"An expert knows more and more about less and less until he or she knows everything about nothing."This saying is apparently a comedic amplification of a shorter one used by William Warde Fowler (1847-1921) of Oxford University in a 1911 book review in the journal,

• Astrophysics (Including Astrophysics of Galaxies; Cosmology and Nongalactic Astrophysics; Earth and Planetary Astrophysics; High Energy Astrophysical Phenomena; Instrumentation and Methods for Astrophysics; Solar and Stellar Astrophysics)

• Condensed Matter (Including Disordered Systems and Neural Networks; Materials Science; Mesoscale and Nanoscale Physics; Other Condensed Matter; Quantum Gases; Soft Condensed Matter; Statistical Mechanics; Strongly Correlated Electrons; Superconductivity)

• General Relativity and Quantum Cosmology

• High Energy Physics (Including High Energy Physics-Experiment High Energy Physics-Lattice High Energy Physics-Phenomenology High Energy Physics-Theory)

• Mathematical Physics

• Nonlinear Sciences (Including Adaptation and Self-Organizing Systems; Cellular Automata and Lattice Gases; Chaotic Dynamics; Exactly Solvable and Integrable Systems; Pattern Formation and Solitons)

• Nuclear Experiment

• Nuclear Theory

• Physics (Including Accelerator Physics; Applied Physics; Atmospheric and Oceanic Physics; Atomic and Molecular Clusters; Atomic Physics; Biological Physics; Chemical Physics; Classical Physics; Computational Physics; Data Analysis, Statistics and Probability; Fluid Dynamics; General Physics; Geophysics; History and Philosophy of Physics; Instrumentation and Detectors; Medical Physics; Optics; Physics and Society; Physics Education; Plasma Physics; Popular Physics; Space Physics)

• Quantum Physics

*A milestone in physics specialization was reached in 1970 when Physical Review, the pre-eminent physics journal published since 1893, split into four journals; namely, A (Atomic, Molecular, and Optical Physics), B (Condensed Matter, Structure, Phase Transitions, Nonordered Systems, Magnetism, Superconductivity, Superfluidity, Electronic Structure, Semiconductors, Surfaces, Low Dimensions)), C (Nuclear Physics), and D ((Particles, Fields, Gravitation, and Cosmology)).An additional journal, E (Statistical, Nonlinear, and Soft Matter Physics), was added in 1993.Annual online subscriptions are presently $70 for each.(Modified Wikimedia Commons image by Inductiveload.)*

My favorite polymath is Carl Friedrich Gauss (1777-1855), who made significant contributions to mathematics and many fields of physics. In common with many polymaths, Gauss's mental abilities were apparent at a very early age, as an interesting anecdote from the history of mathematics illustrates.[2] As the story goes, Gauss was an eight year old elementary school student who was given a lengthy mathematics problem by his teacher. It's a common tactic for teachers to assign such "busy work" problems when they need time to grade homework or tests. The assignment, sure to take the students quite a lot of time, was to add together all the numbers from one to a hundred. Gauss immediately arrived at the correct answer, 5050, with no written calculation. Gauss had realized that it was possible to pair the numbers, 1 with 99, 2 with 98, etc., to give you 49 sums of a hundred each. Adding the additional 100 and 50 immediately gave the answer, 5050.[2] There's a general formula for calculating the sum of such a XX up to the number

This gives us Gauss' answer when

*Carl Friedrich Gauss (1777-1855).Gauss has a plethora of things named after him, including a unit of magnetism, the gauss, that's been superseded by the tesla.As I wrote in an earlier article (Great Circle Routes, June 25, 2018), Gauss did an experiment in the 1820s to determine whether space was curved; that is, non-Euclidean.He used his era's version of a laser theodolite, called a heliotrope, to survey a triangle between three mountains - Brocken, Hohenhagen, and Inselberg.This triangle had sides of length 69, 85 and 107 kilometers, but even such a large triangle has the now known difference in the sum of angles from 180 degrees of just 0.1 picoradians.(An 1840 oil portrait by the danish painter Christian Albrecht Jensen (1792-1870), via Wikimedia Commons)*

Fourier transforms are ubiquitous in signal analysis, and they are most importantly used to calculate the frequency spectrum of time-series signals. Everyone desires more speed in calculations, and the Fourier transform was improved by an algorithm called the Fast Fourier transform (FFT), the first such example of which was the Cooley–Tukey FFT algorithm, named after its creators, James Cooley (1926-2016) and John Tukey (1915-2000).[3] Interestingly, the mathematics behind the Cooley-Tukey FFT was anticipated by Gauss around 1805 and published in 1866 after his death.[4-5] Gauss developed this mathematics in calculating the orbits of the asteroids, Pallas and Juno. This work of Gauss even predated that of Joseph Fourier (1768-1830), who published preliminary results in his 1807 Mémoire sur la propagation de la chaleur dans les corps solides (Treatise on the propagation of heat in solid bodies), with a more complete treatment in his Théorie analytique de la chaleur (Analytical theory of heat) in 1822. Fourier did not know of Gauss' work, since it was only published in 1866.

*A trigonometric identity from the paper of Carl Friedrich Gauss in the section containing mathematics relevant to the Fourier transform. (Via the University of Göttingen.[4])*

Author and journalist, Andrew Robinson, has published a review of the recent book, "The Polymath: A Cultural History from Leonardo da Vinci to Susan Sontag," by Peter Burke (b. 1937), an emeritus professor of cultural history at Cambridge University.[6-7] The appendix to Burke's book lists 500 "western polymaths," among whom are counted Charles Darwin, Alan Turing, Linus Pauling, and Thomas Young.

*Linus Pauling graciously autographed a reprint of one of his papers that he sent to me. Like Pauling, I also published a paper on the ^{57}Fe isotope of iron.[8]*

The myriad interests of da Vinci are known to most people. However, Thomas Young, who was regarded as the greatest polymath since da Vinci, is known mostly to physicists; and, then, just for his famous double slit experiment that demonstrated the wave nature of light. Young's tombstone at Westminster Abbey remembers his being "eminent in almost every department of human learning."[6] Another of Young's accomplishments was in his initial deciphering of the text of the Rosetta Stone as a clue to understanding Egyptian hieroglyphics. Burke identifies the principal characteristics of polymaths as "unusual powers of concentration, capacious memory, speed, imagination, restlessness, industriousness, and obsession with not wasting time."[6] These characteristics likely arise during a person's early years; so, I don't think polymath coaching of adults would ever work.

- A Specialist Knows More and More About Less and Less, Quote Investigator.
- Clever Carl, NRICH team of the Millennium Mathematics Project, February, 2011.
- J.W. Cooley and J.W. Tukey, "An algorithm for the machine calculation of complex Fourier," Math. Comput., vol. 19 (1965), pp. 297-301. A PDF file is available here.
- Carl Friedrich Gauss, "Theoria interpolationis methodo nova tractata", Werke, Band 3, Königliche Gesellschaft der Wissenschaften, Göttingen, 1866), pp. 265-327 (Via the University of Göttingen.
- James W. Cooley and John W.Tukey, "On the Origin and Publication of the FFT Paper," Citation Classics, Current Contents, vol. 33, nos. 51-52 (December 20-27, 1993), pp. 8-9.
- Andrew Robinson, "A history of insatiable intellectuals," Science, vol. 369, no. 6507 (August 28, 2020), p. 1064, DOI: 10.1126/science.abb7546
- Peter Burke, "The Polymath: A Cultural History from Leonardo da Vinci to Susan Sontag, Yale University Press, September 8, 2020, 352 pp. (Via Amazon).
- D.M. Gualtieri, W. Lavender and S. Ruby,
^{57}Fe-YIG: Narrow Xray Linewidth Epitaxial Layers on Gd_{3}Ga_{5}O_{12}, J. Appl. Phys, vol. 63, no. 8 (1988), pp. 3795-3797, https://doi.org/10.1063/1.340617.