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Rules ocsasionally useful to men in particular callings and pur
suits of life.
§ 1. Involution.
INVOLUTION, or the raising of powers is the multiplying of any given nunber into itself continually, a certain number of times. The quantities in this way produced, are called POWERS of the given number. Thus, 4X4=16 is the 2d. power, or square
of 4. 4X4X4=64 is the 3d. power, or cube of 4. 4X 4X4 X 4—256 is the 4th. power, or biquadrate of 444 The given number, (4) is called the first power ; and the small figure, which points out the order of the power, is called the Index or the Exponent.
§ 2. Evolution.
EVOLUTION, or the extraction of roots, is the operation by which we fird any root of any given number.
The root is a number whose continual multiplication into itself produces the power, and is denominated the square, cube, biquadrate, or 2d, 3d, 4th, root, &c. accordingly as it is, when raised to the 2d, 3d, 4th, &c. power, equal to that power. Thus, 4 is the square root of 16, because 4X4316. 4 also is the cube root of 64, because 4 X 4X4364 ; and 3 is the square root of 9, and 12 is the square root of 144, and the cube root of 1728, because 12 X 12 X 12—1728, and so on.
To every number there is a root, although there are numbers the precise roots of which can never be obtained. But, by the help of decimals, we can approximate towards those roots, to any necesssry degree of exactness. Such roots are called Surd Roots, in distinction from those, perfectly accurate, which are called Rational Roots.
The square root is denoted by this character ✓ placed before the power; the other roots by the same character, with the index of the root placed over it. Thus, the square root of 16 is expressed ✓ 16, and the cube root of 27 is
✓ 27, &c.
When the power is expressed by several numbers with the sign + orbetween them, a line is drawn from the top of the sign over all the parts of it;
bi in pl
thus, the second power of 21–5 is v212-5,and the 3d. power of 5678
is 56 +8, &c.
The second, third, fourth, and fifth powers of the nine digits may be seen in the following
Biquadrates or 4th. Powers. 116 81 256 6251296 2401 4096
or 5th. Powers. 13212431024 31251777616807 32768 59049
§ 3. Ertraction of The Square Hoot.
To extract the square root of any number, is to find another number which multiplied by, or into itself, will produce the given number; and after the root is found, such a multiplication is a proof of the work.
1. « DISTINGUISH the given number into periods of two figures each, by putting a point over the place of units, another over the place of hundreds, and so on, which points shew the number of figures the root will consist of.
%. “ FIND the greatest square number in the first, or left hand period, place the root of it at the right hand of the given number, (after the manner of a quotient in division) for the first figure of the root, and the square number, under the period, and subtract it therefrom, and to the remainder bring down the next period for a dividend.
3. “Place the double of the root, already found, on the left hand of the dividend for a divisor.
4. “Seek how often the divisor is contained in the dividend,(except the right hand figure) and place the answer in the root for the second figure of it, and likewise on the right hand of the divisor : multiply the divisor with the figo ure last annexed by the figure last placed in the root, and subtract the product from the dividend : To the remainder join the next period for a new dividend.
5. “ DOUBLE the figures already found in the root, for a new divisor, (or bring down your last divisor for a new one, doubling the right hand figure of it) and from these, find the next figure in the root as last directed, and continue the operation in the same manner, till you have brought down all the periods.
"Note 1. IF, when the given power is pointed off as the power requires, the left hand period should be deficient, it must nevertheless stand as the first period.
6 NOTE 2. If there be decimals in the given number, it must be pointed both ways from the place of units : If, when there are integers, the first period in the decimals be deficient, it may be completed by annexing so many cyphers as the power requires : And the root must be made to consist of so many whole numbers and decimals as there are periods belonging to each ; and when the periods belonging to the given number are exhausted, the operation may be continued at pleasure by annexing cyphers."
EXAMPLES 1. What is the square root of 729 ?
729(27 the root.
The given number being distinguished into
periods, I seek the greatest square number in 47)329
the left hand period (7) which is 4, of which 329
the root (2) being placed to the right hand of
the given number, after the manner of a 000
quotient, and the square number (4) subtractPROOF.
ed from the period (7) to the remainder (3) 27
I bring down the next period (29) making 27
for a dividend, 329. Then the double of the
root (4) being placed to the left hand for a 189
divisor, I say how often 4 in 32 ? (excepting 9 54
the right hand figure) the answer is 7, which I
place in the root for the second figure of it, 729
and also to the right hand of the divisor, then multiplying the divisor thus increased by the figure (7) last obtained in the root, I place the product underneath the dividend, and subtract it therefrom, and the work is done.
Of the reason and ncture of the various steps in the extraction of she
SQUARE Root The superficial content of any thing, that is, the number of square feet, yards, or inches, &c. contained on the surface of a thing, as of a table or floor, a picture, a field, &c. is found by multiplying the length into the breadth. If the length and breadth be equal, it is a square, then the measure of one of the sides as of a room, is the root,of which the superficial content in the floor of that room, is the second power. So that having the superficial contents of the floor of a square room, if we extract the square root, we shall have the length of one side
On the other hand, having the length of one side of a square room, if we multiply that number into itself, that is raise it to the second powe er, we shall then have the superficial contents of the floor of that room.
The extraction of the square root, therefore has this operation on numbers, to arrange the number of which we extract the root into a square form. As if a man
of that room.
should have 625 yards of carpeting, 1 yard wide, if he extract the square root of that number (625) he will then have the length of one side of a square room, the floor of which, 625 yards will be just sufficient to cover.
To proceed then to the demonstration
EXAMPLE 2. SUPPOSING a man has 625 yards of carpeting, 1 yard wide; what will be the length of one side of a square room, the floor of which his carpeting will cover ?
The first step is to point off the number into periods of two figures each. This determines the number of figures of which the root will consist, and is done on this principle, that the product of any two numbers can have at most but so many places of figures as there are places in both the factors, and at least; but one less, of which any person may satisfy himself at pleasure.
The number being pointed off, as the rule directs, we find we have two periods ; consequently, the root will consist of two figures. The greatest square númber in the left band period (6) is 4, of which two is the root; there.
fore, 2 is the first figure of the root, and as it c. is certain we have one figure more to find in
the root, we may for the present supply the place of that figure by a cypher, (20) then 20 will express the just value of that part of the root now obtained. But it must be remem. bered, that a root is the side of a square of equal sides.
Let us then form a square, A; Fig. I. each side of which shall be supposed 20 yards. Now the side a b of this square, or either of the sides, shews the root, 20, which
we have obtained. 6.
To proceed then by the rúle, "place the square number underneath the fie riod, subtract, and to the remainder bring down the next period.” Now the square number (4) is the superficial content of the square A-made evident thus, each side of the square A, measures 20 yards, which number multiplied into itself, produces 400, the superficial contents of the square A ; also, the square number, or the square of the figure 2 already found in the root, is 4, which placed under the period (6) as it falls in the place of hundreds, is in reality 400; as might be seen also by filling the places to the right hand with cyphers, then 4 subtracted from 6 and to the remainder (2) the next period (25) being bro't down, it is plain, the sum 625 has been diminished by the deduction of 400, a number equal to the superficial contents of the square A.
HENCE, F'ig. I exhibits the exact progress of the operation. By the operation, 400 yards of the carpeting have been disposed of, and by the figure is seen the disposition made of them.
Now the square A, is to be enlarged by the addition of the 225 yards which remain, and this addition must be so made that the figure, at the same time, shall continue to be a complete and perfect square. If the addition be made to one side only, the figure would lose its square form ; it must be made to two sidrs; for this reason the rule directs, “place the double of the root already found on the left hand of the dividend for a divisor.” The double of the root is just equal to two sides b c and c d of the square, A, as may be seen by what
Now if the sides bc and c d of the square A, Fig. II. is
the length to which the re201 20
maining 225 yards are to be 20
added, and the divisor ( 4 tens) 5
is the sum of these two sides, 400
it is then evident, that 225 di100
vided by the length of the two sides, that is by the divi
sor (4 tens) will give the 20
supp $ SCOZ
breadth of this new addition
of the 225 yards to the sides The square A =400 yds.
b c and c d of the square, A. Cef=100% Cgh=100
But we are directed to “except the right D
hand figure,” and also to “place the quotient
figure on the right hand of the divisor ;” the Proof, 625 yde, reason of which is, that the additions, Cef
and C g h to the sides bc and cd of the
square, A, do not leave the figure a complete square, but there is a deficiency, D, at the corner. Therefore, in dividing, the right hand figure is excepted, to leave something of the dividend, for this deficiency; and as the deficiency, D, is limited by the additions Cefand Ceh, and as the quotient figure (5) is the width of these additions, consequently
equal to one side of the square, D; therefore, the quotient figure (5) piaced to - the right hand of the divisor ( 4 tens) and multiplied into itself, gives the con
tents of the square, D, and the 4 tens to the sum of the sides, bc and cd of the addition Cef and Cgh, multiplied by the quo'ient figure, (5) the width of those additions, give the contents Cef and C g h, which together subtracted from the dividend, and there being no remainder, shew that the 225 yards are disposed in these new additions Çef, C gh, and D, and the figure is seen to be continued a complete square.
CONSEQUENTLY, fig. II. shews the dimensions of a square room, 25 yards on a side, the floor of which 625 yards of carpeting, 1 yard wide will be sufficient to cover.
The proof is seen by adding together the different parts of the figure.
Such are the principles, on which the operation of extracting the square root is grounded.