August 2i, 1917] 



SCIENCE 



189 



In the detailed application of this principle, 

 there is one point where confusion might 

 arise, though it can readily be avoided. It 

 is the anomalous behavior of the unit, radian, 

 which appears as a perfectly respectable \mit 

 when an angular velocity is converted from 



— : — to ^^ but does not appear when the 

 min rad 



same angular velocity is foimd from - . This 

 anomaly is the only one of its kind, and is 

 not nature's fault, but o\ir own. If we de- 

 fine angle as degree of opening, to be meas- 

 ured in units of the same kind, the substitu- 

 tion method outlined above is the most nat- 

 ural method of converting say — ^ to -^. If, 



mm sec 

 on the other hand, we define angle as a mere 

 ratio of are to radius it is necessarily a pure 

 number (like a sine or a tangent). If we 

 swap horses in midstream, we shall either 

 miss this unit later or else see it fioating up 

 where we do not expect it. This means we 

 must insert or rub out the imit radian when- 

 ever it is convenient to do so. Fortunately 

 angle is the only quantity treated in such a 

 way. 



For the sake of such mathematical purists as 

 may not approve of the above on . philosoph- 

 ical grounds, a few words should be inserted 

 here on the meaning of the term " multipli- 

 cation." In elementary arithmetic it means 

 merely repeated addition, but with the intro- 

 duction of irrational nmnbers the term is ex- 

 tended by mathematicians to an operation 

 that is not strictly repeated addition. The 

 plan here advocated extends the notion of 

 multiplication still further, to cover a physical 

 combination of concrete quantities. In gen- 

 eral the definition of multiplication in each 

 individual case amounts to translating into 

 algebra the ordinary verbal definition of the 

 compound quantity involved (area, velocity, 

 work, etc.). This extension is made practic- 

 able by the fact that the operation thus de- 

 fined obeys the same logical postulates as the 

 corresponding algebraic operation on pure 

 numbers. In other words, the machinery of 

 mathematics can be applied not merely to 

 numbers, but to any group of concepts and 



operations satisfying the same postulates. 

 This fact is accepted intuitively by most stu- 

 dents; and incidentally the emphasis it puts 

 on the definitions prevents most of the well- 

 known confusion between acceleration and 

 velocity, power and work, and so on. 



To sum up, it seems to me after several 

 years' experience with this system, that it has 

 the following important advantages: (1) It 

 treats equations as neat shorthand statements 

 about real physical things and emphasizes the 

 esthetic side of mathematics in general; (2) 

 It provides an enlarged principle of dimen- 

 sions by which equations may be checked 

 during computation; and (3) It removes com- 

 pletely all restrictions on the units to be used 

 and enables the student to concentrate his at- 

 tention on the facts of nature without the dis- 

 turbing influence of arbitrary rules. 



David L. Webster 

 Jeiteeson Physical Labokatort, 

 Cambeidge, Mass. 



SPECIAL ARTICLES 



ON THE SWELLING AND " SOLUTION " OF 



PROTEIN IN POLYBASIC ACIDS 



AND THEIR SALTS 



There are available only scattered observa- 

 tions on the absorption of water by proteins 

 in the presence of various polybasic acids and 

 their salts. In order to obtain fm'ther exiDeri- 

 mental data in this field, we undertook a 

 rather detailed study of this problem during 

 the past year; As examples of proteins, dried 

 gelatin discs and powdered fibrin were used. 

 For the polybasic acids we chose phosphoric, 

 citric and carbonic. In connection with the 

 swelling of gelatin, we studied also its " solu- 

 tion." The general results of our experiments 

 may be summed up as follows. 



The amounts of water absorbed by gelatin 

 from equimolar solutions of monosodium, 

 disodium and trisodium phosphate depend not 

 only upon which of these salts are present, but 

 upon their concentration. Gelatin absorbs 

 but little more water in a solution of mono- 

 sodium phosphate than it does in pure water. 



