Let us take a flower of five parts, with five leaves to the calyx, five petals, &c. We should suppose it to be upon the two fifths arrangement; but this would bring each petal exactly over a leaf of the calyx, instead of between them. Botanists have been much puzzled to explain this, to find any satisfactory mode of bringing the habitual position of the floral organs into accordance with that of the leaves, so that both might be expressed by the same law. When this is done, the proof of the doctrine that the leaves and floral organs are homologous parts will be complete. Prof. Braun, who, with Dr. Schimper, first developed these laws of phyllotaxis, and who has worked them out very elaborately in all their manifold applications, by introducing the term prosenthesis to express the singular deviation, or pushing forward by half an interval, which appears to take place in passing from each floral verticil to the next, has only stated the exception, without explaining it. The only attempt at explanation that I am aware of, which is at all satisfactory, has recently been made by M. Adrien de Jussieu. Instead of assuming that the trimerous flower is actually formed on the plan 1-3, and the pentamerous flower on the plan 2-5, he projects them both upon the 5-13 arrangement, and shows that this gives the floral organs in threes or in fives, so nearly in their position of regular alternation, that, with the tendency of all growing parts to develop on the side of least pressure, they could hardly fail to fall into that position. It is well known that in irregular flowers, organs are often suppressed or rendered abortive on the side where most pressed, and are pushed out of place towards the side of least resistance. [Prof. Gray illustrated this view by a diagram, and showed upon the proposed plan, how little displacement was required to bring about the actual alternation.] MATHEMATICAL INVESTIGATION OF THE FRACTIONS WHICH OCCUR IN PHYLLOTAXIS. BY PROF. B. PEIRCE. THE Association may wonder what a mathematician can have to do with Botany, and what right he has to discuss such a subject as vege table morphology. But let me assure you that the geometer is some. what omnivorous in intellect, and although he has lived and thriven for centuries upon the sun and moon, the planets and comets, and other such inorganic food, he is already aspiring to a vegetable diet, and may ere long be whetting his teeth for flesh and blood. But, in the present case, the botanists have provoked the invasion by undertaking to demonstrate that plants grow according to exact mathematical laws. They have presumed to measure with minute accuracy, and exact measurement must open the path to geometry. They have dared to use our numbers and fractions, and we must reclaim them with interest. Upon the principal points, which have occurred to me in the investigation of the curious fractions of vegetable morphology, I apprehend that I have been anticipated by Bravais. But it may be that I have developed the subject more distinctly than he has done, and am therefore disposed to present my views to the Association. I do not regard these fractions as isolated and independent of each other; but all of them seem to me to be approximations, more or less accurate, to one and the same fraction, or rather to several fractions of one series. It is as if in the forms of vegetable growth, there had been one great thought underlying the whole structure. The thought has in it an element of infinity, but the mode of expres. sion is necessarily finite. It is everywhere partially developed, with more or less approach to perfection in different plants. This very defect of expression has enabled us to discern and comprehend the divine idea with our finite capacities. Had it been fully expressed, we should not probably have discovered it. Most certain is it that if the infinite fraction had been introduced into the creation, we could not have detected it; for the infinite series would not have been completed, even though the tree had grown to heaven itself. The fraction cannot be fully written out, but the law of its formation is easily perceived. It is in which n is any positive n + a 1 integer, while a has a peculiar value. The quantity a is an infinite continued fraction, formed wholly of units; thus, The successive approximations to the value of a, are obtained in the way familiar to mathematicians by stopping at the different points of the downward train of units. They correspond identically to the fractions which have been observed in the finite forms of the vegetable universe. They are, 1, 2, 3, 8, 6, fs, Hi, 31, &c. §, 13, 11, 5 8 Duration of Graptolites in Geological Periods, and their Value in the Identification Duration and Variation of Winds, Instrument for Measuring. Prof. J. H. Coffin. 386. Earth's Surface, Erosions of. President Hitchcock. 148. Economic Values of British and American Coals. Prof. W. R. Johnson. 221. 66 of a Plate of Zinc buried in the Earth, Experiments on. Prof. Loomis. Electro-Dynamic Forces. Prof. Lovering. 278. Elephant, Fossil Remains of, found in Vermont. Prof. Agassiz. 100. 196. Embryology of Ascidia, and the Characteristics of New Species from the Shores of Massachusetts. Prof. Agassiz. 157. Embryology of Cephalopoda. Prof. Agassiz. 409. Erratic Phenomena of the White Mountains. Prof. Guyot. 308. Erosions of the Earth's Surface. President Hitchcock. 148. Establishment of an Astronomical Journal in the United States. Prof. J. S. Hubbard. Prof. S. S. Haldemann. 426. 378. Ethnology, Linguistic, with Illustrations. 196. Fat, Intestinal, of a Large Sea Turtle, Investigation of. Dr. C. Linck. 384. Fibrolite, of Bournon, Identity of with Kyanite. Prof. B. Silliman, Jr. 111. Fluvicola Herricki, Metamorphosis of. Prof. J. D. Dana. 194. Dr. Le Conte. 272. Fluids, Circulation of, in Insects. Prof. Agassiz. 140. Forces, Electro-dynamic. Prof. Lovering. 278. a New Demonstration of the Parallelogram of. Prof. B. Peirce. 221. a new Demonstration of the Parallelogram of. J. D. Oliver. 221. Fossil Remains from Broome County, N. Y. W. C. Redfield. 255. Myliobates, New Species of, from the Eocene of South Carolina. Dr. R. Fossil Crinoids of Tennessee. Prof. G. Troost. 59. Remains of an Elephant found in Vermont. Prof. Agassiz. 100. Fossils, New, from the Cretaceous, Eocene, and Pliocene, of South Carolina, Ala- Fractions which occur in Phyllotaxis, Mathematical Investigation of. Prof. B. Fresh-water Cottus of North America, Monograph of. Ch. Girard. 412. Gadolonite in New York. Prof. B. Silliman, Jr. 134. Gaussian Tables. Dr. B. A. Gould, Jr. 362. General Tables for the Reduction of the Apparent Places of Stars. Prof. Hub- bard. 274. Geodesy, Results obtained in, by the Application of the Theory of Least Squares. Prof. A. D. Bache. 102. Geological Periods, Duration of Graptolites in, and their Value in the Identification 351. Geological Structure of Keweenaw Point. Dr. C. T. Jackson. 288. Dr. C. T. Jackson. 283. to the Slaty Cleavage of Rocks. Gorgonia Antipathes, Chemical Examination of. Prof. B. Silliman, Jr. 130. Green Picrolite and Slaty Serpentine, from Lancaster County, Pa., Analysis of. Habits of the Amphiumaina in a State of Captivity. Dr. J. L. Le Conte. 194. Hawaii, Isolation of Volcanic Action in. Prof. J. D. Dana. 95. History of Phalangopsis, a Genus of Orthoptera, with three New Species, two of Hydroid Polypi, Structure and Homologies of the Radiated Animals with Refer- Identity of Sillimanite of Bowen, of Bucholzite of Brande, and of Fibrolite of Improvements in the Application of the Magnetic Telegraph to Astronomical Indianite of Bournon, and the American Mineral of the Same Name. Prof. B. Infusoria, Process for Detecting the Remains of, in Sedimentary Deposits. Prof. J. W. Bailey. 409. Insects, Circulation of Fluids in. Prof. Agassiz. 140. Instrument for Determining the Variation and Duration of Winds. Prof. J. H. Coffin. Instruments, Meteorological. Prof. Guyot. 274. Instrument for Determining the Quantity of Ozone. Prof. Horsford. 127. 386. Intestinal Fat of a Large Sea Turtle, Investigation of. Dr. C. Linck. 384. Peirce. 444. Investigation of the Intestinal Fat of a Large Sea-Turtle. Dr. C. Linck. 384. Islands, Sandwich, Plants of the Order Composite from. Prof. Gray. 397. Jamaica, Distribution of the Testaceous Mollusca in. Prof. C. B. Adams. 147. Hubbard. 378. Jura, Upheaval of. Prof. Guyot. 115. Kentucky, Discovery of a New Cave in. Dr. L. Feuchtwanger. 355. Dr. C. T. Jackson. 288. Kirkwood's Analogy. Dr. B. A. Gould, Jr. 363. Lake Superior, Mirage on. Dr. C. T. Jackson. 143. 66 Prof. J. S. Geology, Mineralogy, and Mines of. Dr. C. T. Jackson. 283. Lake and River Terraces of the United States and Europe, Origin of. Prof. H. D. Rogers. 239. Larva of Physocalus Inflatus. Prof. S. S. Haldeman 347. Least Squares, Results obtained in Geodesy by the Application of the Theory of. Lecture-Rooms, Application of the Principles of Acoustics to the Construction of. Leptanide of the Silurian Period. Prof. James Hall. 347. Leucine and its Homologous Relations. Level, Fiord's Evidence of a Change of. T. S. Hunt. 231. W. O. Ayres. 402, Prof. J. D. Dana. 194. Linguistic Ethnology, with Illustrations. Prof. S. S. Haldeman. 426. Localities, Mineral, in Worcester and Franklin Counties, Mass., Description of. Magnesia, Bisilicate of, of Dr. Thomson. Prof. B. Silliman, Jr. 109, Magnesium, Ammonio-chloride of. Dr. C. Linck. 235. Magnetic Telegraphs, Improvement in the Application of, to Astronomical Obser- Mammalia, Young, Zoological Character of. Prof. Agassiz. 85. Mastodon Angustidens. Dr. John C. Warren. 93. Mathematical Investigation of the Fractions which occur in Phyllotaris. Prof. B. Peirce 444. Medical Geology. I. A. Lapham. 406. Meridian, Prime. Lieut. C. H. Davis. 78. R. A. Marr, U. S. N. 335. (6 American Prime. Prof. I. F. Holton. 381. |