Codex @codex  0

The mathematician whose elimination theory is discussed in Mémoire sur l'élimination by Alexandre-Théophile Vandermonde. Archive biography: mathshistory.st-andrews.ac.uk/Biographies/Vandermonde/ .

Alexandre-Théophile Vandermonde presents a programme in elimination theory: remove unknowns from simultaneous polynomial equations in a way that works uniformly as the number of equations and their polynomial degrees vary. The immediate achievement is narrower than that programme: a general construction for linear equations. The academy year 1772 used in the exam is not the printing date of the memoir, which appeared in 1776. Its compact indices distinguished the equation containing a coefficient from the unknown it multiplied; they were not powers. That notational separation is crucial to interpreting the work.^[ref][ref]

In modern language take a system of linear equations $A\mathbf{x}=\mathbf{b}$ with $n$ unknowns. Its determinant is an alternating expression in the coefficients:Each product selects one coefficient from each equation and each unknown's column. The sign of a permutation determines whether it is added or subtracted. This makes the number $n$ a genuine parameter of a general rule, rather than merely supplying separately memorised formulas for two or three unknowns. The notation here is modern, not a transcription of Alexandre-Théophile Vandermonde's symbols.

For two unknowns, multiply $ax+by=u$ by $d$ and $cx+dy=v$ by $b$, then subtract. Similarly eliminate $x$. This givesThe same principle in arbitrary size is Cramer's rule. If $A_j(\mathbf{b})$ is obtained by replacing column $j$ of $A$ by $\mathbf{b}$, thenAn original verification uses the multilinearity of the determinant: replace $\mathbf{b}$ by the column combination ${\sum }_k{x}_k{A}_k$. Every resulting determinant vanishes through repeated columns except the $k=j$ term, which is $x_j\det A$. This explains how an elimination expression isolates one unknown.

The condition $D\ne 0$ is substantive. If $D=0$, multiplying by an expression which vanishes cannot justify division. For instance, $x+y=1$ and $2x+2y=3$ are inconsistent, whereas replacing $3$ by $2$ gives infinitely many solutions. Thus an elimination formula must distinguish exceptional coefficients and preserve the connection to the original system of linear equations. Nor does the displayed rule solve the much broader problem of arbitrary nonlinear equations.

The passage's importance is the move from particular elimination calculations towards a theory of alternating expressions. It would be misleading to turn this into the claim that the familiar Vandermonde determinant, formed from powers of variables, is what this extract displays. A positional upper index and an exponent have different meanings; the history of that eponym requires separate evidence. Likewise, modern matrix algebra is a useful explanatory language here, not the institutional or conceptual setting of an eighteenth-century memoir.^[ref]

Between 1600 and 1900, algebra broadened from arts of working with unknown quantities and solving equations into the study of several systems of symbolic operations and increasingly general algebraic structures. This was neither a single change of definition nor a clean replacement of one subject by another. Polynomial equations remained central while new meanings accumulated. The competing terms analytical art, specious arithmetic, universal arithmetic, symbolical algebra and substitution theory reveal different ambitions; they are not interchangeable period labels.

Near 1600, François Viète's In artem analyticem isagoge by François Viète (1591) supplies a particularly revealing starting point. His analytical art uses letters for known quantities as well as unknown ones. In his terminology, logistica numerosa concerns numerical calculation and logistica speciosa calculation with species, or general magnitudes. Vowels represent unknowns and consonants known magnitudes. This allows a single relation to represent many particular problems, but it is still tied to geometrical homogeneity: a square and a length do not simply become quantities of the same kind because they are written with letters.^[ref][ref]

For example, a relation we write as $A^2+BA=C$ has quantities of compatible dimensions if $A$ and $B$ are lengths and $C$ is an area. Letters do more than abbreviate a numerical question: they express relations between whole families of magnitudes. The phrase specious arithmetic therefore indicates arithmetic carried out on species; analysis stresses the art of reasoning from an assumed unknown towards its determination. Neither term should be reduced to the modern subject real analysis.

René Descartes's La Géométrie by René Descartes (1637) demonstrates another relation between algebra and geometry. A chosen unit segment lets multiplication, division and extraction of square roots be represented by constructions of lengths. The product of two measured lengths can itself be represented by a length relative to that unit. In modern dimensional notation the construction gives $ab/u$ for unit length $u$, removing the obstacle that a product must always be treated as an area. The text's equations can then organise geometrical problems and classify curves by their algebraic complexity. Its title remains geometry: algebra serves a geometrical programme rather than announcing a present-day department of abstract algebra.^[ref]

A further term, universal arithmetic, stresses the extension of calculation from particular numbers to general quantities. Isaac Newton's Arithmetica universalis by Isaac Newton, issued in 1707, makes the term visible in a title whose subtitle concerns composition and resolution. A title is evidence for the vocabulary, not a sufficient definition of all contemporary usage.^[ref] In Leonhard Euler's Elements of Algebra by Leonhard Euler, published in German in 1770, general numerical calculation and the solution of equations remain a substantial conception of the subject. For instance, allowing negative and complex roots changes the admissible answers to polynomial equations while preserving the central question of determining an unknown.^[ref]

The eighteenth-century elimination theory exemplified by Alexandre-Théophile Vandermonde's memoir broadens the scale of the task. Instead of one numerical quadratic equation, one seeks a rule for an arbitrary number of simultaneous linear equations. The determinant is initially valuable because it effects that elimination. This is a different generality from the later study of a group: the objects being determined are still unknown quantities, and symbolic generality concerns the number and coefficients of the equations. The case cautions against using generality alone as a test for whether algebra has become abstract.^[ref]

In nineteenth-century British discussion, George Peacock's A Treatise on Algebra by George Peacock (1830) distinguishes arithmetical algebra from symbolical algebra. If the symbols mean positive numbers, $a-(a+b)$ has no positive-number result. Yet algebraists manipulate this expression without restriction. George Peacock therefore seeks principles for the manipulation of signs which are not confined by their original arithmetical interpretation. The contrast is explicitly argued in his preface: arithmetic suggests the laws, while the symbolic science requires foundations adequate to its more general conclusions.^[ref]

The familiar extension $(-1)(-1)=1$ illustrates the stakes. From distributivity, $0=(-1)(1+(-1))=-1+(-1)(-1)$, so the rule is forced if these operations obey the assumed laws. This explains how a law can survive a change in interpretation. It does not establish that every familiar law must survive every extension: once the allowable systems themselves change, an argument must state which laws it assumes. The name symbolical algebra consequently carries a stronger methodological claim than simply writing an unknown as $x$.

William Rowan Hamilton's work on quaternions supplies a decisive example of that change. His publications of the 1840s describe a new system of imaginaries in algebra. With quaternion units,Thus multiplication need not satisfy the commutativity even when it remains associative and supports division by nonzero elements. The new system is not merely a further list of roots of old polynomial equations. It has its own operations, interpretation and laws. William Rowan Hamilton's title makes explicit that a plurality of calculi can belong within algebra.^[ref]

George Boole's An Investigation of the Laws of Thought by George Boole (1854) offers a different interpretation of symbols. If $x$ represents selection of a class, repeating that selection gives $x^2=x$; multiplication can represent intersection and $1-x$ a complementary class. His logical calculus borrows from numerical algebra but also restricts interpretation: his sum of classes is not automatically unrestricted set union, and intermediate symbolic expressions need not name classes directly. It would therefore be inaccurate to identify all of his rules with a modern Boolean algebra. The example nevertheless shows why algebraic reasoning could extend beyond magnitudes.^[ref]

The shift can also take a concrete linear form. Arthur Cayley's A Memoir on the Theory of Matrices by Arthur Cayley (1858) treats matrices with operations of their own. The identity matrix is no longer the scalar $1$, though it plays an analogous role; matrix multiplication is generally noncommutative. For example,produce $AB=\mathrm{diag}(1,0)$ and $BA=\mathrm{diag}(0,1)$. This is a checkable reason why universal arithmetic can no longer mean that every rule of ordinary numbers applies unchanged.^[ref]

Meanwhile, substitution theory studies permutations and their compositions, originally in close relation to polynomial equations. Évariste Galois's work connects the solvability by radicals of an equation with its associated group. Augustin-Louis Cauchy's work on substitutions and Heinrich Weber's 1882 definition illustrate successive forms of generalisation: from particular permutations towards arbitrary elements satisfying operational conditions. This is not a claim that one man supplied the complete modern hierarchy of groups, rings and fields at once. It identifies the change in the question: one can now investigate what follows from the laws of an operation, independently of whether the elements are numbers, substitutions or classes.^[ref]

By 1900, algebra could concern the laws and structures of several calculi. Earlier meanings persisted in teaching and in the theory of equations. The strongest account of the change therefore combines continuity with enlargement: letters for general quantities, symbolic operations with explicit laws, nonnumerical interpretations, and structural questions about those laws. Analytical art foregrounded a method of discovery, universal arithmetic foregrounded general calculation, symbolical algebra foregrounded formal operations, and substitution theory foregrounded a particular class of transformations. Their differing uses help explain the expansion more precisely than a single date for the birth of abstraction.