and 1. Wherefore the four roots of the transformed equation xa—6x2—16x+21=o, are −2+ √−3... −2−√−3...3 and 1; but since z=x+1, by adding unity to each of these roots, we shall have the four roots of the given equation z1 —4 z3—8z+32 =o, as follows; z= −1 + √−3, z=—1— √−3, 2=4, and z= 2, as was required". 2. Given z1-4 z3 — 3 z3 —4 z+1=0, to find the values of z. 3. To find the roots of x*-3x2-4x-3=0. Ans. x= 1+√3 2 and 2 57. EULER'S RULE FOR BIQUADRATIC EQUATIONS'. RULE I. Let x-ax2-bx-co, be a general biquadratic equation wanting its second term, and let f=2,8=1+ 1, α C 16 4 II. With these values of f, g, and h, let the cubic equation z3 —ƒz3 +gz—h=o be formed, and let its three roots (found by any of the preceding methods) be p, q, and r. III. Then will the four roots of the proposed biquadratic be as follows, viz. 1 8 When b is positive When b is negative. 8 1st root, x=√p+ √9+ √r | x= 2nd root, x=√p+ √9−√r | x= √p-√9+ √r u This rule applies to that case only in which two of the roots are possible, and two impossible. ▾ The learned and venerable Leonard Euler, joint Professor of Mathematics at the University of Petersburg, was the inventor of this method, which he first published in the 6th volume of the Petersburg Commentaries for the year 1738; and afterwards in his Algebra, translated from the German inte French, in 1774, and lately into English. EXAMPLES.-1. Given x1-25 x2 +60 x—36=0, to find the four roots. Here a=25, b=-60, and c=36; wherefore f=( =) and h= ; consequently by substituting 16 4 these values in the cubic equation z3—fz2+gz―h=o, it becomes The three roots of this equation being found, will be z= 2. Given x-5x+4=0, to find the roots. +2,-1, and -2. 25 4 25 3 4 25 = -6 4 Ans. x+1, 3. Given x-3 x3-36 x2 +68x+ 240=0, to find the roots. Ans. x=-2,-5,+4, and+6. 4. Find the roots of x+x3- 29 x2 —9x+180=0. Ans, x= 3,4,-3, and -5. 5. Find the roots of y-4 y3-19 y'+46y+120=0. 58. SIMPSON'S RULE FOR BIQUADRATIC EQUATIONS *. This method supposes the given biquadratic to be equal to the difference of two assumed squares; thus, * This rule was first given by Mr. Thomas Simpson, Professor of the Mathematics at the Royal Military Academy, Woolwich; and published in the second edition of his Algebra, about the year 1747: it is in some instances preferable to either of the preceding methods, and some trouble is saved by it, as here we are not under the necessity of exterminating the second term from the complete biquadratic equation, which in the preceding rules is indispensable. RULE I. Let x1+ax3+bx2 +cx+d=o, be the proposed 1 equation, and equal to the difference x2+ +ax ax +A2 — Br+C3. Bx+d2. II. Square the two latter quantities, making the difference of the squares equal to the proposed equation, and you will have x2+ax3 +2Ax2 1 +— a2x2 +ɑAx+A2 = x2+ax3 + bx2+cx+d=o. 4 - B2x2-2 BCx-C2 III. Make the coefficient of x in each term on one side of the equation, equal to the coefficient of the same power of ≈ on the other; then will First, 24+——ao—B2=b, or 24+ 1 a2—b― B2. Secondly, a4-2 BC=c, or aA—c=2 BC. Thirdly, AC2=d, or A2—d=C2. IV. Multiply the first and last of these equations together, and the product (B2C2) will evidently be equal to (4B2C2) 1 one fourth the square of the second; that is, 24+ — q2—b.4 VI. Find the root or value of A in this cubic equation, by any of the foregoing methods; which being done, B and C will 1 likewise be known, since B=√24+— ·ab, and C= 3 1 aA-c 2 B VII. And since the proposed quantity x2+ax2+bx2+cx+d 1 is equal to nothing, its equal x2+ ·ax+ A 2 — Bx+C2 will 2 likewise be equal to nothing; wherefore it follows, that VIII. Extract the square root from both sides of this equation, A; which equation solved, gives x=+ B a+ +4 1 -aB+—— B2+C−A; wherein all the four roots of 4 the given equation are exhibited, according to the variations of the signs". EXAMPLES.-1. Given x — 6 x3 — 58 x2-114x-11=0, to find the values of x. Here a=-6, b=-58, c=-114, and d=-11, whence k 1 ac—d=) 182, l=(— c2 + d.—a2 —b=) 2513; whence by =ac 4 l=o, it becomes A3 +29A' +1824—1256=0, the root of y Dr. Hutton remarks, that Mr. Simpson has subjoined an observation to this rule, which has since been proved to be erroneous; namely, that "the value of A, in this equation, will be commensurate and rational, (and therefore the easier to be discovered,) not only when all the roots of the given equation are commensurate, but when they are irrational, and even impossible; as will appear from the examples subjoined." This, continues the Doctor, is a strange reason for Simpson to give in proof of a proposition: and it is wonderful that he fell on no examples that disprove it, as the instances in which his assertion holds true, are very few indeed in comparison with those in which it fails. Math. Dict. vol. I. p. 211. When either A=0, B=0, or C=0, the roots of the proposed biquadratic will be obtained by the resolution of a quadratic only. Simpson's Alg. 6th edit. p. 155. Besides the rules by Ferrari, Des Cartes, Euler, and Simpson, two other rules for the solution of biquadratics have been discovered: one by La Fontaine, of the Royal Academy of Sciences at Paris, and inserted in the Memoirs of that learned society for 1747; and the other by Dr. Edward Waring, Lucasian Professor of Mathematics at Cambridge, in a profound work, entitled, Meditationes Algebraica, published in the year 1770. Attempts have not been wanting to discover methods of resolving equations of the higher orders, but they have hitherto been unsuccessful; no general rule for the solution of adfected equations above the fourth power, has yet been discovered. aA-c which (found by Cubics) is 4=A; whence B=(√2A+—a2· 1 -b 1 =) 5/3, C=( a=c=) 3/3, and r=+== B- a+ 2 B 2 4 11.761947, or 3.101693, or +2.830127+/-1.1865334798, for the four roots; the two latter, expressed by the double sign, are impossible. 2. Let the roots of x1—6 x3 +5 x2+2x−10=0, be found. Ans. x=5, −1, 1+ √/−1, and 1—√/−1. 3. Given x-12x-17=0, to find the values of x. 2.0567, or .6425, or .7071±√−4 7426406. Ans. x= 4. Given x1-25 x2 +60 x=-36, to find the roots. Answer x=3, 2, 1, and —6. 5. Given ƒa—ƒ3 + 2x2−3x+20=0, to find the roots. RESOLUTION OF EQUATIONS BY 59. The foregoing rules require for the most part great labour and circumspection, and after all, they are applicable z Methods of approximating to the roots of numbers, were employed as early as the time of Lucas de Burgo, who flourished in the 15th century; but the first who are known to have applied the doctrine to the resolution of equa. tions, were Stevinus of Bruges, and Vieta, a celebrated mathematician of Lower Poitou; the former in his Arithmetic, printed at Leyden, in 1585, and in his Algebra, published a little later; and the latter in his Opera Mathematica, written about the year 1600, and published by Van Schooten, in 1646. Their methods, although in some respects improved by Oughtred in his Key to the Mathematics, 1648, were still very tedious and imperfect: to remedy these defects, Sir Isaac Newton turned his attention to the subject, and it is to his successful application to this branch, that we are principally indebted for a general, easy, and expeditious method of approximating to the roots of all sorts of adfected equations, as may be seen in his tract De Analysi per Equationes numero terminorum infinitas, 1711, and elsewhere. Dr. Halley invented two rules for the same purpose, one called his rational theorem, and the other, his irrational theorem, both of which are still justly esteemed for their utility. This necessary part of Algebra is likewise indebted to the labours of Wallis, Raphson, De Lagni, Thomas Simpson, and others; whose methods have been given by various writers on the subject. |