Febeuaey 26, 1897.] 



SCIENCE. 



353 



ticularly rieli in examples of methods of calcu- 

 lating which have long since disappeared from 

 our arithmetics, aud, as the author points out, 

 some of these are, by no means, inferior to those 

 now used. Such examples make the history 

 of arithmetic very real to one. The sections 

 entitled ' Causes which checked the growth of 

 demonstrative ilrithmetic in England,' 'Re- 

 forms in arithmetical teaching,' and ' Arithme- 

 tic in the United States,' show forcibly the 

 stagnation which results in regarding it not as 

 a demonstrative science, but merely as an art 

 of calculation. 



The accounts of modern synthetic geometry 

 aud of non-Euclidean geometry (pp. 252-275) 

 seem well chosen. It is necessary for teachers 

 of geometry to have a broader view of their sub- 

 ject than is afforded by the typical text-book. 



Having called attention to some of the merits 

 of Professor Cajori's work, it is unfortunately 

 necessary now to note some of its defects. The 

 inconvenient method of introducing an abbre- 

 viation, the first time a work is cited, to be used 

 for it subsequently, we trust will in future edi- 

 tions be remedied by a table at the end of the 

 volume. It is confusing, if one is not certain of 

 their identity, to have ' Ptolemj' ' and ' Ptole- 

 mseus ' used indiscriminately. In the statement 

 that " 1^2 cannot be exactly represented by any 

 number whatever" (p. 51), the word rational 

 has, of course, inadvertently been omitted. 

 Foot-note 3, p. 72, is very indefinite in its pres- 

 ent form. Referring to remarks at the top of 

 page 74, we quite agree with the author that 

 rigor in geometry demands the proof of the pos- 

 sibility of all constructions before they are used. 

 For example, that the circumference of a circle 

 admits of being divided into any number of 

 equal parts should be shown (which involves 

 no diflBculty) before considering regular in- 

 scribed polygons in general. The example of 

 the text leads one to suppose that rigor demands 

 our ability to construct (subject, in fact, to the 

 arbitrary condition of having only ruler and 

 compass) every inscribed polygon we may wish 

 to use. 



The material of the volume in places shows 

 lack of coordination and incomplete moulding 

 into an organic whole. One feels at times lost 

 in a maze of fact. We are given part of the 



biography of Leonardo of Pisa on page 119 

 and part on page 134. The origin of the word 

 ' sine ' is found on page 124 and again on page 

 130. On page 75 and again on page 78 we are 

 told of the tomb of Archimedes. 



In the foot-note 1, page 160, the conclusion 

 that the base of Napier's logarithms is e~^ is 

 erroneous, and it does not follow from what 

 precedes it. If we define the logarithm of x 

 with respect to the constant base 6, by the equa- 

 tion x = h '"St *, then the numbers discovered by 

 Napier are not logarithms ; but if b is not re- 

 stricted to be constant, the above equation de- 

 fines Napier logarithms when 



(Hagen, Synopsis der hoeheren Mathematik I. , 

 p. 107.) To define the base of Napier's loga- 

 rithms as the number whose logarithm is unity 

 is in this case misleading. The term is, how- 

 ever, so used by Cantor (Geschichte der Mathe- 

 matik, II., p. 672), who gives its value to be 



l\io / 



lo'-^ c^ 



E. M. Blake. 



Pdedue University. 



Die Bedingungen der Fortpflanzung bei einigen 



Algen und Pilzen. Von De. George Klebs. 



Jena, Gustav Fischer. 1896. Pp. i.-}-543, 



3 plates. 



This work of Dr. Klebs' is an important con- 

 tribution to the physiology of reproduction. As 

 its title indicates, the experiments were con- 

 ducted for the purpose of determining the con- 

 ditions of reproduction in certain algae and 

 fungi. A preliminary account of some of this 

 work has been published in earlier contribu- 

 tions. The earlier experiments have been am- 

 plified and extended to a number of additional 

 plants, and the present work details carefully 

 his later experiments and presents the philos- 

 ophy and deductions of all his work upon this 

 topic. It is a remarkable work, alike for the 

 painstaking conduct of the experiments, the 

 precautions against error, the important results 

 obtained and the cautious generalizations upon 

 the relations of the different kinds of reproduc- 

 tion to environment. Not only is the work 

 one of great interest to the student of develop- 



