By: Kimberly Eckard

1545: Jerome Cardan publishes his Ars Magna, containing what is usually taken to be the first publication of the idea of a complex number. With that said, it took more than two centuries for complex numbers to be accepted as legitimate mathematical entities.

1687: Isaac Newton publishes his Principia Mathematica, in which he discusses the idea of a parallelogram of forces. Newton did not have the idea of a vector. He was, however, getting close to the idea, that forces, because they have both magnitude and direction, can be combined, or added, so as to produce a new force.

1799: Caspar Wessel publishes a paper in the memoirs of the Royal Academy of Denmark in which he lays out for the first time the geometrical representation of complex numbers.

1799: Carl Friedrich Gauss works out the geometrical interpretation of complex quantities. He made crucial use of complex numbers to prove the Fundamental Theorem of Algebra.

1805: The birthday of William Rowan Hamilton in Dublan, Ireland. As you will see, he will later play a key role in the development of vectors.

1806: Jean Robert Argand publishes the geometrical interpretation of complex numbers.

Hamilton 1827: August Ferdinand Möbius publishes The Barycentric Calculus, in which he introduced directed line segments that he denoted by letters of the alphabet, vectors in all but the name.

1828: Hamilton publishes an impressive work on optics entitled A Theory of Systems of Rays. This paper firmly established Hamilton’s reputation. In it Hamilton included some of his own methods for working with systems of linear equations.

1831: Carl Friedrich Gauss publishes the geometrical justification of complex numbers, which he had worked out in 1799.

1837: Hamilton publishes a long paper interpreting complex numbers as ordered couples of numbers. He showed that the complex numbers could be considered abstractly as ordered pairs (a,b) or real number. In the paper, Hamilton also mentions his hope to publish a “Theory of Triplets,” i.e., a system that would do for the analysis of three- dimensional space what imaginary numbers do for two-dimensional space.

1843: On October 16th, Hamilton’s his search ends with his discovery of mathematical entities he calls “quaternions.” His quaternions were written, q = w + ix + jy + kz, where w, x, y, and z were real numbers. He soon 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 called a vector.

1844: Hermann Grassmann composes The Calculus of Extension. In the book Grassmann expanded the conception of “vectors” from the familiar two or three dimensions to an arbitrary number, n, of dimensions which greatly extended the ideas of space.

Grassmann 1846: Hamilton publishes a paper in which he introduces the terms scalar and vector, referring respectively to the real and the imaginary parts of his quaternion.

1852: Matthew O’Brien of King’s College, London publishes a paper on a system of vector analysis, which was developed, it seems, partly in terms of Hamilton’s quaternions.

1865: Death of William Rowan Hamilton, who by this time had published 109 of the 150 papers that had been published on quaternions.

1870: Benjamin Pierce expanded on what he called “this wonderful algebra of space” in composing his Linear Associative Algebra, a work of totally abstract algebra. In this publication, working from Hamilton’s discovery of the possibility of new algebras, lays out and classifies 162 different algebras.

1873: James Clerke Maxwell publishes his Treatise on Electricity and Magnetism in which he emphasized the importance of the Doctrine of Vectors as a mathematical method of thinking.

Maxwell 1877: Death of Hermann Grassman, whose achievements extended far beyond mathematics.

1878: William Kingdon Clifford publishes his Elements of Dynamic in which he broke down the product of two quaternions into two very different vector products, which he called scalar product (now known as the dot product) and the vector product (now known as the cross product).

1881: Josiah Willard Gibbs prints the first half of his Elements of Vector Analysis, which presents what is essentially the modern system of vector analysis.

Gibbs

1884: Gibbs publishes the second half of his Elements of Vector Analysis, which concentrates on the more advanced parts of vector analysis, including vector functions of such a nature that a function of the sum of any two vectors is equal to the sum of the functions of the vectors.

1893: Oliver Heaviside publishes the first volume of his Electromagnetic Theory, which contains “The Elements of Vectorial Algebra and Analysis,” a 173-page presentation of the modern system of vector analysis.

Heaviside 1901: Edwin B. Wilson publishes the first book on modern vector analysis, Vector Analysis: A Text Book for the Use of Students of Mathematics and Physics and Founded upon the Lectures of J. Willard Gibbs. The book was essentially Gibbs’ notes assembled by Wilson, one of Gibbs’ last graduate students.

1903: Death of Josiah Willard Gibbs.

1907: Pavel Osipovich Somoff publishes in Russian the first book in that language on vector analysis.

1909: Joseph George Coffin publishes his An Introduction to Vector Methods and Their Various Applications to Physics and Mathematics.

Bibliography:

Crowe, Michael J.. "A History of Vector Analysis." 2002. Pages 1-17. 26 February 2008 .

Larson, Ron, Bruce H. Edwards, and David C. Falvo. Elementary Linear Algebra. 5th ed.. Boston: Houghton Mifflin Company, 2004.

"History of Vectors." Vectors. 26 Feb 2008 .