May 7, 1920] 



SCIENCE 



465 



SimuUaneous Differentials (or Correspond- 

 ing Fluxions) are Limits of EquimuUiples of 

 SimuUaneous and Decreasing Differences. 



As we have seen, Newton also made this 

 definition in " Quadrature of Curves," essen- 

 tially as Hamilton gathered it from the 

 "Principia." Many better mathematicians 

 than myself, or than Professor Huntington, 

 have, in fact, examined this definition care- 

 fully, and have found it to be rigorous, simple, 

 and of great generality. 



The infinitesimal method of Leibniz is to be 

 found essentially in Newton's first tract " De 

 analysi per aequationen . . .," which Newton 

 himself later rejected as illogical. A third 

 method of explanation is that of Lagrange, 

 which consists in assuming (for independent 

 variables), dx = Ax, dy = Ay, and for a depend- 

 ent 'variable z dz^ principle part of Aa;, which 

 Lagrange proposed to determine as the terms 

 of first degree in the expansion of z ■-{- Az in 

 ascending powers of Ax, Ay. Newton's dz is 

 the same, if we put dx = Ax, dy = Ay. The 

 adoption of the derivative method, led to de- 

 vices to obtain the same significance of dz by 

 derivatives, without assuming expansions in 

 series. These involve various logical difficul- 

 ties, especially when there are several inde- 

 pendent variables. Also the differentials ap- 

 pear to change their values by changing 

 the independent variables, whereas, Newton's 

 method shows that for every equation be- 

 tween the variables, there exists (if difl^er- 

 entiation be possible) a definite corresponding 

 equation between their differentials, irrespec- 

 tive of the choice of independent variables. 



Unquestionably, there has been a long con- 

 tinued propaganda, fostered at bottom to pro- 

 tect the claims of Leibniz, and aided by the 

 inertia of established usage, to keep the meth- 

 ods of Newton in abeyance. Imagine, if the 

 nationalities of these men had been reversed, 

 the number of pamphlets that would have 

 exploited the matter, and the number of text- 

 books in that method which would years ago 

 have been published. 



Arthur S. Hathaway 



EosE Polytechnic Institute 



CARBON DIOXIDE AND INCREASED CROP 

 PRODUCTION 



To THE Editor of Science: In 1912, at the 

 International Congress of Chemists held in 

 New York, Professor Ciamician, of the Uni- 

 versity of Bologna, presented a paper on the 

 "Photochemistry of the Future," in which, 

 among other things, the suggestion was made 

 that crop production might be increased by 

 increasing the concentration of carbon dioxide 

 in the air. Of course, the idea underlying 

 such a suggestion is that since the carbon 

 dio.'^ide of the air is a necessary constituent 

 in the synthesis of carbohydrate by the plant, 

 and since, furthermore, the percentage of the 

 gas in the air is comparatively small, any in- 

 crease in the amount of carbon dioxide may 

 tend to increase the amoimt of carbohydrate 

 produced. 



That such is actually the case has been 

 found by a number of German chemists, ac- 

 cording to the Berlin correspondent of the 

 N. Y. Tribune (April 4). "Working in green- 

 houses attached to one of the large iron com- 

 panies in Essen, and utilizing the carbon 

 dioxide (freed from impurities) obtained from 

 the blast furnaces, the yield of tomatoes was 

 increased 175 per cent, and cucumbers 70 per 

 cent. Further experiments in the open air, 

 on plots around which punctured tubes were 

 laid, and through the latter of which the 

 carbon dioxide was sent, gave increases of 

 150 per cent, in the yield of spinach, 140 per 

 cent, with tomatoes and 100 per cent, with 

 barley. Benjamin Harrow 



STRUCTURAL BLUE IN SNOW 



To THE Editor of Science : The recent bliz- 

 zard began here with a heavy downpour of rain 

 on the evening of March 5, which later turned 

 into a glistening snow that was shattered by 

 the furious wind and formed a crystalline- 

 looking glittering coherent mass whose struc- 

 ture was maintained by the low temperature 

 (about 20° F.). 



When the sun finally came out on Saturday 

 afternoon, I noticed that the shadows of the 

 trees and the shadow masses of the distant 

 snow, appeared unusually hlue, and that the 



