add1-continued-fractions

Additions Chapter I — On Continued Fractions

Summary: Lagrange’s first Addition develops the full theory of continued fractions: their genesis from any real number by iterated integer-part extraction, their convergents and best-approximation property, the alternation and error-bound theorems, and worked applications to calendar reform and the digits of $\pi$.

Sources: additions-1

Last updated: 2026-05-10

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Preface (pp. 412-413)

Lagrange opens with a survey of indeterminate (Diophantine) analysis: Bachet and Fermat extended Diophantus, and Euler renewed the subject in the 18th century. Lagrange composes the Additions to make Euler’s Elements of Algebra a more complete reference for indeterminate problems. He announces the four-part programme:

1. Continued fractions (Chapter I) — a tool indispensable for indeterminate problems in integers, scarcely treated in arithmetic and algebra texts.
2. General methods for indeterminate equations of the first and second degree — both rational and integer cases.
3. A theory of similar functions whose products remain similar — useful for indeterminate problems where standard methods fail.
4. Various complementary remarks.

This chapter occupies Articles 1–22 (pp. 414–435).

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Definition (Art. 1)

A continued fraction is any expression of the form

$\alpha +\frac{b}{\beta +\frac{c}{\gamma +\frac{d}{\delta +\cdots }}}$

with $\alpha ,\beta ,\gamma ,\delta ,\dots$ and $b,c,d,\dots$ integers (positive or negative). Lagrange specializes immediately to the unit-numerator form (the only form of analytic utility):

$\alpha +\frac{1}{\beta +\frac{1}{\gamma +\frac{1}{\delta +\cdots }}}$

written compactly as $[\alpha ;\beta ,\gamma ,\delta ,\dots ]$ in modern notation.

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Historical Note (Art. 2)

• Brouncker (mid-17th century) was the first to use continued fractions; he expressed the ratio of the circumscribed square to the area of the circle (i.e., $4/\pi$) as

$1+\frac{1}{2+\frac{9}{2+\frac{25}{2+\cdots }}}$

• Wallis in Arithmetica Infinitorum showed Brouncker’s fraction equals his own product $\frac{3\cdot 3\cdot 5\cdot 5\cdot 7\cdot \cdots }{2\cdot 4\cdot 4\cdot 6\cdot 6\cdot \cdots },$ and gave the general method of reducing CFs to vulgar fractions.
• The deeper properties (convergent law, best-approximation) belong to Huygens, who used them in designing the gear ratios of his planetary automaton.

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Genesis from a Real Quantity (Art. 3)

Given any quantity $a$, let $\alpha$ be the nearest integer below $a$. Then $a-\alpha$ is a fraction less than 1, so $1/(a-\alpha )=b>1$. Take $\beta$ the nearest integer below $b$, and continue. This produces the chain

$a=\alpha +\frac{1}{b},b=\beta +\frac{1}{c},c=\gamma +\frac{1}{d},\dots$

and substituting backward expresses $a$ as a continued fraction. Sign convention: if all $\alpha ,\beta ,\gamma ,\dots$ are taken as the integer below (so $\alpha \le a<\alpha +1$), then $\beta ,\gamma ,\delta ,\dots$ are all positive. Taking some integers from above flips signs.

Termination criterion (Art. 3, end): The expansion terminates iff $a$ is rational; for irrational or transcendental $a$, the CF is infinite.

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Reduction Rule for Vulgar Fractions (Arts. 4–5)

To reduce $A/B$ to a continued fraction:

1. Divide $A$ by $B$, quotient $\alpha$, remainder $C$;
2. Divide $B$ by $C$, quotient $\beta$, remainder $D$;
3. Divide $C$ by $D$, quotient $\gamma$, remainder $E$; …

Continue until the remainder is 0. The quotient sequence $\alpha ,\beta ,\gamma ,\dots$ is the CF. This is exactly the Euclidean algorithm — see ch1.3.7-greatest-common-divisor.

Worked example (Art. 5): $1103/887$. Divisions yield quotients $1,4,9,2,1,1,4$, so

$\frac{1103}{887}=1+\frac{1}{4+\frac{1}{9+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4}}}}}}.$

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Sign Transformations (Arts. 6–7)

Choosing integer parts above the value yields all-negative denominators; mixed choices give mixed signs. The transformation

$\mu +\frac{1}{-\nu +\frac{1}{\pi +\cdots }}=\mu -1+\frac{1}{1+\frac{1}{\nu -1+\cdots }}$

removes a negative sign at the cost of inserting a unit denominator. A unit denominator anywhere in the CF signals that the preceding integer-part choice was not the nearest integer; the simplification reduces the term count.

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Approximating Irrationals (Art. 8)

Irrational quantities are first expressed in decimals (with both the truncated and the next-larger value, to get reliable bounds). Using Ludolph’s value of $\pi$ to 35 digits:

$\pi =3.14159265358979323846264338327950288\dots$

Lagrange computes the partial quotients

$\frac{\text{Circumference}}{\text{Diameter}}=[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,2,1,84,2,1,1,15,3,13,1,4,2,6,6,1].$

The unit denominators can be eliminated by Article 7’s transformation (giving a CF with negative terms).

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Application to Roots of Equations (Art. 9)

Given an equation in $x$, find the integer $\alpha$ nearest to a root, substitute $x=\alpha +1/y$, get a new equation in $y$ with a root $>1$, find its integer part $\beta$, substitute $y=\beta +1/z$, and so on. The root is

$x=\alpha +\frac{1}{\beta +\frac{1}{\gamma +\cdots }}.$

Terminates iff the root is rational; otherwise infinite. See approximation-methods for the broader context of numerical root-finding in Elements of Algebra.

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Convergents and Their Recurrence (Art. 10)

Let $p_n/q_n$ denote the rational obtained by truncating the CF at depth $n$. Lagrange writes these as $A/A^{\prime },B/B^{\prime },C/C^{\prime },\dots$ with the recurrences

A = \alpha, & A' = 1, \\ B = \beta A + 1, & B' = \beta, \\ C = \gamma B + A, & C' = \gamma B' + A', \\ D = \delta C + B, & D' = \delta C' + B', \\ \vdots\, & \vdots \end{array}$$ In modern notation: $p_n = a_n p_{n-1} + p_{n-2}$, $q_n = a_n q_{n-1} + q_{n-2}$. The sequence $A/A', B/B', C/C', \ldots$ converges to $a$. If $a$ is rational $V/V'$, this sequence terminates at $V/V'$ itself. --- ## The Cross-Product Identity (Art. 12) Cross-multiplying consecutive convergents, $$BA' - AB' = 1, \quad CB' - BC' = -1, \quad DC' - CD' = 1, \quad ED' - DE' = -1, \ldots$$ i.e., consecutive convergents satisfy $p_n q_{n-1} - p_{n-1} q_n = (-1)^{n-1}$. Two consequences: 1. **Convergents are in lowest terms**: if $p_n$ and $q_n$ shared a factor $> 1$, that factor would divide $\pm 1$ — impossible. 2. **No fraction with smaller denominator can lie strictly between two consecutive convergents** (Lagrange's optimality lemma; see [[best-rational-approximations]]). --- ## Error Bounds and Alternation (Arts. 13–14) The error in approximating $a$ by $C/C'$ satisfies $$\frac{1}{C'(D' + C')} \;<\; \left| a - \frac{C}{C'} \right| \;<\; \frac{1}{C' D'}.$$ The convergents are **alternately less than and greater than** $a$ (when all partial quotients are positive). Equivalently, $a$ always lies strictly between two consecutive convergents. --- ## Best Approximation Property (Arts. 15–16) Each convergent $C/C'$ approximates $a$ more closely than any other rational $m/n$ with $n \le C'$. The principal series naturally splits into: - *Increasing series* (convergents less than $a$): $\frac{A}{A'}, \frac{C}{C'}, \frac{E}{E'}, \ldots$ - *Decreasing series* (convergents greater than $a$): $\frac{B}{B'}, \frac{D}{D'}, \frac{F}{F'}, \ldots$ See [[best-rational-approximations]]. --- ## Intermediate (Semi-)Convergents (Arts. 17–18) Between two consecutive principal convergents $\frac{A}{A'}$ and $\frac{C}{C'}$ (with $C = \gamma B + A$, $C' = \gamma B' + A'$, and $\gamma > 1$), one can insert $\gamma - 1$ **intermediate fractions** $$\frac{B + A}{B' + A'}, \frac{2B + A}{2B' + A'}, \ldots, \frac{(\gamma-1)B + A}{(\gamma-1)B' + A'}$$ forming an arithmetic progression of numerators and denominators. These share the best-approximation property *within their series* (all less than $a$, or all greater) but not necessarily across series. See [[semi-convergents]]. --- ## Example 1 — The Solar Year and Calendar Reform (Art. 20) Per M. de la Caille, the tropical year is $365^\mathrm{d}\, 5^\mathrm{h}\, 48'\, 49''$, exceeding 365 days by $5^\mathrm{h}\, 48'\, 49''$. The ratio $24^\mathrm{h} : 5^\mathrm{h}\, 48'\, 49''$ equals $86400/20929$. Reducing to a CF gives quotients $$4, 7, 1, 3, 1, 16, 1, 1, 15$$ and convergents $$\frac{4}{1}, \frac{29}{7}, \frac{33}{8}, \frac{128}{31}, \frac{161}{39}, \frac{2704}{655}, \frac{2865}{694}, \frac{5569}{1349}, \frac{86400}{20929}.$$ Each fraction $p/q$ states: intercalate $p$ leap days every $q$ years. - $4/1$ → 1 day every 4 years (the **Julian calendar**, slightly too many) - $29/7$ → 7 days every 29 years (slightly too few) - $33/8$ → 8 days every 33 years (slightly too many) - … The Gregorian rule of 97 days every 400 years (i.e., $400/97$) is **not** among Lagrange's convergents — it derives from Copernicus's slightly different year-length estimate $365^\mathrm{d}\, 5^\mathrm{h}\, 49'\, 20''$. Lagrange notes that, by the optimality theorem, $161/39$ is more accurate than $400/97$ relative to de la Caille's value. See [[calendar-approximations]]. --- ## Example 2 — Convergents to $\pi$ (Art. 21) Using the CF from Article 8, the principal convergents to $\pi$ are $$\frac{3}{1}, \frac{22}{7}, \frac{333}{106}, \frac{355}{113}, \frac{103993}{33102}, \frac{104348}{33215}, \frac{208341}{66317}, \ldots$$ Each $p/q$ approximates $\pi$ more closely than any rational with denominator $< q$. The famous $22/7$ (Archimedes) and $355/113$ (Tsu Chung-chih) both appear. Error bounds: $|\pi - 3| < 1/7$, $|\pi - 22/7| < 1/(7 \cdot 106)$, etc. Splitting into the two monotone series gives the intermediate-fraction families - *less than $\pi$*: $\frac{3}{1}, \frac{25}{8}, \frac{47}{15}, \frac{69}{22}, \frac{91}{29}, \frac{113}{36}, \ldots, \frac{333}{106}, \frac{688}{219}, \ldots$ - *greater than $\pi$*: $\frac{4}{1}, \frac{7}{2}, \frac{10}{3}, \frac{13}{4}, \frac{16}{5}, \frac{19}{6}, \frac{22}{7}, \frac{355}{113}, \ldots$ --- ## Scholium (Art. 22) Lagrange credits Wallis (in the appendix to Horrox's posthumous works) for the first published treatment, and Huygens (in *Opera Posthuma*, treatise *Descriptio Automati Planetarii*) for the principal discoveries. He cites Euler's Petersburg Commentaries (Vols. IX, XI old; IX, XI new) and Berlin Memoirs of 1767–1768 as the modern continuation, and notes John Bernoulli's application of CFs to tables of proportional parts in the *Recueil pour les Astronomes*, Vol. I. --- ## Concept Pages - [[continued-fractions]] — definition, generation, termination - [[convergents]] — recurrences, key identity, error bounds - [[semi-convergents]] — intermediate fractions - [[best-rational-approximations]] — Lagrange's optimality theorem - [[calendar-approximations]] — Julian/Gregorian intercalation as a CF problem ## Related pages - [[ch1.3.7-greatest-common-divisor]] — the Euclidean algorithm underlying CF reduction - [[approximation-methods]] — Euler's numerical root-finding methods (CF root extraction sketched here) - [[ch1.4.16-approximation]] — recurrence-series root approximation - [[ch1.3.12-infinite-decimal-fractions]] — the alternative decimal representation of fractions - [[infinite-series]] — alternative way to express irrationals