DEMONSTRATION

Of the Reason and Nature of the various steps in the operation of extracting the CUBE ROOT.

ANY solid body having six e qual sides, and each of these 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 multipli ed 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.

B

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 a cubic body, whose length, breadth, and thickness are equal. Let us then form a cube, Fig. I. 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.

6000 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, & 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. 1. 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 square 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 remain; 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 "multiply the square of the quotient by 300," the object of which is, to find the superficial contents of the three faces a,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.

Side A B=207

Side AF20 of the face, a.

Superficial content —400 j

3

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

THE two sides A B and 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 the face, a multiplied by 3, will give the contents of a, c, and b.

2 quotient figure

2

4 the square of 2

300

The triple square 1200—the super ficial 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 anexed 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, & 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 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. 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 sum of the three lines, or sides against which

2 quotient, 30

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 sides; the length of each side is 20 feet, therefore 20

3

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

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 pro duct, as if the true value of the quotient, 20, had been multiplied by 3 alone. 60 the triple quotient. 1200 the triple square.

We now have

{

THE Sum of which, 1260 is the divisor, equal to 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.

5824 subtrahend:

20

20

20

1200 triple square.
4 last quotient figr.

FIG II. exhibits the additions made to the squares 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 subtrahend, which subtrahend is the number of solid feet, contained in all the additions to the cube, by the last fig

ure 4.

THEREFORE, the rule directs, "multiply the triple square by the last quotient figure."

THE triple square, it must be remem bered, is the superficial contents of the faces a, c, and b, which multiplied by 4, the depth now added to those 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, a, b, of Fig. I. a depth of 4 feet on each.

FIG. III.

4n

60 triple quotient.

THEN,"multiply the square of the last quotient figure by the triple quotient." This is to fill the deficiencies n, n, n, Fig. II. Now these deficiencies are limited in length,by the length of the sides (20) and the triple quotient is the sum of the length of the deficiencies. They are limited in width by the last quotient figure (4) the square of which gives the area, or superficial contents at one end, which multiplied into their length, or the triple quotient, which is the same thing, gives the con-tents of those additions 4n4, 4n,

16 square of the last quotient figure. An, Fig. III.

360

60

960 feet disposed in the deficiences, between the additions to the squares, a, c, b. Fig. III. exhibits these deficiencies supplied, 4n4, 4n, 4n, and discovers another deficiency where these approach together, of a corner wanting to make the figure a complete cube.

F4n

44

Now the sum of these additions make the subtrahend, which subtract from the dividend, and the work is done.

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

FIGURE IV. shews the pile which 13824 solid blocks of one foot each, would make when laid together. The root (24) shews the length of a side. Fig. I. shews the pile which would be formed by 8000 of those blocks, first laid together; Fig. II. and Fig. III. shew the changes which the pile passes thro' 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.

Feet.

8000 Contents of Fig. I.

4800 addition to the faces or squares a, c, and b, Fig. 11.

960 addition to fill the deficiencies n, n, n, Fig. III,

64 addition at the corner e, e, e, Fig. IV. where the additions which fill the deficiencies n, n, n, approach together.

13824 Number of blocks, or solid feet, all which are now disposed in Fig. IV. forming a pile, or solid body of timber, 24 feet, on a side.

Such is the demonstration of the reason and nature of the various steps in the operation of extracting the cube root. Proper views of the Figures, and of those steps in the operation illustrated by them, will not generally be acquired without some diligence and attention, Scholars, more especially will meet with difficulty. For their assistance, small blocks might be former of wood in imitation of the Figures, with their parts in different pieces. By the help of these, Masters, in most instances, would be able to lead their pu pils into right conceptions of those views, which are here given of the nature of this operation.

3. WHAT is the cube root of 21024576 ?

Answer, 276.

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