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 60-to 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. 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 faces a 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. 1 a depth of 4 feet on each. 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 contents of those additions 4n4, 4n, 4n, Fig. III. 16 square of the last quotient." 360 60 960 feet disposed in the deficiencies, between the additions to the squares a cb, 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. FIG. IV. 20 4n Lastly, "Cube the last quotient figure." This is done to fill the deficiency, Fig. III. left at one corner, in filling up the other deficiencies, nnn. This corner is limited by those deficiencies on every side, which were 4 feet in breadth, consequently the square of 4 will be the solid content of the corner which in Fig. IV. e e e is seen filled. 44 F4n Now the sum of these additions make the subtrahend, which subtract from the dividend and the work is done. 16 4 64 feet is the corner ee e, where the additions n nn, 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. Fig. III. and Fig. IV. shew the changes which the pile passes 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. Feet. 8000 Contents of Fig. I. 4800 addition to the faces or square a, c, and b, Fig. II. 64 addition at the corner, e, e, e, Fig. IV where the additions 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 or attention. Scholars more especially will meet with difficulty. For their assistance, small blocks might be formed 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 pupils into the right conceptions of those views, which are here given of the nature of this operation. 3. What is the cube root of 21024576? Ans. 276. |