526 PROCEEDINGS OF THE ACADEMY OF 



but extensive reference to these authorities would be inconsistent with our 

 present limits : many of them, either directly or indirectly, advocate the pos- 

 sibility of a mathematical explanation of the cause of organic forma. 



Professor Bronn* considers that there is an inconsistency in supposing the 

 organic world alone to be derived from a direct act of creation, whilst all the 

 rest is born and perishes from the effect of general forces eternally immanent 

 in matter. He concludes that all species of animals and vegetables were 

 originally created by a natural force, at present unknown that they do not 

 owe their origin to a successive transformation of a few primitive forms and 

 that this force held a most intimate and necessary relation to the forces and 

 events which have controlled the development of the surface of the globe. He 

 thinks that such a hypothetical force would be in entire harmony with the 

 whole economy of nature, and that the hypothesis would not only permit the 

 belief in a Creator presiding over the development of organic nature by means 

 of an intermediate force, but that this conception is more sublime than the 

 idea of a direct supervision, by the Creator, of the succession of plants and 

 animals. Professor Bronn also considers the fundamental form of a plant to 

 be that of an egg placed upright. Investigation of the relation between natu- 

 ral and mathematical ovoid forms might furnish a test for the correctness of 

 this idea, or, if it is well founded, assist in explaining its application. 



Some mathematical writers treat as an evident proposition the ultimate 

 connection between mathematics and the explanation of natural processes. f 

 Fechner undoubtedly encourages this idea, and even proposes, more or less 

 definitely, the adoption of a mathematical classification in physiognomy, cra- 

 niology, and ethnology. J 



Lotze, on the other hand, takes the opposite extreme. In one of his more 

 skeptical passages he compares the attempt to discover the laws of organiza- 



* Essai d'une ileponse a la question de Prix, &c. Comptes Rondus, vol. 51, p. 511. 



f The principles of mechanics must be of the greatest importance for all branches of natural 

 science, (as Aristotle was aware,) because, according to our conception of the changes of the ma- 

 terial world, they must be referred to motion. Dr. H. Burhenne, Grundriss der Hoeheren Anal} sis, 

 Cassel.1849, p. 84. 



Dr. Zeising, and others whom we have cited, refer at length to the works of Pythagoras, Plato, 

 and Aristotle, in order to show that the ancients regarded numbers as in some mysterious sense 

 the principia of the universe. The Pythagorean quaternary, as improved by Plato, consists of 

 the celestial numbers 1, 3, 7, 9, of which the sum is 20, and of the terrestrial series 2, 4, 6, 8, 

 whose sum is likewise 20. These two together make the sacred quaternary 40. The number 

 5, which is not in the quaternary, but is the middle of the whole series from 1 to 9, represents 

 the Nous, or supreme intelligence. According to Montucla. these numbers and the idea of their 

 mystic importance were derived from the Egyptians. The ancient Chinese also venerated the 

 Pythagorean quaternary, and ascribed its invention to the emperor Fo-hi (2900 B. C.) Ko-hi was 

 the inventor of the binary arithmetic, ot which he left the notation in the Cova. or Figure of 

 Eight. M. Iluc relates that the Chinese still venerate a mysterious book, called the book of 

 Changes, y-King. The meaning of this book has long been lost. From M. Hue's description of the 

 64 whole and broken lines of this book, and from Leibnitz's description and interpretation of the 

 Cova, I have little doubt that the y-King pertains to the arithmetical system recorded in the Cova. 

 The tradition of the Chinese, that the y-King is capable of explaining all things, may, therefore, 

 indicate that the ancient Chinese were not unaware of the importance of number in the order of 

 the universe, and that their sages had conceived the idea of a mathematical explanation of Ma- 

 ture, as clearly as such an idea could be conceived in advance of the science of physical mathe- 

 matics: possibly they progressed no further than to incorporate the Cova in their religious mys- 

 teries. Montucla, Histoire des Mathematiques, vol. i. p. 122. Chinese Empire, by 31 Due, 

 London, 1855, vol. i. p. 124. Leibnitz, Memoire de 1'Acad. Franchise, vol. xviii. 1703, p. 85. Dr. 11. 

 Burhenne, Grundriss der Hoeheren Analysis, Cassel,1849, p. 84. 



J Ueber die Mathematische Bebandlung OrgaDischer Gestalten und Processe. Verh. d. Koenl. 

 Saechs. Gisellsch. Mathematisch-Physische K!., Jarhgang 1849. 



Mr. Hay has published a method for defining geometrically the shape of the human head and 

 the proportion of its parts. The method is founded on a system of triangles, of which the angles 

 have certain ratios manifested in the vibrations of musical strings. See D. It. Hay on the Beau- 

 Ideal Head of Ancient Greek Art. Trans. Soc. of Arts, vol. i. part 2, New Series, 1847-8. 



The same author has written several works on, the Beautiful in Form. His Natural Principles 

 of Beauty (London and Edinburgh, 1852) gives a concise explanation of his geometrical construction 

 of the human figure. The same subject is differently treated bv Dr. A. Zeisiug, Neue Lchre von 

 den Prnportionen des Menschlichen Koerpers; Leipzig, 1854. The student of Morphology will be 

 interested in comparing with these works, Die aus der Arithmetic und Geometric herausgeholtcn 

 Gruende iur Menschlichen Proportion; Georg Lichtensteger, Nuremberg, 174ti. 



[Dec. 



