6. Deduct the subtrahend from the resolvend; to the difference annex the next period of the given number; then proceed as before to find a divisor, &c.

EXAMPLES. What is the cube of 1000? Answer 10.

D. ................................3375......15 Answer.

.17

D°..
D°.

42

Do...

...2.571282.

.25 ......2.924018.
..3.476027.

A SIMPLER METHOD OF EXTRACTING THE CUBE ROOT.

RULE 1. By trial, find the nearest cube to the given number, and call it the assumed cube.

2. Then, as the sum of the given number, and double the assumed cube, is to the sum of the assumed cube, and double the given number, so is the root of the assumed cube to the root required, nearly.

3. Assume the cube of the root last found as a new assumed cube, and proceed as before, by which a root will be found approximating still more nearly to the real root. This is a sufficiently exact method for all general purposes. The oftener the operation is repeated, the more exact will be the result.

EXAMPLE. Required the cube root of 128000.

It lies between 50 and 51.

Assumed cube 125000. Root 50.

As 128000+twice 125000 or 125000+twice 128000 or 378000 : 381000 50-so 50 to 50.4 nearly. Cube of 50.4 128024, and

:

064 exceeding 128000 by 24.064 only.

LOGARITHMS.

LOGARITHMS are a series of numbers, or rather roots of numbers, calculated in order to facilitate those operations, which cannot be performed, without extreme labour and delay, by common arithmetic.

By means of a table of logarithms, multiplication is performed by addition, and division by subtraction.

The integer prefixed to a logarithm is called its index; thus 2 is the index, the logarithm 2.2081725.

The logarithm of 10 is 1; of 100, is 2; of 1000,3; of 10000, 4, &c. When the number for which a logarithm is wanted lies between 1 and 10; 10 and 100; 100 and 1000, &c. a reference must be made to a table of logarithms.

The index of the logarithm of any integer or mixed number is always 1 less than the number of integer places in the natural number. Thus, between 100 and 1000, it is 2; 1000 and 10000, 3, &c.

The index is generally omitted in tables for the sake of brevity.

To find the Logarithm of any mixed Decimal Number.

RULE. Find the logarithm as if it were a whole number, and prefix the index of the integer part.

What is the logarithm of 259.7, is 41447, to which, if the index be prefixed, the logarithm is 2.41447.

TO FIND THE LOGARITHM OF A VULGAR FRACTION.

SUBTRACT the logarithm of the denominator from the logarithm of the numerator, borrowing 10 in the index, when the denominator is the greatest, the remainder is the logarithm required.

What is the logarithm of ?

Logarithm of 5=69897

9=95424

9.74473 Answer.

MULTIPLICATION BY LOGARITHMS.

ADD the logarithms together of the multiplier and multiplicand, the sum is the logarithm of the answer required.

Multiply 9 by 253.

Logarithm of 9.95424

253 40312

1.35736

35736 is the logarithm of 2277, the Answer.

DIVISION BY LOGARITHMS.

SUBTRACT the logarithm of the divisor from the logarithm of the dividend; the difference is the logarithm of the quotient.

Divide 477 by 3.

Logarithm of 477 .67852

3 47712

20140

2014 logarithm of 159, the Answer.

INVOLUTION BY LOGARITHMS.

MULTIPLY the logarithm of the root by the index of the power to which it is to be raised; the product is the logarithm of the answer.

Required the 5th power of 11.

Logarithm of 11=1.4139

X 5

5.20695

20695 is the logarithm of 161051, the Answer.

EVOLUTION BY LOGARITHMS.

DIVIDE the logarithm of the given number by the index of the power; the quotient is the logarithm of the root.

EXAMPLE. What is the cube root of 156252 ?

Logarithm of 15625=4.19382. 4.19382-3=1.397606. 1.397606= logarithm of 25, the Answer.

RULE OF THREE BY LOGARITHMS. RULE. Add together the logarithms of the 2d from their sum, deduct the logarithm of the 1st. the logarithm of the answer.

EXAMPLE. If 110 give 19, what will 94 give? 110: 94: 19:

Logarithms 2.04139. 1.97313: 1.27875

1.97313

3.25188

Deduct 2.04139

Difference 1.21049

and 3d numbers, and The difference will be

.21049 logarithm of 16,22, the Answer,

SECTION II.

OF

PRACTICAL GEOMETRY AND MENSURATION.

DEFINITIONS.

THE first definition in Geometry is a Point, or dot, which is abstractedly considered as having no parts or magnitude; neither length, breadth, nor depth.

A Line is considered as length without breadth.

A Superficies, or surface, is an extension, having only length and breadth.

A Body or Solid, is a figure of three dimensions; namely, length, breadth, and thickness.

Hence surfaces are the extremities of solids; lines the extremities of surfaces; and points the extremities of lines.

Lines are either right, or curved, or mixed of these two.

A right or straight line is one which lies evenly between its extreme points, and is the shortest distance between those points.

A curve continually changes its direction between its extreme points. Parallel lines are those which have no inclination towards each other; or which, being every where equi-distant, would never meet, although ever so far produced.

An angle is the inclination or opening between two lines, having different directions, and meeting in a point; hence a Plane

Angle is a space or corner formed by two straight lines meeting each other.

When a straight line AD standing upon another CB, makes angles ADC, ADB, on each side, equal to one another; each of these angles is called a Right Angle; and the line AD is said to be Perpendicular to the line CB.

B

An angle is usually expressed by three letters; that placed at the angular point being always in the middle: as D is the angle of ABC.

An Obtuse Angle is that which is greater than a right

angle, as ABC.

An Acute Angle is that which is less than a right angle, as DBC.

C

By an ANGLE of ELEVATION is meant the angle contained between a line of direction, and any plane on which the projection is supposed to