PROOF OF FERMAT'S THEOREM X ID f A ll- ''an h -, - - ^. i '! ' ''.,, PROOF OF FERMAT'S THEOREM AND McGINNIS' THEOREM OF DERIVATIVE EQUATIONS IN AN ABSOLUTE PROOF OF FERMAT'S THEOREM; REDUCTION OF THE GENERAL EQUATION OF THE FIFTH DEGREE TO AN EQUATION OF TIHE FOURTH DEGREE; AND SUPPLEMENTARY THEOREM S BY MICHAEL ANGELO McGINNIS PUBLISHIED BY L. C. HJORTH & SONS, CIVIL ENGINEEIRS MEADOWDALE, WASHINGTON, U.S.A. COPYRIGHTED) BY MICHAEL ANGELO McGINNIS JANUARY, 1913. Xemritob Aoress J. S. Cushinr Co.- Berwick & Smith Co. Norwood, Mass., U.S.Ao )etbfcation BORN upon what was once French soil, this little volume is gratefully dedicated to the memory of the immortal Fermat, whose worthy achievements in the field of mathematical research have added much to the knowledge and happiness of the human race. He bequeathed to posterity a new-born star that will continue to shine when the last sun goes down to rise no more. While Scientific France reveres and does honor to his memory, all nations and all peoples claim him; for such men belong to no nation and to no clime. They are the stars and suns that light the world, revealing the hidden links of Truth in that endless chain which binds man to his Creator, and each newly discovered link is but a step upward, drawing him closer and nearer to his God.. Scientific Truths are immortal things emanating from Divinity. They are thought vibrations proceeding from the First - the Only - SupremelyPerfect Cause of all things that be, and can not be received and reflected by unsound and impure minds. Fermat was truly one of God's most perfect, purest souls, toiling in the field of mathematics, in which he saw the laws of God shimmering in the sunbeam, vibrating upon the bosom of ocean, and eddying in every gust of wind. All praise to the name and memory of Pierre Fermat. THE AUTHOR. PREFACE THIS little volume contains my proof of EFermat's Theorem, - "The sum of no two powers except squares is itself a power of the same degree." My attention was first called to the existence of this now celebrated problem (judging from the number of proposed solutions arriving almost daily in G6ttingen) by reading a synopsis of a lecture on "Mathematical Research" delivered in New York by Professor G. A. Miller of Urbana, Illinois, and published in the "Literary Digest " June 29, last. The copy of the "Digest" was given to me by Mr. John F. Dwyer of Jefferson City. I am, therefore, indebted to Professor Miller for his lecture, and to Mr. Dwyer for the copy of the " Literary Digest." The proof speaks for itself, and presents to my mind many new and important ideas, one of which I shall take the liberty to mention. It is this: If the dimensions of or in space are not infinite, then, the number must be limited to three; and the assumed fourth dimension is purely imaginary from a mathematical standpoint, and can not exist either in Time or Space, - but may exist independently of Time and Space! (In the analysis and proof of this proposition will be found the link that connects pure mathematical reasoning with that of abstract philosophy, thus giving to the latter a true place and meaning in Science.) The " Theorem of Derivative Equations," in its application, furnishes an absolute proof that the sum of the nth vii viii powers of two numbers can not be an nth power of a third number when n > 2. I have not had the time nor convenience to test the "Theorem of Derivative Equations " in the Equations of Curves; but I am satisfied it will add new light and meaning to the theory of derivatives found in the Calculus. The proofs of Theorems I and II I believe to be well established; and in the linking of the proofs (38-39), additional proof is given to the proof of the proposed theorems of the writer in an absolute and final proof of Fermat's Theorem. All theorems and remarks in the supplementary addition form no part of the proof of Fermat's Theorem. These theorems are the result of the author's successful effort (after seven years of toil) to solve the general equation of the third degree without recourse to '" Cardan's ingenious device." A true general solution of the Cubic is the Key to general solutions of all degrees. In 1900, 1 succeeded in reducing the general equation of the sixth degree to an equation of the fifth degree, and discovered that all even degree equations can be reduced to an equation one degree lower. But, I found that odd degree equations would not yield to the same method of attack. The general equation of the nth degree, when n is even, has (n-1) algebraically symmetrical functions; and, when n is odd, it has n such functions. An algebraically symmetrical function is a function of the roots of an equation that can be algebraically expressed in terms of the coefficients of the equation. Thus: If a, b, c, d are the roots of the biquadratic, then, we have: ab+ed, ac +bd, and ad+be. These three functions will form a cubic equation. The algebraically symmetrical functions of the sixth are given in the latter part of this volume. ix The reduction of odd degree equations depends upon the proof of Theorems I, II, and III in the supplementary addition (42). The proofs of Theorems I and III are so extensive that I am unable to present them here. (If I live, and opportunity offers, they shall be published.) Due credit must be given to L. C. Hjorth & Sons, who deem this little volume of sufficient scientific value to finance the publication of the same for the sole benefit of the author and in the interest of science, - and thus expressing their love for the advancement of science, and their lack of purely selfish motives. To Rev. Henry A. Geisert of Jefferson City is due the credit of assisting the author in many ways, and interesting himself in the speedy publication of the work. And while we thus feel grateful to those who lend assistance in whatever form, we must needs remember that, although the problem was promulgated by a Frenchman, it finds its proof, if to be found at all, in the generosity of the great German thinker and scholar, Dr. Paul Wolfskehl, who, by his munificent gift, has invited the scholars of the civilized world to seek the proof. There can be but little doubt, if indeed any, but that Dr. Wolfskehl saw that many new and valuable ideas would find existence in the investigation of the proof of Fermat's Theorem that would prove to be more valuable to humanity in the advancement of science than what might be expected from the mere proof itself. He shall always be considered by those who seek truth for its own sake as one of the world's most worthy benefactors —one who loved his fellow man, and truth, better than gold. I study the science of mathematics for the truths it contains. It makes man better, nobler. It makes him love truth, and despise falsehood, and hate injustice. And x knowing that "in science there are no social, religious, or international boundary lines,"- I turn to Germany, the land of thinkers and scholars, for a verdict through her Royal Academy of Scientific Researches of Gottingen. I shall be satisfied if it be found that I discovered but one new and valuable truth in the boundless field of mathematical research in which I feel naught - but a weak, devoted student. MICHAEL ANGELO McGINNIS. KANSAS CITY, Mo., U.S.A. January, 1913. CONTENTS SECTION ARTICLES 1. FERMAT'S THEOREM....... 2 2. PROPOSITIONS AND THEOREMS BY THE AUTHOR UPON WHICH PROOF OF FERMAT'S THEOREM DEPENDS. 2 3. INTRODUCTORY PROOFS...... 3-18 4. PROFo OF THEOREM I.... 19-28 5. PROOF OF THEOREM 11... 29-31 6. McGINNIS' THEOREM OF DERIVATIVE EQUATIONS IN PROOF OF FERMAT'S THEOREM.... 32-38 7. LINKING OF THE PROOFS OF THEOREM II, AND McGINNIS' THEOREM OF DERIVATIVE EQUATIONS IN A FINAL AND ABSOLUTE PROOF OF FERMAT'S THEOREM...... 38-39 8. FINAL ARGUMENT AND SUMMATION OF PROOFS,. 40-42 9. SUPPLEMENTARY THEOREMS..... 42 10. ALGEBRAICALLY SYMMETRICAL FUNCTIONS OF TlHE SEXTIC....... 42 11. REDUCING THE QUINTIC TO A BIQUADRATIC 43 xi PROOF OF FERMAT'S THEOREM 1. Fermat's Theorem. THE SUM OF No Two POWERS EXCEPT SQUARES IS ITSELF A POWER OF THE SAME DEGREE. 2. The proof of " Fermat's Theorem" depends upon the proof of the following propositions and theorems: (a) To divide any number into two such parts that the sum of the squares of the parts will equal the square of a third number. (6) To divide a given straight line into two such parts that the sum of the squares described upon the parts will equal the square described upon a third line. (c) From the sum of the solid contents of two cubes to construct a third and perfect cube. (d) To construct a triangle two of whose sides are known, and its third side the nth root of the sum of the nth powers of its two given sides. (e) Theorem I. If the quantities a and /3 are increasing functions such that a = 1, 2, 3, 4,. for all numbers, and / = 2, 3, 4, 5,... for all numbers except 1, and / > a, at all times, when combined with a, then if a remains permanent while 3 assumes all possible values of the nth degree greater than a, the nth root of the sum of a + /3n is incommensurable when n > 1. (f) Theorem II. If the quantities a and 3 represent positive, integral, or fractional numbers, then the nth root of the sum of a" + /3 is incommensurable when n > 2. (g) McGinnis' Theorem of Derivative Equations. The general equation of the nth degree has n derivative equations 1 2 of the (n - 1)th degree, and each derivative equation contains at least one root of the equation from which it is derived. (h) Theorem III. The (n- 1)th root of n is incommensurable when n > 2. (i) Theorem IV. The nth root of the sum of an +- an is incommensurable when / > a, and n >1. And the (an)th root of the sum of an + /an is incommensurable when 3 > a, and n > 1. 3. To Prove (a)-Art. 2. As the sum of the squares of two numbers is to be considered, we must arrange or combine all numbers in pairs taken two at a time. That is: the square of every conceivable number must be added to the square of every other conceivable number except itself. (1) Assume that the quantities a and, are increasing functions such that (2) a = 1, 2, 3, 4, 5,..., for every conceivable number; (3) and /3= 2, 3, 4, 5, 6,..., for every conceivable number except 1. We will then have for the sum of the squares of any two numbers (except like numbers), the following formulae: [ 1 + 22, 1 + 32, 1 + 42,..., to 1+32m, (4) The sum of J 22+ 3& 22+42, 22+52,..., to 222+,2 a2 +/^g2- 3242+ 32 + 52, 32 + 62,.., to 32 + i2 and so on, for the sum of the squares of any two numbers, in which the square of every conceivable number is added to the square of every other conceivable number except itself. (The sum of two like powers as a2+ a2 will be considered later on.) 3 Let us assume that it is possible for the following equality to exist, viz.: (5) a2 +2 = y2. (6) Assume that / > c, at all times, when combined. (7) If the sum of a2 + 2 is a power of the second degree, it will be possible to construct a right triangle whose sides will be a, 3, and y, -r being the square root of the sum of a2 + 32. (8) As /3> a, y > 3. (The square root of the sum of the squares of two numbers or quantities is greater than the greater number.) (9) a + / > y. (The sum of two sides of a triangle is greater than the third side.) (See any geometry for proof.) (10) Assume that 7-/3=A/3. Then A -/3+/3=y. Let A/3 =x. Then, x + / = y. We now have the following equations: (11) (x + 3)2 == 2 = + 32 = (3 + 3)2. Expanding (x + 3)2 we have (12) x + 2 /3 + /32 = 2 = (A 3 +/3)2 = a2 +/2. By transposition of a2 + 32 for 72 we have (13) x2 + 2 3xa - 2 = 0. Dividing the roots of (13) by /, we have, writing y for the new value of x, being -x = A = y (14) 2 + 2 y - = 0, (15) +y +1 i[ + 4 a2 Transposing - in (14) to the right of the sign of equality and adding 1 to both sides of the equation and taking the square root of both members. a2 As a general rule the sum of + 1 is not a power of the second degree; and equation (14), having a perfect power for its absolute term, and its sign minus, and all the other coefficients of the equation whole numbers, and their sign plus, and the coefficient of the highest power unity, can not have a rational fraction for one of its roots, unless when the sum of a2 + 32 is a perfect square. This will occur in all cases where the values of a and 3 are amenable to Law I. Thus: When a = 3, or 5, and =-=4 or 12, a2 +32 = 72= 32+42 = 52; 52+122 132. The numbers 3, 4, and 5, and 5, 12, and 13 represent in each case the three sides of a right triangle. (16) We may now write the general formula for all such numbers - the sum of the squares of. which are powers of the same degree: (n 3)2 + (n 4)2 = (n 5)2; and (n 5)2 + (n 12)2 - (n 13)2. n may represent any positive or negative number, integral or fractional. (17) Any number that is exactly divisible by 7, or 17, can be separated into whole numbers, the sum of the squares of which will be a power of the second degree. Thus: To divide 14 and 34 into two such parts that the sum of the squares of the parts will be a perfect square. SOLUTION: 14 -- 7 - 2 = n in formula.. 2 3 = 6, and 2 x 4 = S. 62 + 82 = 102. 34 - 17 = 2 = n..-.2x 5=10, and 2 x 12 =24. 102 + 242 = 262. If the number to be divided is not an exact multiple of 7 or 17 (or the sum of any two numbers, the sum of the 5 squares of which is a power of the second degree), we will obtain fractional results. LAW I. When the square of the sum of two numbers is diminished by twice their product and there remains a perfect square, then, the sum of the squares of such numbers will be a power of the second degree. And, geometrically: If the square described upon the sum of two lines is diminished by twice the rectangle of the lines and there remains a perfect square, then, the sum of the squares described upon such lines will be equal to the square described upon a third line. In all other cases imperfect powers of the second degree are the results obtained. It is therefore possible to construct the proposed triangle when the sum of the numbers or lines complies with Law I. It is therefore possible, in special cases, for the following equality to exist: a2 + 32 = y2. Therefore, Fermat's Theorem holds good for powers of the second degree. REMARK. Further discussion of the sum of a2 + /f2 will be given in the Theorem of Derivative Equations. 13. To prove the proposed theorem of Fermat for the sum of any two powers of the third degree, we build the following formula: 1 +23, 1 -33, 1 +43, 1 +/3,, The sum of CC+IB= 1 23 + 33, 23 + 43 23 + 53 23 + 3m, The sum of a8 + 3 = 3+, 33s+ 4, 3 8+ 53, $a +8,. 3a + /3.~ and so on, ad infinitum, for the sum of any two powers of the third degree, in which the cube of every conceivable number is added to the cube of every other conceivable number, except itself. 6 14. The sum of any two like powers of the nth degree cannot be equal to the nth power of a third number. (1) The proof of the above proposition can be easily demonstrated as follows: (2) Assume that a3= 1, 23, 33, 43, 53,... for the cubes of all numbers. (3) a3 + 3 = 2 O3. Then the cube root of the sum of 3 + a = the cube root of 2 3, which is ac- 2. In this case, n = 3; when n =4, then, we have ac/2; when n = 5, we have a-/2, and so on, ad infinitum. We can then write the general expression for the nth root of the sum of any two like powers, as follows: 1 1 (4) (c + a) = a(2)n. And as the nth root of 2 is incommensurable for all values of n>2, it follows that the sum of any two like powers can not be a power of the same degree when n > 2. 15. The sum of two cubes is often equal to the square of a third number. Thus, 1 + 23 = 32, and 43 + 83 =242. 4 + 8 is but a multiple of 1 + 2. When it is found that the sum of two cubes is equal to the square of a third number, then, we may write a general formula that will embrace all such numbers, viz.: (n2 1)3+ (n2 2)3= (n3 x 3)2 for (1 + 23 = 32). In general, (Ot2a)3 + (n2P)3 (n3 x d)2 in which a + /3 = d2. n = 1, 2, 3,.. for all numbers. 16. We now lay down the law for the sum of two cubes which will equal a perfect square: LAW II. When the cube of the sum of any two numbers is diminished by three times their product into their sum, and there remains a perfect square, then, the sum of the cubes of such numbers will be a power of the second degree. Thus: ( +, + (a + 0)3 - 3 a/3(( + /3) = 72. When such proves to be the case, a+ /3 = 72. REMARK. The sum of three cubes is often equal to a fourth cube. Thus: 33 + 43 + 53 = 63. Often the sum of two cubes is equal to the sum of two other cubes. Thus: 93 + 103 =123 + 1I; and (123 + 3) -- 13 =23 + 53; and (93 + 103) - 19 = 33 + 43. But we can not find from the formula (13), the sum of two cubes which will give us a power of the third degree; and as the number of combinations is infinite, it is impossible to build an arithmetical solution that will prove in any way to be general in character. 16 —. It is useless and a waste of time to carry on a discussion of the proof of " Fermat's Theorem " from a purely arithmetical standpoint. To establish the truth of " Fermat's Theorem," we must found, or build, a proof that is arithmetical, geometrical, and algebraic in character. 17. To Prove (c), Art. 2. Let a and b represent the sides of two cubes the solid contents of which are, respectively, a3 and b3. We are to prove that the sum of (I) a3 + 3 = c3 (assumed to be a rational quantity or number). Assume that the quantities a and b represent positive, integral, or fractional numbers; and that each represents the side of a cube. Then will the cube root of the sum of their cubes be a rational number or quantity which we designate by e? If the sum of a3 + h3 be a power of the same degree, then it will be possible to construct a third cube whose side is assumed to be c from the sum of a3 + b3. 8 Assume that b> a. Then c > b. (The cube root of the sum of two numbers is greater than the greater number.) But c < a + b. (The cube root of the sum of two numbers is less than their sum.) Assume that the difference between c and b is Ab (read "increment" b.) Let b + Ab = c. Let Ab = x. We will now begin to build the cube whose side is assumed to be equal to e, by increasing the cube whose side is b. To build a cube by additions to a given cube, we must make additions upon three sides to preserve the uniformity of the cube throughout. As the cube to be increased in size is the cube, 63, whose side is b, the solid contents of one of the additions will be: 6 x b x Ab = x 6 x x = b2x. And the three sides will contain 3 6x b = 3 bx. But we see the cube is not complete, for we have three spaces yet to fill, having for the dimensions of each, b in length, and Ab wide, and Ab deep. The solid contents of one of these additions will be: b x Ab x Ab = b x x x= 6x2, and the three additions will contain 3 x bx2 = 3 bx2. The cube is now examined, and we discover that, in order to complete it, we must insert in one corner which is vacant, a little cube, Ab long, wide, and deep =x long, wide, and deep. Its solid contents will be: Ab x Ab x Ab = A3b3 = x x x x= x3. The cube is now examined, and found to be complete, at least in appearance. The sum of these additions added to the cube, b3, forms the algebraic equation: (1) X3+ 3 bx2+ 3 b2 + 3= c3= (a3 + 3). (2) x8+ 3 bX2 + 3 12x = a3. (Transposing 63, and taking (a3 + b3) for C3.) (3) x3+3 bx2+3 b2 - a3=0. (Transposing a3 in (2).) 9 (4) y3 + 3 y+ 3 y - ^ =0. (Dividing the roots of (3) x Ab by b and writing y for the new value of x, = = A.) b b a3 As b > a, -< 1. Therefore y < 1. Is y a rational fraction? If it be a rational fraction, then c is a rational function of the sum of (a' + b3). That is: The cube root of the sum of (a3+ b3) is equal to c. If this shall prove to be true, then, "Fermlat's Theorem " will not hold good for powers of the third degree. Continuing the discussion: Assume that b = a. Then, - =1. We will, in that case, have the equation, (5) y3 + 3y2+ 3 y - 1 = 0, an equation of the third degree, in which all the coefficients of y are whole numbers and plus, and the last, or absolute term, unity, and its sign minus, and the coefficient of y3, unity. By the "Sturm Theorem" or the " Horner Method," it is found that the real value of y (which is the one we seek) lies between 0 and 1. It is therefore fractional. But it has been proven that such an equation can not have a rational fraction for one of its roots. (See any Complete Algebra for proof.) Therefore, y in (5) is incommensurable. And when a3 the absolute term of (5) becomes less than - 1, as b, y is also incommensurable in (4). Therefore, y in (2) and (3) is incommensurable. Therefore, c is incommensurable. If we multiply or divide an incommensurable number or quantity by a rational number or quantity, the result will be an incommensurable number or quantity. 10 REMARK. It will be shown farther on that the general equation an [(Y ~- 1)n - 1] 0= can not have a rational fraction for one of its roots when n > 2. Proposition (e), Art. 2, is, therefore, impossible. Hence the truth of "Fermat's Theorem," that the sum of no two powers of the third degree can be a power of the same degree. 18. THE UNIT 1 IS THE UNIVERSAL POWER OF ALL DEGREES. 19. To prove Th. I, Art. 2. Demonstration. (1) It is assumed that a = 1, 2, 3,..* for all numbers. (2) It is assumed that f = 2, 3, 4,... for all numbers except 1. (3) It is assumed that 3 > a, at all times, when combined with a. Then we are to prove that the nth root of the sum of a + 43n is incommensurable when n > 1. We will now construct the general formula upon which our proof must rest, viz.: (4) The sum of ( 1 + 2n, J + 43n, 2 + 3", + 4n, + 5n, 3 + 4, + 5n, + 6n, a + 3" = 45 4, + 5 6, + 7, |5 + G6n, - 7n, + 8n, +- 5n + 6n, + 8, + 9n, + -.. to 1 + 3'm, +... to 2 +/S, +... to 3 + /3,nm +... to 4 q- 3nm, +... to 5 +/3nn", and so on, in which the first power of number is added to the nth power of number except itself. every conceivable every conceivable 11 (5) Let us assume that it is possible for the following equality to exist, viz.: a + 3n = 7y. If y be a rational quantity, or number, it will be possible to construct a triangle whose sides will be a, /, and 7,- y being the nth root of the sum of a + n. (6) As 3 > a, 7 > /3. (The nth root of the sum of two numbers is greater than the greater number.) But 7 < a + /3. (The sum of any two sides of a triangle is greater than the third side.) (7) Assume that 7- ~ = A/3. Then, AB + /3 =r. Let x = A/3. Then, (8) x+/3=. We now have the equations, (9) (x + 1)n= = +n= 7 a = t ( + (A/3 3) Expanding ( + /3)n by the "Binomial Theorem," we have (10) xn + n/3Xn-1 + n(n - 1),2x-2 +... + n/n- 1 + /n 1 x 2 n- 1x2 a= 77== + /3. Transposing a + /3 for 7n, we have (11) "n+x3n-1+ n(n1 )/32x-2+... +,n-1x-a=0. 1x2 As 3 > a, x <a. Then, x< 1, 2, 3, 4, **. for all values of a. Assume that a = 1. If n be of odd degree, the equation will contain at least one real root with a sign contrary to its absolute term which is unity and minus. One value of x will therefore be real and plus, because all the coefficients of x in (11) are whole numbers, or represent whole numbers, and the coefficient of xn is unity. As x < a< 1, it is therefore fractional. But, it has been proven that such an equation can not have a rational 12 fraction for one of its roots. (See any Complete Algebra for proof.) (12) x is therefore incommensurable in (11) when a_1. (13) y is therefore incommensurable in all the foregoing equations. It is therefore impossible to construct the proposed triangle when a = 1. We therefore conclude that the nth root of the sum of 1 + = 1 2 1 + 32n,. 1 + f.n, is incommensurable in each and every combination when />1, and n>1. (14) Therefore, if 1 be added to the nth power of any number greater than 1, the nth root of their sum can not be a power of the same degree when n> 1. (15) Let a= 2 in (11). We then have x< 1i<l. x is therefore fractional; and for reasons heretofore given is an incommensurable number or quantity. And as y depends for its value upon the value of x, y is also incomnmensurable, and it will be impossible to construct the proposed triangle when a= 2. We therefore conclude that the nth root of the sum of 2 + /n = 2 + 3n, 2 + 4n, 2 + 5n,.. 2 + 38m is, in each and every combination, an incommensurable number or quantity. (16) Therefore, if 2 be added to a perfect power of the nth degree greater than 2, their sum can not be a power of the same degree when n > 1. (17) Let a= 3. Then - <1<. x<1.. <l Theren,3n -1 fore, x, in (11), will be incommensurable for reasons heretofore given. Therefore, the nth root of the sum of 3 +,n = 3 + 4n, 3 + 5,... 3 + /n is, in each and every 13 combination an incommensurable quantity or number. The triangle is therefore impossible when a = 3, and n > 1. (18) We therefore conclude that in the general formula, (4), x is incommensurable, and the proof of Theorem I is established, viz.: (19) x< 1, and 1>, 2 -. am in (11), and there2n\ 3n, 4n,... Pm fore incommensurable for reasons heretofore given. (20) COROLLARY. The nth root of the sum of - + 1 is incommensurable for all values of a and, when/38 > a, is incommensurable for all values of a and /3, when /3 > a, and n >1. 20. From the general equation (11-19) we may write the following general equations of the second, third, fourth, and fifth degrees, and so on to the nth degree equation in which a= 1, 2, 3, 4,... for all numbers, and in which /3> a at all times, when combined with a. (1) 2 + 2 (2, 3, 4,... )x- a =0. (2) 3 + 3 (2, 3, 4, 5,... ) x2 + 3 (22,2, 42, 52,... 2) - = 0. (3) X4 + 4 (2, 3,, 4,... /n) X3 + 6 (22, 32, 42, 52,... / 2) X2 + 4 (23, 33, 43, 53,... /3m) X- = O. (4) 5 + 5 (2, 3, 4, 5,... /m) 4 + 10 (2, 32, 42, 52,... /32) X + 10 (23, 33, 43, 53,... /S3) X + 5 (24, 34, 449 54... 4m)X - C-Z O. 14 (5) +n n(2, 3, 4, 5,... m) xn+ n - ) (22, 32, 42, 52,... 12m) xn-2 + n(n-1 )(n- 2) (23, 33, 43, 53,..3m)Xn-3 1x2x3 - n (n -1)(n - 2) (n- 3) (24, 34, 44 54. 1 4 ) Xn-4 I h 1, b -, 4- t- ~ ~ ~ i m x * * e i + n (2n-1, 3s-1, 4n-1, 5n-1,... 3n~n lm) a = O. (6) It has been proven that x is incommensurable in (11-19); it is therefore incommensurable in all of the foregoing equations for all values of a when 3> a and n>l. It is therefore incommensurable when n = 2, 3, 4, 5, 6,..., for the indicated powers of all degrees. (7) n + nn- n (n —1) n-2+... +ny +l 1 + 1. Dividing the roots of (11-19) by /3, and writing y for the new value of x, being x = y = = A; and transferring a before division to the right of the sign of equality, and after division, adding 1 to both sides of the equation, we have, by taking the nth root of both members: 1 (8) Y+l [ +lj, an incommensurable quantity or number for all values of a and 1, -3 > a, and n > 1. 21. It has been established in the proof of Theorem I, that the nth root of the sum of 1 + /3n is an incommensurable quantity or number when n = 2, 3, 4, 5,... for the indicated powers of all degrees..'. if 1 + /3n is incommensurable for its nth root, when n > 1, it will be, according to our hypothesis, incommensurable when n> 2, 3, 4 *... 15 22. We will now construct a series of dual powers from the formula (I) (1 + 2n, 1 + 3, 1 + 4n, 1 + 5n,... to 1 + /3) viz.: (a) (1 + 2n) x 1, 2n 3n, 4n, 5n,... an,. (b) (1 + 3n) x 1, 2n, 3n, 4n, 5n,... an. (e) (1 + 4n) x 1, 2n, 38, 4n, 5n,... a. (d) (1 + 5n) x 1, 2, 3n, 4n, 51,... a. (e) (1 + 3n ) x 1, 2n, 3n 4n, 5n2,....m 23. The foregoing formule may be grouped as follows: (a) 1+ 2n (b) 1 + 3 2n + 4n 2n + 6, 3n + 6' 3" + 9n 4n + 8n 4" + 12n 5n + 10n 5n + 15n an m + (aCm/m) a nm + (arn/m)n (e) 1 + 4n- (d) 1+ 5n... 2n + 8n 2n+ 1 0n... 3n + 12n 3n + 15n.. 4n + 16,. 4n 7+ 20n... 5n + 20n 5n + 25n... In a, n + (a~ )?, aM + (amom)7 am + (am3m)4 24. It has been proven that the nth root of the sum of 1 + "8 is incommensurable when n > 1; therefore, the sum of the power multiples of 1 + /n is incommensurable when n= 2, 3, 4, 5, *.. for the indicated powers of all numbers, etc. 16 25. It has been proven by Theorem I that if the first power of any number is added to the nth power of a greater number, their sum can not be a power of the nth degree when n > 1. 26. Letting or assuming that a = 1, 2, 3, 4, 5,... for all conceivable numbers, then, certain values of a to the first power will represent the nth power of other values of a. To illustrate: when a = 8, then, 8 is the third power of a certain value of a, which is 2. It has been proven (Theorem I) that the sum of 8 + 9n can not be a power of the nth degree-in other words: Their sum can not be equal to the nth power of a rational number. Therefore, 2 + 93 can not be a power of the third degree when n >1. When a = 27. The sum of 27 + 28n has been proven can not be a power of the nth degree. Therefore, as 33= 27, the sum of 33+283 can not be a power of the third degree. And 216 + 2173 = 63 +2173 can not be a power of the third degree, and so on, when n > 1. 27. We can now write the following general formula for all such numbers, viz.: (1) ac + (an (+ 1) which can be illustrated arithmetically as follows: When n = 2, n= 3, n = 4, 1 + (12 + 1)2 1 +(1 3+ 1)3 1 + (1 + 1)4 22 + (22 + 1)2 23 - (23 + 1)3 24 + (24+1)4 32+ (32+ 1)2 33 + (338 1)3 a2rn + (a2m + 1)2 a3m + (a3m + 1)3 a4m + (at47n + 1)4 and so on for the sum of the nth powers of all numbers in the form of an + (a" + 1)n which are proven by Theorem I can not be a power of the nth degree. 17 28. From the foregoing proofs we lay down the following arithmetical, geometrical, and algebraic law - LAw 1II. If 1 be added to the nth power of an integer or to the nth power of a fraction whose value is less than 1, then, in either case, their sum can not be a power of the nth degree when n > 2. Again: if 1 be subtracted from the nth power of an integer greater than 1, their difference can not be a power of the nth degree when n > 1. 29. To prove Theorem II, Art. 2. (1) Demonstration. Assume that 3 > a, and that both a and 3 are increasing functions such that a = 1, 2, 3, 4, 5,.*. for all numbers; and that = 2, 3, 4, 5, 6,.. for all numbers, except 1; and that / > a, at all times, when combined with cc. (2) We may now build the general formule upon which our proof must rest, viz.:(General Formulce I) 1 + 2', 1 + 3n, 1 - 4n,... to 1 + 3'n", 2 n+ 3n 2n _ 4n 2n J 5n... n 2_ +nm The sum of an 3n= 3n+4n, 3n+5, 3n+6n,. 3n + and so on, ad infinitum, in which the nth power of every conceivable number is added to the nth power of every conceivable number, except itself. REMARK. P3, represents a number of infinite value. (3) Let us consider the possible existence of the equality: (4) an + 3n _ =yn If the sum of an + fn be a power of the nth degree, it will be possible to construct a triangle, or a series of triangles, from the general formule (I). 18 (5) Assume that a and /3 represent, at all times, and for all values of n, two known sides of the proposed triangle; and let it also be assumed that the third side is the nth root of the sum of the nth powers of its two known sides. (6) The three sides of the triangle will then be represented by a, /, and y, - y being the assumed nth root of the sum of ac + /3. (7) a = 1, 2, 3,..., for all numbers, by hypothesis, = 2, 3, 4, *.., for all numbers, by hypothesis, except 1. (8) 3 > a, by hypothesis. (9) y > 3 —the nth root of the sum of the nth powers of two numbers is always greater than the greater number. (See any geometry or algebra for proof.) (10) a +/>7y. (The sum of any two sides of a triangle is greater than the third side.) (11) Assume that y - / = A/ (read " increment / "). (12) Let A/3=x. Then x+/3= y, and, also, A/3 +/3=y. From (12) we have the equations (13) (x + /3)n = n = (A/ + /)n = a + /. Expanding (x + 9)n, we have (14) n + naxn- + + (n 2xn-2 + + n/3n — ) +n" 1x2 = =,n + = "= (A/3 + 3)". Transposing a" + /3" for 7" and omitting (A/3 + /3)n = (x - /3)", we have the equation, (15) xn +-n3Xn-1 - n (n — 1) 2Xn-2 +.....** n-_l- n -O. 1x2 Dividing the roots of (15) by /3, we have (writing y for the new value of x which is - = A = y) la/3 19 (16) y" + ny-l n (n1) n- +...+ny — 0 1x2 an f.n Transferring - to the right of the sign of equality, and adding 1 to both sides of the equation, and then taking the nth root of both members, we have the equation (17) y+l==[ + 1 (18) y + 1 is therefore incommensurable. (Th. I, and Law 111-28.) (19) y is therefore incommensurable. (Th. I, and Law III-28.) (20) - +1 I is therefore incommensurable. (Th. I, and Law 111-28.) (21) y is therefore incommensurable. (22) x is therefore incommensurable, in all the foregoing equations. Therefore, it is impossible to construct the proposed triangle, because its third side, 7, can not be measured. Therefore, y being an incommensurable quantity or number, the sum of n + -3n can not be a power of the nth degree when n > 2. Hence the truth of Theorem II. Therefore, PFermat's Theorem holds good for the sum of any two powers of the nth degree when n > 2. 30. We may now write the general equations embraced within the proof of Fbermat's Theorem, viz.:I. ~n(A + l)n= an +. II. 8"(a + l)n- 1 = o. 20 III. (A + l)n- 1 When n is of infinite value, the equality exists; but when n has a finite value, the equality is impossible. (See final argument (39-40).) IV, [n(A + 1)in - /3n] - e = O [(X + /)n,, - n = 0. (x = A3). an v. [(A + 1) -] - = = [( y+I)n-]- =0. We now lay down the following unimpeachable laws from an algebraic and geometrical standpoint, viz.: (1) The general equation (IV), having for its absolute term a perfect power of the nth degree and its sign minus, can not have a rational quantity, or number, for one of its roots when n > 2. (2) The general equation (V), having for its absolute term the nth power of a fraction whose value is less than 1, and its sign minus, can not have a rational fraction for one of its roots when n > 2. The proof of the above laws, we believe, is well established in the proofs of Theorems I and II. REMARK. In the foregoing equations (IV and V), if n be of odd degree, the equation will contain but one real plus root, and the reinaining roots will be imaginary. But when n is of even degree, the equation will contain two real roots of the same numerical value, but of opposite signs. The remaining roots will be imaginary. 31. We will introduce something new in the field of mathematical research as a further and absolute proof of the impossibility to find for the sum of any two powers of the nth degree, an nth power of a third number when n > 2. And unto this end we submit the following theo 21 rem which is the first of a number of theorems proven by the writer in general and logarithmic solutions for equations of the third, fourth, fifth, and sixth degrees (not yet published). 32. McGINNIS' THEOREM OF DERIVATIVE EQUATIONS (g- 2). The general equation of the nth degree has n derivative equations of the (n-1)th degree; and each derivative equation contains at least one root of the equation fromn which it is derived. (1) Assume that f(x) =0 is an equation of the nth degree, whose coefficients we will represent by A, B, C, D, E, *.., following the general law in the formation of equations. Designate the n derivatives as follows: (2) f' (X), f / (f)),fI(X), f4(X),/ f (),.. fn(X). From the theorem we build the following formulae: f(x) = -(Xj), and fl(x) - q(xl) = 0. f2(x) = (x2), and f2(X) - +(X2) = 0. f3(x) = O(x3), and f3(x) - (r(x) = 0. (3) f(4(x) = (X4), and f4() - (X4) = 0. f(xz) = (x,5), and f5(x) - O(x5) = 0. fn(x) = b(~n), and fn(x) - cb(x) = 0. (Read "the first derivative of function x equals function x ".) (4) Applying formule (3) to equations of the third, fourth, and fifth degrees, we have the following group of derivative equations: 33. The general equation of the third degree may be written for solution as follows: (1) x3+Ax2 +Bx + C=0. 22 From formula (3-32), we have fl(x)= 3 xl2 + 2 Ax1 + B = O(x) = (x2 - X1) (x3 - Xi), I. f2(x)= 3 X22 + 2 Ax2 + B = 0(X2)=(X1 - X2) (X -X2), f3(x) = 3 x3 + 2 Ax3 + B = (x3)= (X - X) (X2- X3). The general equation of the fourth degree may be written for solution as follows: (2) x4 + Ax3 + Bx2 + Cx + D = O. From formulae (3-32), we have fl(x) = 4 13 + 3 Ax12 + 2 Bx + C = (X2 - x1) (x3 - X1) (X4 - x1)= = (1), f2(x)= 4 x23 + 3 Ax22 + 2 Bx2 + C II. = (X1 -- X2) (X3 - X2) (X4 - X2) = 0 (X2), f3(x) = 4 X33 + 3 Ax32 + 2 Bx3 + C = (l - x3) (X2 - X3) (x4 - x3)= (3), f(x) = 4 x43 + 3 Ax42 + 2 Bx4 + C = (z - X4) (X2 - x4) (3 - X4) = 0 (X4). The general Quintic may be written for solution as follows: (3) x5 + Ax4 + Bx3 + Cx2 + Dx +- E= 0. From (3-32), we have fl(x)=5 X14 + 4 Axi13 + 3 Bxl2 + 2 CxI + D =(X2 - i) (X3 - X1)(X4 - X1)(x5 - X1)=(X1), f2() = 5 x24 + 4 Ax23 + 3 Bx2 + 2 Cx2 + D = (1i -- X2) (X3- X2) (X4 - X2) (X - X2) = (), I f3(X) -= 5 3+4 Ax33 + 3 Bx32 + 2 Cx3 + D =(x1 - ) (X2 - x23) (X4 — X) (X5 - X3)= x (X)), f4() = 5 X44 +- 4 Ax43 -- 3 Bx4 + 2 Cx4 + D = (1 - X4) (2 - X4) (X3 - X4) (X5 - 4) = ~(X4), f5(x) = 5 x54 4 Ax5- + 3 Bx52 + 2 Cx5 + D = (Xz - X5) (X2 - X$) (X - X5) (X4 - X5) = (5). The general equation of the nth degree may be written for solution as follows: (4) xn + Axn-1 + Bxn-2 + Cxn-3 + *-. + Dx + E= 0. 23 From (3-32), we have fl(x) = nxln-l + (n - 1)AXlf-2 +(n - 2)BxZ'l-3 +.. + D = (2- ) (X3 - 1)... (nX,- X= z(Xz). IV............ 1( (X) -nx1- + (n - 1)Ax^n-2 + + D = (xX~=1 (a — Xn) (X2-X^) (X3-Zn).. = (X). 34. The algebraic product of the differences of the roots of an equation of the nth degree is expressed as follows: (1) xn + Az-l + Bx-2 + C + x... *Dx + E= 0, which represents the general equation of the nth degree. From the formule IV-(33), we have (2) [(X((x)) (+(X2))... ((xn))]) ' 2+t(A, B, C )... E)' = (Z - xl)(8 - X)... (n - 1) (x3 -2) (X4 - x2)... (Xn - X2)(X4 -X) (x5 - x3)... (- x3).. (, - x-1). In the cubic (3) x3+- Ax2 + Bx C= 0, we have (4) [((~ 1))(( 2))( 2 = [18 ABC+ AB2-4 A3C- 4 B3 -27 612] = (X2 - X1) (x3 - X)(X3 - X2) To illustrate numerically, take the cubic (5) x3 + 15 x2 + 68 x + 8- = 0. Substituting in (4) the numerical values of A, B, C, we have (6) [18 x 15 x 68 x 84 + (15 x 68)2 -(153 x 84)4 -4 x 683- 27 x 842]i = (400) = 20 = (x2 - xl) (x, - x,)(x3 - x2) The differences of the roots of the cubic (5) will form the cubic (7) Y - 21 y + 20 = 0. 24 The product of the differences of (7) will be [4(21)3 - 27(20)2]2 = (26244) =_ 162. If we remove the second term of the cubic we will have two cases: 1st, y3 +py ~ q = 0. 2d, y3-py~q=0. From the 1st: (4p3 + 27 q2)> = product of differences of roots, and is always an imaginary quantity. From 2d: [4p3 - 27 q2] = product of the differences of roots. 35. The value of the derivative is the product of the algebraic differences of the roots of the equation, in which every value of x is subtracted, in turn, from every other value of x, the factors of each product being, in number, one less than the degree of the equation. We will also take notice that, when any two values of x are algebraically equal, the value of such function will be zero, and, in which case, the derivative function is equal to zero; and the derivative equation will contain such common root. We also notice that if all the roots of an equation are algebraically equal, the derivative equation will contain (n - 1) roots of the original equation. 36. We shall now apply formulae (3-32) to obtain, if possible, the nth root of the sum of an + 3n. (1) It is assumed that the quantities a and 3 represent whole numbers. (Using the same notation as in prior proofs.) (2) It is assumed that 3 > cc, at all times, when combined with a. (3) If the sum of the nth powers of a and f be a power of the nth degree, then, it will be possible to construct a 25 triangle, two of the sides of which are known (a and 8), and its third side, the nth root of the sum of a" + 3n. (4) Assume that it is possible for the following equality to exist, viz.: (5) an+ /n= n. (6) It is assumed that 3 > a... 7> /3. (The nth root of the sum of two nth powers is greater than the greater number.) But, a + /3> y. (The sum of any two sides of a triangle is greater than the third side.) (See any geometry for proof.) (7) Assume that - /3 = A/3. Then, A/3 f/3 = r. Let x = A3. Then, x + /3 =. We now have, from our hypothesis, the following equations: (8) (x + 3)n = =yn _= n +3r = (A/3 + ( 3)n. Expanding (x + /3)", we have (9) xn + n/3;xn- + n(n -1) 82Xn-2 +... + n/3nlx + /3 1x2 = y = " + /n (A3 + 3)n. By transposition, taking a" + 3n for yn, we have (10) xn+-nxn-l-+ n /32-2+.. +n3nlx - a-nO. 1x2 Dividing the roots of (10) by /, and writing y for the new value of x, which is- =A = y, we have (11) y"n nyn-1 + (n- "ny - " = 0. 1x2 /" 26 Taking the first derivative of (11), we have (12) nym-1 + n(n - l)yn-2 + n(n 1 ~ ) yn- +2 + n = -(yl) = (Y2 - y1) *.. (y - y).~ Dividing (12) by n, the coefficient of yn-l, we have, (13) yn-1 + (n - i)yn-2 + (n 1(n - 2) 1 = (y2Y- 1)(Y3- Y1).. (Yn- Y1). n Taking the (n — 1)th root of both members of (13), we have, 1 (14) y 1 = r(Y2-y1)(3 -Y1) (4 Y- ) (Yn -Y)]1..~-L (n)n-i (n)n-1 Multiplying by (n)n-1, and dividing by y +1, we have, (15) [njn-1-= [Q (Y2- qY1) (Y — Yl) (g4 - ) (Yn- Yl) iy+1 +1 As the (n - 1)th root of n is incommensurable for all values of n > 2, y is incommensurable. It is, therefore, impossible to construct the proposed triangle, because its third side, y, can not be measured, can not be accurately determined. Therefore, the sum of no two nth powers, when n > 2, can be a power of the same degree. Therefore, " Fermat's Theorem " holds good for the sum of any two powers of the nth degree when n > 2. 27 37. Continuing the discussion, we can, as an illustration of the foregoing proofs, write the following equations: When n = 2, we have, from (12-36), (1) 2 y + 2 = (y) = Y2 - 1 or yl - y (2)... 1 =a +- +( 'I Y2 Y 1 /= 2 )2 1a2 A. or - ( +1 =Yl-Y2. If a2 + b2 be a perfect square, then y is rational. This will occur when the sum of ao + complies with Law 1-3. When n = 3, we have ~: ---a-= (3) (3)= KY)0(Y [(.y2- y1)(y3-,yl)] ( 4(1) ^+1) 3 y+1 When n = 4, we have -a4 -- -=:-, (4) (4) = (y) I [2 Y- 3( - yl)(y4- 1) ] y + +-1 y+l When n = 5, we have a: ' —t+ —5; (5) (5)= - byl),); and so on, for equations of all 7+1; degrees. As the square root of 3, the cube root of 4, and the fourth root of 5 are all incommensurable, it follows that the fifth root of 6 and the sixth root of 7 are likewise incommensurable. Therefore, in general, the (n —l)th root of n is incommensurable when n > 2. Therefore, 28 the sum of no two powers of the nth degree can be a power of the same degree when n > 2. Hence, the truth of Fermat's Theorem- The sum of no two powers except squares is itself a power of the same degree. 38. Linking of the Proofs of Theorems I, 17, and MeGinnis' Theorem of Derivative Equations in a Final and Absolute Proof of Fermat's Theorem. Repeating from (11)-36. Transferring to the right of the sign of equality, and adding 1 to both sides of the equation, and then taking the nth root of both members, we have (1) Y La.^ - This is what occurs in the proof of Th. II; and it is clearly evident that the y + 1 in (14-15-36), is identical with the y + in (1). Therefore, both theorems and proofs are thus linked together, and we can write, with absolute certainty, the following equations of equality: (2) (3) (4) 1 1, +1 _ ___n1 n(n (n)= 1 y+l (From equation 16-29.) (From equation 14-36.) (From (3) by multiplication and division.) (From (2) and (4).) (5) (n)n- _= (Y). -- n a1 a + 1 29 (6) (n)nj + )n = (y). (From (5).) (7) n(Yl)=n + 1. (From (6).) Equation (7) gives us a general expression for the value of the b(Yl) in terms of the coefficients of the equation from which it is derived. To illustrate: From (12-36), we have, (8) ny-1 + n(n -l)yn-2+ n(n- 1)n- 2) Y +.+n 1x2,-n = ==(y1+) = n + 1 = (- ) (Y3 1)... (Y-1i) Dividing (8) by n, we have, n-l (9) ^-i+ (-l)yn-2+ - 1 +1 )8n n= +1~ (10).*. y +l - +. (Taking the (n- l)th root +n '. J of both sides of (9).) 1 1 (11) 1()l = n + lI. (From (2) and (3).) 39. Final argument and closing remarks in the proofs of Fermat's Theorem. 30 Taken from the foregoing proofs: 1. The nth root of the sum of an + 38 > /. 2. The nth root of the sum of an + f < a + /3. 3. Assume that the nth root of the sum of a" + /3 is equal to A/M +- f. Viewing assumption (3) from any mathematical standpoint it must be admitted that the assumption is not only allowable and well taken, but it is arithmetically, geometrically, and algebraically true. It then follows that it is possible for the following equality to exist: 1 4. [n +/ n] n= A/ + P. If we raise both sides of (4) to the nth power, we have 5. ~ +/ n = (A/ + )n. The question may now be asked: How is it that in (4) the equality expresses an incommensurable quantity or number equal to an incommensurable quantity or number, but when both sides of the equation (4) are raised to the nth power we have equation (5), which is the nth power of an incommensurable quantity or number equal to a rational or commensurable number or quantity, which, we say, is absurd? But when and how is it absurd? Let us assume that n is of infinite value; then we must admit that the equality does exist, because infinity can not be measured. Let us assume that n = 3, a finite quantity, or value. Then 6. (a3 +/ 3)3= +/3. 7. a3 + /3 = (A/ + /)3. (Raising both sides of (6) to the third power.) What have we done? Why this change from an in 31 )mmensurable to a commensurable quantity? Surely, the sum of a3 + /3 can be measured. What interpretation can we now place upon it? This alone can be given: If n be infinite in (A/3 + 3)1 = an + /3, then both sides of the equation are infinite, and therefore incommensurable. But when we place an arbitrary limiting value upon n, we simply place a limit upon (A3 + /3)n. Therefore, in the equality (A/3 + )3 = a3 + 33, the sum of a3 + /3 is the limit to which (A/3 +- )3 approaches, and which it can only reach when n = 2, and then in special cases only (see Law I), but never when n> 2. Hence, the assumption is not absurd, but the equality is impossible when n > 2. And it is this impossible equality which we have proven to exist in (ano + f n= yn) that proves the truth of bermat's Theorem. 40. (1) It has been proven by Theorem I that a + /n can not be a power of the nth degree, because the nth root of the sum of a + /3 is incommensurable when n > 1. (2) It has been proven by Theorem I that the nth root of the sum of - + 1 is incommensurable, and therefore /3n the sum of -- + 1 can not be a power of the nth degree when n > 1. (3) It has been proven by Theorem II that the nth root of the sum of ac +~ /1 is incommensurable when n > 2; and therefore the sum of aO + i3n can not be a power of the same degree when n > 2. (4) It has been proven by Theorem II that the nth root of the sum of - + 1 is incommensurable when n > 2; gln and therefore the sum of a + 1 can not be a power of the same degree when n > 2. 32 (5) It has been proven by the Theorem of the Derivative Equations that for all positive values of a and /, the sum of a + f3n can not be equal to the nth power of a rational number or quantity when n > 2, - because the (n - 1) tl root of n is incommensurable for all values of n except 1 and 2. 41. REDUCING THE QUINTIC BY THE THEOREM OF DERIVATIVE EQUATIONS. The general Quintic may be written for solution as follows: (I) x5 + mx4 + nx3 + ox2 + px + q = 0. Multiply (I) by x+ -t = 0, and we have (II) x6 + m 5 + (M-2-+ Nx4+ o 5 K W lx 5 +K +O) + + -p4 2 x+ + qx + '-=0. Taking the first derivative of (II), we have (III) 6x5+ 6m a4+(4 +4n) 3+ 3-f+3o).T + (2 m+2p x + —+q 5 5 4 m5 m20 _ m3o 3125 2-5 + 125 5 By transposition and division, we have, from (III), (I)T rx + (j/2 m 2 n\ /n z + O (IV) X5 + Tmx4 + -5 + X3 + + 2 (5 3 06 + 0_o 2 fmoi o mjm wMn m130 2 m + + ]- 5 _ +i 3x-lo 750 150 9'75 33 Dividing (IV) by x + = 0, we have 4 mx3 + 9q2\ 2222 m3 0 mrni (V) x+ 3 25 +(75 2 30) m+ + _ mo 2 nm 0 150 3 30 1875 Equation (V) contains four true derivative roots of (I); and will contain four roots of (I), if all the roots are algebraically equal. (4m i5 m2o m3o mp> (5 )(3125 +25 + -125 5 which is the absolute term of (I) when the second term is removed, and is equal to \Any / q on o- e - e n d (4 mx5 Any equation of the nth degree can be reduced by this method to an equation of the (n - l)th degree. Further discussion of the value of such derivative equations in general solutions for equations of all degrees can not be given here. (VI) All equations can be reduced directly to an equation of one degree lower, and depends upon the proof of Theorems I, II, and III. (42) 42. Supplementary theorems proven by the writer in general and logarithmic solutions for equations of the third, fourth, fifth, and sixth degree: (I) From the general equation of the nth degree there can be derived an equation of the nth degree which will contain one root in common with the equation from which it is derived. 34 (II) When two equations of the nth degree have one root in common, such equations, by subtraction, reduce to an equation of the (n - 1)th degree; and such reduced equation will contain such common root. COROLLARY. If two equations of the nth degree have one root in common, and the algebraic sum of their roots equal, such equations by subtraction will reduce to an equation of the (n-2)th degree, and such reduced equation of the (n - 2)th degree will contain such common root. (III) The general equation of the nth degree can be reduced to an equation of the (n - )th degree whether n be of odd or even degree and greater or less than 4. (IV) All even degree equations contain (n- 1) algebraically symmetrical functions, and can be reduced to an equation one degree lower. (V) The algebraically symmetrical functions of the sixth degree equation are as follows: Let x = a, b, c, d, e, f, the six values of x in an equation of the sixth degree. Then the functions will be: (1) ab + ed + ef. (2) ac + de + If. (3) ad+ be + cf. (4) ae + be + df. (5) af + bd + ce. These five functions will form an equation of the fifth degree, the first three terms of which I remember to be: y5 -By4+ (AC- 4D)y-.... The sixth = x6 + Ax5 + Bx4 + Cx3 + Dx2 + Ex + F = 0.