356 



SCIENCE. 



[N. S. Vol. VII. No. 167, 



the author's brief but weighty paper on Tro- 

 choids which was presented before the Ameri- 

 can Association for the Advancement of Sci- 

 ence a few years since. We cannot forbear 

 to dwell upon the superb illustrations, which 

 make the book a portfolio of art. The author 

 is particularly happy in deciding conflicts of 

 nomenclature, as where he refuses to follow 

 Javary (g 508) in calling the geodesic on a cone 

 a conical helix. 



The author has been extraordinarily pains- 

 taking in the proof-reading, and the book is 

 practically free from error. A few trifles have 

 been noticed : Page 156, § 433, first line, for 

 •* prism' read ' cylinder.' Page 171, §442, first 

 line, for 'axes' read 'bases.' Page 37, sixth 

 line from below, for 90° read 9°. Page 67, 

 § 194, seventh line, for 9 read 6. 



The slip on page 55, § 166, in stating the 

 brachistochrone and tautochrone properties of 

 the cycloid, is so evidently a reference to a re- 

 versed or inverted line inadvertently omitted 

 that it also is trivial. As the briefest hint of 

 contents by chapters : I., definitions. II., free- 

 hand sketching. III., draughtsman's outfit. 

 IV., use of instruments. V., higher plane 

 curves. VI., conventional representations. VII., 

 lettering. The treatment of lettering is par- 

 ticularly full and 64 alphabets are given. 

 VIII., copying processes. IX., Descriptive 

 •Geometry of Monge. X., projections, intersec- 

 tions, development of surfaces, with applica- 

 tions to elbowjoints, blast pipes, arch construc- 

 tions, etc. XI., trihedrals. XII., projection of 

 sphere. Here the now disused orthographic 

 projection is somewhat condensed, but the 

 stereographic, which is used, is treated at 

 compensatory length. XIII., shades and shad- 

 ows. XIV., perspective. XV. and XVI., 

 isometric and clinographic projection, with 

 applications ; also crystals in oblique projection. 

 XVII., bridge details, toothed gearing, etc. Out 

 of a host of beautiful figures we may mention 

 92 as particularly efficient in teaching homology 

 or complete plane perspective. 



It is a particular pleasure to welcome the 

 book, because it is on just the lines where Eng- 

 lish and American mathematics has hitherto 

 been sterile. 



Even now the tremendous, the fundamental 



importance of von Standt'sgeometry of position, 

 the pure projective geometry, both for science 

 and philosophy, is realized by few. For example, 

 in the Bolyai type of non-Euclidean geometry, 

 not only is the straight line infinite, but also it 

 has two distinct points at infinity ; it is never 

 closed, even by points at infinity. Writing in 

 1835, even the superhuman penetration of 

 Lobachevski attributed this essential openness 

 to the straight in itself. In the introduction to 

 his ' New Elements of Geometry,' he says : "I 

 consider it unnecessary to analyze in detail other 

 assumptions too artificial or arbitrary. Only 

 one of them still deserves some attention, 

 namely, the passing over of the circle Into a 

 straight line. Moreover, here the fault is visible 

 from the beginning in the violation of continuity, 

 when a curve which does not cease to be closed, 

 however great it may be, must change immedi- 

 ately into the most infinite straight line, since 

 in this way it loses an essential characteristic. 



In this regard the imaginary geometry [the 

 non- Euclidean geometry] fills out the interval 

 much better. When in it we increase a circle 

 all whose diameters come together at a point; 

 finally we so attain to a line such that its nor- 

 mals continually approach, although they no 

 longer can cut one another. This characteristic 

 does not pertain to the straight, but to the 

 curve which, in my paper ' On the Foundations 

 of Geometry,' I have called circle-limit.'" 



Of course, it was not until in the next decade 

 (1847) that von Standt published his immortal 

 'Geometric der Lage,' but long afterward 

 Helmholtz suffers still more seriously for lack 

 of the pure projective geometry, treating the 

 projective questions which necessarily came up 

 in his extended optical researches, sometimes by 

 means and methods of his own make, sometimes 

 only by general reasonings. 



Again, in Mind (1876) Helmholtz misses thus 

 a fundamental difference. He says, p. 315 : "It 

 is, in fact, possible to imagine conditions for 

 bodies apparently solid such that the measure- 

 ments in Euclid's space become what they 

 would be in spherical or pseudospherical space. 

 * * * Think of the image of the world in a con- 

 vex mirror. * * * Now Beltrami's representa- 

 tion of pseudospherical space in a sphere of 

 Euclid's space is quite similar, except that the 



