5IO 



NA TURE 



[March 29, 1900 



only about one-third that suggested by Prof. Langley. 

 With the view of obtaining a more correct determination, 

 Prof. Miiller has proposed, in connection with Dr. Kempf, 

 to observe the brilliancy of a star simultaneously, as seen 

 at sea-level and on the top of a neighbouring mountain, 

 thus getting different lengths of atmospheric path ; but the 

 few observations that have hitherto been made with this 

 object do not point to any trustworthy conclusion. 



In the section devoted to the description of photometric 

 apparatus, one looks for completeness rather than 

 originality, and though Dr. Miiller may have omitted 

 some of the forms that have been recommended, yet so 

 many examples have been illustrated and described, one 

 would willingly hope that he has exhausted the entire 

 catalogue. Practically there are only two principles 

 which underlie every form of photometric measurement — 

 extinction, and equalisation of light — but ingenuity has 

 applied these two principles in every variety of method. 

 This variety suggests great dissatisfaction with existing 

 apparatus, and insinuates that a simple, convenient photo- 

 meter, that would inspire confidence and ensure accuracy, 

 has yet to be invented. After all, the eye is the real 

 photometer ; and it may be that here the inherent diffi- 

 culty, physiological rather than mechanical, exists, though 

 attempts to displace the eye, and to substitute the photo- 

 graphic film, have not as yet led to more satisfactory 

 results. We cannot follow Dr. Miiller through the 

 various arrangements of diaphragms and the employment 

 of absorbing media, by which the problem has been 

 sought to be solved on the principle of extinction. 

 Neither can we mention the many advocates of the 

 method of equalisation, who have pressed into the service 

 of photometry rotating discs, reflecting spheres, and other 

 ingenious devices, while the applications of polarised 

 light in various forms are legion. The special depart- 

 ment of spectral photometry receives new additions to its 

 apparatus every month ; but if any one imagines that he 

 has a new scheme to add to the many already proposed, 

 we advise him to study the critical remarks of our author 

 before venturing on a ground which is already strewn 

 with failures, or, at least, doubtful successes. 



The third and concluding section is not the least in- 

 teresting. Here it is the author's purpose to show what 

 has been accomplished by the application of photo- 

 metrical methods to the planets and satellites, to the 

 comets and stars. We not only get an historical ac- 

 count of what has been effected, and are thus reminded 

 of the energy that astronomers of this century mainly 

 have devoted to this subject, but are able to appreciate 

 the degree of success that has attended their efforts, and 

 see in what direction future enterprise is possible. From 

 an observer of Dr. Mtiller's experience we get many 

 critical remarks of great value, and are able to see the 

 exact position in which celestial photometry stands. 

 Perhaps, in the single direction of the application of 

 photography, something more might have been added ; 

 but it must be remembered that the author's colleague. 

 Dr. Scheiner, has already fairly occupied this ground, 

 and some large catalogues, like the " Southern Durch- 

 musterung " of Dr. Kapteyn, had not appeared at the time 

 of publication. In this section, where so much is im- 

 portant, it is difficult to select any one topic for comment ; 

 but special mention should be made of the references to 

 NO. 1587, VOL. 61] 



the observation of variable stars, and the relations existing? 

 between eye-estimations of star magnitude and photo- 

 metric measurement. With this last subject is necessarily 

 connected the difficult question of standards and units of 

 measurement. Over such subjects, to which may be 

 added the desirability of attaining uniformity of scale, 

 some little controversy has hitherto ruled, and perhaps 

 is not yet definitely settled, owing to the diversity in the 

 instruments employed, and the systematic errors intro- 

 duced, it may be, by abnormal vision. On the whole, Dr, 

 Miiller preserves a judicial dignity and impartiality, but 

 some of the strictures concerning the Southern Catalogues 

 produced under Harvard methods and auspices are, we 

 think, undeserved. The lower altitude of the Pole at 

 Arequipa did undoubtedly give rise to great difficulties, 

 but any want of continuity, supposing it to exist, between 

 the Northern and Southern Catalogues can hardly be 

 ascribed to the method of observation. Dr. Miiller's own 

 method of establishing numerous points of reference, 

 scattered over the sky, to serve as standard comparison 

 stars, has much to recommend it, but it is a plan that 

 involves long preparation before the work of real cata- 

 loguing can begin. How far the increased accuracy of 

 the work repays for this preliminary labour, and to what 

 extent other astronomers will avail themselves of these 

 scattered standards, rather than to trust to a single star, 

 are points that can hardly be decided yet ; but every 

 attempt that aims at additional accuracy must be 

 welcomed, and on the side of accuracy we believe that 

 the Potsdam photometric measures stand unrivalled. 



AN ILLUSTRATED HISTORY OF 

 MA THE MA TICS. 

 Histoire des Mathematiques. Par J. Boyer. Pp. xii -f 

 260. (Paris : G. Carr^ et C. Naud, 1900.) 



TO write within the compass of less than 240 octavo 

 pages an intelligible outline of the history of 

 mathematics from the earliest times to the present day is 

 a task so difficult that its accomplishment might reason- 

 ably be thought impossible. Yet by adopting a consistent, 

 if modest, programme, and frankly accepting its necessary 

 limitations, M. Boyer has achieved a considerable 

 measure of success ; possibly as much as the scope of 

 his essay could allow. A reader whose mathematical 

 knowledge is only moderate, or even elementary, will 

 obtain from this book some idea, trustworthy so far as it 

 goes, of the contributions made to the science of mathe- 

 matics by the ancient Babylonians, Egyptians, Indians 

 and Greeks ; he will be able to follow the feeble course 

 of geometry and analysis through the Middle Ages, and 

 appreciate to some extent the glorious renascence of the 

 sixteenth and seventeenth centuries ; and he will acquire 

 some conception of the work of the giants, from Newton, 

 to Laplace, who established the primary landmarks of 

 modern mathematics. 



Perhaps the most satisfactory chapters in the book are 

 xiv.-xvii., which cover the period from Descartes to 

 Laplace. It deserves to be specially remarked that M. 

 Boyer discusses the invention of the infinitesimal calculus 

 in an impartial and charitable spirit, and that he does 

 substantial justice to the merits of Cauchy and Monge, 

 who are not always appreciated as they deserve, even by 

 their own countrymen. 



