By P. J. Federico
The current essay stems from a historical past of polyhedra from 1750 to 1866 written a number of years in the past (as a part of a extra common paintings, no longer published). such a lot of contradictory statements concerning a Descartes manuscript and Euler, by way of a number of mathematicians and historians of arithmetic, have been encountered that it used to be determined to put in writing a separate research of the appropriate a part of the Descartes manuscript on polyhedra. The meditated brief paper grew in measurement, as just a particular remedy will be of any worth. After it used to be accomplished it turned glaring that the whole manuscript might be taken care of and the paintings grew a few extra. the end result provided here's, i'm hoping, an entire, actual, and reasonable remedy of the whole manuscript. whereas a few perspectives and conclusions are expressed, this can be in simple terms performed with the proof earlier than the reader, who may perhaps draw his or her personal conclusions. i want 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 recommendations. i'm specially indebted to Dr. Hess, of the Leibniz-Archiv, for his counsel in reference to the manuscript. i've been vastly helped in getting ready the interpretation ofthe manuscript via the collaboration of a Latin student, Mr. Alfredo DeBarbieri. assistance from librarians is fundamental, and i'm indebted to a couple of them, during this kingdom and in another country, for finding fabric and providing copies.
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Extra resources for Descartes on Polyhedra: A Study of the De Solidorum Elementis (Sources in the History of Mathematics and Physical Sciences)
Neque hic gnomon cum numeris convenit ut in prioribus. ) 28 The physique which consists of J2 pentagonal and 20 hexagonalfaces. has ninety facets. 60 angles and 32 faces. Its gnomon has JJ pentagonal and 18 hexagonal faces. minus seventy six radices. plus forty eight angles. Gnomon F+ F- R+A, eleven+ 18- 76+48, fifty five+ 108 - 152+48, 132+270 - 228 +48, zero I 60 282 106 II. Translation and remark §28 determine 28 28a The characters at the left facet of the road within the manuscript have been washed out and uncertain. (Nor does the gnomon believe the numbers as within the preceding).
Ninety two Descartes algebraic symbolism of 31, 123 n. 20 De Solidorum Elementis. ms. of five, 6,8-10,43,64, 122 n. four (date of) four, 30-32 and Diophantus one hundred thirty n. \00 and Euler's Theorem on polyhedra 3,63, 72-80 and Faulhaber 5,30, 116-118, 123 n. 17, 124 n. 23, a hundred thirty five n. 121 and figurate numbers 86, 88, 89, ninety four, ninety five, 107, 108, a hundred and twenty Geometrie 31, sixty two, 123 n. 20, 124 n. 20, 124n. 25, 131 n. IIO, 131 n. 113 lifetime of five, 30, 32 pointed out 44,45. 46,49, fifty two. fifty eight. 60, 61,62,68,91,96,112,115,118, 119. 122 n. 14, 124 n. 24, 126 n. 36, 126 n. 38, 133 n.
I, 121 n. four 144 Index of individuals Descartes (cont. ) and Pappus 131 n. 113 Regulae advert Direclionern Ingenii forty six, 119, 123 n. 20, 124 n. 20, 126 n,40, one hundred thirty n. a hundred, 131 n. llO, 131 n. 113 terminology of 35, 38, forty, fifty one, fifty six, seventy two, 80,88,94, 126 n. 37, 131 n. 112 Theorem of 63,64,72,75, 128 n. fifty nine topology sixty four, seventy eight, 128 n. sixty two use of cossic symbols by means of 31, 32, eighty four Dickson, L. E. 83,87, 118, 119, a hundred thirty n. 102, a hundred thirty five n. 122, a hundred thirty five n. 123, one hundred thirty five n. 124 Diophantus 83,85,86,130 n. IOO, a hundred thirty n. 104 Drabkin, I. E. eighty three, one hundred thirty n. 102, one hundred thirty n. 106, 131 n.
For that reason its gnomon consists of two hexagonal and three triangular faces, minus 6 radices, plus 2 angles: Gnomon F+ three+ nine+ 18 + 30 + forty five + F212 30 fifty six ninety - R + A, zero 6 + 2, 12 + 2, 12 18 + 2, forty four 24 + 2, 108 30 + 2, 215 23a of those now the diversities are hence outlined, first I, II. Translation and remark §23 one zero one I II 32 sixty four 10 S 21 ~ 107 \ 161 \ 32 forty three fifty four. reviews. the current paragraph starts the respect of the Archimedean or semiregular solids, which occupies the remaining ofthe manuscript.
The perimeters of the triangle are a, b, c, equivalent to the face angles of the cast attitude. With element A as pole build its polar nice circle (equator); that's, a airplane is drawn via zero, the heart of the field, perpendicular to the road OA, and the intersection of this aircraft with the field is a brilliant circle with pole A. Arc b'c' is a part of this nice circle. equally, the polar nice circle of aspect B supplies arc a 'c' and that of 5. §9: external strong attitude forty-one element C provides arc a'b'. Triangle a'b'c' shaped by way of those 3 arcs is the polar triangle of triangle ABC and its good attitude Oa'b'c' the polar (supplement, external) stable attitude of OABC.