31. There is a number, one third, one fourth, and one fifth of which added together are 94; what is the number?

32. What is that number a fifth of which exceeds a sixth of it by 4?

33. What number is that of which a fourth part exceeds a seventh part by 9?

34. In a certain orchard there are apple, peach, and pear trees; the apple trees are two more than half the whole; the peach trees are one third of the whole, and are 14 less than the apple trees; the rest are pear trees; how many are there of each kind, and how many in all?

SECTION XIX.

SOLID MEASURE.

Whatever has length and breadth, and thickness is a solid. A block of wood 1 inch long, 1 inch high, and 1 inch wide, is a solid inch. A block 1 foot long, 1 foot wide, and 1 foot high, is a solid foot. A block 1 yard long, 1 yard wide, 1 yard high, is a solid yard.

1. How many solid inches are there in a block 3 in. long, 2 in. wide, and 1 in. high?

2. How many in a block 4 in. long, 3 in. wide, and 2 in. high?

3. How many solid feet are there in a block 5 feet long, 3 feet wide, and 2 feet high?

4. How many solid feet in a block 7 feet long, 2 feet wide, and 2 feet high?

When a solid has its length, height, and breadth, equal to each other, it is called a cube; and the linear measure of its length, height, or breadth is called the root of the cube. We have seen what is a cubic inch, a cubic foot, and a cubic yard.

Suppose now we have a pile of cubic inch blocks, and we wish to construct from them a cube, each of whose dimensions shall be 2 inches; we will first take 2 blocks, and place them down side by side; this will be as long as the required figure, but it will not be wide enough, nor high enough; to make it

wide enough, we will place 2 more blocks down by the side of the former; the figure now contains 4 cubic inches, and is 2 inches long, and 2 inches wide, but it is only 1 inch high. To make it 2 inches high, we must place upon this another layer of 4 blocks arranged just like the former. The figure will then be 2 in. long, 2 in. wide, and 2 in. high; it contains 8 cubic inches, and is the cube of 2.

5. How many blocks will you require, and how will you arrange them to make the cube of 3?

6. How many blocks will you require, and how will you arrange them to make the cube of 4?

7. How many blocks will you require, and how will you arrange them to make the cube of 5?

The cube when expressed in numbers is the same as the 3d power of the root. It is found by taking the root 3 times as a factor. Thus the 3d power of 2 is 2×2×2=8. The 3d power of 3 is 3×3×3=27. Of 4, is 4X4X4=64. Of 5, is 5X5X5=125.

of any

number.

In this way we may find the 3d power 8. How many blocks, of a cubic foot each, will it take to form a cubic solid, 6 ft. on a side?

9. How many blocks of a cubic foot each will it take to form a cubic solid of 7 feet each way?

10. How many cubic feet will it take to form a cube of 8 feet?

11. How many cubic feet will it take to form a cube of 9 feet?

12. How many cubic feet will it take to form a cube of 10 feet?

13. How many cubic inches are there in a cubic foot?

14. A pile of wood 8 feet long, 4 feet high, and 4 feet wide, makes a cord; how many cubic feet are there in a cord?

15. A pile of wood, 4 feet long, 4 feet high, and 1 foot thick, makes what is called a cord foot; how many cubic feet are there in a cord foot?

16. How many cord feet are there in a cord?

17. There is a pile of wood 40 feet long, 4 feet wide, and 5 feet high; how many cords does it contain?

18. There is a stick of hewn timber 25 feet long, 1 foot wide, and 1 foot thick; how many cubic feet does it contain?

19. There is a tree from the but-end of which a stick may be hewn 13 feet long, 2 feet wide, and 2 feet thick; how many cubic feet will it contain?

20. It is estimated that 50 feet of hewn timber weigh a ton; if 50 cubic feet weigh 20 cwt. net weight, what will 1 foot weigh?

21. If you divide a cubic inch into blocks measuring an inch each way, how many such will there be in a cubic inch? 22. How many cubic half inches are in a cubic inch?

23. If you divide a cubic inch into cubes of of an inch each, how many such will there be?

24. How many cubic quarter inches are there in a cubic inch?

25. How many cubic inches are there in a cube of one inch and a half?

26. If a man digs a cellar at the rate of of a dollar for a cubic yard, what will the job come to, if the cellar is 18 feet long, 12 feet wide, and 6 feet deep?

27. A stone-layer agreed to build a solid wall 30 feet long, 4 feet thick, and 6 feet high, for 2 dollars a cubic yard; what did the wall cost?

CONSTRUCTION OF THE CUBE.

We have seen that the third power, or cube of any number, is obtained by taking the number three times as a factor; the product is the cube, or third power.

In this way the cube of any number whatever may be obtained. There is another way, however, of constructing the cube, the knowledge of which is very important in the operation of extracting the cube root.

Suppose we wish to find the cube of 5. Instead of taking 5 three times as a factor, thus, 5×5X5=125, we will regard the number 5 as consisting of two parts, 3 and 2. We will call 3 the first part, and 2 the second part of 5.

We will begin by making the cube of the first part 3, thus, 3X3X3=27.

We will regard this as a cube of 3 inches, that is, 3 in. long, 3 in. wide, and 3 in. high; and represent it by the following figure.

The question now is, how shall we enlarge this cube of 3, so as to make it the cube of 5? It is evident, it must be 2 in. longer, 2 in. broader, and 2 in. higher than it now is. We

in to tube will begin then by putting 2 layers of inch blocks on the front side, 2 ed layers on the right side, and 2 layers on the top. The figure thus enlarged is not a cube. There are several places not filled up. It is nearer the cube of 5 than it was before; 8 but something more must be added. Before making that addition, howwe have done. The figure is the cube of 3, which is the first part of 5. To this there are 3 equal additions made. Each of these additions is 3 in. square, and 2 in. thick. Now 3 is the first part of 5; each addition, therefore, contains the square of the first part, 3, multiplied by the second part, 2, or 32X2; therefore the three additions will be 3 times the square of the first part multiplied by the second.

ever, let us see what we

The whole figure, therefore, after these three additions are made, contains 33+3 times 32X2.

We will now see what additions must next be made to the figure.

There are 3 places that need filling up, each 3 in. long, 2 in. wide, and 2 in. high. Each of these new additions is 2 in. square, and 3 in. long. It consists, therefore, of the first part of 5 multiplied by the sq. of the second; and the three together are 3 times the first part multiplied by the sq. of the second. There is one addition wanting to complete the cube; that is at the corner. It must be 2 in. long 2 in. wide, and 2 in. high; that is, the cube of 2; or, in other words, the cube of the 2d part.

Remembering that the two parts of 5, as here divided, are 3 and 2; the printed figure is the cube of the first part, the first addition is 3 times the square of the 1st part, multiplied by the 2d; the 2d addition is 3 times the first part, multiplied by the sq. of the 2d; the 3d addition is the cube of the 2d part. Let the letter a stand for the first part, 3; and the letter b, for the 2d part, 2. The printed figure will then be a3; the first addition, 3a2b; the second addition, 3ab2; the third addition, b3. The whole cube, therefore, will be a3+3a2b+ 3ab2b3. Observe that the letters and numbers are to be multiplied together though there is no sign of multiplication

between them, as 3a2b is three times the square of ag multiplied by b.

These are called the four terms of the cube, when the root is in two parts.

If we express the above in the numbers for the cube of 5, it will stand thus

33+3×32×2+3×3×22+23.

1. What number makes the 1st term of this cube?
2. What number forms the 2d term?
3. What number forms the 3d term?
4. What number forms the 4th term?
5. What do all the 4 terms amount to?

6. Which of the four terms contains the third power of the first part? Which contains the 2d power of the first part? 7. Which contains the first power of the first part? Which term contains the 1st power of the second part? Which the 2d power? Which the 3d?

8. If the 4th term of the above cube were not given, how could you determine from the others what it must be?

9. If the 3d term were gone, how could you restore it? If the 2d was gone how could you restore it?

10. If you divide the number 5 into the two parts, 4 and 1, and express the cube according to the above rule, what will the 1st term be? What will be the 2d term?

the 3d term? What the 4th?

What will be

Remember here, that all powers of one are one, more nor less.

Divide the number 6 into 4+2, and form the cube according to the above rule.

11. What will the 1st term be?

The 4th?

The 2d? The 3d?

12. What will they all amount to Multiply 6 into itself 3 times, thus, 6×6×6, and see if it amounts to the same.

13. Divide 6 into the parts 5+1, What will be the 1st term? The 2d? What do they all amount to?

and form the cube. The 3d? The 4th?

14. There is a cube in 4 terms, the first two terms of which are 33+3×32×1; what must be the 3d term? What the 4th term? What is the number of the cube? What is the