Name: A treatise on electricity and magnetism
ISBN: 9780198503736

A treatise on electricity and magnetism

A Treatise on Electricity and Magnetism is a two-volume treatise on electromagnetism written by James Clerk Maxwell in 1873. The work is described as both "celebrated" and "rambling and difficult", while admired as a pioneering mathematical source to understand electromagnetic theory. The four famous Maxwell's Equations were derived from its content in the next decade. Maxwell was revising the Treatise for a second edition when he died in 1879. The revision was completed by William Davidson Niven for publication in 1881. A third edition was prepared by J. J. Thomson for publication in 1892.

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adrian_bApr 2026

Phase angle is not dimensionless.

I have explained in another comment why this is so.

This misconception about angles being dimensionless quantities is unfortunately taught in many school textbooks and it cripples the thinking about physics of many people.

The foundation of physics is the theory of the measurement of physical quantities. However, when teaching physics frequently this is taught either badly or not at all, instead of following classical examples, like that of James Clerk Maxwell, who started his Treatise on Electricity and Magnetism with an exposition of the theory of the measurement of physical quantities that was complete and up-to-date for that time.

The plane angle a.k.a. phase angle not only is not a dimensionless quantity, but its unit of measurement plays a crucial role in determining the other base units of any modern system of units of measurement.

The reason why the unit of plane angle is so important is that plane angle is the only fundamental continuous quantity for which it is possible to approximate measurement by counting, because the numeric value of a plane angle measured in cycles can be decomposed in a sum of a discrete quantity, the integer part of the numeric value, with a continuous quantity, the fractional part of the numeric value. The value of the discrete quantity that is the integer part of the numeric value of an angle measured in cycles can be obtained by counting.

This unique property is the cause that for the highest possible precision in measurements all continuous quantities are converted by various methods into phase angles, before the analog-to-digital conversion.

The fact that plane angle has a natural unit, which is the cycle (or its integer multiples or submultiples) is exploited in defining almost all units of other continuous quantities.

For instance, the unit of length is said to be the wavelength of a certain wave. Because wavelength is a physical quantity equal to the ratio between length and plane angle, the previous definition stated correctly says that the unit of length is equal to the unit of plane angle multiplied with a certain wavelength (i.e. 1 meter = 1 cycle multiplied by a wavelength measured in meter per cycle). Similarly for many other continuous quantities, whose units are also derived in one form or another from the unit of plane angle, because only it can be defined intrinsically, instead of being derived from other units.

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rramadassAug 2026

Well said.

Thanks for calling out these sort of posers and charlatans on HN. We should not tolerate these people if we are to discuss/argue/motivate interesting/hard subjects productively.

I automatically discount anybody on HN (until i have looked at their profile/comment history/any personal bio websites etc.) who claim they have read/studied a) Euclid's Elements b) Newton's Principia c) Maxwell's Treatise on Electricity and Magnetism d) Einstein's 1905 Annus Mirabilis papers. e) Principia Mathematica by Russell/WhiteHead f) Godel's Theorem g) Bourbaki's mathematics books etc. etc. They might have browsed it out of curiosity but that is not the same as reading/studying it.

Actual conceptual mathematics/science is intrinsically hard even ignoring the archaic language/notations.

As a good example; the Nobel-prize winning physicist S.Chandrasekhar wrote Newton's Principia for the Common Reader where he explains a subset of the principia (only dealing with gravitation) using modern notation and language. He himself found it quite hard and thus the "common reader" in the title is somebody who has had a good course in calculus and has the motivation to put forth the effort in understanding it.

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gjm11Jul 2026

I'm not sure that's right. The conjecture is (claimed to be, by the people who explicitly made it) simply a reformulation of a claim made by Maxwell in his "Treatise on Electricity and Magnetism" of 1873.

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dualvariableJul 2026

From the intro to the paper:

> In J. C. Maxwell’s 1873 treatise on electricity and magnetism he discusses the number of equilibria of the electric field generated by n point charges [5, §113]. Apparently unaware of this, M. Morse and S. S. Cairns in 1969 posed the problem of finding an upper bound for the number of equilibria [6, p. 293]. The first general bounds were supplied by A. Gabrielov, D. Novikov, and B. Shapiro in [3] who, based on their reading of [5, §113], formulated the ‘Maxwell conjecture’ which states that if the critical points of the electrostatic potential generated by n point charges are all non-degenerate then their number cannot exceed (n − 1)^2. These bounds were later improved by V. Zolotov in 2023 [8] and further improved by H. Edelsbrunner, C. Fillmore, and G. Oliveira in 2026 [2]. Maxwell’s bound is trivially achieved for n = 2 but it is not known even for n = 3 if 4 is the maximum number, except in the case of equal charges [7]. Further related problems in classical electrostatics are discussed in [1].

And reference 3:

> [3] A. Gabrielov, D. Novikov, and B. Shapiro, Mystery of point charges, Proc. Lond. Math. Soc. (3), 95 (2007), pp. 443–472.

This is pretty niche and the conjecture was only proposed about 20 years ago. It was actually not conjectured by Maxwell himself.

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momoschiliSep 2024

Going to need some citations here since the texts that I'm familiar with from that time period are "A Treatise on Electricity and Magnetism" by Maxwell (mid-late 1800s) and "A History of the Theories of Aether and Electricity" by E. T. Whittaker, neither of which mentions anything of the sort. I suspect you are choosing from texts that at the time likely would not have been considered academic or standard.

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segfaultbuserrJun 2024

> Maxwell was also not really taking advantage of them much and always split them up into scalar and vector part.

Did Maxwell actually use quaternions? If I recall correctly, at least in A Treatise on Electricity and Magnetism, quaternions were not actually used. Instead, he did most things in Cartesian coordinates, and all equations were applied to a vector's x, y, z components tediously. But many sources claimed Maxwell used quaternions, including quotes from Lord Kelvin. My reading on this part of history is limited, so my guess is that he did use them in personal research or in later papers. On the other hand, some other physicists of the same era used quaternions extensively, including applying them to Maxwell's electromagnetism, that is a sure fact...

Coincidentally, A Treatise on Electricity and Magnetism was written as an overview all electromagnetic phenomena as a whole, so it paid very little special attention to the generation and transmission of electromagnetic waves. Combining that with its difficult math, the book would puzzle physicists for another decade before they see the light from the book, and made it a rather curious period of history in electromagnetism.

> I've been trying to find these 4 equations in Heaviside's writing but so far have not been successful.

In 1885, Heaviside published Electromagnetic Induction and Its Propagation in The Electrician, and formulated what he called the "Duplex Form" of Maxwell's equation. This was a long series of papers published in several months, and later republished in Electric Papers, Volume I. Basically, following his physical intuition, he felt that electric and magnetic fields should be symmetric and generate each other, and that should be directly highlighted in equations.

The logic of the paper went like the following.

First, he started with a definition of electric current [1]:

    C = kE
    D = cE / 4π
    Γ = C + D

in which, E denotes electric force, C denotes conduction current, k denotes specific conductivity constant, D denotes displacement current, and c denotes dielectric constant. Finally, Γ denotes true electric current, which is the sum of the conduction and displacement terms.

Next, a definition of magnetic current [2]:

    B  = µH
    G  = Ḃ / 4π = µḢ / 4π
    G' = gH + µḢ / 4π

H denotes magnetic force, B denotes magnetic induction, µ denotes permeability, G denotes magnetic current, Ḃ and Ḣ are derivatives of B and H (Newton's notation). Hypothetically, suppose that magnetic monopoles exist (Heaviside did so), G' would denote the "true magnetic current", with an extra conduction term gH, where g is a constant similar to k.

Then, he introduced the concepts of divergence and curl, and their physical significance [3]. After more discussion and derivation, he finally wrote [4]:

    curl (H - h)  = 4πΓ = 4πkE + cĖ
    -curl (e - E) = 4πG = 4πgH + µḢ

in which, e and H denote impressed electric and magnetic forces to take static fields into account. Finally, since magnetic monopoles don't exist, he made g = 0, but kept this term in the equations for symmetry and elegance. [0]

This is the core of Heaviside's Duplex Form of Maxwell's equations. one can clearly see the co-evolution of electric and magnetic fields, and is the precursor of the modern Maxwell's equations as we know today in its vector calculus formulation. As far as I know, his treatment of "physical" vectors as first-class objects is his original invention (independently invented by Gibbs as well), although the concepts themselves came from quaternions.

This is not a complete summary, as he continued his analysis in a series of publications.

A good book on this part of history is Oliver Heaviside: the life, work, and times of an electrical genius of the Victorian age, by Paul J. Nahin.

[0] So the claim "Maxwell's equations need modifications if magnetic monopole has been discovered" is historically inaccurate, it should rather be, "be restored to Heaviside's original form."

[1] Electric Papers, Volume I, Page 429, https://archive.org/details/electricalpapers01heavuoft/page/...

[2] Page 441: https://archive.org/details/electricalpapers01heavuoft/page/...

[3] Page 443: https://archive.org/details/electricalpapers01heavuoft/page/...

[4] Page 449: https://archive.org/details/electricalpapers01heavuoft/page/...

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