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SCIENCE. 



[Vol. IV., No. 85 



Professor Mansfield Merriman, the author of the 

 well-known treatise on 'Least squares,' proposed a 

 criterion for the rejection of doubtful observations, 

 founded upon Hagen's demonstration of the law of 

 frequency of error, which was simpler than Pierce's 

 or Chauvenet's. It involves, however, a determina- 

 tion of what is the unit of increment between errors 

 of different sizes, a thing difficult to determine in 

 very many cases. Professor Harkness, of the Naval 

 observatory, thought that in the case of a criterion for 

 the rejection of doubtful observations, — upon which 

 the most eminent mathematicians disagreed, — prac- 

 tically every one was a law unto himself. He noted 

 the rather doubtful method of taking a large number 

 of shots on a target-board as a good illustration of 

 the law of frequency of error, especially in any such 

 case as that of long-distance shooting, where, on ac- 

 count of the varying character of the wind, the skil- 

 ful marksman will frequently change his rifle-sights 

 an amount corresponding to twenty or thirty feet on 

 the target, and yet make a complete series of bull's- 

 eyes, or very close to it. Professor Rogers, of the 

 Harvard college observatory, expressed his disbelief 

 in the efficacy of least squares to tell the truth, illus- 

 trating it by several cases. For rejecting discordant 

 observations, he referred to the late Professor Win- 

 lock's method of determining the personal habit and 

 accuracy of each observer as a means of getting an 

 empirical criterion. He closed with an expression of 

 the opinion that the method of least squares was a 

 method of ' covering a multitude of sins.' Professor 

 Pickering said that we must have some criterion, and 

 every one would practically use one of some kind. 

 He referred to his plan of using ' average deviation ' 

 as easier to compute than ' probable error,' and con- 

 sidered five times the average deviation a good limit 

 for the rejection of discordant results. Professor 

 Stone, of the University of Virginia, referred to the 

 very common case of only three, four, or five observa- 

 tions of a star, where the data are not sufficient to 

 apply any criterion, and to the advisability, when it 

 was possible, of making more observations to settle 

 the question. Another speaker referred to the im- 

 portance of a special search for systematic abnormal 

 errors. Professor Rogers referred to the uncertainty 

 of trusting to the impressions upon one's senses, and 

 said that in nine cases out of ten, where he thought 

 he had observed a transit over a particular wire too 

 early or too late, it would come out just the other 

 way. Professor Hough, director of the Dearborn ob- 

 servatory, thought an observer generally incapable of 

 judging or weighting his observations according to 

 his impressions. In the case of uncertain conditions, 

 like an unsteady atmosphere, he thought it best to 

 quit work and wait for better. Professor Frisby, of 

 the Naval observatory, emphasized the danger of re- 

 jecting observations, or forming any arbitrary limit 

 for this purpose. • Professor Langley, director of the 

 Allegheny observatory, hoped that further experience 

 would be given upon this question of trusting one's 

 own impressions in rejecting or weighting observa- 

 tions, as it was an exceedingly interesting and im- 

 portant one. Professor Merriman, the author of the 



paper, referred to the importance of eliminating all 

 sources of systematic errors so far as possible, and of 

 separation into groups, for separate discussion, in 

 order to discover such errors. Professor Rogers re- 

 ferred to the various values of the solar parallax 

 which had been deduced in one way or another by 

 least squares, and another speaker referred to the 

 hidden sources of error which least squares could not 

 deal with. Professor Paul, assistant astronomer at 

 the Naval observatory, said the method of least 

 squares was hardly receiving fair treatment in the 

 discussion, and thought the difficulty was that half or 

 three-quarters of those who used the method failed to 

 bear in mind the theory on which it rested: that it 

 only applied to purely accidental errors; whereas in 

 * more than half the cases it is actually applied to 

 errors distributed round a point which is continually 

 moving or jumping, clue to systematic sources of 

 error or sudden disturbance, and that no attempt is 

 made to discover and eliminate these systematic or 

 sudden-jumping errors, but least squares is applied 

 indiscriminately to the whole, with a sort of blind 

 faith that it will bring good results out of poor ob- 

 servations, and make it all right somehow. He said 

 that, intelligently applied, the method not only gave 

 the most probable result, but furnished the only 

 measure of the exactness of the observations so far 

 as accidental errors were concerned, and at the same 

 time the most effective method of discovering these 

 hidden sources of systematic error. Professor Stone 

 illustrated this by the case of combining many series 

 of comet-observations, made at different observa- 

 tories, into one orbit, without attempting to discover 

 any systematic errors in the series of the different 

 observers. The discussion was closed by Professor 

 Eddy with remarks upon the necessity of some cri- 

 terion dependent upon the results themselves, and 

 independent of the observer's arbitrary judgment. 



Professor Pickering then read another paper upon 

 systematic errors in stellar magnitudes, showing, 

 without any question, that the magnitudes of all 

 the star-catalogues from that of Ptolemy down to the 

 great work of Argelander in the Durchmusterung — 

 all depending upon eye estimates — are systematically 

 affected by being in, or close to, the Milky Way; they 

 all being estimated too faint, and the error amount- 

 ing to about half a magnitude in the Milky Way 

 itself. This arises from the brightness of the back- 

 ground upon which the star is viewed. In the Har- 

 vard photometry measures, this source of error is 

 avoided; since, in the comparison of each star with 

 the pole-star, the two fields are superposed, and their 

 added brightness affects both stars alike. 



Prof. M. W. Harrington, director of the Ann Arbor 

 observatory, read a paper upon the asteroid ring. 

 He showed that the representative average orbit 

 would be an ellipse of small eccentricity, with semi- 

 major axis equal to about 2.7 times that of the earth, 

 and inclined to the plane of the ecliptic about 1°; 

 and that, in the progressive discovery of these small 

 bodies, the average mean distance had gradually in- 

 creased, but now seemed to have reached its limit. 

 On the assumption that the surfaces of all the aster- 



