14, and this doubles the 6. The reason of proceeding as above will appear from 23 23 69 46_ 529(23 4 43)129 129 the composition of an ordinary product. Thus, 23 x 23 is the sum of the four products 20 x 20, 3 x 20, 20 x 3, and 3x3, i.e., 20 2, twice 20 x 3, and 3x3. Taking away, then, the square of 20, i.e., 400, the remainder must be the sum of 2 x 20 x 3, and 3x3, i.e., 3 times the sum of 2 x 20 and 3, which we obtain by the method adopted. If there is a remainder, ciphers may be taken down in pairs, and as many decimal places obtained as we please. In this case there must always be a re mainder, since no unit multiplied by itself produces ciphers. After getting half the decimal places required, we may proceed by contracted division ( 25). To extract the square root of a vulgar fraction, we find the roots of the numerator and denominator separately. For, since x = T, the square root of T 9 g- must be f-. If either term of the fraction is not a complete square, it should be reduced to a decimal. Thus, N /^- = ,J 5 ^/Q 50 = 7071067812 nearly. 31. To extract the cube root of a given number, point off the number from the units place into periods of three figures : write under 73402752(428 64 40 2 40 = 480014402 3 = 240 2 2 = 4 5044 10088 3 =5292004314752 8x3= 10080 8 2 = 64 5393444314752 the first period the greatest cube con tained in it, subtract, and annex the next period to the remain der ; then, regarding the root found as tens, multiply the square of it by 3 for a trial divisor, and divide by this for the next figure of the root ; to the trial divisor add three times the product of the two root numbers (the first being tens), and also the square of the last root figure ; multiply the sum by the last root figure; subtract the product from the number obtained by taking down last period ; annex the next period to the remainder, and proceed as before. For demonstration of the reason of this process, see ALGEBRA, vol. i. p. 528. It depends on the form of the product obtained in raising a number to its third power. Thus, 24 x 24 x 24 is 24 times 20 2 + 2 x 20 x 4 + 4 2, which will be found to be 20 3 + 3 x 20 2 x 4 + 3 x 20 x 4 2 + 4 3. And this is 20 3 added to 4 times 3 x 20 2 + 3 x 20 x 4 + 4 2 , which agrees with the process described above. II. PRACTICAL ARITHMETIC. 32. Having explained in the foregoing sections the various operations of arithmetic, we now proceed to con sider them in their combinations and practical applications. What has been said up to this point refers to numbers merely as numbers, or numbers in the abstract. Now they are to be regarded as applied to particular things, or repre senting particular magnitudes. Numbers so regarded are called concrete; and we now treat of concrete as dis tinguished from abstract, arithmetic. Concrete numbers frequently represent not so much number as quantity. To form a distinct and accurate idea of 5 Ib of tea bought for 15s., it is not necessary to think of the tea as divided into five portions, or as paid for with fifteen pieces of money. It would be found extremely (indeed intolerably) incon venient to have to make all payments, great and small, by means of one particular species of coin, or to serve out all quantities of goods, using only one kind of weight or measurement. Various monies, weights, and measures are therefore in customary use, this or that being employed in each particular case according to circumstances. When these measures are of the same kind, differing only in the unit of one of them being so many times the unit of another, they are said to be of different denominations ; as, pounds, shillings, pence ; or again, yards, feet, inches. In addition and subtraction, the quantities added and subtracted must be either abstract numbers or concrete quantities of the same kind. In multiplication, the multi plicand may be concrete, but the multiplier is regarded in the process as abstract. If 20 men, for example, receive 5 each, the 5 is not multiplied by 20 men, but taken 20 times, the number of times merely corresponding to the number of men. In division, when the dividend is con crete, the divisor may be abstract, giving a concrete quotient of the same kind as the dividend, or concrete giving an abstract quotient. Thus, 100 may be divided into 20 parts, giving 5 as quotient, or it may be divided into parts of 5 each, giving as quotient the abstract number 20, i.e., containing 5 20 times. A fraction is strictly abstract, though we often write | and the like for of 1. 33. The following are the tables of monies, weights, and Tabhs. measures in common use : l I. Money. 4 farthings = 1 penny, d. 12 pence =1 shilling, s. 20 shillings 1 pound, or L II. Avoirdupois Weight. 16 drams, drs. =1 ounce, oz. 16 ounces =1 pound, Ib. 28 pounds = 1 quarter, &amp;lt;JT. 4 quarters = 1 hundred weight, cict. 20 hundredweights 1 ton. III. Troy Weight. 24 grains, grs. 1 penny weight, dwt. 20 pennyweights = 1 ounce, oz. 12 ounces =1 pound, Ib. IV. Length. 12 inches, in. =1 foot, ft. 3 feet 54 yards V. Surface. 144 square inches = 1 square foot. 9 sq. feet =1 sq. yard. yards =1 sq. pole. sq. poles =1 rood, ro. roods =1 acre, ac. 40 4 = 1 yard, yd. = 1 pole or perch, po. = 1 furlong, fur. = 1 mile, mi. VI. Solidity. 1728 cubic inches = 1 cubic foot. 27 cubic feet =1 cubic yard. VII. Capacity. 4 gills =1 pint, pt. 2 pints = 1 quart, qt. 4 quarts 1 gallon, gal. 2 gallons = 1 peck, pk. 4 pecks n 1 bushel, bus. 8 bushels = 1 quarter, qr. VIII. Time. 60 seconds, s. = 1 minute, m. 60 minutes =1 hour, hr. 24 hours I day. 365 days 1 year, yr. 40 poles 8 furlongs On these tables we make the following remarks : I. The guinea is 21 shillings, and the crown 5 shillings. Sales are often made in guineas, though the coin is not now used. Farthings are always written as fractions of a penny. lO^d. ^q. means tenpeuce three farthings and seven-eighths of a farthing. III. Apothecaries weight agrees with this, except that the ounce (J) is divided into 8 drams (3), and the dram into 3 scruples (3) of 20 grains. 1 By an Act of Parliament, passed in 1824, the imperial standard weights and measures are connected in the following way with the mean solar day, the length of which is fixed invariably : The yard of 36 inches is determined from the length of a pendulum, vibrating once in a second, which, in the latitude of London, is 39 13929 inches ; the pound troy of 5760 grains, from a cubic inch of distilled water, weigh ing 252 458 grains ; and the gallon as being the space occupied by 10 Ib avoirdupois (i.e., 70,000 grains troy) of distilled water, equivalent to 277 274 cubic inches, all these verifications being made with ther mometer at 62 Falir. and barometer at 30 inches.