September 6, 19 17] 



NATURE 



iimensions. In that case there is no necessity to 

 -mploy sliding prisms and scale, or equivalent, as the 

 part of the field where the coincidence occurs depends 

 upon the distance of the object, and thus a scale of 

 listance at the focus of the eyepiece is all that is 

 needed. Of all methods of using the angle of parallax 

 :o find the distance, the most attractive is one proposed 

 )V a workman in the Zeiss works, and which, after 

 nuch difiiculty in its elaboration had been overcome, 

 A as shown to the present writer by the late Dr. Czap- 

 >ki at the Paris Exhibition of 1900. In this instrument 

 he right and left beams are received by the right and 

 '■ft eyes respectively of the observer, and owing to the 

 iistance between the two beams entering the instru- 

 nent a superstereoscopic view of the object is seen. 

 At the same time each eye sees in the field of view a 

 scale of distance, but the two scales are differently 

 ruled in such manner that the eyes combine them 

 stereoscopically and the scale of distance appears pro. 

 jected away into space. It was fascinating to sweep 

 this scale past more or less distant buildings and see 

 the divisions of the distance scale pass behind or m 

 front of the different objects, or to look up the Eiffel 

 Tower and tickle the members of the framework with 

 the nearer divisions. For the purpose of aircraft 

 range-finding this method, on account of its speed, 

 would appear to have great advantages, and even if it 

 does not equal in accuracy the more deliberate methods 

 [ of other range-finders, this cannot be of consequence 



when the range is changing at so high a rate. Some 

 liscussion of this type of range-finder by Prof. Cheshire 

 would have been very valuable. The number of the 

 German patent is 82,571, and the date July, 1895. A 

 description is to be found in the second volume of the 

 collected papers of Ernst Abbe, published by Gustav 

 Fischer in the year 1906. 



Returning now to the question of the limitation of 

 accuracy, the figures quoted as having been obtained on 

 a Barr and Stroud instrument are important and sur- 

 prising. The base of the instrument was three yards, 

 but the diameter of the object glasses is not stated. 

 Using an optically prepared artificial object, the accu- 

 racy of setting obtained by an experienced and highly 

 skilled observer was such that the mean error was 

 about one-fifth of a second of arc, i.e. an angle with a 

 circular measure of one divided by a million. When it 

 is remembered that the defining power of a telescope 

 as measured by the diameter of the star image is 

 about 45 seconds of arc divided by the aperture in 

 inches, this is equivalent to saying that the aligning 

 power of this range-finder is equal to the separating 

 power of a perfect telescope of about 22-in. aperture, 

 and that irrespective of the length of its base. Or if, 

 as is likely, the aperture is about 2 in., the aligning 

 power is more than ten times the possible separating 

 power. Similarly, on multiplying by the magnifying 

 power, it appears that the aligning power of 

 the unaided eye is in the neighbourhood of 

 3 seconds of arc, which is still more surprising 

 when it is remembered that the separating power 

 is certainly insufficient to divide 60 seconds. It would 

 be interesting to ascertain what is the aiming power 

 of a good billiard player when, for instance, the object 

 ball is near the striking ball and far from the pocket, 

 but when, nevertheless, with this coefficient against 

 him, he can time after time drive the ball clean into 

 the pocket. That, whatever it is, must be very great, 

 but it must be exceeded by the aligning power of the 

 eve in the comfortable use of a good range-finder. 

 Figures such as are here given must be realised before 

 the skill and marvellous attainment"of the designer 

 and constructor of the modern range-finder can be 

 appreciated. There is much more in this pamphlet 

 that it would be interesting to follow if .space were 

 available. C. V. Boys. 



NO. 2497, VOL. 100] 



THE RELATIONS OF MATHEMATICS TO 



THE NATURAL SCIENCES. 

 T)Y a happy coincidence, the addresses of the retiring 

 ■'-' presidents of two leading mathematical societies, 

 delivered almost simultaneously, follow similar lines, 

 although from somewhat different angles of view, 

 and are of unusual interest for the man of science 

 whose surmises regarding natural phenomena receive 

 their ultimate justification from mathematical reason- 

 ing. Such a man has had cause more and more in 

 recent years to deplore the divorce between the more 

 striking mathematical developments of the present time 

 and those which are urgently necessary as an inspira- 

 tion to progress in his own work. For, as the two 

 presidents point out, the insistent call for help to the 

 pure mathematician has now begun, though perhaps 

 reluctantly, to take shape even from the biological 

 sciences. 



Prof. E. VV. Brown, in his address to the twenty- 

 third annual meeting of the American Mathematical 

 Society, selected the subject the title of which we 

 have borrowed, and indicated somewhat precisely the 

 types of work really needed from the pure mathe- 

 matician in this regard, and their capacity for furnish- 

 ing a fruitful field of research of great interest to 

 any willing investigator. Sir Joseph Larmor, in his 

 address to the London Mathematical Society in 

 November, 19 16, limited his detailed remarks more 

 especially to the scope and limitations of the har- 

 monic analysis associated with the name of Fourier. 

 The problems connected with periodic phenomena 

 were evidently predominant also in the mind of Prof. 

 Brown during the preparation of his address, and 

 the necessity for a Fourier type of treatment 

 of such problems renders the two addresses com- 

 plementary in the regions in which they are not closely 

 parallel. 



We may turn, in the first place, to the more general 

 point of view present in both addresses, and outlined 

 in greater detail in Prof. Brown's. Pure mathematics 

 is a science or an art which is self-contained, and 

 requires for its development no external inspiration. 

 Applied mathematics is an aid towards the develoo- 

 ment of the natural sciences, and in fact of all in- 

 vestigations which depend on deduction from exact 

 statements. Such statements are, of course, founded 

 not on axioms, but on physical laws which sum up 

 the results of series of experiments, and these laws 

 no longer, as in the past, serve to suggest suitable 

 axioms and profitable lines of development of pure 

 mathematics as an art. So large a body of doctrine, 

 in fact, has pure mathematics become that isolation 

 is marked among its many branches, and one mind 

 can no longer be fully conversant with each of them. 

 The task of our presidents, in attempting a fusion 

 between pure and applied mathematics, becomes more 

 and more difficult. 



Prof. Brown points out one fundamental difficulty 

 in the lack of standardisation of mathematical sym- 

 bols. In spite of the fixed character of the underlying 

 principles, such a symbol as (i) may still denote a 

 number, operator, group, function, axiom, or con- 

 vention, and any of these may have special limita- 

 tions for the purpose in view. He suggests that the 

 task of a reader of several members should be facilitated 

 by extending the principle now used in the case of the 

 special type adopted to represent vectors. Such a pre- 

 arranged system would have special advantages in the 

 subsequent compilation of any future mathematical 

 encyclopaedia. Prof. Brown pleads also for an ex- 

 tension of the growing practice, even at the cost of 

 artistic appearance, of printing a summary at the 

 end of each published paper. 



These and other purely mechanical aids to the 



