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The nature of the quaternion

An Erratum to this article was published on 01 December 2008

I regard it as an inelegance, or imperfection, in quaternions, or rather in the state to which it has been hitherto unfolded, whenever it becomes or seems to become necessary to have recourse to x, y, z, etc.” William Rowan Hamilton (quoted in a letter from Tait to Cayley)

Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell.” William Thomson, first baron Kelvin, 1892

…quaternions appear to exude an air of nineteenth century decay, as a rather unsuccessful species in the struggle-for-life of mathematical ideas. Mathematicians, admittedly, still keep a warm place in their hearts for the remarkable algebraic properties of quaternions but, alas, such enthusiasm means little to the harder-headed physical scientist.” Simon L. Altmann, 1986 [1]


Some of the confusions concerning quaternions as they are employed in spacecraft attitude work are discussed. The order of quaternion multiplication is discussed in terms of its historical development and its consequences for the quaternion imaginaries. The different formulations for the quaternions are also contrasted. It is shown that the three Hamilton imaginaries cannot be interpreted as the basis of the vector space of physical vectors but only as constant numerical column vectors, the autorepresentation of a physical basis.

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Correspondence to Malcolm D. Shuster.

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Shuster, M.D. The nature of the quaternion. J of Astronaut Sci 56, 359–373 (2008).

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  • Geometric Algebra
  • Spacecraft Attitude
  • Quaternion Multiplication
  • Attitude Matrix
  • Quaternion Space