Mathematics

van der Waerden and the Ascending Chain Condition

 van-der-Waerden in 1930
B. L. van der Waerden in 1930 (fl. 1903-1996)

A truism in mathematics, and perhaps most subjects, is that its disseminators are as important as its creators. Niccolò Tartaglia (1499-1557) comes to mind, a great and prolific encyclopedist whose works were consulted in his own day and for generations to come. Perhaps the premiere example is Frans van Schooten and his associates, who did so much to organize and present Descartes' deep but scattered and somewhat opaque geometric approach in La Geometrie. van Schooten's "appendices" amplified La Geometrie massively, the 1659-61 edition becoming a textbook for Leibnitz and Newton and the rock that calculus was built on. Of course the disseminators are mathematicians too, among the select few who expand the new path as well as trod it ("Cartesian" coordinates appear nowhere in Descartes, for example, but were introduced by his acolytes). They cast a great net, subsuming existing mathematics into the new framework as they organize and simplify, all the while spreading the word and making it available to a broad audience of practitioners, many having already worked obscurely to clear patches of brambles and more than ready for the big breakthrough.

Resurrecting Tartaglia

 Tartaglia - Questi
Tartaglia on the cover of his Quesiti (1546).

Niccolò Tartaglia (1499-1557) was badly used by Cardano and his disciples and has been deprecated ever since, as if to prove the old adage about hating those the most whom you've unjustly injured the most. His partial solution of the cubic equation, expropriated by Cardano, should have propelled him into the first rank of mathematicians of his day[1]. Instead, it's been used to run him down, the supposedly small-minded foil to the great Cardano. In fact, the impoverished, self-trained Tartaglia was highly productive and exemplified the forward-looking mathematizing approach to the new science leading in direct line to Galileo and the modern approach generally.[2] First and foremost, Tartaglia was a maestro d'abaco, that is, a teacher of the new art of calculation with pen and paper. This tradition centered in vernacular commercial schools outside the universities, sidestepping the barren Boethian preoccupations of "mathematicians" debating endlessly in Latin quaestios about such matters as the true nature of even and odd numbers and what that might mean for the cosmos.

Gerbert's Abacus Illustrated

 Gerbert's abacus
Gerbert's abacus with representative numbers

Gerbert of Aurillac was a Benedictine monk from this small village in south-central France who was born about 950 AD and died in 1003. He was a great teacher and scholar, politician and church leader. He became Pope Sylvester II in 999 and served in that role till his death. He was the first French pope and the only mathematician to become pope. And mathematician he was, sent as a young man by his abbot to study in Christian Spain, bordering Al-Andalus, then a shining light of Islamic civilization. The first documented appearance of the Hindu-Arabic numerals in Christian Europe was in 976 in a manuscript by Vigila, a monk in the vicinity of Barcelona, then a Christian outpost in northern Spain near where Gerbert resided from 967 to 970. Gerbert brought the Hindu-Arabic numerals he encountered in Spain back to Christian Europe for an abacus / counting board he invented which was subsequently named after him.

Pascal on Sums of Powers of Integers

 Blaise Pascal
Blaise Pascal (1623–1662)

Pascal's celebrated paper Traité du triangle arithmétique (Treatise on the Arithmetic Triangle)[1] was written in 1654 and published in 1665. Though since named for him because of this work, "Pascal's" Triangle was long known in Europe before 1650 and in other civilizations too[2]. The Traité contains a short appendix, Potestatum Numericarum Summa (The Sum of Powers of Numbers)[3], showing how to derive polynomial expressions for sums of powers of integers like:
\begin{equation}{\sum_{k=1}^n k^2 = {1^2 + 2^2 + 3^2 + \cdots + n^2} = {{1 \over 6} n (n + 1)(2n+1)} = {{1 \over 3}n^3 + {1 \over 2}n^2 + {1 \over 6}n}.}\tag{1} \end{equation}

CORDIC in C and Javascript

 How CORDIC works
How CORDIC trig works, courtesy of Wikipedia.

CORDIC is is a complex of fast algorithms to calculate transcendental functions using only table lookup, addition and bit shifting. Here I take up Volder's original scheme from 1959 to calculate sines and cosines quickly (CORDIC stands for COordinate Rotation DIgital Computer). My original article from 1992 holds up reasonably well, The CORDIC Method for Faster sin and cos Calculations, except for the Borland graphics — nice pre-Windows, but long-gone. Converting to Visual C requires only one change to the core routines related to moving from 16-bit to 32-bit integers (see comment in code). I've bundled up the source files for anyone who wants to take them into a Visual C Win 32 application, and here is a minimal C implementation. Wikipedia has a first-class article on this subject with lucid explanations and many pertinent references. Volder himself published an illuminating retrospective in 2000, explaining that the initial motivation was digitizing the navigation system of Convair's B-58 jet bomber (Volder worked for Convair).

Hilbert Proves that e Is Transcendental

 David Hilbert in 1912
David Hilbert, 1862-1943 (photo 1912)

It's easy to prove that \( e \) is irrational, but takes more work to prove that it's transcendental, meaning not the root of any equation:

\[ \begin{equation}{a_0 + a_1 x + \cdots + a_{n-1} x^{n-1} + a_n x^n = 0, \hspace{16pt} a_i \in \mathbb{Z}.}\tag{1} \end{equation} \]

Put otherwise, it is never the case that \( a_n e^n + a_{n-1} e^{n-1} + \cdots + a_1 e + a_0 = 0 \; \) for any choice of integers \( a_0, a_1, \cdots, a_{n-1}, a_n \). We already have the result for \( n = 0 \) since \( e \) is not an integer, and for \( n = 1 \) since \( e \) is not the root of any equation \( bx - a = 0 \) for integers \( a \) and \( b \) (simply restating that \( e \) is irrational). In 1840, Liouville proved that \( e \) is not the root of a quadratic equation with integer coefficients, but the final object is to prove that \( e \) never satisfies \( (1) \) for any choice of integers \( a_i \) and any \( n = 0, 1, 2, 3, 4, \ldots \).

Liouville Proves That e Is Not the Root of a Quadratic Equation

 Liouville on e

More properly, \( e \) is not the root of a quadratic equation with integer coefficients, which is the same as saying rational coefficients, because denominators can be cleared. This is sometimes stated as \( e \) is not a quadratic irrationality. Liouville proved the theorem in his journal in 1840.[1] It was a step towards proving that \( e \) is transcendental, meaning that it is not the root of any polynomial equation with integer coefficients. Apparently Liouville regarded it as such[2], but couldn't push through the general proof. Hermite was the first to prove that \( e \) is transcendental in 1873.[3] Hilbert simplified the proof[4], among others. But let's get back to the quadratic case and Liouville's proof. Here it is in its entirety, translated into English:

Fourier's Proof that e Is Irrational

 de Stainville's Melanges

Janot de Stainville ascribes this proof to Joseph Fourier.[1] Start with:

\[ \begin{align*}
e &= \sum_{n=0}^\infty{1 \over n!}\\
&= 1 + {1 \over 1!} + {1 \over 2!} + {1 \over 3!} + {1 \over 4!} + \cdots\\
&\lt 1 + 1 + {1 \over 2} + {1 \over 2^2} + {1 \over 2^3} + \cdots\\
&= 3.
\end{align*} \]

The reason for the inequality is that each factorial is greater than a product of as many twos as there are factors in the factorial, discounting the one; for example, \( 4! = 1 \cdot 2 \cdot 3 \cdot 4 \gt 1 \cdot 2 \cdot 2 \cdot 2 = 2^3. \) The final step results from summing the geometric series. It follows that \( 2 \lt e \lt 3 \) and in particular, that \( e \) is not a whole number.

Eratosthenes Measures the Earth

 Eratosthenes Measures the Earth
Eratosthenes Measures the Earth

Like Aristotle[1] before him and in keeping with virtually all educated opinion in ancient Greece[2], Eratosthenes (c. 276 BC - c. 194 BC) assumed the earth was spherical. He set out to measure its size using a sound method that has stood the test of time. His results were good (about \( 5.4\% \) too high) and no one did better for over a thousand years. His near correct result was ignored well into the modern era, including by Columbus. Snellius and Picard finally nailed down the earth's circumference in the seventeenth century using Eratosthenes' experimental design and taking advantage of advances in mathematics and instrumentation that had accrued over the intervening \( 1900 \) years.

Chinese Remainder Theorem Calculator

 Chinese Remainder Theorem example

A system of three congruences is shown on the right, but start with the simpler system:
\[ \begin{align*}
x &\equiv 1 \hspace{-.6em} {\pmod{2}}\\
x &\equiv 2 \hspace{-.6em} {\pmod{3}}.
\end{align*} \]
Values congruent \( \text{mod} \; 6 \) are certainly congruent \( \text{mod} \; 2 \) and \( \text{mod} \; 3, \) so in looking for an \( x \) solving both congruences simultaneously, it suffices to consider congruence classes \( \text{mod} \; 6 \) and in particular their smallest positive residues, namely \( 0, 1, 2, 3, 4, 5. \) We're seeking an odd number among those \( 6 \) since \( x \equiv 1{\pmod{2}}, \) one that is also congruent to \( 2 \; \text{mod} \; 3. \) \( x = 1 \) won't do, since \( 1 \equiv 1{\pmod{3}} \) neither will \( x = 3, \) since \( 3 \equiv 0{\pmod{3}}. \) \( x = 5 \) is the solution, since it satisfies both congruences, and it is the only solution \( \text{mod} \; 6. \)

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