0CT0BEE9, 1896.] 



SCIENCE. 



533 



William L. Root ; and in Physics, George K. 

 Burgess, William D. Coolidge and Ralph R. 

 Lawrence. 



Dr. E. Lesser has been appointed associate 

 professor of dermatology at Berlin and Dr. 

 Chermak to the chair of comparative anatomy 

 and embryology at Dorpat. Dr. Winkler, pro- 

 fessor of chemistry, has been appointed director 

 of the School of Mines at Freiberg i. S., and 

 Dr. Godschmidt has been promoted to an as- 

 sistant professorship of chemistry in the Uni- 

 versity of Heidelberg. 



DISCUSSIOlf AND CORRESPONDENCE. 

 THE STRAIGHT LINE AS A MINIMUM LENGTH. 



To THE Editor of Science : In looking over 

 the beautiful new text-book of geometry by 

 Profs. Phillips and Fisher one meets with the 

 following proposition of spherical geometry : 



The shortest line that can be drawn on the sur- 

 face of a sphere between two points is the arc of a 

 great circle, not greater than a semi-circumference, 

 joining these points. 



The demonstration given is one which has 

 been given before. It appears, for example, in 

 the treatise of Chauvenet (1869) and also in 

 that of George Bruce Halsted (1885). In con- 

 nection with this demonstration, the reader can 

 hardly escape noticing that every step of it ap- 

 plies equally well to plane geometry. In fact, 

 it is perfectly easy for any student of Euclid's 

 Elements to construct, step by step, a precisely 

 similar proof of the corresponding proposition 

 of plane geometry : 



The shortest line that can be drawn between any 

 two points is the straight line which joins them. 



The definition of a straight line given by Profs. 

 Phillips and Fisher, therefore, embodies a state- 

 ment capable of deduction from the geometrical 

 axioms by a chain of logical reasoning, and 

 as a definition, is on strictly scientific grounds, 

 quite indefensible. 



Upon examining Prof. Halsted' s book, the 

 definitions of which more closely conform to 

 the Euclidean models, one naturally wonders 

 why this demonstration, even more simple in 

 plane than in spherical geometry, has been in- 

 troduced only in connection with spherical ge- 

 ometry ; and one is led to inquire at how early 



a point the proposition of plane geometry could 

 properly be introduced. 



In attempting to establish between any two 

 lines a relation of equality or inequality, we find 

 ourselves compelled to start from the following 

 principles : The whole is greater than any of its 

 parts; The whole is equal to the sum of all its 

 its parts ; Lines which may be placed so as to co- 

 incide are equal. Using these principles alone, 

 it is evident that we cannot compare every two 

 arbitrary lines in magnitude. In any such 

 comparison we must be able to place one of the 

 lines, or portions of it, in complete or partial 

 coincidence with the other. No direct com- 

 parison can be instituted, for example, between 

 a straight line and a line no part of which is 

 straight. For the purposes of the proposition 

 in question, therefore, it is necessary to make 

 the distinct assumption, that the magnitude of ev- 

 ery line is comparable with the magnitude of every 

 other line, and between these magnitudes there ex- 

 ists a relation either of equality or of inequality ; 

 or else, what is better, to await the method of 

 limits and the development, by means of it, of 

 metrical ideas, not only for straight lines, but 

 also for curves. Prof. Halsted, accordingly, in 

 spite of his apparent lateness in introducing the 

 proposition, is guilty of an error in theory. 

 He has attempted to give a complete discussion 

 of a proposition, and appears to believe that he 

 has done so, when in reality assumptions addi- 

 tional to those previously made must be intro- 

 duced before such a discussion can be under- 

 taken. 



It seems worth while to make these criticisms, 

 because the two books above referred to are at 

 other points remarkable for their scientific ac- 

 curacy, and are of so high an order of excel- 

 lence generally that the student may not read- 

 ily appreciate the existence of such errors as 

 occur, Thomas S. Fiske. 



September 30, 1896. 



'A CURVE-TRACING TOP,' AND A CURIOUS OP- 

 TICAL ILLUSION. 



Editor of Science : If Prof. Barus will use 

 a smoked glass for his curve-tracing top to spin 

 on, he will get more beautiful tracings than 

 with any lead pencil arrangement. Then let 

 him flow it over with thin demar varnish, and 



