There is a circle whose diameter is 4 inches; I demand the diameter of a circle 3 times as large?

4x4=16; and 16x3=48; and 48=6.928+ inches, Answer. PROB. XI. To find the diameter of a circle equal in area, to an ellipsis, (or oval) whose transverse and conjugate diameters are given.*

KULE. 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?

43×36 1728, and 1728=41.569+the Answer. Note. The square of the hypothenuse, or the longest side of a right angled triangle, 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?

36×36=1296; and 24×24=576. Then, 1296–576=720, and √720=2633+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 the top of it over to the hither edge of the water, is 256 feet. What is the breadth of the stream, provided the land on each side of the water be level?

256×256=65536; and 75=75=5625: Then, 65536-5625= 59911 and 59911=244·76+ and 244·76—=222·76 feet, Ans.

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?

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?† 88-32 leagues, Answer. PROB. XVI. Given the sum of two numbers, and the difference of their squares, to find those numbers.

RULE. Divide the difference of their squares by the sum of the numbers, and the quotient will be their difference. The two num

* The transverse and conjugate are the longest and shortest diameters of an ellipsis; they pass through the centre, and cross each other at right angles, and the diameter of the equal circle is the square root of the product of the diameters of the ellipsis.

†The square root may in the same manner be applied to navigation; and, when deprived of other means of solving problems of that nature, the following proportion will serve to find the course.

As the sum of the hypothenuse (or distance) and half the greater leg (wheth er difference of latitude or departure) is to the less leg; so is 86, to the angle opposite the less leg.

bers may then be found, from their sum and difference, by Prob. 4, page 57.

Ex. The sum of two numbers is 32, and the difference of their squares is 256, what are the numbers?

Ans. The greater is 20. The less 12.

PROB. XVII. Given the difference of two numbers, and the difference of their squares, to find the numbers.

RULE. Divide the difference of the squares by the difference of the numbers, and the quotient will be their sum.

ceed by Prob. 4, p. 57.

Then pro

Ex. The difference of two numbers is 20, and the difference of their squares is 2000; what are the numbers?

Ans. 60 the greater. 40 the less. Examples for the two preceding problems.

1. A and B played at marbles, having 14 apiece at first; B having lost some, would play no longer, and the difference of the squares of the numbers which each then had, was 336; pray how many did B lose? Ans. B lost 6.

2. Said Harry to Charles, my father gave me 12 apples more than he gave brother Jack, and the difference of the squares of our separate parcels was 288; Now, tell me how many he gave as, and you shall have half of mine.

Ans. Harry's share 12.

Jack's share

EXTRACTION OF THE CUBE ROOT.

6.

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 fi gure 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 divi dend.

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.

The reason of pointing the given number, as directed in the rule, is obvious from Coroll. 2, to the Lemma made use of in demonstrating the square root. The process for extracting the Cube Root may be illustrated in the same manner as that for the Square Root. Take the same number 37, and multiply

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.

as before, collecting the twice 21 into one sum, as they belong to the same place and the operation will be simplified, 37 3=50653.

3X321893×3a ×7 3x30218900=3×302X7
4413×3×72 3x30 4410=3x30x7

3X3

As 27 or 27000 is the greatest cube, its root is 3 or 30, and that part of the cube is exhausted by this extraction. Collect those terms which belong to the same places, and we have 32 ×7=63, and 2×32x7=126, and 63+126=3×32 X7=189; and 2×3×72=294 and 3×72=147, and 294+147=441=3x3x72, for a dividend, which divided by the divisor, formed according to the rule, the quotient is 7, for the next figure in the root. And it is evident on inspecting the work, that that part of the cube not exhausted is composed of the several products which form the subtrahend, according to the rule. The same may be shown in any other case, and the universality of the rule hence inferred. The other method of illustration, employed on the square root, is equally applicable in this case.

2

37-30-7, and 30+7=3024-2x30x7+72

30-+7 the multiplier.

3032x302X7+30×73

302 x7+2×30×724-73

373=50€53=3034-3×502×7+3×30×72+73(304-7=37

Divisor3x302+3x30

303

)3X302×7+3×30×72473 dividend.

3x302×7+3X30X72+73 subtrahend.

It is evident that 303 is the greatest cube. When its root is extracted, the next thice terms constitute the dividend; and, the several products formed by means of the quotient or second figure in the root, are precisely equal to the remaining parts of the power, whose root was to be found.

The arithmetical demonstrations of the Rules for extracting either the square or cube root, are not only more consistent with the plan of an Arithmetick than demonstrations on the figure, called a square, and the solid, called a cube, but they are much more readily understood by those unaccustomed to the mathematical consideration of solid bodies.

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.

2.Div. 1689750)14161824—2d. Divid. 75×75x300

75X30

125

78875-1st. Subtra.

1687500-2d. Trip.sq. 2250 2d. do. quo.

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 denomi nator 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 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.

EXAMPLE.

Required the Cube Root of 16194277 ?

It may not be amiss to remark here, that the denominators, both of the square and cube, shew how many numbers they are denominators to, that is, what numbers are contained between any square or cube number and the next succeeding square or cube number, exclusive of both numbers, for a complete number, of either, leaves no fraction, when the root is extracted, and consequently has no use for a denominator, but all the numbers contained between them have occasion for it: Suppose the square root to be 12, then its square is 144, and the denominator 24, which will be a denominator to all the succeeding numbers, until we come to the next square number, riz. 169, whose root is 13, with which it has nothing to do, for between the square numbers 144 and 169 are contained 24 numbers excluding both the square numbers. It is the same 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 succeeding numbers, until we come to the next cube number, viz. 343, whose root is 7, with which it has nothing to do, as ceasing 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.