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6. Multiply 1of 4' 5" and 7f. 8' 6" together. Prod. 79f. 11' O’ 6” 6iv. 7. Multiply 5f. 4' 8" by 9s. 8' 6". Prod. 52f 3' 9" 8”. 8. Multiply 311.f. 7' 5" by 36f. 4' 11". Prod. 11345f. 11' 1" 5” 71v. 9. What will a mahogany side-board cost, which is 5 feet 11 inches long, and 3 feet 7 inches wide throughout, at 4s. 6d. per square foot 2 Ans. 4l. 15s. 4d.4. 10. What sum will pay for new glazing a hall window containing 60 squares, each 1.f. 2' 3" long, and 11'5" wide, at 3s. Sd. per square foot? Ans. 12l. 8s. 6d.;. 11. What is the price of a marble slab, the length of which is 5 feet 7 inches, and breadth 3 feet 8 inches, at 9s. per square foot 2 Ans. 91.4s. 3d. 12. A room which is 66 feet 10 inches about, was wainscotted 3 feet 11 inches upwards from the floor; what did it come to at 2s. 7d.). per square foot? Ans. 341.7s. 1d.4. 13. A drawing-room which is 32 feet 8 inches long, and 25 feet 9 inches wide, is surrounded with a cornice 3+ inches wide, the gilding of which cost 31, 5s. 6d. required what sum was charged per square foot? Ans. 1s. 11d. for. 14. The paving of a brew-house, 24 feet 11 inches long, and 34 feet 6 inches broad, cost 7s. 9d. per square yard; what did the whole amount to? Ans. 37 l. Os. 2d. 15. The expense of digging, planting, and manuring a kitchen-garden amounted to 14l. 1s. 8d. how much is that per square yard, supposing the length to be 109 feet 6 inches, and the breadth 58 feet 6 inches 2 Ans. 4d.4. 16. What sum must I pay for painting a room 48 feet 10 inches about, and 9 feet 10 inches high, at 2s. 8d.: per square yard 2 Ans. 7 l. 5s. 7d.4.

INVOLUTION “.

261. The power of any number is the product that arises by multiplying that number into itself; and the product (if necessary) into the given number; and this product (if necessary) into the given number, and so on continually. 262. The product arising from one multiplication, is called the square, or second power of the given number; the product arising from two successive multiplications, is called the cube, or third power; the product of three multiplications, the fourth power; of four multiplications, the fifth power, and so on. 263. The power of any number is denoted by a small figure, called the inder or exponent of the power, placed over, and a little to the right of, the given number. Thus 3° denotes the second power, or square of 3, the small 2 being the inder or earponent of the second power; 7° denotes the third power of 7, where 3 is the index; 27 denotes the fifth power of 21, where 5 is the index, &c. 264. Involution teaches to find the powers of any given number. 265. To involve whole numbers or decimals to any power. RULE I. Multiply the given number into itself for the square, and this product into the given number for the cube, and so on continually for the higher powers; observing, that to obtain any power, the number of successive multiplications will always be one less than the index of the required power. II. If there are decimals in the number given to be involved, mark off the decimals in each product, according to the rule for multiplying decimals, Art. 224.

* The name Involution is derived from the Latin involvo, to wrap or fold in. The number to be involved is called the root of the proposed power; the number arising from the involution is called the power of the given root. The terms square and cube are applied to certain numbers, because they arise by processes similar to the known method of computing the capacity of those figures: and because the second power is called a square, and the third power a cube, the second root is named the square root, and the third root the cube root ; the fourth power is sometimes called the biquadrate, (bis quadratus,) and the fourth root the biquadrate root. Particular names for other powers and roots are to be found in old books, but they are now seldom used ; see the note on Art. 52. part 3.

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ExAMPLEs. 1. What is the fourth power of 12? Operation. # = 1st power. Erplanation.

- Here the index of the required power being 4, 144 = 2nd power. three multiplications are necessary: the first pro12 duces the square, or second power; the second pro-- duces the cube, or third power; and the third pro1728 = 3d power. duces the biquadrate, or fourth power, as was re19 quired.

20736 = 4th power.

2. Involve 2.3 and 103 each to the fifth power.

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3. Involve 234 to the square. Square .54756. 4. What is the cube of 54 Cube 157464. 5. Involve 100.2 to the third power. Third power 1006012.008. 6. Involve 94.75 to the fourth power. Fourth power 80596628.44140625.

266, To involve a simple fraction to any power.

RULE. Involve the numerator and denominator each separately to the given power, and the results will be the respective terms of a new fraction, which will be the power required. - R 4

2 4 7. Involve or to the square, and TE to the cube.

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216 3 10. Required the biquadrate of T' Biquadrate #

267. To involve a mixed number to any power.

RULE I. Either reduce the given mixed number to an improper fraction, by Art. 172; involve both terms of this fraction by the last rule; reduce the resulting improper fraction to its proper terms, by Art. 173, and the result will be the power. Or,

H. Reduce the fractional part of the given number to a decimal, Art. 233. subjoin this to the whole number, and involve the result to the given power, by Art. 265.

ll. Involve 24 to the second power.

Thus, Art. 172. 23 = 3 × 4 + 3++ 3 – #.

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required, or, Secondly, 24 = 2.75 by Art. 233. Then 3.75)2= 2.75 x 2.75 = 7.5625, the power, as before; for this decimal .5625 being reduced to a vulgar fraction, (Art. 232.) will = or as above.

12. Involve # to the cube. ir

2+ 35

Thus, Art. 178. 4; T 66.

353 42875

Then: = 3;

13. Required the square of 14. Square 23. 14. What is the cube of 2+ 2 Cube 2034.

15. To find the biquadrate of 104. Biquad. 11401??.

= the cube required.

EVOLUTION .

268. The root of any number is that which being multiplied once, or oftener continually into itself, will produce the said number.

269. A number which being multiplied once into itself produces the given number, is called the square root of that number; a number which multiplied successively twice, produces the given number, is called the cube root of that number; if it produces the given number by three successive multiplications, it is called the fourth, or biquadrate root of that number, and so on.

270. The root of any number is denoted either by a radical sign y, with a small figure expressive of the number of the root

* The name Evolution is derived from the Latin evolvo, to unfold. With respect to the operation, the evolution of roots consisting of only one figure is merely a simple mechanical process, the reason of which immediately appears: but when the root consists of several figures, the grounds of the rule by which it is extracted are by no means obvious. A respectable writer, whose name would do honour to these pages, observes, that “any person who can extract the “ square and cube root in Algebra, will not be at a loss to demonstrate the “rules of square and cube root” in Arithmetic; “and to those who cannot, “ a demonstration would be of little or no use.” The truth is, that the common rules for the extraction of roots, either in Algebra or Arithmetic, as far as I have been able to learn, have never yet been demonstrated independently, and without supposing that both the root and its power are previously known: having the root given, its power, although unknown, is easily obtained by multiplication; but the root being unknown, cannot be obtained from the power by the converse operation of division, because the divisor is not known ; hence it appears, that the rules for Evolution were first discovered mechanically, or by dint of trial; and the only proof that they are true is, that the number arising from their operation, being involved, produces the given power. See the note on Art. 57. part 3. Mr. Wood has shewn very clearly in the extraction of the cube root, how the several steps in the Arithmetical and Algebraic operations respectively coincide with each other. Elem, of Algeb, pp. 62, 68. Third JEdit.

The square root of any number, is a mean proportional between unity and that number; the cube root is the first of two mean proportionals between unity and the given number; the biquadrate root is the first of three mean proportionals between unity and the given number; and in general, if between unity and any power there be taken mean proportionals in number one less than the index of that power, the first of these will be the root required.

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