Pebeuaey 13, 1920] 



SCIENCE 



167 



Pell, in 1673, is clarifying. Leibniz was a poli- 

 tician, not a mathematician, and worked and 

 wrote for the power and prestige of Germany. 

 To this end he founded the Berlin Academy of 

 Science, and was perhaps the first to inaugu- 

 rate that system of espionage on scientific 

 work in foreign countries by which the use- 

 fulness and credit of as much of that work as 

 possible might be transferred to Germany. 



It may be urged that calculus has been 

 benefited by the interference of Leibniz. 

 This is true as to notation, but it has been 

 harmful as to the theory and understanding 

 of the subject. On the one hand we have an 

 illogical infinitesimal method, on the other an 

 incomplete derivative one in protest of the 

 first, whose rival expounders reason along dif- 

 ferent lines, and hardly understand each other. 

 Newton substitutes one rigorous theory, 

 broader than either of these, neglecting no 



Starting from given corresponding values, x, 

 Vj z, the actual variables are corresponding 

 increments to these with a common -first value, 

 0; and starting with any corresponding incre- 

 ments, Ax, A2/, Aa, we form an ideal variation 

 in the same ratio, A'a; = Nl^x, A'y = NAy, 

 A'z = NAz, where the common multiplier N, 

 varies. This is the familiar law of uniform 

 variation between two sets of values of the 

 variables, and the symbols A'x, etc., are not 

 limited to small values but vary from to 

 00, as iV so varies, however small Ax, etc., 

 may be. 



Such A'a;, A'y, A'z are approximate fluxions; 

 and the exact fluxions dx, dy, dz, are limits of 

 these for lim. Ax = 0, lim. Ay = 0, lim. Az = 0. 

 For example, let z ^xy, then Az = yAx -\- 

 (x -|- Ax) Ay, and multiply both members by N. 



A'z = yA'x -f- (a; -f Ax) A'y, 

 whence by limits, dz = ydx -{- xdy. 



Az^^shaded area, A's^sbaded »rea. dz'shaded iir«». 



We may illustrate the three variations geometrically: 



(1) Actual. (2) In the Same Eatio. (3) In the First Eatio. 



quantity, however small, leaving no unex- 

 plained symbol, and yet of an arithmetical 

 character of the utmost simplicity. A free 

 translation of his definition in " Quadrature 

 of Curves," is as follows: 



In their highest possible approximation, 

 fluxions are quantities in the same ratio as 

 the smallest possible corresponding increments 

 of variables, or, in a form of exact statement, 

 they are in the first ratio of nascent incre- 

 ments. 



Thus fluxions, or differentials, are inter- 

 preted as ordinary arithmetical increments, 

 but in a variation defined as in the first ratio, 

 or, as the variables hegin to increase, or, in 

 the instantaneous state, which are all one. 



Arthur S. Hathaway 

 Rose Polytechnic Institute 



SCIENTIFIC BOOKS 



REPORT OF THE CANADIAN ARCTIC EXPEDI- 

 TION, 1913-18 



Shortly after the return of the Southern 

 Party of the Canadian Arctic Expedition with 

 their collections in the fall of 1916, steps were 

 taken to arrange for the publication of the 

 scientific results of the expedition. Although 

 the general direction of the operations of the 

 expedition had been under the Department of 

 the Naval Service, most of the scientific men 

 on the expedition were under the Geological 

 Survey, of the Department of Mines, the col- 



