6. A wall is 36 feet high, and a ditch before it is 27 feet wide; what is the length of a ladder, that will reach to the top of the wall from the opposite side of the ditch? Ans. 45 feet. NOTE. A Figure of three sides, like that formed by the wall, the ditch and the ladder, is called a right angled triangle, of which the square of the hypothenuse, or slanting side, (the ladder) is equal to the sum of the squares of the two other sides, that is, the height of the wall and the width of the ditch.

7. A line of 33 yards 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?

Ans. 26,83+ yards.

8. Glasgow is 44 miles west from Edinburgh; Peebles is exactly south from Edinburgh, and 49 miles in a straight line from Glasgow; what is the distance between Edinburgh and Peebles?

Ans. 21,5+miles.

$ 4. EXTRACTION OF THE CUBE ROOT.

To extract the Cube Root of any number is to find another number, which multiplied into its square shall produce the given number

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."

Of the reason and nature of the various steps in the operation of extracting the CUBE ROOT..

Any solid body having six equal sides, and each of the sides an exact square is a CUBE and the measure in length of one of its sides is the root of that cube For if the measure in feet of any one side of such a body be multiplied three times into itself, that is, raised to the third power, the product will be the number of solid feet the whole body contains.

And on the other hand, if the cube root of any number of feet be extracted, this root will be the length of one side of a cubic body, the whole contents of which will be equal to such a number of feet.

Supposing a man has 13824 feet of timber, in distinct and separate blocks of one foot each; he wishes to know how large a solid body they will make when laid together, or what will be the length of one of the sides of that cubic body? To know this, all that is necessary is to extract the cube root of that number, in doing which I propose to illustrate the operation.

20

OPERATION.

In this number, pointed off as the rule directs, there are two periods: of course there will be two figures in the root.

The greatest cube in the right hand period (13) is 8, of which 2 is the root, therefore 2 placed in the quotient is the first figure of the root, and as it is certain we have one figure more to find in the root, we may for the present supply the place of that one figure by a cypher (20) then 20 will express the true value of that part of the root now obtained. But it must be remembered, that the cube root is the length of one of the sides of the cubic body, whose length, breadth and thickness are equal. Let us then form a cube, Fig. 1, each side of which shall be supposed 20 feet; now the side A B of this cube, or either of the sides, shews the root (20) which we have obtained.

8000 feet the solid contents of the Cube.

The Rule next directs, subtract the cube thus found from the said period, and to the remainder bring down the next period, &c. Now this cube (8) is the solid contents of the figure we have in representation. Made evident thus-Each side of this figure is 20, which being raised to the 3d power, that is the length, breadth and thickness being multiplied into each other, gives the solid contents of that figure-8000 feet, and the cube of the root, (2) which we have obtained, is 8, which placed under the period from which it was taken as it falls in the place of thousands, is 8000, equal to the solid contents of the cube A B C D E F, which being subtracted from the given number of feet, leaves 5824 feet.

Hence, Fig. I. exhibits the exact progress of the operation. By the operation 8000 feet of the timber are disposed of, and the figure shews the disposition made of them, into a solid pile, which measures 20 feet on every side.

Now this figure or pile is to be enlarged by the addition of the 5824 feet, which remains, and this addition must be so made, that the figure or pile shall continue to be a complete cube, that is, have the measure of all its sides equal. To do this the addition must be made equally to the three different squares, or faces a, c and b.

The next step in the operation is, to find a divisor; and the proper divisor will be, the number of square feet contained in all the points of the figure, to which the addition of the 5824 feet is to be made.

Hence we are directed to "multiply the square of the quotient by 300,' the object of which is to find the superficial contents of three facesja, c, b, to which the addition is now to be made. And that the square of the quotient, multiplied by 300 gives the superficial contents of the faces a, c, b, is evident from what follows:

The triple square 1200=the superficial contents of the faces a, c, and b.

The two sides A B & A F of the face a, multiplied into each other, give the superficial content of a, and as the faces, a, c, and b, are all equal, therefore the content of face a, multiplied by 3, will give the contents of a, c, and b.

The triple square 1200 the superficial contents of the faces a, c, and b.

Here the quotient figure 2 is properly, two tens, for there is another figure to follow it in the root, and the square of 2, standing as units, is 4, but its true value is 20 (the side A B) of which the square is 400, we therefore lose two cyphers, and these two cyphers are annexed to the figure 3-Hence it appears that we square the quotient with a view to find the superficial content of the face or square a, we multiply the square of the quotient by 3, to find the superficial contents of the three squares) a, c, and b, and two cyphers are annexed to the 3, because in the square of the quotient two cyphers were lost, the quotient requiring a cypher before it in order to express its true value, which would throw the quotient (2) into the place of tens, whereas now it stands in the place of units. Now when the additions are made to the squares a, c, and b, there will evidently be a deficiency, along the whole length of the sides of the squares, between each of the additions, which must be supplied before the figure can be a complete cube. These deficiencies will be 3, as may be seen, Fig. II. nnn.

X

Therefore it is, that we are directed, "multiply the quotient by 30, calling it the triple quotient."

The triple quotient is the sum of the three lines, or sides, against which are the deficiencies n n n, all which meet at a point, nigh the centre of the figure. This is evident from what follows,

The deficiencies are three in number, they are the whole length of the sides, the length of each side is 20 feet,

Therefore 20
3

Triple quotient 60to the length of 3 sides where are deficiencies to be filled.

2 quotient.
30

Triple quotient 60 equal the length of 3 sides, &c.

Here as before, the quotient lacks a cypher to the right hand, to exhibit its true value; the quotient itself is the length of

one of the sides, where are the deficiencies; it is multiplied by 3 because there are 3 deficiencies, and a cypher is annexed to the 3 because it has been omitted in the quotient, which gives the same product, as if the true value of the quotient 20, had been multiplied by 3 alone,

We now have

}

1200 the triple square.
60 the triple quotient.

The sum of which, 1260 is the divisor, equal the number of square feet contained in all the points of the figure or pile, to which the addition of the 5824 feet is to be made,

OPERATION, continued.
13824(24 the root.

This Figure in the root (4) shews the depth of the addition on every point where it is to be made to the pile or figure, represented, Fig. I.

FIG. II. exhibits the additions made to the squares a cb, by which they are covered or raised by a depth of 4 feet.

The next step in the operation is to find a subtrahend, which subtrahend is the number of solid feet contained in all the additions to the cube, by the last figure 4.

Therefore the rule directs, multiply the triple square by the last quotient figure. The triple square it must be remembered, is the superficial contents of the facestui c and b, which multiplied by 4, the depth now added to these faces, or squares, gives the number of solid feet contained in the additions by the last quotient figure 4.

4800 feet, equal the addition made to the squares, or faces, a, c, b, of Fig. I. a depth of 4 feet on each.