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EVOLUTION.

I LXXXV. Q. What number, multiplied into itself, will make 12 that is, what is the first power or root of the square number 16?

A. 4.
Q. Why?
A. Because 4 X 4 = 16.

Q. What number multiplied into itself three times, will make 27? that is, what is the 1st power or root of the cubic number 27?

A. 3.
Q. Why?
A. Because 3 X3 X3=27.

Q. What, then, is the method of finding the first powers or roots of 2d, 3d, &c., powers called?

A. Evolution, or the Extraction of Roots.

Q. In Involution we were required, with the first power or root being given, to find higher powers, as 21, 3d, Rec., powers; but now it seems, that, with the 2d, 3d, &c., powers being given, we are required to find the lst power or root again; how, ther., does Evolution differ from Involution ?

A. It is exactly the opposite of Involution.
Q: How, then, may Evolution be defined ?
A. It is the method of finding the root of ai y

number. 1. What is the square root of 144? A. 12. Q. Why? A. Because 12 X 12=144. 2. What is the cube root of 27 ? A, 3. Q. Why? A. Because 3 X3X3= 27. 3. What is the biquadrate root of 81? A. 3. Q. Why? A. Because 3 X3X3 X3=81.

We have seen, that any number may be raised to a perfect power by Invo lution ; but there are many numbers of which precise roots cannot be obtained, as, for instance, the square root of 3 cannot be exactly determined, there being no nur

er, whích, by being multiplied into itself, will make 3. By the aid of decinals, however, we can come nearer and nearer, that is, approximate towards the root, to any assigned degreo of exactness. Thoso numbers, whose roots cannot exactly be determined, are called Surd Roots, and those, whose roots can exactly be determined, are called Rational Roots.

To show that the square root of a number is to be extracted, we profix this character, . Other roots are denoted by the same character with the index of the required root placed before it. Thus, „ 9 signifies that the square root of 9 is to be extracted; 3727 signifies that the cube root of 27 is to be ex wacted ; 464=the 4th root of 64.

When we wish to express the power of several numbers that are connected together by these signs, +,

,-, &c., a vinculum or parenthesis is used, drawn fro:n the top of the sign of the root, and extending to all the parts of it; tho ine cube root of 30 -- 3 is expressed thus, /30 — 3, &c.

EXTRACTION OF THE SQUARE ROOT.

any number?

LXXXVI. Q. We have seen (LXXXV.) that the root of any number is ils 1st power ; also that a squaro is the 2d power: what, then, is to be done, in order to find tho Ist power; that is, to extract the square root of

A. It is only to find that number, which, being multiplied into itself, will produce the given number.

Q. We havo seeni ( LXXIX.) that the process of finding the contents of a square consists in multiplying the length of one side into itself; when, thea, the contents of a square are given, how can we find the length of cach side! or, to illustrate it by an example, If tho contents of a square figuro be 9 feet, wbai must be the length of each side?

A. 3 feet.
Q. Why?
A. Because 3 feet x 3 feet = 9 square feet.

Q. What, then, is the difference in contents botween a square figure whore sides are each 9 feet in length, and one which contains only 9 square feet?

A. 9 X 9=81-9=72.

Q. What is the difference in contents between a square figure containing 3 square feet, and one whose sides are each 3 feet in length ?

A. 6 square feet.

Q. What is the square root of 144? or what is the length of each sido of a figure, which contains 144 square feet ?

A. 12 square feet
Q. Why?
A. Because 12 x 12=144.
Q. Ilow, then, may we know if the root or answer be right?

A. By multiplying the root into itself; if it prosluces the given number, it is right.

Q. If a square garden contains 16 square rods, how many rods does it mean ure on each side ? and why?

A. 4 rods. Because 4 rods x 4 rods = 16 square mnds 1. What is the square root of 64? and why? 2. What is the square root of 100? and why? 3. What is the square root of 49? and wby? 4. Extract the square root of 144. 5. Extract the square root of 36. 6. What is the square root of 3600? 7. What is the square root of ,25?. A. ,5. 8. What is the sijuare root of 1,44? A. 1,2. 9. What is the value of 25 ? 'or, what is the square root of 35 1 10. What is the value of ,4? A. ,2. 11. What is the square root of Į ? A. H. 12. What is the value of w? A. . 13. What is the square rooi of of }? A.. 14. What is the square root of 6? no== {=24, Ana 15. What is the value of veof? A.. 16. What is the square root of 301?

17. What is the difference hetu cen the square root of 4 and the square a 4. or, which is the same thing, what is the difference between 4 and 4*?

~4=2, and 42 = 16; then, 16 - 2= 14, tras

18. What is the difference between 9 and 92 ?
19. What is thu difference between 16 and x 9?
20. What is the difference between po't and ?? A. O.

21. There is a square room, which is calculated to accommodate 100 scho! ang; how many can sit on one side ?

3. If 400 boys, having collected together to perform somo military evolutions, should wish to march through the town in a solid phalanx, or square body, of how many must the first rank consist ?

23. A general has 400 men; how many must be place in rank and file to form them into a square ?

24. A certain square pavement contains 1600 square stones, all of the same size; I demand how many are contained in one of its sides? A. 10.

25. A man is desirous of making his kitchen garden, containing 24 acres, or 400 rods, a complete square ; what will be the length of one side?

26. A square lut of land is to contain 22 acres, or 3600 rods of ground; but, or the sake of fruit, there is to be a smaller square within the larger, which is o contain 225 rods : what is the length of each side of both squares ?

A. 60 rods the outer, 15 rods the inner

Exercises for the Slate. 1. If a square field contains 6400 square rods, how many rods in length does it measure on each side ? A. 80 rods.

2. How many trees in each row of a square orchard, which contains 2500 trees? A. 50 trees.

3. A general has a brigade consisting of 10 regiments, each regiment of 10 companies, and each company of 100 men : how many must be placed in rank and file, tú form them in a compleie square ? A. 100 men.

4. What is the square root of 2500? A. 50.
5. What is the 1st power of 1000000? : A. 1000.
6. What is the value of v360000? A. 600.

7. What is the difference between the square root of 36 and the square ed 36 ? A. 1290.

8. What is the difference between 4900 and 49002 ? A. 24009930. 9. What is the difference between 81 and 81%; A. 6552. 10. What is the difference between use and 147?? v=X, and = 1; then, &- 128= =3f, Ans

11. What is the difference between 16 and 162? A. 356.
12. What is the amount of 4 and 9? A. 5.
13. What is the sum of 4 and 92: A. 83.
14. What is the amount of ~ 304 and 2724? A. 22.

15. What is the length of one side of a square garden, which contains 126 square rods? in other words, what is the square root of 1296 ?

In this example, we have a little difficulty in ascertaining the root. This, perhaps, may be ohviated by examining the igure on the following page, (which is in the form of tho garden, and supposed to contain 1296 squaro rods,) and carefully noting down the operation us we proceed. OPERATIONS.

In this example, we know that 1st.

2d.

the root, or the length of one side Square Rods. Square Rods. of the garden, must be greater 30 ) 1296 ( 39 3) 1296 ( 36 than 30, for 30o = 900, and less 900

9

than 40, for 402 = 1600, which is

greater than 1296 ; thereforo, wo 60 + 6 = 66 ) 396( 6 66 ) 396 take 30, the loss, and, for conve 396

396

nienco'sake, write it at the lost of 1296, as a kind of divisor, likewize at the right of 1296, in the

1296

O roda

6 rods.

30 rods.

30 rods.

6

FIG.

form of a quotient in division 30 rods.

6 rods.

(See Operation Ist.); then, sub

tracting the square of 30, = 900 30

6 sq. rods, from 1296 sq. rods, leaves B 6

396 sq. rods. 180

36

The pupil will bear in mind, that the Fır, on the left is in the form of the garden, and contains

the same number of square rods, А

D viz. 1296. This figure is divided

into parts, called A, B, C, and D.

It will be perceived, that the 900 Rods,

square rods, which we deducted, 30, length of A. 30 aro found by multiplying the 30, breadth of A.

length of A, being 30 rods, by the

breadth, being also 30 rods, that 900, sq. rods in A. 180 is, 30o = 900.

To obtain the square rods in B, C, and I, the remaining parts

of the figure, we may multiply 30 rods.

6 rods the length of each by the breadth

of each, thus ; 30 X 6= 180, 6 X 6 = 36, and 30 x 6= 180; then 180 +36 + 180 = 396 square rods; or, add the length of B, that is, 30, to the lengih of D, which is also 30, making 60; or, which is the same thinwe may double 30, making 60; to this add the length of C, 6 rods, and the sum is 66. Now, to obtain the square rods in the whole length of B, C, and D, we multiply their length, 6 rods, by th adth of each side, thus, 66 X 6=396 square rode, the same as before.

We do tho same in the operation ; that is, we first double 30 in the quo tient, and add the 6 rods to the sum, making 66 for a divisor; next, multiply 66, the divisor, hy 6 rods, the width, making 396 ; then, taking 396 from 356 leaves 0.

The pupil will perceive, the only difference between the Ist and 2d opera tion (which sec) is, that in the 21" we neglect writing the ciphers at the right of the numbers, and use only the siguticant figures. Thus, for 30 + 6, wa write 3 (tens) and 6 (units), which, joined together, make 36 ; for 900, we writo 9 (hundreds). This is obvious from the fact, that the 9 retains ito place under the 2 (hundreds). Instead of 60+ 0, we write 66. Omitting the ciphers in this manner cannot reasonably make any difference, and, in fact, it does not, for the result is the same in both.

By neglecting the ciphers, we may, perhaps, be at a loss, sometimes, to do termine where we must place the square number. In the last example, we knew where the square of the root 3 (tens) = 9 (hundreds) should be placed, for the ciphers, at the right, indicate it; but had those ciphers been dropped we should, doubtless, have hesitated in assigning tho 9 its proper place. The difficulty will be obviated by observing what follows.

The square of any number never contains but twice as many, or at least but one figure less than twice as many, figures as are in the root. Thus, the square of the root 30 is 900; now, in 900 there are but three figures, and in 30, two figures; that is, the square of 30 contains but one figure more than 30. We will take 99, whose square is 4801, in which there are four figures, and in ita root, 99, but two; that is, there are exactly twice as many figures in the square 9801 as are in its root, 99. This will be equally true of any numbers whatevek

Hence, to know where to place the several square numbers, we may point off the figures in the given number into periods of two figures each, commencing with the units, and proceeding towards the test. And, sinco the value of both whole numbers and decimals is determined by their distance from the unita' place, consequontly, when there are decimals in the given number, wo may begin at the units' place, and point off the figures towards the right, in the samo manner as we point off whole numbers towards the left.

By each of the preceding operations, then, we find that the root of 1993 b *8or, in other words, the length of each side of the garden in 36 soda

PROOF. This work may now be proved by adding together all the square pods contained in the several parts of the figure, thus :

A contains 30 x 30 = 900 square rods.
B . 30 X 6 = 180

Or, by Involation,
6 X 6= 36

36 x 36 = 1296, Ans., as before 30 X 6= 180

1296 square rods.

From these iHustrations we derive the following

RULE. Point off the given number into periods of two figures each, by putting

a dot over the units, another over the hundreds, and so on; and, if there are decimals, point them in the same manner, from units towards the right hand These dots show the number of figures of which the root will consist.

Find the greatest square nuinber in the left-hand period, and write its root as a quotient in division; subtract the square number from the left-hand period, and to the remainder bring down the next right-hand period for a dividend.

Double the root (quotient figure) already found, and place it at the left of the dividend for a divisor.

Write such a figure at the right hand of the divisor, also the same figure in the root, as, when multiplied into the divisor thus increased, the product shall be equal to, or next less than the dividend. This quotient figure will be the second figure in the root.

Note.-The figure last described, at the right of the divisor, in the second operation, is the 6 rods, the width, which wo add to 60, making 66; or, ting the 0 in 60, and annexing 6, then multiplying 66 by 6, we wrote the 6 in the quoticnt, at the right of 3, making 36.

Multiply the whole increased divisor by the last quotient agure, and write the product under the dividend.

Subtract this product from the dividend, and to the remainder bring down the next period, for a new dividend.

Double the quotient figures, that is, the root already found, and continue the operation as before, till all the periods are brought down.

omit

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