# 1. Objective and scope We formalize the claim: > Adding circular motions (epiciclos) progressively improves the approximation > of a periodic planetary trajectory. Mathematically, this is the statement that a periodic planar motion can be approximated arbitrarily well by finite sums of uniform circular motions. This claim depends only on periodicity and regularity, not on heliocentrism or geocentrism. --- # 2. Mathematical setting We work in the complex plane $\mathbb C \simeq \mathbb R^2$. A uniform circular motion of radius $R$ and angular frequency $\omega$ is $$ z(t)=R\,e^{i(\omega t+\phi)}. $$ A general **epicyclic model** with finitely many cycles is $$ z_N(t)=c_0+\sum_{k=1}^{N} c_k e^{i\omega_k t}, \qquad c_k\in\mathbb C. $$ This expression captures the full geometric content of the Ptolemaic construction. --- # 3. Periodicity and angular parametrization Let $z(t)$ be a bounded planar motion with fundamental period $T$. Define the angular variable $$ \theta := \frac{2\pi}{T}t \in \mathbb T:=\mathbb R/2\pi\mathbb Z. $$ Then $z(t)=z(\theta)$ where $z:\mathbb T\to\mathbb C$ is $2\pi$-periodic. --- # 4. Fourier decomposition of periodic motion If $z\in L^2(\mathbb T)$, it admits the Fourier expansion $$ z(\theta)=\sum_{k\in\mathbb Z} \hat z_k e^{ik\theta}, \qquad \hat z_k=\frac{1}{2\pi}\int_0^{2\pi} z(\theta)e^{-ik\theta}\,d\theta. $$ The partial sums $$ z_N(\theta)=\sum_{|k|\le N} \hat z_k e^{ik\theta} $$ satisfy $$ \|z-z_N\|_{L^2}\xrightarrow[N\to\infty]{}0 $$ by completeness of the trigonometric system [1]. **Interpretation.** Each term $\hat z_k e^{ik\theta}$ represents a uniform circular motion. Hence a truncated Fourier series is exactly an epicyclic model. --- # 5. Keplerian motion as a periodic function Consider planar Keplerian motion with semi-major axis $a>0$, eccentricity $e\in[0,1)$, and mean motion $n>0$. Define the mean anomaly $$ M:=nt. $$ The eccentric anomaly $E(M)$ satisfies Kepler’s equation $$ M=E-e\sin E. $$ With the focus at the origin, the position is $$ z(M)=a(\cos E-e)+i\,a\sqrt{1-e^2}\sin E. $$ --- # 6. Regularity of the Kepler map Since $$ \frac{dM}{dE}=1-e\cos E>0 \quad (e<1), $$ the implicit function theorem ensures that $E(M)$ exists, is unique, and is $C^\infty$ and $2\pi$-periodic. Consequently, $$ z(M)\in C^\infty(\mathbb T)\subset L^2(\mathbb T). $$ --- # 7. Harmonic expansion of Keplerian motion Because $z(M)\in L^2(\mathbb T)$, it admits a Fourier series $$ z(M)=\sum_{k\in\mathbb Z} c_k e^{ikM}, $$ with convergence in $L^2$. Smoothness implies rapid decay of $|c_k|$ and uniform convergence. The coefficients $c_k$ can be written explicitly using Fourier–Bessel expansions involving Bessel functions $J_k(ke)$ [2,3]. --- # 8. Truncation and approximation by epiciclos Define the truncated approximation $$ z_N(M)=\sum_{|k|\le N} c_k e^{ikM}. $$ Then $$ \|z-z_N\|_{L^2}^2=\sum_{|k|>N}|c_k|^2 \xrightarrow[N\to\infty]{}0. $$ Thus, increasing the number of cycles (epiciclos) strictly improves the approximation in a well-defined norm. --- # 9. Structural equivalence We may state precisely: > The Ptolemaic epicyclic construction is mathematically equivalent to a > truncated Fourier representation of a periodic orbital motion. Keplerian motion corresponds to the infinite-harmonic limit of this representation, with coefficients fixed by dynamical laws rather than empirical adjustment. The distinction is epistemic, not structural: - Empirical fitting of coefficients versus analytic determination. - Finite truncation versus infinite completion. --- # 10. Conclusion 1. The intuition that adding cycles improves accuracy is mathematically correct. 2. Fourier analysis proves and formalizes this intuition. 3. Keplerian motion lies in the closure of the space spanned by epicycles. 4. Following the Ptolemaic construction and abstracting it rigorously leads naturally to Keplerian motion. The representational idea is the same; only the mathematical language evolved. --- # References [1] J. Fourier, *Théorie analytique de la chaleur*. Paris: Firmin Didot, 1822. DOI: 10.3931/e-rara-8845 [2] J. L. Lagrange, “Sur l’altération des moyens mouvements des planètes,” *Mémoires de l’Académie Royale des Sciences de Berlin*, 1781. DOI: 10.3931/e-rara-14297 [3] V. I. Arnold, *Mathematical Methods of Classical Mechanics*. Springer, 1978. DOI: 10.1007/978-1-4757-1693-1 [4] O. Neugebauer, *A History of Ancient Mathematical Astronomy*. Springer, 1975. DOI: 10.1007/978-3-642-61981-8