## Lectures on Quaternions

*"Time is said to have only one dimension, and space to have three dimensions. ... The mathematical quaternion partakes of both these elements; in technical language it may be said to be 'time plus space', or 'space plus time': and in this sense it has, or at least involves a reference to, four dimensions. And how the One of Time, of Space the Three, Might in the Chain of Symbols girdled be." — William Hamilton*

FIRST EDITION IN ORIGINAL CLOTH of Hamilton’s *magnum opus*. Hamilton’s invention of quaternions marked "a turning point in the development of mathematics" that "made possible the creation of the general theory of relativity" (Pickering; *PMM*).

"The achievement in pure mathematics for which [Hamilton] is best remembered now is the invention of quaternions, a linear algebra of rotations in space of three dimensions. Quaternions were the first non-commutative number system system to be investigated in detail, and Hamilton's discovery that a consistent and useful system of algebra could be constructed without obeisanse to the commutative law of multiplication was comparable in importance to the invention of non-Euclidian geometry. Quaternions led to vector analysis, and were eventually superceded by the later, which has become of the greatest importance in mathematical physics and was developed by Ritman and Christoffel into tensor analysis. This made possible the creation of the general theory of relativity" (*Printing and the Mind of Man* 334).

Hamilton's invention "was the first example of a coherent, significant mathematical system that preserved all of the laws of ordinary arithmetic, with the exception of commutativity... [His] ideas had an enormous influence on the gradual introduction and use of vectors in physics. Hamilton used the name scalar for the real part a of the quaternion, and the term vector for the imaginary part bi + cj + dk, and defined what are now known as the scalar (or dot) and vector (or cross) products. It was through successive work in the 19th century of the Britons Peter Guthrie Tait, James Clerk Maxwell, and Oliver Heaviside and the American Josiah Willard Gibbs that an autonomous theory of vectors was first established while developing on Hamilton's initial ideas" (*Britannica*). Recently, quaternions have found new life in the modeling of the three-dimensional rotations involved in modern computer graphics.

"Hamilton's achievement in constructing quaternions is of considerable historical interest. It marked a turning point in the development of mathematics, involving as it did the introduction of noncommuting quantities into the subject matter of the field as well as an exemplary set of new entities and operations, the quaternion systerial agency, harnessing and directing that agency--domesticating it" (Andrew Pickering, "Concepts and the Mangle of Practice: Constructing Quaternions").

Although Hamilton had previously published several articles on quaternions, it wasn't until the appearance of his book, *Lectures on Quaternions*, that his invention was given full expression. PMM 334.

*Provenance*: Light pencil ownership signature on half-title (dated 1858) of A.H. Buchanan, almost certainly Andrew Hays Buchanan, professor of mathematics at Cumberland University (1853-1911) and considered “Cumberland’s greatest mathematician” (*A History of Cumberland University 1842-1935*).

Dublin: Hodges and Smith, 1853. Octavo, original cloth; custom cloth box. Scattered foxing (as usual); fading to spine and general light wear to cloth. RARE.

Price: $5,600 .