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I'm very interested to know how the concept of vectors came in mathematics and physics. How were vector quantities discovered in physics, and how and by whom were they developed?

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You can see :

We have to take into account some "precursors", like the representation of the parallelogram of the velocities and of the forces, already present into ancient Greek science (see the pseudo-Aristotle, Mechanica) and some attempts by Leibniz with his Analysis situs.

The key achievement was the geometrical representation of complex numbers, discovered independently in 1831 by Wessel, Gauss, Argand, etc.

Then we have William Rowan Hamilton with the quaternions and finally the modern system of vector analysis developed by Josiah Willard Gibbs with his lectures of 1881 and 1884 [published as E.B.Wilson, Vector Analysis, a text-book for the use of students of Mathematics and Physics, founded upon the Lectures of J.Willard Gibbs, (1901)] and Oliver Heaviside, with his 1882-83 paper on electromagnetism published on The Electrician.

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  • $\begingroup$ The complex numbers were discovered much earlier than 1831. Euler, for example, published $e^{ix} = \cos x + i\sin x$ in 1748. $\endgroup$
    – David Hammen
    Mar 23, 2015 at 3:56
  • $\begingroup$ Yes, there's a long history of use of parallelogram rules for things like motions and velocities, and later, forces. Newton's earliest draft De motu gives a parallelogram rule as if it already had something near to axiomatic status. $\endgroup$
    – terry-s
    May 13 at 17:46
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I agree with Mauro's answer, and want to add two interesting details.

  1. The first "use of vectors" in science that I know was the famous theorem of Apollonius which says that epicycles are equivalent to excentrics (in the description of the motion of celestial bodies). This theorem is reproduced in Ptolemy, with a proof, and credited explicitly to Apollonius. Ptolemy was not very generous in giving credits, so I supose this is really due to Apollonius. Translated in modern language the theorem says that vector addition is commutative. (It is amazing that in the ancient times this "evident" thing required a proof).

  2. Another curious thing is that quaternions were introduced much earlier than vectors, and later introduction of vectors to physics (by Gibbs) generated a substantial controversy. This controversy is nicely described in this paper:

http://www.ams.org/journals/bull/2008-45-01/S0273-0979-07-01182-2/S0273-0979-07-01182-2.pdf

  1. A also want to mention that mathematicians (like Grassmann, Cayley and Silvester) routinely used vectors long before physicists.

Remark. The paper I mentioned above is a part of the Gibbs Lectures collection, which on my opinion is a great source on the history of Math and Science. All old lectures are published in the AMS Bulletin which is freely available. Unfortunately they discontinued publication of these lectures, so the recent ones do not seem to be available at all.

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