216 JOURNAL OF THE, WASHINGTON ACADEMY OF SCIENCES VOL. 13, No. 10 
It is evident that if the solutions of these two angles are tabulated cor- 
responding to all the usual values of each of the three sides of the triangle, 
there will be no need for any calculation, and, hence, that a simplification 
is afforded in obtaining the data for drawing the Sumner line by the new 
tables of simultaneous hour angles and azimuths in which are given the 
values of the hour angle and azimuth that a celestial body would have at 
stated true altitudes above the horizon of an observer situated in successive 
degrees of latitude ranging from 60° north to 60° south of the equator. 
Mr. C. V. Hopason presented a paper on Precise measurement of dis- 
tances on the earth. The paper was illustrated by lantern slides and was 
discussed by Messrs. Wm. Bow1z, Rupr, Pawnine, Tuckerman, Hum- 
PHREYS, HAWKESWORTH, FERNER, and GISsH. 
Author’s abstract: The first part of the paper was a brief résumé of the 
earlier methods of precise distance measurements with reference to the 
approximate accuracy obtained with each. Lantern slides were shown of 
the contact bar, duplex bar, wire bar and tapes, with references to the prin- 
cipal mechanical and theoretical features of each, and their advantages and 
disadvantages. 
A recent extension of the principles of precise base measurement to pre- 
cise traverse was then discussed. About 3000 miles of precise traverse has 
been executed in the United States by the Coast and Geodetic Survey in 
regions where triangulation would be very expensive. The accuracy of 
this traverse, as measured by closure in position, is usually from 1—70,000 
to 1-100,000. The cost will range from $38.00 to $60.00 per mile, and the 
speed with the single party from 60 to 120 miles per month. 
As an indication of the accuracy which may be obtained with invar tapes, 
reference was made to a precise base being measured in California where 
the accuracy desired is represented by an actual error of 1 part in 1,000,000 
with a probable error of about 1 part in 4,000,000. Such accuracy necessi- 
tates unusual refinement in methods, some of which consist of special stand- 
ardizations of tapes and bars, corrections for the temperature of the springs 
and the balances, investigations as to the amount of error resulting from 
slight errors in tension, temperature and inclination of tapes, and special 
field methods to limit such errors. 
Mr. R. W. G. Wyckorr presented two papers, (1) on Atomic radii, 
and (2) on Crystal structure of the alums. The papers were illustrated by 
lantern slides and were discussed by Messrs. TucKERMAN, WILLIAMSON, 
SosMANn, and GIsH. 
Author’s abstract: A comparative study has been made of the best available 
crystal structure data to see whether they are in agreement with the hypoth- 
esis of constant atomic radii advanced two years ago by W. L. Bragg. Ac- 
cording to this hypothesis if a definite size is assigned to the atoms of the 
various elements, crystals can be built up by packing together these atomic 
spheres. The existing data show clearly that this idea is not satisfied by 
the better information now at hand. They do, however, conform quite 
exactly to the rule that in isomorphous crystals composed of only two kinds 
of atoms the interatomic distances have additive properties which can be 
illustrated through a summing up of “atomic radii.”” The rule often holds 
approximately between the most electropositive and electronegative atoms 
of a crystal but deviates widely in its application to the other atomic separa- 
tions. In crystals of different sorts the effective volume of an atom depends 
both upon the nature of the other atoms with which it is associated and the 
manner of their distribution about it. These data seem to show furthermore 
