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 
 
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 PLANE AND SOLID GEOMETRY 
 
 By G. A. WENTWORTH 
 Revised by GEORGE WENTWORTH and DAVID EUGENE SMITH 
 
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 publication of this revision marks an epoch in the teaching of 
 mathematics. New text, new cuts, and new binding have in- 
 creased the pedagogical efficiency of the book without impair- 
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 No untried experiments have been permitted in this work. 
 The book stands for sound educational policy. Its success has 
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 sents the science of geometry in a clear, interesting, and usable 
 manner. It represents the best thought of the best teachers in 
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 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 
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