January 13, 1922] 



SCIENCE 



31 



will gradually condense and become more and 

 more conspicuous as an illustration of what 

 can be predicted from precise data. 



Photographic parallax determinations seem 

 to be relegating all previous results to the 

 discard, but parallax observers might as well 

 hurry and get these results while thej' are still 

 valuable, as the spectroscopic method though 

 at present dependent upon the trigonometric 

 results for a basis need not always remain so, 

 and the mere possibility of interferometer 

 measures of parallaxes should be enough to 

 dampen one's enthusiasm for undertaking too 

 large a program of safe and sane trigono- 

 metric determinations. 



One of the striking differences between ob- 

 servers and experimenters is their use of the 

 method of least squares. I have heard a young 

 physicist state that he had been advised against 

 taking a course in least squares, because he 

 would never have occasion to use that subject 

 in physics. The answer is that both he and 

 his adviser have probably used the method a 

 great deal, without being aware of it. Experi- 

 menters as a rule do not repeat measures 

 enough to get many residuals — one astronomer 

 has said that he wants at least fifty observa- 

 tions to determine a reliable probable error — 

 but the method of least squares is by no means 

 as limited in its usefulness as might be 

 imagined. It is striking in how many fields 

 of exact science the discussion of measurements 

 takes the directions of a graphical exhibition of 

 the results. The experimenter gets some 

 measures which he puts on a graph exhibiting, 

 say, the dependence of one variable upon an- 

 other. Through a series of plotted points he 

 proceeds to draw a curve; but how does he 

 draw this curve Just what does he try to do 

 when he makes a smooth line pass through a 

 series of points Even for the simplest case of 

 a straight line if you ask a student what he 

 does, he may say that he tries to draw the line 

 as "near as possible to all of the points," what- 

 ever they may mean, or he may try to have as 

 many points on one side as on the other side 

 of the line. It is very doubtful if by intuition 

 be will draw that line which makes the sum of 

 the squares of the residuals a minimum, and it 

 is difficult to see how he is to fit any curve to 



observations without using some of the prin- 

 ciples of the method of least squares. 



In passing it might be noted that some 

 authors still persist in publishing curves with- 

 out representing the observed points on which 

 these curves are based. Such a suppression of 

 evidence should not be countenanced, especially 

 as the graph of the original observations gives 

 any one else such a convenience test of the relia- 

 bility of the curves. 



An application of the method of least 

 squares which is of the utmost importance to 

 the experimenter is in tlie law of propagation 

 of error. The well known relation 



(dX\2 / dX\2 



d^J ^■'^+ (rJ ^^+ - - 



where It is the probable error of X, a function 

 of several measui'ed quantities, X^, X„, . . . ., 

 is not only useful for determining the probable 

 error of a result, but is even more important 

 in planning a program of observation or of 

 experimentation. Where several quantities 

 enter into a determination there is no object in 

 spending time or effort in the wrong place, and 

 one wonders at the tremendous amount of mis- 

 directed effort which is constantly being 

 wasted because of investigators measuring and 

 being careful about the wrong thing, when an 

 elementary acquaintance with this formula 

 would show them which of the various sources 

 of error was contributing most to the inaccu- 

 racy of the result. Another advantage of the 

 method of least squares is that it enables a 

 number of unknown quantities to be disentan- 

 gled from a mass of data where it has been 

 impossible for the experimenter to differen- 

 tiate with respect to one variable at a time. In 

 astronomical practice this is too elementary 

 even to mention, but it is amazing how 

 physicists and others can get along without 

 knowing how to proceed when the conditions 

 are such that they can make only indirect 

 observations on several quantities. It is, of 

 course, the safest practice to measure directly 

 the quantity sought, and to vary but one thing 

 at a time when that is possible, but an experi- 

 menter may find advantage in knowing how to 

 derive several unknowns simultaneously. 



However, with all of the advantages of the 

 method of least squares, it is not so seldom 



