112 619. Example 1. At 11. per C, what is that for 3 tb? By Reduction of Fractions 1. of ai and 3 tb of a tb; and IC in the raction of a tb=1 Hence the Stating would be, if 42 tb : 44::tb: the Answer. By Multiplication of Fractions x =+4=4; and, by Divifion of Fractions, 2+12=274 by Abbreviation of a £. by Valuation 10 Pence. 7 28 28 28 XI 28 777 By the second Method above-mentioned, we have 112×3×2=672 the Denominator, and 1 × 4×7 28 the Numerator; and the Anfwer 23 as before. Hence plainly appears the Agreement of the two Methods. 620. Example 2. Admit a Dog is pursuing a Hare that is go Yards a Head of him; and that, for every Yard the Hare runs, the Dog runs 2 Yards: Quare, How many Yards the Hare will run, be fore the Dog gets up with her? T ༧ 1 .. 2 Solution. In the Time that the Dog runs 2 Yards, the Hare runs 1 Yard, by the Queftion; ., in the Time that the Hare runs Yard, the Dog gains upon her 2 II Yard; hence the Question will now be to this Purpose, if, whilft the Dog gains on the Hare Yard, the Hare runs 1 Yard, how many Yards will the Hare have run, when the Dog hath gained 50 Yards upon her, or, in other Words, hath caught her? Hence, 1 being, the Stating will be, as the Anfwer, which is thus found, 2 × 1 × 50 = 100 for the Numerator, and 3 x IX I 3 for the Denominator; and ... the Answer of a Yard 33 Yards. But, if it had been required to find how many Yards the Dog must run to overtake the Hare, the Stating would have been, as the Yards the Dog muft run; .' 2 × 5 × 50 = 500 for the Numerator, and 3 × 2 × 1 =6 for the Denominator; and.. the Dog must run 500 of a Yard 83 Yards. And the Truth of these Operations may be easily proved thus: By the Queftion, the Dog muft run 50 Yards more than the Hare, 3 2 5 ••50 Τ : but 1 but 83 Yards 33 Yards 50 Yards, for Proof. 621. Scholium. In the Rule of Three, &c. in Order to avoid Fractions, as much as might be, we generally bring the middle Term into the lowest Species; but, as the Learner is now supposed acquainted with Valuation of Fractions, he may many Times fave fome Trouble by putting it down in the Species: given in the Question, if it be but one; and, if more than one, by reducing no lower, than the leaft Species mentioned in the Question; and then, when we have found the Answer in that Species, if there remain any Fraction, we can by Valuation find its Value, in the inferior Species. Take an Example. In Article 198. it was required to find the Value of 2437 #b, at 13s. per C. This ftated will be, if 112 lb: 135. 2437: 2437x13 s. = 3163=2825.97 = 147. 25.972, and by Valuation 97 of a Shilling = 10d. 1gr. 4, and the Anfwer is 141. 25. 10d. I gr. 4, and this Fraction may be abbreviated to 4. 6 6 112 S.= IN 622. TNVOLVE the Numerator for a Numerator, and the Denominator for a Denominator. The Reason of this is evident, from Involution of whole Numbers, and Multiplication of Fractions. 623. Example. What is the Square of? Solution. 2 x 2 = 4 the Numerator, and 3 x3 = 9 the Denominator, and fo the Square of 34. 2 'CHAP. 624. 1. CHAP. L.. EVOLUTION of FRACTIONS. IF F the Fraction, whofe Root is to be found, is an immediate Power of fome Root, (formed as is fhewn in Involution of Fractions) its Root may be found by extracting the Root of the Numerator for a Numerator, and the Root of the Denominator will be the Denominator, as is manifeft. 2. But fometimes the Fraction propofed is not an immediate Power, but equal to fuch a Power of the Root; then we must reduce the given Fraction to its lowest Terms, and find the Root as above directed. 3. But if it fhould fo happen, that the proposed Fraction, when reduced to its loweft Terms, cannot have the perfect Root of both its Numerator and Denominator found, then we may be affured, that it is neither an immediate Power, nor an Equivalent to one; and in fuch Cafe muft turn the Fraction into. a Decimal, and fometimes be contented with an Approximation of its Root; this we shall illuftrate, when when we treat of Decimals. 625. Example 1. Extract the Square Root of . Solution. The Square Root of 4= 2, of 9 = 3; the required Root is. 93; 1250 626. Example 2. Extract the Cube Root of Solution. It does not admit of the true Root, as it ftands here, but by Abbreviation it is; now the Cube Root of 82; and of 1255; the Root is 3. CHAP. CHAP. LI. Of POSITION by FRACTIONS. 627. I fuppofed fuch Numbers as might avoid Frac TN Pofition, it may be observed, we always tions in the Operation; because the Operation would be more fimple, and the Learner was not fuppofed at that Time to understand the fundamental Rules of Fractions. But, as fometimes it may happen that fuch Numbers are not easily thought on, perhaps it may not be ufelefs to give an Example folved by a fractional Operation. 628. Example. Let it be required to give a Solution to Question 2. Art. 506. 35 Solution. Here we put any Number, as 1d, for a Share; then 4 Men muft have 1x44; the Captain, and the Boy, the Sum of 4, 1, and, = 5 ÷ + ÷ = 5 Now as : 1d. 510SS = 3752 d. 19 as in Art 506; and the remaining Part of the Solution is the fame as in that Article. : 2 6 x 3 X 51055 629. We think it needlefs to apply Fractions to any more Rules of Vulgar Arithmetic; because, if the Reader rightly understands what has been already laid down, he cannot, when Occafion requires, be at a Lofs to apply it to any other Rule in common Arithmetic. MA Mathematical ESSAYS. 1. ESSAY II. Containing DECIMal Arithmetic. 'CHA P. I. NOTATION of DECIMALS. A Decimal Fraction (from Decimus Lat.) is a Fraction whofe Denominator is 10, or 100, or 1000, or 10000, &c. For here we fuppofe any Integer to be divided into 10 Parts; and each of thefe into 10 Parts, making in the Whole 100 Parts; and each of these laft Parts into 10 Parts, making in the Integer 1000 Parts; &c. at Pleasure; and any Number of thefe Parts are called Decimal Parts, and are the Numerator of the Fraction, by which we would express how many Parts we would fignify; and the Number of Parts into which the Integer is divided, is the Denominator. 2. Hence these Parts may be expreffed as in Vulgar Fractions; but, for the more ready Management, the Denominators are omitted, and only the Nume rators I. |