In general we may say, that there can be hardly any doubt about 

 the fact that the remarkable symmetry of the physical properties 

 of solid crystallized matter, must find its primary cause in the 

 special symmetry-properties of the smallest constituting particles 

 themselves and in the characteristic symmetry of the moving systems, 

 we call atoms. However, the modern views on the intimate structure 

 of these atoms, with their complicated architecture of electrons 

 rotating round a minute centre (Rutherford, Bohr, Debije, etc.), 

 are at this moment still too hypothetical and uncertain to allow more 

 detailed attempts here to be made towards the solution of the nume- 

 rous problems associated therewith. This must be postponed as a 

 task for the future, to be begun as soon as more exhaustive and certain 

 information about the structure of the atoms shall be available. 



18. Finally a few remarks on another subject. In the preceding 

 paragraphs we have not dealt with the symmetry in the arrangement 

 of numerical data as they are often found as the result of statistic 

 investigations on a great number of facts, because this subject is, 

 properly speaking, merely a chapter of pure mathematics. 



That there are often to be detected symmetrical arrangements of 

 numbers in cases x ) of numerical arrangement, where series of such 

 data are considered, to which the calculus of probabilities (fre- 

 quency-curves, etc. 2 ) can be applied, is a wellknown fact (binomial 

 coefficients, etc.). This symmetry manifests itself for instance in the 

 numbers obtained by Gr. Mendel in his famous researches con- 

 cerning the heredity of properties in plant-hybrids, and in the 

 corresponding work of several other investigators. 3 ) 



Instances of this kind may easily be augmented; however, it is 

 not our purpose to go into details here, but simply to draw the 

 attention of the reader also to these occurences, which represent 

 more especially a chapter of the general theory of numbers. 



About the symmetrical arrangement of some organs in plants, 

 - a problem which is closely related to the kind of problems men- 

 tioned here, we will say something at the end of the next chapter. 



*) Cf. also: A. Sommerfeld, Die Naturwissenschaften, 8, 61, (1920); V. Gold- 

 schmidt, Ueber Harmonie im Weltraum, O s t w a 1 d's Ann. der Naturphilos., 5, 51 . 



2 ) J. C. Kapteyn, Skew frequency-curves in Biology and Statistics, Gro- 

 ningen, (1916); on symmetrical probability-curves, cf. also: H. Reichenbach, 

 Zeits. f. Philosophic und philos. Kritik, 161, (1917). 



3) Gr. Mendel, Versuche iiber Pjlanzenhybriden, Verh. naturf. Verein. Brunn 

 4, 347, (1865); Ostw., Klass. d. ex. Wiss. No. 121, (1901), p. 17; Cf. ajso: 

 J. Tammes, Rec. des Trav. botan. Neerl., 8, 232, (1911). 



