Summary of book
The book has eight chapters: the first on the origins of vector analysis including Ancient Greek and 16th and 17th century influences; the second on the 19th century William Rowan Hamilton and quaternions; the third on other 19th and 18th century vectorial systems including equipollence of Giusto Bellavitis and the exterior algebra of Hermann Grassmann.
Chapter four is on the general interest in the 19th century on vectorial systems including analysis of journal publications as well as sections on major figures and their views (e.g., Peter Guthrie Tait as an advocate of Quaternions and James Clerk Maxwell as a critic of Quaternions); the fifth chapter describes the development of the modern system of vector analysis by Josiah Willard Gibbs and Oliver Heaviside.
In chapter six, "Struggle for existence", Michael J. Crowe delves into the zeitgeist that pruned down quaternion theory into vector analysis on three-dimensional space. He makes clear the ambition of this effort by considering five major texts as well as a couple dozen articles authored by participants in "The Great Vector Debate". These are the books:
- Elementary Treatise on Quaternions (1890) Peter Guthrie Tait
- Elements of Vector Analysis (1881,1884) Josiah Willard Gibbs
- Electromagnetic Theory (1893,1899,1912) Oliver Heaviside
- Utility of Quaternions in Physics (1893) Alexander McAulay
- Vector Analysis and Quaternions (1906) Alexander Macfarlane
Twenty of the ancillary articles appeared in Nature; others were in Philosophical Magazine, London or Edinburgh Proceedings of the Royal Society, Physical Review, and Proceedings of the American Association for the Advancement of Science. The authors included Cargill Gilston Knott and a half-dozen other hands.
The "struggle for existence" is a phrase from Charles Darwin’s Origin of Species and Crowe quotes Darwin: "…young and rising naturalists,…will be able to view both sides of the question with impartiality." After 1901 with the Gibbs/Wilson/Yale publication Vector Analysis, the question was decided in favour of the vectorialists with separate dot and cross products. The pragmatic temper of the times set aside the four-dimensional source of vector algebra.
Crowe's chapter seven is a survey of "Twelve major publications in Vector Analysis from 1894 to 1910". Of these twelve, seven are in German, two in Italian, one in Russian, and two in English. Whereas the previous chapter examined a debate in English, the final chapter notes the influence of Heinrich Hertz' results with radio and the rush of German research using vectors. Joseph George Coffin of MIT and Clark University published his Vector Analysis in 1909; it too leaned heavily into applications. Thus Crowe provides a context for Gibbs and Wilson’s famous textbook of 1901.
The eighth chapter is the author's summary and conclusions. [2] The book relies on references in chapter endnotes instead of a bibliography section. Crowe also states that the Bibliography of the Quaternion Society, and its supplements to 1912, already listed all the primary literature for the study.
Summary of reviews
There were significant reviews given near the time of original publication. Stanley Goldberg [3] wrote "The polemics on both sides make very rich reading, especially when they are spiced with the sarcastic wit of a Heaviside, and the fervent, almost religious railing of a Tait." Morris Kline begins his 1969 review [4] with "Since historical publications on modern developments are rare, this book is welcome." and ends with "the subtitle [,The Evolution of the Idea of a Vectorial System,] is a better description of the contents than the title proper." Then William C. Waterhouse—picking up where Kline's review left off—writes in 1972 "Crowe's book on vector analysis seems a little anemic in comparison, perhaps because its title is misleading. ... [Crowe] does succeed in his goal of tracing the genealogy of the 3-space system, concluding that it was developed out of quaternions by physicists." [5]
Karin Reich wrote that Arnold Sommerfeld's name was missing from the book. [6] As assistant to Felix Klein, Sommerfeld was assigned the project of unifying vector concepts and notations for Klein's encyclopedia.
In 2003 Sandro Caparrini challenged Crowe’s conclusions by noting that "geometrical representations of forces and velocities by means of directed line segments...was already fairly well known by the middle of the eighteenth century" in his essay "Early Theories of Vectors". [7] Caparrini cites several sources, in particular Gaetano Giorgini (1795 — 1874) and his appreciation in an 1830 article [8] by Michel Chasles. Caparrini goes on to indicate that moments of forces and angular velocities were recognized as vectorial entities in the second half of the eighteenth century.
Oliver Heaviside FRS was an English self-taught mathematician and physicist who invented a new technique for solving differential equations, independently developed vector calculus, and rewrote Maxwell's equations in the form commonly used today. He significantly shaped the way Maxwell's equations are understood and applied in the decades following Maxwell's death. His formulation of the telegrapher's equations became commercially important during his own lifetime, after their significance went unremarked for a long while, as few others were versed at the time in his novel methodology. Although at odds with the scientific establishment for most of his life, Heaviside changed the face of telecommunications, mathematics, and science.
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields, and fluid flow.
Josiah Willard Gibbs was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in transforming physical chemistry into a rigorous inductive science. Together with James Clerk Maxwell and Ludwig Boltzmann, he created statistical mechanics, explaining the laws of thermodynamics as consequences of the statistical properties of ensembles of the possible states of a physical system composed of many particles. Gibbs also worked on the application of Maxwell's equations to problems in physical optics. As a mathematician, he invented modern vector calculus.
In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group representation theory.
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative.
Hermann Günther Grassmann was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mathematical work was little noted until he was in his sixties. His work preceded and exceeded the concept which is now known as a vector space. He introduced the Grassmannian, the space which parameterizes all k-dimensional linear subspaces of an n-dimensional vector space V.
The nabla is a triangular symbol resembling an inverted Greek delta: or ∇. The name comes, by reason of the symbol's shape, from the Hellenistic Greek word νάβλα for a Phoenician harp, and was suggested by the encyclopedist William Robertson Smith to Peter Guthrie Tait in correspondence.
Alexander Macfarlane FRSE LLD was a Scottish logician, physicist, and mathematician.
In abstract algebra, the algebra of hyperbolic quaternions is a nonassociative algebra over the real numbers with elements of the form
Alexander McAulay was the first professor of mathematics and physics at the University of Tasmania, Hobart, Tasmania. He was also a proponent of dual quaternions, which he termed "octonions" or "Clifford biquaternions".
Cargill Gilston Knott FRS, FRSE LLD was a Scottish physicist and mathematician who was a pioneer in seismological research. He spent his early career in Japan. He later became a Fellow of the Royal Society, Secretary of the Royal Society of Edinburgh, and President of the Scottish Meteorological Society.
In mathematics and physics, vector notation is a commonly used notation for representing vectors, which may be Euclidean vectors, or more generally, members of a vector space.
A treatise is a formal and systematic written discourse on some subject concerned with investigating or exposing the principles of the subject and its conclusions. A monograph is a treatise on a specialized topic.
Vector Analysis is a textbook by Edwin Bidwell Wilson, first published in 1901 and based on the lectures that Josiah Willard Gibbs had delivered on the subject at Yale University. The book did much to standardize the notation and vocabulary of three-dimensional linear algebra and vector calculus, as used by physicists and mathematicians. It was reprinted by Yale in 1913, 1916, 1922, 1925, 1929, 1931, and 1943. The work is now in the public domain. It was reprinted by Dover Publications in 1960.
William Rowan Hamilton invented quaternions, a mathematical entity in 1843. This article describes Hamilton's original treatment of quaternions, using his notation and terms. Hamilton's treatment is more geometric than the modern approach, which emphasizes quaternions' algebraic properties. Mathematically, quaternions discussed differ from the modern definition only by the terminology which is used.
In mathematics, quaternions are a non-commutative number system that extends the complex numbers. Quaternions and their applications to rotations were first described in print by Olinde Rodrigues in all but name in 1840, but independently discovered by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. They find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations.
The Quaternion Society was a scientific society, self-described as an "International Association for Promoting the Study of Quaternions and Allied Systems of Mathematics". At its peak it consisted of about 60 mathematicians spread throughout the academic world that were experimenting with quaternions and other hypercomplex number systems. The group's guiding light was Alexander Macfarlane who served as its secretary initially, and became president in 1909. The association published a Bibliography in 1904 and a Bulletin from 1900 to 1913.
Felix Klein's Encyclopedia of Mathematical Sciences is a German mathematical encyclopedia published in six volumes from 1898 to 1933. Klein and Wilhelm Franz Meyer were organizers of the encyclopedia. Its full title in English is Encyclopedia of Mathematical Sciences Including Their Applications, which is Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen (EMW). It is 20,000 pages in length and was published by B.G. Teubner Verlag, publisher of Mathematische Annalen.
Thomas W. Hawkins Jr. is an American historian of mathematics.
Michael J. Crowe is Rev. John J. Cavanaugh Professor Emeritus in the Program of Liberal Studies and Graduate Program in History and Philosophy of Science at the University of Notre Dame. He is best known for writing the influential book A History of Vector Analysis. After the Great Vector Debate of the 1890s it was generally assumed that quaternions had been superseded by vector analysis. But in his book, published in 1967, Crowe showed how, contrarily, vector analysis directly stemmed from the quaternions. In 1994 a new edition was published.
