582 



NATURE 



[June 29, 191 1 



trials; thus in his examples he works to seven places 

 of decimals, and wc have not noticed a case in which 

 more than seven trials are required. The one. unfor- 

 tunately fatal, objection, is that he has ignored the 

 iiiiditions of the problem; it is as thouj^h the value of 

 TT were found from the perimeter of a regular polygon 

 of 2" sides. By takinp^ n large enough, we can get 

 by Euclidean construction a value as near t as we 

 please; but it is needless to say that this is not what 

 is meant by "squaring the circle" with rule and com- 

 pass. 



Dr. Whiteford's method is sufticiently ingenious 

 to deserve a brief description. Let .AP be an arc of 

 a circle, of which AOB, COD are perpendicular 

 diameters; then if through P a line PQRS be drawn 

 cutting the circle in R, and the diameters CD, AB in 

 O, S, so that QS = AB, then the angle ROB is one- 

 third of the angle AOP. This theorem was known 

 to the Greeks, and, in fact, led to the invention of 

 the conchoid of Nicomedes as an auxiliary curve for 

 the trisection of the angle. If we take Q anywhere 

 on CD and produce PQ to S, making^ C)S = AB, the 

 locus of S is a conchoid, and its intersection with AB 

 ^ives the solution of the problem. Now Dr. White- 

 ford gives a construction for a point on AB, which is 

 in no case very far from S, and thence obtains a 

 sequence of points which have S for their limiting 

 point. After translating^ his construction into trigono- 

 metry, he gives fifty-one worked-out examples, so 

 variously distributed over the quadrant as to leave no 

 doubt that his sequence does actually converge in every 

 case, though, as might be expected, he makes no 

 attempt to prove thig fact. 



It is easy to draw the obvious moral that it is waste 

 of time to attack a mathematical problem without 

 completely understanding what the problem is. How- 

 ever, this is perhaps ungracious ; let us rather con- 

 clude by admitting that Dr. Whiteford has added one 

 more to the fairly numerous approximate solutions of 

 this celebrated problem. G. B. M. 



LILIE^'TIIM.'S WORK ON AVIATION. 

 Birdflight as the Basis of Aviation : a Contribution 

 towards a Systetn of Aviation, Compiled from the 

 Results of Numerous Experiments made by O. and 

 G. Lilicnthal. By Otto Lilienthal. With a bio- 

 graphical introduction and addendum by Gustav 

 Lilienthal. Translated from the second edition by 

 A. W. Isenthal. Pp. xxiv+ 142 + viii plates. 

 (London : Longmans, Green, and Co., 191 1.) Price 

 9s. net. 



THE interest of this book lies chiefly in the bio- 

 graphical part. The two Lilienthals, born in 

 the town of Anklam, were from childhood devoted 

 to mechanics, and also to natural history. Otto 

 gained practical knowledge as a mechanic in the 

 works of Schuarlzkopf in Berlin, while his brother 

 was educated as an architect. Watchin^j the flight of 

 the storks which abounded about their home seems 

 to have eiven them their first taste for the problems 

 which afterwards occupied so much of their atten- 

 tion. 



NO. 2174, VOL. 86] 



One curious passage (p. 103) is here quoted. After 

 a picturesque description of the storks following the 

 harvesters to pick up field mice and such " small 

 deer," and a reference to the high regard in which 

 the birds are held by the farmers, we read :— 



"Thus it is not to be wondered at if the farmers, 

 above whose homesteads these birds with a span of 

 two metres hold great flight meetings every year, 

 evince great interest in the art of flight, though they 

 do not wish it to be known, fearing ridicule. Never- 

 theless, from no other trade or profession have so 

 manv inquiries for light engines— for a secret pur- 

 pose—reached the author, as from farmers." 



The theoretical part of the book is not of any great 

 value, and some of the diagrams (notably those on 

 pp. 45 and 56, of the flow of a stream past an obstacle) 

 are very far from representing- the actual facts. 



Lilienthal seems to have attached great importance 

 to curved, as against plane, and flapping, as against 

 fixed wings. Flapping wings are really the more 

 efficient of the two, but not for the reasons given. 



Difficulties of mechanical construction have hitherto 

 prevented their adequate trial on a large scale, but 

 it would be too much to say that they are imprac- 

 ticable. 



The merit of the curved wing section now generally 

 in use is not that it gives an increased lift or efficiency, 

 but that the efficiency does not vary so rapidly with 

 the change of inclination to the air current as happens 

 when the surfaces are planes. 



It is seldom that the enthusiasm, mechanical ability, 

 knowledge, and money necessary for experiments in a 

 new field are found in the possession of one and the 

 same individual. Lilienthal had the two first, but 

 was hampered as regards the others. This book, how- 

 ever, which is well translated, should be read as 

 giving a very interesting account of the work of a 

 tvpical "inventor" of the best class. 



A. Mallock. 



OUR BOOK SHELF. 

 A Systematic Handbook of Volumetric Analysis; or. 

 the Quantitative Determination of Chemical Sub- 

 stances bv Measure, Applied to Liquids, Solids, and 

 Gases. By F. Sutton. Tenth edition. Revised 

 throughout, with numerous additions, by W. L. 

 Sutton and A. E. Johnson. Pp. xiv + 621. (Lon- 

 don : J. and A. Churchill, 191 1.) Price 21s. net. 

 A BOOK that has been the standard text-book on 

 its subject for nearly fifty years, and continues to 

 enjoy that position, calls for little remark as edition 

 after edition is issued. Every student of chemistry 

 knows, or ought to know, his "Sutton," as he knows 

 his "Fresenius," and it would therefore be super- 

 fluous to attempt to describe its character and scope. 

 The author, being now eighty years old, has placed 

 the preparation of this tenth edition entirely in the 

 hands of the two editors named in the title. There 

 seems to be everv evidence that the editors have done 

 their work thoroughlv and judiciously. They say that 

 a good deal of obsolete matter has been deleted, and 

 we naturally turned to some of those classical methods 

 that have served so well in the past, such as Bunsen's 

 method of gas analysis, and have been almost 

 wholly replaced, rather than superseded, by processes 

 that are more suitable for technical work. But the 



