Giuseppe Peano’s Formulario Mathematico is a working encyclopedia of mathematics written in a stripped-down Latin he called “latino sine flexione” — Latin without grammar, its inflections pared away so that the language behaves like a symbol system. The volume gathers definitions, theorems, proofs, historical citations, and a biographical dictionary of mathematicians into one continuous symbolic apparatus, and it is meant to be read as much for its notation as for its results. This fifth edition, prepared in Turin in 1908, presents itself as the fifth tome of a larger project Peano had been building for years: a complete formal language for the whole of mathematics.
Peano’s governing conviction is stated plainly in the preface. Every real advance in mathematics, he argues, has come from the introduction of ideographic signs. The Indo-Arabic ciphers, the signs +, −, ×, =, √, the letters e and π, the integral and differential marks — each arrival made a body of reasoning shorter, clearer, and more capable of calculation. Archimedes, working with Greek numerals, labored to reach two decimal places of π; Indian and European mathematicians with better signs reached four, then fifteen, then thirty-two. Fewer symbols, and better ones, mean a more perfect system.
From Ciphers to a Universal Script
The Argument for Symbols
Peano opens the fifth edition with a preface in his own Latin without inflection, and his first claim is historical: every advance in mathematics has come from the introduction of ideographic signs. The Indo-Arabic ciphers, brought into Europe around 1200, show why. They are brief to write, they shrink the vocabulary (no separate word is needed for ten or twenty or a hundred), and above all they make calculation possible. Archimedes, labouring with Greek numerals, reached two decimal places of π; Aryabhata, working with the new ciphers, reached four; European mathematicians around 1600 reached fifteen and then thirty-two. The same reasoning, Peano insists, applies to every symbolic system. The signs +, −, ×, =, √, e, π, ∫, d built algebra itself, and much of what Euclid wrote in words becomes shorter, freer of dead terminology, and open to new theories once modern symbols replace the prose. A symbolic proposition can be read aloud in words, but ordinary treatises carry far more words than symbols—sum, product, factor, coefficient—and these words drag their definitions and their obscurities with them. Worse, some words answer to nothing real: “irreducible fraction” names no class, since 2/3 = 4/6; “binomial” and “monomial” name pseudo-entities, since 2 + 3 = 5; “exponent” fails because 2² = 4². Words about formal properties cannot be turned into symbols at all. Conversely, language often lacks the right word: “derivata” and “integrale” have no vulgar equivalent, and their meaning comes only from definition. Geometric calculus, Peano continues, arrived around 1800 through the vector, an entity answering to no older word and rendering the old nomenclature unnecessary; its notations still vary, chiefly because authors introduce symbols without studying the notations already in use.
A Dictionary of Mathematicians
The bibliography that follows is a roll-call of the discipline, each entry compressed to dates, places, posts, and works. Cayley studies at Trinity and practises as a notary before returning to Cambridge; Cesàro teaches at Naples; Cotes, Newton’s disciple, edits the second Principia and dies at thirty-three. Cramer writes on algebraic curves; Darboux holds chairs at the Collège de France and the Sorbonne; De Lagny, Delambre, De L’Hôpital, De Morgan pass in brief. Descartes is educated at La Flèche, serves as a soldier in the Netherlands, wanders Europe, settles at Amsterdam, and dies at Stockholm at the Queen of Sweden’s invitation. Dini, Diophantus, Dirichlet, Euclid, Euler—disciple of Jacob Bernoulli, member of the St. Petersburg Academy, president of the Berlin Academy, recalled to Russia—each receive their line. Fermat’s works appear at Paris in 1891; Fourier, a tailor’s son, teaches at the École polytechnique, goes to Egypt with Bonaparte, and serves as prefect of Isère. Fresnel is a bridge engineer; Gauss studies at Braunschweig and Göttingen and directs the observatory there from 1807. Galileo, Gergonne, Genocchi, Sophie Germain, Girard, Grassmann—theologian turned schoolmaster—Gregory, Hamilton of Dublin, Harriot, who sails to Virginia with Raleigh, Hérigone, Hermite: the list runs on, a whole tradition reduced to its essentials.
From Word Elements to the Primitive Ideas of Number
The main text begins with language itself. Peano treats the sign “-” as a mark of union between two elements, its value following from arithmetic, and calls the common element of two words the theme, index, prefix, or suffix. He tracks “is” across tongues—Latin est, English is, German ist, Russian est, Sanskrit as—and shows “es” surviving in essentia and hidden in ontologia. Rules of sound-shift follow: e becomes i in Germanic under given conditions; Latin ae becomes e everywhere but German; the ending -a replaces -o in Spanish and Italian. Word after word is dissected—aequale, deduzione, ipotesi, classe, sillogismo, logica, associativo (Hamilton, 1841), commutativo and distributivo (Servois, 1815)—with Leibniz’s occasional Latin “to” noted, and the Egyptian origin of numerals and of the signs + and × recorded. Then the arithmetic proper: N, 0, and + are taken as primitive, their meaning fixed by five propositions—zero is a number, successors are numbers, induction, equal successors imply equal numbers, zero follows nothing. Peano justifies each by counter-example: the twenty-four hours satisfy all but the last; an anti-periodic series fails equality of successors; the rationals fail induction.
The Arithmetic of Signed Numbers and Fractions
Peano opens the mathematical core with the relative numbers—integers carrying a sign—and sets down the conventions of equality, negation, and subtraction as numbered propositions, each followed by a bracketed proof. Two relative numbers are equal, he explains, when for every absolute number permitting the operation, subtracting each from that number yields the same result. From this come the lemmas that −a = −b implies a = b, and that a = b gives a − b = 0. Subtraction itself enters by definition. Multiplication follows in its own section: a×0 = 0, 1×a = a, commutativity, associativity, distributivity, and the zero-product law stating that ab = 0 forces a = 0 or b = 0. A run of polynomial identities comes next—a(b−c)+b(c−a)+c(a−b) = 0, and expressions built on (a−b)(b−c)(c−a)—some credited to Euler and Legendre. Powers with natural exponents bring the sign rules for even and odd exponents, and the binomial expansions (a+b)² = a²+2ab+b² and (a+b)(a−b) = a²−b².
Rational numbers he treats as operations: b/a means multiply by b and divide by a, so that a fraction, like a signed number, is an operation rather than a thing. Common usage writes 2/5 for “divide by 5, multiply by 3,” but Peano inverts the order to keep the operation feasible. He reaches back to Ahmes, the Egyptian reckoner of about 3000 BC, citing the Rhind papyrus, and notes that the scribes suppressed the numerator when it was one. Equality of fractions, a/b = c/d exactly when ad = bc, is credited to Euclid. Addition, multiplication, order, and roots of rationals follow, thick with citations from Euclid and Pappus to Cauchy and Legendre.
Limits, Logarithms, and Sums
The real numbers are defined as the class of upper limits of classes of rationals. Quadratic equations arrive with quotations from Leonardo of Pisa and Brahmagupta in translation, and Tartaglia’s verse on the cubic. Logarithms, credited to Neper, come with a three-decimal table. The summation sign X(f, a–b) gathers the values of a function over a class, and the arithmetic series, the sums of squares—attributed to Aryabhata and Alqachani—and geometric sums are worked out.
A Catalogue of Attributions
The formulas continue in a dense chain, each tagged to its historical author: Archimedes, Ibn al-Banna, Alqachani, Fermat (with a Latin quotation on sums of squares), Wallis, Jacob Bernoulli and his table of relations among the sums s₁ through s₇. Number-theoretic functions and combinations draw on Pascal, Wallis (quoted on the periodic return of digits in decimal fractions), Herigone, Euler, Vandermonde, Cayley, Abel, Lagrange, Cesàro, Dixon, Johann Bernoulli, and Hatzidakis. The binomial formula cites the medieval Arithmetica integra and Pascal. Propositions on primes and factorials run through Kummer, Legendre, Gauss, Glaisher, and others. Finite differences, with the Δ sign from Euler, lead to the Bernoulli numbers, defined by A₀ = 1, A₁ + ½ = 0, and listed as fractions up to B₆₀, with Euler’s notation and Binet’s adoption noted.
Cardinality and the Diagonal Argument
Cauchy supplies the mean; Cantor and Bernstein the cardinal number. Infinite numbers N₁ and N₀ appear, with countability, and Cantor’s results that the square of N₁ has the cardinality of N₁, that the reals match N₁, and the rationals N₀—quoted in French on algebraic numbers paired one-to-one with the positive integers. A diagonal argument then shows that the reals between 0 and 1 cannot be serialized.
Determinants, Complex Numbers, and the Lexicon
Limits, derived classes, intervals, and probability (after Moivre) give way to complex numbers and determinants, the latter credited to Leibniz, Cramer, Laplace, Cauchy, Binet, Vandermonde, Smith, and Sylvester. Linear substitutions invoke Gauss, Boole, Grassmann, Cayley, and Peirce; the imaginary unit i draws on Bombelli, Euler, and Gauss. There follows a bilingual lexicon, Latin and Italian with English, German, French, Spanish, and Russian equivalents, tracing etymologies through Greek and Sanskrit—functione, operatione, variabile, limite, logarithmo (with Neper’s “characteristica” and “mantissa”)
Defining the Limit
Peano begins with the notion of a finite quantity, defined by the condition that a succession has a limit equal to that quantity: for any positive h there is an index s beyond which every term differs from u by less than h. The defining formula contains letters absent from the defined term, and these must be apparent — bound by the sign of inversion — as l and s are here. The limit of a succession follows: x is the limit when every positive h is answered by an index s after which every term exceeds h, with the analogous definition for limit −2. A necessary and sufficient condition for a finite or infinite limit is that the maximum and minimum limits coincide. Bolzano (1811) is cited in German and Latin: a series whose later terms differ from the nth by less than any assigned magnitude possesses one determinate magnitude to which its terms continually approach.
History and Etymology of the Limit
The idea of limit, though not the word, appears in Euclid’s measurement of the pyramid and in Archimedes; the concept reduces to the earlier symbol l. Wallis’s definition agrees with the particular case of monotone functions, but the complete definition of lim dates only from around 1860. Cauchy’s class Lm (1821) is cited, along with his remark that lim(sin 1/x) admits infinitely many values between −1 and +1. Peano traces “limite” to limes, the oblique case of lime, and notes that the common writing lim f y conceals an apparent letter and is an incomplete symbol. Definitions follow for constant, increasing, and decreasing functions, with etymologies for consta, crescente, and decrescente.
Theorems on Limits and Series
An increasing succession has a limit equal to the superior limit of its values, always determined, finite or infinite. Propositions on limits of sums, products, quotients, powers, and roots carry attributions to Cauchy, Bernoulli, Legendre, Tchebychef, and Cesaro; examples show functions with two limits, functions with none, and the function equal to 0 at rationals and 1 at irrationals. A series is defined as the limit of partial sums; it converges when that limit is a quantity, and the remainder after n terms is the difference between the total and the nth partial sum. Etymologies for series, converge, adverge, and diverge are given, with three distinct senses of divergence in different authors. Convergence requires lim u = 0, though not sufficiently — the harmonic series refutes the converse — and the necessary and sufficient condition is stated.
The Number e
Every value of (1+1/m)^m is less than every value of (1+1/m)^(m+1), so the upper limit of the first class is at most the lower limit of the second; since the ratio between corresponding values approaches 1, the two limits coincide. Setting m=1.5 yields the integer part of e; from m=30 one gets bounds tied to the value of one franc at compound interest over twenty and twenty-one years. Napier invented e as the base of a logarithmic system; Cotes called it the modular ratio; Euler denoted it by e and computed it. Defined as a limit of (1+x/m)^m, the exponential yields e^x = 1 + x/1! + x²/2! + …, and summing ten terms determines e within 10⁻⁸. Hermite proved e transcendental; Euler showed e and its square irrational; Lambert showed e^x irrational for rational x. Mercator’s series for logarithms follows, giving log 2, log 5, and log 10 = 2.30258509, whose reciprocal is the modulus of decimal logarithms. Cotes’s continued fraction for e — quotients 2, 1, 2, 1, 1, 4, 1, 1, 6, … — is quoted. Defining cos x and sin x as real and imaginary parts of e^(ix), every trigonometric formula derives from the exponential; the addition formulas are proved, with Abū’l-Wēfā and Ptolemy cited, and π/2 is defined analytically as the least positive root of cos x = 0.
The Derivative and Its Rules
The derivative of f in field u at x is the limit of (fy−fx)/(y−x) as y tends to x; the field must appear in the notation because the derivative can depend on it, as the modulus function shows, with derivative 1 on the positive side and −1 on the negative. Peano warns against decomposing Df|x into Df times x, and against substituting numbers into incomplete formulas such as Dx²=2x. The usual rules follow — sum, constant multiple, product, quotient, chain, and inverse function — together with D(xⁿ)=nxⁿ⁻¹, extended to rational exponents.
Taylor’s Theorem and the Calculus of Small Differences
Peano opens with the theorem that bears Taylor’s name, stated in limit form: if a function has derivatives up to order n at a point x, then the difference between f(y) and the polynomial f(x) + (y−x)Df(x) + … divided by (y−x)ⁿ tends, as y approaches x, to the nth derivative over n!. He coins the phrase “infinitesimal of order n” for a function of y whose quotient by (y−x)ⁿ tends to a finite limit, and reads the theorem as a decomposition of f(y) into f(x) plus successive infinitesimals of rising order. Three proofs follow, the third credited to Bernoulli, with citations to Genocchi, Cesàro, and Mathesis. From there he passes to maxima and minima: a null first derivative with positive second derivative at an interior point guarantees a neighbourhood on which f(x) is the minimum; a negative second derivative gives a maximum. Power series come next, with the radius of convergence built from the reciprocal of the upper limit of the nth root of |uₙ|, and the coefficient of xⁿ identified as the nth derivative at zero over n!. A historical excursus reproduces Bernoulli (1694), Taylor (1715), Maclaurin (1742), and Arbogast (1800); Bernoulli claims priority, Taylor’s theorem being “not materially different,” and Peano, citing Pringsheim, judges the rehabilitation of Taylor’s claims to have been made “in vain.” Lagrange and Cauchy supply forms of the remainder. Interpolation, finite differences, and Newton (1686) lead to the theorem that the nth incremental ratio equals some value of the nth derivative over n!. Integer functions, partial fractions, Legendre polynomials, and Bernoulli numbers round out the algebraic side, while for complex functions the increment ratio is only a mean between derivative values, and the hodograph enters (Möbius 1843, named by Hamilton 1846).
Tangents, Curvature, and Torsion
The differential geometry of curves is laid out in definitions: tangent line, normal plane, osculating plane, axis, centre and radius of curvature, curvature, and torsion, the last credited to Frenet (1847). Peano’s sign convention is memorably concrete — torsion is positive in the hop and the cucumber, negative in the vine and the European screw, while screws made in China and Japan have positive torsion. Cartesian and polar coordinates are treated, and the linear equation Df = af yields f(x) = f(0)e^{ax}.
Differential Equations and the Motion of Bodies
The linear first-order equation Df·x = af·x + b, with constant a and b, is integrated in a worked derivation covering the general exponent and the case n = 1. A heavy point with constant acceleration g obeys D²p = g and traces p = p₀ + tDp₀ + ½gt², a parabola — Galileo’s Dialoghi of 1633 is cited for the projectile compounded of uniform horizontal and naturally accelerated downward motion. For a central force directed toward a fixed point o, the triangle o, p, p+Dp keeps constant magnitude, plane, and sense, and conversely; Newton’s Principia, Book 1, is invoked for areas proportional to the times. A heavy point in a resisting medium, D²p = g − aDp, follows an exponential or logarithmic curve with an asymptote (Newton 1687, Book 2). The equation D²f = a²f is solved in the stated form, and D²p = −a²(p−o) gives an ellipse about o whose period 2π/a is independent of the initial data — isochronous motion, matching Newton’s Principia Book 1, Prop. X. For a substitution a of order n, Dx = ax has solution x = (x₀)e^{at}, the exponential being reducible by the characteristic equation to a polynomial of degree n−1 in a. Partial derivatives D₁f, D₂f are adopted in place of other authors’ incomplete notations; the chain rule, the interchange D₁D₂f = D₂D₁f under continuity, and a Taylor expansion in (hD₁ + kD₂) follow. For a function of a complex variable the derivative is defined as the linear transformation whose remainder over |y−x| vanishes, its elements being the partial derivatives (Jacobi 1811, Grassmann 1862).
The Integral from Archimedes to Euler
Archimedes’ limit argument for the spiral, the barycentre of a triangle, and the parabola opens the account, followed by Cavalieri (1639) on the doubling, tripling, and quadrupling of lines, squares, and cubes, and his probable inference to higher powers. Fermat (edited by Mersenne 1644)
Existence Proofs and the Axiom of Choice
The passage opens with a proof that a bounded, closed class can be found inside each of a nested sequence of classes, and that a common element exists across all of them. Peano then attributes a related theorem to an unnamed author who divides n-dimensional space into cubes of side 1, finds a cube sharing an element with every class fr, then subdivides into cubes of side 1/2, 1/4, and so on, always finding a partial cube with the same property; the shrinking sequence converges to a limit point satisfying the thesis. This style of demonstration, he notes, appears in Bolzano (1817), Cauchy’s Cours d’Analyse (1821), and frequently in Weierstrass. The objection arises that choosing a cube among infinitely many requires a general law of election, which is not licit without one. Peano therefore supplies such a law and writes an explicit expression for the limit point, giving an expression of an individual belonging to the class whose existence he proves.
Closed Classes, Limes, and Distributive Properties
The class common to every class of the sequence f₀ f₁ f₂… is closed. Peano generalizes: if f is a class of complexes of order n and a function of positive quantities, and if for every positive h the class fh is non-empty and bounded, and if for every h−k the limit class of fh is contained in fk, then an element common to all fh exists. Dropping the boundedness hypothesis, he writes fh as a general limit class; dropping the requirement that each class contain the following ones, he considers the logical sum. The class in the second member he calls the “limes” (Lm) of the sequence f. When every class consists of a single individual, this yields the definition of limit for real quantities; the general definition already occurs in his Geometry and is adopted for the definition of tangent figure.
A property ω of classes of complexes of order n is called distributive when, if the logical sum of two classes u and v has ω, then one of u and v has it, and conversely. If some bounded class s has a distributive property, there exists an individual in the limit class of ω, and a z such that every sphere centered at z with arbitrary radius r has property ω. This form occurs in Cantor’s Ueber unendliche Punktmannichfaltigkeiten (1884). The proposition “class ω contains infinitely many individuals” is distributive; hence if s is bounded with infinitely many individuals, there exists an element z near ω such that every sphere centered at z contains infinitely many individuals of s.
The General Derivative
Given a function f, a complex of a real variable, Dg f x — the “general derivative of f for value x” — indicates the limit class of the incremental ratio of f for values x and y as y varies in the field where f is defined and tends to x. Every complex function of a real variable, for x in its field of variability and in its derived field, always has a general derivative, which is a non-null class. The operation D is linked to Dg as lim is to Lm: if the class Dfx consists of a single individual, that individual is the derivative of f. Rules of derivation subsist with little modification when Dg is substituted for D, or Lm for lim: if the general derivatives of two functions are bounded, the general derivative of the sum is contained in the sum.
Dirichlet’s Integral and the Limit of an Integral
A quotation attributed to Dirichlet (1829) states that whatever the function f(x), provided it remains continuous between 0 and A (A positive and at most equal to π) and increases or decreases from the first limit to the second, the integral of (sin x)/x · f(x) ends by differing constantly from (π/2)f(0) by a quantity less than any assignable number, when the upper limit is increased beyond every positive bound. A second Dirichlet citation concerns a limit of a sum.
§35 treats the limit of an integral. Given a class of quantities, a point in it, an interval, and a real continuous function of two variables, if one integrates f(x,y) over y in the interval, the limit of the integral as x varies in the class and tends to a equals the integral where a replaces x. This expresses the commutative property of the operations limit and integral. The proof uses the modulus of the difference of integrals being less than the integral of the modulus of the difference, uniform continuity, and the length of the integration interval. A corollary: the integral of a continuous function of two variables, made with respect to one, is a continuous function of the other. Continuity is assumed for the composite; continuity “in the divided sense” for x and for y does not suffice.
Words That Name Nothing
The preface carries an argument that runs underneath the whole book: ordinary mathematical language is full of words that answer to no symbol, and some of them answer to nothing at all. Peano’s example is the “irreducible fraction.” Since 2/3 = 4/6, and the first is irreducible while the second is reducible, the phrase has the grammar of a class of fractions without being one — it fails the basic property of equality. Likewise “binomial” and “monomial” in 2 + 3 = 5, or “exponent” in 2² = 4². These are pseudo-entities, and the words that denote them can be suppressed along with their definitions and the obscurity those definitions breed. Where common language has too many words, it also has too few: “derivative” and “integral” look like ordinary words but take their meaning from written definitions, not from etymology.
This is why the Formulario is organized as a dictionary as much as a treatise. Its lexicon gives each headword with equivalents in English, German, French, Spanish, Italian, and Russian, traces its roots, and compares it to Greek and Sanskrit — an etymological apparatus in service of notational hygiene. The book’s primitives are chosen with the same care. Number, zero, and successor are taken as undefined and fixed by five propositions, each shown necessary by a counter-example: a clock-face series of hours satisfies all but the axiom that zero follows no number; a finite series of three terms satisfies all but the axiom that every number has a successor. Geometry begins from point and a four-place relation among points, because a point, Peano says, cannot be logically defined — the idea comes from the physical world.
The Notation as the Argument
What holds the book together is not a thesis but a method: say everything in symbols, and let the symbols carry the proof. The derivative is defined as a limit of an incremental ratio, and Peano insists on writing the field of variability into the notation, because the derivative of the modulus function at zero is 1 on one side and −1 on the other. He warns against the “incomplete symbol” — the formula Dx² = 2x found in textbooks, into which one cannot simply substitute numbers. The same discipline governs the treatment of limits, where the definition is parsed to show which letters are real and which are bound, and of the alternating product, where Peano adopts ∧ (for “alterno”) over Grassmann’s bracket because it is clearer. Even the quaternion gets its vocabulary audited: “scalar” comes from the Latin scala, not from English, whatever some physicist may think.
History enters as evidence rather than ornament. Nearly every proposition carries an attribution — Euclid quoted in Greek, Fermat in French, Rolle’s cascade in his own words, Galileo on the parabola, Newton on orbits and areas. The biographical dictionary runs from Cayley and Cesàro back through Euler, Gauss, and Descartes to Diophantus and Euclid, each with dates, posts, and principal works. The effect is a mathematics presented as a cumulative human construction, in which each sign has a discoverer and each theorem a date.
A Language for Calculation
The book’s legacy lies less in any single result than in the idea that drove it: that a fully symbolic language is not a convenience but a condition of rigor, and that such a language can be built, taught, and translated. Peano’s latino sine flexione is the extreme case of his own principle — a natural language reduced toward notation, its grammar stripped to make it unambiguous. The Formulario was overtaken by the great collaborative projects of twentieth-century mathematics, but its instinct survives wherever notation is treated as thought rather than shorthand, and wherever a definition is expected to earn the words it replaces.
