3. Three merchants gained, by trading, $1920, of which A took a certain sum B took three times as much as A, and C four times as much as B; what share of the gain had each? A. A's share was $120; B's, $360, and C's, $1440.

4. A person, having about him a certain number of crowns, said, if a third, a fourth, and a sixth of them were added together, the sum would be 45; how many crowns had he? A. 60.

5. What is the age of a person, who says, that if of the years he hea lived be multiplied by 7, and 3 of them be added to the product, the sum woald be 292? A. 60 years.

6. What number is that which, being multiplied by 7, and the product divided by 6, the quotient will be 14? A. 12.

DOUBLE POSITION.

XCIV. This rule teaches to solve questions by means of two suf posed numbers.

In Single Position, the number sought is always multiplied or livided by going proposed number, or increased or diminished by itself, or some known part of itself, a certain number of times. Consequently, the result will be propor.. tional to its supposition, and but one supposition will be necessary; but, in Double Position we employ two, for the results are not proportional to the suppositions.

1. A gentleman gave his three sons $10000, in the following manner : to the second $1000 more than to the first, and to the third as many as to the first and second. What was each son's part?

Let us suppose the share of the first, 1000

Then the second 2000
Third 3000

Total, 6000

This, subtracted from 10000, leaves 4000

The shares of all the song will, if our supposition be correct, amount to $10000; but, as they amount to $6000 only, we call the error 4000.

Suppose, again, that the share of the first was 1500

Then the second 2500

We perceive the error in this case to be $2000.

The first error, then, is $4000, and the second $2000. Now, the difference between these errors would seem to have the same relation to the difference of the suppositions, as either of the errors would have to the difference between the supposition which produced it and the true number. We can easily make this statement, and ascertain whether it will produce such a result: As the difference of errors, 2000: 500, difference of suppositions: either of the errors, (say the first) 4000: 1000, the difference between its supposition and the true number. Adding this difference to 1000, the supposition, the amount is 2000 for the share of the first son; then $3000 that of the second, $5000 that of the third, Ans. For 2000+3000+5000=10000, the whole estate. Had the supposition proved too great, instead of too small, it is manifest that we must have subtracted this difference.

The differences between the results and the result in the question are called errors: these are said to be alike, when both are either too great or too sina unlike, when one is too great, and the other too small.

From these illustrations we derive the following

RULE.

1. Suppose any two numbers, and proceed with each according to the manner described in the question, and see how much the result of cach differs from that in the question.

II. Then say, As the difference of the errors: the difference of the suppositions :: either error: difference between its sup vosition and the number sought.

More Exercises for the Slate.

2. Three persons disputing about their ages, says B, "I am 10 years older than A;" says C, "I am as old as you both:"now what were their several ages, the sum of all of them being 100? Ans. A's, 20; B's, 30; C's, 50.

3. Two persons, A and B, have the same income; A saves of his yearly; br P, by spending $150 per annum more than A, at the end of 8 years finds himself $400 in debt; what is their income, and what does each spend per an

num ?

First, suppose cach had $200; secondly, $300; then the crrors will be 400 and 200. A. Their income is $400; A spends $300, B $450.

4. There is a fish whose head is 8 feet long, his tail is as long as his head and half his body, and his body is as long as his head and tail; what is the whole length of the fish?

First, suppose his body 30; secondly, 28; the errors will then be 1 and 2.

A. 32 feet

5. A labourer was hired 80 days upon this condition,-that for every day ho was idle he should forfeit 50 cents, and for every day he wrought he should ro ceive 75 cents; at the expiration of the time he received $25; now how many days did he work, and how many days was he ille?

A. Ic.worked 52 days, and was idle 28

MISCELLANEOUS EXAMPLES.

1. There is a room, one side of which is 20 feet long and 8 feet high, how many square feet are centained in that side?

160 sq. N., Ans.

Hypothearse

B

20

This side is a regular parallelogram (¶ LXXIX.); and, to find the squaro contents, we have seen that we must raultiply the length by the breadth; hus. 20 ft. 8 A. But, had we been required to find the square contents of half of this parallelogram, as divided in tho figure on the left, it is plain that, if we should multiply (20) the whole length by of (8) the width, or, it this case, the height, the product would be the square contents in this half, that is, in the figure BCD; thus, of 84; then, 4X2080 eq. ft., which is precisely of 160, the square contents in the whole figure. The haif BCD is called a triangle, because it has, as you see, 3 sides und Jangles, and because the line BC falls perpendicularly on CD); the angle at C is called a right angle; the whole angle, then, BC 15 may properly be called a right-angled triangle.

D

Base. 20.

C

*The difference of the errors, when alike, will be one subtracted from the other ; when unlike, one added to the other.

The line BC is called a perpendicular, CD the base, and D B the hypothe

nuse.

Note. Both the base and perpendicular are sometimes called the legs of the triangle.

Hence, to find the area of a right-angled triangle ;Multiply the length of the base by the length of the per pendicular; the product will be the arca required.

2. What is the area of a triangular piece of land, one side of which is 10 rods, and the distance from the corner opposite that side to that side 20 rods ' Ans. 20 X 40= 400 rods Note. To find the area of any irregular figure, divide it inta triangles.

Hypothcause.

50.

Base. 40.

A

Perpendicular. 30.

In any right-angled triangle, it has been ascer tained, that the square of the hypothenuse is equal to the sum of the squares of the other two sides. Thus, in the adjacent figure, 40= 1600, and 30° = 900; then, /900+160050, the hypothenuse

I. Hence, to find the bypothenuse, when the legs are given;

Add the squares of the two legs together, and extract the square root of their sum.

II. When the hypothenuse and one leg are given, to find the other leg ;~

From the square of the hypothenuse subtract the square of the given leg, and the square root of the remainder will be the ether

3. A river 80 yards wide passes by a fort, the walls of which are 60 yards high; now, what is the distance from the top of the wall to the opposite bank of the river?

In this example we are to find the hypothenusc. Ans. 100 yards.

4. There is a certain street, in the middle of which, if a ladder 40 feet long bo placed, it will reach a window 24 feet from the ground, on either side of said street; what is the width of the street?

In this example, we are to find the length of the base of two triangles, and then the sum of these will be the distance required. Ans Gi feet.

5. There is a cortain elm, 20 feet in diameter, growing in the centre of a circalar island; the distance from the top of the tree to the water, in a straight line, is 120 feet; and the distance from the foot 90 feet; what is the height of the tree?

As the tree is 20 feet in diameter, the distance from its centre to the water is the length of the base, that is, 1090 100 feet. A. 66,332 . t.

6. Two ships sail from the same port; one goes due north 40 leagues, the ther due oast 30 leagues; how far are they apart?

Wo are here to find the hypothenuse. A. 50 leagues

7. A man, in a hunting excursion, shot a aquirrol from the top of a stately oak, BO feet high, its diameter being 6 feet, the person stood 19 paces from the tree (3 fort being equal to one pace;; now, how far was it from the squirrel to the ace where the hunter stood, when ho ischarged his piece? A. 100 ft.

80 ft.

19 races

8. What is the circumference of a wheel, the diameter of which is 8 feet?

The circumference of a circle is greater, you are sensible, than the diameter, being a little more than 3 times, or, more accurately, 3,141592 times the diameter. A. 25,13+ft.

9. What is the diameter of a wheel, or circle, whose circumference is 12 feet? A. 4 ft., nearly.

10 If the distance through the earth be 8000 miles, how many miles around ? A. 25132,7 miles, nearly.

11. What is the area or contents of a circle, whose diameter is 6 feet, and As circumference 19 feet?

NOTE. The area of a circle may be found by multiplying salf the diameter by half the circumference, or by multiplying he square of half the diameter by 3,141592. A. 281 ft.

12. What is the area of a circle, whose diameter is 20 feet?

10'100X3,141592314,1592, Ans. 13. What is the diameter of a circle, whose area is 314,1592? A. 20 ft. 14. What is the area, or square contents, of the earth, allowing it to be 8000 miles in diameter, and 25000 in circumference?

NOTE. The area of a globe or ball is 4 times as much as the area of a circle of the same diameter; therefore, if we multiply the whole circumference into the whole diameter, the product cill be the arca. A. 200000000.

15. What are the solid contents of a globe or ball 12 inches in diameter? The solid contents of a globe are found by multiplying its

arca by of its diameter. A. 904+ solid inchos

16. What are the solid contents of a round stick of timber, 10 inches in di ameter, and 20 feet long?

In this example, we may first find the area of one end, as before directed for a circle; then multiply by 20 feet, the length. A. 11 feet, nearly.

Note. Solids of this form may be called cylinders

17. What are the solid contents of a cylinder 4 feet in diameter, and 10 feet long? A. 125 feet.

When solids, being either round or square, taper regularly till they come to a point, they contain just as much as if they were all the way as large as they are at the largest end.

When solids decrease regularly, as last described, they are called pyramids. When the base is square, they are called square pyramids; when triangular, triangular pyramids; and when round, circular pyramids, or cones.

Hence, to find the solid contents of such figures;

Multiply the area of the largest end by of the perpendicular height.

What are the solid contents of a cone, the height of which is 30 feet, and its base 8 feet in diameter? A. 502,6+ft.

18. There is a pyramid, whose base is 3 feet square, and its perpendicular height 9 feet; what are its solid contents? A. 32x3=27 ft.

19. What is the length of one side of a cubical block, which contains 9261 solid feet? A. 21 ft.

20. In a square lot of land, which contains 2648 acres, 3 roods, and 1 rod, what is the length of one side? A. 651 rods.

21. A grocer put 5 gallons of water into a cask containing 30 gallons of wine worth 75 cents per gallon; what is a gallon of this mixture worth?

A. 64 cts. 22. The first term of a geometrical series is 4, the last 56984, and the ratio 6, what is the sum of all the terms? A. 08380.

23. "The great bet, and when it will be paid. The public mind has been considerably anused for a few days past with a singular bet, said to have been made between a friend of Mr. Adams and a friend of Gen. Jackson, on the eastern shore of Maryland. The bet was, that the Jackson man was to receive from the Adams man 1 cent for the first electoral vote that Jackson should receive over 130, 2 cents for the second, 4 for the third, and so on, doubling for every successive vote; and the Adams man was to have one hundred dollars if Jackson did not receive over 130 votes. According to the present appearances, Jackson will receive 173, 43 over 130, and the sum the Adams man will have to pay, in that event, will be $87960930222,07.

"But the joke does not appear to be all on the Jackson man's side. The money is to be counted, and it will take a pretty long lifetime of any common man to count out the shiners.' Let's see:-allowing that a man can count sixty dollars a minute, and that he continues to count without ceasing, either to sleep, to take refreshment, or to keep the Sabbath, it will take him twenty-seven hundred and eighty-nine years, nearly; but allow him to work eight hours a day, and rest on the Sabbath, he will be occupied 9789+ years; so that the Adams man, when he is called upon for the cash, may tell his Jackson friend, Sit down, sir; as soon as I can count the money you shall have it; even the banks take time to count the money, you know."

To those Teachers who adopt this work, the author takes pleasure In recommending Mr. Shaw's Visible Numerator. This apparatus is exceedingly simple in its construction; so much so, that "every one is surprised that it has not been thought of before." It consists of a series of blocks, admirably adapted, by their comparative size, to convey to the mind of the pupil the relative value of the different orders of units, and to develope in the same simple manner, the true principles on which the rul of Arithmetic are founded.

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