index

Elements of Algebra — Wiki Index

Wiki for Elements of Algebra by Leonhard Euler (with additions by Lagrange).

Part I, Section I — Simple Quantities (Chapters 1–10)

Chapter pages

• ch1.1.1-mathematics-in-general — Defines magnitude, number, and algebra as the foundation of mathematics
• ch1.1.2-signs-plus-minus — Introduces + and − signs, positive/negative quantities, and integers
• ch1.1.3-multiplication-simple-quantities — Multiplication notation, commutativity, and sign rules
• ch1.1.4-integers-and-factors — Prime and composite numbers; prime factorisation
• ch1.1.5-division-simple-quantities — Division, remainders, and sign rules; introduces need for fractions
• ch1.1.6-properties-integers-divisors — Residue classes, divisibility, table of divisors 1–20
• ch1.1.7-fractions-in-general — Fractions as quotients; proper/improper fractions; infinity
• ch1.1.8-properties-of-fractions — Equivalent fractions; reduction to lowest terms
• ch1.1.9-addition-subtraction-fractions — Adding/subtracting fractions via common denominator
• ch1.1.10-multiplication-division-fractions — Multiplying and dividing fractions; inversion rule

Concept pages

• magnitude — The foundational concept: anything capable of increase or diminution
• positive-negative-numbers — Sign rules, integers, and the number line
• prime-numbers — Numbers with no factors other than 1 and themselves
• fractions — Rational numbers arising from non-exact division
• infinity — Euler’s treatment of infinity via fractions and division by zero

Part I, Section I — Powers, Roots, and Logarithms (Chapters 11–23)

Chapter pages

• ch1.1.11-square-numbers — Defines squares, squares of fractions, and the double sign of square roots
• ch1.1.12-square-roots-irrational-numbers — Introduces square roots of non-squares, surds, and radical arithmetic
• ch1.1.13-imaginary-quantities — Introduces imaginary numbers as square roots of negative quantities
• ch1.1.14-cubic-numbers — Defines cubes and their sign behavior
• ch1.1.15-cube-roots — Introduces cube roots and cube-root irrationals
• ch1.1.16-powers-in-general — Formalizes general powers, exponents, zero powers, and negative exponents
• ch1.1.17-calculation-of-powers — Derives the laws for multiplying, dividing, and re-raising powers
• ch1.1.18-roots-and-powers — Generalizes roots of every order and distinguishes even from odd roots
• ch1.1.19-fractional-exponents — Rewrites radicals as fractional exponents and simplifies surds
• ch1.1.20-methods-of-calculation — Connects the algebraic operations and motivates logarithms
• ch1.1.21-logarithms-in-general — Defines logarithms and proves their basic laws
• ch1.1.22-logarithmic-tables — Specializes logarithms to base 10 and sketches table construction
• ch1.1.23-expressing-logarithms — Explains decimal logarithms, characteristics, and table use

Concept pages

• square-roots-and-irrational-numbers — Square roots as the source of Euler’s first irrational quantities
• irrational-numbers — Determinate magnitudes not expressible by integers or fractions
• imaginary-numbers — Even roots of negative quantities and their algebraic use
• cube-roots — Third roots, including the real cube roots of negative numbers
• powers-and-exponents — Unified notation and laws for positive, zero, negative, and fractional exponents
• roots — Roots of all orders as inverse operations to powers
• logarithms — Exponents of a fixed base that turn products into sums
• decimal-fractions — Place-value notation used to express logarithms in tables

Part I, Section II — Compound Quantities (Chapters 1–4)

Chapter pages

• ch1.2.1-addition-compound-quantities — Addition of multi-term expressions by preserving signs and combining like terms
• ch1.2.2-subtraction-compound-quantities — Subtraction by changing the signs of the subtracted expression
• ch1.2.3-multiplication-compound-quantities — Distributive multiplication of expressions and classic identities
• ch1.2.4-division-compound-quantities — Division termwise for simple divisors and long division for exact compound cases
• ch1.2.5-infinite-series — Resolving fractions into infinite power series via repeated long division; geometric and alternating series
• ch1.2.6-squares-of-compound-quantities — Binomial and polynomial square identities with numerical applications
• ch1.2.7-extraction-of-roots-compound — Algorithm for extracting square roots of compound quantities and numbers; radical notation for non-perfect squares
• ch1.2.8-calculation-irrational-quantities — Arithmetic of surds: addition, subtraction, multiplication, and rationalization of denominators
• ch1.2.9-cubes-and-cube-root-extraction — Binomial cube identity and algorithm for extracting cube roots of compound quantities and numbers
• ch1.2.10-higher-powers-compound — Higher powers of binomials, Pascal’s triangle, and the direct binomial coefficient formula
• ch1.2.11-transpositions-and-binomial-proof — Permutation-based proof of binomial coefficients; factorials and multinomial coefficients
• ch1.2.12-irrational-powers-infinite-series — Fractional exponents in the binomial series: series for square and cube roots with iterative approximation
• ch1.2.13-negative-powers-infinite-series — Negative exponents: infinite series for $1/(a+b{)}^m$; verified by multiplication

Concept pages

• compound-quantities — Expressions made of several terms treated as single algebraic quantities
• like-terms — Terms with the same literal part that can be combined
• algebraic-identities — Binomial square, binomial cube, difference of squares, sum/difference of cubes; general binomial theorem
• division-of-compound-quantities — Euler’s exact-division procedure for compound divisors
• infinite-series — Sums of infinitely many terms; geometric series, binomial series, and irrational/negative-power series
• rationalization — Eliminating radicals from denominators by multiplying by conjugate expressions
• pascal-triangle — Triangular table of binomial coefficients; rows sum to powers of 2
• binomial-theorem — General formula $(a+b{)}^n$ for integer, fractional, and negative $n$
• factorials — $n!$ as a count of permutations; used to derive binomial and multinomial coefficients

Part I, Section III — Ratios and Proportions (Chapters 1–13)

Chapter pages

• ch1.3.1-arithmetical-ratio — Defines arithmetical ratio as the difference between two numbers; translation and scaling properties

• ch1.3.2-arithmetical-proportion — Equality of two differences; sum of means = sum of extremes; finding the fourth term

• ch1.3.3-arithmetical-progressions — Four parameters of a progression; formulas to find any one from the other three

• ch1.3.4-summation-arithmetical-progressions — Sum formula $S=n(a+z)/2$; sums of naturals and odds; table of sums by difference

• ch1.3.5-polygonal-numbers — Triangular, square, pentagonal, hexagonal numbers and the general $m$-gonal formula

• ch1.3.6-geometrical-ratio — Geometrical ratio as quotient; antecedent/consequent; types of ratio; scaling invariance; reduction to lowest terms

• ch1.3.7-greatest-common-divisor — Euclidean algorithm for GCD with proof; worked examples

• ch1.3.8-geometrical-proportions — Product of extremes = product of means; transpositions; Rule of Three; combining proportions

• ch1.3.9-observations-proportion-utility — Currency exchange applications; Rule of Reduction (Double Rule of Three)

• ch1.3.10-compound-relations — Compound ratios; duplicate/triplicate ratios; areas, volumes, falling bodies, diamonds; Rule of Five

• ch1.3.11-geometrical-progressions — Geometrical progressions: sum formula $s=a(b^n-1)/(b-1)$; infinite decreasing and alternating progressions

• ch1.3.12-infinite-decimal-fractions — Converting vulgar fractions to decimals; repeating decimals and their conversion back via the 9s-denominator rule

• ch1.3.13-calculation-of-interest — Compound interest as a geometrical progression; logarithmic computation; annual additions/withdrawals; present value and annuities

Concept pages

• ratios — Arithmetical ratio (difference) vs geometrical ratio (quotient); Euler’s terminological conventions
• arithmetical-proportion — Equality of two arithmetical ratios; means, extremes, and the fourth-term rule
• arithmetical-progressions — Sequences with constant common difference; four-parameter system and sum formula
• polygonal-numbers — Figurate numbers as partial sums of progressions; triangular through general $m$-gonal formula
• geometrical-ratio — Geometrical ratio $a:b=a/b$; types, scaling invariance, reduction to lowest terms
• greatest-common-divisor — Euclidean algorithm and proof; coprime numbers; connection to fraction reduction
• geometrical-proportion — $ad=bc$ condition; transpositions; Rule of Three; combining proportions
• compound-relations — Products of ratios; duplicate/triplicate ratios; geometry, physics, and Rule of Five
• geometrical-progressions — Sequences with constant ratio; finite and infinite sum formulas; relation to repeating decimals and compound interest
• repeating-decimals — Periodic decimal expansions; period length; conversion to fractions via $10^k-1$ denominators
• compound-interest — Exponential principal growth $(1+p/100{)}^{n}a$; annual additions/withdrawals; present value

Part I, Section IV — Equations (Chapters 1–16)

Chapter pages

• ch1.4.1-solution-of-problems-in-general — Algebra defined as finding unknowns from knowns; equations introduced; degree classification; transformation rules
• ch1.4.2-resolution-simple-equations — Rules for isolating $x$ in any linear equation: transposing, clearing fractions, radicals, and exponential unknowns
• ch1.4.3-solution-of-questions — Twenty-one worked word problems: partition sums, inheritance chains, progressions, trade, the identical equation, tenth-part inheritance
• ch1.4.4-resolution-two-or-more-equations — Simultaneous linear equations: substitution, elimination, general 2×2 formulas, 3-variable elimination, auxiliary-sum trick
• ch1.4.5-pure-quadratic-equations — Pure (incomplete) quadratics; $x=\pm \sqrt{c/a}$; three cases by sign; five worked problems
• ch1.4.6-mixt-quadratic-equations — Completing the square; quadratic formula; ten commercial word problems
• ch1.4.7-extraction-roots-polygonal-numbers — Inverting polygonal formulas via quadratics; root formulas and integrality conditions for each polygonal class
• ch1.4.8-extraction-square-roots-binomials — Square roots of binomial surds; $a^2-b=c^2$ condition; application to quartic equations
• ch1.4.9-nature-quadratic-equations — Factored form, Vieta’s formulas, sign rules, discriminant, and finding the second root by polynomial division
• ch1.4.10-pure-cubic-equations — Pure (incomplete) cubics $x^3=a$; three roots (one real, two imaginary); five worked problems
• ch1.4.11-complete-cubic-equations — Complete cubics; Vieta for cubics; rational root theorem; sign rule; six worked problems
• ch1.4.12-cardanos-rule — Proof of no fractional roots; derivation of Cardan’s formula; $x^2$-elimination substitution; casus irreducibilis
• ch1.4.13-quartic-equations — Pure, incomplete, and symmetric quartics; Vieta for degree 4; Descartes’ sign rule; palindromic special cases
• ch1.4.14-bombelli-rule — Bombelli’s difference-of-squares reduction of quartic to cubic; worked examples
• ch1.4.15-new-method-quartic — Euler’s radical-ansatz method $x=\sqrt{p}+\sqrt{q}+\sqrt{r}$; removing the cubic term; irrational-root example
• ch1.4.16-approximation — Newton-like linearization and recurrence-series methods for numerical root-finding; Fibonacci sequence; degree 5+

Concept pages

• equations — Definition, degree classification, transformation rules, the identical equation, and multi-variable setup
• systems-of-linear-equations — Substitution and elimination methods; Cramer-like 2×2 formulas; auxiliary-sum trick for symmetric systems
• quadratic-equations — Pure vs. mixt classification; quadratic formula; two-root theorem; discriminant
• cubic-equations — Pure, complete, and general cubics; three roots; Vieta for cubics; rational root theorem; Cardan’s formula
• completing-the-square — Adding $\frac{1}{4}{p}^2$ to turn $x^2+px$ into a perfect square; derivation of the quadratic formula
• vieta-formulas — Sum and product of roots; holds for quadratics and cubics; even for imaginary roots
• discriminant — Sign of $a^2-4b$ determines real vs. imaginary roots; imaginary roots are not approximable
• square-roots-of-binomials — Extracting $\sqrt{a\pm \sqrt{b}}$ as $\sqrt{(a+c)/2}\pm \sqrt{(a-c)/2}$ when $a^2-b=c^2$
• rational-root-theorem — Rational roots must divide the constant term; proved via Vieta; extended to quartics
• cardanos-rule — Derivation and use of Cardan’s formula for the depressed cubic; casus irreducibilis
• quartic-equations — Four roots, Vieta for degree 4, Bombelli and radical-ansatz solution methods
• bombelli-rule — Bombelli’s resolvent cubic and difference-of-squares decomposition for quartics
• descartes-sign-rule — Sign changes = positive roots; successions = negative roots
• approximation-methods — Newton-like linearization and recurrence-series ratio methods for numerical root-finding

Part II — Indeterminate Analysis (Chapters 1–15)

Chapter pages

• ch2.0.1-indeterminate-equations-first-degree — Indeterminate analysis introduced; iterative reduction and Euclidean-table method for $ax+by=c$ in integers; impossibility condition $\gcd (a,b)\mid c$
• ch2.0.2-regula-caeci — Regula Caeci: two equations in three or more unknowns; feasibility bounds; alloy and purchase problems
• ch2.0.3-compound-indeterminate-equations — Compound indeterminate equations where $y$ is linear and $x$ appears to higher degree; divisor conditions; preview of rationalization of surds
• ch2.0.4-surd-rationalization — Four rules for making $\sqrt{a+bx+c{x}^2}$ rational; Pythagorean triples; bootstrap from known solutions; impossibility preview
• ch2.0.5-impossibility-quadratic-squares — Residue-class analysis mod 3, 4, 5, 7 to prove that certain $a{t}^2+b{u}^2$ can never be squares
• ch2.0.6-integer-solutions-quadratic-squares — Seed solution plus Pell pair $m^2=a{n}^2+1$ yields infinite integer families for $a{x}^2+b=y^2$; triangular, pentagonal, hexagonal squares
• ch2.0.7-pell-equation-method — Pell’s iterative descent for solving $m^2=a{n}^2+1$; closed forms for $a$ near a square; table for $a=2$ to $100$
• ch2.0.8-cubic-surd-rationalization — Two-term and three-term root ansätze for $\sqrt{a+bx+c{x}^2+d{x}^3}$; bootstrap from a known seed; finitely-many-solution phenomena (e.g. $1+x^3$ at $x=0,-1,2$)
• ch2.0.9-quartic-surd-rationalization — Three subclasses for $\sqrt{a+bx+c{x}^2+d{x}^3+e{x}^4}$ by which end is a square; up to six new values per pass; $(1+y)/(1-y)$ trick; degree-5 frontier
• ch2.0.10-cubic-formula-as-cube — Three subclasses for $\sqrt[3]{a+bx+c{x}^2+d{x}^3}$; sum-of-two-cubes impossibility cited from Fermat/Euler; biquadrate sketch; free parametric cubes from squared-factor formulas
• ch2.0.11-quadratic-form-factorization — Factoring $a{x}^2+bxy+c{y}^2$ via imaginary factorization; Brahmagupta-Fibonacci identity; sums of two squares; proto-genus theory I × II = I
• ch2.0.12-quadratic-form-as-power — Transforming $a{x}^2+c{y}^2$ into squares, cubes, biquadrates, and fifth powers via $n$-th-power-of-conjugate ansatz; odd powers always work, even powers need a seed
• ch2.0.13-impossibility-biquadrate-sums — Infinite descent proof that $x^4\pm y^4\ne \square$ (Fermat’s Last Theorem at $n=4$); derived impossibilities; contrast with $x^4-2y^4=\square$ which has infinite Pell-driven family
• ch2.0.14-questions-squares — Seventeen showcase Diophantine problems on squares; theorem $a\pm x$ both squares ⇒ $a$ a sum of two squares; impossibility of $p^2+q^2\&p^2+3q^2$ both squares
• ch2.0.15-questions-cubes — Cube-formula questions; Euler’s proof of FLT $n=3$ via $\mathbb{Z}[\sqrt{-3}]$ + descent; theorem $x^3\pm y^3\ne 2z^3$; parametric family for $x^3+y^3+z^3=v^3$

Concept pages

• indeterminate-analysis — Branch of algebra seeking integer solutions when equations are fewer than unknowns
• linear-diophantine-equations — $ax+by=c$ in integers; Euler’s reduction algorithm; solvability condition; simultaneous congruences
• regula-caeci — The Rule of False for two equations with three or more unknowns; feasibility condition; alloy and purchase applications
• pythagorean-triples — Integer triples $p^2+q^2=r^2$; general parametric formula $(2mn,n^2-m^2,n^2+m^2)$ derived from rationalizing $\sqrt{1+x^2}$
• quadratic-residues — Remainders of perfect squares modulo 3, 4, 5, 7, $d$; used to prove impossibility of Diophantine equations
• pell-equation — $m^2=a{n}^2+1$ in integers; Pell’s descent method; minimal solutions; role in generating infinite solution families
• sum-of-two-cubes — Fermat’s Last Theorem at $n=3$: $t^3+x^3=z^3$ has no nontrivial integer solutions; cited by Euler to bound $1+x^3$ as a cube
• brahmagupta-fibonacci-identity — $(p^2+c{q}^2)(r^2+c{s}^2)=x^2+c{y}^2$; derived via complex factorization; basis for sums of squares closure and Pell composition
• sums-of-two-squares — Numbers expressible as $x^2+y^2$; Fermat’s two-square theorem (primes $\equiv 1(\mathrm{mod}4)$); multiplicative closure
• fermats-last-theorem-n3 — $x^3+y^3\ne z^3$ proved by Euler 1770 via $(t+u\sqrt{-3}{)}^3$ and infinite descent
• fermats-last-theorem-n4 — $x^4+y^4\ne z^2$ proved by infinite descent; the only FLT case Fermat himself proved
• three-cubes-as-cube — Four-parameter family for $x^3+y^3+z^3=v^3$; generates $3^3+4^3+5^3=6^3$, $9^3=8^3+6^3+1^3$, etc.
• infinite-descent — Fermat’s proof technique of constructing strictly smaller solutions to derive contradiction

Lagrange’s Additions — Chapters I–IX

Chapter pages

• add1-continued-fractions — Preface to the Additions and full theory of continued fractions: definition, generation, convergent recurrences, alternation, error bounds, best approximation, and worked examples on calendar reform and π
• add2-arithmetic-problems — Four Diophantine-minimum problems via CFs: best $|p-aq|$ approximation, homogeneous-form minimization, binary quadratic forms, and the periodicity theorem; Pell solvability (Art. 37) and Euler’s $\sqrt{N}$ table (Art. 41)
• add3-integer-linear-equations — Integer solutions of $ax-by=c$ via continued fractions; second-to-last convergent of $b/a$ gives the seed; Bachet de Méziriac (1624) credited
• add4-integer-method-linear-y — General polynomial Diophantine equations linear in one variable: resultant divisor scan; finite candidate set except in the constant-denominator case
• add5-rational-quadratic-surds — First systematic decision procedure for $\sqrt{a+bx+c{x}^2}$ rational: descending sequence $A,A^{\prime },A^{\prime \prime },\dots$ via $z=nq-A{q}^{\prime }$; success or certified impossibility in finitely many steps
• add6-double-triple-equalities — Diophantine simple/double/triple equalities; linear cases reduce to a simple equality; quadratic doubles produce a quartic with no general method
• add7-integer-quadratic-method — Direct general method for $x^2-A{y}^2=B$ in integers: reduce $\to C{y}^2-2nyz+B{z}^2=1$, solve via continued fractions or Lagrange reduction, propagate via Pell; refutes Euler’s induction rule (Art. 84)
• add8-pell-method-critique — History of Wallis-Brouncker-Fermat method; counterexample $p^2-6q^2=1$ showing the algorithm fails if approximation directions are mixed
• add9-norm-forms-composition — Multiplicative closure of the symmetric form ${\prod }_i(x+{\alpha }_{i}y+{\alpha }_i^{2}z+\cdots )$ for any-degree polynomial; quadratic, cubic, quartic norm forms; anticipates Gauss composition

Concept pages

• continued-fractions — Nested expansion $\alpha +1/(\beta +1/(\gamma +\cdots ))$; terminates iff value is rational; built by iterated integer-part extraction (= Euclidean algorithm)
• convergents — Truncations $p_n/q_n$ of a continued fraction; satisfy $p_n=a_n{p}_{n-1}+p_{n-2}$; in lowest terms; alternate around target with error $\sim 1/q_n^2$
• semi-convergents — Intermediate fractions inserted between principal convergents when partial quotient $>1$; best approximations within their monotone series
• best-rational-approximations — Lagrange’s theorem: every convergent $p_n/q_n$ beats every rational $m/n^{\prime }$ with $n^{\prime }<q_n$; sharpened to the $|p-aq|$ form in Add. II
• calendar-approximations — Tropical-year ratio $86400/20929$ as a CF; convergents yield Julian, Persian, and other intercalation rules; critique of the Gregorian $400/97$
• binary-quadratic-forms — $A{p}^2+Bpq+C{q}^2$ minimization via the $P^k,Q^k$ recursion driven by the CF of a root; three regimes by sign of $B^2-4AC$
• periodicity-quadratic-irrationals — Lagrange’s theorem: the CF of any quadratic irrational is eventually periodic; pigeonhole proof via bounded $P^k,Q^k$
• square-root-continued-fractions — Euler’s table of CF expansions for $\sqrt{N}$, $N\le 99$; period parity governs whether $p^2-N{q}^2=-1$ is solvable
• lagrange-reduction-algorithm — Descending procedure for $A{p}^2+B{q}^2=z^2$: residue substitution $z=nq-A{q}^{\prime }$ with $|n|\le A/2$ and $A\mid n^2-B$; strict descent via infinite-descent termination
• double-and-triple-equalities — Diophantine terminology: simple/double/triple equalities; reach and limits of Lagrange’s algorithm; degree-4 frontier
• wallis-brouncker-method — Historical method for Pell’s equation: Fermat → Brouncker → Wallis → Euler; equivalent to the CF expansion of $\sqrt{A}$; Wallis’s “proof” of solvability is petitio principii
• norm-forms — Symmetric product ${\prod }_i(x_0+{\alpha }_i{x}_1+\cdots +{\alpha }_i^{n-1}{x}_{n-1})$ over roots of a degree-$n$ polynomial; the field norm $N_{K/\mathbb{Q}}$ in power-basis form
• composition-of-forms — Bilinear law $\mathbf{x}\star {\mathbf{x}}^{\prime }$ on norm-form indeterminates such that $N(\mathbf{x})N({\mathbf{x}}^{\prime })=N(\mathbf{x}\star {\mathbf{x}}^{\prime })$; quadratic and cubic explicit formulas