ART. 277. All circles are to each other as the squares of their diameters, semidiameters, or circumferences.

All similar triangles and other rectilineal figures are to each other as the squares of their homologous or corresponding sides.

ART. 278. To find the side, diameter, or circumference of any surface, which is similar to a given surface.

RULE. ·State the question as in Proportion, and square the given sides, diameters, or circumferences, and the square root of the fourth term of the proportion will be the required answer.

ART. 279. To find the area of any surface which is similar to a given surface.

RULE.-State the question as in Proportion, and square the given sides, diameters, or circumferences, and the fourth term of the proportion is the required answer.

EXAMPLES FOR PRACTICE.

Ex. 1. I have a triangular piece of land containing 65 acres, one side of which is 100 rods in length; what is the length of the corresponding side of a similar triangle containing 32 Ans. 70.71 rods.

acres?

OPERATION.

65:32:1002:5000; 5000-70.71+ rods. 2. I have a board in the form of a triangle, the length of one of its sides is 16 feet. My neighbor wishes to purchase one half the board; at what distance from the smaller end must it be divided parallel to the base or larger end? Ans. 11.31+ feet.

3. There is a triangular piece of land, the length of one side of which is 11 rods; required the length of the corresponding side of a similar triangle containing three times as much. Ans. 19.05+ rods.

4. The diameter of a circle is 6 feet, and its area is 28.3 feet; what is the diameter of a circle whose area is 42.5 feet? Ans. 7.35+ feet.

5. If an anchor, which weighs 2000lb., requires a cable 3 inches in diameter, what should be the diameter of a cable, when the anchor weighs 4000lb.? Ans. 4.24 inches. 6. A rope 4 inches in circumference will sustain a weight of 1000lb.; what must be the circumference of a rope that will sustain 5000lb ? Ans. 8.94 inches.

7. There is a triangle containing 72 square rods, and one of

QUESTIONS. Art. 277. What proportion do circles have to each other? Art. 278. What is the rule for finding the side, diameter, &c., of a surface similar to a given surface? - Art. 279. What is the rule for finding the area of a surface similar to a given surface?

its sides measures 12 rods; what is the area of a similar triangle whose corresponding side measures 8 rods?

Ans. 32 rods.

8. A gentleman has a park, in the form of a right angled triangle, containing 950 square rods, the longest side or hypothenuse of which is 45 rods. He wishes to lay out another in the same form, with an hypothenuse the length of the first; required the area. Ans. 105.55-+ square rods.

9. If a cylinder 6 inches in diameter contain 1.178+ cubic feet, how many cubic feet will a cylinder of the same length contain, that is 9 inches in diameter ? Ans. 2.65+ feet.

10. If a pipe, 2 inches in diameter, will fill a cistern in 201 minutes, how long would it take a pipe, that is 3 inches in diameter ? Ans. 9 minutes.

11. A tube of an inch in diameter will empty a cistern in 50 minutes; required the time it will empty the cistern, when there is another pipe running into it of an inch in diameter? Ans. 62 minutes. ART. 280. To find the side of a square that can be inscribed in a circle of a given diameter.

HYPOTH

ence.

A square is said to be inscribed in a circle when each of its angles or corners touches the circumferIt may be conceived to be composed of two right angled triangles, the base and perpendicular of each being equal, and their hypothenuse the diameter of the circle, as seen in the diagram. Hence the RULE.-Extract the square root of half the square of the diameter, and it is the side of the inscribed square.

BASE.

EXAMPLES FOR PRACTICE.

1. What is the length of one side of a square that can be inscribed in a circle, whose diameter is 12 feet?

Ans. 8.48 feet. 2. How large a square stick may be hewn from a round one, which is 30 inches in diameter ?

Ans. 21.2+ inches square. 3. A has a cylinder of lignum-vitæ, 19 inches long and 11 inches in diameter; how large a square ruler may be made Ans. 1.06 inches square.

from it?

QUESTIONS. Art. 280. When is a square said to be inscribed in a circle? Of what may the inscribed square be conceived to be composed? What part of the circle is the hypothenuse of the two triangles? What is the rule for finding the side of the inscribed square?

EXTRACTION OF THE CUBE ROOT.

ART. 281. The CUBE ROOT is the root of any third power, and is so called because the cube or third power of any number represents the contents of a cubie body, of which the cube root is one of its sides.

ART. 282. To extract the cube root is to find a number, which, being multiplied into its square, will produce the given number. The following numbers in the upper line represent roots, and those in the lower line their third powers or cubes. Roots, 1 2 3 4 5 Cubes,

1

6

7 8 9

10

8 27 64 125 216 343 512 729 1000

It will be observed that the cube or third power of each of the numbers above contains three times as many figures as the root, or three times as many wanting one, or two at most. Hence,

To ascertain the number of figures in the cube root of any given number, it must be divided into periods, beginning at the right, each of which, excepting the last, must always contain three figures, and the number of periods will denote the number of figures, which the root will contain.

Ex. 1. I have 17576 cubical blocks of marble, which measure one foot on each side; what will be the length of one of the sides of a cubical pile, which may be formed of them? Ans. 26 feet.

It is evident that the number of blocks or feet on a side will be equal

to the cube root of 17576. (Art. 281.) Beginning at the right hand, we divide the number into periods, by placing a point over

the right hand figure of each period. We then find the greatest cube number in the left hand period 17 (thousands) to be 8 (thousands), and

QUESTIONS. Art. 281. What is the cube root, and why so called? - Art. 282. What is meant by extracting the cube root? How many more figures in the cube of any number than in the root? How do you ascertain the number of figures in the cube root of any number? What is found by extracting the cube root of the number in the example? What is first done after separating the number into periods?

its root 2, which we place in the quotient or root. As 2 is in the place of tens, because there is to be another figure in the root, its value is 20, and it represents the side of a cube (Fig. 1), the contents of which are 8000 cubic feet; thus 20 X 20 X 20 = 8000.

Fig. 1.

20

20

=

We now subtract the cube of 2 (tens)

8 (thousands) from the first period, 17 (thousands), and have 9 (thousand) feet remaining, which, being increased by the next period, makes 9576 cubic feet. This must be added to three sides of the cube, Fig. 1, in order that it may remain a cube. To do this, we must find the superficial contents of the three sides of the cube, to which the additions are to be made. Now, since one side is 2 (tens) or 20 feet square, its superficial contents will be 20 X 20 = 400 square feet, and this multiplied by 3 will be the superficial contents of three sides; thus, 20 X 20 X 3 = 1200, or, which is the same thing, we multiply the square of the quotient figure, or root, by 300; thus, 22 X 300 = 1200 square feet. Making this number a divisor,

20

=

we divide the dividend 9576 by it, and obtain 6 for the quotient, which we place in the root. This 6 represents the thickness of each of the three additions to be made to the cube, and their superficial contents being multiplied by it we 20 have 1200 X 6 = 7200 cubic feet for the contents of the three additions, A, B and C, as seen in Fig. 2.

Having made these additions to the cube, we find that there are three other deficiencies, n n, o o, and rr, the length of which is equal to one side of the additions, 2 (tens) or 20 feet; and their breadth and thickness, 6 feet, equal to the thickness of the additions. Therefore, to find the solid contents of the additions, necessary to supply these deficiencies, we multiply the product of their length, breadth and thickness by the number of additions; thus, 6 X 6 X 20 X 32160, or, which is the same thing, we multiply the square of the last quotient figure by the former figure of the root, and that product by 30; thus, 62 X 2

QUESTIONS. What is done with this greatest cube number, and what part of Fig. 1 does it represent? What is done with the root? What is its value, and what part of the figure does it represent? How are the cubical contents of the figure found? What constitutes the remainder after subtracting the cube number from the left hand period? To how many sides of the cube must this remainder be added? Why? How do you find the divisor? What parts of the figure does it represent? How do you obtain the last figure of the root? What part of Fig. 2 does it represent? Why do you multiply the divisor by the last quotient figure? What parts of the figure does the product represent? What three other deficiencies in the figure?

X 30 2160 cubic feet for the contents of the additions ss, u u, and vv, as seen in Fig. 3.

Fig. 3.

These additions being made to the cube, we still observe another deficiency of the cubical space xxx, the length, breadth, and thickness of which 20 are each equal to the thickness of the other additions, which is 6 feet. Therefore, we find the contents of the addition necessary to supply this deficiency by multiplying its length, breadth, and thickness together, or cubing the last figure of the root; thus, 6 X 6 X 6 = 216 cubic feet for the contents of the

26

RULE.

Fig. 4.

The cube is now complete, and if we add together the several additions that have been made to it, thus, 7200+ 2160 +2169576, we obtain the number of cubic feet remaining after subtracting the first cube, which, being subtracted from 26 the dividend in the operation, leaves no remainder. Hence, the cubical pile formed is 26 feet on each side; since 26 X 26 X 26 17576, the given number of blocks, and the sum of the several parts of Fig. 4. Thus, 8000+ 7200 + 2160 +216 = 17576. Hence the following

=

1. Separate the given number into periods of three figures each, by placing a point over the unit figure, and every third figure beyond the place of units.

2. Find by the table the greatest cube in the left hand period, and put its root in the quotient.

3. Subtract the cube, thus found, from this period, and to the remainder bring down the next period; call this the dividend.

4. Multiply the square of the quotient by 300 for a divisor, by which divide the dividend and place the quotient, usually diminished by one or two units, for the next figure of the root.

5. Multiply the divisor by this last quotient figure, and write the product under the dividend; then multiply the square of the last quotient figure by the former quotient figure or figures, and this product by 30, and place the product under the last; under all, set the cube of the last quotient figure, and call their sum the subtrahend.

QUESTIONS. How do you find their contents? What parts of Fig. 3 does the product represent? What other deficiency do you observe? To what are its length, breadth and thickness equal? How do you find its contents? What part of Fig. 4 does it represent? What is the rule for extracting the cube root?