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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.<In artem analyticem isagoge by François Viète>[1591, opening analytical programme and rules of species; terminology and notation discussed in the MacTutor Viète biography]{ref}<François Viète by John Joseph O'Connor and Edmund Frederick Robertson>[Discussion of known and unknown symbols and dimensional homogeneity]{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>.<La Géométrie by René Descartes>[Book I, operations on lines and choice of unit; Book II, treatment of curves]{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.<Arithmetica universalis by Isaac Newton>[1707 title page and original-edition record, New York Public Library]{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.<Elements of Algebra by Leonhard Euler>[1770 original publication and later English edition; sections on operations and equations]{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.<Mémoire sur l'élimination by Alexandre-Théophile Vandermonde>[Opening programme and positional coefficient notation]{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.<A Treatise on Algebra by George Peacock>[1830 preface, especially the discussion of a−(a+b) and arithmetic as a science of suggestion]{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,
$$
\boxed{i^2=j^2=k^2=-1,\qquad ij=k,\qquad ji=-k.}
$$
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>.<On Quaternions, or on a new System of Imaginaries in Algebra by William Rowan Hamilton>[Philosophical Magazine series beginning in 1844, archived by Trinity College Dublin]{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.<An Investigation of the Laws of Thought by George Boole>[Chapters II–III, laws of class symbols and the special law x²=x]{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,
$$
A=\begin{pmatrix}0&1\\0&0\end{pmatrix},\qquad
B=\begin{pmatrix}0&0\\1&0\end{pmatrix}
$$
produce $AB=\operatorname{diag}(1,0)$ and $BA=\operatorname{diag}(0,1)$. This is a checkable reason why universal arithmetic can no longer mean that every rule of ordinary numbers applies unchanged.<A Memoir on the Theory of Matrices by Arthur Cayley>[1858, opening definitions and operations]{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.<The abstract group concept by the MacTutor History of Mathematics Archive>[Nineteenth-century development from substitutions to abstract definitions]{ref}
\b[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.
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