
History of Vectors The parallelogram law for the addition of vectors is so intuitive that its origin is unknown. It may have appeared in a now lost work of Aristotle (384322 B.C.), and it is in the Mechanics of Heron (first century A.D.) of Alexandria. It was also the first corollary in Isaac Newton’s (16421727) Principia Mathematica (1687). In the Principia, Newton dealt extensively with what are now considered vectorial entities (e.g., velocity, force), but never the concept of a vector. The systematic study and use of vectors were a 19^{th} and early 20^{th} century phenomenon. Vectors were born in the first two decades of the 19^{th} century with the geometric representations of complex numbers. Caspar Wessel (17451818), Jean Robert Argand (17681822), Carl Friedrich Gauss (17771855), and at least one or two others conceived of complex numbers as points in the twodimensional plane, i.e., as twodimensional vectors. Mathematicians and scientists worked with and applied these new numbers in various ways; for example, Gauss made crucial use of complex numbers to prove the Fundamental Theorem of Algebra (1799). In 1837, William Rowan Hamilton (18051865) showed that the complex numbers could be considered abstractly as ordered pairs (a, b) of real numbers. This idea was a part of the campaign of many mathematicians, including Hamilton himself, to search for a way to extend the twodimensional "numbers" to three dimensions; but no one was able to accomplish this, while preserving the basic algebraic properties of real and complex numbers. In 1827, August Ferdinand Möbius published a short book, The Barycentric Calculus, in which he introduced directed line segments that he denoted by letters of the alphabet, vectors in all but the name. In his study of centers of gravity and projective geometry, Möbius developed an arithmetic of these directed line segments; he added them and he showed how to multiply them by a real number. His interests were elsewhere, however, and no one else bothered to notice the importance of these computations. After a good deal of frustration, Hamilton was finally inspired to give up the search for such a threedimensional "number" system and instead he invented a fourdimensional system that he called quaternions. In his own words: October 16, 1843, which happened to be a Monday, and a Council day of the Royal Irish Academy – I was walking in to attend and preside, …, along the Royal Canal, … an undercurrent of thought was going on in my mind, which at last gave a result, whereof it is not too much to say that I felt at once the importance. An electric circuit seemed to close; and a spark flashed forth, … I could not resist the impulse … to cut with a knife on a stone of Brougham Bridge, as we passed it, the fundamental formulae…. Hamilton’s quaternions were written, q = w + ix + jy + kz, where w, x, y, and z were real numbers. Hamilton quickly realized that his quaternions consisted of two distinct parts. The first term, which he called the scalar and "x, y, z for its three rectangular components, or projections on three rectangular axes, he [referring to himself] has been induced to call the trinomial expression itself, as well as the line which it represents, a VECTOR." Hamilton used his "fundamental formulas," i^{2} = j^{2} = k^{2} = ijk = 1, to multiply quaternions, and he immediately discovered that the product, q_{1}q_{2} =  q_{2}q_{1}, was not commutative. Hamilton had been knighted in 1835, and he was a wellknown scientist who had done fundamental work in optics and theoretical physics by the time he invented quaternions, so they were given immediate recognition. In turn, he devoted the remaining 22 years of his life to their development and promotion. He wrote two exhaustive books, Lectures on Quaternions (1853) and Elements of Quaternions (1866), detailing not just the algebra of quaternions but also how they could be used in geometry. At one point, Hamilton wrote, "I still must assert that this discovery appears to me to be as important for the middle of the nineteenth century as the discovery of fluxions was for the close of the seventeenth." He acquired a disciple, Peter Guthrie Tait (18311901), who in the 1850s began applying quaternions to problems in electricity and magnetism and to other problems in physics. In the second half of the 19^{th} century, Tait’s advocacy of quaternions produced strong reactions, both positive and negative, in the scientific community. At about the same time that Hamilton discovered quaternions, Hermann Grassmann (18091877) was composing The Calculus of Extension (1844), now well known by its German title, Ausdehnungslehre. In 1832, Grassmann began development of "a new geometric calculus" as part of his study of the theory of tides, and he subsequently used these tools to simplify portions of two classical works, the Analytical Mechanics of Joseph Louis Lagrange (17361813) and the Celestial Mechanics of Pierre Simon Laplace (17491827). In his Ausdehnungslehre, first, Grassmann expanded the conception of vectors from the familiar two or three dimensions to an arbitrary number, n, of dimensions; this greatly extended the ideas of space. Second, and even more generally, Grassmann anticipated a good deal of modern matrix and linear algebra and vector and tensor analysis. Unfortunately, the Ausdehnungslehre had two strikes against it. First, it was highly abstract, lacking in explanatory examples and written in an obscure style with an overly complicated notation. Even after he had given it serious study, Möbius was not able to understand it fully. Second, Grassmann was a secondary school teacher without a major scientific reputation (compared to Hamilton). Even though his work was largely ignored, Grassmann promoted its message in the 1840s and 1850s with applications to electrodynamics and to the geometry of curves and surfaces, but without much general success. In 1862, Grassmann published a second and much revised edition of his Ausdehnungslehre, but it too was obscurely written and too abstract for the mathematicians of the time, and it met essentially the same fate as his first edition. In the later years of his life, Grassmann turned away from mathematics and launched a second and very successful research career in phonetics and comparative linguistics. Finally, in the late 1860s and 1870s, the Ausdehnungslehre slowly began to be understood and appreciated, and Grassmann began receiving some favorable recognition for his visionary mathematics. A third edition of the Ausdehnungslehre was published in 1878, the year after Grassmann died. During the middle of the nineteenth century, Benjamin Peirce (18091880) was far and away the most prominent mathematician in the United States, and he referred to Hamilton as, "the monumental author of quaternions." Peirce was a professor of mathematics and astronomy at Harvard from 1833 to 1880, and he wrote a massive System of Analytical Mechanics (1855; second edition 1872), in which, surprisingly, he did not include quaternions. Rather, Peirce expanded on what he called "this wonderful algebra of space" in composing his Linear Associative Algebra (1870), a work of totally abstract algebra. Reportedly, quaternions had been Peirce’s favorite subject, and he had several students who went on to become mathematicians and who wrote a good number of books and papers on the subject. James Clerk Maxwell (18311879) was a discerning and critical proponent of quaternions. Maxwell and Tait were Scottish and had studied together in Edinburgh and at Cambridge University, and they shared interests in mathematical physics. In what he called "the mathematical classification of physical quantities," Maxwell divided the variables of physics into two categories, scalars and vectors. Then, in terms of this stratification, he pointed out that using quaternions made transparent the mathematical analogies in physics that had been discovered by Lord Kelvin (Sir William Thomson, 18241907) between the flow of heat and the distribution of electrostatic forces. However, in his papers, and especially in his very influential Treatise on Electricity and Magnetism (1873), Maxwell emphasized the importance of what he described as "quaternion ideas … or the doctrine of Vectors" as a "mathematical method … a method of thinking." At the same time, he pointed out the inhomogeneous nature of the product of quaternions, and he warned scientists away from using "quaternion methods" with its details involving the three vector components. Essentially, Maxwell was suggesting a purely vectorial analysis. William Kingdon Clifford (18451879) expressed "profound admiration" for Grassmann’s Ausdehnungslehre and clearly favored vectors, which he often called steps, over quaternions. In his Elements of Dynamic (1878), Clifford broke down the product of two quaternions into two very different vector products, which he called the scalar product (now known as the dot product) and the vector product (today we call it the cross product). For vector analysis, he asserted "[M]y conviction [is] that its principles will exert a vast influence upon the future of mathematical science." Though the Elements of Dynamic was supposed to have been the first of a sequence of textbooks, Clifford never had the opportunity to pursue these ideas because he died quite young. The development of the algebra of vectors and of vector analysis as we know it today was first revealed in sets of remarkable notes made by J. Willard Gibbs (18391903) for his students at Yale University. Gibbs was a native of New Haven, Connecticut (his father had also been a professor at Yale), and his main scientific accomplishments were in physics, namely thermodynamics. Maxwell strongly supported Gibbs’s work in thermodynamics, especially the geometric presentations of Gibbs’s results. Gibbs was introduced to quaternions when he read Maxwell’s Treatise on Electricity and Magnetism, and Gibbs also studied Grassmann’s Ausdehnungslehre. He concluded that vectors would provide a more efficient tool for his work in physics. So, beginning in 1881, Gibbs privately printed notes on vector analysis for his students, which were widely distributed to scholars in the United States, Britain, and Europe. The first book on modern vector analysis in English was Vector Analysis (1901), Gibbs’s notes as assembled by one of his last graduate students, Edwin B. Wilson (18791964). Ironically, Wilson received his undergraduate education at Harvard (B.A. 1899) where he had learned about quaternions from his professor, James Mills Peirce (18341906), one of Benjamin Peirce’s sons. The Gibbs/Wilson book was reprinted in a paperback edition in 1960. Another contribution to the modern understanding and use of vectors was made by Jean Frenet (18161990). Frenet entered École normale supérieure in 1840, then studied at Toulouse where he wrote his doctoral thesis in 1847. Frenet's thesis contains the theory of space curves and contains the formulas known as the FrenetSerret formulas (the TNB frame). Frenet gave only six formulas while Serret gave nine. Frenet published this information in the Journal de mathematique pures et appliques in 1852. In the 1890s and the first decade of the twentieth century, Tait and a few others derided vectors and defended quaternions while numerous other scientists and mathematicians designed their own vector methods. Oliver Heaviside (18501925), a selfeducated physicist who was greatly influenced by Maxwell, published papers and his Electromagnetic Theory (three volumes, 1893, 1899, 1912) in which he attacked quaternions and developed his own vector analysis. Heaviside had received copies of Gibbs’s notes and he spoke very highly of them. In introducing Maxwell’s theories of electricity and magnetism into Germany (1894), vector methods were advocated and several books on vector analysis in German followed. Vector methods were introduced into Italy (1887, 1888, 1897), Russia (1907), and the Netherlands (1903). Vectors are now the modern language of a great deal of physics and applied mathematics and they continue to hold their own intrinsic mathematical interest. 
