Pi: A Source Book

Example of a round field of diameter 9 khet. What is its area?

The mathematicians of ancient Egypt approximated the area of a circle by a square with astonishing accuracy. The way to find this approximation is not handed down. In this paper a conjecture is given which seems to be much more simple than earlier attempts.Die Mathematiker des alten Ägypten approximierten mit erstaunlicher Genauigkeit die Kreisflache durch ein Quadrat. Es ist nicht überliefert, wie diese Approximation entstanden ist. In der vorliegenden Arbeit wird darüber eine Vermutung mitgeteilt, die wesentlich einfacher ist als bisherige Erklärungsversuche.

The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference, of the circle.

Let p _N and P _N denote half the lengths of the perimeters of the inscribed and circumscribed regular N-gons of the unit circle. Thus % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa % aaleaacaaIZaaabeaakiabg2da9iaaiodadaGcaaqaaiaaiodaaSqa % baGccaGGVaGaaGOmaiaacYcacaaMi8UaamiCamaaBaaaleaacaaIZa % aabeaakiabg2da9iaaiodadaGcaaqaaiaaiodaaSqabaGccaGGSaGa % aGjcVlaadchadaWgaaWcbaGaaGinaaqabaGccqGH9aqpcaaIYaWaaO % aaaeaacaaIYaaaleqaaaaa!4981! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${p_3} = 3\sqrt 3 /2,{\kern 1pt} {p_3} = 3\sqrt 3 ,{\kern 1pt} {p_4} = 2\sqrt 2 $$, and P₄ = 4. It is geometrically obvious that the sequences {p _N } and {P _N } are respectively monotonic increasing and monotonic decreasing, with common limit π. This is the basis of Archimedes’ method for approximating to π. (See, for example, Heath [2].) Using elementary geometrical reasoning, Archimedes obtained the following recurrence relation, in which the two sequences remain entwined: 1a% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaac+ % cacaWGqbWaaSbaaSqaaiaaikdacaWGobaabeaakiabg2da9maalaaa % baGaaGymaaqaaiaaikdaaaWaaeWaaeaacaaIXaGaai4laiaadcfada % WgaaWcbaGaamOtaaqabaGccqGHRaWkcaaIXaGaai4laiaadchadaWg % aaWcbaGaamOtaaqabaaakiaawIcacaGLPaaaaaa!45AC! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$1/{P_{2N}} = \frac{1}{2}\left( {1/{P_N} + 1/{p_N}} \right)$$1b% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa % aaleaacaaIYaaabeaakmaaBaaaleaacaWGobaabeaakiabg2da9maa % kaaabaWaaeWaaeaacaWGqbWaaSbaaSqaaiaaikdacaWGobaabeaaki % aadchadaWgaaWcbaGaamOtaaqabaaakiaawIcacaGLPaaaaSqabaaa % aa!4026! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${p_2}_N = \sqrt {\left( {{P_{2N}}{p_N}} \right)} $$

This paper discusses the method of Liu Hui (3rd century) for evaluating the ratio of the circumference of a circle to its diameter, now known as π. A translation of Liu’s method is given in the Appendix. Also examined are the values for π given by Zu Chongzhi (429–500) and unsurpassed for a millenium. Although the method used by Zu is not extant, it is almost certain that he applied Liu’s method. With the help of an electronic computer, a table of computations adhering to Liu’s method is given to show the derivation of Zu’s results. The paper concludes with a survey of circle measurements in China.

For example, let there be two different circles, the circles ABG, DEZ [see Fig. 38]. Let their diameters be BG, the diameter of circle ABC, and EZ, the diameter of circle DEZ. I say, therefore, that

The main theorems of the foregoing article are collected below; underneath each theorem, its enunciations in the original śloka form and in English are given. In Theorems 3–12, C denotes the circumference of a circle whose diameter is D. The abbreviations employed to denote the references are all those of the article.

After 1650, analytic methods began to receive more attention and to replace geometric methods based on the writings of the ancients. This was due partly to the acceptance into geometry of those algebraic methods that Descartes and Fermat had introduced, and partly to the still very active interest in numerical work—interpolation, approximation, logarithms—a heritage of the sixteenth and early seventeenth centuries. This tradition was strong in England, where Napier and Briggs had labored.

Hinc fequitur, quod Si ex Tabellae prop. 184. locis vacuis unus quilibet numero noto fuppleatur, erunt & reliqui omnes cogniti.

Circumferential ad diametrum rationem investigare; & ex datis inscriptis in dato circulo invenire longitudinem arcuum quibus ille subtenduntur.

The formula for π mentioned in the title of this article is (1) % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qadaWcaaqaaiabec8aWbqaaiaaisdaaaGaeyypa0JaaGymaiabgkHi % Tmaalaaapaqaa8qacaaIXaaapaqaa8qacaaIZaaaaiabgUcaRmaala % aapaqaa8qacaaIXaaapaqaa8qacaaI1aaaaiabgkHiTmaalaaapaqa % a8qacaaIXaaapaqaa8qacaaI3aaaaiabgUcaRiabl+Uimjaac6caaa % a!45F6! $$ \frac{\pi }{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots . $$

William Jones (1675-1749) was largely a self-made mathematician. He had considerable genius and wrote on navigation and general mathematics. He edited some of Newton’s tracts.

If ?% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgU % caRiaadgeadaWgaaWcbaGaamOEaaqabaGccqGHRaWkcaWGcbWaaSba % aSqaaiaadQhaaeqaaOWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaam % 4qamaaBaaaleaacaWG6baabeaakmaaCaaaleqabaGaaG4maaaakiab % gUcaRiaadseadaWgaaWcbaGaamOwaaqabaGcdaahaaWcbeqaaiaais % daaaGccqGHRaWkcqWIVlctcqGH9aqpdaqadaqaaiaaigdacqGHRaWk % cqaHXoqycaWG6baacaGLOaGaayzkaaWaaeWaaeaacaaIXaGaey4kaS % IaeqOSdiMaamOEaaGaayjkaiaawMcaamaabmaabaGaaGymaiabgUca % Riabeo7aNjaadQhaaiaawIcacaGLPaaadaqadaqaaiaaigdacqGHRa % WkcqaH0oazcaWG6baacaGLOaGaayzkaaGaeS47IWeaaa!61C9!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$1 + Az + B{z^2} + C{z^3} + D{z^4} + \cdots \ = \left( {1 + \alpha z} \right)\left( {1 + \beta z} \right)\left( {1 + \gamma z} \right)\left( {1 + \delta z} \right) \cdots $$, then these factors, whether they be finite or infinite in number, must produce the expression % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgU % caRiaadgeacaWG6bGaey4kaSIaamOqaiaadQhadaahaaWcbeqaaiaa % ikdaaaGccqGHRaWkcaWGdbGaamOEamaaCaaaleqabaGaaG4maaaaki % abgUcaRiaadseacaWG6bWaaWbaaSqabeaacaaI0aaaaOGaey4kaSIa % eS47IWeaaa!46FD!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$1 + Az + B{z^2} + C{z^3} + D{z^4} + \cdots $$, when they are actually multiplied. It follows then that the coefficient A is equal to the sum % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey % 4kaSIaeqOSdiMaey4kaSIaeq4SdCMaey4kaSIaeqiTdqMaey4kaSIa % eyicI4Saey4kaSIaeS47IWeaaa!445C!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\alpha + \beta + \gamma + \delta + \in + \cdots $$. The coefficient B is equal to the sum of the products taken two at a time. Hence % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaiabg2 % da9iabeg7aHjabek7aIjabgUcaRiabeg7aHjabeo7aNjabgUcaRiab % eg7aHjabes7aKjabgUcaRiabek7aIjabeo7aNjabgUcaRiabek7aIj % abes7aKjabgUcaRiabeo7aNjabes7aKjabgUcaRiabl+Uimbaa!529F!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$B = \alpha \beta + \alpha \gamma + \alpha \delta + \beta \gamma + \beta \delta + \gamma \delta + \cdots $$. Also the coefficient C is equal to the sum of products taken three at a time, namely % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiabg2 % da9iabeg7aHjabek7aIjabeo7aNjabgUcaRiabeg7aHjabek7aIjab % es7aKjabgUcaRiabek7aIjabeo7aNjabes7aKjabgUcaRiabeg7aHj % abeo7aNjabes7aKjabgUcaRiabl+Uimbaa!50DC!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$C = \alpha \beta \gamma + \alpha \beta \delta + \beta \gamma \delta + \alpha \gamma \delta + \cdots $$. We also have D as the sum of products taken four at a time, and E is the sum of products taken five at a time, etc. All of this is clear from ordinary algebra.

Démontrer que le diametre du cercle n’eft point à fa circonférence comme un nombre entier à un nombre entier, c’eft là une chofe, dont les géometres ne feront gueres furpris. On connoit les nombres de. Ludolph, les rapports trouvés par Archimede, par Metius ctc. de même qu’un grand nombre de fuites infinies, qui toutes fe rapportent à la quadrature du cercle. Et fi la fomme de ces fuites eft une quantité rationelle, on doit affez naturellement conclure, qu’elle fera ou un nombre entier, ou une fraction très fimple. Car, s’il y falloit une fraétion fort compofée, quelle raifon y auroit-il, pourquoi plutôt telle que telle autre quelconque? C’eft ainfi, par exemple, que la fomme de la fuite % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgatCvAUfeBSn0BKvguHDwzZbqegSSZmxoasaacH8srps % 0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr % 0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaci % GacaGaaeqabaWaaeaaeaqbaOqaamaalaaabaGaaGOmaaqaaiaaigda % caGGUaGaaG4maaaacqGHRaWkdaWcaaqaaiaaikdaaeaacaaIZaGaai % OlaiaaiwdaaaGaey4kaSYaaSaaaeaacaaIYaaabaGaaGynaiaac6ca % caaI3aaaaiabgUcaRmaalaaabaGaaGOmaaqaaiaaiEdacaGGUaGaaG % yoaaaacqGHRaWkcaGGMaGaam4Baiaac6caaaa!529C!$$\frac{2}{{1.3}} + \frac{2}{{3.5}} + \frac{2}{{5.7}} + \frac{2}{{7.9}} + \& o.$$ eft égale à l’unité, qui de toutes les quantités rationelles eft la plus fimple. Mais, en omettant alternativement les 2, 4, 6, 8 &c. termes, la fourme des autres % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgatCvAUfeBSn0BKvguHDwzZbqegSSZmxoasaacH8srps % 0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr % 0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaci % GacaGaaeqabaWaaeaaeaqbaOqaamaalaaabaGaaGOmaaqaaiaaigda % caGGUaGaaG4maaaacqGHRaWkdaWcaaqaaiaaikdaaeaacaaI1aGaai % OlaiaaiEdaaaGaey4kaSYaaSaaaeaacaaIYaaabaGaaGyoaiaac6ca % caaIXaGaaGymaaaacqGHRaWkdaWcaaqaaiaaikdaaeaacaaIXaGaaG % 4maiaac6cacaaIXaGaaGynaaaacqGHRaWkcaGGMaGaam4yaiaac6ca % aaa!54BB!$$\frac{2}{{1.3}} + \frac{2}{{5.7}} + \frac{2}{{9.11}} + \frac{2}{{13.15}} + \& c.$$ donne l’aire du cercle, lorsque le diametre eft = 1. Il femble done que, fi cette Comme étoit rationelle, elle devroit également pouvoir êrre exprimée par une fraεtion fort fimple, telle que feroir 3/4 ou 4/5 &c. En effet, le diametre étant = 1, le rayon = l/2, le quarré du rayon = 1/4, on voit bien que ces expreffions étant aufi fimples, elles n’y mettent point d’obftacle. Et comme il s’agit de tout le cercle, qui fair une efpece d’unité, & non de quelque Seεeur, qui de fa nature demanderoit des fraεtions forr grandes, on voit bien, qu’encore à cet égard on n’a point fujet de s’attendre à une fraεtion fort compofée. Mais comme, après la fraεtion 1/1 1/4 trouvèe par Achimede, qui ne donne qu’un à peu près, on paffe à celle de Metius, 3/4 5/5 5/2, qui n’eft pas non plus exεate, & dont les nombres font confidérablement plus grands, on doit être fort porté à conclure, que la fomme de cette fuite, bien loin d’être égale à une fraεtion fimple, eft une quantité irrationelle.

By 1750 the number π had been expressed by infinite series, infinite products, and infinito continued fractions, its value had been computed by infinite series to 127 places of decimals (see Selection V.15), and it had been given its present symbol. All these efforts, however, had not contributed to the solution of the ancient problem of the quadrature of the circle; the question whether a circle whose area is equal to that of a given square can be constructed with the sole use of straightedge and compass remained unanswered. It was Euler’s discovery of the relation between trigonometric and exponential functions that eventually led to an answer. The first step was made by J. H. Lambert, when, in 1766–1767, he used Euler’s work to prove the irrationality not only of π, but also of e.

Étant donné un nombre quelconque de quantités numériques α₁, α₂,..., α _n , on sait qu’on peut en approcher simultanément par des fractions de même dénominateur, de telle sorte qu’on ait % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0x % c9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8fr % Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHXo % qydaWgaaWcbaGaaGymaaqabaGccqGH9aqpdaWcaaqaaiabfU5amnaa % BaaaleaacaaIXaaabeaaaOqaaiabfU5ambaacqGHRaWkdaWcaaqaai % abgkGi2oaaBaaaleaacaaIXaaabeaaaOqaaiabfU5amnaakaaabaGa % eu4MdWealeqaaaaaaaa!4578!$${\alpha _1} = \frac{{{\Lambda _1}}}{\Lambda } + \frac{{{\partial _1}}}{{\Lambda \sqrt \Lambda }}$$, % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0x % c9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8fr % Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHXo % qydaWgaaWcbaGaaGOmaaqabaGccqGH9aqpdaWcaaqaaiabfU5amnaa % BaaaleaacaaIYaaabeaaaOqaaiabfU5ambaacqGHRaWkdaWcaaqaai % abgkGi2oaaBaaaleaacaaIYaaabeaaaOqaaiabfU5amnaakeaabaGa % eu4MdWealeaacaWGUbaaaaaaaaa!466E!$${\alpha _2} = \frac{{{\Lambda _2}}}{\Lambda } + \frac{{{\partial _2}}}{{\Lambda \sqrt[n]{\Lambda }}}$$,................,% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0x % c9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8fr % Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHXo % qydaWgaaWcbaGaamOBaaqabaGccqGH9aqpdaWcaaqaaiabfU5amnaa % BaaaleaacaWGUbaabeaaaOqaaiabfU5ambaacqGHRaWkdaWcaaqaai % abgkGi2oaaBaaaleaacaWGUbaabeaaaOqaaiabfU5amnaakeaabaGa % eu4MdWealeaacaWGUbaaaaaaaaa!4713!$${\alpha _n} = \frac{{{\Lambda _n}}}{\Lambda } + \frac{{{\partial _n}}}{{\Lambda \sqrt[n]{\Lambda }}}$$∂₁∂₂, ..., ∂ _n ne pouvant délasser une limite qui dépend seulement. de n. C’est, comme on voit, une extension du mode d’approximation résultant de la théorie des fractions continues, qui correspondrait au cas le plus simple de n = 1. Or, on peut se proposer une généralisation semblable de la théorie des fractions continues algébriques, en cherchant les expressions approchées de n fonctions φ₁(x), φ₂(x),..., φ _n (x) par des fractions rationnelles % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0x % c9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8fr % Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaa % qaaiabfA6agnaaBaaaleaacaaIXaaabeaakmaabmaabaGaamiEaaGa % ayjkaiaawMcaaaqaaiabfA6agnaabmaabaGaamiEaaGaayjkaiaawM % caaaaacaGGSaWaaSaaaeaacqqHMoGrdaWgaaWcbaGaaGOmaaqabaGc % daqadaqaaiaadIhaaiaawIcacaGLPaaaaeaacqqHMoGrdaqadaqaai % aadIhaaiaawIcacaGLPaaaaaGaaiilaiaac6cacaGGUaGaaiOlaiaa % cYcadaWcaaqaaiabfA6agnaaBaaaleaacaWGUbaabeaakmaabmaaba % GaamiEaaGaayjkaiaawMcaaaqaaiabfA6agnaabmaabaGaamiEaaGa % ayjkaiaawMcaaaaaaaa!570B!$$\frac{{{\Phi _1}\left( x \right)}}{{\Phi \left( x \right)}},\frac{{{\Phi _2}\left( x \right)}}{{\Phi \left( x \right)}},...,\frac{{{\Phi _n}\left( x \right)}}{{\Phi \left( x \right)}}$$, de manière que les développements en série suivant les puissances croissantes de la variable coïncident jusqu’à une puissance déterminée xM. Voici d’abord, à cet égard, un premier résultat qui s’offre immédiatement. Supposons que les fonctions φ₁(x), φ₂(x),..., φ _n (x) soient toutes développables en séries de la forme α+βx+γx2 et faisons % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0x % c9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8fr % Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHMo % GrdaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH9aqpcqqHBoatcaWG % 4bWaaWbaaSqabeaacaWGTbaaaOGaey4kaSIaeuOKdiKaamiEamaaCa % aaleqabaGaamyBaiabgkHiTiaaigdaaaGccqGHRaWkcaGGUaGaaiOl % aiaac6cacqGHRaWkcqqHAoWscaWG4bGaey4kaSIaamitaaaa!4E59!$$\Phi \left( x \right) = \Lambda {x^m} + {\rm B}{x^{m - 1}} + ... + {\rm K}x + L$$.

Bei der Vergeblichkeit der so ausserordentlich zahlreichen Versuche**), die Quadratur des Kreises mit Cirkel und Lineal auszuführen, hält man allgemein die Lösung der bezeichneten Aufgabe für unmöglich; es fehlte aber bisher ein Beweis dieser Unmöglichkeit; nur die Irrationalität von π und von π2 ist festgestellt. Jede mit Cirkel und Lineal ausführbare Construction lüssst sich mittelst algebraischer Einkleidung zurückführen anf die Lösung von linearen und quadratischen Gleichungen, also auch auf die Lösung einer Reihe von quadratischen Gleichungen, deren erste rationale Zahlen zu Coefficienten hat, während die Coefficienten jeder folgenden nur solche irrationale Zahlen enthalten, die durch Auflösung der vorhergehenden Gleichungen eingeführt sind. Die Schlussgleichung wird also durch wiederholtes Quadriren übergeführt werden können in eine Gleichung geraden Grades, deren Coefficienten rationale Zahlen sind. Man wird sonach die Unmöglichkeit der Quadratur des Kreises darthun, wenn man nachweist, dass die Zahl πüberhaupt nicht Wurzel einer algebraischen Gleichung irgend welchen Grades mit rationalen Coefficienten sein kann. Den dafür nöthigeu Beweis zu erbringen, ist im Folgenden versucht worden.

Für die Ergebnisse der in der genannten Abhandlung1 mitgetheilten Untersuchungen des Ilrn. Lindemann, namentlich für den darin enthaltenen, bis dahin vergebens erstrebten Beweis, dass die Zahl π keine algebraische Zahl und somit die Quadratur des Kreises auf constructivem Wege unausführbar ist, hat sich in den weitesten Kreisen ein so lebhaftes Interesse kundgegeben, dass ich glaube, es werde eine möglichst elementar gehaltene, nur auf allbekannte Sätze sich stützende Begründung der Lindemann’schen Theoreme, wie ich sie im Nachstehenden zu geben versuche, zahlreichen Mathematikern willkommen sein.2

Man nehme an, die Zahl e genüge der Gleichung nten Grades % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgU % caRiaadggadaWgaaWcbaGaaGymaaqabaGccaWGLbGaey4kaSIaamyy % amaaBaaaleaacaaIYaaabeaakiaadwgadaahaaWcbeqaaiaaikdaaa % GccqGHRaWkcqWIVlctcqGHRaWkcaWGHbWaaSbaaSqaaiaad6gaaeqa % aOGaamyzamaaCaaaleqabaGaamOBaaaakiabg2da9iaaicdaaaa!48A9!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$a + {a_1}e + {a_2}{e^2} + \cdots + {a_n}{e^n} = 0$$, deren Coefficienten a, a₁, ..., a _n ganze rationale Zahlen sind. Wird die linke Seite dieser Gleichung mit dem Integral % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qmaeaacq % GH9aqpaSqaaiaaicdaaeaacqGHEisPa0Gaey4kIipakmaapedabaGa % amOEamaaCaaaleqabaGaeqyWdihaaOWaamWaaeaadaqadaqaaiaadQ % hacqGHsislcaaIXaaacaGLOaGaayzkaaWaaeWaaeaacqaHvpGzcqGH % sislcaaIYaaacaGLOaGaayzkaaGaaiOlaiaac6cacaGGUaWaaeWaae % aacaWG6bGaeyOeI0IaamOBaaGaayjkaiaawMcaaaGaay5waiaaw2fa % amaaCaaaleqabaGaeqyWdiNaey4kaSIaaGymaaaakiaadwgadaahaa % WcbeqaaiabgkHiTiaadQhaaaGccaWGKbGaamOEaaWcbaGaaGimaaqa % aiabg6HiLcqdcqGHRiI8aaaa!5CC5!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\int_0^\infty = \int_0^\infty {{z^\rho }{{\left[ {\left( {z - 1} \right)\left( {\phi - 2} \right)...\left( {z - n} \right)} \right]}^{\rho + 1}}{e^{ - z}}dz} $$ multiplicirt, wo ρ eine ganze positive Zahl bedeutet, so entsteht der Ausdruck % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaape % dabaGaey4kaScaleaacaaIWaaabaGaeyOhIukaniabgUIiYdGccaWG % HbWaaSbaaSqaaiaaigdaaeqaaOGaamyzamaapedabaGaey4kaScale % aacaaIWaaabaGaeyOhIukaniabgUIiYdGccaWGHbWaaSbaaSqaaiaa % ikdaaeqaaOGaamyzamaaCaaaleqabaGaaGOmaaaakmaapedabaGaey % 4kaScaleaacaaIWaaabaGaeyOhIukaniabgUIiYdGccqWIVlctcqGH % RaWkcaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaamyzamaaCaaaleqaba % GaamOBaaaakmaapedabaaaleaacaaIWaaabaGaeyOhIukaniabgUIi % Ydaaaa!5857!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$a\int_0^\infty + {a_1}e\int_0^\infty + {a_2}{e^2}\int_0^\infty + \cdots + {a_n}{e^n}\int_0^\infty $$ und dieser Ausdruck zerlegt sich in die Summe der beiden folgenden Ausdrücke: % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGqb % WaaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaamyyamaapedabaGaey4k % aSIaamyyamaaBaaaleaacaaIXaaabeaakiaadwgadaWdXaqaaiabgU % caRiaadggadaWgaaWcbaGaaGOmaaqabaaabaGaaGimaaqaaiabg6Hi % LcqdcqGHRiI8aaWcbaGaaGimaaqaaiabg6HiLcqdcqGHRiI8aOGaam % yzamaaCaaaleqabaGaaGOmaaaakmaapedabaGaey4kaSIaeS47IWKa % ey4kaScaleaacaaIWaaabaGaeyOhIukaniabgUIiYdGccaWGHbWaaS % baaSqaaiaad6gaaeqaaOGaamyzamaaCaaaleqabaGaamOBaaaakmaa % pedabaGaaiilaaWcbaGaaGimaaqaaiabg6HiLcqdcqGHRiI8aaGcba % GaamiuamaaBaaaleaacaaIYaaabeaakiabg2da9iaadggadaWgaaWc % baGaaGymaaqabaGccaWGLbWaa8qmaeaacqGHRaWkcaWGHbWaaSbaaS % qaaiaaikdaaeqaaOGaamyzamaaCaaaleqabaGaaGOmaaaakmaapeda % baGaey4kaSIaeS47IWKaey4kaSIaamyyamaaBaaaleaacaWGUbaabe % aakiaadwgadaahaaWcbeqaaiaad6gaaaGcdaWdXaqaaiaac6caaSqa % aiaaicdaaeaacaWGUbaaniabgUIiYdaaleaacaaIWaaabaGaaGOmaa % qdcqGHRiI8aaWcbaGaaGimaaqaaiaaigdaa0Gaey4kIipaaaaa!7984!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\begin{gathered} {P_1} = a\int_0^\infty { + {a_1}e\int_0^\infty { + {a_2}} } {e^2}\int_0^\infty { + \cdots + } {a_n}{e^n}\int_0^\infty, \hfill \\ {P_2} = {a_1}e\int_0^1 { + {a_2}{e^2}\int_0^2 { + \cdots + {a_n}{e^n}\int_0^n . } } \hfill \\ \end{gathered} $$.

A circular area is equal to the square on a line equal to the quadrant of the circumference; and the area of a square is equal to the area of the circle whose circumference is equal to the perimeter of the square.

This paper has grown out of a number of requests for information over a number of years, by students and others, concerning some supposed action taken by the Indiana State Legislature with regard to fixing the value of pi, that is, the result of dividing the length of the circumference of a circle by the length of its diameter, at a certain value that was different from the true value. Of course the interest in and wonder at such an action lies in the presumption of a group of supposedly fairly well educated men to attempt to legislate upon something not in the realm of legislation.

I have long been interested in the notorious attempt to legislate a legal value of n. I have read several articles [2,5,6,7,8,10] on the history of this attempt and it has been mentioned in the popular press recently [1,9]. From these it is clear that only Greenblatt [5] and Hallerberg [7] have tried to understand the obscure content of the proposed bill. Greenblatt found four different values of n given in it! Hallerberg has tried to understand how some of the values came about.

Let PQR be a circle with centre 0, of which a diameter is PR. Bisect PO at H and let T be the point of trisection of OR nearer R. Draw TQ perpendicular to PR and place the chord RS = TQ.

If we suppose that (1)% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaig % dacqGHRaWkcaWGLbWaaWbaaSqabeaacqGHsislcqaHapaCcaWGobGa % amOBaaaakiaacMcacaGGOaGaaGymaiabgUcaRiaadwgadaahaaWcbe % qaaiabgkHiTiaaiodacqaHapaCcaWGobGaamOBaaaakiaacMcacaGG % OaGaaGymaiabgUcaRiaadwgadaahaaWcbeqaaiabgkHiTiabek7aIj % abec8aWjaad6eacaWGUbaaaOGaaiykaiaac6cacaGGUaGaaiOlaiab % g2da9iaaikdadaahaaWcbeqaaiaadshaaaGccaWGLbWaaWbaaSqabe % aacqGHsislcqaHapaCcaWGobGaamOBaiaac+cacaaIYaGaaGinaaaa % kiaadEeadaWgaaWcbaGaamOBaaqabaGccaGGUaGaaiOlaiaac6caca % GGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6caaaa!66E4!$$(1 + {e^{ - \pi Nn}})(1 + {e^{ - 3\pi Nn}})(1 + {e^{ - \beta \pi Nn}})... = {2^t}{e^{ - \pi Nn/24}}{G_n}......... $$ and (2)% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaig % dacqGHsislcaWGLbWaaWbaaSqabeaacqGHsislcqaHapaCcaWGobGa % amOBaaaakiaacMcacaGGOaGaaGymaiabgkHiTiaadwgadaahaaWcbe % qaaiabgkHiTiaaiodacqaHapaCcaWGobGaamOBaaaakiaacMcacaGG % OaGaaGymaiabgkHiTiaadwgadaahaaWcbeqaaiabgkHiTiabek7aIj % abec8aWjaad6eacaWGUbaaaOGaaiykaiaac6cacaGGUaGaaiOlaiab % g2da9iaaikdadaahaaWcbeqaaiaadshaaaGccaWGLbWaaWbaaSqabe % aacqGHsislcqaHapaCcaWGobGaamOBaiaac+cacaaIYaGaaGinaaaa % kiaadEgadaWgaaWcbaGaamOBaiaacYcaaeqaaOGaaiOlaiaac6caca % GGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6caaaa!6723!$$(1 - {e^{ - \pi Nn}})(1 - {e^{ - 3\pi Nn}})(1 - {e^{ - \beta \pi Nn}})... = {2^t}{e^{ - \pi Nn/24}}{g_{n,}}........$$ then G_n and g_n can always be expressed as roots of algebraical equations when n is any rational number.

There is a well-known story about the late Archbishop Temple, that he once had to listen to a sermon by a youthful and inexperienced clergyman, and to dine with him afterwards; the young man, by way of making conversation during the meal, ventured to remark, “I think, my lord, that I chose a good text for my sermon”. Instantly there came the grim reply, “There was nothing wrong with the text”. It may be that the consequence of my having selected a seductive title which does not possess a very close connection with the actual subject of my address will be that, when we adjourn presently, I may get the impression that my audience has consisted entirely of archbishops.

In the December 1938 issue of this Monthly, D. H. Lehmer gave a very comprehensive list of formulas for computing π. He rightly chose formulas (23) and (32) as the best self checking pair, with (18) a good substitute for (23).

An important source of solutions for the quadrature of the circle has been the use of Gregory’s series % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaqGHbGaaeOCaiaabogacaqG0bGaaeyyaiaab6gacqaHhpWypaWa % aWbaaSqabeaapeGaaeiiaiaab2cacaqGGaGaaeymaaaakiaabccaca % qG9aGaaeiiaiabeE8aJ9aadaahaaWcbeqaa8qacaqGGaGaaeylaiaa % bccacaqGXaaaaOGaaeiiaiaab2cacaqGGaWaaSaaa8aabaWdbiaabg % daa8aabaWdbiaabodaaaGaeq4Xdm2damaaCaaaleqabaWdbiaabcca % caqGTaGaaeiiaiaabodaaaGccaqGGaGaae4kaiaabccadaWcaaWdae % aapeGaaeymaaWdaeaapeGaae4maaaacqaHhpWypaWaaWbaaSqabeaa % peGaaeiiaiaab2cacaqGGaGaaeynaaaakiaabccacaqGTaGaaeiiai % aac6cacaGGUaGaaiOlaaaa!5BED! $$ {\text{arctan}}{\chi ^{{\text{ - 1}}}}{\text{ = }}{\chi ^{{\text{ - 1}}}}{\text{ - }}\frac{{\text{1}}}{{\text{3}}}{\chi ^{{\text{ - 3}}}}{\text{ + }}\frac{{\text{1}}}{{\text{3}}}{\chi ^{{\text{ - 5}}}}{\text{ - }}... $$ in certain identities, for instance in those expressing 1/4 π as a sum of nrctangents, of which the following, due to Euler, is typical:% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs % 0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaai % aabeqaamaabaabauaakeaadaWcaaqaaiaaigdaaeaacaaI0aaaaiab % ec8aWjabg2da9iaaisdacaaMb8UaciyyaiaackhacaGGJbGaaiiDai % aacggacaGGUbGaaGzaVpaalaaabaGaaGymaaqaaiaaiwdaaaGaeyOe % I0IaciyyaiaackhacaGGJbGaaiiDaiaacggacaGGUbGaaGzaVpaala % aabaGaaGymaaqaaiaaiEdacaaIWaaaaiabgUcaRiGacggacaGGYbGa % ai4yaiaacshacaGGHbGaaiOBaiaaygW7daWcaaqaaiaaigdaaeaaca % aI5aGaaGyoaaaacaGGUaaaaa!63AC!$$\frac{1}{4}\pi = 4\arctan \frac{1}{5} - \arctan \frac{1}{{70}} + \arctan \frac{1}{{99}}. $$

Let π=a/b, the quotient of positive integers. We define the polynomials % MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGMb % GaaiikaiaadIhacaGGPaGaeyypa0ZaaSaaaeaacaWG4bWaaWbaaSqa % beaacaWGUbaaaOGaaiikaiaadggacqGHsislcaWGIbGaamiEaiaacM % cadaahaaWcbeqaaiaad6gaaaaakeaacaWGUbGaaiyiaaaacaGGSaaa % baGaamOraiaacIcacaWG4bGaaiykaiabg2da9iaadAgacaGGOaGaam % iEaiaacMcacqGHsislcaWGMbWaaWbaaSqabeaacaGGOaGaaGOmaiaa % cMcaaaGccaGGOaGaamiEaiaacMcacqGHRaWkcaWGMbWaaWbaaSqabe % aacaGGOaGaaGinaiaacMcaaaGccaGGOaGaamiEaiaacMcacqGHsisl % cqGHflY1cqGHflY1cqGHflY1cqGHRaWkcaGGOaGaeyOeI0IaaGymai % aacMcadaahaaWcbeqaaiaad6gaaaGccaWGMbWaaWbaaSqabeaacaGG % OaGaaGOmaiaad6gacaGGPaaaaOGaaiikaiaadIhacaGGPaGaaiilaa % aaaa!6D57!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\begin{gathered} f(x) = \frac{{{x^n}{{(a - bx)}^n}}}{{n!}}, \hfill \\ F(x) = f(x) - {f^{(2)}}(x) + {f^{(4)}}(x) - \cdot \cdot \cdot + {( - 1)^n}{f^{(2n)}}(x), \hfill \\ \end{gathered}$$</EquationSource></Equation> the positive integer <Emphasis Type="Italic">n</Emphasis> being specified later. Since <Emphasis Type="Italic">n!f(x)</Emphasis> has integral coefficients and terms in <Emphasis Type="Italic">x</Emphasis> of degree not less than <Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">f(x)</Emphasis> and its derivatives <Emphasis Type="Italic">f<Superscript>(<Emphasis Type="Italic">i</Emphasis>)</Superscript>(x)</Emphasis> have integral values for x=0; also for <Emphasis Type="Italic">x</Emphasis>=&#x03C0;<Emphasis Type="Italic">=a/b</Emphasis>, since <Emphasis Type="Italic">f(x) =f(a/b&#x2212;x)</Emphasis>. By elementary calculus we have <Equation ID="Equb"><EquationSource Format="MATHTYPE"><![CDATA[% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca % WGKbaabaGaamizaiaadIhaaaWaaiWaaeaacaWGgbGaai4jaiaacIca % caWG4bGaaiykaiGacohacaGGPbGaaiOBaiaadIhacqGHsislcaWGgb % GaaiikaiaadIhacaGGPaGaci4yaiaac+gacaGGZbGaamiEaaGaay5E % aiaaw2haaiabg2da9iaadAeacaGGNaGaai4jaiaacIcacaWG4bGaai % ykaiGacohacaGGPbGaaiOBaiaadIhacqGHRaWkcaWGgbGaaiikaiaa % dIhacaGGPaGaci4CaiaacMgacaGGUbGaamiEaiabg2da9iaadAgaca % GGOaGaamiEaiaacMcaciGGZbGaaiyAaiaac6gacaWG4baaaa!63CA! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\frac{d}{{dx}}\left\{ {F'(x)\sin x - F(x)\cos x} \right\} = F''(x)\sin x + F(x)\sin x = f(x)\sin x$$ and 1% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qmaeaaca % WGMbGaaiikaiaadIhacaGGPaGaci4CaiaacMgacaGGUbGaamiEaiaa % dsgacaWG4bGaeyypa0ZaamWaaeaacaWGgbGaai4jaiaacIcacaWG4b % GaaiykaiGacohacaGGPbGaaiOBaiaadIhacqGHsislcaWGgbGaaiik % aiaadIhacaGGPaGaci4yaiaac+gacaGGZbGaamiEaaGaay5waiaaw2 % faaaWcbaGaaGimaaqaaiabec8aWbqdcqGHRiI8aOWaa0baaSqaaiaa % icdaaeaacqaHapaCaaGccqGH9aqpcaWGgbGaaiikaiaadIhacaGGPa % Gaey4kaSIaamOraiaacIcacaaIWaGaaiykaaaa!60A7!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\int_0^\pi {f(x)\sin xdx = \left[ {F'(x)\sin x - F(x)\cos x} \right]} _0^\pi = F(x) + F(0)$$.

Early in June, 1949, Professor John von Neumann expressed an interest in the possibility that the ENIAC might sometime be employed to determine the value of π and e to many decimal places with a view toward obtaining a statistical measure of the randomness of distribution of the digits, suggesting the employment of one of the formulas: π/4 = 4 arctan 1/5 − arctan 1/239π/4 = 8 arctan 1/10 − 4 arctan 1/515 − arctan 1/239π/4 = 3 arctan 1/4 + arctan 1/20 + arctan 1/1985 in conjunction with the Gregory series % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qaciGGHbGaaiOCaiaacogacaGG0bGaaiyyaiaac6gapaGaaGjbV-qa % caWG4bGaeyypa0ZaaabCa8aabaWdbiaacIcacqGHsislcaaIXaGaai % yka8aadaahaaWcbeqaa8qacaWGUbaaaaWdaeaapeGaamOBaiabg2da % 9iaaicdaa8aabaWdbiabg6HiLcqdcqGHris5aOGaaiikaiaaikdaca % WGUbGaey4kaSIaaGymaiaacMcapaWaaWbaaSqabeaapeGaeyOeI0Ia % aGymaaaakiaadIhapaWaaWbaaSqabeaapeGaaGOmaiaad6gacqGHRa % WkcaaIXaaaaOWdaiaac6caaaa!565B! $$ \arctan \;x = \sum\limits_{n = 0}^\infty {{{( - 1)}^n}} {(2n + 1)^{ - 1}}{x^{2n + 1}}. $$

3000 B. C. The Pyramids (Egypt). 3 1/7(3.142857.....). The sides and heights of the pyramids of Cheops and of Sneferu at Gizeh are in the ratio 11:7, which makes the ratio of half the perimeter to the height 3 1/7. In 1853, H. C. Agnew, Esq., London, published a letter from Alexandria on evidence of this ratio found in the pyramids being connected or related to the problem of the quadrature of the circle.

The aim of this paper is to determine an explicit lower bound free of unknown constants for the distance of π from a given rational or algebraic number.

In his historical survey of the classic problem of “squaring the circle,” Professor E. W. Hobson [1]* distinguished three distinct periods, characterized by fundamental differences in method, immediate aims, and available mathematical tools.

The following comparison of the previous calculations of π performed on electronic computers shows the rapid increase in computational speeds which has taken place.

The computation of Euler’s constant, γ, to 3566 decimal places by a procedure not previously used is described. As a part of this computation, the natural logarithm of 2 has been evaluated to 3683 decimal places. A different procedure was used in computations of γ performed by J. C. Adams in 1878 [1] and J. W. Wrench, Jr. in 1952 [2], and recently by D. E. Knuth [3]. This latter procedure is critically compared with that used in the present calculation. The new approximations to γ and In 2 are reproduced in extenso at the, end of this paper.

In a recent paper [1] methods were introduced for investigating the accuracy with which certain algebraic numbers may be approximated by rational numbers. It is the main purpose of the present paper to deduce, using similar techniques, results concerning the accuracy with which the natural logarithms of certain rational numbers may be approximated by rational numbers, or, more generally, by algebraic numbers of bounded degree.

Statement of Results.—Schmidt5 proved that for almost all real numbers α, the number of solutions in integers p, q of the inequalities % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadg % hacqaHXoqycqGHsislcaWGWbGaaiiFaiaabccacqGH8aapdaWcgaqa % aiaaigdaaeaacaWGXbGaaeiiaiaabggacaqGUbGaaeizaiaabccaca % qGXaGaeyizImQaamyCaaaacqGHKjYOcaWGcbaaaa!49B6!$$|q\alpha - p|{\text{ }}{1 \mathord{\left/{\vphantom {1 {q{\text{ and 1}} \leqslant q}}} \right.\kern-\nulldelimiterspace} {q{\text{ and 1}} \leqslant q}} \leqslant B$$ is asymptotic to a constant times log B.One might conjecture that the classical numbers (e.g., algebraic numbers, e, π) behave like almost all numbers. Machine computations1 were carried out for some of these numbers, and they seemed to bear out such a conjecture. Also, Lang3 has proved that the estimate is valid when α is a real quadratic irrationality.

While Liouville gave the first examples of transcendental numbers, the modern theory of proofs of transcendency started with Hermite’s beautiful paper “Sur la fonction exponentielle” (Hermite, 1873). In this paper, for a given system of distinct complex numbers ω₀, ω₁, ..., ω _m , and of positive integers ϱ₀, ϱ₁,... , ϱ _m with the sum σ, Hermite constructed a set of m + 1 polynomials% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBa % aaleaacaaIWaaabeaakiaacIcacaWGAbGaaiykaiaacYcacaaMi8Ua % amyqamaaBaaaleaacaaIXaaabeaakiaacIcacaWGAbGaaiykaiaacY % cacaaMi8UaaiOlaiaac6cacaGGUaGaaiilaiaayIW7caWGbbWaaSba % aSqaaiaad2gaaeqaaOGaaiikaiaadQfacaGGPaaaaa!4AD0!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${A_0}(Z),{\kern 1pt} {A_1}(Z),{\kern 1pt} ...,{\kern 1pt} {A_m}(Z)$$ of degrees not exceeding σ − ϱ₀, σ−ϱ₁, ..., σ − ϱ _m , respectively, such that all the functions% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBa % aaleaacaWGRbaabeaakiaacIcacaWGAbGaaiykaiaadwgadaahaaWc % beqaaiabeM8a3jaadYgacaWG6baaaOGaeyOeI0IaamyqamaaBaaale % aacaWGSbaabeaakiaacIcacaWGAbGaaiykaiaadwgadaahaaWcbeqa % aiabeM8a3jaadUgacaWG6baaaOGaaGjcVlaacIcacaaIWaGaeyizIm % Qaam4AaiabgsMiJkaadYgacqGHKjYOcaWGTbGaaiykaaaa!547B!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${A_k}(Z){e^{\omega lz}} - {A_l}(Z){e^{\omega kz}}{\kern 1pt} (0 \leqslant k \leqslant l \leqslant m)$$ vanish at z = 0 at least to the order σ + 1. On putting z = 1, these formulae produce simultaneous rational approximations of the numbers 1, e, e2, ..., em that are so good that they imply the linear independence of these numbers and hence the transcendency of e.

The French naturalist Comte de Buffon (1707–1788) is known to mathematicians for two contributions—a translation into French of Newton’s Method of Fluxions, and his “Essai d’arithmétique morale,” which appeared in 1777 in the fourth volume of a supplement to his celebrated multi-volume Histoire naturelle.

One of the most famous of all numbers is that universally designated today by the lower cast Greek letter π; it represents, among other things, the ratio of the circumference to the diameter of a circle. It has enjoyed a long and interesting history, and over the years it has received ever better approximations.

The lemniscate constants, and indeed some of the methods used for actually computing them, have played an enormous part in the development of mathematics. An account is given here of some of the methods used—most of the derivations can be made by elementary methods. This material can be used for teaching purposes, and there is much relevant and interesting historical material. The acceleration methods developed for the purpose of evaluating these constants are useful in other problems.

A new formula for n is derived. It is a direct consequence of Gauss’ arithmetic geometric mean, the traditional method for calculating elliptic integrals, and of Legendre’s relation for elliptic integrals. The error analysis shows that its rapid convergence doubles the number of significant digits after each step. The new formula is proposed for use in a numerical computation of n, but no actual computational results are reported here.

Let f(x) be one of the usual elementary functions (exp, log, artan, sin, cosh, etc.), and let M(n) be the number of single-precision operations required to multiply n-bit integers. It is shown that f(x) can be evaluated, with relative error O(2−n), in O(M(n)log (n)) operations as n → x for any floating-point number x (with an n-bit fraction) in a suitable finite interval. From the Schönhage-Strassen bound on M(n), it follows that an n-bit approximation to f(x) may be evaluated in O(n log2(n) log log(n)) operations. Special cases include the evaluation of constants such as π, e, and eπ. The algorithms depend on the theory of elliptic integrals, using the arithmetic-geometric mean iteration and ascending Landen transformations.

At the “Journées Arithmétiques” held at Marseille-Luminy in June 1978, R. Apéry confronted his audience with a miraculous proof for the irrationality of ζ(3) = 1−3+2−3+3−3+... . The proof was elementary but the complexity and the unexpected nature of Apéry’s formulas divided the audience into believers and disbelievers. Everything turned out to be correct however. Two months later a complete exposition of the proof was presented at the International Congress of Mathematicians in Helsinki in August 1978 by H. Cohen. This proof was based on the lecture of Apéry, but contained ideas of Cohen and Don Zagier. For a more extensive record of this little history I refer to A. J. van der Poorten [1]. Apéry’s proof will be published in Acta Arithmetica.

The board of programme changes informed us that R. Apéry (Caen) would speak Thursday, 14.00 “Sur l’irrationalité de ζ(3).” Though there had been earlier rumours of his claiming a proof, scepticism was general. The lecture tended to strengthen this view to rank disbelief. Those who listened casually, or who were afflicted with being non-Francophone, appeared to hear only a sequence of unlikely assertions.

We describe several new algorithms for the high-precision computation of Euler’s constant γ = 0.577 .... Using one of the algorithms, which is based on an identity involving Bessel functions, γ has been computed to 30,100 decimal places. By computing their regular continued fractions we show that, if γ or exp(γ) is of the form P/Q for integers P and Q,then |Q| > 1015000

R. Apéry [1] was the first to prove the irrationality of % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOTdO3aae % WaaeaacaaIZaaacaGLOaGaayzkaaGaeyypa0ZaaabCaeaadaWcaaqa % aiaaigdaaeaacaWGUbWaaWbaaSqabeaacaaIZaaaaaaaaeaacaWGUb % Gaeyypa0JaaGymaaqaaiabg6HiLcqdcqGHris5aaaa!4401!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\zeta \left( 3 \right) = \sum\limits_{n = 1}^\infty {\frac{1}{{{n^3}}}} $$.

While creationists’ beliefs are being weighed by an Arkansas judge, a sister organization has evolved, if you will, hundreds of miles away in the hallowed halls of Emporia State University.

In the first book of Kings, we find a description of the temple built by King Solomon in which the measurements of the various parts are stated. In Chapter 7 v. 23 the ‘molten sea’, a large basin containing water in which the priests washed their hands and feet before performing the rites, is described. The verse states (in the A.V.)

In a recent investigation of dihedral quartic fields [6] a rational sequence {a _n } was encountered. We show that these a _n are positive integers and that they satisfy surprising congruences modulo a prime p. They generate unknown p-adic numbers and may therefore be compared with the cubic recurrences in [1], where the corresponding p-adic numbers arc known completely [2]. Other unsolved problems are presented. The growth of the a _n is examined and a new algorithm for computing a _n is given. An appendix by D. Zagier, which carries the investigation further, is added.

The arithmetic-geometric mean of two numbers a and b is defined to be the common limit of the two sequences % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaaca % WGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaWaa0baaSqa % aiaad6gacqGH9aqpcaaIWaaabaGaeyOhIukaaaaa!3E85!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\left\{ {{a_n}} \right\}_{n = 0}^\infty $$, and % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaaca % WGIbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaWaa0baaSqa % aiaad6gacqGH9aqpcaaIWaaabaGaeyOhIukaaaaa!3E86!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\left\{ {{b_n}} \right\}_{n = 0}^\infty $$, determined by the algorithm 0.1% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGHb % WaaSbaaSqaaiaaicdaaeqaaOGaeyypa0JaamyyaiaacYcacaaMc8Ua % amOyamaaBaaaleaacaaIWaaabeaakiabg2da9iaadkgacaGGSaaaba % GaamyyamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccqGH9aqp % daqadaqaaiaadggadaWgaaWcbaGaamOBaaqabaGccqGHRaWkcaWGIb % WaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaGaai4laiaaikda % caGGSaGaaGPaVlaadkgadaWgaaWcbaGaamOBaiabgUcaRiaaigdaae % qaaOGaeyypa0ZaaeWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaOGa % amOyamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaamaaCaaale % qabaWaaSGbaeaacaaIXaaabaGaaGOmaaaaaaGccaGGSaGaaGPaVlaa % d6gacqGH9aqpcaaIWaGaaiilaiaaigdacaGGSaGaaGOmaiaacYcacq % WIMaYsaaaa!6508!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\begin{gathered} {a_0} = a,\,{b_0} = b, \hfill \\ {a_{n + 1}} = \left( {{a_n} + {b_n}} \right)/2,\,{b_{n + 1}} = {\left( {{a_n}{b_n}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}},\,n = 0,1,2, \ldots \hfill \\ \end{gathered} $$. Note that a₁ and b₁ are the respective arithmetic and geometric means of a and b, a₂ and b₂ the corresponding means of a₁ and b₁, etc. Thus the limit 0.2% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaabm % aabaGaamyyaiaacYcacaWGIbaacaGLOaGaayzkaaGaeyypa0ZaaCbe % aeaaciGGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHEisPae % qaaOGaaGPaVlaadggadaWgaaWcbaGaamOBaaqabaGccqGH9aqpdaWf % qaqaaiGacYgacaGGPbGaaiyBaaWcbaGaamOBaiabgkziUkabg6HiLc % qabaGccaaMc8UaamOyamaaBaaaleaacaWGUbaabeaaaaa!52CB!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$M\left( {a,b} \right) = \mathop {\lim }\limits_{n \to \infty } \,{a_n} = \mathop {\lim }\limits_{n \to \infty } \,{b_n}$$ really does deserve to be called the arithmetic-geometric mean of a and b. This algorithm first appeared in a paper of Lagrange, but it was Gauss who really discovered the amazing depth of this subject. Unfortunately, Gauss published little on the agM (his abbreviation for the arithmetic-geometric mean) during his lifetime. It was only with the publication of his collected works [12] between 1868 and 1927 that the full extent of his work became apparent. Immediately after the last volume appeared, several papers (see [15] and [35]) were written to bring this material to a wider mathematical audience. Since then, little has been done, and only the more elementary properties of the agM are widely known today.

We produce a self contained account of the relationship between the Gaussian arithmetic-geometric mean iteration and the fast computation of elementary functions. A particularly pleasant algorithm for x is one of the by-products.

We produce more elementary algorithms than those of Brent and Salamin for, respectively, evaluating ex and π. Although the Gauss arithmetic-geometric process still plays a central role, the elliptic function theory is now unnecessary.

The nature of the number π has intrigued mathematicians since the beginning of mathematical history. The most important properties of π are its irrationality and transcendence, which were established in 1761 and 1882, respectively. In the twentieth century the focus has been on a different sort of question, namely whether π, despite being irrational and transcendental, is normal.

For a time I stood pondering on circle sizes. The large computer mainframe quietly processed all of its assembly code. Inside my entire hope lay for figuring out an elusive expansion. Value: pi. Decimals expected soon. I nervously entered a format procedure. The mainframe processed the request. Error. I, again entering it, carefully retyped. This iteration gave zero error printouts in all — success. Intently I waited. Soon, roused by thoughts within me, appeared narrative mnemonics relating digits to verbiage! The idea appeared to exist but only in abbreviated fashion — little phrases typically. Pressing on I then resolved, deciding firmly about a sum of decimals to use — likely around four hundred, presuming the computer code soon halted! Pondering these ideas, words appealed to me. But a problem of zeros did exist. Pondering more, solution subsequently appeared. Zero suggests a punctuation element. Very novel! My thoughts were culminated. No periods, I concluded. All residual marks of punctuation = zeros. First digit expansion answer then came before me. On examining some problems unhappily arose. That imbecilic bug! The printout I possessed showed four nine as foremost decimals. Manifestly troubling. Totally every number looked wrong. Repairing the bug took much effort. A pi mnemonic with letters truly seemed good. Counting of all the letters probably should suffice. Reaching for a record would be helpful. Consequently, I continued, expecting a good final answer from computer. First number slowly displayed on the the flat screen — 3. Good. Trailing digits apparently were right also. Now my memory scheme must probably be implementable. The technique was chosen, elegant in scheme: by self reference a tale mnemonically helpful was ensured. An able title suddenly existed — “Circle Digits”. Taking pen I began. Words emanated uneasily. I desired more synonyms. Speedily I found my (alongside me) Thesaurus. Rogets is probably an essential in doing this, instantly I decided. I wrote and erased more. The Rogets clearly assisted immensely. My story proceeded (how lovely!) faultlessly. The end, above all, would soon joyfully overtake. So, this memory helper story is incontestably complete. Soon I will locate publisher. There a narrative will I trust immediately appear, producing fame. THE END.

In a recent work [6], Borwein and Borwein derived a class of algorithms based on the theory of elliptic integrals that yield very rapidly convergent approximations to elementary constants. The author has implemented Borweins’ quartically convergent algorithm for 1/π, using a prime modulus transform multi-precision technique, to compute over 29,360,000 digits of the decimal expansion of π. The result was checked by using a different algorithm, also due to the Borweins, that converges quadratically to π. These computations were performed as a system test of the Cray-2 operated by the Numerical Aerodynamical Simulation (NAS) Program at NASA Ames Research Center. The calculations were made possible by the very large memory of the Cray-2.Until recently, the largest computation of the decimal expansion of π was due to Kanada and Tamura [12] of the University of Tokyo. In 1983 they computed approximately 16 million digits on a Hitachi S-810 computer. Late in 1985 Gosper [9] reported computing 17 million digits using a Symbolics workstation. Since the computation described in this paper was performed, Kanada has reported extending the computation of π to over 134 million digits (January 1987).This paper describes the algorithms and techniques used in the author’s computation, both for converging to π and for performing the required multi-precision arithmetic. The results of statistical analyses of the computed decimal expansion are also included.

More than 200 million decimal places of π were calculated using arithmetic-geometric mean formula independently discovered by Salamin and Brent in 1976. Correctness of the calculation were verified through Borwein’s quartic convergent formula developed in 1983. The computation took CPU times of 5 hours 57 minutes for the main calculation and 7 hours 30 minutes for the verification calculation on the HITAC S-820 model 80 supercomputer with 256 Mb of main memory and 3 Gb of high speed semiconductor storage, Extended Storage, for shorten I/O time.

Pi, the ratio of any circle’s circumference to its diameter, was computed in 1987 to an unprecedented level of accuracy: more than 100 million decimal places. Last year also marked the centenary of the birth of Srinivasa Ramanujan, an enigmatic Indian mathematical genius who spent much of his short life in isolation and poor health. The two events are in fact closely linked, because the basic approach underlying the most recent computations of pi was anticipated by Ramanujan, although its implementation had to await the formulation of efficient algorithms (by various workers including us), modern supercomputers and new ways to multiply numbers.

This talk revolves around two focuses: complex multiplications (for elliptic curves and Abelian varieties) in connection with algebraic period relations, and (diophantine) approximations to numbers related to these periods. Our starting point is Ramanujan’s works [1], [2] on approximations to π via the theory of modular and hypergeometric functions. We describe in chapter 1 Ramanujan’s original quadratic period-quasiperiod relations for elliptic curves with complex multiplication and their applications to representations of fractions of „ and other logarithms in terms of rapidly convergent nearly integral (hypergeometric) series. These representations serve as a basis of our investigation of diophantine approximations to π and other related numbers. In Chapter 2 we look at period relations for arbitrary CM-varieties following Shimura and Deligne. Our main interest lies with modular (Shimura) curves arising from arithmetic Fuchsian groups acting on H. From these we choose arithmetic triangular groups, where period relations can be expressed in the form of hyper-geometric function identities. Particular attention is devoted to two (commensurable) triangle groups, (0,3;2,6,6) and (0,3;2,4,6), arising from the quaternion algebra over Φ with the discriminant D = 2.3.

The year 1987 was the centenary of Ramanujan’s birth. He died in 1920 Had he not died so young, his presence in modern mathematics might be more immediately felt. Had he lived to have access to powerful algebraic manipulation software. such as MACSYMA, who knows how much more spectacular his already astonishing career might have been.

Gregory’s series for π, truncated at 500,000 terms, gives to forty places % MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinamaaqa % habaWaaSaaaeaacaGGOaGaeyOeI0IaaGymaiaacMcadaahaaWcbeqa % aiaadUgacqGHsislcaaIXaaaaaGcbaGaaGOmaiaadUgacqGHsislca % aIXaaaaaWcbaGaam4saiabg2da9iaaigdaaeaacaaI1aGaaGimaiaa % icdacaaIWaGaaGimaiaaicdaa0GaeyyeIuoakiabg2da9iaabodaca % GGUaGaaeymaiaabsdacaqGXaGaaeynaiaabMdacaaIWaGaaeOnaiaa % bwdacaqGZaGaaeynaiaabIdacaqG5aGaae4naiaabMdacaqGZaGaae % OmaiaabsdacaaIWaGaaeinaiaabAdacaqGYaGaaeOnaiaabsdacaqG % ZaGaae4maiaabIdacaqGZaGaaeOmaiaabAdacaqG5aGaaeynaiaaic % dacaqGYaGaaeioaiaabIdacaqG0aGaaeymaiaabMdacaqG3aaaaa!67A2!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$4\sum\limits_{K = 1}^{500000} {\frac{{{{( - 1)}^{k - 1}}}}{{2k - 1}}} = {\text{3}}.{\text{14159}}0{\text{65358979324}}0{\text{4626433832695}}0{\text{2884197}}$$

In December 1987 J. P. Bézivin and Ph. Robba found a new proof of the Lindemann-Weierstrass theorem as a by-product of their criterion of rationality for solutions of differential equations. Let us recall the Lindemann-Weierstrass theorem, to which we shall refer as LW from now on.

Following the success of “Circle Digits”, my prose mnemonic for the first 402 digits of it that was published in The Mathematical Intelligencer in 1986, I began work on several even more ambitions mnemonics. I realized that, in esssence, the construction of a literary π mnemonic is a (rather difficult) form of constrained writing. Constrained writing is the art of constructing a work of prose or poetry that obeys some artificially-imposed constraint. (For example, the French novel La Disparition by George Perec does not contain the letter e.)

We give algorithms for the computation of the d-th digit of certain transcendental numbers in various bases. These algorithms can be easily implemented (multiple precision arithmetic is not needed), require virtually no memory, and feature run times that scale nearly linearly with the order of the digit desired. They make it feasible to compute, for example, the billionth binary digit of log(2) or π on a modest work station in a few hours run time.We demonstrate this technique by computing the ten billionth hexadecimal digit of π, the billionth hexadecimal digits of π2, log(2) and log2(2), and the ten billionth decimal digit of log(9/10).These calculations rest on the observation that very special types of identities exist for certain numbers like π, π2, log(2) and log2(2). These are essentially polylogarithmic ladders in an integer base. A number of these identities that we derive in this work appear to be new, for example the critical identity for π: % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgatCvAUfeBSn0BKvguHDwzZbqegSSZmxoasaacH8srps % 0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr % 0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaci % GacaGaaeqabaWaaeaaeaqbaOqaaiabec8aWjabg2da9maaqahabaWa % aSaaaeaacaaIXaaabaGaaGymaiaaiAdadaahaaWcbeqaaiaadMgaaa % aaaaqaaiaadMgacqGH9aqpcaaIWaaabaGaeyOhIukaniabggHiLdGc % caGGOaWaaSaaaeaacaaI0aaabaGaaGioaiaadMgacqGHRaWkcaaIXa % aaaiabgkHiTmaalaaabaGaaGOmaaqaaiaaiIdacaWGPbGaey4kaSIa % aGinaaaacqGHsisldaWcaaqaaiaaigdaaeaacaaI4aGaamyAaiabgU % caRiaaiwdaaaGaeyOeI0YaaSaaaeaacaaIXaaabaGaaGioaiaadMga % cqGHRaWkcaaI2aaaaiaacMcacaGGUaaaaa!629A! $$\pi = \sum\limits_{i = 0}^\infty {\frac{1}{{{{16}^i}}}} (\frac{4}{{8i + 1}} - \frac{2}{{8i + 4}} - \frac{1}{{8i + 5}} - \frac{1}{{8i + 6}}).$$