1.24 Sir AY. R. Hamilton on Roberts Construction 



culars they have not been adopted, while in others they have 

 been outgrown — I trust he will either discuss them from a purely 

 abstract point of view, or at any rate select some other and 

 more willing antagonist with whom to fight his battle. 

 I am, Gentlemen, 



Your obedient Servant, 



William Odling. 



I 



XXIY. On Roberts Construction of the Heptagon. By Sir Wil- 

 liam Rowan Hamilton, LL.D., M.R.I. A. } F.R.A.S., fyc. } 



Andrews Professor of Astronomy in the University of Dublin, 

 and Royal Astronomer of Ireland*. 



N a recent Number of the Philosophical Magazine, obser- 

 vations were made on some approximate constructions of 

 the regular heptagon, which have recalled my attention to a very 

 remarkable construction of that kind, invented by a deceased pro- 

 fessor of architecture at Dresden, Friedrich Gottlob Robert, who 

 came to conceive, however, that it had been known to the Egyp- 

 tians, and employed by them in the building of the temple at 

 Edfu. Rober, indeed, was of opinion that the connected triangle, 

 in which each angle at the base is triple of the angle at the ver- 

 tex, bears very important relations to the plan of the human 

 skeleton, and to other parts of nature. But without pretending 

 to follow him in such speculations, attractive as they may be 

 to many readers, I may be permitted to examine here the accu- 

 racy of the proposed geometrical construction, of such an isosceles 

 triangle, or of the heptagon which depends upon it. The close- 

 ness of the approximation, although short of mathematical rigour, 

 will be found to be very surprising. 



2. Roberts diagram is not very complex, and may even be 

 considered to be elegant ; but the essential parts of the construc- 

 tion are sufficiently expressed by the following formulae : in which 

 p denotes a side of a regular pentagon ; r } r 1 the radii of its in- 

 scribed and circumscribed circles ; r" the radius of a third circle, ' 



* Communicated by the Author. 



f The construction appears to have been first given in pages 15, 16 of a 

 quarto work by his son, Friedrich Rober, published at Dresden in 1854, 

 and entitled Beitrdye zur Erforschuny der geometrischenGrundformen in den 

 alien Ternpeln Aegyptens, und deren Beziekuny zur alten Naturerkenntniss. 

 It is repeated in page 20 of a posthumous work, or collection of papers, 

 edited by the younger Rober, and published at Leipzig in 1861, entitled 

 Elementur -Beitrdye zur Bestimmuny des Naturgesetzes der Gestaltuny und 

 des Widerstandes, und Anwenduny dieser Beitr'dge auf Natur und alte 

 Kunstyestultuny , von Friedrich Gottlob Rober, ehemaligen Koniglich- 

 Sachsischen Professor der Baukunst und Land-Baumeister. Both works, 

 and a third upon the pyramids, to which I cannot at present refer, are 

 replete with the most curious speculations, into which however I have 

 above declined to enter. 



