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Practical Exercises in Involution and Evolution.
I. A figure of three sides is called a triangle, and, if one of its corners, or angles, should be a square corner, like the angle at B in the annexed figure, it is called a right angle; and the figure is called a right angled triangle, and the two sides adjoining the right angle are said to be perpendicular to each other. The side A C, opposite the right angle, is called the hypothenuse. It is shown by Geometry, that the square of the hypothenuse is equal to the sum of the squares of the other two sides. It follows that the difference between the square of the hypothenuse and that of either of the other sides is equal to the square of the remaining side, since, if 9=4+5, then 9–5=4, and 9–4=5.
II. The round line which forms the boundary of a circle is called its circumference. Any straight line which passes through the centre, or middle point, of a circle, and is terminated in both ends by the circumference, is called its diameter. Now, we also learn by Geometry, that the areas, or contents, of circles, are not in proportion to their diameters, but to the squares of these diameters. Thus, a circle of 6 inches, or 6 feet, in diameter is 4 times as large as one of 3 inches, or 3 feet, in diameter, because the square of 6  is 4 times as large as the square of 3 .
1. If a square field contain 2304 square rods, how many rods does it measure on each side; in other words, what is the square root of 2304 ?
See Definitions 1 and 2, Involution, p. 162.
2. If each side of a square field be 48 rods long, how many square rods does it contain ?
3. There are two square fields; the side of one being 20, and of the other 40 rods long. How many square rods in each, and how many times is the one larger than the other ?
4. If the sides of one square field be twice as long as that of another square field, how many times is the larger greater than the less ?
O In order that the pupil may have a clear conception of this fact, let him halve a straight line upon his slate, and form squares of the half and of the whole lines, the one within the other.
5. If each side of a square field measure 25 rods, what will be the length of the side of a square field containing 4 times as many square rods?
6. If the side of a square field measure 50 rods, what will be the length of the side of another square field which contains exactly one-fourth of the number of square rods?
7. A square mile contains 640 acres. How many acres are contained in two miles square? How many square miles in two miles square ? Ans. to 1st question, 2560 acres.
8. A certain square orchard contains 1600 trees. The trees are in rows, two rods apart each way, and on every side the orchard fence is a rod from the trees. How many acres are there in the orchard, if 160 square rods make an acre ?
9. How many trees can be placed in rows in a square field containing 40 acres, the trees to be two rods apart each way, and the outer rows exactly a rod from the fence ?
10. A carpenter has a wooden square, one side of which is 4 feet long, and the other 3 feet long. What is the length of a board which will just reach from one end to the other ? See Definition 1, above.
11. One of the sides of a carpenter's square is 4 feet long, and a board 5 feet long just reaches from one end of it to the other. What is the length of the other side of the square ?
12. A wall is 32 feet high, and a ditch before it is 24 feet wide ? What is the length of a ladder that will reach from the top of the wall to the opposite side of the ditch ?
13. If a ditch be 24 feet wide, what is the height of a wall that can just be reached by a ladder 40 feet long?
14. If the ladder be 40 feet, and the wall 32 feet, what is the width of the ditch ?
15. If a ladder 60 feet long be so placed as to reach a window on one side of a street 36 feet from the ground, and, by turning it over to the other side of the street, without moving its foot, it just reaches a window 48 feet from the ground, what is the width of the street ?
If the pupil be at a loss here, let him draw on his slate a horizontal line, to represent the width of the street, and two lines perpendicular to its ends, to represent the walls of the opposite houses, and then complete the figure by another line for the ladder, placing its foot not far from the middle of the street.
16. A certain street is 84 feet wide. How far from the middle of the street must a ladder 60 feet long be placed so as to reach to the top of a wall of the height of 48 feet on one side of the street ?
17. The distance across a building between the outer edges of the plates on which the rafters rest is 32 feet, and the height of the ridge above the beam on which they stand is 12 feet. Required the length of the rafters if they project one foot beyond the walls.
Ans. 21 feet. 18. There is a building 30 feet in length, and 22 in width, and the eaves project beyond the walls one foot on every side. The roof terminates in a point at the centre of the building, and is there supported by a post, the top of which is ten feet above the beams on which the rafters rest. What is the distance from the foot of the post to the corners of the eaves ? the length of a rafter reaching to the middle of one side ? of a rafter reaching to the middle of one end ? of a rafter reaching to the corners of the eaves ?
Ans. in order, 20 ft.; 15-62+ft.; 18-86+ft; and 23-36-+-ft.
19. What is the distance from the centre to each corner of a square field containing 1600 square rods?
Ans. 28-28 +rods. 20. A society of men raised $576 for a certain purpose, each man contributing as many dollars as there were men. What was the number of the society?
Ans. 24. 21. At another time, their treasurer informed the same society that their funds had been reduced by payments to $39. Whereupon all present made a new contribution, each paying as many dollars as there were members present, when, on a fresh count, the funds amounted to $400. How many members were present ?
Ans. 19. 22. If the diameter of a circle be 2 feet, what will be the diameter of one 4 times as large ?
Ans. 4 feet. 23. What is the distance measured through the centre of a cube from one corner to its opposite corner, the side of the cube being 3 feet?
Ans, 5-196 feet.
24. There are two circular pieces of land, the one 100 feet, the other 20 feet in diameter. How many times is the one greater than the other ?
Ans. 25 times. 25. What is the superficies of one side of a cubical block containing 6859 solid inches, and what is the superficies of the whole block ? Ans. to the last question, 2166 square in.
26. How many times is a globe 2 feet in diameter greater than one that is 1 foot in diameter ?
Ans. 8 times. 27. If a globe of silver, 1 inch in diameter, is worth $6, what is the value of a globe 12 inches in diameter ?
Ans. $10368. 28. Find the sum of the roots of all the perfect squares oontained between 1 and 100 inclusive.
Ans. 55. 29. Find the sum of the numbers whose square roots are surds between 1 and 20 inclusive.
Ans. 180. 30. Find the sum of the numbers whose cube roots are surds between 1 and 20 inclusive.
Ans. 201. 31. Find the sum of 64, 343, 64, 2564, and 83.
MULTIPLICATION AND DIVISION BY EASY NUMBERS.
1. Multiplication by Division. 1. Multiply 2647938 by 5 by division (See Oral Arithmetic, Chap. I., Sect. XVIII., 3, 4, 5, p. 60), and prove by multiplication.
2. Multiply 7946287 by 5 by division, and prove as above.
3. Multiply 1678432 by 25 by division (See Oral Arithmetic, pp. 61, 10), and prove as above.
4. Multiply 6238937 by 25 by division, and prove. 5. Multiply 421695 by 25 by division, and prove. 6. Multiply 2834926 by 25 by division, and prove. 7. Multiply 7394845 by 25 by division, and prove. 8. Multiply 84739284 by 125 by division (125=1900),
9. Multiply 3462845 by 125 by division, and prove. 10. Multiply 64834921 by 125 by division, and prove. 11. Multiply 1346824 by 125 by division, and prove.
2. Multiplication by Subtraction. 12. Multiply 87649324 by 9 by subtraction.
Suggestive Questions. If a number has been taken 10 times, when it ought to have been taken only 9 times, how many times has it been taken too many? How can the error be rectified?
Then 87649324 =10 times by position,
leaves 788843816= 9 times.
But it is unnecessary to use so many figures, as it may plainly be seen that the process consists simply in subtracting the right hand figure from 0, every other figure from the figure at its right, and lastly 0 from the left hand figure.*
On the same principle, a number may be multiplied by 99, 999, or any number of nines, by subtraction, by supposing as many ciphers to be annexed to the multiplicand as there are nines in the multiplier, and then subtracting the original multiplicand from this product. For instance, if 999 be the multiplier, take each of the three figures at the right from zero, every other figure from the third figure on its right, and zero from the three figures at the left, the reasons for which will distinctly appear from an inspection of the following example.
13. Multiply 47368259 by 999 by subtraction, and prove by adding the complement.
Hence 47368259 =1000 times by position,
leaves 47320890741 =999 times.
14. Multiply 67245896 by 99 by subtraction, and prove by adding the complement.
15. Multiply 34 by 9999 by subtraction, and prove by adding complement.
16. Multiply "246 by 9999 by subtraction, and prove by adding complement.
* In the above and in the succeeding exemplification the pupil should omit the 2d line, which is only inserted here to show the reason for the process.