74 



NATURE 



[November 24, iJ 



solution can be found, »>. 5, ii = i ii +2 1 1 +2,'ii = (by 

 the table) 1 1 1 +2 '6 + 2/66= (by their table of halves) 

 1/11 + 13+133. 



On the right half of Plate xvi. is another somewhat 

 similar but very much shorter table in a single column, 

 giving some subdivisions of fractions which might prove 

 useful in calculations, the results of course being ex- 

 pressed in divisions of unity, the table reading — 



2/jof2/3=l/3 +1/9 

 I '3 „ 2/3=1/6 +1/18 

 2/3 „ 1/3=1/6 +1/18 

 2/3 .. 1/6 = 1/12+1/36 

 2/3 ,, 1/2=1/3 

 and so on. To the left of the table is a rule for finding 

 2/3 of a fraction, which reads as follows — 



"To t^ke 2 3 of a fraction. When it is said to thee, 

 'What is 2 3 of I 5 ?' make thou its double and its six 

 times. That is its 2 3. Thus is it to be done for every 

 fraction that occurs." 



The scribe evidently meant 



2/3 ^ '/S = 1/5x2 + 1/5 X 6= 1/10+ 1/30 

 This general rule is exceedingly interesting, as it is the 

 only one that occurs on the papyrus. In fact, the rest 

 of the purely arithmetical part of the papyrus (Plates 

 vii.-xii.) is thrown into the form of problems which are 

 worked out with the object of being of practical use. 



We have not space to do more than indicate the 

 principal contents of the rest of the papyrus. Parts ii. 

 and iii. (Plates xii.-xiv.) deal with the measurements of 

 volume and area, in which given problems are worked 

 out in estimating the amount of grain that can be stored 

 in cylindrical and rectangular spaces of given proportions ; 

 in others the scribe has worked out the superficial area 

 of fields of various shapes, the linear measurements of 

 which are given ; while diagrams are in some cases 

 drawn of the shape of the fields. In this section of the 

 work Dr. Budge remarks that the scribe has made a good 

 many mistakes. Part iv. (Plate xv.) deals with the 

 measurements of pyramids, of which six examples, with 

 five figures, are given. The first on the page has the 

 angle of the lower half of the southern stone pyramid at 

 Dahshur, four have the same angle as that of the second 

 pyramid at Cllzeh, while the angle of the last is that of 

 the pyramid at Medimi. Part v. (Plates xvi.-xx.) gives a 

 worked out series of practical problems dealing with the 

 reckoning of farm produce, the division of food among 

 workmen, the cost of food for birds and animals, &c. 

 They are all questions of practical interest, worked out, 

 more or less, by rule of thumb. As Dr. Budge remarks 

 in his Introduction to the Papyrus : — 



"None of the examples or problems indicate that the 

 E^'yptians had any deep theoretical knowledge of 

 arithmetic or geometry, but all of them show that they 

 were very ready in making practical calculations." 



Since the late Dr. Samuel Birch published an account 

 of the contents of this document in the Zeitschrift fiir 

 Aegyptische Sprache in 1868, the Rhind Papyrus has 

 formed the subject of much discussion among both 

 mathematicians and Egyptologists. A large body of 

 students will, therefore, be grateful to the Trustees of the 

 British Museum for placing in their hands the actual text 

 of the papyrus in the form of a facsimile, which in beauty 

 and accuracy of reproduction leaves nothing to be 

 desired. 



NO. 1517, VOL. 59] 



NEW ACCOUNT OF THE CHARACE.E OF 



EUROPE. 



Synopsis Characcarum europcartim . By Dr. Walter 

 Migula. Pp. 176. Illustrated by woodcuts. (Leipzig 

 1898.) 



THIS is an abridgment of the author's large anci 

 elaborate work on the subject, which forms vol. v. 

 of the new edition of " Rabenhorst's Kryptogamen-Flor,-» 

 von Deutschland, Oesterreich und der Schweiz." The 

 latter is by far the most extensive and minute account 

 of the European Characeas which has yet appeared ; but 

 the present work, containing as it does almost all the 

 illustrations, and most of the letter-press information in a 

 condensed form, will, we think, be found the more useful 

 for all ordinary purposes. 



The first eighteen pages are taken up with an account 

 of the structure of the Characea.', with illustrations, and 

 achapter on collecting, examining, and preserving speci- 

 mens. These are followed by a key to the genera and 

 species. At the end there is a list of books and memoirs 

 dealing with the Characea^, also one of published sets of 

 specimens. 



The remainder of the book consists of a tolerably 

 complete description of the genera and species, with 

 figures of nearly all the latter. The arrangement fol- 

 lowed is almost entirely that of Braun's " Fragmente " ; 

 but the anomalous species Lyc/mot/iamnus stc/liger of 

 Braun is elevated to a separate genus, Tolype/lopsis, 

 Migula {C/iara, sect. Tolypellopsis, Leonh.), and Cliara 

 rudis, C. horrida, C. crassicaulis, and C. delicatula, 

 which Brai^ regarded as sub-species, are treated as 

 species. The author has gone to extreme lengths in the 

 diflferentiation of varieties. For instance, C. aspcra is 

 credited with thirty-nine named forms, Cfragilis wjth 

 thirty-seven, C. oinita, C. cera/op/iylln, C. intermedia 

 and C. contraria have each more than twenty ; while 

 C.fcetida (excluding C. crassicaulis) has actually sixty- 

 nine ! The majority of these forms are of the author's 

 own describing. From the larger book we learn that 

 many of them are based on plants from single localities, 

 and, judging by the citations, it is fair to suppose in some 

 instances from individual specimens. Considering the 

 extreme variation to which the vegetative parts of these 

 plants are subject, it seems to us a great pity to multiply 

 names by distinguishing every trivial form. 



In the nomenclature of the species Dr. Migula has I 

 implicitly followed Braun, in several instances quite dis- 

 regarding the accepted rules. It would, we think, have 

 been as well to have mentioned some of the more gener- 

 ally known synonyms. 



The book is, unfortunately, entirely in German. As it 

 deals with the Charace;o of the whole of Europe, it would 

 have been more widely appreciated had the key and the 

 diagnoses of the genera and species, at least, been in 

 Latin. 



As we have remarked, the illustrations are the same as 

 those in the larger work. They are on the whole decidedly 

 good, especially those of the magnified portions of the 

 plants, and several of the species are more completely 

 figured than in any earlier book. Some of the represent- 

 ations of the entire plants are somewhat crude and 

 inartistic, and th.it of Nitclla transluans is very poor ; 



