i86 



NATURE 



[October 15, 1914 



find the moon's place at any date we have then only 

 to insert that date and to perform the necessary 

 numerical calculations. This is not done directly, on 

 account of the labour involved. What are known as 

 "tables of the Moon's Motion" are formed. These 

 tables constitute an intermediate step between the 

 theory and the positions of the moon which are printed 

 in the " Nautical Almanac." Their sole use and 

 necessity is the abbreviation of the work of calcula- 

 tion required to predict the moon's place from the 

 theoretical values which have been found. For this 

 reason, the problem of producing efficient tables is not 

 properly scientific ; it is mainly economic. Neverthe- 

 less, I have found it as interesting and absorbing as 

 any problem which involves masses of calculation is 

 to those who are naturally fond of dealing- with arith- 

 metical work. My chief assistant, Mr. H. B. Hedrick, 

 has employed his valuable experience in helping- me 

 to devise new ways of arranging the tables and making 

 them simple for use. 



A table is mainly a device by which calculations 

 which are continually recurring are performed once 

 for all time, so that those who need to make such 

 calculations can read off the results from the table. 

 In the case of the moon, the tables go in pairs. Each 

 term in the moon's motion depends on an angle, and 

 this angle depends on the date. One table gives the 

 value of the angle at any date (a very little calculation 

 enables the computer to find this), and the second 

 table gives the value of the term for that angle. As 

 the same angles are continually recurring, the second 

 table will serve for all time. 



We can, however, do better than construct one table 

 for each term. The same angle can be made to serve 

 for several terms, and consequently one table may be 

 constructed so as include all of them. In other words, 

 instead of looking out five numbers for five separate 

 terms, the computer looks out one number which gives 

 him the sum of the five terms. The more terms we 

 can put into a single table the less work for the 

 astronomer who wants the place of the moon, and 

 therefore the more efficient the tables. A still better 

 device is a single table which depends on two angles, 

 known as a double-entry table ; many more terms can 

 usually be included in this than in a single-entry 

 table. The double interpolation on each such table is 

 avoided by having one angle the same for many 

 double-entry tables and interpolating for that angle on 

 the sum of the numbers extracted from the tables. 



The problem of fitting the terms into the smallest 

 number of tables is a problem in combinations — some- 

 thing like a mixture of a game at chess and a picture- 

 puzzle, but unlike the latter in the fact that the inten- 

 tion is to produce ease and simplicity instead of diffi- 

 culty. This work of arrangement is now completed, 

 and, in fact, about five-sixths of the calculations neces- 

 sary to form the tables are done ; more than one-third 

 of the copy is ready for the printer, but, owing to the 

 large mass of the matter, it will take from two to 

 three years to put it through the press. The cost of 

 performing the calculations and printing the work has 

 been met from a fund specially set aside for the pur- 

 pose by Yale University. 



A few statistics will perhaps give an idea of our 

 work. Hansen has 300 terms in his three co- 

 ordinates, and these are so grouped that about a 

 hundred tables are used in finding a complete place 

 of the moon. We have included more than looo terms 

 in about 120 tables, so that there are on the average 

 about eight terms per table. (In one of our tables we 

 have been able to include no fewer than forty terms.) 

 Each table is made as extensive as possible in order 

 that the interpolations — the bane of all such calcula- 

 tions — shall be easy. The great majority of them 

 involve multiplications by numbers less than 100. 

 NO. 2346, VOL. 94] 



There are fewer than ten tables which will involve 

 multiplications by numbers between 100 and 1000, and 

 none greater than the latter number. The computer 

 who is set to work to find the longitude, latitude, 

 and parallax of the moon will not need a table of 

 logarithms from the beginning to the end of his work. 

 The reason for this is that all multiplications by three 

 figures or less can be done by Crelle's well-known 

 tables or by a computing machine. But Mr. Hedrick 

 has devised a table for interpolation to three places 

 which is more rapid and easy than either of these 

 aids. It is, of course, of use generally for all such 

 calculations, and arrangements are now being made 

 for the preparation and publication of his tables. The 

 actual work of finding the place of the moon from the 

 new lunar tables will, I believe, not take more time 

 — perhaps less — than from Hansen's tables, so soon 

 as the computer has made himself familiar with 

 them. Fortunately for him, it is not necessary to 

 understand the details of their construction ; he need 

 only know the rules for using them. 



I am now going to show by means of some dia- 

 grams the deviations of the moon from its theoretical 

 orbit, in which, of course, errors of observation are 

 included. The first two slides exhibit the average 

 deviation of the inoon from its computed place for the 

 past century and a half in longitude.^ The averages 

 are taken over periods of 414 days and each point of 

 the continuous line shows one such average. The 

 dots are the results obtained by Newcomb from 

 occultations ; the averages for the first century are 

 taken over periods of several years, and in the last 

 sixty years over every year. In both cases the same 

 theory and the same values of the constants have 

 been used. Only one empirical term has been taken 

 out — the long-period fluctuation found by Newcomb 

 having a period of 270 years and a coefficient of 13'. 

 I shall show the deviations with this term included, in 

 a moment. 



The first point to which attention should be directed 

 is the agreement of the results deduced from the 

 Greenwich meridian observations and those deduced 

 from occultations gathered from observatories all over 

 the world. There can be no doubt that the fluctua- 

 tions are real and not due to errors of observation. 

 A considerable difference appears about 1820, for 

 which I have not been able to account, but I have 

 reasons for thinking that the difference is mainly due 

 to errors in the occultations rather than in the meri- 

 dian values. In the last sixty years the differences 

 become comparatively small, and the character of the 

 deviation of the moon from its theoretical orbit is well 

 marked. This deviation is obviously of a periodic 

 character, but attempts to analyse it into one or two 

 periodic terms have not met with success ; the number 

 of terms required for the purpose is too great to allow 

 one to feel that they have have a real existence, and 

 that they would combine to represent the motion in 

 the future. The straight line character of the devia- 

 tions is a rather marked peculiarity of the curves. 



The actual deviations on a smaller scale are shown 

 in the next slide ; the great empirical term has here 

 been restored and is shown by a broken line. The 

 continuous line represents the Greenwich meridian 

 observations; the dots are Newcomb 's results for the 

 occultations before 1750, the date at which the meri- 

 dian observations begin. With a very slight amount 

 of smoothing, especially since 1850, this diagram may 

 be considered to show the actual deviations of the 

 moon from its theoretical orbit. 



The next slide shows the average values of the 

 eccentricity and of the position of the perigee.^ The 



1 Monthly Notices, R.A.S., vol. Ixxiii.. plate 22. . . r u 



2 Tables II. and III. of a paper on " The Perigee and Eccentricity of the 

 Moon," Monthly Notices, R.A.S., March, 1914. 



