I' PART. V. Of the Indices, or Exponents, of Fan Unit be Multiplyed by any Quantity a, and the Product a by a, and that Product aa by a, and that Product aaa by a, &c; the feveral Products are the 1st, 2d, 3d, 4th, &c. Powers of a. Formerly the Manner of Writing the Root and the several Powers of a was as follows. Root or ift. Power, Square or 2d. Power, Cube or 3d. Power. But, of late, the Powers of a are more ufually defigned in the following Manner, viz. The Figures 1, 2, 3, 4, &c. writ Superior to 4, and fhewing its Powers, are called Indices or Exponents. -It is evident that the Powers of a in (or the Terms of) the foregoing Series, are in a continal Geometrical Proportion, whofe Ratio is the 1ft. Power or Root a, and the Indices of thofe Powers in a continued Arithmetical Proportion, whofe comnon Excefs is 1, by the Definitions of both Proportions. Now, Now, fince the Exponent of each Power of the foregoing Series is, by the common Excefs of the Indices, to wit by 1, more than the Index of the next foregoing Power, it must follow of Course, if the Series be continued backward, that the feveral Terms of it will be found (by Subtracting from the Index of the Root a' the common Excefs of the Indices, to wit 1, and from the Remainder o the faid common Excefs, and from this Remainder 1, the faid common Excess, and from this Remainder 2 the faid common Excefs, &c.) = &c. a-3, a, a—2, ao, a2, a2, a3, a2, &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 that Series be continued backward, the feveral Terms of it will be found (by Dividing the Root a by the Ratio a, and the Quotient 1 by the faid Ratio, and this Quotient by the faid Ratio, and this Quotient a a3 I a by the said Ratio, &c.) = &c. 11, 1, 1, 1, a, a2, a3, aa, &c. From the two last Paragraphs, it is manifeft that ao is = 1, alfo = ——, alfo a—2 = = —, also a—3 = &c. And that the Series being continued backward and forward, may be writ I I I 3 Thus, &c. a―s, a—4, a—3, a—2, a—1‚a°, a', a3, a3, aa, as &c. Or thus, &c., 2 −, I, a, a2, a3, a‡, as, &c. From what hath been faid, may be deduced and demonstrated the two Fundamental Rules for all Operations relating to the Exponents of Powers. The 1ft. of which is, that the Exponent of the Product of any two Terms of a Series in (viz. of fuch a Series as the laft but one) is equal to the Sum of the Exponents of those two Terms. And the 2d. is, that the Exponent of the Quotient of any two Terms in a Series in is had by Subtracting the Exponent of the Divifor from the Dividend's Exponent. The first Rule may be prov'd by Algebraical Multiplication, by the help of what hath been already faid in this Part V. So the Exponent of the Product of a and at is prov'd to be 6410, by Multiplying a by at; for a6 × at is = a1o a6+4. In like manner the Exponent of the Product of a and a3 may be proved to be 4+3=-1; thus → = is = is (by what is abovefaid) ——, and — × a3 = I a4 a== ; but is = 4'; confequently ponent of a4 × a3. a -I is the Ex And the fecond Rule muft follow of courfe from the Firft, or may be proved by Algebraical Divifion; fo the Exponent of asa is prov'd to be 6-42, by Dividing ao by a* ; for a6 ÷ at ===== = a2. Also the Exponent of the Quotient a6 of a3 Divided by as is prov'd to be 3—5—— 2 ; thus a3 I a2 a3 as ; a (by what has been prov'd in this Part V.) In like manner the Exponent of the Quotient of a- Divided by a is prov'd to be = 4+3=1; thus a― (by what has been said in this Part) is = ——, and The Method of Raising any Quantity to any required Power, or Extracting any Root out of it, is eafily deduc'd from the two foregoing Rules; thus the Square of as a x as is = a3+s =a6, and 6 = 3 x 2; that is, the Exponent of the Square is double the Exponent of the Root; or if you fuppofe as to be the Root, then 43 will be the Square-Root, wherefore fince 36) you may conclude that the Exponent of the SquareRoot is half the Index of the Root. Alfo the Cube of y= y3 × y3 × 93 = y3 +3+3 = y9, and 93x3, which fhews that хуз the Exponent of the Cube is treble the Exponent of the Root ; or that the Exponent of the Cube-Root is one third Part of the Exponent of the Root. Alfoy-xy-1xy is = y=y-3. And Univerfally the Exponent of the m-Power is m times the Exponent of the Root; and the Exponent of the m-Root (or the Exponent of the Root. I m Hence the m Power of the m Root is the Root; that is the Square of the Square-Root, or the Cube of the CubeRoot, &c. is equal to the Root. By By the laft Paragraph but one, it is evident that ✔a, ja, Ja, &c. are equal to a3, a3, a, &c. refpectively; for at being the Root, half its Index, to wit, the half of I must be the Index of the Square-Root; alfo of the Index of the Root must be the Index of the Cube-Root; &c. The Index of the Square-Root, Cube-Root, &c. may be otherwise discovered, thus ; Let it be required to find the Square-Root a3. au Then aux au= 4; that is the Square of the Square-Root is equal to the Root; wherefore (by the first Rule) z + u = 3, and ; confequently is the Square-Root of a3. Again let it be required to find the Cube-Root of a→ Suppose ita"; then a" × a” × a" a3w (by the ift Rule) isa; wherefore 3w 2, and w; confequently a is the Cube-Root of a→2. &c. a-2. So that this Remark is only a Confirmation of the above mentioned Paragraph. The Sum of what hath been faid in relation to the Indices or Exponents of Powers is, putting am and an equal to any two Quantities, that Ift. am × an = am+n 2d. ama2 = am-n 3d. am rais'd to the nth Powergmu 4th. The z-Root of am m Here follow more Varieties, which, in reality, are Included in the four laft Articles, fince m and n are Univerfal, and confequently may be equal to any Quantities Whole or Fracted, Affirmative or Negative. a-qxa+p=a-q+p a-q÷ap =a-q-p apa-q=ap+9 a-q rais'd to p Power a-qP a-qxa-ra-q-r a-qara-q+r the r-Root of a−9 is = a PART I. PART VÍ. Of Surd-Roots. CHA P. I. Definitions. 1.W Hen any Number or Quantity hath its Root propos'd to be Extracted, and yet is not a true figurate Number of that Kind; that is, if its Square-Root being demanded, it felf is not a 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 can never be exactly Extracted; and fuch Roots are ufually defign'd with their Indices, or Radical Signs: So the Square-Root of 2 is writ thus 23, or thus 2; for 'tis evident that the SquareRoot of 2 is not a whole Number; neither is it a Fraction, because the Square of any Fraction is also a Fraction (by 16. 8. Eucl. El.); and confequently it is not expreffible by any Rational Number. Alfo the Cube-Root of b is writ thus b (= b 3), or thus 3/ ¿•, &c. 2. But, altho' these Surd-Roots (when truly fuch) are inexpreffible by Rational Numbers, they are notwithstanding capable of Algebraical Operations. 3. Surds are either Simple, which are exprefs'd by one fingle Term as b,c,d, &c. or Compound, which are form'd by the Addition and Subtraction of Simple Surds, as √5+ √2, I 2 5 3 — 2 31‚ àa3 + b3 — ca3 ; &c. or elfe Univerfat as 7+2 d which fignifies the Cubick-Root of that Number, which is the Refult of adding 7 to the Square-Root of 2; 8r. 4. Surds are alfo Commenfurable or Incommenfurable. Commenfurable Surd. Roots are fuch, whofe Ratio or Proportion to one another, may be exprefs'd by Rational Numbers or Quantities: And fuch Surd-Roors whofe Ratio cannot be exprefs'd by Rati onal Numbers or Quantities are call'd Incommenfurable. CHAP. |