“366 Foragn Laterature and Science. 
The work under consideration contains a description of a 
variety of solids hitherto unnoted, and a number of new and 
remarkable properties of those solids that have been long 
known. In tracing the properties of the platonic bodies, the 
author shows that they naturally divide themselves into two 
series, each consisting of five solids; and, what is remarka- 
ble, that each individual solid, in one of the series, is to be 
found in great abundance among crystals, whereas not a sin- 
gle individual in the other series has ever been found among 
such productions. The first he calls the natural, the other 
the artificial series. ‘These two series bear a strong resem- 
blance to each other; inasmuch as the last in each series 
contains all the foregoing in the same series: the angular 
points of the contained solids may be traced out in the sur- 
face of the last solid : and what perhaps is equally remark- 
able is, that the whole of the solids composing the natural 
series are commensurable with each other when the first 
four are contained in the last, and that they are to each oth- 
er as the numbers 1, 3, 4,6 and 8. There is another solid 
whose extremities may be traced out in the surface of the 
last of the natural series, which solid the author calls a cu- 
boctahedron ; this solid, though it is commensurable with 
the rest, is not simple, being as 525: consequently it is 
somewhat less than the fourth, being to it as 80: 81. The 
author has combined the solids belonging to the natural se- 
ries in pairs, in every possible manner, and given the ratios 
of their volumes in two tables; he has likewise given the 
ratios of a number of remarkable lines in or upon the solids, 
and has shown how each may be extracted from the others. 
The ratios between the members of the artificial series 
appear to be mcommensurable, except in one instance, 
on which account they make a very striking contrast with 
the natural series, which are all commensurable. He 
afterwards describes five distinct dodecahedrons, which all 
admit of indefinite variation, and which, with the two before 
described, make seven; the whole of which are shewn to 
be singularly related with the cube. ‘The descriptive part 
is followed by a series of demonstrations contained in fifty- 
three theorems, concluding with an appendix by Dr. Roget. 
containing a demonstration of the relations subsisting be- 
tween the numbers of the artificial series, and likewise be- 
tween their faces and their axes. ‘The whole is illustrated 
