Name: A history of vector analysis
ISBN: 9780486679105

What Hacker News says

rramadassOct 2025

Suggestions for books/articles from a couple of my previous comments;

https://news.ycombinator.com/item?id=45110857

https://news.ycombinator.com/item?id=45088830

The OP's article though simple, still does not really explain things intuitively. The key is to understand the concept of a Vector from multiple perspectives/coordinate systems and map the operations on vectors to movements/calculations in the coordinate space (i.e. 2D/3D/n-space). Only then will Vector Spaces/Matrices/etc. become intelligible and we can begin to look at Physical problems naturally in terms of vectors/vector calculus.

The following are helpful here;

1) About Vectors by Banesh Hoffmann.

2) A History of Vector Analysis: The Evolution of the Idea of a Vectorial System by Michael Crowe.

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chadcmulliganFeb 2020

I recommend this a lot "A History of Vector Analysis: The Evolution of the Idea of a Vectorial System (Dover Books on Mathematics)" by Michael Crowe. It's a page turner.

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chadcmulliganJan 2020

There's also a great book "A History of Vector Analysis: The Evolution of the Idea of a Vectorial System" by Crowe which covers vectors from Complex numbers to Gibbs vectors and includes Hamilton and the competitor at the time Grassman Algebra, both the basis for geometric algebra.

Its one of the only maths history books I couldn't put down.

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jacobolusNov 2016

Here’s a comment I made elsewhere recently:

* * *

You might start with teasers like Hestenes’s papers [“Reforming the Mathematical Language of Physics”][oersted], [“Grassmann’s Vision”][gvision], etc.

For a summary from a mathy perspective, try Chisolm’s [book-like thingy on arxiv][chisolm]

If you want a whole book of concrete problems to solve at an advanced undergraduate physics student level, try Hestenes’s [New Foundations for Classical Mechanics][nfcm]. This is the best source I’ve seen anywhere about understanding complex rotations in mechanics problems.

If you want to solve some plane geometry problems, this thing looks fairly accessible [Treatise of plane geometry through the geometric algebra][planegeo] (I haven’t looked too closely).

Two websites are [geocalc.clas.asu.edu][asu] and [geometry.mrao.cam.ac.uk][cambridge]. Also see the [link page at The Net Advance of Physics][netadvance], and this [mirror of Lounesto’s site][lounesto] (he passed away a while back). Lounesto liked to publish [collections of counterexamples][counterexamples].

There are a number of journals, conferences, etc. Try a google search for “Clifford Algebra”.

If you want an introductory undergraduate textbook which tries to teach both geometric algebra and traditional matrix algebra, you could look at [MacDonald][]’s [Linear and Geometric Algebra][laga]; I don’t think Hestenes is really on board with this approach, but there’s not too much else pitched at a similar audience. MacDonald also has a book [Vector and Geometric Calculus][vagc] which I suspect will be substantially easier to work through than Hestenes and Sobczyk’s [Clifford Algebra to Geometric Calculus][cagc]. (I haven’t looked at either of MacDonald’s books)

If you’re interested in geometric modeling for robotics, computer graphics, computer vision, or similar, check out [these papers][invkinematics] and then look at the book [Geometric Algebra for Computer Science][gacs]. The “conformal geometric algebra” model proposed there is pretty neat. Also see [these papers][unifalg] about related topics.

If you’re interested in crystallography, check out the papers [“Point Groups and Space Groups in Geometric Algebra”][crystalsymmetry] and [“The Crystallographic Space Groups in Geometric Algebra”][crystalga].

If you’re interested in Lie theory or representation theory, check out this paper [“Lie Groups as Spin Groups”][lgasg].

If you’re a physicist / physics student / electrical engineer / etc., try the book [Geometric Algebra for Physicists][gap] or perhaps [Understanding Geometric Algebra for Electromagnetic Theory][gaet]

For a slightly different perspective, take a look at [Sobczyk][]’s [New Foundations in Mathematics: The Geometric Concept of Number][nfm]. Sobczyk has some interesting papers about representing geometric algebras using (real or complex-valued) matrices, which might be helpful if you want to write fast numerical code on current computers, which have been optimized to do matrix math.

If you’re interested in the history, I recommend Crowe’s 1967 [History of Vector Analysis][hva].

I believe you can find the collected mathematical papers of Clifford online if you do a google search, or buy a used copy of a nice version published by Chelsea in the 1960s; the recent AMS reprint is awful quality.

Grassmann’s two mid-19th century Ausdehnungslehre books have been relatively recently translated into English, as has Peano’s late-19th century book Geometic Calculus, all three by Kannenberg.

[oersted]: http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

[gvision]: http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf

[chisolm]: https://arxiv.org/abs/1205.5935

[nfcm]: http://geocalc.clas.asu.edu/html/NFCM.html

[planegeo]: http://web.archive.org/web/20011215062737/http://campus.uab....

[asu]: http://geocalc.clas.asu.edu/

[cambridge]: http://geometry.mrao.cam.ac.uk/

[netadvance]: http://web.mit.edu/redingtn/www/netadv/Xgeomealge.html

[lounesto]: https://users.aalto.fi/%7Eppuska/mirror/Lounesto/

[counterexamples]: https://users.aalto.fi/~ppuska/mirror/Lounesto/counterexampl...

[MacDonald]: http://faculty.luther.edu/~macdonal/

[laga]: http://faculty.luther.edu/~macdonal/laga/index.html

[vagc]: http://faculty.luther.edu/~macdonal/vagc/

[cagc]: http://geocalc.clas.asu.edu/html/CA_to_GC.html

[invkinematics]: http://geocalc.clas.asu.edu/html/InvariantKinematics.html

[gacs]: http://www.geometricalgebra.net

[unifalg]: http://geocalc.clas.asu.edu/html/UAFCG.html

[crystalsymmetry]: http://geocalc.clas.asu.edu/pdf/crystalsymmetry.pdf

[crystalga]: http://geocalc.clas.asu.edu/pdf/CrystalGA.pdf

[lgasg]: http://geocalc.clas.asu.edu/pdf/LGasSG.pdf

[gap]: http://geometry.mrao.cam.ac.uk/2007/01/geometric-algebra-for...

[gaet]: http://www.wiley.com/WileyCDA/WileyTitle/productCd-047094163...

[Sobczyk]: http://www.garretstar.com

[nfm]: http://www.springer.com/us/book/9780817683849

[hva]: https://en.wikipedia.org/wiki/A_History_of_Vector_Analysis

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jacobolusAug 2014

For those interested in the history, I highly recommend Crowe’s book A History of Vector Analysis. It’s written a bit from the perspective of the dominant “modern” system of vectors from Gibbs/Heaviside, so it probably somewhat undersells the significance/usefulness of both Hamilton’s quaternions and Grassmann’s vectors (or what Hestenes calls “geometric algebra”), but the historical and biographical details are fascinating.

Also recommended is all of Hestenes’s work (e.g. A New Foundation for Classical Mechanics, his Oersted Medal lecture, various papers), as well as the book Geometric Algebra for Computer Science by Dorst, Fontijne, & Mann. The latter is a bit light on the mathematical formalities but gives a good introduction that could get someone started using geometric algebra in practice in e.g. robotics, computer graphics, etc.

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mslaAug 2023

“Symmetrical equations are good in their place, but ‘vector’ is a useless survival, or offshoot from quaternions, and has never been of the slightest use to any creature.”

— William Thompson, 1st Baron Kelvin (better known as Lord Kelvin) Letter to G. F. FitzGerald (1896) as quoted in A History of Vector Analysis: The Evolution of the Idea of a Vectorial System (1994) by Michael J. Crowe, p. 120

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