The corresponding values of the fraction itself of vegetable growth are, when n is 2, 5 8 1, 3, 1, 13, 2T, 11, 31, 16, &c. which is the series usually observed. The values, when n is 3, are 4. 4. Fr, 18, 27, 14, &c., which are also found in nature, but much more rarely than the preceding row of fractions. These, if I do not misunderstand the botanists, are the only fractions which directly occur in the ordinary forms of growth. The few other fractions, which have been observed, are the products of indirect species of development which are rarely exhib ited; but even these fractions are included in the general form. They correspond to the cases in which n is 4, 5, or 6. would give a good arrangement, infinity of other fractions. The same series of fractions in all This expression of the forms of vegetable growth in a fraction, leads to the inquiry why this particular, fraction was selected. It obviously gives an admirable packing to the leaves, as Dr. Gray has shown, and this may justly be regarded as a sufficient a priori reason for its adoption. It must be admitted, however, that other excellent systems of packing might be devised, which would appear to be very little inferior to this; the fraction and the same might be said of an exclusive adoption of one and the vegetable forms, must therefore be sought, partly in the desire to construct a system sufficiently simple and uniform to admit of human investigation and study, and at the same time so complicated and varied as to command his wonder and admiration, and partly in the uniformity of the laws of vegetable growth through the machinery of which it must be introduced and developed. It cannot be supposed that the leaves attain their proper position through a voluntary selec tion of the widest spaces. They all start to grow at once, and each phyton occupies at the outset the very place to which it belongs. It seems to me most probable that beneath this series of fractions lies the fundamental law of organic action, and the simplicity of the series confirms this opinion. I must now take the liberty to draw the attention of the Association to another domain of the physical universe, in which there are distinct traces of these same fractions. They are approximate expressions of the relative times of rotation of the successive planets of the solar system. Thus the ratio of the mean motion of each planet to that of the next inner planet is nearly equal to some one of these fractions. This is so manifest, that all the great inequalities of long period which occur in the solar system depend upon these ratios, and they are interwoven with all the most important irregularities of motion of the primary planets. Whence could this extraordinary coincidence have arisen but from the action of a single mind? and what does it indicate but that the same Word which created the planet, is expressed in the plant? May I close with the remark, that the object of geometry in all its measuring and computing, is to ascertain with exactness the plan of the great Geometer, to penetrate the veil of material forms, and disclose the thoughts which lie beneath them? When our researches are successful, and when a generous and heaven-eyed inspiration has elevated us above humanity, and raised us triumphantly into the very presence, as it were, of the divine intellect, how instantly and entirely are human pride and vanity repressed, and by a single glance at the glories of the infinite mind, we are humbled to the very dust. INDEX. Acoustics, Application of the Principles of to the Construction of Lecture Rooms. Prof. Henry. 432. Action, Volcanic, Isolation of in Hawaii. Prof. J. D. Dana. 95. Alps, Central, Erratic Phenomena of. Prof. Guyot. 311. Structural Features of the Apallachians compared with those of. Prof. H. Altona Observatory, Correspondence in Relation to. 65. American and British Coals, Economic Values of. Prof. W. R. Johnson. 221. Ammonia, Moisture and Organic Matter in the Atmosphere. Prof. Horsford. 124. 235. Amphiumaina, Habits of in a State of Captivity. Dr. J. L. Le Conte. 194. Analogy, Kirkwood's. Dr. B. A. Gould, Jr. 363. in the Periods of Rotation of the Primary Planets. S. C. Walker. 207. of the Ribbon Structure of Glaciers to the Slaty Cleavage of Rocks. Prof. H. D. Rogers. 181. of the Morphology of Cells to that of the Ovum. Dr. W. I. Burnett. 261. Prof. B. Silliman, Jr. 134. Analytical Notation, Geometrical Interpretation of. J. Patterson. 369. Animals, Coral, Structure of. Prof. Agassiz. 68. แ Radiated, Structure and Homologies of, with Reference to the Systematic Animal Morphology. Prof. Agassiz. 414. Anthracite Coal, Soda in the Ashes of. Prof. E. N. Horsford. 233. Appallachians, Structural Features of, compared with those of the Alps. Prof. H. D. Rogers. 113. Apparent Places of the Stars, General Tables for the Reduction of. Prof. Application of the Principles of Acoustics to the Construction of Lecture Rooms. Application of the Magnetic Telegraph to Astronomical Observations, Improve- 274. Application of the Theory of Least Squares, Comparison of the Results obtained in Applications of Phyllotaxis. Prof. Gray. 438. Aqueous Action. Prof. H. D. Rogers. 239. Ascidia, Embryology of, and Characteristics of New Species from the Shores of Ashes of Anthracite Coal, Soda in. Prof. E. N. Horsford. 233. Astronomical Journal, Establishment of in the United States. Prof. J. S. Hubbard. Astronomical Observations, Improvements in the Application of the Magnetic Tele- Atlantic Ocean, Winds and Currents of. Lieut. Maury. 160. Prof. E. N. Horsford. 124. Axis of Subsidence and Trend of Islands in the Pacific. Prof. J. D. Dana. 321. Batrachians, Urodellian. Prof. Baird. 402. Bisilicate of Magnesia of Dr. Thomson, and Boltonite of Shepard. Prof. B. Silliman, Jr. 109. Boltonite, Remarks on. L. Saemann. 105. of Shepard and Bisilicate of Magnesia of Dr. Thomson. Prof. B. Bone Caves of Pennsylvania. Prof. Baird. 352. Bones of the Dinornis Nova Zealandia. Prof. Chase. 267. Bournon, Count, Indianite of, and the American Mineral of the same Name. Prof. B. Silliman, Jr. 131. Brachiopoda of the Silurian Period, particularly the Leptanidæ. Prof. James British and American Coals, Economic Values of. Prof. W. R. Johnson. 221. Bucholzite of Brande, Sillimanite of Bowen, and Fibrolite of Bournon, Identity of Canada, Geology of. T. S. Hunt. 325. Mineral Waters of. T. S. Hunt. 233. Caves, Bone, of Pennsylvania. Prof. Baird. 352. Cave, New, Discovery of in Kentucky. Dr. L. Feuchtwanger. 355. Cephalopoda, Embryology of. Prof. Agassiz. 409. Change of Level, Fiord's Evidence of. Prof. J. D. Dana. 194. Character, Vegetable, of Xanthidium. Prof. Agassiz. 89. Zoological, of Young Mammalia. Prof. Agassiz. 85. Characteristics of New Species of Ascidia from the Shores of Massachusetts. Chemical Examination of Gorgonia Antipathes. Prof. B. Silliman, Jr. 130. Cholera and Electricity, Connection between. Dr. Hare. 201. Circulation of the Fluids in Insects. Prof. Agassiz. 140. Classification, Natural, of Curves. Rev. Thomas Hill. 101. |