May 4, 191 1] 



NATURE 



;o7 



ORDERS OF INFINITY. 

 Orders of Infinity: the "' InfinitdtcalciiV of Paul du 

 Bois-Reymond. By G. H. Hardy, F.R.S. Pp. iv4- 

 62. (Cambridge : University Press, 19 lo.) Price 

 2S. 6d. net. 



THE subject of this tract has been hitherto inac- 

 cessible to English readers, and it is not alto- 

 gether easy to give a brief account of its contents. 

 Perhaps the simplest method is to start from the 

 familiar facts that log x and e^ both tend to infinity 

 with A,-, but in very different ways, namely, (log x)/x^ 

 tends to zero, however small 6 may be, and e^jx^ 

 tends to infinity, however large A may be (it is under- 

 stood that 5, A do not vary with x). These results 

 would be expressed in du Bois-Reymond 's notation by 

 the symbols 



log j; ^ JT* and e^ ^ a "^ ; 



in addition to these two symbols, du Bois-Reymond 

 used another to imply that the ratio of two functions 

 /, <fi lies between finite limits (when x tends to in- 

 finity). . But later writers have found it convenient to 

 make the notation rather more precise, and to write 



to imply the relation just mentioned, and further to 

 use /~</> to imply that lim {ff4>) = i; there are also 

 other sub-cases for which reference must be made to 

 Mr. Hardy's tract. 



The ideas mentioned above lead very naturally to 

 the logarithmic scale of infinity, represented by the 

 sequence of functions 



. . ., /^x, l<yX, lyX, X, e^x, e^v, ^^x, . . . 

 where 



and 



/,.:r=log (/n-i-r), /^x^log x, 



enX= (en-ix), e^x=e'. 

 This scale has the property that any element tends 

 to infinity more slowly than any positive power 5 of 

 the following element, and more rapidly than any 

 positive power A of the preceding^ element ; and it 

 is possible to utilise the scale to classify all ordinarv 

 functions of analysis. Mr. Hardy has considered (pp. 

 16-36) the question as to what more or less arti- 

 ficial functions do and do not fall into place in the 

 scale; we must content ourselves with mentioning 

 the comparatively simple function "^x^jv ! obtained by 

 selecting certain terms from the exponential 

 series ; this does fit into the scale if v 

 takes the values i, 4, 9 . . . {v=n^) but does not if v 

 has the values i, 8, 27, . . . {v = n% nor if v=i, 2, 4, 8, 



. . . (i' = 2»). 



In Appendix ii. (pp. 48-57) Mr. Hardy has made a 

 most interesting summary of recent results in analysis, 

 in which du Bois-Reymond's ideas have proved help- 

 ful, and readers who are less interested in logic than 

 in results may be advised to turn to this appendix 

 first. Appendix iii. gives a large variety of numerical 

 results to emphasise the amazing rapidity of increase 

 of the logarithmic scale at the upper end, and its 

 corresponding slowness of increase at the lower end. 



We may, perhaps, quote the largest number 

 suggested by physical considerations, namely, the 

 NO. 2166, VOL. 86] 



number of molecules in the earth, which is found 

 to be of the order 



I'oSx lo'i or 42 ! or i?.,(477). 

 This is considerably larger than the number of 

 sodium wave-lengths which light could traverse in 

 geological time, a number of the order 30 ! or e,(4'32). 

 Both of these numbers are quite small when expressed 

 in terms of second order exponentials, and are far 

 smaller than o^- the largest number expressible in 

 terms of three digits ; this last number contains 

 369,693,100 digits when written at full leng-th, and 

 (printed with 16 digits to the inch) would cover more 

 than 350 miles. T. J. I'a. B. 



MAYA ASTRONOMY. 

 The Numeration, Calendar Systems, and .istronomiccd 



Knowledge of the Mayas. By C. P. Bowditch. 



Pp. xviii + 346 + xix plates. (Cambridge : Univer- 

 sity Press, 1910. Privately printed.) 

 THIS volume offers to the reader "a statement of 

 the knowledge which we possess of the numera- 

 tion, calendar, and astronomical attainments of [a] 

 wonderful people." 



The sources of information are primarily the records 

 of the Mayas themselves; secondarily, the writings of 

 Spaniards and others about the Mayas. The first 

 source may be subdivided into : — 



(i) The books of Chilan Balam. 



(ii) The codices. 



(iii) Inscriptions'. 



The Book of Chilan Balam of Mani (Mani being 

 the name of a village added to the title for the pur- 

 poses of identification) was composed before 1595. 

 All books of this class were copies of older manu- 

 scripts, with occasional addenda of current interest. 

 They were regarded with superstitious veneration by 

 the village to which they belonged. 



The codices, of which three are extant, contain 

 accounts of Mayan histories, ceremonies, sacrifices, 

 and calendar. 



Bishop Landa, at Mani, in 1562, carried out so far 

 as possible a "general destruction of everything which 

 related to the ancient life of the nation." He was a 

 Franciscan friar, and subsequently became Bishop of 

 Yucatan. He was a .sincere friend and protector to 

 the natives ; he has preserved the Maya alphabet, 

 and with it the key to the inscriptions, a service 

 which "wipes out over and over again his faults* 

 which were those of the century." 



The Mayas used a series of t\\<ni\ day-names in 

 an invariable order, Kan, Chiccli.iii, \; ., the first fol- 

 lowing the last without a break, just as the days of 

 the week do with us. The period of twenty days is 

 referred to as a month. 



In addition to this, a device was used of counting 

 up to thirteen, and then beginning- again, so that the 

 complete cycle becomes 260 days ; just as we should 

 have a complete cycle of 210 days il every month 

 contained thirty days, and if it were usual to name 

 only the day of the week and the day of the month 

 without naming the month. The series of 260 days 

 is called Tonalamath. 



