π

1

Dase (1824-61), 200 places, using the formula=tan-1+tan

1

4

11⁄2 +tan-1/ +tan-1; Richter, who extended the value to 500 decimal places, and Shanks, who carried it to 700 decimal places. These efforts are of value chiefly in showing the superiority of the modern over the ancient methods. Practically, as the late Professor Newcomb remarked, "ten decimals are sufficient to give the circumference of the earth to the fraction of an inch, and thirty decimals would give the circumference of the whole visible universe to a quantity imperceptible with the most powerful microscope." The results of these extended computations revealed nothing concerning the real nature of л, nothing as to whether it is rational or irrational, and nothing as to its possible transcendental character.

The foundation for the solution of the problem as to the nature of was furnished by Euler in connection with the formulas involving e, the base of the so-called Naperian logarithms, although first used as a base in the tables of John Speidell, published in London in 1619. Starting with Maclaurin's formula,

x2

ƒ(x) =ƒ (0) +ƒ′(0) · x+ƒ” (0) · 1 ·2+ƒ'''(0) · 1 ·2·3

x x2 x3

+ + 1 1.2

it is evident that

er = 1 +

1.2.3

+.

all being convergent series. It was by the help of these series that Euler (1707-83) showed that

an expression involving perhaps the five most interesting quan-
tities in mathematics. It is by means of this equation that
the transcendent nature of 7 was proved about a century
and a half after Euler's discovery.

Euler also gave numerous other relations between e and л,
and expressed in various ways the values of these numbers in
infinite series and products, and as continued fractions. For
example, he showed that the following relations exist:

The third period in the history of the study of begins with the work of the German mathematician Johann Heinrich Lambert (1728-77). In his treatise on the quadrature and rectification of the circle (1766) he set forth two fundamental propositions, viz.:

1. If x is a rational number, not 0, then er cannot be rational;

2. If e is a rational number, not 0, then x cannot be rational.

He reached these conclusions by starting with Euler's expression for (e-1), viz.:

and from these continued fractions he drew the conclusions stated, the proof not being rigorous. For the special case of

TC

x= we know that tan-1, whence he asserted that a can4

not be rational. The failure of Lambert to prove that the continued fraction

m

n+m'

n' +m"

n'+.

is irrational, the number of terms being infinite, m, m'...

and n, n' being integral, and

m m'
n' n'

being less than 1, was remedied by Legendre (1752-1833), who supplied the proof in his Eléments de géométrie (1794). With Legendre's work, therefore, the proof of the irrationality of may be said to have been settled, and to this he added a proof of the irrationality of 2.

The next noteworthy step was taken by Liouville (1809–82) in 1840, when he showed that e cannot be the root of a quadratic equation with rational coefficients, or, in other words, that if a, b, c are rational, ae2+be+c=0 is impossible. This was the first successful attempt toward verifying what Legendre had stated to be probable, that is of such a nature that it cannot be classed among algebraic numbers, that is, that it is not the root of any algebraic equation with a finite number of terms with rational coefficients. The question then, as it stood after the contribution of Liouville, was twofold: Of what, if any, algebraic equations with a finite number of terms with rational coefficients can e and be roots? Is it not possible to find numbers that are not roots of an algebraic equation of this kind? Legendre was the first to express the doubt contained in the second part of this question, and the doubt became a certainty when Liouville proved, in 1844, the existence of non-algebraic numbers and justified the division of numbers into algebraic and transcendental.

As the result of a careful investigation of the exponential function Hermite succeeded in proving, in 1873, that the

number e is transcendental, and Lindemann, in 1882, succeeded in proving the same for, basing his proof upon the labors of Hermite. Lindemann proved essentially that in an equation of the form ao +α1е” +α2е2 +ager +... =0, the exponents and coefficients cannot all be algebraic numbers. It therefore follows that in the Euler equation, 1+eiz=0, where the coefficients are algebraic, the exponent in is not algebraic, and hence is transcendental. While we shall not follow Lindemann's proof exactly, it is nevertheless necessary, as a preliminary to considering the nature of 7, to prove that e is a transcendental number.

3. The transcendence of e. Since Hermite first proved that e is transcendental others have materially simplified his treatment of the problem. The contributions of Hilbert, Hurwitz, and Gordan were published in the Mathematische Annalen in 1893. The Gordan proof was still further simplified by Weber in his Algebra, and later in the Encyklopädie der ElementarMathematik (1903), and Enriques, in his Fragen der Elementargeometrie (German edition, 1907), presents it in its latest form. To the last-named work the basis of the following proof is due but the proof has been materially simplified, chiefly through the kind assistance of Professor E. V. Huntington, of Harvard University, who planned the treatment for e, and who made the suggestion of using the cubic instead of the general equation, and of distinctly setting forth the three

lemmas.

To prove that e is a transcendental number means that it must be shown that e is not a root of any algebraic equation with rational coefficients. In other words, it must be shown that it is impossible to have a general equation of the form

Co+C1e+C2e2+...+Cnen = 0),

(1) where n is any positive integer, and where the coefficients Co, C1,.. ..., are any rational numbers, including 0, except that Co and C, cannot be 0, since this would change the degree of the equation.

n

In order to simplify the proof it is proposed to take a cubic