## On Quaternions; or on a new System of Imaginaries in Algebra in The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Volume 25, 3rd Series, no. 163, 166 & 169, 1844

London: Taylor and Francis, 1844. 1st Edition. FIRST EDITION, FIRST PRINTING of William Hamilton's seminal On Quaternions, "a turning point in the development of mathematics" and one that "made possible the creation of the general theory of relativity" (Pickering, PMM 334). "On of the most imaginative mathematicians of the nineteenth century, Hamilton was considered by many the equivalent in intellect to Newton; his math changed the course of modern science" (Hankins, Sir William Rowan Hamilton, xv). Hamilton's first 3 papers on quaternions are included.

"Hamilton's discovery that a consistent and useful system of algebra could be constructed without obeisance to the commutative law of multiplication was comparable in importance to the invention of non-Euclidean geometry. Quaternions led to vector analysis...which has become of the greatest importance in mathematical physics and was developed by Riemann and Christoffel into tensor analysis. This then made possible the cretaion of the general theory of relativity" (ibid).

Hamilton was searching for an extension of the complex number system to use in geometric optics; instead, he discovered hyper-complex numbers consisting of 4 components. In short, quaternions are 4 dimensional numbers.

"The most important change in the concept of 'number'...came after Hamilton's discovery in 1843 of a completely new number system. Hamilton had noticed that coordinatizing the plane using complex numbers (rather than simply using pairs of real numbers) vastly simplified plane geometry. He set out to find a similar way to parametrize three-dimensional space. This turned out to be impossible, but led Hamilton to a four-dimensional system, which he called the quaternions. These behaved much like numbers with one crucial difference: multiplication was not commutative, that is, if q and q' are quaternions, qq' and q' q are usually not the same.

"The quaternions were the first system of hypercomplex numbers, and their appearance generated lots of new questions. Were there other such systems? What counts as a number system? If certain 'numbers' can fail to satisfy the commutative law, can we make numbers that break other rules?

"In the long run, this intellectual ferment led mathematicians to let go of the vague notion of 'number' or 'quantity' and to hold on, instead, to the more formal notion of an algebraic structure. Each of the number systems, in the end, is simply a set of entities on which we can do operations. What makes them interesting is that we can use them to parametrize, or coordinatize, systems that interest us. The whole numbers, for example, formalize the notion of counting, while the real numbers parametrize the line and serve as the basis for geometry" (Gowers, Princeton Companion to Mathematics, 81).

Since the late 20th century, quaternions have been widely employed...Quaternions are used in computer graphics, computer vision, robotics, control theory, signal processing, altitude control, physics, bioinformatics, molecular dynamics, computer simulation and orbital mechanics. Due to their speed, compactness, and reliability, it is common for spacecrafts to be commanded [by] quaternions" (Pickover, 232). Item #430

CONDITION & DETAILS: London: Taylor & Francis. (8.5 x 5.5 inches; 213 x 138mm). Complete. [viii], 522, [4]. In-text illustrations throughout. Hamiltons papers: pp. 10-14; 241-246; 489-495. Ex-libris bearing a rather handsome early paper label on the spine and a discreet stamp on the title page. Solidly and tightly bound in three quarter brown calf over marbled paper boards. Some scuffing and rubbing at the edge tips and along the spine. Gilt-ruled and lettered at the spine. Very slight age toning within; largely clean and bright. Very good condition.

Price: \$1,500.00

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