CHAPTER LX¶
On Four-Sided (Quadrilateral) Lateral Columns and the Five Species of Quadrilaterals (Plates XLVII–XLVIII).
The second species of lateral columns comprises the four-sided or quadrilateral columns (colonne laterate quadrilatere, Plates XLVII–XLVIII), which have two parallel quadrangular bases (upper and lower) surrounded by four rectangular or parallelogrammic lateral faces.
Because these columns vary according to the shape of their quadrangular bases, Your Highness must recall from the beginning of the first book of Euclid (as preserved in the Arabic-Latin tradition of Campanus) that all rectilinear four-sided plane figures are divided into four regular/parallelogrammic species and a fifth irregular class (illustrated by the five marginal diagrams 0053.g2.png and 0054.g1.png–0054.g5.png):
The Five Species of Rectilinear Quadrilaterals ([I.Def.22])
Figure A — Quadrato (Square): Has all four sides equal and all four angles right angles (tetragonon).
Figure B — Tetragono Longo (Oblong Rectangle): Has all four angles right angles, and opposite sides equal and parallel, but is longer than it is wide (heteromekes).
Figure C — Elmuaym or Rombo (Rhombus): Has all four sides equal and opposite sides parallel, but its angles are not right angles (rhombos).
Figure D — Simile al Elmuaym or Romboide (Rhomboid / Parallelogram): Has only opposite sides equal and parallel, and its angles are not right angles (rhomboeides).
Figure E — Elmuariffe (Trapezia / Irregular Quadrilaterals): All other four-sided figures beyond the first four species—having neither all sides equal nor opposite sides both equal and parallel—are called elmuariffe in the Arabic-Latin geometry (that is, irregulari or trapezia, trapezia).
According to all these variations of their bases, quadrilateral columns may be equilateral or inequilateral, rectangular or oblique; yet in every case their altitude (altezza) is understood as the perpendicular distance (equidistantia) between their upper and lower bases, and every such column is bounded by 6 quadrilateral surfaces (2 bases and 4 lateral faces), 8 solid angles, and 12 edges.