THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES FAMOUS PROBLEMS ELEMENTARY GEOMETRY THE DUPLICATION OF THE CUBE THE TRISECTION OF AN ANGLE THE QUADRATURE OF THE CIRCLE AN AUTHORIZED TRANSLATION OF F. KLEIN'S VORTBAGE UBER AUSGEWAHLTE FRAGEN DER ELEMENTARGEOMETRIE AUSGEARBEITET VON F. TAGERT BY WOOSTER WOODRUFF BEMAX PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF MICHIGAN AND DAVID KHiKXE SMITH ;>^>I: OF MAIIU.M in. > i.\ TEAI-HERS COI.LKOK. CuLriiuiA UMVLKSIIV GINX AND COMPANY BOSTON NM.\V YoltK CHICACiO I.OXIKN ATLAXTA DALLAS COLL'-MHUS SAN KUAXCISCO COPYRIGHT, 1897, BY WOOSTEB WOODRUFF BEMAN AITO DAVID EUGENE SMITH ALL RIGHTS RE8ERYED 316.9 atfrtnacum GINN AND COMPANY- PRO- PRIETORS BOSTQN U.S.A. s i\ f Library i Jo *3 PREFACE. THE more precise definitions and more rigorous methods of demonstration developed by modern mathematics are looked upon by the mass of gymnasium professors as abstruse and excessively abstract, and accordingly as of importance only for the small circle of specialists. With a view to counteract- ing this tendency it gave me pleasure to set forth last summer in a brief course of lectures before a larger audience than \ usual what modern science has to say regarding the possibility ' of elementary geometric constructions. Some time before, I ^ had had occasion to present a sketch of these lectures in an * Easter vacation course at Gottingen. The audience seemed *k to take great interest in them, and this impression has been confirmed by the experience of the summer semester. I ven- ture therefore to present a short exposition of my lectures to the Association for the Advancement of the Teaching of Math- ematics and the Natural Sciences, for the meeting to be held at Gottingen. This exposition has been prepared by Oberlehrer Tagert, of Ems, who attended the vacation course just men- tioned. He also had at his disposal the lecture notes written out under my supervision by several of my summer semester students. I hope that this unpretending little book may con- tribute to promote the useful work of the association. F. KLEIN. GOTTINGEN, Easter, 1895. TRANSLATORS' PREFACE. AT the Gottingen meeting of the German Association for the Advancement of the Teaching of Mathematics and the Natural Sciences, Professor Felix Klein presented a discus- sion of the three famous geometric problems of antiquity, the duplication of the cube, the trisection of an angle, and the quadrature of the circle, as viewed in the light of modern research. This was done with the avowed purpose of bringing the study of mathematics in the university into closer touch with the work of the gymnasium. That Professor Klein is likely to succeed in this effort is shown by the favorable reception accorded his lectures by the association, the uniform commen- dation of the educational journals, and the fact that transla- tions into French and Italian have already appeared. The treatment of the subject is elementary, not even a knowledge of the differential and integral calculus being required. Among the questions answered are such as these : Under what circumstances is a geometric construction pos- sible ? By what means can it be effected ? What are tran- scendental numbers ? How can we prove that e and TT are transcendental ? With the belief that an English presentation of so impor- tant a work would appeal to many unable to read the original, vi TRANSLATOR'S PREFACE. Professor Klein's consent to a translation was sought and readily secured. In its preparation the authors have also made free use of the French translation by Professor J. Griess, of Algiers, following its modifications where it seemed advisable. They desire further to thank Professor Ziwet for assist- ance in improving the translation and in reading the proof- sheets. August, 1897. W. W. BEMAN. D. E. SMITH. CONTENTS. INTRODUCTION. MM PRACTICAL AND THEORETICAL CONSTRUCTIONS .... 2 STATEMENT OF THE PROBLEM IN ALGEBRAIC FORM ... 3 PART I. The Possibility of the Construction of Algebraic Expressions. CHAPTER I. ALGEBRAIC EQUATIONS SOLVABLE BY SQUARE ROOTS. 1^4. Structure of the expression x to be constructed ... 5 5, 6. Normal form of x ........ 6 7, 8. Conjugate values 7 9. The corresponding equation F(x) = o .... 8 10. Other rational equations f(x) = o . . . . . .8 11, 12. The irreducible equation 0(x) = o . . . . 10 13, 14. The degree of the irreducible equation a power of 2 . .11 CHAPTER II. THE DELIAN PROBLEM AND THE TRISECTION OF THE ANGLE. 1. The impossibility of solving the Delian problem with straight edge and compasses ....... 13 2. The general equation x 3 = X 13 3. The impossibility of trisecting an angle with straight edge and compasses 14 CHAPTER III. THE DIVISION OF THE CIRCLE INTO EQUAL PARTS. 1. History of the problem 16 2-4. Gauss's prime numbers ....... 17 5. The cyclotomic equation ....... 19 6. Gauss's Lemma ......... 19 7, 8. The irreducibility of the cyclotomic equation . . .21 viii CONTENTS. CHAPTER IV. THE CONSTRUCTION OF THE REGULAR POLYGON OF 17 SIDES. PAGE 1. Algebraic statement of the problem 24 2-4. The periods formed from the roots . . . . .25 6, 6. The quadratic equations satisfied by the periods . . 27 7. Historical account of constructions with straight edge and compasses ......... 32 8, 9. Von Staudt's construction of the regular polygon of 17 sides 34 CHAPTER V. GENERAL CONSIDERATIONS ON ALGEBRAIC CONSTRUCTIONS. 1. Paper folding ......... 42 2. The conic sections 42 3. The Cissoid of Diocles ........ 44 4. The Conchoid of Nicomedes ...... 45 6. Mechanical devices , 47 PART II. Transcendental Numbers and the Quadrature of the Circle. CHAPTER I. CANTOR'S DEMONSTRATION OF THE EXISTENCE OF TRANSCENDENTAL NUMBERS. 1. Definition of algebraic and of transcendental numbers . 49 2. Arrangement of algebraic numbers according to height . 50 3. Demonstration of the existence of transcendental numbers 53 CHAPTER II. HISTORICAL SURVEY OF THE ATTEMPTS AT THE COM- PUTATION AND CONSTRUCTION OF TT. 1. The empirical stage 56 2. The Greek mathematicians ...... 56 3. Modern analysis from 1670 to 1770 58 4, 5. Revival of critical rigor since 1770 ..... 59 CHAPTER III. THE TRANSCENDENCE OF THE NUMBER e. 1. Outline of the demonstration 61 2. The symbol hr and the function 0(x) .... 62 3. Hermite's Theorem 65 CONTENTS. lx CHAPTER IV. THE TRANSCENDENCE OF THE NUMBER TT. PAGE 1. Outline of the demonstration 68 2. The function ^(x) 70 3. Lindemann's Theorem ....... 73 4. Lindemann's Corollary 74 5. The transcendence of IT . . . . . . 76 6. The transcendence of y = e x . . . . . . .77 7. The transcendence of y = sin J x ..... 77 CHAPTER V. THE INTEGRAPH AND THE GEOMETRIC CONSTRUCTION OF 7T. 1. The impossibility of the quadrature of the circle with straight edge and compasses ....... 78 2. Principle of the integraph 78 3. Geometric construction of IT . . . . . . .79 INTRODUCTION. THIS course of lectures is due to the desire on my part to bring the study of mathematics in the university into closer touch with the needs of the secondary schools. Still it is not intended for beginners, since the matters under discussion are treated from a higher standpoint than that of the schools. On the other hand, it presupposes but little preliminary work, only the elements of analysis being required, as, for example, in the development of the exponential function into a series. We propose to treat of geometrical constructions, and our object will not be so much to find the solution suited to each case as to determine the possibility or impossibility of a solution. Three problems, the object of much research in ancient times, will prove to be of special interest. They are 1. The problem of the duplication of the cube (also called the Delian problem). 2. The trisection of an arbitrary angle. 3. The quadrature of the circle, i.e., the construction of TT. In all these problems the ancients sought in vain for a solution with straight edge and compasses, and the celebrity of these problems is due chiefly to the fact that their solution seemed to demand the use of appliances of a higher order. In fact, we propose to show that a solution by the use of straight edge and compasses is impossible. 2 INTRODUCTION. The impossibility of the solution of the third problem was demonstrated only very recently. That of the first and second is implicitly involved in the Galois theory as presented to-day in treatises on higher algebra. On the other hand, we find no explicit demonstration in elementary form unless it be in Petersen's text-books, works which are also noteworthy in other respects. At the outset we must insist upon the difference between practical and theoretical constructions. For example, if we need a divided circle as a measuring instrument, we construct it simply on trial. Theoretically, in earlier times, it was possible (i.e., by the use of straight edge and compasses) only to divide the circle into a number of parts represented by 2", 3, and 5, and their products. Gauss added other cases by showing the possibility of the division into parts where p is a prime number of the form p = 2 2 ^ -(- 1, and the impos- sibility for all other numbers. No practical advantage is derived from these results; the significance of Gauss's de- velopments is purely theoretical. The same is true of all the discussions of the present course. Our fundamental problem may be stated : What geometrical constructions are, and what are not, theoretically possible ? To define sharply the meaning of the word "construction," we must designate the instruments which we propose to use in each case. We shall consider 1. Straight edge and compasses, 2. Compasses alone, 3. Straight edge alone, 4. Other instruments used in connection with straight edge and compasses. The singular thing is that elementary geometry furnishes no answer to the question. We must fall back upon algebra and the higher analysis. The question then arises : How INTRODUCTION. 3 shall we use the language of these sciences to express the employment of straight edge and compasses ? This new method of attack is rendered necessary because elementary geometry possesses no general method, no algorithm, as do the last two sciences. In analysis we have first rational operations : addition, subtraction, multiplication, and division. These operations can be directly effected geometrically upon two given seg- ments by the aid of proportions, if, in the case of multiplica- tion and division, we introduce an auxiliary unit-segment. Further, there are irrational operations, subdivided into algebraic and transcendental. The simplest algebraic opera- tions are the extraction of square and higher roots, and the solution of algebraic equations not solvable by radicals, such as those of the fifth and higher degrees. As we know how to construct Vab, rational operations in general, and irrational operations involving only square roots, can be constructed. On the other hand, every iii<li/-i<ln<d geometrical construction which can be reduced to the intersection of two straight lines, a straight line and a circle, or two circles, is equivalent to a rational operation or the extraction of a square root. In the higher irrational operations the construction is therefore impossible, unless we can find a ivay of effecting it by the aid of square roots. In all these constructions it is- obvious that the number of operations must be limited. We may therefore state the following fundamental theorem : Tin- necessary and sufficient condition that an until /jtic expres- sion, can be constructed with straight edge and compasses is that it can be derived from the known quantities by a finite number of rational operations and sqiiare roots. Accordingly, if we wish to show that a quantity cannot be constructed with straight edge and compasses, we must prove that the corresponding equation is not solvable by a finite number of square roots. 4 INTRODUCTION. A fortiori the solution is impossible when the problem has no corresponding algebraic equation. An expression which satisfies no algebraic equation is called a transcenden- tal number. This case occurs, as we shall show, with the number TT. PART I. THE POSSIBILITY OF THE CONSTKUCTION OF ALQEBKAIC EXPRESSIONS. CHAPTER I. Algebraic Equations Solvable by Square Roots. The following propositions taken from the theory of alge- braic equations ar| probably known to the reader, yet to secure greater clearness of view we shall give brief demon- strations. If x, the quantity to be constructed, depends only upon rational expressions and square roots, it is a root of an irreducible equa- tion f (x) = 0, whose degree is always a power of 2. 1. To get a clear idea of the structure of the quantity x, suppose it, e.g., of the form a + Vc + ef + c-f Vb p + Vq Va + Vb V7 where a, b, c, d, e, f, p, q, r are rational expressions. 2. The number of radicals one over another occurring in any term of x is called the order of the term ; the preceding expression contains terms of orders 0, 1, 2. 3. Let fji designate the maximum order, so that no term can have more than radicals one over another. 6 FAMOUS PROBLEMS. 4. In the example x = V2 + V3 + V6, we have three expressions of the first order, but as it may be written x = V2 + V3 + V2 V3, it really depends on only two distinct expressions. We shall suppose that this reduction has been made in all the terms of x, so that among the n terms of order /* none can be expressed rationally as a function of any other terms of order p or of lower order. We shall make the same supposition regarding terms of the order p. 1 or of lower order, whether these occur ex- plicitly or implicitly. This hypothesis is obviously a very natural one and of great importance in later discussions. 5. NORMAL FORM OF x. If the expression x is a sum of terras with different denom- inators we may reduce them to the same denominator and thus obtain x as the quotient of two integral functions. Suppose VQ one of the terms of x of order p. ; it can occur in x only explicitly, since /* is the maximum order. Since, further, the powers of VQ may be expressed as functions of VQ and Q, which is a term of lower order, we may put _ a + b VQ ~ c + d VQ' where a, b, c, d contain no more than n 1 terms of order fi, besides terms of lower order. Multiplying both terms of the fraction by c d VQ, VQ disappears from the denominator, and we may write _(ac-bdQ)+(bc-ad)VQ_ /? r c 2 -d 2 Q rpvy, where a and ft contain no more than n 1 terms of order /A. For a second term of order /u, we have, in a similar manner, x = ^ -f- fa VQi, etc. ALGEBRAIC EQUATIONS. 7 The x may, therefore, be transformed so as to contain a term of given order p. only in its numerator and there only linearly. We observe, however, that products of terms of order p, may occur, for a and /3 still depend upon n 1 terms of order p.. We may, then, put a = a n a 12 and hence x = (a n + a 12 V QO + (/? n + /? 12 V&) VQ. 6. We proceed in a similar way with the different terms of order p. 1, which occur explicitly and in Q, Q 1? etc., so that each of these quantities becomes an integral linear func- tion of the term of order /x 1 under consideration. We then pass on to terms of lower order and finally obtain x, or rather its terms of different orders, under the form of rational integral linear functions of the individual radical expressions which occur explicitly. We then say that x is reduced to the normal form. 7. Let m be the total number of independent (4) square roots occurring in this normal form. Giving the double sign to these square roots and combining them in all possible ways, we obtain a system of 2 m values which we shall call conjugate values. We must now investigate the equation admitting these conjugate values as roots. 8. These values are not necessarily all distinct ; thus, if we have x = \J a + Vb + \Ja Vb, this expression is not changed when we change the sign of Vb. 8 FAMOUS PROBLEMS. 9. If x is an arbitrary quantity and we form the poly- nomial F (x) = (x Xi) (x x 2 ) . . . (x x 2m ), F (x) = is clearly an equation having as roots these con- jugate values. It is of degree 2 ra , but may have equal roots (8). The coefficients of the polynomial F (x) arranged with respect to x are rational. For let us change the sign of one of the square roots ; this will permute two roots, say X A and xy, since the roots of F (x) = are precisely all the conjugate values. As these roots enter F (x) only under the form of the product (x X A ) (x xy), we merely change the order of the factors of F (x). Hence the polynomial is not changed. F (x) remains, then, invariable when we change the sign of any one of the square roots ; it therefore contains only their squares ; and hence F (x) has only rational coefficients. 4 10. When any one of the conjugate values satisfies a given equation with rational coefficients, f (x) = 0, the same is true of all the others. f (x) is not necessarily equal to F (x), and may admit other roots besides the x/s. Let Xi = a + (3 VQ be one of the conjugate values ; VQ, a term of order p. ; a and ft now depend only upon other terms of order \t. and terms of lower order. There must, then, be a conjugate value Let us now form the equation f (x t ) = 0. f (x x ) may be put into the normal form with respect to VQ, ALGEBRAIC EQUATIONS. 9 this expression can equal zero only when A and B are simul- taneously zero. Otherwise we should have i.e., VQ could be expressed rationally as a function of terms of order p, and of terms of lower order contained in A and B, which is contrary to the hypothesis of the independence of all the square roots (4). But we evidently have hence if f (x x ) = 0, so also f (x/) = 0. Whence the following proposition : If x x satisfies the equation f (x) = 0, the same is true of all the conjugate values derived from Xj by changing the signs of the roots of order p.. The proof for the other conjugate values is obtained in an analogous manner. Suppose, .for example, as may be done without affecting the generality of the reasoning, that the expression x x depends on only two terms of order /*,, VQ and VQ'. f (x^ may be reduced to the following normal form : (a) f ( Xl ) = p -f q VQ + r V^X + s VQ VQ 7 ^ 0. If Xj depended on more than two terms of order //., we should only have to add to the preceding expression a greater num- ber of terms of analogous structure. Equation (a) is possible only when we have separately (b) p = 0, q = 0, r = 0, s = 0. Otherwise VQ and VQ' would be connected by a rational relation, contrary to our hypothesis. Let now VR, VR', ... be the terms of order /x 1 on which xj depends ; they occur in p, q, r, s ; then can the quantities p, q, r, s, in which they occur, be reduced to the 10 FAMOUS PROBLEMS. normal form with respect to VR and VR' ; and_if, for the sake of simplicity, we take only two quantities, VR and we have and three analogous equations for q, r, s. The hypothesis, already used several times, of the inde- pendence of the roots, furnishes the equations (d) K = 0, A = 0, n = Q, v = 0. Hence equations (c) and consequently f (x) = are satisfied when for Xi we substitute the Conjugate values deduced by changing the signs of VR and VR'. Therefore the equation f (x) = is also satisfied by all the conjugate values deduced from K^by changing the signs of the roots of order p 1. The same reasoning is applicable to the terms of order p. 2, f*. 3, ... and our theorem is completely proved. 11. We have so far considered two equations F (x) = and f (x) = 0. Both have rational coefficients and contain the Xj's as roots. F (x) is of degree 2 ra and may have multiple roots ; f (x) may have other roots besides the x/s. We now introduce a third equation, </> (x) = 0, defined as the equation of lowest degree, with rational coefficients, admitting the root Xj and conse- quently all the x t 's (10). 12. PROPERTIES OF THE EQUATION <f> (x) = 0. I. < (x) is an irreducible equation, i.e., <f> (x) cannot be resolved into two rational polynomial factors. This irreduci- bility is due to the hypothesis that < (x) = is the rational equation of lowest degree satisfied by the x t 's. For if we had *(*)=* 00 x(*). ALGEBRAIC EQUATIONS. 11 then < (Xi) = would require either if/ (x x ) = 0, or % (*i) = 0> or both. But since these equations are satisfied by all the conjugate values (10), < (x) =0 would not then be the equa- tion of lowest degree satisfied by the Xi's. II. <f> (x) = has no multiple roots. Otherwise < (x) could be decomposed into rational factors by the well-known meth- ods of Algebra, and <f> (x) = would not be irreducible. III. < (x) = has no other roots than the x/s. Otherwise F (x) and < (x) would admit a highest common divisor, which could be determined rationally. We could then decompose <f> (x) into rational factors, and < (x) would not be irreducible. IV. Let M be the number of x/s which have distinct values, and let X l5 X 2 , . . . X M be these quantities. We shall then have <f> (x) = C (x x t ) (x . x 2 ) . . . (x X M ). For < (x) = is satisfied by the quantities x t and it has no multiple roots. The polynomial < (x) is then determined save for a constant factor whose value has no effect upon <f> (x) = 0. V. <f> (x) = is the only irreducible equation with rational coefficients satisfied by the x t 's. For if f (x) = were another rational irreducible equation satisfied by x : and consequently by the Xj's, f (x) would be divisible by <f> (x) and therefore would not be irreducible. By reason of the five properties of < (x) = thus estab- lished, we may designate this equation, in short, as the irre- ducible equation satisfied by the x/s. 13. Let us now compare F (x) and <(x). These two poly- nomials have the x/s as their only roots, and <f> (x) has no multiple roots. F (x) is, then, divisible by < (x) ; that is, 12 FAMOUS PROBLEMS. F! (x) necessarily has rational coefficients, since it is the quo- tient obtained by dividing F (x) by </> (x). If F x (x) is not a constant it admits roots belonging to F (x) ; and admitting one it admits all the x/s (10). Hence F! (x) is also divisible by < (x), and If F 2 (x) is not a constant the same reasoning still holds, the degree of the quotient being lowered by each operation. Hence at the end of a limited number of divisions we reach an equation of the form F,, _ ! (x) = Ci 4> ( x )> and for F (x), The polynomial F (x) is then a power of the polynomial of minimum degree <f> (x), except for a constant factor. 14. We can now determine the degree M of <(x). F (x) is of degree 2 m ; further, it is the vth power of < (x). Hence 2 m = vM. Therefore M is also a power of 2 and we obtain the following theorem : The degree of the irreducible equation satisfied by an expres- sion composed of square roots only is always a power of 2. 15. Since, on the other hand, there is only one irreducible equation satisfied by all the x/s (12, V.), we have the converse theorem : If an irreducible equation is not of degree 2 A , it cannot be solved by square roots. CHAPTER II. The Delian Problem and the Trisection of the Angle. 1. Let us now apply the general theorem of the preceding chapter to the Delian problem, i.e., to the problem of the duplication of the cube. The equation of the problem is manifestly x 3 = 2. 3. This is irreducible, since otherwise V2 would have a rational value. For an equation of the third degree which is reducible must have a rational linear factor. Further, the degree of the equation is not of the form 2 h ; hence it cannot be solved by means of square roots, and the geometric con- struction with straight edge and compasses is impossible. 2. Next let us consider the more general equation x 3 = A., X designating a parameter which may be a complex quantity of the form a + ib. This equation furnishes us the analyt- ical expressions for the geometrical problems of the multi- plication of the cube and the trisection of an arbitrary angle. The question arises whether this equation is reducible, i.e., whether one of its roots can be expressed as a rational func- tion of X. It should be remarked that the irreducibility of an expression always depends upon the values of the quan- tities supposed to be known. In the case x 3 = 2, we were dealing with numerical quantities, and the question was whether v2 coiild have a rational numerical value. In the equation x 3 = X we ask whether a root can be represented by a rational function of A. In the first case, the so-called 14 FAMOUS PROBLEMS. domain of rationality comprehends the totality of rational numbers ; in the second, it is made up of the rational func- tions of a parameter. If no limitation is placed upon this parameter we see at once that no expression of the form in which <j> (A) and i/r(A) are polynomials, can satisfy our equation. Under our hypothesis the equation is therefore irreducible, and since its degree is not of the form 2 h , it can- not be solved by square roots. 3. Let us now restrict the variability of A. Assume A = r (cos < + i sin <) ; whence VA = Vr Vcos <f> + i sin <j>. Our problem resolves itself into two, to ~~X extract the cube root of a real number and also that of a complex number of the form cos <j> -\- i sin <, both numbers being regarded as arbitrary. We shall treat these separately. I. The roots of the equation x 3 = r are 3 / 3 / 3 / Vr, c Vr, e 2 Vr, representing by c and e 2 the complex cube roots of unity - 1 + i V3 . _ - 1 - i V3 C C Taking for the domain of rationality the totality of rational functions of r, we know by the previous reasoning that the equation x 8 = r is irreducible. Hence the problem of the multiplication of the cube does not admit, in general, of a construction by means of straight edge and compasses. II. The roots of the equation x 3 = cos <f> -\- \ sin <f are, by De Moivre's formula, THE TRISECTION OF THE ANGLE. 15 - i SID.-, x 2 = cos X 3 = COS 2-7T , <6 -I- (- i sm : i sin These roots are represented geometrically by the vertices of an equilateral triangle inscribed in the circle with radius unity and center at the origin. The figure shows that to the root x x cor- responds the argument ~. o the equation Hence x = cos FIG. 2. i sin <f> is the analytic expression of the problem of the trisection of the angle. If this equation were reducible, one, at least, of its roots could be represented as a rational function of cos <f> and sin <, its value remaining unchanged on substituting </> + 2?r for <. But if we effect this change by a continuous variation of the angle <, we see that the roots x l5 x 2 , x 3 undergo a cyclic permutation. Hence no root can be represented as a rational function of cos </> and sin <. The equation under consideration is irreducible and therefore cannot be solved by the aid of a finite number of square roots. Hence the trisection of the angle cannot be effected with straight edge and compasses. This demonstration and the general theorem evidently hold good only when < is an arbitrary angle ; but for certain spe- cial values of <f> the construction may prove to be possible, e.g., when < = . CHAPTER III. The Division of the Circle into Equal Parts. 1. The problem of dividing a given circle into n equal parts has come down from antiquity ; for a long time we have known the possibility of solving it when n = 2 h , 3, 5, or the product of any two or three of these numbers. In his Disquisitiones Arithmeticae, Gauss extended this series of numbers by showing that the division is possible for every prime number of the form p = 2^ + 1 but impossible for all other prime numbers and their powers. If in p = 2^ -J- 1 we make p = and 1, we get p = 3 and 5, cases already known to the ancients. For p = 2 we get p = 2 22 + 1 = 17, a case completely discussed by Gauss. For p. = 3 we get p = 2 ?3 + 1 = 257, likewise a prime num- ber. The regiilar polygon of 257 sides can be constructed. Similarly for /A = 4, since 2^ + 1 = 65537 is a prime number. /* = 5, /A = 6, /i = 7 give no prime numbers. For /u. = 8 no one has found out whether we have a prime number or not. The proof that the large numbers corresponding to /x = 5, 6, 7 are not prime has required a large expenditure of labor and ingenuity. It is, therefore, quite possible that p. = 4 is the last number for which a solution can be effected. Upon the regular polygon of 257 sides Eichelot published an extended investigation in Crelle's Journal, IX, 1832, pp. 1-26, 146-161, 209-230, 337-356. The title of the memoir is : De resolutione algebraica aequatlonls x 257 = 1, sine de divisions circuli per bisectionem anyuli septies repetitam in paries 257 inter se aequales commentatio coronata. THE DIVISION OF THE CIRCLE. 17 To the regular polygon of 65537 sides Professor Hermes of Lingen devoted ten years of his life, examining with care all the roots furnished by Gauss's method. His MSS. are preserved in the collection of the mathematical seminary in Gottingen. (Compare a communication of Professor Hermes in No. 3 of the Gb'ttinger Nachrichten for 1894.) 2. We may restrict the problem of the division of the circle into n equal parts to the cases where n is a prime num- ber p or a power p a of such a number. For if n is a com- posite number and if /* and v are factors of n, prime to each other, we can always find integers a and b, positive or nega- tive. such that -, ii 1 = a /A + bv ; 1 a . b whence = | . fLV V fl To divide the circle into p.v = n equal parts it is sufficient to know how to divide it into p. and v equal parts respectively. Thus, for n = 15, we have 1 _2_3 15~3 5' 3. As will appear, the division into p equal parts (p being a prime number) is possible only when p is of the form p = 2 h + 1. We shall next show that a prime number can be of this form only when h = 2 M . For this we shall make use of Fermat's Theorem : If p is a prime number and a an integer not divisible by p, these numbers satisfy the congruence p 1 is not necessarily the lowest exponent which, for a given value of a, satisfies the congruence. If s is the lowest exponent it may be shown that s is a divisor of p 1. In particular, if s = p 1 we say that a is a primitive root of p, 18 FAMOUS PROBLEMS. and notice that for every prime number p there is a primitive root. We shall make use of this notion further on. Suppose, then, p a prime number such that (1) P = 2 h + 1, and s the least integer satisfying (2) 2 8 = + 1 (mod. p). From (1) 2 h < p ; from (2) 2"> p. .-. s>h. (1) shows that h is the least integer satisfying the congruence (3) 2 h = 1 (mod. p). From (2) and (3), by division, 2 8 ~ h = 1 (mod. p). .-.(4) s-h^h, s<2h. From (3), by squaring, 2 a = 1 (mod. p). Comparing with (2) and observing that s is the least expo- nent satisfying congruences of the form 2* = 1 (mod. p), .-. s = 2h. We have observed that s is a divisor of p 1 = 2 h ; the same is true of h, which is, therefore, a power of 2. Hence prime numbers of the form 2 h -}- 1 are necessarily of the form 4. This conclusion may be established otherwise. Sup- pose that h is divisible by an odd number, so that h = h'(2n + l); then, by reason of the formula THE CYCLOTOMIC EQUATION. 19 p = 2 h ' (2n + 1) + l is divisible by 2 h> + 1, and hence is not a prime number. 5. We now reach our fundamental proposition : p being a prime number, the division of the circle into p equal parts by the straight edge and compasses is impossible unless p is of the form p = 2 h + 1 = 2^ + 1. Let us trace in the z-plane (z = x + iy) a circle of radius 1. To divide this circle into n equal parts, beginning at z = 1, is the same as to solve the equation z" 1 = 0. This equation admits the root z = 1 ; let us suppress this root by dividing by z 1, which is the same geometrically as to disregard the initial point of the division. We thus obtain the equation z n-l_j_ z n-2 +> .. +2 + 1=0, which may be called the cyclotomic equation. As noticed above, we may confine our attention to the cases where n is a prime number or a power of a prime number. We shall first investigate the case when n = p. The essential point of the proof is to show that the above equation is irreducible. For since, as we have seen, irreducible equations can only be solved by means of square roots in finite number when their degree is a power of 2, a division into p parts is always im- possible when p 1 is not equal to a power of 2, i.e., when p gfe 2 h + 1 = 2^ + 1. Thus we see why Gauss's prime numbers occupy such an exceptional position. 6. At this point we introduce a lemma known as Gauss's Lemma. If F(z) = z m + kz m ~ l + Bz m ~ 2 +. . . + Lz+ M, 20 FAMOUS PROBLEMS. where A, B, . . . are integers, and F(z) can be resolved into two rational factors f (z) and <f> (z), so that F (z) = f (z) ^ (z) = (z'"' + ai z m - 1 + a 2 Z ra '- 2 + . . .) then must the a's and /3's also be integers. In other words : If an integral expression can be resolved into rational factors these factors must be integral expressions. Let us suppose the a's and /3's to be fractional. In each factor reduce all the coefficients to the least common denom- inator. Let a and b be these common denominators. Finally multiply both members of our equation by a b . It takes the form a b F(z) = fi (z) fa (z) = (a z m ' + a^"'- 1 + . . .) (boz^' + biz- 11 - 1 +...). The a's are integral and prime to one another, as also the b's, since a and b are the least common denominators. Suppose a and b different from unity and let q be a prime divisor of a b . Further, let a { be the first coefficient of f a (z) and b k the first coefficient of fa (z) not divisible by q. Let us develop the product f a (z) </>! (z) and consider the coefficient of z m ' + m " - * - k . It will be 1+2k b k _ 2 + . . . According to our hypotheses, all the terms after the first are divisible by q, but the first is not. Hence this coefficient is not divisible by q. Now the coefficient of z m'+m"-i-k j n ^ g rst member is divisible by a b , i.e., by q. Hence if the identity is true it is impossible for a coefficient not divisible by q to occur in each polynomial. The coefficients of one at least of the polynomials are then all divisible by q. Here is another absurdity, since we have seen that all the coefficients are THE CTCLOTOMIC EQUATION. 21 prime to one another. Hence we cannot suppose a and b different from 1, and consequently the a's and /3's are in- tegral. 7. In order to show that the cyclotomic equation is irre- ducible, it is sufficient to show by Gauss's Lemma that the first member cannot be resolved into factors with integral coefficients. To this end we shall employ the simple method due to Eisenstein, in Crelle's Journal, XXXIX, p. 167, which depends upon the substitution z = x + l. We obtain All the coefficients of the expanded member except the first are divisible by p ; the last coefficient is always p itself, by hypothesis a prime number. An expression of this class is always irreducible. For if this were not the case we should have f (x + 1) = (x m + a^- 1 + . . . + a_, x + a m ) where the a's and b's are integers. Since the term of zero degree in the above expression of f (z) is p, we have a m b m - = p. p being prime, one of the fac- tors of a m b m ' must be unity. Suppose, then, Equating the coefficients of the terms in x, we have 22 FAMOUS PROBLEMS. The first member and the second term of the second being divisible by p, a m _ib m - must be so also. Since b m - 1, a m _x is divisible by p. Equating the coefficients of the terms in x 2 we may show that a m _ 2 is divisible by p. Similarly we show that all of the remaining coefficients of the factor x m -f- a 1 x m ~ 1 + . . . + a m _i x -f- a m are divisible by p. But this cannot be true of the coefficient of x m , which is 1. The assumed equality is impossible and hence the cyclo- tomic equation is irreducible when p is a prime. 8. We now consider the case where n is a power of a prime number, say n = p a . We propose to show that when p > 2 the division of the circle into p 2 equal parts is impos- sible. The general problem will then be solved, since the division into p a equal parts evidently includes the division into p 2 equal parts. The cyclotomic equation is now z-1 It admits as roots extraneous to the problem those which come from the division into p. equal parts, i.e., the roots of the equation p _ .. Suppressing these roots by division we obtain as the cyclotomic equation. This may be written Z P(P-1) _|_ Z P(P~2) _|_ _j_ 2 p _J_ J __ Q Transforming by the substitution z = x + l, we have (x + 1)PCP-D + ( X + l)p<p-2) + . . . + ( X THE CYCLOTOMIC EQUATION. 23 The number of terms being p, the term independent of x after development will be equal to p, and the sum will take the form where ^ (x) is a polynomial with integral coefficients whose constant term is 1. We have just shown that such an expres- sion is always irreducible. Consequently the new cyclotomic equation is also irreducible. The degree of this equation is p (p 1). On the other hand an irreducible equation is solvable by square roots only when its degree is a power of 2. Hence a circle is divisible into p 2 equal parts only when p = 2, p being assumed to be a prime. The same is true, as already noted, for the division into p a equal parts when a > 2. CHAPTER IV. The Construction of the Regular Polygon of 17 Sides. 1. We have just seen that the division of the circle into equal parts by the straight edge and compasses is possible only for the prime numbers studied by Gauss. It will now be of interest to learn how the construction can actually be effected. The purpose of this chapter, then, will be to show in an elementary way how to inscribe in the circle the regular poly- gon of 17 sides. Since we possess as yet no method of construction based upon considerations purely geometrical, we must follow the path indicated by our general discussions. We consider, first of all, the roots of the cyclotomic equation x 16 +x ls +. . . + x 2 +x+l = 0, and construct geometrically the expression, formed of square roots, deduced from it. We know that the roots can be put into the transcendental form < = cos - + i sin j- (K = 1, 2, . . . 16) ; and if 2-n- . . . 2-n- = cos +i sin, that Geometrically, these roots are represented in the complex plane by the vertices, different from 1, of the regular polygon of 17 sides inscribed in a circle of radius 1, having the origin THE REGULAR POLYGON OF 17 SIDES. 25 as center. The selection of e x is arbitrary, but for the con- struction it is essential to indicate some e as the point of departure. Having fixed upon e^ the angle corresponding to K is K times the angle corresponding to e^ which completely determines C K . 2. The fundamental idea of the solution is the following : Forming a primitive root to the modulus 17 we may arrange the 16 roots of the equation in a cycle in a determinate order. As already stated, a number a is said to be a primitive root to the modulus 17 when the congruence a 8 = + 1 (mod. 17) has for least solution s = 17 1 = 16. The number 3 pos- sesses this property ; for we have 3 13 = 12 3 14 = 2 , 3 15 EEE 6 3 16 = 1J Let us then arrange the roots e K so that their indices are the preceding remainders in order e 8> *9> e 10j C 13) e 55 e !5> c ll> C 16j e !4? C 8) C 75 C 4) 12f 2? C 6 , Cj. Notice that if r is the remainder of 3* (mod. 17), we have 3*= 3 3 5 = 5 3 9 =14 02 Q O - i7 3 6 =15 3 10 EE 8 3 3 = 10 3 r =ll 3 U = 7 3 4 = 13 3 8 = 16 3^= 4 whence c r = e = If r r is the next remainder, we have similarly , __ , 8 K+1 - /, 3 K \3 - / \3 e r i ( e l ) (*r) Hence in this series of roots each root is the cube of the preceding. Gauss's method consists in decomposing this cycle into sums containing 8, 4, 2, 1 roots respectively, corresponding to the divisors of 16. Each of these sums is called a period. 26 FAMOUS PROBLEMS. The periods thus obtained may be calculated successively as roots of certain quadratic equations. The process just outlined is only a particular case of that employed in the general case of the division into p equal parts. The p 1 roots of the cyclotomic equation are cyclic- ally arranged by means of a primitive root of p, and the periods may be calculated as roots of certain auxiliary equa- tions. The degree of these last depends upon the prime fac- tors of p 1. They are not necessarily equations of the second degree. The general case has, of course, been treated in detail by Gauss in his Disquisitiones, and also by Bachmann in his work, Die Lehre von der Kreisteilung (Leipzig, 1872). 3. In our case of the 16 roots the periods may be formed in the following manner : Form two periods of 8 roots by taking in the cycle, first, the roots of even order, then those of odd order. Designate these periods by x x and x 2 , and replace each root by its index. We may then write symbol- ically x : = 9 + 13 + 15 + 16 + 8 + 4+ 2 + 1, x 2 = 3 + 10 + 5 + 11 + 14 + 7+12 + 6. Operating upon x x and x 2 in the same way, we form 4 periods of 4 terms : yi = 13 + 16+ 4+ 1, y 2 = 9 + 15+ 8+ 2, y 8 = 10 + ll+ 7+ 6, y 4 = 3+ 5 + 14 + 12. Operating in the same way upon the y's, we obtain 8 periods of 2 terms : 2 1 = 16+1, 2 E = H+ 6, z 2 = 13 + 4, 2 6 = 10+ 7, 2 S = 15 + 2, z 7 = 5 + 12, z 4 = 9+8, Zg = 3 + 14. THE EEGULAR POLYGON OF 17 SIDES. 27 It now remains to show that these periods can be calculated successively by the aid of square roots. 4. It is readily seen that the sum of the remainders corre- sponding to the roots forming a period z is always equal to 17. These roots are then e r and ei 7 _ r ; 2-7T 2-7T e r = cos r + i sin r , 2ir 2ir V = 17 _ r = cos (17 - r) ^ + i sm (17 r) , 2?r . . 2ir = cos r i sm r . Hence 27T e r + e r - = 2 COS r . Therefore all the periods z are real, and we readily obtain 2,r = 2 cos-, z s = 2 cos 4r 17' z 6 = 2 cos ' -2 2 2?r 17' z 7 = 2 cos 5 rr -2 8 27r cos 17 , z s = 2cos 2^ 17" Moreover, by definition, *i = zi + z 2 + z 3 + z 4 , x 2 = z 5 + z 6 + z 7 + z 8 , Yl = Zl + Z 25 Y2 = Z-3 + Z4> Y3 = Z5 + Z6> Y4 = Z7 + Z 8- 5. It will be necessary to determine the relative magnitude of the different periods. For this purpose we shall employ the following artifice : We divide the semicircle of unit radius into 17 equal parts and denote by Si, S 2 , S 17 the distances 28 FAMOUS PROBLEMS. of the consecutive points of division A t , A 2 , . A 17 from the initial point of the semicircle, S 17 being equal to the diam- eter, i.e., equal to 2. The angle A K Ai 7 has the same measure as the half of the arc A K 0, which equals 2/<7T TT . Hence 34 (17 K)T = 2c S 34 That this may be identical with 2 cos h , we must have 4h = 17 K, K = 17 4h. Giving to h the values 1, 2, 3, 4, 5, 6, 7, 8, we find for K the values 13, 9, 5, 1, 3, 7, 11, 15. Hence z i Sis, Z 2 = Si, Z S = S 9 , Z 4 ~ Z 5 = S 7 , Z 6 == Sn, Z 7 == S 3 , z 8 = S 5 . The figure shows that S K increases with the index ; hence the order of increasing magnitude of the periods z is Z 6> Z 8) Moreover, the chord A^A^+p subtends p divisions of the semi- circumference and is equal to S p ; the triangle OA K A K + p shows that and a fortiori Calculating the differences two and two of the periods y, we easily find THE REGULAR POLYGON OF 17 SIDES. 29 Yi y 2 = Sis + Si S 9 + S 15 > 0, yi y 3 = S 13 + $! + S 7 + S n > 0, yi y4 = Sis + Si + S 3 Si > 0, y 2 y s = S 9 Sis -f S 7 + S u > 0, y 2 y* = S 9 Sis + S 3 S 5 < 0, y s y* = S 7 Sn + S 3 S 5 < 0. Hence y 3 < y 2 < y* < yi- Finally we obtain in a similar way x 2 < x t . 2rr 6. We now propose to calculate z l = 2 cos . After mak- ing this calculation and constructing z 1? we can easily deduce the side of the regular polygon of 17 sides. In order to find the quadratic equation satisfied by the periods, we proceed to determine symmetric functions of the periods. Associating Zi with the period z 2 and thus forming the period y l5 we have, first, zi + z 2 = yi- Let us now determine z^. . We have where the symbolic product xp represents C K ' C P K + p' Hence it should be represented symbolically by K + p, remem- bering to subtract 17 from K -J- p as often as possible. Thus, z 1 z 2 = 12 + 3 + 14 + 5 = y 4 . Therefore Z! and z 2 are the roots of the quadratic equation (0 z 2 - yi z + y 4 = 0, whence, since z l > z 2 , _ yi yi 2 ~ 2 30 FAMOUS PROBLEMS. We must now determine y x and y 4 . Associating y x with the period y 2 , thus forming the period x 1? and y s with the period y 4 , thus forming the period x 2 , we have, first, Yi -f y 2 = x lt Then, yi y 2 =(13 + 16 + 4 + l) (9 + 15 + 8 + 2). Expanding symbolically, the second member becomes equal to the sum of all the roots ; that is, to 1. Therefore y A and y 2 are the roots of the equation (,) y-x iy -l = 0, whence, since y t > y 2 , Similarly, y and y s y 4 = - 1. Hence y 8 and y 4 are the roots of the equation (V) y 2 -x 2 y-l = 0; whence, since y 4 > y 8 , _x 2 + Vx 2 2 + 4 x 2 -Vx 2 2 Y4- ~ -, y* = - It now remains to determine X! and x 2 . Since x x + x 2 is equal to the sum of all the roots, Xl + X 2 = 1. Further, xix 2 = (13 + 16 + 4 + 1 + 9 + 15 +8 + 2) (10 + 11 + 7 + 6 + 3 + 5 + 14 + 12). Expanding symbolically, each root occurs 4 times, and thus = 4. THE REGULAR POLYGON OF 17 SIDES. 31 Therefore x t and x 2 are the roots of the quadratic () x 2 +x-4 = 0; whence, since x t > x 2 , _-l + Vl7 _ - 1 - Vl7 xi- 2 ' * 2 ~ 2 Solving equations , 17, i/, in succession, z x is determined by a series of square roots. Effecting the calculations, we see that z t depends upon the four square roots Vl7, V Xl 2 + 4, Vx 2 2 + 4, V yi ' 2 -4y 4 . If we wish to reduce z l to the normal form we must see whether any one of these square roots can be expressed rationally in terms of the others. Now, from the roots of (rj), Vx t 2 + 4 = y! y 2 , Vx 2 2 + 4 = y 4 y,. Expanding symbolically, we verify that (yi Ys) (Y4 Ys) = 2 (xj x a ),* * (yi - y 2 ) (y 4 - y) = (13 + 16 + 4 + 1 - 9 - 15 - 8 - 2) (3 + 6 + 14 + 12-10-11-7-6) = 16+ 1 + 10+ 8- 6- 7- 3- 2 + 2+ 4+13+11- 9-10- 6- 5 + 7 + 9 + 1 + 16-14-15-11-10 + 4+ 6 + 15+13-11-12- 8- 7 -12-14- 6- 4+ 2+ 3+16+15 - 1- 3-12-10+ 8+ 9+ 5+ 4 - 11 - 13 - 5 - 3 + 1 + 2 + 15 + 14 - 5 - 7 - 16 - 14 + 12 + 13 + 9 + 8 = 2(16 + 1 + 8 + 2 + 4 + 13 + 15+9-10-6-7-3-11-5-14 -12) = 2(Xi - x g ). 32 FAMOUS PROBLEMS. that is, _ _ Vx t 2 + 4 Vx 2 + 4 = 2 Vl7. Hence Vx 2 3 -f- 4 can be expressed rationally in terms of the other two square roots. This equation shows that if two of the three differences y,. y 2 , y 4 y 3 , *i *2 are positive, the same is true of the third, which agrees with the results ob- tained directly. Replacing now x x , y x , y* by their numerical values, we obtain in succession 1 4- Vl7 + \/34 - 2 Vl7 ~~~ - 1 - Vl7 + V34 + 2 V17 " 4 1 + Vl7 + V34 2 Vl7 Zi = 8 68 fl2Vl7 16V34+2V17 2(1 Vl7)V34 2V17 The algebraic part of the solution of our problem is now completed. We have already remarked that there is no known construction of the regular polygon of 17 sides based upon purely geometric considerations. There remains, then, only the geometric translation of the individual algebraic steps. 7. We may be allowed to introduce here a brief historical account of geometric constructions with straight edge and compasses. In the geometry of the ancients the straight edge and com- passes were always used together ; the difficulty lay merely in bringing together the different parts of the figure so as not to THE REGULAR POLYGON OF 17 SIDES. 33 draw any unnecessary lines. Whether the several steps in the construction were made with straight edge or with com- passes was a matter of indifference. On the contrary, in 1797, the Italian Mascheroni succeeded in effecting all these constructions with the compasses alone ; he set forth his methods in his Geometria del compasso, and claimed that constructions with compasses were practically more exact than those with the straight edge. As he ex- pressly stated, he wrote for mechanics, and therefore with a practical end in view. Mascheroni's original work is difficult to read, and we are under obligations to Hutt for furnishing a brief resume in German, Die Mascheroni' schen Constructionen (Halle, 1880). Soon after, the French, especially the disciples of Carnot, the author of the Geometrie de position, strove, on the other hand, to effect their constructions as far as possible with the straight edge. (See also Lambert, Freie Perspective, 1774.) Here we may ask a question which algebra enables us to answer immediately : In what cases can the solution of an algebraic problem be constructed with the straight edge alone? The answer is not given with sufficient explicitness by the authors mentioned. We shall say : With the straight edge alone we can construct all algebraic expressions ivhose form is rational. With a similar view Brianchon published in 1818 a paper, Les applications de la theorie des transver sales, in which he shows how his constructions can be effected in many cases with the straight edge alone. He likewise insists upon the practical value of his methods, which are especially adapted to field work in surveying. Poncelet was the first, in his Traite des proprietes projectives (Vol. I, Nos. 351-357), to conceive the idea that it is sufficient to use a single fixed circle in connection with the straight lines 34 FAMOUS PROBLEMS. of the plane in order to construct all expressions depending upon square roots, the center of the fixed circle being given. This thought was developed by Steiner in 1833 in a cele- brated paper entitled Die geometrischen Constructionen, ausge- fuhrt mittels der geraden Linie und eines festen Kreises, als Lehrgegenstand fur hohere Unterrichtsanstalten und zum Selbst- unterricht. 8. To construct the regular polygon of 17 sides we shall follow the method indicated by von Staudt (Crelle's Journal, Vol. XXIV, 1842), modified later by Schroter (Crelle's Jour- nal, Vol. LXXV, 1872). The construction of the regular polygon of 17 sides is made in accordance with the methods indicated by Poncelet and Steiner, inasmuch as besides the straight edge but one fixed circle is used.* First, we will show how with the straight edge and one fixed circle we can solve every quadratic equation. At the extremities of a diameter of the fixed unit circle (Fig. 4) we draw two tangents, and select the lower as the 4 axis of X, and the diameter perpendicular to it as the axis of Y. Then the equa- tion of the circle is Let y(y-2)=0. px + q= FIG. 4. be any quadratic equation with real roots x t and x 2 . Required to construct the roots /! and x 2 upon the axis of X. Lay off upon the upper tangent from A to the right, a seg- 4 ment measured by - ; upon the axis of X from 0, a segment * A Mascheroni construction of the regular polygon of 17 sides by L. Ge'rard is given in Math, Annalen, Vol. XL VIII, 1896, pp. 390-392. THE REGULAR POLYGON OF 17 SIDES. 35 measured by ; connect the extremities of these segments by P the line 3 and project the intersections of this line with the circle from A, by the lines 1 and 2, upon the axis of X. The segments thus cut off upon the axis of X are measured by X! and x 2 . Proof. Calling the intercepts upon the axis of X, X! and x 2 , we have the equation of the line 1, 2x + x 1 (y-2)=0; of the line 2, 2x + x 2 (y-2) = 0. If we multiply the first members of these two equations we get x 2 + ^- 2 x (y-2) + x -^ (y -2)* = as the equation of the line pair formed by 1 and 2. Subtract- ing from this the equation of the circle, we obtain x(y-2) + ^ (y-2)'-y(y-2) = 0. This is the equation of a conic passing through the four intersections of the lines 1 and 2 with the circle. From this equation we can remove the factor y 2, correspond- ing to the tangent, and we have left x + X -f (y-2)-y = 0, which is the equation of the line 3. If we now make *i H~ x a P an d X i x 2 = q> we get and the transversal 3 cuts off from the line y = 2 the seg- 36 FAMOUS PROBLEMS. Thus the merit - , and from the line y = the segment - correctness of the construction is established. 9. In accordance with the method just explained, we shall now construct the roots of our four quadratic equations. They are (see pp. 29-31) () x 2 + x 4 = 0, with roots Xx and x 2 ; x x > x 2 , 0?) y 2 xjy 1 =0, with roots yi and y 2 ; yi > y 2 , (V) y 2 X 2y 1 = 0, with roots y 3 and y 4 ; y 4 > y 3 , () z 2 y t z + y 4 = 0, with roots z and z 2 ; z > z 2 . These will furnish z: = 2cos^, whence it is easy to construct the polygon desired. We notice further that to construct z v it is sufficient to construct We then lay off the following segments : upon the upper tangent, y = 2, 444 ~ 4 ' 7' 7' 7 ; AI Ag jri upon the axis of X, This may all be done in the following manner : The straight line connecting the point -f- 4 upon the axis of X with the point 4 upon the tangent y = 2 cuts the circle in JL o FIG. 5. THE REGULAR POLYGON OF 17 SIDES. 37 two points, the projection of which from the point A (0, 2), the upper vertex of the circle, gives the two roots x x , x 2 of the first quadratic equation as intercepts upon the axis of X. 4 To solve the second equation we have to lay off above x i and -- below. *1 To determine the first point we connect K! upon the axis of X with A, the upper vertex, and from 0, the lower vertex, draw another straight line through the intersection of this line with the circle. This cuts off upon the upper tangent 4 the intercept . This can easily be shown analytically. x i The equation of the line from A to ^ (Fig. 5), 2x + Xl y = 2 Xl , and that of the circle, x 2 + y(y-2) = 0, give as the coordinates of their intersection 4 Xl 2 Xl 2 The equation of the line from through this point becomes 4 cutting off upon y == 2 the intercept . x i We reach the same conclusion still more simply by the use of some elementary notions of projective geometry. By our construction we have obviously associated with every point x of the lower range one, and only one, point of the upper, so that to the point x = oo corresponds the point x' = 0, and con- versely. Since in such a correspondence there must exist a 38 FAMOUS PROBLEMS. linear relation, the abscissa x' of the upper point must satisfy the equation f const. x X Since x' = 2 when x = 2, as is obvious from the figure, the constant = 4. __L o +1 y< FIG. 6. To determine upon the axis of X we connect the point x i 4 upon the upper with the point + 1 upon the lower tan- gent (Fig. 6). The point thus determined upon the vertical 4 diameter we connect with the point above. This line cuts off upon the axis of X the intercept . For the x i line from 4 to + 1, intersects the vertical diameter in the point (0, ). Hence the equation of the line from to this point is *i 5y 2x lX = 2, and its intersection with the lower tangent gives . x i The projection from A of the intersections of the line from 1 4 to with the circle determines upon the axis of X the x i *i two roots of the second quadratic equation, of which, as THE REGULAR POLYGON OF 17 SIDES. 39 already noted, we need only the greater, y x . This corres- ponds, as shown by the figure, to the projection of the upper intersection of our transversal with the circle. Similarly, we obtain the roots of the third quadratic equa- tion. Upon the upper tangent we project from the inter- section of the circle with the straight line which gave upon the axis of X the root + x 2 . This immediately gives the 4 intercept , by reason of the correspondence just explained. -4 FIG. 7. If we connect this point with the point where the vertical diameter intersects the line joining 4 above and -f- 1 below, we cut off upon the axis of X the segment , as desired. x 2 If we project that intersection of this transversal with the circle which lies in the positive quadrant from A upon the axis of X, we have constructed the required root y 4 of the third quadratic equation. We have finally to determine the root z x of the fourth quad- 4 y ratic equation and for this purpose to lay off above and - below. We solve the first problem in the usual way, by pro- jecting the intersection of the circle with the line connecting A with + y x below, from upon the upper tangent, thus 4 obtaining . For the other segment we connect the point -f- 4 above with y 4 below, and then the point thus determined 40 FAMOUS PROBLEMS. 4. upon the vertical diameter produced with . This line cuts y . off upon the axis of X exactly the segment desired, . For the line a (Fig. 8) has the equation Z, 0, FIG. 8. It cuts off upon the vertical diameter the segment The equation of the line b is then _T7' and its intersection with the axis of X has the abscissa . yi If we project the upper intersection of the line b with the 2ir circle from A upon the axis of X, we obtain z l = 2 cos . If we desire the simple cosine itself we have only to draw a diameter parallel to the axis of X, on which our last projecting o ray cuts off directly cos . A perpendicular erected at this point gives immediately the first and sixteenth vertices of the regular polygon of 17 sides. The period Zj was chosen arbitrarily ; we might construct in the same way every other period of two terms and so find the remaining cosines. These constructions, made on separate figures so as to be followed more easily, have been combined in a single figure (Fig. 9), which gives the complete construc- tion of the regular polygon of 17 sides. THE REGULAR POLYGON OF 17 SIDES. 41 FIG. 9. CHAPTER V. General Considerations on Algebraic Constructions. 1 . We shall now lay aside the matter of construction with straight edge and compasses. Before quitting the subject we may mention a new and very simple method of effecting cer- tain constructions, paper foldiny. Hermann Wiener has shown how by paper folding we may obtain the network of the regular polyhedra. Singularly, about the same time a Hindu mathematician, Sundara Kow, of Madras, published a little book, Geometrical Exercises in Paper Foldimj* (Madras, Addison & Co., 1893), in which the same idea is consider- ably developed. The author shows how by paper folding we may construct by points such curves as the ellipse, cissoid, etc. 2. Let us now inquire how to solve geometrically prob- lems whose analytic form is an equation of the third or of higher degree, and in particular, let us see how the ancients succeeded. The most natural method is by means of the conies, of which the ancients made much use. For example, they found that by means of these curves they were enabled to solve the problems of the duplication of the cube and the trisection of the angle. We shall in this place give only a general sketch of the process, making use of the language of modern mathematics for greater simplicity. Let it be required, for instance, to solve graphically the cubic equation x 8 + ax 2 + bx + c = 0, or the biquadratic, x 4 +ax 3 + * See American edition, revised by Beman and Smith, The Open Court Publishing Co., Chicago, 1901. ALGEBRAIC CONSTRUCTIONS. Put x 2 = y ; our equations become 43 and xy + ay + bx + c = y 2 + axy + by + ex + d = 0. The roots of the equations proposed are thus the abscissas of the points of intersection of the two conies. The equation x 2 = y represents a parabola with axis vertical. The second equa- tion, xy + ay + bx + c = 0, represents an hyperbola whose asymptotes are parallel to the axes of reference (Fig. 10). One of the four points of inter- FIG. 10. FIG. 11. section is at infinity upon the axis of Y, the other three at a finite distance, and their abscissas are the roots of the equa- tion of the third degree. In the second case the parabola is the same. The hyper- bola (Fig. 11) has again one asymptote parallel to the axis of X while the other is no longer perpendicular to this axis. The curves now have four points of intersection at a finite distance. 44 FAMOUS PROBLEMS. The methods of the ancient mathematicians are given in detail in the elaborate work of M. Cantor, Geschichte der Mathematik (Leipzig, 1894, 2d ed.). Especially interesting is Zeuthen, Die Kegelschnitte im Altertum (Kopenhagen, 1886, in German edition). As a general compendium we may men- tion Baltzer, Analytische Geometrie (Leipzig, 1882). 3. Beside the conies, the ancients used for the solution of the above-mentioned problems, higher curves constructed for this very pur- pose. We shall mention here only the Cissoid and the Conchoid. The cissoid of Diodes (c. 150 B.C.) may be constructed as follows (Fig. 12) : To a circle draw a tangent (in the figure the vertical tangent on the right) and the diameter perpendicular to it. Draw lines from 0, the vertex of the circle thus determined, to points upon the tangent, and lay off from upon each the segment lying between its intersection with the circle and the tangent. The locus of points so deter- mined is the cissoid. To derive the equation, let r be the FIG. 12. radius vector, the angle it makes with the axis of X. If we produce r to the tangent on the right, and call the diameter of the circle 1, the total segment equals cos 0' circle is cos 0. The difference of the two segments is hence The portion cut off by the r, and r = cos Q cos = sin 2 6 cos 0' ALGEBRAIC CONSTRUCTIONS. 45 By transformation of coordinates we obtain the Cartesian equation, (x 2 -f y 2 ) x y 2 = 0. The curve is of the third order, lias a cusp at the origin, and is symmetric to the axis of X. The vertical tangent to the circle with which we began our construction is an asymp- tote. Finally the cissoid cuts the line at infinity in the cir- cular points. To show how to solve the Delian problem by the use of this curve, we write its equation in the following form : We now construct the straight line, This cuts off upon the tangent x = 1 the segment A, and intersects the cissoid in a point for which y _ A 3 l-x~ This is the equation of a straight line passing through the point y = 0, x = 1, and hence of the line joining this point to the point of the cissoid. This line cuts off upon the axis of Y the intercept \ 3 . We now see how ^2 may be constructed. Lay off upon the axis of Y the intercept 2, join this point to the point x = 1, y = 0, and through its intersection with the cissoid draw a line from the origin to the tangent x = 1. The inter- cept on this tangent equals ^2. 4. The conchoid of Nicomedes (c. 150 B.C.) is constructed as follows : Let be a fixed point, a its distance from a fixed 46 FAMOUS PROBLEMS. line. If we pass a pencil of rays through and lay off on each ray from its intersection with the fixed line in both directions a segment b, the locus of the points so determined is the conchoid. According as b is greater or less than a, the origin is a node or a con- jugate point ; for b = a it is a cusp (Fig. 13). Taking for axes of X and Y the perpendicular and paral- lel through to the fix-<i line, we have whence The conchoid is then of the fourth order, has a double point at the origin, and is composed of two branches having for common asymptote the line x = a. Further, the factor (x 2 + y 2 ) shows that the curve passes through the cir- cular points at infinity, a mat- ter of immediate importance. We may trisect any angle by means of this curve in the following manner : Let <= MOY (Fig. 13) be the angle to be divided into three equal parts. On the side OM lay off OM = b, an arbitrary length. With M as a center and radius b describe a circle, and through M perpendicular to the axis of X with origin draw a vertical line representing the asymptote of the conchoid to be constructed. Construct the FIG. 13. ALGEBRAIC CONSTRUCTIONS. 47 conchoid. Connect with A, the intersection of the circle and the conchoid. Then is /_ AOY one third of Z. <f>, as is easily seen from the figure. Our previous investigations have shown us that the prob- lem of the trisection of the angle is a problem of the third degree. It admits the three solutions Every algebraic construction which solves this problem by the aid of a curve of higher degree must obviously furnish all the solutions. Otherwise the equation of the problem would not be irreducible. These different solutions are shown in the figure. The circle and the conchoid intersect in eight points. Two of them coincide with the origin, two others with the circular points at infinity. None of these can give a solution of the problem. There remain, then, four points of intersection, so that we seem to have one too many. This is due to the fact that among the four points we necessarily find the point B such that M B = 2 b, a point which may be determined without the aid of the curve. There actually remain then only three points corresponding to the three roots furnished by the algebraic solution. 5. In all these constructions with the aid of higher alge- braic curves, we must consider the practical execution. We need an instrument which shall trace the curve by a con- tinuous movement, for a construction by points is simply a method of approximation. Several instruments of this sort have been constructed ; some were known to the ancients. Nicomedes invented a simple device for tracing the conchoid. It is the oldest of the kind besides the straight edge and compasses. (Cantor, I, p. 302.) A list of instruments of more recent construction may be found in Dyck's Katalog, pp. 227-230, 340, and Nachtrag, pp. 42, 43. PART II. TRANSCENDENTAL NUMBERS AND THE QUADRATURE OF THE CIRCLE. CHAPTEK I. Cantor's Demonstration of the Existence of Transcendental Numbers. 1. Let us represent numbers as usual by points upon the axis of abscissas. If we restrict ourselves to rational numbers the corresponding points will fill the axis of abscissas densely throughout (iiberall dicht), i.e., in any interval no matter how small there is an infinite number of such points. Neverthe- less, as the ancients had already discovered, the continuum of points upon the axis is not exhausted in this way ; between the rational numbers come in the irrational numbers, and the question arises whether there are not distinctions to be made among the irrational numbers. Let us define first what we mean by algebraic numbers. Every root of an algebraic equation a co n + a^ 11 - 1 + + a n _ lW + a n = with integral coefficients is called an algebraic number. Of course we consider only the real roots. Rational numbers occur as a special case in equations of the form a w -f- a t = 0. 50 FAMOUS PROBLEMS. We now ask the question : Does the totality of real algebraic numbers form a continuum, or a discrete series such that other numbers may be inserted in the intervals? These new numbers, the so-called transcendental numbers, would then be characterized by this property, that they cannot be roots of an algebraic equation with integral coefficients. This question was answered first by Liouville (Comptes rendus, 1844, and Liouville's Journal, Vol. XVI, 1851), and in fact the existence of transcendental numbers was demon- strated by him. But his demonstration, which rests upon the theory of continued fractions, is rather complicated. The investigation is notably simplified by using the developments given by Georg Cantor in a memoir of fundamental impor- tance, Ueber eine Eigenschaft des Inbegriffes reeller algebra- ischer Zahlen (Crelle's Journal, Vol. LXXVII, 1873). We shall give his demonstration, making use of a more simple notion which Cantor, under a different form, it is true, sug- gested at the meeting of naturalists in Halle, 1891. 2. The demonstration rests upon the fact that algebraic numbers form a countable mass, while transcendental numbers do not. By this Cantor means that the former can be arranged in a certain order so that each of them occupies a definite place, is numbered, so to speak. This proposition may be stated as follows : The manifoldness of real algebraic numbers and the mani- foldness of positive integers can be brought into a one-to-one correspondence. We seem here to meet a contradiction. The positive inte- gers form only a portion of the algebraic numbers ; since each number of the first can be associated with one and one only of the second, the part would be equal to the whole. This objection rests upon a false analogy. The proposition that the part is always less than the whole is not true for TRANSCENDENTAL NUMBERS. 51 infinite masses. It is evident, for example, that we may establish a one-to-one correspondence between the aggregate of positive integers and the aggregate of positive even num- bers, thus : 1 2 3 n 2 4 6 2n In dealing with infinite masses, the words great and small are inappropriate. As a substitute, Cantor has introduced the word power (Mdchtigkeit), and says : Two infinite masses have the same power when they can be brought into a one-to-one cor- respondence with each other. The theorem which we have to prove then takes the following form : The aggregrate of real algebraic numbers has the same power as the aggregate of positive integers. We obtain the aggregate of real algebraic numbers by seek- ing the real roots of all algebraic equations of the form a o) n + a^"- 1 + + a n _ lW -f a n = ; all the a's are supposed prime to one another, a positive, and the equation irreducible. To arrange the numbers thus obtained in a definite order, we consider their height N as defined by (aj representing the absolute value of a^ as usual. To a given number N corresponds a finite number of algebraic equations. For, N being given, the number n has certainly an upper limit, since N is equal to n 1 increased by positive numbers ; moreover, the difference N (n 1) is a sum of positive numbers prime to one another, whose number is obviously finite. 52 FAMOUS PROBLEMS. N n la,! tail Kl I..I w EQUATION. *<N) BOOTS. 1 1 1 x = 1 2 2 1 2 2 -1 1 1 zl=0 + 1 2 1 3 1 3 4 2 2 1 2zl = 1 2 1 1 2 z2=0 2 2 2 + 2 1 1 1 1 3 1 4 1 4 12 -3 3 1 3x 1 =0 - 1.61803 2 2 - - 1.41421 1 3 x3 = 0.70711 2 3 - 0.61803 2 1 -0.33333 2 1 2x 2 1 = + 0.33333 1 2 + 0.61803 1 1 1 x 2 x - 1 = + 0.70711 1 2 X 2 - 2 = + 1.41421 3 2 + 1.61803 1 1 + 3 1 1 1 1 4 1 Among these equations we must discard those that are reducible, which presents no theoretical difficulty. Since the number of equations corresponding to a given value of TRANSCENDENTAL NUMBERS. 53 N is limited, there corresponds to a determinate N only a finite mass of algebraic numbers. We shall designate this by <(N). The table contains the values of <(!), <(2), <(3), <(4), and the corresponding algebraic numbers w. We arrange now the algebraic numbers according to their height, N, and the numbers corresponding to a single value of N in increasing magnitude. We thus obtain all the algebraic numbers, each in a determinate place. This is done in the last column of the accompanying table. It is, therefore, evident that algebraic numbers can be counted. 3. We now state the general proposition : In any portion of the axis of abscissas, however small, there is an infinite number of points which certainly do not belong to a given countable mass. Or, in other words : The continuum of numerical values represented by a portion of the axis of abscissas, however small, has a greater power than any given countable mass. This amounts to affirming the existence of transcendental numbers. It is sufficient to take as the countable mass the aggregate of algebraic numbers. To demonstrate this theorem we prepare a table of algebraic numbers as before and write in it all the numbers in the form of decimal fractions. None of these will end in an infinite series of 9's. For the equality 1 = 0.999- -9- shows that such a number is an exact decimal. If now we can construct a decimal fraction which is not found in our table and does not end in an infinite series of 9's it will certainly be a transcendental number. By means of a very simple process indicated by Georg Cantor we can find not only one but infinitely many transcendental numbers, even 54 FAMOUS PROBLEMS. when the domain in which the number is to lie is very small. Suppose, for example, that the first five decimals of the num- ber are given. Cantor's process is as follows. Take for 6th decimal a number different from 9 and from the 6th decimal of the first algebraic number, for 7th decimal a number different from 9 and from the 7th decimal of the second algebraic number, etc. In this way we obtain a decimal fraction which will not end in an infinite series of 9's and is certainly not contained in our table. The proposition is then demonstrated. We see by this that (if the expression is allowable) there are far more transcendental numbers than algebraic. For when we determine the unknown decimals, avoiding the 9's, we have a choice among eight different numbers ; we can thus form, so to speak, 8" transcendental numbers, even when the domain in which they are to lie is as small as we please. CHAPTER II. Historical Survey of the Attempts at the Computation and Construction of ir. In the next chapter we shall prove that the number TT belongs to the class of transcendental numbers whose exis- tence was shown in the preceding chapter. The proof was first given by Lindemann in 1882, and thus a problem was definitely settled which, so far as our knowledge goes, has occupied the attention of mathematicians for nearly 4000 years, the problem of the quadrature of the circle. For, if the number TT is not algebraic, it certainly cannot be constructed by means of straight edge and compasses. The quadrature of the circle in the sense understood by the ancients is then impossible. It is extremely interesting to follow the fortunes of this problem in the various epochs of science, as ever new attempts were made to find a solution with straight edge and compasses, and to see how these neces- sarily fruitless efforts worked for advancement in the mani- fold realm of mathematics. The following brief historical survey is based upon the excellent work of Rudio : Archimedes, Huygens, Lambert, Legendre, Vier Abhandlungen iiber die Kreismessung, Leipzig, 1892. This book contains a German translation of the investigations of the authors named. While the mode of presentation does not touch upon the modern methods here discussed, the book includes many interesting details which are of practical value in elementary teaching. 56 FAMOUS PROBLEMS. 1. Among the attempts to determine the ratio of the diameter to the circumference we may first distinguish the empirical stage, in which the desired end was to be attained by measurement or by direct estimation. The oldest known mathematical document, the Rhind Papyrus (c. 2000 B.C.), contains the problem in the well- known form, to transform a circle into a square of equal area. The writer of the papyrus, Ahmes, lays down the following rule : Cut off of a diameter and construct a square upon the remainder ; this has the same area as the circle. The value of TT thus obtained is (^) 2 = 3.16 , not very inaccurate. Much less accurate is the value TT = 3, used in the Bible (1 Kings, 7. 23, 2 Chronicles, 4. 2). 2. The Greeks rose above this empirical standpoint, and especially Archimedes, who, in his work nwXov /ner/aT/o-is, com- puted the area of the circle by the aid of inscribed and cir- cumscribed polygons, as is still done in the schools. His method remained in use till the invention of the differential calculus ; it was especially developed and rendered practical by Huygens (d. 1654) in his work, De circuit magnitudine inventa. As in the case of the duplication of the cube and the fcrisection of the angle the Greeks sought also to effect the quadrature of the circle by the help of higher curves. Consider for example the curve y = sin 1 x, which repre- sents the sinusoid with axis vertical. Geometrically, TT appears as a particular ordinate of this curve ; from the standpoint of the theory of functions, as a particular value of our transcendental function. Any apparatus which describes a transcendental curve we shall call a transcendental appara- tus. A transcendental apparatus which traces the sinusoid gives us a geometric construction of TT. In modern language the curve y = sin~ 1 x is called an THE CONSTRUCTION OF it. 57 integral curve because it can be defined by means of the integral of an algebraic function, dx /dx Vl-x 2 The ancients called such a curve a quadratrix or ovcra. The best known is the quadratrix of Dinostratus (c. 350 B.C.) which, however, had al- ready been constructed by Hippias of Elis (c. 420 B.C.) for the trisection of an angle. Geometrically it may be denned as follows. Having given a circle and two perpendicular radii OA and OB, two points M and L move with constant velocity, one upon the radius OB, the other upon the arc AB (Fig. 14). Starting at the same time at and A, they arrive simultaneously at B. The point of inter- section P of OL and the parallel to OA through M describes the quadratrix. From this definition it follows that y is proportional to 0. 7T Further, since for y = 1, B = we have =fy; and from = tan~ x - the equation of the curve becomes x y TT - FIG. 14. It meets the axis of X at the point whose abscissa is y x = lim - - , for y = ; 7T tan-y 58 hence FAMOUS PROBLEMS. 2 x = 7T According to this formula the radius of the circle is the mean proportional between the length of the quadrant and the abscissa of the intersection of the quadratrix with the axis of X. This curve can therefore be used for the rectifica- tion and hence also for the quadrature of the circle. This use of the quadratrix amounts, however, simply to a geo- metric formulation of the problem of rectification so long as we have no apparatus for describing the curve by continuous movement. Fig. 15 gives an idea of the form of the curve with the branches obtained by taking values of 6 greater than TT or FIG. 15. less than IT. Evidently the quadratrix of Dinostratus is not so convenient as the curve y = sin" 1 x, but it does not appear that the latter was used by the ancients. 3. The period from 1670 to 1770, characterized by the names of Leibnitz, Newton, and Euler, saw the rise of modern analysis. Great discoveries followed one another in such an almost unbroken series that, as was natural, critical rigor fell into the background. For our purposes the development THE COMPUTATION OF if. 59 of the theory of series is especially important. Numerous methods were deduced for approximating the value of TT. It will suffice to mention the so-called Leibnitz series (known, however, before Leibnitz) : This same period brings the discovery of the mutual depend- ence of e and TT. The number e, natural logarithms, and hence the exponential function, are first found in principle in the works of Napier (1614). This number seemed at first to have no relation whatever to the circular functions and the number TT until Euler had the courage to make use of imagi- nary exponents. In this way he arrived at the celebrated formula e^ = cos x + i sin x, which, for x = TT, becomes e i7r = 1. This formula is certainly one of the most remarkable in all mathematics. The modern proofs of the transcendence of TT are all based on it, since the first step is always to show the transcendence of e. 4. After 1770 critical rigor gradually began to resume its rightful place. In this year appeared the work of Lambert : Vorlaufige Kenntnisse fur die so die Quadratur des Cirkuls suchen. Among other matters the irrationality of TT is dis- cussed. In 1794 Legendre, in his Elements de geometrie, showed conclusively that TT and 7r 2 are irrational numbers. 5. But a whole century elapsed before the question was investigated from the modern point of view. The starting- point was the work of Hermite : Sur la fonction exponentielle (Comptes rendus, 1873, published separately in 1874). The transcendence of e is here proved. 60 FAMOUS PROBLEMS. An analogous proof for TT, closely related to that of Hermite, was given by Lindemann : Ueber die Zahl TT (Mathematische Annalen, XX, 1882. See also the Proceed- ings of the Berlin and Paris academies). The question was then settled for the first time, but the investigations of Hermite and Lindemann were still very complicated. The first simplification was given by Weierstrass in the Berliner Berichte of 1885. The works previously mentioned were embodied by Bachmann in his text-book, Vorlesungen iiber die Natur der Irrationalzahlen, 1892. But the spring of 1893 brought new and very important simplifications. In the first rank should be named the memoirs of Hilbert in the Gottinger Nachrichten. Still Hilbert's proof is not absolutely elementary : there remain traces of Hermite's reasoning in the use of the integral But Hurwitz and Gordan soon showed that this transcen- dental formula could be done away with (Gottinger Nach- richten ; Comptes rendus ; all three papers are reproduced with some extensions in Mathematische Annalen, Vol. XLIII). The demonstration has now taken a form so elementary that it seems generally available. In substance we shall follow Gordan's mode of treatment. CHAPTER III. The Transcendence of the Number e. 1. We take as the starting-point for our investigation the well-known series which is convergent for all finite values of x. The difference between practical and theoretical convergence should here be insisted on. Thus, for x = 1000 the calculation of e 1000 by means of this series would obviously not be feasible. Still the series certainly converges theoretically ; for we easily see that after the 1000th term the factorial n ! in the denominator increases more rapidly than the power which x n occurs in the numerator. This circumstance that - has for n! any finite value of x the limit zero when n becomes infinite has an important bearing upon our later demonstrations. We now propose to establish the following proposition : The number e is not an algebraic number, i.e.,sui equation with integral coefficients of the form is impossible. The coefficients Q may be supposed prime to one another. We shall use the indirect method of demonstration, show- ing that the assumption of the above equation leads to an absurdity. The absurdity may be shown in the following 62 FAMOUS PROBLEMS. way. We multiply the members of the equation F(e) = by a certain integer M so that MF(e)=MC +MC ie +MC 2 e 2 + - -+MC n e n = 0. We shall show that the number M can be chosen so that (1) Each of the products Me, Me 2 , Me n may be sepa- rated into an entire part M K and a fractional part C K , and our equation takes the form M F(e) = MC + MA + M 2 C 2 + + M n C n + dei + C 2 2 + -f C n c n = ; (2) The integral part MCO+MA + - -+M n c n is not zero. This will result from the fact that when divided by a prime number it gives a remainder different from zero ; (3) The expression ClC! + C 2 C 2 + ' ' ' + C n C n can be made as small a fraction as we please. These conditions being fulfilled, the equation assumed is manifestly impossible, since the sum of an integer different from zero, and a proper fraction, cannot equal zero. The salient point of the proof may be stated, though not quite accurately, as follows : With an exceedingly small error we may assume e, e 2 , ' ' e n proportional to integers which certainly do not satisfy our assumed equation. 2. We shall make use in our proof of a symbol h r and a certain polynomial <(x). The symbol h r is simply another notation for the factorial r I Thus, we shall write the series for e x in the form TRANSCENDENCE OF THE NUMBER e. 63 The symbol has no deeper meaning ; it simply enables us to write in more compact form every formula containing powers and factorials. Suppose, e.g., we have given a developed polynomial We represent by f(h), and write under the form c r h r , the sum Cl -l + c a -2! + c 8 -3!+- - + c n -n! But if f (x) is not developed, then to calculate f (h) is to develop this polynomial in powers of h and finally replace h r by r !. Thus, for example, the c' r depending on k. The polynomial <(x) which we need for our proof is the following remarkable expression .[(1 x)(2 x)- -(n-x)p *( X ) = XP ~ (p-l)l where p is a prime number, n the degree of the algebraic equation assumed to be satisfied by e. We shall suppose p greater than n and |C |, and later we shall make it increase without limit. To get a geometric picture of this polynomial <f> (x) we con- struct the curve y = <M. At the points x = 1, 2, n the curve has the axis of X as an inflexional tangent, since it meets it in an odd number of points, while at the origin the axis of X is tangent without inflexion. For values of x between and n the curve remains in the neighborhood of the axis of X ; for greater values of x it recedes indefinitely. 64 FAMOUS PROBLEMS. Of the function <j> (x) we will now establish three important properties : 1. x being supposed given and p increasing without limit, <(x) tends toward zero, as does also the sum of the absolute values of its terms. Put u = x (1 x) (2 x) (n x) ; we may then write which for p infinite tends toward zero. To have the sum of the absolute values of < (x) it is suffi- cient to replace x by |x| in the undeveloped form of <(x). The second part is then demonstrated like the first. 2. h being an integer, <f> (h) is an integer not divisible by p and therefore different from zero. Develop <(x) in increasing powers of x, noticing that the terms of lowest and highest degree respectively are of degree p 1 and np + p 1. We have r=np+p 1 ,,,_V c'x"-' , c"x" , HP+P-I = 2, v "5=iji+gp55i + *fr=3)i' r=p 1 Hence r=np+p 1 *(h) = 2 c r h'. r=p-l Leaving out of account the denominator (p 1) !, which occurs in all the terms, the coefficients c r are integers. This denominator disappears as soon as we replace h r by r!, since the factorial of least degree is h 1 *" 1 = (p 1) !. All the terms of the development after the first will contain the factor p. As to the first, it may be written (1-2-8... n)p.(p-l)! (p-1)! and is certainly not divisible by p since p > n. Therefore < (h) = (n !) (mod. p), and hence h0, TRANSCENDENCE OF THE NUMBER e. 65 Moreover, < (h) is a very large number ; even its last term alone is very large, viz.: 3. h being an integer, and k one of the numbers 1, 2 < (h -f- k) is an integer divisible by p. We have <(h + k) = 2c r (h + k) r =^c' r h r , a formula in which we are to replace h r by r ! only after hav- ing arranged the development in increasing powers of h. According to the rules of the symbolic calculus, we have first _ - (p-1)! One of the factors in the brackets reduces to h ; hence the term of lowest degree in h in the development is of degree p. We may then write r=np+p 1 The coefficients still have for numerators integers and for denominator (p 1)!. As already explained, this denomi- nator disappears when we replace h r by r ! . But now all the terms of the development are divisible by p; for the first may be written ( l) k P kP- 1 [(k 1) ! (n k) !]" p ! (p-1)! = ( l) kp k^ 1 [(k - 1) ! (n k) !]" p. < (h -f- k) is then divisible by p. 3. We can now show that the equation F(e)=C +C 1 e + C 2 e 2 +- +C n e" = is impossible. 66 FAMOUS PROBLEMS. For the number M, by which we multiply the members of this equation, we select < (h), so that * (h) F (e) = C < (h) + d4> (h) e + C 2 < (h) e 2 + + C D <t> (h)e n . Let us try to decompose any term, such as C k < (h) e k , into an integer and a fraction. We have e* 4 (h) = e k c r h r . r Considering the series development of e k , any term of this sum, omitting the constant coefficient, has the form h r k h r k 2 h r k r h r k r+1 Replacing h r by r !, or what amounts to the same thing, by one of the quantities rh- 1 , r(r l)h*-- -, r(r !) 3 h 2 , r(r-l)- 2 h, and simplifying the successive fractions, h r = , ~i J The first line has the same form as the development of (h + k) r 5 in the parenthesis of the second line we have the series k k 2 f 7+T + (r + l)(r + 2)~ l whose terms are respectively less than those of the series .. The second line in the expansion of e k h r may therefore be represented by an expression of the form q r , k ' e k k r , q r>k being a proper fraction. TRANSCENDENCE OF THE NUMBER e. 67 Effecting the same decomposition for each term of the sum e k 2 c r h r r it takes the form e k 2 c r h r = c r (h + k) r + e* q r , k c r k r . r r r The first part of this sum is simply <j> (h -|- k) ; this is a number divisible by p (2, 3). Further (2, 1), tends toward zero when p becomes infinite : the same is true a fortiori of q r , k c r k r , and also, since e k is a finite quantity, r of e k q r?k c r k r , which we may represent by e k . r The term under consideration, C k e k < (h), has then been put under the form of an integer C k <(h + k) and a quantity C k e k which, by a suitable choice of p, may be made as small as we please. Proceeding similarly with all the terms, we get finally + C n e n . It is now easy to complete the demonstration. All the terms of the first line after the first are divisible by p ; for the first, | C 1 is less than p ; < (h) is not divisible by p; hence C <(h) is not divisible by the prime number p. Consequently the sum of the numbers of the first line is not zero. The numbers of the second line are finite in number ; each of them can be made smaller than any given number by a suitable choice of p ; and therefore the same is true of their sum. Since an integer not zero and a fraction cannot have zero for a sum, the assumed equation is impossible. Thus, the transcendence of e, or Hermite's Theorem, is demonstrated. CHAPTER IV. The Transcendence of the Number it. 1. The demonstration of the transcendence of the number TT given by Lindemann is an extension of Hermite's proof in the case of e. While Hermite shows that an integral equa- tion of the form C + C 1 e + C 2 e 1 '+- + C n e" = cannot exist, Lindemann generalizes this by introducing in place of the powers e, e 2 sums of the form 1 + e 2 + ' ' ' + e v e where the k's are associated algebraic numbers, i.e., roots of an algebraic equation, with integral coefficients, of the degree N ; the I's roots of an equation of degree N', etc. Moreover, some or all of these roots may be imaginary. Lindemann's general theorem may be stated as follows : The number e cannot satisfy an equation of the form (1) C +C 1 (e kl + e k2 +- - + e k ") + C,(e !l + e 2 + - + e v )+- - = where the coefficients C t are integers and the exponents k u li, * ' are respectively associated algebraic numbers. The theorem may also be stated : The number e is not only not an algebraic number and there- fore a transcendental number simply, but it is also not an interscendental * number and therefore a transcendental number of higher order. * Leibnitz calls a function x\ where X is an algebraic irrational, an interscendental function. TRANSCENDENCE OF THE NUMBER it. 69 Let ax" + a^"- 1 + + a N = be the equation having for roots the exponents k } ; bx N ' + biX"'- 1 + - + b N , = that having for roots the exponents Ij, etc. These equations are not necessarily irreducible, nor the coefficients of the first terms equal to 1. It follows that the symmetric functions of the roots which alone occur in our later developments need not be integers. In order to obtain integral numbers it will be sufficient to consider symmetric functions of the quantities aki, ak 2 , ak N , bl b b! 2 , bl N ., etc. These numbers are roots of the equations y" + atf*- 1 + a 2 ay N - 2 + + a H a*" 1 = 0, Y N/ + biy"'- 1 + b 2 by N '- 2 + + b^b 1 *'" 1 = 0, etc. These quantities are integral associated algebraic numbers, and their rational symmetric functions real integers. We shall now follow the same course as in the demonstra- tion of Hermite's theorem. We assume equation (1) to be true ; we multiply both members by an integer M ; and we decompose each sum, such as M(e kl + e k2 +- - + e k >0, into an integral part and a fraction, thus Our equation then becomes C M + C 1 M 1 + C 2 M 2 + - + C 2 e 2 + ' = 0. 70 FAMOUS PROBLEMS. We shall show that with a suitable choice of M the sum of the quantities in the first line represents an integer not divisible by a certain prime number p, and consequently different from zero ; that the fractional part can be made as small as we please, and thus we come upon the same contra- diction as before. 2. We shall again use the symbol h r =r! and select as the multiplier the quantity M =^(h), where ^(x) is a gene- ralization of <(x) used in the preceding chapter, formed as follows : [(li - x) (I, - x) (I N - - x)] b* b*'* - b""" where p is a prime number greater than the absolute value of each of the numbers C , a, b, , a K , b w ,, and later will be assumed to increase without limit. As to the factors a* p , b N ' p , , they have been introduced so as to have in the development of i/r(x) symmetric functions of the quantities ak!, ak 2 , -, ak K , HI, b! 2 , , bl,,, that is, rational integral numbers. Later on we shall have to develop the expressions The presence of these same factors will still be necessary if we wish the coefficients of these developments to be integers each divided by (p 1) !. 1. </f(h) is an integral number, not divisible by p and con- sequently different from zero. TRANSCENDENCE OF THE NUMBER it. 71 Arranging \l/ (h) in increasing powers of h, it takes the form r=np+Jt'p+ =2 C r h'. r=p-l In this development all the coefficients have integral numer- ators and the common denominator (p 1) !. The coefficient of the first term h"" 1 may be written p-y (aki ak 2 ak > .) p a N ' p a s " p (bli b! 2 bl x ,) p b Np b*" p - ( l) lf P +N 'p + '"(a N a N - 1 ) p a I '' p a N ' p - (b N ,b N '- 1 ) p b Np b N "P If in this term we replace \\*~ l by its value (p 1) ! the denominator disappears. According to the hypotheses made regarding the prime number p, no factor of the product is divisible by p and hence the product is not. The second term c p h p becomes likewise an integer when we replace h p by p! but the factor p remains, and so for all of the following terms. Hence i/^(h) is an integer not divis- ible by p. 2. For x, a given finite quantity, and p increasing without limit, i/f (x) = c r x r tends toward zero, as does also the sum We may write <p-l -i \ I 21 SI D D (^l ^~ ^/\"2 ^/ * * \"N " ^/ Since for x of given value the expression in brackets is a con- stant, we may replace it by K. We then have a quantity which tends toward zero as p increases indefinitely. FAMOUS PROBLEMS. The same reasoning will apply when each term of ^(x) is replaced by its absolute value. 3. The expression ty (k,, + h) is an integer divisible by p. v=l We have (k K -k,-h)p The i/th factor of the expression in brackets in the second line is h, and hence the term of lowest degree in h is h p . Consequently r=jrp+N'p+ whence v=H r=Np+u'p+ X^(k, + h)= 2 C' r h r . v=l r=p The numerators of the coefficients C' r are rational and integral, for they are integral symmetric functions of the quantities akj, ak 2 , , ak N , bli, b! 2 , , bl N ., and their common denominator is (p 1) ! . If we replace h r by r I the denominator disappears from all the coefficients, the factor p remains in every term, and hence the sum is an integer divisible by p. Similarly for We have thus established three properties of \f/ (x) analogous to those demonstrated for < (x) in connection with Hermite's theorem. TRANSCENDENCE OF THE NUMBER it. 73 3. We now return to our demonstration that the assumed equation (1) C +C 1 (e kl +e k2 +- + e k ")+C 2 (e ll + e 1 '+- -e v )+- -=0 cannot be true. For this purpose we multiply both members by \l/ (h), thus obtaining C ^(h) + C 1 [e k V(h) + e k V(h) + - - - + e k ^(h)] + - . = 0, and try to decompose each of the expressions in brackets into a whole number and a fraction. The operation will be a little longer than before, for k may be a complex number of the form k = k' + i k". We shall need to introduce | k | = + Vk' 2 +k" 2 - One term of the above sum is e k ^ (h) = e k c r h r = c r e k h r . r r The product e* h r may be written, as shown before, The absolute value of every term of the series k k 2 is less than the absolute value of the corresponding term in the series k k k 2 ^ ^ I + 2! 4 Hence k k 2 I / I -1 \ / I O\ I r + 1 (r k k 2 <e' k ' q, )k being a complex quantity whose absolute value is less than 1. We may then write FAMOUS PEOBLEMS. By giving k in succession the indices 1, 2, N, and form- ing the sum the equation becomes = S 0<r + h ) + ! |k "' S cXq r , k J. i/=l v=l r Proceeding similarly with all the other sums, our equation takes the form (2) Co * (h) 4- C/'SV (k, + h) + C 2 " 2 *(r +)+" y=l =! + c3 N Se! k ,lc r k,q r>ki/ + C 2 *f 2 el', c r l',.q r , 1( ,+ =0. v=l r v =l r By 2, 2 we can make 2|c r k r | as small as we please by taking r p sufficiently great. Since | q r _ k | < 1, this will be true a fortiori of 2 c r k r q r , k r and hence also of Since the coefficients C are finite in value and in number, the sum which occurs in the second line of (2) can, by increasing p, be made as small as we please. The numbers of the first line are, after the first, all divis- ible by p (3), but the first number, C $(h), is not (1). Therefore the sum of the numbers in the first line is not divisible by p and hence is different from zero. The sum of an integer and a fraction cannot be zero. Hence equation (2) is impossible and consequently also equation (1).* 4. We now come to a proposition more general than the preceding, but whose demonstration is an immediate conse- * The proof for the more general case where Co = may be reduced to this by multiplication by a suitable factor, or may be obtained directly by a proper modification of f (h). TRANSCENDENCE OF THE NUMBER it. 75 quence of the latter. For this reason we shall call it Linde- mann's corollary. The number e cannot satisfy an equation of the form (3) C' +C' 1 e kl + C' s e I '+- = <), in which the coefficients are integers even when the exponents k u Ij, are unrelated algebraic numbers. To demonstrate this, let k 2 , k 3 , , k N be the other roots of the equation satisfied by k x ; similarly for I 2 , I 3 , , I M -, etc. Form all the polynomials which may be deduced from (3) by replacing ^ in succession by the associated roots k 2 , , l t by the associated roots I 2 , Multiplying the expres- sions thus formed we have the product a = 1, 2, , N = Q, + d (e kl + e k2 H ----- h e k ") + C 2 (e kl+kj + e k2+ks -\ In each parenthesis the exponents are formed symmetrically from the quantities ki, l t , , and are therefore roots of an algebraic equation with integral coefficients. Our product comes under Lindemann's theorem ; hence it cannot be zero. Consequently none of its factors can be zero and the corollary is demonstrated. We may now deduce a still more general theorem. The number e cannot satisfy an equation of the form CW + C ( 1 1) e k +C ( 2 >e 1 + - - = where the coefficients as well as the exponents are unrelated algebraic numbers. For, let us form all the polynomials which we can deduce from the preceding when for each of the expressions C a \ we substitute one of the associated algebraic numbers C ( f, C<?>, C^. 76 FAMOUS PROBLEMS. If we multiply the polynomials thus formed together we get the product " a = 1, 2, , N 0=1,2, -,, 7 = 1,2, n *, V. c _i_ r P k -i- r p 1 -i- ^->0 \^ ^'k*' i^ v^jC ^^ + C k , k e k+k +C k ,,e k+1 + where the coefficients C are integral symmetric functions of the quantities <?J>, C< 2 >, -, CW, and hence are rational. By the previous proof such an expression cannot vanish, and we have accordingly Linde- mann's corollary in its most general form : The number e cannot satisfy an equation of the form C +C 1 e k +C a e 1 +- - = where the exponents k, I, -as well as the coefficients C , Ci, are algebraic numbers. This may also be stated as follows : In an equation of the form C +C 1 e k +C 2 e 1 +- - = the exponents and coefficients cannot all be algebraic numbers. 5. From Lindemann's corollary we may deduce a number of interesting results. First, the transcendence of TT is an immediate consequence. For consider the remarkable equa- tion 1 4- e iir = 0. TRAJfSCSNDEXCX OF THE NUMBER it. 77 The coefficients of this equation are algebraic ; hence the exponent \TT is not. Therefore, TT is transcendental. 6. Again consider the function y = e 1 . We know that 1 = e. This seenis to be contrary to our theorems about the transcendence of e. This is not the case, however. We must notice that the case of the exponent was implicitly excluded. For the exponent the function ^r(x) would lose its essential properties and obviously our conclusions would not hold. Excluding then the special case (x =0, y = 1). Lindemann's corollary shows that in the equation y = e x or x = log e y, y and x. i.e., the number and its natural logarithm, cannot be alge- braic simultaneously. To an algebraic value of x corresponds a transcendental value of y, and conversely. This is certainly a very remarkable property. If we construct the curve y = e x and mark all the algebraic points of the plane, i.e., all points whose coordinates are alge- braic numbers, the curve passes among them without meeting a single one except the point x = 0. y = 1. The theorem still holds even when x and y take arbitrary complex values. The exponential curve is then transcendental in a far higher sense than ordinarily supposed. 7. A further consequence of Lindemann's corollary is the transcendence, in the same higher sense, of the function y = sin" 1 x and similar functions. The function y = sin" 1 x is defined by the equation We see, therefore, that here also x and y cannot be algebraic simultaneously, excluding, of course, the values x = <>. y 0. We may then enunciate the proposition in geometric form : The curre y = sin" 1 x, like the curve y = e x . no algebraic point of the plane, except x = 0. y = 0. CHAPTER V. The Integraph and the Geometric Construction of n. 1. Lindemann's theorem demonstrates the transcendence of TT, and thus is shown the impossibility of solving the old problem of the quadrature of the circle, not only in the sense understood by the ancients but in a far more general manner. It is not only impossible to construct TT with straight edge and compasses, but there is not even a curve of higher order defined by an integral algebraic equation for which TT is the ordinate corresponding to a rational value of the abscissa. An actual construction of TT can then be effected only by the aid of a transcendental curve. If such a construction is desired, we must use besides straight edge and compasses a " transcendental " apparatus which shall trace the curve by continuous motion. 2. Such an apparatus is the integraph, recently invented and described by a Russian engineer, Abdank-Abakanowicz, and constructed by Coradi of Zurich. This instrument enables us to trace the integral curve Y = F(x)=/f(x)dx when we have given the differential curve y = f(x). For this purpose, we move the linkwork of the integraph so that the guiding point follows the differential curve ; the tracing point will then trace the integral curve. For a fuller description of this ingenious instrument we refer to the original memoir (in German, Teubner, 1889 ; in French, Gauthier-Villars, 1889). GEOMETRIC CONSTRUCTION OF it. 79 We shall simply indicate the principles of its working. For any point (x, y) of the differential curve construct the auxiliary triangle having for vertices the points (x, y), (x, 0), (x 1, 0) ; the hypotenuse of this right-angled triangle makes with the axis of X an angle whose tangent = y. Hence, this hypotenuse is parallel to the tangent to the inte- gral curve at the point (X, Y) corresponding to the point (x, y). FIG. 16. The apparatus should be so constructed then that the trac- ing point shall move parallel to the variable direction of this hypotenuse, while the guiding point describes the differential curve. This is effected by connecting the tracing point with a sharp-edged roller whose plane is vertical and moves so as to be always parallel to this hypotenuse. A weight presses this roller firmly upon the paper so that its point of contact can advance only in the plane of the roller. The practical object of the integraph is the approximate evaluation of definite integrals ; for us its application to the construction of ir is of especial interest. 3. Take for differential curve the circle 30 FAMOUS PROBLEMS. the integral curve is- then Y =f Vr* x 2 dx = ~ sin" 1 - + o ^ ~ * 2 - *. T Zi This curve consists of a series of congruent branches. The points where it meets the axis of Y have for ordinates +^, -.. - 2 ' Upon the lines X = r the intersections have for ordinates 'V 4 ' If we make r = 1, the ordinates of these intersections will determine the number IT or its multiples. It is worthy of notice that our apparatus enables us to trace the curve not in a tedious and inaccurate manner, but with ease and sharpness, especially if we use a tracing pen instead of a pencil. Thus we have an actual constructive quadrature of the circle along the lines laid down by the ancients, for our curve is only a modification of the quadratrix considered by them. ANNOUNCEMENTS THE TEACHING OF GEOMETRY By DAVID EUGENE SMITH, Professor of Mathematics in Teachers College, Columbia University. I2mo, cloth, 339 pages, $1.25 ^ I A HE appearance of this latest work upon the teaching of A mathematics by Professor Smith is most timely and can- not fail to have a powerful influence not merely upon the work in geometry, but upon secondary education in general. The mathematical curriculum has been so severely attacked of late that a clear and scholarly discussion of the merits of geometry, of the means for making the subject more vital and more attrac- tive, of the limitations placed upon it by American conditions, and of the status of the subject in relation to other sciences, will be welcomed by all serious teachers. The work considers in detail the rise of geometry, the chang- ing ideals in the teaching of the subject, the development of the definitions and assumptions, and the relation of geometry to algebra and trigonometry. It takes up in detail the most im- portant propositions that are considered in the ordinary course, showing their origin, the various methods of treating them, and their genuine applications, thus giving to the teacher exactly the material needed to vitalize the work in the high school. Great care has been taken in the illustrations, particularly with respect to the applications of geometry to design, to men- suration, and to such simple cases in physics as are within the easy reach of the student. The work cannot fail to set the standard in geometry in this country for years to come, and to stimulate teachers to the holding of higher ideals and to the doing of stronger work in the classroom. 1961 GINN AND COMPANY PUBLISHERS PLANE AND SOLID GEOMETRY By G. A. 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The school that uses it will not have to change books within a year or so in order to avoid the dangers attend- ant upon the use of some eccentric text. The publishers are confident that in presenting this work to the educational public they are offering the most usable textbook in geometry that has yet appeared, and one that will meet with the approval of all teachers who stand for sound scholarship and for educational progress. I2 3 K GINN AND COMPANY PUBLISHERS University of California SOUTHERN REGIONAL LIBRARY FACILITY Return this material to the library from which it was borrowed. 2 7 890 g & Matiierr.rtital ces ihrary ACK JUL72