8. Glasgow is 44 miles weft from Edinburgh: Peebles is exactly fouth from Edinburgh, and 49 miles in a straight line from Glafgow; 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 put-⚫ ting 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 faid period, and to the remainder bring down the next period, and call this the dividend.

4. Multiply the fquare of the quotient by 300, calling it the triple fquare, and the quotient by 30, calling it the triple quotient, and the fum of these call the divifor.

5. "Seek how often the divifor may be had in the dividend, and place the refult in the quotient.

6. "Multiply the triple fquare by the laft quotient figure and write the product under the dividend; multiply the fquare of the laft quotient figure by the triple quotient, and place this product under the laft; under all, fet the cube of the last quotient figure, and call their fum the fubtrahend.

7. "Subtract the fubtrahend from the dividend, and to the remainder bring down the next period for a new dividend, with which proceed as before, and fo on till the whole be finifhed.

NOTE. "The fame rule must be obferved for continuing the operation, and pointing for decimals, as in the fquare root."

X®

Of the Reafon and Nature of the various fleps in the operation of extrading the

CUBE ROOT.

Any folid body having fix equal fides, and each of thefe fides an exact Square, is a CUBE, and the measure in length of one of its fides is the root of that cube. For if the measure in feet of any one fide of fuch a body be multiplied three times into itself, that is, raised to the third power, the product will be the number of folid 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 fide of a cubic body, the whole contents of which will be equal to fuch a number of feet.

2. Suppofing a man has 13824 feet of timber, in diftinct and feparate blocks of one foot each; he wishes to know how large a folid body they will make when laid together, or what will be the length of one of the fides of that cubic body?

To know this, all that is neceffary is to extract the cube root of that number, in doing which I propofe to illuftrate the 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 firft figure of the root, and as it is certain we have one figure more to find in the root, we may for the prefent fupply 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 fides of the cubic body, whofe length, breadth, and thickness are equal. Let us then form a cube, Fig. I. each fide of which fhall be fuppofed 20 feet; now the fide A. B. of this cube, or either of the fides, fhews the root, (20) which we have obtained.

8000 fect the folid contents of the cUBE.

The Rule next directs, fubtract the cube, thus found, from the faid period and to the remainder bring down the next period, &c. Now this cube (8) is the folid contents of the figure we have in representation. Made evident thus-Each

fide 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 folid 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 the ufands, is 8000, equal to the folid contents of the cube ABCDEF, which being fubtracted from the given number of feet, leaves 5824 feet.

Hence Fig. I, exhibits the exact progrefs of the operation. By the operation 8000 ft. of the timber are difpofed of, and the figure fhews the difpofition made of them, into a fquare folid pile which measures 20 feet on every fide.

Now this figure or pile is to be enlarged by the addition of the 5824 feet, which remains; and this addition must be fo made, that the figure or pile, fhall continue to be a complete cube, that is have the measure of all its fides equal.

To do this the addition must be made equally to the three different squares, or faces a, c and b.

The ne,xt step, in the operation is, to find a divifor ; and the proper divifor will be, the number of fquare 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 " multiply the fquare of the quotient by 300,” the object of which is, to find the fuperficial contents of the three faces, a, c, b, to which the addition is now to be made. And that the fquare of the quotient, multiplied by 300 gives the superficial contents of the faces a, c, b, is evident from what follows.

Side A B-201
Side A F-20
Superficial contents — 400

3

2 quotient figure.

2

of the face, a.

The triple fquare 1200=the fuperficial contents of the faces, a, c, and b.

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

4 the fquare of 2

300

The triple fquare of 1200 the fuperficial 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 fquare of 2, ftanding as units, is 4, but its true value is 20 (the fide A B) of which the fquare is 400, we therefore lofe two cyphers, and these two cyphers are annexed to the figure 3.-Hence it appears, that we fquare the quotient, with a view to find the fuperficial content of the face, or fquare a; we multiply the fquare of the quotient by 3, to find the fuperficial contents of the three squares, a, c, and b, and two cyphers are annexed to the 3, because in the fquare of the quotient two cyphers were loft, the quotient requiring a cypher before it in order to exprefs its true value which would throw the quotient (2) into the place of tens, whereas now it ftands 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 fides of the fquares 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 feen, Fig. II. n. n. n.

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

The triple quotient is the fum of the three lines, or fides 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 3 in number, they are the whole length of the fides; the length of each fide is 20 feet, therefore

20 3

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

2 quotient. 30

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

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

itfelf, is the length of one of the fides, where are the deficiencies; it is multiplied by 3, because there are 3 deficiencies, and a cypher is annexed to the 3 becaufe it has been omitted in the quotient, which gives the fame product, as if the true value of the quotient, 20, had been multiplied by S alone.

We now have

The fum of which,

1200 the triple fquare.

60 the triple quotient.

1260 is the divifor, equal the number of fquare feet contained, in all the points of the figure or pile, to which the addition of the 5824 feet is to be made.

4800

960

64

582 fubtrahend.

0000

FIG. II

20

120C triple fquare.

20

4 la quotient figure.

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

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

The next step in the operation is to find a fubtrahend which fubtrahend is the number of folid feet contained in all the additions to the cube, by the laft figure 4.

Therefore, the rule directs, " multiply the triple fquare by the laft quotient figure." The triple fquare, it must be remembered, is the fuperficial contents of the faces a, c, and b, which multiplied by 4, the depth now added to thofe faces, or fquares, gives the number of folid feet contained in the additions by the laft quotient figure 4.

4800 feet, equal to the addition made to the fquares, or faces, a, c, b, of Fig

I. a depth of 4 feet on each.

4

20

FIG. III.

F4n

4n

60 triple quotient.

Then," multiply the fquare of the laf quotient figure by the triple quotient." This is to fill the deficiencies, n, n, n, Fig. II. Now thefe deficiencies are limited in length, by the length of the fides (20) and the triple quotient is the fum of the length of the deficiencies. They are limited in width by the last quotient figure (4) the fquare of which gives the area, or fuperficial contents at one end, which multiplied into their length, or the triple quotient, which is the fame thing, gives the contents of thofe additions 4n4, 4n, 4n.

16 fquare of the last quotient figure, Fig. III.

360

60

960 feet difpofed in the deficiencies, between the additions to the fquares, a,c, b, Fig. III. exhibits thefe deficiencies fupplied, 4n4, 4n, 4n, and difcovers another deficiency where thefe approach together, of a corner waniing to make the figure a complete cube.

FIG IV.

4 n

20

20

Laftly, "cube the left quotient figure." This is done to fill the deficiency Fig. III. left at one corner, in filling up the other deficiencies, n, n, n. This corner is limited by thofe deficiencies on every fide, which were 4 feet in breadth, confequently, the fquare of 4 will be the folid content of the corner, which in Fig. IV. e, e, e, is feen filled.

20

F4n

44

16

4

Now the fum of thefe additions make the fubtrahend, which fubtract from the dividend, and the work is done.

64 feet difpofed in the corner, e, e, e, where the additions n, n, n, approach together.

FIGURE IV. fhews the pile which 13824 folid blocks of one foot each, would make when laid together. The root (24) thews the length of a fide. Fig. I. fhews the pile which would be formed by 8000 of thofe blocks, firit laid together: Fig. II. and Fig. III. fhews the changes which the pile paffes through in the addition of the remaining 5824 blocks or feet.

PROOF. By adding the contents of the first figure, and the additions exhibited in the other figures together.