February 8, 1918] 



SCIENCE 



145 



sent the table, as in the expression " Thogh 

 Argus (that is, al-Khowarizmi) the noble 

 covnter Sete to rekene in hys counter." The 

 story is connected with Fitz-Neal's " Dvalogua 

 de Scaccario " of 1178 and the court of the 

 exchequer, with backgammon, and with divers 

 other ramifications. The counter went by 

 various names, such as Rechenpfennig, Zahl- 

 pfeunig, and Raitpfennig in Germany; pro- 

 jectiles and abaculi as well as calculi in 

 Latin; jetons, gectz, getoirs, giets, and the 

 like in France; Leggelt and Werpgeld in Hol- 

 land; and jettons and Venetian money as 

 well as counters in England. 



It is the field of counters that Professor 

 Barnard has made his own in the monu- 

 mental and sumptuous work under review. 

 For many years he has been collecting spec- 

 imens of counters of the various European 

 countries. He has examined upwards of 40.- 

 000 specimens and has in his own cabinet 

 some 7,000, most of those described being in 

 this collection. With great care he has select- 

 ed typical specimens, the choice being deter- 

 mined by their historical importance, artistic 

 merit and general value. Fifty-nine Early 

 English jettons are described, nineteen Italian, 

 three hundred seventy-two French, ninety 

 Low Country, one hundred twenty-two Ger- 

 man, and four Portuguese. Each specimen 

 is illustrated photographically and each is 

 described with all the care of a trained nu- 

 mismatist. Nothing could be more satisfac- 

 tory to the collector or the student, and it 

 would be difficult to suggest a single partic- 

 ular in which the descriptive material could 

 be improved. 



To the historian of mathematics one of the 

 items of greatest value is the set of photo- 

 graphs of reckoning tables at Basle and 

 Niirnberg, of reckoning cloths at Munich, of 

 illustrations from manuscripts and early 

 printed books, and of Dutch jetton cylinders, 

 together with a description of each. There 

 is also a valuable list of one hundred fifty- 

 nine extracts from English inventories with 

 references to counting boards, thus showing 

 that these devices were common from the 

 fourteenth to the sixteenth century at least. 



The first two of these extracts, of date 1321 

 and 1337, respectively, make mention of 

 " Camera qusedam cum mensa quadrata ad 

 calculandum " and of " Unum computator- 

 iimi," while other interesting items refer to 

 *' the cheker," " counting-bord,'' " uniim 

 scaccarium," " cowntynge horde or table," 

 " accomptyng horde," " unam mensam vo- 

 catam a coimter," and " the counterborde in 

 the Hall." 



Perhaps the most important part of the 

 work from the standpoint of the historian is 

 that on " the methods of casting with jettons " 

 (pages 254-319). Here Professor Barnard 

 has given a very satisfactory summary of the 

 more important European works on the sub- 

 ject, such as those by Recorde, Awdeley, 

 Reisch, Cusanus, Siliceus, Kobel, de Moya, 

 and Trenchant. There should also be men- 

 tioned as of great value to students the 

 bibliography of upwards of six hundred titles. 



Taken as a whole the work may be safely 

 characterized in superlatives. Such an elab- 

 orate treatise on any special field of the his- 

 tory of mathematics has never before ap- 

 peared, nor are we likely soon to see another. 

 The infinite pains taken by the author in his 

 research, the munificence shown in the publi- 

 cation itself, and the fact that a mass of 

 technical material is presented in a style that 

 makes every page readable, all combine to 

 render the work unique in its way. No 

 library of reference can afford to be without 

 the book, and students of the history of mathe- 

 matics should add it to their personal li- 

 braries as soon as they can arrange to do so. 

 It will be the classic upon the subject for 

 generations to come. 



David Eugene Smith 



The Nature of Solution. By H.\rry C. Jones. 



New York, D. Van Nostrand Co. 1917. 



23 x 15 cm. ; pp. xxiv + 380. 



The present work is not a text-book, but a gen- 

 eral discussion of some of the more important 

 properties of solutions, true and colloidal. It is 

 therefore written in a non-mathematical, indeed, 

 largely in a semi-popular style. It is hoped that 

 this work may interest students of the various 

 branches of science to go on into the real physical 



