NATURAL SCIENCES OP PHILADELPHIA. 531 



optical experiments, might result from the vibrations of other substances 

 which in their vibrations may follow the same or similar laws. This may 

 possibly be the reason of some resemblances of the kind we shall now consider. 

 Construction of the Hyper aster. Figures resembling star-fish may be derived 

 from the ellipse by a construction similar to that given for Fig. 3. Both con- 

 structions can be included under a general mathematical formula.* To con- 

 struct the hyperaster with five points, make the ellipse (Fig. 5) with the 

 semi-axis F B equal to the short radius F B (Fig. 6) of the star, and with 

 the longer semi-axis F A equal to the long radius F A of the star. Then, be- 

 ginning at B, proceed as for the construction of Fig. 3, except that the angle 

 B F P of Fig. 6 is to be always taken equal to tioo-ffths of B F P in Fig. 5. 

 When the radius F P of Fig. 5 has passed through a revolution of 90, it will 

 coincide with F A, and P will then fall upon A. During the same time, the 

 radius F P of Fig. 6 will pass over two-fifths of 90, or 36, and will reach A. 

 When the radius of Fig. 5 reaches F C, it will have passed over 180, and in 

 Fig. 6 the radius, then at C, will have passed over two-fifths of 1S0, or 72, 

 which is the fifth part of the circumference of the circle. The arm B A C F of the 

 star is therefore derived from the semi-ellipse B A C. A repetition of the same 

 process will derive the next arm of the star from the semi-ellipse C D B ; and 

 so on, until the five arms of the star are completed.! By means of this con- 

 struction, star-fish or other organic bodies resembling them can be imitated. 

 Returning to Figs. 2 and 3, it will be observed that, starting at B, the entire 

 Fig. 3 is generated from the semi-ellipse B A C, in the same way that the arm 

 B A C F of Fig. 6 is derived from the semi-ellipse B A C, Fig. 5. Viewed in this 

 manner, the egg, Fig. 3, appears a one-armed star-fish. Whether or not this 

 conception may have any significance in nature, it appears remarkable to 

 find two different organic forms thus classed under the same mathematical 

 formula. Some of the figures known as acoustic figures, produced by the vi- 

 bration of elastic plates, can also be imitated. Figures resembling Fig. 6 are 

 given by Chladni in his treatise. Possibly the acoustic figures might be pro- 

 duced on a scale sufficiently large to test their agreement with the mathemat- 

 ical figures, by measurement ; and hence it could be, perhaps, determined 

 whether these truly represent the former. 



Interesting resemblances can be traced between the optical and acoustic 

 figures, and between these and curves similar in their construction to those we 

 have described, if not always precisely of the same construction. The follow- 

 ing is of the same general construction as the previous. By taking the ellipse 

 Fig. 2, and making the angle B F P in the derived figure always equal to one- 

 half of the same in the ellipse, we derive a curve similar to Fig. 7. Figs. 8 

 ?.nd 9 represent an optical figure and an organic form, having a resemblance 

 to this. J 



* Studies in Organic Morphology, Formula 2, p. 32. We propose to call the curve whose equa- 

 tion is p = the elliptoaster, because the equation resembles that of the ellipse, and the 



curve itself may represent a star. The name hyperaster may be given to the curve whose radius 

 is a power or root of the radius of the elliptoaster. 



+ In actual constructions, it will be sufficient to derive one arm of the star, and then, by means 

 of tracing-paper, to dispose five such arms around the centre F. Stars of any desired uumler rl 

 points may be thus constructed ; the angle B F P of the star must be to the angle BFPof the 

 ellipse as the number 2 is to the number of points in the star. 



% See Encyclopedia Britannica, Boston ed., art. Optics, p. 672, for Fig. 9. For Fig. 8 see Zeitechr. 

 fur Wiss. Zoolc.gie, Leipzig, 1854, vol. v. Plate XIV. Fig. 3-i. These resemblances could be followed 

 tea greater extent. The writer has collected many drawings of mathematical lines, organic ob- 

 jects, optical, acoustic, and electric figures, but must omit further notice of them on the present 

 occasion. By large collections of this kind, and by diligent comparison of their materials, some- 

 thing may, perhaps, be elicited which will establish a reliable foundation for the study of Organic 

 Morphology as a mathematical science. 



1862.] 



