AI 中文总结
本文综述拉格朗日、切比雪夫、达布、米尔诺关于地理地图构造的成果，完整证明达布与米尔诺的结论，明确最优地图特征，且米尔诺定理的证明适用于更一般情形。
AI 中文摘要
拉格朗日、切比雪夫和达布分别于1779年、1856年和1911年发表了标题同为《论地理地图的构造》的文章。1969年，米尔诺撰写了一篇提及切比雪夫论文的文章，对该论文进行了新的表述与证明。本文综述了上述所有论文的成果，阐释了各文包含的核心思想并指出了它们之间的关联。我们完整给出了达布和米尔诺的结论的证明，二者均旨在明确并证明切比雪夫的结果，但内容有所不同。尽管切比雪夫未明确说明“最优”的含义，他与达布、米尔诺的结论一致：最优地理地图的特征是其保形因子在表示区域的边界上为常数。我们对米尔诺定理的表述与证明适用于比他所给出的更一般的情形。本文最终版本将收录于《艺术与科学数学手册（第二版）》，由Bharath Sriraman编辑，施普林格出版社2027年出版。
英文摘要
Lagrange, Chebyshev, and Darboux, in 1779, 1856, and 1911, respectively, wrote articles all bearing the same title, \emph{On the Construction of Geographical Maps}. In 1969, Milnor wrote a paper in which he refers to Chebyshev's paper, of which he provides a new formulation and proof. In the present article, we review the results of all these papers, explaining the main ideas they contain and pointing out connections between them. We give complete proofs of the statements by Darboux and Milnor, both of which aim to make explicit and provide a proof of Chebyshev's result, but whose contents are different. Although Chebyshev did not state explicitly what the word ``best'' means, his conclusion, like that of Darboux and of Milnor, is that a best geographical map is characterised by the fact that its conformal factor is constant on the boundary of the region represented. Our statement and proof of Milnor's theorem work in a more general setting than the one he gives. The final version of this paper will appear in the Handbook of Mathematics in the Arts and Sciences (second edition), ed. Bharath Sriraman, Springer, 2027.
