Vector Algebra

The evolution of vector analysis is an amazing story of intrigue, discovery, jealousy, prior art, coincidence and human weaknesses, all of which are covered in Michael Crowe’s excellent book A History of Vector Analysis [6]. No one mathematician can claim that he or she discovered vectors—although the word vector was coined by the Irish physicist and mathematician, William Rowan Hamilton [1805–1865]. However, long before Hamilton was born, the German diplomat and natural philosopher, Gottfried Wilhelm Leibniz [1646–1716], had written to Christian Huygens in 1679:

I have discovered certain elements of a new characteristic which is entirely different from algebra and which will have great advantages in representing to the mind, exactly and in a way faithful to its nature, even without figures, everything which depends on sense perception. [7]

Although Leibniz had developed a system for annotating line segments with letters, he failed to show how such geometric elements could be manipulated arithmetically. We had to wait for the German mathematician, August Ferdinand Möbius [1790–1868], to publish hisDer barycentrische Calcul in 1827, which contained the foundations of vectors in the form of barycentric coordinates [8]. In 1843 Möbius published Die Elemente der Mechanik des Himmels where he showed how directed line segments could be added and subtracted.