PROB. XI. To find the diameter of a circle equal in area, to an ellipsis, (or oval) whose transverse and conjugate diameters are giv.

en.*

RULE. Multiply the two diameters of the ellipsis together, and the square root of that product will be the diameter of a circle equal to the ellipsis.

Let the transverse diameter of an ellipsis be 48, and the conjugate 36: What is the diameter of an equal circle?

48x36=1728, and

1728=41.569+ the Answer. Note. The square of the hypothenuse, or the longest side of a right angled triangle, (by 47th B. 1. Euc.) is equal to the sum of the squares of the other two sides; and consequently the difference of the squares of the hypothenuse and either of the other sides is the square of the remaining side.

PROB XII. A line 36 yards long will exactly reach from the top of a fort to the opposite bank of a river, known to be 24 yards broad. The height of the wall is required?

36x36=1296; and 24×24=576. Then, 1296-576-720, and 720-26-83+yards, the Answer.

PROB. XIII. The height of a tree growing in the centre of a circular island 44 feet in diameter, is 75 feet, and a line stretched from What the top of it over to the ther edge of the water, is 256 feet. is the breadth of the stream, provided the land on each side of the water be level?

256x256=65536; and 75x75=5625: Then, 65536-5625-59911 and/59911=244·76+ and 244·76-44-222.76 feet, Answer.

PROB. XIV. Suppose a ladder 60 feet long be so planted as to reach a window 37 feet from the ground, on one side of the street, and without moving it at the foot, will reach a window 23 feet high on the other side; I demand the breadth of the street?

60X60-3600. 37x37=1369. 23×23=529: Then, 3600-1369 =2231, and 2231=47·23+, and 3600-529-3071, and 3071= 55.41+, then, 47.23+55.41=102.64 feet, the Answer.

PROB. XV. Two ships sail from the same port; one goes due north 45 leagues, and the other due west 76 leagues: How far are they asunder?†

45x45-2025.

76X76-5776. Then,

√7801-88.32 leagues, the answer.

5776+2025=7801

and

EXTRACTION

* The tranfverfe and conjugate are the longest and shortest diameters of an el ripfis; they pafs through the centre, and cross each other at right angles

The fquare root may in the fame manner be applied to navigation; and, when deprived of other means of folving problems of that nature, the following proportion will ferve to find the course.

As the fum of the hypothenuse (or distance) and half the greater leg (whether difference of latitude or departure) is to the less leg; fo is 86, to the angle oppofite the less leg.

EXTRACTION OF THE CUBE ROOT.

A cube is any number multiplied by its square. To extract the cube root, is to find a number which, being multiplied into its square, shall produce the given number.

FIRST METHOD.

RULE.

1. Separate the given number into periods of three figures each, by putting a point over the unit figure, and every third figure beyond the place of units.

2. Find the greatest cube in the left hand period, and put its root in the quotient.

3. Subtract the cube, thus found, from the said period, and to the remainder bring down the next period, and call this the dividend.

4. Multiply the square of the quotient by 300, calling it the triple square, and the quotient by 30, calling it the triple quotient, and the sum of these call the divisor.

5. Seek how often the divisor may be had in the dividend, and place the result in the quotient.

6. Multiply the triple square by the last quotient figure, and write the product under the dividend; multiply the square of the last quotient figure by the triple quotient, and place this product under the last; under all, set the cube of the last quotient figure and call their sum the subtrahend.

7. Subtract the subtrahend from the dividend, and to the remainder bring down the next period for a new dividend, with which proceed as before, and so on till the whole be finished.

Note. The same rule must be observed for continuing the operation, and pointing for decimals, as in the square root.

+ The reasou of pointing the given number, as directed in the rule, is obvious from Gorell. 2, to the Lemma made use of in demonstrating the square root.

Ꮓ

To find the true denominator, to be placed under the remainder, after the operation is finished.

In the extraction of the cube root, the quotient is said to be squared and tripled for a new divisor; but is not really so, till the triple number of the quotient be added to it; therefore when the operation is finished, it is but squaring the quotient, or root, then multiplying it by 3, and to that number adding the triple number of the root, when it will become the divisor, or true denominator to its own fraction, which fraction must be annexed to the quotient, to complete the

root.

Suppose the root to be 12, when squared it will be 144, and multiplied by 3, it makes 432, to which add 36, the triple number of the root, and it produces 468 for a denominator.*

SECOND METHOD.
RULE.

1. Having pointed the given number into periods of three figures each, find the greatest cube in the left hand period, subtracting it therefrom

It may not be amifs to remark here, that the denominators, both of the fquare and cube, fhew how many numbers they are denominators to, that is, what numbers are contained between any fquare or cube number and the next fucceeding fquare or cube number, exclufive of both numbers, for a complete number, of either, leaves no fraction, when the root is extracted, and confequently has no ufe for a denominator, but all the numbers contained between them have occafion for it-Suppofe the fquare root to be 12, then its fquare is 144, and the denominator 24, which will be a denominator to all the fucceeding numbers, until we come to the next fquare number, viz. 169, whofe root is 13, with which it has nothing to do, for between the fquare numbers 144 and 169 are contained 24 numbers excluding both the fquare numbers. It is the fame in the cube; for, suppose the root to be 6, the cube number is 216, and its denominator 126 will be a denominator to all the fucceeding numbers, until we come to the next cube number, viz. 343, whofe root is 7, with which it has nothing to do, as ceafing then to be a denominator; for between the cube 343 and 216 are 126 numbers, excluding both cubes. And so it is with all other denominators, either in the square or cube.

therefrom and placing its root in the quotient; to the remainder bring down the next period and call it the dividend.

2. Under this dividend write the triple square of the root, so that units in the latter may stand under the place of hundreds in the former; and under the said triple square, write the triple root, removed one place to the right hand, and call the sum of these the divisor.

3. Seek how often the divisor may be had in the dividend, exclusive of the place of units, and write the result in the quotient.

4. Under the divisor write the product of the triple square of the root by the last quotient figure, setting down the unit's place of this line, under the place of tens in the divisor; under this line, write the product of the triple root by the square of the last quotient figure, so as to be removed one place beyond the right hand figure of the former; and, under this line, removed one place forward to the right hand, write down the cube of the last quotient figure, and call their sum the subtrahend.

5. Subtract the subtrahend from the dividend, and to the remainder bring down the next period for a new dividend, with which proceed as before, and so on until the whole be finished.

FIRST METHOD BY APPROXIMATION.

RULE.

1. Find, by trial, a cube near to the given number, and call it the supposed cube.

2. Then as twice the supposed cube, added to the given number, is to twice the given number, added to the supposed cube, so is the root of the supposed cube, to the true root, or an approximation to it, 3. By taking the cube of the root, thus found, for the supposed cube, and repeating the operation, the root will be had to a greater degree of exactness,

EXAMPLE.

It is required to find the cube root of 54854153 ? Let 64000000=supposed cube, whose root is 400 ; Then, 64000000

54854153

2

2

182854153)69483322400(379=root nearly.

=

Again, let 54439939 supposed cube, whose root is 379.
Then, 54439939 54854153

108879878

2

2

109708306

1. Divide the resolvend by three times the assumed root, and reserve the quotient.

2. Subtract one twelvth part of the square of the assumed root from the quotient.

3. Extract the square root of the remainder.

4. To this root add one half of the assumed root, and the sum will be the true root, or an approximation to it; take this approximation as the assumed root, and, by repeating the process, a root farther approximated will be found, which operation may be farther repeated,

as