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1 2 3
1
2
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<9
tttnmum*
THE
AccomrxAiirT'S guide,
ELEMENTAR
IN CANADA.
.■■^^'.
^'
"N
\ '•'
BY WILLIAM MOBRbL • \'^5 "J^ /j^.
"le^^
PUBLISHED
BT THE AUTHORITr OF THE PARLUMENT OW
LOWER CANADA.
QUEBEC :
1833.
. m^ III » .i ii r' i»iiiwp
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Having encountered many difficulties during ten
years as a Teacher in Quebec for want of a pro-
per initiating book of arithmetic, I was induced to
compile the following' work, which it is believed is
suited to supply the existing demand of Ele-
mentary schools in that department.
My object has been to select only that which is
useful and esssential to a good Accountant, Mea-
surer and Book-Keeper. The arrangement is
designed to meet the present state of our schools;
and to facilitate the acquisition of common Arith-
metic in the shortest time of which the study
admits. ' /
The public are indebted to the liberality of tlie
Provincial Legislature for the sum oi fifty poundi^
which they voted toward the expenses of the pub-
lication, that the price might be so reduced, that
all the- children may be supplied with a copy. If
the work should advance the cause of Education
in our Elementary schools, the beneficence of the
Parliament will be repaid, and the object of th«
Compiler will be fiilly attained^
WiLLUU Morris.
(luAu, IZlhJmtt 1833.
CONTENTS.
w t
, t?> fj ' f rof * nn *;> r. i m H
ruiinereUoB and Notation, . % a'(u>v. n T ^^^ ^
3
7
Addition, . • r • * • rt 'r* i^ •!' v*
Subtraction, . . ; ^ •^;»''^';}lol cm?1 r>h,|, ^
MuhipKcation, . . . '■ i'^'f . 'J > II
Division, . • ... • • • • 17
Shop Accounts, • • • ,g/;tl^'>: • 22
Beduction, . • • *! ini^v^ifni hn* • 27
The Rule of Three Direct, • '}J-^'.mU* Un\* 40
The Rule of Three Inverse, •' 4^ r^^y • 46
The Double Rule of Three, . .^ , ^ . / 44
Simple Interest, • . »,*';"• • •' *®
Compound Interest^ . *}4jii« t^ ^ ^^
Vulgar Fractions, . ;•,-•. ♦ .. ,• * • 4^
Decimal Fractions, ' . " "i^ 'i^^ ^^'^^^^l ^ 66
Duodecimals, or Cross MukiplicatioH^' ' ' t''i'>> . 61
Involution, . • . . * • lii r: 62
Evolution, • . . 'a/f ^lis i/uW ,tif »■ W
Practical Geooietir, • ; * .>'<fii'!l)»'^H ^^
Mensuration of Superficee, . Av'^UhuM^^ -J*^ ^
laen3uration of Solids,
Artificers* Work,
U*Olt'j3' 71B.ttiCiif5tH 'it .
dd
(f Iff ;/*"?$?;> :«1JJii
110
Book Keeping by Single Entm .; ' 'I.;!. '^ *^i Ha
.iisaV^iSC ,-iiniwP
t
:\
ARITHMETIC.
Question. What is Arithmcftic ?
Answer. Arithmetic is the art of reckoning by num-
bers ; and consists of five principal rules ; namely.
Notation or Numeration, Addition, Subtraction, Multi-
plication, and Division.
Q. How are numbers expressed ) .
A. All numbers are expressed by the ten following
figures :
1 98 4569 89 10 O
•m, or unit, two, Unree, four, fire, six, MTen, eight, nine, ten, cypher.
anrBCB&Anaxr axtd xroTATzoir.
Q. What is Notation?
A. Notation is the method of writing down a number
in figures.
Q. What is Numeration ?
A. Numeration is the art of reading a number ex-
pressed in figures.
Q. How must numbers be read ?
A. From the lefl hand toward the right hand.
, Q. How does the value of figuTies increase ?
.^ A. In a ten-fold proportion from the right hand toward
the left ; thus, the first figure on the right hand signifies
soniany units; in the second place it represents so
many tens ; and, in the third place, so many hundreds.
Q. yf\ki is the use of the cypher ?
A. The cipher serves to bring figures to their proper
places, hy supplying vacant places. iThus, 7, seven ;
70« SQi^nty; 7Q0, seven hundred; 770, seven hundred
and seventy ; 777) ^even bundre4 and seventy-aeyeD.
'K0t*
i
.
'
M ,
2 NUMERATION AND NOTATION.
Q. Repeat your
B H i B
9. 8 7, 6
A. Units, one ; tens, twenty-one ; hundreds, three
hundred and twenty-one ; thousands, four thousand three
hundred and twenty-one ; tens of thousands, fifty-four
thousand three hundred and twenty-one; hundr«)ds of
thousands, six hundred and fifly-four thousand, three
hundred and twenty-one; millions, seven million, six
hundred and filly-four thousand, three hundred and twenty-
one ; tens of millions, eighty seven millions, six hundred
and fifty-four thousand, three hundred and twenty-one ;
hundreds of millions, nine hundred and eighty-seven mil-
lions, six hundred and fifly-four thousand, three hundred
and twenty-one.
Examples.
Write down in Figurea the following numbers :
1. Twenty-three f Ans. 88.
2. Two hundred and fifly-four.
3. One thousand eight hundred and thirty-two.
4. Twenty-fiv«) thousand, eight hundred and fifly-siz*
6. One hundred and twenty-threethousand, one hun-
dred and twenty-three.
6. Eight hundred thousand, seven hundred and six.
7. Four millions, nine hundred and forty-one ^usand
four hundred.
8. Twenty-seven millions, one hundred fifty-seven
thousand, eight hundred thirty-two.
9. Seven hundred and twenty-two milfioni, tflB hmi-
^red Ihirty-oiie thousand, five hundred and leur.
« ^
,_ _.. . — , ... ._ ^
1 •
ADDITION. A
10. Six hundred and two millions, two hundred iad
Ion thousand, five hundred.
Write doi
11. 36.
12. 69
18. 172
14. 909
m in ffordi the following numbers :
Ans. Thirty five. I
16. 2017 18. 2071909. |
16. 20760 19. 70064008.
17. 764068 20. 123466789. j
v.>
Q. What is Addition?
A. Addition teaches to add two or more sums toge-
ther, to make one whole or total sum.
Q. Repeat your Addition Table.
A. One and one are two, 1 and 2 are 3, 1 and 3 are 4,
and so on.
Addition Table.
%
lane
1
2
8
4
^
6
7
8
9
10
1=
2
3
4
6
6
7
2 „
=
3
4
6
6
7
8
9
10
11
3 „
4 „
=
4
5
6
7
8
9
10
11
12
6
6
7
8
9
10
11
12
13
5 „
=
6
7
&
d
10
11
12
13
14
6 „
=
7
6
9
10
11
12
13
14
15
7 „
=
8
9
10
11
12
13
14
15
16
8 „
s:
9
10
11
12
13
14
16
16
17
9 „
10
11
12
13
14
15
16
17
18
A, •'■*(<WW-
ADDITIOfr.
k '
Q. Repeat jour
MOMWr T,
A. 4 Farthings make 1
12 Pence make 1
and 20 Shillings make 1
NoTB.— £ atandf Tor pounds.
$. stands for shilling!,
and d. standi for panes.
I signifies one farthing.
iXiB.
penny,
shilling,
pound sterling.
i signifies two farthings or a halfpenny.
I signifies three farthings.
!
fit ^
HI
Q. Repeat the
following
Tables*
rARTHINOS.
PENCE.
SHILLINGS.
qra, d.
d..
s. d.
d.
».
d.
8,
£ s.
4 make 1
12mak6l
70 make 5
10
20 make 1
6 .. 1*
18 ,
, 1 6
72 ,
, 6
30 ,
, 1 10
6 „ li
20 ,
, 1 8
80 ,
. 6
8
40 ,
, 2
7 „ 1£
24 ,
, 2
84 ,
♦ 7
50 ,
, 2 10
8 „ 2
30 ,
, 2 6
90 ,
, 7
6
60 ,
, 3
9 M 2i
36 ,
, 3
96 ,
» 8
70 .
, 3 10
10 „ 2i
40 ,
, 3 4
100 ,
. 8
4
80 ,
, 4
11 » 2|
48 ,
, 4
108 ,
, 9
90 ,
, 4 10
12 ., 3
60 ,
. 4 2
110 ,
, 9
2
100 ,
, 6
60 ,
, 5
120 ,
, 10
Rule* — First place the numbers of a like denomina-
tion under each other; that is to say, pounds under
pounds, shillings under shilUngs, pence under penqe, and
larthings under farthings.
FartbingS^ column. — Add up this column, be-
ginnins at the bottom, and, by the help of the farthing
table, nnd how many pence it contains ; if there be any
fhrthinss over, set them down under this column» anidi
carry tSe pence to the next column.
ADDITION.
"^
ING8.
£
».
el
1
10
2
2
10
3
3
10
4
4
10
5
nina-
inder
>and
. be-
lling
any
and
J*-
Pence coLUMii.*-A<ld up Ihii eolimm wi«H the
peoM whieh you carried (rom the flurthinp, and by the
help of the pence table And how many ahillhige it con-
taint ; if there be any pence overt Mt them down under
thia cokmuHf and carry the ahilNngi to the neit column*
SllillillSB^ coLUMH. — Add up the unita of thia
column to the top, then deaeend by the next column to
the bottom, counting every one ten-— then, by the help of
the diilling table, find how many pounds it contains ; if
there be any ahillinga over, set them down under this
column, and carry tM pounds to the next column.
Pounds^ oolumit. — Add up each column, and if
the sum exceed 10, put down the excess, or what is over,
and carry one ; but if it exceed 20, put down the excess,
and carry 2 ; if 30, cany 3 ; 40, carry 4, and so on*
Examples
1. Add together
£ s.
24 13
1 12
132 2
60 15
Ans.
iB219 8
2J
Faitlllngs*-*! and 3 are 4, and 2 are 6, and 1 are
7 — t farthings are IJd. ; set down j, and cany 1 to the
pence. - -
Pence* — l carried and 2 are 3, and 5 are 8, and 4
are 12, and 2 are 14 — 14d. are 1«. 2d. ; set down 2d.»
and carry 1 to the shillings.
Sbllllng8« — 1 carried and 5 are 6, and 2 are 8, and
2 are 10, and 3 are 18, and 10 are 28, and 10 are 38, and
10 are 43— 43«. are £2 38, ; set down 8, and cany 2 to
the pounds.
1*
\
\
.ADDITION.
: P011]ldS«-*2 dttrried and 2 are 4, and 1 are 6, and4
are 9; set down 9; 6 and 3 are.9, and 2 are 11; set
down 1, and cany 1 to the next column ;r 1 carried and 1
aie 2, set down 2. Total in words* Two huncjtared and
nineteen poundfli three shillings, and two-pence three
farthings.
2.
5.
£ 8.
d.
£ 8.
d.
£ t.
d.
26 12
H
3. 27 11
^
4. 30 12
4i
10 13
S
29 10
^
22 14
2i
12 3
13 14
3*
14 15
8
11 2
2
10 4
H
19 10
3|
£ 8.
d
£ 8,
d.
£ 8.
d.
121 13
3|
6. 241 14
H
7. 372 10
H
133 10
H
253 11
m
121 11
3i
120 ,12
n
142 17
2:
327 16
H
102 14
6|
240 13
2
172 14
3
8. Place the following sums of money properly under
each other, and add them toffether, namely, jS178 3^. 4^(2.
^27 59. 7d.+£23^ 4«. eid,+£ZO 2s. 3d.
Ans. ig469 lbs. 8|d.
9. Add together jei734 I6s, 2ld.+£l2d Us. l^d.+
£14 Is, 10|d.+J£239 189. 6^d. Ans. iS2112 10«. 9d.
10. Add together £3109 0». lld.H-£798 13». 4^^.+
£9146 139. 7d.+£874 O9. 8d.+£9146 39. 4d.+£8749
139. 5d.+£8735 199. 9d, Ans. £40560 59. O^d.
11. Add together £38456 139. 2|d.+£1403 lOt.
4id.+iS130 09. 10j|d.+ iSl23 I89. 2ld.+£21 49.
9d.+£218 29. 7|d.+ £241 I9. 3d.
Ans. £45589 119. 3i(r.
12. Add together £30745 179. 4|{2.+£3170 O9;
7d.+£ 21074 IO9. 0d.+£753 O9. Od.4~£ 39875 I5.
10d.+£ 29 199. 11 jd. Ans. £ 95648 99. 4^6.
13. Suppose that A is indebted to B £34 139. 7d.,
and to C £ 1730, to D £ 9 199. 2d., to £ £ 134 O9. 7d.,
flUBT&AC^nOll*
t
to F 17f. 2d., and to 6 9(2. What is the amoont of A'a
whole debt? Ana. £1909 lU.Zd.
14. ,Sappoae that B owea A iS75 17s. ; C owes 15«.
Bd,; D owes iB2l 13s. 6|dt; £ owes 9|il.r F owes
£ 796 1^. Sd. ; and G owes £ 17 ISU. lOd. What is due
to A bj all of them t Ans. £9i2Qa. lOd.
16; A owes to B for tea £ 13 lOa. ; for cheese £ 17
13«. 5d ; for cotton £ 208 17s. ; for Indian chintz £86
and 7d. ; for his acceptance of a bill iS 300 ; for factor-
age £ 15 17«. SJd. ^ tuso for insurance and other charges
£ 30 10s. 4ld* How much is A's whole debt to B ?
Ans. jS672 8s. 84d.
16. Acorn factor pays for wheat jS37 15«. Sd.; for
rye iS 11 16«. 3d. ; for oats £96 and 7^ ; for barley
£5Z 12s* ; also for peas and beans iS 10 ; he has also
paid for carriage and other petty charges j^ 3 17t. 5|d. ;
and for insurance iSll 3fd. Now, supposing his com-
mission on the whole is j£7 3s. 0|d., for how much must
he draw upon his employer to dear the account ?
Ans. jf231 6s. 4^
17. A merchanthas in cash ;f 148 17s. 8d. ; wine to
the value of ;f718 lis. 8d. ; rumd?398 18«. 5|d. ; brandy
^178 19s. 1 Id. ; gin £918 13s. 1 Id. ; tea j£518 1 Is. lid. ;
sugar £316 199. B^d. ; various other goods £317 19s.
8d. What is the worth of his stock ?
Ans. £3616 12«. lid.
18. A bankrupt owed to one of his creditors £784 18s.
lid.; to anottier £315 17ir. 8d. ; to another £88 Os.
ll|d. ; to another £778 15«. 8d. ; xo another £785
18«. ll^d.; to another £13 8«. 6|d.;'to another £67
18*.; to another £318; to another £164 11 d. Re-
quired his whole debt ? Ans. £3296 18«. 8^^.
Mi
Q. What is subtraotioni
A. Subtraction tesvdies to take a less number from »
greater* and shows the remaindert or difference.
hi
i!^ r
M *
111 •
1. lip.
B BUBTKACnOll.
Q. RapettjTOQr
Sabtraetlon Table*
1
2
3
4
5
6
7
8
from
t»
*t
It
''■/"■■
n
II
11
T
1
1
2
3
2
II
3
4
"5
1
4
5
1
6
7
6
7
8
-6
8
9
i
■5
1
9
T
ll
9
10
10
11
3
4
5
6
11
12
4
5
6
7
8
9
10
11
12
13
5
6
7
8
P
10
11
12
13
14
6
7
8
9
10
11
12
13
14
15
7
8
9
8
9
10
9
10
11
10
11
11
12
12
13
13
14
14
16
16
16
16
17
16
17
18
9
12
18
14
15
Rule* — Set the less number under the greater, ob-
serving to write pounds under pounds, shillings under
shillings, pence under pence, and farthings under far-
things, as in addition ; then begin at the right hand, or
the Farthings, and subtract each number of the under line
from that of the like name in the upper : but if it be too
great, subtract it from the value of one of the next higher
f^Kaev add the remainder to the upper number, and write
« tlM^in|«p>below ; ap.d in this case cariy one to the under
figl^ ^ the next name.
£xamples«
10 90 la 4
1.
Ans.
£ 8. d.
From 438 15 7^
Take 278 17 9)
^169 17 9|
SUBTRACTION.
Fartlltllp^.— IV t< 2 farthingf from 1 farthing I
cannot, 2 farthings from 4 farthings and 2 remain — 2 and
1 are 3 farthings, set do?m }, and carry 1 to the pence.
Pence* — 1 carried and 9 are 10, take 10 from 7, 1
cannot, 10 from 12 and 2 remain— 2 and 7 are 9 ; set
down 9, and carry 1 to the shillings.
Shillings.— 1 carried and 17 are 18 ; take 18 from
15 I cannot ; 18 from 20 and 2 remain — 2 and 15 are
17 ; set down 17 and carry 1 to the pounds.
Pounds* — 1 carried to 8 are 9 ; take 9 from 8 I
cannot, 9 from 10 and 1 remains — 1 and 8 are 9 ; set
down 9 and carry 1 to the next column ; 1 carried and 7
are 8 ; take 8 from 3 I cannot ; 8 from 10 and 2 re-
main — 2 and 3 are 5 ; set down 5, and carry 1 to the
next column ; 1 carried and 2 are 3— take 3 from 4, and
1 remains ; set down 1.
Remainder in words, one hundred and fifty-nine pounds*
seventeen shillings and nine-pence three farthings.
2. From ^547 13 10 3. From iS7864 17 4|
Take 326 10 9 Take 5412 11 l|
4. „ JS 21384 2 7i
. „ 10120 1 2j
5. „ iB 721384 3 7^
„ 120123 4}
6.
£53907 11 54
21302 id 10)
7. „ JB 38597 12 IJ
„ 13270 10 8|
6. „ £32975 16 4) 9. „ £57384 13 7
12264 17 9| „ 27172 18 10|
)»
10. „ £75432 3 8| 11. ,,
, „ 14129 1 74
;fc'37921 10 2|
12737 8 li
1
10
SUBTRACTION*
12. Froin;e37205 13 9\
Take 17921 17 9
13. From ;675082 4}
Take 17392 16 6}
14.
'
15.
•«
>»
^£12764 19 7
139 11 10
£9999
0}
. •
■ s
16. What is the difference between £73 0«. 6ld, and
£19 13s. lOd. Ans. £53 6«. l^d.
17. A lends to B £100. How much is B in his debt,
after A has received £73 12^. A^d, 1 Ans. £26 7«. l^d,
18. Subtract £17 2«. 6d. from iS500, and tell me the
remainder ? Ans. £482 17«. 6d.
19. If £482 179. 6d. be taken from £500, what will
be the remainder? Ans. £17 2«. 6d.
20. If I owe my friend £ 700, and I pay him i& 50 2«.
9|d. on account, what will remain due ?
Ans. J&649 17». ^d.
21. What is the difference between £99 19«. ll|({.
and £100? Ans. 4.
22. Take one farthing from £100.
^ Ans. £99 19«. il^d.
23. A merchant has in cash £474 89. 9d. ; goods,
value £3443 15^. ; a house worth £713 II9. ; a ship
£574; another £315; debts due to him £957 I89.
Hid. Now he owes to A £115 7a. 8d. ; to B £327
18s. 4|(i. ; to C £74 139. 4d. I demand his net stock?
^ Ans. iS 5960 149. 3|^.
24. A borrowed from B, at sundry times, the following
sums, viz. £781, £63 159., £52 IO9., £565 ; and has
paid ab follows, at differert times in cash, £330 IO9.,
£54 13». 4d., £67 IO9. ; in goods £54 I89. 6d., £73
159. Sd. ; by a draft on John Steele, £63. What is A
still due? Ans. £817 179. 6d.
25. Suppose that my rent for half a year is £10 129. ;
and that I have laid out, for the land tax, 149. 6cl., and for
several repairs £ 1 39. 3;|d., what have I to pay of my
half year's rent.' Ans. £8 149. 2jd[.
o|
1<
■' '<"
i
^d. and
6«. 7|d.
bis debt,
7a.7ld.
I roe the
17*. 6d.
^at will
' 28. 6d,
^50 28.
fa, 2id.
. ll^rf.
ina, j.
. Hid.
goods,
a ship
7 I8s.
£327
stock?
^ 3|rf.
owing
id has
) 10*., .
, £73 ^
t is A
*. 6d.
12*. ;
id for
»f my
2ji
Q. What is Multiplication
A. Multiplication teaches
of two Numbers given, as ofte
less; and compendiously perfori
additions.
Q. Repeat your
eater
in the
of many
Mnltiplieation Table.
14
16
as
IS 34
34
34
27
33
36
IS
3128
33
44
48
35
55
60
35
43
42
40 48
54
73
16 18
34 37
33 36
40 45
48 54
49
56
6677
84
8 9 10
56. 63
64
79
73 81 9U
so! 90100
88' 99
20
30
40
50
60
70
80
96108
110
120
11
13
93 34
33
44
55
66
77
88
99
110
121
133
36
48
13
60
73
84
96
i08
120
133
144
96
39
52
65
78
91
104
117
130
143
156
14
38
42
56
70
84
112
126
140
154
168
15
30
45
60
75
90
105
120
135
150
165
180
16
32
48
80
06
112
144
160
176
192
17
34
51
68
85
102
119
136
18
36
54
73
00
108
136
144
153163
170|180
1671198
204216
10
38
57
11
95
114
133
152
171
190
■b
20
40
60
80
100
120
140
160
180
•200
200830
338340
Rule 1. — ^When the multiplier is not greater than 12,
write it under the pence of the multiplicand, and in multi-
plying, put down the overplus of farthings, pence, and
shillingSff and carry as in addition.
Examples. *'
£ t. d.
1. MulUply 14 16 H MuHipKcand
hf 7 Multiplier
;ei03 16 2j P»>duct
fi
■■^»JM^*,^.^ -«^^t» ,.^j^
ik.
:^f
■*«•■*
■*-,.
12
MULTIPLICATION.
»? :
- *»
Farthings. — 7 times l are7— 7 farthings are Ijcf.;
set down |, and carry 1 to the pence.
Pence*— 7 times 7 are 49, and 1 carried are 50 —
60d. are 4f. 2d.; set down 2, and carry 4 to the
shillings.
ShlUingrs*—''^ times 16 are 112, and 4 are 116 —
116a. are £5 169.; set down 16 and carry 5 to the
pounds.
PonndS. — 7 times 4 are 28, and 5 carried are 33 ;
set down 3, and carry 3. — 7 times 1 are 7, and 3 carried
are 1.Q ; set down 10. Product in words, one hundred
and three pounds, sixteen shillings, and two-pence three
farthings.
£ 8. d. £ 8, d,
2. Multiply 124321 2 4| 3. MoUipiy 23434 5 5^
by 2 by 2
4.
n
234204 4 24 5.
3
T
135246 5 41
3
6.
432510 5 3} 7.
4
t*
274321 6 2|
4
8. „ 34523 12 6|
5
f*
9.
I*
273534 13 3|
5
10. „ 417383 11 3^ 11. „ 543210 14 4;
6 ' .. 6
»»
: •
12. „ 350214 15 4^ 13.
7
ft
„ 215438 16 2i
7
MULTIPLICATION.
18
£ 9. d, £ i. d
14. Mdtipiy 521403 6 7} 15. MiUUpiy 488025 7 10)
hj £ by 8
5
41
3
6
2|
4
3
3i
5
4
4J
6
6
I
16. „ 378210 10| 17. „ 321457 17 41
9 H 9
n
18. „ 527032 7 3j 19. „ 382721 14 3|
10 „ 10
»»
rt
20. „ 387204 15 2J 21.
'■9
22. „ 521432 13 4| 23.
!♦ 12
„ 432579 10 4
11
„ 732173 4 10)
It la
To multiply by any number greater than 12, observe
the following
Rule* — Multiply the top line by 10, and that product
again by the same number, until you have as many lines
as there are figures in the multiplier ; then multiply the
1st line by the last figure, the second line by tne 2nd
figure, and so on ; add these products together, and the
sum will be the product of the number given.
4
m\
.-.■i
: I
i-ii
14
■fh ,t
^
MULTIPLICATION.
Examples.
r^ « .*.
i
24. What 18 the price of 345 yards of cloth, at £2 12«.
7}d. per yard?
lit line £2 12 7^X5
10
2nd line
drd line
26 6 3 x4
10
"B^M^lM--'^-* 4
r « k'
H*: .'
263 2 6 x3
3
it, •
789 7 6 price of 300 yaMs.
"105 5 price of 40
13 3 1} price of 5
»»
^,\ ' Ans.
£907 15 7i price of 345 yards.
i*<
■ li
For 13, multiply by 10, and add 3 times the top line.
14, multiply by 10, and add 4 time&' the top line, &c.
24, multiply by 10 and by 2, and add 4 times the top
line.
35, multiply by 10 and by 3, and 5 times the top line.
46, multiply by 10 and by 4, and 6 times the top line.
127, by 10 and by 10, twice the second line and 7
times the top line.
394, by 10 and by 10, 3 times the 3rd line, 9 times
■r:-^ *--- the 2nd, and 4 times the top line.
K. B."- The pupil should be exercised for a few minutes in the
plan abore, before he enters on the following questions :—
What is the price of
25. 13 ib of sugar, at 1«. 3(2. per lb? Ans. 16*. 3d.
26. 14 moidores, at £1 7a, each? Ans. £18 18«.
27. 15 pistoles, at 17». ed. each? Ans. £13 2«. 6d.
28. 16 cwt of cheese, at £1 18«. Sd. per cwt. ?
Ans. £30 18«. ed.
MULTIPLICATION.
15
79, 18 cwt of tobacco, at £5 llf. 4d. percwtT
Ana. £100 4f.
30. 20 cwt. of hops, at £4 7t. 2d, per cwt ?
Ana. £87 8f. 4d.,
31. 21 cwt. of hemp, at JSl 12«. per cwt.?
Ana. ^33 12t.
82. 22 toua of hay, at £1 2«. per cwt?
Ans. ;f24 4«.
33. 25 yda. of broad cloth, at 9«. 2d. per yard ?
Ana. iSll 9a. 2d,
34. 28 yda. of auperfine do. at 19«. 4d, per yard?
Ana. £27U,4d.
35. 32 yda. aerge, at 3«. 7d. per yard ?
Ana. £5 14«. 8(2.
36. '48 acrea of land, at £2 3a, per acre?
Ana. £103 4«.
37. 66 gallona of rum, at 8a. lOd. per gal. ?
Ana. £29 3«.
38. 84 qra. of wheat, at ^1 12«. 8cl. per qr. ?
Ana. £137 48.
39. 106 qra. of barley, at 149. 7^(2. per qr. ?
Ana. £77 8a.0id.
40. 127 cwt. of hopa, at £3 Oa. 2d. per cwt. ?
Ana. ^382 la. 2d.
41. 224 ft, of tea, at 79. 3|d. per lb?
Ana. jf81 8a, 8d.
42. 336 fb of do. at 59. 2|(i. per ib?
Ana. jf87 179.
43. 532 firkina of butter, at ^2 159. 6d. per firkin?
Ana. ^1476 69.
44. 941 cwt. of augar, at £7 Oa. 4d. per cwt. ?
Ana. ^6602 139. 8d,
45. 3918 yda. of brown cloth, at 129. 6d. per yard ?
Ana. ^2448 159.
46. 6874 aeta of bucklea, at 159. 6d. per aet?
Ana. ^^5327 79.
47. 9674 yda. of velvet, at 149. lOd. per yard ?
Ana. £7174 179. Sd.
48. 10,000 yds. of ahalloon, at 1 Hd. per yard ?
Ana. £479 39.4(1.
16
MULTIFUOATIOlf.
Rnle 3* — To mulUpI/ a whole number by ft number
eonsi^ og of two or more figures. Place the multiplier
undo the multiplicand, then multiply by each figure se-
parately, observing to put the first ngure of every product
under its multiplier. Add these products together, and the
sum will be the total product required.
liXamples*
49. Multiply 472035 by 20034? Ans. 9456749190.
Multiply 472035 Multiplicand
by 20034 Multiplier %
1888140
1416105
944070
.04;
Ans. 9456749190 Product.
Ii.
ir
50. Multiply 273580961 by 23? Ans. 6292362103.
51. Multiply 402097316 by 195? Ans. 78408976620.
52. MulUply 82164973 by 3027?
Ans. 248713373271.
V 53. Multiply 16358724 by 704006 ?
Ans. 11516639848344.
54. How many letters are there in a page of a book
which contains 45 lines, each line 59 letters ?
Ans. 2655 letters.
55. How many grains of wheat will fill 987 bushels,
when 1 bushel contains 675000 ?
Ans. 666225000 grains.
56. How many strokes does the hammer of a clock
strike in a year of 365 days, at 156 strokes in a day?
V Ans. 56940 strokes.
57. How many feet will reach from Quebec to Mon-
treal, if the distance be 180 miles, and 5280 feet in a
»inile? Ans. 950400 feet.
DIVISION.
17
Bivisioir.
Q. What is Division ?
A. Division is the method of finding how often one
number is contained in another. The first number is
called the Divisor, the second the Dividend, and the re-
sult the Quotient.
Q. Repeat your ' ,
DiTision Table.
2 into
•
2
"3
1
i
;
6
8
1
n
6
9
12
8
12
16
20
24
^
32
36
40
44
t
s
10
10
15
20
25
30
35
40
45
50
55
60
i
3
<e
12
__
18
24
30
36
42
48
54
60
66
72
'5
t-
14
21
28
35
42
49
56
63
70
77
84
■
16
24
40
48
56
64
Ti
80
88
96
i
s
•a
o>
18
27
36
45
"54
63
~72
~81
"90
"99
108
i
20
lo
40
50
60
"to
80
90
100
no
120
1
■a
wm
32
I3
44
"55
8
S
a
©»
24
36
"48
60
PS
26
39
65
i
g
28
"42
56
"to
'S
>n
30
45
60
75
8
"48
64
"so
1
a
34
51
I5
i
a
CD
36
54
"72
90
38
"57
1
40
3 ..
4 »
4
76
M
5 „
5
10
12
14
16
18
20
If
IB
•21
'24
27
30
95
100
6 „
6
66
77
88
99
110
121
132
72
84
96
108
120
132
144
78
91
104
117
130
143
156
84
"98
112
126
140
154
168
90
105
120
135
96
112
128
144
102
108
114
120
7 .,
7
119
136
126
144
133
140
160
8 „
8
9 „
9
153
162
171
180
10 „
10
150
165
18'/
160
176
192
170
187
204
180
190
20C
11 ,.
11
22
■24
33
36
198
216
209
220
«2 „
12
228
240
Rule !• — ^When tho divisor is not greater than 12,
place the divisor on the left hand of the dividend, with a
curve line between them ; then find how often the divisor
is contained in the dividend, and place the numbers un-
der the figures divided ; observing to reduce the remain-
der in each name, if any, into the next inferior denomi-
nation, adding the given number of that name, and so con-
tinue to divide in the same manner to the lowest name
placing the last remainder, if any, on the right.
'*■' -T^.y:
*2
"■J :
18
PIVISIOlf.
) !
i
Bxamplet*
1. Difide £6207 3«. SJd by 6.
£ «. d.
6) 6207 8 8^
Ana.
£1241 8 8]— 2 over.
Pounds* — 5 into 6, once and 1 over — set down I,
and carry 10—10 and 2 are 12, 5 into 12, twice and 2
over— -set down 2, and carry 20 — 20 and are 20, 6 into
20, 4 times — set down 4 — 6 into 7, once and 2 over — set
down 1, and carry 40 to the shillings.
Shillings*— 40 carried and 3 are 43, 5 into 43,
8 times and 3 over — set down 8, and carry 36 to the
pence.
Pence*— 36 carried and 8 are 44, 5 into 44, 8 times
and 4 over — set down 8, and carry 16 to the farthings.
Farthinfl^S* — 16 carried and 1 are 17, 5 into 17,
3 times and 2 over— set dowa |, and 2 over on the
right.
£ ». d. £ ». d.
9* 2)2468 10 4 8. 2)26845 4 10
4.
3)36390 12
9
6.
4)48408 16
8
8.
5)56126 6
4i
10.
tf^-iMSg 6
3
^^^ M*
5.
3)43687 2 3
7.
4)57385 b 1
9.
5)62716 7 3i
11.
6)84037 7 4i
12.
DIVISION.
£ 9. d. £ i. d-
7)834576 2 8} It. 7)321407 4 6]
14. 8)123729 7 4 15. 8)78600« 7 8}
16. 9)387642 6 7i 17. 9)3072i ? t '}
18. 10).>27343 12 lOJ 19. 10)321785 5 8|
20. 11)387503 16 8J 21. 11)87927 9 5J
22. 12)32040 4} 23. 12)87980 lo;.
Rule 2* — To divide by any number greater than
12. First ; draw a curve line on each side of the dividend,
and put the divisor on the left hand side ; make a small
table, by multiplying the divisor by the 9 digits, 1,2, 3,
&c. respectively placing the products with their multi-
pliers in horizontal rows under the divisor. 2d. If the
first figure of the dividend be greater than the first
figures of the divisor, count off as many figures from the
led hand of the dividend as there are figures in the
divisor; but if the first figures be less^ count 1 figure
more from the dividend for the first member. Look in the
tible for that product which is next less than the first
member, and place it under the said member, and the
figure which stands on the same line with the product,
must be placed on the right hand of the dividend for the
1st quotitnt figure. Subtract the said product from the
first member of the dividend, and bring down the next
figure of the dividend to the remainder for a second mem-
ber; [N'oceed with this member the same as before, and
so contmioe till all the figures of the pounds are brou^t
down. Multiply (Im remainder from the pounds, if any.
20
DIVISION.
by 20, adding in the shillings of the dividend, and divide
as before. Multiply the remainder from the shillings, if
any, by 12, adding in the pence of the dividend, and divide
again. Multiply the remainder from the pence, if any, by
4, adding in the farthings, and when divided once more
the operation will be finished.
Examples.
24. Divide ,^375683 17». 3irf. by 234.
DIVISOR. DIVIDEND. (QUOTIENT.
£ 8, d, £ 8. d.
1 . . 234) 375683 17 3J (1605 9 8|-32 rem.
234»»»
1416
1404
" wl^
1 .
. 234
2 .
. 468
3 .
. 702
4 .
. 936
6 .
'. 1170
6 .
. 1404
7 .
. 1638
8 .
. 1872
1
9 .
.2106
1283
1170
113 remainder from the pounds.
• 20 ■ •
234) 2277 (9*.
2106
171 remainder from the shillings.
12 '
234) 2055 {Sd,
1872
;<*■
183 remainder from the pence.
4
234) 734 (I
702
V -
f,
32 remainder.
Ans. JCieOS 0«. 8}d.=32 remain.
DIVISION^
21
25. Divide £14693 4f. e\d. by 13 ?
Ans. £1130 4*. lli(/.-TV
' 26. Divide £17934 10#. TJd. by 14?
Ans. £1281 9d.~t^*
27. Divide £37846 17*. lOfd. by 16 ?
Ans. £2365 8*. 7}(f.-||.
2S. Divide 5 7384 19*. l^d, by 23?
Ans. £2494 19*. ll}d.-,V
29. Divide £138457 14*. 2Jd. by 67 ?
Ans. £2429 1*. 7}(f.--if
30. Divide £137586 13*. 5^^. by 124 ?
Ans. £1109 11*. 4}-y',V
* 31. Divide £321204 19*. lljd. by 674 ?
Ans. £476 11*. 3irf.— 14|.
32. Divide £1875486 13*. 5Jrf. by 5374?
Ans. £348 19*. 10(i.~|f^|.
33. Divide £49 14*. 6cl. equally between 39 men?
Ans. £1 5*. 6cf.
34. If 27 cwt. of sugar cost £47 18*. 9(^., what cost
1 cwt. ? Ans. £1 15*. 3|(^.— 15 over.
35. If 72 yds. of cloth cost £85 5*. 6d., what cost
lyard? Ans. £l3*.8Jd.
36. A prize of £7257 3*. 6d. is to be equally divided
amongst 500 sailors. What is eacQ man's share ?
Ans. £14 10*. 3|(|.
37. If a gentleman's income be £500 a year, what is
he worm each day, counting 365 days in a year ?
Ans, £1 7*. 4|d.
38. If a farm of 57 acres is let at £55 4*. 4|ci., what is
the rent per acre? Ans. 19*. 4}</.
39. If 6 men's wages for a year be £577 11*., what
does each man earn per day ? Ans. 4*. 6d.
■r^-<. -, :
ii: t-
fj •!%
Wis
'iifiV
' '4'
22
SHOP ACCOUNTS.
ssov Aoooims. .
Q. How are Shop Accounts calculated?
A. Shop accounts are generally calculated by Multi-
plication or Division ; but when any doubts of correct-
ness exist* accountants use both methods, in order to
prove their work.
Q. What is meant by aliquot parts t
A. One number is said to be an aliqu ot or even part of
another, when it divides it without a remainder : thus, 3 is
an aliquot part of 12, because 3 divides 12 without a re-
mainder; the following is a
Money Table of Aliquot Parts.
'O
00*
•
•
1^
¥^
w^
^
•^
PM
0«
h
SS
o
o
O
o
i(^. IS
i
OR, A J
OR, ftJir
l9.
IS Vt
1 »
I
♦♦ ^v
» ^h
1«. 3d.
„tV
1 M
ft
i» TI
n ^i«r
1 8
,.tV
u •*
?»
1
»» 7
n 16 ff
2
,»TT
2 „
»»
" i
M ViT
2 6
I
»» T
3 „
tt
„ i
» tV
3 4
"t
4 „
»»
::l
»» tV
4
»» T
6 „
»»
»f tV
5
»> ''
8 „
»»
»» »♦
I
M 3T
6 8
»»
|io „
t«
M »»
10
Rule 1 • — If the given price or rate be an aliquot
part of a penny, shilling, or pound, divide the quantity by
the aliquot part, which gives the answer in pence, shil-
lings, or pounds respectively : if the answer be found in
shillings, divide by 20, to bring it to pounds ; but if the
answer be in pence, divide by 12, and then by 20.
SHOP ACCOUNTS.
2S
K. B.'It ii expected that the pupil will work each of the fol-
lowing questions by multiplication and division, as in the folloiiin§
Examples.
1. What cost 100 ib of sugar, at 4d, per lb ?
BT DIVISION.
8,
4<i.i8|=:100ibat4d.
BY MULTIPLICATION.
£ a. d.
% 4
I
3 4
10
Ans.
£1 13 4
: . ii
-.,,-*.:.
2. 16
3. 26
4. 37
6. 49
6. 67
7. 68
8. 74
9. 89
10. 90
11. 100
12. 24
13. 36
14. 48
15. 60
16. 72
17. 80
18. 84
19. 90
20. 100
<7.
yds.
at
SI
»»
Of
»»
1
i«
H
f»
2
«»
3
»»
4
»f
6
)*
8
f»
1
»«
1
i»
1
3
»»
2
»i
2
6
*t
3
4
»»
4
«t
5
,,6 8
„ 10
iCl
20)0 33 4
Ans. £1 13 4
a
£
a.
d
Ans.
4
„
1
1
„
3
1
„
6
li
„
9
6
»,
17
*,
1
4
8
»,
2
4
6
„
3
„
4
3
4
„
1
4
„
2
5
„
4 16
„
7
10
„
12
1,
16
„
21
„
30
f«
60
Rule 2« — If the given price be not an aliquot part
of a penny, shilling, or pound, take an aliquot part less
than the price ; and if the remainder be not an aliquot
*■!
S4
SHOP ACCOUNTS.
part, take a part less tlian it, and so on, till all the parts
together be equal to the giTen price ; then add the values
of all these parts together, and the sum will be the whole
value at the given price.
Examples.
1. What cost 95 yds. at 6<. 6d* per yard?
BY DIVISION.
95 at 6a. 6d,
£
BT MULTIPLICATION,
£0 6 6X5
10
3 6
9
. , ^9 5
1 12
6
Ans. £30 17
6
8,
5
1
d.
is 1=23 15
is 4= 4 15
6isJ= 2 7 6
6 6 s £30 17 6 Ans.
!a
at Ofd. p^ryard?
t*
If
It
II
If
II
II
II
II
II
II
II
II
II
«i
II
>•
M
IP.
1 .-r
■'•'■ l\l
/•,
-r t&*
. - 1
>^ £ 8,
d.
Ans
7
6}
#
13
2}
II
1 10
3
fH
4 11
lOj
n
.^ 6 4
n
' m-'
« 7 12
^
«
9 16
11
31 18
r
60 2
1
II
69 15
4
II
86
10
II
103 1.5
0}
II
123 18
7j
II
213 17
II
308 18
^
II
463 15
3
f*
607 11
8i
II
943 16
Hi
•9
1997 18
4
SHOP ACCOUNTS.
25
Rule 3* — When the quantity given consists of
several denominationsi as Tons, Hundreds, Quarters,
and Pounds ; multiply the price by the first or highest
name, and take parts for the lower denominat'ions, as
follows .
Weight Table of Aliquot Part8.
\ •
X,
.13 ■ ^
d9
•
•
a
2
w^
h
h
(*t
pit
o
O
O
o
OJib
IS
1 s=
Jh
=
tIt
=
ll^T
Oi „
1 «
-* «
4 „
7 „
»»
»»
u
»»
ft
ft
tt
tV
1
j
=
TTlTt
1
8 „
tf
ft
ft
=
Vr
=
shf
14 „
»>
ft
i
=
i
~~
TiiT
16 „
11
ft
1
=
xiiT
Iqr.
M
f)
tt
i
=
1
2qrs.
ft
ft
ft
2
=
I
Icwt
»
tt
ft
Tff
a M
»»
t»
It
tV
4„
»»
tf
ft
i
5 „
♦»
ft
ft
I
10 „
»♦
ft
ff
JL
JV'iJ '
1 .1.
^
3
SHOP ACCOUNTS.
' '}
Examples.
1. At i^3 17*. 6d. per cwt.,what is the value of 25
cwt. 2qrs. I41b of tobacco ?
£ *. d.
3 17 6
6
JB19 7 6
5
t-^^:^^"^'
— — Cwt. qr. ft).
9« 17 6 price of 25
2qrs.=J of lcwt.= 1 18 9 price of 2
141b =i of 2qrs. = 9 8^ price of 14
Ans. £99 5 11^ price of 25 2 14.
2. What cost 13cwt. Iqr. 7ib of molasses, at £l I2s.
4d. per cwt.? Ans. £21 10s. d^d.
3. What cost IScwt. 2qr. 81b of pearl ashes, at £ 2 5s.
6d. per cwt. ? Ans. £ 42 5s,
4. What cost 21cwt. 3qr. 14lb of starch, at £2 16s.
8d. per cwt.? Ans. iSei 19s. 7d.
5. What cost 32c wt. Iqr. 161b of soap, at £3 5».
lid. per cwt.? ^ Aus. i2l06 15s. 2|d.
6. What cost 43cwt. 2qr. 21ib of madder, at £3 19».
4d, per cwt. ? £1 73 6s. 1 0^ d.
7. What cost 17cwt. Iqr. lift of cheese, at £3 14s.
8d. per cwt. ? Ans. £64 15s. 4d.
8. What cost 85c wt. Iqr. lOlb of butter, at £4 6s. 4d.
per cwt.? Ans. i:368 7s. 7jd.
9. What cost 72cwt. Iqr. 181b of hops, at £4 5s. 8d.
per cwt. ? Ans. iS 310 3s. 2d.
10. What cost 27cwt. 2qrs. 151b of raisins, at ie2 6s.
8d. per cwt.? Ans. £64 9s. 7d.
11. What cost 78cwt. 3qrs. 12ft» of currants, at ^22
17*. 9d. per cwt ? Ans. £227 14».
REDUCTION.
27
12. What cost 56c wt. Iqr. 17Jb of sugar, at i^2 15f.
9d. per cwt. ? Ans. .£157 4*. 4^^.
13. What cost 97cwt. 151b of tobacco, at ^^3 17*. \0d.
per cwt. ? Ans. £378 3d.
14. What cost 37cwt. 2qrs. 131t» of sugar, at i:4 14*.
6d. per cwi.? Ans. .£177 14«. 8jci.
15. What cost 15cwt. Iqr. lOlb of sugar, at c£3 14*.
6d. per cwt. ? Ans. £ 57 2s. 9d.
16. What cost 172cwt. 3qrs. 121b of madder, at .£4
15*. 4d. per cwt. ? Ans. £823 19«. O^d.
17. What cost 53cwt. I7tb of soap, at £3 1 1*. 6d. per
cwt.? Ans. £190 4d.
18. What cost 45tons 17cwt. 2qrs. of iron, at ^7
18«. 4d. per ton? Ans. .£363 3*. 6id.
nuBvcTzosr.
Q. What is Reduction ?
A. Reduction is the changing or reducing monies,
weights, and measures, &c. out of one denomination into
other numbers of another denomination, but equal to the
same in value.
Q. How are all great names brought into small ?
A. Multiply by so many of the less as make one of the
greater.
Q. How are all small names brought into great?
A. Divide by so many of the less as make one of the
greater. *
Money.
2 Farthings make 1 Halfpenny.
4 Farthings make 1 Pei\ny.
,.^. 12 Pence make 1 Shilling. ^
^^ 20 Shillings make 1 Pound.
Pounds multiplied by 20, are shillings ;
Shillings multiplied by 12, are pence ;
Pence multiplied by 4, are farthings ;
^'€ Fence multiplied by 2, are halfpence.
I i
•*«
28
REDUCTION.
Farthings divided by 2, are halfpence ;
Farthings divided by 4, are pence ;
Pence divided by 12, are shillings;
Shillings divided by 20, are pounds.
^Examples*
-^
1. In <£21 109. 6|d, how
many shillings, pence, and
farthings ?
£ 8. d,
21 10 6^
Multiply by 20 and add in
the 10a.
Shillings=430
Multiply by 12 and add in
the 6d.
Pence=5166
Multiply by 4 and add in
thejd.
rartblnci = 20666
2. In 20666 farthings,
how many pence, shillings,
and pounds t
4) 20666 farthings
12) 5166} pence
20) 430 6| shillings.
JS21 10 6J
Proof of the first.
£ s, d,
A. 5166d[.430«.21106}.
A. 430«. 5166(1. 20666qrs.
3. In £8, how many shillings ? Ans. 160«.
4. In 160 shillings, how many pounds? Ans. BQ.
5. In i^ 12, how noany farthings? Ans. 11520far.
6. In 11520 farthings, how many pounds ? Ans. iS12.
7. \n£\l ba, S^d, how many farthings ?
Ans. 16573.
8. In 16573 farthings, how many pounds ?
Ans. £11 bs^ Z\d,
9. In £36 7». 9d., how many pence ?
Ans. 8733 pence.
10. In 8733({., how many pounds ? Ans. jf 36 7«. Ocf.
11. £^lb 175. lOjd., how many farthings ? ^
Ans. 360859.
12. In 360S59 iarthings, how many pounds ?
Ans. iS376 17«. 10j<;.
¥«
\
^w Troy weight.
29
Id. In 100 crowns of 5«. eachthow many farthingst
Am. 24000 far.
14. In 36 guineas of 21«. each, how many pence ?
Ana. 9072d.
15. In 9072d., how many guineas of 2l9. each?
Ans. 36 guineas.
•■■-%^M ■ .m'nl
VaOT WBXOBT.
24 Grains make 1 Pennyweight.
20 Pennyweights make 1 Ounce.
12 Ounces make 1 Pound.
Pounds troy multiplied by 12, are ounces.
Ounces „ multiplied by 20, are pennyweights.
Pennyweights multiplied by 24, are grains.
j
Grains divided by 24, are pennyweights.
Pennyweights divided by 20, are ounces.
Ounces divided by 12, are pounds troy.
^*v
Examples.
1. In lib troy, how many
grains ?
lib.
12
12 ounces.
20
240 pennyweights.
24
960
480
Ans. 5760 grains.
2. In 5760 grains how
many pounds troy 1
24) 6760 (grains.
48
I 240
96
96
20) 240 pennyweights.
12) 12 ounces.
Ans. 1 pound troy.
3*
<U I.
II
f
■M
t
I
(
h
\
l«
AVOIEDUPQIS WEIGHT.
do
9. Id %Xtt§. how inanj grams t Am. 1 1520 grv.
, 4. Reduce 11520 grains into pounds? Ans. 2ftȤ>
5. Uow many grains are there in STfts. t
Ans. 213120 grs.
6. In 213120 grains, how many lbs. 7 Ans. 37tts.
7. lo 484% lloz. ITdwts. 23 grs., bow many
grains? Ans. 2793561 grs.
8. Reduce 2793551 grs. into pounds ?
Ans. 4841b. lloz. ITdwts. 23 grs.
AVOniDUVOZS WBZOBV.
16 Drams make 1
16 Ounces make 1
14 Pounds make 1
2 Stone or 28ib make 1
4 Qrs. or 1121b make 1
20 Hundred wt. make 1
Ounce.
Pound.
Stone.
Quarter.
Hundred wt.
Ton.
Tons multiplied by 20, are hundreds.
Hundreds multiplied by 4, are quarters.
Quarters multiplied by 28, are [jounds.
Pound multiplied by 16, are ounces.
Ounces multiplied by 16, are drams.
Drams divided by 16, are ounces.
Ounces divided by 16, are pounds.
Pounds divided by 28, are quarters.
Quarters divided by 4, are hundreds.
Hundreds divided by 20» are tons»
I
I
AVOIRDUPOIS WEIGHT.
81
Examplet*
1. In 1 tODf how manj
dnunft
1 ton.
SO
20 hundreds.
4
80 quarters.
38
' «40
160
2240 pounds.
16
13440
2240
35840
le
215040
35840
Ans. JS573440 drams.
a. Reduce 573440 drams
into tons.
16) 673440 drams.
48
35840
93
80
134
128 ,
64
64
16) 35840 ounces.
28) 2240 pounds;
20)
Ans.
80 quarters.
20 hundreds.
1 ton.
3. In 15 tons how many pounds ? Ans. 33600]bs.
4. Reduce 3d6001bs. into tons. Ans. 15 tons.
5. In 27cwt. 2qrs. 12Ib» how many &s. 1
Ans. 3092ibs.
6. Reduce 3092ibs. into cwts.
Ans. 37cwt. 2qrs. 12ib.
7. In 3 qrs. 14!b» how many ounces ?
Ans. 1568 ounces.
1
m
82
I't CLOTH MEASURe.
8. Reduce 1568 ounces into quarters.
Afiii. Sqrs. 14tbs.
9. In 35 tons, 17cwt. Iqr. 23ib. 7oz. ISdrs., how
many drams ? Ans. 20571005drs.
10. Reduce 20571005drs. into tons.
Ans. 25ts. 17cwt. Iqr. 23Ib. 7oz. iSdrs.
o&OTB miAsraB.
V
li.
2| Inches make 1 Nail.
4 Nails make 1 Quarter of a yard.
3 Quarters make 1 Flemish ell.
4 Quarters make 1 Yard.
5 Quarters make 1 English ell. J
6 Quarters make 1 French ell.
Yards multiplied by 4, are quarters.
t., Quarters multiplied by 4, are nails.
Nails divided by 4, are quarters.
Quarters divided by 4, are yards.
,..•... .1.,
Exainplesi
1. In 1 yard, how many
nails ?
1 yard.
4
4 quarters.
4
.^'^•''m
•- Ans. 16 nalL
2. Reduce 16 nails into
yards ?
Ans.
4) 16 nails.
4) 4 quarters.
1 yard.
\X i^ .*.f$. *-: < , »i
•5;V; ,"^^1 .\-^V S (!)
LONG MEASURE.
3. In 87 yds.f how many imilt ? Ans. 592 naili.
4. How many yds. are in 592 aails ? Ans. 37 yds.
5. Reduce 15 yds. 3qrs. 1 n. to nails. ? Ans. 253 nails.
0. How many yds. aro there in 253 nails ?
Ans. 15yds. Sqrs. Inl.
7. In 73 ells Flemish, how many qrs. ? Ans. 219qrs.
8. In 73 ells English, how many qrs. ? Ans. 36iSqrs.
9. In 73 ells French, how many qrs. ? Ans. 438qr8.
10. Reduce 352 nails into ells English ?
Ans. 17 ells, Sqrs.
m
XiONa
MSASvaa.
12 Lines
make I Inch.
12 Inches
make 1 Foot.
3 Feet
make 1 Yard.
5| Yards
make I Pole or Rod
40 Poles
make 1 Furlong.
8 Furlong
9 make 1 Mile*
3 Miles
make 1 League.
69^ Miles
make 1 Degree.
.'■i'
I
m
Leagues multiplied by 3, are miles— miles multiplied
by 8, are furlongs — furlongs multiplied by 40, are poles —
poles multiplied by 5J, are yards — yards multiplied by 3,
are feet — feet multiplied by 12, are inches.
Inches divided by 12, are feet — feet divided by 3, are
yards — ^half-yards divided by 11, are poles — poles divided
by 40, are furlongs — furlongs divided by 8, are miles-
miles divided by 3, are leagues.
:;|f
'1#
> i
•f I
:<%/" .d^k'
.'tl I
34
LONG MEASURE.
Examples.
1. In 1 mile, how many
inches ?
1
. 8
8 furlongs.
40
J of=320 poles.
1600
160
1760 yards.
3
5280 feet.
12
Ans. 63360 inches.
2. In 63360 inches, how
many miles?
12) 63360 inches.
3)
6280 feet.
5J
2
1760 yards.
2
11)
3520
40)
320 poles.
■ 8)
8 furlongs.
Ans.
1 mile.
3. In 273 miles, how many inches ?
Ans. 17297280 inches.
4. Reduce 17297280 inches, how many miles ?
Ans. 273 miles.
5. Reduce 5 m. 6 fur. 3 yds. into inches ?
Ans. 364428 inches.
6. In 364428 inches, how many miles ?
Ans. 5m. 6f. 3yds.
7. Reduce 2m. If. 8pls. 3yrds. 2inch. into inches ?
Ans. 136334 inches.
8. In 136334 inc\ i, how many miles ?
Ans. 2m. If. 8 p. 3yds. 2 inch.
LAND MEASURE.
i LAND BIBASUllB.
35
144 Square inches make 1 Square Foot.
9 Square feet make 1 Square Yard.
30^ Square yards make 1 Sq. Pole or Perch.
40 Poles make 1 Rood.
4 Roods, or 10 chains make 1 Acre.
Acres multiplied by 4, are roods — roods multiplied by
40, are perches.
Perches divided by 40, are roods — roods divided by 4,
are acres.
^ V examples*
1. In 1 acre, how many
perches ?
1 acre.
4
;ofe
4 roods.
40
Ans. 160 perches.
2. In 160 perches, how
many acres?
40) 160 perches.
4) 4 roods.
Ans. 1 acre.
8. In 15 acres, how many poles or perches ?
Ans. 2400 poles.
4. How many acres are there in 2400 poles ? .
Ana* 15 acres.
5. Reduce 27&. Ir. 32p. into pole^.
Ans. 4392 poles
6. Reduce 4392 poles into acres. Ans. 27a. Ir. 32p
LZanD XMBASVMI.
2 Glasses make 1 Gill. .
4 Gills make 1 Pint.
2 Pints make 1 Quart
4 Quarts make 1 Gallon.
%
Ki
'H
I
^
I
'.li |,
>if
'4 'i
n
it
At
(
36 LIQUID MEASURE.
42 Gallons make 1 Tierce.
63 Gallons make 1 Hogshead.
126 Gallons make 1 Pipe. ;^. ,
252 Gallons make 1 Tun.
Tuns multiplied by 4, are hogsheads — tuns multiplied
by 2, are pipes or butts — pipes multiplied by 2, are hogs-
heads — ^hogsheads multiplied by 63, are gallons — gallons
multiplied by 4, are quarts— quarts multiplied by 2, are
pints — pints multiplied by 4, are gills.
Gills divided by 4, are pints — pints divided by 2, are
quarts— quarts divided by 4, are gallons — gallons divided
by 63, are hogsheads — hogsheads divided by 2, are
pipes — hogsheads divided by 4, are tuns.
XSxamples*
r
1. In 1 tun, how many
glasses ?
1 tun.
4
4 hogsheads.
262 gallons.
4
1008 quarts.
9
2016 pints.
4
8064 giUs.
2
Ans. 16128 glaise^.
I
2. In 16128 glasses, how
many tuns ?
.Xi':.
2) 16128 glasses.
4) 8064 gills.
2) 2016 pinti.
4) 1008 quarts.
68)
4)
Ans.
hi:'-'
252 gallons.
252 (4
4
1 tUD.
it
DRY MEASURE.
87
3. In 19 hogsheads, how many pints ?
Ans. 9576 pints.
4. How many hogsheads are there in 9576 pints ?
Ans. 19hhd.
5. Reduce 13 1. 1 p. Ihhd. 17 gal. 5pts. into pints.
Ans. 27S61 pints.
€. Reduce 27861 pints into tuns.
Ans^ 13t. Ip. Ihhd. 17gal. 5pt8.
;5vaOft^
,1*?*^ Us--
av* '
2 Pints make 1 Quart.
4 Quarts make 1 Gallon.
2 Gallons make 1 Peck.
4 Pecks make 1 Bushel.
8 Bushels make 1 Quarter.
5 Quarters make 1 Load.
2 Loads make 1 Last.
•1 .3
Lasts multiplied by 80, are bushels-^bushels multiplied
by 4, are pecks — pecks divided by 4, are bushels — bush-
els divided by 80, are lasts.
Examples.
1. tni 1 fast,' how many
pecks?
1 last.
2. In 320 pecker, how
many lasts?
'XIR
-^ ' - -
- 80
4) 320 pecks.
Ans.
80 bushels.
320 pecks.
80) 80 bushels.
Ans. 1 last.
3. In 128 bushels, how many pecks ?
Ans. 512 pecks.
.1
'-"4
h
'h
i
I'd
^1
I'
I
n
*ik
Tl
TIIIE.
4. In 612 pecki, how many bushels ?
Ans. 128 bushels.
5. In 20 lasts,) 3 bush. 3 pecks, how many pecks ?
Ans. 6415 pecks.
6. Reduce 6415 pecks into lasts?
Ans. 201. 3 b. 3pks.
Villi a.
1 Minute.
1 Hour.
1 Day.
1 Week.
1 Lunar month.
60 Seconds . . . make
60 Minutes . • make
24 Hours . . . make
7 Days . . make
4 Weeks . • make
18 Lunar months, \
12 Calendar months, or > make 1 Common year.
365 Days, . • . j
365d. 5h. 48m. 48s. . make
365d. 6h. . , • • make
366 Days . • . make
1 Solar year.
1 Julian year.
1 Leap year.
Tears multiplied by 365^, are days — days multiplied by
24, are hours — ^hours multiplied by 60, are minutes —
minutes multiplied by 60, are seconds — seconds divided
by 60, are minutes — minutes divided by 60, are hours —
hours divided by 24, are days — days divided by 365J,
are years.
i f
>'^J.\,h
' (ii.
Y
I
.^?lA.
.x.f...^ ■^mmm^-l ,• ' ;|i'?:ftt "ifii lu S.
TIME.
Exampleg.
1. In 1 year, how many 2. In 31557600 secondfl,
seconds ?
V
•v.
1 year.
365J
365} days.
24
1460
730
6 is the} of 24.
8766 hours.
60
625960 minutes.
As. 31657600 seconds.
how many years ?
60) 31557600 sec.
60) 525960 min.
24) 8766 hn.
365}) 365} days.
4 4
1461)
Ans.
1461
1461
(I
1 year.
3. In 13 years, how many days ? Ans. 4748} days.
4. Reduce 4748} days into years. Ans. 13 years.
5. In 28 years, how many hours? Ans. 245448 hours.
6. Reduce 245448 hours into years. Ans. 28 years.
7. In lOyrs. 26days. 12h., how many minutes?
' ^^ Ans. 5297760 min.
8. In5297760minutes, how many years?
Ans. lOyrs. 26ds. 12fas.
9. How many days since the birth of Christ, this year
being 1832? Ans. 669138 days.
10. How many years are there in 669138 days?
Ans. 1832 years.
NoTi.— In quettiont which are perfonatd br Multiplication and
DiTision, the operatioa mar often be abridf ed» by nmpljr adding or
lubtractinf a part of the given number^uiin,
To reduce sterling into cunrtncy, multiply by 60, imI
diride by 64r-»or add ^* ^? |t»i* j^uwrjt'n tnn bk ^m-
u
i3 f>
ft
•8?
i
*f:'
u
I
40
THE RULE OP THREE DIRECT.
To reduce currency into Bterling* multipl/ by 54, and
divide by 60 — or deduct •j'y.
To reduce guineas into pounds, add ■^^,
To reduce pounds into guineas, deduct ^\.
To reduce ells English into yards, add ^, '-^-'i^*.
To reduce yards into ells English, deduct j, &c. &e«
!Examples*
1. In ^9 ster. how many
pounds currency X
I) i£9 sterling.
f,% 'i
h
Ans. ^10 currency.
2. In ;f 10 currency, how
many pounds sterling?
y^^) £10 currency.
1
Ans. £9 sterling.
3. In i£l8 ster. how many pounds currency?
Ans. .£20 curr.
4. Reduce <£20 curr. into sterl. Ans. £ 18 sterl.
5. How much currency must be paid for an English
bill of ^100 ? Ans. iClll 29. ^^dL
6. How much sterl. is equal to £11 1 2^. 2|d. curr. ?
^^ Ans. £100 sterl.
*7. How much must be paid in Quebec, to receive in
London <£18Q? Ans. £200.
8. How much sterl. must be paid in London to receive
in Quebec £200 ? Ans. jf 180.
9. In 20 guineas how many pounds? Ans. £21.
10. Reduce £21 into guineas. Ans. 20gs.
11. In 20 ells English, how many yards ?
Ans. 25yard8.
1.3. ReduCQ 25 yards into ells English. Ans. ^OellSf
^^•i«t ,,"
VBB ILir&B or TBBJBII 9X1L1ICT.
H-S!i«f
'^ Q. What is taught by the Rule of Threat '***^""'
A. The Rule of Three teaches, by three numbers given
to find a fba^th, which shall have the same proportion to
the third, as the second has to the first. ^^ n\ m*\mf
n
THK RULB or TRRIE DIRECT.
41
Q. WbMi if the proportiott Mud to be ^reet ?
A* Direct proportion require! iho fourth term to he
grMUwr than the itcimd, when the third ib greater than the
firat ; or the fourth to be leu than the secondi when the
third is lesa than the firat
Role*— First state the question ; that is, place the
numbers in such order, that the first and third be of one
kind, and the second the same as the number required :
then bring the first and third numbers into one name, and
the second into the lowest term mentioned. Multiply the
second and third numbers together, and divide the pro-
duct by the first; the quotient will be the answer to the
question in the same denomination you left the second
number in*
.■*»*rp »y,X ;.
£xamples«
J. 1. If one ib of sugar cost 4|d., what will 54ib cost t
.H ^i .1^.1/. . If 1 -41 • 64
18
l4 -i^k
• .;«5«» I
b2Xr>*'|*
^3 v^f V farthmgs
. K' Or 7,: - ^ *
IS 432
54
• "*< •
4) 972 farthings.
"J
12) 243 pence.
.4' }'- ' ^'' ■
. i
2,0) 2,0 3
. Ans.
£10 3
IT <* '
•4
S
mi
■i' >
u
1 •
m
^ 2. If 4 yards of doth cost 3«., what will 24ydfl. cost?
Ana. 18*.
, 8. If 24 yds. of cloth cost 18»., what will 4yd8. cost?
« Ana. 3#.
4»
't.4
42
THE RULE OF THREE DIRECT.
I
i!
4. If I buy 4 yds. of oloth for 3«., how many yardi will
IBt, buvl Ana. 24yd8.
5. If 24 ydi. cost 18*., how many will I get for da, 1
Ans. 4yds.
6. If 1 yard cost 159. 6c2., what will 32yds. cost ?
Ans. iS24 16«.
( 1 7. If 82yds. cost jf 24 16«., what is the value of 1 yrd. ?
Ans. 15«. 6d.
8. What will aScwt. Sqrs. 141b of tobacco come to» at
15^(f.perib? Ans. £187 Z8,dd.
9. Bought 27|yd8. of muslin, at 6a. 9}d. per yrd.,
what is the amount of the whole ? Ans. £9 5«. Of^tL . J.
10. Bought 17cwt. Iqr. 141b of iron, at Sid per lb.
what was t^ price of the whole ? Ans. x26 7«. 0^,
11. If coffee is sold for b\d, 'per ounce, what will be
the price of 2cwt. 1 Ans. £82 2a, 8d,
12. How many yards of cloth may be bought for ;£21
lit. Hd„ when SJyds. cost £2 14«. 3d. ?
Ans. 27yds. 3qrs. l-^jvl.
13. If lowt. of Cheshire cheese cost £1 149. Bd., what
roust I give for S^ft,'i Ans. 1«. Id.
14. If a gentleman's income be £ 500 a year, and he
spend 199. id. per day, what is his annual saving ?
Ans. £ 147 39. 4d.
15. If 504 Fleir^jsh ells, 2qrs. cost iS283 179. 6d.,
what is the cost of 14yds. ? Ans. £ 10 109.
16. If 1 English ell, 2qrs. cost 49. 7(2., what will
39}yds. cost at the same rate ? Ans. £5 ds. 5^d, . 4.
17. If 27yds of Holland cost iS5 129. 6d., how many
English ells can I buy for iS 100 ? Ans. 384 ells.
18. A draper bought 420yds. of broad cloth, at the
rate of 149. 10|d. per ell English, what was the whole
amount ? Ans. jS250 59.
19. What must be paid fbr 7 casks of prunes, each
weighing 2cwt. Iqr. I4]b» at £2 199. 8d. per cwt. ?
Ans. Jg49 1l9. Uj^d.
20. At £\9 199. ll|d. the ton, what will 19 tons»
19c wt. 3qrs. 27\fb come to»at that rate ?
,f. . Ans. ^99 19». 6|f f f^.
It;
u
(^
THE RULE or THEEE INVRftSE.
43
^
„_I
,ai.
T ^. _ 4 (■ a. < k.^ tf .^M-WlK ^-«^^^i '
'*iu^ t I I
* VBB BiVAS or TB&mi xanramsa.
.♦^
Jl^ Q. TV bat 18 Inverse proportion ?
A. Inverse proportion requires the fourth term to be
Ui» than the second^ when the third is greater than the
first ; or the fourth to be greater than the second, when
the third is less than the first.
Rule* — State the question, and reduce the terms as
in the rule of three dirwt ; then multiply the first and
second terms together, and divide their product by the
third; the quotient witl be the answer* aa in the last
rule.
Examples*
1. If 8 men can do a piece of work in 12 days, in bow
many days can 16 men do the same I
m. d. m.
If 8 : 12 : : 16
;>;r»^ r^flt j — ■ i a'.
m*'> < ' 16) 96 (6 day». Ans. 'mo
W. 96
•J
■\'f
i
i
2. If 54 men can build a hou9G in 90 days, how many
men can do the same in 50 days ? Ans. 97} men.
3. How many sovereigns, of 208, each, are equivalent
to 240 pieces of 129. each ? Ans. 144.
' 4. How many yards of stuff three quarters wide, are
equal in measure to 30yds. of f^ quarters wide ?
Ans. 50yds.^
5. If I lend a friend i&200 for 12 months, how long
ought he to lend me £150 ? Ans. 16 months.
6. If for 24«. I have 12001b carried 36 miles, what
weight can I have carried 24 miles for the same money ?
Ans. 18001b.
7. If I have a right to keep 45 sheep on a common 20
days, how long may I keep 50 upon it ? Ans. 18 days..
-4 DOUBLE EVLE OF TBEEE. r
8. If 1000 foldMra have proviiions for 8 months, how
maax mutt be eent Awa/» uai the prevkioM may last
Smooths? Ans. 400.
9. A courier makes a journey^in 34 davs» by traveling
12 hours a day : how roan^ days will he be in gofaig the
•ame joumey* traveling 16 hours a day ? Ans. 18 da vs.
10. How much will line a cloke* which is made of 4*
yards of plush* 7 quarters wide, the stuff for the lining
being but 3 quarters wide ? Ans. 9| yards..
TMM BOVBU mu&a or
Q. What is the Double Rule of Three ?
A. The Double Rule of Three has five terms given»
three of supposition and two of demand, to find a aixth,
in the same proportion with the terms of demand, as that
of the terms of supposition. It is performed by two stat-
ings of the single rule of three.
Rule* — Put the terms of demand one under another
in the third place ; the terms of supposition in the same
order in the firtt place, except that which is of the same
kind as the term required, which must be in the second
place. Examine the statings separately, using the middle
term in each, to know if the proportion is direct or tn-
ver«e. When the stating is direct, mark ihe first term with
an asterisk : when inverse, mark the third term*, then
multiply the marked terms together for a divisor, and
multiply oU the other terras for a dividend; divide, and
the quoti«;:t will be the answes..
If
J/ \
s'^?s,i&
'I Sf*''
DOUBLE RULE OP THREE.
45
Examples.
1. If 14 horses eat 56 bushels of oats in 16 days, how
many bushels will serve 20 horses 24 days ?
* Examine the sttUngs, thee:
1ft.
If 14 honei eat 56 bushels, 20
horses, being more, will eat
more ; the stating is, there*
fore, Direct.
" 2nd.
If 16 days consume 56 bushels,
24 days, being more, will
consume more; the slating
is, therefore, Direct
14
16
84
14
66
20
1120
24
4480
2240
26880(1206. Ans.
224
J^m^
448
448
2. If 8 men in 14 days can mow 112 acres of grass*
how many men can mow 2000 acres in 10 days ?
Ans. 200 men.
3. If iSlOO in 12 months gain £6 interest, how much
will £75 gain in 9 months ? Ans. £3 7«. 6d,
,4. If £100 in 12 months gain £6 interest, what prin-
cipal will gain £ 3 7«. 6d, in 9 months ? Ans. i)75.
$. If JS 100 gain £ 6 interest in 12 months, in what
time will £ 75 gain £ 3 7«. 6d» interest ? Ans. 9 months.
6. If a carrier charges £ 2 28. for the carriage of
3cwt., 150 miles ; how much ought he to charge for the
carriage of7cwt. 3qrs. 141b., 50 miles?
Ans. £1 16«. 9d.
7. If 40 acres of grass be mown by 8 men in 7 days,
how many acres can be mown uy 24 men in 28 days ?
Abb. 480.
i
111
'* ' r
*■ A
•J
'i) K\
M
46
SIMPLE INTBRE8T.
8. If £2 will pay 8 men for 6 days* work, how much
will pay 32 men for 24 dayi' work ? Ana. £9S St.
9. If I pay i6 14 10«. for the carriage of 60cwt. 20
miles, what weight can I have carried 30 miles for £6 8«.
9d, ? Ans. 15 cwt.
10. If 144 threepenny loaves serve 18 men for 6 days,
how many fourpenny loaves will serve 21 men for 9 days ?
Ans. 189.
■j^.
,j
SZMV&a ZSVTB&BST.
Q. What is Simple Interest ?
A. Simple Interest is the premium allowed for the
loan of money for a given time.
The money lent, is called . . The Principal.
The premium for the loan of £ 100 J j^ ^ ^^
for 1 year, is called . . j '^
The Principal and Interest together, \ ja ^,„umni
is called ... ] *
Rule* — To find the interest of any sum of money,
for any given time ; multiply the principal by the rate per
cent, and the product divided by 100 will give the interest
for i year ; multiply the interest for 1 year by the number
of years given, to which add the interest of tbe given
months and days, if any, which may be found by tdcing
the aliquot parts of a year's interest, and their sum will be
the interest required.
vif
^h
m,:-
8IMPLB INTESMT.
Examplef.
1. What if the interest of £384 2«. 6d. for 6 yn. 7 m.
15 dayi, at the rate of 5 per cent, per annum t
£ t, d,
384 2 6 Principal.
6
1,00)
iB 19,20 12
20
6
f. 4,12
d. 1,50
4
'
gr. 2,00
»
iB19 4
6
;.ui^
«• r,-
• ■ -
96 74 interest for 5 jrean.
6 mo. I year ss 9 1 j3 ^ Interest for 6 montha.
lmo.Iof6m.^ 1 12 Interest for 1 „
15dys.| oflm.ss 16 Interest for 15 days.
Ana. Jf 108 8} Intst. for yiB.5 7 15
o
*;y.
m
2. What ii tiie interest of ie375 for 1 year, at 6 per
cent per annum 7 Ans. i618 1 St.
3. What is the interest of £945 lOs. for 1 year, at £4
per cent, per Annum ? Ans. £ 37 16». 4}<ii
4. What is Uto interest of £ 547 16«. at £ 5 per cent
per annum, for 3 yeara ? Ans. £82 3«. Bd,
f. Whatisthamterestofie857Sf,ld. at£4percent
per unum, for 1 year and 9 nontha ? Ans. «18 l|(l.
■m
48
COMPOUND INTEREST.
i
6. What is the interest of £ 479 5«. for 5} years, at £b
per cent, per annum f Ans. jS 125 169. 0}d.
7. What is the amount of £ 576 2«. 7d. in 7^ years, at
£A\ per cent, par annum ? Ans. j&764 1». SJcf.
8. What is the interest of £730 for 7yrs. 7 mo. 15
days, at 4 per cent, per annum ? Ans.
•Mk
ooacpovm zotb&bsv.
Q. What is Compound Interest ?
A. Compound Interest is that which arises from both
the principal and interest ; ^hat is, when the interest of
money, having become due, and not being paid, is added
to the principal, and the subsequent interest is computed
on the amount
Rule*— ^Compute the first year's interest, which add
to the principal ; then find the interest of that amount,
which add as before, and so on for the number of years.
Subtract the given sum from the last amouniy and the re-
mainder will be the compound interest. ^
examples*
1. What is the Compound Interest of ^^720, for 4
years, at 5 per cent, per annum ?
of 100 £ 8, d,
36
5 is tV) 720
1st years pnncip.
1st year's interest.
ii:
f
Vtr) 766
37 16
2nd year's princip.
2nd year's interest.
♦»»{ B Hi M'y% ^'^) 793 16
*iii f<l%\m;. 39 13
3rd year's princip.
9} 3rd year's interest.
-M
833 9
41 13
876
720
9J 4th year's princip.
6} 4th year's interest.
3 J the whole Amount. "•'
given sum subtracted.
*m:)
•TKI
Am. 166 3 3^ Compound Interest reqd
VULGAI^ FRACTIONS. ^^
49
2. What is the compound interest ofiSSOO, forborn
3 jears, at £5 per cent, per annum ? Ans. ;f78 16«. 3d.
3. What is the amount of £400 in 3} years, at £5 per
cent per annum, compound interest ?
Ans. £474 Us, 6|(/.
4. What will j€650 amount to in 5 years, at 5 per cent,
per annum, compound interest ? Ans. iS829 lU. 7Jd.
5. What is the amount of jf550 10«. for 3| years, at
£6 per cent, per annum, compound interest ?
Ans. £ 675 68, 6d,
6. What is the compound interest of ^674 for 4 years
and 9 months, at ^6 per cent, per annum ?
Ans. £243 188. Sd,
•Tl.i t»l
'.MC
m:,:
*j*Aa:;i
tnriiaAB. ruAcvzonrs.
Q. What is a fraction ?
A. A fraction is a part of a thing, and is expressed or
written with two numbers, with a line between them ; the
upper number is called the numerator, and the lower Uie
denominator— thus, ^^ geToX^ir.
Q. What do the numerator and denominator show ?
A. The denominator shows into how many equal parts
the whole thing is divided ; and the numerator expresses
how many of tiiese parts the fraction contains ; thus the
fraction -^^ of a shilhng, shows the shilling to be divided
into 12 parts, and die numerator 1, e:ipresse8 1 of these
parts, or 1 penny, ^». signifies 2d. ; j%^z=dd, ; y«j=z:4(/.,
and so on for any other quantity ; thus A of a foot signi-
fies 2 inches, 1^=3 inch. -^=^4 inch., &c.
Q. What is a proper fraction?
A. A proper fraction, is one whose numerator is /m«
than the denominator . as }, }, |, |^, &c.
Q. What is an improper fraction ? , , , , r ,
^. ^ A. An improper fraction, is one whose numerator is
either equal to, cr greater than ita denominator : as |, |, |,
i
'If
i
50
ADDITION OF FRACTIONS.
Q. What IB a mixed number T *
A. A mixed mimber is composed of a whole number
and a fraction : as If, 17}, 8f]-, &c.
.i'
ABBiTzov or raAonoxrs.
Rule 1 • — ^When the fractions have the same com-
mon denominator, add their numerators toffether, and
place the sum over the denominator, if less 5ian it ; but
if it be grtater, divide the sum by the denominator, set
down the remainder, and carry the quotient to the whole
numbers, if any.
Note. — ^If the remaining fraction can be divided by any number
that can be discovered by inspection, divide it, and it wiu give an
tquivaUni flraction in lower terms.
' -
! ;; Examples.
1. Add UT^yyds. 13t»y,
Yds.
14^.
"A-
24A
17tVi 24^ together. ' »i >f
* * 3 f Numerators.
. r Ans. yds. 69J
10) 15 (li yanb. ' ;;'
10 '
•
i. Add 14), 16}, 20} yards together.
Ans. 5I|yds.
, 3. Add 18{, 12|, 19j, 20} yds. together.
Atas. 71yds.
4. Add 142^, 6^, 20|t, 81|{ yds. togetfier. "
Ans. 250{yds.
AM>ITI0N OF FRACTIONS*
51
5. Add together 27^, 18^, ^\, 187 yds. together.
Ans. 252f|yd8.
6* Add 17^, 87/r* 146|^ 50}|, and d|f
Ans. 305}fyds.
1 /it*
Rule 2* — ^When the fractions have not a common
denominator, arrange the denominators in a line, and
divide any two or more of them by any common divisor,
placing the quotients and the undivided numbers below ;
proceed with them in the same manner, and repeat the
process till there remain not any two numbers common-
surable; then multiply the undivided numbers by the
quotients and divisors, and the last product will be the
least common denominator — then divide this common
denominator by each of the denominators of the given
fraction, and multiply the quotient by the numerator, net-
ting down the ^^luct opposite each fraction — add Uiese
products toge.r > ind proceed as in the last examples.
.-j-j --
Examples.
7. Add together 12^ yds. 14|, 15j, 18}, and 17tV
60 C. den.
jrds.
12i
(To (U>d tha laut Com. Donon.)
2) 2 . 3 . 6 . 4 . 10
••^kfS
3) 1 . 3 . 3 . 2 . 5
2
5
2
141 .
18| .
17iV
V 30 , ,
45
6
— Au. 77^1 yds. 60) 11 l(myds.
' i^i hiv
10
3
30
2
60
■h
a-rii
3) H-\h
C.p. 60 ^mi
ji^.
, i
ft .d
■:t :
^
i
II
I
t
1
i
»ii<
\^Q
52
SUBTRACTION OF 7RACTT0NS.
8. Add 40}, 27|, 34|, 48^, and 39) yaiils.
Ans. 185^ yd^.
9. Add 150^ lb. 139|, 162|, and ITOjlh. together.
Ans. 623,Vlb.
10. Add 16^, 1^ ^, 13j, 20j, 25^^, 30|, and ll^Ib.
Ans. 136;{lb.
11. Add 124|, lOlf, 79}, and 17 together.
Ans. '
12. Add 132}5, 507J, 384|. and 18,^.
Ans.
■J
h
sirBTiLjt.oTzoar or nuLoszoxni.
Rule* — Prepare the fractions the same as for Addi-
tion, when necessary ; then subtract the one numerator
from the other, and set the remainder over the common
denominator, for the difference of the fraction sought.
Note. When the numerator of the fractional part in the niMr».
htnd is greater than the other numerator, lubtract it from the com*
mon denominator, and add the remainder to the other numerci tqr— <
set down the fraction, and ^rry 1. —.,...-..
• ► <*,? « ■ »■ ■' *
examples*
1. From 18} yards of cloth take I4g yards,
yds. ""f K ,
From 18} . . . 4
Take 14| . . .6.^.4.
f r
Ans. 3| yds. }
2. What is the difference between 20} and 16) ?
Ans. 4|.
3. What is the difference between 37f and 29| t
Ans. 8).
4. From 26^ take 18). Ans. 8^.
6. If 59/t be taken from 102), what will remain ?
Ans. 42).
MULTIFUOATiON OF ffEAOflONS.
93
6. Lent £123^ and racttved £ a7)f wliat ia yet due ?
r iCAns. 96^.
' 7. Borrowed jS87^, and paid 84) ; what do I yet
owet Ana. jf2|f.
uvibTzvuoATzoir or mAonom.
" R^le* — Multiply all the numeratora together for a
nume) vtor, and all the denominatora together for a deno-
minator, which will give the product required.
Note.— If there be a mixed number, multiply the whole number
by the denominator of the fraction, adding in the numerator, and
place the sum over the denominator.
Examples. V f
1. Required the product of 3| and
a*
..I
10) 28 (2f Ans.
20 ^i'
^2.
3.
4.
5.
off.
6.
7.
8.
Ana. J.
Ans. /y.
Ans. j"^.
9.
Required tbo product of J and |,
Required the product of i^ and |.
Required the product of ^, |, and |f. ^^.
Required the continued product of ft 3^, 5, and |
Ana. 42.
What is the \ of £20? Ans. £6 J3«. 4</.
What is the \ off of } of & 20 ?Ana. jg 5.
What ia the f of a guinea of 2U. ? Ans. 189. 4)<*.
How much ia the I of I of-} of f of an apple ?
Ana. J of an apple.
6*
i
nn
M
54
^:«^0 DIVISION OF PRACTIONS; 'f^*
10. How much ii lh« } of 1 cwt weight ?
Aim. 3 qra. 31 lb.
U. Multiply 14|V feet by 8j feet? Ana. 127f f feet.
ess
mrv.dzov or nuLOSzovs.
Rule* — Prepare the given numbers, if they require
it, by the last rule ; then multiply the numerator of the
dividend by the denominator of the divisor for a numera-
tor ; and multiply the numerator of the divisor by the de-
nominator of the dividend for a denominator.
E3(ample8«
\» Divide f by ^.
ih
18) 75 (4iAn8.
72
4) ^
i 1
2. It is required to divide || by |.
3. It is required to divide j^ by j.
4. It is required to divide y by |.
5. It is required to divide f by y.
6. It is required to divide ^ by |.
7. It is required to divide ^ by |.
8. It is required to divide -j^ by 3.
9. It is required to divide | by 2.
10. It is required to divide 7^ by 9f .
11. It is required to divide f of i by
12. It is required to divide 5205} by
Ans. |.
Ans. ^,
Ans* l\,
Ans. -fj,
Ans. 4.
Ans. \^^
Ans. yV«
Ans. ^,,
Ans. iL
4of7|.
Ans. j^j,
I of 91.
Ami. 71)..
.ft-LjOJ) '**•■■ i'^' •! •fe«i»^
i-'?.
RULB OP THRU.
55
Role* — Reduce the whole* or mixed numbergr, if
any, to fractions — then state the question as in the liulie
of Three, in whole numbers i then having considered
whether Uie question is dirtct or inverut perform the
operation* according to the rule already given under the
head of each respectively.
£xample3.
1. IfS^i yards of cloth cost iS4|, what cost 18) yards?
"* yds. £
3i : 4) : : 18|
•:'m4'^
i
¥
¥
7
3
91
13
6
273
91
105
1183
2
jl*;^
.hhi^C-i ''■^■'^L
^•m-Kji^-^- _*- ^>
105) 2366 (^22 lOf. Sd. Ans.
210
266
210
56
20
105)
1120
105
70
12
106)
840
840
(10*.
(8d.
ill
56
DECIMAL rBACTIONS.
^ i. IfiofayaideoitiBftWhatwiUiVdrtjrinleMtT
Ana, 15ff.
3. Iff yrd. coat iS}, what will H T^- 908$ T
JUs, lit. S<f.
4« If } of a yaid cost 7|«., what will 101 yrd. cost ?
,, Ana. £4 199. lO^iT.
5. If } lb. cost !«., how much will f «. buy T
Aos. 1^ R>.
6. If 48 men can build a wall in 24| days, how many
men can do the same in 192 days ? Ans. 6^ men.
7. If } yrd. of Holland cost J6|, what will 12f ells
cost at the same rate ? Ans. £7 6|a. . |.
8. If 3} yards of cloth« that is 1| yard wide, be suffi-
cient to make a cloak, how much that is | of a yard wide,
will make another of the same size t Ans. 4^ yds.
9< If 12| yards of cloth cost 15«. 9d., what will 48 J
yards cost at the same rate ? Ans. ^3 9}</. . •^,
10. If 25^8, will pay for the carriage of 1 cwt. 145;|
miles, how far may 6| cwt. be carried for the same
money? Ans. 22^ miles.
11. If /^ of a cwt. cost JC14 4^., what is the value of
7} cwt. ? Ans. ;6118 69. Sd,
12. What quantity of shalloon that is | of a yard wide,
will line 7} yards of cloth, that is 1| yrd, wide t
< > Ans, 15 yds.
Q. What is a Decimal Fraction ?
A. A decimal fraction is that which has for its denomi-
nator 1, with as many ciphers annexed as the numerator
has places ; and it is usually expressed by setting down
the numerator only, with a point before it, on the lefl
blind. Thus, fy is .4« and f^^ is .24, and j^^^ is .074,
&c. &c.
;VM5,)- vfed
ADDITIOK OP DECIMALS.
57
ABBRZON 09 BBOZBKA&I.
Role* — Set down the numbers in such order that the
separating points may stand exactly under each other, then
add up the columns, and place the point in the sum directly
under the other points.
liXamples.
1 Add together 14.25+121.372+107.3804-26.
14 . 25
121 . 372
107 . 380
.26
Ans. 243 . 262
2. \Vhat is the sum of 276. +39.213+72014.9+417.
and 5032? Ans. 7779.113.
3. Add together 7530.+ 16.201+3 -0142+957.13.
Ans. 8506.3452.
4. Add together 1.5+85.07+121.321+23.17.
Ans. 181.081.
HI
i
SUBTmAOTZOBT or BBOZaiEALS.
' Rule* — Place the numbers under each other as in
addition ; then subtract and point off the decimals as in
the last rule.
Examples.
1. What is the difference between 91.73 and 3.138.
91 . 73 .
2 . 138 . 'titl
Ans. 89 . 592 the diflbrenee.
M
wA
■"(,
m
58
DIVISION OF DECIMALS.
2. Find the difference between 1.9185 and 2. 73.
Am. 0.8115.
8. Subtract 4.90142 from 214.81.
Ana. 209.90858.
4. Find the difference between 2714 and *916.
Ans. 2713.084.
wvxinvuojLVzoir or dboziba&b.
'•■ Rule* — Place the factors under each other, and mul-
, tiply them together. Then point off in the product just as
many places of decimals as there are decimals in both
factors. But if there be not so many figures in the pro-
duct, supply the defect by prefixing cyphers.
Examples.
. 1. Multiply 24.5 by 1.6.
24
1
5
6
147 .
245
L -^
*l^-^.^^'
An9f 39 . 20 the Product.
■ i. KS
*.- ««?
2. Multiply 79 . 347 by 23 . 15. Ans. 1836 . 88305.
8. Multiply . 63478 by . 8204. Ans. . 520773512,
4. Multiply. 385746 by. 004(54. Ans. .00178986144.
T
Hulc* — Divide as in the rule of division, and point
off in the quotient as many places for decimals, as the
decimal pltces in the dividend exceed those in the
divisor.
REDUCTION OF DECIMALS.
59
Sxamples*
1. Divide 34.80 by 1.6.
1.5) 34.80 (23.2 Ans.
30
48
45
"lo
J**
8. Divide 123.70536 by 54.25.
3. Divide 12 by .7854.
4. Divide 4195.6i3 by 100.
Ans. 2.2803.
Ans. 15.278.
Ans. 41.9568.
RUDVoTXoir or dbozhaui.
TO REDUCE A FRACTION TO A DECIMAL.
Rtlle* — Divide the numerator by the denominator
annexing cyphers to the numerator as far as necessaryi so
shall the quotient be the decimal required.
fiXamples.
1 . Reduce | to a decimal.
4) 100
.25 Ans.
.4.
* '. > ' ,.
XM A
,m 2. Reduce 1 to a decimal.
' 3. Reduce } to a decimal.
4 4. Reduce | to a decimal.
.^ 6. Reduce A to a decimal.
Ans. •6.*'
Ans. !75'.
Ans. •«5.
Ans.^lS.
0,
/
M
^iC.
00
REDUCTION or DECIMALS.
6. Reduce jfj to a decimal. Ana. .031360.
7. Reduce 9«. to the decimal of a pound. Ana. '46.
8. Reduce 9d. to the decimal of a shilling. Ana. "76.
9. Reduce J to the decimal of a penny. Ana. *26.
TO RBDCCE A DECIMAL TO ITS PROPER VALUE.
Rule. — Multiply the given decimal by the number of
parts of the next mferior denominationt cutting off the
decimals from the product ; then multiply the remainder
by the next inferior denomination ; thus proceedingt till
you have taken in the least known parts of an integer.
Examples.
1. What is the value of .8323 of a ;f 7
8323
20
9. 16.6460
12
■r ft i ->•■
v.. m. ..f . /, .r( , «^- '''•'''*20 Ana. 16». 7}d.
.*■ •ti'irj/i-i !
}008
•» 1
a. What is the value of *775 of a £ ? Ans. 15«. 6<f.
3. What is the value of *626 shillings 1 Ans. 7id,
4. What is the value of £-8635 ? Ans. 17«. 3-24d.
6. What is the value of ^ew lb. troy? .
Ans. 6oz. 12d. lV'}(44gr8.
6. What is the value of '6876 yds. ? Ans. 2 qrs. 3 nla.
7. What is the value of '6626 of a foot?
Ana. 6} inches.
8. What is the value of '626 of a cwt. ?
^;i '*/ Ans. 8qra. 141b.
DUODECIMALS.
1350.
•45.
•75.
•25.
.*.'
61
• Q. What VI Duodecimals ?
A. Duodecimals is a rule used by workmen aod arti-
ficers, in computing the contents of their work.
Rule* — Under the multiplicand write the correi-
pondinip terms of the multiplier ; multiply by the feet in
the multiplier, observing to carry one for every twelve,
from each lower denomination to the next superior ; in the
same manner multiply by the inches in the multibiier, set-
ting the result from each term one place farther to the
ri^ht ; proceed in like manner with the remaining 'leno-
mmator, and the sura of the products will be the total
product.
Examples.
1. Multiply 7 ft. 9 inch, by 3 ft. 6 inch.
. 'm, ' 7 9
3 5 t ii:u{'^ • i ,. k
23 3
3 10 6
ii
/ • ,
feet 27 1 6 Ans.
fl. in. ft. in.
ft. ir . ?;^.
2. Multiply 8 5 by 4 7 Ans. 38 11
3. Multiply 9 8 oy 7 6
4. Multiply 8 1 by 3 5
6. Multiply 7 6 by 5 9
6. Multiply 4 7 by 3 10
; 7. Multiply 76 7 by 9 *8
8. Multiply 97 8 by 8 9
9. Multiply 7ft. 5 in. 9pt. 7
by 3 ft. 5in. 3pt S
10. Multiply 10 ft. 4 in. 5pt. \
by 7ft. Sin. 6pt. )
„ 72 6 «
„ 27 7 6
,, 43 1 6
„ 17 6 10
„ 730 7 8
„ 854 7
i»
t*
25 8 6
f» m
2 8
79 11 6
I
62
INVOLUTION.
v.»>.
znvo&wxoifa
Q. What is Involution ?
A. Involution is the multiplying any number by itself,
and that product by the former multiplier ; and the pro-
ducts which aiise are called powers.
Table of the Squares and Cubes of the
nine digits.
Roots
1
1
2
3
4
5 6
•
7
8
9
Squares. . .
1
4
9
16
25
36
49
64
81
Cubes
1
8
27
64
•
125 216
343
512
729
• Examples.
1. What is the fifth power of 8 ?
*i|X'
4:
\
■P
■n,
M'^ " \-
8 Root or 1st power.
8
64 Square or 2nd power.
8
■"^ 612 Cube or 3rd power.
8
4090 Biquadrate or 4th power.
8
32768 Sursolid or 5th power.
2. lYhatis the square of .085?
3. What is the cube of 25.4 ?
4. What is the biquadrate of 1 .2 ?
Ans. .007225.
j^. 16387.064.
Ans. 2.0736.
Q
THE SQUARE EOOT.
Bvosunoir.
Q. What is Evolution? # >
A. Evolution is the method of finding the first powers
or iroots of any given numbers : and it is commonly called
extraction of the square* cube, biquadrate, sursolid,
roots, &c.
i- f
TBS SQVA&B ROOT.
Q. What is the extraction of the Square Root ?
A. To extract the square root, is to find out such a
number as being multiplied by itself, the product will be
equal to the given number.
Rule* — Divide the given number into periods of two
figures each, beginning at the right hand ; find the great-
est square in the first period on the lefl hand, and set its
root on the right hand of the given number, after the man-
ner of a quotient figure in division ; then subtract the
square thus found from the said period, and to the remain-
der annex the two figures of the next following period, for
a dividend — double the root above mentioned for a divisor,
and find how often it is contained in the said dividend,
exclusive of the right hand figure, and set that quotient
figure both in the quotient and divisor. Multiply the whole
augmented divisor by this last quotient figure, and sub-
tract the product from the said dividend, bringing down
to it the next period of the given numbers for a new
dividend, and proceed as before.
NoTB.— Whtv, thcfigaret of the whole number are ezhanited. If
there be a remainder, peiiodi of cipben may be uied at pleaaure to
continue the extraction { but tba figures produced in the quotient
will be deeimala*
iT^.^W
' V,
M
THE CUBE ROOT.
Examples.
1. What is the square root of 119025?
119025
(345 the root. Ans.
*'■'
,..,,.
9
64)
290
256
685)
■ 1 * k.
3425
3425
2.
3.
4.
5.
6.
7.
8.
9.
10.
What
What
What
What
What
What
What
What
What
is the square root of 106929 ?
is the square root of 22071204 ?
is the square root of 17.3056?
is the square root of 000729 ?
is the square root of 3 ? Ans.
is the square root of 5 ? Ans.
is the square root of 6 ?
is the square root of 10 ?
is the square root of 12 ?
Ans.
Ans.
Ans.
Ans.
Ans. 327.
Ans. 4698.
Ans. 4.16.
Ans. .027.
1.732050.
2.236068.
2.449489.
3.162277.
3.464101.
TBB CVBB ROOT.
Q. What is the extraction of the Cube Root ?
A. To extract the cube root» is to find out a number,
ivhich being multiplied into itself, and then into that pro-
duct, producetb the given number.
Rule* — Divide the given number into periods of
three figures each, beginning at units' place; find the
greatest cube in the first period, and subtract it there-
from ; put the root in the quotient, and bring down the
iiffures in the next period to the remainder, for a JRe-
solvend ; multiply the square of the root found by 300,
for a divisor, and annex to the root the number of times
which that is contained in the Reaolvend ; add 30 times
THE CUBE ROOT.
the preceding figure, or figures, multiplied by the last,
and the square of the last, to the divisor ; and multiply
the sum by the last for a Subtrahend : subtract it from
the Re9olvendi and repeat the process as far as necessary.
'1^
)ds of
id the
there-
vn the
a JRe-
300,
times
times
Examples.
1. What is the cube root of 99252847?
.15^
99252847
4'=64 (463
4^X800=4800) 35252 Resolvend.
j-fy*.^
720=4X30X6
36=6'*
4800
5556
6
33336 Subtrahend.
46''x300=634800) 1916847 Resolvend.
, ' 4140=46x30x3
9=3«
634800 ^'
-■*»
638949
3
1916847 Subtrahend.
/^
f "S-
U!i
r
i .1' ...
2. What is the cube root of 389017 ? Ans. 73.
3. What is the cube root of 673373097125 ?
Ans. 8765.
4. What is the cube root of 12-977875 ? Ans. 2.35.
6*
lip, I
'J
>•■
PRACTICAL GEOMETRY*
.fr-'tf'
t r
h
PBJionoA& OBOBsaTrnv.
Practical Geometry is a mechanical method of de»
scribing mathematical figures by means of a ruler and
compasses, or other instruments proper for the purpose.
1. A point is that which has no parts, or
dimensions; as A.
2. A line is length without breadth ; and its
bounds or extremes are points.
3. A right, or straight Une, is that which
lies evenly between its extreme points; A.
as A B.
4. A superficies is that which ,
has length and breadth only ;. and
its bounds or extremes are tines ;
as A BCD.
.A
.B
I
5. A plane, or plane superficies, is that which is every
where perfectly flat and even.
6. A body or solid, is that which has
length, breadth, and thickness, and its
bounds or extremes are superficies ; as
ABCD.
7. A plane recHlinetU
angle is the inclination or
opening of two right lines»
which meet in a point with-
out cutting each other; as
ABC.
8. One right line is said to
be perpendicular to another,
when the angles on each side of
it are equal. Thus A B is perp.
to C D. ,
%
#
PRACTICAL OEOHETRT«
67
9. A right angle is that which
is formed by two right lines, that
are perpendicular to each other;
as ABC.
B
10. An acute angle is that which is
less thaii e right angle ; as A B C.
11. An obtuse angle is that which
is greater than a right angle ; as
ABC.
12. A circle is a plane figure, formed
by the revolution of a right line about
one of its extremities, which remain ^1
fixed ; a&j ^
IS, The centre of a circle is the point 0, about which
it is described ; and th<3 circumference is the line or
boundary A B C A, by which it is contained.
"' 14. The radiu3 of a circle is a right
line drawn from the centre to the cir-
cumference ; as A.
'n
15. The diameter of a circle is a
right line passing through the centre,
and terminated on each side by the
circumference ; as A B.
;i^
^"^1
PRACTICAL OEOMBTRY.
,-y
16. An are of a circle is any part
its periphery, or circumference ; as A D. (
17. A chord is a right line which joins the /
extremities of an arc ; as A B. \
*
18. All rthne figures, bounded by three right Hues, are
called iricn^^!efi; nnd receive different denominations
according to th^ nature ot their sides and angles*
'J
•■i«'
19. An equilateral irianf^le
is that which has all its sides
•equal ; as A B C.
20. An isosceles triangle is that which haB
only two of its sides equal , as A B C.
<i
21. A sccAene triangle is that which
has all its three sides unequal; as
ABC.
H''''<^' r-
PRACTICAL GEOMETRY.
Cft
22. A right-angled triangle is that
which has one right angle ; the side
opposite to the right angle being
called the hypothenuse, and the other
two sides the legs ; as A B C.
I:
*■
7
t,',.
23. An obtuse-angled triangle is that
which has one obtuse angle ; as A B C,
and C is the obtuse angle.
24. An acute-angled triangle is that
which has all its angles acute; as
ABC.
25. All plane figurest bounded by four right Knes, aro
called quadrangles^ or quadrilaterals; and receive dif-
ferent names according to the nature of their sides and
angles. ' ,
26. A square is a quadrilateral, whose
sides are all equal, and its angles all
right angles ; as A B C D.
«*,>
27, A rhombus is a quadrilateral,
whose sides are all equal, but its
angles are not right angles; as
AB C D.
70
PRACTICAL GEOMETRY.
28. A parallelogram is a
. quadrilateralt whose opposite
sides are parallel; as A £
C D.
1
I
I I
29. A rectangle is a parallelo-
gram, whose angles are all right
angles ; as A B C D.
As
'Jfrr'
30. A rhomboid is a paral- A^
lelogram, whose angles are /
not right angles ; as A B /
C D. B^
/
31. All other four-sided figures, besides these, are
called trapeziums.
32. Parallel right lines are such
as are every whore at an equal dis- -^
tance from each other ; thus A B is q
parallel to CD.
3
.D
33. An angle is usually de*
noted by three letters, the one
which stands at the angular
point being always to be read
in the middle; as A B C, a-
CBD, DB£, &c.
PRACTICAL GEOIIETEY.
71
■ ■:rr
<
Problem 1*
To divide a given line A B into two equal parta.
»' ,
7K
v»J
>f!
-B
1. From the points A and B, as centres, with any dis-
tance greater than half of A B, describe arcs cutting each
other in n and m.
2. Through these points draw the line ncmi and the
point c, where it cuts A B, will be the middle of the linot
as required.
* j^ Problem 2.
To divide a given angle ABC into two equal parts*
■%}■
^-s-#( :*Jt'.-
1. From the point B, with any radius, describe the arc
A C ; and from the points A, C, with the same, or any
other radius, describe arcs cutting each other in n.
2. Then through the point n, draw the line B n^ and it
will bisect the angle A B C, as was required.
irii^:
72
PRACTICAL GEOMETRY,
Problem 3*
From a given point C, in a given right line A B, to
erect a perpendicular.
Case !• — When the point is in or near the middle of
the line.
M
■>^-
'.*■ ' V
^.-
^<SI^-
vv
-^ki'.
^f^
1. On each side of the point c take any two equal dis-
tances c ritcm,
2. From n and m, with any radius greater than n c or
m Cf describe arcs cutting each other in S.
3. Then, through the point S draw the line S c, and it
will be the perpendicular required.
Case 2* — ^When the point is at, or near the end of
the line.
1. Suppose c to be the given point, then take any
point 0, out of the line, and with the radius or distance
c, describe the arc mc n, cutting A B in m and c.
^. Through the centre o, and the point m, draw the
line mon, cutting the arc m c n in n.
3. Then from the point n draw the line n c, and it will
be the perpendicular required. w
PEAOTICAL QT ^MintT. IS
From a given poiat Ct witkcMita gifan lino ▲ Bt to lat
fall a peqModicular.
/^
I
'.I
.?l is Oliil P/).
if.M'f.'iixj "."1 J. nci^^ * r
1. From the point C, with any radius, describe the are
n m, cutting A B in n and m.
2. From the points n m, with the same or any other
radius, describe two arcs cutting each oAer in S.
3. Then through the poinU C, 3« draw the lijM C D S,
and C D will be the perpendicular required.
Problem $•
'J v/jni> 'ii'v';''j .i
At a given point £, to make an angle equal to a given
angle ABC.
<tmL
t^-'i'^H
"fiUil •'?>ti>liif!
b ^H'i '.'j^' '1'» '>ff'
^^ 1. From the point B, with any radius, describe the arc
n m, cutting the legs B A, B C m the points m n.
»si
I
'«jl
i J
i'
74
PRACTICAL 0£OMET|tY«
2. Draw the line E D ; and from the point E, with tbf
same radius as before, describe the arc r «.
3. Take thedifConce m r, on the fonnmr arc, and appljr
it to the arc r «, from r to ».
4. Then through the points E, S, draw the line E Ft
and the angle D £ F will b^ equal to the angle ABC,
f
Problem 6. - -^
as was required.
To draw a line parallel to a given line A B.
1. From any tWd poinfe r, !», in the line A B with any
radius describe the arcib am.
2. Then draw the line C D, to touch these arcs, with-
out cutting them, and it will be parallel to A B, as was
required.
T^roblem 7. r \' ^ dt^«
To divide a given line A B into any proposed number
of equal parts.
I
#^ — ^^.
/
/ ' "-^- ' ^
/
1. From one end A of the line draw A''in, making any
angle with A B ; and from the other end B, draw B n,
making an equal angle A iB n. .
■ f^fe- O. eJi|Wl »»M ^i-^ .iy^ r«^ «
Ui.
PRACTICAL OEOMlRTRY.
75
2. In each of the linet A m, B n, beginning at A and
B, set off as many equal parU, of any convenient length,
M A B is to be aivided mto.
3. Then join the points A, 5 ; 1 , 4 ; 2, 3, &€., and A B
will be divided as was required.
xt
Problem 8.
To find the centre of a given circle, or of one ab^tdj
described.
y
v ,
f»» a ./ a
• Jt'ifJ .ill Ifi/^ I. ims
. .OI- Mi'twii*rr-A
1. Draw any dhord A B« and bisect it with the perpen-
dicular C D.
2. Bisect C D, in like manner, with the chord E F, ind
their intersection, o, will be the centre required ; obsei*v-
ing that the bisection of the chords, and the raising of the
perpendicular, may be performed as in problems Ist
Aod drd. . ^
/ A
St- —
'A
'^il
1?J
u
"311
ifcl
^C
75
fRAOTICAL OSOMETRT.
■ Problem 9rvriafc5«]f^f"'"
To describe the dircumference 9f a circle through an/
three ^ven points A, B, C, provided the/ are not in a
right line. ;^J^
vhftytff; oaa '\h 10
4<33 mil bfui .;T
1. From the middle point B, draw the lines or chords
B A, B C ; and bisect them perpendicularly, with lines
meeting each other in O.
2. Then from the point of intersection, O, with the
distance O A, O B, or O C, describe the circle ABC,
and it will be that required.
Problem lO,
'1 Upon a given right Une A B, to make an equilateral
triangle, ' Wr * ■ '^ '*sf»
vtds 10 KHiiaLM -Mil biifi ,8i ./ . \ i j'4u*i)ouai(f vHlrii>ifi *itu
1. From the poinU A and B, with a radiua equal to
A B, describe aros cutting each other in C.
2. Draw thb lines A CJB C, and the agui« A C B will
be the triangl6 .equired.
PRACTICAL cmbMtttiit.
77
Problem 11.
To make a triangle^ the three sides of which shall be
respectively equal to thffis ^ven lines, A, B, C.
vIX
; .-. /lv;j;ix;,_;^J) "j/;
1. Draw a line D E equal to one of the given lines C.
2. From the point D, with a radius equal to A, describe
an arc ; and from the point E, with a radius equal to B,
describe another arc, cutting the former in F.
3. Then draw the lines D F, E' F, and D F E, will be
the triangle required.
^-k
Problem 12.
Upon a given line A li to describe a square.
&di il^ bm f-Am^am
'\ ,-,',
.J^ . A.
iMn 'A\ vMnl .1:
\S. i
'\r'> h From the point B, draw B C perpendicular, and
equal to AB.
2. From the points A and C, with the radius A B, or
C B, describe two arcs cutting each other in B, and draw
the lines A D, C D, and the figure A B C D will be the
square required.
7*
■I.
%\
'•''M
il
78
PRACTICAL OFOMETRY.
Problem 13.
lo A given triangle, A B C, to inscribe a circle.
-.•* t^* r**" f:'«'*-y»*
i u;--pj -f^:,-7>2-^- . -/
Bisect any two of the angles, as A and B, with the
lines A and B o,
2. Then from the point of intersection o, let fall the
perpendicular o n upon either of the sides, and it will be
the radius of the circle required. ,jf ,, Arf,t(T . I
In a given circle, to inscribe any regular polygon.
n»?! ni)'/l^^ 'i a.
1. Draw the diameter A B, which divide into as many
equal parts as the figure has sides.
2. From the points A, B^ as centres, and with the
radius A B, describe arcs crossing each other in C.
3. From the point C, through the second division of
the diameter, draw the line C D ; then, if the points
A, D, be joined, the line A D will be the side of the
polygon required.
NoTi;.-— It ii to be obienred that in the construction here giTea,
A D ii the side of a pentagon, or a figure of Jive sides.
PRACTICAL GEOMETRY.
79
W
Problem 15.
On a giTen line A B, to make a regular pentagon.
i
1. Produce A B towards n, and at the point B make
the perpendicular B m equal to A B.
2. Bisect A B in r, and from r as a centre, with the
radius r m describe the arc m, n, cut ing the produced
line A B in n.
3. From the points A and B, with the radius A n, de-
scribe arcs cutting each other in D, and from the points
A D and B D, with the radius A B, describe arcs cutting
each other in C and E.
4. Then if the line B C, C D, D E, and E A be drawn*
ABODE will be the pentagon required.
(itlU -^V
Problem 16.
On a given line A B to make a regular hexagon.
1. From the points A, B, as centres, with the radius
A B, describe arcs cutting each other in ; and from the
point 0, with the diiL^tance A or OB, describe the
circle A B C D E F.
2. Then if the line A 6 be applied six times round tlie
circumference, it will form the hexagon requhred.
m
Il
1 V
80
P&ACTICAL GEOMETRY.
Problem 17.
Ib a given ciicle to iascribe a regidar heptagon. >^
.'4-/f« il inioij: ^3 •^^^^:::i^
• 1, From any point A in the circumference, with the
radius A of the circle; describe the arc B O C, cutting
the circumference in B and C. '
2. Draw the chord B C, cutting A in D, and B D,
or C D, carried seven times round the circle from A, will
form the heptagon required.
'^^'''^^^^'^^•-r:, Problem 18.
On a given line A B, to form a regular octagon.
.HO^li< ^^iflf 1<\V2,
1. On the extremities of the given line A B ere^^i the
indefinite perpendiculars A F and B E.
2. Produce A B both ways to m and. n, and bisect the
angles m A F and n B £ with the lines A H and B G.
tj»
'IX
aiicj"*..')
PRACTICAL QEOMETRT*
91
3. Make A H and B C each eoual to A B, and draw
H 6, C D parallel to A F or B £, and make each of
them equal to A B.
4. From G, D, as centres, i?:th a radiua eqa&i to
A B, describe arcs crossing A F, B £« in F and E ; then
if 6 F, F E and £ D be drawn, ABCDEFGUwiU
be the octagon required.
... A.
Problem 19.
\
^ ■
- k
M
To make an angle of any proposed number of degrees.
So flJ^flMwS)?) '-'"•
1. Draw any line A B ; and having taken the first 60
degrees from Uie scale of chords, describe, with radius,
the arc n m.
2. Take in like manner the chord of the proposed
number of degrees from the same scale, and apply it
from n to m.
3. Then, if the lino A C be drawn from the poiat A
through m, the angle CAB will be that required.
\f
\
r.
"«C.-
■"^^
. fi
V K-t, 1-^
82
PRACTICAL GEOMETRY.
fT f
Problem 20*
HV a.Vi X
n
' To find a risht line that shall be Qearly equal io.^aj
given arc A D B of a circle. 4, n /,
^1 nty^ifyo ftiO f» J
.';'U'>''"i > vydiu
iljufi 06 ;;ii.^(.} oT
1. Divide the chord A B into four equal parts, nnd set
^ne of the parts A C, on the arc from B to D.
2. Draw C D, and the double of this line will be equal
to the arc A D B nearly.
Note. — If a right line be made equal to 3^ times the diameter of
a circle, it will be equal to the circumference nearly.
Upon a given line A B> to describe az^ oval, or a i%ur6
resembling an ellipse.
■ 7;5cvt>pij h» irmituiti
-T .«» ot i\ moi\
i i:u M, -uU 1.;. ■. /'\ /'^ \-U'U ,if'MYV ,8
'Mlt -itt i*^m!'5fiJ
1. Divide A B into three equal parts A C, C D, D B ;
and from the points C, D, with the radii C A, D B, d^
scribe the circles A G D E and C H E F.
2. Through the intersections m, n, and centres €, D,
draw the lines m H, E n ; and from the points n, m, with
.Ka.
m
m
.H.^ PRACTICAL GEOlfETRY.
83
loh
the radii n E, tn H, describe the arcs £ F» H 6, and A O
U B F £ will be the oval required*
. i^_. ,^ Or ihuii -^VtB't-^
The transverse and conjugate diameters being given.
t*, IV U01S
y <iH vi imp***;-'!
IV-v.'vI|-jH
1!. Draw the transverse ^na conjugate diameters, A B^
C £>, bisecting each other perpendicularly in the centre 0«
2. With the radius A O, and centre C, describe area
cutting A B in F/; and these two points will be the foci
of[ the ellipse. ft^-— < — — — j -" " " ' " " " ^'^'
n /3. Take any number of points n,«, &c. in the tr«>.ns-
^Verse diameter A B, and with the radiii An, ftB, and
centres F /, describe arcs intersecting each other in
.;, ,4^ Through the points S, S, &c. draw the curve
A^ C S 1! Dt and it will be the circumference required ;
or, having found the foci, F/, as before, take a thread, of
the length of the transverse diameter, and put it round two
pins fixed at the points F/,* then stretch the thread F S/
to its greatest extent, and it will reach to the point S in
the curve; and by moving a pencil round within tha
.thread, keeping it always stretched, it will trace out the
curve required.
m ?*i->d.:ft vj:.: i£Rt nt
iRVB %nun w* ' I •''^
i
I
ml
^:i:
84
IIKNSORATIOlf or 8UPSRnCIE8.
i U
xursmuinoif or ■rmmriozBS.
Q. VHiat is Mensuration of Superfkies ?
A. Mensuration of superficies teaches how to find the
area or superficial content of any figure, without anj
regard to its thickness. ^^ .
Rule* — ^To multiply the side by itself gives the area.
.1 .*
Examples*
' 1. WhatistheareaofthesquareABCDiWhoMfide
18 8 feet?
.«!
^
> No».— Iftachofthe
. tides of the square A B
3 feet C D be divided into 8
3 P*i^t*t '^^^ ^* opposite
" ■ , points be connected, it
A«. o <i« A will appeer obvious that
ATM, USq.II. the square A BCDwiU
contain 9 somJI squares,
I or square feet.
\ 2. What is the area of a square whose side is 4 feet?
t i H Ans. 16 feet.
3. How many square yards in a square, whose side is
20 feet? Ans. 44yds. 4 feet.
4. How many square yards in a squr»re, whose side is
31 feet? Ans. 106yds. 7 feet.
5. How many acres in a square field, whose side is 50
poles? Ans. 15 ac. 2 roods, 20 poles.
6. How many acres in a square field, whose side is 80
poles? Ans. 40 acres.
MENSURATION OF SUPERFICIES.
or A mBOTAXVOXiB.
85
Rule. — Multiply the length by the breadth, gives the
area.
Examples*
1. Required the area of the rectangle A B C D, whoso
length A B is 13.75 chains, and breadth B C 9,6 chains.
13.75=A B.
u.>' ■«* ...y. ''4v^'
t . '
9.5
6875
12375
10)
130.625
acres
13.0625
^ 4
roods
0.2500
40
poles
10.0000
n
Ans. 13 ac. Oro. 10 po,
2. Required the area of a rectangular board, whose
length is 12 feet, and breadth 2 feet. Ans. 24 feet.
3. Required the area of a rectangular board, whose
length is 12| feet, and the breadth 10 inches.
Ans. 10/^ sq. feet.
4. Required the superficial content of a rectangular
field, whose length is 12.25 6hains, and bieadth 8.5
chains. Ans. 10 ac. 1 ro. 26 po.
5. How many square yards of painting in a partition,
whose length is 20 feet, and height 8 feet ?
Ans. 17 yds. 7 feet.
6. What is the area of a rectangle, whose length is 14
feet, 6 inches, and breadth 4 feet, 9 inches ?
Ans. 68 .ft. 126 sq. in.
h
Wi
^^%
^1
■11
n
86
MENSURATION OP SUPERFICIES*
or iL ABOMBVf.
Rllle* — Multiply the side by the pcrpcndic>a]ar
breadth) and the product will be the area.
J.
£xample8«
* !^ *
sfn;
1. Required the area of the rhombus A B C D, whose
side A B is 12 ft. 6 in. and its perpendicular breadth D £
9ft. 3in.
ft. in.
i 12 6=A B
' 9 3=D£
112 6
3 1 6
•W
Ans. Feet 115 7 6 p.
2. What is the area of a rhombus, whose side is 14
feet, and perpendicular breadth 5 feet ? Ans. 70 feet.
3. Required the area of a rhombus, the length of whose
side is 12 ft. 9 in. and its height 10 ft. 6 in.
;: Ans. 133 ft. 10 in. 6 p.
4. Required the content of a rhombus, the side being
4 ft. 10 in. and the perpendicular 18 inches.
Ans. 7 ft. 3 in.
5. Find the content of a piece of land in the form of a
rhombus, its length being 6.20 chains, the perpendicular
5.45 ch. Ans. 3 ac. 1 ro. 20 po.
.i^-^:.
-M
''■K'
*^%fe,^.
¥-Al
le
MENSURATION OF SUPERFICIES.
87
;if
or
BHOMBOZO.
R\ille.— Multiply the length by the perpendicular
heigh', gives the area.
Examples.
1. What is the area of the rhoml •
length A B is 10.52 chains, and its p<
D £ 7.63 chains ?
t
^, whose
jlt height
10.52=A B
7.63
3156
6312
7364
10);80.2676
acres 8.02676
4
roods 0.10704
40
Ans. 8a. Or. 4 p.
poles 4.28160
2. Required the area of a rhomboid, whoso length is
10.51 chains, and breadth 4.28 chains.
Ans 4 ac. 1 ro. 39 po.
3. What is the area of a rhomboid, whose length is
7 ft. 9 in. and height 3 ft. 6 in. ? Ans. 27 ft. 18 sq.- in.
4. How many square yards in a rhomboid, whose
length is 37 ft. and height 5 ft. 3 in. ?
Ans. 21/j sq. yds.
^.
>.
o \>.^.
IMAGE EVALUATION
TEST TARGET (MT-3)
1.0
I.I
^1^ 1^
us
140
:
1— III— llll^
<
6"
».
Hiotographic
Sdeaces
Corporalion
\
,V
\\
23 VVIST JMAIN STRUT
WIBSTEt,N.Y. 145S0
(716) t72-4S03
■45
^
88
MENSURATION OF SUPERFICIES.
or A
Rule 1* — Multiply the boUie by the perpendicular
height, and take half the product for the area. ^
Examples*
1. Required the area of the triangle ABC, whose base
A B is 10 ft. 9 in. and its perpendicular height D C 7 (I
^ ft. in.
S ' 10 9— 10.76=AB
7 3= 7.25=C D -
ni.
Vj:
?:^^t
p»*<mU-«» vi**
?...
5375
2150
7625
\ 2) 77.9375
feet 38.96875
144
\
&\
~.-t»'«»*.'A-.i-)r-'fc.
387500
387500
96875
aq. in. 139.50000
Ans. 38 ft. 139 sq. in.
2. What is the area of a triangle, whose base is 20 ft.
and the perpendicular 5 feet .? Ans, 50 ft.
3. How many square yards in a triangle, whose base is
40 feet, and the perpendicular 30 feet ?
Ans. 66|sq. yrds.
4. Required the area of a triangle, whose base is 12.25
ohains, and height 8.5 chains. Ans. 5 uc* 33 po.
Rule 2«-*When the three sides are given : Add all
the three sides together, and take half that sum ; next sub*
MENSURATION OF SUPERFICIES.
89
tnct each side severallj from the said half sum, then mul-
tiply that half sum and the three remainders together, and
extract the square root of the product for the area of the
triangle.
( Examples.
1. Required the area of the triangle ABC, whose
three sides B C, C A, A B, are 13. 14. and 15 feet re-
spectively.
13=BC
14=CA
■ ^ 16=AB
— 21 21 21
2) 42 13 14 15
half sum 21 8i2, Tf^*
6M
r
8 1st rem.
168 ^ ^
7 2nd rem.
\
'rta.
iw
1176
6 3rd rem. -
7056
64 (84 sq. feet.
164) 656
656
2. Required the area of a triangle whose sides are 20,
30 and 40 feet. Ans. 290.4737 sq. feet.
i 3. Required the area of an equilateral triangle, each of
whose equal sides is 25 chains. Ans. 27.0632 acres.
4. Required the area of an isosceles triangle, whose
base is 20, and each of its equal sides 15. ^ ns. 1 1 1.803.
5. Required the area of a right-angled .rii^ngle whose
hypotenuse is 50 and the other two sides oO ami 40.
' Ans. 600.
f . ■• ■■■■ 8*
(M
MENSURATION OF SUPERFICIES.
6. How many acres aro there in the triangle, whose
three sides aro 380, 420, and 765 yards.
Ans. 9 ac. ro. 38 po.
OF A T&APSZZUM.
Rule*— Add the two perpendiculars together, multi-
lily that sum by the diagonal, and take haf the product
for the area.
Examples*
1. Required the area of the trapezium A B C D, whose
diagonal A C is 84, the perpendicular B £ 28, and the
perpendicular D F 21. i*
28=B£
21r=DF /
49
84
■jf-
X
196
392
' '* -■ • »-...**#»,+,. ,s^ ,*M* «- <.*.■!*' «y«»*«**r*
2) 4116
Ans. 2058 area of the trar \BCD.
2. Required the area of a trapezium whose diagonal is
40 and the two perpendiculars 20 and 10. Ans. 600.
3. Required the area of a trapezis^m whose diagonal is
80|, and the two perpendiculars 24^ and 30 j^^.
Ans. 2197.65.
4. What is the area of a trapezium, whose diagonal is
108 fl. 6 in. and the perpendiculars 56 ft. 3 in. and 60 ft.
9 in. ? Ans. 6347 ft. 36 sq. in.
6. How many square yards of paving are there in a
trapezium, whose diagonal is 65 feet, and the two perpen-
diculars, 28 and 33| feet respectively ?
Ans. 222.083 sq. yds.
'*«5
.^
MEiNSURATION OF SUPERFICIES. W
6. How many acres are there in the trapezium, whose
diagonal is 4.75 chains, and the two perpendiculars fall-
ing on it, 2.25 and 3.6 chains respectively ?
Ans. 13ac. 3ro. 23 po.
or RBOVXJkB POXtTOOm.
Rule* — ^When the side and perpendicular are given,
multiply the length of one of the sides by the number of
sides which the Sgure contains, then multiply that product
by the perpendicular, and take half of the last product for
the area ; but if the side only be given, square the side,
and multiply it by the tabular number found opposite its
name in the following table, and the product will be
the area.
No. of
NAMES.
TABULAR
SIDES.
NUMBER.
3
Trigon ....
0.4330127
4
Tetragon . . .
1.0000000
5
Pentagon . . .
1.7204774
6
Hexagon . . .
2.5980762
7
Heptagon . . •
3.6339126
8
Octagon . . .
4.8284272
9
Nonagon . . .
6.1818240
10
Decagon . . .
7.6942088
U
Hendecagon . .
9.3656411
12
Dodecagon . .
4
11.1961524
£.■
03
MENSURATION OP SUPERFICIES
m..
Examples.
u f .
1. Required the area of the regular pentagon ABC
D £t one of whose sides being 25 feet, and the perpendi-
cular O P from its centre 17.2 feet.
Side A B=r 25
No. of sidesss 5
125
Perpr. P=17.2
250
875
:-•' : V
Or, ihiit :.
25 side A B.
25
125
50
625
1.72 UbaUr auBbw..
2) 2150.0
Ans. 1075 fbet. "*'
J -*--;■,* lA b«t>iv)'£f/
1250
4375
625
Ans. 1075.00
2. Required the area ofa hexagon, one of whose equal
sides is 14.6 feet» and the perpendicular from the centre
12.64 feet? Ans. 553.632 sqr. feet.
3. Required the area of a heptagon, one of whose equal
sides is 19.38» and the perpendicular from the- centre 20.
Ans. 1356.6.
4. Required the areu of an octagon^^ one of whose equal
sides is 9.941, and the perpendicular from the centre 12.
Ans. 477.168.
5. Required the erea of a regular octagon, each of
whose equal sides is 16. Ans. 1236.0773.
6. Required the area of a regular decagon, each of its
equal sides being 20}. Ans. 3233.491125.
OF za&BavziAR BzaaT-ZiZNSD rzouRxis.
Rule* — Divide the figure into triangles and trape-
iaums» and fmd the areas of each of them separately, then
MENSURATION OF SUPERFICIES.
93
add these areas together, and their sum will give the area
of the whole figure.
«
. .1.. Examples.
1. Required the area of the irregular right-lined figure
A B G D £ F, the dimensions of which are as follows :
F B=20.75, F C=27.48, E C-18.5, B n= 14.25,
£ jn=:9.35, D r=12.8, and A «=8.6.
-^ <» •^.s.li
■.1 'YU
...^^'
/
. -i
•r A B F.
»f D E C.
20.75X8.6 =178.450 -7-2=89.225 .r«
18.5 X 12.8 =236.80 ^^2=118.40 am
14.25-^9.85=23.60X27.48=648.5280-1-2=324.264 .re.
of F B C E. _- ^— ^
Abs. 531.889 trM
«ri B C D £ F.
2. Required the area of an irregular hexagon, like that
in the last example, supposing the dimensions of the dif-
ferent lines to be the halves of those before given.
OF
OIROZJB.
"Rule 1* — Multiply the diameter by 3. 1416 — gives
. the circumference.
2. Divide the circumference by 3.1416 — gives the
diameter.
3. Square the diameter, and multiply it by .7854 —
gives the area.
4. Square the circumference, and multiply it by
.07958 — gives the area.
5. Divide the area by .7854, and extract the square
root— "gives the diameter.
6. Divide the area by .07958, and extract the square
root— gives the circumference.
m
ii
94
MENSURATION OF SUPERFICIES.
Examples*
... ^ :>
1. What is the circumference and area of a circle* whose
diameter is 3 feet ?
3.1416
3 f«et dlamcUr.
Ant. 9.4248 fMt einumf.
.7854
Au. 7.0686tq«ft«arM.
2. If the diameter be 26, what is the circumference ?
Ans. 81.6816.
3. If the circumference be 75» what is the diameter ?
* Ans. 23.873.
4 What is the circumference, when the diameter is 7 ?
- Ans. 21.9912.
5. What is the diameter of a circle, whose circumfer-
ence is 50? Ans. 15.9156..
6. What is the area of a circle, whose diameter is 5}
feet? Ans. 23.758350 feet.
' 7. How many square yards are there in a circle, whose
circumference is 10| yds.? Ans. 9.19646375.
8. How many square yards are there in a circle, whose
radius is 15^ feet ? Ans. 81.1798.
9. How many square feet are there in a circle, whose
circumference is 20y*7 yards ? Ans. 289.36.
^ 10. It is required to find the radius of a circle, whose
area is an acre. Ans. 39^ yrds.
Rule 7. — To find the length ofany arc of a circle :
Multiply the chord of half the arc by 8, from that product
subtract the chord of the whole arc, and one-third of the
remainder will be the lengths of the arc nearly ; or multi-
ply the decimal .01745 by the degrees in the given arc,
and that product by the radius of the circle, for the length
of the arc.
MENSURATION OF SUPERFICIES.
dd
Examples.
' 1. The chord A B of the
whole are A B C is 48.74,
and the chord A C of half
the arc 30.25, what is the
length of the arc ?
■-'{
242.00
48.74
3) 193.26
jV->
AnS. 64.42 length of the tK A B C.
2. Required the length
of the arc A B, which con-
tains 60 degrees, and the
radius B O of the circle
being 7 feet.
,i ■
^''^,
1.04700
7
Ans. 7.32900 fMt tba m a b.
-•..•»♦■/>
'' 3. The chord of the whole arc is 30, and the cL.nl of
half the arc is 17, what is the length of the arc ?
Ans. 35^.
4. The chord of the whole arc is 50|, and the chord of
half the arc is 30f, what is the length of the arc ?
Ans. 64.6.
5. What is the length of an arc of 30 degrees, the
radius of its circle being 9 feet ? Ans. 4.7115.
6. What is the length of an arc of 12^ degrees, the
radius being 10 feet? Ans. 2.1231.
Rule 8* — To find the area of a sector of a circle:
Multiply the radius of ^e circle, by half the length of the
A
/■/ H
96
MENSURATION OF SUPERFICIES.
arc of the sector, and the product will be the area : or, as
260 degrees is to the number of degrees in the arc of the
sector, so is the area of the circle to the area of the sector.
Examples.
1. What is the area of
the sector O B C A 0, the
radius A being 10 feet,
and the arc A CB, 18 feet?
1st Example.
10=AO
9=iAGB >
SOfMtuMorOBG AO.
2. Required the area of
the sector, the arc A G B
being 30 degrees, and the
diameter 3 feet.
"'r'
2d Example.
.7854X 9=7.0686=»1!;^:.*«
Ana. no : 80 1 : 70686 : .SBtOS WM .f the MCtort
hi
8. What is the area of a sector, whose arc contains 18
degrees, Uie diameter being 3 feet ? Ans. .35343.
4. What is the area of a sector, whose radius is 10,
and arc 20 ? Ans. 100.
5. Required the area of a sector, whose radius is 25,
and its axz containing 147° 29'. Ans. 804.3986.
Rule 9. — To find the area of a segment of a circle :
Find the area of the sector, having the same arc as the
segment ; then find the area of the triangle formed by the
chord of the segment and the two radii of the sector, and
the sum or difference of these areas, according as the seg-
ment is greater or less than a semicircle, will be the area
of the segment required.
MENSURATION OF SUPCRIiCIES.
97
Examples.
1. The chord A B is 24, and the height or versed sine
C D of half the arc A C B is 5 ; what is the area of the
segment A B C A ?
^■* ,
First find the requintes thus :
y/ A D»+D C« :s:A C 13 the chord of half the arc.
A C>^5=C £ 33.8 the diameter of the circle
C £ 4-2=C O 16.9 the radius of the circle.
AC X8— AB-r3a:A C B 26.666 the length of the arc.
CO-CD=*DO 11.9 the perp. of the A A OB.
D 0=11.9 ^, ,
A B= 12 i L*
142. Sana of ^ CAB.
13.333halfthearc ACB
16.9 the radius.
/
119997 /
79998
13333
From 225.3277 «if,;'» B C A 0.
Take 142.8 ««ofu.. ^, A B.
Hem. 82.6277 •:;^'^ A B C A.
Mgnmil
r«(.t"3»wi
2. Required the area of a segment, its chord being 12,
and the radius of the circle 10. Ans. 16.3504.
3. What is the area of a segment, whose versed sine m
18, and the diameter of the circle 50 ? Ans. 636.375.
4. Required the area of a segment, whose chord is 16,
and the diameter 20. Ans. 44.728.
9
A
06
MENSURATION OF 6UPERPICIC8.
6. What is tho area of a leffment of a circle, wboM
arc is 60", and the diameter of the circle 10 feet ?
Ana. 2.2647.
6. Required the orea of the segment of a circle, whose
▼ersed sine is 5, and the diameter of the circle 20 feet.
Ans. 61.4184.
r
or AN BXAZPSIf OS OVAXi.
Rule* — Multiply the transverse diameter by the con-
gregate, and this product again by .7854, and the result
will be the area.
t
Examples*
1. Required the area of an ellipsis, whose tranverse
diameter A B is 24, and the congregate C D, 18.
A B 24 SB tosMvtrM diMMltr.
G D 18^^ coajogtu tltamtttr.
192
24
482
.7864
1.728
2160
8456
3024
Ans. 389.2928 ss >morik«tiiipri«.
S. Required the area of an ellipse, whose transrerse
and conjugate diameters are 70 and 50. Ans. 2748.9.
3. Find the erea of an oyalt whose two axes are 24
and 18. Ans. 889.2928.
•ft
m.N8UBATIOIf or aOLIIMU
Qi Whtt if Meaauration of Mlidi ?
A. Mensuration of loUdi teaches how to find the whole
eapecity or content of any solid, considered under the
triple dunensions of length, breadth, and thicknesa.
Deflnitioiu*
Q. What is a cube ?
A. A cube is a solid contained
by six equal square sides, or faces,
as A D.
Q. What is a parallelo- /
piped? "^i^
A. A parallelopiped is a
solid contained by six rectan-
gular plane faces, every oppo-
site two of which are equal
and parallel ; aa A D.
n
Q. What is a prism? >
A. A prism is a solid, whose ends are
two equal, parallel, and similar plane
figures, and its sides parallelograms; as
A B C D £ A.
100
MENSURATION OF SOLIDS.
Q. What 18 a oylinder ?
A. A cylinder is a solid, whose surface
is circular, and its ends two circular planes
as AC.
i,»
Q. What is a pyramid ?
A. A pyramid is a solid, whose sides
are all triangles, meeting in a point as
the vertex, and the base any plane
figure ; as A B C D £ A.
Q. What is a sphere ?
A. A sphere or globe is a solid
described by the revolution of a
semicircle about its diameter,
which remains fixed, as A B C D.
OF A OUBS, PBZSSS OH OlTXiXNDBR. ; ?
Rule !• — To find the solid content of a cube,
prism, or cylinder : find the area of the base, or end, and
multiply it by the height or length of the figure, gives the
solid content.
Rule 2* — To find the superficial content of a cube,
prism, or cylinder : Multiply the perimeter or circum-
ference of the base or end of the figure, by the height or
length of the cube, prism, or cylinder, to which product
add the areas of the two ends, if required, gives the whole
surface of the figure.
MENSURATION OF SOLIDS.
101
£xamples«
1. What is the solid and superficial content of a paral*
lelopiped A B C G H £, whose length A B is 8 feet, its
breadth A £ 4^ leet, and its depth A D 6} feet.
H a
4e kreadlh
••^ A E.
6.75 'i'^
225
315
270
9(«ic« Ihc
brtadih.
1 q e twice tiM
22.5 iMriawtof.
8
30.375 area of the end.
8 length A B.
B 180.0 "^/•»
60.75
Ans. 240.75 !i".S:''
Am. 243.000 feet solid content.
•conual.
3. What id the superficial content of a cube, the length
of each side being 20 feet ? Ans. 2400 feet.
3. What is the superficial content of a triangular prism,
whose length is 20 feeif and each side of its end or base
18 inches? ' Ans. 91.948 feet.
4. Find the convex surface of a round prism, or cylin-
der, whose length is 20 feet, and the diameter of its base
2 feet. Ans. 125.664.
5. What is the solid content of a centre, whose side is
24 inches? Ans. 13824.
6. How many cubic feet are in a block of marble, its
length being 3 ft. 2 in., breadth 2 ft. 8 in., and thickness
2ft. 6 in.? Ans. 21^ feet.
7. Required the solidity of a triangular prism, whose
length is 10 feet, and the Uiree sides of its triangular end
or base at 3, 4, 5 feet. Ans. 60.
8. Required the content of a round pillar or cylindw,
whose length is 20 feet, and circumference 5 ft. 6 in.
Ans. 48.1459.
OF
PTAAmD OR ooira.
Rule 1 • — To find the solid content of a pyramid or
cone : Multiply the area of the base by one-third of the
perpendicular height, gives the solid content.
9*
ri'i
102
MENSURATION OF SOLIDS.
Rule 2* — To find the superficial content of a pyra*
mid or cone : Multiply the perimeter of the base by half
the slant height, to which add the area of the ba&e, if re-
quisite, gives the superficial content.
• '^ Examples.
1. What is the solid and superficial content of the cone
A B C, the diameter of whose base A B is 6 feet, and the
slant height A C or B C is 10 feet ?
M'^*
C V.i.''.
'^.'i:. ;"!',i=^.
v5< ■;;. .1
in-i '■'
_, f\;j}-
AC«=10xlO=100
^ABs: ax 3= 9
'■.» ' ri
.i''^'. ^'i
V* 91
^i pevp. o€s= 9.539a
WA.
.7854
AB*=:6X6= 36
47124
23562
8,1416
6
18.8496 circumfer.
5 i slant ht.
94.2480 arMofthabedy.
Si8.2744 arMofthabut.
Ans^ 19^^5224 feetNp>eMta«t
28.2744 area of the base,
multiplied by 3.1797 | of the perp. C. ^
Ans. 89.9041 feet solid coat.
«r-
MENSURATION OF SOLIDS.
103
2. What is : erficial content of a triangular pyramid,
the slant heigbi .:«eing 20 feet, and each side of the base
3 feet? Ans. 90 feet.
8. Required the surface of a cone, the slant height
being 50 feet, and the diameter of the base 8| feet.
^ Ans. 667.69.
4. Requi.ed the solidity of a square pyramid, each side
of its base being 30, and its perpendicular height 25.
Ans. 7500.
5. Find the solid content of a triangular pyramid, whose
perpendicular height is 30, and each side of the base 3.
Ans. 38.97117.
6. What is the content of a pentagonal pyramid, its
height being 12 feet, and each side of its base U feet ?
Ans. 27.5276.
7. Required the content of a cone, its height being 10|
feet, and the circumference of its base 9 feet.
Ans. 22.56093.
1
or THB
FBirSTUM OF A
OR ooira.
Rule 1* — To the areas of the two ends of the frus-
tum add the square root of their product, and this sum
being multiplied hj a tl^rd of the height, will give the
iolidity.
IB
Rule 2* — Multiply the sum of the perimeterr or
circumference of the two ends, by the slant height of the
frustum, and \M the product will give the superficial
eoDtent.
r^ -- •-, —
J ■•
,»^:tftf«
. ii .'Vi
« .:tUii '<■•
-'^A^OF^^'
104
MENSURATION ' OF SOLIDS.
Examples.
1. Required the solid and superficial content of the
frustum of the cone £ A B D, the diameter of whose
greater end A B is 5 feet, that of the less end £ D 3 feet,
and the perpendicular height, S S, 9 feet.
Note 1. — When the
■ItDt keigklU notgiven,
it may be found by ex-
tracting the aquare root
of theium of the iquarea
of the perpendicular
height and diySbreBce of
the radii ;
.7854
5x5= 25
39270
15708
areaof AB==:19.6350
Tktu:
3x3=
.7854
9
A8=ai-ES5=U=«l
9»=9X9=81
areaof£D= 7.0686
19.6350
7.0686
V» 82
9.0553 aleiit bt.
A 8=5 X 3.1416=!15.7080
ED=3X 3.1416= ^'*^^
25.1328 w'"«rfe«'*.
9.0553 alant bt.
^« 138.79196100
11 'YOI Muarerootef
l.#01 protluct.
19.6350 arMofA B.
7.0686 "i^ of ED.
Aim. 226.58504 aup con.
38.4846
\ of perp. 3
Ans. 115.4538 MUdeontMit.
■■* ;t
■0.
,5 H
Note 2.— To the superficial content add the areai of both endi,
if the whole surface be required.
2. What is the solidity of the frustum of a cone, the
diameter of the greater end heing 4 feet, and of the less
end 2 feet, and the altitude 9 feet? Ans. 65.9736.
MENSURATION OF SOLIDS.
105
3. What in the solidity of the fruatoin of a sqare pyni-
at of
dide of the
end
inches
the less end 15 inches^ and the height 60 inches ?
Ans. 9.479 cu. ft.
4. What is the solidity of the frustum of an hexagonrvl
pyramid, the side of whose greater end is 3 feet, that of
the less end 2 feet, and the length 12 feet 7
Ans. 197.463776 cu. ft.
5. What is the superficial content of the frustum of a
square pyramid, whose slant height is 10 feet, one side of
the greater end being 3 ft. 4 in. and of the less end 2 ft.
2 in.? Ans. 110 ft.
6. To find the convex surface of the frustum of a cone,
the slant height of the frustum being 12^ feet, and the
circumference of the two ends 6 and 8.4 feet.
Ans. 90 feet.
4
OF JL 8PBSBB OR OZiOBS.
Rule 1 • — To find the solid content of a sphere :
Cube the diameter, and multiply it by .5236, and the pro-
duct will be the solidity.
'' Rule 2. — To find the superficial content of a
sphere : Square the diameter, and multiply it by 3.1416,
or multiply the diameter by the circumference — either of
these methods will give the superficial content.
^Examples*
1. What is the solid and superficial content of the
sphere A D E B, whose diameter A £ is 17 inches ? «
A^ 17'=289
17^=4913 .^^^^^^ 3.1416
.5236
29478-*^!
14739
9826
24565
Ans. 2572.4468 solid inches.
,
1734
B
289
1156
289
867
Ans.
907.9224 ;:„»;:
if
i'
I
KM
MENSURATION OF 80LID8.
2. What if the loUdity of a tphertf whose diameter is
l^feet? Ans. 1.2411.
3. What is the solidity of the earth, supposing it to b«
perfectly spherical, its diameter being 7957} miles ?
Ans. 263858149120 cu.m.
• 4. What is the convex superficies of a sphere, whose
diameter is 1| feet, and the circumference 4.1888 feet ? r
Ans. 5.58506 sq. feet.
5. If the diameter, or axis of the earth, be 7957 j miles ;
what is the whole surface, supposing it to be a perfect
sphere? Ans. 198944286.35235 sq. miles.
is >
or TBB SfiOMSKT OF A svBsma.
Rule 1 • — To three times the square of the radius of
its base add the square of its height ; and this sum mul-
tiplied by the height, and the product again by .5236, will
give the solidity.
Rule !2«— -Multiply the circumference of the whole
sphere by the height of the segment, and the product will
be the sup trficial content.
'■': i.yi..
i i ^. ^ . V-J
. '*■
I': '
'}■/'■■' ■
^j»-.. -* ~.n4.^^^
a<v^
V
»r *^^f-:**ifi •»■ fc'.^^i
MENSURATION OF SOLIDS.
107
V *.
Examples.
1. Wnat is the solid and superficial content of the
segment C A B, the radius A n being 7 inches, the height
C n 4 inches, f*nd the diameter of the whole sphere C D
16 J inches ?
An=7
7
49
#
147
16=n C»
163
43snC.
C Ds=16.25
3.1416
9760
1625
6500
1625
4875
51.05.1000
4
t;^
I
652
X .6236
1.341.3872 inch, solid content.
Ans. 204.204 •~^ti"J-
2. What is the solidity of the segment of a sphere, the
diameter of whose base is 20, and its height 9 ?
Ans. 1795.4244.
3. What is the content of a spherical segment, whose
height is 4 inches, and the radius of its base 8 ?
Ans^ 435.6352 cu. in.
4. What is the superficial content of the segment of a
sphere, whose height is 4} incheS) and the diameter of the
whole sphere is 21 inches ? Ans. 296.8612 sq. in.
5. What is the convex surface of a spherical zone,
whose breadth is 4 inches, and the diameter of the
sphere* from which it was cut, 25 inches .'
Ans 314.16 sq. incbef.
IV
y
n
I'
106
MENSURATION OF SOLIDS.
or TBB ASOnXiAB BODZS8.
Tho whole number of regular bodies which can possi-
bly bo formed is five, viz. :
1. The Tetraedroni which has four equal triangular
faces.
2. The HexaedroTit which has six equal square faces.
3. The Octaedron, which has eight equal triangular
faces.
4. The Dodecaedron, which has twelve equal penta-
gonal faces.
5. The Icosaedron, which has twenty equal triangular
faces. I * .
Note. — If the following figures be made of pasteboard, and the
lines be cut half through, so that the parts may be turned up and
glued together) they willreprescnt the^oe regular bodies
•^ ■>V." A*^-^
TETRAEDRON.
" " i*'t^..k-'\. .'
HEXAEDRON
d-
J rS- J .
■ i 4rif
OCTAEDROM.
na.fc
:, n-^ -* 1
.t 1 .';::, it-
>.|i "fo DODECAEDROrV
.-..*. '^^
ICOSABDROIT. 'H
MENSURATION OF SOLIDS.
100
Table
or THS SURrACEB AND SOLIDITIES O^ tAK RCQULAR
BODIES, WHEN TUE LINEAR
EDGE IS 1.
No. of
SIDES.
4
6
8
12
20
NAMES.
1
SURFACES.
SOLIDITIES.
Tetraedron . .
Hdxasdroa
Octaedron
Dodecaedron .
Icosaedron . . .
1.73205
6.0000
3.46410
20.64578
8.66025
0.11785
1.0000
0.47140
7.66312
2.18169
Rule 1 • — To find the superficies : Multiply the
square of the linear edge by the tabular area, opposite its
name, and the product will be the superficial content.
K-
Rule 2« — To find the solidity : Multiply the cube
of the linear edge, b)' the tabular solidity, opposite its
name, and the product will be the solid content.
.•lb
;i^
lit
liXampleso
1. What is the superficial and solid content of a
Tetraedron, whose linear edge is 4 ?
Tabular area 1.73205
4«= 16
Tab* solidity 0.11786
1039230
173205
Ana. 17.71280,-Pi,.
m
64
47140
70710
7.54240 -"1
•onUat.
/
m
110
artificers' work.
2. Required the solidity of a tetraedron, whose linear
edge is 6. Ana. 25.452.
3. What is the superficial content of an octaedron,
whose linear side is 4 ? Ans. 65.4256.
4. The linear side of a dodecaedron being 3, what is
the solidity? Ans. 206.90424.
5. What is the solid content of ao icosaedron, whose
sMeisS? - Ans. 58.90563
I
Artificers estimate, or compute the value of their works
by different measures, viz. :
Olazingt Maaoni? flat work, and sotne parts of joiners'
work, are computed at so much per square foot,
Painter8\ Pl€uterers\ Paver8\ ana some descriptions
of joiners' work, are estimated by the square yard,
Roofst Floorst PartitionSf &c., by the square of 100
feet.
Bricklayers' work, by the square rod, containing 272 J
feet. !
NoTB 1.— The roof of a house it said to be of a true pUchf when
the rafters are 1 of the breadth of the building. In this case, there-
fore, the breadth must be accounted equal to the breadth and half
breadth of the building.
NoTK 2.— Bricklayers compute their work at the rate of a brick
and a half thick ; therefore, if the thickness of a wall be more or
lest, it must be reduced to the standard thickness by roultiplyiug
the area of the wall by the number of half bricks in the thickness,
and dividing the product by 3.
i:t
. .^ lExamples. \
1. A certain house'jias 3 tiers of windows, 3 in a tier,
the height of the first tier being 7 feet 10 inches, the
second 6 feet 8 inches, and the third 5 feet 4 inchet ;
artificers' work.
in
and the breadth of each window is 3 feet 11 inches
What will the glazing cost at 14 d. per s<iuarc feet.
A. in.
19 10
11
ft.
in.
7
10
6
8
5
4
19
1 n th* whole
3
11
3
11
n lilt wliolt
-^ brwdtb.
in. 218
C=i= 9
3=;:|= 4
2
11
11 6"
a. iJ233
6 at
l^iV 11
rf.2=i= 1
valuaof 6"=
13
18 10
0^
d.
Ads. £13 11 1
2. What is the price of 8 squares of glass, each mea-
suring 4 feet 10 inches long* and 2 feet 11 inches broad,
at ^\d. per square feet ? Ans. iS 1 1S#. 9d.
3. What is the value of 8 squares, each measuring 3
feet by 1 feet 6 inches, at 7}({. per square foot ?
Ans. £1 3«. 3d'.
4. What is the price of a marble slab, 5 feet 7 inches
long, and 1 foot 10 inches broad, at 69. per square foot ?
Ans. jS3 \a. hd.
5. What will be the expense of ceiling a room, the
length of which is 74 feet 9 inches, and the width 11 feet
6 inches, at 3«. lOjd. per square yard ?
Ans. ^18 105. \d,
6. What will the paving of a court-yard cost, at 4|(f.
per square yard, the length being 58 feet 6 inches, and
the breadth 54 feet 9 inches ? Ans. £7 0». lOd.
7. The circuit of a room is 97 feet 8 inches, and the
height 9 feet 10 inches, what is this charge for painting it,
at 25. 8}d per rquare yard ? Ans. i&14 II5. 2d.
%
r
112
ARTIFICERS* WORK.
8. What is the expense of a piece of wainscot 8 feet 3
inches long, and 6 teet 6 inches broad, at Of. 7\d. per
square yard? Ans. £l Ids. 5d.
9. In a piece of partitioning 173 feet 10 inches long,
and 1 feet 7 inches in height, how many squares ?
Ans. 18aqr. 39 ft. 8' 10"
10. If a house measures wiihin the walls 52 feet 8 inches
in length, and 30 foot 6 inches in breadth, the roof being
of a true pitch ; what will it cost rooting at lOt. 6<i. per
square? Ans. 4^12 \2$. W^d.
11. How many rods are there in a wall 62j feet long
14 feet 8 inches high, and 2 J bricks thick, counting each
rod 279 feet? Ans. 5 rods 167 ft.
,-^-.*>
.- %
^.i ,!♦'' '■•'■ ■' -■" •
•h
■I-'
'■■.- ■r.)^<-
•i ,
'y^
S^ >*i ;^' ^
'^7/^'
1* - .■
iff
f
.
A"' ^.
* , . >■ ''
' '' '■ I'li-i' !
U '1
its
per
:he8
5ing
per
ong
iach
I
BOOK IkKKPING,
BY SINGI. r KMHY.
Question. What is Book knrping ?
Jlnswer. Book keeping is the tirt of recordm g pocuniary
or commci'cial traiisactiuns in a regular and H^r.euiutic
manner.
Q What are the names of the books u»i. <i(^ kc| t T
^ The Day-Boukf the Cush-Book, the /^edger, and
ihe -idl-Book. ,
Q. What is the Day-Book ?
.^. The Day-Bouk contains first an inventor- of i\.e
existing state of the merchants' at}airs : after wl 'h ire
entered, in the regular order of time, the daily transac-
tions of goods bought and sold, where it must be obs- rved
that Dr. or Debtor, is placed alter the name of au\ per-
son for money or goods which he receives ; and ( . or
Creditor, for whatever the merchant receives from hun.
Q. What is the Cash-Book I
A. The Cash-Book contains the particulars of all n o-
ney transactions. Cash is debited, on the left hand si( >,
to all sums received ; and credited, on the right hand siil •,
by all sums paid. The excess of the Dr. side above the
Cr. shows the balance or amount of cash in chest.
Q. What is the Ledger?
A, In the Ledger are collected the dispersed accounts
of each person from the Day-Book and Cash-Book, and
entered in a concise manner in one folio ; the sums in
which he is Dr. being arranged on the left hand, and
those in which he is Cr. on the right hand page of the
folio ; the balance of each is ascertained by taking the
difference between the Dr. and Cr. sides.
Q. What is the Bill-Book I
Ji. In the Bill-Book are copied the particulars of ail
Bills *sj ilxchange, whether Receivable or Payable.
10*
m
ji^
DAY BOOK
Inventory.
Jmn'y i. j have in ready money,
Biilff receivable, No. t, on S. Johnston, due
291 h iiist.
Tea, 3 chests, wt. 2cwt. 3qrs. lOIb. at 68. 2d.
per lb.
Raw Sugar, 2hlids. wt. 27ct. 3qr. 181b. at
3£. 14s. 8d. per cwt.
James Taylor owes me on bond, dated Au-
gust I4j 1828, with interest, at £5 per Ct.
per ann.
tf
I owe as folloi¥s;
John Herdson, a balance of accounts,
Bernard Mason, for purchase of my house,
by auction, to be paid Isl Feb. next, £800
Duty on do. at £6 per ct. 40
Bills payable, viz. No. 1, Wm. Homes' bill
to H. Williams or order, accepted by me,
due I9th inst
Allen, Wild, St Co. Leeds, Cr.
By 3 pieces superfine blue cloth, each 36 yds
at 25s. 6d. per yard,
„ 2 piecr t narrow brown, 84 yds. at 43. 9d.
„ Wrappers,
£.
1500
24
9S
104
70
1796
8.
»
3
1
4
n
8
d.
»
8
>»
8 Bernard Masun,
lo 2 s». raw sugar at 9i per lb.
To 3i lbs. green lea, at 8s. 6d. per lb.
„ 3| yds. blue cloth, at 2ds.
Dr,
37
840
45
922
>i
10
15
If
9
10
»»
10
137
19
157
14
19
5
18
»»
6
1
1 7
5 6
7114
2
"9i
8.
d.
»
*f
3
8
1
»»
4
>t
>«
»
9
10
If
»»
15
10
t
U
»
19
»»
6
6
18
6
2
7
7i
o
9i
DiAY ROOK.
115
J«n> ».
10
»»
14
»
19
u
»
23
23
Samiul Fletcher, Cr.
^y Keni hops, let. Ick- 5lbs. at £5 7t.0d.
„ Worcester, do. 12 at & 11 6
Six innnths' credit, or £ 6 per Ct. diicoont
for present payment.
Simmonds & Co. Liverpool,
Qy yellow soap. 2cwt. at 76c.
„ 12doz. candies, at Ss. 6d.
4 doz. mould do. at lis. 3d.
Cc.
i»
William Tomlingon,
To narrow cloth, 7 yds. at 5s. 6d.
„ callicoi 15 yds. at Os. 8^ per yd.
Dr
Hazard and Jones,
To 1^ St. yellow soap, at 9d. per lb.
^ St. mottled do. at 9^d.
9 lb. candies, at 9d.
Dr
>»
»»
3 lb. moulds, at Is. 0||d.
Jameit Sanderson,
By goods, as per invoice,
Cr.
Hazard and Jones,
To 17.^ lbs. loaf sugar, at Is. Id.
„ 12 ll)s. raw, do. at lOd.
„ li lbs. Congou tea, at 78. 6d.
„ i lb. Hyson, at 12s.
Dr.
£. j •■
6 IS
8
15
d.
6
3
5 9
7 12
6 2
14
19
1119
10
9
15
0! 5
Oi 6
Oi 3
1 II
n
31 .fohn Hcrdson,
'lo Hops, 10 lbs.
,, i| ream cap paper.
Br.
at Is. Id. p.
at 7d. per quire,
O'lO
6
7i
9
9
6
Hi
10
10
ic| 8
*
I
k
116
DAY BOOK.
Feb. 1
10
10
12
William Tomlinson,
To 2 8k yellow soap, at 9d. per lb.
„ 6 lbs. mould candles, at ]«. Id.
„ Id lb(. lump sugar, at la. O^d.
Dr
Tames Taylor, Dr
To half a year's interest on £70, at 6 per cent,
per ann.
William Tomlinson, Dr.
To 1 piece sup. blul cloth, ^6 yds. at 27s. 6d.
For bill at 1 month.
James Sanderson, Cr,
By cheese, 25Ct. 3qrs. 171bs. at £3, 2s. 6d
per cwt. ~
Allen, Wild, & Co. Leeds, Dr.
To my acceptance of their Bill at 2 mon. >
drawn 3d Jan. B. P. Book, No. 2, 5
Oats' Purveyance, in partnership with J.
Henderson, Dr.
To cash for oats purchased by me.
Do. do. by Henderson,
Do. for warehouse room, &c.
Cr.
By cash received for oats sold,
Do. do. by J. H.
Dr.
To Profit, i (£42, 13s. S^d.) being my share, ?
„ J. H. i (£42, 13 6i) being his share, 5
X
i.
1
6
16
2 4
15
49 10
80
80
26
449
18
II
I 13
477 4
507 9
65
d.
6
8
21
562i 1 1
3
8
11
2
8
10
85 6 11
DAY BOOK.
117
1832.
Feb. 12
John Henderson,
To cash advanced him on oatt' concern,
„ Oats sold and reeceived for by him,
By oats purchased by him,
„ his sh^re or profit,
£. i«.
Dr.
t
433 17
56
^'i
Cr.
488
19
449
42
!3
491
13
d.
3
H
11
*
f .^.'iP^^Fy
I
i
CASH BOOK.
^
V
(
' 1832.
Ca8H, Dr. F
£. 8
d.
1332. Contra. Cr.lF
. £. 8
.!d.
Jaa.l
To stock,
1500-
1
Jun. 9.
By S. Fletcher, paid
1
" Bills RcceivaUle,
him bill No. 1.
24 3; 9
No. 1, on S. John-
'IIisBCc't.l5l.5s.9d.
son,
24 :
1 9
less ISs.Sd.
" 9
" S. Fletcher, cheque
Discount,! 41. 10s. 6d
_ —
■ —
on Smith &, Co.
9K
5 3
Diif. sou Dr. side,
91.138.3d.
^,^ ^
. ^
.'
" 19
"Bills payable No. 1.
/
W. Holmes's,
45 lOI —
- , >:
,■ '
" 31
" T. Henderson, bal.
■
i'.
of accU. (ab. 4d.)
37
5 6
Bulauco,
142C17 9
Feb.l
1533 1
7~
Feb.l.
1533 17
To balance, per op-
By Bernard Mason,
posite.
14261
7 9
(total, e:m. Se. 2id.
" *
" W. Tomlinson,
abt. 5s. 2}d.)
832
(«b8t.3i,)
41
3
" 8
" W. Tomlinson, pd.
" J. Taylor, i year's
to J. Sims, by his or-
int. on 701.
11
5
der,
10
" W. Tomlincon, bill
on Jones Sc Co. Lon.
" 16
James Sanderson,
881. Is. 8|d. abtmt.
!
No. 2, duo 10 May,
60
Is. 8}d.
GS
t
I
Should have been at
1
Irao.deljit him with
Balance,
565 1
2 1U
i
discount 10a.
— -
— —
His account 491. 10,
diff. lOi. see Or.
'
side.
— -
- —
" 12
To Hazard & Jones's
Assignee, composi-
tion on 31. 15b. 6d. at
I
1
i
128. 6d in the £•
1
Loss, 11. 88.3}.
2
7 2k
'■
1495
12 Tu
- ' *
1495
12 lU
Index to the Ledger*
A. Allen, Wild, & Co.
B. Bills payable.
F. Fletcher, Samuel.
H. Heudervon, John.
Hazard & jonefl.
M. Maaon, Bernard.
O. Oats' Purveyance.
S. Slock,
Simmonds & Co.
Sanderson, Jamet,
T. Taylor, James,
Tomlinson, Wm.
1832.
Jan. 1
LEDGER.
'^'
6
9
lU
1633. 1 Stock,
J8n.l
Dr.
Feb.l2
1832.
Jan. 1
1832.
Jan.31
To mindriei, amount
of my debta,
" Balance acc't«
James Tavlor, Dr.
To money on bond,
" Haifa yr'a interest,
£■
922
460
JOBN HCNDERSONiDr.
To cash, 371. 5s. 6d.
abt. 4d.
Feb.l2
1838.
Jan. 8
" Sundries,
" Do. on oats' concern,
^ance,
70
1
71
13
18
•1
488
d.
10
0*
1833.
Jan.l
Contra,
1833.
Feb. 4
Cr.
By sundries, amoant
of property.
Cr.
491
Bernard Masoit, Dr.
To sundries,
'• Cash, 833, abat.
5s. Sid
1831. Bills Payable, Dr
Jan.l9Tocas]i,
Feb.l8
10
8
8
Jli
8J
Contra,
By cash for interest,
" Balance,
£• s.
1796
d.
9
cb
1
1332.
Jan.l
Feb.l2
Balunce,
flfo.
1833. ALLBit,Wii.o,&Co.
Dr.
Feb.lOlTo bills payable,
" ISj" Balance, ffo
7
832
840
45
80
14
5
00
10
00
9i
3i
Contra, Cr.
By balance of acc'ts.
By sundries on oats*
concern.
70
"ti
37
401
"491
n
15
10
13
8*
13
84
1832.
Jan. 1
Contra, Cr.
By purchase of house,
and duty, due Feb-
ruary 1st.
80
77
157
00
18
18
1832.
Jan. 1
Feb.l0
..-I—
1832.
Jan. 5
Contra, Cr.
ByHolmea'tb01,No.l
Allen, Wild, & Co's.
bill. No. 3, bb
840
I40
45
10
80
Contra,
By cloth,
Cr.
157
18
157
18
i
120
LEDGER.
1833. 'Samuel Fletcuir,
Dr
Jan. 9 To bill «nd diacount
ou bopi, c. b.
Feb.l2
1832.
Jan.U
Feb. 1
1833.
SiMMONOS & Co. Dr,
To balance, ffo.
Wm. ToMLiNfoir, Dr
To good*,
Do.
To cloth,
Coib, 101. diac. 10s.
(Iazard Sc Jones, Dr.
Jan. 19 To soap and candles,
" 28 " Sugar and tea,
14
1830.
Feb.lO
1830.
Feb.l3
1832.
Feb.l2
Ja's Sanderson, Dr
To cash, 881. abat.
Ic. 8).
Oats' Purveyanre iu
Co. with J. Herdson,
Dr.
To 8undrio8,
ProHi,
it0 8elf,421.13i.5|fd.
JT.H. 42l.13i.5id
CfENERAL Dr.
To cash in hand,
James Taylor owes me
s.
19
19
19
49
10
60
88
477
8.'
1833. Contra,
Jan. 9
Cr.
56-2
8}
11 10
5fir) 12
70
635
12
Jan. 10
By hops,
Choquo on Smith and
Co.
Contra,
By foods.
Cr.
1832.
Feb.l
1832.
Fcbl2
Contra.
By casli.
Abatement,
BybHI,
Cr.
c.b.
c.b
lit
»u
1832.
Jan. 23
Feb.lO
1832
Feb.12
Contra, Cr.
By cash .for composi-
tion.
Remainder lost.
CbNTRA,
By goods.
" Choose,
Cr.
£
15
9
34
14
Contra.
By sundries,
Cr
1832. Balance, Cr.
Feb.12 By T. Herdson, I owe,
Bills payHlilc,
Allen, Wild, & Co.
Simmonds &, Co.
Stock acc't. debited.
60
60
s. d.
5
13
19
19
13
13
9
3
3i
3J
7
80
88
5C2
15
2J
3*
11
6
2*
8i
10
562 11 10
1 17
80 "
77
14
460
635
II
6
04
m
15
9
24
14
5
13
13
60
3i
60
2*
3*
15
7
80
3
16
6
2i
88
84
5C2
11
10
562
11
10
1
80
77
H
46U
17
»>
18
19
18
4i
»»
6
Oi
635
12
lU
•*#