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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. 4..."
Qú. 7. An Indian of the name of Sessa having invented the game of chess, fhewed it to his prince Shehram, who was so pleased with the invention, that he bade Seffa fay what he would have as a reward fór bis ingerruicy. Seffa requefted i grain of wheat for the firit (qırare on the chefs-board, 3 grains of wheat for the second square, 4 for the third, & for the fourth, and so on, 'doubling the quantity of grains of wheat for every succeeding square: Aow the whole number of squares on the chess-board is 64. Suppofing a bushel to contain 640,000 of these grains, how many thips would it require to export the whole, quantity of wheat, cach ship being 100 tons burden? Anf7,205,759,403 ships, and about of a ship.
There are many other questions in progression which favour more of curiosity than real utility. This rule was much admired formerly, before the nature of numbers was so well understood as at present, on account of its furprising increasing power : it is, however, of little use, except in calculating tables, &c.
BEFORE the learner proceeds to extraction of roots, be Thould understand the nature of involution, or the raising of a number to any power required.
A power is the produ&t of any number multiplied into itself any certain number of times.
If a number be multiplied into itself only once, the produe is called the second power, or the square ; thus 4 is the second power, or square of 2, being the produ&t of 2, multiplied by 2 ; and 9 is the second 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 itself, &c. the product is called the fourth, fifth, fixth, or feventh power respectively.' Example 1. What is the fifth power of 6?
6 Fractions are raised to pow.
216 third power
6 ers in the fairie manner as whole
1296 fourth power numbers; multiplying them as
6 taught in multiplication of
7770 fifth power fractions."
First nine Powers of each of the nine single Numbers. A2d| 3d 41k 5th
64 128 256 319 | 27
243 729 2187 6501 19083 4116 04 | 250 | 1024 | 4096 16384 65536 51-5125| 6253125 | 15625 78125
390625 1953125 636 2 16 1290 7770 46656 279936 | 1679610 10077096 7 +93432401110807117046 623543 / 5704801 1 40353607 81041512 1090 2765262144 09710216777216 : 34217724 98117296561 590191531441/4782909 4301072 11387420484
The use of this table is, to find any power less than the Toth 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 279936.
The root of any number is that number which, being mul. tiplied into itself a certain number of times, will produce the given number; thys 2 is the square root of 4, because 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 exaâly found, though by decimal fractions we mag approximate towards it to any degree of exa&tness; but the power of a given number can always be found exactly.
Those roots, which only approximate towards the truth, are called furd roots, and those which perfeâly express the root of the given power are called rational roots : thus the square root of 5 'is a surd root, as it cannot be exactly expressed; but the cube root of 8 is a rational root, being exactly 2.
To extract the Square Root.
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 greatest square root of the first periad, and place it on the right hand of the given number, like a que. tient in division,
3. Subtract the square of this root from the aforesaid period, and to the remainder bring down the next period, and call it the resolvend.
4. Double the figures is the quotient for a new divisor to the resolvend, and find how often this divisor is contained in the resolvend, exclusive of the unit figure, and place the answer both in the quotient, and on the right hand of the divisor.
5. Multiply the divisor thus increased by the last quotient figure, and subtract the product from the resolvend, and to the remainder bring down the next period for a new resolvend.
6. Double the figures in the quotient for a new divisor to this last resolvend, and seek how often it is contained therein, exclusive of the unit figure, placing the answer both in the quotient and divisor as before: and proceed in the same man, ner through each new resolvend, and the figures placed in the quotient will be the answer, or square root of the given number.
Example 1. What is the square root of 5934096.7
4/176 4831 1740
31 1449 48661 29196
6 29196 The Answer
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 subtract from this first period 5, and to the remainder i I bring down the next period 93 ; thus I have 19 for a new resolvend (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 seeking how often it is contained in 19, the resolvend, 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 subtract from the laft refolvend, and to the remainder 17 I bring down the next period 40; thus I have 174 for a new resolvend, and 48 the double of the figures in the quotient for a VOL. I.
new divisor; then I seek how often the divifor 48 is coutained in 174, and the answer 3 I place both in the quotient and divisor as before, and multiply the divisor also thereby, and the product 1449 placed under and subtracted from the resolvend, there remains 291, to which bringing down the last period 96, I have 2919 for the last resolvend, and the double of 243 or 486 for a divisor: this last quotient is 6, which I place in the quotient and divisor, and multiply the divisor thereby, and the product is equal to the last resolvend : thus 2436 is the answer.
Qu. 2. What is the square root of 152399025? - Answer 12345.
Qu. 3. What is the square root of 22071204? - Answer 4698.
To extract the Cube Root.
Rule 1. Divide the given number into periods of three figures each, by placing a point over every third figure, beginning with the unit figure
2. Find the greatest cube in the first period, and place its cube root on the right hand of the given number.
3. Subtract the cube from the said period, and to the remainder bring down the next period, which is called the refolvend.
4. Place three times the root under the resolvend, and also three times the square of the root, the latter being removed dne place to the left ; these two added together form a divisor: '':: 1 5. Seeko how often this divisor is contained in the resolve end, exclufive of the place of units, and place one less than the auswer itr the quotient. 'pen 6. Under
the divifor place the cube of the last quotient figure, the square of it multiplied by three times the rest of the quotient cand three times the root multiplied by the fquare of the quotient; each of the two latter numbers being ramoved one place farthực. towards the left hand than the