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NOTES ON ARITHMETIC 
 
 FOR THE USE OF 
 
 THE CADETS 
 
 OF IHE 
 
 |lopI Pilitarg College d Canada. 
 
 BT 
 
 EDGAR KENSINGTON, 
 
 MAJOR ROVAIi AKTILLERY. 
 
 PROFESSOR OF MATHEMATICS AND ARTILLERY. 
 
 Price, ^0 Cents. 
 
 KINGSTON : 
 
 WILLIAM BAILIB, PRINTER. 
 
 1880. 
 
 \ 
 
 N 
 
 I 
 
 J 
 
■I 
 
 v*^ ' 
 
 ^:^ 
 
 ^rv^*^ 
 
NOTES ON ARITHMETIC 
 
 FOE THE USE OF 
 
 m 
 
 y.^y/ 
 
 THE CADETS 
 
 OF 'HIE 
 
 |l«pl Pilitarg Ctllegt af Canak. 
 
 / 
 
 BY 
 
 ^'^ 
 
 %'^ 
 
 EDGAR KENSINGTON, 
 
 MAJOn ROYAL AUTIIXERT. 
 
 PROFESSOR OF MATHEMATICS AND ARTILLERY. 
 
 
 Price, 40 Cei^s. ^0 '1^'% 
 
 ^"^l!^ 
 
 KINGSTON : 
 
 Wn,UAM BAILIE, FltlNTER. 
 1880. 
 
o 
 
 i 
 
 ' -\ 
 
 ,g 
 
 
NOTES. 
 
 ■nnm 
 
 (1) A number is divisible by 2 if the last fiL'ure is divisible ^™P'« 
 
 1 __ „ ° properties ot 
 
 numbers. 
 
 by 2. 
 
 A iiniuber is divisible by 4 if the last two figures are 
 
 divisible by 4. 
 A minibcr is divisible by 8 if the last thl-ee figures are 
 
 divisible by 8, and so on. 
 A number is divisible by 3 if the sum of the digits is 
 
 divisible by 3. 
 A number is divisible by 9 if the sum of the digits is 
 
 divisible by 9. 
 A number is divisible by 11 if the sum of the digits in 
 
 the odd |)laces eqtials the sum of the digits in the 
 
 even places or difters by 11 or a multiple of 11. 
 
 (2) 13]3 
 109 
 
 14]2 
 196 
 
 15]2 
 223 
 
 16]3 
 256 
 
 17]2 
 289 
 
 18]3 
 324 
 
 Note particularly the squares of 13, 17, 19. 
 
 23 
 
 8 
 
 33 43 
 
 53 
 125 
 
 63 
 
 "216 
 
 2T I 64 
 
 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024. 
 Note particularly 2''=32 ; 2^^=1024. 
 
 3, 9, 27, 81, 243, 729. 
 5, 25, 125, 625, 3125. 
 7, 49, 343. 
 
 (3) i/T= 1-414 V'3'= 1-732 
 
 KB 
 
 343 
 
 19]3 
 361 
 
 83 
 512 
 
 201^ Squares to 
 400. 
 
 400 
 
 9 Cubes to 1000 
 
 729 
 
 Powers of 2. 
 
 Powers of 3. 
 Powers of 5. 
 Powers of 7. 
 
 VT= 2-236 
 
 V 1 -96 = 1 -4 f/ 2-89 =1 >' 4-84 = 22 
 
 Ex en Sin 4no— ^ _ ^2 1-414 _-_ 
 
 i-x. ^^i; Din. 40 = --__=:::; = = -=-707 aDDrox. 
 
 V 2 '2 2 
 
 Approximate 
 roots. 
 
 Ex. (2) Sin. 60° = 
 
 4/ 3 1-732 
 
 = •866 
 
 Ex. (3) Sin. 18" = i^^-Hl = 1^^ = .309 
 4 4 
 
 approx. 
 approx. 
 

3 
 
 (4). It is generally best not to multiply out factors. Es 
 pecially in tlie case of fractions reduced to a coimnon de- 
 nominator, the denominator should at first he left in factors. 
 (5). The simple rules of Arithmetic refer only to abstract 
 numbers, no unit being given. An aljstract number is 
 always a ratio. 
 
 Ex. (I). 3 is the ratio f just as much as I is a ratio. 
 (6). A concrete lunuber consists of two parts multiplied 
 together, viz., a certain si)ecified unit and an alistraet num- 
 ber denoting how many times the unit is to lie taken. 
 
 Ex. (1). The concrete number 3 horses = 3 X one 
 horse. One horse is here the unit, 3 the 
 abstract number. 
 Ditferent numbers or ratios may express tlie same con- 
 crete number or quantity, provided ditferent units are em- 
 ployed. 
 
 Ex. (2). Forty minutes = 40 X !""• = | X l^""' 
 (7). Concrete numberx, of like natuve onlij, can he added 
 or subtracted. First, bring both to the same denomination ; 
 then add or subtract. 
 
 Ex. (1). 3 horses cannot be added to 4 hours. 
 3 feet added to 4 inches = 40 iiu-hes. 
 
 (8). Concrete numbers can be midtipUcd toijdher in cer- 
 tain cases only ; namely, when the multiplication of the 
 units gives an intelligible result. 
 
 There are two most important elementary cases, 
 
 1st. Length X length = surface. 
 
 '2nd. Length X length X length = volume; also, of 
 cource, surface X length = volume. 
 
 1st. Consider what is meant by multiplication of one 
 inch by one inch. 
 
 An inch multiplied by an abstract number means that 
 number of inches added to one another endways. 
 
 An inch nniltiplied by an inch means an inch added to 
 itself not lengthways but broadways as many times as its 
 
 Niuubt'i's ill 
 factitrM. 
 
 Alistnic't 
 nuiiibiTH. 
 
 C'oncri'ti! 
 luinibfrs. 
 
 Atlilition 'ir 
 
 subtrnctioii 
 
 of ('(jiicri'ti' 
 
 nuiiit)';ri<. 
 
 MuUiiilii-iu'i, 
 
 of Concrete 
 
 iiuiiil)i'rs. 
 
 M 
 a 
 
. ,' f 
 
 I 
 
 i 
 
 i 
 f 
 
 1 
 
 I 
 
 
 . . . \ 
 
 f 
 
 J-- a, •■ 1 7,»S J 
 
 (•.'• '^ ""*' 
 
 ;iw»r 
 

 ^■*f*:fii'^ 
 
 * : IS t? 
 i 
 
 brciidth is co:itained in its length. Thus the result is a 
 square inch. 
 
 It may be ol)jecte(l that a line has no thickness, but this 
 means no appreciable thickness, no thickness that can be 
 reckoned on, in fact a thickness that is less than any that 
 can be described or imagined. 
 
 To multiply length by length, the units must be multi- 
 plied and the numbers must be multiplied. 
 
 Ex. 3 inches X i inches = 12 square inches. 
 2 yards X 3 inches = 21G square inches. 
 
 li square feet. 
 
 Similarly one square inch multiplied by a linear inch, 
 irives one cubic inch. 
 
 (9.) There are three elementary units on which all units 
 einj^loyed in scientitic researches may be made to depend. 
 
 VIZ. 
 
 (I). The unit of length 
 (2). The unit of time. 
 (3). The unit of mass. 
 Some of the other more important units are : 
 
 (*) 
 
 m, . „ , . space 
 
 The unit of velocity = TjT.r;; ; 
 
 Ll 111(3 
 
 s = vt. 
 
 81)306 
 
 (5). The unit ot accelerating torce=rj~-n 2; s=^jt^ 
 
 (6). The unit of weight = mass X force ; W — mf. 
 (7). The unit of momentum = mass X velocity ; 
 
 J/ =1 mv. .„ 
 
 (S). The unit of dynamical energy =mass X vel]^ ; 
 
 Energy =^- = -^ 
 
 (9.) The unit of circular measure for measuring 
 
 arc , , , 
 
 antjles = — r. — = the ratio between two 
 ° radius 
 
 lengths. 
 Of these only No. 4 can be considered here. 
 
 Ex. A man walks at 3 miles per hour for 4 hours ; 
 he walks 12 miles, obtained by multiplying 
 the velocity by the time. 
 
 Units. 
 
rf— 
 
 I 
 
 I. i 
 
 
 1 ' 
 
 i 1 
 
 •,K;.<;"> .; 
 
 

 
 In such examples care must be taken to use the same 
 
 units of length or time throughout ; or, if it be necessary to 
 
 alter them, to perform the necessary arithmetical operation. 
 
 Ex. A locomotive goes 44 feet per second, how many 
 
 miles does it go in an hour ? 
 
 44 
 
 Ans. 
 
 528U 
 N.l). 30 miles per hour 
 44 
 
 X 3600 = 30 miles. 
 
 44 feet per second. 
 
 Here 
 
 5280 
 
 v: i.e. the No. of miles per second. 
 3600 = t : i.e. the No. of seconds the journey lasted. 
 (10). It is evident that correspondmy to every ca^e of Division of 
 maltijdication there is a converse case of division. nnmi^M'rB. 
 
 _ ,^^ surface , , space , . 
 
 hx. (1). 1 — -r = leno;th : \. - = velocity. 
 ^ '' length » ' time •' 
 
 N.B. It will be found a safe rule to put the symbol of 
 
 division for the word " per. " 
 
 „ 3 miles 4*4 feet 
 
 JiiX. Velocity is 3 miles per hour = , , == ^s n 
 
 •' *^ 1 liour 1 sec d 
 
 T^ . . ^o.^ 1 $3.60 $1.20 lOcts. 
 
 rrice is $3.60 per yard = -7 :t = r? — ; = T^ — r 
 
 ^ ^ J 1 yard 1 toot 1 inch 
 
 5 1 
 
 Rate of interest is 5 per cent := -^.rrr = -^ 
 
 Concrete numbers of like nature can he divided, the quo- 
 tient being an abstract number or ratio. 
 
 Ex. (2). How many times will 2ft. 3 in. go into 3 yds.? 
 3 yards _ 9 X 1 ft. 9 _ 
 
 2 i\. 3 in. - 2ix~[ ttr = 2i ^ *' ^•®- *^" 
 abstract nntnl)er, because the concrete unit 
 of length has been divided out. 
 Ex. (3). How many times will a gallon go into a 
 cubic foot ? 
 
 1 cubic ft 
 1 gallon 
 
 n;„„„ \ »i «»bic ft. of water weighs 1000 ozs. ) , 
 
 ^'"'^^ I a gallon of water weigfis 10 lbs. \ "^^^''^^ 
 1 000 X(wlia t holds 1 oz. of water) 
 
 __ _ 1000^ ^ 
 1 6 X 10 X (wliat holds 1 oz. of water) ~ 160 — ^^^' 
 
 .JS* 
 
r rr" 
 
 ! ^ 1 
 
 WW 
 
 ! 5!: 
 
 i - I 
 
 i n 
 
 \^ n 
 
 iii 
 
 nil 
 
/ 
 
 
 ^;-:v;,»-t 
 
 Here also the unit of capacity divides out, leaving an 
 abstract number for the quotient. 
 
 (11), A correct appreciation of units and dimensions is of 
 the greatest importance in Mathematics, and forms one of 
 the simplest aids to memory and a frequent means for de- 
 tecting errors in work. 
 
 See Notes on Algebra, page 3, sec 10. 
 
 (12). A concrete number = the abstract number X the Rules for 
 U7iit. Only concrete numbers of like nature can be added numbers 
 
 or subtracted. summarised. 
 
 Concrete numbers cannot be multiplied tog'ither except in 
 certain cases, of which the simplest are 
 
 Length X length = surface. 
 Surface X length = volume. 
 Velocity X time = distance travelled ; s = vt. 
 
 Concrete numbers cannot be divided except in the corres- 
 ponding cases, and when both are of like nature, in which 
 case the quotient is an abstract number or ratio. 
 
 (13). Troy weiglit, 1 lb. = 12 ozs. =12X20X24 grains Weights. 
 = 5760 grains ; only used for precious metals and special 
 purposes. 
 
 Apothecaries' weight, 1 Ib. = 12ozs. = 12x8x3x20gr8. 
 = 5760 grains ; only used for drugs and chemicals. 
 Note the similarity to Troy weight. 
 
 Avoirdupois weight, 1 lb. = 16 ozs. = 7000 grains Troy ; 
 used for all general purposes. 
 
 Note that the grain is the only connection between this 
 and the former. The pound and ounce are both different. 
 
 N.B.— In England 1 cwt. = 4 X 28 lbs. = 112 lbs. 
 In Canada 1 cwt. = 4 x 25 lbs. = 100 lbs. 
 
 (14). The clue to the measures of length is this : 
 22 yards = 1 chain = 100 links. 
 4840 sq. yd8.=l acre=10 X [22 yards] »=10 square chains. 
 
 Length and 
 surface. 
 
!• I i 
 
 H i 
 
 . I ; 
 
 'i ' 
 
 rr 
 
 .•• - i 1'! 4 ■' 
 
 ( ' 
 
 ( ;' < -1 
 
 i! ,it 
 
 

 .V^' 
 
 k _ t 
 
 Then it will be seen that 
 
 I pole = 5^ yards = \ chain = 25 links. 
 1 furlong = 220 yards = 10 chains. 
 1 mile = S turlonj'8 = 80 chains. 
 
 Note. Chains, poles, links ; 
 
 cwts., quarters, pounds; 
 dollars, quarters, cents ; 
 
 I exactly correspond, and 
 
 liave the advantage of 
 
 I being decimal measures. 
 
 A ton would correspond to a twenty dollar bill. 
 A mile would correspond to a bill for eighty dollars. 
 Ex, (1). Reduce 3 tons, 17cwt., 3 qrs., 9 lbs. to 
 
 hundredweights. ' ' 
 
 '' 84 
 
 Ans. 77-84 cwt. = 7784 lbs. 
 Ex. (2). Reduce 3 miles, 5 fur., 37 poles, 2Jyd8. to chains. 
 
 240 chains -f- 50 chains + OJ chains -f-TTTSo'chains. 
 
 Ans. 299i + i = 299| chains = 299-375 chains. 
 
 The great use of measurement by chain is that 1 acre=:10 
 square chains. 
 
 Ex. (3). Reduce 37 2-5783 square chains to acres, roods, »fec. 
 
 4 
 
 roods 
 
 poles 
 
 1 
 
 0-3132 
 40 
 
 2528 X ~ 
 4 
 
 or 
 
 632 
 11 
 
 yards 
 fbot 
 
 6952 
 11 
 76472 
 9 
 
 2528 
 ^Oi 
 
 5840 
 
 632 
 
 6 472 
 
 9 
 
 5 8248 
 I 12 
 
 9,9376 
 I 12 
 
 inches 119'25l2~ 
 
 This reads as follows : 
 
 The vertical line serves to mark the position of the deci- 
 mal point after dividing by 10. 
 
 :.„.* 
 
n Mi; 
 
 ! 1 
 
 !iM 
 
 I f! 
 
 1 I . 
 
 
 I'- 
 
 i ' 
 
 ;v • -:;»-:. 
 
 ;i 'K. ' i.u;- 
 
 
 1,1 
 
8 
 
 372-5783 sq. cliains=37-25783 acres. 
 
 ucres roods. 
 
 = 37 
 
 1-03132 
 
 rood poles. 
 
 1 1-2528 
 
 rood pole si|. \ih. 
 
 1 1 7-6472 
 
 rood i>ole sii. j-ds. st]. ft. 
 
 1 1 7 5-8248 
 
 rood poll? sq. yds. sq. ft. S(\. in. 
 
 117 5 119-2512 
 
 (lo). Break nu the multiplier into simple factors, if Compound 
 ^ ^ * Multiplicat'n 
 
 acres 
 
 =37 
 
 neros 
 
 =37 
 
 litres 
 
 =37 
 
 iieres 
 
 = 37 
 
 possible, and multiply by them in the most convenient order, 
 Ex. (1). Multiply 2 miles, 2 fur., 13 poles, oj yds. by 308. 
 
 3087=4 X 7 X 11. 
 Multij)ly first by 11 to get rid of yards and fractions, for 
 
 11 half yards = 1 pole .-.11 x 3^ yards = 7 poles. 
 Multiply next by 4 to get rid of poles. 
 
 2 : 
 
 2 
 
 : 13 : 
 
 11 
 
 25 
 
 1 
 
 : 30 
 4 
 
 
 100 ^ 7 
 miles 706: 1 f. Ans. 
 Much time, trouble and consequent risk of error may be 
 saved by a little consideration of this kind. 
 
 Secondly, if the multiplier cannot be broken into simple 
 factors, the best way is generally as follows : 
 
 Multiply by as many 10s as necessary, and each line by 
 
 the corresponding unit. 
 
 ». d. 
 13 : 3i 
 
 26 ; 12 ; U " 
 
 266 : 9 : 2 
 
 Ex. (2). Multiply 2 
 
 by 239. 
 
 
 
 4 = 200 X 2 
 9 = 30 X 
 7i = 9 X 
 
 13 : 
 
 do. 
 
 do. 
 
 (/. 
 
 532 : 
 3 X 2n^i line = 79 : 
 9 X 1st line = 23 : 
 
 18 : 
 
 18 : 
 
 19 : 
 
 H 
 
 636 : 
 
 16 : 
 
 84 
 
 
 In special cases other methods of breaking up the multi- 
 plier may be found more convenient. 
 
i M 
 
 1 :' 
 
 I 
 
 Ul 
 
 
 w 
 
 ' ' J 
 
 ,1,. umimmmmmemm- 
 
r 
 
 (16). No sucli process as the latter can be adopted for di- 
 vision ; but it is generally' best to break up the divisor into 
 factors, when possible, so as to work by simple division in- 
 stead of by long division. 
 
 £ ». d. 
 Ex. Divide 2 : 13 : 3^ by 308; 308 = 4 x 7 x 11 
 4 ) 2 : 13 ; 3^ 
 
 7) 13 : 3f for the remainder 3^ divided by 4 is f. 
 
 47 47 . 
 
 11) 1 : lOgg- for 5| i.e. -y remams to be divided by 7 
 
 Compound 
 Division. 
 
 616 
 The fractions must at each step be left in their lowest terms. 
 
 If any of the fractions is improper, i.e. > 1, it is a sure 
 sign that there is n mistake. 
 
 (17). It is usual, in reducing fractions to their lowest *3i. C. M. 
 terms, to divide out any simple factor of both numerator 
 and denominator that can be observed, before resorting to 
 the lengthy process of finding the G. C. M. 
 
 In like manner a question in G. C. M. can generally be 
 much shortened by observing such simple factors. 
 
 Rule. Strike out simple factors at any time during the 
 process, retaining tlwse only which divide both divisor and 
 dividend. 
 
 These latter fractions multiplied by the final result give 
 the required G. 0. M. 
 
 Ex. (1). Worked at length by the ordinary process. 
 
 10395)16819(1 
 10395 
 
 0424)10395(1 
 0424 
 
 3971)6424(1 
 3971 
 
 2453)3971(1 
 2453 
 
 352)583(1 
 352 
 
 1518)2453(1 
 1518 
 
 231)352(1 
 231 
 
 935)1518(1 
 935 
 
 121)231(1 
 121 
 
 iio 
 
 583)935(1 
 5^ 
 352 
 
 IfglHIIMKtMitK 
 
10 
 
 It id unnecessary to work further, for tlie result \\i!l 
 
 .!vidcntly he 11, heciiuse 11 is the (r.C.M. of -| Yl^ '■ 
 
 It is shewn in "Notes on Algehra", \k 1.'), that this G. CM. 
 
 the G. C. M. ot every pair, v,z. : -^ ^^.^^.^ ^ ; - ^^^.^. ^ ; 
 
 \ 3971 ) . . ( n52 ) ( 231 ) \ 121 ) 
 I 0424 f *-^^'' '^'^'•' \ 583 f ' i 352 ) ' ( 231 ) 
 
 Iftlien at any time (lurin<x the process, the (f. CM •>[' 
 
 any pair can he seen or ohtained by a direct method, tliis 
 
 is the required G. CM. 
 
 V ■ , \ ;^J'>2 = 11 X 32) ,, . . 
 
 r or instance - -„., _ .. ' ... ,- ; tins is easy to ^ee ; 
 
 now 32 and 53 are prime to one another, therefore 11 nnist 
 he the G. C M. 
 
 Same Example worked by the above rule. 
 5)10395 16819, fur there is no 5 in 10819. 
 
 9)2079 for there is no 9 or 3 or 10819. 
 
 3[23T 
 " 77 = 7X11 
 11 is the only taetor of 10395 that divides 10819. There- 
 fore 11 is the G. C. M. 
 
 Ex. (2). 110)1335510; 8)115872 
 3)12141 1 14484 
 3)4047; 4)4828 
 1207)1349: 1207(17 
 
 120" 
 . 2)T42| 
 
 71 which divides 1207 exactly. 
 .-. G. C. M. = 2 X 3 X 71. 
 
 Here by striking out simple factors, the original luim- 
 bers are reduced to ■! 19^7 [ whose G. (X M. is 71. 
 
 Observe that tlie next pair is | ^^^r. t which is sinipli- 
 
 lied again by striking out the factor 2, which is not contain- 
 ed in 1207. 
 
 2 divides both. 
 
 3 " 
 
11 
 
 Ex. (3). Find the G. C. M. of 805, 1311, 1978. 
 Strike cut every tiut t which is not contained in every 
 one. Thus S05 reduces to 23 a prime number ; if then 
 there if any G. C, M, it can be no other than 23. 
 
 5)S4)5 
 
 3)13)1 
 
 2)1978 
 
 7)l6l 
 
 437(19 
 
 989(43 
 
 23 
 
 23 
 
 1 
 
 92 
 
 m 
 
 69 Ans. 23 
 
 It shouUl be noticed, witliout dividing, that 23 divides 437 
 and 989. 
 
 The rule given above is intended to supplement, not to 
 s\ipersede, the ordinary rule. 
 
 Lengthy processes should always be avoided, if possible, 
 l)ecau8e they lead to error. 
 
 (18). In the ordinary process the rule is to divide by 
 prime numbers only. 
 
 The rule may be shortened if composite divisors are used 
 with care. 
 
 Rule. When using composite divisors, divide each num- 
 ber hy as many of the factors of the divisor as possible. 
 
 Rule. Shorten the process at any time by crossing out 
 any number which is contained by another number in the 
 same row. (Shewn below in block type). 
 
 Ex. 10) 72, 30, 36, 220, 275, 99, 162 
 
 12) 36, 3, 
 
 22, 
 
 55, 99, 
 
 81 
 
 33)3, 
 
 ll> 
 
 55, 33, 
 
 27 
 
 Au8. 10 X 12 X 33 X 5 X 9 = 600 X 297 = 178200 
 The result is the same as the following : 
 
 5) 72, 30, 36, 220, 275, 99, 162 
 
 2) 72, 
 
 6, 
 
 44, 
 
 55, 99, 162 
 
 3) 36, 
 
 
 22, 
 
 55, 99, 81 
 
 2) 12, 
 
 
 22, 
 
 55, 33, 27 
 
 2) 6, 
 
 
 ll> 
 
 55, 33, 27 
 
 11)3, 
 
 
 
 55, 33, 27 
 
 5, 3, 27 
 
 L. CM. 
 
 «i>n»MMHV9v<'w3cn&#«)%*< 
 
^ 
 
 • iC.'i 
 
 .dK 
 
 [ < 
 
 I- - ■ «.. 
 
 'M'l 
 
^t^?ii-i]aateiu :V-i— 3. 
 
 ^' 
 
 
 12 
 
 There is a far shorter and more practically useful inethod 
 of ascertaining the L. C. M. in factors, at sight. 
 
 Consider the highest power of each of the prime factors 
 of all the given numhers. 
 Thus, in the above example. 
 
 Highest power of 2 is 8 contained in 72 
 " " 3 " 81 " 162 
 
 " " 5 " 25 « 275 
 
 n " U " 275 and others. 
 
 .-. L.C.M. = 8 X 81 X 25 X 11 = 200 x 891 = 178200. 
 The least cointnou multiple is generally recpiired for the 
 addition of fractions. 
 
 (19). If the above nunibers represent the denominators 
 of several fractions, mnltiply the numerators as follows : 
 That over 72 i.e. 8x9 by 9 X 25 x 11 
 
 30 " 2 X 3 x 5 '■' 4 X 27 X 5x11 
 30 "4X9 " 2 X 9 X 25 X 11 
 
 220 " 4 X 5 xll " 2 X 81 X 5 
 275 " 25 xll " 8 X 81 
 
 and so on ; that is, in each case, by the remaining factors 
 of the L. C. M. 
 
 (20). Generally reduce mixed numbers to improper frac- 
 tions. Except; for addition or subtraction, keep thenj as 
 mixed numbers. 
 
 (21). Put the symbol of multiplication for the word "of." 
 (22). Avoid cnminy out so much as to make the work 
 difficult to look over. Rather take two or three extra steps, 
 (23). Invert and multiply. 
 
 (24). Avoid the clumsy process of bringing the nnmera- 
 tor a common denominator and likewise the denominator. 
 
 But multiply numerator and denominator by sncli a 
 number as will clear away the compound fractions. 
 
 Ex m i--i_+i_4-i2 + 9_ 1 
 
 "■■■ •■ 
 
 If this elementary form of complex fraction is well un- 
 (lerstood, others of greater complexity will cause no serious 
 
 Addition of 
 Fractions. 
 
 Mixed 
 nunibHrH, 
 
 Compound 
 Fractions. 
 
 Siniplificati'n 
 of Fractions. 
 
 Division of 
 Fractions. 
 
 ('omplex 
 Fractions. 
 
•(.i "'• 
 
 t'l ; 
 
 L.,.._ 
 
¥ 
 
 13 
 
 difficulty. Care must always be taken to distinguish the 
 central line of the fraction. 
 
 6+4+3 
 Ex m - *+*+* _ 12 
 
 ^13 1 6x7x9 
 
 12 
 
 5 x7x3 _ 105 _ 17 
 8x11 ~"8S"- % Ans. 
 
 Note.— Fur 5 X 7+5x9+7x9 = 5x16+63 = 80+63 
 Considerations of this kind hicilitate mental calculation 
 
 1 
 
 1 
 
 Ex. (3). , , X - -, = — X =- ^-- X ' - 
 
 ^^ 14. L 11 , , 4 - 11^^37 -3T Ans. 
 
 1 _ 1 .33 
 1+36^ 
 (25). Ex. (1). Reduce 6«- 7f- to the fraction of r- 9^'- 
 
 ^^- ^'- = 7«T^ X r- 9"- 
 
 .-. The Answer is ^^ = ^1 ><i^ = '^1 . 
 <«• 9d. 93 X l'<- 651 ^ 
 
 Fractions of 
 Concrete 
 numbers. 
 
 ns. 
 
 These are examples of dividing concrete numbers of like 
 nature to obtain ratios or abstract numbers. 
 
 Ex. (2). Reduce 3 roods, 27* poles to the fraction of an acre. 
 
 S^^27^-_^ liH ^ ?9i = 59 
 1 acre 160 32 "64 
 
 Ex. Redu.ie 1 oz. Troy to the fraction of 1 oz. Avoirdupois. 
 
 r'- Ty. _ 480g'-^- 16 Xi8__ 192 
 l^'^Av. TOOOK"" 700 ~175"^"'' 
 16" 
 
 This is the same thing as saying that an ounce Troy and 
 an ounce Avoirdupois are in the ratio 192 : 175. 
 
 1: 
 
 r 
 
 ^« 
 
'<.u: f 
 
"f 
 
 u 
 
 (26). Tlie rules for decimals follow from the rules for Decimal 
 vulgar fractious, of which they are only a particular case. ^■"""''""s. 
 
 (27). Only fractious having the same denominator can be Addition or 
 at once added or subtracted. Sulnraction. 
 
 Therefore i)lace the decimal points in a vertical row. 
 
 For there are then the same number of 10« in the denom- 
 inators. 
 
 (28). In multiplying fractions we multiply the numer- Multipllcafn 
 atators tor a new numerator, and the denominators for a 
 new denominator. 
 
 Iii decimals the first is done by the actual multiplication 
 the second I)y pointing off as many places of decimals as 
 are in the multiplier and multiplicand together. 
 
 (29). Here also the division of tlie numerators gives the Division 
 actual figures; the subtraction of the number of decimal 
 places in the divisor from the number in the dividend gives 
 the numl)er in the quotient. 
 
 Or, it may be easier as follows : 
 State the sum ready for dividing. 
 
 Mark ofi'from the dividend as many decimal places as 
 there are in the divisor, (adding cyphers if necessary). 
 
 Divide up to this point, then put the decimal point, and 
 continue if necessary. 
 
 This is practically the same as multiplying both divisor 
 and dividend by a certain number of 10b, which does not 
 afreet the result. 
 
 Ex. (1). •37)-00i01813(-00049 
 148 
 333 
 
 A us. 
 
 Ex. (2). •037)181-300(4900 Ans. 
 148 
 
 33^ 
 
 These two examples include all possible cases. 
 
 ■I 
 
 -"TiTurTrmtmrniaam ijw 
 
.i«»S|»^«» l"'" 
 
 
 Uvlu'v 
 
 '■J 
 
 .-,, ip. 
 
 I 
 
_liU.J JIU.I11II 
 
 1 
 
 T = -3 ; 
 
 15 
 
 1 1 1 
 
 4-= -25; 0-= 125; J^= 0625 
 
 1.1.1 
 -;r=l«; ir= 1; t:; = 083 
 
 12 
 
 Simple 
 
 Fractions 
 
 expressed as 
 
 Decimals. 
 
 1 
 
 —•2 : ^= 04 ; — = 008 ; I5r,= 0016 
 
 125 
 
 25 
 
 1 
 
 Y= -142857 
 
 2 
 
 y= -285714 
 
 3 
 
 J= -428571 
 
 4 
 
 y= -571428 
 
 y= -714285 
 
 6 
 
 -7= -857142 
 
 y= -999999 = 1 
 
 625" 
 
 n=09 
 
 3 . . 3 . . 9 . . 10 . 
 jj=-18; -z=27; JY=:-81; Ji=-90 
 
 Tlie above decimals should all be carefully examined, as 
 it frequently saves much laborious work, if the source of a 
 circulating decimal is recognized. 
 
 It must be observed that all vulgar fractions can be ex- 
 pressed as decimals which will terminate, if the denominator 
 contains no other prime factor than 2 and 5, but which will 
 otherwise recur. 
 
 (31). It is most important to observe that -9 = 1. 
 
 Hence it is plain that -239 = '24. 
 
 Observe that when a single figure recurs it is due to a 
 denominator 3 or 9. 
 
 When two figures recur and their sum equals 9, such 
 as -09; -27; -QO, &c., &c., division by 11 has been the 
 cause. 
 
 Circulating 
 Decimals, 
 
! 
 
 
 t i 
 
 j^ 
 
 ■ja.i-iJiii- : .11. iJ J»^-^ ■ 
 
 T>; 
 
 II 
 
■^ 
 
 T^: 
 
 16 
 
 Ex. Reduce 25-354 to a vulgar fraction. 
 
 Multiply numerator and denominator l)^' 11 
 . 95.QH4 _ 278-9 2789 
 
 o«3-9 39 
 
 or = 25-jj = 25 no 
 
 6 
 
 or it may be written as 25-3 -j-t- 
 
 (32) The number 7 deserves particular notice, for when 
 1^3 
 7 ' T' T' '^"•' ^'"'^ expressed as decimals, the same 
 
 six lignrcs ai^ always involved in the circulating portion, 
 atid always in the same order. These figures should always 
 be recognized when met with. 
 
 The No. 142857 x 7 = 999999. 
 
 This No. is composed of 14, 28, 57 ; 28 being twice 14, 
 and 57 being 1 more than twice 28 (the 1 being in reality 
 (tarried from 112, the double of 56). 
 
 Thus it should be easily remembered. 
 
 3 40876 ^ 276 
 
 Ex. Reduce 5839428571 =58-39 y^=- 
 
 700 " *'''- ^^700" 
 
 Recurring 
 
 or 
 
 Circulating 
 
 Decimals 
 
 This is a much simpler result than could be directly ob- 
 tained by any general method. 
 (33). Let N be the circulating decimal. 
 Multiply N by as many lO^ as there are decimal places 
 to the end of the first recurring section ; and again by as Decimals 
 many IQi as there are decimal figures to the beginning of*''U""PS8ed as 
 the same section ; (i.e. in following Ex. by lO^ & lO^) and ^'"*'''''"''- 
 subtract. Thus the circulating part vanishes. 
 Ex. Let N = 802U83 
 .-. 105 X N = 100000 N :^ 802083-083 
 
 10" X N = 100 N == ^C02-083 
 
 801281 
 
 .-.99900 N = 801281 .-.Nrr 
 Or, setting aside the whole number, N = 8~-~ — 
 
 99900 
 
 I 
 
 i 
 
t r"' '"• . 
 
 agiil ' ^J:: 
 
 !H 
 
 i ■- 
 
 is ; ' 
 
 'I' ]■'.' 
 1 1 
 
 \ 
 
 M: 
 
 ?U i 
 
T 
 
 T^f 
 
 TCPVMUPI^ 
 
 17 
 Ilenee tlie Rule. 
 
 Subtract t/ie unn-recnrriuij portinn (iiicliuliug intci^er it' "•'■ 2 
 'm\)ft'om the whole and divide hy m inamj 9» as there are '*'' '"'"'^• 
 recurrhuj ,fi<jiirctt, f(>Ui>ii\d by 'i.s many cyphers as non- 
 recxiri'huj places of decimals. 
 
 Thus tlie above example 58-394-iS57i becomes 
 5S39422732 „ 39428532 
 99999900 '^'"^^99999900 V 
 
 Those would l)e found, after much trouldo, to reduce to 
 10219 _ , 69 , 
 
 175 ^^'"''^''^700 ol)tained so easily above. 
 
 (34). Rif.K. Place the dc(!imals in order for addition. 
 Mark off by a vertical line all non-recurring fio-ures. 
 
 Complete each recurring section to the right of the ver- 
 tical line. 
 
 Continue each recurring decimal by a complete section 
 at u time until all terminate at a second vertical line. 
 
 Add, taking care to carry the correct number; and the 
 result between the vertical lines will recur, 
 
 2-4: 7847841 
 1-2 2026261 
 
 .Addition or 
 Sul)tra(.'tion. 
 
 Ex. (1). Add j \^ I i.e. 
 
 ( 2-4 784 
 11-2 26 ^'•^• 
 
 f 
 
 3-704741 i Ans. 
 Ex. (2.) Subtract the same numbers 1-2522158. 
 
 It may be observed that the number of recurrin<T fitrures 
 in the answer is the L. C. M. of the numbers of recurring 
 figures in the separate decimals. 
 
 (35). Recurring decimals may be multiplied or divided, Multii.licafn 
 care being taken to make the proper carryings. «nf' Division. 
 
 Never multiply or divide by a recurring <lecimal ; it is 
 most dangerous as well as cnnd)rous. 
 
 In such cases transform, if necessary, to a vulgar fraction. 
 Ex. Multiply 3-437 by 1-571428 i.e. by If 
 3-437 
 
 n 
 
 7)37-8ii 
 
 5-4015873 Ans. 
 
 I 
 
Hi 
 
 Sill 
 
 '■I:'' 
 
 
 
T 
 
 '\f 
 
 18 
 
 (36). The last figure of a non-ter.ninating deci.nal mmt 
 be taken as near as possible to the correct result, i.e. 1 nuist 
 be acMcd to it if followed by more than 5. 
 
 Ex. 7T written to 5 places is 314159 
 ; " 4 " 31416 
 
 ** 3 « 3142 
 
 " 2 « 3-14 ^ 
 
 buftw^Ts lZ"''T' ^T}^' ^^'' ^''''''' '« t^'« «-»•-^ D->'-'"f a 
 
 out theie is a shorter method. Concrete 
 
 number. 
 
 Ex. Reduce 4^. Tfl- to the decimal of a pound. 
 
 12) , •• -'; . 
 
 20)4^625 Sliilliiigs. 
 £•23125 Ans. 
 
 Ex. (2). Reduce 11 seconds to the decimal of 5 days. 
 
 60) 11 seconds. 
 60) -183 minutes. 
 1 2)-0030B hours. 
 
 2)-000254629 
 5)J00127314814 days. 
 ^000025462T~ Ans. 
 
 in one''''"''^ ^^'^ '"'' *'^'' '^'^''''*"' ''''"''^ ^^ ''^"^ '"^''^ ^^^''3' 
 
 Ex. (2). Reduce 6 furlongs, 41 yds. to the decimal of a mile. 
 
 6 furlongs 41 yards. 
 20) 
 
 11 )205 
 8 )6-1863 fa rlontTS. 
 
 •7732954 of a mile. Ans. ' ^^^ 
 
 1 
 
?\f 
 
 !:l; 
 
 MM 
 
 m 
 
 m 
 
 
 t;;' 
 
 ^u. 
 
 
 ! 't 
 
i. i* 
 
 ' . ♦ 
 
 < * 
 
 19 
 
 (3S). Tliere are three type cases of sums to be considered. 
 First. A compaund concrete number multiplied by a 
 large abstract number. Ex. £2 : 13 : 3 J x 239. See Art. 15. 
 
 239 
 
 2 
 
 239 
 
 _2 
 
 £ 478 
 I38 4d = ix2 159: 6: 8 
 -239 X id= __^9^11^ 
 
 636: 16: 8^ 
 Ans. 
 
 9. £ 478 
 
 10 is'^ 119: 10 
 2«:6*l.i 29:17:6 
 6'1- 1 5 : 19 : 6 
 3<l. ^ 2 : 19 : 9 
 ¥■1 9: Hi 
 
 63'6 : 16 : Si Ans. 
 Seoimd. Find the value of 9y'ls. 2ft- lO'"- at b'^- 7i'l- per 
 
 yard. That is, multiply 9y<l3. 2ft. lo'm. x -VJ-*-- 
 
 The unit of length divides out .•. the Answer is in £. s. d. 
 viz.: 5s.7i^l. X (9| + JJ) 
 
 Place the money at the head of tlie column and break uj> 
 the quantity into aliquot parts. 
 
 8. d. 
 
 5: 7i 
 J) 
 
 2 : 10: 7i = value of 9y'is. 
 lft.6'«>=iy'i. 2: 9f = " ift. 6"'- 
 
 1ft. =|y<l. 1 :ioi 
 
 £2: 15: Hi Ans. 
 Third. Calculations of the following kind frequently 
 occur in logarithmic work, and "practice" is far the most 
 convenient method. 
 
 If -0003567 corresponils to 1 minute, what corresponds 
 to 17 seconds. 
 
 •0003567 
 
 iRsec Lmin. 
 
 osec. Lmin. 
 
 — 80 
 
 8917 
 
 1189 
 
 •0001011 Ans. 
 Care must be takeu not to add in the top line. 
 
 Practice, 
 
 n 
 
,i-'^K ,-.?=;'i' Wi':? 
 
 '^r 
 
 a 
 

 20 
 Ex. Find W, sin. 19<) 10' 37". 
 
 Diff. = 3633 
 
 log. sin. 19o 10' = 9-5162936 
 
 30" 
 
 r 
 
 18165 
 
 3633 
 
 605 
 
 .-. log. sin. 19o 10' 37" = 9-5165176 
 
 This is specially useful when several logarithms have to 
 l>e added together. 
 
 Kx. Add log. 296-|-log. sin. 32'> 18' 53"-f-log. sec. 54o 41 • 28" 
 log. 296 = 2-4712917 
 
 log. sin. 32o 18' =9-7278277 D = 1998 
 
 t 80 999 
 
 20 666 
 
 8 999 
 
 log. sec. 54041' =10-2380008 D = 1784 
 
 20 594:7 
 
 6 178i4 
 
 2 59|5 
 
 .-. required log. = 22 •4373799 
 
 (39). State the sum (without calculation if possible) in HuL-stor 
 the form of proportion, so as to agree with the wordincr. KuleofTliree 
 
 The third term must be of the same nature as the 
 answer. 
 
 The f rst and second terms must be of the same nature, 
 and when brought to the same denomination, may be treat- 
 ed as abstract numbers, because the concrete unit divides 
 out. 
 
 Consider whether the answer should be greater or less 
 than the third term, and arrange the first two terms ac- 
 cordingly. 
 
 greater : less : : greater : | ... "^- I 
 
 less : greater : : less : | ., "^- I 
 
 ( if greater, j 
 
 Reduce by dividing out common factors from the Ist and 
 2nd or Ist and 3rd terms. 
 
► 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 ; 
 
 
 
 
 
 
 
 
 
 
 
 ; 
 
 ( 
 
 
 
 
 /"■_ • 
 
 ' i' ■ "'■" ' ' -■- . 
 
 ^r 
 
 
 
 
 
 1 
 
 1 
 
 
 
 -'l^. 
 
 
 
 • 
 
 
 ■ 
 
 
 
 ! r 
 
 ■■ -• '■■ :{'■' 
 
 
 
 
 
 
 
 If 
 
 > 
 
 
 1 . 
 
 
 
 
 
 
 
 
 *^ 
 
 
 i 
 
 H 
 li 
 
 i| 
 k ■ 
 
 ^¥: 
 
 '■ »^-.',. ■^.'- .' 
 
 \ ■ 
 
 
 V 
 
 
 j, 
 
 
 
 
 
 1 . 
 
 -', ' 
 
 
 
 
 
 
 
 
 
 
 
 r- 
 
 
 * 
 
 
 
 
 
 
 
 
 ■ 
 
 
 ii. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ' 
 
 
 
 h 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 1, 
 
 
 
 
 
 
 
 
 
 
 • 
 
 I;' 
 
 1 
 
 
 V 
 
 
 
 
 * 
 
 
 
 
 
 
 
 
 
 fnaAiKIB 
 
 
 X 
 
 
 
21 
 
 m 
 
 Multiply 2iul and 3r<l mid divide by let. 
 Ex. If 10 men can mow a field in 12 days, in how many 
 days will 15 men mow it? Kvidently in less days. 
 
 tnt'ii. men. tiny". 
 
 15 : 10 : : 12 
 3 2 
 1 2 4 
 
 2 ' ' "-^^- ='- ' , 
 
 8 days. Ans. 
 (40). When the rosnlt depends on several independent Double Uul. 
 proportions, consider each projjortion separately ; treating "* ''''"■'"^• 
 the others as if they were constant. 
 
 Multiply all the first terms for a new first term, and all 
 the second terms for a new second term, and proceed as for 
 Simple Rule of Tiiree. 
 
 The concrete units divide out from all the first and 
 second terms as before. 
 
 Ex. If it cost $84 to keep 3 horses for 7 months, what 
 will it cost t(; keep 2 horses for 1 1 months i 
 
 84 
 
 horses 3 
 
 months 7 
 
 "21 
 
 less for fewer horses. 
 
 more for longer time. 
 
 : 84 
 
 2 ; 
 11 
 22 
 
 4 
 
 $SS Ans. 
 
 (41). Interest, Present Worth and Discount are all to be interest, 
 worked by Rule of Three. „ t>i8count, 
 
 It !^2 IS the interest ot $100 tor any length of time, then 
 $2 is the discount of $102 for the same time, and $100 is 
 the present worth of $102. 
 
 If this is clearly understood there is no difiicultyin work- 
 ing examples. 
 
 The statements are as follows : Ex. 2 percent for 3 rnos. 
 
 i its interest 
 
 \ 3 
 
 100 
 102 
 102 
 
 .:{ 
 
 its discount 
 
 2 
 
 ) its present 
 1 ^^^ 
 
 '. : The sum : its Interest. 
 > : : The sum : its Discount. 
 worth I .. Thesnm-.iU Present Worth 
 
 .a 
 
 If 
 
r 
 
 ■^" 
 
 « 
 
 i 
 
 ' 
 
 ai'ii 
 
 
m 
 
 . 
 
 22 
 
 The interest of $100 for any fixed time and rate is the Inti-n'st, 
 time nuiltiplied by the rate. Hence the rules. Present wonU 
 
 I^or Interest. — Multiply by the rate and time and divide 
 by 100. 
 
 J'^or Discovjit. — Multiply as before and divide by (100 
 -\- the multiplier). 
 
 J^oi' Present worth. — Same division, but multiply by 100. 
 
 The above rule for interest is better, for more reasons 
 tlian one, than the method usually taught on the American 
 Continent, which is only applicable to a Decimal coinage. 
 The above rule is applicable to every coinage, and is fre- 
 (juently shorter even for a Decimal coinage. 
 
 Ex. (1). Interest on $556.50 for 1^ years at 8 per cent. 
 
 
 
 HerefxH ^^ 
 
 1 
 
 
 
 
 
 
 100 ~ 100 ~ 
 
 10 
 
 
 
 
 
 
 .*. Interest = 55.65 
 
 An 3. 
 
 
 
 
 Ex. 
 
 (2). 
 
 Interest on £1534 6s 3d f 
 11 
 168178 8 9 
 15 48 
 5 85 
 .-. Interest =r £168 15 
 
 )r If years 
 
 d. 
 5-85 
 
 at 8 
 
 per 
 
 cent 
 
 = £168 15 5il '^"^• 
 20 
 
 Ex. (3). Discount on $487 due 5 mos. hence at 7 jier (ient. 
 
 5 X7 35 ^, ■ „^ , 35 
 
 _____ _ Discount on 100-|-|^ 
 
 Interest on $100 : 
 35 
 
 35 
 
 .-. 100 + J2 
 
 1235 
 
 247 
 
 12 
 35 
 
 7 
 
 : '- 487 
 
 __7 
 
 247)3409($13.80|~ 
 247 
 939 
 741^ 
 1980 
 1976 
 
 Ans. 
 
 40 
 .-. The Tresent Worth = %^nz.\^f 
 
Miiii l aiiiH j 1. mtf - 
 
 pj 
 
^ 
 
 23 
 
 Ex. (4), Present worth of $340 due 5 mos. hence at 6 per cent. 
 ^""^ 100 : : 340 
 
 100 + 
 
 205 
 41 
 
 12 
 
 100 
 
 680 
 
 13600 
 
 123 
 
 180 
 128 
 
 on 
 
 )|331.70|i- 
 
 Ans. 
 
 70 
 41 
 
 2yo 
 
 287 
 30 
 
 11 
 
 ThecH6connt=|8.294j 
 
 (42). If Ji stock is quoted at a sum diiferent from 100. 
 Tlie interest will be found in the same manner, but the 
 divisor will be the sum quoted, instead of 100. 
 
 Ex. Find the yearly income from $1008 invested in the 
 
 per cents at 84. 
 
 1008 X 6 ^^^ . 
 Income = — g. = $84 Ans. 
 
 Brokerage increases the cost of the investment. 
 
 diminishes the proceeds of sale. 
 (43). The following Algebraical formulse may be advan- 
 tageously inserted here : — 
 
 Let P = Principal or Present Worth. 
 
 1 = Interest for whole time. 
 M = P + I = Amount. 
 
 D ^ M — P = Discount on M. 
 Hence it is evident that I on P = D on M. 
 Let r = rate ])er cent. 
 
 R = 1 -f- ~ = one dollar plus its interest for 1 year. 
 Let n = time in years. 
 
 Stocks. 
 
 Brokerage. 
 
 Intureat. 
 
 .-. I 
 
 = 
 
 Pin 
 100 
 
 
 
 .-.M 
 
 := 
 
 P + 
 
 1 = 
 
 P 1+ "'I 
 
 ^ ' 100 J 
 
 .-. P 
 
 = 
 
 
 M 
 
 _ 100 M 
 
 1 
 
 + 
 
 rn 100+ rn 
 
 100 
 
 .-. D 
 
 = 
 
 M- 
 
 P 
 
 Mm 
 ~ 100 + rn 
 
 Hence the Rule. Interest. 
 
 Hence the Rule. Present worth 
 
 Hence the Rule. Discount. 
 
««■ — '-■ - - 
 
 
 I 
 

 24 
 
 (44). All interest and disconnt onglit, in justice, to be cal- Compound 
 cnlated at compound interest. Interest. 
 
 Let Mj, Mg, &e., M„ mean the amount at the end of 
 1,2, tfec, n years respectively. 
 
 .-. M, = P I I + j,'„) j = P R = new principal. 
 
 .••M3 = Pr{i+-LJ ^PKXR=PR^ 
 
 .-. iM„=PRn 
 
 Quostiouh in conipouiid interest should generally lie cal- . 
 
 •!ulated as logaritlims. 
 
 To find present worth and discount : 
 
 P = - 
 
 ^" R" — 1 ( 1 1 
 
 .•.d = m-p = m.-V=m{i-j^„1 
 
 (45). Let P he the annuity. Present 
 
 A. . 1 ,. T-, ( n-l n-2 ) Value of an 
 
 mount at end ot n years = P|riri_ii} Annuity 
 
 1)11 1 I J calculated at 
 
 = i' X u 
 
 ,'. Present Yalue = 
 
 R-1 
 
 P, 
 
 (Prove by dividing out), SnTS' 
 R"-l 
 
 R-l 
 
 R" "■(R-l)R" 
 .*. Present Value of a perpetual annuity =P. 
 
 P loop 
 
 ~R-l-'r 
 
 i.e — the sum whose interest is the annuity. 
 
 (4f)). These are generally examples of Rule of Three, Percent. 
 100 being the 2nd or 3rd term of the proportion. 
 
 (47). Rule. As the sum of the given parts is to any one Proportional 
 of them, so is the entire quantity to be divided to the cor- **"'"^* 
 responding part of it. 
 
 See examples in any book on Aritlurtetic, 
 
 (48). It will be observed that the method of pointing squaro Root^ 
 adopted on the American continent differs from the old 
 method in use in England. 
 
 There is no apparent advantage gained by the alteration, 
 but it introduces confusion into the minds of those who 
 meet with both systems. 
 
, I 
 
 f,; 
 
 * J 
 
 te jur O MBHW l TC M WW liM 
 
4i 
 
 25 
 
 The learner must Ijear in inind that the dots merely serve S<iunri' Ko.it. 
 to mark off sections of two figures from the right of the in- 
 tegral portion towards the left, and again from the decimal 
 portioti towards the right. 
 
 Ex. 13 47|28'59 63 2. 
 
 There will be as many figures in the Answer as there are 
 sections. 
 
 Tlie ol)ject of tlio method is to find the nearest square 
 number successively below 13; 1347; 134728; 134738-59, 
 tfec, and depends upon the Algebraical formula. 
 
 (« -f *)2 = ^2 + 2 rti + ft8 = a 8 + J (2 a + J) 
 where a stands for the root already found, and I for the 
 next figure. 
 
 «2=900 
 2a +ft 
 
 66 
 
 13|47i28-59,63!2 
 9 
 
 )447 
 396 
 
 (367054. Ans. 
 
 727 
 
 )5128 
 5089 
 
 73405 
 
 )395963 
 367025 
 
 28938 
 
 An additional number of figures, one less than the num- 
 ber already found, can be obtained by common division. 
 That is, in the above Example a total of seven figures 
 could be obtained by dividing 3959632 by 7340. See Tod- 
 hunter's large Algebra. 
 
 (49). The above remarks, with reference to pointing, ap- Cube Root. 
 \)\y also to cube root, but the sections comprise 3 figures each. 
 
 Rule. Mark off every three figures both ways from the 
 decimal point. 
 
 Find the nearest cube number not greater than the left 
 section. 
 
 Subtract this cubed number and brins down the next 
 
 three figures. 
 
 Place the number in the answer. 
 
 Place " 3 X the number '' in the first column. 
 
 Place " 3 X the square of the number " in the second 
 
n»-»^- 
 
 ^^i 
 
 t\ 
 
 •««»wvSt»>-<^wJW**^*»'*^' 
 
 i^,,i4#i<*a^ 9 ^ ^fttKwMMmwmr*^ * * * ' 
 
 „| l,llMfl»lM -•■'"' I. II 'Ml«fl'1l 
 
 iwi fi iii i ii w ii N f y i-. 
 
%■/ 
 
 36 
 
 column ami place two cyphers on the right. This is tlie '^'"i"' H- 
 trial divisor, which is not ver}' reliable till after two or 
 three fiynres of the an^iwer have been obtained. 
 
 Place in the answer the probable quotient. 
 
 Place this number in the 1st column, on the right. 
 
 Add in to the 2nd column the product of the number and 
 the 1st column. 
 
 This is the real divisor. 
 
 Divide and bring down the ne.\t three figures. 
 
 To obtain a third figure we get a new first column by 
 trebling the last tiguro (i.e. by trebling the whole answer 
 already found). 
 
 We get a new 2nd column by adding in the square of the 
 last fiiruro found to the two last rows in the 2nd column, 
 and placing two cy]>herson the right as before (i.e. by treb- 
 ling the sqiuire of the whole answer already found). 
 
 The proof depends on the Algebraical formula. 
 {a +6)3 = a^ + 3aH + 3ah^ _±b^ 
 
 = a» + ^ (3'*^ + ^ "3'* + *)• 
 Whenever a cypher is found in the answer, plsvce one 
 
 cvi)her on to the 1st column, two cyphers to the 2nd column 
 
 and bring down the next three figures. 
 
 264|609!28S(642 Ans. 
 216 
 
 Ex. lS4=3ti+/> 10800 = 3«2 
 
 73()=/X3«+''>) 
 
 1922 
 
 Ex. 2101 
 
 21030 
 
 1 1530 •'««' divisor. 
 10 
 
 48609 
 46144 
 
 1228800= 
 3844 
 
 =3(.«+/>)2 
 
 2405288 
 
 1232644 
 
 
 2465288 . 
 
 1470000 
 2101 
 
 
 344-5(7-0102nL"nrly. Ans. 
 343 
 
 1472101 
 1 
 
 1500000 
 1472101 
 
 14742030000 
 
 27899000000 
 
 Additional figures, two less than those already found, can 
 be obtained by common division. Thus a total of 6 figures 
 can bo obtaitied in the last example by using the last real 
 divisor. See Todhunter's Algebra. 
 
T 
 
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