The History of Mathematics: A Brief Course

262 9. MEASUREMENT FIGURE 14. Finding the Sun's elevation at a given hour. In this relation, ÌF is the sine of MS, that is, the ...

INDIA 263 the square root of the product is the area. In our terms this rule says that the area of a quadrilateral of sides a. ...

264 9. MEASUREMENT rectangle that is shaded dark, which lies inside these two isosceles triangles but outside the squares of sid ...

4 FIGURE 16. Frustum of a pyramid with its corners removed and side "ramps" folded up to form a cross-shaped prism. 9.4. Explain ...

266 9. MEASUREMENT 0;9,36 from [0; 15], leaving 0;5,24. What is the square root of 0;5,24? The lower end of the beam is [0;18] f ...

QUESTIONS AND PROBLEMS 267 FIGURE 17. A disk cut into sectors and opened up. are approximations for. How does this problem illus ...

268 9. MEASUREMENT Compare this list with Aryabhata's list, and note the systematic divergence. These differences should be appr ...

Chapter 10. Euclidean Geometry We shall divide the history of Greek mathematics into four periods. The first period, from about ...

270 10. EUCLIDEAN GEOMETRY use of the sexagesimal system of measuring angles and in Ptolemy's explicit use of Mesopotamian astro ...

THE EARLIEST GREEK GEOMETRY 271 plan, since one could not get into the Pyramid to measure the distance from the center to the ...

272 10. EUCLIDEAN GEOMETRY 1.3. Pythagorean geometry. Euclid's geometry is an elaboration and system- atization of the geometry ...

THE EARLIEST GREEK GEOMETRY 273 b FIGURE 1. Left: turning a triangle into a rectangle. Right: turn- ing a rectangle into a squ ...

274 10. EUCLIDEAN GEOMETRY have a unique solution. Were the additional conditions imposed simply to make the problem determinate ...

THE EARLIEST GREEK GEOMETRY 275 FIGURE 3. Hippocrates' quadrature of a lune, acording to Simplicius. doing so, first invoking ...

276 10. EUCLIDEAN GEOMETRY gave it a more exotic origin. In The Utility of Mathematics, the commentator Theon of Smyrna, who liv ...

THE EARLIEST GREEK GEOMETRY^277 FIGURE 5. The three conic sections, according to Menaechmus. The three surfaces intersect in a ...

278 10. EUCLIDEAN GEOMETRY Axis S R Q Ï (Vertex) R Q FIGURE 6. Sections of a cone. Top left: through the axis. Top right: perpen ...

THE EARLIEST GREEK GEOMETRY 279 A Ε d^2 = 2AE v^2 =w(w + 2d) {x + y)^2 = 2(w + d)^2 ÷^2 + y^2 = (w + d)^2 + V^2 2xy = d^2 FIGU ...

280 10. EUCLIDEAN GEOMETRY FIGURE 8. The quadratrix of Hippias. the regular heptagon. Surprisingly, however, the regular 17-side ...

THE EARLIEST GREEK GEOMETRY 281 m Β Ã Æ FIGURE 9. Pappus' construction of a neusis using a rectangular hyperbola. catch-all ca ...