November 24, 1904] 



NA TURE 



75 



that with no further elaboration of the furnace than 

 a short firebrici-c arch at the fore-part (illustrated in 

 Fig. 15) they will perform their work very efficiently, 

 and with practically no smoke when using a bitu- 

 minous coal. 



A chapter is devoted to the chemistry of the com- 

 bustion process. In referring to the hydrogen in fuels, 

 the statement occurs, " it is generally assumed to be 

 present combined with carbon to form hydrocarbons. 

 The most important of these for the fuel user are 

 — methane, ethylene and acetylene." A small amount 

 of at least the first may be present in coal, but are we 

 to assume the authors to mean that these are the 

 important hydrocarbons existing in the coal before it 

 has been heated ? 



In view of Bone's work (mentioned In a short foot- 

 note) it is a pity the authors did not revise their theory 

 to account for the formation of smoke, seeing that the 

 book was not published until a twelvemonth after Bone 

 and Wheeler's paper appeared in the journal of the 

 Chemical Society (August, 1903), and Armstrong's 

 paper in the same number, in which it is definitely 

 stated " neither hydrogen nor carbon being burnt 

 preferentially." J. S. S. B. 



SCHOOL MATHEMATICS. 

 New School Arithmetic. Part ii. By Charles Pendle- 



bury, assisted by F. E. Robinson. Pp. vi + 207 ^^ 



468 + xliv. (London : George Bell and Sons, 1904.) 



Price 2S. 6d. 

 New School Arithmetic. By Charles Pendlebury, 



assisted by F. E. Robinson. Pp. xvii + 468 + xliv. 



(London : George Bell and Sons, 1904.) Price 45. 6d. 

 New School Examples in Arithmetic. By C. Pendle- 

 bury, assisted by F. E. Robinson. Pp. xiii + 223 + 



xliv. (London : George Bell and Sons, 1904.) 



Price 3s. 

 .4 School Geometry. Part vi. By H. S. Hall and 



F. H. Stevens. Pp. iv + 347 to 442 + iv. (London: 



Macmillan and Co., Ltd., 1904.) Price is. 6d. 

 Theoretical Geometry for Beginners. Part iv. By 



C. H. Allcock. Pp. 224. (London : Macmillan and 



Co., Ltd., 1904.) Price is. 6d. 

 Elementary Plane Geometry. By V. M. Turnbull. 



Pp. vi+136. (London: Blackie and Son, Ltd., 



1904.) Price 2S. 

 Mathematical Problem Papers. By the Rev. E. M. 



Radford. Pp. vi + 203. (Cambridge: University 



Press, 1904.) Price 4s. 6d. net. 

 "pART II. of Messrs. Pendlebury and Robinson's 

 A " New School .Arithmetic " has followed quickly 

 on the publication of part i., and this excellent text- 

 book is now complete. The second part is concerned 

 mainly with the application of arithmetic to the trans- 

 actions of commerce, dealing with such subjects as 

 interest, discount, commission, stocks and shares, profit 

 and loss, &c. Ratio and proportion find a place, and 

 they are illustrated largely by this class of problem. 

 The authors devote a little space to the training of 

 youths in computations suitable to experimental work 

 in the laboratory. Thus we find that algebraical 

 symbols are freely introduced, and chapters are given 

 NO. 1830. VOL 71] 



on averages, approximations, graphs, elementary 

 mensuration, and logarithms. This portion of the 

 book might well have been extended even at the 

 expense, if necessary, of some of the chapters relating 

 to purely business matters. But the subjects treated 

 are very numerous, affording considerable ground 

 for selection, and many teachers will no doubt, and 

 with advantage, omit some of the technical com- 

 mercial chapters. At every stage examples are 

 introduced in great abundance, the answers to 

 which extend to nearly fifty pages. The book con- 

 cludes with a collection of test papers, and a large 

 number of miscellaneous problems. Parts i. and ii. 

 are published separately, and also in one volume. The 

 examples and answers may also be obtained without 

 the other text. .Altogether the book is one that de- 

 serves, and will no doubt obtain, an extended 

 circulation. 



With the issue of part vi. of Messrs. Hall and 

 Stevens's " School Geometry," this popular text-book 

 must now be nearing its completion. The present 

 section corresponds, substantially, with Euclid, Book 

 xi., 1-2 1, and it further deals with the mensuration of 

 the simpler geometrical solids. In establishing the 

 theorems of pure solid geometry, the authors follow 

 Euclid rather closely, but there are some useful addi- 

 tions. Thus it is shown how a point in space is located 

 by means of rectangular coordinates; but it is not 

 shown how position and form may be exhibited 

 graphically by means of projections. In dealing with 

 areas and volumes, elementary trigonometry is used. 

 The prismoidal formula is also introduced, but its value 

 is scarcely made sufficiently manifest, and it is not 

 shown how to deal approximately with irregularly 

 shaped figures, by means of Simpson's or other rules. 

 The book is printed in very distinct type, and the 

 figures and diagrams are beautifully designed and 

 executed. The subject-matter is presented and de- 

 veloped in the clear and attractive style which is always 

 found in the authors' text-books, and is illustrated by 

 well chosen examples. 



Part iv. of Mr. Allcock's " Theoretical Geometry for 

 Beginners " treats, in the first instance, of ratio and 

 proportion, with geometrical applications. The pro- 

 positions correspond roughly with Euclid, Book vi., 

 but the style of proof is different. The reader is first 

 introduced to the conception of ratio and proportion by 

 means of numerical and algebraical examples, and his 

 knowledge of arithmetic and algebra is drawn upon 

 in establishing some preliminary theorems, which are 

 subsequently used in demonstrating the various 

 theorems. The latter half of the book is devoted to 

 modern geometry, including chapters on harmonic 

 pencils, the complete quadrilateral, poles and polars, 

 centres of similitude, inversion, maxima and minima, 

 and envelopes. Some numerical examples are given 

 at intervals, but, as the title implies, the propositions 

 and the exercises thereon are almost entirely confined 

 to deductive geometry, and from this point of view the 

 treatment is eminently satisfactory. The book is got 

 up and printed in a way that leaves nothing to be 

 desired. 



The " Elementary Plane Geometry " by Mr. Turn- 



