29. The diameter of a 91b. iron shot being 4 inches, what is the weight of a shot 6 inches in diameter?

Ans. 30lb.

N. B. It is proved by geometry, that the cubic contents (and consequently the weights) of similar solids are directly proportional to the cubes of their like sides or diameters.

30. What is the diameter of a 481b. iron shot?

31.

What is the diameter of a 24lb. shot?

Ans. 6.99 inches.

Ans. 5.55 inches.

32. A lead ball whose diameter is 4 inches weighs 177b. nearly; hence it is required to find the diameter of a nusket ball whose weight is

an ounce?

Ans. •656 of an inch.

33. If the depth of a barrel which holds 80%. of gunpowder be 20 inches, what is the depth of another barrel of similar dimensions which holds three times that quantity?

Ans. 28.84 inches.

34. If a musket barrel which carries an ounce ball (656in. in diam. is 3 feet in length; what would be the diameter of the bore, and

length of a similar barrel for a pound ball, allowing of an inch for

1. If the first term of an arithmetical progression be, the common difference, and the number of terms 50, what is the last term?

Ans. 243.

2. If the first and last terms of an arithmetical series be 18 and 2, and the number of terms 9, what is the common difference?

3. Required 3 arithmetical means between 1 and 2?

Ans. 2.

Ans. 14, 14, 13.

4. If the first term of an arithmetical progression be 0, the last term 10, and the number of terms 20, what is the sum?

Ans. 100.

5. Suppose a triangular battalion to consist of 20 ranks, the first rank being 1 man, the next 4, the third 7, the fourth 10, and so on; what is its strength?

Ans. 590 men.

6. If a detachment march 32 miles at the rate of 4 miles the first hour, and 1 mile the last, in what time did they perform the journey supposing each hour's march was successively diminished by the same distance, and what was that distance?

Ans. 13 hours.

And the decrease im. per hour.

7. It is found that a heavy body near the earth's surface descends by its own weight (from rest) the space of 16 feet in the first second of time, 48 in the next second, 80 feet in third second, and so on constitutin series in arithmetical progression, whose first term s ΤΣ feet, and common difference 32 jeet; now according to this late, how far would a heavy body descend in 10 seconds?

Ans. 1608 feel.

Geometrical Progression.

1. If the first term be 1, the ratio or multiplier 3, and the number of terms 10, what is the last term?

2.

Ans. 295241.

Let the first term be 9, the ratio or divisor 1, and number of terms 8, what is the last term?

Ans. 128

3. Suppose the first term is 100, the ratio or multiplier 1·05, and the number of terms 8, what is the last term? In other words-What is the amount of 1004. in 7 years, at 5 per cent. per annum compound interest?

Ans. £140710042265625.

4. Let the extremes be 6 and 24, and number of terms 3; required the middle term? Or, what is the mean proportional between 6 and 24 ? Ans. 12.

5. Required a geometrical mean between 10 and 20?

Ans. 14 1421356 nearly.

6. If the first term is 22, last term 1305013, and the number of terms 4; what is the ratio, and the two mid lle terms?-Or let it be required to find 2 geometrical means between 22 and 1305018?

Ans. 39 ratio.

And the middle terms 858 and 33462.

7. Required two geometrical mean proportionals between 10 and 100 ?

8. Suppose the musket cartridges necessary for an army to be counted at 16 times; the first count being 3, the next 6, the third 12, the fourth 24, and so on; what is the whole number of cartridges?

Ans. 196605.

9. What would be the produce (or last crop) in 10 years from a grain of wheat, the increase or crop being constantly sown, and each grain producing yearly an ear of 40 grains, supposing 7000 grains to weigh a pound, and 60lb. to the bushel?

Ans, 3120761904grs. 62 bush.

10. Required the sum of the progression, TN, TOOO, TOOO, &c. continued ad infinitum, (the ratio or divisor being 10, and last terin 0)? Ans.

11. What is the sum of the series, I ,, &c. continued ad infinitum?

Ans. 1;

12. The sum of three continued proportionals being 100, and the ratio of the first to the third as 1 to 4, what are the 3 numbers?

Ans. 147, 284, 574,

13. Suppose the ratio of the first to the 3d. as 2 to 3, required the three numbers?

Ans. 26 8475, 32.8813, 40.2712, nearly,

14. To divide 100 into 5 continued proportionals, the ratio of the first to the 5th being as 16 to 81?

Ans. 71, 117, 17, 25411, 3887

OF LOGARITHMS,

154. LOGARITHMS are a set of numbers so contrived, that the products in multiplication, and the quotients in division, are obtained by means of addition and subtraction only.

155. Or, Logarithms are a series of numbers in arithmetical progression corresponding to another series of numbers in gcometrical progression.

Thus if 1 be the first term of a geometrical progression, and 2 the ratio or multiplier, the terms will be

1, 2, 4, 8, 16, 32, 64, 128, &c.

(146) or 1°, 2, 2, 23, 24, 25, 26, 27, &c.

And the arithmetical series of indices or exponents

0, 1, 2, 3, 4, 5, 6, 7, &c,

are the logarithms of the corresponding terms of the geometrical series or powers of the ratio 2.

1, 2, 4, 8, 16, 32, 64, 128, &c. numbers.

0, 1, 2, 3, 4, 5, 6, 7, &c. logarithmns.

156. Now the sums and differences of the indices or logarithms answer to the products and quotients of the corresponding terms or numbers.

Thus 23 make 5 the index or logarithm answering

to 32.

(111.) And the product of 4 and 8 (the terms corresponding to 2 and 3) make 32.

Again, the difference of the indices or logarithms 7 and 4 is 3, the index or logarithm of the term or number 8.

And the quotient of the corresponding terms, or 128 divided by 16 is 8.

Therefore the products and quotients of the mbers in the geometrical progression are found by taking the su as or differences of the corresponding indices or logarithms.

157. But the indices 0, 1, 2, 3, 4, 5, 6, 7, &c. may denote the powers of any other number or ratio; consequently different ratios or geometric progressions give different systems of logarithms.

Thus if 1 be the first term, and 10 the ratio of a geometrical progression, the terms will be

1, 10, 100, 1000, 10000, 100000, &c.

or 1°, 101, 102, 103, 10,

105,

&c.

And the indices 0, 1, 2, 3, 4, 5, &c. are the logarithms of

the corresponding terms or numbers, as before.

1, 10, 100, 1000, 10000, 100000, &c. numbers.

L, 1, 2, 3, 4,

5, &c. iogarithms.