3. How many times does a clock usually strike in 12 hours? Ans. 78. 4. A man on a journey travelled 20 miles the first day, 24 the second, and continued to increase the nuner of miles by every day for 10 days trave?

5. A firmer bought 20 cows, and the first, 4 for the second, and so on, proportion from the first to the last. for the whole?

CASE II.

How far did he Ans, 380 miles. gave 2 dollars for giving in the same What did he give Ans. $120.

When the two extremes and the number of terms are given to find the common difference.

RULE.

Siltract the less extreme from the greater, and divide the remainder by one less than the number of termsthe quotient will be the common difference.

EXAMPLES.

1. The extremes eing 3 and 19, a d the number of terms 9, what is the common difference?

Common difference 5 Answer.

3. If the extremes le 10 and 70, and the number of terms 21, what is the common difference, and the sum of the series? Ans. com. diff. 3, and the sum, 840.Į 4. A certain debt can be paid in one year, or 52 veeks, by weekly payments in Arithmetical Progression, the first payment being 1 dollar, and the last 103 do!lars. What is the common difference of the terms?

Ans. $2.

5. A debt is to be discharged at 16 several payments] in Arithmetical Progression; the first payme t to le 20 follars, and the last 110 dollars. What is the common difference?

Ans. $6.

GEOMETRICAL PROGRESSION. Geometrical Progression is the increase of any series of numbers by a common multiplier, or the decrease of any series by a common divisor; as 3, 6, 12, 24, 48; and 48, 24, 12, 6, 3. The multiplier or divisor by which any series is increased or decreased, is called the ratio. CASE I.

་་

To find the last term and sum of the series.

RULE

Raise the ratio to a power whose index is one less than the number of terms given in the sum. Multiply the product by the first terni, and the product of that multiplication will be the last term: then multiply the last term by the ratio, subtract the first term from the product, and divide the remainder by a number that is one less than the ratio-the quotient will be the sum of the series.

EXAMPLES.

1. Bought 12 yards of calico, at 2 cents for the first yard, 4 cents for the second, 8 for the third, &c.: what was the whole cost?

NOTE.-The number of terms 12, and the ratio 2.

1st.

term 2 1st. power.

1024 10th. power.

2

2

2. Bought 10 lbs. of tea, and paid 2 cents for the first pound, 6 for the second, 18 for the third, &c. What did

the whole cost?

Ans. $590.48.

3. The first term in a sum is 1, the whole number of terms 10, and the ratio 2; what is the greatest term, and the sum of all the terms? Ans. The greatest term is 512, and the sum of the terms 1023.

4. What debt may be discharged in 12 months, by paving 1 dollar the first month, 2 dollars the second month, 4 the third month, and so on, each succeeding payment being double the last; and what will be the mount of the last payment?

Ans. The debt is $1095, and the last payment $2048. 5. A father whose daughter was married on a newrear's day, gave her one cent, promising to triple it on the first day of each month in the year: what was the amount of her portion? Ans. $2657.20.

6. One Sessa, an Indian, having invented the game of chess, shewed it to his prince, who was so delighted with it, that he promised him any reward he should ask ; upon which Sessa requested that he might be allowed ne grain of wheat for the first square on the chess board, 2 for the second, 4 for the third, and so on, doubling coninually, to 64, the whole number of squares.-Now, supposing a pint to contain 7680 of these grains, and one quarter or 8 bushels to be worth 27s. 6d., it is required to compute the value of all the wheat?

Ans. £34481488206.

7. What sum would purchase a horse with 4 shoes, and eight nails in each shoe, at one farthing for the first nail, half penny for the second, a penny for the third, &c., loubling to the last? Ans. £1473324. 5s, 34d.

8. A merchant sold 15 yards of satin, the first yard for 1s. the second for 2s. the third for 4s, the fourth for Ss. &c.; what was the price of the 15 yards?

Ans. £1638, 7s.

9. Bought 30 bushels of wheat, at 2d. for the first bushel, 4d. for the second, 8d. for the third, &c.; what does the whole amount to, and what is the price per bushel on an average?

Ans.

(£3947848. 10s. 6d. Amount.
£298261. 12s. 4d. per bushel.

PERMUTATION.

Permutation is used to show how many ways things may be varied in place or succession,

RULE.

Multiply all the terms of the series continually, from 1 to the given number inclusive; and the last product will be the answer required.

EXAMPLES.

1. How many changes can be made with 8 letters of the alphabet?

1x2×3×4x5X6X7X8=40320 Answer.

2. In how many different positions can 12 persons place themselves round a table?

1×2×3×4×5×6×7×8×9×10×11×12=

479001600 Ans. 3. How many changes may be made with the alphaAns. 620448401733239439360000.

bet?

SKETCH OF MENSURATION,

OR PLANES AND SOLIDS.*

Planes, surfaces, or superficies, are measured by the inch, foot, yard, &c., according to the measures used by different artists. A superficial foot is a plane or surface of one foot in length or breadth, without reference to thickness. Solids are measured by the solid inch, foot, yard, &c.; thus, 1728 solid inches, that is 12x12x12 make one cubick or solid foot. Solids include all bodies which have length, breadth and thickness,

ARTICLE I.

To measure a square having equal sides.

RULE.

Multiply any one side of the square by itself, and the product will be the area, or superficial contents, in feet, yards, or any other measure, according to the measure used in measuring the sides.

*Planes are the same as superficies, or surfaces.

EXAMPLES.

Let A, B, C and D represent a square, having equal sides each measuring 20 feet. Multiply the length of one side by itself, thus

20 feet

20 feet

20

A

20

Ans. 400 square feet.

2. How many square rods are there in a field 30 rods! Ans. 8100 square rods.

square?

ARTICLE II.

To measure the plane or surface of a parallelogram.

RULE.

Multiply the length by the breadth-the product will be the superficial contents.

EXAMPLE.

Let A, B, C and D represent a parallelogram whose length is 40 yards, and breadth 15 yards.

Ans. 600 square yards.

2. How many square feet are there in the floor of a room 36 feet long and 16 feet wide? Ans. 576 square

ft.

3. I engage to give a plasterer 15 cents per square yard, for plastering the walls and ceiling of a room 30 feet long, 15 feet wide, and 9 feet high. How much will his work come to? Ans. $21.00.

Note. The contents of boards and other articles which are measured by feet, &c., may be easily found by Duodecimal Fractions.

ARTICLE III.

To measure the plane or surface of a triangle.

RULE.

Multiply the base by half the perpendicular, if it be a right angled triangle, and the product will be the area, or superficial contents; or multiply the base and perpendicular together, and half the product will be the area.