Any two sides of a right angled triangle given, to find the third

side.

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.

EXAMPLES.

1. The top of a castle from the ground is 45 yards high, and is 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.

2. 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.

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.

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.

N. B. 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 battle, or to find the number of ranks and files.

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

EXAMPLES.

1. An army consists of 331776 men, I desire to know how many in rank and file? Ans. 576. 2. 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.

To find the area of a piece of land in form of a triangle. RULE. Add together the three sides, from half their sum subtract each side, and note the remainder, then multiply the half sum by one of those remainders, and that product by another remainder; the square root of the last product will be the area.

EXAMPLE.

Suppose a triangular piece of land, whose sides are 24, 16, and 12 rods; what is the area?

24+16+12=52÷2-26 for half.

26-24 2 then 26X2X10X14-7280, the square root of which is 85,32+ rods.

26-16-10

26-12-14

Multiplying the longest side by half of the nearest distance, to its opposite angle; or, multiplying the longest side by the nearest distance to its opposite angle and taking half of the product, gives the area.

EXAMPLE.

If the three sides of a piece of land in form of a triangle be 15, 14, and 13 rods, required the area?

The nearest distance would be 11,2 rods.

5,6

Or, 11,2 15

half
longest side 15

as before.

If the three sides be 120,5, 112,6 and 90,3 rods, required the area?

Ans. 4832,7 rods 30 acres and 32,7 perches. Any irregular four-sided piece of land may be divided into two triangles by a diagonal line, and a five-sided piece into three triangles by two diagonals. If the length of the sides be agreed on, there can be no dispute on the admeasurement, as all who are acquainted with the rule will agree in the result.

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. 1. Point every third figure of the cube given, beginning at the unit's place, seek the greatest cube to the first point and subtract it therefrom, put the root in the quotient, and bring down the figures in the next point to the remainder for a resolvend.

2. Find a divisor by multiplying the square of the quotient

by 3. See how often it is contained in the resolvend, rejecting the units and tens, and put the answer in the quotient.

3. To find the subtrahend. 1. Cube the last figure in the quotient. 2. Multiply all the figures in the quotient by 3, except the last, and that product by the square of the last. 3. Multiply the divisor by the last figure. Add these products together, gives the subtrahend, which subtract from the resolvend; to the remainder bring down the next point, and proceed as before.

Roots. 1. 2. 3. 4. 5. 6. 7.
Cubes. 1. 8. 27. 64. 125. 216. 343.

EXAMPLE.

8.

9.

512. 729.

Another new and more concise method of extracting the CUBE ROOT.

RULE. 1. Point every third figure of the cube given, beginning at the unit's place, then find the nearest cube to the first point, and subtract it therefrom, put the root in the quotient, bring down the figures in the next point to the remainder for a resolvend.

2. Square the quotient and triple the square for a divisor -as, 4x4x3=48. Find how often it is contained in the resolvend, rejecting units and tens, and put the answer in the otient.

3. Square the last figure in the quotient put it on the right hand of the divisor:

As 6×6=36 put to the divisor 48-4836.

4. Triple the last figure in the quotient, and multiply by the former, put it under the other, units under the tens, add them together, and multiply the sum by the last figure in the quotient, subtract that product from the resolvend, bring down the next point, and proceed as before.

5556 X 6 = 33336

Square of 46-2116×3—6348 div.

Square of 3-9 put to 6348=*634809

3x3x46= 414

1916847

638949x3=1916847

2. What is the cube root of 389017? 3. What is the cube root of 5735339?

Ans. 73. Ans. 179.

When the given number consists of a whole number and decimal together, make the number of decimals to consist of 3, 6, 9, &c. places, by adding ciphers thereto, so that there may be a point fall on the unit's place of the whole number. 4. What is the cube root of 12,977875? Ans. 2,35. 5. What is the cube root of 36155,027576? Ans. 33,06+

To extract the cube root of a vulgar fraction.

RULE. Reduce the fraction to its lowest terms, then extract the cube root of the numerator and denominator for a new numerator and denominator; but if the fraction be a surd, reduce it to a decimal, and then extract the root from it.

When the quotient is 1, 2, or 3, there must be a cipher put to supply the place of tens,

To extract the cube root of a mixed number.

RULE. Reduce the fractional part to its lowest terms, and then the mixed number to an improper fraction, extract the cube roots of the numerator and denominator for a new numerator and denominator; but if the mixed nur.aber given be a surd, reduce the fractional part to a decimal, annex it to the whole number, and extract the root therefrom.

1. If a cubical piece of timber be 47 inches long, 47 inches broad, and 47 inches deep, how many cubical inches does it contain? Ans. 103823.

2. There is a cellar dug that is 12 feet every way, in length, breadth, and depth, how many solid feet of earth were taken out of it? Ans. 1728.

To find the side of a cube that shall be equal in solidity to any given solid, as a globe, cylinder, prism, cone, &c. RULE. The cube root of the solid content of body given is the side of the cube of equal solidity.

EXAMPLE.

any solid

If the solid content of a globe is 10648, what is the side of a cube of equal solidity?

Ans. 22.

The side of the cube being given, to find the side of that cube, that shall be double, treble, &c. in quantity to the given cube.

RULE. Cube the side given, and multiply it by 2, 3, &c. the cube root of the product is the side sought.

EXAMPLE,

There is a cubical vessel, whose side is 12 inches, and it