THE EXTRACTION OF THE SQUARE RO o t. O extract the Square Root of any Number, is to find out a Number, which being multiplied by itfelf, the Product fhall be equal to the given Number. Thus, fuppofe 9 the given Number; then will the Root of it be 3; becaufe 3 multiplied by 3 will be equal to 9, the given Number. And this may be geometrically demonftrated by the following Figure, where each Side contains 3 equal Parts, by which the great Square A B CD is divided into 9 little Squares. The Extraction of the Square Root, therefore, is, by having the Number of little Squares given, that are contained in a greater Square; to find out how many of the lefs Squares make one Side of the greater. In order to extract the Square Root of any Number, it will be neceffary to have by Heart the following Squares, whofe Roots are one Figure. Root 1 2 3 4 5 6 7 8 9 Square 1 49 16 25 36 49 64 81 By this Table you perceive the Square of 1 is 1; the Square of 2 is 4; the Square of is 9; and fo of the reft. This being done, we may proceed to extract the Square Root of any given Number by the following General Rule. First, fet down the given Number; then make a Dot over the Place of Units, and so on, over every second Figure; *(towards the Left Hand in whole Numbers; but towards the Right Hand in Decimals :) and as many Dots as there are in the given Number, (which is ufually called the Refolvend) fo many Figures will there be in the Root. Next, feek the nearest Square to the first Period or Dot, on the Left Hand; whofe Root fet in the Quotient, and its Square place under the firft Period; then fubtract, and to the Remainder bring down the next Period, (as in Division) which will form a Dividend. Now, double the Root you put in the Quotient for a Divifor, and place it on the Left Hand of the Dividend; then feek how oft this Divifor can be had in all the Figures of the Dividend except the laft; fet it in the Quotient, and alfo on the Right Hand of the Divifor. Multiply this increafed Divifor, by the Figure laft put in the Quotient, fet the Product under the Dividend, and fubtract it therefrom. Laftly, to this Remainder bring down another Period for a new Dividend; then, double all the Figures in the Quo * The Reason for pointing every fecond Figue is, because the Square of the greatest Number under 10 can confist but of two Places. tient for a new Divifor; divide with the fame Care as before; and fo proceed till all the Periods are brought down, and the Operation be finished. A few Examples will make this plain and easy. To prove if you have performed the Operation right, multiply the Root by itfelf; to which Product add the Remainder if there be any; then, if this Sum be equal to the Refolvend, or Number given, it is right, otherwise not; and the Error must be fought out by performing the Operation over again. What is the Square Root of 17956? Refolvend .17956 (134 true Root. I 23).79 264) 1056 If any Thing remains after all the Periods are brought down, its Value may be found to what Exactnefs we please, by adding two Cyphers at a Time to the Remainder; and for every Pair of Cyphers fo added, we shall have one Decimal Place in the Root; as in this Example. Note. In this Example, we have added two Pair of Cyphers; therefore we have two Decimal Places in the Root. And thus, by adding Cyphers by Pairs, we may carry on the Work to what Number of Decimals we please; but still fomething will remain. All fuch Numbers are called Surds; and their Roots can never be perpectly found. If the Number to be extracted is either mixt, or a Decimal; only make the Number of Decimal Places even, (if they are not fo already) by adding a Cypher, (or Cyphers) that the Point may fall on the Units Place of the whole Number. Here we added a Cypher to make the Decimal Places even; and because there was a large Remainder, we annexed two more. Then, as there are three Points over the Decimal, we cut off three Places in the Quotient for Decimal Parts; the other two are Integers or whole Numbers. |