2. And if at a 126 4; then 814*+ 81a 10279, and one of the Values of a is 3.33333 +3 }· N.B. Tho' I have not hitherto in this Part IV, in Dividing any Refolvend by its refpective Divifor, pursued such Divifion beyond finding the first Significant Figure of the Quotient, or of the Value of y. Yet when the Divifor once takes place, it may be continued to as many, or almost as many Figures, as the next preceeding x bath of the first Figures of the Root fought; as in the following Examples. Example 1. If a3 231; Quære a proxime. Suppofe xa; then (a3 =) x3 + 3x2y + 3xy2 + y2 = 231. If as Example 2. 104 10000; Quære a proxime. Suppofe x+y=a; then (a1 =) x2 + 4x3y +6x2y2 + 4xy3 +» — 104 =) — 10X 1oy. Tho' the Method us'd in these two laft Examples is better (as being more Expeditious) than that us'd in the foregoing Examples; yet ftill a better Method (which is the fame with Mr. Raphfon's, and by him call'd the Converging Series) may be deduced from this; thus in Extracting the Cube-Root of 231, in Ex. 1. You may readily fee that each Divifor is 3x2, and that each Refolfrom the time the Divifor takes place, or vend is = 231-*, and confequently, ferves to discover the firft Figure of the 231. x3 ; 3x2 This is the most natural the Roots of Equations. way of raising Raphfon's Theorems for Extracting and therefore the Theorem true Value of y, each y= In like manner in Example 2. viz. d4 - A is y = * 3x2 10000, each Divifor is 410, each Resolvend 10, each Resolvend 10000 x+-† Iox; and from the time the Divisor takes place, each 10000 x4 +10x ; and therefore the Theorem for find y = 4x3 10 = ing the Value of a in this Equation. viz. at pag is y = 9 - x++ px. And in this or the like Manner Theorems may 4x3 P be rais'd for Evolving all poffible Equations. Wherefore in Extracting any Root of any Equation, whether Simple or Adfected, Let x be taken the Value of the 1st. Figure, or ift and 2d Figures of the Root fought, then, by its ref K pective pective Theorem, find the Value of the firfty to fo many Figures as you think to be the true ones; the Sum of which, and that of x call x (i. e. your next x), with which proceed, by its respective Theorem, to find the Value of the next y, and fo on. Thus, if the Cube Root of 231 is to be Extracted by this Me thod, let 1ft x be taken =6; then 1ft .13; therefore 6 +. 13 (=1ft x = (A)— A-x3-231-216 3x2 1fty)=6.13=2d x, 108 then 2d y Then 3dy= 231-230.9997244553 + =.00000243966; 6.13579 (2dx+zdy)=3d x. therefore 6. 13579243966 (=3dx + 3d y) is very near = Cube-Root of 231. You may fee by this Example, and by as many more as you are pleas'd to try, that from the time the Divifor takes place, each Renewal doubles entirely, or almoft, the true Figures in the laft x; and of confequence a few renewals afterwards will ferve to Extract any Surd Root required to very many places of Figures. of PART V. the Indices, or Exponents of Powers. 1F Fan Unit be Multiplied by any Quantity a, and the Product a by a, and that Product aa by a, and that Product aaa by 4, &c. the feveral Products are the ft. 2d. 3d. 4th, &c. Powers of a. The Antients Manner of Writing the Root; and several Powers of a was as follows. Root or ift. Power, Square or 24. Power, Cube or 3d. Power, But Des Cartes chofe to defign the faid Powers of a in the following Manner, (which, for many Reasons, is the best), viz. The Figures 1, 2, 3, 4, &c. writ over a, and shewing its Powers, he call'd Indices or Exponents. It is Evident that the Powers of a in for the Terms of) the foregoing Series, are in a continued Geometrical Proportion, whose Ratio is the ft. Power or Root a, and the Indices of the faid Powers in a continued Arithmetical Proportion, whofe common excess is (by the Definitions of both Proportions). Now fince the Exponent of each Power of the foregoing Series is, by the common excess of the Indices, to wit by 1, more than the Index of the next foregoing Power, it must follow of K 2 courfe |