Makoh 23, 1900.] 



SCIENCE. 



469 



show their connection with tt and applied them 

 to explain the rule of signs, the roots of + 

 and — , the logarithm of quantities, the funda- 

 mental rules of quaternions and vector-anal- 

 ysis, the analogy of the circle to the hyperbola, 

 and imaginary intersections. 



E. D. Preston, 



Secretary. 



THE TEXAS ACADEMY OF SCIENCE. 



The regular monthly meeting of the Texas 

 Academy of Science was held in the Chem- 

 ical lecture room of the University of Texas 

 on Friday evening, February 16th, President 

 Simonds in the chair. 



Dr. H. Y. Benedict, Instructor in Mathe- 

 matics and Astronomy in the University, spoke 

 on ' Astronomy in the XIX. Century ' which 

 took the form of a summary of achievements in 

 that branch of science during the past hundred 

 years. First he considered the additions to in- 

 struments, the invention of which had materi- 

 ally assisted in the refinement and increase of 

 our knowledge, special mention being made of 

 the spectroscope, camera, photometer, etc. 



Additions to the solar system constituted the 

 next topic. These were both numerous and 

 important, including one planet — Neptune — 

 its satellite, the two satellites of Mars, the fifth 

 of Jupiter, Hyperion and the ninth of Saturn, 

 and the two of Uranus. 



The cometary and periodic movements of 

 meteorites were mentioned as were also the 

 measurements of the parallaxes of the fixed 

 stars. 



The last topic, the ' Nebular Hypothesis and 

 the Theory of Tidal Evolution,' aroused con- 

 siderable enthusiasm. Among those who took 

 part in the discussion were Professors Halsted, 

 Harper, Ellis and Mezes. Throughout the pa- 

 per it was made plain that in the astronomical 

 work of the century Americans had played a 

 leading part. 



Dr. L. B. Dickson, Associate Professor of 

 Mathematics, presented a paper entitled ' An 

 Elementary Account of the Greater Problems 

 solved by the Modern Group Theory.' Among 

 the examples chosen to illustrate the definition 

 of a group, the most elementary was that of 

 the three distinct rotations of a plane triangle 



into itself, viz, the right-hand and the left- 

 hand rotations through 120° and the zero- rota- 

 tion called the identity. Denoting them by R, 

 L, I, respectively, it is seen by inspection that 



E'- = L, L'^ = R, RL = LR = B' = L' = I, 



so that the result of applying any one of the 

 three and afterwards any one of the three is 

 equivalent to applying some single one of the 

 three. The corresponding permutations of the 

 three vertices give rise to a ' group of substitu- 

 tions.' 



The connection between rational, integral 

 functions and groups of substitutions was illus- 

 strated by several examples. The solution of 

 the quadratic and cubic equations was made to 

 depend upon the determination of very simple 

 non-symmetric functions of their roots. After 

 indicating the impossibility of solving by rad- 

 icals the general equation of degree n>4, the 

 question of the solution of special equations 

 was declared to be a problem capable of most 

 direct answer by employing the group of the 

 equation, as developed by Galois. A problem 

 in mathematics usually depends upon the solu- 

 tion of an algebraic equation or upon the inte- 

 gration of a differential equation. By consid- 

 ering the group of the equation, finite in the 

 former case and continuous in the latter, we 

 are able to decide whether or not the equation 

 can be solved by radicals or integrated by 

 quadratures. More generally it tells what 

 series of simple problems may be taken in 

 place of the original problem. The group of a 

 problem not capable of such a reduction to a 

 chain of simpler problems is called simple. 

 To borrow a chemical term, these simple groups 

 are the ' elements,' to which any problem in its 

 final analysis is to be reduced. The present 

 state of our knowledge of these elements was 

 discussed both for finite groups and for con- 

 tinuous groups. 



Professor Thos. U. Taylor, M.C.E., read an 

 abstract of his report to the U. S. Geological 

 Survey on ' The Silting up of Lake McDonald, 

 Austin, Texas.' This is the body of water re- 

 tarded by the erection of the great dam across 

 the Colorado River. A comparison of the 

 cross-sections of the lake at sixteen different 

 stations, averaging \\ miles apart, for 1893 



