mon difference, and I added to the quotient will be the number of terms.

Example 2. What is the number of terms of that progref fion whose extremes are 3 and 67, and common difference 2?

Thus 64, the difference of the extremes, divided by 2, the common difference, quotes 32, to which adding 1 there is 33, the number of terms; as may be seen annexed.

In

67

}}

3

extreme

2)64 difference

32

I

33

number of terms.

every arithmetical progreffion the sum of any two terms is equal to the fum of any two other terms, taken at an equal distance from the former, and on oppofite fides: thus, in the foregoing progreffion, 15 and 19 is equal to 23 and 11; viz. 34; and 31 and 33 is equal to 23 and 41, viz. 64. And the double of any one term is equal to the fum of any two terms taken on each fide of it, and at an equal distance from it; thus the double of 25 is equal to 19 and 31, viz. 50. How many ftrokes does an English clock strike in 8 days?-Anf. 1248 ftrokes.

Qu. 3.

Qu. 4. If a traveller go a journey of 10 days, travelling 3 miles the first day, and increafing three miles every day, how many miles will he travel in the 10 days? and how many mites the laft day?-Anf. 165 miles in the whole, and 30 miles the laft day.

Geometrical Progreffion.

When any series of numbers increase or decrease by a conftant multiplication or divifion, they are faid to be in geome trical progreffion.

Thus the numbers 4, 8, 16, 32, 64, &c. and 243, 81, 27, 9, 3, 1, are in geometrical progreffion, the former increasing by multiplying each preceding number by 2, and the latter degreafing by dividing each preceding number by 3. 5

Thus,

Thus, in geometrical progreffion, the numbers are increased by multiplication, and decreased by divifion; whereas in arithmetical progreffion they are increased by addition and decreased by subtraction.

The number in geometrical progreffion by which the feries are multiplied or divided, is called the ratio; whereas in arithmetical progreffion the number by which the feries are increased or diminished is called the common difference.

In geometrical progreffion, as in arithmetical progreffion, having any three of the following terms, the other two may be readily found.

Having the firft and last term and the ratio, to find the fum of the terms.

Rule. Multiply the last term by the ratio, and from the product fubtract the first term, and the remainder divided by one lefs than the ratio will quote the fum of the feries.

Example 1. What is the fum of the series of a geometrical progreffion whofe extremes are 1 and 65536, and the ratio 4?

Here the last term is multiplied by the ratio 4, the first term 1 fubtracted from the product, and the remainder divided by 3, which is 1 lefs than the ratio, and the quotient 87381 is the fum of the feries.

Qu. 2. What is the sum of a series of numbers, in geometrical progreffion, whofe extremes are 1024 and 59049, and the ratio 3?-Answer 88061,

PROBLEM II.

Having the firft term and the ratio, to find any other term required.

Rule. Write down a few of the leading terms of the feries, and place their indices over them, beginning with a cypher *.

Then add any of those indices together, which will make an index an unit lefs than the number which expreffes the place of the term required.

Multiply the terms of the feries together belonging to those indices, and the product will be a dividend, to be divided by the product of the firft term, multiplied by a number an unit efs than the number of terms multiplied; and the quotient will be the answer.

Example 3. What is the laft term of that geometrical series whose first terin is 3, ratio 2, and number of terms 10?

Indices o, I, 2, 3, 4, 5, 6,

*The indices of the terms in this rule are numbers expreffing the place of each term according to the natural order of numbers; thus the index of the firft term is o, that of the fecond term 1, the third term 2, the fourth 3, &c. ; but when the firft term of the feries is equal to the ratio, the indices must begin with an unit, and the product of the terms will be the answer, without dividing by the product of the firft term, as directed above.

In this example, the indices 4 and 5 added together are 9, which is less than the number of the term fought; then the term belonging to these indices, 48 and 96, are multiplied together for a dividend, the firft term 3 multiplied by lefs than the number of terms, which are multiplied together, is the divifor; and the quotient 1536 is the answer.

Note. The number of terms multiplied together in this example are 2; therefore 3 the first term multiplied by 1 less than 2, or 1 only, produces 3.

Example 4. What is the 15th term of a series whofe first term is 3, and ratio 3?

I, 2, 3, 4, 5, 6, Indices

3, 9, 27, 81, 243, 729, Leading terms

729

243

2187

2916

1458

177147

81

6

4

5

15 index to the 15th term

177147 1417176

14348907 the 15th term, or Anfwer.

Here, as the first term is equal to the ratio, the indices must begin with 1, instead of o; and as many indices must be taken as will make the entire index to the term fought, viz. 15, and the product of the terms must not be divided, as in the former cafe.

Qu. 5. What is the laft term of that geometrical series where the first term is 1, ratio 2, and number of terms 23? -Anf. 4194304.

Qu. 6. A perfon having an elegant houfe to difpofe of, offered it to fale on the following terms: for the first window the purchaser was to pay 1 farthing, for the fecond window 2 farthings, for the third window 4 farthings, and fo on, dou

bling the price for every fucceeding window; there were 32 windows in the house: what would be the price of the house at that rate?-Anf. 4,473,9241. 55. 4d.

Qu. 7. An Indian of the name of Seffa having invented the game of chefs, fhewed it to his prince Shehram, who was fo pleased with the invention, that he bade Seffa fay what he would have as a reward for his ingenuity. Seffa requested i grain of wheat for the firft fquare on the chefs-board, z grains of wheat for the fecond fquare, 4 for the third, 8 for the fourth, and so on, doubling the quantity of grains of wheat for every fucceeding fquare: now the whole number of fquares on the chefs-board is 64. Suppofing a bufhel to contain 640,000 of thefe grains, how many fhips would it require to export the whole quantity of wheat, each ship being 100 tons burden?-Anf. 7,205,759,403 fhips, and about of a fhip.

There are many other questions in progreffion which favour more of curiosity than real utility. This rule was much admired formerly, before the nature of numbers was fo well understood as at prefent, on account of its furprising increasing power it is, however, of little ufe, except in calculating tables, &c.

SECT. XXI.

OF EVOLUTION, OR THE EXTRACTION OF ROOTS,
SQUARE AND CUBIC.

BEFORE the learner proceeds to extraction of roots, he should understand the nature of involution, or the raising of a number to any power required.

A power