162 



SCIENCE. 



[N. S. Vol. XIX. No. 474. 



angles of the one and those of the other are 

 equal each to each, or else can be regarded 

 as the difference of polygons capable of 

 this mode of decomposition. But we must 

 observe that an analogous condition does 

 not seem to exist in the case of two equiv- 

 alent polyhedrons; so that it becomes a 

 question whether or not we can determine 

 the volume of the pyramid, for example, 

 without an appeal more or less disgTiised 

 to the infinitesimal calculus. ' It is, there- 

 fore, not certain that we can dispense with 

 the axiom of Archimedes in the measure- 

 ment of volumes. Moreover, Professor 

 Hilbert has not attempted it." 



Professor Halsted, in the paper in ques- 

 tion, has attacked the problem in the fol- 

 lowing manner: 



The product of an altitude of a tetrahe- 

 dron by the area of its base is the same 

 whichever of the four faces may be chosen 

 as base. This product is, therefore, a 'nat- 

 ural invariant' of the tetrahedron and 

 may be designated as its volume, except 

 that in order to adjust the conception to 

 our ordinary numerical scale the factor 

 one third is arbitrarily introduced. After 

 defining a transversal partition of a tetra- 

 hedron as one made by a plane through 

 an edge and a point of the opposite edge, 

 it was shown that, however this solid be 

 cut by a plane, the partition can be ob- 

 tained as a result of successive transversal 

 partitions, using not more than tM'o other 

 planes. 



The above being explained, it was shown 

 that the volume of any tetrahedron is 

 equal to the sum of the volumes of all 

 tetrahedrons which result from any set of 

 transversal partitions. This need not be 

 assumed as self-evident, but may be dem- 

 onstrated as a necessary consequence of 

 the so-called 'betweenness' assumption 

 with reference to three co-straight points. 

 Similar principles were deduced for poly- 

 hedrons in general, and by their use a gen- 



eral theory of volume was built up with- 

 out reference to the ordinary notions of 

 ratio and eommensurability. The same 

 method of treatment may be applied to 

 figures in hyperspace of any order. 



Lines on the Pseudosphere and the Syn- 

 tractrix of Revolution: E. L. Hancock, 

 Purdue University, Lafayette, Indiana. 

 The lines of the pseudosphere are re- 

 viewed and those of the syntractrix of rev- 

 olution studied. The latter surface S^ is 

 defined as the surface generated by the 

 revolution of the curve C^ about its asymp- 

 tote; Cj being determined by laying off a 

 constant distance d on the tangents of the 

 tractrix. 



The geodesic, asymptotic and loxodro- 

 mic lines on 8-^ are worked out and studied 

 by classifying the surfaces according as 



d = 2c, 



c being the constant of the tractrix. When 

 d=2c it happens that the geodesic lines on 

 8^ are all real ; while for d < 2c they are 

 real or imaginary according as 



i r2rf2 I 

 ="|d2— 4ed|' 



K being a constant of integration. 



The loxodromie lines of the syntractrix 

 of revolution are represented in the plane 

 by the same system of straight lines as rep- 

 resent the loxodromie lines of the pseudo- 

 sphere. 



The Rotation Period of the Planet 8aturn: 



Professor G. W. Hough, Director of 



Dearborn Observatory, Evanston, Ills. 



In 1877 Professor Asaph Hall, then at 



the U. S. Naval Observatory, observed a 



spot near to Saturn's equator and by its 



means determined- the period of the 



planet's rotation. From that time on, 



until the recent opposition, no well-defined 



spot has been visible. On June 23, 190.3, 



however. Professor E. E. Barnard, of the 



