. 99 9 . VII.-MEASURES OF SURFACE, OR SQUARE MEASURE. 144 square inches make 1 square foot. 9 square feet 1 square yard. 304 square yards 1 square rod, pole, or perch. 40 square perches 1 rood. 4 roods, or 160 square perches 1 acre, 10 square chains, or 100,000 sq. links 1 acre. 640 acres 1 square mile. -100 square feet 1 square of flooring. 1 square rod of brick work. make 1 cubic foot. 27 cubic feet 1 cubic yard. IX.-MEASURES FOR LIQUID AND DRY GOODS. 4 gills make 1 pint. 4 pecks make 1 bushel. 2 pints 1 quart. 1 quarter. 1 sack. 2 gallons 1 peck. 1 ton. It may be well to notice here, that the avoirdupois pound contains 7,000 grains, of which 5,760 make a pound troy; so that 144 pounds avoirdupois are equal to 175 pounds troy. The ounce troy exceeds the ounce avoirdupois by 423 grains. The gallon contains 10 pounds avoirdupois of distilled water, and its solid measure is 277 cubic inches and 274 thousandths of an inch. Reduction.–Arithmetic is now to be applied to concrete quantities, such as those named in the foregoing tables : hitherto its operations have been confined to abstract numbers. The name reduction is given to the methods by which quantities are changed to others of the same values but of different denominations; as, for instance, the changing, or reducing, pounds to shillings, pence, to farthings, yards to miles, minutes, hours, &c. to years, and so on. There are two rules for such reductions: the one applying when the quantity is to be converted from a higher to a lower denomination,-—as, for instance, from pounds to pence; and the other applying when the change is to be from a lower denomination to a higher, as from pence to pounds I. To reduce a Quantity to one of Lower Denomination. RULE.-From the table see how many of the next lower denomination make 1 of the higher; multiply by this number : the product will be the number of quantities of the next lower denomination. If d. any of the lower denomination be connected with the 136 8 48 proposed quantity, the number of these must be added in 20 with the product. 2728 numb.of shillings. Suppose, for example, we have to reduce £136 8s. 42d. to. 2728 pence. Then, as 20s. make £1, we multiply the number 136 by the number 20, adding in the number 8: the product 32740 numb. of pence. is 2728, the number of shillings. Again, since 12 pence makc 18., we multiply this last number by 12, taking in the 130963 numb. of farthgs. 4: the product is 32740, the number of pence. And lastly, multiplying this by 4, because 4 farthings make ld., and taking in the 3, the number of farthings, we get 130963 for the number of farthings required. * A cube is a solid of six equal square faces, like a common die. If the edge of this figure be 1 inch, the solid is a cubic inch, while each face is a square inch. In a similar solid, of which the edge is 1 foot, there are 1728 of the smaller cubes, or cubic inches. 4 a d. b. m. You perceive here that, although we have been dealing with concrete quantities, yet, after all, our operations are performed entirely with abstract numbers. We do not multiply £136 by 20, because we should then get £2728 for the product; much less do we multiply by 20 shillings (as some books direct us to do), for to attempt to multiply by shillings is to attempt an absurdity : “20 shillings times 136 pounds,” is a mode of expression as ridiculous as it is meaningless. As a second example, let it be required to reduce 217 days 14 hours and 36 minutes to minutes. Since 24 hours make one day, we multiply the number 217 by 24; and in adding in the 14, we include the units in the units' 217 14 36 amount of the product that is, in the first result of the first partial 24 product,--and the tens (1) in the first result of the second partial product. We thus get 5222, the number of hours ; this number we multiply by 60, 872 435 because 60 minutes make 1 hour, and we add in the 36,-units with units, and tens with tens, as before. and we thus find the number of 5222 minutes to be 313356. 60 Sometimes we have to multiply by a fraction, as, for instance, when 313356 perches of length are to be reduced to yards; for you see by the table that 54 yards make 1 perch: also, in reducing square perches to square yards, we have to multiply by 301, the number of square yards in 1 square perch. Now, to multiply by { means simply to take half the multiplicand, that is, to divide it by 2; and to multiply by \, means to take a fourth part, Linear perches. or to divide the multiplicand by 4. This is certainly a departure 2) 248 from the primitive meaning of the word multiply; but it is sanc 5} tioned by common practice. It is customary to speak of two 1240 and-a-half times this, or three-and-a-quarter times that; and so 124 for . on: thus, two-and-a-half times 4 we know to mean 10; and two 1364 yards. and-a-quarter times, 9. The way to introduce such fractional parts in the arithmetical operation will be sufficiently seen from the two Square perches. examples worked in the margin; the first being to reduce 248 4) 248 linear perches to linear yards, and the second to reduce 248 square 30 perches to square yards. If the number of perches had been 249, 7440 the multiplier I would have given 1241, and the multiplier \, 623. By aid of the tables, which ought, indeed, to be committed to memory, you will easily be able to show the truth of the following 7502 sq. yds. statements, namely : 62 for ^ (1.) 138. 4d. = 160d. (2.) £32 ls. 6d. = 7698d. (3.) £5 12s. 4 d.=4914 farthings. (4.) 27 cwt. 2 qr. 22 1b. = 3102 lb. (5.) 17 lb. 6 oz. 14 dwt. troy=4214 dwt. (6.) 131 mls. 3 fur. 10 per. 3 yds.=231278 yds. (7.) 29 days 3 hours 21 min.=41961 min. (8.) 37 acres 3 roods 12 perches = 183073 yds. (9.) 2391 gals. = 7664 gills. (10.) 327 square perches =98914 sq. yards. (11.) 263 tons 18 cwt. 3 gr. 21 lb.=591211 lb. To reduce a Quantity to one of Higher Denomination. RULE.—Find by the table how many of the given denomination makc 1 of the next higher, and divide by this number; the quotient will express how many of the next higher denomination are in the proposed quantity. In like manner, divide by the number expressing how many of the new denomination make 1 of the next higher to it; and so on, till the required denomination is reached. Suppose, for instance, we had to find how many pounds there were in 4) 2640397 2640397 farthings. Dividing the number of farthings by 4, we get the number of pence — namely, 660099, and one 12) 660099... d. farthing over. Dividing the number of pence by 12, we get 2,0) 5500,8...3d. the number of shillings-namely, 55008, and three pence over; and lastly, dividing by 20, we get the number of pounds – £2750 88. 3 d. namely, 2760, and 8s. over. Consequently, in the proposed number of farthings, there are £2750 8s. 3 d. Again: let it be required to convert 591241 lb. into tons, cwt., &c. As 28 lb. make 1 qr., the next higher denomination to pounds, we divide 7) 591241 first by 28, or by 7 and by 4, the two factors of 28, as it is better to use short division: we thus get 21115, the 4) 84463 number of quarters, with 21 lb. over. This number, 4)21115...21 lb. divided by 4, gives the number of cwt.-_namely, 5278 and 3 yrs. over: and lastly, dividing by 20, the num2.0) 5278...3 qrs. ber of cwt. in 1 ton, we get finally 263, the number 263 t. 8 cwt. 3 qr. 21 lb. of tons : so that there are 263 tons 8 cwt. 3 qr. 21 lb. in 591241 lb. All this is so easy and obvious that I am sure I need not occupy space with any more worked-out examples. I shall merely give one cautionary direction—it is this: that when you have to divide by 5}, bring both this divisor and the dividend into halves ; that is, double both; making the divisor 11, instead of 5}; but remember that the remainder will be so many halves. In like manner, when you have to divide by 304, bring all into quarters ; that is, divide 4 times the dividend by 121, which is 4 times 304; remembering, however, that the remainder will be quarters ; so that a fourth part of the number, which is the remainder, will be the number Square yards. 2463 of wholes. See the operation in the margin, where the factors of 121, 4 viz., 11, 11, are used to get the quotient by short division. This quotient shows that there are 81 square perches, and 51 quarter-yards 11) 9852 over; that is, 127 square yards : the result would therefore be written, 11) 895...7 81 square perches, 124 square yards. The examples given at page 20 may be employed for exercise in this 81...51 rule, by taking in each the quantity on the right of the sign of equality, and converting it into that on the left; but two or three others are added here : (1.) 28635 seconds=7 h. 57m. 15 sec. (2.) 10085760 gr.=1751 lb. troy: (3.) 633600 inches=10 miles. (4.) 024 yds.=225 mi. 4 fur. 26 per. 1 yd. (5.) 91476 sq. ft.=2 ac. O rds. 16 per. (6.) 100000 cubic in.=2 cub. yds. 3 ft. 1504 in. The four fundamental operations of arithmetic may now be applied, in order, to compound quantities; that is, to concrete quantities, of several denominations. Addition of Compound Quantities. To add together a set of concrete quantities of different denominations, the rule is as follows : RULE.- Arrange the quantities to be added one under another, so that all in the same vertical column may be of the same denomination. Add up the quantities of lowest denomination : find how many of the next denomination are contained in the sum : put the remainder under the column, and carry the quotient to the next column. : Proceed in this way, from column to column, till all have been added up. £ s. d. 17 9 3 42 13 43 16 10 21 7 2 91 1 18 10/ 85 14 6 The principle of this rule is too obvious to require any explanation; the carryings merely transfer the quantities of advanced denominations to the columns in which those denominations are arranged, just as in the addition of abstract numbers. Thus the sums of money in the margin are arranged so that the denomination farthings forms one column, the denomination pence the next, shillings the next, and pounds the next. The sum of the farthings' column is 10 farthings, in which are contained 2 pence, and there are 2 farthings, or over; this is therefore put down, and the 2 pence carried to the pence column; the sum of this column is 30 pence, that is, 2s. 6d. the 6d. is put down, and the 28. carried to the shillings' column, the sum of which is 54s., that is, £2 148.; we therefore say 14, and carry 2; and this 2 added in with the pounds' column, makes the amount of that column £85; therefore the sum of the whole is £85 148. 6d. It may be noticed that in adding up the shillings' column of an account, the best way is to disregard the tens in that column till all the units have been added; then, having reached the top unit-figure, to proceed downwards taking in every ten that appears. Thus, in the present example, the sum of units' column of shillings is 24; so that, proceeding downwards, taking in each ten as we meet with it, we say 34, 44, 54; so that the sum is 54s., or £2 148. lb. oz. dr. 8 13 11 9 10 13 4 6 9 11 11 15 7 3 8 Two other examples are here annexed; the one in Avoirdupois weight, and the other in Time. In the former the sum of the drams is found to be 56 dr.; by reduction, we find that in these drams there are 3 oz. 8 dr., we therefore put down 8 dr. and carry the 3 oz. to the next column, which gives 46 oz., or 2 lb. 14 oz.; writing down the 14 oz., and carrying the 2 lb. to the column of lbs., we get 41 lb. for the sum of this column: therefore the whole sum is 41 lb. 14 oz. 8 dr. d. h. m. s. 34 13 9 15 18 9 0 37 27 21 11 19 14 18 23 4 10 7 14 16 13 14 21 19 119 11 19 50 41 14 8 In the next example the column of seconds amounts to 110 seconds, that is, to 1 minute 50 seconds: the 50 seconds is put down, and the 1 minute carried to the next column, the amount of which is 79 minutes, that is, 1 hour 19 minutes, 19 and carry 1: the hour column amounts to 83 hours, or 3 days 11 hours; 11 and carry 3 to the day's column, the amount of which is 119: therefore the whole amount is 119 days, 11 hours, 19 minutes, 50 seconds. Subtraction of Compound Quantities.-The subtraction of concrete quantities, of different denominations, is effected by the following rule: RULE. Place the less of the two quantities under the greater, arranging the denominations as in addition. Commence with the lowest denomination, and subtract, if the upper number be sufficiently great; if not, increase it by as many as will make 1 of the next denomination, and then subtract, taking care afterwards to carry 1, as in subtraction of abstract numbers: and proceed in like manner with each denomination till the subtraction is finished. £ 8. d. 124 16 9 75 19 33 In this way the difference between £124 16s. 91d. and £75 198. 3d. is found, as in the margin. Since 3 farthings cannot be taken 48 17 5 from 2 farthings, we increase the 2 farthings by 4 farthings, or 1d., and say 3 from 6 and 3 remain, that is, d. carry 1: 4 from 9 and 5 remain: 19 from 36 (increasing the 16s. by 20s.*) and 17 remain; carry 1 : 6 from 14 and 8 remain; carry 1: 8 from 12 and 4 remain: therefore the difference is £48 17s. 5ąd. m. fur. per. yd. 43 0 1 1 24 6 21 2 Again, suppose we have to subtract 24 miles, 6 furlongs, 21 perches, 2 yards from 43 miles, 1 perch, 1 yard. Then, having placed the quantities as in the margin, and seeing that the 1 yard is too small, we increase it by 1 perch, that is by 5 yards, and subtract 2 yards from 64 yards, we thus get the remainder 4 yards; and carry 1: and as 40 perches make 1 furlong, we subtract 22 from 41, and get 19 for remainder carrying 1 to the 6 we subtract 7 from 8-the furlongs in 1 mile-and get 1 for remainder and carrying 1 to the 4, it merely remains to subtract 25 from 43: the complete remainder is therefore 18 miles, 1 furlong, 19 perches, 4 yards. 18 1 19 4 : If the complete remainder in an operation of this kind be added to the compound quantity immediately above it, that is to the subtractive quantity, the sum will be equal to the upper row, that is, to the quantity which has been diminished: so that we may prove in this way the correctness of the subtraction. The following examples, if thus worked and proved, will afford exercise both in subtraction and addition : (1.) Subtract £374 11s. 8d. from £920 17s. 7 d. (2.) Subtract £173 9s. 4 d. from £200. (3.) Subtract 8ìb. 4oz. 23 gr. from 23lb. 11oz. 21 gr. (4.) Subtract 342 mls. 6 fur. 4 per. 4 yds. from 687 mls. 3 für. 1 per: (7.) Subtract 121 sq. yds. 7 ft. 132 in. from 237 sq. yds. 3 ft. 101 in. Multiplication of Compound Quantities. From the nature of multiplication, it is plain that a concrete quantity can be multiplied only by an abstract number; indeed, whatever be the multiplicand, the multiplier, which simply denotes how many times the former is to be taken, must necessarily be a mere number. Strange to say, however, books on arithmetic, of the most recent date, are to be found, in which the multiplication together of concrete quantities is insisted upon, and pretended to be taught. People have disputed over and over again about the product of £19 19s. 11 d., multiplied by itself! They might as well have disputed about the multiplication of Cheapside by Lombard Street; or, as Mr. Walker pithily expresses it, about multiplying "5lbs. of beef by 3 bars of music." This last operation, palpably absurd as the thing is, the arithmeticians referred to would not for a moment hesitate to undertake, provided the beef and music occurred in a rule-of-three question, as indeed they very well might; for they refer to the rule-of-three in justification of such a process.‡. Is it not ridiculous to appeal to a rule instead of to reason and common sense, in a subject which professes * Instead of thus increasing the upper term by the unit of next higher denomination, the learner will find it a little easier to subtract at once from this unit, expressed in the lower denomination, and to add the remainder to the term above: thus we may say, 19 from 20, 1; and 16 make 17. + "Philosophy of Arithmetic," p. 58. Paganini was a very wonderful performer on the violin. Many people would have given a good deal of beef for a few bars of his music. Suppose, in time of need, he had exchanged 11 bars for 5lb., how many lb. might have been exchanged, at the same rate, for 3 bars? This is a rule-of-three question, and there are plenty of books (Walkingame, for instance) that would direct the following stating:Bars of music 11 lb. of beef. bars of music. lb. of beef. And to get this 1 4-11 lb. of beef, they would direct the beef and the music, in the second and third terms, to be multiplied together! The author of this, when learning arithmetic (?), would have proceeded to incorporate the beef and music, without the slightest compunction. |