course, if the faid Series be continued backward, that the several Terms of it will be found (by Subftracting the said common Excels of the Indices, to wit 1 from the Index I of the Root a', and from the Remainder o the faid common Excefs, and from that Remainder 1, the faid common Excefs, and from that Remainder -, the faid common Excefs, &c.) =&c, a—', a—', a—', a°, a', a2, a3, a1, &c. And fince each Power of the faid foregoing Series is the Product of the next foregoing Power, and of the common Ratio; of Confequence, if the faid Series be continued backward, the feveral Terms of it will be found (by Dividing the Root a by the Ratio a, and the Quotient by the faid Ratio; and that Quotient by the said Ratio, and that Quotient a by the said Ratio,&c.) Hence 'tis manifeft that a° is 1, also 4-1 I allo a I a3 ==‚· also 4-2 =&c. And that the said Series being con tinued backward and forward, may be writ Thus &c. 4-1, a-2, a-1, a°, a', a', a3, -&c. From what has been faid, may be deduc'd the two Fundamental Rules for all Operations relating to the Exponents of Powers. The firft of which is that the Exponent of the Product of any two Terins in the said Series in ÷ is equal to the Sum of the Exponents of the faid two Terms. And the 2d. is, that the Exponent of the Quotient of any two Terms in the faid Series in is had by Substracting the Exponent of the Divifor from the Dividends Exponent. For the ft. Rule may be prov'd by Algebraical Multiplication, by the help of what has been already faid in this Part V. So the Exponent of the Product of 46 anda is prov'd to be = 6 + 4 o, by Multiplying a' by at; fòr ao xat is a3° — a6 +4. In like manner the Exponent of the Product of a-4 and a3 may be prov❜d to be -- 4+3= 1; thus 4-4 is (by what has been prov'd in this Part V.) = the Exponent of a-4 X a3: and Xa3 = a3 4-1; confequently And the zd. Rule muft follow of Courfe from the 1ft, or may be prov'd by Algebraical Division; fo the Exponent of 45 44 is prov'd to be=6-4-2, by Dividing a by at; for aʻ➡ a1 = a. Also the Exponent of the Quotient of a3 Divided by a3 is prov'd to be 3-5= 2; thus a3 - a2 a-2 (by what has been prov'd in this Part V.) In like manner the Exponent of the Quotient of a-4 Divided by a-3 is prov'd to + 3=- 1, thus a-4 (by what has been faid in this be 4 a3 I that is 4-1 ; &c. ? - Or I may prove that as a, a, a3, &c, is a Series of Terms in increafing, fo a2, a', ao, a—ı, a−2, a—3, &c. is a Series of Terms in Decreafing; thus, = Suppose the faid Series of Terms in Decreafing to be a2, a', a", a”, α, aï, &c. refpectively; then a', a', a", are in that is 42 a! :: But the Product of the Means is equal to the Product of the Extreams; that is a' x a' a2 Xa"; therefore (by what has been already said in this Part V.) 1+1=2+u; confequently uo. Again, fince a', 4°, a", are in; then a.. 4° : : 4° . . a; therefore o to it, and . In like manner x will be prov'd The manner of raifing any given Quantity to any required Power, or Extracting any Root out of it, is easily deducible from the two foregoing Rules; thus the Square of a3 is a3× a3 =a3+3 4, and 63x2; that is, the Exponent of the Square is double the Exponent of the Root; or if you fuppofe a to be the Root, then a3 will be the Square-Root; wherefore (fince 63 2) you may conclude that the Exponent of the Square-Root is half the Index of the Root. Allo the Cube of y3 y3 x y3 xy3 — y3+3+3, and 93 × 3, which fhews that the Exponent of the Cube is triple the Exponent of the Roor, or that the Exponent of the Cube-Root is one third Part of the Exponent of the Root. Alfoy-1.xy-1xy-1 isy-1-1-1= . And universally the Exponent of the m Power, is m times the Exponent of the Root, and the Exponent of the m-Root (or Power) Hence the m- Power of the m - Root is the Root; that is the Square of the Square Root, or the Cube of the Cube Root, &c. is equal to the Root. is By the laft Paragraph but one it is evident that 2√«‚3√a‚*↓a,&c. are -a a *, &c. refpectively; for a being the Root, half its Index, towit the half of muft be the Index of the Square Root; allo of the Index of the Root must be the Index of the Cube-Root; &c. The Index of the Square Root, Cube-Roor, &c. may be otherwise discover'd. Thus Let it be required to find the Square Root of a3. Then a" Xaa3; that is, the Square of the Square-Root is the Root; wherefore (by ift Rule) u+u=3, and u=2, confequently a is the Square Root of a3 Again, let it be required to find the Cube-Root of 4−2. Suppose ita", then awa" × awa3 (by the 1st Rule) isa2; wherefore 3 w= 2 and w 2. Confequent ly a is the Cube Root of 4-2. &c. So that this Remark is only a Confirmation of the foregoing univerfal Rule. The Sum of what has been faid in relation to the Indices or Exponents of Powers is, Putting amany Quantity whatsoever, and 4" the fame or any other Quantity. Here follow more Varieties, which in reality are included in the 4 laft foregoing Articles, fince m and n are Universal, and confequently either or both of them may be equal to any Quantities Whole or Fracted, Affirmative or Negative. PART VI. Of Surd-Roots. WE CHAP. I. Definitions. Hen any Number of Quantity hath its Root propos'd to be Extracted, and yet is not a true figurat Number of that kind; that is, if its Square Root being demanded, it felf is not à true Square; if its Cube-Root being required, it felf be not a true Cube; &c. that Root is called a Surd Root, because it is inexpreffible by any known way of Notation, otherwile than by its Index or Radical Sign; fo the Square Root of 2 can be only 2 writ thus 2 or thus 2; for 'tis evident that the Square Root √2; of 2 is not a whole Number; neither is it a Fraction, because the Square of a Fraction is also a Fraction, (by the 16. 8. Eucl. El.) and consequently it is not expreffible by any rational Number. Alfo the Cube-Root of b2 can be only writ thus b3 (=b3) thus b. &c. 2. But altho these Surd-Roots (when truly fuch) are inexpreffible by rational Numbers, they are notwithstanding capable of Algebraical Operations, as fhall be hereafter fhew'd. Surds are either Simple, which are express'd by one single Term as b3, 3√c, bd\2, &c; or Compound, which are form'd by the Addition and Subftraction of Simple Surds, as √5 + √2, 5 213, d3±3+b3— c3, &c. or elfe univerfal as 7+2 which fignifies the Cubick-Root of that Number which is the refult of adding 7 to the Square-Root of 2, &c. Surds are alfo Commenfurable or Incommenfurable. CommenfurableSurd-Roots are fuch, whofe reafon or proportion to one another, may be expreft by rational Numbers or Quantities. And fuch Surd Roots whofe proportion cannot be expreft by rational Numbers or Quantities are call'd Incommenfurable. СНАР. |
