By George Bruce Halsted
MATHEMATICAL MONOGRAPHS. EDITED through MANSFIELD MERRIMAN AND ROBERT S. WOODWARD. No. 2. artificial PROJECTIVE GEOMETRY. by means of GEORGE BRUCE HALSTED, PROFESSOR OF arithmetic IN KKNYON collage. FOURTH version, ENLARGED. FIRST THOUSAND. big apple JOHN WILEY SONS. LONDON CHAPMAN corridor, restricted. 1906. EDITORS PREFACE. the amount known as larger arithmetic, the 1st variation of which used to be released in 1896, contained 11 chapters via 11 authors, every one bankruptcy being self reliant of the others, yet all supposing the reader to have at the least a mathematical education such as that given in classical and engineering faculties. The ebook of that quantity is now discontinued and the chapters are issued in separate shape. In those reissues it's going to regularly be discovered that the monographs are enlarged by way of extra articles or. appendices which both magnify the previous presentation or list fresh advances. This plan of e-book has been prepared so one can meet the call for of academics and the benefit of sessions, however it can also be proposal that it can end up positive to readers in particular traces of mathematical literature. it's the purpose of the publishers and editors so as to add different monographs to the sequence now and then, if the decision for a similar turns out to warrant it. one of the subject matters that are into consideration are these of elliptic capabilities, the speculation of num bers, the gang thought, the calculus of diversifications, and non-Euclidean geometry potentially additionally monographs on branches of astronomy, mechanics, and mathematical physics might be integrated. it's the desire of the editors that this type of ebook may well are inclined to advertise mathematical research and study over a much broader box than that which the previous quantity has occupied. December, 1905. iii AUTHORS PREFACE. guy, imprisoned in a bit physique with short-arm fingers rather than wings, created for his assistance a mole geometry, a tactile area, codified via Euclid in his immortal components, whose basal precept is congruence, dimension. but guy is not any mole. countless feelers radiate from the win dows of his soul, whose wings contact the fastened stars. The angel of sunshine in him created for the information of eye-life an inde pendent method, a radiant geometry, a visible area, codified in 1847 via a brand new Euclid, by way of the Erlangen professor, Georg von Staudt, in his immortal Geometric der Lage released within the old fashioned and historic Nurnberg of Albrecht Dtirer. Born at the twenty fourth of January, 1798, at Rothenburg ob der Tauber, von Staudt used to be an aristocrat, factor of the union of 2 of the few regicrenden households of the then nonetheless unfastened Reichsstadt, which 4 years later closed the 630 years of its well known exist ence as an self sufficient republic. This production of a geometry of place disembarrassed of all volume, utterly non-metric, neither absolutely nor negatively quantitative, resting solely on kinfolk of scenario, takes as element of departure the since-famous quadrilateral building. To-day it has to be reckoned with from the abstractest domain names of philosophy to the bread-winning marts of utilized technological know-how. hence Darboux says of it it kind of feels to us that less than the shape first given it via von Staudt, projective geometry needs to turn into the mandatory better half of descriptive geometry, that it's known as to renovate this geometry in its spirit, its approaches, its applica tions. KENYON collage, GAMBIER, OHIO, December, 1905. CONTENTS. advent web page 7 artwork. i. the weather AND PRIMAL kinds eight 2. PROJECTING AND slicing 10 three. components AT INFINITY 10 four. CORRELATION AND DUALITY 12 five. POLYSTIMS AND POLYGRAMS 12 6. HARMONIC components 15 7. PROJECTIVITY 18 eight. CURVES OF the second one measure 20 nine. POLE AND POLAR 25 10. INVOLUTION 26 eleven. protecting CONIC levels 29 12. heart AND DIAMETER .... 32 thirteen. aircraft AND element DUALITY 34 14. governed QUADRIC SURFACES 36 15. CROSS-RATIO forty two 1 6. HOMOGRAPHY AND RECIPROCATION forty five 17. TRANSFORMATION. PENCILS AND levels OF CONICS .... fifty four INDEX fifty nine artificial PROJECTIVE GEOMETRY...