Charles Davies.

School arithmetic. Analytical and practical online

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should have one third, and the mother the remainder. Now it so
happened that they both returned : how mustthe estate be divided
to fulfill the father's intentions ?

71. Two persons depart from the same place, one travels 82,
and the other 36 miles a day : if they travel in the same direction,
how far will they be apart at the end of 19 days, and how far if
they travel in contrary directions ?

72. In what time will $2377.50 amount to $2852.42, at 4 per
cent, per annum ?

73. What is the height of a wall, which is 14^ yards in length,
and -fo of a yard in thickness, and which has cost $406, it having
been paid for at the rate of $10 per cubic yard ?

74. What will be the duty on 225 bags of coffee, each weighing
gross 160 Ibs., invoiced at 6 cents per Ib. ; 2 per cent, being the
legal rate of tare, and 20 per cent, the duty ?

75. Three persons purchase a piece of property for $9202 ; the
first gave a certain Bum ; the second three times as much ; and
the third one and a half time as much as the other two: what
did each pay ?

76. A reservoir of water has two pipes to supply it. The first
would fill it in 40 minutes, and the second in 50. It has likewise
a discharging pipe, by which it may be emptied when full in 25
minutes. Now, if all the pipes are opened at once, and the water
runs uniformly as we have supposed, how long before the cistern
will be filled?

77. A traveller leaves New Haven at 8 o'clock on Monday
morning, and walks towards Albany at the rate of 3 miles an
hour : another traveller sets out from Albany at 4 o'clock on the
same evening, and walks towards New Haven at the rate of 4
miles an hour ; now, supposing the distance to be 130 miles,
where on the road will they meet ?



MENSURATION. 303



MENSURATION.

315. A triangle is a portion of a plane
bounded by three straight lines. BC is
called the base ; and AD, perpendicular to
BC, the altitude.

316. To find the area of a triangle.
The area or contents of a triangle is equal

to the product of half its base by its altitude
(Bk. IV. Prop. VI).*

EXAMPLES.

1. The base, BC, of a triangle is 40 yards, and the perpendicu-
lar, AD, 20 yards ; what is the area ?

2. In a triangular field the base is 40 chains, and the perpendi-
cular 15 chains : how much does it contain ? (ART. 110.)

3. There is a triangular field, of which the base is 35 rods and
the perpendicular 26 rods : what are its contents ?




317. A? square is a figure having four equal sides,
and all its angles right angles.



318. A rectangle is a four-sided figure like a
square, in which the sides are perpendicular to each
other, but the adjacent sides are not equal.

319. A parallelogram is a four-sided figure
which has its opposite sides equal and parallel, but
its angles not right angles. The line DE, perpendi-
cular to the base, is called the altitude.



320. To find the area of a square, rectangle, or parallelogram,



Multiply the base by the perpendicular height, and the product
will be the area. (Book IV. Prop. V).

EXAMPLES.

1. What is the area of a square field of which the sides are
each 33.08 chains ?

2. What is the area of a square piece of land of which the
sides are 27 chains?

3. What is the area of a square piece of land of which the sides
are 25 rods each ?

* All the references arc to Davies' Legendre.




304: MENSURATION.

4. What are the contents of a rectangular field, the length of
which is 40 rods and the breadth 20 rods ?

5. What are the contents of a field 40 rods square ?

6. What are the contents of a rectangular field 15 chains long
and 5 chains broad ?

7. What are the contents of a field 27 chains long and 9 rods
broad ?

8. The base of a parallelogram is 271 yards, and the perpendi.
cular height 360 feet : what is the area ?

321. A trapezoid is a four-sided figure
ABCD, having two of its sides, AB, DC,
parallel. The perpendicular CE is called
the altitude.

322. To find the area of a trapezoid.

Multiply half the sum of the two parallel sides "by the alti-
tude, and the product will be the area. (Bk. IV. Prop. VII.)

EXAMPLES.

1. Required the area of the trapezoid ABCD, having given

AB=321.51/., DC=214.24/*., and CE=171.16/^.

2. What is the area of a trapezoid, the parallel sides of which
are 12.41 and 8.22 chains, and the perpendicular distance between
them 5.15 chains '?

3. Required the area of a trapezoid whose parallel sides are 25
feet 6 inches, and 18 feet 9 inches, and the perpendicular distance
between them 10 feet and 5 inches.

4. Required the area of a trapezoid whose parallel sides are
20.5 and 12.25, and the perpendicular distance between them
10.75 yards.

5. What is the area of a trapezoid whose parallel sides are 7.50
chains, and 12.25 chains, and the perpendicular height 15.40 chains V

6. What are the contents when the parallel sides are 20 and 32
chains, and the perpendicular distance between them 26 chains ?

323. A circle is a portion of a plane
bounded by a curved line, called the circum-
ference. Every point of the circumference is
equally distant from a certain point within
called the centre : thus, C is the centre, and
any line, as ACB, passing through the centre,
is called a diameter.

If the diameter of a circle' is 1, the circumference will be
3.1416. Hence, if we know the diameter, ^ce may find the circum-
ference by multiplying by 3.1416 ; or, if we know the circumference.,
we may find the diameter by dividing by 3.1416.




MENSURATION. 305

EXAMPLES.

1. The diameter of a circle is 4, what is the circumference ?

2. The diameter of a circle is 93, what is the circumference ?

3. The diameter of a circle is 20, what is the circumference ?

4. What is diameter of a circle whose circumference is 78.54 ?

5 What is the diameter of a circte whose circumference is
11052.1944?
6. What is the diameter of a circle whose circumference is 6850 ?

324. To find the area or contents of a circle.

Multiply the square of the diameter by the decimal .7854 (Bk. V.
Prop. XII. Cor. 2).

EXAMPLES.

1. What is the area of a circle whose diameter is 6 ?

2. What is the area of a circle whose diameter is 10?

3. What is the area of a circle whose diameter is 7 ?

4. How many square yards in a circle whose diameter is 3i feet ?

325. A sphere is a figure terminated
by a curved surface, ull the parts of which
are equally distant from a certain point
within called the centre. The line AB
passing through its centre C is called the
diameter of the sphere, and AC its radius.

o~6. To find the surface of a sphere,

Multiply the square of the diameter by
3.1416 (Bk. VIII. Prop. X. Cor).

EXAMPLES.

1. What is the surface of a sphere whose diameter is 12 ?

2. What is the surface of a sphere whose diameter is 7 ?

3. Required the number of square inches in the surface of a
sphere whose diameter is 2 feet or 24 inches.

327. To find the contents of a sphere,

Multiply the surface by the diameter and divide the product by 6;
the quotient mil be the contents. (Bk. VIII. Prop. XIV. Sch. 3.)

EXAMPLES

1. What are the contents of a sphere whose diameter is 12 ?

2. What are the contents of a sphere whose diameter is 4 ?

3. What are the contents of a sphere whose diameter is 14i7i. ?
4 What are the contents of a sphere whose diameter is Gfl. ?

20




306



MENSURATION.




328. A prism is a figure whose ends are equal
plane figures and whose faces are paralelograms.

The sum of the sides which bound the base is
called the perimeter of the base, and the sum of the
parallelograms which bound the solid is called the
convex surface.



329. To find the convex surface of a right prism,

Multiply the perimeter of the base by the perpendicular height, and
thegtroduct will be the convex surface (Bk. VII. Prop. I).

EXAMPLES.

1. What is the convex surface of a prism whose base is bounded
by five equal sides, each of which is 35 feet, the altitude being 26
feet?

2. What is the convex surface when there are eight equal sides,
each 15 feet in length, and the altitude is 12 feet ?

330. To find the solid contents of a prism.

Multiply the area of the base by the altitude, and the product will
be the contents (Bk. VII. Prop. XIV).

EXAMPLES.

1. What are the contents of a square prism, each side of the
square which forms the base being 15, and the altitude of the
prism 20 feet ?

2. What are the contents of a cube each side of which is 24
inches ?

3. How many cubic feet in a block of marble of which the
length is 3 feet 2 inches, breadth 2 feet 8 inches and height or
thickness 2 feet 6 inches ?

4. How many gallons of water will a cistern contain whose di-
mensions are the same as in the last example ?

5. Required the contents of a triangular prism whose height is
10 feet, and area of the base 350 ?



331. A cylinder is a figure with circular
ends. The line EF is called the axis or alti-
tude, and the circular surface the convex sur-
face of the cylinder.




MENSURATION.



307



332. To find the convex surface,

Multiply the circumference of the base by the altitude, and
the product ivill be the convex surface. (Bk. VIII. Prop. I.)

EXAMPLES.

1 What is the convex surface of a cylinder, the diameter of
whose base is 20 and the altitude 50 ?

2. What is the convex surfa'ce of a cylinder, whose altitude is
14 feet and the circumference of its base 8 feet 4 inches ?

3. What is the convex surface of a cylinder, the diameter of
whose base is 30 inches and altitude 5 feet ?

333. To find the contents of a cylinder,

Multiply the area of the base by the altitude : the product will be
the contents. (Bk. VIII. Prop. II).



EXAMPLES.

1. Required the contents of a cylinder of which the altitude is
12 feet and the diameter of the base 15 feet ?

2. What are the contents of a cylinder, the diameter of whoso
base is 20 and the altitude 29?

3. What are the contents of a cylinder, the diameter of whose
base is 12 and the altitude 30 ?

4. What are the contents of a cylinder, the diameter of whose
base is 16 and altitude 9 ?

5. What are the contents of a cylinder, the diameter of whose
base is 50 and altitude 15 ?



334. A pyramid is a figure formed by
several triangular planes united at the
same point S, and terminating in the
different sides of a plain figure as
ABCDE. The altitude of the pyramid
is the line SO, drawn perpendicular to
the base.



335. To find the contents of a pyramid,

Multiply the area of the base by one-third of the altitude.
(Bk. VII, Prop XVII).




308



MENSUEATlQJS'.



EXAMPLES.

1. Required the contents of a pyramid, of which the area of the
base is 95 and the altitude 15.

2 What are the contents of a pyramid, the area of whose base
is 260 and the altitude 24 ?

3. What are the contents of a pyramid, the area of whose base
is 207 and altitude 18?

4 What are the contents of a pyramid, the area of whose base
is 403 and altitude 30 ?

5. What are the contents of a pyramid, the area of whose base
is 270 and altitude 16?

6. A pyramid has a rectangular base, the sides of which are 25
and 12 : the altitude of the pyramid is 36 : what are its con-
tents ?

7. A pyramid with a square base, of which each side is 30, has
an altitude of 20 : what are its contents ?




336. A cone is a figure with a circular
base, and tapering to a point called the
vertex. The point C is the vertex, and the
line CD is called the axis or altitude.



337. To find the contents of a cone,

Multiply the area of the base ly one-third of tJie altitude.
(Bk. VIII. Prop. V.)

EXAMPLES.

1. Required the contents of a cone, the diameter of whose base
is 5 and the altitude 10.

2. What are the contents of a cone, the diameter of whose base
is 18 and the altitude 27 ?

3. What are the contents of a cone, the diameter of whose base
is 20 and the altitude 30 ?

4. What are the contents of a cone, whose altitude is 27 feet
and the diameter of the base 10 feet ?

5. What are the contents of a cone, whose altitude is 12 feet
and the diameter of its base 15 feet ?



GAUGING 309

GAUGING-.

338. The mean diameter of a cask is found by adding to tho
head diameter, two thirds of the difference between the bung and
head diameters, or if the staves are not much curved, by adding"
six-tenths. This reduces the cask to a cylinder. Then, to find
the solidity, we multiply the square of the mean diameter by the
decimal .7854 and the product by the length. This will give
the solid contents in cubic inches. Then, if we divide by 231,
we have the contents in gallons. (Art. 114).

Multiply the length by the square of the OPERATION.
mean diameter, then by the decimal .7854, Ixd 2 X '- 7 - 8 w 5 -^-
and divide by 231. ' J x d 2 x .0034.

If, then, we divide the decimal ,7854 by 231, the quotient car-
ried to four places of decimals is .0034, and this decimal multi-
plied by the square of the mean diameter and by the length of the
cask, will give the contents in gallons.

339. Hence, for gauging or measuring casks, we have the fol.
lowing

HULE. Multiply the length "by the square of the mean diameter ;
then multiply ly 34 and point off four decimal places, and the pro-
duct icill then express gallons and the decimals of a gallon.

1. How many gallons in a cask whose bung diameter is 36
inches, head diameter- 30 inches, and length 50 inches ?

We first find the difference of the diameters, OPERATION.

of which we take two thirds and add to the 3630= 6

head diameter. We then multiply the square 2 O f 6 = 4

of the mean diameter, the length and 34 3 Q()4-4. 4
together, and point off four decimal places

in the product. 34 =1156

2. What is the number of gallons in a -. Qr r 7
cask whose bung diameter is 38 inches, head lyo.O^roc.
diameter 32 inches, and length 42 inches ?

3. How many gallons in a cask whose length is 36 inches, bung
diameter 35 inches, and head diameter 30 inches ?

4. How many gallons in a cask whose length is 40 inches, head
diameter 34 inches, and bung diameter 38 inches?

5 A water tub holds 147 gallons ; the pipe usually brings in
14 gallons in 9 minutes : the tap discharges at a medium, 40 gal-
Jons in 31 minutes. Now, supposing these to be left open, and
the water to be turned on at 2 o'clock in the morning ; a servant
at 5 shuts the tap, and is solicitous to know at what time the tub
will be filled in case the water continues to flow.



310



APPENDIX,



FORMS RELATING TO BUSINESS IN GENERAL.



FORMS OF OKDERS.

MESSRS. M. JAMES & Co.

Please pay John Thompson, or order, five hundred
dollars, and place the same to my account, for value received.

PETER WORTHY.
Wilmington, N. 0., June 1, 1855.

MR. JOSEPH RICH,

Please pay, for value received, the bearer, sixty-one
dollars and twenty cents, in goods from your store, and charge the
same to the account of your

Obedient Servant

JOHN PARSONS.
Savannafi, Ga., July 1, 1855.



FORMS OF RECEIPTS.

Receipt for Money on account.

Received, Natchez, June 2d, 1855, of John Ward, sixty dollars
on account.

$60,00 JOHN P. FAY.

Receipt for Money on a Note.

Received, Nashville, June 5, 1856, of Leonard Walsh, six hun-
dred and forty dollars, on his note for one thousand dollars, dated
New York, January 1, 1855.

$640,00 J. N. WEEKS.

NOTES.

1. A NOTE, or as it is generally called, a promissory note, is a
positive engagement, in writing, to pay a given sum at a time
specified, either to a person named in the note, or to his order, or
to the bearer.

2. By mercantile usage a note does not really fall due until the
expiration of 3 days after the time mentioned on its face. The
three additional days are called days of grace.



APPENDIX. 311

When the last day of grace happens to be Sunday, or a holiday,
such as New Years, or the Fourth of July, the note must be paid
the day before : that is, on the second day of grace.

3. There are two kinds of notes discounted at banks : 1st. Notes
given by one individual to another for property actually sold
these are called business notes, or business paper. 3d. Notes made
for the purpose of borrowing money, which are called accommo-
dation notes, or accommodation paper. Notes of the first class are
much preferred by the banks, as more likely to be paid when they
fall due, or in mercantile phrase, " when they come to maturity."

FORMS OP NOTES.

No. 1. Negotiable Note.



$25,50. . Providence, May 1, 1856.

For value received I promise to pay on demand, to Abel
Bond, or order, twenty-five dollars and 50 cents.

REUBEN HOLMES.



Note Payable to Bearer.
No. 2.



$875,39. St. Louis, May 1, 1855.

For value received I promise to pay, six months after
date, to John Johns, or bearer, eight hundred and seventy-five
dollars and thirty-nine cents.

PIERCE PENNY.



Note by two Persons.
No. 3.



$659,27. Buffalo, June 2, 1856.

For value received we, jointly and severally, promise to
pay to Richard Ricks, or order, on demand sis hundred and fifty-
nine dollars and twenty-seven cents.

ENOS ALLAN.

JOHN ALLAN.



Note Payable at a Bank.

$20,25. Chicago, May 7, 1856.

Sixty days after date, I promise to pay John Anderson,
or order, at the Bank of Commerce in the city of New York,
twenty dollars and twenty-five cents, for value received.

JESSE STOKES.



312 APPENDIX.



REMARKS RELATING TO NOTES.

1. The person who signs a note, is called the drawer or maker
of the note ; thus, Reuben Holmes is the drawer of Note No. 1.

2. The person who has the rightful possession of a note, is
called the holder of the note.

3. A note is said to be negotiable when it is made payable to
A B, or order, who is called the payee (see No. 1). Now, if Abel
Bond, to whom this note is made payable, writes his name on the
back of it, he is said to endorse the note, and he is called the en-
dorser ; and when the note becomes due, the holder must first
demand payment of the maker, Reuben Holmes, and if he declines
paying it, the holder may then require payment of Abel Bond, the
endorser.

4 If the note is made payable to A B, or bearer, then the
drawer alone is responsible, and he must pay to any person who
holds the note.

5. The time at which a note is to be paid should always be
named, but if no time is specified, the drawer must pay when re-
quired to do so, and the note will draw interest after the payment
is demanded.

6. When a note, payable at a future day, becomes due, and is
not paid, it will draw interest, though no mention is made of inter-
est.

7. In each of the States there is a rate of interest established by
law, which is called the legal interest, and when no rate is speci-
fied, the note will always draw legal interest. If a rate higher
than legal interest be taken, the drawer, in most of the States, is
not bound to pay the note.

8. In the State of New York, although the legal interest is 7
per cent, yet the banks are not allowed to charge over G per cent,
unless the notes have over 63 days to run.

9. If two persons jointly and severally give their note, (see No.
3,) it may be collected of either of them.

10. The words "For value received" should bo expressed in
every note.

11. When a note is given, payable on a fixed day, and in a spe-
cific article, as in wheat or rye, payment must be offered at the
specified time, and if it is not, the holder can demand the value in
money.

A BOND FOR ONE PERSON, WITH A CONDITION.

KNOW ALL MEN BY THESE PRESENTS, THAT I, James
Wilson of the City of Hartford and State of Connecticut, am held
and firmly bound unto John Pickens of the Town of Waterbury,
County of New Haven and State of Connecticut, in the sum of



APPENDIX.



313



Eighty dollars lawful money of the United States of America, to
be paid to the said John Pickens, his executors, administrators, or
assigns : for which payment well and truly to be made J bind
myself, my heirs, executors, and administrators, firmly by these
presents. Sealed with my Seal. Dated the Ninth day of March,
one thousand eight hundred and thirty-eight.

THE CONDITION of the above obligation is such, that if tlio
above bounden James Wilson, his heirs, executors, or administra-
tors, shall well and truly pay or cause to be paid, unto the above-
named John Pickens, his executors, administrators, or assigns, the
just and full sum of

[Here insert the condition.]

then the above obligation to be void, otherwise to remain in full
force and virtue.



Sealed and delivered in
the presence of

John Frost, )
Joseph Wiggins,)



James Wilson.




NOTE. The part in Italic to be filled up according to circum-
stance.

If there is no condition to the bond, then all to be omitted after
and including the words, " THE CONDITION, &c."



BOOK-KEEPING.

PERSONS transacting business find it necessary to wiite down
the articles bought or sold, together with their prices and the
names of the persons to whom sold.

BOOK-KEEPING is the method of recording such transactions in a
regular manner.

COMMON ACCOUNT BOOK.

The following is a very convenient form for book-keeping, and
requires but a single book. l"c is probably the best form of a com-
mon Account Book.



J. BELL. DR J. BELL. CR.


1846.




$


c.


1846.




*|.


June 1

11 6
July 9


To 5 cords of wood,
at $1,75 per cord,
To 1 day's work,
To 4bn. of rye, at 62
cents per bu.


8
1

2


75

00

48


July (
" 1C
u 20
Aug. 1


By shoeing horse,
u mending sleigh,
" ironing wagon,
" Cash to balance,


100

325
512
386






12


23






12i23



314



ANSWERS.





p.


EX.


ANS.


EX.


ANS


EX


ANS.


EX.


ANS.


24.
24.


9

10


577
7689


11

12


502616
799999


13
14


43 cts.
73 cts.


15




|888






20.
25.


17

18


4083
6846


19
20


9798
8601


21

22


7032
979


23


559






2Ve

27.
27.
27.


5
6

7
8


12089
26901
28637
203933


9
10
11
12


23272 -
233642
247481
1994439


13
14
15
16


175874
172775
98967
10742750


28.
28.
28.


20
21
22


787676921
100570011
15371781930


23
24
25


26754
730528

7047897


2fc

27


25687540
297303078


29.
29.
29.


28
29
30


13115375
3942805S
140700034


31
32
33


1819857171537
1105354
1079167


34




1118969






30.|| 1


365 || 2


5567 ||3| 16375||4|421||5|392||6


34671660


31.
31.
31.


7
8
9


82869
2576406
270


10
11


4596-119*
J4239052<
( 453090S


) 12
) 13
>




1287462
1665400






32.
32.
32.


14

15
16


50994
143985
2728116


17
18
19


5990267
6644374
7685134


20
21


23191876
23191876








37.
37.
37.
37.


9
10
11
12


260822
2935621
50391719
28443


13
14
15
16


99246591
999999
776462
18561747


17

18
19




4244083
8013105

52528






38.
38.
38.


1
2
3


10 -
45
$1115


4 23^

5

5 6


t
f


7
8
9


62

785608
37


10
11
12


175502
696

2687


39.
39.
39.
39.


13
14
15
16


250-$1500

26
1860805


17
18
19
20


239

1759
55


21
22
23
24


190
$4020-1340
2769818
94



ANSWERS.



315



p.


EX.


ANS.


EX.


ANS.


EX.


ANS


EX.


ANS.


40.
40.


25
26


145

168


27
28


168
137


29
30


15914260
20463760


31


2769818



40.|| 1 | 29045


| 2


$418


! 3 | $714


4 | $5795


41.


5


$390


fc


! $919


11


230-527


14


11854617


41.


^


$224980


{


) 55


12


19553068












41.


7


$1706


1C


) 28223


13


$3818












47.


9 936


11


$298


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$28511


_


47.


10 $1236


12


35688


14


$6578







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3


7913576


12


65948806


21


764819895290424


49.


4


2537682


13


36914176


22


6241519790


49.


5


4280822


14


85950000


23




105062176


49.


6


19014604


15


3320863272


24




601380780


49.


7


85564584


16


816515040


25


4984155396


49.


8


2183178497


17


68959488


26




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49.


9


93939864472


18


35843685


27




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49.


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395061696


19


267293339604


28


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49.


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51.


2 274032


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553248



52.
52.
52.
52.
52.
52.


1
2
3
4
5
6


2540
64800
7987000
98400000
375000
67040000


7
8
1
2
3
4


214100
87200000
1833600
4368560000
148512000000
1315170000000


5
6
7
8
9


25
196

10
521


9175000000
0310474010
1484000
9215040000
0018850000








53. |


1|480||2|4415||3|168||4


$291 || 5| 2214-123 || 6 | 11680


54.
54.
54.
54.


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Online LibraryCharles DaviesSchool arithmetic. Analytical and practical → online text (page 22 of 24)