THE 



PHILOSOPHICAL MAGAZINE 



AND 



ANNALS OF PHILOSOPHY. 



[NEW SERIES.] 



MARCH 1828. 



XXVI. On the Regular or Platonic Solids. By Davies 

 Gilbert, F.R.S. fyc. 



To Mr. Richard Taylor. 

 Sir, 

 T^HE following trifle, extracted from my memorandum-book 

 A of many years standing, may perhaps (as a mere curiosity) 

 be favoured with a place in your Journal. It exhibits an easy 

 method of deriving these bodies from a modern discovery in 

 spherics ; and it illustrates the extreme generalization attached 

 to algebraic expressions. 



Montucla and other writers on mathematics, attribute to 

 Albert Girard, about the year 1629, the discovery of this very 

 curious property of the sphere : — That if the whole surface of 

 the sphere, or what may be considered as its subtense from 

 the centre, be represented by eight right angles ; then will the 

 subtense from the centre of any figure, on the surface of the 

 sphere bounded by arches of great circles, equal the excess of 

 the spherical angles above the plane angles formed by the 

 chords of the sides. A discovery applied in recent times with 

 great advantage to all geodetical operations conducted on an 

 extensive scale. 



Now as three planes at the least must meet at each solid 

 angle, or point of every solid ; it is obvious that no regular 

 equal and similar planes of any figures exceeding the pentagon 

 in number of sides, can possibly meet in one point or solid 

 angle. Thus the angle between two sides of an hexagon 

 being 1^ right angle, three of these would make up 4 right 

 angles, and consequently the sides would expand themselves 



NewSei'ies. Vol. 3. No. 15. March 1828. Y into 



