No. 149.] 209 



always been measured by cubes ; and the la^ s of geometrical 

 operation recognize but one kind of unit, and that is alwiiys a 

 perfect cube. The law/ of geometry make lines and surlaces to 

 be as truly solids^ that is, to consist of cubic quantity^ iis the solid 

 itself. Lines and surfaces are n > part or property of the solid, 

 but separate and distinct quantities applied to the solid by the 

 laws of geometry for the purposes of measurement; and those 

 laws make every surface to have the thickness of the cunic unit 

 used in the calculation, and every line to have the breadth and 

 thickness of the cubic unit. 



Mr. Smith then proceeded to demonstrate the truth of his 

 theory by the vieasurement of water in diiferent gecmelrical forms. 

 For this purpose he had an apparatus constructed of tin vessels, 

 upon the scale of a cubic inch for the unit. This apparatus con- 

 sisted of sphere?, tetrahedron^, cubes, octahedrons, cylinders 

 and cones; and one inch being taken for the unit, the surfaces 

 consisted of pans one inch in depth, exactly covering every face 

 of each solid ; and the lines consisted of tubes one inch square, 

 that is, having the breadth and thickness of the cubic unit. The 

 proper diameter of the quantity of extension of all solids, ac- 

 cording to Mr. Smith, is an inscribed sphere ; and computing 

 any solid in numbers., if diameter is one, the contents or solidity 

 equals one-six h of the surface; if diameter is three, solidity 

 equals half the surface; if diameter is six, solidity equals the 

 surface, &c. Mathematicians have always said there is no geo- 

 metrical agreement between the solid and its surface because the 

 surface has no thickness ; but in every instance Mr. Smith show- 

 ed that, if the diameter was one inch, the solid six times filled 

 just filled its surface pans ; if the diameter was three inches, the 

 solid twice filled just filled its surface pans; and if the diameter 

 was six inches, the solid once filled just filled its surface pans. 

 This law extended to the cylinder and the cone as perfectly as 

 to the cube or the tetrahedron. He also poured solids into lines 

 and surlaces into lines, showing in every instance exactly the 

 same agreement between them in cubic quantity that is known to 

 exist in arithmetical yiumbers; for instance, if an octahedron, 

 whose diameter is wie, be calculated in numbers, the contents or 



[Assembly, No. 149.] 



