464 



SCIENCE. 



[N. S. Vol. XV. No. 377. 



to the calculation and design of tank bottoms. 

 We are warned, however, in the introduction not 

 to expect 'elaborate calculations and deduc- 

 tions based upon problematical theories and 

 conditions,' but only 'such facts as may have 

 been verified, freed, as nearly as may be pos- 

 sible, from the tons of mathematical rubbish,' 

 etc. The following are, presumably, some of 

 the 'verified facts': On page 59 it is stated 

 that 'the moment of forces about a point may 

 hold each other and establish equilibrium of 

 the body, even though the forces themselves 

 fail to balance.' Also that 'the direction of 

 the resultant of two forces is exerted in a line 

 bisecting the original angle at which the forces 

 met, and the extent of the force exerted by 

 this resultant is the difference between that 

 offered by the two or more original forces, or 

 the moment of those forces.' Again, in Chapter 

 VIII., in the analysis of the stresses in a four- 

 post tower, scarcely any of the stresses have 

 been correctly determined. The tower legs are 

 straight and have an inclination of one in ten ; 

 the wind bracing is of the usual type, consist- 

 ing of horizontal struts and diagonal tie rods. 

 The method of calculating the compression in 

 the struts is as follows : " The inclination of 

 the column being one in ten, one-tenth of the 

 load is transferred to the horizontal member 

 as compression-stress, and the remaining nine- 

 tenths is distributed at the base of the column 

 to the foundation." The column stress being 

 133.9 tons, the thrust against the strut is 

 therefore 13.39 tons ; but, since the thrust from 

 each of the two opposite columns is 13.39 tons, 

 the strut must be designed to resist twice that 

 or 26-78 tons! The stress in the strut 'in 

 transferring the wind stress as tensile stress' 

 is not considered, this member being designed 

 only for the compression as above found, 

 together with the stresses due to its own 

 weight. In finding the wind stresses in the 

 diagonals of the upper panel, the stress in each 

 is taken at one-eighth of the total wind pres- 

 sure on the tanli, presumably because there are 

 eight diagonals in the top story of the tower. 

 In this way the stress is computed to be about 

 eight tons, with an assumed wind pressure of 

 seventy tons, whereas the correct stress is 

 about thirty-two tons. • Finally the wind stress 



in each column is taken as constant from top 

 to bottom. 



These and other illustrations which could 

 be given suggest that it might have been better 

 to admit some of the 'mathematical rubbish' 

 so carefully excluded. 



F. E. T. 



Oeornetric Exercises in Paper Folding. By T. 

 SuNDAEA Kow. Edited and revised by Pro- 

 fessors W. W. Beman and D. E. Smith. 

 Published by the Open Court Publishing 

 Company, Chicago. 1901. Pp. x + 148. 

 In the author's preface to this little work, 

 dated from Madras, India, 1893, the double 

 purpose is set forth 'not only to aid the teach- 

 ing of geometry in schools and colleges but 

 also to afford mathematical recreation to young 

 and old, in an attractive and cheap form.' 

 Without attempting to develop a geometry as 

 rigidly confined to folding as the Euclidean 

 is to compass-and-ruler work, it is shown how 

 a large number of interesting metrical and 

 positional relations can be illustrated without 

 the use of instruments other than a penknife 

 and scraps of paper, the latter for setting off 

 equal lengths on folds. Sheets of paper 

 adapted to the work accompany the book, and 

 the allusions in the text to certain kinder- 

 garten 'gifts' imply the pupil's possession of 

 an equipment of elementary geometric forms. 

 The processes are based on the principle of 

 congruence. 



The first nine chapters are devoted to the 

 regular polygons of Euclid's first four books, 

 and to the nonagon. Beginning with the fold- 

 ing of the fundamental square, and progressing 

 through equilateral and other triangles, the 

 Pythagorean theorem and consequent proposi- 

 tions are reached, with certain puzzle squares 

 based thereon. In Chapter X. progressions — 

 arithmetic, geometric and harmonic — are 

 neatly illustrated, as also the summation of 

 certain series. This section is enlivened by the 

 insertion of the legend regarding the duplica- 

 tion of the cube. It would have been an 

 appropriate place to refer to the adaptation 

 of the cissoid and conchoid of Chapter XIV. 

 to the same problem. 



In Chapter XL the numerical value of n 



