An ANALYTICAL SOLUTION of certa infinitefimal Equations, tranflat out of the Latin, from the Philo phical Transactions of the acute M thematician Mr. ABRAHAM I MOIVRE, F. R. S. ET n be any Number, the unknown Qua or the fought Root of the Equation, and let any Quantity likewise known, or, as the Mathemat call it, Homogeneum Comparationis; and let the lation of these be expreffed among themselves b Equation, viz. It is manifeft from the Nature of this Series, t fome unequal Number be taken for n (viz. an Int for it matters not whether it be Affirmative or Nega then the Series will be limited, as above, whofe Root Inn I-nn nn + - nn nn nx+ ·nx3-|-· X •nx5+ X 2x3 2×3 4×5 2X3 4×5 nx7, &c. a, its Root will be, viz. be unequal, the And here it is to be noted, that if Sign of the Root found, must be contrary to it, let this Equation be proposed, viz. 5x➡20x3† 16x5 = 6, whence n=5, and a 6, the Root will be=√6+√ 35 ความ 12; or because 6+√ 35 = 11.916, its Loga5/6+√ 35 rithm will be 1.6761304, and its 0.2152561, the Arithmetical Compliment 9.7847439, the Numbers of thefe Logarithms are 1.6415, and 0.6091 respectively, whofe half Sum is — 1.1253*. But if it fhould happen, that a is less than Unity, then the fecond Form of the Root, which is neater for our Purpose, is to be chose above the reft; fo if the Equation fhould be, viz, 5x —20x3 + 16x5 61 64 and indeed if the Fifth Root can be extracted by any Means, the true and poffible Root will appear, although the very Expreffion itself seems to be impoffible, and the Root of the Binomial 61 64 +/ 375 4096 -15, and likewise the Root of the Binomial is + 61 64 375 is ——15, the half Sum of which Bi4096 nomials is = =x. But if this Extraction cannot be had, or feems to be more difficult, it may very neatly be performed by a Table of Natural Sines, after the following Manner. 61 To the Rad. 1. let a be of a certain Arc, which will be Part (because n = =0.95112, the Sine 64 72° 23', whose fifth 5) is 14° 28', the Sine of 0.24981 = nearly, and fo we may proceed to Equations to a more fuperior Kind. A METHOD of approximating in extracting the Roots of Equations in Numbers. N Philofophical Tranfactions, No. 210, the late Dr. thod of extracting the Roots of adfected Equations of the common Form in Numbers. This Method proceeds by affumiug the Root defired, nearly true to one or two Places in Decimals (which is done by Geometrical Conftruction, or fome other convenient Way) and correcting the Affumption, by comparing the Difference between the true Root, and the affum'd, by Means of a new Equation, whofe Root is the Difference, and which he fhews how to form from the Equation propofed, by fubftituting the Value of the Root fought, partly in known, and partly in unknown Terms. In doing this he makes ufe of a Table of Products (which he calls Speculum Analyticum) by which he computes the Co-efficients in the new Equation for finding the Difference mentioned. This Table, I obferved, was form'd in the fame Manner from the Equation propofed, as the Fluxions are, taking the Root fought for the only flowing Quantity, its Fluxion for Unity, and after every Operation dividing the Product fucceffively by the Numbers 1. 2. 3. 4. &c. Hence I foon found, that this Method might eafily and naturally be made applicable, not only to Equations of the common Form (viz. fuch as confift of Terms, wherein the Powers of the Root fought are pofitive and integral without any Radical Sign) but alfo to all Expreffions in general, wherein any Thing is proposed as given, which by any known Method might be computed; if vice verfa, the Roots were confidered as given: Such as are all Radical Expreffions of Binomials, Trinomials, or of any other Nomial, which may be computed by the Root given, at leaft by Logarithms, whatever be the |