REVIEWS 343 



equality" of non-coinitial vectors is explained (pp. 21-27), and the use of the 

 vector algebra described further treated (cf. pp. 37-40). Finally the book contains 

 a treatment of vector equations of straight lines and planes (pp. 40-41), some 

 examples (pp. 41-49), a proof of Pascal's theorem (pp. 49-52), and a treatment 

 of ranges and pencils of conies (pp. 52-66). There is an Appendix (pp. 67-76) 

 and an Index. 



The author remarks (p. 2) that it will not be necessary to subdivide the 

 contents of the book into a formally logical array of numbered definitions, axioms, 

 lemmas, theorems, and corollaries, and, from a formal point of view, objections 

 might possibly be made to this exposition. But it must be acknowledged that 

 of late years the tendency to put mathematical work into formal shape succeeded 

 in hiding under a cloak of pedantry the essential nature of any advances that 

 may have been made by the symbolists. This is one of the many points for 

 which all mathematicians will value Dr. Silberstein's excellent and stimulating 

 work. 



In conclusion, it may be mentioned that the promise made in the note on 

 p. 2 has been fulfilled in Dr. Silberstein's " Further Contributions to Non-Metrical 

 Vector Algebra" in the Philosophical Magazine for July 1919. 



Philip E. B. Jourdain. 



ASTRONOMY 



Navigation. By Harold Jacoby, Rutherford Professor of Astronomy in 

 Columbia University. Second Edition. [Pp. xi+ 350, with 22 figures.] 

 (New York : The Macmillan Company, 1918. Price lis. 6d. net.) 



The study of navigation received a tremendous impetus in the United States, 

 after their entry into the war, and many professional astronomers were employed 

 instructing officers for the mercantile marine. The volume under review, the 

 first edition of which appeared in 1917, is an outcome of this development. It is 

 an admirable little volume, written with remarkable clearness, and with a full 

 explanation of the practical methods. Formal mathematical or astronomical 

 knowledge is not assumed, and though some readers may object to the quoting 

 in the text, without proof, of various formulae, those who have mathematical 

 ability can either prove the formulas for themselves or find proofs in the usual 

 textbooks. That a second edition is 'already called for shows that the volume has 

 met a real need. The contents include an account of methods of dead reckoning 

 without and with logarithms, descriptions of the compass and sextant, details of 

 coastwise navigation, an explanation of the pages of the Nautical Almanac, and a 

 very lucid account, first of the methods of the older navigation (by noon-sights 

 and time-sights), and then of the methods of the newer navigation (by the Sumner- 

 line, including Saint Hilaire's improvement). The latter possesses an enormous 

 advantage over the older methods, yet the writer was surprised to learn recently 

 that on some, at any rate, of the largest vessels in the British Mercantile Marine, 

 the old. time-sight and noon-sight methods are still employed. This volume can 

 be thoroughly recommended to those who desire a clear exposition of the new 

 methods. The final chapter is entitled, " A Navigator's Day at Sea," and gives an 

 account of an imaginary voyage from New York to Colon with all the observations 

 made en route worked out in detail. 



At the end of the volume are given certain (abridged) nautical tables with a 

 view to making the volume complete in itself, so that with its aid a vessel could 

 be navigated without other books or tabular works excepting only the Nautical 



