Descartes on Polyhedra: A Study of the De Solidorum by P. J. Federico

By P. J. Federico

The current essay stems from a background of polyhedra from 1750 to 1866 written a number of years in the past (as a part of a extra basic paintings, now not published). such a lot of contradictory statements concerning a Descartes manuscript and Euler, by way of quite a few mathematicians and historians of arithmetic, have been encountered that it used to be made up our minds to put in writing a separate research of the proper a part of the Descartes manuscript on polyhedra. The reflected brief paper grew in dimension, as just a unique remedy might be of any price. After it was once accomplished it grew to become obvious that the total manuscript will be taken care of and the paintings grew a few extra. the outcome offered this is, i am hoping, a whole, exact, and reasonable remedy of the total manuscript. whereas a few perspectives and conclusions are expressed, this can be simply performed with the proof earlier than the reader, who might draw his or her personal conclusions. i need to specific my appreciation to Professors H. S. M. Coxeter, Branko Griinbaum, Morris Kline, and Dr. Heinz-Jiirgen Hess for analyzing the manuscript and for his or her encouragement and proposals. i'm particularly indebted to Dr. Hess, of the Leibniz-Archiv, for his guidance in reference to the manuscript. i've been tremendously helped in getting ready the interpretation ofthe manuscript by way of the collaboration of a Latin student, Mr. Alfredo DeBarbieri. assistance from librarians is integral, and i'm indebted to a couple of them, during this kingdom and in a foreign country, for finding fabric and delivering copies.

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In (c) a point P is taken inside the angle AOB and perpendiculars PC and PD drawn to OB and OA, respectively, forming angle CPD. The four interior angles of the quadrilateral thus formed add up to 4to; two of these are right angles and hence the remaining two, a and /3, add up to two right angles. These figures are used later for analogy. ] c B B o (a) A A D (b) o ~ D A (e) Figure 3 (5) Exterior polygon angle sum. Proclus also gives the sum of the exterior angles ofa polygon. J1 His proof is quite simple and is illustrated by Fig.

Consider the exterior solid angles constructed at each vertex of the polyhedron in the manner described first in Section 5, paragraph 9 (p. 40). This is analogous to the construction of the exterior angles of a polygon as shown in Fig. 4(b) of Section 5 (p. 38) which is similar to P6lya's Fig. 7. The exterior solid angles are closed by describing a portion of a sphere about each, forming each exterior solid angle into a sector of a sphere (as in Fig. 4(b) arcs of circles are drawn forming each exterior angle into a sector of a circle).

Except where otherwise indicated or obvious from the context, the discussion is limited to things which, it is believed, would have been known generally to mathematicians at the time of the manuscript, and to explanations of these things. (In the explanations and derivations we do not limit ourselves to seventeenth century terminology). ( I) Polygons. " Euclid (I def. 19)29 used "rectilinear figures," defined as those contained by straight lines, and distinguished as trilateral, quadrilateral and multilateral.

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