5. James lent William a sum of money on interest, and in 10 years it amounted to $1,600; what was the sum lent? A. $1,000.

6. Three merchants gained, by trading, $1,920, of which A took a certain sum, B took 3 times as much as A, and C four times as much as B; what share of the gain had each?

A. A, $120; B, $360; C, $1,440. 7. 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.

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

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

DOUBLE POSITION.

A. 12.

CX. 1. DOUBLE POSITION teaches to solve questions by means of two supposed numbers.

2. In SINGLE POSITION, the number sought is always multiplied or divided by some proposed number, or increased or diminished by itself, or some known part of itself, a certain number of times. Consequently, the result will be proportional 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.

3. A gentleman gave his three sons $10,000, 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 1,000

Then the second=2,000

Third 3,000

Total, 6,000

The shares of all the sons will, if our supposition be correct, am't to 10,000; but, as they amount to $6,000 only we call the error 4000.

This subtracted from 10,000, leaves 4,000] Suppose again, that the share of the first was 1,500 Then the second=2,500

Third=4,000

We perceive the error in this

8,000 case to be 2000.

2,000

4. The first error, then, is $4,000, and the second $2,000. 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:

5. As the difference of errors, 2,000: 500 difference of suppositions:: either of the errors (say the first,) 4,000: 1,000, the difference

between its supposition and the true number. Adding this difference to 1,000, the supposition, the amount is 2,000 for the share of the first son: then $3,000 that of the second, $5,000 that of the third, Ans. For 2,000+3,000+5,000=10,000, the whole estate.

6. 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 small; unlike, when one is too great, and the other too small.

RULE.

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

8. Then say, as the difference* of the errors: the difference of the suppositions:: either error: difference between its supposition and the number sought.

9. Three persons disputing about their ages, says B, years older than A;" says C, "I am as old as you both :" were their several ages, the sum of them all being 100?

"I am 10 now, what

A. A's, 20; B's, 30; C's, 50.

10. Two persons, A and B, have the same income: A saves of his yearly; but B, 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 annum?

A. A's income $400; A spends $300; B $450. 11. 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. A. 64 feet.

12. A laborer contracted to work 80 days for 75 cents per day, and to forfeit 50 cents for every day he should be idle during that time. He received $25: now how many days did he work, and how many days was he idle? A. 52 days; idle 28.

MENSURATION.

CXI. 1. MENSURATION is the measuring of Surfaces and Solids.

O F ANGLES.

2. AN ANGLE is the inclination or opening of two lines that meet each other, as in the Figures on next page. The point of intersection is called the Angular point ; and in common language, the Angle.

3. An ANGLE is greater or less, not according to the length of the

CXI. Q. What is Mensuration? 1. An Angle? 2. The point of intersection? 2. How is the size of an angle determined? 3.

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

lines, but according as they are more or less inclined or opened; thus, the angle at C, below, is the greatest of the three.

4. A RIGHT ANGLE is one formed by a line drawn perpendicular to another: as A, in Fig. 1.

5. OBLIQUE ANGLES are those formed by oblique lines, and are either Acute or Obtuse; as B and C.

6. An OBTUSE ANGLE is greater, and an ACUTE ANGLE is less than a right angle.

OF TRIANGLES.

7. A TRIANGLE is a plane* figure that has three sides and three angles; as in the following Figures.

8. AN EQUILATERAL TRIANGLE has three equal sides. [Fig. 4.j 9. AN ISOSCELES TRIANGLE has two equal sides. [Fig. 5.] 10. A SCALENE TRIANGLE has three unequal sides. [Fig. 6.] 11. A RIGHT-ANGLED TRIANGLE has one right angle. [Fig. 7.] 12. AN OBTUSE-ANGLED TRIANGLE has an obtuse angle. [Fig. 6.] 13. AN ACUTE-ANGLED TRIANGLE has three acute angles. [Fig. 4.] 14. In a right-angled triangle the longest side is the HYPOTHENUSE, and the other two sides the LEGS, or the BASE and PERPENDICULAR. In other triangles the longest side is usually considered the Base.

15. In every right-angled triangle,-The square of the hypothenuse is equal to the sum of the squares of the other two sides; as, 502= 40+302. [Fig. 8.]

FIG. 8.

Hypothenuse 50

Baze 40

icular 30

Perpendicula

16. Hence, to find the different sides, we may proceed as follows: To find the hypothenuse.-Add the squares of the two legs together, and extract the square root of that sum. To find either leg. From the square of the hypothenuse subtract the square of the given leg, and the square root of the remainder will be the other leg.

12.

Q. How is a right angle formed? 4. What are oblique angles? 5. Obtuse? 6. Acute? 6. What is a triangle? 7. An equilateral triangle? 8. An Isosceles? 9. Scalene? 10. Right angled triangle? 11. Obtuse angled triangle? Acute angled triangle? 13. What are the names of the sides in a right angled triangle? 14. How are each found? 16. On what principle is each operation based? 15.

* PLANE, [L. Planus.] An even or level surface, like plain in common language. An instrument used in smoothing boards.

The learner will discover the application of the rule better by drawing a triangle on his slate, like Fig. 8, and noting the sides which are intended to correspond with those which are given in the question. Indeed, without some such illustration, he will scarcely be able to apply the rule at all, except in cases where the particular sides are designated.

17. Required the hypothenuse of a right-angled triangle whose legs are 24 and 32 feet. A. 40 feet. 18. Required the base of a right-angled triangle the other sides of which are respectively 15 and 25 feet. A. 20 rods. 19. Suppose a lot of land lies in the form of a right-angled triangle, and that the longest side is 100 rods and the shortest 60; what is the distance around it?

A. 240 rods. 20. 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. A. 100 yards.

21. A ladder 40 feet long may be so placed upon one side of a street that it will reach a window 32 feet from the ground; and without moving it at the bottom, it will reach a window on the other side 24 feet high; what is the width of the street? A. 56 feet.

22. There is a certain elm, 2 feet in diameter, growing in the centre of a circular island, the distance from the top of the tree in a straight line to the water is 120 feet, and the distance from the foot 89ft.; what is the height of the tree? (89+1 base.) A. 79.372ft.+.

23. When two ships, which sailed from the same port, have gone, one due north 40 leagues, and the other due east 30 leagues, how far are they apart then? A. 50 leagues.

97

FIG. 9.

B

E 76 F

24. There are three towers, A, B, and C, standing in a direct line, the heights of which are 64, 90, and 50 feet respectively. The distance between the top of the tower A and that of B, is 97 feet, and the distance between the bottom of the tower B and that of C is 76 feet. From these data please inform me what are the several distances from the top of A to the bottom of B, from the top of B to the bottom of A, from the bottom of A to the bottom of B, from the bottom of B to the top of C, from the bottom of C to the top of B, and from the top of B to the top of C. A. D E, 93.45+; A E, 113.26+; B D, 129.74+; C E, 90.97+; B F, 117.79+; B C, 85.883+. 25. A castle wall there was, whose height was found To be 100 feet from th' top to th' ground;

Against the wall a iadder stood upright,
Of the same length the castle was in height:
A waggish youngster did the ladder slide

(The bottom of it) 10 feet from the side;

Now I would know how far the top did fall,

By pulling out the ladder from the wall! A 6 in nearly.

26. A gentleman has a garden in the form of an equilateral triangle, the sides of which are each 50 feet; at each corner of the garden stands a tower; the height of the tower A is 30 feet, that of B 34 feet, and that of C 28 feet. At what distance from the bottom of each of these towers must a ladder of the same length with each side be placed, that it may just reach the top of each tower, allowing the ground of the garden to be horizontal ?

A. 40ft.; 36.66ft.+; 41.42ft.

O F SURFACES.

27. A QUADRILATERAL has four sides and four angles. PARALLELOGRAM is a general name for all quadrilateral figures, that have at least their opposite sides and angles equal; as below.

28. A SQUARE is a quadrilateral that has its sides equal and its angles right angles, as Fig. 10.

29. A RECTANGLE is a quadrilateral that has its angles right angles, and only its opposite sides equal, as Fig. 11.

30. RHOMBOID is a quadrilateral which has its opposite sides and angles equal, two of its angles being acute, and two obtuse, as Fig. 12. 31. A RHOMBUS is a quadrilateral that has its sides equal, but its angles like those of a rhomboid, as Fig. 13.

32. A POLYGON is a rectilineal figure of more than four sides, which when they are all equal, form regular polygons. Squares are also regular figures; so are Triangles, when they are equilateral.

33. The PERIMETER of any plane rectilineal* figure, is the entire distance round it; and is found by adding together all the sides that bound it.

34. A CIRCLE is a plane figure bounded by a curved line, called the Circumference or Periphery; which is every where equally distant from a certain point within it, called the CENTRE. An ARC is any part of the circumference.

35. The DIAMETER of a circle, is a straight line drawn through the centre and terminating in the circumference on each side. A

Fig. 14.

Radius

Diameter

Circumference

Q. What is a Quadrilateral? 27. Parallelogram? 27. Square? 28. Rectangle? 29. Rhomboid? 30. Rhombus? 31. Polygon? 32. What other regular figures are mentioned? 32. What is the perimeter of a figure? 33. How is it found? 33. What is a circle? 34. Circumference? 34. An Arc? 34. Diameter? 35. Chord? 35. Segment? 35.

* RECTILINEAR or RECTILINEAL, [L. rectus, right or straight, and linea, a line.) Consisting of right or straight lines; straight.