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On the Marinus--Ptolemy and Delisle--Euler conical maps

Hideki Miyachi, Ken'Ichi Ohshika, Athanase Papadopoulos

TL;DR

This paper draws a historical and mathematical connection between Marinos–Ptolemy and Delisle–Euler cartography, showing that two centuries of practical mapmaking culminate in a shared conical projection framework. It frames the problem of projecting a sphere to the plane, reviews the specific Marinos–Ptolemy and Delisle–Euler constructions, and explains how Euler’s methods were later justified using differential geometry and quasiconformal theory. Through modern distortion analyses and the Tissot indicatrix, the authors demonstrate that the Delisle–Euler projection is optimal within a defined conical family, bridging ancient craftsmanship and contemporary mathematics. The work underscores historical continuity in mathematics and the value of rigorous, quantitative validation of long-standing cartographic techniques.

Abstract

We examine connections between the mathematics behind methods of drawing geographical maps due, on the one hand to Marinos and Ptolemy (1st-2nd c. CE) and on the other hand to Delisle and Euler (18th century). A recent work by the first two authors of this article shows that methods of Delisle and Euler for drawing geographical maps, which are improvements of methods of Marinus and Ptolemy, are best among a collection of geographical maps we term ``conical''. This is an instance where after practitioners and craftsmen (here, geographers) have used a certain tool during several centuries, mathematicians prove that this tool is indeed optimal. Many connections among geography, astronomy and geometry are highlighted. The fact that the Marinos--Ptolemy and the Delisle--Euler methods of drawing geographical maps share many non-trivial properties is an important instance of historical continuity in mathematics.

On the Marinus--Ptolemy and Delisle--Euler conical maps

TL;DR

This paper draws a historical and mathematical connection between Marinos–Ptolemy and Delisle–Euler cartography, showing that two centuries of practical mapmaking culminate in a shared conical projection framework. It frames the problem of projecting a sphere to the plane, reviews the specific Marinos–Ptolemy and Delisle–Euler constructions, and explains how Euler’s methods were later justified using differential geometry and quasiconformal theory. Through modern distortion analyses and the Tissot indicatrix, the authors demonstrate that the Delisle–Euler projection is optimal within a defined conical family, bridging ancient craftsmanship and contemporary mathematics. The work underscores historical continuity in mathematics and the value of rigorous, quantitative validation of long-standing cartographic techniques.

Abstract

We examine connections between the mathematics behind methods of drawing geographical maps due, on the one hand to Marinos and Ptolemy (1st-2nd c. CE) and on the other hand to Delisle and Euler (18th century). A recent work by the first two authors of this article shows that methods of Delisle and Euler for drawing geographical maps, which are improvements of methods of Marinus and Ptolemy, are best among a collection of geographical maps we term ``conical''. This is an instance where after practitioners and craftsmen (here, geographers) have used a certain tool during several centuries, mathematicians prove that this tool is indeed optimal. Many connections among geography, astronomy and geometry are highlighted. The fact that the Marinos--Ptolemy and the Delisle--Euler methods of drawing geographical maps share many non-trivial properties is an important instance of historical continuity in mathematics.

Paper Structure

This paper contains 10 sections, 3 theorems, 6 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The problems of cartography
  3. Marinos and Ptolemy: Some biographical markers
  4. The Marinos--Ptolemy map
  5. Delisle, Euler and geography
  6. The Euler-Delisle projection
  7. About terminology in geography: conical and other projections
  8. The Tissot indicatrix of a geographical map
  9. Comparison between conical projections: dilatation and quasiconformal distortion
  10. In guise of a conclusion

Key Result

Proposition 2.1

There is no map from any nonempty open subset of the sphere to the Euclidean plane that preserves distances up to scale.

Figures (7)

  • Figure 1: Figure for \ref{['prop:27']}.
  • Figure 2: A 15th century reconstruction of Ptolemy's world map. The British Library, Harley MS 7182.
  • Figure 3: A map of the Russian Empire: this is the last map of the Atlas Russicus (Saint Petersburg, 1745), and it is drawn using Delisle's method. Bibliothèque Nationale de France, Département Cartes et Plans.
  • Figure 4: Map of the Northern Pacific, representing the Eastern part of Asia and the Northern part of America, from Euler's Atlas Geographicus (Berlin, 1753) and it is drawn using Delisle's method.
  • Figure 5: A figure extracted from Euler's memoir Euler-Delisle in which he explains Delisle's projection. The reader may refer to that memoir or to the paper Charitos-Papa for the explanations.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3