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Reduce 13s. 4 d. to Reduce 9s. 9d. 2

a Decimal,

12s. is .6

Is. is .05

to a Decimal. 8 s. is .4 Is. is .05

4 d. is .017

6 d. is .025 3d. is .0125

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39. is .003

.4905

It may be obferved, in Example 2, that, for four Pence, which is fixteen Farthings, I fet down 17, because it is more than three Pence.

In the fame Manner may the Parts of any Integer be reduced to a Decimal with little Trouble, by obferving, that .5 is the Half of any Integer, .25 the Quarter, .125 the Eighth, and fo on But, as this Way is not quite exact in fome Inftances, I would recommend it to the Reader to proceed by the ufual Method of dividing the Numerator by the Denominator, in order to reduce a Vulgar Fraction to a Decimal.

To extract the Square Root of any Number.

B

EFORE we. inftruct the Learner to extract the Square Root, it will be proper to explain what is meant by a Square Number, and what by the Root of fuch Square. That Number is called a Square which is produced by multiplying any Number into itself: Thus, if a 6 be multiplied by itself, the Product is 35, which is therefore a Square Number; and 6 is called the Root or Square Root of 36, becaufe 6 times 6 is 36: In the fame Manner, 9 is a Square, the Root of which is 3, for 3 times 3 is 9: So is 16 a Square Number, whofe Root is 4.

If a Piece of Board, Stone, or any other Matter, be cut exactly fquare, and each Side is four Inches, then is the Area or Content fixteen Inches; that is, there will be fixteen fquare fuperficial Inches in the Board, as may be feen by the following Figure.

Each

Each Side being divided into four Parts, and Lines drawn from thofe Divifions, the whole Square will be divided into fixteen little Squares: From whence it appears, that the Square Root of any Number is the Length of the Side of a Square, whose Area, or fuperficial Content, is equal to the given Square Number.

If any Square Number is multiplied by its Root, the Product is called the Cube; which is the Content of a Solid, whofe Length, Breadth, and Thicknefs are equal to the Root, Thus, if a Piece of Timber was cut exactly cubical, each Side being four Inches, the Content of the Cube would be fixty-four Inches, fixty-four being the Cube of four; which four Inches are the Length, Breadth, and Thickness, or Depth of the Piece.

Thus having explained what a Square and a Cube are, I fhall next give the Squares and Cubes of all the nine Digits, in the folllowing Table.

A Table of the Squares and Cubes of the nine Digits.

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In the upper Row are placed the nine Digits, under each of which are placed their respective Squares and Cubes; therefore, when the Square or Cube of any Number under 10 is required, they may be found in this Table by Inspection: Thus, the Square of 5 is 25, and of 7 is 49: The Cube of 6 is 216, and the Cube of 9 is 729; and fo of the reft. And, in the fame

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Manner,

Manner, if the Roots of any of the Squares in the Table are required, they are found the fame Way: But, in Cafe the Square Root of any Number not to be found in the Line of Squares in the Table is required, this must be done by the following Method.

Article 49. When a Number is given for the Square Root to be extracted, place a Dot over the Units Place; then, miffing one Figure, put another Dot over the third Figure, or Place of Hundreds; and thus continue to put a Dot over every other Figure to the End; and, as many Dots as are thus placed over the Number, of fo many Figures will the Root confift.

Exam. 1. Extract the Square Root of 61504.

This Number being dotted as above directed, it appears, that the Root confifts of three Figures or Places: To find the first of the three, look in the foregoing Table, in the middle Row, or Row of Squares, for the Number pertaining to the firft Dot to the left Hand, which in this Example is 6; if it is not there, take the next lefs Square, which in this Case is 4, whofe Root is 2; now draw a crooked Line to the right Hand of the Figures, and place the Root 2 as a Quotient in Divifion; then put 4, the Square of 2, under the 6, and fubtract it, fetting down 2, the Remainder, underneath, as in Divifion Then take down the two Figures pertaining to the next Dot, viz. 15, and annex them to the 2 that remained, and this is called the Refolvend; now draw a crooked Line to the left Hand of the Refolvend, and double the Root in the Quotient for a Divifor; then place this Divifor to the left of the crooked Line, leaving Room fufficient for another Figure to be put betwixt that and the crooked Line: Then the Work will ftand thus,

61504 (2

4

. 4) 215

Then find how many times the Divifor 4 can be had in 215 the Refolvend, (but with this Restriction, that the Figure expreffing the Number of times it may be had being placed to the right Hand of the 4, to make the Divifor com

plete,

a

plete, and multiplied by the fame Figure in the Root, may not exceed the Refolvend) and I find it will be 4 times; fet down 4 in the Root, and alfo in the Divifor; then multiplying 442 this increased Divifor, by the 4 laft placed in the Root, the Product 176 must be placed under the Refolvend 215, and fubtracted from it: To the Remainder 39 bring down the next Square, that is, the laft two Figures, thus,

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Draw a crooked Line before 3904, the new Refolvend, and double 24, the Figures in the Root, and place them as a Divifor; then find how many times the Divifor 48 can be had in 3904, (but with the fame Reftriction as before, that the Number of times it may be had, being placed to the right Hand of the 48, to make the Divifor complete, and multiplied by the fame Figure in the Root, may not exceed the Refolvend) which is 8 times; fet down 8 in the Root, and alfo in the Divifor; then multiplying 488, the Divifor thus increased, by the 8 laft placed in the Root, the Product is 3904, which being fubtracted, there remains o: Therefore 248 is the true Root fought. The whole Operation is as follows:

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The Way to prove if the Work is right, is, to multiply the Root into itfelf: Thus 248, being multiplied by itself, produces 61504, the given Number, N 2

Exam.

Exam. 2. Extract the Square Root of 56644 (z

4

4) 166

In this Example, the firft Square is 5, and the next less Number in the Line of Squares in the Fable is 4, whofe Root 2 must be placed in the Root, and its Square 4 under the 5 then fubtract, and there remains 1; to which bring down the next Period 66, as taught in the former Example: Double the Root 2 for a Divifor, and place 4 its Double to the left of the crooked Line; then ask how many times 4 the Divifor can be had in 166, (but with the fame Reftriction as before, that the Number of times it may be had, being placed to the right Hand of the 4, to make the Divifor complete, and multiplied by the fame Figure in the Root, may not exceed the Refolvend) and I find it will be 3 times; then, putting 3 in the Root, and alfo in the Divifor, multiply 43, the increafed Divifor, by 3, and the Product is 129; which place under the Refolvend, and fubtract as before,

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Thus having taken down the laft Period 44, and placed it

the right of 37 the Remainder, the Refolvend is 3744 i and 23, the Figures in the Root, being doubled for a Divifor, is 46, which is placed to the left of the crooked Line: Now inquire how often 46 can be had in 3744, (under the fame Restriction as before, that, the Figure being placed in the Roor, and to the right Hand of the Divifor, and multiplying the increafed Divifor by the Figure in the Root, it may not exceed the Refolvend) and I find it will be 8 times; place 8 in the Root, and alfo on the right Hand of the Divisor, betwixt that and the crooked Line: Laftly, multiply 468, the Divifor thus increased, by 8, the Figure laft placed in the Root, and the Product is 3744; then fubtract, and there remains o: Therefore 238 is the true Root of the given Number, as appears by the Work.

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