The difference in form and content suggests that the Polygonal Numbers was not part of the larger work.

They contain, however, a fragment of a separate tract on Polygonal Numbers.

This may be rectangular in shape (" straight " shed), containing a series of parallel tracks on which the engines stand and which are reached by means of points and crossings diverging from a main track outside; or it may take a polygonal or circular form (round house or rotunda), the lines for the engines radiating from a turn-table which occupies the centre and can be rotated so as to serve any of the radiating lines.

An air-tube consists of an epithelium of large polygonal cells with a thin basement-membrane externally and y a chitinous layer internally, the lastnamed being continuous with the outer cuticle.

The chitinous layer is usually strengthened by thread like thickenings which, in the region close to the outer opening of the tube, form a network enclosing polygonal areas, but which, through most of the tracheal system, are arranged spirally, the strengthening thread not forming a continuous spiral, but being interrupted after a few turns around the tube.