Adhémar Barré de Saint-Venant on Flow

If the velocities [of water in rivers] remained constant in each point of the traversed space, the surface of the liquid would look like a plate of ice and the herbs growing at the bottom would be equally motionless. Far from that, the stream presents incessant agitation and tumultuous, disordered movements, so that the velocities change in an abrupt and most diverse manner from one point to another and from one instant to the next. As noted by Leonardo da Vinci, Venturi, and especially Poncelet, one can perceive eddies, large and small, with a vertical mobile axis. One can also see, at the surface, bouillons, or eddies with a nearly horizontal axis, that constantly surge from the bottom and thus form genuine ruptures, with the intertwining and mixing motions that M. Boileau observed in his experiments.

Adhémar Barré de Saint-Venant, 1872.

Fechner-Weber

The Fechner-Weber law states that for a sensation’s intensity to increase in an arithmetic progression, the stimulus must increase in a geometric progression. This relation describes sensory perceptions and physical stimuli for hearing. Human hearing can detect noise so quiet that the eardrum moves less than an atom’s width, and noise 10 trillion times more powerful. Similarly, the faintest star we can see is about 10 trillion times less powerful than the Sun. In a noisy environment, we can distinguish quieter sounds, such as conversation or a dropped coin. Senses respond not to the absolute increase in a stimulus, but to fractional increase. Ernst Heinrich Weber showed in 1846 that the change in a person’s perception of weight was proportional to the logarithm of any increase. For acoustics, as a stimulus becomes ten times stronger, the perceived increase only doubles. Gustav Fechner elaborated on Weber’s discovery in 1860. The Fechner-Weber law can be experienced by halfing noise. If our sense of hearing was linear with respect to intensity, that would reduce the noise by half. However, although the sound power in the room has halved, the difference in the volume we hear is barely noticeable.

References:

Weber, E.H., 1851. Annotationes anatomicae et physiologicae: Tractatus de motu iridis summa doctrinae de motu iridis. Koehler.

Fechner, G. T. (1860). Elemente der Psychophysik.

Liouville and Numbers

In 1844, Joseph Liouville demonstrated that the decimal representations of certain numbers were infinitely long and lacked pattern. This idea, which suggests that numbers do not necessarily have an exact and finite value, was first proposed by Greek philosopher Zeno in the 5th century BCE. Zeno’s paradoxes are based on the infinite divisibility of space. The resolution to these paradoxes was not found until the development of calculus by Leibniz. They demonstrated that an infinite geometric series can converge, balancing the infinite number of “half-steps” traveled with the increasingly short amount of time needed to cross the decreasing distances.

A rational number is a number that can be expressed as a fraction, p / q, where p and q are integers. Familiar examples of irrational numbers include pi and root 2. Transcendental numbers, a subset of irrational numbers, cannot be expressed using algebra; they are not roots of a polynomial with rational coefficients. While Liouville failed to prove that e is a transcendental number, he did construct an infinite class of transcendental numbers using continued fractions. In 1851, he produced an example of a transcendental number now known as Liouville’s constant, an infinite string of zeros and ones with a one positioned at every value of the exponential factorial, n!. In 1873, e was shown to be transcendental, and pi was proven to be so in 1882. Most numbers are transcendental; those with definable patterns are in the minority.

References

  • Liouville, J. (1851). Sur des classes très étendues de quantités dont la valeur n’est ni algébrique, ni même réductible à des irrationnelles algébriques. Comptes Rendus de l’Académie des Sciences, 32, 135.
  • Hermite, C. (1873). Sur la fonction exponentielle. Comptes Rendus de l’Académie des Sciences, 77, 18-24, 74-79, 226-233, 285-293.
  • Lindemann, F. (1882). Über die Zahl π. Mathematische Annalen, 20(2), 213-225.

Daniel Bernoulli on Jean le Rond d’Alembert

I have seen with astonishment that apart from a few little things there is nothing to be seen in his hydrodynamics but an impertinent conceit. His criticisms are puerile indeed, and show not only that he is no remarkable man, but also that he never will be.

Daniel Bernoulli on Jean le Rond d’Alembert

On the Computer

Down the rabbit hole on digital calculations. Computers, as programmable tools, trace their origins to the 1800s. Joseph Jacquard’s loom, which used punched cards for pattern storage, indirectly influenced the field (1800s). C. Babbage, inspired by the loom, built the Difference Engine in 1822 for mathematical calculations and later designed the Analytical Engine, the first computer with memory and programmability [2]. Lord Kelvin in the 1870s created an analog computer for tide calculation, notable for its accuracy and long-term use [3]. In some places, still used today for its simplicity and accuracy.

References

  • Jacquard, J. (1800s). Programmable loom development.
  • Babbage, C. (1822). Difference Engine and Analytical Engine.
  • Kelvin, Lord (1870s). Analog computer for tides.

Complex Polynomials

The Fundamental Theorem of Algebra states that the field of complex numbers is algebraically closed, implying that every polynomial equation of degree n has n roots within the complex numbers, with at least one being a solution where the polynomial evaluates to zero. Historically, the theorem’s origin traces back to the conjectures by Albert Girard in 1629 and René Descartes in 1637, though neither provided proofs including complex numbers. A first significant proof attempt was made by Jean d’Alembert (my personal hero) in 1746. The most notable advancements were made by Carl Friedrich Gauss, who, in 1799, offered a proof that, despite its gaps, marked a significant step forward in understanding the theorem. Gauss’s efforts were refined by Jean Robert Argand in 1814, who introduced an existence proof that set the stage for more concrete proofs, including those by Gauss himself in subsequent years using Euler’s earlier work. The theorem’s proof evolution highlights the gradual understanding and acceptance of complex numbers in solving polynomial equations, an understanding put further by Hellmuth Knesser (1940).

References

  • Girard, A. (1629). Invention Nouvelle en l’Algèbre.
  • Descartes, R. (1637). La Géométrie.
  • d’Alembert, J. (1746). Recherches sur le calcul intégral.
  • Gauss, C. F. (1799). Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse.
  • Argand, J. R. (1814). Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques.

Personal Equation

1796, Maskelyne, Astronomer Royal of Britain, dismissed his assistant for what he believed to be persistent inaccuracies in the timing of observations, specifically delays of approximately half a second. This decision unknowingly set the stage for the development of an important concept in measurement science: the personal equation. Maskelyne’s published their mixed observational results. After Maskelyne’s death, the issue was investigated further by F. Bessel, a German astronomer. Bessel’s research revealed a consistent measurable difference in the timing of observations made by different astronomers. He termed this difference the “personal equation,” highlighting its role in scientific measurements.

The concept of the personal equation stimulated significant research in the newly emerging field of psychology in the late 19th century. Researchers focused on understanding individual differences in reaction times, which could affect observational accuracy. The term personal equation entered general use, applied to any personal influence on objective situations. This illustrates how a specific scientific concept can expand beyond its original context to gain wider societal acceptance.

Important References

  • Astronomische Beobachtungen auf der Koniglichen Universitiits-Sternwarte F. W. Bessel, 8. Abtheilung voe 1. Januar bis 31. December, 1822.

Backus on IBM / Fortran

Much of my work has come from being lazy. I didn’t like writing programs, and so, when I was working on the IBM 701, writing programs for computing missile trajectories, I started work on a programming system to make it easier to write programs.

John Backus, 1979, Interview IBM Think Magazine

Origins of Graph Theory

Graph theory emerged in the 18th century, connecting geometry with fields like topology and set theory. Leonhard Euler formulated graph theory during his time in Königsberg, now Kalingrad. His seminal work began with the 1736 paper, “The Seven Bridges of Königsberg.”

Residents of Königsberg enjoyed crossing the city’s seven bridges in one outing – considered impossible, but not proven. Euler approached this challenge not by traditional geometry or algebra, but through an abstraction where cities and bridges became vertices and edges in a graph. This approach showed that a path crossing each bridge once was impossible if more than two vertices had an odd number of edges.

Euler’s insights created the foundation for graph theory, now used in various applications from computer algorithms for facial recognition to network and process design. Or perhaps a bit of frustration by myself in graduate school studying math in the physics department at Penn State. Crossing all the bridges became possible later on in Kalingrad due to war, as they were partly destroyed.

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