392 



SCIENCE 



[N. S. Vol. XL. No. 1029 



observations. In this way a mueli closer 

 agreement between theory and observation 

 has been obtained, and the initial position 

 and velocity of the moon at a given date 

 are now known with an accuracy compara- 

 ble with that of the theory. I shall shortly 

 return to this problem and exhibit this 

 degree of accuracy by means of some dia- 

 grams which will be thrown on the screen. 



I have spoken of the determination of 

 these initial values as if it constituted a 

 problem separate from the theory. Theo- 

 retically it is so, but practically the two 

 must go together. The increase in accuracy 

 of the theory has gone on successively with 

 increase in accuracy of the determination 

 of these constants. "We do not find, with a 

 new theory, the new constants from the 

 start, but corrections to the previously 

 adopted values of these constants. In fact, 

 all the problems of which I am talking are 

 so much interrelated that it is only justi- 

 fiable to separate them for the purposes of 

 exposition. 



Let us suppose that the theory and these 

 constants have been found in numerical 

 form, so that the position of the moon is 

 shown by means of expressions which con- 

 tain nothing unknown but the time. To 

 find the moon's place at any date we have 

 then only to insert that date and to per- 

 form the necessary numerical calculations. 

 This is not done directly, on account of the 

 labor 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 "Nau- 

 tical Almanac. ' ' Their sole use and neces- 

 sity is the abbreviation of the work of 

 calculation 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. 



Nevertheless, I have found it as interest- 

 ing and absorbing as any problem which 

 involves masses of calculation is to those 

 who are naturally fond of dealing with 

 arithmetical 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 cal- 

 culations 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 continu- 

 ally recurring, the second table will serve 

 for all time. 



"We can, however, do better than con- 

 struct one table for each term. The same 

 angle can be made to serve for several terms 

 and consequently one table may be con- 

 structed so as to include all of them. In 

 other words, instead of looking out five 

 numbers for five separate terms, the com- 

 puter 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 de- 

 vice 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 



