History of Mathematics: A Supplement

Springer Science & Business Media, 10. des. 2007 - 274 sider
1 An Initial Assignment I haven’t taught the history of mathematics that often, but I do rather like the course. The chief drawbacks to teaching it are that i. it is a lot more work than teaching a regular mathematics course, and ii. in American colleges at least, the students taking the course are not mathematics majors but e- cation majors— and and in the past I had found education majors to be somewhat weak and unmotivated. The last time I taught the course, however, themajorityofthestudentsweregraduateeducationstudentsworkingtoward their master’s degrees. I decided to challenge them right from the start: 1 Assignment. In An Outline of Set Theory, James Henle wrote about mat- matics: Every now and then it must pause to organize and re?ect on what it is and where it comes from. This happened in the sixth century B. C. when Euclid thought he had derived most of the mathematical results known at the time from ?ve postulates. Do a little research to ?nd as many errors as possible in the second sentence and write a short essay on them. Theresponsesfarexceededmyexpectations. Tobesure,someoftheund- graduates found the assignment unclear: I did not say how many errors they 2 were supposed to ?nd. But many of the students put their hearts and souls 1 MyapologiestoProf. Henle,atwhoseexpenseIpreviouslyhadalittlefunonthis matter. I used it again not because of any animosity I hold for him, but because I was familiar with it and, dealing with Euclid, it seemed appropriate for the start of my course.

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Innhold

 Introduction 1 Annotated Bibliography 11 Foundations of Geometry 41 The Construction Problems of Antiquity 87 Conic Sections 100 Quintisection 110 Algebraic Numbers 118 Petersen Revisited 122
 Descartes Rule of Signs 196 De Guas Theorem 214 Concluding Remarks 222 Some Lighter Material 225 A Poetic History of Science 229 Drinking Songs 235 Concluding Remarks 241 A Small Projects 247

 Concluding Remarks 130 A Chinese Problem 133 132 The Cubic Equation 147 Examples 149 The Theorem on the Discriminant 151 The Theorem on the Discriminant Revisited 156 Computational Considerations 160 One Last Proof 171 Horners Method 175
 Inscribing Circles in Right Triangles 248 cos9 249 Using Polynomials to Approximate π 254 π a la Horner 256 Parabolas 257 Root Extraction 260 The Growth of Science 261 Index 263 Opphavsrett