314 TRANSACTIONS OF SECTION A. 
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 interesting and absorbing as any problem which involves masses of calcula- 
tion is to those who are naturally fond of dealing with arithmetical work. 
My chief assistant, Mr. H. B. Hedrick, has employed his valuable experi- 
ence 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 to 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—something like a mixture of a game at chess and a 
picture-puzzle, but unlike the latter in the fact that the intention is to produce 
ease and simplicity instead of difficulty. This work of arrangement is now 
completed and, in fact, about five-sixths of the calculations necessary to form 
the tables are done; over 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 
purpose 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 over 1,000 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 less than forty terms.] Each table is made as extensive as possible 
in order that the interpolations—the bane of all such calculations—shall be 
easy. The great majority of them involve multiplications by numbers less 
than 100. There are less than ten tables which will involve multiplications by 
numbers between 100 and 1,000 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, as 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 diagrams the deviations of the 
