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IN MEMORIAM 
FLOR1AN CAJORI 




TUB 



WESTERN CALCULATOR, 



NEW AND COMPENDIOUS SYSTEM 



PRACTICAL ARITHMETIC; 

CONTAINING 

THE ELEMENT \RY PRINCIPLES AND RULES OP CALCULATION IN 
WHOLE, MIXED, AND DECIMAL NUMBERS, 

ARRANGED, DEFINED, AND ILLUSTRATED, 

IN A PLAIN AND NATURAL ORDER 5 

ADAI'TEP TO THE USE OF SCHOOLS, THROUGHOUT THE WESTERN COUNTRY 
AND PRESENT COMMERCE OF THE UNITED STATES. 

% 

IN EIGHT PARTS. 



BY J. STOCKTON/ A.M. 



PITTSBURGH : 

PRINTED AND PUBLISHED BY JOHNSTON & STOCKTON 
MARKET STREET. 



STEREOTYPED BY J. HOWE- 

1880. 



Entered according to IK af the Congress, in the year 1832, bv 
JOCNSTON & STOCKTWR, jfr Clerk's office of the District Court of 
the Western District W ' . 






PREFACE. 



AMONG the many systems of Arithmetic now used in our American 
schools, though each has its individual merit, yet all contain many 
things which are either entirely useless, or *of but little value to most 
beginners. 

It is to be regretted also, that in most of these systems, even if? 
those parts which are valuable and important, the authors appear not 
to have been sufficiently aware of giving a plain and natural arrange, 
ment and system to the whole. The;e is not that visible connexion * 
/ between the parts, which enables the attentive pupil to discover, as he ] 
/ progresses, that he is learning a system, and not a number of separate * 
^ and unconnected rules. 

In many things, also, more attention has been given to gratify the 
Jr^quirifjff of tliR jirn/ffipMt^thnn to furnish plain, but necessary instruc- 
tion to the beginner. The age, capacity, and progress of the scholar 
are also overlooked ; and a mode ^f instruction _too_ learned^ and too 
elaborate, is pursued. It is forgotten how difficult even the most sim- 
ple parts are to a young mind ; nor are the instances few, in which 
even the variety~of ways laid down, in which the same question may be 
solved, leaves the learner perplexed, and swells the size of the work. 

To remedy, in some measure, these defects, and to furnish our nu- 
merous schools, in the western country, with a plain and practical trea- 
tise of Arithmetic, compiled and printed among ourselves, thereby 
saving a heavy annual expense in the purchase of such books, east of 
the rnnnnt?iins T and {ikewi^ tliP ./..arringp tbageof^liave been the mo- 
lives which induced the compiler to undertake this work. 

In it the following objects have been steadily kept in view : 

1st. Plainness and simplicity of style, so that nothing should be in- 
troduced above the common" capacities of scholars, at the early age in 
which they are generally put to the study of Arithmetic. 

2d. A natural and lucid arrangement of the whole, as a systelii, in 
which the connexion and dependence of all the parts may be easily 
discovered and understood. To accomplish this object, the work ia 
divided into eight parts, following each other, in what appears to tnc 
compiler the natural and simple divisions of the science. Each of 
these parts is again divided into sections, following the same connected 
arrangement. In each of these sections, the rules are expressed in a 



M305992 



IV PREFACE. 

short and plain manner, and each rulo is illustrated, with a few easy 
and familiar examples, gradually proceeding from that which is sim- 
pie, to such as are more abstruse and difficult. 

3d. Clearness and precision in the definitions, directions, and exam- 
ples. Carefully explaining every technical term when first used, and 
thereby guarding against ambiguity and uncertainty. 

4th. Brevity in each part, so that every thing useless, or unimport- 
ant, may be excluded ; in order that the work may find its way into 
schools at the cheapest rate ; that parents, when examining the school- 
books of their children, may not find, whilst one part is worn out, the 
other is untouched, and half the price of the book entirely lost. 

How far these objects have been obtained, must be left to the deci- 
' sion of time. Should the work be found to aid the progress of scholars, 
in acquiring a practical knowledge of this useful science to save ex- 
penses in a book so many of which are required ; and be found a use- 
ful assistant to merchants, mechanics, and farmers, as well as in some 
degree to lessen the labor of teachers in this branch of the sciences ; 
the compiler will have obtained his object. 



NOTICE TO THE FOURTH EDITION. 

THE favorable reception, and wide circulation, of the former edi- 
tions of the Western Calculator, stimulate the author to make it still 
more deserving of public patronage. 

He has, therefore, at the suggestion of several respectable teachers, 
given sundry additional questions to some of the rules ; and also some 
other alterations, which several years' experience in teaching has 
pointed out. 

The greatest care has been taken to prevent errors from appearing. 
in this edition. 

Pittsburgh, February 1, 1823. 



APHORISMS 

FOR THE 

SCHOLAR'S CAREFUL CONSIDERATION AND ATTENTION. 



KNOWLEDGE is the chief distinction between wise men 
and fools ; between the philosopher and the savage. 

The common and necessary transactions of business can- 
not be conducted with profit or honesty, without the know- 
ledge of Arithmetic. 

He who is ignorant of this science must often be the 
dupe of knaves, and pay dear for his ignorance. 

Banish from your mind, idleness and sloth, frivolity and 
trifling ; they are the great enemies of improvement. 

Make study your inclination and delight ; set your hearts 
upon knowledge. 

Accustom your mind to investigation and reflection ; de- 
termine to understand every thing as you go along. 

Commit every rule accurately to memory, and never resP 
satisfied until you can apply it. 

As much as possible do every thing yourself; one thing 
found out by your own study, will be of more real use than 
twenty told you by your teacher. 

Be not discouraged by seeming difficulties ; patience and 
application will make them plain. 

Endeavor to be always the best scholar in your class, 
and to have the fewest mistakes, or blots, in' your book. 

" The wise shall inherit honor, but shame shall be the 
promotion of fools." 

A2 



EXPLANATION 

OF THE 

SEVERAL CHARACTERS EMPLOYED FOR THE SAKE OF 
BREVITY, IN THIS TREATISE. 



Two parallel lines, signifying equality: as, 100 cents= 

1 dollar ; that is, 100 cents are equal to 1 dollar. 

-f- Signifying more, or addition: as, 6 + 4=10; that is, 6 
arid 4 added make 10. This character is called Plus. 

A single line, signifying less, or subtraction : as, 6 4= 

2 ; that is, 6 le^s 4 is equal to two. This character is 
called Minus. 

x Signifying Multiplication: as, 2X 4=8 ; that is, 2 mul- 
tiplied by 4 is equal to 8. 

-I- Signifying Division: as, 6-f-3=2 ; that is, 6 divided by 

3 is equal to 2. 

:: : Signifying Proportion : as, 2 : 4 :: 6 : 12 ; that is, as 
2 is to 4, so is 6 to 12 ; or, that there is the same propor- 
tion between 6 and 12, as there is between 2 and 4. 

v/ or V Signifying the square root of the number before 
which it is placed : as, \/64=8 ; that is, the square root 
of 64 is 8. 

<X Signifying the cube root : as, y/64=4 ; that is, the cube 
root of 64 is 4. 

A Vinculum, or chain : denoting the several quanti- 
ties over which it is placed, are to be considered as one 
simple quantitv 



THE 

WESTERN CALCULATOR* 



PART I 
ARITHMETIC IN WHOLE NUMBERS. 



ARITHMETIC is the art, or science, of computing by num- 
bers, and is generally divided into five j^incipal parts, or 
primary rules : viz. Numeration, Additioi* Subtraction, 
Multiplication, and Division. 



SECTION 1. 

OF NUMERATION. 



NUMERATION (or, as it is often called, Notation) is the art 
of expressing any given or supposed number, by the ten 
following characters : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The 
first of these is called a cipher, the rest are termed digits or 
figures. 

These nine digits are divided into three periods, three in 
each period. The first period includes units, tens, and hun- 
dreds. The second period includes thousands, tens of thou- 
sands, hundreds of thousands. The third period includes 
millions, tens of millions, hundreds of millions. 

Note. The cipher is also called nought and zero. They are all 
Arabic characters. 



NUMERATION. 

The relative value of each period, and the different fig- 
ures ID each period, may be learned from the following 

TABLE. 

3d period. 2d period. 1st period. 




9 
9 
9 


8 

8 


7, 










9 


8 


7, 


6 








9 


8 


7, 


6 


5 






9 


8 


7, 


6 


5 


4, 




9 


8 


7, 


6 


5 


4, 


3 


9 


8 


7, 


6 


5 


4, 


3 2 


9 


8 


7, 


6 


5 


4, 


321, 



After the foregoing table, with the preceding definitions 
and explanations, are well explained by the teacher, and 
accurately committed to memory by the pupil, let him next 
proceed 

To write Numbers, 

Observing carefully the following 

RULE. 

Write down first, the given sum, in such figures as ex- 
press its value, and then supply the deficiencies therein with 
ciphers 

Application. 

Write down in figures the following numbers. 

1. Sixteen. 

2. Forty -nine. 



ADDITION. 9 

3. Three hundred and eighty-five. 

4. Two thousand six hundred and ten. 

5. Sixty-four thousand, five hundred and thirty-six. 

6. Two hundred and fifty-three thousand, eight hundred 
and forty-two. 

7. Five millions, six hundred thousand and six. 

8. Ninety millions, three hundred and five. 

9. Eight hundred and twenty-nine millions, six thousand 
and two. 

Write dcnvn in words at length the following sums : 
5, 17, 35, 458, 6829, 72348, 384721, 2683200, 50678024. 

Numbers are also expressed by letters, and are called 
numeral letters, or Roman numbers. Thus, 

12.3456 7 8 9 10 11 20 30 
1, II, III, IV, V, VI, VII, VIII, IX, X, XI, XX, XXX, 

40 41 50 60 70 80 90 100 200 500 1000 
XL, XLI, L, LX, LXX, LXXX, XC, C, CC, D, IVL 

When a letter of less value stands before one of a greater, 
it diminishes, but when placed after, it increases, the value 
of the greater. 



SECTION 2. 

OF ADDITION. 

ADDITION is of two kinds, viz. Simple and compound. 

Simple addition teacheth to collect two or more numbers 
of the same denomination into one sum : as, 6 dollars and 
4 dollars, make 10 dollars. 

RULE 1. Write the different numbers in the given sum in 
such a manner, that the units may stand under units, tho 
tens under tens, the hundreds under hundreds, &c. till the 
whole is set down. 

2. Begin with the column of units, and add it into OIK 
sum, carrying to the next column, one for every ten, and 
set down the remainder directly underneath ; proceed in the 
same manner fnnn tens to hundreds, &c. till all is finished. 



10 ADDITION. 

Prove the work by reckoning downwards as well as up- 
wards, and if the amounts be equal the work is right. 



EXAMPLES. 



Dollars 
2465 
4832 
6143 



Yards 

2. 468256 
348928 
764182 



13440 added upwards. 



13440 added downwards. 



Feet 

3. 647502434 
861948260 
959276398 



4. 258335091 5. 


237680923 


6. 919283746 


138097653 


423315687 


213536978 


573217809 


098172635 


321325687 


532458976 


523516533 


978562313 


532175633 


653213563 


321897553 


249753290 


327865309 


213587921 




7. 5643218624 


8. 4 


9. 6856789436 


135940536 


45 


40590428 


42006302 


456 


36491 


9580469 


4567 


2849653210 


550214 


45678 


540 


32651 


456789 


34906 


4168 


4567890 


3458000 


324 


45678901 


300 


68 


456789012 


9 


4 


4567890123 


35 




10. 3683678048934 


11. 


595536210486 


2864948946496 




376891345613 


8498649476828 




765248567446 


3646280568245 




684720476828 


6421424H78427 




852134567812 


3678156496862 




236889634567 


76545S4964859 




335678902345 



SUBTRACTION. 11 

Application. 

1. Add 125-f 23 + 16 -f 2060 + 8009574 + 6. 

^Iws. 8011804. 

2. Add one hundred and twenty-nine, six hundred and 
fifty-four, eight thousand and seventy, ten thousand, and 
four millions. Ans. 4018853. 

3. If I have received 125 dollars from A, 286 from B, 
29 from C, 672 from D ; how much have I received from 
all four? Ans. 1112. 

4. Bought 60 barrels of flour from one man for 480 dol- 
lars, 75 barrels from another for 675 dollars, 220 from an- 
other for 2200 dollars, and 126 from another for 1386 dol- 
lars ; how many barrels of flour had I, and how much did 
they cost me ? Ans. 481 barrels, and cost 4741 dolls. 

5. A farmer raised in one year 297 bushels of wheat, 
125 of rye, 754 of corn, 127 of barley, and 245 of oats ; 
how many bushels did he raise in all ? Ans. 1548. 

6. Add one thousand two hundred and nine, four hun- 
dred and seventy-six, eight thousand and seventeen, three 
millions, one hundred and nineteen thousand, two hundred 
and twenty-one together. Ans. 3128923. 

7. James was born in the year 1811 ; in what year will 
be Le 21 years old? Ans. 1832. 

8. A father bequeathed to his 5 sons the following sums, 
viz : to George he gave 3560 dollars, to William 3240, to 
Samuel 2850, to Henry 2555, and to Thomas 2226 ; how 
much did he bequeath in all ? Ans. 14431 dolls. 



SECTION 3. 
OF SUBTRACTION. 

SUBTRACTION is either simple or compound. 
Simple subtraction is the taking a less number from a 
greater, and thereby finding the difference. 

RULE. 

Place the less number under the greater, with units under 
units, tens undei tens, &c. ; begin with the units, and take 
the under figure r rom the upper, and then proceed with t ic 



12 SUBTRACTION. 

tens, &c. in the same manner. But if the under figure is 
the greatest, then suppose ten added to the upper figure, and 
take the lower from that number, carrying 1 to the next 
place. Or, take the lower figure from 10 and add the upper 
one to the remainder. 

PROOF. 

Add the remainder to the less number, and that will equal 
the greater. 



From 446875296 
Take 234521173 



Rem. 212354123 



Proof 446875296 

From 76542189768 
Take 32478127130 



EXAMPLES. 



From 86250732493, 
Take 37014921872 



Rem. 49235610621 



Proof 86250732493 

From 5417630912 
Take 27096470 



From 90621247680 
Take 34567892000 



From 100000000 
Take 09999999 



Application. 

1. What was the age of a man in the year 1818, who 
was born in 1777 ? Ans. 41 years old. 

2. A merchant owes 5648 dollars, and pays thereof 3460 ; 
how much is yet to pay? Ans. 2188 dolls. 

3. D having on hand 1260 barrels of flour, sells to A 
320, and to B ^435 ; how many barrels are yet unsold ? 

Ans. 505 barrels. 

4. From six thousand take six hundred, and tell what 
remains. Ans. 5400. 

5. Suppose a boy had 145 cents given him at one time, 
15 at another, and 40 at another ; and he gave 35 cents for 
;i penknife, Mf) Inr a slate, (M lor paper, and 30 for ap l les ; 
!io\v many rents has he left? Ans. 106 cents. 



MULTIPLICATION. 13 

SECTION 4. 

OF MULTIPLICATION. 

MULTIPLICATION is either simple or compound. 
Simple multiplication is a compendious way of adding 
numbers of the same denomination into one sum. 

The number to be multiplied, is called the multiplicand. 
The number multiplied by, is called the multiplier. 
The amount produced, is called the product. 
The multiplier and multiplicand are often called factors. 

MULTIPLICATION TABLE. 



fl| 2] 3| 4| 5| 6| 7| 8| 


9| 10J 11| 12] 


2| 4| 6| 8|10J12|14|16| 


18| 20| 22 24 


3| 6| 9|12|15|18|21|24| 


27| 30| 33| 36 


4! 8|12!l6|20|24|28j32| 


36| 40| 44| 48 


5|10|15|20|25|30|35|40| 


45| 50| 55 60 


6|12|18|2430|36|42|48| 


54| 60| 66| 72 


7|14|21|2835|42|49|56| 


63| 70| 77| 84 


8|16|24|32|40|48|56|64j 


72| 80| 88| 96 


9|18|27|36|45|54|63|72| 


81| 90| 99|108 


10|20|30|40|50|60|70|80| 


90|100|110]120 


11|22|33|44|55|66|77|88| 


99|110|121|132 


12|*4|36|48|60|72|84|96|108|120|132144 



This table must be committed to memory, with great care 
and accuracy, till it can be used without difficulty or hesita- 
tion by the scholar. 

Case 1. 
When the multiplier does not exceed 12. 

RULE. 

Place the multiplier under the multiplicand ; units under 
units, and tens under tejis, and then multiply as the table 
directs, taking care to carry 1 for every 10. 

EXAMPLES. 

46274963 24639576 3675432568 
24 S 



0-25*9920 



14 MULTIPLICATION. 

246H5761 4708-44753 964703024 

a 5 6 



74057343 



74020005 2901946808 246354276 
8 9 11 



Case 2. 

When the multiplier exceeds 12. 
RULE. 



Multiply by each figure in the multiplier separately, be- 
ginning with units, taking care to set the first figure in each 
product directly under its own multiplier. Then add as if. 
addition. 



EXAMPLES. 



2345601 68523047653 

234 2367 



9382404 
7036803 
4691202 

.,48870634 



PROOF. 



Method 1. Change the multiplier and multiplicand ; and 
then, if right, the product from this multiplication will he 
equal to the first. 

Method 2. Cast the nines out of each factor separately . 
set dovvn the remainders and multiply them together ; casi 
rhe nines out of this product, and note the remainder ; then 
;ast the nines out of the product, and if right, th* 3 two last 
remaindors will he equal. 



MULTIPLICATION. 16 

EXAMPLES. 

Method 1. Method 2. 

246 425 425 
425 246 246 



1230 2550 2550 
492 1700 1700 
984 850 850 



104550 104550 104550 6 

Note. This last method is not absolutely certain ; yet the probability 
is so great, that in general it may be relied on. 

3. Multiply 5221 by 145 Ans. 757045 

4. 23430 230 5388900 

5. 3800920 80750 306924290000 
S. 89536925 735 65809639875 
7. 78965987 5893 465346561391 

8. What will 75 bushels of wheat come to at 1,15 cents 
per bushel ? Ans. 86 dolls. 25 cents. 

9. Bought 3950 Ibs. of coffee, at 29 cts. per Ib. what must 
1 pay? Ans. 1145 dolls. 50 cents. 

10. There are 12 pence in one shilling. How many are 
there in 40 ? Ans. 480 pence. 

Case 3. 

When the multiplier is the exact product of any two fac- 
tors in the multiplication table. 

RULE. 

Multiply the given sum by one of these ; and that produci 
multiplied by the other, will give the number required. 

EXAMPLE. 

1. Multiply 4236 by 16. 

41 

4X4=16. 



Product 67776 J 

2. Multiply 871075 by 21 Ans. 18292575 

3. 2453642 36 88331112 

4. 43102 64 2758528 

5. 23645 144 3401880 
tt 12071 99 



48.00 
3600 by 400 
44000 550000 
663000 60000 


Ans. 4800 
1440000 
24200000000 
39780000000 



16 MULTIPLICATION. 

Case 4. 

When there are ciphers at the right of one or both the 
factors. 

RULE. 

Omit them in the operation, but annex them to the product 

EXAMPLE. 

1. Multiply 240 by 20. 24.0 

2.0 



2. 
3. 
4. 

Note. When the multiplier is 10, the product will be found by add- 
ing one cipher to the multiplicand ; if 100, add two ciphers ; if 1000 
add three ; &c. 

EXAMPLE. 

1. Multiply 200 by 10 Ans. 2000 

2. 462 100 46200 

3. 879 1000 879000 

Application. 

1. A gentleman owes 25 laborers 15 dollars each ; how 
much does the whole come to ? Ans. 375 dolls. 

2. A saddler owes his journeyman for 43 days' work, ai 
125 cents per day ; how much does he owe him in all ? 

Ans. 53 dolls. 75 cts. 

3. A merchant buys 440 yards of muslin at 32 cents per 
yard ; how much does the whole cost? Ans. 140 dolls. 80 c. 

4. A farmer sells 60 bushels of wheat at 125 cents per 
bushel ; 40 bushels of rye at 85 cents ; 34 of corn at 50 
cents ; how much is he to receive for each, and how much 
does the whole amount to 7 

Ans. 75,00 cents for the wheat, 34,00 cents for the rye 
17,00 cents for the corn ; and the. whole amounts to 126,00 
<-<-nts, or 126 dollars. 

.1. A dollar is equal to 10 dimes, and a ciime is equal to 
10 cents ; how many dimes and cents are there in 100 dol- 
Ans. 1000 dimes, and 10,000 cents. 

6. llo\v_ many panes of glass are then: in a house that 
has 32 windows, 20 of which have 2 4 lights <--ach, and the 
r<st hav*' J - : ca<-h .' Ann. 096 panes. 



17 

7. What sum is equal to 7525 multiplied by 125? 

Ans. 940625. 

8. A has 250 dollars, B has three times as many, and C 
has four times as many as B ; how man) 7 dollars have B and 
C each, and how many have they altogether ? 

Ans. B has 750 dolls. C. 3000 dolls, altogether $4000. 



SECTION 5. 

OF DIVISION. 

Division is either simple or compound. 
Simple division is finding how often one number is con- 
tained in another of the same name, or denomination. * 
The number given to be divided, is called the dividend. 
The number given to divide by, is called the divisor. 
The result, or answer, is called the quotient. 

Case 1. 
When the divisor does not excised 12. 

RULE. 

Find how often the divisor is contained in the first figure 
or figures in the dividend, under which set the result, if any 
remain, conceive it as so many tens added to the next figure, 
and then proceed in the samp manner. 

Division is proved by multiplying the quotient by^the di- 
visor, and adding the remainder, if any : the amount will 
equal the dividend. 

EXAMPLES. 

Divisor 2)46578238 3)672245139 4)4756394344 



Quotient 23289119 
2 



Proof 



46578238 



224081713 
3 

672245139 



5)97036142 8)37846210 



12)64381259 



6)3824966 7)46825486 9)8297463813 



18 



DIVISION. 



Case 2. 
When the divisor exceeds 12. 

RULE. 

Begin with as many of the first figures in the dividend as 
will contain the divisor. Try how often the divisor is con- 
tained therein, and set the result in the quotient. Subtract 
the product of the divisor multiplied by the quotient figure 
from the dividual above, to this remainder annex the rvxr 
figure in the dividend for a new dividual, arid so proceed tii : 
all the figures in the dividend are brought down. 

Note. A dividual is when one or more figures of the dividend, (in the 
operation of long- division) are divided separately from the rest. 



EXAMPLES. 



Divis. 42)9870 (235 Quot. 
84 42 



41)94979 (2316 
82 41 



147 
126 

210 
210 



470 . 
940 

9870 proof. 



123 



67 
41 



2316 
9264 

94956 
23 rem. 



269 94979 pr. 
246 

23 Rem, 



3. Divide 29687624 
4. 47989536925 


Quotient. 

by 64 Ans. 463869 
735 65291886 


and 8 Rem. 
715 


5. 


4917968967 


2359 


2084768 


1255 


6. 


5374608 


671 


8009 


569 


7. 


19842712000 


175296 


113195 


81280 


M. 


5704392 


108 


52818 


43 



Case 3. 

When the divisor is the exact amount of any two factors 
in the table. 

RULE. 

Divide the given sum by any one of these, and the quo- 
tient by the other. 



DIVISION. 
EXAMPLE. 

Divide 9870 by 42. 

6 ) 9870 

7 ) 1645 

235 Ans. 

Case 4. 
Vhen one or more ciphers stand on the right of the divisor. 

RULE. 

Omit them in the operation, cutting off from the right of 
the dividend as many figures, taking care to annex them to 
the remainder. 

EXAMPLE. 

1 . Divide 2564 by 200. 

2.00)25.64 



12 1 Rem. 
64 



Quot. 12 164 Rem. 

2. Divide 87654 by 600 Ans. 146 54 Rem. 

3. 28347 ' 80 354 27 

4. 137000 1600 85 1000 

Note. When the divisor is 10 the quotient will be had by cutting off 
one figure from the right of the dividend, when the divisor is 100 cut 
off two figures, when it is 1000 cut off three figures, &c. When the 
figures cut off from the right of the dividend are digits, they are to be 
considered as so much of a remainder. 

EXAMPLE. 

I. Divide 5640 by 10. 

1.0)564.0 Ans. &64. 



L\ Divide 25654 byj 100 Ans. 256 54 Rem. 

'3. 876029 1000 876 29 

4. 800000 10000 80 

Application. 

1. Several boys went to gather nuts, and collected 4275 : 

when they had divided them, each had 855 ; how many 

boys were in company? Ans. 5. 



20 DIVISION. 

2. If 2072 apple trees were planted in 28 rows, how 
many would there be in each row ? Ans. 74. 

3. If 45000 dollars were divided among 75 persons, how 
many would each one receive? Ans. 600. 

4. Into how many parts must I divide the number 8164. 
so that each part may be 27, leaving the remainder 10? 

' Ans. 302. 

5. There is a certain number, to the double of which if 
you add 12, then 5 times that sum will equal 150 ; what is 
that number. Ans. 9. 

6. A father dying, left 13440 dollars to be divided among 
his 6 sons in the following manner, viz. to the eldest one- 
fourth part, to the second one-fifth, to the third one-sixth, to 
the fourth one-seventh, to the fifth one-eighth, and to the 
youngest the remainder ; what was each son's share ? 

Ans. 1st 3360, 2d 2688, 3d 2240, 4th 1920, 
5th 1680, 6th 1552 dolls. 

7. What number^ if multiplied by 72084, will make 
5190048? Ans. 72. 

8. A, B, and C, engage to do a piece of work for 228 dolls* 
which together they accomplish in 40 days : now it was 
previously agreed that A should have 1 cents per day more 
than B, and" B 10 cents more than C ; what was each man's 
share? Ans. A 80, B 76, C 72 dolls. 

9. A man on counting his money, found he had an equal 
number of half eagles, (5 dollar pieces) half dollars, and 
quarter dollars, and that the whole amounted to 1437 dol- 
lars 50 cents ; how many pieces of each kind had he ? 

Ans. 250 of each kind. 

10. The crew of an armed ship, consisting of the cap- 
tain, mate, and 40 men, took a prize worth 4550 dollars 
now it was agreed that the captain should have 6 shares, 
the mate 4, and each seaman 1 share ; what did each one 
receive ? Ans. The capt. 546 dolls, the mate 364, and 

each seaman 91 dolls. 

As but few examples are given under each of the foregoing rules, it 
is recommended that every teacher add as many similar ones, as may 
be found necessary to make the pupil well acquainted with their appli 
cation, and both expert and accurate in working such questions as 
properly belong to these rules. Every experienced teacher is well 
aware that until this knowledge is obtained by the scholar, every at- 
tempt at any thing farther is only a waste of time and money. When 
this knowledge is once acquired, the future progress of the scholar will 



FEDERAL MONEY. 21 

oe pleasant and rapid. The teacher will then be justly rewarded lor his 
labor and trouble in this part, by the approbation of parents, and the 
gratitude of his scholars, who will have acquired the necessary qualifi- 
cations (accuracy and expertness) for the great variety of studies and 
avocations in future life, which require the aid of arithmetic and math- 
ernatics. 



PART II. 
ARITHMETIC IN MIXED OR COMPOUND NUMBERS. 



SECTION I. 

FEDERAL MONEY 

Is so called from its being the general currency establish- 
ed by the Federal, or United States' government, and is justly 
considered superior to every other kind of currency now in 
use for its simplicity and plainness. 

Standard weight as establieh- 
ed by law. 

Its denominations are, dwt. gr. 

10 Mills make 1 Cent 

10 Cents 1 Dime 1 16 f, > gn 

10 Dimes or 100 Cents 1 Dollar 17 ' l| < 
10 Dollars 1 Eagle 11 4| 5 n } , 

i Eagle 5 14J- $ b 

From this table it will readily be seen, that addition, sub- 
traction, multiplication, and division of federal money may 
be performed as if they were whole numbers. It will also be 
seen that to reduce any number of mills to cents, it is only 
necessary to point, or cut off the last figure, as 100 mills 
= 10,0 cents; an^l cents in the same way to dimes, as 100 
eents=:10,0 dimes, and dimes to dollars, as 100 dimes=:10,0 
dollars, and dollars to eagles, as 100 dollars .10,0 eagles; 
and also that eagles may be brought to dolls, and dolls, to 
dimes, &c. by adding a cipher to each one, as 10 E.= 100 
dolls. = 1 OOOd. = 1 OOOOc. 1 OOOOUm. 

In all calculations in federal money, according to com- 
mon custom, it is usual to omit the names of eagles, dimes, 



22 FEDERAL MONEY. 

and mills, and only to reckon by dollars and cents; the 
eagles being considered as so many 10 dollars, the dimes as 
so many 10 cents, and the mills as fractional parts of the 
cent. See the following 

. TABLE. 



i s 



Si Dolls. 

1, 2 5 1,25 cents 

3 4, 1 2 i 34,12 and a fourth cents 

4 5 6, 2 5 i 456,25 and a half cents 
8 2 6 4, 7 5 | 8264,75 and three-fourth cents 
Note. 1. In addition, subtraction, multiplication, and division of 
federal money, if the sums are dollars only, the amount, remainder, 
product, or quotient, will be dollars; but when the sums consist of 
dollars and cents, or cents only, the two first right hand figures arc 
cents, and all the rest are dollars. 

2. When fractions of cents are used according to the above table, 
every four of them make one cent : in adding, or subtracting these, 
we carry one for every four ; and in multiplying, the upper figure, called 
the numerator, is to be multiplied by- the multiplier, and divided by the 
lower figure, called the denominator. 

EXAMPLES OF ADDITION. 

Edd cm DC DC 

25,6,4,8,2 5675,25 53258,75* 

24.7.6.2.4 2386,63 93620,33^ 

63.8.1.3.5 3972,80 30176,56$ 

92.2.3.4.6 7285,75 . 27532,35 



206,4,5,8,7 



SUBTRACTION. 

E d d c m DC D 

83,6,5,3,5 
32,9,3,7,5 



50,7,1,0,0 



FEDERAL MONEY. 23 

MULTIPLICATION. 

Edd cm DC DC 

23,6,3,5,7 2637,25 6378,75 

369 



70,9,0,7,1 

DIVISION. 

Edd cm DC DC 

2)63,3,8,6,2 5)3632,75 8)82750,33 



31,6,9,3,1 



Promiscuous Questions. 

1. Add 25 eagles, 62 dollars, 8 dimes, 75 cents, and 5 
mills. Ans. 3.13d 55c 5m. 

2. A person deposited at bank 1055 dollars in notes, 260 
dollars in gold, 3650 dollars in silver, and 2,50 cents : how 
much is the amount? Ans. 4967d 50c. 

3. Bought a barrel of sugar for 39 dollars 87 cents, a 
bag of coffee for 22 dollars 18J cents, and a pound of tea 
for 2 dollars 12^ cents ; how much do they all cost? 

Ans. 64d 18jc. 

4. Bought goods to the amount of 645 dollars 95 1 cents, 
and paid at the time of purchase 350 dollars ; how much re- 
mains to be paid ? Ans. 295d 95 f. 

5. A man lent his friend 1000 dollars, and received at 
sundry payments, first 160 dollars 25 cents, second 285 
dollars 66-J cents, third 300 dollars 28| cents ; what remains 
yet to be paid? Ans. 253d 79fc. 

6. What is the product of 102 dollars 19 cents, multiplied 
by 120? Ans. 12262d 80c. 

7. What will 16 barrels of flour amount to, at 4 dollars 
50 cents per barrel ? Ans. 72d. 

8. How much will 132 pieces of calico come to, at 11 
wllars 37 i cents a piece? Ans. 2293d 50c. 

9. What is the quotient of 6022 dollars 50 cents, divided 
by 5? Ans. 1204d 50c. 

10. A butcher bought 18 beef cattle for 252 dollars 90 
cents; how much did he pay for each? Ans. 14d 05c. 

11. Bought 45 yards of linen for 22 dollars 50 cents, 
what was the price of one yard? Ans. 50cU- 



24 COMPOUND ADDITION. 

12. If 25 men expend 15555 dollars 50 cents in the erec- 
tion of a bridge, how much has each one to pay, if the 
shares are equal ? Ans. 622d 22o 

Having treated of federal money separately, inasmuch as it requires 
to be well understood, seeing it is the general currency in the United 
States; we now proceed to the other parts of mixed numbers, or as they 
are frequently termed, divers denominations. 



SECTION 2. 

OP COMPOUND ADDITION. 

COMPOUND Addition is the collecting together, and thereby 
ascertaining the amount of several quantities, of divers de- 
nominations. 

RULE. 

Place the numbers in such a manner, that all of the same 
denomination may stand directly under each other, then be- 
ginning with the lowest denomination, add as in whole num- 
bers, carry at that number which will make one of the next 
greater; set down the remainder (if any) and so proceed till 
nil are added. 

PTCOOF as in simple addition. 

ENGLISH MONEY. 

The denominations are, pounds, shillings, pence and far- 
things, and are 
Thus valued : 
4 farthings (marked qr.) make 1 penny (marked) d. 

12 pence 1 shilling *. 

20 shillings 1 pound . 

TABLE OF SHILLINGS. 

s. . s. 

20 shillings make 1 00 

30 .... 1 10 

40 .... 2 00 

50 .... 2 10 

60 .... 3 00 

70 .... 3 10 

80 .... 4 00 

90 . . . 4 10 

100 . 5 00 



PENCE TABLE. 



d. 

20 pence make 

30 . . . 

40 ... 

50 ... 

no . . . 

70 ... 

90 ... 

00 ... 

100 ... 



*. d. 

1 8 

2 6 

3 4 

4 2 

5 

5 10 

6 8 

7 6 
a 4 



COMPOUN D A DDITIO3N . 25 

EXAMPLE* 

. s. d. . s. d. qr. . s. d. qr. 

12-*6 11 8 35678 11 9 \ 2368 17 5 4 

9462 8 4 37562 18 7 $ 3969 19 11 | 

3215 10 6 63497 15 10 i 9386 14 6 $ 



.13934 10 6 



raor WEIGHT. 

This weight is used for jewels, gold, silver, and liquors. 

The denominations are, pounds, ounces, pennyweights, and 
grains. 

Thus valued. 

24 grains (gr.) make . 1 pennyweight dwt. 

20 pennyweights . . 1 ounce oz. 

12 ounces .... 1 pound ft. 

EXAMPLES* 

ib.oz.dwt.gr. lb. oz. dwt. gr. lb.oz.dwt.gr. 

4 10 15 16 5 8 11 16 45 17 11 

8 6 10 11 9 10 15 21 9 6 12 9 

6 9 14 23 6 11 18 17 18 11 19 23 



20 3 1 2 



AVOIRDUPOIS WEIGHT 

This weight is used for heavy articles generally, and all 
metals but gold and silver. 

Tlie denominations are, tons, hundreds, quarters, pounds 
1 ounces, and drams. 

Thus valued. 

16 drams (dr.) make ... 1 ounce oz. 

16 ounces 1 pound lb. 

28 pounds 1 quarter qr. 

4 quarters, (or 112 lb.) . . 1 hundred cwt. 

20 hundreds 1 ton T. 

C 



2f> COMPOUND ADDITION 

EXAMPLES. 

T. cwt. qr. Ih. oz. dr. T. cwt. " qr. Ib. oz. dr. 

10 16 2 24 9 12 856 12 3 19 11 10 

15 11 1 15 12 9 537 19 1 23 8 9 

85 8 3 19 13 13 638 10 2 21 12 6 

18 15 1 14 10 8 897 19 3 27 15 15 



APOTHECARIES' WEIGHT. 

This weight is used by apothecaries in mixing medicines, 
.' "t they buy and sell by avoirdupois. 

The denominations are, pounds, ounces, drams, scruples, 
and grains. 
Thus valued. 

20 grains (gr.) make ... 1 scruple sc. or 9 

3 scruples 1 dram dr. 3 

8 drams , . 1 ounce oz. 3 

12 ounces 1 pound Ib. ft 

EXAMPLES. 

Ib. oz. dr. sc. gr. Ib. oz. dr. sc. gr. 

* 5 2 1 16 17 5 7 2 14 

3 11 7 2 19 80 3 2 1 16 

6 8 6 1 12 85 10 3 2 5 

5 2 4 2 9 36 6 2 1 15 



CLOTH MEASURE. 

By this measure, cloths, ribands, &c. are measured. 
The denominations are, Ells French, Ells English, Elh 
Flemish, yards, quarters, and nails. 

Thus valued. 
4 nails (na.) make ... 1 quarter qr. 

4 quarters I yard yd. 

3 quarters i Kll Flemish E. FL 

5 quarters 1 Mil Kn^iish E. En. 

quarters i Kll French E. Fr. 

EXAMPLES. 

Yd. qr. na. E. 1?l. qr. nn. E. Fr. qr. na. E. En. qr. na 

56 2 2 HO 2 3 16 4 2 53 4 3 

sG 13 18 1 2 17 5 1 53 3 ? 

33 3 2 36 2.1 80 2 2 32 2 1 

1*8 2 1 :W 21 13 3 3 81 



COMPOUND ADDITION. 27 

LONG MEASURE. 

By this, lengths and distances are measured. 

The denominations are, degrees, leagues, miles, furlongs, 
poles, rods or perches, yards, feet, inches, and barley- 
corns. *y 

Thus estimated. 

3 barleycorns (be.) make 1 inch in. 

12 inches 1 foot ft- 

3 feet 1 yard yd. 

5 yards (or 16J feet) . 1 rod, pole, or perch P. 

40 poles rods, or perches, > fo , /wn 
(or 220 yards) $ 

8 furlongs (or 320 poles, ? -, -, a/ r 

?-r^n i \ t 1 mi l e * 

or 17 GO yds.) $ 

3 miles 1 league L. 

60 geographic, or ) ., , 

flft-f * * miles 1 degree deg. 

69i statute ^ 

360 degrees make a circle, or the circumference of the 

earth. 
A hand is a measure of 4 inches, and a fathom of 6 feel. 

EXA3fPLES. 

deg. m. fur. po. yds. ft. in. be. L.' M. fur. yds. ft. in. 

50 30 5 15 2 2 9 2 5 2 6 75 2 11 

60 25 7 12 4 1 10 1 3 1 4 95 1 9 

75 35 2 92281 2 1 3 15 2 8 

20 55 6 8 1 1 11 2 125 200 1 6 



LAND MEASURE. 

By this the quantity of land is estimated. 

The denominations are, acres, roods, perches, yards, anl 
feet. 

Thus rated. 

9 feet (ft.) make . . . . . 1 yard yd. 

30^ yards ..._....! perch P. 

40 perches 1 rood R. 

4 roods (or 160 perches) . . 1 acre A. 



28 COMPOUND ADDITION. 

EXAMPLES. 

A. R. P. A. R. P. A. R. P. 

25 3 20 265 2 15 246 3 29 

33 1 16 375 1 29 762 1 12 

33 2 34 860 3 39 632 2 11 

68 1 39 632 2 20, 357 3 20 



CUBIC, OR SOLID MEASURE. 

By this, wood and other solid bodies are estimated. 
The denominations are, cords, tons, yards, feet, and inches 

Thus rated : 

1728 inches (in.) make ... 1 foot ft. 

27 feet ....... 1 yard yd. 

40 feet of round timber, or > ^ tQQ y 

50 feet of hewn timber $ 
128 feet 1 cord cor. 

EXAMPLES. 

Co. ft. in. T. ft. in. T. ft. in. 

4 112 1260 6 39 1384 23 *12 1400 
6 84 1500 2 26 526 68 45 1600 
8 127 1700 8 18 260 82 49 1700 

5 63 403 3 12 1100 96 18 50 



TIME. 

This relates to duration. 

The denominations arc, years, months, iceeks, days, hours, 
minutes, and seconds. 

The relative differences are these. 

60 seconds (sec.) make ... 1 minute mi. 

(>0 minutes 1 hour h. 

24 hours 1 day d. 

7 d.'iys 1 week w. 

4 weeks 1 month M. 

12 months, 5'^ weeks, or 365 > . . vr 
days and 6 hours \ 

Note. The solar year, according to the most .xnrt observation, con 
tains 3G5dav.s, 5 JuMirs, 4ft minutes, 57 seconds. 



COMPOUND ADDITION 21) 

TTie number of days in the 12 calendar months, is thus found: 

Thirty days are in September, 
April, June, and November ; 
February hath twenty-eight alone, 
And all the rest have thirty-one. 

JNote. Every fourth year is called bissextile, or leap-year, in which 
February has twenty-nine <days. 

EXAMPLES. 

Y. M. da. h. mi. sec. Y. da,, h. mi. sec. 

22 10 25 16 34 55 4 350 15 19 5 

34 6 16 20 48 33 2 268 13 54 38 

46 9 13 23 59 59 6 350 22 50 50 



MOTION. 

This relates to the measure of circles. 

The denominations are, circles, (or revolutions) signs, de- 
grees, minutes, and, seconds. 
The relative differences are, 

60 seconds (sec.) make - 1 minute mi. or ' 

60 minutes ..... 1 degree deg. or 

30 degrees - - - . - 1 sign sig. 

12 signs (or 360 degrees) 1 circle 

EXAMPLES. 

sig. deg. mi. sec. sig. ' " 

2 24 48 58 3 20 30 40 

2 29 59 59 2 25 35 45 

3 21 20 20 3 2d 38 58 



LIQUID MEASURE. 

This is used for measuring wine, spirits, cider, beer,*&c. 
The denominations are, tuns, pipes or butts, hogsheads, bar- 
rels, gallons, quarts, pints, and gills. 

Thus estimated. 

4 gills (gi.) make - - 1 pint pt. 

2 pints - - - - 1 quart qt. 

4 quarts 1 gallon gal. 

63 gallons - - - - 1 hogshead hhd. 

2 hogsheads 1 pipe, or butt pi. bt 

2 pipes (or four hogsheads) 1 tun T. 

Note. By a law of Pennsylvania, 32^ gallons make a barrel, and 
16 gallons make a half barrel. 

C2 



30 



COMPOUND SUBTRACTION. 
EXAMPLES. 



T.hhd.gal. qt. pt. 

4 3 53 2 I 

6 2 25 3 1 

8 1 62 1 1 



T. hhd. gal. 
24 2 33 
19 3 54 
34 1 50 



DRY MEASURE. 

This is used for measuring grain, salt, fruit, &c. 
Tht denominations are, bushels, pecks, quarts, and pints. 

Thus estimated. 

2 pints (pt.) make - - 1 quart qt. 

8 quarts - - - 1 peck P. 

4 pecks (or 32 quarts) 1 bushel bu. 



bu. P. qt. 

25 2 4 

36 3 6 

34 1 2 

78 2 7 



EXAMPLES. 

bu. P. qt. 

256 3 6 

243 1 6 

468 3 1 

584 2 7 



bu. P. qt. 

34156 3 7 

2003 1 2 

90 3 6 

4809 



SECTION 3. 
OF COMPOUND SUBTRACTION. 

COMPOUND SUBTRACTION is the taking a lesser numher 
from a greater, of divers denominations, and thereby finding 
l he difference. 

RULE. 

Set down the lesser number under the greater, as in com- 
pound addition. Then beginning with the lowest number, 
subtract as in subtraction of whole numbers : when the 
lower number is greater than the upper, take it from as 
many of that denomination as will make one of the greater, 
and to the remainder add the upper number, set down the 
amount and carry one to the next, and so proceed till all are 
subtracted. 



COMPOLLN 1> SUBTRACTION 

PROOF. 
Add the remainder to the lower line. 

EXAMPLES. 

. s. d-. 
From 25(T 15 6 
Take 129 12 8 $ 



31 



T. cwt. qr. Ib. oz. dr. 
From 246 15 2 18 11 5 
Take 89 16 1 24 8 15 



Rem. 127 2 9 } 



Proof 256 15 6 



mi. fur. P. ft. in. be. 
From 250 4 24 10 61 
Take 125 6 30 5 10 2 



D. h. mi. sec. 
From 325 18 30 24 
Take 236 20 45 50 



bn. , P. qt. pt. 
From 204 2 6 1 
Take 150 3 2 



sig. deg. mi. sec. 
From 6 16 32 29 
Take 3 . 24 16 48 



T. hhd. gal. qt. pt. 
From 50 2 45 2 1 
Take 20 3 60 3 



A. R. P. 
From 1658 2 Vj 
Take 1249 3 ?4 



Promiscuous Questions in Compound Addition and 
Subtraction. 

1. A merchant bought five pieces of linen, containing as 
follows: No. 1, 36 yards 3 quarters 2 nails; No. 2, 45 
yards 1 quarter 3 nails ; No. 3, 48 yards 2 quarters 1 nail ; 
No. 4, 52 yards 3 nails ; No. 5, 64 yards 2 quarters ; how 
many yards were in all ? Ans.. 247 yds. 2qr. Ina. 

2. Sold 5 head of beef cattle, at the following prices, viz : 
the first far 67. 2s. 4d. the second for 5Z. 10s. 9%d. /he third 
for 11. the fourth for SI. 10s. 6d. the 5th for Ql. 2s. 6d. and 
received. 2.21. 10s. 6d. in ready payment, and a note for the 
remainder ; how much did the cattle cost, and for how much 
was the note given ? 

Ans. The cattle cost 367. 6s. l^d. and the note wastfor 
13/. 15*. l%d. 



32 COMPOUND MULTIPLICATION. 

3. A silversmith bought 26lb. 9oz. lOdwt. of silver, and 
. wrought up ISlb. \6dwt. Wgr. how much has he left ? 

Ans. Sib. Soz. ISdwt. llgr. 

4. A physician bought Qlb. Woz. 6dr. 2sc. (apothecaries' 
weight) of medicine, and has used 4Z&. Soz. 4dr. Isc. I7gr. 
what quantity has he yet remaining ? 

Ans. 2lu. 5oz. 2dr. Osc. 3gr. 

5. William was born on the 15th day of January, 1816, 
at 6 o'clock in the morning, and Charles was born on the 
20th of March, 1817, at 9 in the evening ; how much older 
.6 William than Charles 1 Ans. 1 year, 2mo. 5d. 15h. 

ft An innkeeper bought four loads of hay, weighing as 
lollowing, viz. first load, 19 hundred 2 quarters and 14 Ib. 
second load, 16 hundred 3 quarters 18 Ib. third load, 22 
hundred and 24 Ib. fourth load, 24 hundred and 1 quarter 
how much hay in all ? Ans. 4 tons 2 hundred. 

7. From a piece of broadcloth which at first measured 
55 yds. I sold to A 5J yds. to B 6|, to C 7|, to D a quan 
tity not recollected, and to E just half as much as to D ; on 
Pleasuring the remainder, I found there was 20J yds. left ; 
how many yards did D and E each receive 1 

Ans. D 10, and E 5 yds. 

8. A wine merchant bought 1 pipe 2 hhds. and 3 qr. 
casks of wine, each 26 galls. ; of these he sold 1 hhd. and 

-jr. casks; he also found that the pipe had leaked 17 galls, 
the remaining hogshead 11, and the cask 5-J ; how many 
gallons did he buy, and how many had he left ? 

Ans. Bought 330 galls, left 18l galls. 

9. Bought 4 pieces of cloth, the two first measured 9 Ells 
Fr. 3 qr. 2 na. each, and the two last 8 Ells Fr. 2 qr. 3 na. 
each, of these I sold 40 5 yards ; how much have I left ? 

Ans. 13y. 2 qr. 2 na. 



SECTION 4. 
OF COMPOUND MULTIPLICATION. 

COMPOUND MULTIPLICATION teaches to multiply any given 
quantities or numbers of divers denominations. 

Case 1. 
When the multiplier does not excer<i 12. 



COMPOUND MULTIPLICATION 
RULE. 



33 



Begin by multiplying the lowest number first, as in inte- 
gers ; divide the product by that number which will make 
one of the greater. Set down the remainder, if any, and 
carry the quotient to the next number. 



KXAMPLES. 



Ib. 

14 


. 5. d. 
24 10 6 


cjr. T. cwt. qr. Ib. oz. dr. 
\ 48 14 1 4 12 11 
2 3 


ft. 
1 
11 


2 ) 49 1 1 




24 10 6 


4 Proof. 


oz. dwt. gr. 
4 11 11 
5 


bu. pc. qt. hhd. gal. qt. 
24 3 7 25 48 3 

8 




* 

8 


mi. fur. p. 
24 6 34 

8 


yds. y/. in. be. A. R. 

24 282 89 3 
6 


P. 

26 
9 




to/. 

48 


y?c. #. 
3 6 
11 


d. h. mi. sec. Y. _m. w* d. 
84 19 38 15 125 8 3 4 
9 12 





Application. 



I. 5 yards of cloth at 

.2. 9 ' do. at 

3. 11 bushels of flax-seed at 

4, 12 do. clover-seed at 



. s.d. 
264 

1 2 \>\ 
12 9j 

2 4 2| 



Ans. 



. 
11 

10 

7 



s. d. 

11 8 
2, 84 
8i 



26 10 6 



Case 2. 

When the multiplier exceeds 12, uut is the exact produd 
of any two factors in the table- 1 . 



34 COMPOUND MULTIPLICATION. 

RULE. 

Multiply the given sum by any one of these, in the same 
nanner as above, and the product by the oilier. 

EXAMPLES. 

. s. d. qr. 

Multiply 12 8 6 by 18=3x6 
3 



37 5 6 | 
6 

223 13 4 
Application. 

1. Multiply 4T. Scwt. Iqr. 16ZZ>. Soz. Wdr. by 36. 

Arts. 150 T. 2cwt. Iqr. lib. 6oz. 8dr. 

2. 120Z. (5s. 9d. by 24. A/is. 238SZ. 2s. Od. 

3. 24 T. 4e-7/tf. 2<p% lib. by 48. 

Aws. 1162T. Wcwt. Oqr. OZ/;. 

Case 3. 

When the multiplier is not the exact product of any two 
factors. 

RULE. 

Multiply as in the last case, by any two factors that will 
come the nearest to the multiplier, but less ; and add for the 
deficiency. 

EXAMPLES. 
bll. pC. qt. 

Multiply 12 2 4 by 17 

4x4+1 = 17 



50 2 
4 

202 = 16 
12 2 4= 1 



214 ^ 4 17 



3. Multiply H/>. 4/>. I2?ni. f>v. by 2JK 

AH*. 237 />. I/?. 507/w. 



COMPOUND DIVISION. 35 

Case 4. 

When the multiplier exceeds the product of any two fac- 
tors in the table. 

RULE. 

Multiply by the units figure, as in case 1, and set down 
the amount; again multiply the given sum b}* the figure of 
tens, and that product by 10, and place tins amount under 
the first ; again multiply by the figure of hundreds, and the 
product by 10 and 10, which set down under the other pro- 
ducts- in the same way for thousands, by three tens, &c. 

EXAMPLES. - 

s. d. 2626 

Multiply 2 6 by 245 4 2 
5 

10 50 

12 6 first, or units product 10 10 

500 second, or tens do. 

25 third, or hundreds do. 500 2100 

10 

Ans. 30 12 6 total. 

25 

. s. d. . s. d. 

2. Multiply 14 6 by 240 Ans. 174 

3. 123 117 130 3 3 

4. 126 275 309 7 6 



SECTION 5. 

OF COMPOUND DIVISION. 

COMPOUND DIVISION, teaches to divide any sum or quan- 
tity of divers denominations. 

Case 1. 
When the divisor does not exceed 12. 

RULE. 

Divide the highest, or left-hand denomination; if any re- 
mains, multiply by that number which will reduce it to the 
next highest, add this product to the second, then divide as 
Df.'ibre, and so proceed till all are divided. 



30 COMPOUND DIVISION. 

PROOF By compound multiplication. 

EXAMPLES. 

. s. d. qr. . s. d. qr 

2)465 10 6 i 3)563 15 4 



Quotient 232 15 3 
2 



Proof 465 10 6 

T. cwtf. f/r. Ib. yds. ft. in. 

6)91 16 1 14 5)960 1 9 



T. Mid. gal. qt. . w. d. h. mi. sec. 

8)468 1 48 3 10)30 6 18 48 50 

Case 2. 

When the divisor is the exact product of any two factors 
in the table. 

RULE. 

Divide first by one as above, and the quotient by the 
other. 

EXAMPLES. 

6)224 12 6 by 30=6x5 



5) 37 8 9 



Quotient 


7 9 


9 




. 


s. 


d. 




. 


s. 


d. 


2. Divide 


134 


18 


S 


by 44 


Ans. 3 


1 


4 


3. 


9H4 








144 


6 


16 


8 


4. 


474 








72 


6 


11 


8 



Case 3. 

When the divisor is not the exact product of any two (ac- 
tors in the table, or eA,:ee.ds them. 

RULE. 

Divide the highest denomination in the ^iv<-n sum, in the 
.s.-iuic manner as in case 2, of whole numbers; reduce I lit 
remainder, if any, to the next lower denomination, adding it 



COMPOUND DIVISION. 37 

to the number of the same denomination in the given sum ; di- 
vide this in the same manner, and so proceed till all are divided. 

EXAMPLES. 

1. Divide 264Z. 10s. 7$d. by 25. 

25)264/. 10s. l^d. (Wl.lls.l^d. Ans. 
25 



20 

25)290(11 
25 

~40 
25 

15 
12 

25 ) 187 ( 7 
175 

Is 

4 

25)50(2 

50 
. s. d. . s. d. 

2. Divide 409 13 9 by 345 Ans. 139 

3. 232 4 9 524 8 l(5i 

4. 3236 12 4J 654 4 l3 llj 

5. 132 8 68 1 18 10 
Promiscuous Questions, for exercise, in the foregoing rules 

of Compound Addition, Subtraction, Multiplication and 
Division. 

1. What is the value of 672 yards of linen at 2s. 5d. per 
yard? Ans. 81Z. 45. 

2. A goldsmith bought 11 ingots of silver, each of which 
weighed 4db. loz. I5dwt. 22gr. how much do they all 
weigh? Ans. 45Z&. loz. 15dwt. 2gr. 

3. Bought 8 loads of hay, each weighing 1 ton 2 hundred 
3 quarters 16 pounds; how much hay in all? 

Ans. 9 ton 3 hun. 161b. 

4. Divide 9 ton 3 hundred 161b. into 8 shares. 

Ans. 1 ton 2 hun. 3qr. 16lb. 

5. Bought 15 tracts of land, each containing 300 acres 
2 roods and 20 perches ; what is the amount of the whole ? 

Ans. 4509 acres 1 qr. 20 rods. 
D 



38 COMPOUND DIVISION. 

6. Divide a tract of land containing 4509 acres 1 rood 
and 20 perches equally among 15 persons; what is each 
one's share? Ans. 300 acres 2 roods 20 perches. 

7. Bought 179 bushels of wheat for 201 dollars 37^ cts. 
what is it per bushel? Ans. I doll. 12^ cents. 

8. If a man spends 7 pence per day, how much will it 
amount to in a year? Ans. 10Z. 12s. lid. 

9. What is the value of 1000 bushels of coal at 10| cents 
per bushel? Ans. 105 dolls. 

10. Bought 135 gallons of brandy at 1 dollar arid 62 
cents per gallon, which was sold for 2 dollars and 5 cents 
per gallon ; required the prime cost, what it was sold for, and 
the gain? Ans. prime cost 219 dolls. 37^ cts. 

sold for 276 dolls. 75 cts. gain 57 dolls. 37 cts. 

11. If 27 cwt. of sugar cost 47/. 12s. W$d. what cost 1 
cwt. ? * Ans. ll. 15s. 3^d. 

12. Suppose a man has an estate of 9708 dollars, which 
he divides among his four sons : to the eldest he gives f , and 
to the other three an equal share of the remainder ; what is 
the share of each? 

Ans. eldest son, 3883 dolls. 20 cents, other sons, each 
1941 dollars 60 cents. 

13. A dollar weighs lldwt. Sgr. what will 45 dollars 
weigh at that rate? ^Ans. 39oz. 

14. An eagle of American gold coin should weigh lldwt. 
tigr. now 150 were found to weigh 84oz. 7dwt. 20gr. how 
much was this over or under the just weight ? Ans. &gr. over. 

15. What cost 2^- cwt. of sugar, at 13 cents 3 mills per 
pound ? Ans. 37 dolls. 24 cts. 

16. A merchant deposited in bank 35 twenty-dollar notes, 
63 eagles, 284 dolls. 642 half dolls. 368 qr. dolls. 256 
twelve and a half cent pieces ; he afterwards gave a check 
to A for 560 dolls, and another to B for 820 dolls. ; what 
sum has he still remaining in bank? Ans. 679 dolls. 

17. A merchant bought a, piece of broad cloth containing 
,o yards, at 4 dolls. 66 cts. per yard ; of this he found 4 
yards were so damaged that he sold them at half price ; 8 
yards he sold at 5 dolls. 50 cents per yard : on the whole 
piece he gained IM) dolls. 56 cents ; at what rate did he sell 
the remainder? Ans. (> dollars per yard. 

18. Five travellers, upon leaving a tavern in I ho morning, 
icund they were charged 12 cents eaeh HM- their beds, 4 
limes flint sum for their spnpor rind break (hst. 75 eenls fnr 



RKDIJCTJOJV. 39 

liquor among them all, 25 cents each ibr hay ; the remain- 
der of their bill, which amounted to 6 dollars, was for oats 
at 2 J cents per quart; how many gallons of oats had they, 
and how much had each man to pay ? 

Ans. 8| gallons ; each paid 120 cents. 

19. A laborer engaged to work for 75 cents per day, 
working 8 hours each day, or 8 hours for a day's work ; but 
being industrious he worked 12 hours 25 minutes each day 
for five days, and then 11 hours 30 minutes for 9 days more ; 
what sum is he entitled to receive for his services ? 

Ans. 15 dollars 52 cents 3 mills-f 

20. If 25 hhds. contain 1534 galls. 1 qt. and 1 pt. of 
brandy, each an equal quantity, how much is there in each 
hogshead? Ans. 61 galls. 1 qt. 1 pt. 

21. If a man do 114 hours 45 minutes' work in 9 days, 
how long did he work each day? Ans. I2h. 45mi. 

22. Divide 180 dollars among 3 persons A, B, and C; 
give B twice as much as A, and C three times as much as B. 

C A 20 dolls. 
Ans. I B 40 
f C 120 



SECTION 6. 

OF REDUCTION. 

REDUCTION is the changing of numbers from one denomi- 
nation to another, without altering their real value. Thus. 
1 -dollar, if reduced to cents, will be 100 cents, which in 
their real value are equal to 1 dollar: or, 3 feet reduced 
to yards, is one yard, which is still the same length as the 
3 feet. 

RULE. 

If the reduction is from a higher to a lower denomina- 
tion, multiply ; but if from a lower to a higher denomina- 
tion, divide by as many of the next less as make one of the 
greater ; adding the parts of the same denomination to the 
product as it descends ; and setting down the remainders as 
il ascends. 

Reduction ascending, and descending, mutually prove each 
t.lher. 



40 



REDUCTION. 





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.REDUCTION. 41 

A TABLE of other Foreign Coins, &c. with their value in Federal 
Money , as established by a late act of Congress. 

E. d. d. c. m. \ E. d. d. c. m. 

Pound Sterling' . 4, 4 4 4 t The Guilder of the 



Pound of Ireland . 4, 1 

Pagoda of India . 1, 9 4 

Tale of China . 1, 2 4 

Millree of Portugal 1, 2 4 

Ruble of Russia . 0, 6 6 

Rupee of Bengal . 0, 5 5 5 



U. Netherlands 0, 3 9 

Mark Banco of Ham- 
burg 0, 3 3 5 

Livre Turnois of 

France 0, 1 3 5 

Real Plate of Spain 0, 1 



MONEY. 

Cents are reduced to pence by subtracting one-tenth of 
their number. Pence are reduced to cents by adding one- 
ninth of their number. 

Pence are to cents as 9 is to 10, and to mills as 9 to 100. 
This only applies where the dollar passes at 7$. Qd. or 90 
pence. 

1. Reduce 100 cents to pence. 

10) 100 cents 
10 

90 pence. Arts. 

2. Reduce 90 pence to cents. 

9 ) 90 pence 
10 

100 cents. Ans. 

3. Reduce 1251. 10s. 6%d. to farthings. 

125Z. 105. 
20 



2510 shillings 
12 



30126 pence 
4 



120506 farthings. Ans. 



4. Reduce 120506 farthings to pounds. 

Ans. 1251. 105. 

5. Reduce 260 cents to pence. Ans. 234d. 

D2 



42 REDUCTION. 

6. Reduce 480Z. 1 9s. 9d. to cents. 

Ans. 128263 cents. 

7. Reduce 4658 pence to pounds. Ans. 191. 8s. 2d. 
S. Reduce 648 pence to cents. Ans. 720 cents. 

9. Reduce 720 cents to pence. Ans. 648 pence. 

10. Reduce 24235 half- pence to pounds. 

Ans. 507. 9s. 9%d. 

11. How many pounds, Pennsylvania currency, in 216 
French crowns ? Ans. 897. 2s. 

12. In 29Z. 17s. how many cents and dollars? 

A?i. 7960 cents 79 dolls. 60 cts. 

13. In 375Z. Pennsylvania currency, how many dollars ! 

Ans. 1000 dolls. 

Note. To bring pounds (Penn. currency) to dollars, multiply by 8 and 
divide by 3 ; and dollars to pounds, multiply by 3 and divide by 8. 

TROY WEIGHT. 

1. Reduce 115200 grains to pounds. Ans. 201. 

2. Reduce 30Z&. to grains. Ans. 172800r. 

3. Reduce 45648 pennyweights to ounces. 

Ans. 22S2oz. Sdwt. 

4. Reduce 4.lb. Soz. I5dwt. 20g?\ to grains. 

Ans. 2726Qgr 

5. Reduce 27260 grains to pounds. 

Ans. 4Z6. Soz. 15dwt. 20gr. 

6. In 24 spoons, each weighing Sdwt. 6gr. how many 
grains ? Ans. 4752gr. 

AVOIRDUPOIS WEIGHT. 

1. Reduce 3 tons to pounds. Ans. 6720Z6. 

2. Reduce 2867200 drams to tons. Ans. 5 tons. 

3. Reduce 5 tons to drams. Ans. 2867200dfr. 

4. In 6 barrels of flour, each weighing Icwt. 3qr. Low 
many pounds? Ans. 1176/6. 

5. In \ficwt. 2qr. 14Z&. how many pounds? 

Ans. 18627A. 

6. In a load of hay weighing 2876Z&. how many hun- 
dreds? Ans. 25cwt. 2qr. 207ft. 

APOTHECARIES 1 WEIGHT. 
1. Reduce 15Z&. to scruples. Ans. 4320*?. 



REDUCTION. 43 

2. In a bottle containing 3lb. of calomel, how many 
grains? Ans. IT2$Qgr. 

3. In 2%lb. of drugs, how many parcels, each 16 drams ? 

Ans. 15 parcels. 

4. In 576000 grains, how many pounds ? 

Ans. 100Z&. 

CLOTH MEASURE. 

1. Reduce 250 yards to nails. Ans. 4000 nails. 

2. In 8642 nails, how many Ells English ? 

Ans. 432 Ells E. 2 nails. 

3. In 324 Ells French, how many yards ? 

Ans. 486 yards. 

4. In 16 bales of cloth, each measuring 36 Ells Flemish, 
how many yards.? Ans. 432 yards. 

LONG MEASURE. 

1. Reduce 260 miles to inches. 

Ans. 16473600 inches. 

2. Reduce 11 miles 7 furlongs 38 perches 2 yards 2 feet, 
to barley-corns. Ans. 2280060&C. 

3. Reduce 1267200 feet to geographical degrees. 

Ans. 4 degrees. 

4. Reduce 3 leagues 2 furlongs 110 yards 1 foot 5 inches, 
to inches. ' Ans. 590057 inches. 

5. How many inches will reach round the world, at 60 
miles to a degree? Ans. 1368576000 inches. 

o 

LAND MEASURE. 

1. Reduce 25 acres to perches. Ans. 4000 perches. 

2. Reduce 176000 perches to acres. 

Ans. 1100 acres. 

3. A tract of land containing 640000 perches is to be 
iivided into 400 equal shares ; how many acres will be in 

rach share ? Ans. 1 acres. 

4. In 10 acres, how many square inches ? 

Ans. 62726400 inches. 



CUBIC, OR SOLID MEASURE. 
1. Reduce 3200 feet of wood to cords. 



Ans. 25 cords. 



44 REDUCTION. 

2. In. 20 tons of square timber, how many feet? 

Ans. 1000 feet. 

3. In 30 tons of round timber, how many inches ? 

Ans. 2073600 inches. 

Note. The cubic feet of any circular body, such as grindstones, &c 
is found in the following manner. Add halt' the diameter to the whole 
diameter ; multiply the amount by the aforesaid half^ and this product 
by the thickness; this will give the contents in cubic inches; divide 
these by 1728, and the quotient will be the cubic feet. 

4. In a grindstone 48 inches diameter and 6 inches thick, 
how many feet ? 

48 diameter 
24 half do. 

72 
24 

288 
144 

1728 
6 



1728 ) 10368 ( 6 cubic feet. Ans. 
10368 

5. In a millstone 4 feet 6 inches diameter, and averaging 
18 inches in thickness, how many cubic feet ? 

54 
27 

81 
27 

567 
162 

2187 
18 



17406 
2187 

1 728 ) 39366 ( 22 feet 1350 inches. Ans. 
3456 

4806 
3456 

1350 



DECIMAL ARITHMETIC. 46 

TIME. 

1. Reduce 8 weeks 2 days 6 hours 20 minutes, to minutes. 

Ans. 83900 minutes. 

2. Reduce ten years to seconds. 

Ans. 315576000 sec. 

3. How many days since the commencement of the 
Christian era to the present time, 1823 '? 

Ans. 665850 days 18 hours. 

4. How many seconds in a week ? Ans. 604800 sec. 

LIQUID MEASURE. 

1. Reduce 4 tuns to pints. Ans. 8064 pints. 

2. Reduce 4032 pints to hogsheads. Ans, 8 hhds. 

3. Reduce 38 hogsheads to pints. Ans. 19152 pints. 

DRY MEASURE. 

1. Reduce 78 bushe's 3 pecks 7 quarts to pints. 

Ans. 5054 pints. 

2. Reduce 2196 pints to bushels. Ans. 346w. Ipc. 2qt. 



PART III. 
DECIMAL ARITHMETIC. 

DECIMAL ARITHMETIC is a plain and easy method of dis- 
covering the value of an unit, or one, divided into any given 
number of parts. Thus, if 1 dollar is divided into 10 equal 
parts, any one of these parts will be one-tenth, 2 will be 
two-tenths, 3, three-tenths, &c. Again, if 1 dollar is divided 
into a hundred equal parts, any one of these will be one- 
hundredth, 2, two-hundredths, &c. 

The number of parts into which the unit is divided is 
called the denominator, and any number of these parts less 
than the whole is called the numerator, and which always 
stands over the denominator ; thus, 
2 numerator 
10 denominator 

is read two-tenths ; and these two so placed constitute 
what is termed a fraction. In decimal fractions, the de- 
nominator can only be an unit, with one or more ciphers 



46 ADDITION OF DECIMALS. 

added thereto ; as y 5 -, y 2 ^, T 2 / F V The numerators of these 
are usually written without their denominators, and are dis- 
tinguished from whole numbers, by prefixing a point called 
the separatrix, as ,5 ,25 ,'225. 

Ciphers placed to the right hand of decimals, make no 
change in their value, for ,5 ,50 ,500, &c. are decimals of 
the same value, each being equal to J. But when prefixed 
to the decimal, they decrease the value in a tenfold propor- 
tion. Thus, ,5 ,05, ,005, have the same proportion to each 
other a,s 5, 50, 500, have in whole numbers. 

This is made plain by the following 

TABLE. 
Integers. Decimals. 



5 

5 ,5 

500 ,05 

5000 ,005 

50000 ,0005 

500000 ,00005 

5000000 ,000005 



all's 11 

.2 13 w 3 "2 < ,2 

=3 5 G Q 5 C -^ 




SECTION I. 

ADDITION OF DECIMALS. 

RULE. 

SET down the numbers according to their value, viz. 
units under units, tenths under tenths, &c. Then add as 
in addition of whole numbers, and place the point in the 
amount exactly und^i those in the given sum. 



SUBTRACTION OF DECIMALS 47 

EXAMPLES. 

2468,5036 3460000,0000643 

521,0428 460000,000643 

32,0004 3400,3680005 



3021,5468 



3. Add 283,604 + 490006,003275 -f 21,05 -f 1,2 + 
6200,3476. Ans. 496512,204875. 

4. Add ,246 + ,012 + ,02 + ,6 + ,41 3 + ,5. Ans. 1,791. 

5. Add 25,52 + 225,005 + ,0035 + 844 -f2,2 + 300,825 
+ ,00005. Ans. 1397,55355. 

6. Add one hundred and twenty-five, and five-tenths, + 
ten thousand, and five millionths, + fifteen, and seventy-two 
thousandths, + two, and one hundredth. 

A.ns. 10142,582005. 

7. Add five, and four-tenths, + fifteen and four hundredths, 
+ one hundred, and four thousandths, + six thousand and 
four hundred thousandths, + ninety-three thousand eight 
hundred and eighty, and four ten thousandths. 

Ans. 100000,44444. 



SECTION 2. 
SUBTRACTION OF DECIMALS. 

RULE. 

SET the less under the greater, with the points as in addi- 
tion, and place the point in the remainder, in the same 
manner. 

EXAMPLES. 

Prom 6432,50437 From 848,045 From 15,6547 

Take 369,95429 Take 162,54936^ Take 7,35 



Kern. 6062,55008 



4. From 45,005 take 23,65^82. Ans. 21,35018. 

5. From six hundred and twenty, and two-tenths, take 
wo hundred and two thousandths. Ans. 420,li/ k rt 



48 MULTlPLICATIGiN OF DECIMALS. 

6. From 5 take ,10438. Ans. 4,89562. 

7. From 2 take ,00002. Ans. 1,99998. 

8. From sixteen take sixteen thousandths parts. 

Ans. 15,984. 



SECTION 3. 

MULTIPLICATION OF DECIMALS. 

RULE. 

MULTIPLY as in whole numbers, and from the product 
point ofF as many on the right-hand for decimals, as there 
are in both the factors. If the whole product should be too 
few, then must ciphers be added on the left of the product 
till an equal number is had. 



EXAMPLES. 



Multiply 29,831 
by ,952 



59662 
149155 
268479 

Product 28,399112 



24,021 
4,23 


22,2043 
,12345 


72063 
48042 
96084 


1110215 

888172 
666129 
444086 
222043 


101,60883 



4. Multiply ,385746 by ,00463 

5. 158,694 23,15 

6. ,024653 ,00022 



2,741120835 

Ans. ,00178600398 

3673,7661 

,00000542360. 



7. Multiply twenty-five and four hundredths, by two thou- 
sandths. Ans. ,05008. 

8. Multiply six hundred and forty-five, and three thou- 
sandths, by five millionths. Ans. ,003225015. 

Note. The product of any number when multiplied by a decimal 
only, will be less than the multiplicand, in llrt PUIIW proportion as the 
multiplier is les than one. 



MULTIPLICATION OF DECIMALS. 49 

CONTRACTION IN MULTIPLICATION OF DECIMALS. 

If only a limited number of decimals is sought for, instead 
of retaining the whole product, obtained by the foregoing 
method, work by the following 

RULE. 

1. Set the multiplier, in an inverted order, under thn 
multiplicand, placing the units figure of the multip'ier under 
the lowest decimal place in the multiplicand, that is wished 
to be retained. 

2. In multiplying, omit those figures in the multiplicand 
which are on the right of the multiplying figure, but to the 
first figure in each line of the product, add the carriage 
which would arise from the multiplication of the omitted 
figures, carrying one from 5 to 15, 2 from 15 to 25, 3 fron 
25 to 35, &c. Place the first figures in each product di- 
rectly under each other, and add as in addition. 

Note. If you would be absolutely certain that the last figure retained 
is the nearest to the truth, work for one place more than you wish to 
retain* 

EXAMPLES. 

1. Multiply 34,6733 by 3,1416, retaining four decimal 
places in the product. 

34,6733 
61413 inverted 



1040199 
34673 
13869 

347 

209 

108,9296 Arts. 

2. Multiply ,78543 by ,346787, retaining five decimal 
places in the product. 

34G787 Or thus, ,78543 

34587 inverted 787643,0 

24275 235G3 

2774 3142 

173 471 

14 55 

1 6 



.27237 4ns 427237 



50 DIVIS[OIN OF DECIMALS. 

3. Multiply 23,463 by 2,34, retaining three decimals. 

Ans. 54,903. 

4. Multiply 234,216 by 2,345, retaining two decimals. 

Ans. 549,23. 

5. Multiply 3,141592 by 52,7438, retaining four deci 
mals. Ans. 165,6995, 



SECTION 4 
DIVISION OF DECIMALS. 

RULE. 

DIVIDE in the same manner as in whole numbers, and 
point off on the right of the quotient as many figures for de- 
cimals, as the decimals in the dividend exceed those in the 
divisor. When the decimals in the divisor exceed those in 
the dividend, let ciphers be added to the dividend, till they 
equal those in the divisor. And if there be a remainder, let 
ciphers be annexed thereto, and the quotient carried on to 
any degree of exactness. 

EXAMPLES. 

29,831 ) 2' ,d991 12 ( ,952 24,021 ) 101,60883 ( 4,28 
' 268479 96084 



155121 55248 

149155 48042 



59662 72063 

59662 72063 



Nate. When the divisor is 10, 100, 1000, &c. the division is perform- 
ed by pointing off as many figures in the dividend for decimals, aa 
there are ciphers in the divisor. 

10 ) I 685,6 

Thus, 6856 divided by 100 > is < 68,56 

1000 > ( 6,856 

3. Divide 65321 by 23,7 Ans 2756,16 + 

4. 234,70525 64,25 3,653 

5. 10 3 3,3333 + 

6. 9 ,9 10 

7. ,00178600398 ,00463 ,385746 



DIViSIOiN OF DECIMALS. fj 1 

8. Divide ,2327898 by 2,46 Ans. ,09463 

9. ' ,2327898 ,09463 2,46 
10. * ,000162 ,018 ,009 

CONTRACTION IN DIVISION OF DECIMALS. 

When only a limited number of decimals in the quotient 
is sought for, work by the following 
RULE. 

1. Take as many figures only on the left hand side of 
the divisor, as the whole number of figures sought for in the 
quotient, and cut off the rest. 

2. Make each remainder a new dividend, and for a new 
divisor, point off one figure continually from the right hand 
of the former divisor, taking care to bring in the increase, 
or carriage of the figures so cut off, as in multiplication. 

Note. When the whole divisor does not contain as many figures as 
are sought for in the quotient, proceed as in common division, without 
cutting off a figure, till the figures in the divisor shall equal the re- 
maining figures required in the quotient, and then begin to cut off as 
above directed. 

EXAMPLES. 

1. Divide 14169,206623851 by 384,672258, retaining 
four decimal places in the quotient, or in all six quotient 
figures. 

3.8.4,6.7.2|258 ) 14169,206623851 ( 36,8345. Ans. 
1154017 



262903 
230803 

32100 
30774 



1326 
1154 

172 
153 

19 
19 



52 REDUCTION OF DECIMALS. 

2. Divide ,07567 by 2,32467, true to four decimal places, 
or three significant figures, the first being a cipher. 

2,3.2|467 ) ,07567 ( ,0326. Ans. 
697 

59 
46 

13 
14 

3. Divide 5,37341 by 3,74, true to four decimal places. 

3,74 ) 5,37341 ( 1 ,4367. Ans. 
374 

1633 
1496 

1374 
1122 



252 
224 

28 
26 

4. Divide 74,33373 by 1,346787, true to three decimal 
places. Ans. 55,193. 

5. Divide 87,076326 by 9,365407, true to three decimal 
places. Ans. 9,297. 

6. Divide 32,68744231 by 2,45, true to two decimal 
olaces. Ans. 13,34. 

7. Divide ,0046872345 by 6,24, true to five decima 
places. Ans. ,00075 



SECTION 5. 
REDUCTION OF DECIMALS. 

Case 1. 
To reduce a vulgar fraction to a decimal. 



REDUCTION OF DECIMALS. 53 

RULE. 

Annex one or more ciphers to the numerator, and divide 
by the denominator; the quotient will be the answer in 
decimals. 

EXAMPLE. 

1. Reduce to a decimal. 

4)1,00 



,25 Ans. 

2. Reduce to a decimal. Ans. ,5 

3. - J to a decimal. ,75 

4. - -J to a decimal. ,875 

5. - -Jj to a decimal. ,04 

6. - |j to a decimal. ,95 

7. - T 6 3 of a dollar to cents. ,40 cts. 

Case 2. 

To reduce numbers of different denominations to a deci- 
mal of equal value. 

RULE. 

Set down the given numbers in a perpendicular column, 
having the least denomination first, and divide each of them 
by such a number as will reduce it to the next name, annex- 
ing the quotient to the succeeding number ; the last quotient 
will be the required decimal. 

EXAMPLE. 

1. Reduce 17s. 8%d. to the decimal of a pound. 



4 
12 
20 



3 

8,75 
17,720166 

,8864583 + Ans. 



2. Reduce 195. to the decimal of a pound. Ans. ,95 

3. - 3d. to the decimal of a shilling. ,25 

4. - 3d. to the decimal of a pound. ,0125 

5. - kcwt. 2qr. to the decimal of a ton. ,225 

6. - 2qr. 14/6. to the decimal of a cwt. ,625 

7. - 3qr. 3na. to the decimal of a yard. ,9375 

E2 



54 REDUCTION OF DECIMALS. 

Case 3. 
To reduce a decimal to its equal value in integers. 

RULE. 
Multiply the decimal by the known parts of the integer 

EXAMPLE. 

1. Reduce ,8864583 of a pound to its equivalent value 
in integers. 

,8864583 
20 



s. 17,7291660 
12 

d. 8,7499920 
4 

qr. 2,9999680 

It is usual when the left hand figure in the remaining 
decimal exceeds five, to expunge the remainder, and add 
one to the lowest integer. Thus, instead of 17s. 8d. 2,999, 
die. we may say 17s. 8%d. Ans. 

2. What is the value of ,75 of a pound ? Ans. 15s. 

3. What is the value of ,7 of a pound troy? 

Ans. Soz. Sdwt. 

4. What is. the value of ,617 of a cwt. ? 

Ans. 2qr. 13Z&. 1 oz. W + dr. 

5. What is the value of ,3375 of an acre? 

Ans. I rood, 14per. 

6. What is the value of ,258 of a tun of wine ? 

Ans* Ihhd. 2 -{-gals. 

7. What is the proper quantity of ,761 of a day? 

Ans. ISh. 15?ni. 50,4src. 

i. What is the proper quantity of ,7 of a Ib. of silver? 

Ans. Soz. Sdwt. 

: What is the proper quantity of ,3 of a year? 

Ans. lQ9d. 13ft 

10. What is the difference between ,41 of a day and ,16 
of an hour ? Ans. 97*. 40wi. 48sec. 

11. What is the sum of ,17T. 19^. ,l7qr. and lib. 1 

Ans. Zcwt. 2qr. 



DECIMAL KHACT1O.NS. 55 

Promiscuous Questions in Decimal Fractions. 

1. Multiply ,09 by ,000. Ans. ,00081. 

2. In ,36 of a ton (avoirdupois) how many ounces ? 

Ans. 12902,402. 

3. What is the value of ,9125 of an ounce troy? 

Ans. 18dwt. 6gr. 

4. Reduce ^j to a decimal. Ans. ,0127 nearly. . 

5. Reduce 2oz. I6dwt. 20gr. to the decimal of a pound 
troy. Ans. ,2368 -f nearly. 

6. What is the length of ,1392 of a mile? 

Ans. 1 fur. 4 per. 3 yds nearly. 

7. What multiplier will produce the same result, as mul- 
tiplying by 3, and dividing the product by 4 ? 

Ans. ,75. 

8. What decimal of Icwt. is 6lb. Ans. ,0535714. 

9. What part of a year is 109 days 12 hours? 

Ans. ,3. 

10. In ,04 of a ton of hewn timber, how many cubic 
inches? Ans. 3456. 

11. What is the value of T 3 T of a dollar divided by 3? 

Ans. 6f cents. 

12. What is the value of ,875 of a hhd. of wine? 

Ans. 55 gal. qt. 1 pt. 

13. What divisor, true to six decimal places, will produce 
the same result as multiplying by 222 ? 

Ans. ,004504. 

14. In ,05 of a year, how many seconds, at 365 days 6 
hours to the year? Ans. 1577880. 

15. What number as a multiplier will produce the same 
result as multiplying by ,73 and dividing first by 3, and the 
quotient by ,25 ?" Ans. ,973^. 

16. What is the difference between ,05 of a year, and ,5 
of an hour ? Ans. 2i. 2d~ ISh. 42m. 

17. In ,4 of a ton, ,3 of a hhd. and ,8 of a gallon, how 
many pints ? Ans. 964. 

18. How many perches in ,6 of an acre; multiplied by 
X)2? Ans. 1,92.* 

19. What part of a cord of timber is 1 cubic inch? 

Ans. ,000004 -f 

20. What part of a circle is 28 deg. 48 minutes ? 

ATU. ,08. 



66 SINGLE RULE OF THREE DIRECT 

PART IV. 

PROPORTIONS. 

Tins part of arithmetic which treats of proportions is 
very extensive and important. By it an almost innumerable 
variety of questions are solved. It is usually divided into 
three parts, viz. Direct, Inverse, and Compound. The first 
of these is called the Single Rule of Three Direct, and 
sometimes by way of eminence the Golden Ride. The sec- 
ond is called the Single Rule of Three Inverse: and the 
last is called the Double Rule of Three. In all these, cer- 
tain numbers are always given, called data* by the multipli- 
cation and division of which, the answer in an exact ratio 
of proportion to the other terms is discovered. 



SECTION 1. 

SINGLE RULE OF THREE DIRECT 

IN this rule three numbers are given to find a fourth, that 
shall ha\e the same proportion to the third, as the second 
has to the first. 

If by the terms of the question, more requires more, or 
less requires less y it is then said to be direct, and belongs to 
this rule. 

In stating questions in this rule, the middle term must 
always DC of the same name with the answer required; the 
last term is that which asks the question, and that which is 
of the same name as the demand, the first. When the ques- 
tion is thus stated, reduce the first and third terms to the 
lowest denomination in either ; and the middle term (if com- 
pound) to its lowest, and proceed according to the following 

RULE. 

Multiply the second and third terms together, and divide 
the product by the first ; the quotient will be the fourth term, 
or answer, in the same name with the second. 



SINGLE RULE OF THREE DIRECT. 5? 

PROOF. 

Invert the question, making the answer the first term ; the 
result will be, the first term in the original question. 

Note. 1. After division if there be any remainder, and the quotient 
be not in the lowest denomination, it must be reduced to the next less 
denomination, dividing as before, till it is brought, to the lowest denomi- 
nation, or till nothing" remains. 

2. When any of the terms are in federal money, the operation is con- 
ducted in all respects as in simple numbers, taking care to place the 
separatrix bet'ween dollars and cents, according to what has already 
been laid down in federal money and decimal fractions. 

EXAMPLE. 

1. If 8 yarcls of cloth cost 32 dollars, what will 24 yards 
cost ? 

Yds. D. Yds. D. Yds. D. 

As 8 : 32 :: 24 Proof. As 96 : 24 :: 32 

24 32 

128 48 

64 72 

8)768 96)768(8 

768 
96 Ars. 

2. When sugar is sold at 12 dollars 32 cts. per cwt. whni 
will Wlb. cost? Ans. I doll. 76 cts. 

3. What is the amount of 3 cwt. of coffee at 36 cents per 
pound? Ans. 120 dolls. 96 cts. 

4. What will 4 pieces of linen come to, containing 23, 
24, 25, and 27 yards, at 72 cents per yard ? 

Ans. 71 dolls. 28 cts. 

5. What will bcwt. 2qr. Sib. of iron come to at 48 cents 
for 4/6. ? Ans. 61 dolls. 44 cts. 

6. What will V2Slb. of pork come to at 8 cts. per pound? 

Ans. 10 dolls. 24 cts. 

7. If 9 dozen pair of stockings, cost 68 dollars 40 cent? 
what will 3 pair cost ? Ans. 1 doll. 80 cts. 

8. If 20 bushels of oats cost 9 dollars 60 cents, what will 
three bushels come to? Ans. 1 doll. 44 cts. 

9. A merchant bought a piece of cloth for 16 dollars 50 
cents, at 75 cents per yard ; how many yards were there iii 
the piece ? Ans. 22 yds. 

10. If 17 cwt. 3qr. 17 Ib. of sugar cost 320 dollars 80 cts. 
what must be paid for 6oz. ? An*. 6 o*nts. 



58 SINGLE RULE OF THREE DIRECT 

11. If 9,7 Ib. of silver is worth 97 dollars, what is the 
value of l,5oz. 1 Ans. 1 doll. 25 cts. 

12. If 125,5 acres are sold for 627,5 dollars, what will 
4,75 acres cost ? Ans. 23 dolls. 75 cts. 

13. If 1,5 gallons of wine cost 4 dollars 50 cents, what 
will 1,5 tuns cost? Ans. 1134 dolls. 

14. How many reams of paper at 1 dollar 66 cents, 1 
dollar 97 cents, and 2 dollars 31 cents per ream may be 
purchased for 528 dollars 66 cents, of each an equal num- 
ber ? Ans. 89 reams of each sort. 

15. When iron is sold for 224 dollars per ton, what will 
Iqr. 1Mb. cost? Ans. 4* dolls. 20 cts._ 

16. A merchant paid 1402 dollars 50 cents for flour,- at 5 
dollars 50 cents per barrel ; how many barrels must he re- 
ceive? Ans. 255 barrels. 

17. A man has a yearly salary of 1186 dollars 25 cents, 
how much is it per day ? Ans. 3 dolls. 25 cts. 

18. A man spends 2 dollars 25 cents per day, and saves 
378 dollars 75 cents at the end of the year, what is his 
yearly salary? Ans. 1200 dolls. 

19. What will 4T. Wcwt. Iqr. I2lb. of hay come to at 
1 dollar 12 cents per cwt.? Ans. 101 dolls. 20 cts. 

20. How much will a grindstone 4 feet 6 inches diameter, 
stfid 9 inches thick, come to at 1 dollar 10 cents per cubic 
foot? Ans. 12 dolls. 53 cts. 

21. \\hat will a grindstone 28 inches diameter, and 3,5 
inches thick, come to at 1 dollar 90 cents per cubic foot ? 

Ans. 2 dolls. 26 cts. 

22. At 221. Ss. per ton, what will 203 T. Scwt. 3qr. Mb. 
of tobacco come to ? Ans. 4558Z. 3s. 

23. If 850 dolls. 50 cents is paid for 18 pieces of cloth 
at the rate of 11 dollars 25 cents for 5 yards, how many 
yards were in each piece, allowing an equal number to each 
piece? Ans. 21 yds. 

24. If 124 yards of muslin cost 1Z. 17$. 6rf. what is it 
pe> yard? Ans. 3s. 

25. If a staff 4 feet long cast a shadow (on level ground) 
7 feet long, what is the height of a steeple whose shade at 
the same time, is 218 feet 9 inches? A?is. 125 feet. 

26. If 4292 dollars 32 cents are paid for 476 acres 3 
roods 28 perches of land, how much is it per acre ? 

Ans. 9 dollars. 

27. If a man's annual income be 1333 dollars, and he 



SLN(iLK lUiJJ-: OK TUkiaO INVERSK. 59 

expend daily 2 dollars 1-4 cents, how much will lie save at 
'the end of the year 1 Ans. 551 dolls. 90 cts. 

28. If 321 bushels of wheat cost 240 dollars 75 cents, 
what is it per bushel ? Ans. 75 cts. 

29. If l yard of cloth cost 2 dollars 50 cents, what will 
1 quarter 2 nails come to ? Ans. 62^ cts. 

30. Bought 3 pipes of wine, containing 120-|, 124, and 
126| gallons, at 5s. 6d. per gallon ; what do they cost ? 

Ans. 102Z. Is. lOicZ. 

31. A sets out from a certain place and goes 12 miles a 
day ; 5 days after, B sets out from the same place, the same 
way, and goes 16 miles a day; in how many days will he 
overtake A? Ans. 15 days. 

32. If I have owing to me 1000Z. and compound with my 
debtor, at, 12s. Qd. per pound, how much must I receive? 

Ans. 625Z. 

33. If 365 men consume 75 barrels of pork in 9 months, 
how many will 500 men consume in the same time ? 

Ans. 10241 barrels. 

34. How much land at 2 dollars 50 cents per acre, musl 
f>e given in exchange for 360 acres at 3 dollars 75 cents 1 

Ans. 540 acres. 

35. If the earth, which is 360 degrees in circumference, 
turns round on its axis in 24 hours, how far are the inhabit- 
ants at the equator carried in 1 minute, a degree there being 
69! miles ? Ans. 17 miles 3 fur. 



SECTION 2. 

SINGLE RULE OF THREE INVERSE. 

IF in any given question, more requires less, or less re- 
quires more, the proportion is inverse, and belongs to this 
rule. 

Having stated the question, as in the rule of three direct, 
proceed according to the following 

RULE. 

Multiply the first and second terms together, and divide 
the product by the third ; the quotient will be the answer, in 
the same name as the second. 



60 SINGLE RULE OF THREE INVERSE 

EXAMPLE. 

1. If 20 men can build a wall in 12 days, how long will 
it require 40 men to build the ^ame ? 

M. d. M. d. M. d. 

As 20 : 12 :: 40 Proo/ As 6 : 40 :: 12 
12 40 



40 ) 240 ( 6 days. Ans. 12 ) 240 ( '20 men. 
240 24 





2. If 60 men can build a bridge in 100 days, how long 
will it require 20 men to build it ? Ans. 300 days. 

3. If a wall 100 yards long requires 65 men 4 days, in 
what time would 5 men complete it ? Ans. 52 days. 

4. If a barrel of flour will last a family of six persons 24 
days, how long would it last if 3 more were added to the 
family? Ans. 16 days. 

5. If 5 dollars is paid for the carriage of \cwt. weight, 
150 miles, how far may Sent* weight be carried for th^ 
same money ? Ans. 25 miles. 

6. If a street 80 feet wide and 300 yards long, can be 
paved by 40 men in 20 days, what length will one of 60 
feet wide be paved by the same men in the same time ? 

Ans. 400 yards. 

7. If a field that is 30 rods wide and 80 in length, con- 
tain 15 acres, how wide must one be to contain the same 
quantity, that is but 70 rods long ? Ans. 3412. 4/7. 8&iri. 

S. If a board be ,75 of a foot wide, what length must it 
be to measure 12 square feet? Ans. 16 feet. 

9. How much cloth 1,25 yards wide, can be lined by 
42,5 yards of silk that is ,75 of a yard wide ? 

Ans. 25,5 yards. 

10. If 10 men could complete a building in 4,5 months, 
what time would it require if 5 more were employed ? 

Ans. 3 months. 

11. In what time will 600 dollars gain 50 dollars, when 
80 dollars would gain it in 15 years? fins. 2 years. 

12. If a traveller can perform a journey in 4 days, when 
the days are 12 hours long, what time will lw require when 
thf! days arn 1 (> hours long ? _ Ans. 3 days. 

I ;*. Suppose 100 nvn in a garrison are supplied 



SINGLE RULE Oi TliUKE INVERSE. 61 

provisions for 30 days, how many men must be sent out if 
they would have the provisions last 50 days ? 

Ans. 160 men. 

14. Lent a friend 292 dollars for six months ; afterwards 
J borrow from him 806 dollars ; how long may I keep it to 
balance the favor? Ans. 2 months 5 days. 

15. 1200 men stationed in a garrison, have provisions for 
9 months, at the rate of 14 ounces per day; how long at 
the same allowance will the same provisions last if they are 
reinforced by 400 men ? And also what diminution must be 
made on each ration, that the provisions may last for the 
same time ? Ans. 6| mo. at the same allowance 

3J oz. deduction to last for the same time. 

16. If a piece of land 40 rods in length and 4 in breadth, 
make an acre, how wide must it be if it is but 25 rods long? 

Ans. 6| rods. 

17. How much in length that is 3 inches broad, will make 
a square foot ? Ans. 48 inches. 

18. If a pasture field will feed 6 cows 91 days, how long 
will it feed 21 cows ? Ans. 26 days. 

19. There is a cistern having 1 pipe, which will empty it 
in 10 hours ; how many pipes of the same capacity will 
empty it in 24 minutes ? Ans. 25 pipes. 

20. How many yards cf carpeting that is half a yard 
wide, will cover a floor that is 30 feet long and 18 feet wide? 

Ans. 120 yards. 

21. What is the weight of a pea to a steelyard, which 
being suspended 39 inches from the centre of motion, will 

l 

Ans. 4Z6. 

22. A and B depart from the same place, and travel the 
same road ; but A goes 5 days before B at the rate of 20 
miles a day, B follows at the rate of 25 miles a day ; in 
what time, and at what distance, will he overtake A? 

Ans. 20 days, and 500 miles. 

The following- rule, if adopted, will suit for the stating of all questions 
in single proportion, whether direct or inverse. 

GENERAL RULE, 

Place that number for the third term, which signifies the 
same kind, or thing, as that which is sought ; arid consider 
whether the number sought will be greater or less ; if great- 

F 



equipoise 2QSlb. suspended at the draught end f of an inch ? 



f>2 SLXGLK RULF. OF THRRK. 

er, place the least of the other terms for the first, but if less 
place the greater for the first term, and the remaining one 
for the second. 

Multiply the second and third terms together, and divide 
the product by the first ; the quotient will be the answer 
required. 

EXAMPLES. 

1. If 30 horses plow 12 acres, how many will 40 horses 
plow in the same time ? 

k. h. acr. 

Direct Proportion. 30 : 40 : : 12 

12 



30)480(16 acres. A us. 

2. If 30 horses plow 12 acres in 10 days, in how many 
days will 40 horses plow the same quantity ? 
h. h. D. 

Inverse Proportion. 40 : 30 : : 10 

10 



40)300(7,5 days. Ans. 

3. If 800 soldiers in a garrison have provisions sufficient 
for 2 months ; how many must depart that the provisions 
may last them for 5 months ? Ans. 480. 

1. Bought a hogshead of Madeira wine for 119 dollars, 
nine gallons of which leaked out ; what was the remainder 
sold at per gallon, to gain 12 dollars on the whole 1 

-Ans. 2 dolls. 42-J-fcts. 

5. If 225 pounds be carried 51*3 miles for 20 dollars, 
how many pounds may be carried 64 miles for the same 
money? Ans. 1800Z6. 

6. If 87 dolls. 50 cents be assessed on 1750 dolls, what 
is the tax of 10 dolls, at the same rate? Ans. 50 cts. 

Promiscuous Questions in Direct and Inverse Proportion. 

1. Suppose a man tnivels to market with his wagon 
loaded, at the rate of 2- miles .n hour, and returns with \\ 
empty at the rate of 3 miles an hour ; how long will he be 
in performing a journey, going and returning, to a place 
123 miles distant? Ana. 84J-J hoars. 



SINGLE HULK OF TlfREK. (>3 

2. A lent B 1000 dollars fur IS!) days, without interest 
how long should B 'end A iif>0 dollars to requite the favor? 

Ans. 290fg days. 

3. Bought 14 casks of butter, each weighing Icwt. Iqr. 
4/6. at 12 dollars 60 cents per cwt. ; what did they come to, 
and how much per Ib. ? 

Ans. 226 dolls. 80 cts. whole cost ; 1 1 cts 2 \ m. per Ib. 

4. Sold 4 chests of tea, each weighing Icwt. Qqr. 14//>. 
the first for 80 cents per Ib. the second for 90 cents, the 
third for 1 doll. 5 cents, and the fourth for 1 doll. 25 cents ; 
how many pounds of tea were there, what was the average 
price, and what did the whole come to ? 

Ans. 504//?. average 1 doll. come to 504 dolls. 

5. When flour is sold at 2 dolls. 24 cents per cwt. what 
will be the first cost of one dozen of rolls, each weighing 5oz. 
allowing the bread to be in proportion to the flour, as five is 
co four ? Ans. 6 cents. 

6. If a merchant bought 270 barrels of cider for 780 dol- 
lars, and paid for freight 37 dolls. 70 cents, and for other 
charges and duties- 30 dolls. 60 cents ; at what must he sell 
it per barrel to gain 143 dolls. Ans. 3 dolls. 67^ 4 T cts. 

7. If half a ton of hay was equally divided among 80 
horses, how much must be given to 7 ? Ans. 3qr. 14Z&. 

8. Suppose the circumference of one of the larger wheels 
of a wagon to be 12 feet, and that of one of the smaller 
wheels 9 feet 3 inches ; in how many miles will the smaller 
wheel make 1000 revolutions more than the larger? 

Ans. 7 m. 5 fur. 34 yds. 1 ft. 7 T \ in. 

9. If a man perform a journey in 18 days, when the days 
are 15 hours long, how many days will it require to per- 
form the same journey when the days are only 12 hours 
long? Ans. 22^ days. 

10. A merchant bought a piece of broadcloth measuring 
42 yds. for 191 dolls. 25 cents; 15 yards of this being 
damaged, he sells it at two-thirds of its cost ; the residue he 
is willing to sell so as to gain 1 doll, per yard on the whole 
piece ; at what rate must he sell the remainder ? 

Ans. 6 dolls. 86 T 4 T cents per yd. 

11. If 60 yards of carpeting will cover a floor that is 30 
feet long and 18 broad, what is the width of the" carpeting ? 

Ans. 3 feet. 

12. ff a piece of -land be 40 rods in length, how wide 
must it be to contain 4 arres ? Ans. 1 6 rods. 



64 SINGLE RULE OF THREE. 

13. Suppose a large wheel, in mill work, to contain 70 
cogs, and a smaller wheel, working in it, to contain 52 cogs; 
in how many revolutions of the greater wheel, will the les- 
ser one gain 100 revolutions? Ans. 2'88. 

14. The number of pulsations in a healthy person is, say 
70 in a minute, and the velocity of sound through the air is 
found to be 1142 feet in a second : now I counted 20 pulsa- 
tions between the time of observing a flash of lightning from 
a thunder cloud, and hearing the explosion of the thunder; 
what was the distance of the cloud ? 

Ans. 3 m. 5 fur. 145 yds. 2j ft. 

15. A merchant bought 5 pieces of cloth, of different 
qualities, but of equal lengths, at the rate of 5, 4, 3, 2, and 1 
doll, per yd. for the different pieces ; the whole came to 532 
dolls. 50 cents ; how many yards did each piece contain ? 

Ans. 35 yds. 

16. What principal will gain as much in 1 month, as 127 
dollars would gain in 12 months? Ans. 1524 dolls. 

17. If a pair of steelyards be 36 inches in length to the 
centre of motion, the pea 5 Ib. and the draught end \ inch 
in length, what weight, will they draw? Ans. 360 Ib. 

18. Supposing the nbovp steelyards would only Hrow 00 
Jb., what is the length of the draught end ? 

Ans. 2 inches. 

19. If 1 yard of cloth cost 2 dolls. 71 cts. 1J mills, what 
will 67i yards come to at the same rate? 

Ans. 183 dolls. 4 c. 6 m. 

20. If a man's income be 16s. 5d. l-J^qr. per day, what 
is it per annum ? Ans. 300Z. 

-21. How many pieces of wall paper that is 3 qrs. wide. 
and 11 yards long, will it require to paper the walls of a 
room that is 25 feet long, 15 wide, and 10- high, allowing a 
reduction of j\ for doors and windows? Ans. lQ-f T . 

22. The length of a wall being tried by a measuring line, 
appears to be 1287 feet 4 inches; but on examination the 
line is found to be 50 feet 10 inches in length, instead of 
50 feet its supposed length; rrquirod the true length of the 
wall? Ans. 1309 feet 10} J inch. 

23. If a dealer in liquors use, instead of a gallon, a mea- 
sure which is deficient by half a pint, what will be the true 
measure of 100 of these false gallons? Ans. 93j galls. 



DOUBLE RULK OF THREK DIRECT. 66 

SECTION 3. 

THE DOUBLE RULE OF THREE. 

THE DOUBLE RULE OF THREE, or as it is often called, 
Compound Proportion, is used for solving such questions as 
have five terms given to find a sixth. In all questions be- 
longing to this rule, the three first terms must be a supposi- 
tion, the two last a demand. 

RULE FOR STATING. 

1. Set the two terms of the supposition, one under the 
other. 

2. Place the term of the same kind with the answer 
sought in the second place. 

3. Set the terms of the demand, in the third place, ob- 
serving to place the correspondent terms of the supposition 
and demand in the same line. Consider the upper and lower 
extremes, with the middle terms, separately, as in the single 
rule of three ; if both lines are direct, then the question will 
be in direct proportion ; but if either lines are inverse, then 
will the question be in inverse proportion. 

When the question is in direct proportion, multiply the 
product of the two last terms by the middle term for a divi- 
dend, and multiply the two first terms for a divisor; the quo- 
tient will be the answer in the same name with the middle 
term. 

But if the proportion be inverse, transpose the inverse 
terms and proceed in the same manner as in direct proportion. 

DIRECT PROPORTION. 

EXAMPLE. 

1. It' 6 men in 8 days earn 100 dollars, how much will 
12 men earn in 24 days ? 

dolls. 
6 men ) , mn 5 12 men 



>t \ 
f ' 



8 days f ' J ) 24 days 

48 288 

100 



48) 28800 (600 dolls. AM. 

288 

00 
F2 



06 DOUBLE RULE OF THREE INVERSE 

2. If 10 bushels of oats suffice 18 horses for 20 days, 
how many bushels will serve 60 horses 36 days ? 

Ans. 60 bushels. 

3. If 56 pounds of bread will suffice 7 men 14 days, how 
much bread will serve 21 men 3 days ? Ans. 36 pounds. 

4. If 8 students spend 384 dollars in 6 months, how much 
will maintain 12 students 10 months? Ans. 960 dollars. 

5. If 20 hundred weight is carried 50 miles for 25 dol- 
lars, how much must be given for the carriage of 40 hun- 
dred weight 100 miles? Ans. 100 dollars. 

6. If 14 dollars interest is gained by 700 dollars in t> 
months, what will be the interest of 400 dollars for 5 years '.' 

Ans. 80 dollars. 

7. If 4 men can do 12 rods cf ditching in 6 days, how 
many rods may be done by 8 men in 24 days ? 

Ans. 96 rods. 

INVERSE PROPORTION. 

EXAMPLE. 

1. If 4 dollars pay 8 men for 3 days, how many days 
must 20 men work for 40 dollars ? 

days. 

As 4 dolls. ( q J 40 dolls. 

8 men. \ ) 20 men. 

Here the lower line is inverse, which transposed will stand 
thus : <t 

days. 

As 4 dolls. ( o ) 40 dolls. 

20 men. { ) 8 men. 

80 320 

3 

80 ) 960 ( 12 days. Ans. 
80 

160 
160 

2. If 4 men are paid 24 dollars for 3 days work, how 
many days may 16 men be employed for 3^4 dollars? 

Ans. 12 days. 

JJ. If 4 men arr {-aid '24 dollars f<;r 3 days work, hou 
many mon may !/o rmployrfl 16 days for 96 dollar? 

Ana. 3 men. 



DOUBLE KULK OF THRICE. (57 

4. If 7 men can reap 84 acres of grain in 12 days, bow- 
many men can reap 100 acres in 5 days ? Ans. 20 men. 

5. If 7 men can reap 84 acres of grain in 12 days, how 
many days will it require 20 men to reap 100 acres ? 

Ans. 5 days. 

6. If 40 cents are paid for the carriage of 200 pounds 
for 40 miles, how far may 20200 pounds be carried for 60 
dollars 60 cents ? Ans. 60 miles. 

7. If 5 men spend 200 dollars in 22 weeks and 6 days, 
now long will 300 dollars support 12 men ? 

Ans. 14 weeks 2 days. 

Promiscuous Questions. 

1. If 12 oxen in 8 days eat 10 acres of clover, how many 
acres will serve 24 oxen 48 days? Ans. 120 acres. 

2. A person having engaged to remove 8000 weight 15 
miles in 9 days ; with 18 horses, in 6 days, he removed 
4500 weight ; how many horses will be necessary to re- 
move the rest, in the remaining 3 days ? Ans. 28 horses. 

3. If the carriage of 9 hogsheads of sugar, each weighing 
12 cwt., for 60 miles, cost 100 dollars, what must be paid 
for the carriage of 50 barrels of sugar, each weighing 2,5 
cwt., 300 miles? Ans. 578 dolls. 70 + cents. 

4. If 1 pound of thread make 3 yards of linen, 5 quar- 
ters wide, how many pounds of thread will it require to 
make a piece of linen 45 yards long and 1 yard wide ? 

Ans. 12 Ib. 

5. If a footman travels 240 miles in 12 days, when the 
days are 12 hours long ; in how many days will he travel 
720 miles, when the days are 16 hours long? 

Ans. 27 days. 

6. A perch of stone measures 16^ feet long, 1^ foot broad, 
and 1 foot high ; at 1 doll. 25 cts. per perch,. what will a 
pile of stone *come to, which measures 30 feet long, 26 feet 
broad, and 4 feet high ? . Ans. 177 dolls. 27 -f cts. 

7. How many cords are there in a pile of wood 200 feet 
long, 10 feet high, and 36 feet broad ; the cord measuring, 
according to law, 8 feet in length, 4 in breadth, and 4 in 
height ? Ans. 562J cords, 

8. If 3 pounds of cotton make 10 yards of cloth, 6 qr. 
wide, how many pounds will it take to make a piece 100 
yards long and 3, qr. wide ? Am. 15 Ib 



68 PRACTICE 

9. If 24 men buiid a wail 200 feet long, 8 feet high, and 
6 feet thick, in 80 days, in what time will 6 men build one 
20 ft. long, ft. high, and 4 ft. thick ? Ans. 16 days. 

10. If a family of 9 persons spend 450 dolls, in 5 months, 
how much would they spend in 8 months, if 5 more were 
added to the family'? Ans. 1120 dollars. 

11. If a baker's bill for a family of 8 persons amounts to 
11^: dolls, in a month, when flour is at 10 dolls, per barrel, 
what will it amount to in 6 months if 4 more are added tc 
the family, and flour is at 11 dollars per barrel ? 

Ans. Ill dolls. 37 cts. 

12. If a cellar which is 22,5 feet long, 17,3 feet wide, and 
10,25 deep, is dug by 6 men in 2,5 days, working 12,3 
hours each day ; how many days of 8,2 hours will it re- 
quire 9 men to dig one which is 45 feet long, 34,6 wide, and 
12,3 feet deep ? Ans. 12 days. 



PART V. 
MERCANTILE ARITHMETIC. 



SECTION 1. 

OF PRACTICE. 

PRACTICE, so called from its frequent use in business, is 
only a contraction of the preceding rules of proportion. 
By it a compendious way is given of finding the price of 
any given quantity of goods or other articles of trade, when 
the price of 1 is known. 

Case 1. 

When the price consists of dollars, cents, and mills. 
Reduce the given quantity by multiplication, as in whole 
numbers, and point off from the right of the product for 
mills and cents, according to the rules in federal money. 
Or, multiply by the dollars only, and take aliquot or frac- 
tional parts for the cents and mills. 



PRACTICE 

TABLE. 

50 cents is of a dollar. 144 cents is 
33J . . . , 124 . 
25 . i . . 11J . 

20 . i . 10 

ief . ; t 5 

EXAMPLE. 

1. W^hat will 175 pounds of tea come to at 1 dollar 30 
cents and 5 mills per Ib. ? 



T'O 



130,5 
175 

6525 
9135 
1305 

228375 



cts. 
Or, 25 
5 
5 m. 



i 
J 

TV 



175 
4375 

875 
875 

228375 



2. 
3. 

4. 
5. 
6. 

7. 
8. 



228 dolls. 87 cts. 5 mills. 
Z). c. m. D. c. 7 

250 yards at 1,75 Ans. 437,50 



201 do. 4,20 884,20 

2210 do. 1,10 2431,00 

421 do. 2,41,5 1016,71,5 

625 do. 25 156,25 

8275 do. 4,4 864,10 

8275 do. 5 41,37,5 

Case 2. 

When the price is the fractional part of a dollar, or cent, 
such as J of a dollar, f of a cent, multiply the quantity by 
the numerator, and divide the product by the denominator ; 
the quotient will be the answer. 

EXAMPLE. 

1. What will 375 yards of muslin cost at } of a dollar 
per yard ? 

375 at | 
3 



4 ) 1125 



cwt. qr 
2. 4 1 
8. 12 2 
4. 14 2 



281,25 Ant. 
Ib. D. c. m. 

14 of sugar at of a doll, per Ib. Ans. 122,50 
13 of spice at f do. do. 942 

7 of lead at J do. for Mb. 285,42,5 



70 PRACTICE. 

Application. 

1. Bought 6 hogsheads of tobacco, each weighing 12,5 
cwt. at f of a doll, per pound ; what did it cost ? 

Ans. 3150 dolls. 

2. A gentleman bought a vessel of 60 tons burden, and 
gave at the rate of 2J eagles per ton ; what did the vessel 
cost 1 Ans. 1560 dolls. 

3. A carpenter bought 12650 feet of boards at 10 dol- 
lars per thousand ; what did they cost him ? 

Ans. 137 dolls. 56 cts. 8 m. 

Case 3. 

When the price and quantity given are of several denom- 
inations, multiply the price by the integers, or whole num- 
bers, and take aliquot parts for the rest. 

Table of a hundred weight. 



56 Ib. is of a Cwt. 

28 4 

16 . . 



141b. is 4 of a Cwt. 
A 



EXAMPLE. 



1. Bought Wcwt. Iqr. IQlb. of tobacco at 12 dollars 44 
cents per hundred weight ; what did it cost ? 

12,44 
16 



Iqr. is 1 4 I 19904 
1Mb. I 4 I 311 

1777} 

2039274 Ans. 203 dolls. 92 cts. 7| ms. 

2. ITcwt. 3qr. I9lb. of sugar at 10 dollars 94 cents per 
nundred weight. Ans. 196 dolls. 4cts. 

3. 5c*ot. Iqr. Qlb. of tobacco at 13 dollars 41 cents per 
hundred weight. Ans. 70 dolls. 40 cts. 2 m. 

4. 7&vt. Qqr. I9lb. of sugar at 15 dolls. 5 mills per hun- 
dred weight. Ana. 107 dolls. 58 cts. 



PRACTICE 



71 



Case 4. 

When the price consists of pounds, shillings, pence, and 
farthings. 

1. Reduce the given price to dollars and cents, (see re- 
duction of money, page 41) and then proceed according to 
the foregoing cases. Or, 

2. Multiply by the integers, and take aliquot parts for the 
remainder. 



of a shilling. 





TABLE. 


,?. d. 




J. 


10 is 


i of a pound. 


6 


68 


J 


4 


5 




3 


4 


f 


2 


34 


i 




26 


i 




2 


T V 




1 8 


Jy 





EXAMPLES. 

1. What will 4548 yards come to at Is. fid. per yard ? 
Is. 6d.= 18d. 20 cents= } of a dollar. 
cts. 
20 4548 



909,60 dollars. Ans. 

Or, 6d. | i | 4548 at 1 shilling, will be the same num- 
2274 oer of shillings. 



2|0 ) 682,2 shilling* 

.341 2 = 909 dolls. 60 cents. 



473 yards at 6 

397 do. 3 

\5\lb. of coffee at 1 
658/6. of tea at 12 
745 yds. of cloth at 16 
969* do. 19 

3715 do. 9 



4. 
5. 
6. 
7. 
8. 
9. 4567 



d. 


. 


s. 


d. D. c. 


8 


Arc*. 157 


13 


4=420,44$ 


4 


66 


3 


4 = 


8 


13 


5 


5r= 




394 


16 







596 





0= 


11 


964 


19 


a 


4i 


1741 


8 


ij 



do. 



19 114 4557 



9 i = 



72 



PRACTICE. 



Case 5. 

When both the price of the integer and the quantity are 
of different denominations, multiply the price by the inte- 
gers of the quantity, and take parts of the price for those 
of the integer. 



EXAMPLES. 



1. What will 45cu?i. 2qr. 14Z&. of sugar come to at 37. 

7 s. 9d. per hundred weight? 



5s. is 
2 

6d. is 
3 


t 

TV 
i 

i 

i 

i 


45 2 14 
379 


135 
11 5 
4 10 
1 2 6 
11 3 
1 13 104 
8 54 


2qr. is 
1Mb. 



Or thus, 



. 

3 



d. 
9 
5X9=45 



16 18 



152 8 9 
2qr. is 4 1 13 104 
14Z6. i 8 54 



.154 11 1 Ans. 



.154 11 1 



2. 37 T. llcwt. 2qr. 14Z6. of hemp at 89Z. 65. Sd. per 
ton. Ans. 3370Z. 13s. 2d. 

3. 3T. I2cwt. Sqr. 27lb. of sugar at bZ. 11s. 5d. per cwt. 

Ans. 6251. 11s. Wd. 

4. IT. Icwt. 2qr. 2llb. of rice at 3Z. 17s. 6d. per cwt. 

Ans. 5601. 13s. 3Jd. 

5. 476 acres 3 roods 28 perches of land at 3Z. 7s. lid. 
per acre. Ans. 1619Z. 11s. l$d. 

6. 640 acres 2 roods 20 perches at 10 dollars 55 cents 
per acre. Ans. 6758 dolls. 594 ct s- 

7. 229 acres 3 roods 18 perches at 18 dollars 50 cents 
per acre. Ans. 4252 dolls. 454 cts. 

9. I2cwt. Qqr. lib. at 6 dollars 34 cents per cwt. 

Ans. 76 dolls. 47 cts. 6 m. 

9. llcwt. 3qr. 24JZ&. at 14 dollars per cwt. 

Ans. 251 dolls. 56 cts. 24 m. 

10. 16 acres 3 roods 25 perches, at 125 dollars 50 cents 
per acre. Ans. 2121 dolls. 73 cts. 3-f m. 

11. 25ctrf. 3gr. 14Z/>. at 3Z. 17*. 6d. per cwt. 

AIM. 1007. 5*. 3j<f. 



TARE AND TRET. 73 

SECTION 2. 

Allowance on the weight of goods, called 
Tare and Tret. 

Tare, is an allowance made for the weight of the barrel, 
box, trunk, &c. in which goods are packed. 

Tret, is an allowance to retailers, for waste in the sales 
of their commodities. 

Gross, is the whole weight of the .goods with the barrel, 
box, &c. in which they are put up. 

Neat, is the weight of the goods after all allowances are 
deducted. 

RULE. 

Find the amount of the tare, and subtract it from the grosi 
weight, the remainder will be the neat. 

EXAMPLE. 

1. What is the neat weight of a hogshead of tobacco, 
weighing gross I2cwt. 3qr. I2lb. tare 14Z&. per cwt. 
Ib. cwt. qr. ib. 

14 12 3 12 gross 

12 1 2 12 tare 



168= the tare of I2cwt. Am. 11 1 neat 
10 8oz. = the tare of 3qr. 
1 Soz.~ the tare of I2lb. 



180= Irw*. 2$r. 12Z&. 

2. What will 3 barrels of sugar come to, weighing as fol- 
<ows: viz. No. 1, 2cwt. Iqr. 25/6. No. 2, 2cwt. 2qr. No. 3, 
2cwt. 2llb. tare 2llb. per barrel ; at 12 dollars 50 cents per 
cwt.? An,s. 82 dolls. 47 cts. 7 m. 

3. At 45 cents per pound, wnat wil 14 barrels of indigo 
come to, weighing as follows : 

cwt. qr. Ib. Ib. 

No. 1, 3 3 2 Tare 29 
No. 2, 4 1 10 38 

\^ 3, 4 o ]9 32 

No. 4, 4 00 35 



Am. 760 dolls. 95 cts. 



74 INTEREST. 

4. Bought 2 hogsheads of sugar, weighing as follows 
viz. No. 1, llcict. Iqr. lllb. tare 112Z6. No. 2, I2cwt. 2qr 
rare 74ZZ>. at 16 dollars 80 cents per cwt. neat; for which 
I gave 18 barrels of flour at 4 dollars 50 cents per bbl. and 
I % ton of iron at 120 dollars per ton; what was the balance 
still due? Ans. 112 dolls. 65 cts. 

5. What is the neat weight of 12 barrels of potash, each 
weighing ^cwt. 2qr. 26lb. tare I2lb. per cwt. ; and what 
will it come to at 9 dollars per cwt. ? 

Ans. SQcwt. 2qr. 23lb. and comes to 456 dolls. 34| cts. 

6. Sold a hogshead of sugar, weighing 6cwt. gross, tare 
100Z&. tret 4Z&. per 104, for 82 dolls. 50 cents ; what was it 
sold for per pound? Ans. 15 cents. 

7. In I20cwt. 3qr. gross, whole tare llllb. tret 4Z6. per 
104, how much is the neat weight in pounds, and what the 
amount at 73 cents per pound? 

Ans. 12833,6Z&. and comes to 9368 dolls. 53 cts. 

8. Bought 9 hogsheads of sugar, each weighing 6cwt. 
'2qr. I2lb gross ; tare lllb. per cwt. what is the neat weight, 
and wh?' Joes it amount to at 16 dollars per cwt. ? 

Ans. 5Qcwt. Iqr. 22lb. amounts to 807 dolls. 14 cts. 

9. Sold 27 bags of coffee, each 2cwt. 3qr. lllb. gross; 
ea re l'3ib. per cwt. tret 4Z&. per 104; what is the neat 
weight, and what will it co*ne to at 32 cents per pound ? 

Ans. 66cwt 2qr. lllb. and comes to 2386 dolls. 88 cts. 



SECTION 3. 

OF INTEREST. 

I NT FOREST is a compensation allowed for the use of money, 
ibi a givim time; and is generally throughout the United 
States fixed by law at the rate of 6 dollars for every 100, 
;-K>;- annum. 

1. Thr sum of w < .ney at interest, is called the Principal. 

^. The sum per -nt. agreed on, is called the Rate. 

3. Tho principal id interest added together, is called the 
A/Hf*itn.t. * 

Interest is firher. nfflc or compound. 



SIMPLE INTEREST. 75 

SIMPLE INTEREST. 

SIMPLE INTEREST is a compensation arising from the 
principal only. 

Case 1. 

When the given time is one or more years, and the prin- 
cipal dollars only. 

RULE. 

Multiply the given sum by the rate per cent. ; the product 
will be the interest for one year in cents, which multiplied 
by the number of years, will be the answer required. 

EXAMPLES. 

1. What is the interest of 454 dollars for one year, at 6 
per cent. ? 

454 
6 



2724 cents. Ans. 27 dolls. 24 cts. 
2. Required the interest of the same sum for 5 years, at 
the same rate ? 

454 




2724 
5 

13620 cents. Ans. 136 dolls. 20 cts. 
3. Required the amount of the same sum for 5 years, at 
he same rate ? 

454 
6 



2724 
5 

13620 interest. 
4.5400 principal. 



59020 amount. Ans. 590 dofts. 20 cts. 
4. What is the interest of 200 dollars for 2 years, at 6 
per cent ? Ans. 24 dolls. 



76 SIMPLE INTEREST. 

5. What is the interest of 1260 dolls, for 4 years, at 7 
per cent. ? Ans. 352 dolls. 80 cts. 

6. What is the amount of a note for 560 dollars for 3 
years, at 8 per cent. 1 Ans. 694 dolls. 40 cts. 

7. W T hat sum must be given to discharge a bond given for 
4520 dollars, on which there is 6 years interest at 5 per 
cent. ? Ans. 5876 dolls. 

Note. When the rate per cent, contains a fraction, such as , ^, f 
the principal must be multiplied by the fraction, as well as the whole 
number : this may be done either by adding- the parts of \, i, &.c. of 
the principal to the product of the whole number; or reduce the frac- 
tion to a decimal. See case 1, in reduction of decimals. 

8. W r hat is the amount of 400 dollars for 2 years, at 6| 
per cent. 1 Ans. 452 dolls. 

9. What is the interest of 4925 dollars lor y years, at 7^ 
per cent. ? Ans. 3324 dolls. 37 cts. 5 m. 

10. What is the amount of 2500 dollars for 1 year, at 7$ 
per cent. ? Ans. 2693 dolls. 75 cts. 

Case 2. 

When the principal is dollars and cents, or dollars, cents, 
and mills, and the time years only. 

RULE. 

Multiply the given sum by the rate per cent, and divide 
the product by 100 ; or, what is the same, point off two 
figures on the right of the product ; .the quotient, or remain- 
ing figures, will be the answer, in the same name with the 
lowest denomination in the principal. 

EXAMPLES. 

1. What is the interest of 264 dollars 50 cents for 1 year, 
at 6 per cent. ? 

264,50 
6 



cents 1587,00 Ans. 15 dolls. 87 cts. 
2. What is the interest of 468 dollars 22 cents and 5 
mills for 1 year, M 8 per cent. 
468,225 



mills 37458,00 Ans. 37 d.|ls. 45 cts. 8 m. 



SIMPLE INTEREST. ^ 77 

3. What is the interest of 364 dollars 50 cents for 5 years, 
at 6 per cent, per annum ? 

36450 Or, 36450 

6 30= product of the rate 

and time. 

218700 cents 10935,00 
5 



cents 10935,00 Ans. 109 dollars 35 cents. 

4. What is the amount of a note for 1260 dollars 50 
cents and 5 mills for 3 years, at 7^ per cent, per annum? 

Ans. 1544 dolls. 11 cts. 8-f ms. 

5. What sum will discharge a bond given for 630 dollars 
50 cents, on which there is 5 years interest at 8 per cent, 
per annum ? Ans. 882 dolls. 70 cts. 

6. What is the difference between the interest of 1274 
dollars 64 cents 6 mills for 3 years, at 7J per cent, per an- 
num, and the interest of 3462 dollars 84 cents, for 4 years, 
at 3^: per cent, per annum ? 

Ans. The lattep is 163 dolls. 37 c. 3,85 m. the greater. 

7. A gave B his bond for 3422 dolls. 25 cents, to be paid 
in the following manner, viz. one-third at the end of one 
year, one-third at the end of two years, and the remainder 
at the end of three years, with interest from the date, at 6 
per cent, per annum ; what will be the annual payments, 
and what the whole amount ? 

Ans. 1st payment 1209 dolls. 19,5 cts. 2d, 1277 dolls. 64 
cts. 3d, f346 dolls. 8,5 c. ; whole amount 3832 d. 92 c. 

Case 3. . 

When the principal is dollars, cents, &c. and the time is 
years and months, or months only. 

RULE. 

Multiply by half the number of months in the given time, 
when the rate is 6 per cent, per annum : but if the rate per 
cent, be more or less than 6 per cent., multiply the given 
number of months by the rate, and divide the product by 
12 ; the quotient will be the rate for the time; the principal 
multiplied by this rate, will give the interest required. 
G2 



78 ., SIMPLE INTEREST 

EXAMPLES. 

1. What is the interest of 650 dollars for 8 months, at 6 
per cent, per annum ? 

650 

4 half the months 



cents 2500 Ans. 26 dollars. 

2. What 13 the interest of 860 dollars for 1 year and 6 
months, at 6 per cent, per annum ? 
860 

9 half the months 



cents 7740 Ans. 77 dolls. 40 cents. 
3. What is the interest of 420 dollars for 9 months, at 8 
per cent, per annum ? 

9 months 420 

8 per cent. 6 



12 ) 72 2520 Ans. 25 dolls. 20c. 

6 the rate for the time. 

4. What is the amount of a note for 724 dollars, with 18 
months interest due thereon, at 4 per cent, per annum ? 

Ans. 767 dolls. 44 cts. 

5. What is the interest' of 240 dollars for 15 months, at 
7 2 per cent, per annum ? A.ns. 22 dolls. 50 cts. 

6. What is the interest of 1260 dollars for 4 months, at 
G per cent, per annum ? Ans. 27 dolls. 30 cts. 

Case 4. 

When the principal is dollars, cents, &c. and the time is 
months and days, or days only. 

RULE. 

Find the interest for the given months by the last case, 
find take aliquot parts for the days. 

Note. In calculation of interest, 30 days make a month. 

Or, multiply the given sum when the rate is 6 per cent. 
i)\ the number of days, and divide the product by 60 ; the 
<]ii<>;ient is the interest required. 

Note. Though both the foregoing methods are considered sufficiently 
exart for common business, by merchants and jiccomptants generally, 



SIMPLK INTKRKST. 7<J 

yet as this is only allowing 360 days in the year, and not 365, the true 
time ; if, therefore, the principal is large, on which interest is due, and 
greater exactness is required, then find the interest of the given sum 
tor 1 year, and proceed according to the single rule of three. As 365 
days : to the interest for one year :: the given number of days : the an 
swer. Or by the double rule of three, find the fixed divisors, which fbi 
5 per cent, is 7300, fer 6 per cent, is 6083, for 7 per cent. ^214 ; multi- 
ply the principal by the days, and divide by these divisors according to 
the rate per cent, required. 

EXAMPLES. 

1. What is the interest of 260 dollars for 5 months aid 
20 days, at 6 per cent, per annum ? 

260 
2 

\ month I i |5,2G interest ibr 4 months 
15 days [ i UfSO interest for 1 do- 
5 j |4 I 65 interest for 15 days 

2 1,6 interest for 5 do- 

7,36,6 Ans. 7 dolls. 36 cts. 6 m. 

2. What is the interest of 450 dollars for 36 days, at ti 
per cent, per annum ? 

450 6,0 ) 1620,0 

36 

2,70 Ans. 2 dolls. 70 cts. 
2700 
1350 

16200 

3. What is the interest of 564 dollars for 44 days, at 6 
per cent, per annum ? 

564 
6 

days days 

Ai 365 : 3384 :: 44 
44 



13536 
13536 

365 ) 148896 ( 4,079, Ans. -i dolls. 07 cts. 9 -f m. 
1460 

2896 
2555 

3410 
3285 

125 



80 SIMPLE INTEREST. 

4. What is the interest of 960 dollars for 70 days, at 6 
per cent, per annum ? 
960 

" 70 



6083 ) 67200 ( 1 1 + dolls. Ans. 
6083 

6370 
6083 

287 

5. What is the interest of 12000 dollars for 40 days, at 
7 per cent, per annum 1 Ans. 92 dolls. 6 cts. 

6. What is the interest of 8400 dollars for 20 days, at 5 
,,er cent, per annum ? Ans. 23 dolls. 

D. c. days D.c.m. 

7. 517,90 for 84 at 6 per cent, per annun*. Ans. 7,15,1 

e. 73,41 27 33 

9. 225,24 40 .... 1,48,1 

10. 1200,00 80 - 15,78,1 

11. ^62,19 254 123,68,8 

12. 1733,97 102 - 29,07,5 

A TABLE, 

knowing the number of day s+ from any day in any mojtth, to the same 
day in any other month through the year. 



From jJa.|Fb.|Mr.|Ap.|Ma. Ju. |Jly.|Au.| Se.|Oc.|No.|De. 



ToJan.|365|334|306|275|245|214|184|153|122| 92| 61| 31 



Feb. | 31j365|337|306[276|245|215|184|153|123| 92| 62 



59| 28|365|334|304|273|243|212!lftl|151|120| 90, 



April | 90| 59| 31|365|335|304|274',243|212|182|151|121 



May |120| 89| 61| 30|365J335|304|273|242|212|181|151 



|151|120|' 92 1 61| 31|365J885|304!273|243|212|182 



July |181|150|122| 91| 61| 30|365|334|303|273|242|212 



Au ? ust|2121181|153|122| 92 1 61 1 31|365|334|304|273(243 



sept. |243|212|184|153|123| 92| 62| 31|365|335|304|274 



Oct. |273|242|214|183|153|122| 92| 61 1 30|365|334|3Q4 



|304|273|245|214|184|153|123| 92| 61 1 31;365|335 



|334|303|275|244|214|183|153|122| 91 1 61| 30|365 



Suppose the number of days between the 10th of April 
and l()th of Ortohpr wpre rerjuirorl ; under the column of 



SlMl'LK 1NTKRKST. 8) 

April at the top of the table, look for October, and you find 
=188, the number required. 

Ii the days in the given months be different, their differ- 
ence must be added or subtracted, to or from the tabular 
number. Thus, from the 10th of April to the 20th of Oc- 
tober, is 183 + 10 = 198 days. And from the 20th of April 
to the 10th of October, 18310=173 days. 

If the time exceed a year, 365 days must be added for 
each year. 

Case 5. 

When the amount, rate, and time are given to find the 
principal. 

RULE. 

As the amount of 100 dollars, at the rate and time given, 
is to 100 dollars, so is the amount given to the principal re- 
quired. 

EXAMPLE. 

1. What principal being put to interest for 9 years, at 5 
per cent, per annum, will amount to 725 dollars ? 
9 



45 
100 

As 145 : 100 :: 725 : 500 Ans, 

2. What principal being put to interest for 12 years, at 
ft per cent, per annum, will amount to 2752 dollars? 

Ans. 1600 dolls. 

3. Received 728 dollars as payment in full for a note 
with 5 years interest thereon, at. 6 per cent, per annum ; for 
how much was the note given? Ans. 560 dolls. 

4. What sum put to interest for 4 years, at 7 5 per cent. 
per annum, will amount to 1 638 dollars ? 

Am. 1260 dolls. 

5. Received 2000 dollars as payment in full for a bond, 
with 5 years interest thereon, at 5| per cent, per annum ; 
what principal did the bond contain ? 

Ans. 1553 dolls. 39 cts. 8 T Jm. 



82 SIMPLE INTEREST 

Case 6. 

When the amount, time, and principal are given to find 
the rat3. 

RULE. 

1. As the principal is to the interest for the whole time 
so is 100 dollars to its interest for the same time. 

2. Divide the interest so found by the time, and the quo- 
tient will give the rate per cent. 

EXAMPLE. 

1. At what rate of interest per cent, will 500 dollars 
amount to 725 dollars in 9 years ? 

725 
500 

D. D. D. D. 

225 As 500 : 225 :: 100 : 45 
9)45 

5 per cent. Ans. 

2. Paid 858 dollars in full for a note given for 650 dollars, 
with 4 years interest due thereon; what was the rate per 
cent, per annum charged on said note ? Ans. S per cent. 

3. At what rate per cent, will 1600 dollars amount to 275*2 
dollars in 12 years? Ans. 6 per cent. 

4. At what rate per cent, will 640 dollars amount to 860 
dolls. 80 cents in 6 years? Ans. 5f per cent. 

5. At what rate per cent, will 12000 dollars amount to 
50100 dollars in 15 years? Ans. 4 percent. 

Case 7. 

When the principal, amount, and rate are given to find 
the time. 

RULE. 

Find the interest of the principal for one year. And then s 
as the interest for 1 year, is to 1 year, so is the whole inter- 
est to the time required. 

KX AMPLE. 

1. In what time will 500 dollars amount to 725 dollars 
at 5 per cent, per annum? 
500 
5 

D. Y. D. Y. 



!, acres! for 1 year 25,00 As 25 : 1 :: 22,) : <) Ans. 
Amount 725 
Principal 500 

Whole interest 22o 



Sl^lPLK INTKRKST. 83 

2. ln what time will 050 dollars amount to 910 dollars, 
at S per cent, per annum 1 Ans. 5 years. 

3. In what time will 1600 dollars amount to 2080 dollars 
at 6 per cent, per annum ? Ans. 5 years. 

Case 8. 

When the principal is in English money, viz. pounds-, 
shillings, and pence, and the interest required either in Fed- 
eral or English money. 

RULE. 

Reduce the English to Federal money, and find the in 
terest by the~preceding rules. 

EXAMPLE. 

1. What is the interest in Federal money, of 325Z. 105. 
English money, for 5 years, at 6 per cent, per annum ? 
. s D. c. 
325 10=1445,22 

30=the time multiplied by the rate. 



4335660 Ans. 433 dolls. 56 cts. 6m 

2. What is the amount of a note for 640/. 3s. 6d. with 3 
years interest due thereon, at 5 per cent, per annum, in Fed- 
eral money? Ans. 3268 dolls. 73 cts. 3|i m. 

3. What is the interest of 1374Z. ls.-9d. for U ycar : . aJ 
5f per cent, per annum? Ans. 1157. 185. Or/. 

Case 9. 

Computing interest on bonds, notes, &c. on which differ- 
ent payments have been made. 

RULE I. 

Find the interest of the principal from the time the inter- 
est first commenced, to the time of the first payment made ; 
add that interest to the principal, and subtract from the 
amount the payment made ; the remainder forms a new 
principal ; on which proceed in the same manner, till all the 
payments are brought in. 

Note 1. When a payment alone, or in conjunction with any pre- 
ceding- payment, is. less than the interest due at the time, then no cal- 
culation must be made ; but these lesser payments added to the next 

2. By this rule, no part of the interest ever forms a part of the prin- 
cipal carrying interest, the payments being first applied to discharge 
the interest. 



84 SIMPLE INTEREST. 

EXAMPLES. * 

1. A has B's note for 1000 dollars, dated 1st January 
1816, payable in 18 months, with interest from the date, at 
6 per cent, per annum. On which the following payments 
are endorsed, viz. 

1816. July 1. Rec'ci on the within note 230 dollars 

1817. Jan. 1. Rec'd - 300 
March 1. Rec'd ... 4 
April 1. Rec'd - 250 

What was the balance due on the 1st of July, 1817, when 
the whole note is payable ? 

D. c. 

Principal at interest from January 1, 1816, 1000 00 
1816, July 1. Interest (6 months) - 30 

1030 
Paid same date 230 



Remainder for a new principal - 800 

1817, January 1. Interest (6 months) '24 

824 
Paid same date ..... 300 



Remainder for a new principal - ^ 524 

March 1 . Paid 4 dollars less than the interest 

and not to be calculated. 
April 1. Interest (3 months) - 7 86 

531 86 
Paid same date 250 + 4= - . . 254 



277 86 
July 1. Interest (3 months) - 4 17 

Balance due Ans. 282 03 

RULE II. 

Multiply the principal by the number of days, till the first 
payment is made ; the remaining principal by the number 
of days, between the first and second payment, &c. till aU 
the payments are made ; divide the whole amount by 60 ; 
the quotient will give the interest required. 

Note. By this method the interest is generally calculated among 
merchants. 



INTEREST. 85 

1. A bond was given by B to C, for 2400 dollars, payable 
in 2 years, with interest from the date. Dated July 1, 1815. 
On this bond the following payments are eridgrsed ; viz. 
May 1, 1316, 900 dollars ; October 1, 1816, 450 dollars , 
January 1, 1817, 620 dollars. Required the amount due on 
the 1st of May, 1817? 

1815. July 1. Principal 2400 dollars. 

1816. May 1. 2400 multiplied by 304 is 729600 

Paid 900 



Oct. 1. 1500 .... 153 . 229500 

Paid 450 



1917. Jan. 1. 1050 . .-. . 92 . 96600 

Paid 620 



May 1. 430 .... 120 . 51600 

Interest 184 55 

. Divide by 6,0 ) 110730,0 

Balance 614 55 Ans. 

Interest 184,55 

2. A note was given by A to B, for 1800 dollars, dated 
1st January, 1820, with interest from the date. On which 
the following payments are endorsed, viz. April 1, 1821, 
700 dollars; January 1, 1822, 400 dollars; July 1, 1822, 
500 dollars. Required the amount due on the 1st of Janu- 
ary, 1823 ? Ans. 414 dolls. 16 cts. 6 m. 



COMPOUND INTEREST 

Is a compensation allowed not only for the principal, but 
also for the interest as it becomes due. 

RULE. 

Add the simple interest of the given sum for one year to 
the principal. This amount forms a new principal for the 
second year, and so on for any number of years required. 
Subtract the first principal from the last amount, the re- 
mainder will be the compound interest required. 

H 



COMPOUND INTEREST. 



EXAMPLE. 

1. What is the compound interest of 500 dollars for 3 
years, at 6 per cent, per annum I 
dolls. 

500 1st principal 
6 

30,00 interest 
500 



530,00 2d principal 
6 



31,80,00 
530 

561,80 3d principal 
6 



33,70,80 
561,80 

595,50,8 last amount 
500 

95,50,8 Ans. 95 dolls. 50 cents, 8 mills. 

Compound Interest may be more expeditiously calcufctted by the follow- 
ing Table, in which the amount of one dollar for any number of years 
under 30 is shown, at the rates of 5 and 6 per cent, per annum, com- 
pound interest. 



Years 


5 Ra 


tes 6 


Years 


5 Ri 


ites 6 


1 


1.05000 


1.06000 


16 


2.18287 


2.54035 


2 


1.10250 


1.12360 


17 


229201 


2.69277 


3 


1.15762 


1.19101 


18 


2.40662 


2.85434 


4 


1.21550 


1.26247 


19 


2.52695 


3.02559 


5 


1.27628 


1.33822 


20 


2.65329 


3.207 1.5 , 


6 


1.34009 


1.41852 


21 


2.78596 


3.39956 


7 


.40710 


1.50363 


22 


2.92526 


3.60353 


8 


.47745 


1.59384 


23 


3.07152 


3.81i>75 


9 


.55132 


1.68948 


33 


3.22510 


4.04898 


10 


.62889 


1.79064 


25 


3.38635 


4.29187 


11 


.71034 


1.89829 


26 


3.55567 


l.r,ii)38 


12 


.79585 


2.01219 


27 


.T73345 


1234 


13 


1.88565 


2.l32;j^ 


28 


3.920 J 3 


5.11168 


14 


1.97993 


2.26090 


29 


4.11613 


5.41838 


15 


2.07892 


2.39655 


30 


4.32194 


5.74349 



To find the compound interest of any sum by this labic, 
multiply the figures opposite the number of years, under the 
rate percent, b r 'n prim-ipal ; the product will be 

tbt: amount required ; from this subtract the principal, tht. 
remainder will b'' th< 



INSURANCE COMMISSION. AND BROKAGK. 87 

EXAMPLE. 

2. What is the compound interest of 1000 crbllars for W 
years, at 6 per cent, per annum ? 

1,59384 the tabular number for the time 
1,000 the principal 



1593,84000 
1000 

593,84 the interest. Ans. 593 dolls. 84 cts. 

3. What is the amount of 1500 dollars for 5 years, at 5 
pej* cent, per annum? Ans. 1914 dolls. 42 cts. 

4. What is the compound interest of 4500 dollars for 16 
years, at 6 per cent, per annum ? 

Ans. 6931 dolls. 57 cts. 5 m. 

5. A has B's note for 650 dollars, payable at the end of 
20 years, at 6 per cent, per annum, compound interest ; 
what sum will it require to discharge the note, at the expira- 
tion of the given time ? 

Ans. 2084 dolls. 63 cts. 4 m. 

6. A father left a legacy of 8000 dollars at compound in- 
terest, 6 per cent, per annum, to be equally divided among 
his three sons, when the youngest, who was 4 years old, 
should arrive at the age of 21 ; what will be each one's 
share? Ans. 7180 dolls. 72 cts. each share. 



SECTION 4. 

Insurance, Commission, and Brokage. 

INSURANCE is a premium given for insuring the owners of 
property against the dangers and losses to which it is liable, 
or indemnifying for its loss. 

The instrument of agreement by which this indemnity is 
secured is termed the policy of insurance. 

Commission is a compensation allowed to merchants and 
others for buying, selling, and transporting goods, wares, 



88 INSURANCE A\V> COMMISSION. 

Brokage is an allowance given to brokers for exchanging 
money, buying and selling stock, &c. 

The method of operation in all these is the same as in 
simple interest, 

EXAMPLES, 

INSURANCE. 

1. What is the premium of insuring 1260 dollars, at 5 
per cent ? 

1260 
5 



63,00 Ans. 63 dolls. 

D. c. 

2. 1650 dollars at 15 per cent. - Ans. 255 75 

3. 4500 25 - - - 1125 00 

4. What sum must a policy be taken out for, to cover 
900 dollars, when the premium is 10 per cent? 

100 policy 
10 premium 

90 sum covered 
As 90 : 100 :: 900 : 1000 dolls. Ans. 

5. What sum will it require to cover a policy of insurance 
for 4500 dolls, at 25 per cent 1 Am. 6000 dolls. 

6. What sum witl.it require to cover a policy of insurance 
for 560 dollars, at 9 per cent 7 Ans. 615 dolls, 38$ cts, 

COMMISSION. 

1. What is the commission on 850 dolls, at 5 per cent. 
850 
5 



42,50 Ans. 42 dolls. 50 cts. 

2. What is the commission on 1260 dollars, at 6 per 
rent? Ans. 75 dolls. 60 cts. 

D. c. 

3. 2550 dollars at 4 per cent. - - Ans. 102 00 

4. 26342 H 790 26 

5. 6422 } 48 10 J 



BROKAGE Bl'YINU AND SELLING STOCKS. 89 

6. A commission merchant receives 1260 dollars to fill 
an order, from which he is instructed to deduct his own 
commission of 5 per cent, how much will remain to satisfy 
the order? 

100 

5 per cent. 

As 105 : 100 :: 1260 : 1200 dolls. Ans. 

7. A commission merchant has received 4120 dollars 
\vith instructions to vest it in salt at 8 dollars per harrel ; de- 
ducting from it his commission of 3 per cent, how many 
barrels of salt can he purchase ? Ans. 500 barrels. 

BROKAGE. 

1. What is the brokage on 1000 dollars, at lj per cent? 
1000 
H 



1000 
500 

15,00 Ans. 15 dolls. 

2. What is the brokage on 1625 dollars 50 cents, at 3^ 
per cent? Ans. 54 dolls. 18 cts. 

3. 1868 dollars at 2 per cent. Ans. 46 dolls. 70 cts. 

4. 560 6 38 60 



SECTION 5. 

BUYING AND SELLING STOCKS. 

Stock is a fund vested by government, or individuals in a 
corporate capacity, in banks, turnpike roads, bridges, &c. 
the value of which is subject to rise and fall. 

RULE. 

Multiply the given sum by the rate per cent, and divide 
the product by 100 

H2 



90 REBATE OR DISCOUNT. 

EXAMPLES. 

1. What is the amount of 1650 dollars, United States 
bank stock, at 125 per cent, or 25 per cent, above par? 
1650 
125 



6250 
3300 
1650 

1,00 ) 2062,50 Ans. 2062 dolls. 50 cts. 
Or thus, 25 is i)1650 

412 50 

2062 50 AM. 

D. D. c. 

1500 bank stock at 110 per cent. Ans. 1650 00 

1686 128 . 2158 08 

4. 25000 108 - - 27000 00 

5. 1260 90 - - 1134 00 

6. 9254 84 - 7773 36 

7. 1518 83| 1271 32$ 



SECTION 6. 
REBATE OR DISCOUNT, 

Is a reduction made for the payment of money before il 
tecomes due. It is estimated in such a manner, as that 
the ready payment, if put to interest at the same rate and 
time, vvould amount to the first sum. Thus, 6 dollars is the 
discount on 106 dollars for 12 months, at 6 per cent, leaving 
100 dollars the ready payment, which, if put to interest for 
the same rate and time, would regain the 6 dollars discount. 

RULE. 

As 100 dollars and the interest for the given time, is to 
100 dollars, so is the given sum to its present worth. Sub 
tract the present worth from tho given sum, and the remain 
dor is the discount. 



REBATE OR DISCOUNT. 91 

^ EXAMPLE. 

1. What s the discount of 1696 dollars, due 12 months 
hence, at 6 per cent, per annum ? 

As 106 : 100 :: 1696 : 1600 
1600 



96 dolls. Ans. 

2. What is the present worth of 2464 dollars, due 1 year 
and 6 months hence, discounting at the rate of 8 per cent. 
nor annum] Ans. 2200 dolls. 

3. A has B's note for 1857 dollars 50 cents, payable 8 
months after date ; what is the present worth of said note, 
discounting at the rate of 5J per cent, per annum? 

Ans. 1791 dolls. 80 cts. 

4. What reduction must be made for prompt payment of 
a note for 650 dollars, due 2 years hence, 7 per cent, per 
annum being allowed for discount ? 

AILS. 79 dolls. 83 + cts. 

5. What is the present worth of 5150 dollars, due in 4^ 
months, discounting at the rate of 8 per cent, per annum, 
and allowing 1 per cent, for prompt payment ? 

Ans. 4950 dolls. 

Note. Discount and interest are often supposed to be one and the 
same thing; and in business, the interest for the time is frequently 
taken for the discount, and it is presumed neither party sustains any 
loss. This however is not true, for the interest of 100 dollars for 12 
months, at 6 per cent, is 6 dollars, whereas the discount for the same 
sum, at the same rate and ti'ne, is only 5 dollars 66 cents, making a 
difference of 34 cents for every 100 dollars for 1 year at 6 per cent. 
The following 1 examples will show the difference. 

EXAMPLES. 

1. What is the discount of 1272 dollars, due in 12 
months, discounting at 6 per cent, per annum ? 

.As 106 : 100 :: 1272 : 1200 

discount 72 dolls. 

2. What is the interest of the same sum, for the same 
time and rate ? 

1272 D. c. 
6 . Interest 76 32 
Discount 72 



76,32 interest, 



Difference 4 32 



92 BANK DISCOUNT. 

8. What is the difference between the interest and dis- 
count on 7280 dollars, for 18 months, at 8 per cent, per an- 
num? Ans. 93 dolls. 60 cts. difference. 

Note. But when discount is made for present payment, without re- 
gard to time, the interest of the sum as calculated for a year, is the 
discount. 

EXAMPLE. 

1. How much is the discount of 260 dollars at 5 per 
cent? 

260 
5 



13,00 Ans. 13 dollars. 

2. What is the discount on 1650 dollars, at 3 per cent! 

Ans. 49 dolls. 50 cte. 

3. What sum will discharge a bond for 2464 dollars, on 
which a discount of 8 per cent, is given ? 

Ans. 2266 dolls. 88 cts. 



SECTION 7. 

BANK DISCOUNT. 

BANK discount is the interest which banks receive for the 
use of money loaned by them for short periods. And as 
banks from long established custom, give three days over 
and above the time limited by the words of the note, called 
days of grace ; and as the day of the date, and the day of 
payment are both calculated, which makes the time 4 days 
longer than expressed in the note, so interest must be calcu 
lated on these days in addition to the regular interest on the 
given sum, for the specified time. 

RULE. 

Add 4 to the number of days specified in the note, multi- 
ply the given sum by this number, and divide the product by 
60. Or, 

Multiply the given sum by half t)ie number of days, and 
divide bv 30. 



EQUATION OF PAYMENTS. f 98 

Note. When the cents in the given sum are less than 50, the bank 
loses the interest on them, but when they are more than 50 they 
charge interest for one dollar. 



EXAMPLES. 



1. Required the discount of 1500 dollars for 60 days. 
1500 Or, 1500 

64 32= half the days 



6000 3000 

9000 4500 



6,0)9600,0 3,0)4800,0 



16,00 Ans. 16 dolls. 16,00 

2. What is the discount of 250 dollars for 30 days ? 

Ans. 1 doll. 41 1 cts. 

3. What is the discount of 600 dollars for 90 days ? 

Am. 9 dolls. 40 cts. 

4. What is the discount of 1260 dollars 40 cents for 60 
days 1 Ans. 13 dolls. 44 cts. 

5. What is the discount of 2649 dolls. 75 cents for 60 
days ? Ans. 28 dolls. 26 cts. 4 m. 

Form of a note offered for discount. 

Pittsburgh, July , 1832. 
Dollars 



Sixty days after date, I promise to pay A. B. or order, at 
the bank of , the sum of dollars, without de- 
falcation ; value received. J. P. 



SECTION 8. 

EQUATION OF PAYMENTS. 

EQUATION of payments is the finding the mean time, for 
the payment of two or more sums of money payable at dif- 
ferv.~t times. 

RULE. 

Multiply each sum by its own time. Add the products 



94 FELLOWSHIP. 

into one sum and divide this amount by the whole debt ; t/ie 
quotient will be the mean time. 

EXAMPLE. 

1. A owes B 600 dollars, of which 200 is to be paid at 
4 months, 200 at 8 months, and 200 at 12 months; but they 
agree to make but one payment ; when must that paymetit 
be made? 

200 X 4= 800 

200 X 8=1600 

200X12 = 2400 



600 ) 4800 ( 8 months. Ans. 

4800 



2. A merchant has owing to him from his friend, the sum 
of 3000 dollars, to be paid as follows, viz. 500 dollars at 2 
months, 1000 dollars at 5 months, and t le rest at 8 months ; 
but they agree to make one payment of the whole ; whit 
will be the mean time of payment? Ans. 6 months. 

3. A buys of B 50 acres of land, for which he agrees to 
pay 1000 dollars at the following times, viz. 200 dollars at 
5 months, 300 dollars at 8 months, and the rest at 10 months, 
but an equation of payments is afterwards agreed upon ; 
when must the payment be made? 

Ans. 8 months 12 days. 

4. C owes D 1400 dollars, to be paid in 3 months, but D 
being in want of money, C pays him 1000 dollars at the ex- 
piration of 2 months ; how much longer than 3 months may 
he in justice defer the payment of the rest ? 

Ans. 2% months 



SECTION 9. 

FELLOWSHIP. 

'SHH' loaches to find tho profit or loss arising f o 
different partners in trade, in proportion to the cap: J or 
stock each has advanced. 

Fellowship is either nngle or compound. 



SINGLE FELLOWSHIP. 95 

SINGLE FELLOWSHIP, 

Is when the stocks employed are different, but the time 
alike. 

RULE. 

Find the amount of the whole stock employed ; and then 
(by proporfton) as the whole stock is to the whole gain or 
loss, so is each partner's stock to his share of the gain or 
loss. 

EXAMPLE. 

1 . Two merchants join their stock in trade ; A puts in 
eOO dollars, and B puts in 400 dollars, and they gain 250 
dollars ; what part belongs to each ? 

A 600 A 1 nnn - n 600 to A's share 150 ^ 

B 400 ; 400 to B's 100 ! ^^ 

1000 250 J 

2. Three merchants enter into partnership in trade ; A 
advanced 7500 dollars, B 6000, and C 4500, with this they 
gained 5400 dollars ; what was each partner's share ? 

C A 2250 dolls, 
Ans. I B 1800 
I C 1350 

3. A bankrupt is indebted to A 1291 dollars 23 cents, to 
B 500 dollars 37 cents, to C 709 dollars 40 cents, to D 228 
dollars ; and his estate is worth but 2046 dollars 75 cents ; 
how much does he pay per cent, and how much is each 
creditor to receive ? 

D. c. 

(A receives 968 42$ 
B 375 27 | 

C 532 05 

D 171 

4. Three men, A, B, and C, rent a farm containing 585 
acres 2 roods and 34 perches, at 600 dollars per year, oJ 
which A pays 180 dollars, B 195, and C 225, and they 
agree that the larm shall be divided in proportion to the 
rents; how many acres must each man have? 

A. R. P. 
; share is 175 2 34j 
Ans. { B's 190 1 17^ 

219 2 22] 



i A's share is 175 
, { B's 190 

f C's 219 



96 S1JNGLE FELLOWSHIP. 

5. Three merchants freighted a ship with 2160 barrels of 
flour, of which 960 barrels belonged to A, 720 barrels to B, 
and 480 barrels to C ; but on account of stormy weather 
they were obliged to throw 900 barrels overboard ; how 
many barrels did each man lose? 

i A lost 400 barrels. 
Ans. IE $00 
(C 200 

6. Three merchants join stock in trade ; A put in 1260 
dollars, B 840 dollars, and C a certain sum ; and they gained 
825 dollars, of which C took for his part 275 dollars ; re- 
quired A and B's part of the gain, and how much stock C 
put in? 

( A gained 330 dolls. 
< Ans. <B 220 

( C's stock was 1050 

7. Four men traded with a stock of 800 dollars, and they 
gained in two years time twice as much, and 40 dollars over ; 
A's stock was 140 dollars, B's 260, C's 300 ; required D's 
stock, and what each gained ? 

(D's stock was 100 dolls. 
A's gain was 287 
B's 533 

C's 615 

D's 205 

8. Three butchers lease a pasture field for 96 dollars, into 
which they put 300 beef cattle ; of these 80 belonged to A , 
100 to B, and 120 to C ; how much had each to pay ? 

Ans. A 25 dolls. 60 cts. B 32 dolls. C 38 dolls. 40 cts. 

9. A father left an estate of 5000 dollars to his three sons, 
in such a manner that for every 2 dolls, that A gets, B shall 
have 3, and C 5 ; how much did each son receive ? 

Ans. A gets 1000 dolls. B 1500, C 2500. 

10. A, B and C put in money together, A put in 20 dolls 
B and C together put in 85 dollars ; they gained 63 dolls, of 
which B got 21 dollars ; what did A and C gain, and how 
much did B and C separately put in ? 

{A gained 12 dolls. 
C 30 

B put in 35 
C 50 



COMPOUND FELLOWSHIP. 97 



COMPOUND FELLOWSHIP. 

COMPOUND FELLOWSHIP is when both the stocks 'and 
times are different. 

RULE. 

Multiply each partner's stock by the time it is employed, 
add all the products into one sum; then say, as the sum of 
the products is to the whole gain or loss, so is each part- 
ner's stock multiplied by the time, to his share of the gain 
or loss. 

EXAMPLE. 

1. Three merchants entered into trade; A put in 2500 
dollars for 4 months, B 3000 dollars fur 6 months, and C 
4000 dollars for 8 months, and they gained 1200 dollars; 
what is each man's share of the gain ? 

D. M. 

A 2500x4=10000 
B 3000X6=18000 
04000X8=32000 



Sum 60000 

10000 : 200 A's share ) 

As 60000 : 1200 :: < 18000 : 360 B's V Ans. 
32000 : 640 C's > 



1200 proof. 

2. Three merchants enter into partnership for 16 months; 
A put into stock at first 600 dollars, and at the end of 8 
months, 200 dollars more ; B put in at first 1200 dollars, 
but at the end of 10 months, was obliged to take out 600 
dollars; C put in at first 1000 dollars, and at the end of 12 
months put in 800 more ; with this stock they gained 2300 
dollars ; what was each man's share ? 

C A's share is 560 dolls. 
Ans. 1 B's 780 

( C's 960 

3. A and B join stock in trade ; A put in 600 dollars or. 
the first of January ; B advanced on the first of April a sum 



98 COMPOUND FELLOWSHIP. 

which entitled him to an equal share of the profit at the end 
of the year ; required the sum B put in ? 

Ans. 800 dollars. 

4. D put in stock 1800 dollars ; E at the end of 4 months 
agrees to advance such a sum as at the end of the year wili 
entitle him to an equal share of .the profits; what sum must 
E advance? Ans. 2700 dollars. 

5. Two gentlemen, A and B, hired a carriage in Pitts- 
burgh to go to Philadelphia, and return, for 160 dollars, 
with liberty to take in two others by the way. When at 
Philadelphia they took in C, and afterwards, 100 miles from 
Pittsburgh, they took in D. Now allowing it to be 300 miles 
from Pittsburgh to Philadelphia, and also that each man pays 
in proportion to the distance he rode ; it is required to tell 
how much each must pay ? 

!A pays 60 dolls. 
B 60 

C 30 

D 10 

160 proof. 

6. Three graziers hired a piece of pasture ground for 145 
dolls. 20 cents ; A put in 5 oxen for 4 months, B put in 8 
oxen for 5 months, and C put in 9 oxen for 6 months ; how 
much must each pay ? 

Ans. A pays 27 dolls. B 48 dolls, and C 70 dolls. 20 cts. 

7. A, B, and C have received 665 dollars interest; A put 
in 4000 dolls, for 12 months, B 3000 for 15 months, and C 
5000 for 8 months ; how much is each man's part of the in- 
terest ? 

( A 240 dolls. 
Ans. I B 225 
f C 200 

HI. Three merchants lost by somo dealings 263 dollars 90 
-outs ; A's stock was 580 dolls, for 6 months, B's 580 dolls, 
for 9J months, and C's 870 dolls, frr 8| months ; how much 
is each man's part of this loss ? 

A's loss 59 dolls. 15 cts. 
Ans. B's 86 45 

C's 118 30 



PROFIT AND LOSS. 99 

SECTION 10. 

PROFIT AND LOSS. 

BY this rule we discover what has been gained or lost 
on the purchase and sale of goods, and merchandise of every 
kind. 

RULE. 

Prepare the question by reduction when necessary, and 
then work by the Rule of Three or Practice, as the nature 
of the question may require. 

EXAMPLE. 

1. Bought 360 barrels of flour for 6 dollars 25 cents per 
barrel, and sold it for 7 dollars 50 cents per barrel; what 
is the profit on the whole ? 

D. c. 

7 50 

6 25 



1 25 gain per barrel. 
B. D.c. B. D. 

As 1 : 1 25 :: 360 : 450 Ajis. 

2. Bought a piece of cloth for 1 doll, and 20 cents pei 
yard, and sold it again for 1 dollar 50 cents a yard ; what 
is the gain per cent 1 Ans. 25 per cent. 

3. Bought a piece of linen containing 42 yards for 21 
dollars, and sold it at 66 cents per yard ; what is the gain 
or loss on the whole piece ? 

Ans. 6 dolls. 72 cents gain. 

4. A merchant bought 6 barrels of whiskey containing 
32 gallons each, for 96 dollars ; while in his possession he 
l ost 12 gallons by leakage, the residue he sold for such a 
sum as gained him 12 dollars on the whole; how much per 
gallon did he buy and sell for ? 

Ans. Bought for 50 cents, and sold for 60 cents per gall. 

5. Bought 120 doz. of knives for 20 cents each knife, 
and sold them again for 17 cents each, what was the loss on 
the whole? Ans. 43 dolls. 20 cts. 



100 PROFIT AND LOSS 

6. A merchant gave 149 dollars for 100 yards of cloth ; 
at how much per yard must he sell it to gain 51 dollars on 
the whole 1 Arts. 2 dollars. 

7. Bought a chest of tea at 1 dollar and 25 cents per 
pound, but finding it to be of an inferior quality, I am will- 
ing to lose 18 per cent, by it ; how must I sell it per pound ? 

Ans. 1 doll. 24 cents per Ib. 

8. A merchant bought 20 dozen of wool hats at 90 cents 
per hat ; at what rate must he sell them again to gain 20 per 
cent, and how much does he gain on the whole 1 

Ans. he must sell at 1 dollar 8 cents per hat, and gains 
48 dollars 20 cents. 

9. A trader bought a hogshead of rum of a certain proof, 
containing 115 gallons, at 1 dollar 10 cents per gallon ; how 
many gallons of water must he put into it to gain 5 dollars, 
by selling it at 1 dollar per gallon 1 

Ans. 16i gallons. 

10. A merchant bought 4 hundred weight of coffee for 
134 dollars 40 cents, and was afterwards obliged to sell it 
at 25 cents per pound ; what was his loss on the whole, and 
how much on each pound ? 

Ans. 5 cents, loss on each pound, and 22 dollars 40 
cents on the whole. 

11. If by selling 360 yards of broadcloth for 1728 dollars, 
there is gained 20 per cent, profit, what did it cost per yard ? 

Ans. 4 dollars. 

12. A merchant laid out 1000 dollars on cloth, at 4 dol- 
lars per yard, and sold it again at 4 dollars 90 cents per 
yard ; what was his \yhole gain ? Ans. 225 dolls. 

13. A sells a quantity of wheat at 1 dollar per bushel, 
and gains 20 per cent. ; shortly after he sold of the same to 
the amount of 37 dollars 50 cents, and gained 50 per cent. ; 
how many bushels were there in the last parcel, and at what 
rate did he sell it per bu >hel ? 

Ans. 30 bushels, at 1 doll. 25 cents per bushel. 

14. A trader is about purchasing 5000 galls, of whiskey, 
which he can have at 48 cents per gallon in ready money, 
or 50 cents with two months credit ; which will be the most 
profitable, either to buy it on credit, or by borrowing the 
money at 8 per cent, per annum, to pay the cash price? 

Ans. he will gain 68 dollars by paying the cash. 

15. A butcher bought 12 head of beef cattle of equal 



BARTER. 101 

weight, for 240 dollars, which he sells again for 4 cents per 
pound ; what ought each one to weigh, that the butcher may 
have the hides and tallow as clear gain ? 

Ans. 4:cwt. Iqr. 24Z&. 



SECTION 11. 

BARTER. 

BARTER is the exchanging of one commodity for another 

at the rates agreed upon by their owners. 

$ 

RULE. 

Proceed by the rules of reduction and proportion, as the 
nature of the question may require. 

EXAMPLE. 

1. How many yards of linen at 50 cents per yard must 
be given for 6^ yards of broadcloth, at 4 dollars 50 cents 
per yard ? 

4,50 dollars 
6* 



2700 
112J 

28,12$ 

c. yd. D. c. yds. 

As 50 : 1 :: 28,12* : 56% Ans. 

2. A has 320 bushels of salt at 1 dollar 20 cents per 
bushel, for which B agrees to pay him 160 dollars in cash 
and the rest in coffee at 20 cents per pound ; how much cof- 
fee must A receive? Ans. 1120 Ib. 

3. How much rye at 70 cents per bushel must be given 
for 28 bushels of wheat, at 1 dollar 25 cents per bushel ? 

Ans. 50 bushels. 

4. A barters 319 Ib. of coffee at 23* cents per pound, 
with B for 250 yards of muslin ; what does the muslin cost 
A per yard 1 Ans. 30 cents nearly. 

5. C has flour at 5 dollars per barrel, which he barters 

12 



102 EXCHANGE. 

to D at a profit of 20 per cent, for tea which cost 1 dollar 
25 cents per pound ; at what rate must D sell the tea to 
make the barter equal? Ans. I doll, 50 cts. per Ib. 

6. A has cloth which cost him 2 dollars 50 cents per yard, 
but in trade he must have 2 dollars 80 cents ; B has wheat 
at 1 dollar 20 cents per bushel ; at how much per bushel 
should he sell to A, to make the barter equal ? 

Ans. I doll. 34f cents. 

7. P has 240 bushels of rye which cost him 90 cents per 
oushel ; this he barters with Q at 95 cents per bushel for 
wheat which stands Q 99 cents per bushel ; how many bushels 
of wheat is he to receive in barter, and at what price, that 
their gains may be equal ? 

Ans. 218/y bushels, at 1 doll. 4| cts. per bushel. 

8. A gives B in barter 26lb. 4oz. of cinnamon, at 1 dol- 
lar 28 cents per pound, for rice at 6 cents per pound ; how 
much rice must A receive ? Ans. 5 cwt. 

9. C and D barter ; C has muslin that cost him 22 cents 
per yard, and he puts it at 25 cents ; D's cost him 28 cents 
per yard ; at what price must he put it to gain 10 per cent, 
more than C ? Ans. 34|| cents per yard. 

10. A buys 250 barrels of flour from B, at 6 dollars 25 
cents per barrel, in payment B takes 4 cwt. of coffee at 30 
cents per pound, 64 pounds of tea at 1 dollar 75 cents per 
Ib. 25 yards of broadcloth at 6 dollars per yard, 206 dollars 
10 cents in cash, and the balance in salt, -at 8 dollars per 
barrel ; how many barrels of salt must B receive ? 

Ans. 120 barrels. 



SECTION 12. 

EXCHANGE. 

EXCHANGE is the reducing the money, coin, &c. of one 
tate or country to its equivalent in another. 

Par is equality in value ; but the course of exchange is 
often above or below pcu . 

Agiu is a term sometimes used, to express the difference . 
between bank and current money. 

Case 1. 
To change the currency of one state into that of another. 



EXCHANGE 



RULE. 

Work by the Rule of Three ; or by the theorems in the 
following table : 



* The New England 
and Maine. 
Note. In some part* 
Louisiana, Mississippi, I 


GQ 

O 

f ? 

. - 

~ p 
p 


r 

OKI 

5 2 
CL ^ 


^i 3 

o' co 

!> 

"~i P 

CD 3 

R?' 

^1 


II 

5* P~* 
- 

02 


ri 

3 p: 
Cta 

CD 


[TABLE, exhibiting 
Theorems fo 




p 




P CD 










.0- 


sr 




CD 
02 


i ,-*JL^ 


K 


>l * 
Is 3 


i 


l^ 3 

CD o- 


O 02 

P C 
CD cT 


t 


|f|.^ 


s ^** 


11 B 


o '->-' 




01 ^T 


co r 


^ 2 s^ s 


^^ S 


P CD p 








* P 








cr 


jj* Q 


P^ O 




^. re a ^ 


Ct5 ^5 


0> " 


a 


<-^ 






p. ^ s" 3" 





| 1" 3' 











50 ^^f 


f! 








o 4 S 


-<i 










S ' 


|| 1 


?^^ 


| 


< 


o 

CD t> 


^&^J 
a^&'S s 
** S 5 




j| 1 


P 


t * J^ 
O5 p 


Sf 




*m 
Ril- 


ii 


*-"* JO 


0- 










^ c^ 


o> i sr 












C^ 


"S _ 










> . 


53 S* 


s'S ; 
' ll I 


BQ 

0^ rj- (0 


QO O^ 


D 

H-* p- 

CJ P- 


O 

g > 


New York 
and 
Vorth Carotin 


Z. f? 

^ a- 


si 










s> 


^ "* 


"i 




^H 




=!- CC 


|; 


&1 


1 1 


E^ 


^2- 


.j. ^ 


? g" 


fifi 


3 


^ 


i^ 


- 


oi' 0|- 


8 S 


^'^S, 


4. 


r T 




MH-P- 






s' 


S 



The value of a dollar in any state is found, either opposite 
to that state, or under it in the table. 



104 EXCHAN^K. 



EXAMPLE. 

1. What is the value of 480Z. Pennsylvania currency in 
North Carolina? 





s. d. 


s. 


. 


. 


As 


7 6 


: 8 :: 


480 


: 512 Ans. 






. 






Or, 




480 








Add V 


t = 32 







512 Ans. 

2. What is the value of 256Z. New York currency in 

Pennsylvania? Ans. 240Z. 

How much South Carolina currency is equal to 1500Z. 

of New Jersey? Ans. 933Z. 6s. 3d. 

4. What sum New York currency is equal to 180Z. in 
Massachusetts? Ans. 240Z. 

1. How much Virginia currency will purchase a bill for 
280Z. South Carolina ? Ans. 360Z. 

6. A bill of exchangj being remitted from Rhode Island 
__ Kjouth Carolina for 304Z., what is its value in the currency 
of the latter? Ans. 236Z. 85. 



Case 2. 

To change the currency of the different states to Federal 
money. 

RULE. 

Divide the given sum, reduced to shillings, six-pences, or 
pence in a dollar, as it passes in each state. 

EXAMPLES. 

1. Change 127Z. 12s. New England money to dollars and 
cents. 

127/. 12s.=2552 shillings. 
The dollar, New England, is 6s. ) 2552 

425,33 
Ans. 425 dolls. 33 cts. 



FOREIGN KXCHAIS'GK. I0i 

2. Change 37/. 10s. Pennsylvania currency, to dollars. 
37 L 10s. 1500 six-pences. 

7s. 6d. or 15 six-pences make a dollar Pennsylvania cur- 
rency ; hence 1500 ~ 15= 100 dolls. Ans. 



Or, 37Z. 10s. = 9000 pence= 100,00 cents. Ans. 

3. Change 2251. 12s. New York currency to Federal 
money. 

225L 12s. 4512 shillings -h 8 the dollar New York cur- 
rency=564 dollars. Ans. 

4. A bill of exchange for 468Z. 9s. 6d. Virginia curren- 
cy, is remitted to Philadelphia ; what is its value in Federal 
money? Ans. 1563 dollars 25 cts. 

5. A merchant deposited in the United States branch 
bank at Pittsburgh, the sum of 750Z. 10s. Pennsylvania cur- 
rency, for what sum may he draw for in Federal money ? 

Ans. 2001 dollars 33^ cents. 

Note. Federal money being now generally introduced into mrrc;;r 
tile business throughout, the United States, has nearly superseded ti": 
use of the above rules of exchange between the different States. 



Case 3, 
FOREIGN EXCHANGE, 

Accounts are kept in Kngland, Ireland, and the West 
India Islands, in pounds, shillings, pence, and farthings 
though their intrinsic value in these places is different. 

A TABLE 

Of different Moneys^ as they are denominated and valued 
in different countries. 

REAT BRITAIN, IRELAND, AND THE WEST INDIES. 

4 farrhings =1 penny 

12 pence 1 shillino 

20 shillings I pound 



106 FOREIGN EXCHANGE. 



FRANCE. 


12 


Deniers 


= 1 Sol 




20 


Sols 


1 Livre 




3 


Livres 


- 1 Crown 


- 


SPAIN. 


4 


Marvadies Vellon, or 
Marvadies of Plate 


= 1 Quarta 




8* 
34 


Quartas, or 
Marvadies Vellon 


1 Rial Vellon 




16 
34 


Quartas, or 
Marvadies of Plate 


1 Rial of Plate 




8 


Rials of Plate 


1 Piaster, Pezo, or 


Dollar 


5 


Piasters - 


1 Spanish Pistole 




2 


Spanish Pistoles 


- 1 Doubloon 




ITALY. 


12 


Deniers 


= 1 Sol 




20 


Sols 


1 Livre 




5 


Livres 


1 Piece of Eight at 


Genoa 


6 


Livres 


1 Ditto at Leghorn 


6 


Solidi 


- 1 Gross 




24 


Grosses - 


1 Ducat 




PORTUGAL. 


400 


Reas 


= 1 Crusadoe 




1000 


Reas 


- 1 Millrea 




HOLLAND. 


8 


Penning - 


= 1 Groat 




2 


Groats 


- 1 Stiver =2d. 




6 


Stivers 


I Shilling 




20 


Stivers 


1 Florin, or Guilder 


3i 


Florins - 


1 Rix Dollar 




6 


Florins 


- 1 . Flemish 





5 Guilders - 1 Ducat 

DENMARK. 

16 Shillings - = 1 Mark 

6 Marks ~ - 1 Rix Dollar 
32 Rustics - 1 Copper Dollar 

6 Copper Dollars - - 1 Rix Dollar 

RUSSIA. 

18 Pennins - = 1 Gros 

30 Gros - 1 Florin 

3 Florins - 1 Rfx Dollar 

2 Rix Dollars - 1 Gold Puoai 



FOREIGN EXCHANGE ,10? 



RULE, 

In exchanging of foreign moneys, work by the R lie of 
Three, or by Practice ; and for exchanging foreign moneys 
to Federal, work by the table in page 40. 

EXAMPLES. 

1. Philadelphia is indebted to London 1749/. currency, 
what sum sterling must be remitted, when the exchange is 
65 per cent. ? 

. . . . 

As 165 : 100 :: 1749 : 1060 sterling. Ans. 

2. London is indebted to Philadelphia 1060Z. sterling; 
what sum Pennsylvania currency must be remitted, the ex- 
change being 65 per cent, as above ? 

. . . . 

As 100 : 165 :: 1060 : 1749 Ans. 

Or, 50 J 1060 

10 i 530 

5 | 106 

53 

1749Z. Ans. 

3. Baltimore, Oct. 1, 1817. 
Exchange for 1260Z. 10s. sterling. 

Thirty days after sight of this my first of exchange, sec- 
ond and third of like tenor and date not being paid, pay to 
A. B. or order, twelve hundred and sixty pounds ten shil- 
lings sterling, value received, and place the same to account, 
as per advice from 

P S n. 

W. L. merchant, London. 

What is the value of this bill in Federal money ? 

1260Z. 1 Os. = 1260,5.x by 444 cents=5596 dolls. Oa 
cents. Ans. 



108 FOREIGN EXCHANGE. 



4. London, January 1, 1818. 

Exchange for 5596 dolls. 62 cts. Federal money. 
Thirty days after sight of this my second of exchange, 
first and third of the same tenor and date not paid, pay to 
J. B. or order, five thousand five hundred and ninety -six 
dollars sixty-two cents, value received, and place the same 
to account, as per advice from 

S.'S. 

Mr. T. L. merchant, Baltimore. 

How much sterling is the above bill, 4,44 cents to the 
pound? 

444)5596,62(1260 
444 



1156 

888 

2686 
2664 

222 

20 



4440(10 Ans. 1260Z. 10*. 
4440 



5. A merchant of Philadelphia receives from his cor- 
respondent in Dublin, a bill of exchange for 540Z. 15s. Irish 
currency ; what is its value in Federal money ? 

Ans. 2217 dolls. 7 cts. 

6. A merchant in Philadelphia draws on his correspond- 
ent in Dublin for the balance of an account amounting to 
2217 dolls. 7 cents ; what sum Irish currency mast be re- 
mitted to satisfy the draft? Ans. 540/. 15s. 

Note. In tiPrie last examples tha course of exchange is considered 
;IH 1 cin{r at jKir: when the exch.-mge is nbove or below par, the per 
cent, must be added or subtracted, as the case requires. 

7. In a settlement between A of London and B of Phila- 
delphia, B is indebted to A in the sum of 3207. sterling, what 
sum must be remitted by B to A to settle the balance, the 
uxv hange being 12J per cent, from the United States to 
(iivat Britain? Ans. 1598 dolls. 40 cts. 



ALLIGATION. 109 

8. C of New Yo] ;v remits 3259 dollars to his correspond- 
ent in Dublin, to be placed to his account ; for what sum 
Irish currency, must he receive credit, the course of ex- 
change being 8 per cent, in favor of Ireland ? 

Ans. 7361. nearly. 

Note. The par ?f exchange between the United States of America 
and most other trading- countries, may be found by the table in page 40. 



SECTION 13. 

ALLIGATION. 

ALLIGATION is a rule for finding the prices, and quantity 
of simples in any mixture compounded of those things.* 

Case 1. 

To find the mean price of any part of the composition, 
when the several quantities and prices are given. 

RULE. 

As the sum of the whole quantity, is to its total value, so 
is any part of the composition, to its value. 

EXAMPLE. 

1. A merchant mixed 2 gallons of wine at 2 dollars per 
gallon, 2 at 2 dollars 50 cents, and 2 at 3 dollars ; what is 
one gallon of this mixture worth ? 
gal. 

2 at 2,00^400 
2 at 2, 50 = 500 
2 at 3,00 = 600 



6 1500 

G. D. c. G. D.c. 

As 6 : 15,00 :: 1 : 2,50 Ans. 

2. A grocer mixed 20 Ib. of sugar at 10 cents per Ib. 30 
Ib. at 15 cents, and 40 Ib. at 25 cents; what is one pound 
of this mixture worth? Ans. 18-1- cts. 

3. A trader mixes 10 bushels of salt at 150 cents, 20 at 

K 



110 ALLIGATION. 

160 cts. and 30 at 170 cts. per bushel; at what rate can he 
afford to sell one bushel of this mixture? Ans. 1G3-J cts. 

4. If 4 ounces of silver at 75 cents per ounce, be melted 
with 8 ounces at 60 cents per ounce, what is the value of 
one ounce of this mixture ? Ans. 65 cents. 

Case 2. 

To find what quantity of several simples must be taken 
at their respective rates, to make a mixture worth a given 
price. 

RULE. 

Place the rates of the simples under each other, and link 
each rate which is less than the mean rate, with one or more 
that is greater. The difference between each rate and the 
mean price set opposite to the respective rates with which it 
is linked, will be the several quantities required. 

Note. 1. If all the given prices be greater or less than the mean 
rate, they must be linked to a cipher. 

2. Different modes of linking will produce different answers. 

EXAMPLES. 

1. How many pounds of tea at 150, 160, and 200 cents 
per pound, must be mixed together, that 1 pound may be 
sold for 180 cents? 

t 150x 20 at 150 cents } 

Mean rate 180 2 160\) 20 at 160 V Ans. 

( 200^ 30 + 20=50 at 200 ) 

2. How many gallons of wine at 3, 5, and 6 dollars per 
gallon, must be mixed together, that one gallon may be 
worth 4 dollars? 

Ans. 3 gallons at 3 dolls. 1 gallon at 5 dolls, and 1 
gallon at 6 dollars. 

3. How many bushels of rye at 40 cents per bushel, and 
corn at 30 cents, must be mixed with oats, at 20 cents, to 
make a mixture worth 25 cents per bushel ? 

t 20^rx 15 f 5 C 6 bushels of rye 

1. Ans. 25 ? 30-J ) 5 2. Ans. < 6 do. corn 

( 40-^ 5 (24 do. oats 

4. A grocer has four several sorts of tea, viz. one kind 
at 120 cents, another at 110 cents, another at 90 cents 




ALLIGATION. Ill 

and another at 80 cents per pound, how much of each sort 
must be taken to fnake a mixture worth 1 dollar per pound ? 
' 2 at 120 cents. ( 3 at 120 cents. 

110 2 ^ ns >2 110 

(3 80 

f 1 at 120 cents. 

( 1 80 
f 2 at 120 cents. 
6. Ans. ) j* *g0 
(3 80 

Note. From this last example it is manifest that a great many dif- 
ferent answers may result to the same question, according to the va- 
rious modes of linking the numbers together. 

Case 3. 

When the rate of all the simples, the quantity of one of 
them, and the compound rate of the whole mixture are given, 
to find the several quantities of the rest. 

RULE. 

Arrange the mean rate, and the several prices, linked to- 
gether as in case 2, and take their difference. 

Then, as the difference of the same name with the quan- 
tity given, 

Is to the rest of the differences respectively : 

So is the quantity given, 

To the several quantities required. 

EXAMPLE. 

1. A grocer would mix 40 pounds of sugar at 22 cents 
per pound, with some at 20, 14, and 12 cents per pound ; 
how much of each sort must he take to mix with the 40 
pounds, that he may sell the mixture at 1 8 cents per pound ? 
f!2 N 41b. 

is) 14 V ~ 2 

^ 20_ / / 4 

. 22 6 against the price of the given quantity. 
As 6 : 40 :: 4 : 26,66 Ib. at 12 cents.) 

6 : 40 :: 2 : 13,33 do. 14 } Ans. 

and 26,66 do. 20 



112 ALLIGATION. 

2. How much wheat at 48 cents, rye at 06 cents, and 
barley at 30 cents per bushel, must be mixed with 24 bush- 
els of oats at 18 cents per bushel, that the whole may rate 
at 22 cents per bushel ? Ans. 2 bushels of each. 

3. How much gold at 16, 20, and 24 carats fine, and how 
much alloy must be mixed with 10 ounces of 18 carats fine, 
that the composition may be 22 carats fine ? 

Ans. 10 oz. of 16 carats fine, 10 of 20, 170 of 24, 
and 10 of alloy. 

Case 4. 

When the price of all the simples, the quantity to be mix- 
ed, and the mean price are given, to find the quantity of each 
simple. 

RULE. 

Find their differences by linking as before : 
Then, as the sum of the differences, 
Is to the quantity to be compounded ; 
So is the difference opposite to each price, 
To the quantity required. 

EXAMPLE. 

1. How much sugar at 10, 12, and 15 cents per pound, 
will be required to make a mixture of 40 pounds, worth 1 3 
cents per pound ? 



8 sum of the different simples. 
As 8 : 40 :: 2 : 10 lb. at 10 els. ) 

8 : 40 :: 4 : 20 do. 15 \ Ans. 
and 10 do. 12 > 

2. How much golJ of 15, of 17, of 18, and of 22 carats 
fine, must be mixed together to form a mixture of 40 ounces 
of 20 carats fine? 

Ans. 5 oz. of 15, of 17, and of 1 8, and 25 oz. of 22. 

3. How many gallons of water must be mixed wkh wine 
at (> dollars per gallon, to fill a vessel of 70 #,i lions, so that 
it may he sold without loss at 5 dollars per gallon? 

4/7A-. 1 1 07) lions of water. 



VULGAR FRACTIONS. 1 1 3 

PART VI. 
VULGAR FRACTIONS. 

A VULGAR FRACTION is any supposed part or parts of an 
unit, and is represented by two numbers placed one above 
the other, with a separating line between them ; thus, | one- 
fifth, four-ninths. 

The number above the line is called the numerator, and 
that below the line the denominator. Thus, 

4 numerator 6 
&c. 

9denomina. 10 

The denominator shows how many parts the unikpr inte- 
ger is divided into, and the numerator shows how many of 
those parts are contained in the fraction. 

Vulgar fractions are either proper, improper, compound, 
or mixed. 

A proper fraction is when the numerator is less than the 
denominator, as -J, f, j, |f , &c. 

An improper fraction, is when the numerator is eitner 
equal to, or greater than the denominator, as f , f, 2 T , &c. 

A compound fraction is a fraction of a fraction, as f of $, 
f of/ of jf,&c. 

A mixed fraction is a whole number and fraction united, 
as 8|, 4J, 120f , &c. 



SECTION 1. 

Reduction of Vulgar Fractions. 

Case 1. 
To Reduce a Vulgar Fraction to its lowest terms. 

RULE. 

Divide the greater term by the less, and that divisor by 
the remainder, till nothing be left, the last divisor is the 
K2 



114 REDUCTION OF VULGAR FRACTIONS. 

common measure, by which divide both parts of the frac- 
tion : the quotient will be the answer. Or, 

Take aliquot parts of both terms continually, till the frac- 
tion is in its lowest terms. 

Note. 1. 1.' the common measure when found is 1, the fraction ia 
already in its lowest terms. 

2. Ciphers to the right of both the terms may be cut off thus, 

w=f 

EXAMPLES. 

1. Reduce J| to its lowest terms. 
36 ) 48 ( i 
36 

Common measure 1 2 ) 36 ( 3 
36 

12)36(3 

36 3 2 div. 6 div. 

Ans. 36 18 3 

12)48(4 Or, = =- Ans. 

48 48 24 4 

2.. Reduce ^f to its lowest terms. Ans. f 

3. Reduce T 7 ^ to its lowest terms. | 

4. Reduce fVVVV to i ts lowest terms. f 

5. Reduce T y T to its lowest terms. | 

6. Reduce gW/T to its lowest terms. J 

Case 2. 
To reduce a mixed number to an improper fraction. 

RULE. 

Multiply the whole number, by the denominator of the 
fraction, and add the numerator to the product, for a new 
numerator, under which place the given denominator. 

EXAMPLE. 

1. Reduce 8$ to an improper fraction. 
8 
4 

3-35 

- Ans. 
4 



REDUCTION OF VULGAR FRACTIONS. 115 

2. Reduce 12 T f to an improper fraction. Ans. 2 j~? 

3. Reduce 183 -/ T to an improper fraction. *ff 8 

4. Reduce 514 T \ to an improper fraction. 8 T | 9 

5. Reduce 68425? to an improper fraction. a 7 3 T 7 3 

Case 3. 

To reduce an improper fraction to a whole or mixed 
number. 

RULE. 

Divide the numerator by the denominator ; the quotient 
will be the answer required. 
Note. This case and case 2, prove each other. 

- EXAMPLE. 

1 . Reduce \ 5 to its proper terms. 

4)35(8| Ans. 
32 

3 

2. Reduce 3 f T 8 to its proper terms. Ans. 183^ 

3. Reduce 2 \ 6 5 to its proper terms. 352i 

4. Reduce 3 T 6 y to i ts proper terms. 56y 

5. Reduce 8 y| 9 to its proper terms. 514 T 5 

Case 4. 

To reduce several fractions to others that shall have one 
common denominator, and still retain the same value. 

RULE. 

Reduce the given fractions to their lowest terms, then 
multiply each numerator into all the denominators, but its 
own. for a new numerator ; and all the denominators into 
each other for a common denominator. 

EXAMPLE. 

1. Reduce t, |, and |, to a common denominator. 
1X3X412V 
2X2X4 16> numerators. 
3X2X318) 
2X3X424 common denominator. 



116 



REDUCTION OF VULGAR FRACTIONS. 



2. Reduce f , |, and -, to a common denominator. 

Ana. ?&, V&, || 
8. Reduce i, -|, T 4 j, and f , to a common denominator. 



Case 5. 

To reduce several fractions to others, retaining the same 
value, and that shall have the least common denominator. 

RULE. 

Divide the given denominators by any number that will 
divide two or more of them without a remainder ; set the 
quotients and undivided numbers underneath ; divide these 
numbers in the same manner, and continue the operation, 
till no two numbers are left capable of being lessened ; the 
product of these remaining numbers, together with the di- 
visor or jdivisors, will give the least common denominator. 

Divide the common denominator, so found, by each par- 
ticular denominator, and multiply the quotient by its own 
numerator for a new numerator, under which place the com- 
mon denominator. 

EXAMPLE. 

1. Reduce |, J, j-, and f , to the least common denom- 
inator. 

3)2 368 



2)2 128 

111 4 X~2x 3=24 common denominator. 
'2)24 



Divisors 



12X1 = 12 
3 8x2=16 
6 4X5=20 
^8 3X7=21 

Then, if, ||, ff, f i Ans. 

2. Reduce f , f , T \, and vV, to the least common donom. 
mator. Ans. A n -, rV 5 o> T V 



REDUCTION OF VULGAR FRACTIONS. _ 117 

3. Reduce |, f , T 4 5 -, and |, to the least common denomi- 
nator. Ans. J-f , H, H> If 

Case 6. 
To reduce a compound traction to a single one. 

RULE. 

Multiply all the numerators together for a new numerator, 
and all the denominators for a new denominator. 

Note. Such figures as are alike in the numerators and denominators 
may be cancelled. 

EXAMPLE. 

1. Reduce f of J of to a single fraction. 

2 X 3 X 4 24 2 Or cancelled. 

= - Ans. 2 3 4 2 

3X4X5=60 5 _ _ _ = _ Ans. 

3 4 5 5 

2. Reduce J of of T \ to a single fraction. 

Ans. fi 

3. Reduce of f of J to a single fraction. 

Ans. ^ 

4. Reduce f of ^ of f J to a single fraction. 

Ans. T Vo 
Case 7. 

To reduce a fraction of one denomination to the fraction 
.of another, but greater, retaining the same value. 

RULE. 

Make it a compound fraction, by comparing it with all 
the denominations between it and that to which it is to be 
reduced ; reduce this fraction to a single one. 

EXAMPLE. 

1 . Reduce f of a penny to the fraction of a pound. 
5 X 1 X 1 5 



6 X 12 X 20 1440 

2. Reduce of a pennyweight to the fraction of a pound 
troy Ans. J T 

% Reduce T 9 :i - of a pint of wine to the fraction of a hogs- 
head. Ans. ^fa 



118 REDUCTION OF VULGAR FRACTIONS. 

4. Reduce TT of a minute to the fraction of a day. 



TJ84 

Case 8. 

To reduce tne fraction of one denomination to the frac- 
tion of another, but less, retaining the same value. 

RULE. 

Multiply the given numerator, by the parts of the denom- 
inator, between it and that to which it is to be reduced, for a 
new numerator, and place it over the given denominate,,-, 
which reduce to its lowest terms. 



1. Reduce TT 5 T -g- of a pound to the fraction of a penny. 

5 X20X12 1200 5 

1440X1 X 1 H40 6 

2. Reduce -$~ of a pound troy to the fraction of a pen- 
nyweight. Ans. j 

3. Reduce 7 ~ of a hogshead to the fraction of a pint. 

Ans. ^ 

4. Reduce yj-g-4 f a day to tne fraction of a minute. 

Ans. VT 
Case 9. 

To- find the value of a fraction in the known paits of an 
integer. 

RULE. 

Multiply the numerator by the known .parts of the inte- 
ger, and diviilo by the denominator. 

EXAMPLE. 

1. What is the value of f of a pound sterling? 
20 shillings = 1 pound. 
2 

3 ) 40 

13 4 Ans. 13*. 4ct 

t. Red are J of a pound troy tc iu proper quantity. 

Ans. 7oz. 4dwt. 



REDUCTION OF VULGAR FRACTIONS. 11D 

3. Reduce f of a mile to its proper quantity. 

Ans. 6fur*. Wp. 

4. Reduce T 3 F of a day to its proper time. 

Ans. 7h. I2min. 

5. What is the value of f of a dollar. Ans. 80 cts. 

Case 10. 

To reduce any given quantity, to the fraction of a greater 
Munomination of the same kind. 

RULE. 

Reduce the given quantity to the lowest denomination 
mentioned for a new numerator, under which set the integral 
part (reduced to the same name) for a denominator. 

EXAMPLES. 

1. Reduce 6s. Sd. to the fraction of a pound. 

s d. s. 

68 20 

12 12 

80 1 240 

=- Ans. 

240 3 

2. Reduce 25 cents to the fraction of a dollar. 

25 1 

100 4 

3. Reduce 31 gallons 2 quarts to the fraction of a hogs- 
head. Ans. i. 

4. Reduce 6 hundred weight 2 quarters 18| pounds to the 
fraction of a ton. Ans. J. 

Case 11. 

To reduce a vulgar fraction to a decimal of the same 
value. 

RULE. 

Add ciphers to the right-hand of the numerator, and cfivide 
by the denominator. 



120 ADDITION OF VULGAR FRACTIONS 



EXAMPLE. 

' 1. Reduce f o a decimal fraction of the same value. 
4 ) 300 

,75 Ans. 
2. Reduce ^J to a decimal fraction. Ans. ,85 



SECTION 2. 
ADDITION OF VULGAR FRACTIONS. 

Case 1. 

To add fractions that have the same common denomi- 
nator. 

RULE. 

Add all the numerators together, and divide the amount 
by the common denominator. 

EXAMPLE. 

1. Add T ^, r 5 2, T 7 2> T 9 2 and |J together. 

numerators. 
1 
5 
7 
9 
11 

common denominator 12 ) 33 ( 2| An*. 
24 

9 3 

12 4 

2. Add /y, T 8 T , i j, {, and if together. Anj. 2$ 
8. Add Jf, H, AJ,and JJ together. 3*. 



ADDITION OF VULGAR FRACTIONS. 121 

Case 2. 

To add fractions having different denominators. 
RULE. 

Reduce the given fractions to a common denominator, by 
case 5, and proceed as in the foregoing case. 

EXAMPLE. 

1. Add $, J, and T 9 j together. 

12 

1359 12 18 15 18 18 
---- = -- --- 15 
I 4 8 12 24 24 24 *4 18 

24)63(2} 

48 

15 

2. Add i, i, and together. Ans* 1 & 

3. Add f , f ,3, f and r 8 5 together. 3^ 

Case 3. 

To add mixed numbers. 

RULE. 

Add the fractions as in the foregoing cases, and the inte- 
gers as in addition of whole numbers. 

EXAMPLES. 
I. Add 13^, 9 7 <y and 3/j together. 



2. Add 5|, 6 j and 4 together. 

5|-rr5if common denominator 
6J = 6U 15 



122 SUBTRACTION OF VULGAR FRACTIONS. 

3. Add 1J, j of J, and 9^ together. Ans. 

4, Add I fa 6, of i, and 7i together. 

Case 4. 
To add fractions of several denominations. 

RULE. 

Reduce them to their proper quantities by case 10 in re- 
duction,, and add them as before. 

EXAMPLE. 

1 . Add J of a . and y 3 ^ of a shilling. 

s. d. 

f of a . = 15 6|=|| common denom. 
ft of a ,.-0 3j = A 



9 



Shillings 15 10/3- Ans. 

2. Add i of a yard to f of a foot. Ans. 2 feet 2 inches. 

3. Add i of a day to of an hour. 

Ans. 8 hours 30 minutes. 

4. Add J of a week, J of a day. and \ of an hour to- 
gether. . Ans. 2 days 14 hours 30 minutes. 

5. Add J of a miie, f of a yard, and J of a foot together 

Ans. 1540 yards 2 feet 9 inches. 



SECTION 3. 

Subtraction of Vulgar Fractions. 

RULE. 

PREPARE the fractions as in addition, and sub:ract the 
rower numerator from the upper, and place the rfrilerence 
over the common denominator. 

Note. 1. When the lower numerator is greater than the upper, sub- 
tract it from the common denominator, adding the .nipper numerator to 
*he difference, and carry 1 tcf the units place of t^e integer. 

2. When the fractions are of different integers, find their values, 
separately t and subtract as in compound subtraction of whole tium 



MULTIPLICATION OF VULGAR FRACTIONS. 



123 



EXAMPLES. 



From | 
Take f 



From f 
Take 



From 1= 
Take i= 



Rem. f =i A/is. Rem. 



From ji 
Take J 

Rem. i 



From 
Take 

Rem. 



From f 
Take j 

Rem. TjV 

From 13 
Take 6 



Ans. Rem. 
From |?f 



Rem. : 

From 
Take 

Rem. 



fc Ans. 

- From }| 
Take |i 

Rem. T V 

From 19 T \ 
Take T 7 



Rem. 



From Jof a .=15 6| 
Take ft of a s. = 3J 



Rem. 



15 



From 7 weeks 
Take 9 T V days 



Rem. 



. 7h. 12m. 



SECTION 4. 

Multiplication of \ ulgar Fractions. 

RULE. 

REDUCE the compound fractions to simple ones, and 
mixed numbers to improper fractions, then multiply the nu- 
merators together for a new numerator, and the denomina- 
tors for a new denominator. 



1. Multiply J by 



EXAMPLES. 
2X1=2 1 

- - =- An*. 
3X4=12 



124 DIVISION 7 OF VTLHAR FRACTIONS. 

2. Multiply 4i by | 

2 

9X1= 9 

- - An*. 

2x8=16 

3. Multiply | by f Ans. ^ 

4. ' J of | by . . . - & 

5. H by i - . - 1 j 

T. 48} by 13| - . . . 672y\ 



SECTION 5. 

Division of Vulgar Fractions. 

RULE. 

PREPARE the fractions, if necessary, by reduction ; inrert 
the divisor, and proceed as in multiplication. 

EXAMPLES. 

1. Divide f by j| 

3x3 9 

- -= An*. 

8x2 16 

2. Divide 4i by 1| 

*J H 

2 3 

9X3 27 
5 Then - -==2,7, 

2X5 10 
2 a 



RULE OF THREE IN VULGAR FRACTIONS. 125 

3. Divide f by f Ans. f } 

4. Divide if by f Iff 

5. -Divide li by 4 T \ ft 

6. Divide J by 4 /, 

7. Divide 9| by i of 7 2if 

8. Divide 5205} by f of 91 71* 



SECTION 6. 

The Rule of Three in Vulgar Fractions. 

THE operation of tbe Rule of Three in Vulgar Fractions, 
whether direct, inverse, or compound, is performed <in the 
same manner and agreeably to the principles laid down in 
whole numbers under these rules. 

When the question is in direct proportion, prepare the 
terms by reduction, and invert the first term ; then proceed 
as in multiplication of fractions. 

EXAMPLE. 

1. If i of a yard of cloth cost f of a dollar, what will I 
of a yard come to ? 

yd. D. yd. 
As i : 1 :: | 
4X2X7 = 56 D.c. 

Inverted - - - =2J dollars. Ans. 2 33 J 
1X3X8 = 24 

2. If f of a ton of iron cost 164J dollars, what will ^ of 
a ton come to? Ans. 211 dolls. 28| cts. 

3. A person having f of a coal mine, sells | of his share 
for 171 dollars, what is the value of the whole mine at the 
same rate? Ans. 380 dollars. 

4. At f of a dollar per yard, what will 42 yards come to ? 

Ans. 35 dollars. 

5. A gentleman owning | of a vessel, sells f of his share 
for 312 dollars, what is the whole vessel worth ? 

Ans. 11 70 dollars. 
L2 



126 INVERSE PROPORTION. 

6. If li bushel of apples cost 79 J cents, what will 3f 
bushels cost at the same rate ? - Ans. 202 T \ cents. 

7. If J of a ship be worth 175 dollars 35 cents, what 
part of her may be purchased for 601 dollars 20 cents? 

Ans. 3j 



SECTION 7. 
INVERSE PROPORTION. 

RULE. 

PREPARE the question as in direct proportion, invert the 
third term, and proceed as in multiplication of fractions. 

EXAMPLES. 

1. How much shalloon f yard wide, will lino 4J yards of 
cloth l yard wide? 

H = % Then, as I : f :: f 

4i=4 
Or, inverted f : f : : | = LO 9 Ans. 

2. If 6 hundred weight be carried 22 ^ miles for 25f 
dollars, how far may 1 hundred weight be carried for the 
same money? Ans. 145J miles. 

3. If 12 men can finish a piece of work in 37 f days; 
how long will it take 16 men to do the same work ? 

Ans. 281 days. 

4. A lends to B lOOf dollars for 6| months; what sum 
should B lend to A for 3|- years, to requite his kindness ? 

Ans. 14|f f dollars. 

5. Mow many feet long must a board be, that is | of a 
root wide, to equal one that is 20 feet long, and f of a foot 
wide? Ans. 17 i lert long. 

6. In exchanging 20 yards of cloth of 1 i yard wide, for 
some of the same quality of | yard wide, what quantity of 
the latter makes an equal barter ? 

Ans. 34 yards. 



INVOLUTION. 



12? 



PART VII. 

EXTRACTION OF THE ROOTS, AND COMPARATIVE 
ARITHMETIC. 



SECTION 1. 

Involution, or the Raising of Powers. 

INVOLUTION is the multiplying of a given number by 
itself continually, any certain 'number of times. 

The product of any number so multiplied into itself, is 
termed the power of that number. 

Thus 2x2 = 4= the second power or square of 2* 

2x2x2 = 8= the third power or cube of 2. 
2x2x2x2 = 16= the fourth power of 2, &c. 

The number denoting the power to which any give n sum 
is raised, is called the index or exponent of that powe r. 

If two or more powers are multiplied together, ttu ir pro- 
duct will be that power, whose index is the sum of the expo- 
nents of the factors. Thus 2x 2=4, the 2d powei of 2 , 
4X4=16, the 4th power of 2; 16x16 = 256, tlie 8th 
power of 2, &c. 

TABLE 
Of the first nine powers. 



1 


CU 

rt 
Jf 


i 

o 


o 

1 


1 


1 
S 


1 
1 


ft 

o 

1 

00 


5 


1 


1! 


11 


11 1 


1 


11 1| 


1 


2 


4, 


8| 


16! 32 


64 


128 256| 


512 


3| 


yi 


27| 


81 243 


729| 2187) 6561! 


19683 


4|I6| 


64 j 


256| 1024 


4096| 16384J 65536) 


262144 


5|25|125| 625 


3125| 15625) 78125J' 3J0625| 


1953125 


6|36j 


216|12:)6| 7776, 


46656) 


279936| 167961 6j 


10077696 


7|4i)|343|2401|16807|11764<)| 


823543| 5764801J 


40353607 


8|64|512)4096!32768 


262114|2097152| 16777216J i 34217728 


[ 9(81 72:)|6561I5904 ; ): 


531441 14782)6:) 43046721)3874204891 



123 EVOLUTION. SQUARE ROOT. 

EXAMPLE. 

1. What is the 3d power of 15 ? 

15x15x15=3375 Ans. 
2 What is the 4th power of 35 ? Ans. 1500625. 

3. What is the third power of 1,03 1 Ans. 1,092727. 

4. What is the 5th power of ,029 ? 

Ans. ,000000707281. 

5. What is the 4th power of J 1 Am. -fa 



SECTION 2. 

Of Evolution, or the Extracting of Roots. 

EVOLUTION js the reverse of involution. For as 3x 3=9 
y 3 = 27, the power ; so 27-^-3=93=3, the root of that 
power. Hence the root of any number, or power, is such 
a number as being multiplied into itself a certain number of 
times, will produce that power. Thus, 4 is the square root 
of 16, for 4X4=16; and 5 is the cube root of 125, for 
5x 5x 5=125. 



SECTION 3. 
THE SQUARE ROOT. 

number multiplied once into itself is called the 
square of that number. Hence, to extract the square root 
<;f any number, is to find such a number as being multiplied 
f >y itself will be equal to the ^iven number. 

RUf.R. 

1. Point off the given sum into periods of two figures 
a_, beginning at the right hand. 



SQUARE ROOT. 



12!) 



2. Subtract from the first period on the left, the greatesi 
square contained therein ; setting the root, so found, for the 
first quotient figure. 

3. Double the quotient for a new divisor, and bring down 
trte next period to the remainder lor a new dividual. Try 
how often the divisor is contained in the dividual, omitting 
the units. figure, and place the {lumber, so found, in the quo- 
tient, and on the right of the divisor ; multiply and subtract 
as in division. 

4. Double the quotient for a new divisor ; bring down the 
next period, and proceed as before, till all the periods are 
brought down. When a remainder occurs, add ciphers for a 
new period, the quotient figure of which will be a decimal, 
which may be extended to any required degree of exactness. 

PROOF. * 

Square the root, adding the remainder (if any) to the 
product, which will equal the given number. 

EXAMPLE, 

1. What is the square root of 531441 ? 

53144J ( 729 An*. 

49 

I. double the quotient 14,2 ) 414 

284 



2. double 



do. 144,9) 13041 
13041 
729 

729 



6561 
1458 
5103 


Ans. 327 
2187. 
6561. 
19683. 

4698. 
6031. 

1506,23 -t 
2756,22^4 


531441 proof. 
2. What is the square root of 106929 ? 
3. What is the square root of 4782969 ? 
4. What is the square root of 43046721 ? 
5. What is the square root of 387420489 ? 
6. What is the square root of 22071204 ? 
7. What is the square root of 36372961 ? 
8, What is the square root of 2268741 ? 
9. What is fhe square roof of 7596796 ' 



130 SQUARK ROOT. 

When there are decimals joined to the whole numbers in 
/he given sum, make the number of decimals even by adding 
ciphers, and point off both ways, beginning at the 'decimal 
point. 

10. What is the sqiuire root of 9712,718051 ? 

Ans. 98,553 + 

11. What is the square root of 3,1721812 ? 

Ans. 1,78106 + 

12. What is the square root of 4795,25731 ? 

Ans. 69,247 

13. What is the square root of ,00008836 ? 

Ans. ,0094 

To extract the square root of a vulgar fraction. 

RULE. 

Reduce the fraction to its lowest term ; then extract the 
"Kjuare root of the numerator for a new numerator, and the 
scjuare root of the denominator for a new denominator. 

Nute. When the fraction is a surd* that is, a number whose exact 
ff>ot cannot be found, reduce it to u decimal and extract the root there- 
from. 

EXAMPLES, 

1 . What is the square root of Jf A 7 Ans. J 

"2. What is th<; square root of |Jf A ! f 

3. What is the square root of iff f ? ^f 

Surds. 

4. What is the square root of ffj '! Ans. ,86602 + 

5. What is the square root of f Jf ? ,93309 + 

6. What is ihr square root of f|J ? ,72414 + 

To ertraci thr vjimre root of a mired number. 

RUTJv 

1. Reduce the fractional part of the mixed number to its 
lowest term, and thn mixed number to an improper fraction. 



SQT T ARK ROOT. 131 

2. Extract the roots of the numerator and denominator, 
for a new numerator and denominator. 

If the mixed number given be a surd, reduce the frac- 
tional part to a decimal, annex it to the whole number, and 
extract the square root therefrom. 

EXAMPLES. 

1. What is the square root of 37 ^f? Ans. 6-J 

2. What is the square root of 27 r 9 F ? 5i 

3. What is the square root of 51|}? 7} 

4. What is the square root of 9ff ? 3i 

Surds. 

5. What is the square root of 7 T 9 T ? Ans. 2,7961 + 

6. What is the square root of 8f ? 2,951 9 + 

7. What is the square root of 85-ff ? 9,27 -f 



Any two sides of a right angled triangle given to find the 
third side. 




RULE. 

As the square of the hypothenuse or longest side, is al- 
ways equal to the square of the base and perpendicular, the 
other sides added together ; then it is plain if the length of 
the two shortest sides are given, the square root of both 
these squared and added together, will be the length of the 
third or longest side. 

Again, when the hypothenuse, or longest side, and one of 
the others are given 5 the square root of the difference of the 
squares of these two given sides will be the len ;th of the 
remaining side. 



132 



SQUARE ROOT. 



EXAMPLE. 



1. The wall of a fortress is 3fi feet high, and the ditch 
before it is 27 feet wide: it is required to find the length of 
n ladder that will reach to the top of the wall from tho op- 
posite side of the ditch ? Ans. 45 feet. 




27 ft. ditch 



2. The top of a castle" from the ground is 45 yards high, 
and is surrounded with a ditch 60 yards broad, what length 
must a cable be to reach from the outside of the ditch to the 
top of the castb? Ans. 75 yards. 

8. In a right angled triangle, ABC, the hypothenuse 
line A C is 45 feet, the base A B 27 feet; required the length 
of the perpendicular line B C? Ans. 36 feet. 




B 



4. In a right angled triangle, A B C, the line A C is 75 
feet, B C 45 feet; required the length of the line A B? 

Ans. 60 feet. 

To find the side of a square equal in area to any given ./ 



RULE. 

Extract the square root of the content of the given su 
|H;rfices ; the quotient will give the side of the equal square 
sought. 



SQUARK ROOT 133 

EXAMPLES. 

1. If the content oi a given circle be ICO, what is the 
side of the square equal? Ans. 12,(>4911-{- 

2. If the area of a circle he '2025, what is the side of the 
square equal ? Ans. 45. 

3. It* the area of a circle be 750, what is the side of the 
square equal? Ans. 27,38612 + 

To find tht diameter of a circle of a given proportion 
larger or less than a given one. 

RULE. 

Square the diameter of the given circle, and multiply 
(if greater) or divide (if less) the product, by the number of 
times the required circle is greater or less than t.\c given 



EXAMPLES. 

1. There is a circle whose diameter is 4 feet; what is the 
diameter of one 4 times as large ? Ans. 8 feet. 

2. A has a circular yard of 100 feet diameter, but wishes 
to enlarge it to one of 3 times that area; what wall the 
diameter of the enlarged one measure? Ans. 173,2 -f 

3. If the diameter of a circle be 12 inches, what will be 
the diameter of another circle of half the size ? 

Ans. 8,48 -f inches. 

The area of a circle given to find the diameter. 
RULE. 

Multiply the square root of the area by 1,12837, and the 
produce will be the diameter. 

EXAMPLES. 

1. When the area is 160, what is the diameter? 

Ans. 1 4,272947 -f 

2. What length of a halter will be sufficient to fasten a 
horse from a post in the centre, so that he may be able to 
graze upon an acre of grass, and no more? 

Ans. 7,1364 perches, or 117 ft. 9 inches* 
M 



134 SQUARE ROOT. 

* Application. 

1. If an army of 20736 men is formed into a square 
column ; how many men will each front contain? 

Ans. 144 men. 

2. How many feet of boards will it require to lay the 
floor of a room that is 25 feet square? Ans. 625 feet. 

3. A certain square pavement contains 191736 square 
stones, all of the same size ; how many are contained in one 
of its sides ? Ans. 444. 

4. In a triangular piece of ground containing 600 perches, 
one of the shortest sides measures 40 perches, and the other 
30 ; what is the length of the longest side ? 

Ans. 50 perches. 

5. Two gentlemen set out from Pittsburgh at the same 
time ; one of them travels 84 miles due north, and the other 
50 miles due west ; what distance are they asunder ? 

Ans. 97^ -{- miles. 
ft. What is the square root of 964,5192360241 ? 

Ans. 31,05671. 

7. What is the square root of 1030892198,4001 ? 

Ans. 32107,51. 

As it is probable many teachers rind it difficult to explain 
satisfactorily the reasons and principles upon which the rules 
for the extraction of the roots are founded, I have subjoined 
the following demonstration of the rule for extracting the 
square root ; and which will also serve to show the reason 
of the rules for extracting the roots of the higher powers. 
From what has been already said on this rule, it is sufficiently 
evident that the extraction of the square root has always this 
operation on numbers, viz. to arrange the number of which 
the root is extracted into a square form. Thus, if a car- 
penter should have 625 feet of dressed boards for laying a 
floor ; if he extracts the square root of this number, (625) 
he will have the exact length of one side of a square floor, 
which these boards will be sufficient to make. 

Let this then be the question : Required to find the length 
of one side of a square room, of which 625 square feet of 
boards will be sufficient to lay the floor. 

The first step, according to the rule given, is to point off 
the numbers into periods of two figures each, beginning 



at the unit's place. This ascertains the number of figures 
of which the root will consist, from this principle, that the 
product of any two numbers can have, at most, but so many . 
places of figures, as there are places in both the factors, and 
at least, but one less. 

The number (625) will then have two periods, and con- 
sequently the root will consist of two figures. 

Operation. The last, or left-hand period in this 

. . number is 6, in which 4 is the greatest 

625 ( 2 square, and 2 the root ; hence 2 is the 

4 first figure in the root, and as one fig- 

ure more is yet to be found, we may 

225 for the present supply the place of 

that figure with a cipher (20) ; then 20 
Figure 1. will express the just value of that part 

of the root now obtained. But a root 
is the side of a square, of equal sides. 



A 



20 
20 



Hence, figure 1 exhibits a square, 
each side of which is 20 feet, and the 
a| 400 |b area 400, of which 20 is the root now 

20 obtained. 

As the rule requires, we next subtract the greatest square 
contained in the first period, and to the remainder bring 
down the next period. 4 is the greatest square contained in 
the first period (6), and as it falls in the place of hundreds, 
is in reality 400, as may be seen by filling up the places to 
the right-hand with ciphers ; this subtracted from the (i 
leaves 2, as a remainder, to which if the next period is 
brought down, the remainder will be 225 ; and the original 
number of feet (625) has been diminished by the deduction 
of 400 feet, a number equal to the superficial content of the 
square A. 

Figure 1, therefore, exhibits the exact progress of the 
operation, and shows plainly how 400 feet of the boards 
have been "disposed in the operation thus far, and also that 
225 feet yet remains to be added to this square, by en- 
larging it in such a manner as not to destroy its quadrate 
form, or its continuing a complete and perfect square. 
Should the addition be made to one side, only, the figure 
would lose its square. The addition must be made to two 
sides: accordingly the rule directs to lt double the quotient 



136 



SQUARE ROOT. 



(viz. the root already found) for a new divisor ;" the double 
of the root is equal to two sides of the square A, for the 
double of 2 is 4, and as this 4 falls in the place of tens, 
since the next figure in the root, according to the rule, is to 
be placed before it in the place of units, it is in reality 40, 
and equal to a b and b c which are 20 each. 

Operation continued. Again, as the rule di- 

. . rects, try how often the di~ 

625 ( 25 visor is contained in the 

4 dividual, omitting the units 

figure. 

45)225 The divisor is here 4, 

225 which, as has already been 

shown, is 4 tens, or 40 ; 
000 this is to be divided into 

the remainder 225 ; omit- 
n 5 ting the last figure=220. 
But 40 the sum of the 
two sides b c and c d / to 
which the remaining 225 
is to be added, and the 
square A enlarged, which 
omitting the last figure (5) 
gives 5 for the last quo- 
tient figure ; 5 is the breadth 
of the two parallelograms 
B B, the area of each (5 X 
20 b 5 20) is 100. 

The rule requires us to omit the last figure in the divid- 
ual^ and also, to place the quotient figure, when found, on 
the right of the divisor ', the reasons for which are, that ad- 
ditions of the two parallelograms B B ta the sides of the 
square A (fig. 2) do not leave it a perfect square, a deficiency 
remaining at the corner D ; the right-hand figure is omitted 
to leave something of the dividend for this deficiency. 
And as this deficiency is limited by the two parallelograms 
B B, and the quotient figure (5) is the breadth of these, 
consequently the quotient 5= the length of each of the 
sides of the small square D ; this quotient then being 
placed on the right of the divisor and multiplied into itself 
gives the area of the square I) ; which being added to the 





20 


5 






B 5 


D 5 


en 


d 


100 


25 






c 








A 


B 25 







20 


20 






20 


5 




a 


400 


100 


X 



CUR!-: ROOT. 137 

contents of the two parallelograms B B each (100) 200, 
shows that the remaining 225 feet of boards have kasji dis- 
posed of, in these three additions (B B D) made to the T^st 
square A ; whilst the figure is seen to be continued a corrt* 
plete square. 

Q. E. D. 
PROOF. 

The square A =400 feet 

The parallelograms B B =200 
The square D = 25 

625 feet. 



SECTION 4. 
THE CUBE ROOT. 

THE cube is the third power of any number, and is found 
by multiplying that number twice into itself. As 2 X 2 x 2^. 

To extract the cube root, therefore, of any number, is to 
find another number, the cube of which will equal the given 
number. Thus 4 is the cube root of 64 ; for 4X 4x 4=64. 

RULE. 

1. Point off the given number into periods of three figures 
each, beginning at the units place, or decimal point. These 
periods will show the number of figures contained in the 
required root. 

2. Find the greatest cube contained in the first period, 
and subtract it therefrom ; put the root of this cube in the 
quotient, and bring down the next period to the remainder 
for a new dividual. 

3. Square the quotient and multiply it by 3 for a defective 
divisor; 2x2x3 = 12. Find how often this is contained 
in the dividual, rejecting the units and tens therein, and 
place the result in the quotient, and its square to the right of 
the divisor. 4 x 4=16 put to the divisor 12 = 

M2 



138 



CUBE TtOOT. 



4. Multiply the last tigure in the quotient by the rest, and 
the product by 30 ; add this to the defective divisor, ana 
multiply this sum by the last figure in the quotient, subtract 
that product from the dividual, bring down the next period, 
and proceed as before. 

Note. When the quotient is 1, 2, or 3, put a cipher in the place of 
tens in filling up the square on the right of the divisor. 



EXAMPLE. 

U What is the cube root of 48228544 1 



Greatest cube in 48 is 



Operation. 

48228544(364 Ans. 
27 



21228 

Square of 3 X by 3=27. 1 def. divis. ' 
Square of 6 put to 27 =2736 

6 last -quo. fig. X by the rest 

and 30 = 540 

Complete divisor 3276 J 19656 



Square of 36 X 3=3888. 2 def. divis. 
Square of 4 put to 3888 = 388816 
4 last quo. fig. X by the rest 

and 30 = 4320 



Complete divisor 



393136 



1572544 



1572544 



2. What is the cube root of 13824 T 
3. What is the cube root of 373248 ? 
4. What is the cube root of 5735339 ? 
5. What is the cube root of 84604519 ? 
6. What is the cube root of 27054036008 I 
7. What is the cube root of 122615327232 ? 
8. What is the cube root of 22069810125 ? 
9. What is the cube root of 219365327791 ? 
10. What is the cube root of 673373097125 ? 
11. What is the cube root of 12,977875 ? 
12. What is the cube root of 15926,972504 ? 
13. What is the cube root of 36155,027576 ? 


Ans. 24 
72 

179 
439 
3002 
4968 
2805 
6031 
8765 
2,35 
?Mfi f 
33,06 -f 



CUBE ROOT. 139 

14. What is the cube root of ,053258279 ? Ans. ,376 + 

15. What is the cube root of ,001906624 ? ,124 

16. What is the cube root of ,000000729 ? ,009 

17. What is the cube root of 2 ? 1,25 + 

To extract the cube root of a vulgar fraction. 

RULE. 

Reduce the fraction to its lowest terms ; then extract the 
cube root of the numerator for a new numerator, and the 
cube root of the denominator for a new denominator ; but 
if the fraction be a surd, reduce it to a decimal, and extract 
the root from it for the answer. 

EXAMPLES. 

1. What is the cube root of f f-f 1 Ans. 4 

2. What is the cube root of T 3 /^ ? f 

3. What is the cube root of 4-f f ? f 

Surds. 

4. What is the cube root of 4 ? Ans. ,829 -f- 

5. What is the cube root of f ? ,873 -f 

6. What is the cube root of f ? ,822 -f 

To extract the cube root of a mixed number. 

RULE. 

Reduce the fractional part to its lowest terms, and the 
mixed number to an improper fraction ; extract the cube 
roots of the numerator and denominator for a new numerator 
and denominator ; but if the mixed number given be a surd, 
reduce the fractional part to a decimal, annex it to the whole 
number, and extract the root therefrom. 

^ EXAMPLES. 

1. What is the cube root of 31 J/j ? Ans. 3| 

2. What is the cube root of 12if ? 2J 

3. What is the cube root of 405 r 2 ^ ? 7f 



140 CUBE ROOT. 

Surds. 

4. What is the cube root of 7} 1 Ans. 1,93-f- 

5. What is the cube root of 8f ? 2,057 + 

6. What is the cube root of 9 ? 2,092 + 

To find the side of a cube that shall be equal to any given 
solid, as a globe, a cone, <fyc. 

RULE. 

Extract the cube root of the solid content of any solid 
body, for the side of the cube of equal solidity. 

EXAMPLES. * 

1. If the solid content of a globe is 10648, what is the 
side of a cube of equal solidity ? Ans. 22. 

2. If the solid content of a globe is 389017, wha't is the 
side of a cube of equal solidity? Ans. 73. 

Note. The relative size of different cubical vessels is found by mul- 
tiplying the cube oi' the side of the given vessel, by the proportional 
number, and taking the cube root of the product for the answer sought. 

EXAMPLES. 

1, There is a cubical vessel whose side is two feet I de- 
mand the size of another vessel which shall contain three 
times as much 1 Ans. 2 feet 10 inches and } nearly. 

2. There is a cubical vessel whose side is 1 foot ; re- 
quired the side of another vessel that shall contain three 
times as much ? Ans. 17,306 inches 

Application. 

1. If a ball of 6 inches diameter weigh 32 lb., what will 
one of the same metal weigh, whose diameter is 3 inches ? 

Ans. 4 lb. 

2. What is the side of a cubical mound equal to one 28H 
feet long, 216 broad, and 48 high? An*. 144 feet. 

3. There is a stone of cubic form, which contains 389017 
solid feet ; what is the superficial content of one of its sides ? 

AIM. 5329 feet. 



PROGRESSION. 141 

4. What is the difference between half a solid foot, and a 
solid half foot ? Ans. 3 half feet. 

5. In a cubical foot, how many cubes of 6 inches, and 
how many of 4 are contained therein ? 

Ans. 8 of 6 inches, and 27 of 4 inches. 



SECTION 5. 
OF PROGRESSION. 

PROGRESSION is of two kinds, arithmetical and geometrical. 

Arithmetical progression is when any series of numbers 
increase or decrease regularly by a common difference. As 
1, 2, 3, 4, 5, 6, &c. are in arithmetical progression by the 
continual adding of one ; and 9, 7, 5, 3, 1, by the continual 
subtracting of two. 

Note. In any series of even numbers in arithmetical progression, 
the sum of the two extremes will be equal to the sum of any two terms, 
equally distant therefrom ; as 2. 4. 6. 8. 10. 12., where 2 -f- 12 = 14, so 
4 -f 10 = 14, and 6 -f 8 = 14. But if the number of terms is odd, the 
double of the middle term will be equal to any two of the terms 
equally distant therefrom ; as 3. 6. 9. 12. 15. where the double of 9 the 
middle term = 18, and 3 -f 15 = 18, or 64-12 = 18. 

In arithmetical progression five things must be carefully 
observed, viz. 

1. The first term, 

2. The last term, 

8. The number of terms, 

4. The equal difference, 

5, The sum of all the terms." 

Casel. 

The first term, common difference, and number of terms, 
given to find the last term, and sum of all the terms. 

RULE. 
1 . Multiply the number of terms, less 1 , by the common 



142 PROGRESSION. 

difference, and to the product add the first term, the sum 
will be the last term. 

2. Multiply the sum of the two extremes by the number 
of terms, and half the product will be the sum of all the 
terms. 



1. A merchant bought 50 yards of linen, at 2 cents for 
the first yard, 4 for the second, 6 for the third, &c. increas- 
ing two cents every yard ; what was the price of the last 
yard, how much the whole amount, and what the average 
price per yard ? 

50 number of terms 
1 

49 number of terms less one 
Multiply by 2 common difference 

98 
Add 2 first term 

100 last term 

2 + 100=102 sum of the two extremes 
Multiply by 50 number of terms 



2)5100 



50 ) 25,50 sum of all the terms 



51 cents 

C 100 cents the last yard 
Ans. < 25,50 do. the whole amount 

f 51 do. the average price per yd. 

2. Bought 20 yards of calico at 3 cents for the first yard, 
6 for the second, 9 for the third, &c. ; what did the whole 
cost? Ans. 6 dolls. 30 cents. 

3. If 100 apples were laid two yards distant from each 
"other, in a right line, and a basket placed two yards distant 
from the first apple, what distance must a person travel to 
gather them singly into the basket ? 

Ans. 11 miles, 3 furlongs, 180 vardau 



PROGRESSION. 1 13 

4. A agreed to serve B 10 years, at the rate of 20 dol- 
lars for the first year, 30 for the second, 40 for the third, 
&c. ; what had he the last year, how much for the whole 
time, and what per annum 'I 

Ans. 110 dolls, for the last year, 650 dolls, the whole 
amount, and 65 dolls, per^xnnum. 

5. A sold to B 1000 acres of land, at 10 cents for the 
first acre, 20 for the second, 30 for the third, &c. ; what was 
the^price of the last acre, and what did the whole come to? 

j $ 100 dolls, the last acre, 
' I 50050 do. whole cost. 

Case 2. 

When the two extremes, and number of terms are given, 
td find the common difference. 

RULE. 

Divide the difference of the extremes, by the number of 
terms, less one ; the quotient will be the common difference. 

EXAMPLE. 

1 . A is to receive from B a certain sum to be paid in 1 1 
several payments in arithmetical progression ; the first pay- 
ment to be 20 dollars, and the last to be 100 dollars ; what 
is the common difference, what was each payment, and how 
much the whole debt ? 

Operation, 

100 last term 
20 first term 

No. of terms 11 1 = 10)80 the difference 

8 common difference 
20 -f 100X 5i=660 whole debt 

20 first payment 
20 + 8=28 second do. 
28 -j- 8 =36 third do, &c. 

2. There are 21 persons whose ages are equally distant 
from each other, in arithmetical progression ; the youngea! 



1 44 GEOMETRIC A L PROGRESSION. 

ts 20 years old, and the eldest 60 ; \\ hat is the common dif- 
ference of their ages, and the age of each man ? 

C 2 common difference 
Ans. < 20 + 2 22 the second 

(22 4-2 = 24 the third, &c. 

3. A man is to travel from Pittsburgh to a certain place, 
tn 12 days, and to go but three miles the first day, increas- 
ing each day's journey in arithmetical progression, making 
the last day's travelling 58 miles ; what is the daily increase, 
and what the whole distance I 

* ( 5 miles daily increase 
* < 366 miles whole distance. 



SECTION 6, 

GEOMETRICAL PROGRESSION. 

ANY series of numbers increasing or decreasing by one 
continual multiplier, or divisor, called the ratio, is termed 
geometrical progression; as 2, 4, 8, 16, 32, &c. increase 
by the multiplier 2 ; and 32, 16, 8, 4, 2, decrease, continu- 
ally, by the divisor 2. 

In geometrical progression there are five things to bo 
carefully observed. 

1. The first term, 

2. The last term, 

3. The number of terms, 

4. The ratio, 

5. The sum of all the terms. 

To find the last term, and sum of all the series in gcf>. 
netricnl progression, worh by the folloiring 

RULE. 

1 . R iise the ratio in the given sum, to that power whose 
index 5 hall always bo one less than the number of term? 
give n ; multiply the number so found by the first term, and 
Tho jrviduct will be the last term, or greater extreme. 



GEOMETRICAL PROGRESSION. 145 

2. Multiply the last term by the ratio, from that product 
subtract the first term, and divide the remainder by the 
ratio, less one ; the amount will be the sum of the series, or 
of all the terms. 

EXAMPLE. 

1. Suppose 20 yards of broadcloth was sold at 4 mills 
for the first yard, 12 for the second, 36 for the third, &c. 
what did the cloth come to, and what was gained by the 
sale, supposing the prime cost to have been $15 per yard ? 

Note. In this question observe, the first term is 4, the ratio 3, arid 
the number of terms 20, consequently the ratio 3 must be raised to 
20 1= 19th power. Thus, 

g S S 1 

! I 1 | 

& 3 8 I 

Indices 1234 

Ratio 3 9 27 81 

. 81 

81 

648 

6561= the 8th power 
6561 

6561 
39366 
32805 
39366 

43046721= 16th power 
27= 3d power 



301327047 
86093442 

1162261467=19th power 
X 4 first term 

4649045868=20th or last term 
X3 



13947137604 

4 first term 



Ratio 31=2)13947137600 



69 73568,80,0= sum of the series, or number of 
First cost of the cloth 300,00 mills for which the cloth was sold 

0=gam, 

N 



146 GEOMETRICAL PROGRESSION. 

2. A father gave his daughter who was married on the 
first day of January, one dollar towards her portion, prom- 
ising to double it on the first day of every month for one 
year ; what was the amount of her whole portion 'I 

Ans. 4095 dollars. 

3. A merchant sold 15 yards of satin ; the first yard for 
is. the second for 2s. the third for 4s. &c. in geometrical 
progression ; what was the price of the 15 yards? 

Ans. 1638Z. 7s. 

4. A goldsmith sold 1 pound of gold at 1 cent for the first 
ounce, 4 for the second, 16 for the third, &c. ; what did it 
come to, and what did he gain, supposing he gave 20 dollars 
per ounce ? 

Ans. He sold it for 55924 dollars 5 cents, and gained 
55684 dollars 5 cents. 

5. What sum would purchase a horse with 4 shoes and 
8 nails in each shoe, at one mill for the first nail, 2 mills foi 
the second, 4 for the third, &c. doubling in geometrical pro- 
gression to the last ? 

Ans. 4294967 dollars 29 cents 5 mills. 

t>. What sum would purchase the same horse, with the 
same number of shoes, and nails, at 1 mill for the first nail, 

3 for the second, 9 for the third, &c., in a triple ratio of 
geometrical progression to the last ? 

Ans. 926510094425 dollars 92 cents. 

7. What sum would purchase the same horse, with the 
same number of shoes, and nails, at 1 mill for the first nail, 

4 for the second, 16 for the 3d, &c., in a quadruple ratio of 
geometrical progression to the last ? 

Ans. 6148914691236517 dollars 20 cents 5 mills. 

S. Sold 30 yards of silk velvet, at 2 pins for the first 
yard, 6 for the second, 18 for the third, &c., and these dis- 
posed of at 100U for a farthing ; what did the velvet amount 
to, and what was gained by the sale, supposing the prime 
cost to hav<3 IKJCII 100/. per yard '! 

Amount 214469929/. 5s. 3 



An *' * Gained 214466929/. 5*. Sjd. 



SINGLE POSITION. 147 

SECTION 7. 

OF POSITION. 

POSITION is a rule for finding the true number, by one or 
more false or supposed numbers, taken at pleasure. 

It is of two kinds, viz. Single and Double. 

Single position teaches to resolve such questions as require 
but one supposed number. 

RULE. 

1. Suppose any number whatever, and work in the same 
manner with it as is required to be performed in the given 
question. 

2. Then, as the amount of the errors, is to theN 
sum, so is the given number to the one required. 

PROOF. 

Add the several parts of the result together, and if k 
agrees with the given sum, it is right. 

EXAMPLES. 

1. A schoolmaster being asked how many scholars he had, 
said, if I had as many, half as many, and one quarter as 
many more, I should have 1 32 ; how many had he ? 

Suppose he had 40 As 110: 40 :: 132 : 48 Ana. 

as many 40 p roof ^ 

as many 20 40 

\ as many 10 g. 

US J? 

132 

2. It is required to divide a certain sum of money among 
4 persons, in such a manner that the first shall have , the 
second ^, the third -J, and the fourth the remainder, which 
is 28 dollars ; what was the sum ? 

Suppose 72 As 18 : 72 :: 28 : 112 dolls. AIM. 

i is 24 Proof 112 

i is 18 , . rzr 

1 i 19 * ls d ' 

* 1S _ i is 28 

.54 J- is 18f 

Rem. 18 84 

28 last share. 



148 DOUBLE POSITION. 

3. A, B, and C, buy a carriage for 340 dollars, of which 
A pays three times as much as B, and B four times as much 
as C ; what did each pay ? f A paid 240 dolls. 

Ans. IE 80 

(C 20 

4. What is the sum of which 1, ]-, and , make 148 dol- 
lars? Ans. 240 dollars. 

5. A person having spent ^, and J of his money : had 26f 
dollars left; what had he at first? Ans. 100 dollars. 

6. A, B, and C, talking of their ages, B said his age was 
once and a half the age of A ; C said his was twice and T \ 
the age of both, and that the sum of their ages was 93 ; what 
was the age of * each? i A's age 12 years. 

Ans. < B's 18 
( C's 63 

7. Seven-eighths of a certain number exceeds four-fifths 
by 6; what is that number? Ans. 80. 

8. A gentleman bought a chaise, horse, and harness, for 
360 dollars ; the horse came to twice the price of the har- 
ness, and the chaise to twice the price of the horse and har- 
ness together ; what did he give for each ? 

( 80 dollars for the horse 
Ans. < 40 harness 

( 240 chaise 

9. A gentleman being asked the price of his carriage, an- 
swered that i, i, i, and of its price was 228 dollars ; what 
was the price of the carriage ? Ans. 240 dolls. 

10. A saves of his wages, but B, who has the same sal- 
ary? by spending twice as much as A, sinks 50 dollars a 
year; what is their annual salary? Ans. 150 dolls, each. 



SECTION 8. 

DOUBLE POSITION. 

DOUBLE POSITION is making use of two supposed num- 
bers, to find the true one. 

RULE. 

1. Take any two numbers, and proceed with them ac- 
cording to the conditions of the question, noting the errors 



DOUBLE POSITION. 149 

of the results ; multiply these errors cross-wise, viz. the 
first position by the last error, and the last position by the 
first error. 

2. If the errors be alike, that is, both greater, or both less 
than the given number, take their difference for a divisor, 
and the difference of the products for a dividend : but if they 
are unlike, take their sum for a divisor, and the sum of the pro- 
ducts for a dividend, the quotient will be the answer required 

EXAMPLE. 

1 . A father leaves his estate to be divided among his three 
sons, A, B, and C, in the following manner, viz. A is to 
have one-half wanting 50 dollars, B one-third, and C 10 dol- 
lars less than B ; what was the sum left, and what was each 
son's share? 

Operation. 

1st. Suppose 240 dollars. 
Then 240 -f- 250=70 A's part 
240-+- 3= 80 B's part 
B's share 80 10= 70 C's part 



Sum of all their parts 220 



20 er. too little, 
2d. Suppose 300 dollars. 
Then300-f- 2 50=100 A's part 
300-:- 3= 100 B's part 
B's share 10010= 90 C's part 

Sum of all their parts 290 

10 er. too little, 
errors. 
1st. sup. 240^20=6000 

2d. sup. 300-^-10=2400 



10)3600(360 An*. 
Proof 360-;- 2 50=130 
360-^ 3= 120 
120 10= 110 

360 

N2 



150 DOUBLE POSITION. 

2. A and B have the same income ; A saves the of his, 
but B by spending 30 dollars per annum more than A, at the 
end of 8 years finds himself 40 dollars in debt ; what is their 
income, and how much does each spend per annum ? 

( Their income is 200 dolls, per annum 
Ans. < A spends 175 

( B spends 205 

3. A, B, and C, would divide 100 dollars between them, 
so as B may have 3 dollars more than A, and C 4 dollars 
more than B ; how many dollars must each have ? 

( A 30 dollars. 
Ans. I B 33 
(C37 

4. A, B, and C, built a house which cost 10,000 dollars; 
A paid a certain sum, B paid 1000 dollars more than A, and 
C paid as much as both A and B ; how much did each one 
pay? 

( A paid 2000 dolls. 
Ans. 1 B 3000 
I C 5000 

5. A gentleman has 2 horses and a saddle worth 50 dol- 
lars, which saddle if he put on the back of the first horse, 
will make his value double that of the second; but if he put 
it on the second horse, it will make his value triple that of 
the first ; what is the value of each horse ? 

A $ First horse 30 dolls. 
s ' I Second do. 40 

6. The head of a fish is 9 inches long, and its tail is as 
long as its head and half its body, and its body is as long as 
its head and tail together ; what is its whole length ? 

Ans. 6 feet. 

7. A laborer hired 40 days upon this condition, that he 
should. receive 20 cents for every day he wrought, and for- 
l<'it 10 cents for every day he was idle; at settlement he re- 
ceived 5 dollars ; how many days did he work, and how 
many was he idle? Ans. Wrought 30 days, idle 10 

8. A and B vested equal sums in trade ; A gained a sum 
equal to J of his stock, and B lost 225 dollars ; then A's 
money was double that of B's ; what sum had each vested ? 

Ans. 600 dollars. 

9. Divide 15 into two such parts, so that when the greater 



PERMUTATION. 151 

is multiplied by 4, and the less by 16, the products will be 
equal? Ans. Greater 12, less 3. 

10. A person being asked in the afternoon what o'clock 
it was, answered, that the time past from noon was equal to 
fa of the time to midnight ; what o'clock was it ? 

Ans. 36 minutes past .1 o'clock. 



SECTION 9. 
PERMUTATION. 

PERMUTATION is the finding how many different ways any 
given number of things may be varied in position, or ar- 
rangement. Thus, 123 are six different arrange- 
132 ments made upon the three 
213 figures 1 2 3. 

2 3 1 

3 1 2 
321 

RULE. 

Multiply all the terms from an unit up to the given num- 
ber into one another, and the last product will be the num- 
ber of changes required. 

EXAMPLE. 

1. In how many different positions may 5 persons be 
placed at a table. 

1X2X3X4X5^120 Ans. 

2 How many changes may be rung on 12 bells, and how 
Jong would they be ringing but once over, allowing 10 
changes to be rung in 1 minute, and the year to contain 365 
days 6 hours ? 

Ans. 479001600 changes, and would require 91 years, 

3 weeks, 5 days, and 6 hours. 

3- Seven men not agreeing with the owner of a boarding 
house about the price of boarding, offer to give 100 dollars 
each, for as long time as they can seat themselves every day 
differently at dinner ; this offer being accepted, how long 
may they stay ? Ans. 5040 days, or 13 years, 295 days. 



152 COMBINATION. 

4. What number of variations will the 9 digits admit of? 

Ans. 362880. 

5. How many changes may be made on the 26 letters of 
the alphabet ? 

Ans. 403,291,461,126,605,635,584,000,000. 

Quatril. Trilns. Billions. Millions. Units. 

Note. From the answer to this last question, which amounts to a 
number, of which we cannot form any conceivable idea, we may dis- 
cover the surprising power of numbers, and also the endless variety of 
ideas that may be distinctly communicated by these 26 simple charac- 
ters. It will also be evident from the method of notation here used, 
that a row of figures of any given length whatever, may be numerated, 
though we may be entirely unable to comprehend the amount. 



SECTION 10. 

COMBINATION. 

COMBINATION of quantities, is the showing how often a 
less number of things can be taken out of a greater, and 
combined or joined tdgether differently. 

RULE. 

Take a series of 1, 2, 3, &c. up to the number to be com- 
bined ; take another series of as many places, decreasing by 
unity from the number out of which the combinations are to 
be made ; multiply the first continually for a divisor, and the 
latter fcr a dividend, and the quotient will be the answer. 

EXAMPLE. 

1. How many combinations may be made of 7 dollars 
out of 12? 

1x2x3, &c. up to 7 =5040 divisor. 
Again, 12 the whole number of terms less 7 = 5 
Hence 12 x 11 X 10, &c. down to 5 = 3991680 dividend. 
And 5040 ) 3991680 ( 792 Ans. 

2. How many combinations can be made of 6 letters out 
of 24 of the alphabet ? Ans. 134596. 

3. In how many different ways may an officer select 8 
men out of 30, so as not to make the same selection twice 

Ans. 5H52925 



ADDITION OF DUODECIMALS. 153 

PART VIII. 

MENSURATION. 



SECTION L 

Duodecimals, or Cross Multiplication. 

DUODECIMALS are fractions of a foot, or of an inch, or 
parts of an inch, having 12 for their denominator. Inches 
and parts are sometimes called primes ('), seconds ("), 
thirds ('"), &c. 

The denominations are, 
12 Fourths ("") make 1 Third 
12 Thirds - 1 Second 

12 Seconds 1 Inch in. 

12 Inches 1 Foot ft. 

Nole. This rule is much used in measuring 1 and computing the di- 
mensions of the several parts of buildings ; it is likewise used to find 
the tonnage of ships, and the contents of bales, cases, boxes, &c. 

ADDITION OF DUODECIMALS. 
RULE. 

Add as in compound addition, carrying 1 for each 12 u. 
the next denomination. 

EXAMPLE. 

Ft. in. "" '" "" Ft. in. " 

25 9 3 5 8 244 6 3 10 5 

34 3 9 2 7 355 9851 

28 10 4 8 4 559 10 9 5 8 

64 11 9 7 2 129 5569 

82 7 5 6 8 895 1 10 5 11 

15 3 7 9 10 651 1759 

44 6 11 2 8 555 9 8 5- 5 

22 3 6 1 5 388 11 10 10 9 



318 8 9 8 4 



154 SUBTRACTION OF DUODECIMALS. 

SUBTRACTION OF DUODECIMALS. 
RULE. 

Work as in compound subtraction, borrowing 12 when 
accessary* 

EXAMPLE. 

Ft. in. ' Ft. in. " '" "" 

From 125 4 3 8 2 2756 5780 

Take 68 9 2 10 1 1839 9 5 11 10 



Rem. 56 7 10 1 



3. From a board measuring 35 feet, 9 inches, 2 seconds, 
cut 24 feet, 10 inches, 5 seconds, and 4 thirds; what is left ? 

Ans. Wft. lOin. Ssec. 8'" 

4. A joiner having lined several rooms very curiously 
with costly materials, finds the amount to be, in square 
measure, 803 feet, Z inches, 4 seconds ; but several deduc- 
tions being to be made for windows, arches, &c. those de- 
ductions amounted to 70 feet, 3 inches, 7 seconds, 10 thirds, 
5 fourths ; how many feet of workmanship must he be paid 
for? Ans. 132ft. llin. 8" I"' 7"" 

MULTIPLICATION OF DUODECIMALS. 

Case 1. 
When the feet of the multiplier do not exceed 12. 

RULE. 

Set the feet, or the highest denomination of the multiplier 
under the lowest denomination of the multiplicand, and mul- 
tiply as in compound numbers, carrying 1 for every 12 from 
;m<i denomination to another, and place the result of the 
;owcst denomination in the multiplicand under its multiplier. 

TABLE. 



________ 

2. Feet multiplied by inches, give inches, 



1. Feet multiplied by feet, give feet. 

2. Feet multiplied by inches, give inc^>. 

3. Feet multiplied by seconds, give seconds, &c. 
4 Inches multiplied by inches, give seconds. 



MULTIPLICATION OF DUODECIMALS. 155 

5. Inches multiplied by seconds, give thirds, &c. 

6. Seconds multiplied by seconds, give fourths. 

7. Seconds multiplied by thirds, give fifths, &c. 

PROOF. 

Reduce the given sum to a decimal, or work by the rules 
of practice. 



1. Multiply 



EXAMPLES. 

Ft. in. " Ft. in. 

8 6 9 by 7 3 

7 3 





Ans. 


Ft 


59 
2 


11 3 

1 8 


3 


. 62 


11 


3 


Or practice. 
3 is |i| 8 6 9 

7 


3 






59 
2 


11 
1 


3 

8 


3 


62 


11 


3 A?is. 



7 3 



Proof decimally. 
6 9 = 8,5625 
= 7,25 



Ft. in. 

2. Multiply 9 5 

3. 7 10 

4. 846 



428125 
171250 
599375 
62,078125 

12 

0,937500 
12 

11,250000 
12 

3,000000 

Ft. in. " 

Ans. 36 10 7 

69 10 2 

21 10 5 



Ft. in. 
by 3 11 

by 8 11 

by 2 74 

5. What is the price of a marble slab, whose length is 5 
feet 7 inches, and breadth 1 foot 10 inches, at 1 dollar and I 
50 cents per foot? Ans. 15 dolls. 35^ cts. 

6. There is a house with three tiers of windows, 3 in a 
tier, the height of the first tier is 7 feet 10 inches, of the 
second 6 feet 8 inches, and of the third 5 feet 4 inches, and 
the breadth of each window is ^ ft. 11 inches. ; what will the 
glazing come to at 14 cts. per ft. ? Ans. 32 dolls. 62 cts. . 



156 



MULTIPLICATION OF DUODECIMALS. 



Case 2. 

When the feet of the multiplier exceeds 12. 
RULE. 

Multiply by the feet in the multiplier, and take parts for 
the inches. 

EXAMPLES. 

1. Multiply 84 feet 6 inches, by 36 feet 7 inches and 6 
seconds. 

Operation. 

84 6 



1 

6*cc. 



507 
6 

3042 
42 3 

3 6 



Ans. 3094 9 
Ft. in. Ft. in. 

2. Multiply 76 7 by 19 10 

3. 127 6 by 92 4 

4. 184 8 by 127 6 

Ft. in. sec. 

5. Multiply 311 



4 7 by 36 7 
6x6=36 



Ft. in. sec. 
Ans. 1518 10 10 
11772 6 
23545 
Ft. in. sec. 
5 



(tin. is 



II 



1868 3 
6 

11209 9 " 
155 8 3 6 
25 9 



8 7 



7 ' 
10 4 



21 987 



11402 (I 



1 I 



MULTIPLICATION OF DUODECIMALS. 157 

6. A floor is 70 feet 8 inches, by 38 feet 11 inches ; how 
many square feet are therein 1 

Arts. 2750ft. lin. 4sec. 

7. If a ceiling be 59 feet 9 inches long, and 24 feet 6 
inches broad, how many yards does it contain ? 

Arts. I62yds. 5ft. W$in. 

Note. Divide the square feet by 9, and the quotient will be square 
yards. 

8. What will the paving of a court yard come to at 15 
cents per yard, the length being 58 feet 6 inches, and the 
breadth 54 feet, 9 inches ? 

Arts. 53 dolls. 38 + cts. 

9. What is the solid content of a bale of goods, measur- 
ing in length 7 feet 6 inches, breadth 3 feet 3 inches, and 
depth 1 foot 10 inches? 

Ans. 44/fc. 8in. Ssec. 

Note, To find the cubic feet, or solid content of bales, cases, boxes, 
&c. multiply the length by the breadth, and that product by the thick- 
ness. 

10. A merchant imports from London six bales of the fol- 
lowing dimensions, viz. 

Length. Breadth. Depth. 

Ft. in. Ft. in. Ft. in 

No. 1. 2 10 24 19 

2. 2 10 26 13 

3. 36 22 18 

4. 2 10 28 19 

5. 2 10 26 19 

6. 2 11 28 18 

What are the solid contents, and how much will the freight 
amount to, at 20 dollars per ton of 40 feet ? 

Ans. lift. lin. and freight 35 dollars 79 cts. 

To find a ship's tonnage by Carpenter's measure. 

For single decked vessels, multiply the length, breadth 
at the main beam, and depth of the hold together, and di- 
vide the product by 95 ; but if the vessel be double decked, 
take half the breadth of the main beam for the depth of the 
hold, and work as for a single decked vessel. 

O 



158 MULTIPLICATION OF DUODECIMALS 

. 

EXAMPLES. 

1. The length of a single decked vessel is 60' feet, the 
breadth 20, and depth 10; what is the tonnage? 

Then 60x20x10 = 12000 
And 1 2000 ~ 95 = 1 26 fy tons. An*. 
Or, as 95 : 20x10 :: 60 : 126 r <V Ans. 

2. Required the tonnage of a double decked vessel, whose 
'ength is 90, and breadth 30. 

Then 90 X 30 X 15 (half breadth) =40500. 

And 40500-^-95=426-,^ tons. Ans. 
Or, as 95 : 30 X 15 :: 90 : 426 T \ Ans. 

3. A single decked vessel is 64 feet long, 22 feet broad, 
and 10 feet deep; what is its tonnage? 

Ans. 148 T 4 y- tons. 

4. What will be the tonnage of a double decked vessel 
whose length is 80 feet, and breadth 26 feet ? 

Ans. 284}| tons - 



To find the Government tonnage. 

" If the vessel be double decked, take the length thereof 
from the fore part of the main stem, to the after part of the 
stern post, above the upper deck ; the breadth thereof at the 
broadest part above the main wales, half of which breadth 
shall be accounted the depth of such vessel, and then deduct 
from the length three-fifths of the breadth ; multiply the 
remainder by the breadth, and the product by the depth, 
and divide this last product by 95, for the tonnage. But if 
it be a single decked vessel, take the length and broadth, 
as directed above; deduct from the said length three-fifths 
of the breadth, and take the depth from under the side of 
the deck plank to the ceiling in the hold, then multiply and 
divide as aforesaid, and the quotient shall be deemed the 
tonnage." 



CARPENTERS' OR SLIDING RULE. 159 

SECTION 2. 

The Carpenters' or Sliding Rule. 

THIS rule is not only useful in measuring timber, artifi- 
cers' work, and taking dimensions, but also in ascertaining 
the contents of such work ; it is therefore a rule which all 
mechanics, having any thing to do with mensuration, ought 
to possess and understand. 

It consists of two equal pieces of box, each one foot in 
length, connected together by a folding joint ; in one of these 
equal pieces, there is a slider, and four lines marked at the 
right-hand with the letters A, B, C, D ; two of these lines 
are upon the slider, and the other two upon the rule. Three 
of these lines, viz. A, B, C, are called double lines, because 
they proceed from 1 to 10 twice over ; these three lines are 
alike both in number and division. They are numbered 
from the left-hand towards the right with the figures 1, 2, 3, 

5, 6, 7, 8, 9, then 1, which stands in the middle; the 
numbers then proceed, 2, 3, 4, 5, 6, 7, 8, 9, and 10, which 
stands at the right-hand end of the rule. These numbers 
have no determinate value of their own, but depend upon 
the value you set upon the unit at the left-hand of this part 
of the rule ; thus, if you call it 1, the 1 in the middle will 
be 10, the other figures which follow, will be 20, 30, &c. 
and the 10 at the right-hand will be 100, If you call the 
first or left-hand unit. 10, the middle 1 will be 100, and the 
following figures will' be 200, 300, &c. and the 10 at the 
right-hand end will be 1000. Or if you call the first or 
left-hand unit 100, the middle 1 will be 1000, the following 
figures 2000, 3000, &c. and the 10 at the right-hand 10,000. 
Lastly, as you alter, or number the large divisions, so you 
must alter the small divisions in the same proportion. 

The fourth line D is a single line, proceeding from 4 to 
40 ; it is also called the girt line, from its use in casting up 
the contents of timber. Upon it are marked W G at 17, 15, 
and A G at 18, 95, the wine and gauge points, to make it 
serve the purpose of a gauging rule. 

Tne use of the double lines, A and B, is for working the 
rule of proportion, and finding the areas of plane figures. 
On the other part of this side of the rule there is a table to 



160 CARPENTERS' OR SLIDING RULE. 

ascertain the value of a ton, or 50 cubic feel of timber at all 
prices from 6 pence to 24 pence per foot. 

On the other side of the rule are several plain scales 
divided into 12th parts, marked mrA, ^> i ?, &c. signifying 
that the inch, 5 inch, &c. are divided into 1:2 parts. These 
scales are useful for planning dimensions, that are taken in 
feet and inches. Again, the edge of the rule is divided into 
inches, and each of these into eight parts, representing half 
inches, quarter inches, and half quarters. 

In this description the rule is folded ; but when it is opened 
and the slider drawn out, the hack part will be ibund divided 
like the edge of the rule, so that all together will measure 
3 feet or one yard. 

USE OF THE CARPENTERS' RULE. 
1st. To multiply numbers together. 

EXAMPLE. 

1. Suppose the two numbers 13 and 24. 

Set 1 on B to 13 on A ; then against 24 on B, stands 312 
on A, which is the required product of the two given num- 
bers 13 anu 24. 

Note. In any operation when a number runs beyond the end of the 
line, seek it on the other radius, or other part of the line ; that is, take 
the 10th part of it, or the 100th part of it, &c. and increase the product 
of it proportionally 10 fold, or 100 fold, <fec. 

2. Multiply 12 by 16 Ans. 192. 

3. 35 19 665. 

4. 270 54 14580. 

2d. Division of numbers by the Carpenters' Rule. 
EXAMPLE. 

1. Required to divide 360 by 12. 

Set the divisor 12 on B to the dividend 360 on A ; then 
against 1 on B stands 30, the quotient on A. 

2. Divide 665 by 19. Quotient 35. 

3. 396 " 27. 14,6. 

4. 741 42. 17,6. 

5. 76HO 24. 320. 
Note. In this last example, because 7G80 is not contained on A, one- 

lenth of the number, viz. 768 is taken, to make it fall within the com- 
pass of the scale The quotient of this sum is 32, but as the dividond 
w;is diminished by a division of 1(1, so the juotieiit nmst be multiplied 
uy the Rane number, arid 32 10320. 



CARPENTERS' OR SLIDING RULE. 161 

3d. To square numbers by the Carpenters' Rule. 
EXAMPLE. 

1. Required to square the number 25. 

Set 1 or 100 on B to the 10 on D ; then against every 

number on D stands its square on the line C. Thus against 
25 on D, stands 625, its square on C. 

2. Required the square of 30. Ans. 900. 

3. Required the square of 35. 1225. 

4. Required the square of 40. 1600. 

Note. If the given number be hundreds, &c. reckon the 1 on D for 
100, or 1000, &c. then the corresponding one on C is 10000, or 100000, 
&c. , thus the square of 230 is found to be 52900. 

4th. To find a fourth proportional to three numbers : or to 
perform the Rule of Three by the Carpenters' Rule. 

EXAMPLE. 

1. Required to find a fourth proportional to 12, 28, and 
114. * 

Set the first term 12 on B to the second term 28 on A ; 
then against the third term 114 on B, stands 266 on A, 
which is the fourth proportional sought. 

2. Required the fourth proportional term to the numbers 
25 : 75 : : 100. Ans. 300. 

3. Required the fourth proportional term to the numbers 
27 : 20 : : 73. Ans. 54 2 T 

5th. To extract the Square Root of any number by the 
Carpenters 1 Rule. 

EXAMPLE. 

1. Required the square root of 400, 

Set 1 upon C to 10 upon D ; then against the number 
400 on C, stands its root 20 on D. 

2. Required the square root of 529. Ans. 23. 

3. What is the square root of 900 ? 30. 

4. What is the square root of 300 ? 17,3 -f 

O 2 



162 MEASURfNG OF BOARDS AND TIMBKR. 

SECTION 3. 

Measuring of Boards and Timber. 

1st To find ike superficial content of a board or platik. 

RULE. 

Multiply the length by the mean breadth. When the 
board is broader at one end than the other, add the brewlih 
of the two ends together, and take half the sum for tin..- 
mean breadth. 

By the Carpenters' Rule. 

Set 12 on B to the breadth in inches on A; then against 
the length in feet on B, you will find the superficies on A, 
in feet. 

EXAMPLES. 

1. How many feet are there in a board that is 13 {(-el 
long and 16 inches broad ? 

Operation. 

By duodecimals. By decimals. 

'Ft. in. 13, 

13 1,33 

1 4 



39 

13 39 

440 13 



17 4 Ans. 17,29 Ans. 

By the Carpenters' Rule 
As 12 on B : 16 on A : : 13 on B : 17 on A. Ans. 

2. Required the superficies of a board, whose mean 
breadth is 1 foot 2 inches, and length 12 feet 6 inches? 

Ans. 14 feet 7 inches. 

3. Required the value of 5 oaken planks, at 3 cents pei 
foot ; each of them being 17 feet long, and their several 
breadths as follows, viz. ; two of 13 inches in the middle, 
one of 14 inches in the middle, and the two remaining 
ones, car-h 18 inches at the broader end, and 11:1 inHvs at 
the narrower 1 Ans. 3 dolls. 9i cts. 



MEASURING OF BOARDS AND TIMBER. 163 

2d. Having the breadth of a board or plank in inches, 
tojind how much in length will make afoot^ or any other 
assigned quantity. 

RULE. 

Divide 144, or the area to be cut off, by%ie breadth' in 
inches, and the quotient will be the length in inches. 

EXAMPLE. 

1. How many inches in length will it require to make one 
foot, of a board that is 9 inches broad? 

Operation. 144-1-9=16 inches, the length required. 

2. How many inches in length, of a board that is 23 
inches wide, will make 1 foot? 

Ans. 6,26 -f inches. 

3. From a mahogany plank 26 inches broad, a yard and 
a half (or 13 feet 6 inches) is required to be cut off*; what 
distance from the end must the line be struck ? 

Ans. 74,7692 inches, or 6,23 feet. 

3d. To find the solid content of squared or four sided 
timber. 

RULE. 

Multiply the mean breadth by the mean thickness, and 
that product by the length ; the last product will give the 
solid content. 

Note. 1. If the tree taper regularly from one end to the other, take 
the mean breadth and thickness in the middle ; or take half the sum 
of the dimensions at the two ends, for the mean dimension. 

2. If the piece does riot taper regularly, take several different di- 
mensions, add them all together, and divide their sum by their num- 
ber, for the mean dimension. 

3. The quarter girt is an arithmetical mean, proportional between 
the mean breadth and thickness, that is the square root of the pro- 
duct. 

EXAMPLES- 

1. If a piece of timber be 2 feet 9 inches deep, and 1 foot 
7 inches broad, and the length 16 feet 9 inches,, (or which 



164 MEASURING OF BOARDS AND TIMBER. 

is the same thing,* if the quarter girt be 26 inches, and the 
length 16 feet 9 inches,) how many solid feet are contained 
therein ? 

Operation. 

26 inches quarter girt 16,75 16 feet 9 inches, the length. 

26 676 



156 10050 

52 11725 

10050 

676 square 

144) 11323,00(78,63 + feet. Arts. 
1008 



1243 
1152 

910 
864 

460 
432 

28 rem. 

By the Carpenter's Rule 
As 12 on D : 16f on C : : 26 on D : 78 on C. Ans. 

2. The quarter girt of a piece of squared timber is 15 
inches, and the length 18 feet; required the solidity? 

Ans. 28 J feet. 

3. If a piece of squared timber be 25 inches square at 
the greater end, and 9 inches square at the less, and the 
length be 20 feet ; what is the solid content ? 

Ans. 40,13 feet. 

4. Suppose a piece of squared timber to measure 32 by 
20 inches at the greater end, and 10 by 6 inches ai the less, 
and the length 18 feet; how many feet of timber are con- 
tained therein? Ans. 34,12 -f feet. 

* This is making- use of an arithmetical mean, instead of a geomet- 
rical one, which is not exactly true; it is, however, sufficiently exact 
for common purposes, when the timber is nearly square; the error in- 
creases, the more the breadth and depth differ from each other. 
When greater exact ness is required, multiply the breadth by the 
depth in the middle, and that, product by the length, ibr the true con- 
tent. 



MEASURING OF BOARDS AND TIMBER. 165 

4:th. To find the solid content of round timber. 
RULE. 

Multiply the square of the quarter girt, or of ^ of the 
mean circumference, by the length, for the content. 

Note. 1. To find the quarter girt of round timber, measure round 
the middle with a line, one-fourth part of this is reckoned the quarter 
girt 

2. When the tree is tapering 1 , take either the mean dimension, as in 
squared timber, or girt it at both ends, and take half the sum. If the 
tree is very irregular, divide it into several lengths, add all the girts to- 
gether, and divide the amount by the sum of them, for tho mean girt. 

3. The buyer is allowed to take the girt anywhere between the 
greater end and the middle, if it taper ; an allowance must also be 
made for bark ; one-tenth for oak, but less for ash, beech, &c. 

EXAMPLES. 

1. A piece of round timber being 9 feet 6 inches long,, 
and its mean quarter girt 42 inches, what is the content? 

Operation. 

Decimals. Duodecimals. 

3,542 inches, quarter girt. Ft. in. 

3,5 3 6=42 inches. 
3 6 



175 

105 190 
10 6 



12,25 



9,5 length 12 3 

9 6 



6125 

11025 6 1 
110 3 



116,375 content. 

116 4 6 

By the Carpenters' Rule 

As 9,5 on C : 10 on D : : 3,5 on D : 116J on C ( 4 
Or,9,5 on C : 12 on D : : 42 on D : 116i on C $ 
2. The length of a tree is 24 feet, its girt at the thicker 
end is 14 feet, and at the smaller end 2 feet; what is its 
content? Ans. 96 fert. 



166 CARPENTERS' AND JOINERS' WORK- 

3. If a piece of round timber 18 feet long, measure 00 
inches in circumference, or the quarter girt 24 inches ; how 
many feet of timber does it contain? 

Ans. 72 feet. 

4. If a piece of round timber measure 11 feet 4 inches 
at the larger end, 2 feet 8 inches at the less, and its length 
21 feet, how many feet of timber are contained therein? 

Ans. 64,31 feet. 

5. Required the amount of three pieces of round timb< r 
measuring as follows, viz. 

The first 24 feet long and mean girt 8 feet, 

The second 14* do. do. 3,15 

The third 17i do. do. 6,28 

Ans 147 ft 4. feet. 



SECTION 4. 
OF CARPENTERS' AND JOINERS' WORK. 

To this branch belongs all the wood-work of a house, 
such as framing, flooring, partitioning, roofing, &c. 

Carpenters usually measure their work by the square, 
(consisting of 100 superficial feet) the yard or foot; but 
enriched mouldings, cornices, &c. are estimated by run- 
ning or lineal measure, and some things are rated by the 
piece. 

In measuring of Carpenter's work, the string is made to 
ply close to every part of the work over which it passes. 

Partitions are measured from wall to wall for one dimen- 
sion, and from floor to floor, as far as they extend, for the 
other. 

In framing, no deductions are made for door-ways, fire- 
piacrs, or other vacancies, on account of the additional 
trouble of framing arising from them. 

For stair-case's, take the breadth of all the steps, by 
milking a line ply close over them, from the top to the bot- 
inm, and multiply the length of this line, by the length of 
thr step, for the whole axea. T>y Iho length of a step is 



CARPENTERS' AND JOINERS' WORK. 167 

meant the length of the front, and the returns at the two 
ends ; and by the breadth, is to be understood the girt of its 
two outer surfaces, or the tread and rise. 

The rail of a stair-case is taken at so much per foot in 
length, according to the diameter of the well-hole ; archi- 
trave string boards, by the foot superficial ; brackets and 
strings at so much per piece, according to the workmanship. 

Wainscoting is measured by the yard square, consisting 
of 9 feet. 

Door cases, frame doors, modillion cornices, eaves, frontis- 
pieces, <fec. are generally measured by the foot superficial. 

Joists are measured by multiplying their breadth by their 
depth, and that product by their length. They receive va- 
rious names, according to the place in which they are laid 
to form a floor ; such as trimming joists, girders, binding 
joists, bridging joists, ceiling joists, &c. 

In boarded flooring, the dimensions must be taken to the 
extreme parts ; and the number of squares of 100 feet, are 
to be . calculated from these dimensions. Deductions must 
be made for chimneys, stair-cases, &c. 

In roofing, take the whole length of the timber, for the 
length of the framing, and gird over the ridge from wall to 
wall with a line, for the breadth. This length and breadth 
multiplied together give the content. 

In measuring of roofing for work and materials, all holes 
for chimney-shafts, sky-lights, &c. are included in the mea- 
surement, on account of their trouble and waste of materials ; 
but for workmanship alone, they are generally deducted. 

It is a common rule among carpenters, that the flat of 
any hou.se, and half the flat thereof taken within the walls, 
ks equal to the measure of the roof of the same house ; this 
is, however, only when the roof is the true pitch where 
the length of the rafters are 5 of the breadth of the build- 
ing. The pitch of roofs varies according to the materials 
with whio.h they are covered, and fancy of the builder. 

Weather- boarding, like flooring, is measured by the 
square, arid sometimes by the yard. 



168 CARPENTERS' AND JOINERS' WORK. 

EXAMPLES. 

1. If a floor be 57 feet 3 inches long, and 28 feet 6 inches 
broad, how many squares of flooring does it contain ? 

Operation. 

By decimals. By duodecimals. 
Ft. in. Ft. in. 

57 3=57,25 57 3 

28 6= 28,5 28 6 



28625 456 

45800 114 

11450 28 7 6 



100)1631,625 
16,31,625 



700 



100)1631 7 6 

16,31 7 6 
Ans. 16 squares 31 feet 7 in. 6' 

2. Let a floor be 53 feet 6 inches long, and 47 feet 9 
inches broad, how many squares does it contain ? 

Ans. 25 squares 54 feet. 

3. A floor being 36 feet 3 inches long, and 16 feet 6 
inches broad, what will it cost at 4 dollars and 50 cents per 
square? Ans. 26 dolls. 91 cents. 

4. A room is 35 feet long and 30 feet wide ; there is in it 
a fire-place which measures 6 feet by 4 feet 6 inches, and a 
well-hole for the stairs measures 10 leet 6 inches by 8 feet ; 
what will the flooring come to at 3 dollars and 75 cents per 
square? Ans. 35 dolls. 21 -f cts,. 

5. How many squares are contained in a partition that i* 
82 feet 6 inches long, and 12 feet 3 inches high? 

Ans. 10 squares and 10-ffeet. 

6. If a partition between rooms be in length 91 feet 9 
inches, and its height 11 feet 3 inches; how many squares 
are contained in it, and how much does it come to at 4 dol 
lars and 50 cents per square ? * 

Ans. 10 squares 32 feet, and costs 46 dolls. 44 cts. 



CARPENTERS AND JOINERS WORK- 169 

7. If a house within the walls be 44 feet 6 inches long, 
and 18 feet 3 inches broad ; how many squares of roofing 
will it contain, allowing the roof to be the true pitch ? 

Operation. 

By decimals. By duodecimals. 

Ft. Ft. in. 18 3 

18,25=18 3 the breadth. 44 6 
44,5=44 6 the length. 



72 

9125 72 
7300 



7300 792 
11 0" 



Flat 812,125 9 1 6 

Half 406,062 



Flat 812 1 
-100)1218,187 Half 406 + 



Sum 12,18+ + 100)1218 1 6 



Sum 12,18 

Ans. 12 sq. 18 ft. 

8. What cost the roofing of a house at 1 dollar and 40 
cents per square ; the length within the walls being 52 feet 
8 inches, and the breadth 3-0 feet 6 inches ; the roof being 
of a true pitch ? 

Ans. 33 dollars 73 cents. 

9. Suppose a house measures, within the walls, 40 feet 6 
inches in length, and 20 feet 6 inches in breadth, and the 
roof being a true pitch ; how many squares of roofing does 
it contain, and how much will it cost at 2 dollars 25 cents 
per square ? 

Ans. 12,45375 squares, and costs 28 dolls. 2 + cts. 

Note. All timbers in a roof are measured in the same manner as 
in floors, except king-posts, which are measured by taking their 
breadth and depth at the widest place, arid multiplying these together., 
and the product by the length. 

10. If a room or wainscot, being girt downwards over 



!?(> BIUCKLAYEKS' WORK. * 

the mouldings, be 15 feet 9 inches high, and 126 feet 3 
in* -lies in compass ; how. many yards does that room contain '/ 

Operation. 
By decimals* By duodecimals. 

12V5 ' Ft - in - 

15,75 Ig 3 



63125 63Q 

88375 126 

63125 



12655 189 

63 1 6 

31 6 9 

9)1988,4375 390 



Sum 220,8 

Sum 220 8 

Ans. 220 yds. 8 feet. 

11. If a room of wainscot be 16 feet 3 inches high, and 
the compass of the room 137 feet 6 inches ; how many 
yards are contained in it ? Ans. 248 yards 2 -f feet. 

12. If the window-shutters about a room be 69 feet 9 
inches broad, and 6 feet 3 inches high ; how many yards 
are contained therein, at work and half? 

Ans. 72,656 yards. 

13. What will the wainscoting of a room come to at 80 
cents per square yard, supposing the height of the room, in- 
cluding the cornice and moulding, be 12 feet 6 inches, and 
the compass 83 feet 8 inches ; three window-shutters, each 
7 feet 8 inches by two feet 6 inches, and the door 7 feet by 
3 feet 6 inches ; the shutters and door being worked on both 
sides, are reckoned work and half? 

Ans. 96 dollars 60J cents. 



SECTION 5. 

OF BRICKLAYERS' WORK- 

BKICK WOKK is measure. I ;ind rsfimatr'd in various ways. 
!i somn places walls aiv mrasim-d !>y fho rod square of 
1 i frc'i ; so th.-.t on'- rod in length, and one in breadth 



BRICKLAYERS' WORK. 171 

contain 272,25 square feet ; in other places the custom is to 
allow 18 feet to the rod, that is, 324 square feet. 

In other places they measure by the rod of 21 feet long, 
and 3 feet high, that is, 62 square feet. Again, in other 
places they account 16^ feet long and 1 foot high, that is, 
16^ square feet, a rod or perch; or again, by the yard of 
9 square feet ; and oftentimes the work is estimated at so 
much per thousand bricks. 

When brick work is measured by the rod, or perch, it 
must be estimated at the rate of a brick and a half thick; 
so that if a wall be more or less than this standard thickness, 
it must be reduced to it by the following 

RULE. 

Multiply the superficial content of the wall by the num- 
ber of half bricks in the thickness, and divide the product 
by 3 for the superficial feet in standard thickness. 

EXAMPLES. 

If a wall be 72 feet 6 inches long, and 19 feet 3 inches 
high, and 5 bricks and a half thick ; how many rods of 
brick work are contained therein, when reduced to the 
standard thickness ? 

Operation. 

By decimals. By duodecimals. 

19,25 = the height Ft. in. 

72,5 = the length. 72 6 

"9625 19 3 

3850 648 

13475 - 72 

139^625 1368"" 

11= the thickness. 18 1 6' 

-f-3)15351,875 9 6 

272,25) 5117,291 (18 rods. 1395 7 ^6 

239479 i- 

68,06) 216/79( 3 quarters. -3)15351 10 6 

lalT 272)511^ (18 rote. 

2397 
68) "22f ( 3 quarters 

Tffeet. 

Note. Observe that 68,06 is the one- fourth part of 272,25, and 68 is 
only the one-fourth part of 272. As the number 272^ is an inconve- 
nient number to divide* by, the \ is usually ornitfed, and the content in 
feet divided only by 272 ; the difference being too trifling to be con- 
sidercd in practice. 



172 



BRICKLAYERS' WORK. 



To find fixed divisors for bringing the answer into feet or 
rods of a standard thickness, without multiplying the su- 
perficies by the number of half bricks, <$*c. 

RULE. 

Divide three, the number of half bricks in l, by the 
number of half bricks in the thickness, the quotient will be 
a divisor, which will give the answer in feet. Or if a divi- 
sor is sought for, that will bring the answer in rods at once, 
multiply 272 by the divisor found for feet, and the product 
will be a divisor for rods ; as in the following 

TABLE. 



1 

The thickness of 
the wall. 


2 

Divisors for the 
ansiver in feet. 


3 

Divisors for the 
ansiver in rods. 


1 brick 


1, 


408 


l brick 


1 


472 


2 bricks 


,75 


204 


2J bricks 


,6 


163,2 


3 bricks 


,5 


136 


3i bricks 


,4285 


116,6 


4 bricks 


,375 


102 


4^ bricks 


,3333 


90,6 


5 bricks 


,3 


81,6 


5J bricks 


,2727 


74,18 



Application of the above Table. 

Multiply the length of the given wall bv the breadth , o 
serve the number of half bricks it is in thickness ; and op- 
[>rsite thereto will be found in the second column the di- 
visor to reduce it to feet ana in the third column the divi- 
sor for rods. Thus in the above example 72,5 X by 19,25 
1395,625. 

Ana 1395,620 2727 = 5117+ the number of feet in 
standard nn-asure. 

And i:W5,(>-jr>-74,18^18,8-f th<> num l >cr of rods 



BRICKLAYERS' WORK. 173 

Or, by the Carpenters' Rule 

As the tabular divisor, against the thickness of the wall 
: is to the length of the wall : : so is the breadth : to the 
content. 

As 74,18 on B : 72,5 on A :: 19,25 on B : 18 j on A. Ans. 

To find the dimensions of a building, measure half around 
on the outside, and half on the inside, for the whole length 
of the wall ; this length being multiplied by the height gives 
the superficies. All the vacuities, such as doors, windows, 
window -backs, &c. must be deducted, for materials ; but for 
workmanship alone no deductions are to be made, and the 
measurement is usually taken altogether on the outside. 
This is done in consideration of the trouble of the returns 
or angles. There are also some other allowances, such as 
double measure for feathered gable ends, &c. 

2. How many yards and rods of standard thickness are 
contained in a brick wall, wWbse length is 57 feet 3 inches, 
and height 24 feet 6 inches ; the wall being 2J bricks thick ? 

Ans. 259,74 yards, or 8,58 + rods. 

3. If a wall be 245 feet 9 inches long, 16 feet 6 inches 
high, and 2| bricks thick; how many rods "of brick work 
are contained therein, when reduced to standard thickness ? 

Ans. 24 rods 3 quarters 24 feet. 

4. A triangle gable end is raised to the hei^ of 15 feet 
above the wall of a house, whose width is 45 feet and the 
thickness of the wall 2 bricks ; required the contei^ in rods 
at standard thickness? Ans. 2 rods 18 feet. 

Chimneys by some are measured as if they were solid, de- 
ducting only the vacuity from the hearth to the mantle, on 
account of their trouble. 

But by others, they are girt or measured round for their 
breadth, and the height of their story, taking the depth of 
the jambs for their thickness. And in this case no deduc- 
tion is made for the vacuity from the floor to the mantle-tree, 
because of the gathering of the breast and wings, to make 
-oom for the hearth in the next story. 
P2 



174 MASONS' WORK. 

If the chimney back be a party wall, and the wall be 
measured by itself, then the depth of the two jambs and 
length of the breast is to be taken for the length, and the 
height of the story for the breadth, at the same thickness 
with the jambs. 

Those parts of the chimney-shaft which appear above 
the roof are to be girt with a line round about the least place 
of them for the length, and take the height for the breadth ; 
and if they are 4 inches thick, they are to be accounted as 
one brick work, and if they are 9 inches thick, they are to 
be taken for l brick work, on account of the trouble of 
plastering and scaffolding. 

It is customary in most places to allow double measure 
for chimneys. 



SECTION 6. 



OF MASONS' WORK. 

MASONS' work is measured sometimes by the foot solid, 
sometimes by the foot superficial, and sometimes by the foot 
in length. It is also measured by the yard, and mostly by 
the rod or perch, which is 16 feet in length, 18 inches in 
breadth, and 12 inches in depth. 

Walls are measured by the perch; columns, blocks of 
stone, or marble, &c. by the cubic foot; and pavements, 
slabs, chimney-pieces, &c. by the superficial or square foot. 

Cubic, or solid measure, is always used for materials, but 
square measure generally for workmanship. 

In solid measure, the true length, breadth and thickness 
are taken and multiplied into each other for the content. 

!n superficial measure, the length and breadth of every 
part of the projection, which is seen without the general 
upright face of the building, is taken for the content. 



MASONS' WOHK. 175 



EXAMPLES. 

1. If a wall be 97 leet 5 inches long, 18 feet 3 inches 
high, and 2 feet 3 inches thick, how many solid feel, and 
perches, are contained therein ? 

Operation. 

By decimals. By uuodecimals. 
97,417 length Ft. in. 
18,25 breadth 97 5 
18 3 



4870S5 



194834 776 
779336 97 
97417 24 4 3" 
600 



1777,86025 superficies 1 60 

2,25 thickness 



1777 10 3 

888930125 2 
355572050 

355572050 3555 8 6 
444 5 6 



4000,1855625 solidity. 



in cubic ft. 4000 2094 u. 
4000^-24,75^161,616-f feet. Ans. 

2 How many solid feet and perches are contained in a 
wall 53 feet 6 inches long, 12 feet 3 inches high, and 2 feet 
thick? Ans. 1310,75 feet, and 52,9595 rods. 

3. If a wail be 107 feet 9 inches long, and 20 feet 6 inches 
high, how many superficial feet are contained therein ? 

Ans. 2208 feet 10 inches. 

4. If a wall be 112 feet 3 inches long, and 16 feet 6 
inches high, how many superficial rods, each 63 square 
feet, are contained therein ? Ans. 29 rods 25 feet. 

5. What is a marble slab worth, whose length is 5 feet 7 
inches, and breadth 1 foot 10 inches, at 80 cents per fool 
superficial 1 Ans. 8 dolls. 19 els* 



176 PLASTERERS WOixK 

SECTION 7. 

OF PLASTERERS' WORK. 

PLASTERERS' WORK is principally of two kinds, viz. 
first, plastering upon laths, called ceiling ; and second, plas- 
tering upon walls, or partitions made of framed timber, 
called rendering, which are measured separately. 

Plasterers' work is usually measured by the yard square, 
consisting of 9 square feet ; sometimes it is measured by 
the square foot, and sometimes by the square of 100 feet. 

Enriched mouldings, cornices, &c. are rated by running, 
or lineal measure. In arches, the girt round them multiplied 
by the length, is taken for the superficies. 

Deductions are to be made for doors, chimneys, windows, 
and other large vacuities. But when the windows, or other 
openings, are small, they are seldom deducted, as the plas- 
tered returns at the top and sides are allowed to compensate 
for the vacuity. 

Whitewashing and coloring are measured in the same 
manner as plastering. 

EXAMPLES. 

1. If a ceiling be 59 feet 9 inches long, and 24 feet 6 
inches broad, how many superficial yards of 9 square feet 
does it contain ? 

Operation. 

By decimals. By dtfodecimals. 

Ft. in. Ft. in. 

59 9=59,75 feet 59 9 

24 6= 24,5 do. 24 6 

29875 236 
23900 118 
11950 29 10 6" 
18 



-7-9)1463,875 feet 
Ans. 1 62,65 -f yards 



-r 9) 1463 10 6 



162 5 10 6 An*. 



\VKHJS WORK 177 

2 If the plastered partitions between rooms be 141 feet 6 
inches about, and 1 1 feet 3 inches high, how many yards 
(Jo tiiey contain? Ans. 176,87 yards. 

3. What will the plastering of a ceiling come to at 15 
cents per yard, allowing it to be 22 feet 7 inches long, and 
13 feet 11 inches Lroad ? Ans. 5 dolls. 20 cts. 

4; The length of a room being 20 feet, its breadth 14 
feet 6 inches, and height 1 feet 4 inches ; how many yards 
of plastering does it contain, deducting a fire-place of 4 feet 
by 4 feet 4 inches, and two windows, each 6 feet by 3 feet 
2 inches ? Ans. 73^ T yards. 

5. The length of a room is 14 feet 5 inches, breadth 13 
feet 2 inches, and height 9 feet 3 inches, to the under side 
of the cornice, which projects 5 inches from the wall, on the 
upper part next the ceiling ; required the quantity of render- 
ing and plastering ; there being no deductions but for one 
door, the size whereof is 7 by 4 feet ? 

Ans. 53 yds. 5 ft. of rendering, and 18 yds. 5 ft. ceiling. 

6. The circular vaulted roof of a church measures 105 
feet 6 inches in the arch, and 275 feet 5 inches in length ; 
what will the plastering come to at 12 cents per yard ? 

Ans. 387 dolls. 42 cts. 

7. What will the whitewashing of a room come to at 2 
cents per yard, allowing it to be 30 feet 6 inches long, 24 
feet 9 inches broad, and 10 feet high ; no deductions being 
made for vacuities? Ans. 4 dolls. 13 cts. 



SECTION 8. 

OF PAVERS' WORK. 

PAVERS WORK is measured by the square yard, consist- 
ing of 9 square feet. The superficies is found by multiply- 
ing the length by the I readth. 



178 FA1JNTKRS' WORK. 

EXAMPLES. 

1. What cost the paving of a street 225 feet 6 inches long. 
and 60 feet 6 inches wide, at 30 cents per square yard ? 

By decimals. By duodecimals. 

Ft. in. ^ in. 

225 6=225,5 feet 
60 6= 60,5 do. 



13500 

11275 US 

13530 



9)13642 9 



1515 7 
30 



-7-9 ) 13642,75 superficial ft. 

1515,86 yards. 

30 

26 =the price of 7 ft. 9 in. 

Ans. 454,7580 ~454jeT 

Ans. 454 dolls. 76 cts. 

2. What will the paving of a foot-path come to at 28 
cents per yard, the length being 35 feet 4 inches, and the 
breadth 8 feet 3 inches ? Ans. 9 dolls. 33 cts. 

3. What cost the paving of a court-yard at 38 cents per 
yard, the length being 27 feet 10 inches, and the breadth 14 
feet 9 inches ? Ans. ?7 dolls. 33 cts. 

4. What will be the expense of paving a rectangular 
yard, whose length is 63 feet, and breadth 45 feet, in which 
there is laid a foot-path 5 feet 3 inches broad, running the 
whole length, with broad stones, at 36 cents a yard ; the 
rest being paved with pebbles, at 30 cents a yard ? 

Ans. 96 dolls. 70 cts. 



SECTION 9. 

OF PAINTERS' WORK. 

PAINTERS' WORK is computed in square yards of 9 feet 
Kvery part is measured where the color lies ; and the mea 
suring line is pressed close into all the mouldings, corners, 
&ic. over which it passes. 



PAINTERS' WORK. 179 

Windows, casements, &c. are estimated at so much a 
piece ; and it is usual to allow double measure for carved 
mouldings, &c. 

The value of painting is rated by the number of coats ; 
or whether once, twice, or thrice colored over, and the dif- 
ferent qualities and costliness of the colors. 

EXAMPLES. 

1, How many yards of painting will a room contain 
which (being girt over the mouldings) is 16 feet 6 inches, 
and the compass of the room 97 feet 6 inches ? 

Operation. 

By decimals. By duodecimals. 

Ft. in. 97 6 

97 6=97,5 16 6 

16 6=16,5 



4875 
5850 
975 



-r 9 ) 1608,75 feet -r-9 ) 1608 




Yards 178,6,75 178,6,9 

Ans. 178J yards. 

2. A gentleman had a room painted at 8J cents per yard, 
the measure whereof is as follows, viz. the height 11 feet 

7 inches, the compass 74 feet 10 inches, the door 7 feet 6 
inches by 3 feet 9 inches ; five window shutters, each 6 feet 

8 inches by 3 feet 4 inches; the breaks in the windows 14 
inches deep, and 8 feet high ; the opening for the chimney 6 
feel - inches by 5 feet, to be deducted, the shutters and doors 
are painted on both sides ; what will the whole come to ? 

Ans. 10 dolls. 43 cts. 

3. How many yards of painting are there in a room, the 
length whereof is 20 feet, its breadth 14 feet 6 inches, and 
height 10 feet 4 inches; deducting a fire-place of 4 feet by 
4 feet 4 inches, and two windows, - each 6 feet by 3 feet 2 
inches? Arts. 73^ T yards. 



180 GLAZIERS' WORK. 

4. What cost the painting of a room at 6 cents per yard , 
its length being 24 feet 6 inches, its breadth 16 feet 3 inches, 
and height 12 feet 9 inches ; also the door is 7 feet by 3 
feet 6 inches, and the window shutters of two windows, 
each 7 feet 9 inches by 3 feet *6 inches, but the breaks of the 
windows themselves, are 8 feet 6 inches high, and 1 foot 3 
inches deep ; deducting a fire-place cf 5 feet by 5 feet 6 
inches. A?is. 7 dolls. 66 cts. 9 m. 



SECTION 10. 

OF GLAZIERS' WORK. 

GLAZIEKS compute their work in square feet ; and the di- 
mensions are taken either in feet, inches, and seconds, &c. 
or in feet, tenths, hundredths. 

Windows are sometimes measured by taking the dimen 
sions of one pane, and multiplying its .superficies by the 
number of panes. But more generally they measure the 
length and breadth of the window over all the panes, and 
their frames for the length and breadth of the glazing. And 
oftentimes the work is estimated at so much per pane ac- 
cording to the size. 

Circular, or ovtil windows, as fan-lights, &c. are measured 
as if they were square, taking for their dimensions the 
greatest length and breadth, as a compensation for the waste 
of glass, and labor in cutting it to the proper forms. 

EXAMPLES. 

1. I low many square feet are contained in a window, 
which is 4 feet 3 inches long, and 2 feet 9 inches broad? 

Hv decimals. By duodecimals. 

Ft. in." Ft. in. 

4 = 4, 25 the length 4 3 

2 9 -=2,75 the breadth 2 9 



2125 8 6 

2975 3 2 

sf,0 



11/H75 foot. 



11 S 3 Ans. 



MEASUREMENT OF GROUJND 181 

2. If a window be 7 feet 3 inches high, and 3 feet 5 inches 
broad, how many square feet of glazing are contained 
therein ? . An*. 24 feet 9 inches. 

3. There is a house with three tiers of windows, 7 in a 
tier; the height of the first tier is 6 feet 11 inches, of the 
second, 5 feet 4 inches, and of the third, 4 feet 3 inches ; the 
breadth of each window is 3 feet 6 inches : what will the 
glazing come to at 14^ cents per foot? 

Ans. 58 dolls. 61 cts. 

4. What will the glazing of a triangular sky-light come 
to at 10 cents per foot, the base being 12 feet 6 inches long, 
and the perpendicular height 16 feet 9 inches? 

Ans. 10 dolls. 465 cts. 

5. What is the area of an elliptical fan-light of 14 feet 
inches in length, and 4 feet 9 inches in breadth ? 

Ans. 68 feet 10 inches. 

6. There is a house with three tiers of windows, and & 
in each tier; the height of the first tier is 7 feet 10 inches, 
of the second, 6 feet 8 inches, of the third fret 4 inches 
and the common breadth 3 feet 11 inche? , what will the 
glazing come to at 14 cents per foot? 

Ans. &' <Kis. 87 j cts. 



SECTION 11 
MEASUREMENT OF GKOUND. 

1st. To find the content of a square piece of ground. 

RULE. 

MULTIPLY the base in perches, yards or feet, as the case 
may be, by the perpendicular, and the product will be the 
answer required. 

Note. 1. Any area, or content in perches, being divided by 160, will 
give the content in acres ; the remaining- perches, if more than 40, 
being divided by 40, will give the roods, or quarter acres, and the last 
remainder, if any, will be perches. 

2. Ground is generally measured by chains, of two poles or rods in 
length ; the two pole chain measures 33 feet. Chains of 4 poles are 
sometimes used, and sometimes chains or poles of one rod in length 
only. 

Q 



182 MEASUREMENT OF GROUND 

EXAMPLE. 

1. In a square field, A, B, C, D, each side of which 
measures 40 rods, or poles, how many acres ? 

Operation. 

40 DC 

40 



4,0) 160,0 



10 Acres. 



4 ) 40 A 40 B 

10 acres. Ans. 

2. In a square field, each side of which measures 35 two 
pole chains, how many acres? 

Ans. 30 acres 2 roods 20 perches. 

8. A piece of square ground measures 16^ perches on 
each side ; what is the content in acres ? 

Ans. 1 acre 2 roods 32| perches. 

2d. Tojind the content of an oblong square piece of ground, 
called a parallelogram. 

RULE. 

Multiply the length by the breadth, and the product will 
be the answer. 

EXAMPLE. 

1. There is an oblong square piece of ground, A, B, C, D, 
the longest sides of which measure 64 perches, and the 
shortest sides, or ends, measure 40 ; how many acres does 
it contain ? 

Operation. D C 

64= the length 
40= the breadth 



4,0 ) 256,0 perches 
4)64 



16 Acres. 



64 



16 acres. An*. 



MEASUREMENT oV GROUND. 183 

2. In a piece of ground lying in the form of an oblong 
square, the length measures 120 perches, and the breadth 
84; what is its content in acres? Ans. 63 acres. 

3. A lot of ground lying in the form of an oblong square, 
measures 240 feet in length, and 120 in breadth; what is 
its content in acres '.' 

Ans. acres 2 quarters 25 perches 21 3J feet. 

4. There is an oblong piece of ground, whose length is 
14 two pole chains 25 links, and breadth 8 chains 37 links ; 
how many acres does it contain ? 

Ch. L. Perches. 

8 37 = 17,48 breadth 
14 25= 29 length 

15732 
3496 

4,0)50^,92 
4)12^26" 



3 26,92 

Ans. 3 acres quarters 27 perches nearly. 
Note. The English statute perch is 5 yards, the two pole chain is 
11 yards, or 33 feet, and is divided into 50 links; the four pole chain 
is 22 yards, or 66 feet, and contains 100 links ; hence the length of a 
link in a statute chain is 7,92 inches, and 25 links make 1 rod. And 
consequently, if the links be multiplied by 4, carrying 1 to the chains 
for every 25 links, and the chains by 2, the product will be perches, 
and decimals of a perch. 

5. An oblong piece of ground measures 17 two pole 
chains and 21 links in length, and 15 chains 38 links in 
breadth ; how many acres are contained therein ? 
Ch. L. Ch. L. 

17 21 15 38 

24 24 



34 84 perches, the length. 31 52 p. the breadth. 
Then 34,84x31,52=1098,1568 perches=6 acres 3 qr. 
18,1 5 -f- perches. 
3d. To find the content of a triangular piece of ground. 

RULE. 

Multiply the base by half the perpendicular, or the per- 
pendicular by half the base, or take half the product of the 
base into the perpendicular. 



184 



MEASUREMENT OF GROUND 




B D C 

1. Let A, B, C, be a triangular piece of ground, the 
ongest sio or base B, C, is 24 chains 38 links, and perpen- 
dicular, A D, 13 chains 29 links; how many acres does it 
contain ? 

Operation. 
Ch. L. 

24 38=49,52 perches 
13 28=27,12 



9904 
4952 
34664 
9904 

1342,9824 

Half the sum is 4,0)67,1,4912 perches 
4 ) 16,31 



4 31,4 
Ans. 4 acres roods 31,4 perches. 

2. In a triangular piece of ground, the base or longest 
side measures 75 perches, and the perpendicular 50 ; how 
many acres does it contain ? 

Ans. 11 acres 2 qrs. 35 perches. 

3. How much will a triangular piece of ground come to 
at 45 dollars per acre, the longest side or base of which 
measures 120 perches, and the perpendicular 84 perches. 

Ans. 1417 dolls. 50 cents. 

4. How many superficial yards are contained in a trian- 
gular piece of ground, fho bavc of vhirh measures 140 feet 
and the perpendicular 70 feet'! Ans 544 vards 4 feet. 



MEASUREMENT OF GROUND. 



185 



4ith. To find the content of a piece of ground, in the form 
of an oblique parallelogram. 

RULE. 

Multiply the base into the perpendicular height for the 
content. 

EXAMPLE. 
D C 



A E B 

. Let A, B, C, D, be a piece of ground in the form of 
t~n oblique parallelogram, the base of which, A, B, measures 
44 perches, and the perpendicular, D, E, 40 perches ; how 
many acres does it contain 1 

44 length 
-40 breadth 



4,0)176,0 perches 



4) 44 

11 acres. Ans. 11 acres. 

2. A piece of ground lying in the form of an oblique par- 
allelogram, is found to measure 80 perches along its base, 
and its perpendicular height 24 perches; how many acres 
does it contain? Ans. 12 acres. 

5th. To find the content of a piece of ground bounded by 
four sides, none of which are parallel or equal. 

RULE. 

Find the length of a diagonal line between the two most 
distant corners, and multiply this line by the - sum of the 
two perpendiculars falling from the other corners to that di- 
agonal line, and half the product will be the area. 
Q2 



186 



MEASUREMENT OF GROUND. 




1. Let A, B, C, D, be a field with four irregular and un- 
equal sides, the diagonal line of which, A, C, measures 80 
perches, the perpendicular, B, m, measures 25 perches, and 
the other perpendicular, D, n, 35 perches; how many acres 
does it contain ? 

80 the length of the diagonal line. 

25 -f 35=60 the sum of the two perpendiculars. 

2 ) 4800 



4,0)240,0 perches 



4) 60 

15 acres. An*. 

2. In a field of four unequal sides, the diagonal line be- 
tween the two most distant corners measures 120 rods, and 
the perpendiculars measure, the one 48, and the other 24 
rods ; required the number of acres it contains? 

Ans. 27 acres. 

6/7i. To find the area of a pitce of ground lying in a cir- 
cle y or ellipsis. 

RULE. 

Multiply the square of the circle's diameter, or the pro- 
duet of the longest and shortest diameters of the ellipsis, by 
the derimal number ,7854, the product, will give the area. 

Note. In any circle, the 
Diameter multiplied 
Circumference flivid'-il 



MEASUREMKiVr OF GROUND. 187 

EXAMPLE. 

1. How many acres are contained in a circular piece of 
ground, whose diameter measures 320 perches, or 1 mile ? 
320X320=102400 

,7854 



409600 
512000 
819200 
716800 



4,0 ) 80424,9600 perches 
4)2010,24,9 



502 2 24,9 
Ans. 502 acres 2 qr. 24,9 perches. 

2. A gentleman has an elliptical -yard in front of his 
house, the longest diameter of which measures 30 perches, 
and the shortest 20 ; how much ground is contained therein ? 

Ans. 2 acres 3 qr. 31,2 perches. 

3. How many square yards are contained in a circular 
piece of ground, the diameter of which measures 160 feet? 

Ans. 22 34 + yards. 

From the foregoing simple methods of finding the con- 
tents of ground lying in different forms, it will readily be 
seen, that the content of fields and small pieces ' of land, 
lying in any shape whatever, and bounded by any number 
of sides, may be calculated, without having recourse to the 
more expensive and troublesome practice of employing a 
regular surveyor. No other apparatus than a common rod- 
pole, or line of a known length, is requisite. Pieces of land 
having more than four boundary lines, may be easily divided 
into squares, parallelograms, triangles, (fee. and each calcu 
iated separately by some of the foregoing rules, and then the 
whole amount added into one sum for the content. It is of 
great importance to every practical farmer to know the size 
of the different fields which he cultivates. Besides the satis- 
faction thereof, this knowledge is necessary to enable him to 
regulate the quantity of seed which he should sow, as well 
the price for clearing^ plowing, planting, reaping, &c. 



188 GAUGING. 

SECTION 12. 

OF GAUGING. 

GAUGING is taking the dimensions of a cask in inches, to 
find its content in gallons. 

RULE. 

1. Find the mean diameter, between the head and bung 
diameters, by adding two-thirds of the difference between 
them to the head diameter. If the staves be but little curv- 
ing from the head to the bung, add only six-tenths of this 
difference. 

2. Square the mean diameter, so found, and multiply the 
product by the length of the cask in inches, for the content 
thereof in cubic inches. 

3. Divide the cubic inches, so found, by 294, for wine or 
spirits, and by 359 for ale ; the quot'ent will be the answer 
in gallons. 

EXAMPLE. 

1. How many gallons of wine will a cask contain whose 
bung diameter is 31 inches, head diameter 25 inches, and 
whose length is 3 feet, or 36 inches 1 

Operation. 

31 bung diam. 25 head diameter 

25 head diam. f of 6= 4 two-thirds difference 

6 difference. 29 

29 

261 

58 

841 square of the mean diam. 
36 the length 



5046 
2523 

30276 cubic inches. 
Then 80276 -f- 294 =102ffJ gals. Or, 102 gals. 3 qt. 



GAUGING. 189 

2. The diameter of a barrel at the bung measures 24 
inches, and at the head 18 inches, and its length is 24 
inches ; what is its content in wine measure 1 

Ans. 39f f gals. 

3. How many gallons of spirits will a cask contain, 
whose bung diameter is 36 inches, head diameter 28 inches, 
and whose length is 3 feet 4 inches ? 

Ant. ISlfN, gals. 



4. What is the content, in ale measure, of a barrel whose 
bung diameter measures 18 inches, head diameter 15 inches, 
and whose length is 2 feet 5 inches ? 

Ans. 23i||. gals. 

5. Bought a barrel of ale of the following dimensions, 
viz. bung diameter 22 inches, head diameter 18 inches, and 
length 3 feet ; how many gallons does it contain ? 

Ans. 42|f| gals. 

Of the Gauging or Diagonal Rod. 

The diagonal rod is a square rule, having four faces, be- 
ing commonly four feet long, and folding together by joints. 
This instrument is used for gauging, or measuring casks, 
and computing their contents, and that from one dimension 
only, namely, the diagonal of the cask ; that is, from the 
middle of the bung- hole, to the meeting of the head of the 
cask, with the stave opposite to the bung ; being the longest 
line that can be drawn within the cask from the middle of 
the bung-hole. Am! accordingly one face of the rule is a 
scale of inches, for measuring this diagonal, to which are 
placed the areas in ale gallons, of circles to the correspond- 
ing diameters, in like manner as the lines on the under sides 
of the three slides, in the sliding rule. On the opposite face 
are two scales of ale and wine gallons, expressing the con- 
tents of casks having the corresponding diagonals. And 
these are the lines which chiefly form the difference between 
this instrument and the sliding rule. 

EXAMPLE. 

The rod being applied within the cask at the bung-hole. 



190 MECHANICAL POWERS. 

the diagonal was found to be 34,4 inches ; required the con- 
tent in gallons. 

Now, to 34,4 inches, will be found corresponding on the 
rod, 90 1 ale gallons, and 111 wine gallons, the content re- 
quired. 

Not e. In taking the length of a cask to find the cubic inches, an 
allowance must be made for the thickness of both the heads of 1 inch, 
of 1^ inch, or 2 inches, according to the size of the cask ; and the head 
diameter must always be taken close to the chime. The contents ex- 
hibited by the rod, answer only to casks of the common form. 



SECTION 13. 

OF MECHANICAL POWERS. 

1st. OF THE LEVER. 

To find what weight may be raised or balanced by any 
given power. 

RULE. 

As the distance between the body to be raised, and ful- 
crum, or prop, 

Is to the distance between the prop, and the point where 
the power is applied, 

So is the power to the weight which it will raise. 

EXAMPLE. 

1. If a man weighing 150 Ib. rest on the end of a levei 
12 feet long ; what weight will he bafemce on the other end, 
supposing the prop 1 foot from the weight ? 

Operation. 

12= the length of the lever 
l,5=distance of the weight from the prop 



10,5= the distance from the prop to the man. 
Then, as 1,5 : 10,5 : : 150 : 1050. Ans. 
2. The pea of a pair of steelyards weighing 5 Ib. is re- 
moved 20 inches back from the fulcrum ; what weight will 



MECHANICAL TOWERS. 1^1 

it balance, suspended at I irvh distance on the opposite si le? 

Arts 100 lo. 

*2d. OF THE WHEEL AND AXLE. 

To find what power must be applied at the wheel, to r< \ise 
a given weight suspended to the axle ; or what weight ai 
the axle will be raised by a given power at the wheel. 

RULE. 

As the diameter of the axle : is to the diameter of th< 
wheel : : so is the power applied to the wheel : to the weight 
suspended to the axle. 

EXAMPLE. 

1. It is required to make a windlass in such a manner, 
that 1 Ib. applied to the wheel, shall be equal to 12 lb. sus- 
pended to the axle; now allowing the axle to be 4 inches 
diameter, what must be the diameter of the wheel ? 

lb. in. lb. in. 
As 1 : 4 : : 12 : 48 = 4 feet the diameter of the wheel. Ans. 

2. Suppose the diameter of an axle to be 6 inches, and 
that of the wheel 5 feet; what power at the wheel will bal- 
ance 10 lb. at the axle? Ans. 1 lb. 

3d. OF THE SCREW. 

In the screw there are four things to be considered : viz. 
the power, the weight, the distance between the threads, and 
the circumference. To find any one of these, the other 
three being given, observe the following proportional 

RULE. 

As the distance between the threads of the screw : 
Is to the circumference : : 
So is the power : 
To the weight. 

Note. 1. To find the circumference of the circle described by the 
end oi the lever ; multiply the double of the lever by 3,14150, and the 
product will be the circumference. 

2 It is usual to abate of the effect of the machine for the friction 



192 PROMISCUOUS QUESTIONS. 

EXAMPLE. 

There is a screw whose threads are an inch asunder ; the 
lever by which it is turned is 36 inches long, and the weight 
to be raised a ton, or 2240 Ib. ; what power or force must 
be applied to the end of the lever sufficient to turn the screw 
that is to raise this weight ? 

Thus, the lever 36 X 2=72,and72 X 3,14159^226,194 + 
the circumference. 

Circum. in. Ib. Ib. 

Then, as 226,194 : 1 :: 2240 : 9,903 the power. Ans. 



A COLLECTION OF PROMISCUOUS QUESTIONS, TO EXERCISE 
THE SCHOLAR ON THE FOREGOING RULES. 

1. What is the sum of 2578, added to itself? 

Ans. 5156. 

2. What is the difference between 14676, and the fourth 
of itself! Ans. 11007. 

3. There is the sum of 1468 dollars in three bags; the 
first contains 461, the second 581, how many are in the 
third bag? Ans. 426. 

4. What is the sum of the third and half third of 1 dol- 
lar ? Ans. 50 cts. 

5. What number is that which being multiplied by 45 the 
product will be 1080? Ans. 24. 

6. Required the quotient of the square of 476, divided 
by the half of itself, or its single power? Ans. 952. 

7. A general drawing up his army into a solid square, 
found he had 231 over and above, but increasing each side 
with one soldier, he wanted 44 to complete the square ; how 
many men did his army consist of? Ans. 19000. 

8. What number added to the cube of 21, will make tin- 
sum equal to 113 times 147 ? Ans. 7350. 

9. A person possessed of $ of a ship, sold f of his share 
for 1260 dollars; what was the value of the whole ship at 
Th" sani" rate? Ans. 5040 dolls. 

10. A jruJirdinn paid his ward 3500 dollars for 2500 dol- 
lars, uhieh he had in his hands for 8 years; what rate of 

fiU'rest did lie allow him 7 .!//*. 5 per cent. 



PROMISCUOUS QUESTIONS. 193 

11. A young man received 210 dollars, which was f of 
his elder brother's portion ; now three times the elder bro- 
ther's portion was half of the father's estate ; how much was 
the estate worth? Ans. 1890 dolls. 

12. A broker bought for his principal in the year 1720, 
the sum of 400 dollars capital stock, in the south sea, at 650 
per cent, and sold it again when it was worth but 130 dollars 
per cent. ; how much was lost upon the whole ? 

Ans. 2080 dolls. 

13. A gentleman went to sea at 17 years of age; 8 years 
after he had a son born, who lived 40 years, and died before 
his fether ; after whom the father lived twice 20 years, and 
then died also ; I demand the age of the father when he 
died? Ans. Ill years. 

14. A, B, and C, entered into partnership in trade, A put 
in a sum unknown, B put in 20 pieces of cloth, and C put 
in 500 dollars ; at the end of one year they had gained 1000 
dollars, whereof A received 350 dollars for his share, and B 
400 dollars ; required C's share, how much A put in, and the 
value of B's cloth? 

Ans. C's share 250 dollars, A put in 700 dollars, 
B's cloth was worth 800 dollars. 

15. A captain and 160 sailors took a prize worth 2720 
dollars, of which the captain gets J for his share, and the 
rest is equally divided among the sailors ; what was each 
one's part ? 

Ans. The captain gets 544 dollars, and each sailor 
13 dollars 60 cents. 

16. A lady tells her husband, upon her marriage, that her 
fortune, the interest of which for one year at 6 per cent, 
was 972 dollars, was but the f of the interest of her father's 
estate for three years, at the same rate per cent. ; what was 
the lady's fortune, and what was the value of her father's es- 
tate?" 

Ans. Her fortune was 16,200 dollars, and her father' 
estate was 150,000 dollars. 

17. A stone measures 4 feet 6 inches long, 2 feet 9 inches 
broad, and 3 feet 4 inches deep ; how many cubic feet does 
it contain ? Ans. 41 feet 3 inches. 

18. Suppose I of a' mast or pole stands in the ground, 
12 feet in tne water, and f of its length above the water; 
what is its whole length? Ans. 216 feet. 

R 



194 PROMISCUOUS QUESTIONS. 

19. A gentleman being asked his age, answered, my 
grandfather is 112 years old, and my father 4 o f his age, 
whilst mine is but ^ of my father's ; what was his age ? 

Ans. 21 J years. 

20. A person who was possessed of f share of a copper 
mine, sold 3 of his interest therein for 1710 dollars; what 
was the value of the property at the same rate ? 

Ans. 3800 dollars. 

21. There are two numbers, the one 63, the other half as 
much ; required the product of their squares, and the differ- 
ence of their product and sum 1 

' A $ Product of the squares 3938240,25. 
5 * J Difference 1890. 

22. Two men set out at the same time from the same 
place, but go contrary ways, and each of them travels 34 
miles a day ; required the time in which they will have 
travelled 2000 miles ? Ans. 29 days 9 hours 52ff mi. 

23. If a cannon may be discharged twice with 6 Ib. of 
powder, how many times will 7Cwt. 3qr. I7lb. discharge the 
same piece ? Ans. 295 times. 

24. What number is that, to which if you add f of itself, 
the sum will be 20? Ans. 12. 

25. What number is that, which being divided by j}, the 
quotient will be 21 ? A?is. 15J. 

26. What number is that, which being multiplied by 15, 
the product will be \ 1 Ans. -^\ 

27. W T hat number is that, from which if you take f , the 
remainder will be J? Ans. ff 

28. A gentleman wishing to distribute some money among 
a number of children, found he wanted 8 cents to give them 
:J cents a piece, he therefore gave each 2 cents, and had 
three cents left; how many children were there? Ans. 11. 

29. In what time will 500 dollars amount to 1000, at 6 
per cent, per annum? Ans. 16 years 8 months. 

30. When \ of the members of congress were assembled 
-t 15, there were J-flO absent; how many members were 
in all? Ans. 150. 

31. If the earth be 360 degrees round, each 69 miles, 
how long would it take a man to travel orce round, at 20 
miles a day, admitting there were no obstacles in the way, 
and reckoning 365 i days in the year. 

Ans. 3 years 155J days. 



PROMISCUOUS QUESTIONS. 195 

32. -What is the mean time for paying 100 dollars a; 3| 
months, 150 dollars at 4^ months, and 204 dollars at 5if 
months ? Ans. 4 months 23Jf A days. 

33. If A can do a piece of work alone in 7 days, and B 
do the same in 12, how long will it require them both to- 
gether ? Ans. 4 T V days. 

34. A minor of 14 years of age, had an annuity left him 
of 400 dollars ; this sum his guardian agreed to receive 
yearly, and allow him compound interest at 5 per cent, 
thereon, till he should arrive at 21 years of age ; how much 
must he then receive ? Ans. 3256 dolls. 80 -f cents. 

35. Sold goods to the amount of 700 dollars for four 
months ; what was the present worth, at 5 per cent, simple 
interest? Ans. 688 dolls. 52 -f cents. 

36. Three persons, A, B, and C, purchased a lot in part- 
nership, for which A advanced f , B -f , and C 140 dollars ; 
what sum did A and B pay, and what part of the lot be- 
longed to C ? 

C A paid 267 dolls. 27 + cts. 
Ans. < B paid 305 45J 
( and C had J-J parts. 

37. A gentleman finding several beggars at his door, 
gave to each four cents, and had sixteen left ; but if he had 
given to each six cents, he would have wanted 12 ; how 
many beggars were there ? Ans. 14. 

38. B and C can build a wall in 18 days, but with the 
assistance of A they can do it in 11 days; in what time 
can A do it alone 1 

Suppose the work to consist of 198 parts. 

Then 198-^18=11 parts performed by B and C, in one 
day. 

Again, 198-^11 = 18, performed by A, B, and C, in one 
day. 

But 18 11=7 parts performed by A alone. 
P. D. P. D. h. m. 

And as 7 : 1 : : 198 : 28 3 25f Ans. 

39. Twenty members of congress, 30 merchants, 24 law- 
yers, and 24 citizens, spent at a dinner 192 dollars ; which 
sum was divided among them in such a manner, that 4 
members of congress paid as much as 5 merchants, 10 
merchants as much as 16 lawyers, and 8 lawyers as much 



196 PROMISCUOUS QUESTIONS. 

as 12 citizens; the question is to know the sum of. money 
paid by all the members of congress ; also, by the mer- 
chants, lawyers, and citizens ? 

Ans. The 20 members of congress paid 60 dollars 

the 30 merchants paid 72, the 24 lawyers paid 36 

and the 24 citizens paid 24. 

40. What difference is there between a piece of ground 
28 perches long, by 20 broad, and two others each of half 
those dimensions ? Ans. 1 acre 3 qrs. 

41. Required the dimensions of a parallelogram, con- 
taining 200 acres, which is 40 perches longer than wide ? 

Ans. 200 perches by 160, 

42. How many acres are contained in a square field, the 
diagonal of which is 20 perches more than either of its 
sides? Ans. 14 acres 2 qrs. 11 per. 

43. The paving of a triangular yard, at I8d. per foot, 
came to 100Z. ; the longest of the three sides was 88 feet ; 
what then was the sum of the other two equal sides 1 

Ans. 106,85 feet. 

44. Required the length of a line by which a circle that 
shail contain just half an acre may be laid off? 

Ans. 21% yards. 

45. A ceiling contains 114 yards 6 feet of plastering, and 
the room is 28 feet broad ; what is its length ? 

Ans. 36f feet. 

46. A common joist is 7 inches deep, and 2 thick, but I 
want another just as big again, that shall be three inches 
thick ; what must be its^ether dimensions ? 

Ans. llf inches. 

47. If 20 feet of iron railing weigh half a ton, when the 
oars are an inch and a quarter square, what will 50 feet 
come to at 3 \d. per pound, the bars being but of an inch 
square ? Ans. 20Z. Os. 2d. 

43. A may-pole whose top being broke off by a blast of 
\vind, struck the ground at 15 feet distance from the foot of 
the pole ; what was its whole height, supposing the length 
of the broken piece to be 39 feet ? Ans. 75 ft. 

49. Required a number, from which if 7 be subtracted, arid 
the remainder be divided by 8, and the quotient be multiplied 
by 5, and 4 added to the product, the square root of the sum 
extracted, and three-fourths of that root cubed, the cube di- 
vided by 9, the last quotient will be 24 ? Ans. 103. 



, PROMISCUOUS QUESTIONS. 197 

50. A vintner has a cask of wine containing 500 galls, of 
which he draws 50 galls, and fills it up with water, and re- 
peats the same thing five times ; I demand what quantity of 
wine, and also of water, is then in the cask ? 

Ans. 295 galls. 1 qt. of wine, and 204 galls. 3 qts. 
of water nearly. 

51. Since a pile of wood 4 feet long, 4 feet high, and 8 
feet broad, makes a cord, what part of a cord will be in a 
pile of half the dimensions each way ? Ans. % part. 

The answers to the following questions are designedly omitted, that 
the scholar may be induced to apply to the resources of his own mind 
alone for the solution thereof. Without habits of reflection and inves- 
tigation are acquired, by which he can compare, examine and apply 
the various rules and directions that are contained in this treatise, he 
never can have any good claim to be considered a proficient in arithmetic 

52. A owed B 1864 dollars, for which he gave his note, 
on interest, bearing date April 1st, 1817. 

On the back of the note are the following endorsements*, 
viz. 

Oct. 15th, 1817. Received in cash 225 dolls. 50 cts. 

Jaiv 10th, 1818. Received in cash 150 

Same date, one bag of coffee ; weight 1 Cwt. 22lb. at 29 
cents per pound. 

May 16th. Received 3 ton of iron at 195 dolls, per ton. 

What is the sum due from A to B, on the 1st of August 
1818? 

53. How many cords are there in a pile of wood 36 feet 
long, 6| feet wide, and 8J feet high? 

54. If a man spends 356 dollars 34 cgnts per year, how 
much will it be per day ? 

55. A bankrupt, whose whole property is worth 2564 
dollars 95 1 cents, can pay his creditors but 18 J cents on a 
dollar ; how much does he owe ? 

56. If 8 men spend 20 dollars 50 cents in 30 days, how 
long will 64 men be in spending 100 dollars at the same 
rate ? 

57. A bridge built over a stream in 6 months by 34 men, 
being washed away by a flood, how long time will it take 86 
men to build another in its place, of twice as much work ? 

58. Three gardeners, A, B, and C, having bought a piece 
of ground, find the profits of it to amount to 240 dollars a 
year ; now the sum of money which they gave, was in such 

R 2 



198 PROMISCUOUS QUESTIONS. 

proportion, that as often as A paid 5 dolls. B paid 7, and as 
often as B paid 4 dolls. C paid 6 ; how much must each 
man receive for his share of the profits per annum? 

59. If a county tax of 7 cents and 3 mills per cent, is as- 
sessed on property, how much must that man pay, whose 
property is valued at 8564 dollars 20 cents ? 

60. Suppose a cistern having a pipe which conveys 4 gal- 
lons 2 quarts into it in an hour, and has another that lets 
out 2 gallons 2 quarts and 1 pint in an hour ; in what time 
will it be filled, allowing it to contain 84^ gallons? 

61. What is the length of a lane, which, being 36 fcul 
wide, will contain just one acre of ground ? 

62. If 50 men consume 12 bushels of grain in 30 days, 
how much will 40 men consume in 90 days ? 

63. A gentleman had 18 dollars 90 cents to pay among 
his laborers ; to every boy he gave 6 cents, to every woman 
S cents, and to every man 16 cents; now there were three 
women for every boy, and two men for every woman ; re- 
quired the number of each ? 

64. Two men depart from the same place, and travel the 
same way ; the one travels at the rate of 3 miles an hour, 
for 8 hours every day ; the other goes at the rate of 4J 
miles, for 7 hours each day ; how far are they apart at the 
end of 13 days? 

65. A began to trade on the 1st of January, with a capi- 
tal of 962 dollars ; on the 15th of April following, he took 
in B as a partner, with 1635 dollars ; on the 1st of July, A 
put in 320 dollars more, and 1 month after B drew out i 
of his capital ; on the last day of December, on settling their 
.accounts, they found a gain of 486 dollars 64 cents'; what 

was each partner's share ? 

66. Suppose the Ohio river to be 2500 feet wide, 6 feet 
deep, and runs at the rate of 3 miles an hour ; in what time 
will it fill a cistern of two miles in length, breadth, and 
depth, the mile being 5280 feet? 

67. A sloth was observed climbing a tree at the rate of 
9^ inches every day, but during the night slipped down 6J 
inches ; how long will it be in reaching a limb 45 feet 6 
inches from the ground! 

68. In an orchard of fruit trees, of them bear applet, 
| peaches, cherries, plums, and 46 are pears; how 
many trees does the orchard contain ? 

69. An old soldier lately received a sum of money as 



PROMISCUOUS QUESTIONS. 199 

pension from government : of this sum he paid 94 dollars 
in the payment of debts which he then owed, half of what 
remained he lent to a friend, and the fifth he gave for a suit 
of clothes ; he then found that nine-tenths of his money was 
gone ; what sum did he at first receive ? 

70. What number is that, of which the difference between 
its third and fourth parts is 84 ? 

71. In turning a chaise within a circle of a certain 
diameter, it was discovered that the outer wheel turned 
thrice, while the inner turned twice; now supposing the 
axle-tree 4 feet long, and the wheels of an equal size, the 
length of the circumference described by each wheel is re- 
quired ? 

72. The sum of the sides of an equilateral triangle is 125 
feet ; required the area thereof'/ 

A, in a scuffle, seized on f of a parcel of sugar-plums; B catched three- 
eighths of it out of his hands, and C laid hold on three-tenths more, D ran 
off with all that A had left except one-seventh, which E afterwards secured 
slily for himself: then A and C jointly set upon B, who in the conflict let 
fall he had, which was equally picked up by D and E : B then kicked 
down C's hat, and to work they went anew for what it contained ; of which 
A got i, B A, D two-sevenths, and C and E equal shares of what was left 
of that stock: D then struck f of what A and B last acquired out of their 
hands ; they with difficulty recoved five-eighths of it in equal shares again, 
but the other three carried off one -eighth apiece of the same. Upon this 
they called a truce, and agreed, that the J- of the whole, left by A at first, 
should be equally divided among them How much of the prize, after this 
distribution, remained with each of the competitors ? 

Ans. A got 2863, B 6335, C 2438, D 10294, and E 4950. 
Solution. 

First, f of f = J B's ^ First acquisition 
And y\of |=i C's \ ^ their sum 

Then -| ^=H, or ^ left 
I of if J^- E's first acquisition. 
Also, T V -o = 'iTo D' s * Thus ended the 1st heat. 
Again, i of J = J B's "j 
Retained 1 C's ! Part, at the end of the 

And T Vo+TV=Ht I) ' s | second scuffle. 

Also -V3_. + T i_ = _i_5_L. E > s J 

Proceeding, i of 1 = ^ A's " 

f of !=*'! 

Then ^V + rV + A= T^V to be taken 
from C's. Thus, ^-^ 
and i of T u_ '=j&C 9 a 

And 



'>* 

Their situation 
at the end of 
the 3d attack. 



200 QUESTIONS FOR EXAMINATIONS, 

A. B. 

Further, ^ + ^=J^ and f of &=& lost by A and B. 
Then, t V of ft + iof & ^^i^A's^ p f 
' ' " 



Also, A of /, + i of 

And * of W + riV =T*tt* C's 

i of A+W* 

* Of T'O +TWo 

* Of * = T ' ? 




_.__l_l_ _ _S_4_3J o 

l~3*0 Tl 5 26880 ^ f n ff ot j n of 



r A got 2863 ^ 



So that if the } 6335 

sugar-plums > then <j C 2438 I Arts. 

were 26880, ) B 10294 

IE 4950 J 



QUESTIONS FOR EXAMINATION. 

THIS collection of questions is designed to assist the teacher in l.ie ex- 
amination of his scholars. It will contribute very much to the progress of 
scholars, to assign them a certain number of these questions as lessons, to 
be answered correctly and with facility. Many similar questions will no 
doubt, from time to time, occur to the mind of the teacher, on thf different 
sections, as the scholar proceeds. By accustoming his pupils to answer 
such with ease, not only will his, own burden in teaching be lei^ened, but 
the parents of children, who have been intrusted to his care, will find that 
neither their trouble or expense has been in vain. 

PART I. 

What is Arithmetic ? How many parts does it consist of? 

What are the characters used in arithmetic ? 

\Vhat is numeration ? How are the digits divided ? 

What is the rule for writing numbers? What is simple addition ? 

How do you place numbers to be added ? How is the sum or amount of 
each column to be set down ? Why do you carry at 10, rather than for 
any other number? How is addition proved? What is simple subtrac- 
tion? How must the given numbers be placed ? 

How is subtraction performed ? How is subtraction proved ? 

What is simple multiplication? What are the numbers called? 

In what order are the numbers in multiplication to be placed ? 

IIoxv many cases are there in multiplication? How is the operation to be 
performed in the first and second cases? How is multiplication proved ? 

When there are ciphers on the right-hand of either of the factors, how do 
you proceed ? What is simple division ? 

Wnat are the given numbers called ? How are they to be placed ? 



QUESTIONS FOR EXAMINATION. 201 

How many cases are there in division? 

How is division performed in each case ? When the multiplier is the exact 
product of any two factors, how do you proceed ? 

When there are ciphers on the right of the divisor, how do you proceed ? 

When the divisor is 10, 100, 1000, &c. how do you proceed? 

How is division proved ? 

PART II. 

Federal money, why so called ? 

What are its denominations, and standard weights ? 

How is addition, subtraction, multiplication, and division severally per- 
formed in federal money ? What is compound addition? 

How is compound addition performed \ How is it proved ? 

What are the denominations of English money, and how are they valued ? 

\Vhat articles is troy weight used for? What are its denominations and 
how valued ? What articles is avoirdupois weight used for? 

What are its denominations, and how valued ? What is apothecaries' 
weight used for? What are its denominations and how valued? 

What are the denominations of cloth measure, and how valued ? 

What are the denominations of long measure, and how estimated ? 

What are the denominations of land measure, and how rated ? 

What is cubic measure, what are its denominations, and relative difference ? 

What are the denominations of time, and what their relative differences ? 

What is the exact length of the solar year? 

What are the denominations of motion, and the relative difference? 

For what is liquid measure used, what its denominations, and relative di 
ference ? What are the denominations of dry measure, w 7 hat used for 
and how estimated ? 

What is compound subtraction, and how performed ? 

What is compound multiplication, how many cases, and how performed? 

What is compound division, how many cases, and how performed ? 

What is reduction, and how performed ? How is reduction proved ? 

How are pence reduced to cents, Penn. currency ? How are pounds, shil- 
lings, and pence, reduced to dollars, Pennsylvania currency? 
PART III. 

What is decimal arithmetic, and how distinguished from whole numbers? 

What is the decimal point called? 

What effect has ciphers, placed on the right-hand of the integer, and what 
effect when placed on the left-hand ? llow is addition of decimals per- 
formed ? How is subtraction of decimals performed ? 

How is multiplication of decimals performed ? How is division of decimals 

* performed ? How many cases in reduction of decimals? 

How is a vulgar fraction reduced to a decimal ? 

How are numbers of different denominations reduced to a decimal of 
equal value? How are decimals reduced to their equal value in in- 
tegers ? 

PART IV. 

What is proportion ? Into how many parts is it divided ? 

What are the given terms in proportion called ? 

Wha f is required in the single rule of three direct ? 

How may you know when the question is in direct proportion? 

What is the rule for stating questions in the single rule of three direct? 

How is the operation performed ? How do you prove questions in the single 
rule of three direct ? What is the single rule of three inverse ? 

How may you know when the question is in the single rule of three inverse ? 

How is the operation performed in the single rule of three inverse ? How 
are questions proved in this rule ? What is the double rule of three ? 

How many, and which terms must be a supposition, and how many, and 
which must be a demand ? 

What is the rule for stating questions in the double rule of three ? 

[low is the operation performed in the double rule of three direct? 



202 QUESTIONS FOR EXAMINATION. 

How do you know when the question is in direct proportion, and when in 
inverse? How is the operation performed in inverse proportion ? 
PART V. 

What is practice, and why so called ? How many cases are there in practice ? 

When the price consists of dollars, cents, ana mills, how is the operation 
performed ? When the price is the fractional part of a dollar or cent, 
how is the operation performed ? When the price and quantity given 
are of several denominations, how is the operation performed ? 

When the price consists of pounds, shillings, pence, and farthings, how do 
you proceed ? What is meant by aliquot parts? 

When both the price of the integer and the quantity are of different de- 
nominations, how do you proceed ? 

What is tare and tret, and what is gross and neat? 

How do you work questions in tare and tret? What is interest? 

What is the general rate of interest? What is the sum of money loaned, 
called? What do you understand by the amount? 

How many kinds of interest are there ? What is simple interest ? 

How many cases are there in simple interest? When the given time is 
years and the principal dollars, how is the interest found ? 

When there are cents and mills in the principal, how do you proceed ? 

When the lime is years and months, or months *only, how is the interest 
found ? When the time is months and days, or days only, how is the ope- 
ration performed? How is the interest computed on bonds, notes, &C;? 

What is compound interest, and how is it performed ? What is insurance ? 

What is the instrument of agreement termed ? 

How are the questions in insurance performed ? 

What is commission, and how performed ? What is brokage, and how per- 
formed ? What is stock, and how bought and sold ? 

What is rebate or discount, and what the rule to work questions therein ? 

What is the difference between discount and interest? 

What is bank discount, and how is the discount calculated ? 

What do you mean by the equation of payments? 

How is the mean time found in the equation of payments ? 

What is fellowship, and how many kinds are there ? 

What is single fellowship, and how is the operation performed ? 

What is compound fellowship, and what is the rule for working questions 
therein ? What is profit and loss, and what is the rule of operation 
therein ? What is barter, and how performed ? 

What is exchange, ami of how many kinds ? 

What do you understand by par in exchange, and what by agio? 

How do you reduce the currency of different states to federal money? 

How do you reduce the currency of one state to another, where it is differ- 
ent in them? How are accounts kept in England, Ireland, and how in 
France, Spain, &c. ? What is alligation, and how many cases are therein ? 

How are the operations performed in the first case, second case, &c. ? 
PART VI. 

What is a vulgar fraction, and how many kinds are there ? 

What is a proper fraction, what is an improper fraction, what is a com- 
pound fraction, and what is a mixed fraction? 

What are the numbers above the fine called, and also those below? 

How do you reduce vulgar fractions to their lowest terms ? 

EIow are mixed numbers reduced to an improper fraction ? 

How is an improper fraction reduced to a whole or mixed number? 

How do you reduce fractions to others that shall have a common denomina- 
tor? How do you find the least common denominator? 

How do you find the value of a fraction, in the known parts of an integer? 

How are given quantities reduced to the fraction of a greater denomination ? 

How are vulgar fractions reduced to decimals of the same value ? 

How do you reduce a compound fraction to a single one? 

How are vulgar fractions added, subtracted, multiplied, and divided? 



QUESTIONS FOR EXAMINATION. 203 

How do you perform the single rule of three in vulgar fractions, direct, 
and inverse ? 

PART VII. 

What is involution ? What do you understand by the power of a number ? 

ilow is involution performed ? What is the number denoting the power, 
termed ? What is evolution ? What do you understand by a root ? 

How do you extract the square root ? 

If there be decimals in the given number, how must it be pointed? 

How do you extract the square root of a vulgar fraction? 

How is the square root of a mixed number extracted ? 

Hew do you find the side of a right angled triangle, the other two being 
given f How do you find the side of a square, in any given area ? 

How do you find the diameter of a circle, when the area is given ? 

How do you prove the square root? What is a cube ? 

How do you extract the cube root in whole numbers? 

Ilow do you extract the cube of a vulgar fraction ? 

How do you point off in decimal numbers? How is the cube root of a 
mixed number extracted ? How is the cube root proved ? 

What is progression, and how many kinds ? What is principally to be ob- 
served in arithmetical progression ? How 7 do you find the last term ; and 
sum of all the terms ? How do you find the common difference ? 

vVhat is geometrical progression, and how does it differ from arithmetical ? 

What is principally to be observed in geometrical progression ? 

How do you find the last term, and sum of all the series ? 

What is position, and of how many kinds ? How do you resolve questions 
in singlo position? How is single position proved ? What is double po- 
sition ? What is the rule for working questions in double position ? 

What do you understand by permutation ? How is the number of varia- 
tions found in this rule? What do we learn from the results of this rule ? 

What is combination? How do you find the greatest possible number of 
combinations in any given number? 

PART Vlli. 

VVhat are duodecimals? What are the denominations in duodecimals, and 
what are they termed? How are duodecimals added, subtracted, and 
multiplied ? How do you prove multiplication of duodecimals ? 

How is the solid content of bales, &c. found by duodecimals? 

How do you find a ship's tonnage ? What is the carpenters' rule, and what 
its use ? How do you find the superficial content of boards, &c. ? 

How do you find the solid content of squared timber? 

How do you find the solid content of round timber? 

What things belong to carpenters' work? By what numbers do carpenters 
usually measure their work ? How is brick- work estimated ? 

What is the standard thickness of a brick wall? 

How do you reduce a wall of a different thickness to a standard one ? 

How is masons' work measured ? What kind of measure is used for ma- 
terials? How is the solid content of walls calculated ? 

What is superficial measure ? How is plasterers' work divided ? 

In what manner is plasterers' work measured ? How is white-washing 
and coloring estimated ? How is pavers' work calculated ? 

How do painters compute their work ? In what manner is glaziers' work 
estimated ? How are the contents of squares calculated? 

In what way is the area of an oblong piece of ground ascertained ? 

How do you calculate triangular pieces of ground ? 

How do you calculate a piece of ground lying in the shape of an obiiquo 
parallelogram? How are pieces of ground, oounded by four irregular 
sides, calculated,? How do you calculate the area of a circle? 

What is gauging? How are the contents of casks, &c. calculated? 

How do you calculate the power of the lever? of the axle and wheel ? 

VVhat things are to be considered in finding the power of the screw? 

How do you find the proportion between the weight and the power? 



CONTENTS. 



PART I. 

P^?e 

Numeration 7 

Addition 9 

Subtraction 11 

Multiplication 13 

Division 17 

PART II. 

Federal Money 21 

Compound Addition 24 

Compound Subtraction 30 

Compound Multiplication .... 32 

Compound Division 35 

Reduction 39 

PART III. 

Addition of Decimals 46 

Subtraction of Decimals 47 

Multiplication of Decimals ... 48 

Division of Decimals 50 

Reduction of Decimals 52 

PART IV. 

Single Rule of Three Direct . . 56 
Single Rule of Three Inverse . 59 
Double Rule of Three 65 

PART V. 

Practice 68 

Tare and Tret 73 

Simple Interest 75 

Compound Interest 85 

Insurance, Commission, and 

Brokage 87 

Buying and Selling Stocks . . 89 

Rebate or Discount 90 

Bank Discount 92 

Equation of Payments 93 

Single Fellowship 95 

Compound Fellowship 97 

Profit and Loss 99 

Barter 101 

Exchange 102 

Alligation 109 



PART VI. 

},-.?< 

Vulgar Fractions 113 

Reduction of Vulgar Frac- 
tions '. ib. 

Addition of Vulgar Fractions 120 
Subtraction of Vulgar Fr<ic- 

tions ..; . 122 

Multiplication of Vulgar Frac- 
tions 123 

Division of Vulgar Frac- 
tions 124 

The Rule of Three in Vulgar 

Fractions 125 

Inverse Proportion 12G 

PART VII. 

Involution 127 

Evolution 128 

The Square Root ib. 

The Cube Root 137 

Arithmetical Progression .... 141 

Geometrical Progression .... 144 

Single Position 147 

Double Position 148 

Permutation 151 

Combination 152 

PART VIII. 

Duodecimals 153 

The Carpenters' Rule 159 

Measuring of Boards and 

Timber 162 

Carpenters' and Joiners' Work 166 

Bricklayers' Work 170 

Masons' Work 174 

Plasterers' Work 176 

Pavers' Work 177 

Painters' Work 178 

Glaziers' Work 180 

Measurement of Ground 181 

Gauging 188 

Mechanical Powers 190 

Promiscuous Questions 192 

Questions for Examination . . 200 



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