November 25, 1C09] 



NATURE 



95 



as yet absolutely certain of the developmental origin 

 of many structures in the body, and further research 

 may clear up some of the apparent discrepancies now 

 incidental to the embryogenetic classification. The 

 histogenetic classification itself is not altogether free 

 from the reproach levelled at the embryogenetic one 

 by Prof. Adami. 



As regards the genesis of new growths, the various 

 hypotheses are discussed at considerable length, and 

 an admirable survey of this vexed and complex ques- 

 tion is presented to the reader. Prof. Adami con- 

 siders that no parasitic hypothesis suffices, that 

 Beard's hypothesis of aberrant and misplaced germs 

 and trophoblastic cells (so much in evidence lately in 

 connection with a certain form of treatment) is in- 

 adequate, seeks for an explanation in the hypothesis 

 of a change (? a mutation) in the biological properties 

 of the cells giving origin to tumours, and considers 

 that there is no one specific cause; with all these we 

 cordially agree. 



The concluding portion of the book deals with the 

 regressive changes, the degenerations and infiltra- 

 tions, calcification, pigmentation, &c. The book 

 altogether is an inspiring one, and the careful reader 

 will not only gather what is already known, but will 

 be led to infer in what directions further progress lies. 

 A notable feature of it is the attempt made, usually 

 successfully, to ensure a basis on a sound foundation 

 of general biology. It is carefully and adequately 

 illustrated, and the numerous diagrams and schemata 

 serve to render many of the more abstruse conceptions 

 clear and intelligible. 



A NEW WAY IN ARITHMETIC. 

 Theorie der algchraischen Zahlen. By Dr. Kurt 

 Hensel. Erster Band. Pp. xii + 3So. (Berlin and 

 Leipzig: Teubner, 1908.) Price 14 marks. 

 T X this volume Dr. Hensel gives the first instalment 

 -*- of a treatise on algebraic numbers, embodving an 

 independent method on which he has been engaged 

 for the last eighteen years. Its leading idea may be 

 illustrated by the following example. Let us take the 

 solvable congruence, x- = 2 (mod. 7), the roots of which 

 are .^ = 3 and x = 4. The same congruence can be 

 solved with respect to the moduli y-, y', 7^ &c., and 

 we obtain the solutions, in least positive residues, (3, 

 4). (10, 39), (108, 235), (2 1 16, 285), and so on. Tak- 

 ing the first number in each bracket and expressing it 

 in the septenary scale, only writing the digits in the 

 reverse of the usual order, we obtain the associated 

 solutions, 3, 31, 312, 3126; and it is clear that if 

 x = a^a, . . . a„ is a solution of x-~z (mod. 7"), we 

 can find a number a^a^ . . . (?„<?„ + ,, which is a solu- 

 tion of x-=2 (mod. 7" + '). There is thus a definite 

 sequence of digits, 3, i, 2, 6, . . . a„, . . . such that 

 each is a least positive residue of 7 (or zero), and such 

 that 3126 . . . a„ is a solution of x-~<! (mod. y"}. 

 This sequence may be said to be the symbolical sep- 

 tenary representation of ^/ 2. But conversely we may 

 take any such sequence, a^a^ . . . a„ . . . and define 

 it as a septenary number, in an extended sense. All 

 NO. 2091, VOL. 82] 



such numbers form a corpus, provided we introduce 

 septenary fractions of the same type. Since 

 -i=(7"-i) (mod. 7"), the symbolical form of — i is 

 666 ... or 6; hence every ordinary positive or nega- 

 tive integer or fraction has a symbolic expres- 

 sion which is wholly or partly periodic, e.g. 

 -/3 = (3 + 6)/3 = 32, and so on. Similar results hold 

 for any prime modulus ; when the modulus is com- 

 posite, some curious anomalies occur. 



Now let U'j,'m,, . . . -ji„ be a basis of an algebraic 

 corpus; we may form symbols of the type 

 AiTOj+A2TO„+ . . . +A„7i'„, where Aj.A^, ... A,. 

 are numbers of the kind just described. These new 

 symbols may be called " numbers," and by making 

 use of them Dr. Hensel obtains all the most important 

 known properties of algebraic numbers with surpris- 

 ing facility; he also adds results of his own which 

 are of great interest and beauty. Calling a symbol 

 such as A, a ^-adic number, we may call 

 F(.v) = .'^j.r"+ . . . +\", a /)-adic function; it is 

 shown how to determine, by a finite process, the irre- 

 ducible ^-adic factors of F(x), and by a series of pro- 

 positions we are led up to the remarkable theorem 

 (P-_ 159) that if f{x)=.i^ + a^x>'-'+ . . . +a^, the co- 

 efficients being integral p-adic numbers, and f{x) irre- 

 ducible, then if />« is the highest power of p which 

 divides the discriminant of /(.\), and if a^ is a root of 

 the ordinary equation 



<t>(x)=o 



obtained from f{x) by omitting all the digits of 

 rt„a^, . . . a\ beyond the (5— i)th place, the equation 

 /(.r) = o will have precisely K conjugate roots 

 $i< $-2- ■ ■ ■ ^A expressible as conjugate /-adic numbers 

 m the corpora (a^), (a^) . . . (a,v). This fundamental 

 fact leads to a host of consequences, among 

 them a comparatively simple treatment of a well- 

 known problem, namely, the resolution into their 

 prime ideal factors of the real primes which divide 

 the discriminant of a given corpus. It also leads to 

 a complete theory of congruential roots of unity ; the 

 theory of units in a given corpus is not discussed in 

 this volume. 



On pp. 308 and following will be found a complete 

 solution of the problem of resolving a given real prime 

 into its ideal factors within a given corpus ; this 

 involves the Kronecker method, in which umbrae are 

 used, and probably there is no certain practical way 

 which can dispense with them. As an illustration, it 

 is shown that in the corpus defined by the equation 

 1^ — 0- — 2a-8 = o, the number 2 is the product of three 

 ideal primes, which are actually determined. 



One of the last theorems proved in this volume may 

 be stated in the following terms : — If a corpus is 

 defined by an equation /(.t) = o, which is not Galoisian 

 in the field of ordinary numbers, we cannot make the 

 field Galoisian by the introduction of /i-adic numbers. 



The value of the treatise can hardly be overrated, 

 and its completion will be anxiously expected. It is 

 interesting to compare it with Hensel and Landsberg's 

 treatise on algebraic functions, and observe the points 

 of contact. A special feature is that in the arith- 

 metical work, like the other, there are expansions in 



