A Chronology of e

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Note: This page uses R(n) to indicate the square root of n (this notation was commonly used before the modern day notation was adopted). This page also uses pi for the irrational number 3.14159..., phi for the golden mean (which is the irrational number 1.618...), and ^ to indicate exponents.

1618
A table of natural logarithms appeared in Wright's English translation of Napier's work on logarithms, it was probably created by Oughtred, but he does not mention e. Napier's logarithms are actually logarithms with base 1/e, though he never thought of them in terms of a base.

1624
Briggs gives a numerical approximation for the base 10 logarithm of e, but does not mention e itself.

1661
Huygens studies the area under the rectangular hyperbola xy=1. The area under this curve from 1 to e is 1, which is why e is the base of the natural logarithms, but Huygens does not realize this. Huygens also calculates the logarithm to the base 10 of e to 17 decimal places, but again he fails to realize its importance.

1665
Newton calculates pi to at least 16 decimal places using his own idea, but his results are not published until 1737, after his death. This same year, Newton also comes up with the MacLaurin series for e.

1668
Mercator publishes Logarithmotechnia, which uses the term "natural logarithm" for the first time for logarithms to base e. The number e itself again fails to appear as such and again remains elusively just around the corner.

1683
Jacob Bernoulli investigates the limit of (1 + 1/n)^n as n approaches infinity. He uses the binomial theorem to show that this limits lies between 2 and 3 (the first approximation for e). This is the first time a number is defined by a limiting process.

1690
The number e appears in its own right in a letter from Leibniz to Huygens, although he calls it b.

1727
Euler introduces the symbol e for the base of natural logarithms in a manuscript entitled Meditation upon Experiments Made Recently on Firing of Cannon. The manuscript was not published until 1862.

1737
Euler is able to show that both e and e^2 are irrational and gives several continued fractions involving e. Euler popularizes the use of the symbol pi for the ratio of the circumference to the diameter of a circle. The year before, he had determined a simple series for (pi^2)/6.

1748
Euler publishes Analysis Infinitorum (Analysis of the Infinite) which is an introduction to mathematical analysis. The famous formula e^(i*pi) = -1 appears for the first time in this text, as well as many series for pi and pi^2. A series for e appears, as well as the fact that e is the limit as n approaches infinity of (1 + 1/n)^n. Euler also approximates e to 18 decimal places and gives several continued fractions involving e.

1757
Riccati first notices that the hyperbolic trigonometric functions behave very similarly to the trigonometric functions. He introduces a notation that is only slightly different from what we use today for the hyperbolic sine and cosine.

1761
Lambert proves that pi is irrational. He publishes a more general result in 1768. He also shows that the functions e^x and tanx cannot assume rational values if x is a non-zero rational number.

1814
Barlow produces Barlow's Tables which give factors, squares, cubes, square roots, reciprocals and hyperbolic logs of all numbers from 1 to 10000.

1854
Shanks calculates e to 205 places, but in 1884, Boorman shows that only 187 were correct.

1864
Benjamin Pierce has his picture taken in front of a blackboard with the formula i^-i=R(e^pi) inscribed on it. In 1859, he had tried to introduce new symbols for e and pi, but they did not become popularly accepted.

1873
Hermite publishes Sur la Fonction Exponentielle (On the Exponential Function) in which he proves that e is a transcendental number.

1884
Boorman calculates 346 places of e.

1887
Adams calculates the base 10 log of e to 272 places.

1896
Hadamard and Vallee-Poussin show that the number e is indirectly connected with the prime numbers.

1934
Gelfond and Schneider solve "Hilbert's Seventh problem" independently. They prove that a^q is transcendental when a is algebraic (and not equal 0 or 1) and q is an irrational algebraic number. Gelfond proves that e^pi is transcendental.

Last Modified on Saturday, August 10, 2002.