INSTRUMENTS AND METHODS OF RESEARCH. 117 
ceived its highest development in the Ptolemaic System of 
the Universe? You will recall that Ptolemy, building on 
the suggestions of Apollonius and of Hipparchus, supposed 
a planet to describe an epicycle by a uniform revolution in 
a circle whose center was carried uniformly in an eccentric 
round the earth. By suitable assumptions as to his variable 
factors, he was thus able to represent with considerable accu- 
racy the apparent motions of the planets and to reproduce 
quite satisfactorily other astronomical facts. This was the 
artifice employed by the astronomer of the period before the 
modern and more subtle art of simulating Nature, by the 
sine-cosine method, had become known. 
What seemed so intricate and complex in Ptolemy’s time 
could be expressed in very simple language indeed, when 
a Kepler discovered the true functions as embodied in his 
three fundamental laws. The present method of hiding our 
ignorance of the real law, which seems at times to exert such 
a mesmerizing influence over us as to make us mistake the 
fictitious for the real, reminds one of the old conundrum : 
“Patch on, patch on, hole in the middle; if you guess this 
riddle, I’ll give you a golden fiddle.” 
If the sine and cosine of the angle does not represent the 
curve of observation, patch on a sine and cosine of twice 
the angle; then, if necessary, thrice the angle; next, four 
times, and so ad infinitum! Now guess the riddle! 
Of course I do not mean to discard this useful and in fact 
indispensable tool of research, but simply wish to call atten- 
tion to its limitations and to the importance of not overlook- 
ing the fertile byproducts, the residuals, which, because of 
our neglect of them, may some day rise and smite us in their 
wrath. Each one of us at one time or another has doubtless 
established, by least squares, an empirical formula of some 
kind which so beautifully fits the observations as to make 
us bold and venturesome. Now comes a new observation, 
somewhat outside of the range for which the expression was 
established. Eagerly the test is applied, and we find to our 
chagrin that the formula on which so much work had been 
spent will not fit the new result, and that we have a “coun- 
terfeit” and not the real law. 
