Abstract
The tooth counts in a planetarium are taken to follow from the periods of the bodies. What this paper shows is that a continued fraction fixes them, and that the trade between accuracy and size closes there. No new theorem or law is claimed.
First, expanding the ratio yields a sequence of convergents——$77708431/2640858=\mathbf{29.4254484717}$ expands as $[29,2,2,1,5,1,4,\dots]$, with convergents $29/1$, $59/2$, $147/5$, $\mathbf{206/7}$ (Section 2).
Second, and this is the core. The $\mathbf{206/7}$ that Huygens cut is the $\mathbf{4}$th convergent——with a relative error of $\mathbf{1.061\times10^{-4}}$, and exhausting all denominators up to $7$ returns the same fraction (Sections 2 and 3).
Third, the price of one step better can be measured——the next convergent $1177/40$ shrinks the error to a $\mathbf{7.0}$th while raising the tooth total from $\mathbf{213}$ to $\mathbf{1217}$, a factor of $\mathbf{5.7}$ (Section 4).
Fourth, the convergents are not equally good——relative errors run from $\mathbf{1.446\times10^{-2}}$ at $29/1$ to $\mathbf{8.648\times10^{-4}}$ at $147/5$, shrinking unevenly (Section 2).
Fifth, some numbers resist approximation——the best fraction for the golden ratio with denominator at most $7$ is $8/5$, with relative error $\mathbf{1.115\times10^{-2}}$, some $\mathbf{105}$ times worse (Section 5).
Sixth, the separator is the size of the partial quotients——accuracy jumps just before a large one and stalls where ones run on (Section 5).
Keywords: continued fraction, gear tooth counts, orrery, best rational approximation, Huygens.
Scope (what is not claimed)
No new theorem or law is claimed——the best-approximation property of continued fractions and Huygens’s planetarium are standard. Gear engineering is not discussed——tooth profile, backlash and friction are not treated; only the integers. No historical detail is asserted——how Huygens actually computed is not claimed; only that $206/7$ is a convergent. Modern period values are not used——the period ratio of the time is used as given, and modern values change the expansion. The best-approximation theorem is not proved——denominators up to $7$ were checked exhaustively, nothing more. No clock design is offered——which tooth counts to choose is not the subject. Relation to earlier papers: Paper 213 showed that coming back has distinct roots——the period after which gears return to the same position is fixed by the denominator. Paper 336 showed that the $32768$ in a wristwatch was not chosen by physics——tooth counts likewise are chosen by a habit of approximation rather than by the bodies. Paper 319 showed that a calendar is three approximations stacked——the placing of leap days has the same structure as this sequence of convergents. Paper 176 showed that “small” is two words that do not agree——the trade here is exactly that, between a small error and a small tooth count. Paper 300 showed that sharing a root is decidable——accuracy and tooth count are separate roots under the criterion of what a quantity carries. What is added is expanding the period ratio of the time into convergents, confirming that $\mathbf{206/7}$ is the $\mathbf{4}$th, confirming by exhaustion that no better fraction has denominator at most $7$, measuring the price of one step as a $\mathbf{7.0}$th of the error against $\mathbf{5.7}$ times the teeth, and placing the separator on the size of the partial quotients.
1. Notation and the Question
Setting: a planetarium reproduces ratios of periods by ratios of gear teeth. Since tooth counts are integers, the period ratio must be approximated by a fraction.
The value of the time: the Saturn-to-Earth period ratio Huygens used was $77708431/2640858$, which is $\mathbf{29.4254484717}$.
Observation 1.1 the question is arithmetic, not astronomy. How well can a given ratio be approximated with a small denominator——the bodies do not settle the answer.
Observation 1.2 the denominator becomes a tooth count. More teeth means a larger, heavier instrument——so the trade between accuracy and size sits directly on the denominator.
Question: which fraction should be chosen, and who is choosing?
2. Expanding the Ratio
Computation 2.1: expanding $77708431/2640858$ as a continued fraction, which terminates because the number is rational,
$$[29,\;2,\;2,\;1,\;5,\;1,\;4,\;1,\;1,\;2,\;1,\;6,\;1,\;10,\;2,\;2,\;3]$$Computation 2.2: listing the convergents with their errors.
| convergent | tooth total | error | relative error |
|---|---|---|---|
| $29/1$ | $30$ | $4.254\times10^{-1}$ | $\mathbf{1.446\times10^{-2}}$ |
| $59/2$ | $61$ | $7.455\times10^{-2}$ | $2.534\times10^{-3}$ |
| $147/5$ | $152$ | $2.545\times10^{-2}$ | $\mathbf{8.648\times10^{-4}}$ |
| $\mathbf{206/7}$ | $\mathbf{213}$ | $3.123\times10^{-3}$ | $\mathbf{1.061\times10^{-4}}$ |
| $1177/40$ | $1217$ | $4.485\times10^{-4}$ | $1.524\times10^{-5}$ |
| $1383/47$ | $1430$ | $8.344\times10^{-5}$ | $2.836\times10^{-6}$ |
| $6709/228$ | $6937$ | $9.875\times10^{-6}$ | $3.356\times10^{-7}$ |
Observation 2.3 this is the core. The $\mathbf{206/7}$ Huygens cut is the $\mathbf{4}$th entry of this list——what fixed the tooth counts was the expansion, not Saturn.
Observation 2.4 the shrinking is uneven. From $147/5$ to $206/7$ the relative error falls from $\mathbf{8.648\times10^{-4}}$ to $\mathbf{1.061\times10^{-4}}$, about an eighth, while from $29/1$ to $59/2$ it falls only about a sixth——accuracy jumps where a partial quotient is large.
Observation 2.5 the last convergent returns the original ratio. A rational number’s expansion terminates——a check on the expansion, not a claim.
3. No Better Fraction with Denominator at Most $7$
Computation 3.1: trying every denominator from $1$ to $7$, taking the best numerator for each and comparing errors. The best is $\mathbf{206/7}$ with error $3.123\times10^{-3}$.
Observation 3.2 the continued fraction agrees with exhaustion. Confirming in this case that convergents are best approximations——a check on one instance, not a proof of the theorem.
Observation 3.3 so there is no room for choice. Allow seven teeth and nothing but $206$ can be paired with it——the designer’s discretion enters only in where to stop.
Observation 3.4 the same shape as Paper 336. There the $32768$ of a wristwatch was not chosen by physics——here too the number is chosen by a habit of approximation rather than by the object.
4. The Price of One Step
Computation 4.1: comparing $206/7$ with the next convergent $1177/40$.
| quantity | $206/7$ | $1177/40$ |
|---|---|---|
| error | $3.123\times10^{-3}$ | $\mathbf{4.485\times10^{-4}}$ |
| tooth total | $\mathbf{213}$ | $\mathbf{1217}$ |
Observation 4.2 the error falls to a $\mathbf{7.0}$th and the teeth rise by $\mathbf{5.7}$. Both in one step——the trade can be written as numbers, which is what a designer needs.
Observation 4.3 the shape of Paper 176. There, “small” was two words that do not agree——a small error and a small tooth count cannot both be had, and one must be chosen.
Observation 4.4 teeth translate directly into size. Cutting $1217$ teeth is $5.7$ times the labour of cutting $213$——in seventeenth-century workshops that difference was the real limit.
5. The Separator
Observation 5.1 the separator is the size of the partial quotients. Convergents just before a large one are good, and where ones run on nothing improves——$147/5$ sits before the $5$ in $[29,2,2,1,5,\dots]$, and the next convergent $206/7$ jumps.
Computation 5.2: comparing with the number hardest to approximate. The best fraction for the golden ratio with denominator at most $7$ is $8/5$, with relative error $\mathbf{1.115\times10^{-2}}$.
Observation 5.3 some $\mathbf{105}$ times worse than the Saturn ratio. $\mathbf{1.061\times10^{-4}}$ against $\mathbf{1.115\times10^{-2}}$ at the same denominators——the golden ratio’s expansion is all ones, so stopping anywhere gains little.
Observation 5.4 separate roots under Paper 300. Accuracy carries an error and tooth count carries an integer——that both cannot be reduced at once follows from their carrying different things.
Observation 5.5 the same structure as Paper 319. Placing leap days is likewise choosing a convergent for a ratio of periods——the steps at $4$, $100$ and $400$ years are the marks of stopping partway.
Observation 5.6 what is not said. Gear engineering is not treated——no tooth profile, backlash or friction, and the period value of the time was used as given.
6. Roll Call and Closing
- the ratio is $\mathbf{29.4254484717}$ (Computation 2.1): expanding to $[29,2,2,1,5,\dots]$.
- $\mathbf{206/7}$ is the $\mathbf{4}$th convergent (Observation 2.3): the core.
- relative error $\mathbf{1.061\times10^{-4}}$ (Computation 2.2): tooth total $\mathbf{213}$.
- the shrinking is uneven (Observation 2.4): it follows the quotients.
- the last convergent returns the ratio (Observation 2.5): a check.
- exhaustion also gives $\mathbf{206/7}$ (Computation 3.1): no room for choice.
- one step shrinks error to a $\mathbf{7.0}$th (Computation 4.1): teeth rise $\mathbf{5.7}$ times.
- tooth total $\mathbf{213}$ to $\mathbf{1217}$ (Computation 4.1): the workshop’s limit.
- golden ratio gives $\mathbf{1.115\times10^{-2}}$ (Computation 5.2): $\mathbf{105}$ times worse.
- the separator is the quotients (Observation 5.1): ones stall progress.
- the same structure as a calendar (Observation 5.5): steps are stopping marks.
Closing: The tooth counts in a planetarium are taken to follow from the periods of the bodies. A continued fraction fixes them——the ratio Huygens used, $77708431/2640858=\mathbf{29.4254484717}$, expands as $[29,2,2,1,5,1,4,\dots]$, with convergents $29/1$, $59/2$, $147/5$, $\mathbf{206/7}$, and the fraction he cut is the $\mathbf{4}$th of them. A rational expansion terminates, and that the last convergent returns the ratio checks the expansion. The shrinking is uneven, jumping where a partial quotient is large. With denominator at most $7$ there is nothing better——trying every denominator from $1$ to $7$ returns the same $\mathbf{206/7}$, with relative error $\mathbf{1.061\times10^{-4}}$. This confirms in one instance that convergents are best approximations, without proving the theorem. So allowing seven teeth leaves no choice of numerator, and the designer’s discretion is only in where to stop. As Paper 336 found for the $32768$ of a wristwatch, the number is chosen by a habit of approximation, not by the object. The price of one step can be written down——the next convergent $1177/40$ shrinks the error to a $\mathbf{7.0}$th while raising the tooth total from $\mathbf{213}$ to $\mathbf{1217}$, a factor of $\mathbf{5.7}$. This is the shape of Paper 176: a small error and a small tooth count cannot both be had. Cutting $1217$ teeth is $5.7$ times the labour of $213$, and in seventeenth-century workshops that was the real limit. The separator is the size of the partial quotients——convergents just before a large quotient are good, and where ones run on nothing improves. The golden ratio, whose expansion is all ones, admits only $8/5$ at denominator at most $7$, with relative error $\mathbf{1.115\times10^{-2}}$, some $\mathbf{105}$ times worse than the Saturn ratio. Under Paper 300, accuracy carries an error and tooth count carries an integer, and that is why both cannot shrink together. The calendar of Paper 319 has the same structure, its steps at $4$, $100$ and $400$ years marking where the sequence was stopped. The returning of Paper 213 appears here as the denominator, the period after which the gears realign. To be honest——gear engineering was not treated: no tooth profile, no backlash, no friction, only the integers. How Huygens actually computed is not claimed, only that $206/7$ is a convergent. The period value of the time was used as given, and modern values change the expansion. The best-approximation theorem was not proved, only checked for denominators up to seven, and no advice on choosing tooth counts is offered. What can be said is that a continued fraction, not astronomy, fixed the tooth counts, and that the trade between accuracy and size closes there, and no further.
References
- Brocot, A. (1861). Calcul des rouages par approximation. Revue Chronometrique.
- Hardy, G. H., & Wright, E. M. (2008). An Introduction to the Theory of Numbers (6th ed.). Oxford University Press.
- Huygens, C. (1703). Descriptio automati planetarii. In Opuscula postuma. Leiden: C. Boutesteyn.
- Khinchin, A. Ya. (1964). Continued Fractions. University of Chicago Press.
- Yamagishi, Y. (2026). “Small” Is Two Words, and They Do Not Agree [Paper 176]. Zenodo. 10.5281/zenodo.22097994
- Yamagishi, Y. (2026). Coming Back Has Distinct Roots [Paper 213]. Zenodo. 10.5281/zenodo.22119380
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). A Calendar Is Three Approximations Stacked [Paper 319]. Zenodo. 10.5281/zenodo.22208933
- Yamagishi, Y. (2026). The Number $32768$ Inside a Wristwatch Was Not Chosen by Physics [Paper 336]. Zenodo. 10.5281/zenodo.22681913
Reference
Yamagishi, Y. (2026). What Fixed the Tooth Counts Was Not Astronomy but a Continued Fraction [Paper 539]. YOI: https://www.yamagishiyuuki.com/yoi/539
DOI: 10.5281/zenodo.23014129
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 539. This paper is a preprint and has not been peer reviewed.