The History of Mathematics: A Brief Course

302 10. EUCLIDEAN GEOMETRY FIGURE 18. Finding the surface area of a sphere. that the area of the surface obtained by revolving t ...

ARCHIMEDES 303 FIGURE 19. Volumes of sphere, cone, and cylinder. 1906 and 1908 he journeyed to Constantinople and established ...

304 10. EUCLIDEAN GEOMETRY a cone, the rectangle LGFE generates a cylinder, and each horizontal line such as Ì Í generates a dis ...

APOLLONIUS^305 readable. On the other hand, Apollonius' work is no longer current research, and from the historian's point of ...

306 10. EUCLIDEAN GEOMETRY asymptotes, things necessary for analyzing problems to see what data permit a solution, and the three ...

APOLLONIUS^307 Å Ï ^--Ê - - Ç\. - " " I FIGURE 20. Apollonius' construction of the ellipse. In one sense, this locus definitio ...

308 10. EUCLIDEAN GEOMETRY FIGURE 21. Focal properties of an ellipse. 4.4. Foci and the three- and four-line locus. We are nowad ...

APOLLONIUS^309 FIGURE 22. The basis for solving the four-line locus problem. lines from è to A and à meet the lines through à ...

310 10. EUCLIDEAN GEOMETRY drawn to the other reference line. The commentary on this problem by Pappus, who mentioned that Apoll ...

QUESTIONS AND PROBLEMS^311 FIGURE 23. The four-line locus. If a point moves so that the product of its distances to two lines be ...

312 10. EUCLIDEAN GEOMETRY c D A Β FIGURE 24. Diagonal and side of a square. of incommensurables, is an attempt to bring into sh ...

QUESTIONS AND PROBLEMS 313 A FIGURE 25. Archimedes' trisection of an angle: ZATA = ^ZAAE. 10.15. Show that Archimedes' result on ...

314 10. EUCLIDEAN GEOMETRY 10.19. When the equation y^2 = Cx - kx^2 is converted to the standard form a2 + b^2 ' what are the qu ...

QUESTIONS AND PROBLEMS 315 right-angled cone whose equation is y^2 = zx by the plane χ = 2a — (a^2 z/b^2 ). Then show that by ta ...

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Chapter 11. Post-Euclidean Geometry A certain dullness came over Greek geometry from the beginning of the second century BCE. Th ...

318 11. POST-EUCLIDEAN GEOMETRY 1. Hellenistic geometry Although the Euclidean restrictions set limits to the growth of geometry ...

HELLENISTIC GEOMETRY 319 Â FIGURE 1. Two theorems of Zenodorus. Top: When two regular polygons have the same perimeter, the on ...

320 11. POST-EUCLIDEAN GEOMETRY by the Stoic philosopher Geminus, whose dates are a subject of disagreement among experts, but w ...

HELLENISTIC GEOMETRY 321 FIGURE 2. Heron's proof of his direct method of computing the area of a triangle. gave as an example ...