INVOLUTION.

INVOLUTION is the method of finding the square, cube, &c., of any given number.

When the given number is used twice as factor, the product is the second power, or the square of that number; when three, the third power, or cube; when four times, the fourth power, &c.

The respective powers of numbers are denoted by small figures, called indices or exponents, being placed on the right a little above the line. Thus 4x4=16, is the second power, or the square of 4, and may be written 42, where 2 is the index.

Is the reverse of Involution, and is the method of finding the roots of numbers, as the square root, cube root, &c.

Roots are sometimes denoted by writing the character before the number or power, with the index of the root against it, thus the square root of 80 is 2/80, or only 80 without 2, and the cube root of it is 2/80.

EXTRACTION OF THE SQUARE ROOT.

THIS is to discover such a number as being multiplied once into itself will produce the given number, or square.

ROOTS. 1. 2. 3. 4. 5. 6. 7. 8. 9.
SQUARES. 1. 4. 9. 16. 25. 36. 49. 64. 81.

The following Example with explanation will serve as a Rule.

1. What is the square root of 119025?

1190-25(345 Ans.

9

64)290

256

3425

I divide the given number into periods of two figures each, as in the example, commencing at the unit figure. Then I find the nearest square root, 3, of the first period 11, and put it in the quotient. Next I subtract the square of it, 9, from the first period, and 685)3425 to the remainder annex the next period, 90, for a dividend. Then double the root, 3, for a divisor, which may be called the trial divisor; then 29 divided by 6, gives 4, the next figure in the root; I put this 4 after the 6, and it gives 64 for the true divisor. I then multiply 64 by 4, and subtract the product from the preceding remainder, leaving 34. I now bring down 25, then I bring down 64, doubling the unit figure 4, which is the second figure in the quotient, making 68, which goes 5 times into 342, then I make 5 the unit figure in the divisor, and place it also in the quotient, and by it multiply the divisor, 685.

2. What is the square root of 106929 ? Ans. 327. 3. What is the square root of 2268741? Ans. 1506·23+ 4. What is the square root of 7596796 ?

Ans. 2756.228+

Ans. 6031.

5. What is the square root of 36372961? 6. What is the square root of 22071204? Ans. 4698.

When the given number consists of a whole number and decimals together, make the number of decimals even, by adding ciphers to them, so that there may be a point fall on the unit's place of the whole number.

7. What is the square root of 3271-4207 ?

Ans. 57.19.+

8. What is the square root of 4795·25731 ?

Ans. 69.247.+ 9. What is the square root of 4.372594 ? Ans. 2·091+ 10. What is the square root of 2.2710957 ?

Ans. 1.50701.+

11. What is the square root of '00032754 ?

Ans. 01809.+

12. What is the square root of 1.270054 ?

Ans. 1.1269.+

To extract the Square Root of a Vulgar Fraction.

RULE. Reduce the fraction to its lowest terms: then extract the square root of the numerator for a new numerator, and the square root of the denominator for a new denominator.

If the fraction be a surd (i.e.) a number where a root can never be exactly found, reduce it to a decimal, and extract the root of it.

To extract the Square root of a mixed number.

RULE 1. Reduce the fractional part of the mixed number to its lowest term, and then the mixed number to an improper fraction.

2. Extract the roots of the numerator and denominator for a new numerator and denominator.

3. If the mixed number given be a surd, reduce the fractional part to a decimal, annex it to the whole number, and extract the square root thereof.

19. What is the square root of 51? 20. What is the square root of 27 21. What is the square root of 943 ?

SURDS.

22. What is the square root of 8511? 23. What is the square root of 81⁄2? 24. What is the square root of 63?

THE APPLICATION.

Ans. 7.

Ans. 51.

Ans. 34.

1. There is an army consisting of a certain number of men, who are placed rank and file (that is, in the form of a square, each side having 576 men)-How many does the whole square contain ? Ans. 331776. 2. A certain pavement is made exactly square, each side of which contains 97 feet-How many square feet are contained therein ?

Ans. 9409.

To find a mean proportional between any two given numbers.

RULE. The square root of the product of the given numbers is the mean proportional sought.

3. What is the mean proportional between 3 and 12?

3×12=36, then 36=6 the mean proportional.

4. What is the mean proportional between 4276 and 842 ? Ans. 1897.4. +

To find the side of a square equal in area to any given superficies. RULE. The square root of the content of any given superficies, is the square equal sought.

5. If the content of a given circle be 160—what is the side of the square? Ans. 12.649.+ 6. If the area of a circle be 750-what is the side of the square equal? Ans. 27-38612.+

The area of a circle given to find the diameter.

RULE. AS 355 452, or as 11-273239: the area to the square of the diameter:-or multiply the square root of the area by 1.12837, and the product will be the diameter.

7. What length of cord will be fit to tie to a cow's tail, the other end fixed in the ground, to let her have liberty of eating an acre of grass, and no more, supposing the cow and tail to be 5 yards ? Ans. 6.136+perches.

The area of a circle given to find the periphery or circumference. RULE.-AS 113: 1420, or, as 1 : 12-56637: the area to the square of the periphery;—or, multiply the square root of the area by 3.5449, and the product is the circumference.

8. When the area is 12-what is the circumference? Ans. 12.2799+

9. When the area is 160-what is the periphery?

Ans. 44.839+

Any two sides of a right-angled triangle given to find the third side. 1. The base and perpendicular given to find the hypotenuse. RULE. The square root of the sum of the squares of the base and perpendicular is the length of the hypotenuse.

10. The top of a castle from the ground is 45 yards high, and surrounded with a ditch 60 yards broad; what length must a ladder be to reach from the outside of the ditch to the top of the castle ?

Ans. 75 yards.

11. The wall of a town is 25 feet high, which is surrounded by a moat of 30 feet in breadth-I desire to

know the length of a ladder that will reach from the outside of the moat to the top of the wall? Ans. 39.05+feet.

II. The hypotenuse and perpendicular given to find the base. RULE.-The square root of the difference of the squares of the hypotenuse and perpendicular, is the length of the base.

III. The base and hypotenuse given to find the perpendicular. RULE.-The square root of the difference of the squares of the hypotenuse and base, is the height of the perpendicular.

The two last questions may be varied for examples to the two last propositions.

Any number of men being given, to form them into a square battalion, or to find the number of rank and file.

RULE. The square root of the number of men given, is the number of men either in rank or file.

12. An army consisting of 331776 men-I desire to know how many rank and file?

Ans. 576. 13. A certain square pavement contains 48841 square stones, all of the same size-I demand how many are contained in one of the sides?

Ans. 221.

EXTRACTION OF THE CUBE ROOT.

To extract the cube root is to find out a number, which being multiplied into itself, and then into that product, produceth the given number.

RULE. Divide the given number from the place of units into periods of 3 figures.

Then find the greatest cube in the left hand period; (as 8 in the example below,) subtract it (8) from the first period, and place the root in the quotient (2 in the ex.); the square of this root multiplied by 3, will be the trial divisor for finding the next figure of the root; (22×3

12.) Next multiply the figure or figures of the root, placed in the quotient, by 3, to the product of which annex the next root figure found by the trial divisor; (Ex. 1st, figure 2 in quotient x3=6 to which annex 3 found by the trial divisor 12-63.) Multiply this number by the second new root figure, (Ex. 63x3=189) the product of which place two figures to the right, under the trial divisor, which being added will make the true divisor. (Ex. 1389x3, [or will go 3 times into the resolvend 4487]=4167, the subtrahend, leaving 320, to which bring down the next period, 168.) Then below the true divisor (Ex. 1389) place the square of the last found root figure, (Ex. 32=9) which being added to the two sums above will give the next trial Divisor. For the true divisor proceed as before.