NATURE 



345 



riRiRSnW, AUGUST 



THE DETERMINATION OF ORBITS. 

 Die Bahnhestimmung dcr Hitnmehhurper. By Julius 

 Bauschinger. Pp. xv + 65_j; niit 84 Figuren im 

 Texr. (Leipzig: W'ilhelm Engelmann, 1906.) 

 /lie Gauss-Gibbssche Methode der Bahnbestitntnung 

 dues Himmelskorpers aus drei Bcobachtungen. 

 Mit einem Anhang zum " Grundriss der theo- 

 lelischen Astrononiie." By Prof. Johannes Fris- 

 ihauf. Pp. 47. (Leipzig : Wilhelm Engelmann, 

 '905-) 

 ' I " H D development of convenient and general 

 -1- methods for calculating the orbit of any body 

 aroinid the sun from a limited number of observations 

 constitutes a classical problem in the annals of 

 astronomy. Its history, which has yet to be ade- 

 quately written, now covers a period of rather more 

 than two centuries, and during that time it has 

 attracted the attention of many famous mathematicians 

 whose successes and failures are alike remarkable. 

 At the time of Newton long records had made the prin- 

 cipal features of the orbits of the known planets 

 familiar, and no addition to their number was made 

 within the next hundred years. Hence in the 

 eighteenth century efforts were mainly directed to the 

 determination of the parabolic orbits of comets, '^'et 

 the completely satisfactory solution was deferred until 

 i;97, when Olbers's celebrated work appeared. Why 

 Olbers succeeded when far greater mathematicians, 

 such as Eulir and Lagrange, had met with compara- 

 tive failure is an interesting question. The fact is 

 that the determination of orbits is an art demanding 

 as such a sense of arithmetical technique and not 

 merely an insight into the mathematical principles 

 involved. 



In the nineteenth century, on the other hand, the dis- 

 covery of minor planets, which are now being found 

 at the average rate of one a week, has required 

 general methods of dealing with planetary orbits. The 

 deduction of an orbit from the necessary'three observ- 

 ations has been based mainly on the methods of 

 Gauss's •• Theoria .Motus. " Even in matters of de- 

 tail the variations which have been added have been 

 for the most part slight and unimportant. In a less 

 degree use has been made of the earlier method of 

 Laplace, which has been generally regarded as inferior 

 in practice. In reality the two solutions are essentially 

 equivalent as regards their mathematical foundation, 

 a remarkable theorem due to Lambert standing as the 

 formal connecting link. Again the difference is a 

 matter of technique rather than of principle. 



The determination of orbits, considered in a wide 

 sense, forms a subject so complicated and so closely 

 dependent on other branches of astronomy that com- 

 prehensive treatises serving to bring together what ex- 

 [jerience has shown to be the most practical methods 

 have rendered indispensable service. In England, 

 owing, perhaps, to the too exclusive predominance of 

 one school of thought, little has been contributed to the 

 development of the theory and nothing to its connected 

 NO. 1919, VOL. 74] 



presentment. The well-known treatise of Watson we 

 owe to .America. Of other works, by far the most not' 

 able is that of Oppolzer. LInfortunately, the second 

 volume of this book is now out of print and has be- 

 come scarce. 



In these circumstances a warm welcome must be 

 extended to Dr. Bauschinger's treatise. His position 

 as Director of the Recheninstitut in Berlin, the promi- 

 nent feature of the work of which is the surveillance of 

 the rapidly accumulating multitude of minor planets, 

 leads us to expect an eminently practical treatment, 

 and we are not disappointed. No great originality 

 will be found, nor was it to be looked for, so far as 

 regards the fundamental methods themselves. The 

 aim of the author has evidently been to follow the 

 path which has been proved by experience, and any 

 originality must be sought in the modes of present- 

 ation, which are always elegant, concise, and lucid. 



.\ most imix)rtant feature of a work of this kind is 

 the choice of illustrative examples of actual computa- 

 tions. In both the liberal selection and the arrange- 

 ment of these Dr. Bauschinger has done well. The 

 diagrams are neat and clear. The style of jirinting, a 

 matter of which the importance in the case of a 

 mathematical work can hardly be exaggerated, will 

 bear comparison with the best English examples of a 

 similar class. It is impossible that all errors should 

 have been detected in the course of proof-reading, but 

 though two or three have certainly escaped notice, it 

 is unlikelv that there will be any necessity for a list of 

 corrections such as that inserted in Oppolzer's second 

 volume. 



Some time ago Dr. Bauschinger published a very 

 useful collection of astronomical tables. Frequent 

 reference is made to these in the present work, which 

 is thus relieved of a large amount of additional matter, 

 while the tables themselves are available in a handier 

 form than as an appendix to a bulky volume. As it is, 

 the author has covered the same ground as Oppolzer's 

 " Bahnhestimmung, " and even included some addi- 

 tions within the limits of a single volume. But it is 

 of necessitv a large one, and can scarcely fail to sug- 

 gest the question whether its size could not be reduced 

 bv omissions or compression without prejudice to its 

 utility. At first sight this would certainly seem to be 

 the case. The first part, containing a discussion of 

 astronomical coordinates, is occupied with matter 

 which ought to be accessible in general treatises on 

 practical astronomy. The chapter on the method of 

 least squares might be replaced by simple references 

 to some work devoted to that subject, and what is 

 given in the chapter on mechanical integration ought 

 to be found in treatises on the calculus of finite dif- 

 ferences. But apart from the fact that this supposes 

 the existence of ideal books which have not yet been 

 written, it is a distinct advantage to be saved the 

 trouble of consulting a number of separate works, 

 even when these are at hand. The fuller treatment 

 must be justified by a severely concise and practical 

 discussion of all subordinate topics, and in this respect 

 little fault will be found with Dr. Bauschinger's 

 handling of his material. It is difficult to believe, 



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