Arthur Cayley

How to use this course

Arthur Cayley is the most prolific British mathematician of the nineteenth century, and one of the most prolific of any era. His name sits on a remarkable number of objects that every student meets: the Cayley table of a group, Cayley’s theorem, the Cayley–Hamilton theorem, Cayley graphs, Cayley’s formula for trees, the Cayley transform, the Cayley–Menger determinant, the Cayley–Klein metrics. Many of these were side remarks in papers about something else. This course follows his main lines of work and the life that produced them, much of it in the evenings of a busy London legal career.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements in modern language, proof sketches and exercises with folded solutions. Cayley’s own notation was often heavy; where it helps, the lesson says what he wrote and when, then switches to today’s terms.

Timeline

St Petersburg, London, Cambridge, 1821–1842

Arthur Cayley was born on 16 August 1821 in Richmond, Surrey, while his parents were in England for the summer. His father, Henry Cayley, came from an old Yorkshire family and traded as a merchant in St Petersburg; his mother was Maria Antonia Doughty. Arthur spent his first seven years in Russia. In 1828 the family settled in London, near Regent’s Park. His brother Charles Bagot Cayley later became known as a translator of Dante and Homer.

At fourteen he entered King’s College School in London. He was a quiet, able boy who showed a striking talent for calculation, and he took prizes in chemistry as well as mathematics. His teachers persuaded his father, who had hoped Arthur would join the family business, to send him to Cambridge.

Trinity College

Cayley entered Trinity College, Cambridge, in 1838. The Cambridge mathematical course culminated in the Tripos examination, and serious students worked with private coaches. Cayley’s tutor was George Peacock, a reformer of British algebra, and his coach was William Hopkins, who trained a generation of wranglers. As an undergraduate he read the French analysts, Lagrange and Laplace, and began to publish. His first paper appeared in 1841 in the student-run Cambridge Mathematical Journal; Lesson 04 works through it.

In 1842 he graduated as Senior Wrangler, first in the Tripos, and won the first Smith’s Prize. Trinity elected him to a fellowship the same year.

The young fellow, 1842–1846

A Trinity fellowship gave Cayley a few free years. He used them to write. In this period he published 28 papers in the Cambridge Mathematical Journal alone, on determinants, elliptic functions, analytic geometry of \(n\) dimensions and the theory of linear transformations. He also travelled: a reading party in Scotland, and in 1844 a tour of Switzerland and Italy.

Two strands of his later work started here. In 1843 he published On a theory of determinants, which extended determinants to arrays with more than two indices, the ancestors of today’s hyperdeterminants. In 1845 and 1846 he published papers on linear transformations that asked which polynomial expressions in the coefficients of a form survive a linear change of variables. These papers founded invariant theory (Lesson 07). George Boole had written on the same question, and from 1844 the two corresponded.

In 1845, writing in the Philosophical Magazine, Cayley also described an eight-dimensional algebra extending William Rowan Hamilton’s quaternions. John Graves had found the same algebra in late 1843 and communicated it privately to Hamilton, but Cayley published first, and the octonions are still sometimes called Cayley numbers.

A Trinity fellowship of that era could not be held indefinitely by a layman; renewal required taking holy orders, which Cayley did not wish to do. In 1846 he left Cambridge for the law.

Lincoln’s Inn, 1846–1863

Cayley was admitted to Lincoln’s Inn on 20 April 1846 and called to the bar on 3 May 1849. He specialized in conveyancing, the drafting of deeds for the transfer of property, and built a substantial practice. He treated the law as a way to earn a living that left his mind free; he would take on enough work to be comfortable and no more.

In these fourteen years he wrote roughly 250 mathematical papers. In these years he met James Joseph Sylvester, a fellow mathematician who was then working as an actuary and studying law. The two walked around the courts of Lincoln’s Inn discussing invariants, and their friendship lasted for life. Sylvester invented much of the vocabulary of the subject: the word “matrix” in 1850, and terms such as discriminant and syzygy. Cayley’s papers supplied much of the substance. With Sylvester and the French mathematician Charles Hermite, he made invariant theory the leading branch of algebra in the 1850s.

He was elected a Fellow of the Royal Society in 1852 and received its Royal Medal in 1859. In 1854 he published the first abstract definition of a group, in 1858 the memoir that founded matrix algebra, and in 1859 the sixth memoir on quantics, which showed that Euclidean and non-Euclidean metrics both come from projective geometry. He applied unsuccessfully for chairs at Aberdeen in 1856 and for the Lowndean chair at Cambridge in 1858.

In 1863 Cambridge created the Sadleirian Professorship of Pure Mathematics from a bequest of Lady Mary Sadleir, and Cayley was elected its first holder. The post paid far less than his practice. He accepted at once. On 8 September 1863 he married Susan Moline of Greenwich, and he gave his inaugural lecture on 8 November.

Determinants and the Cayley transform

For the general reader. If you know the distances between every pair of four points in space, you can compute the volume of the tetrahedron they span, and you can tell whether a fifth point is consistent with them. Cayley’s first paper, written as an undergraduate, packaged these facts in a single determinant. Five years later he found a neat way to manufacture every rotation from a skew-symmetric matrix.

Distances and volumes (1841)

Let \(P_1,\dots,P_{n}\) be points in Euclidean space with squared distances \(d_{ij}^2=|P_i-P_j|^2\). Form the bordered matrix

The last statement is the content of Cayley’s 1841 paper On a theorem in the geometry of position: the ten distances among five points in space are not independent. Karl Menger made the determinant the basis of his metric characterization of Euclidean space in 1928, hence the name.

Proof sketch. Translate so that \(P_n=0\) and let \(v_i=P_i\) for \(i<n\). The Gram matrix \(G=(v_i\cdot v_j)\) has \(\det G=((n-1)!\,V)^2\). Since \(d_{ij}^2=|v_i|^2+|v_j|^2-2v_i\cdot v_j\), elementary row and column operations on CM, subtracting the last row and column and using the border of ones to clear the \(|v_i|^2\) terms, reduce it to a block form whose determinant is \((-1)^n 2^{n-1}\det G\).

The Cayley transform (1846)

Euler had found a parametrization of \(3\times3\) rotations by rational functions. Cayley generalized it to any dimension in 1846, in a paper on what he called skew determinants.

Proof. The eigenvalues of a real skew-symmetric matrix are purely imaginary, so \(-1\) is not one of them and \(I+S\) is invertible. The matrices \(I+S\) and \(I-S\) commute, and \((I\pm S)^{T}=I\mp S\). Hence

The determinant is continuous in \(S\), equals \(1\) at \(S=0\), and takes only the values \(\pm1\) on the connected space of skew-symmetric matrices, so it is \(1\) throughout.

The algebra of matrices

For the general reader. Before Cayley, a matrix was a way of writing down the coefficients of a system of equations, and a determinant was a number computed from it. Cayley treated the array itself as a single quantity that can be added, multiplied and even raised to powers, like a number that happens not to commute. Every course in linear algebra now starts from his point of view.

Composition of substitutions

In a French paper of 1855 Cayley introduced a notation for linear substitutions and the rule for “compounding” two of them, together with the inverse matrix. If \(y=Ax\) and \(z=By\), then \(z=(BA)x\) with \((BA)_{ik}=\sum_j B_{ij}A_{jk}\). His 1858 Memoir on the theory of matrices developed the algebra systematically: addition, multiplication by scalars, the zero and unit matrices, the associative law, the failure of commutativity, the transpose, and the inverse \(A^{-1}=\operatorname{adj}(A)/\det A\) when \(\det A\ne0\).

In 1858 Cayley verified this for \(2\times2\) matrices, reported that he had checked the \(3\times3\) case, and declared a general proof unnecessary. William Rowan Hamilton had earlier proved the analogous statement for linear functions of quaternions, hence the double name. The first general proof was given by Georg Frobenius in 1878.

Proof. Let \(B(\lambda)=\operatorname{adj}(\lambda I-A)\). Its entries are polynomials in \(\lambda\) of degree at most \(n-1\), so \(B(\lambda)=\sum_{k=0}^{n-1}B_k\lambda^k\) with constant matrices \(B_k\). The identity \((\lambda I-A)B(\lambda)=p_A(\lambda)I\) gives, comparing coefficients of \(\lambda^k\),

Multiply the equation for \(\lambda^k\) on the left by \(A^k\) and add. The left side telescopes to 0 and the right side is \(p_A(A)\).

Hypercomplex numbers

Cayley saw quaternions as a special algebra of matrices: the map \(a+bi+cj+dk\mapsto\begin{pmatrix}a+bi & c+di\\ -c+di & a-bi\end{pmatrix}\) turns quaternion multiplication into matrix multiplication. His 1845 octonions are the next step of what is now called the Cayley–Dickson doubling: pairs of quaternions \((a,b)\) multiplied by \((a,b)(c,d)=(ac-\bar d b,\ da+b\bar c)\). The result is an eight-dimensional normed division algebra that is no longer associative.

Abstract groups and Cayley graphs

For the general reader. Évariste Galois and Augustin-Louis Cauchy had studied groups of permutations. In 1854 Cayley saw that what matters is only the way elements combine, not what the elements are. Permutations, matrices, quaternion units and symmetries of a figure can all form the same group. This is the abstract point of view of modern algebra, stated decades before the field was ready for it.

The 1854 definition

In On the theory of groups, as depending on the symbolic equation \(\theta^n=1\) (1854) Cayley defined a group as a set of symbols \(1,\alpha,\beta,\dots\), all different, such that the product of any two of them, in either order, is again one of them. He noted that the product is associative but need not be commutative, and that each row and column of the multiplication table contains every symbol exactly once. For a finite set with an identity element these conditions are equivalent to the modern axioms. He tabulated the groups of order 4 and 6, and showed that each order 6 group is either cyclic or isomorphic to the symmetries of a triangle.

Proof. \(\lambda_g\) is a bijection with inverse \(\lambda_{g^{-1}}\). Associativity gives \(\lambda_g\lambda_h(x)=g(hx)=(gh)x=\lambda_{gh}(x)\), so \(g\mapsto\lambda_g\) is a homomorphism. If \(\lambda_g=\mathrm{id}\) then \(g=\lambda_g(1)=1\), so it is injective. In Cayley’s language, each row of the multiplication table is a permutation of the elements.

The 1854 paper was ignored for two decades. Cayley returned to groups in 1878 with several short papers that restated the abstract definition and posed the problem of finding all groups of a given order.

Graphical representation (1878)

In the first volume of the American Journal of Mathematics Cayley showed how to draw a group. Given generators \(S\), the Cayley graph \(\Gamma(G,S)\) has a vertex for each element and an edge \(g\to gs\) for each \(s\in S\), coloured by \(s\). Closed walks correspond to relations: walking around a coloured cycle spells a word equal to 1. The illustration at the top of this page is the Cayley graph of \(S_3\). A century later Cayley graphs became the basic objects of geometric group theory, which studies groups through the large-scale geometry of these graphs.

Quantics and invariants

For the general reader. The discriminant \(b^2-4ac\) of a quadratic tells you whether its roots coincide, and it does not care how you rescale or shift the variable. Cayley asked for all such quantities for polynomials of every degree. He called homogeneous polynomials quantics, and his ten memoirs on them, written from 1854 to 1878, built the theory that dominated nineteenth-century algebra.

Binary forms

A binary form of order \(n\) is \(f(x,y)=\sum_{k=0}^n\binom nk a_k\,x^{n-k}y^k\). The group \(\mathrm{SL}_2(\mathbb C)\) acts on such forms by linear substitution of \((x,y)\), and hence on the coefficients \(a_0,\dots,a_n\). An invariant is a polynomial \(I(a_0,\dots,a_n)\) unchanged by this action. Under \(\mathrm{GL}_2\) it is multiplied by a power of the determinant: \(I(g\cdot f)=(\det g)^{w}I(f)\). If \(I\) is homogeneous of degree \(d\), its weight is \(w=nd/2\).

• Quadratic \(ax^2+2bxy+cy^2\): the invariants are generated by the discriminant \(b^2-ac\), of degree 2.
• Cubic: generated by the discriminant, of degree 4.
• Quartic \(ax^4+4bx^3y+6cx^2y^2+4dxy^3+ey^4\): generated by two algebraically independent invariants

and the discriminant is a constant multiple of \(I^3-27J^2\).

Counting invariants

Let \(p(w;d,n)\) be the number of partitions of \(w\) into at most \(d\) parts, each at most \(n\).

The idea is that monomials \(a_{k_1}\cdots a_{k_d}\) of weight \(k_1+\dots+k_d=w\) span the degree-\(d\), weight-\(w\) space, which has dimension \(p(w;d,n)\). Invariants are the kernel of a differential operator \(\Omega\) in the \(a_k\) that lowers the weight by one. Cayley assumed that \(\Omega\) maps weight \(w\) onto weight \(w-1\) when \(w\le nd/2\); the formula follows by subtracting dimensions. Sylvester proved the surjectivity in 1878. In modern terms it is a statement about weights in the representation theory of \(\mathfrak{sl}_2\).

Finiteness

Cayley used his counting to compute generators and the relations among them, called syzygies. Working from these tables in the 1850s, he drew an incorrect conclusion: that for binary forms of higher order, such as the septic, no finite set of invariants generates all the others. Paul Gordan proved in 1868 that the invariants and covariants of every binary form are finitely generated, and in 1890 David Hilbert extended finiteness to forms in any number of variables by a non-constructive argument. Cayley’s error was natural: the number of generators grows very fast, and the relations among them are subtle.

Cayley also introduced the hyperdeterminant of a \(2\times2\times2\) array, a degree 4 polynomial in eight entries that is invariant under \(\mathrm{SL}_2\times\mathrm{SL}_2\times\mathrm{SL}_2\). It now appears in quantum information theory as a measure of three-qubit entanglement.

The projective metric

For the general reader. Projective geometry studies properties that survive perspective drawing: lines stay lines, but lengths and angles change. Cayley showed in 1859 that length and angle can be recovered inside projective geometry by fixing one special conic, which he called the absolute. Choose it one way and you get spherical geometry; another way and you get the hyperbolic plane. He concluded that projective geometry contains all of geometry.

The absolute

Work in the projective plane with homogeneous coordinates \(x=(x_1:x_2:x_3)\). Fix a non-degenerate symmetric bilinear form \(\Omega(x,y)\); its conic \(\Omega(x,x)=0\) is the absolute. In the Sixth memoir upon quantics (1859) Cayley defined the distance between points \(x\) and \(y\) by

Projective transformations preserving the absolute preserve this distance, so they are the isometries.

Cayley himself did not identify the second case with the non-Euclidean geometry of Lobachevsky and Bolyai. Felix Klein, who learned of Cayley’s idea in 1869–70, made the connection in 1871 and wrote distance as a logarithm of a cross-ratio. The interior of the conic with this metric is the Beltrami–Klein model of the hyperbolic plane, in which geodesics are straight chords. Euclidean geometry is the degenerate case in which the absolute shrinks to a pair of complex points on the line at infinity, the circular points.

Curves, surfaces and the 27 lines

For the general reader. A smooth surface defined by a cubic equation, however curved it looks, always contains exactly 27 straight lines. Cayley and George Salmon found this in 1849, in an exchange of letters. It is one of the first and most beautiful results of enumerative algebraic geometry.

Cayley showed that a cubic surface contains a finite number of lines; Salmon found that the number is 27. Cayley’s paper on the triple tangent planes of cubic surfaces, with Salmon’s contribution, appeared in the Cambridge and Dublin Mathematical Journal in 1849. The modern proof counts dimensions: lines in \(\mathbb P^3\) form a 4-dimensional Grassmannian, and containing a line imposes 4 conditions on a cubic, so one expects finitely many; the number 27 comes from a Chern class computation, or from realizing the surface as the plane blown up at six points. The 27 lines meet in a configuration whose symmetry group is the Weyl group \(W(E_6)\), of order 51840.

Curves through points

In 1843 Cayley published a paper on the intersection of curves. A special case is now known as the Cayley–Bacharach theorem; Isaak Bacharach corrected the general statement in 1886.

Proof sketch. Cubics form a 9-dimensional projective space. Eight points in this position impose independent conditions, so the cubics through them form a pencil, which must be \(\lambda C_1+\mu C_2\). Every member of the pencil vanishes at the ninth point.

Trees and enumeration

For the general reader. A tree is a network with no loops. Cayley counted trees in 1857 because they describe repeated differentiation, again in 1874 because they describe molecules, and in 1889 he published the formula every combinatorics student learns: there are \(n^{n-2}\) trees on \(n\) labelled points. He also posed, in 1878, the question of whether four colours suffice to colour every map.

Carl Borchardt had obtained the result in 1860 by a determinant argument; Cayley’s note A theorem on trees made it widely known, and his name stuck. His own argument was a sketch. The cleanest proof is Heinz Prüfer’s bijection of 1918.

Proof (Prüfer). Given a labelled tree on \(n\ge2\) vertices, repeatedly remove the leaf with the smallest label and record the label of its neighbour, stopping when two vertices remain. This gives a sequence of length \(n-2\) in \(\{1,\dots,n\}\). Conversely, a vertex of degree \(k\) appears exactly \(k-1\) times in the sequence, so the leaves are the labels that do not appear; the smallest one must have been removed first, attached to the first entry, and the tree is rebuilt step by step. The map is a bijection onto all \(n^{n-2}\) sequences.

Rooted trees (1857)

Let \(t_n\) be the number of rooted unlabelled trees with \(n\) vertices and \(T(x)=\sum_{n\ge1}t_nx^n\). Removing the root leaves a multiset of rooted trees, so

Cayley used this recursion to compute \(t_n\). In 1874 he applied related counts to the alkanes \(\mathrm C_n\mathrm H_{2n+2}\), whose carbon skeletons are trees with all degrees at most 4, and so began chemical graph theory.

The Sadleirian professor, 1863–1895

The Sadleirian statutes asked the professor to teach the principles of pure mathematics and to advance the science. Cayley took the second duty more seriously than his audience wished. His lectures followed his current research, while most students wanted preparation for the Tripos, and his classes were small. His influence ran through individuals: Andrew Forsyth, who succeeded him in the chair and edited the last volumes of his papers; Henry Frederick Baker; and Charlotte Angas Scott of Girton College, who studied with him informally and in 1885 became one of the first British women to earn a doctorate in mathematics, before building the mathematics department at Bryn Mawr College.

He lived at Garden House, off Little St Mary’s Lane, from 1864 with his wife Susan and their two children, Henry and Mary. Trinity made him an honorary fellow in 1872 and a full fellow again in 1875. He edited the publications of the Royal Astronomical Society from 1859 to 1881 and refereed a vast number of papers for the Royal Society and the London Mathematical Society, whose president he was from 1868 to 1870. His only book, An Elementary Treatise on Elliptic Functions, appeared in 1876.

In 1881 Sylvester, then professor at Johns Hopkins University in Baltimore, invited him to lecture. From January to May 1882 Cayley taught a course on Abelian and theta functions there. The Royal Society awarded him the Copley Medal the same year. In 1883 he presided over the British Association for the Advancement of Science at Southport; his address surveyed the whole of mathematics and argued that the parallel axiom is part of experience rather than something to be proved. The London Mathematical Society gave him its De Morgan Medal in 1884. He received honorary degrees from Oxford, Dublin, Edinburgh, Göttingen, Heidelberg, Leiden and Bologna, among others.

Death and legacy

Cambridge University Press began publishing Cayley’s Collected Mathematical Papers in 1889. He edited the first seven volumes himself. His health declined over his last years, and he died at Garden House on 26 January 1895, aged 73, after a long and painful illness. He was buried on 2 February in Mill Road Cemetery, Cambridge. Forsyth completed the edition: thirteen quarto volumes containing 967 papers, with an index volume. Contemporaries remarked that he seemed to know all of the mathematics of his day; the American mathematician George Bruce Halsted called him the last man who ever will know everything.

Legacy

• Linear algebra: the matrix as an algebraic object, matrix multiplication and inverse, the Cayley–Hamilton theorem and the Cayley transform, which is still used in numerical methods that preserve orthogonality.
• Group theory: the abstract definition of a group, Cayley tables, Cayley’s theorem, and Cayley graphs, the starting point of geometric group theory and of expander graph constructions.
• Invariant theory: with Sylvester and Hermite, the classical theory of binary forms, whose questions led to Gordan’s and Hilbert’s finiteness theorems and to modern representation theory and geometric invariant theory.
• Geometry: the projective derivation of metrics, which Klein turned into the Erlangen programme; the 27 lines; ruled surfaces; and the Cayley forms that parametrize space curves, precursors of Chow varieties.
• Combinatorics: trees, isomer counting, and the four-colour problem.
• Commemoration: a crater on the Moon bears his name.

Connections

People linked to Arthur Cayley across this collection. A name without a link has no page here yet. See Arthur Cayley in the network of all 337 lives →

Teachers and advisors

• George Peacock
• William Hopkins

Students

• Andrew Forsyth
• Charlotte Angas Scott
• Henry Frederick Baker

Influenced by

• Augustus De Morgan
• George Boole
• Jean-Victor Poncelet
• Julius Plücker

Influenced

• Felix Klein
• Paul Gordan

Colleagues and collaborators

• Charles Hermite
• Gaston Darboux
• George Boole 1844
• Michel Chasles
• William Rowan Hamilton
• George Salmon 1849
• James Joseph Sylvester 1847-1895

Rivals

• Paul Gordan

Friends

• William Rowan Hamilton

Family

• Charles Bagot Cayley

Also mentioned in this life

• Eugenio Beltrami
• János Bolyai
• Augustin-Louis Cauchy
• Leonhard Euler
• Pierre de Fermat
• Fibonacci
• Georg Frobenius
• Évariste Galois
• David Hilbert
• Joseph-Louis Lagrange
• Pierre-Simon Laplace
• Nikolai Lobachevsky
• Blaise Pascal
• Hermann Weyl

Fields

• algebra 29 lives
• invariant theory 7 lives
• group theory 9 lives
• algebraic geometry 17 lives
• combinatorics 9 lives

On CoursesHub

• Group Theory · Physical Mathematics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Arthur Cayley, MacTutor History of Mathematics.
2. Arthur Cayley, Wikipedia.
3. T. Crilly, Arthur Cayley: Mathematician Laureate of the Victorian Age, Johns Hopkins University Press, 2006.
4. A. R. Forsyth, Arthur Cayley (obituary), Proceedings of the Royal Society of London 58 (1895).
5. K. H. Parshall, James Joseph Sylvester: Jewish Mathematician in a Victorian World, Johns Hopkins University Press, 2006.

Mathematics

1. A. Cayley, The Collected Mathematical Papers, 13 vols. and index, Cambridge University Press, 1889–1898. Scans at the Internet Archive.
2. A. Cayley, On a theorem in the geometry of position, Cambridge Mathematical Journal 2 (1841).
3. A. Cayley, Sur quelques propriétés des déterminants gauches, J. reine angew. Math. 32 (1846).
4. A. Cayley, On the triple tangent planes of surfaces of the third order, Cambridge and Dublin Mathematical Journal 4 (1849).
5. A. Cayley, On the theory of groups, as depending on the symbolic equation \(\theta^n=1\), Philosophical Magazine 7 (1854).
6. A. Cayley, A memoir on the theory of matrices, Phil. Trans. Roy. Soc. 148 (1858).
7. A. Cayley, A sixth memoir upon quantics, Phil. Trans. Roy. Soc. 149 (1859).
8. A. Cayley, Desiderata and suggestions, No. 2: The theory of groups, graphical representation, American Journal of Mathematics 1 (1878).
9. A. Cayley, A theorem on trees, Quarterly Journal of Pure and Applied Mathematics 23 (1889).
10. F. Klein, Über die sogenannte Nicht-Euklidische Geometrie, Math. Ann. 4 (1871).
11. B. Sturmfels, Algorithms in Invariant Theory, Springer, 2nd ed. 2008.
12. J. Yue, The 1856 Lemma of Cayley revisited, arXiv:math/0407202.
13. M. Aigner and G. M. Ziegler, Proofs from THE BOOK, chapter on Cayley’s formula, Springer.