A Treatise of Algebra, in Three Parts: Containing I. The Fundamental Rules and Operations; II. The Composition and Resolution of Equations of All Degrees, and the Different Affections of Their Roots; III. The Application of Algebra and Geometry to Each Other. To which is Added an Appendix Concerning the General Properties of Geometrical LinesA. Millar and J. Nourse, 1756 - 432 páginas |
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Página 176
... results , thofe quantities are the limits of that equation . " It is ufeful to obferve , that , in general , " when , by fubftituting any two numbers for x in any equation , the refults have contrary figns , one or more of the roots of ...
... results , thofe quantities are the limits of that equation . " It is ufeful to obferve , that , in general , " when , by fubftituting any two numbers for x in any equation , the refults have contrary figns , one or more of the roots of ...
Página 195
... ( quadratic ) quotient x2 - 4x + 2 = 0 are 2√2 . 3 § 61. If it is required to find the roots of the equation x3 3x2 46x720 , the operation will be thus : Suppof Suppof . Results = x Divifors . 1-1201 , 2 Chap . 6 . 195 ALGEBRA ,
... ( quadratic ) quotient x2 - 4x + 2 = 0 are 2√2 . 3 § 61. If it is required to find the roots of the equation x3 3x2 46x720 , the operation will be thus : Suppof Suppof . Results = x Divifors . 1-1201 , 2 Chap . 6 . 195 ALGEBRA ,
Página 196
... Results = x Divifors . 1-1201 , 2 , 3 , 4 , 5 , 6 , 8 , 10 , 12 , 15 , 20 , 24 , 30 , 40 , 60 , 120 72 1,2,3,4,6,8,9,12,18,24,36,72 . x = -1-301 , 2 , 3 , 5 , 6 , 10 , 15 , 30 . Progreffions . 8345 9234 10 1 2 3 ing their common ...
... Results = x Divifors . 1-1201 , 2 , 3 , 4 , 5 , 6 , 8 , 10 , 12 , 15 , 20 , 24 , 30 , 40 , 60 , 120 72 1,2,3,4,6,8,9,12,18,24,36,72 . x = -1-301 , 2 , 3 , 5 , 6 , 10 , 15 , 30 . Progreffions . 8345 9234 10 1 2 3 ing their common ...
Página 199
... result have among their divisors the arith- metical progreffion m — or , changing the figns , n Where the difference of the terms is m , and the term belonging to the fuppofition of x = o is n . x n , - - n , ― n ; m , n , n + m . $ 64 ...
... result have among their divisors the arith- metical progreffion m — or , changing the figns , n Where the difference of the terms is m , and the term belonging to the fuppofition of x = o is n . x n , - - n , ― n ; m , n , n + m . $ 64 ...
Página 201
... Results . Divifors . Progr . x = I +31 , 3 . 3 3 x 08x3 - 26x2 + 11x + 10 = +101 , 2 , 5,10 . 2 x = 357 - -351 , 5 , 7,35- I The difference of the terms of the laft arith- metical progreffion is 2 , a divifor of 8 , the co- efficient of ...
... Results . Divifors . Progr . x = I +31 , 3 . 3 3 x 08x3 - 26x2 + 11x + 10 = +101 , 2 , 5,10 . 2 x = 357 - -351 , 5 , 7,35- I The difference of the terms of the laft arith- metical progreffion is 2 , a divifor of 8 , the co- efficient of ...
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adeoque affumed alfo arife arithmetical progreffion autem becauſe biquadratic cafe cafu coefficient common meaſure confequently contactus cube cube root cubic equation curvæ curvam curvaturæ curve demonftrated dimenfions diſcover divided divifor ducantur ducta enim equa equal erit ex puncto exponent expreffed expreffions fame manner fecond term feries fhall fide figns fimple equations fince firft firſt firſt term flexus fome fquare root fraction fubftitute fubtract fuch fuppofe furd give greateſt higheſt impoffible integer interfections itſelf laft term laſt leaft lefs lineæ locus metical muft multiplied muſt negative occurrat odd number parabola parallela pofed pofitive poſitive propofed equation puncto punctum quadratic equation quæ quævis quotient recta rectæ refolved refpectively refult repreſent Rule ſhall ſubſtituted ſuppoſe tangentes tertii ordinis thefe theſe thofe thoſe tion unknown quantity vaniſh whence whofe roots whoſe
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Página 74 - AB there be taken more than its half, and from the remainder more than its half, and so on ; there shall at length remain a magnitude less than C.
Página 55 - A privateer running at the rate of 10 miles an hour discovers a ship 18 miles off making way at the rate of 8 miles an hour : how many miles can the ship run before being overtaken ? Ans.
Página 107 - ... -{-24, equal to nothing, according to the propofed equation. And it is certain that there can be no other values of x...
Página 367 - This is a quarto volume, containing xxxii + 38 pages, and three plates of figures. The title points out that the work consists of two parts ; we are principally concerned with the first part : on...
Página 193 - The demonftration is deduced from the laft article, as the 7 8th is from the preceding. CHAP. IX. Of the Methods by which you may approximate to the roots of numeral Equations by their limits. §84. TT 7 HEN any equation is propofed W to be refolved, firft find the limits of the roots (by Chap. 5.) as for example...
Página 22 - Rule. Multiply the numerator of the dividend by the denominator of the divifor, their produit ¡hall give the numerator of the quotient. 'Then multiply t be denominator of the' dividend by the numerator of the divifor, and their predu£f jhall give the denominator.