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I am not sure if this is the right place to ask, but many mathematicians are knowledgeable and interested also in history of math, so here I am.

I am curious to know when the right-hand-rule for vector product was established and used consistently.

I am reading an introductory algebraic geometry text by Victoria Scott from 1894. I have noticed that she seems to use a left hand rule for the computation of vector product: in order to compute an area of a triangle she does a product of the sides (that I assume is the vector product) but, with the order chosen, she should get a negative area. And when she wants a negative area (to remove from another area)… the result should be positive with the modern convention. Since at the end she is just interested in plane geometry, consistency of the signs is the only thing that she needs, and she gets it. But this has aroused my curiosity about the history of the right-hand-rule and the orientation of axes in the euclidean 3-space….

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    $\begingroup$ If get no joy here, hsm.stackexchange.com would be the place to ask ... $\endgroup$
    – J.J. Green
    Commented Sep 27, 2023 at 8:42
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    $\begingroup$ Thank you! I did not know about that wiki, or I would have asked directly there :) $\endgroup$
    – Sofia Tirabassi
    Commented Sep 27, 2023 at 9:19

2 Answers 2

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As far as I know, studying quaternions, I found that the right hand rule was before 1871 because it was in that year that the London Mathematical Society accepted J.C. Maxwell's request to adopt the convention that a system is positively oriented if it rotates in the direction of the earth's rotation as seen from the north pole. Of course, reference was made to the right hand rule.

Reference: Chris Pritchard. "Tendril of the Hop and Tendril of the Vine: Peter Guthrie Tait and the Promotion of Quaternions, Part I". pp 33–34. DOI

It is also mentioned in W. R. Hamilton. "On Quaternions". 1847. Proceedings of the Royal Irish Academy, 3, 1–16.

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    $\begingroup$ interesting, do you have a source? Maxwell's 1871 communication where he introduces the curl does not mention the right-hand-rule clerkmaxwellfoundation.org/… $\endgroup$
    – Carlo Beenakker
    Commented Sep 28, 2023 at 21:32
  • $\begingroup$ Hi, I already edited the post with the reference. $\endgroup$
    – wmora2
    Commented Sep 28, 2023 at 21:42
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    $\begingroup$ Insofar as vector products are concerned, this must be the correct answer (no one even debated their definition before Hamilton, Tait and Maxwell). Yet Wikipedia traces the history even further to Ampère (without a precise reference, but one might perhaps trace one from T. B. Greenslade, Ancestors of the right‐hand rule). $\endgroup$
    – Francois Ziegler
    Commented Sep 29, 2023 at 1:53
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    $\begingroup$ thanks for the references; what I read in Pritchard is the corkscrew rule: "Lay hold of one of these and turn screw wise and you rotate +." This is also mentioned in the Society statement "No arguments in favour of the opposite system being given, the handed system, symbolised by a corkscrew or the tendril of the vine was adopted by the Society." Is there a source where Maxwell refers to the three fingers of the right hand to set the orientation of the axes? [In my education at a Dutch University I actually only learned of the corkscrew rule, "kurketrekker regel".] $\endgroup$
    – Carlo Beenakker
    Commented Sep 29, 2023 at 9:00
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    $\begingroup$ @LSpice Oh I still meant with your hand, like this: en.m.wikipedia.org/wiki/File:Right-hand_grip_rule.svg $\endgroup$
    – Kevin Casto
    Commented Sep 29, 2023 at 16:09
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John Ambrose Fleming is credited with the invention of the right-hand-rule in the context of electromagnetism.
The figure below, illustrating $X=Y\times Z$ in a right-handed coordinate system, is from Fleming's Magnets and electric currents, an elementary treatise for the use of electrical artisans and science teachers (1898).

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    $\begingroup$ "Electrical artisans"! $\endgroup$
    – LSpice
    Commented Sep 28, 2023 at 4:00

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