These two Examples are done in the fame Manner as the former; only it is to be obferved, that, the fourth Example confifting of fix Figures, the Dot does not fall on the last Figure to the left Hand; therefore, the firft Period is 20, whofe next lefs Square in the Table is 16, the Square of 4. Exam. 5. Extract the Square Root of 346921 (589 25 108) 969 1169) 10521 589 589 5301 4712 2945 346921 Proof. Exam. 5. Exam. 5. The firft Square is 34, and the next lefs Square in the Table is 25, whofe Root is 5; therefore put 25 under 34, and fubtract; alfo put 5 in the Root, and proceed as in the former Examples. Exam. 6. Extract the Square Root of 16459249 (4057 16 805) 4592 8107) 56749 4057 4057 Exam. 6. The firft Square being 16, and that being found exactly in the Table, when it is fubtracted, nothing remains; fo that the Refolvend will be only 45, the next Period: When, therefore, the Root 4 is doubled for a Divifor, it cannot be had in 45 (under the Reftriction as in the other Examples); therefore, as the Divifor cannot be had, fet down a Cypher in the Root, and alfo to the right Hand of 8 the Divifor Then bring down the next Period, which is 92, and afk how many times 80, the Double of the Root, can be had in 4592, (with the fame Reftriction as before) and the Answer is 5 times; place 5 in the Root, and alfo to the right Hand of the Divifor; then multiply 805, the increased Divifor, by 5, the Figures laft placed in the Root, and proceed as before. The feventh Example is done in the fame Manner. Exam. 8. The Numbers in the eighth and ninth Examples not being Square Numbers (for there will be a Remainder after the Root is extracted) in order to obtain the Root near the Truth, it is neceffary to annex Cyphers for decimal Places; obferving, that, as many decimal Places as are required in the Root, fo many Pairs of Cyphers must be annexed to the Number; therefore, let the Root be required in each Example to two Places of Decimals, then there must be four Cyphers annexed to each. In thefe Examples the Learner may fee, by the Proofs, that the Root comes out very near the Truth, tho' the Operation is carried no farther than to two Places of Decimals. In Cafes where a greater Exactness is required, more Pairs of Cyphers may be added, to bring it nearer the Truth; for it must be obferved, that, if a Number is not a true Square, fo as to. have no Remainder, the true Root of fuch Number can't be attained, tho' the Operation be carried to ever fo great a Length. The Root 21.38 of the tenth Example, being multiplied by itself, produces 457.1044. Or, if they are Integers only whofe Square Root is to be extracted, and, after all the Periods are taken down, there is a Remainder, you may then take down Pairs of Cyphers at Pleasure, and fo continue the Root to any Number of Decimal Fractions as may be convenient. But, if a mixed Number be given to be extracted, and the Places of Decimals are an odd Number, as three, five, &c. there must be a Cypher annexed to the right Hand of the decimal Places, to make them even: The firft Dot in the Integers must be over the Units Place; therefore, it is best to begin with dotting the Units Place, and then proceed to the reft, whether forward or backward, Exam. II. Extract the Square Exam. 12. Exam. 13. Extract the Square In each of thefe Examples, the Number of decimal Places. are odd; therefore, a Cypher must be added to each, before the Operation is begun; after which, it is the fame as before, The The Proof of thefe laft Examples I have omitted; but the Learner may multiply them for his own Improvement, remembering to add the Remainder. Art. 50. To extract the Cube Root. T HERE are feveral Methods of extracting the Cube Root: The Method by Converging Series is a neat and elegant Manner of Approximation: But the following Rule appears to me the best and most convenient, for the prefent Purpofe, I have yet feen; and for which I am obliged to Mr. Thomas Triplett, Teacher of the Mathematics, at Kingston; who likewife favoured me with a new Method of Cross-Multiplication, which will be explained in its proper Place. Rule. Place a Dot over the Units Place; then, miffing two Figures, put another Dot over the Place of Thoufands; and, miffing two Figures more, put a Dot over the Place of Millions; continue to dot every third Figure, and, as many Dots as are thus placed, of fo many Figures will the Root confift. Then feek in the Table, Page 89, for a Cube Number, in the bottom Line, the next lefs to the first Cube or Period of the given Number on the left Hand; fet the faid Cube Number under the first Period of the given Number, and its Root in the Place of the Root, to the right Hand of the given Number; fubtracting the Cube from the firft Period, and to the Re O mainder |