By Akira Fujiki, etc., Kazuya Kato, T. Katsura, Y. Kawamata, Y. Miyaoka

This quantity files the lawsuits of a global convention held in Tokyo, Japan in August 1990 at the topics of algebraic geometry and analytic geometry.

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However, if we credit the ancient historian Plutarch’s guess at Eratosthenes’ unit of length, we obtain a value for the Earth’s circumference of about 46,250 km (28,738 miles)—remarkably close to the modern value (about 15 percent too large), considering the difficulty in accurately measuring l and α. 36 7 History of Geometry 7 Eratosthenes knew that on a midsummer day the Sun is directly overhead at Syene, as indicated in the figure by the solar rays illuminating a deep well. He also knew the distance between Syene and Alexandria (shown in the figure by the arc l), which, combined with his measurement of the solar angle α between the Sun and the vertical, enabled him to calculate the Earth’s circumference.

After cutting the cylinder along a vertical line and flattening the resulting rectangle, the result was the now-familiar Mercator map. The intense cultivation of methods of projection by artists, architects, and cartographers during the Renaissance eventually provoked mathematicians into considering the properties of linear perspective in general. The most profound of these generalists was a sometime architect named Girard Desargues (1591–1661). Transformation French Circles Desargues was a member of intersecting circles of 17thcentury French mathematicians worthy of Plato’s Academy of the 4th century BCE or Baghdad’s House of Wisdom of the 9th century CE.

Of this preliminary matter, the fifth and last postulate, which states a sufficient condition that two straight lines meet if sufficiently extended, has received by far the greatest attention. In effect it defines parallelism. Many later geometers tried to prove the fifth postulate using other parts of the Elements. Euclid saw farther, for coherent geometries (known as non-Euclidean geometries) can be produced by replacing the fifth postulate with other postulates that contradict Euclid’s choice.