In this example, the indices 4 and 5 added together are 9, which is 1 less than the number of the term fought; then the term belonging to these indices, 48 and 96, are multiplied together for à dividend, the first term 3 multiplied by 1 lefs than the number of terms, which are multiplied together, is the divifor; and the quotient 1536 is the aufwer.

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

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

1, 2, 3, 45 5. 6, Indices

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

729

243

2187

2916

1458

*79 47

6

1:177147

1417176

14348907 the 15th term, or Answer.

Here, as the first term is equal to the ratio, the indices muft 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 ferics. where the firft term is 1, ratio 2, and number of terms 23? -Anf. 4194304.

A

Qu. 6. A perfon having an elegant house 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 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.

Qa. 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 ingeruity. Seffa requested 1 grain of wheat for the first fquare on the chefs-board, z grains of wheat for the second square, 4 for the third, & for the fourth, and fo 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 ships would it require to export the whole, quantity of wheat, each fhip being 100 tons burden?-Anf. 7,205,759,403 ships, and about 4 of a fhip. 号

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 furprifing increafing 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 is the product of any number multiplied into itself any certain number of times.

If a number be multiplied into itself only once, the product is called the second power, or the fquare; thus 4 is the fecond power, or fquare of 2, being the product of 2, multiplied by 2; and 9 is the fecond power of 3. But if a number be multiplied twice into itfelf continually, the product is called the third power, or cube; thus 27 is the cube, or third power of 3; and 64 is the third power of 4.

When a number is multiplied 3, 4, 5, or 6 times into itfelf, &c. the product is called the fourth, fifth, fixth, or feventh power respectively. !

Example 1. What is the fifth power of 6?

Firft nine Powers of each of the nine fingle Numbers.

64

128

256

512

2187

6501

19683

16384

65536

262144

78125 3906251 1953125

2 4 8 16 32 39 271 81 243 729 41664256 | 1024 | 4096 525125 625 3125 15625 636216 1290 7770 46656 279936 | 1679616 10077096 7493432401|16807|117649823543 576480140353607 81641512 4096 12765|2621442097162 167772161 34217728 981 7296561 59049|531441|4782969|43040721|387420484

The

The ufe of this table is, to find any power less than the 1oth power of any number under 10; thus, to find the 7th power of 6, I look in that column which has 7 at the top, aud in that line which has 6 on the left hand, and in the junction of these two lines is 2.79936.

The root of any number is that number which, being mul tiplied into itself a certain number of times, will produce the given number; thus 2 is the fquare root of 4, becaufe 2 multiplied by 2 produces 4; and 3 is the cube root of 27, for 3 multiplied 3 times into itself continually produces 27.

There are many numbers the given roots of which can never be exactly found, though by decimal fractions we may approximate towards it to any degree of exactness; but the power of a given number can always be found exactly.

Thofe roots, which only approximate towards the truth, are called furd roots, and thofe which perfectly exprefs the root of the given power are called rational roots: thus the fquare root of 5 is a furd root, as it cannot be exactly expreffed; but the cube root of 8 is a rational root, being exactly 2.

To extract the Square Root.

Rule 1. Divide the given number into periods of two figures each, by placing a point over every other figure, beginning with that in the place of units.

2. Find the greateft fquare root of the first period, and place it on the right hand of the given number, like a quotient in divifion.

3. Subtract the fquare of this root from the aforefaid period, and to the remainder bring down the next period,

and call it the refolvend.

4. Double the figures in the quotient for a new divifor to the refolvend, and find how often this divifor is contained in the refolvend, exclufive of the unit figure, and place the anfwer both in the quotient, and on the right hand of the divifor.

5. Then

5. Multiply the divifor thus increased by the laft quotient figure, and fubtract the product from the refolvend, and to the remainder bring down the next period for a new refolvend.

6. Double the figures in the quotient for a new divifor to this laft refolvend, and feek how often it is contained therein, exclufive of the unit figure, placing the answer both in the quotient and divifor as before: and proceed in the fame manner through each new refolvend, and the figures placed in the quotient will be the anfwer, or fquare root of the given number.

Example 1. What is the square root of 5934096.2

After dividing the given number into periods of two figures each by the points, I find the nearest square foot of 5 (as that is pointed for the first period), which is 2; I there fore place 2 in the quotient, and the square of 2, which is 4, I fubtract from this first period 5, and to the remainder 1 I bring down the next period 93; thus I have 19 for a new refolvend (for the unit figure 3 is to be neglected), then I take the double of the figure 2 in the quotient for a divisor, and feeking how often it is contained in 19, the refolvend, the answer is 4, which I place in the quotient, and also on the right hand of the divifor 4; then I multiply the divisor thus increased by 4, and the product 176 I place under and fubtract from the laft refolvend, and to the remainder 17 I bring down the next period 40; thus I have 174 for a new resolyend, and 48 the double of the figures in the quotient for a VOL. I. K.k

new