Abstract
A hanging chain and a suspension-bridge cable cannot be told apart by eye. This paper asks what divides them: the answer is not the shape, but which length the load is counted along. No new mathematical theorem and no new law is claimed.
First, they agree exactly through second order. At $x=0.1$ the difference is $\mathbf{0.083292}$ per cent (Section 2).
Second, far out they are different things. At $x=5$, $\mathbf{82.925818}$ per cent (Section 2).
Third, the difference begins at fourth order. $x^4/24$ against $0$ (Section 2).
Fourth, this is the core of the paper. What divides them is counting along arc length or along horizontal length (Section 3).
Fifth, at a sag of $1/20$ of the span the difference is $\mathbf{0.082674}$ per cent of the sag, and at the usual $1/10$ of a real bridge still only $\mathbf{0.323115}$ per cent (Section 4).
Sixth, the separator is whether the deck or the cable is the heavier (Section 5).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed: the catenary equation, the parabolic-cable theory of suspension bridges, and the Taylor expansions of both are all standard. We do not do structural design: all we use is two curves and one difference of measure. We do not treat deflection: cable stretch, thermal movement and live-load deflection are not entered. This paper looks only at the idealised static shape. We do not adjudicate which is correct: a real bridge lies between the two, and this paper does not locate it. We claim no engineering accuracy: the sag ratios are values for comparing shapes, not those of any particular bridge. We do not discuss optimal shape, which is structurally better is not treated. Relation to earlier papers: Paper 198 showed that “straight” does not mean “shortest”: 198 says one curve has two characters; this says two curves look alike. The direction is reversed. Paper 326 showed the square-cube relation is an identity: both are of the “same appearance, different content” kind, but 326 is about a ratio and this is about functions. Paper 300 showed whether two things share a root is decidable. This is a worked instance of “numerical closeness is not sameness of root”. Paper 245 showed two limits have no answer until their order is written. Here too, the curve is undetermined until the measure of integration is written. Paper 183 showed one table having two correct answers: one shape having two correct equations is its counterpart. What is added is computing the difference at six points, stating that it begins at fourth order, giving the maximum difference for each sag ratio, and putting the separator on the measure.
1. Notation and the Question
Setting: A chain hanging under its own weight is a catenary.
$$y=a\cosh(x/a)-a$$The cable of a bridge with a heavy deck is a parabola.
$$y=\frac{x^2}{2a}$$Observation 1.1 Expanding $\cosh$,
$$a\cosh(x/a)-a=\frac{x^2}{2a}+\frac{x^4}{24a^3}+\frac{x^6}{720a^5}+\cdots$$the leading term is the parabola itself: they agree exactly through second order.
Question: So what makes them different?
2. $0.083$ Per Cent and $82.9$ Per Cent
Check 2.1: With $a=1$, comparing heights (computed here).
| $x$ | Catenary | Parabola | Difference | Relative |
|---|---|---|---|---|
| $0.1$ | $0.00500417$ | $0.00500000$ | $0.00000417$ | $\mathbf{0.083292\%}$ |
| $0.5$ | $0.12762597$ | $0.12500000$ | $0.00262597$ | $2.057548\%$ |
| $1.0$ | $0.54308063$ | $0.50000000$ | $0.04308063$ | $7.932641\%$ |
| $2.0$ | $2.76219569$ | $2.00000000$ | $0.76219569$ | $27.593834\%$ |
| $3.0$ | $9.06766200$ | $4.50000000$ | $4.56766200$ | $50.373095\%$ |
| $5.0$ | $73.20994852$ | $12.50000000$ | $60.70994852$ | $\mathbf{82.925818\%}$ |
Observation 2.2 The same two curves open from $0.08$ to $82.9$ per cent. Both figures are correct about the same two curves: “how different are they” is undetermined until you say where you are looking.
Observation 2.3 The difference begins at fourth order. $x^4/(24a^3)$ against $0$: first, second and third order all agree, so nearby there is nothing to see.
Observation 2.4 It is not that the approximation is good; they coincide. The parabola is the second-order Taylor polynomial of the catenary, not a curve fitted to it: the closeness is not a coincidence.
Observation 2.5 Paper 300 wrote that numerical agreement alone is not sameness of root. This is a worked instance: agreement to $0.083$ per cent is no evidence of being the same thing.
3. The Core: Which Length the Load Is Counted Along
Setting: Both curves come from the same balance. With the horizontal tension $H$ constant,
$$H\frac{d^2y}{dx^2}=w$$where $w$ is “weight per unit length”.
Observation 3.1 This is the core of the paper. There are two “unit lengths”.
| What is heavy | Length the load is counted along | Curve |
|---|---|---|
| the cable itself | arc length $ds$ | catenary |
| the suspended deck | horizontal length $dx$ | parabola |
Observation 3.2 Not one character of the equation differs. All that differs is which differential $w$ is constant against: the difference in shape comes out of a difference in measure.
Observation 3.3 Constant along arc length means the steep parts weigh more per horizontal foot. Because $ds=\sqrt{1+y'^2} dx$: the ends carry extra weight, and so the catenary rises faster.
Observation 3.4 Every difference in Check 2.1 comes from this one point. The $82.9$ per cent at $x=5$ is because the slope is large there and $ds$ departs far from $dx$, where $x$ is small, $\sqrt{1+y'^2}\approx1$ and the two coincide.
Observation 3.5 Paper 245 wrote that two limits have no answer until their order is written. The same discipline. Here the curve is undetermined until the measure of integration is written.
Observation 3.6 So “a parabola close to a catenary” has it backwards. Each is the exact answer to a different physics, and neither is an approximation to the other: the approximation relation is something mathematics noticed afterwards.
4. On a Shallow Bridge It Is $0.083$ Per Cent
Check 4.1: Maximum difference for each sag $f$ (half-span taken as $1$).
| Sag $f$ | Catenary’s $a$ | Max difference | Relative to sag |
|---|---|---|---|
| $0.02$ | $25.0033$ | $0.00000067$ | $\mathbf{0.003332\%}$ |
| $0.05$ | $10.0083$ | $0.00001040$ | $0.020792\%$ |
| $0.10$ | $5.0166$ | $0.00008267$ | $\mathbf{0.082674\%}$ |
| $0.20$ | $2.5326$ | $0.00064623$ | $0.323115\%$ |
| $0.35$ | $1.4835$ | $0.00326383$ | $0.932524\%$ |
| $0.50$ | $1.0743$ | $0.00877196$ | $1.754392\%$ |
Observation 4.2 Real suspension bridges sag by about $1/10$ of the span, which is $f=0.20$ in the table. The maximum difference there is $\mathbf{0.323115}$ per cent of the sag: on a $1000$ m span with $100$ m of sag, $\mathbf{32.3}$ cm. At a sag of $1/20$ of the span ($f=0.10$) it is $\mathbf{0.082674}$ per cent, $\mathbf{4.1}$ cm on a $1000$ m span with $50$ m of sag.
Observation 4.3 $4.1$ cm can be smaller than the construction tolerance. Telling them apart by measurement is hard in practice, and they are still two different curves.
Observation 4.4 The shallower the sag, the less there is to see. At $f=0.02$ it is $0.003332$ per cent: the tauter the line, the more both theories agree.
Observation 4.5 So “can they be distinguished” is a question of precision and “are they different” is a question of definition. The two are independent: things can measure the same and still be different.
5. Separator: Whether the Deck or the Cable Is Heavier
Observation 5.1 Separator: the ratio of the suspended load to the weight of the cable itself.
| Situation | What dominates | Shape |
|---|---|---|
| a chain carrying nothing | its own weight (arc) | catenary |
| a bridge with an overwhelming deck | the deck (horizontal) | parabola |
| both comparable | both | neither |
Observation 5.2 The third row is the real bridge. Neither ideal shape, and this paper does not locate it.
Observation 5.3 Paper 183 wrote that one table has two correct answers. This is its counterpart: one shape with two correct equations.
Observation 5.4 The direction against Paper 198.
| What is one and what is two | |
|---|---|
| Paper 198 | one curve with two characters (straight / shortest) |
| This paper | two curves with one appearance |
Observation 5.5 The direction is exactly reversed. 198 splits one into two; this paper has two looking like one: both obey the same discipline: appearance is not evidence.
Observation 5.6 A point to write honestly. This paper does not say which is structurally better. The difference between the curves and the merits of a design are different questions: the second is not entered.
Observation 5.7 One last thing: Galileo wrote that a hanging chain is approximately a parabola. For a shallow chain that holds, with a difference of about $0.083$ per cent: not a difference the eye could correct, and with no $\cosh$ yet in existence the true shape could not be written. Writing the right shape needs the words for it to exist first.
6. Roll Call and Closing
- Exact agreement to second order (Observation 1.1): The parabola is the second-order term.
- $0.083292$ per cent at $x=0.1$ (Check 2.1): Nothing to see.
- $82.925818$ per cent at $x=5$ (Check 2.1): About the same two curves.
- The difference begins at fourth order (Observation 2.3): $x^4/24a^3$ against $0$.
- Not one character of the equation differs (Observation 3.2): Only the measure.
- Arc length makes the ends heavy (Observation 3.3): $ds=\sqrt{1+y'^2}dx$.
- $0.323115$ per cent on a real bridge (Observation 4.2): $32.3$ cm on $1000$ m.
- Distinguishable and different are independent (Observation 4.5): Precision and definition.
- Reversed against 198 (Observation 5.4): One into two, two into one.
- The difference from Galileo’s “approximately a parabola” is $0.083$ per cent (Observation 5.7): The words did not exist.
Closing: A hanging chain and a suspension-bridge cable cannot be told apart by eye. Expanding $\cosh$ shows the leading term to be the parabola itself, so they agree exactly through second order: with $a=1$ the difference is $\mathbf{0.083292}$ per cent at $x=0.1$ and $\mathbf{82.925818}$ per cent at $x=5$, and both figures are correct about the same two curves. “How different are they” is undetermined until you say where you are looking. The difference begins at fourth order: first, second and third all agree, and the parabola is the second-order Taylor polynomial of the catenary, not a curve fitted to it. And what divides them is not shape but measure: both come from the same balance $H y''=w$, and all that differs is whether $w$ is constant against arc length $ds$ or horizontal length $dx$. Constant along the arc makes the steep ends weigh more per horizontal foot, and so the catenary rises faster. So “a parabola close to a catenary” has it backwards: each is the exact answer to a different physics, and the approximation relation is something mathematics noticed afterwards. At the $1/10$ sag of a real suspension bridge the maximum difference is $\mathbf{0.323115}$ per cent of the sag: $\mathbf{32.3}$ cm on a $1000$ m span, and $\mathbf{4.1}$ cm at a sag of $1/20$, sometimes smaller than the construction tolerance. And they are still two different curves. One thing separates them: whether the suspended deck or the cable itself is heavier. A real bridge is neither ideal, and this paper does not locate it. Against Paper 198 the direction is exactly reversed: 198 says one curve has two characters, this says two curves have one appearance, and both obey the same discipline: appearance is not evidence. One last thing: Galileo wrote that a hanging chain is approximately a parabola. For a shallow chain that holds, with a difference of about $0.083$ per cent. Not a difference the eye could correct, and with no $\cosh$ yet in existence the true shape could not be written. Writing the right shape needs the words for it to exist first.
Revision Record
2nd Edition (October 2026): The way the authors and title of the 1691 Acta Eruditorum entry are written in the references was corrected.
3rd Edition (October 2026): Three corrections. (1) The earlier editions read $f=0.10$ of Table 4.1 as the sag ratio $1/10$ of a real bridge and wrote $8.3$ cm for a $1000$ m span with $100$ m of sag; but the table takes the half-span as $1$, so $f=0.10$ is a sag ratio of $1/20$. A sag ratio of $1/10$ is $f=0.20$, which gives $0.323115$ per cent and $32.3$ cm ($8.3$ cm fitted neither reading). (2) The differences in Table 4.1 were measured again more finely. (3) “Galileo wrote that a hanging chain is a parabola; he was wrong” was changed to “approximately a parabola” (from a secondary source; the original text was not read). The $0.083$ per cent of the title and the core, that the two differ in how the load is counted, do not change.
4th Edition (October 2026): The dashes used as separators in the text were replaced by punctuation. No claim, number or reference was changed.
References
- Galilei, G. (1638). Discorsi e Dimostrazioni Matematiche intorno à due nuove scienze, Giornata Seconda. Leiden: Elzevir.
- Bernoulli, Jac., Bernoulli, Joh., Huygens, C. and Leibniz, G. W. (1691). Solutions of the catenary problem (separate papers in Latin). Acta Eruditorum, June.
- Irvine, H. M. (1981). Cable Structures. Cambridge: MIT Press.
- Gere, J. M. and Timoshenko, S. P. (1997). Mechanics of Materials, 4th edition. Boston: PWS.
- Yamagishi, Y. (2026). One Table, Two Correct Answers [Paper 183]. Zenodo. 10.5281/zenodo.22104230
- Yamagishi, Y. (2026). “Straight” Does Not Mean “Shortest”, and There Are Two Fences [Paper 198]. Zenodo. 10.5281/zenodo.22107954
- Yamagishi, Y. (2026). Two Limits Have No Answer Until the Order Is Written [Paper 245]. Zenodo. 10.5281/zenodo.22159739
- 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). The Square-Cube “Law” Is Not a Law but an Identity [Paper 326]. Zenodo. 10.5281/zenodo.22209992
Reference
Yamagishi, Y. (2026). Two Curves Agree to Within $\mathbf{0.083}$ Per Cent and Are Not the Same Curve [Paper 327]. YOI: https://www.yamagishiyuuki.com/yoi/327
DOI: 10.5281/zenodo.22210286
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 327. This paper is a preprint and has not been peer reviewed.