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Elements of Projective Geometry. Translated by Charles Leudesdorf

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Elements of Projective Geometry. Translated by Charles Leudesdorf Handle with EXTREME CARE This volume is damaged or brittle and CANNOTbe repaired! • photocopy only if necessary • return to staff • do not put in bookdrop Gerstein Science Information Centre Digitized by tiie'lnternet Arciiive in 2007 witii funding from IVIicrosoft Corporation littp://www.archive.org/details/elementsofprojecOOcremuoft IB ELEMENTS OP PROJECTIVE GEOMETRY CREMONA HENRY FROWDE Oxford University Press Warehouse Amen Corner, E.C. v-^a ELEMENTS OF PROJECTIVE GEOMETRY LUIGI CREMONA LL.D. EDIN., rOE. MEMB. E. S. LOND., HON. F.E.S. EDIN. HON. MEMB. CAMB. PHIL. SOC. PEOFESSOE OF MATHEMATICS IN THE UNIVEESITY OF EOME TBANSLATED BY CHARLES LEUDESDORF, M.A. FELLOW OF PEMBROKE COLLEGE, OXFOED AT THE CLARENDON PRESS 1885 [ All rights reserved ] ^ 0(3 EIKTRONIC VERSION AVAILABLE ^j^ Nd._:i — AUTHOR'S PREFACE TO THE FIRST editio:n^*. Amplissima et pulcherrima scientia figurarum. At quam est inepte sortita nomen Geometrise ! —NicoD. Frischlinus, Dialog. I. Perspectivse methodus, qua nee inter inventas nee inter inventu possibiles ulla compendiosior esse videtur . —B. Pascal, Lit. ad Acad. Paris., 1654. Da veniam scriptis, quorum non gloria nobis Causa, sed utilitas ofiBciumque fuit.—OviD, e.r Pont., iii. 9. 55. This book is not intended for those whose high mission it is to advance the progress of science ; they would find in it nothing new, neither as regards principles, nor as regards methods. The propositions are all old ; in fact, not a few of them owe their origin to mathematicians of the most remote antiquity. They may be traced back to Euclid (285 B.C.), to Apollonius of Perga (247 B.C.), to Pappus of Alexandria (4th century after Christ); to Desargues of Lyons (1593-1662); to Pascal (1623-1662) ; to De la Hire (1640-1718); to Newton (1642-1 727) ; to Maclaurin (1698-1746); to J. H. Lambert (i 728-1 777), &c. The theories and methods which make of these propositions a homogeneous and harmonious whole it is usual to call modern, because they have been dis- covered or perfected by mathematicians of an age nearer to ours, such as Carnot, Brianchon, Poxcelet, Mobius, Steiner, Chasles, Staudt, etc. ; whose works were published in the earlier half of the present century. Various names have been given to this subject of which we are about to develop the fundamental principles. I prefer * With the consent of the Author, only such part of the preface to the original Italian edition (1872) is here reproduced as may be of interest to the English reader. VI AUTHORS PREFACE TO THE FIRST EDITION. not to adopt that of Higher Geometry [Geometrie snp^rieure, ' hohere Geometrie), because that to which the title higher ' at one time seemed appropriate, may to-day have become very elementary; nor that of Modern Geometry [nenere Geometrie), which in like manner expresses a merely relative idea ; and is moreover open to the objection that although the methods may be regarded as modern, yet the matter is to a great extent old. the title Geometry iiosition Geo?netrie cler Lage) Nor does of ( as used by Staudt'^ seem to me a suitable one, since it excludes the consideration of the metrical properties of figures, I have chosen the name of Projective Geometry f, as expressing the true nature of the methods, which are based essentially on central projection or perspective. And one reason which has determined this choice is that the great Poncelet, the chief creator of the modern methods, gave to his immortal book the title of Traite des proprietes projectives des figures (1822). In developing the subject I have not followed exclusively any one author, but have borrowed from all what seemed useful for my purpose, that namely of writing a book which should be thoroughly elementary, and accessible even to those whose knowledge does not extend beyond the mere elements of ordinary geometry. I might, after the manner of Staudt, have taken for granted no previous notions at all ; but in that ease my work would have become too extensive, and would no longer have been suitable for students who have read the usual elements of mathematics. Yet the whole of what such students have probably read is not necessary in order to understand my book ; it is sufficient that they should know the chief propositions relating to the circle and to similar triangles. It is, I think, desirable that theoretical instruction in * Equivalent to the Deftcriptive Geometry of Catley {Sixth memoir on qualities, of Phil. Trans, the Eoyal Society of London, 1859; P- 9°)- The name Geometrie de position as used by Carnot corresponds to an idea quite different from that which I wished to express in the title of my book. I leave out of consideration other names, such as Geometrie segmentaire and Orc/anische Geometrie, as referring to ideas which are too limited, in my opinion. t See Klein, Ueber die sogenannte nicht-EukUdische Geometrie (Gottinger Nachrichten, Aug. 30, 1871). AUTHOR S PREFACE TO THE FIRST EDITION. Vll geometry should have the help afforded it by the practical constructing and drawing of figures. I have accordingly laid more stress on desc7'ipfk'e properties than on metrical ones ; and have followed rather the methods of the Geometrie der Lage of Staudt than those of the Geometrie superieiire of Chasles*. It has not however been my wish entirely to exclude metrical properties, for to do this would have been detrimental to other practical objects of teaching f. I have therefore intro- duced into the book the important notion of the anJiarmonic ratio, which has enabled me, with the help of the few above- mentioned propositions of the ordinary geometry, to establish easily the most useful metrical properties, which are either consequences of the projective properties, or are closely related to them. I have made use of central projection in order to establish the idea Qiinjinitely distant elements ; and, following the example of Steiner and of Staudt, I have placed the law of duality quite at the beginning of the book, as being a logical fact which arises immediately and naturally from the possibihty of constructing space by taking either the point or the plane as element. The enunciations and proofs which correspond to one another by virtue of this law have often been placed in parallel col imns ; occasionally however this arrangement has been departed from, in order to give to students the oppor- tunity of practising themselves in deducing from a theorem its correlative. Professor Eeye remarks, with justice, in the preface to his book, that Geometry affords nothing so stirring to a beginner, nothing so likely to stimulate him to original work, as the principle of duality; and for this reason it is very important to make him acquainted with it as soon as possible, and to accustom him to employ it with con- fidence. The masterly treatises of Poncelet, Steiner, Chasles, and * Cf. Eeye, Geometrie der Lage (Hannover, t866; 2nd edition, 1877), p. xi. of the preface. t Cf. Zech, Die hdhere Geometrie in Hirer Aiuccndung auf Kegelschnitte und Fldchen zweiter Ordnung (Stuttgart, 1857), preface. viii AUTHORS PREFACE TO THE FIRST EDITION. Staudt * are those to which I must acknowledge myself most indebted ; not only because all who devote themselves to Geometry commence with the study of these works, but also because I have taken from them, besides the substance of the methods, the proofs of many theorems and the solutions of many problems. But along with these I have had occasion also to consult the works of Apollonius, Pappus, Desargues, De la Hire, Newton, Maclaurin, Lambert, Carnot, Brianchon, Mobius, Bellavitis, &c. ; and the later ones of Zech, Gaskin, Witzschel, Townsend, Eeye, Poudra, Fiedler, &c. In order not to increase the difficulties, ah-eady very con- siderable, of my undertaking, I have relieved myself from the responsibility of quoting in all cases the sources from which I have drawn, or the original discoverers of the various pro- positions or theories. I trust then that I may be excused if sometimes the source quoted is not the original onef, or if occasionally the reference is found to be wanting entirely. In giving references, my desire has been chiefly to call the attention of the student to the names of the great geometers and the titles of their works, which have become classical. The association with certain great theorems of the illustrious names of Euclid, Apollonius, Pappus, Desargues, Pascal, Newton, Carnot, &c. will not be without advantage in assist- ing the mind to retain the results themselves, and in exciting that scientific, curiosity which so often contributes to enlarge our knowledge. Another object which I have had in view in giving refer- ences is to correct the first impressions of those to whom the name Projective Geometry has a suspicious air of novelty. Such * PoNCELET, Traits des pro2^riet4s jirojectives des figures QPsiris, 1822). Steinee, Systemafische Enfwickelung der Ahhdngigheit geometrischer Gesfnlten von einander, <fcc. (Berlin, 1832). Chasles, Traite de Gi^ometrie supirieure (Paris, 1852); Traite des sections coniques (Paris, 1865). Staudt, Geomefrie der Lage (Niirnberg, 1847). "j* In quoting an author I have almost always cited such of his treatises as are of considerable extent and generally known, although his discoveries may have been originally announced elsewhere. For example, the researches of Chasles in the theory of conies date from a period in most cases anterior to the year 1830; those of Staudt began in 1831 ; &c. AUTHOR S TREFACE TO THE FIRST EDITION. IX persons I desire to convince that the subjects are to a gi-eat extent of venerable antiquity, matured in the minds of the greatest thinkers, and now reduced to that form of extreme simplicity which Gergonne considered as the mark of perfection in a scientific theory^.
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