440 



MATHKMATICS. ARITHMETIC. 



2728 numb, of thilllng*. 

 12 



32740 numb, of pence. 

 4 



VII I MrxsrEM or SOLID*, OR 



. make 1 cubic f<*'t. 



27 cubic feet 1 ^"'^ i'* 1 ^ 



IX. MEAC*M FOR LIQCID AJTD DRY GOODS. 

 4 gilU ke ! I 1 '"*- 



2 pint* ' M 



4 quirts ,, 1HI;">- 



Sgllon. ,, JJ***- 







! ' ' 



3 OWt Of COals ....,, 1 "*' 



Mxacks ,. I*""- 



It mar be well to notico here, that tho avoirdupoi* 



,1 oonUins 7,000 grain., of which 5,700 make a 



pound troy ; to that 144 pounds avoirdupois are equal to 



i;:. .H.uiili troy. The ounce troy exceed* the ounce 



avoirdupois by 42 ' grains. Tlio gallon contains 10 pound* 



avoirdup. led water, and its solid measure IB 



. ubic inches and -74 thousandths of an inch. 



iu-tii>. Arithmetic is now to be applied to eon- 



iiuantities, such as those named in tho fon-t- 



tables : hitherto ita operations have been confined to 



abftraet numlwrs. The name reduction is given to the 



uxis by which quantities are changed to others of the 



same values, but of different denominations ; as, for in- 



stanco, the changing, or n .lncing, pounds to shillings, 



to farthings, yards to miles, minutes, hours, .to., 

 ars and so o'n. 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 ap- 

 plying when the change is to be from a lower denomina- 

 a higher, as from pence to pounds. 



I. To re Jure a Quantity to one of Lotcer Denomination. 

 RCLK. From the previous table see how many of the 

 ntri lower denomination make 1 of the higher; multiply 

 by this nnmber: the product will be the number of quan- 



* of the next lower denomination. If any of the 

 lower denomination be connected with the pro 

 quantity, the number of those uiunt be added in with the 

 pn-luct. 



Suppose, for example, we have to reduce 136 8s. 4]<i. 

 to peace. Then, as 'Ms. mako 

 l, we multiply the number 

 I'M by the number 20, adding 

 in tho number 8: the product 



niimbrrof ahii: 



Again, since 1 - pence, make Is., 

 we multiply th.- ).i>t iiuml>ci 

 :.v 12. taking in the 

 4 : tho product is 32740, tlio 

 n n nil x-r of pence. And lastly, 

 umltip lying this by 4, because 

 4 farthings make Id., and 

 taking in the 3, the number 



.:i numb, of farthings. 



of farthings, we got 1309G3 for the number of farthings 

 required. 



Although we have been dealing here with concrete 

 q-iiiititu.fi, yet, after all, our operations are performed 

 i-nnn-ly with tiliiti'H-f nunibors. We do not multiply 

 8 by 20, because we should then get 2728 for the 

 product; much less do we multiply by 20 shillings (as 

 iom book* direct us to do) ; for to attriupt to >.. 

 by*Ai//in : ;j is to attempt an absunlity: "20 shillings 

 UN pounds," is a mode of expression as ridiculous 

 as it is meaningless. 



As a second exnm; V. l.-t it Iw required to reduce 217 

 days 14 hours Mia ''>> minute* to minutes. 



urn make 1' day, we multiply tho number 217 

 by 24 . and in adding in the 14, we include the 

 in the units' amount of the product that is, in tlio 

 first remilt of the first partial product and tin 

 (1) in tho first result of the second partial product. Wo 



\ rato to lolld of ! equal Maura fun, lik" common die. If th* 

 *. of t'.i. tnr* b> 1 Inch. lh tdU l> > rxiir Inch ; while r>ch ' i 



. inrh. In a .Imii.r toM. .,f which Iht ni. ii 1 foot, then UK 

 < UM (BAlWr MbM, or cubic tncbw. 



thus got 6222, tho number of hours; this nmnW w* 

 multiply by GO, because CO minutes 

 A. b. m. make 1 hour, and we add in tl>> 

 "17 14 M units with units, and tens with t< us, 

 24 as before : and we thus tii.'l the number 



of minutes to be 31: 



Soiiii-tiinos we have to multiply by 

 a fraction; as, for instance, when 

 MrahM of 1, i.i'th are to be red' 

 ...__' to yards; for you see by the table 



60 that 5J yards make 1 perch: also, in 



reducing tquare perdirs to */ 



313350 yr/n/.i, we have to multiply by . 



the iniinbiT of square yards in 1 

 square perch. Xow, to vmltii>hj by J means simply to 

 take A.I// tho multiplicand ; that is, to ,l',rl,l f it by 2; 

 and to multiply by ;, means to take a fourth )>art, or to 

 divide the multiplicand by 4. This is certainly a de- 

 parture from tlio primitive meaning of the word 

 but it is sanctioned by common i 

 tice. It ia customary to speak of two- 

 and-a-half times this, or thruu-und-a- 

 quarter times that, and so on : thus, 

 two-and-a-half times 4 we know to 

 mean 10; and two-and-a -quarter 

 times, 9. The way to introduce such 

 fractional parts in the arithmetical 

 operation will be sufficiently seen 

 from the two examples worked in 

 the margin ; tbe first being to reduce 

 248 linear perches to linear yards, 

 and the second to reduce 248 si, 

 perches to square yards. If tho 

 unmlier of perches had been 249, 

 the multiplier i would have given 

 1244;, and the multiplier J, G2|. By 

 aid of the tables, which ought, in- 

 deed, to be committed to memory, 

 you will easily be able to show the truth of the following 

 statements, namely : 

 13s. 4rf. = 



Linear prrche 

 2) 248 



1240 

 124 for I 



1364 yards. 



Square perche*. 

 4) 248 



30}; 



7440 



<;_ for ;. 



T")U2 sq. yds. 



.) 32 Is. 6A-7698A 



5 12s. 4jd. = 5394 farthing*. 

 27 cwt. 2 qr. 22 Ib. = 3102 lb. 

 17 Ib. 6oz. 14dwt. coi/=4- > 14 dwt 

 131 miles 3 furlpi. 10 perches 3 yds. = 231278 yds. 

 2! > days 3 hours 21 minutes = 4r.M>l minutes. 

 37 acres 3 roods 12 porches = 183073 yards. 

 239^- gallons =7604 gills. 

 327 square perches 9891 j square yards. 

 (11.) 203 tons 18 cwt. 3 qr. 21 lb. = 591241 lb. 



Ti> reduce a Quantity to one of Higher Denomination. 



RULE. Find by the table how many of the given 

 denomination make 1 of the next higher, and dlvM 

 this number ; the quotient will express how many of die. 

 i:e\t. 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 in- 

 stance, we had to find how 



4) 2040397 

 12) 660099... ;,/. 

 2.0) 5500.8.. .3d. 

 2750 8.-. 



many pounds there were in 2640397 farthings. Divid- 

 ing the number of farthings by 4, we get tho n\n 

 of pence namely, C.lKMl'.i'.t, and one farthing over. 

 Dividing the number of pence by 12, we get the number 

 of shillings - namely, 55008, and three pence over ; and 

 lastly, dividing by' 20, we get the number of pounds 

 namely, 2750, and 8s. over. Consequently, in 

 tho proposed number of farthings, there are i 

 H*. ::;.{. 



A^ain : lot it bo required to convert 591241 lb into 



ewt., <tc. As 28 lb. make 1 qr., the next higher 



denomination to pounds, we divide first by 28, or by 7 



and by 4, the two fuctnri of 28, as it is better to 



use short division: we thus get 21115, the number of 



