Simplifying a Proof of Transcendence for e: A Letter Exchange Between Adolf Hurwitz, David Hilbert and Paul Gordan

Abstract

This article is inspired by a letter exchange between Adolf Hurwitz (1859–1919), David Hilbert (1862–1943) and Paul Gordan (1837–1912) concerning their proofs of the transcendence for Euler’s number e. This correspondence took place in the period from 1892 to 1894 and accompanies the process of developing mathematical conclusions. We analyze the evolution of the three proof variations, in particular with a focus on ideas of simplifying the lines of argumentation. This is contrasted by a look on the later reception of the proofs.

1 A Brief Introduction

A number is called algebraic if it is the root of a not identically vanishing polynomial with rational coefficients; otherwise the number is called transcendental. The first to prove the transcendence of some number was Joseph Liouville (1809–1882) in 1844; his approach was constructive but does not apply to any given significant number.¹ The first who proved that $e=\exp (1)=\sum _{m\geq 0}1/m!$ is transcendental was Charles Hermite (1822–1901) in 1873.² In 1882, The not unrelated case of the transcendence of $\pi$ was solved by Ferdinand Lindemann (1852–1939); his result also gave a negative answer to the classical problem of squaring the circle by ruler and compass only.

In a standard reference for number theory, one can read that the original proofs of Hermite and Lindemann “were afterwards modified and simplified by Hilbert, Hurwitz, and other writers” (see Hardy and Wright 1959, p. 177). In fact, David Hilbert (1862–1943), Adolf Hurwitz (1859–1919), and Paul Gordan (1837–1912) published in 1893 successively proofs of transcendence for Euler’s number e. That these were directly inspired by each other is shown by an accompanying correspondence between the three mathematicians. We consider the proofs as well as the exchange of ideas in the following.

Although there is certainly much to say about the influential mathematicians Hilbert and Gordan and their relationship to each other,³ the focus here is on their common correspondence partner Adolf Hurwitz. In this sense we begin with some introductory biographical notes about him, particularly with regard to his work on the transcendence of e.

2 Historical Context

Throughout his life, the mathematician Adolf Hurwitz maintained an active network of correspondence partners in contemporary mathematics. Both, his scientific estate and his remaining collection of correspondence with several hundred letters with about forty correspondents, testify his silent yet constant involvement in the mathematical community. There seem to have been several reasons for his continuous and extensive letter exchanges: From the very beginning of his higher education Adolf Hurwitz was fortunate that his mathematical talent was recognized by influential and excellent mathematicians. During his school years he received private lessons from Hermann Caesar Hannibal Schubert (1848–1911),⁴ later he became doctoral student of Felix Klein (1849–1925) in Munich and Leipzig, and was supported by the Berlin mathematicians Karl Weierstrass (1815–1897) and Leopold Kronecker (1823–1891). In particular, Weierstrass must have had a large influence on Hurwitz’s occupation with analytic issues and encouraged him finding further fields of research.⁵ By the age of 23, Adolf Hurwitz accomplished his habilitation in Göttingen where again he was promoted by influential mathematicians such as Hermann Amandus Schwarz (1843–1921) and Moritz Abraham Stern (1807–1894).

Hurwitz’s early interest in transcendence is best reflected in his third mathematical diary (Hurwitz 1919, No. 3), which he started in January 1883 and where one can find a detailed reflection on the celebrated theorem of Lindemann–Weierstrass. In his resulting article (Hurwitz 1883), he investigated generalizations by studying arithmetical properties of transcendental functions satisfying a homogeneous linear differential equation. His later wife Ida Samuel-Hurwitz (1864–1951) mentioned in a biographical essay that his work on transcendence was at least one of the reasons why Lindemann became aware of Adolf Hurwitz and initiated the invitation of Hurwitz to an extraordinary professorship at the Albertus University at Königsberg in 1884.⁶ In the eastern province of Prussia his good luck to be surrounded by talented mathematicians unexpectedly continued. It is well known that his students David Hilbert and Hermann Minkowski (1864–1909) became successful mathematicians and the interaction of the three went on for all of their lives.⁷ Hurwitz was indeed not only an interested and helpful mathematical colleague of both but also became a friend. This is revealed by the correspondence of the three and brought to the point in Hilbert’s obituary⁸ for his former teacher:

Here [in Königsberg] I was, at that time still a student, soon asked for scientific exchange and had the luck by being together with Hurwitz to get to know in the easiest and most interesting way the directions of thinking of the at time opposite, however, each other excellently complementing schools, the geometrical school of Klein and the algebraic-analytical school of Berlin. [...] On numerous, sometimes day by day undertaken walks at the time for eight years we have browsed through probably all corners of mathematical knowledge, and Hurwitz with his as well wide and multifaceted as also established and well-ordered knowledge was always our leader.⁹ (Hilbert 1921, p. 162)¹⁰

David Rowe describes the relationship of Hilbert and Hurwitz in Königsberg with the following words: ”Adolf Hurwitz was at the height of his powers and he opened up whole new mathematical vistas to Hilbert who looked up to him with admiration mixed with a tinge of envy” (Rowe 2007, p. 25). Certainly it is difficult to assess Hilbert’s nature or degree of admiration, however, the subsequent letter exchange may provide further insights into their relationship.

When Adolf Hurwitz moved to Switzerland in 1892 as full professor at the Polytechnikum in Zurich (since 1911 named Eidgenössische Technische Hochschule, ETH) the exchange with his colleagues became more important. Only with Hilbert one can find around two hundred exchanged letters in about thirty years. From the first two years in Zurich originates the subsequently discussed correspondence on simplifications of the proofs of transcendence for e.

Throughout his life, Hurwitz remained an active mathematician in research and teaching. He stayed with his wife and their three children in Zurich until the end of his life in 1919.

3 Three Variations of a Proof of Transcendence for e

In this section we focus on the mathematical methods and ingredients of the different approaches taken by David Hilbert, Adolf Hurwitz and Paul Gordan. Therefore, we give and compare the three proofs in detail, starting with some notes on their mathematical and historical background.

The first proof of transcendence for e was given by Hermite (1873). His pathbreaking approach relies on an analogy between classical diophantine approximation (that is approximation of real numbers by rationals) and approximating analytic functions of one variable by rational functions. The central idea is to approximate the exponential function $\exp (x)$ by a rational function in x. For his explicit solution of this problem by using what is now known as Padé approximants¹¹ for the exponential function and a detailed discussion of his transcendence proof for e we refer to Mahler (1976) (in its appendix). In the proofs we are going to analyze below this approximation is realized by the function $F(x)$.

In the late nineteenth century, several mathematicians tried to simplify Hermite’s original, rather lengthy and technical proof as well as the related one for $\pi$, found by Lindemann (1882b,a). Besides Karl Weierstrass’ important contribution on generalizations of their methods,¹² there are Andrei Andreevich Markoff (1856–1922),¹³ as well as Oswald Venske (1867–1939) and Thomas Jan Stieltjes (1856–1894)¹⁴ to be mentioned. The latter two contributions are more relevant since they both focus on e and struggle for a simplification of Hermite’s reasoning. Besides those publications from 1890 there is another paper due to Victor Jamet¹⁵ from 1891 which is very close to Stieltjes’ proof and contains the footnote “On pourra consulter aussi sur une simplification de la methode de M. Hermite une Note de M. Stieltjes (Comptes Rendus, 1890).” Then, in 1893, three simplifications were published, the first by David (Hilbert 1893), followed by Adolf (Hurwitz 1893a), and, finally, Paul (Gordan 1893d). Proofs of transcendence were definitely a hot topic at that time.

Stieltjes’ reasoning relies on an identity due to Hermite, resp. the following variation thereof

$$\displaystyle \begin{aligned} {} \int_0^c e^{-xy}f(x)\,\mathrm{{d}} x=F(0)-e^{-cy}F(c), \end{aligned} $$

(8.1)

where $F(x)=\sum _{m\geq 0}f^{(m)}(x)/y^{m+1}$ and, as usual, $f^{(m)}$ denotes the mth derivative of $f=f^{(0)}$ which is here a suitable polynomial. This simple case of Hermite’s identity can easily be proved by partial integration and induction. In fact, Hilbert’s later proof does not differ too much from Stieltjes’ proof.

Therefore, we begin with a sketch of Hilbert’s proof and afterwards point out the little differences to Stieltjes’ proof. His reasoning is indirect (as well as all the other proofs), so we assume that e satisfies an algebraic equation,¹⁶

$$\displaystyle \begin{aligned} {} 0=\sum_{0\leq j\leq n}a_je^j\qquad \mbox{with}\quad a_j\in\mathbb{Z},\ a_0\neq 0; \end{aligned} $$

(8.2)

in fact, if $a_0$ would vanish, we could divide the equation by e in order to get an algebraic equation for e of smaller degree. Next we define a polynomial

$$\displaystyle \begin{aligned} f(x)=x^\mu \prod_{1\leq c\leq n}(x-c)^{\mu+1}, \end{aligned}$$

where $\mu$ is a positive integer to be chosen later, and multiply the corresponding integral $\int _0^\infty e^{-x}f(x)\,\mathrm {{d}} x$ with the algebraic equation for e. This leads to

$$\displaystyle \begin{aligned} 0=\sum_{0\leq j\leq n}a_je^j\left\{\int_0^j+\int_j^\infty \right\}e^{-x}f(x)\,\mathrm{{d}} x; \end{aligned}$$

we observe that Hermite’s integral (8.1) with parameter $y=1$ appears side by side with an infinite integral. We may rewrite the latter equation as

$$\displaystyle \begin{aligned} 0=A+B, \end{aligned}$$

where

$$\displaystyle \begin{aligned} A=\sum_{1\leq j\leq n}a_je^j\int_0^j e^{-x}f(x)\,\mathrm{{d}} x \end{aligned}$$

and

$$\displaystyle \begin{aligned} B=a_0\int_0^\infty e^{-x}f(x)\,\mathrm{{d}} x+\sum_{1\leq j\leq n}a_je^j\int_j^\infty e^{-x}f(x)\,\mathrm{{d}} x; \end{aligned}$$

here we have split the sum according to certain arithmetical properties we shall investigate now. For this purpose we use the elementary formula

$$\displaystyle \begin{aligned} {} \int_0^\infty x^u e^{-x}\,\mathrm{{d}} x=u ! \end{aligned} $$

(8.3)

which is derived by partial integration and induction (similar to Hermite’s identity (8.1) above). Hence, the expression $a_0\int _0^\infty e^{-x}f(x)\,\mathrm {{d}} x$, appearing in B, is an integer multiple of $\mu !$ but not divisible by $\mu +1$ provided that $\mu$ is chosen as a large integer multiple of $a_0n!$. More precisely, expanding the polynomial $\prod _{1\leq c\leq n}(x-c)^{\mu +1}$, this integral equals

$$\displaystyle \begin{aligned} \begin{array}{rcl} & {}{}{}&\displaystyle {\int_0^\infty e^{-x}x^\mu \prod_{1\leq c\leq n}(x-c)^{\mu+1}\,\mathrm{{d}} x} \\ & =&\displaystyle \int_0^\infty e^{-x}x^\mu\,\mathrm{{d}} x\,\cdot\, \prod_{1\leq c\leq n}(-c)^{\mu+1}+\sum_{k\geq \mu+1} b_k\int_0^\infty e^{-x}x^k\,\mathrm{{d}} x {} \end{array} \end{aligned} $$

(8.4)

with some integer coefficients $b_k$, the sum being finite. In order to show that also the other summands in B are integers we substitute $x=\omega +j$ for $j=1,2,\ldots ,n$, which yields

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} e^j\int_j^\infty e^{-x}f(x)\,\mathrm{{d}} x & =&\displaystyle \int_0^\infty (\omega+j)^\mu\prod_{1\leq c\leq n}(\omega+j-c)^{\mu+1}e^{-\omega}\,\mathrm{{d}}\omega\\ & =&\displaystyle \int_0^\infty \sum_{k\geq \mu+1} b_k\omega^ke^{-\omega}\,\mathrm{{d}} \omega \end{array} \end{aligned} $$

(8.5)

by the binomial theorem with another set of integer coefficients $b_k$. Again in view of (8.3) each of these expressions in B is an integer multiple of $(\mu +1)!$, and therefore B is an integer divisible by $\mu !$ such that

$$\displaystyle \begin{aligned} B/\mu!\equiv \pm a_0(n!)^{\mu+1}\not\equiv 0\bmod\,(\mu+1), \end{aligned}$$

as follows from (8.4).

Now denote by K the maximum of $x\prod _{1\leq c\leq n}\vert x-c\vert$ and by k the maximum of $e^{-x}\prod _{1\leq c\leq n}\vert x-c\vert$, both taken for values x from the closed interval $[0,n]$. Then,

$$\displaystyle \begin{aligned} \vert A\vert \leq \sum_{1\leq j\leq n}\left\vert a_je^j\int_0^j e^{-x}f(x)\,\mathrm{{d}} x\right\vert < \kappa K^\mu, \end{aligned}$$

where $\kappa =\sum _{1\leq j\leq n}\vert a_je^j\vert jk$. Since $K^\mu /\mu !$ are the summands in the convergent exponential series for $\exp (K)$, we may choose $\mu$ in addition to satisfy $\kappa {K^\mu \over \mu !}<1$. Then $B/\mu !$ is an integer not divisible by $\mu +1$, hence $B/\mu !$ does not vanish. Furthermore, $\vert A/\mu !\vert <1$, thus $A/\mu !+B/\mu !\neq 0$, giving the desired contradiction.

Comparing Hilbert’s proof with the one by Stieltjes, one observes that Hermite’s identity appears in disguise: taking $y=1$ in (8.1), we find the corresponding terms $F(0)-e^{-c}F(c)$ as the building blocks of B, and so the parameter y is superfluous in Hilbert’s proof. Moreover, Hilbert omitted a lengthy and technical discussion of a certain integral expression involving a piecewise constant function similar to $A+B$ by using $\mu$ as a parameter and congruence calculus for his arithmetical argument. Another difference is within the polynomial f. Where Stieltjes assumes arbitrary integral roots, Hilbert specifies them to be the integers $c=0,1,2,\ldots ,n$. As a matter of fact, a closeness to Stieltjes’ reasoning is obvious though Hilbert’s presentation is a little shorter, definitely more elegant and more to the point. This is well documented by Hurwitz in his mathematical diary (see Figs. 8.1 and 8.2) and in his correspondence.

Fig. 8.1

An excerpt from Hurwitz’s diary, notebook no. 10 (Hurwitz 1919, No. 10, p. 153), referring to a letter from Hilbert bearing the date 24 October 1892. After a brief outline of Stieltjes’ proof Hurwitz starts his sketch of Hilbert’s reasoning with the words “Hilbert presents [Stieltjes’] proof in a shorter way as follows.” (All diaries are published online on the e-manuscripta.ch platform)

Fig. 8.2

Another excerpt from Hurwitz’s diary, notebook no. 10, on the next page (Hurwitz 1919, No. 10, p. 154). Hurwitz refers to another letter from Hilbert bearing the date 31 December 1892. He writes “Following up on this Hilbert has also given this proof” and continues with a brief sketch of Hilbert’s reasoning. This notebook also contains an early draft of Hurwitz’s proof of the transcendence of e, p. 175

Although Hilbert went beyond Stieltjes in proving the transcendence of $\pi$ by almost similar means, it is noteworthy that he did not refer to Stieltjes in his article at all. Actually, Lindemann is the only mathematician that is mentioned in Hilbert’s article. David Rowe wrote:¹⁷

From one of his letters to Hurwitz, we learn that Hilbert got the initial idea for this new proof by reading a paper on the transcendence of e published by Th.J. Stieltjes in 1890. Stieltjes’ paper was hardly longer than Hilbert’s, and had been published in the widely read Comptes Rendus of the French Academy. Remarkably, Hilbert made no mention of this in his publication; in fact, his notice contains no references to the literature on these problems whatsoever! Was this in the name of simplicity, or perhaps just a sign of supreme intellectual arrogance?

One could take the ambitious Hilbert in shelter by stressing his age and experience as well as the standards of citation at that time (see also Fig. 8.3). Similarly, Stieltjes did not refer to Venske, Hurwitz mentioned only Hilbert and Gordan (although he wrote reviews for the Jahrbuch of both, Venske’s and Stieltjes’ article, namely for the Jahresberichte JFM 22.0435.03 and JFM 22.0437.01, the first one containing several typos), and Gordan cited Hermite, Lindemann, Hurwitz, and Hilbert (see Fig. 8.3). However, as already pointed out by David (Rowe 2007), Felix Klein had the idea to publish the three different proofs of Hilbert, Hurwitz and Gordan in one volume of the well distributed and renowned Mathematische Annalen of which he was the managing editor. Different to the Göttinger Nachrichten¹⁸, where Hilbert and Hurwitz had published their variations, a reader of the Annalen would expect proper references for a hot topic result as these transcendence proofs, however Hilbert and Hurwitz were not much interested in a further publication besides the Göttinger Nachrichten (see Rowe 2007, p. 230, for further details). Gordan’s proof appeared in slightly different form and in French, see Gordan (1893c).¹⁹

Fig. 8.3

Illustration of the citations in publications

We continue with a sketch of Hurwitz’s proof. For $F(x)=\sum _{m\geq 0}f^{(m)}(x)$, differentiation with respect to x shows

$$\displaystyle \begin{aligned} {} {\,\mathrm{d}\over \,\mathrm{d} x}\Big(e^{-x}F(x)\Big)=-e^{-x}f(x), \end{aligned} $$

(1’)

which is the differential analogue of Stieltjes’ variant (8.1) of Hermite’s integral identity (and its proof relies on the product rule which is the counterpart of partial integration that has been used to derive (8.1)). Applying the mean-value theorem from differential calculus, implies the existence of a real number $\xi$ in between 0 and x such that

$$\displaystyle \begin{aligned} e^{-x}F(x)-F(0)=x\left(-e^{-\xi}f(\xi)\right). \end{aligned}$$

Now, for $x=c$ ranging from 1 up to n, we write the intermediate value $\xi$ as $\delta c$ with some $\delta \in (0,1)$ (depending on c) and get

$$\displaystyle \begin{aligned} {} F(c)-e^cF(0) =-ce^{c(1-\delta)}f(\delta c)=:\epsilon_c, \end{aligned} $$

(8.6)

say. For a large prime number p, we consider the polynomial

$$\displaystyle \begin{aligned} {} f(x)={1\over (p-1)!}x^{p-1}\prod_{1\leq c\leq n}(c-x)^p, \end{aligned} $$

(8.7)

which differs only slightly from Hilbert’s polynomial. Then, all the expressions $\epsilon _c$ get arbitrarily small with increasing p. Next we expand $f(x)$ around $x=c$ for $c=1,2,\ldots ,n$ into a power series by applying the binomial theorem, namely

$$\displaystyle \begin{aligned} f(c+h)={h^p\over (p-1)!}\left(b_0+\sum_{k\geq 1}b_k h^k\right), \end{aligned}$$

where the sum is finite and the $b_k$’s are integers depending on c. Comparing with the Taylor series expansion

$$\displaystyle \begin{aligned} f(c+h)=\sum_{m\geq 0}{f^{(m)}(c)\over m!}h^m, \end{aligned}$$

it follows that $f^{(m)}(c)$ vanishes for $m<p$ and

$$\displaystyle \begin{aligned} f^{(m)}(c)={m!\over (p-1)!}b_{m-p}\equiv 0\bmod\,p \end{aligned}$$

for all $m\geq p$. For the root $x=0$ of f, however, the situation is slightly different: here we observe by the same reasoning that $f^{(m)}(0)$ is vanishing for $m<p-1$, is equal to $(n!)^p$ for $m=p-1$, and is an integer multiple of p whenever $m\geq p$. Thus, the numbers $F(c)$ for $c=1,\ldots ,n$ are all integer multiples of p if $p>n$, and $F(0)$ is an integer too but not divisible by p.

Now, assuming an equation of the form (8.2) for e, it follows that

$$\displaystyle \begin{aligned} a_0F(0)+\sum_{1\leq c\leq n}a_cF(c)=\sum_{1\leq c\leq n}a_c\epsilon_c. \end{aligned}$$

Since the left hand side is an integer, the right hand side is an integer too, however, it gets as small as we please by increasing p, so the quantity on the left vanishes. This vanishing contradicts that the integer $a_0F(0)+\sum _{1\leq c\leq n}a_cF(c)$ is not divisible by p when p is chosen to be sufficiently large.

Both proofs so far rely essentially on the differential equation $\exp '=\exp$ for the exponential function (or, equivalently, the integral equation for $\exp$). This kind of reproduction property appears in the proof of Hermite’s identity (8.1) as well as in its differential counterpart (1’). The additional analytic ingredients can also not be called advanced. The estimate of the quantities $\epsilon _c$ in Hurwitz’s proof seems formally easier than bounding the integrals in Hilbert’s reasoning, though it includes the mean-value theorem from differential calculus and the infinitude of prime numbers although Hurwitz (and we as well) did not explicitly mention the latter simple fact at all.

Hurwitz (1893a) contains a footnote:

By the way, Mr. Hilbert, as I learned from him recently, has already occasionally given hints in a lecture how one can avoid the integrals (and at the same time the differentiation) in his proof by replacing the integrals by limit values.²⁰

This idea had been realized by the following third proof published in 1893, found by Paul Gordan, professor in Erlangen and a former colleague of Felix Klein. Jeremy Gray characterizes Gordan as “a master at manipulating long algebraic expressions” (Gray 2018, p. 265).

This ability together with his urge for explicitness can also be found in Gordan’s proof of transcendence. His reasoning relies only on the convergence of the exponential series,

$$\displaystyle \begin{aligned} e^x=\sum_{i\geq 0}{x^i\over i!}, \end{aligned}$$

and a subtle notation, namely denoting $r!$ as $h^r$, where r is any non-negative integer. This allows to write

$$\displaystyle \begin{aligned} {} h^re^x=r!\left\{\sum_{0\leq i\leq r}+\sum_{i>r}\right\}{x^i\over i!}=(x+h)^r+x^ru_r, \end{aligned} $$

(1”)

where

$$\displaystyle \begin{aligned} {} \sum_{0\leq i\leq r}{r!\over i!}x^i=\sum_{0\leq i\leq r}\left({r\atop i}\right)x^ih^{r-i}=(x+h)^r, \end{aligned} $$

(8.8)

(since $({r\atop i})h^{r-i}={r!\over i!(r-i)!}h^{r-i}={r!\over i!}$ and the last equality follows symbolically from the binomial theorem), and

$$\displaystyle \begin{aligned} u_r:=\sum_{i>r}{r!\over i!}x^{i-r}=\sum_{k\geq 1}{x^k\over (r+1)\cdot\ldots\cdot (r+k)}. \end{aligned}$$

In view of the convergence of the exponential series, for $x\geq 0$,

$$\displaystyle \begin{aligned} u_r\leq \sum_{k\geq 1}{x^k\over k!}<e^x; \end{aligned}$$

hence $u_r=q_re^x$ with some real number $q_r$ from the unit interval.

Now, for arbitrary numbers $b_0,\ldots ,b_d$, we deduce from (1”) that

$$\displaystyle \begin{aligned} \sum_{0\leq r\leq d}b_rh^re^x=\sum_{0\leq r\leq d}b_r(x+h)^r+\sum_{0\leq r\leq d}b_rq_re^xx^r, \end{aligned}$$

resp.

$$\displaystyle \begin{aligned} {} e^xf(h)=f(x+h)+e^xg(x), \end{aligned} $$

(8.9)

where

$$\displaystyle \begin{aligned} f(\mathtt{x}):=\sum_{0\leq r\leq d}b_r\mathtt{x}^r\quad \mbox{and}\quad g(\mathtt{x}):=\sum_{0\leq r\leq d}b_rq_r\mathtt{x}^r. \end{aligned}$$

Next we assume that e is algebraic, i.e. an equation of the form (8.2) holds.

Then (8.9) with $x=j$ and $j=0,1,\ldots ,n$ implies

$$\displaystyle \begin{aligned} \begin{array}{rcl} 0& =&\displaystyle \sum_{0\leq j\leq n}a_je^j\cdot f(h)\\ & =&\displaystyle \sum_{0\leq j\leq n}a_jf(j+h)+\sum_{0\leq j\leq n}a_jg(j)e^j=F+G, \end{array} \end{aligned} $$

say.

Now we choose f the same way as Hurwitz did, that is (8.7). For $j=1,2,\ldots ,n$, writing $p!=h^p$, we observe that

$$\displaystyle \begin{aligned} \begin{array}{rcl} f(j+h)& =&\displaystyle {(j+h)^{p-1}\over (p-1)!}\prod_{1\leq c \leq n}(c-j-h)^p \\ & =&\displaystyle -(j+h)^{p-1}{h^p\over (p-1)!}\prod_{1\leq c\leq n\atop c\neq j}(c-j-h)^p\\ & =&\displaystyle -p(j+h)^{p-1}\prod_{1\leq c\leq n\atop c\neq j}(c-j-h)^p. \end{array} \end{aligned} $$

Since by (8.8) every factor is an integer, it follows that $f(j+h)$ is an integer divisble by p. For $j=0$, however, we find similarly

$$\displaystyle \begin{aligned} f(h)={h^{p-1}\over (p-1)!}\prod_{1\leq c\leq n}(c-h)^p=\prod_{1\leq c\leq n}\left(-\sum_{0\leq i\leq p}{p!\over i!}(-c)^i\right), \end{aligned}$$

which is an integer not divisible by the prime p. Hence, F is a non-zero integer.

Since with increasing p the coefficients $b_k$ of f tend to zero (thanks to the factor ${1\over (p-1)!}$), it follows that G gets as small as we please, contradicting $F+G=0$.

Probably, Gordan came to his symbolical proof by introducing formal differentiation in Hurwitz’s reasoning. In fact, this can be seen in (8.8), namely,

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} f(x+h)& =&\displaystyle \sum_{0\leq r\leq d}b_r(x+h)^r=\sum_{0\leq r\leq d}b_r\sum_{0\leq i\leq r}{r!\over i!}x^i \\ & =&\displaystyle \sum_{0\leq m\leq d}\sum_{m\leq r\leq d}b_r{r!\over (r-m)!}x^{r-m}=\sum_{m\geq 0}f^{(m)}(x), \end{array} \end{aligned} $$

(8.10)

which is equal to $F(x)$ in other proofs. Later Heinrich Weber (1842–1913) reinvented in his important monograph (Weber 1899) derivatives without any comment; his intention might have been the counter-intuitive notation Gordan used for circumventing derivatives. Whereas Hilbert and Hurwitz use the differential equation for the exponential function, the convergence of the used exponential series is indeed the only property Gordan needed for his proof.

4 Hessenberg’s Analysis

Concerning Gordan’s symbolic proof, Gerhard Hessenberg (1874–1925)²¹ wrote:

[…] so it is no miracle that G o r d a n ’s symbolism raises commonly some gentle shudder, not least because the imposition to think of $h^\mu$ as $\mu !$ collides with the habit not to think at all while calculating something.²²

In 1912, Hessenberg analyzed the various proofs of transcendence for e and $\pi$ in detail, in some places a little polemic. The only proofs he missed in his careful study are those by Venske and Stieltjes. The first one relies on a certain non-vanishing determinant and is therefore closer to Hermite’s original reasoning whereas Stieltjes’ proof is quite similar to Hilbert’s reasoning (as already pointed out above). According to Hessenberg, the proofs can be dissected into three parts (see Fig. 8.4).

Fig. 8.4

A diagram from Hessenberg (1912, p. 53) illustrating the proof essentials of Hilbert (left), Hurwitz (middle) and Gordan (right). For details Hessenberg refers to various paragraphs. We have adjusted our presentation of the three proofs in question with respect to Hessenberg’s analysis. Here the Gamma-function is mentioned in the upper left with respect to Formula (8.3) in Hilbert’s proof since $\Gamma (u+1)= \int _0^\infty x^u e^{-x}\,\mathrm {d} x$

First of all, there is an arithmetical part linking $f(x)=\sum a_\nu (x-c)^\nu$ with the values $F(c)=\sum a_\nu \nu !$, where the coefficients are all integers. In all three proofs it is shown, by slightly different means, that the numbers $F(c)$ are integers satisfying certain divisibility properties with respect to the multiplicities of the roots of f. It might be worth to notice that Hermite’s line of argument for this purpose differs significantly by a rather lengthy calculation of a certain determinant.

The middle part is the most interesting since this is essentially the only part where the proofs differ. Hilbert’s reasoning relies on integration (and is therefore building on Hermite’s original proof and his identity) by using (8.1) implicitly with $y=1$, resp.

$$\displaystyle \begin{aligned} F(c)=e^c\int_c^\infty e^{-x}f(x)\,\mathrm{{d}} x, \end{aligned}$$

and explicit calculations (by partial integration, resp. the gamma-function; see (8.3), (8.4) and (8.5)).

Hurwitz replaced integration by differentiation and the corresponding formula is indeed simpler. More precisely, he observed ${\,\mathrm {{d}}\over \,\mathrm {{d}} x}\Big (e^{-x}F(x)\Big )=-e^{-x}f(x)$ and applied the mean value theorem in order to obtain (8.6), i.e.,

$$\displaystyle \begin{aligned} F(c)-e^cF(0)=-ce^{-\delta c}f(\delta c) \end{aligned}$$

for some intermediate value $\delta c$ from $[0,c]$. Maybe this formula shows best that the polynomial F works as an approximation for the exponential function. The link to the arithmetical part follows from the Taylor expansion $F(c)=\sum f^{(\nu )}(c)$.

Finally, Gordan argued symbolically without using any integral or differential calculus; even formal differentiation is replaced by using the binomial theorem. Whereas Hilbert and Hurwitz worked with the differential equation for the exponential function in addition with some fundamental results from integral and differential calculus, respectively, the only ingredient needed for Gordan’s argument is the convergence of the exponential series linking both functions F and f. Hessenberg rewrote this shortly by the formulae

$$\displaystyle \begin{aligned} F(c)=\sum p_\nu E_\nu(c),\quad f(c)=\sum p_\nu c^\nu/\nu!; \end{aligned}$$

for the sake of brevity we do not explain the numbers $p_\nu$ and functions $E_\nu$ appearing here though the reader may guess by comparing these expressions with $F(c)=\sum _{m\geq 0}f(c)=f(c+h)$ according to (8.10).

The final third part of Hessenberg’s dissection of the transcendence proofs is analytic and consists of an estimate of the quantities $e^cF(0)-F(c)$. Also here differences appear according to the use of integration, differentiation or just the convergent exponential series.

Hence, and here we follow Hessenberg’s judgement, Hilbert’s proof can be considered purely analytical, Hurwitz’s reasoning is mixed, and Gordan argues purely formal. Moreover, Hessenberg considered also further proofs (e.g. those by Weber, Mertens, Vahlen, and Schottky), however, those are too close to the above discussed ones to be interesting in our framework.

5 Letter Exchanges

In the following we turn to the letter exchange that accompanied the publication of the proofs. These letters give the opportunity to look over the shoulders of mathematicians in their research (although this can only be a brief look, quoting only excerpts of the correspondence). Correspondents were, of course, the three authors Hurwitz, Hilbert and Gordan, supplemented by references to Felix Klein and Charles Hermite. In particular, we consider six related letters from Hurwitz to Hilbert as well as five letters from Hilbert to Hurwitz, furthermore, one letter from Gordan to Hilbert and two letters to Hurwitz, one letter from Hurwitz to Gordan. Besides we give quotes from two letters from Hermite to Hurwitz and one letter to Gordan (see Fig. 8.5).

Fig. 8.5

Illustration of the letters that were exchanged; the thickness of the arrows is related to the number of letters

5.1 Hilbert and Hurwitz

“I draw your attention to a very nice proof of the transcendence for e by Stieltjes (Comptes R. 1890).”²³ (6.9.1892, Adolf Hurwitz to David Hilbert (Hurwitz 1893b, Br. 7))

As mentioned above, Adolf Hurwitz was reviewer of the article by Thomas (Stieltjes 1890) in the annual Jahresberichte. Hilbert’s answer to the brief reference to Stieltjes’ proof from October 24, 1892, can be regarded as the impetus of the three proof variations for the transcendence of e discussed above. Herein the Königsberg mathematician informed Hurwitz about a new discovery which he had investigated while preparing his lecture on integral calculus. Hilbert stated that “Stieltjes’ proof of the transcendence of e can be presented [...], avoiding Hermite’s integrals” (Hilbert 1894, Br. 240).

Although the diary entry quoted above (Fig. 8.1) proves that Hurwitz was interested in Hilbert’s idea, it took some weeks before he commented his proof on 4 December 1892 with a short remark:

Your simplification of Stieltjes’ proof is very nice; I directly wrote it down into my “book of numbers”. One should try to also simplify the proof of the transcendence of $\pi$.²⁴ (Hurwitz 1893b, Br. 8)

After a further simplification by Hilbert, communicated in a letter of 31 December 1892, Hurwitz’s own activity was aroused. Here, Hilbert gave a proof avoiding even “Stieltjes’ point”²⁵ (Hilbert 1894, Br. 243). He chose the function $F(z) = z^\sigma \left \{ (z-1)(z-2)...(z-n)\right \}^{\sigma + 1}$, an explicit simplification, and showed that for sufficiently large $\sigma$ the terms with improper integrals (see B in Section 2) contribute to an integer not equal to zero.

You can see that altogether herewith the poof receives another proof line, in which the inequality to 0 is not shown by the sums of integrals [A] but by the integer number [B]. The use of this conclusion leads also to a simplification for the transcendence of $\pi$, which does not seem to be irrelevant to me.²⁶ (Hilbert 1894, Br. 243)

One may assume that Hurwitz shared his opinion: already some days later, despite of New Year’s Day and holidays, he answered on 10 January 1893 with not only another new proof by his own, but, furthermore, with a certain idea:

Your scientific note concerning the number e was, as you may imagine, very interesting for me. [...] I could not stay calm and discovered a further simplification [...] in a way that one can now give the proof in one of the first lessons of a lecture on differential calculus. (Hurwitz 1893b, Br. 9)

As outlined in Sect. 8.3, Hurwitz’s proof is based on differential calculus and a combination of divisibility and estimates. In the further course of the letter, Hurwitz makes the following suggestion:

Did you already edit your proof? If yes, please write me if you have already sent it to Klein. Then I would let the above mentioned further simplification be printed in a short note subsequent to your work. I would prefer if we would choose the Göttinger Nachrichten. [...] So please answer quickly, also if only with a postcard. It is clear that the punch line can also be applied to $\pi$, however, I have not really thought through this yet.²⁷ (Hurwitz 1893b, Br. 9)

But Hilbert did not agree with his former teacher Hurwitz. Within only three days, he answered:

In fact, I have already worked out my proof for e and $\pi$ in Christmas holidays, therein - in particular in the part concerning $\pi$ - some advantageous and simplifying points arouse, such that the whole thing will be on 4–5 printed pages and my presentation is not even short. Of course, in your proof the integral is avoided; however, if the representation of the proof becomes shorter and clearer, is not yet obvious to me. [In a further of my ideas] the opportunity is given to reduce all to a simple considering of limits and the summation of geometric series. Of course, this thing has to be carefully studied yet. [...] However, it is in my opinion, that the proof with the help of integrals will always be the clearest and most developable one. For p or in my notation $\rho + 1$ I did not choose a prime number, since it is easier to define a number which is divisible by c than a prime number of the needed value. Furthermore, I have given how big $\rho$ is to be chosen and I also put a lot of care on the editing. Later on I please you to give me your opinion also concerning incidental parts. It is strange that I also have chosen the Göttinger Nachrichten; Klein wrote me directly, that he would present the note in this week. However, by no means this does affect your intention to show your proof in the next meeting [...] ²⁸ (Hilbert 1894, Br. 244)

This letter includes a number of interesting details. What had happened? In short time of ten days (including New Year’s Day) Hilbert had completed his proof line and he had sent his paper to Felix Klein, member of the Göttingen Academy of Sciences and therewith able to communicate articles for the Göttinger Nachrichten. Hilbert is not willing to supplement his publication with Hurwitz’s proof and his assessment that his approach will always “be the clearest and most developable” is quite explicit.

When Hurwitz answered one month later (see Fig. 8.6), he is defending his proof:

It’s been a long time since I wanted to answer your kind letter from 13/I, but - how it goes - the answer was shifted from day to day. Today now your note on transcendence, which I have directly dipped into a coffee, appears as guiding hint. You have written the note with Gaussian classicity. I hope that you agree with the note (probably 2 printed pages) in which I wrote down the modification of your proof. Felix Klein showed it 4/II to the Göttingen Society. As advantage of my modification I consider that the proof shows that only the theorem of additivity and the differential equation for $e^x$ (so in last instance only this) imply the transcendence of e, and that the proof relies on an approximate representation of the powers $e, e^2, ...e^n$ by rational numbers with the same denominators. [...] I also had the idea of modifying your proof by replacing the integrals by limits. However it seems that there are some difficulties. [...] Perhaps one is lead to a proof which only uses the definition

$$\displaystyle \begin{aligned} e^z = (1 + \frac{z}{n})^n (n = \infty).\end{aligned}$$

Fig. 8.6

First two pages of the letter from Adolf Hurwitz to David Hilbert, from February 8 or 13, 1893 (Hurwitz 1893b, Br. 10)

²⁹(Hurwitz 1893b, Br. 10)

Hurwitz thus insisted on the “advantage of his modification”. At the same time using terms like “dipping into coffee” and “Gaussian classicity” he expressed joviality towards his former student and successor at the Königsberg University. But when he wrote about his own accepted, shorter (only two pages) paper and finally mentioned that he had had similar ideas as Hilbert and, moreover, even ideas for further simplifications, Hurwitz nearly seems to hit the ball back. Astonishingly, it was taken up by Hilbert who declared in a letter from March 8, 1893: ‘Gordan wrote me a special acknowledgement for my proof of transcendence.”³⁰ (Hilbert 1894, Br. 245) (which indeed had been done in a letter (Gordan 1893a) from 24 February 1893, see Fig. 8.7).

Fig. 8.7

Letter from Paul Gordan to David Hilbert from February 24, 1893: “Your work has given me great pleasure; I congratulate you on it; your proof of the transcendence of e and $\pi$ is surprisingly elegantly conducted”

Some weeks later, the slightly elder informed the younger about a recognition of his proof from the initiator of the proofs of transcendence. In a letter from 8 April 1893 he wrote in a surprisingly comparative manner:

Hermite wrote me in his kind way a letter about my e-proof which is obviously much more convenient to him than yours. He asked me to be allowed to show it to the academy and so it will probably be printed in the next Comptes Rendus [...].³¹ (Hurwitz 1893b, Br. 11)

Indeed, Hermite expressed that after Stieltjes and Hilbert, Adolf Hurwitz had probably said “le dernier mot”³² to the important question of transcendence of e.³³ Explicitly he mentioned:

Studying the fine work of M. David Hilbert I was a bit embarrassed about the essential point at which he established that the integer number [...] is necessarily different to zero. But everything is of extreme clarity in your analysis and what seems to me the foremost remarkable and completely new is to be led to an impossibility regarding integer numbers, when before we encountered another type of contradiction, an integer equal to a quantity decreasing to an inferior.³⁴ (Hermite 1893a, Br. 218)

Although Hermite declared a certain preference for Hurwitz’s proof line, a second letter from 19 April 1893 shows that he considered Hilbert’s work to be mentioned as preparatory:

I made it my duty to fulfill your intention to mention that [your result] was already published in the Nachrichten der Gesellschaften der Wissenschaften zu Göttingen and at the same time I cited the work of M. David Hilbert in the same collection.³⁵ (Hermite 1893a, Br. 217)

Hurwitz’s proof was published on not more than one and a half printed pages in number 116 of the Comptes Rendus³⁶ which covered the period of January to June 1893.³⁷

5.2 Gordan and Hurwitz

The above briefly indicated involvement of Paul Gordan goes in fact much deeper. The Erlangen mathematician was not only curious about Hurwitz’s and Hilbert’s work, he had foremost his own ideas of simplifying the proof itself. Only a short time after his appreciative words to Hilbert, Gordan wrote a letter to Hurwitz on 25 April 1893:

I read your work on the transcendence of the number e with highest interest, however, I believe that some of your conclusions can still be simplified: [...] I beg for your opinion on this.³⁸ (Gordan 1893b, Br. 176)

What follows is the very elegant proof (see Sect. 8.3) based only on the exponential series concluded with the final remark: “Please tell me whether I am right!”. And Adolf Hurwitz approved, answering three days later, on 28 April,

I have no doubt that your new method only relying on the series expansion of $e^x$ for the proof of the vanishing of $F(k) - e^kF(0)$ for $p = \infty$ is completely in order [...] such that one now only has to operate with the simplest tools of analysis. Mister Hermite had asked me if he could show my proof to the French Academy (of course I agreed to this request with many thanks at that time.) At the same time he meant that now the last word about e had been spoken. I see now that Mister Hermite was in a gentle error at this point and I congratulate you that the last word was reserved to you.³⁹ (Hurwitz 1893c, Br. 58)

Indeed, Hermite agreed. In a letter from 6 May 1894 he answered to Gordan’s submission: “Your new proof which you honored me to communicate to me of the transcendence of the number e seems to me of foremost worth of interest and will arise the attention of all geometers.”⁴⁰ (Hermite 1893b, Br. 71)

Returning to the correspondence of Hurwitz and Hilbert, it seems that none of them could give up having “the last word”. In a letter from May 1, 1893, Hurwitz passed on the information to Hilbert about Gordan’s new proof only relying on the series expansion of $e^x$ and refers directly to his remark: “The letter was written quite typically for Gordan. He closed with the words: “Please tell me whether I am right!” ”⁴¹ Half a year later (and after a couple of letters without touching the subject of proofs of transcendence), on 6 January 1894, in a letter concerning his visit to Berlin the younger added: “I met Weierstrass in good health and he said that he read my e and $\pi$-note with pleasure.”⁴² (Hilbert 1894, Br. 150) Hurwitz replied on January 30, 1894:

Your reports from Berlin were of course very interesting to me. So Weierstrass is again completely mobile!⁴³ (Hurwitz 1893b, Br. 17)

6 Further Reception

In view of the various proofs of the three and their ambitions, so explicitly marked in the correspondence, regarding simplicity and capacity for further development of the argumentation, we distinguish in the following their reception in teaching and research (until the 1970s).

We have already mentioned Felix Klein and his role in publishing the three proofs on consecutive pages in an issue of the Mathematische Annalen. One may be curious about his judgement as to which of the given proofs is the most “simple”. In his famous lectures at the Evanston Colloquium , “delivered from Aug. 28 to Sept. 9, 1893, before members of the Congress of Mathematics held in connection with the World’s Fair in Chicago” (a forerunner of the later International Congresses of Mathematicians), Klein chose (in Chapter VII) Hilbert’s “very simple proof”.⁴⁴ In addition, he commented

Immediately after Hurwitz published a proof for the transcendency of e based on still more elementary principles; and finally, Gordan gave a further simplification. [...] The problem has thus been reduced to such simple terms that the proofs for the transcendency of e and $\pi$ should henceforth be introduced into university teaching everywhere.

In the probably first textbook presenting proofs of transcendence for e and $\pi$, namely the Lehrbuch der Algebra (Weber 1899, volume 2, §226) of Heinrich Weber, the author praised the proofs of Hilbert, Hurwitz and Gordan as conducted “with very elementary means and in the simplest way”⁴⁵ (see page 829) and his proof relies mainly on Hurwitz’s reasoning. The later textbook of Edmund Landau (1877–1938) presented a similar proof although without mentioning the simplifiers explicitly but just the names of Hermite and Lindemann (see Landau 1927, pp. 90–93). In his textbook, Oskar Perron (1880–1975) reproduced Hurwitz’s proof, however he writes (see Perron 1921, on page 174)

The first proof of this theorem is by H e r m i t e. The proof has been significantly simplified later by several authors, namely by H i l b e r t and W e b e r, keeping H e r m i t e ’s basic idea, so that it can be made quite elementary today.⁴⁶

The standard reference for number theory for decades, written by Godfrey Hardy (1877–1947) and Edward Wright (1906–2005), gives as well Hurwitz’s proof; as already mentioned in the introduction, the authors remarked that the original proofs of Hermite and Lindemann “were afterwards modified and simplified by Hilbert, Hurwitz, and other writers”.⁴⁷ In the North-American community, the booklet of Ivan Niven (1915–1999) was one of the first sources including proof of transcendences and there also Hurwitz’s proof is reproduced.⁴⁸ The only exception in early sources with a broad readership is the textbook of Alexander Ostrowski (1893–1986) who gave the proof of his mentor Hilbert (see Ostrowski 1954, pp. 85–88).

If we take a look at specialized literature and research in particular, a somewhat different picture emerges. In his important book Transcendental Numbers,⁴⁹ relying on notes from a lecture given at Princeton in 1946, Carl Ludwig Siegel (1896–1981) presents the transcendence results of Hermite, Lindemann, and the more general result of Weierstrass; his reasoning relies on Hermite’s original approach. Siegel wrote at the end of his proofs (on page 30):

It should be mentioned that the preceding proofs of the transcendency of e and $\pi$ [...] are not the simplest to be found in literature. Our proofs are related to the original work of Hermite; however, our procedure in constructing the approximation forms is somewhat more algebraic, and this has been decisive for the generalization which we shall investigate in the next chapter.

Siegel refers to his work on what is now called E-functions (from 1929; see also Shidlovskii (1989)) and later he discusses path-breaking results of his pupil Theodor Schneider (1911–1988), who proved Hilbert’s seventh problem on the transcendence of $\alpha ^\beta$ with algebraic $\alpha ,\beta$ (such that $\alpha \neq 0,1$ and $\beta$ is irrational) in 1934 independently to Aleksandar Gelfond (1906–1968); both reasonings rely on an analytic treatment of complex-valued functions.

That a transcendence problem found its way into Hilbert’s famous list of 23 problems for the twentieth century is, in retrospect, no wonder. But it is remarkable to realize at this point to what extent Hilbert developed in the short time from “Hurwitz’s pupil” in 1892 to one of the few universal mathematicians in 1900, parallel to the “changing of the generations in the German mathematical world” (cf. Gray 2000, p. 33). It is an interesting side note that in the context of his seventh problem (Hilbert 1902) referred to Hurwitz (1883)⁵⁰ indicating the latter one’s early interest in questions around transcendency.

Besides the Gelfond-Schneider theorem another important transcendency result of that time was Schneider’s treatment of abelian functions (in 1937). What we observe here is indeed a return to the roots! In his monograph (Schneider 1957, p. 47), Schneider makes his perspective pretty clear:

The fact that after the above-mentioned proofs of Hermite on the transcendence of e and of Lindemann on the transcendence of $\pi$ numerous further proofs on the same subject have been published, especially in the nineties of the last century, shows that these proofs, although perfect, appeared to be either unsatisfactory with respect to transparency or capable of improvement with respect to the means used. Special emphasis was put on the most possible elimination of analytical means and thus a situation arose which caused G. Hessenberg in 1911 to write his book on the “Transcendence of e and $\pi$” in order to work out clearly the basic ideas of the proofs [...]. Just by elementarization these basic ideas were veiled and the resulting generalizable approaches of Hermite and Lindemann were so narrowed and tailored to the exponential function and its inverse function, that also for this reason further results were not obtained.⁵¹

Schneider’s book was published in the renowned Springer series Die Grundlagen der Mathematischen Wissenschaften in Einzeldarstellungen (as volume LXXXI) and became a standard reference for some period. Concerning the issue of simplification, Baker wrote in (Baker 1975, p. 3) two decades later

The work of Hermite and Lindemann was simplified by Weierstrass in 1885, and further simplified by Hilbert, Hurwitz and Gordan in 1893. We proceed now to demonstrate the transcendence of e and $\pi$ in a style suggested by these later writers.

After giving the proofs, hoewever, he added (on page 8)

The above proofs are simplified versions of original arguments of Hermite and Lindemann and their motivation may seem obscure; indeed there is no explanation a priori for the introduction of the functions I and f. A deeper insight can best be obtained by studying the basic memoir of Hermite where, in modified form, the functions first occured, but it may be said that they relate to generalizations, concerning simultaneous approximation, of the convergents in the continued fraction expansion of $e^x$.

In his lecture notes, Kurt Mahler (1903–1988) discussed the various proofs arising in the 1890s in a final chapter in detail. He begun with the words (see Mahler 1976, p. 213)

We shall here collect a number of such proofs by different mathematicians and explain their relations. Since Siegel’s fundamental paper of 1929 (Chapters 4–9), entirely new and very powerful methods have been introduced into the theory of transcendental numbers. This has the danger that the easier and often very ingenious classical proofs may be forgotten.

Concerning the flexibilities, he later wrote (on page 243)

A. Hurwitz (1893) satisfies the divisibility conditions in a seemingly simpler way by taking his parameter equal to a sufficiently large prime. This is a very convenient choice, but it imposes unnecessary restrictions on the parameter without actually simplifying the proof. The disadvantage of Hurwitz’s choice becomes particularly evident if one wants to derive a measure of transcendence, say for e.

But of course, this was not the intention of Hurwitz and others who struggled for a simple proof; the measure of transcendence is a concept of the twentieth century.

We conclude with a brief evaluation of this compilation of receptions. If the attribute simple or elementary is intended to mean that a certain argument can be accomplished with as few additional aids as possible, then, of course, Gordan’s proof is the simplest. Comparing the proofs of Hilbert and Hurwitz, Hurwitz’s reasoning has to be regarded more simple since Hilbert’s argumentation needs in addition to Hurwitz’s use of the differential equation $\exp '=\exp$ also the fundamental theorem of differentiation and integration as well as partial integration (or basic knowledge of the Gamma-function); on the contrary, Hurwitz applies the mean-value theorem and the infinitude of primes, two results that can be considered more fundamental. This ranking may as well explain the choices of Hurwitz’s proof for general textbooks. If the focus is on research (e.g., the generalization to E-functions), however, then simplification is not of major interest but issues like flexibility or options for generalizations are relevant. In this sense, the generalization by Weierstrass (1885) had proven to be the most fruitful for the advancement of the transcendency proofs of the nineteenth century.