January 12, 1899] 



NA TURE 



243 



All the previous knowledge needed consists of a clear 

 notion of the concepts of point, line and plane, and their 

 fundamental properties, such as the propositions that two 

 points determine a line, two planes meet in a line, &c.^ 

 together with a power of imagining these elements in 

 their mutual relation in space ; for instance, that a line 

 may lie in a plane, or else can cut it in one point only. 



These ideas being in the mind of the reader, nothing 

 more is required, and they will themselves gradually 

 become developed and gain in clearness as he proceeds. 



It goes without saying that such a system rec[uires a 

 number of new concepts ; namely, those of the primitive 

 forms, " the elementary prime forms " in Leudesdorf 's 

 translation of Cremona's "Projective Geometry." These 

 are explained m Lecture i. 



In the second lecture the correlations between the 

 prime forms as given by projections and sections are 

 considered, and these lead to the ideal elements at an 

 infinite distance. By their aid the exceptions due to 

 parallelism are got rid of. At the same time a first 

 instance is given of the leading principle of "correspond- 

 •ence" between the elements in two or more prime-forms. 



In Lecture iii. the principle of duality appears as a 

 natural and simple consequence of the first and funda- 

 mental properties of the elements. 



After these preliminary lectures we come in the next 

 to " harmonic forms," which are deduced by aid of 

 ■certain simple constructions in three dimensions without 

 reference to measurement. There is added a discussion 

 of their metrical properties. 



In the fifth lecture we have the projective properties 

 of primitive forms, and the all-important principle of 

 correspondence between the elements in two of them. 

 It ends with the generation of curves of second order 

 and second class, respectively, by projective ranges and 

 sheaves (flat pencils), the actual construction of points 

 on, or tangents to such curves being given. 



With this the reader has finished the preliminary work. 

 It may at first have appeared to him somewhat tedious, 

 as he is unable to appreciate the usefulness and fertility 

 of the definitions and methods explained. He has 

 acquired many definitions and plenty of new ideas, but 

 not as in Euclid at every step a tangible result. This is 

 now all changed. The next chapter shows that the time 

 and trouble expended are to be amply repaid, for now 

 the study of the curves mentioned begins, and he is told 

 that these curves are the conic sections. Side by side 

 with these go the generation and the properties of cones 

 of the second order, and cones of the second class, but 

 these are at once dropped, as their properties are an 

 immediate consequence of the curves. The latter are 

 now investigated in detail. Their construction is more 

 closely examined, and leads almost at once to the 

 celebrated theorems of Pascal and Brianchon. It also 

 follows that five points or tangents determine a curve of 

 the second order or class. 



The fact that these fundamental properties appear at 

 the very beginning is characteristic of the whole method, 

 and this shows that the method is a natural one. 



In the seventh chapter the more immediate con- 

 sequences of these theorems is considered. The curves 

 are also classified as ellipses, hyperbolas, and paraboke, 

 according to their elements at infinity. 

 NO. 1524, VOL. 59] 



In Chapter viii. the special case of Pascal's theorem 

 when the hexagon is reduced to a quadrilateral leads to 

 the theory of pole and polar, and these are in the next 

 lecture specialised by aid of the lines and the points at 

 infinity so as to give the properties of centres, diameters, 

 and axes of conies. In this lecture, too, the equations of 

 the conies are obtained, and thus the proof is given that 

 the conies studied by aid of coordinate geometry are the 

 same as those here considered. 



Of course the principle of duality is constantly used, 

 and properties of curves of second order and second 

 class are treated simultaneously till in Lecture vii. their 

 identity is proved. Numerous other theorems of conies 

 are given at every stage, but we have only mentioned a 

 few in order to show the sequence in which they follow. 



Of this enough has been done, and the contents of the 

 rest of the book can be dealt with more summarily. 



Lecture x. treats of the ruled cjuadric surfaces ; the 

 ne.xt of projective relations of " elementary forms," 

 viz. the range, the sheaves, curves and cones of second 

 order, and the ruled quadric surfaces. 



These relations lead to the " theory of involution," 

 Lecture xii. The metric properties are considered in 

 Lecture xiii., and applied to establish the focal properties 

 of conies ; whilst in Lecture xiv. problems of the second 

 order are solved, and " imaginary elements " introduced 

 in the manner of Von Standt. By their aid problems 

 are solved by actual construction on the drawing board, 

 in which among the gi%en data are " imaginary" elements; 

 for instance, the problem of drawing a conic which 

 passes through three given points, and the two points in 

 which a given line cuts a given conic in the ease where 

 these points have no actual existence. 



It will be seen that hereby again the greatest generality 

 is obtained, and that the exceptions which would occur it 

 the different case where the line cuts the conic, touches 

 it, or does not cut it, are to be treated separately. 



In the fifteenth and last lecture, principal axes and 

 planes of symmetry, focal axes, and cyclic planes of 

 cones of the second order are dealt with. 



This finishes the first part of Reye's work, which is 

 restricted to the investigation of curves and surfaces 

 generated by projective primitive forms of one dimension, 

 the range of points, the sheaf of lines and the sheaf of 

 planes, that is the range and the flat and axial pencil. 



The projective and primitive forms of two dimensions 

 give rise to quadric and cubic surfaces and twisted cubic 

 curves, whilst the forms of three dimensions, viz. the 

 space with points and planes as elements lead to the 

 theory of co-linear and reciprocal spaces. These form 

 the contents of Vol. ii. The translator does not say 

 anything about it, but as the volume before us is marked 

 both on the title-page and the back as Part i., there 

 seems reason to hope that the translation of the second 

 part will soon follow. 



There is, however, an appendix to Part i., which has 

 still to be mentioned. It contains an account of the 

 principle of reciprocal radii, Dupin's cyclide, ruled surfaces 

 of the third order, cjuadrangles and cjuadrilaterals, which 

 are self-polar with respect to conic sections, and lastly a 

 pretty full, though condensed, account of nets and webs 

 of conic sections. 



The translation is very well done. It has evidently 



