Let us look at the three sets of material which are presented after the exercises with the solid insets, _i.e._, the material for teaching _size_ (the pink cubes), _thickness_ (the brown prisms), and _length_ (the green rods). There is a definite relation between the ten pieces of each series. In the material for length the shortest piece is a _unit of measurement_ for all the rest; the second piece is double the first, the third is three times the first, etc., and, whilst the scale of length increases by ten centimeters for each piece, the other dimensions remain constant (_i.e._, the rods all have the same section).
The pieces then stand in the same relation to one another as the natural series of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
In the second series, namely, that which shows _thickness_, whilst the length remains constant, the square section of the prisms varies. The result is that the sides of the square sections vary according to the series of natural numbers, _i.e._, in the first prism, the square of the section has sides of one centimeter, in the second of two centimeters, in the third of three centimeters, etc., and so on until the tenth, in which the square of the section has sides of ten centimeters. The prisms therefore are in the same proportion to one another as the numbers of the series of squares (1, 4, 9, etc.), for it would take four prisms of the first size to make the second, nine to make the third, etc. The pieces which make up the series for teaching thickness are therefore in the following proportion: 1 : 4 : 9 : 16 : 25 : 36 : 49 : 64 : 81 : 100.
In the case of the pink cubes the edge increases according to the numerical series, _i.e._, the first cube has an edge of one centimeter, the second of two centimeters, the third of three centimeters, and so on, to the tenth cube, which has an edge of ten centimeters. Hence the relation in volume between them is that of the cubes of the series of numbers from one to ten, _i.e._, 1 : 8: 27 : 64: 125 : 216 : 343 : 512 : 729 : 1000. In fact, to make up the volume of the second pink cube, eight of the first little cubes would be required; to make up the volume of the third, twenty-seven would be required, and so on.