University of Montana ScholarWorks at University of Montana Graduate Student Theses, Dissertations, & Professional Papers Graduate School 1930 Projective geometry from 1822-1918 John B. Lennes The University of Montana Follow this and additional works at: https://scholarworks.umt.edu/etd Let us know how access to this document benefits ou.y Recommended Citation Lennes, John B., "Projective geometry from 1822-1918" (1930). Graduate Student Theses, Dissertations, & Professional Papers. 8077. https://scholarworks.umt.edu/etd/8077 This Thesis is brought to you for free and open access by the Graduate School at ScholarWorks at University of Montana. It has been accepted for inclusion in Graduate Student Theses, Dissertations, & Professional Papers by an authorized administrator of ScholarWorks at University of Montana. For more information, please contact [email protected]. PR0JECTI7E GEOMETRY from 1822 to 1918 by J. Burr Lennea Presented, in partial fulfillment of the requirement for the degree of Master of Arts, State Unirer^y of Montana 1930 Approved EXAMHÎIUG COIMITTEE Chairman Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: EP38878 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI OisssrtaMon RjMishing UMI EP38878 Published by ProQuest LLC (2013). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106 - 1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. COEfTSM?S Page Introaiiot ion 1 Chapter I. Greek Geometry with Regard to the Origin of Projective Geometry 4 Chapter II. Geometry in Europe from 1500 to 1800, with Regard to the Origin of Projective Geometry. 12 Chapter III. Projective Geometry from 1822 to 1906. Poncelet 46 Ton Stattdt 88 S tein e r 122 Reye 124 Chapter IT. The Reorganization of Pro­ jective Geometry in the First Part of the Present Century. 131 Chapter T. Resulting conclusions. I. The nature of Intellectual Progress 165 II. The nature, Uses, etc., of the Deductive logical Structure. 173 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. "Bn observant oe q^ne les r^sulta*ts partic-uliers avaient de oornmin entre enz, on est suocesslvement parvenu k des résultats fort étendus, et les sciences mathématiques sont a la fols devenues plus générales et plus sim ples." Laplace tOncted by Steiner on the title page of his "Systematische Bntwiclc/iielung", ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Introduction Although many of the theorems which now form the sub­ ject matter of projective geometry have been known for hun­ dreds of years, and some of them for more than two thousand years, it was not until the nineteenth century that they became unified into a separate, well-defined branch of math­ ematics. It is our object to trace the origin of the methods and propositions which now constitute projective geometry, thus outlining the gradual development of the subject, and to give a picture of modern projective geometry, with special attention to the fundamental significance of the logical pro­ cedure. Thus, it is hoped, we may throw new light upon the early discoveries, revealing more clearly their importance in the history of thought and their position in modern geometry. The subject divides itself naturally into four parts. The first is a brief résumé of what there was in Greek geom­ etry that has become fundamental in projective geometry. The second part covers the period including the seventeenth and sixteenth centuries, when the French and English math­ ematicians were most active in this as in nearly every other branch of mathematics. The third is the account of the jrogress in the nineteenth century, when the subject assumed definite shape and developed more rapidly than in any earlier period. In this period most of the work was done in France and Germany. The fo u rth part is an account of p ro jectiv e Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. geometry in the last thirty years. Since 1900 the whole subject has been given new, rigorous treatment and has been unified in a clear and fundamental way, with full conscious­ ness of the underlying logical foundation. There have been certain major changes in the develop­ ment of geometry which it will be well to enumerate at the o u tse t. One of the most important of these was the swing from the rigorous proofs of the Greeks to the Icgical laxness of the mathematicians of the seventeenth, eighteenth, and early nineteenth centuries, and then back to the very highly de­ veloped rigor of modern p ro jectiv e geometry. Another was the steady change of emphasis from the solution of individual prob­ lems to the development of connected unified theories - that is, the change from the particular to the general. The last Important change was the change from nstrie to non-metric geometry. Greek geometry was essentially metric, whereas mod­ ern projective geometry is essentially non-metric. Completely non-metric geometry is a very recent development in the history of mathematics. Such general tendencies as these give interest and sig­ nificance to the particular historical facts. The facts pre­ sented here have been selected with a view to including all those which were important, especially those which seem to have had an influence on the histoiy of the subject, and to rather rigidly excluding all which from this view point were Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. less relevant. heedless to say, not all the original documents, especial­ ly in regard to the Greeks, are available. And even in the parts dealing with the suhjeot from 1800 to the present time, where the original sources have been used, not all the facts have been considered, since only the major works were studied. It is hoped, however, that nothing of really great significance has been overlooked. The last chapter of this thesis (chapter 7) contains ma­ terial less strictly mathematical and more general in its sig­ nificance. We have seen how painfhlly slow and halting was the development of a subject whiah in the finished state seems relatively simple, and that in spite of the fact that many of the world’s greatest geniuses gave it very insistent and pro­ longed effort. Why did tiiey fail to hit upon some of the later and much more powerful methods? Again the very existence of huge logical structure purely deductive in character is cer­ tainly of general philosophical significance. 7/hat is the nature of such a deductive logical structure, and is it as barren as some have claimed it to be? What is the function in thought of purely deductive logic? If final answers to some of these questions cannot be given, it is believed that they are formulated in a manner to bring out their significance. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter I Plato (430—347 B.C.) is the first of the Greek mathe­ maticians In whose work we find anything related at all di­ rectly to oar subject. It was he who studied conic sections, which foim a considerable part of the subject matter of pro­ jective geometry. There is, however, no evidence that Plato developed many of the properties of these curves. He was the first to femulate the idea of geometrical loci, which is ingortant in all branches of mathematics. Euclid, about 300 B. C., is famous for his "Elements" in which he formulated rather completely the classical Greek geometry, with considerable extension and improvement of his own. This is the basis of ordinary modern elementary geometry. The method of "reductio ad absurdum" which is valuable in projective geometry as everywhere else in logic, was for­ mulated by Euclid. Among his works which have not come down to us there are four books on conic sections, the theory of which he had considerably augmented. Apollonius of Perga, about 247 B. C., perhaps the great­ est of a n the Greek mathematicians, wrote, besides other minor works, a treatise of eight books on conic sections. It was Apollonius who first considered conics as. sections of any obliq[ue cone with a circular base; until then they had been conceived only as sections of right circular cones, and the cutting plane had always been supposed perpendicular to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. one of the lines of the cone passing throngh the vertex. This required, three cones of different vertex angles to form the three conic sections. These curves were designated by the words "section of the acute-angled cone", "section of the right-angled cone", "section of the obtuse-angles cone"; they first took definitely the names ellipse, parabola, and hyperbola in the work of Apollonius, although the words ellipse and parabola were known to Archimedes. The whole of Apollonius’ treatise on conics was based on a unique property of conic sections: Suppose an oblique cone with a circular base to be out by a plane through its vertex and through that diameter of the circle at its base which makes this plane perpendicular to the base.