Descartes on Polyhedra: A Study of the "De solidorum elementis" (Sources in the History of Mathematics and Physical Sciences) [Hardcover]

P. J. Federico (Author)

Product Details

• Hardcover: 145 pages
• Publisher: Springer; 1 edition (December 1, 1982)
• Language: English
• ISBN-10: 0387907602
• ISBN-13: 978-0387907604
• Product Dimensions: 9.1 x 6.8 x 0.6 inches
• Shipping Weight: 12.8 ounces (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #3,858,705 in Books (See Top 100 in Books)

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2 of 2 people found the following review helpful:

5.0 out of 5 stars A charming early Descartes manuscript, March 24, 2006

By 

Viktor Blasjo - See all my reviews
(REAL NAME)   

This review is from: Descartes on Polyhedra: A Study of the "De solidorum elementis" (Sources in the History of Mathematics and Physical Sciences) (Hardcover)

This is a transcription and translation of a short manuscript by Descartes on polyhedra with historical and mathematical commentary. Federico's estimate is that the manuscript was written around 1630 (others have guessed 1620). It was still among Descartes's papers when he died in Stockholm in 1650. His belongings were then shipped to Paris but "when it reached Paris the boat was wrecked; the box of manuscripts fell into the river and was not recovered until three days later ... the papers had to be separated and hung to dry on chords in various rooms about the house". The manuscript was later lost, but Leibnitz had seen it and made copy (facsimiles included here) which was found and published in 1860.

The first part of the manuscript gives a few equations relating basic properties of polyhedra (number of vertices, faces, and plane angles, and the sum of all the plane angles). There is one application: since any polyhedron must obey these equations, one can deduce the nonexistence of impossible polyhedra. Thus Descartes proves that there are only five regular polyhedra. We also look briefly at Euler's work on polyhedra (from 1750 when Descartes's text was unknown and unpublished). Euler recognised that the three basic polyhedral data was the number of vertices, edges and faces. Since Descartes neglected to count edges he missed out on Euler's formula v-e+f=2.

The second part of the manuscript deals with polyhedral figurate numbers. The Greeks studied plane polygonal numbers, but even though the regular polyhedra were their pet objects they did not study polyhedral numbers. As the polygonal numbers, the polyhedral numbers may be regarded as built up layer by layer. In this way Descartes calculates formulas for the figurate numbers in the cases of the regular polyhedra and many semi-regular polyhedra.