The first painter who deliberately rather than instinctively based his compositions on geometrical principles seems to have been Fra Bartolommeo, in his Last Judgment, in the church of St. Maria Nuova, in Florence.  Symonds says of this picture, “Simple figures—­the pyramid and triangle, upright, inverted, and interwoven like the rhymes of a sonnet—­form the basis of the composition.  This system was adhered to by the Fratre in all his subsequent works” (Illustration 54).  Raphael, with that power of assimilation which distinguishes him among men of genius, learned from Fra Bartolommeo this method of disposing figures and combining them in masses with almost mathematical precision.  It would have been indeed surprising if Leonardo da Vinci, in whom the artist and the man of science were so wonderfully united, had not been greatly preoccupied with the mathematics of the art of painting.  His Madonna of the Rocks, and Virgin on the Lap of Saint Anne, in the Louvre, exhibit the very perfection of pyramidal composition.  It is however in his masterpiece, The Last Supper, that he combines geometrical symmetry and precision with perfect naturalness and freedom in the grouping of individually interesting and dramatic figures.  Michael Angelo, Andrea del Sarto, and the great Venetians, in whose work the art of painting may be said to have culminated, recognized and obeyed those mathematical laws of composition known to their immediate predecessors, and the decadence of the art in the ensuing period may be traced not alone to the false sentiment and affectation of the times, but also in the abandonment by the artists of those obscurely geometrical arrangements and groupings which in the works of the greatest masters so satisfy the eye and haunt the memory of the beholder (Illustrations 55, 56).

[Illustration 55:  THE EMPLOYMENT OF THE EQUILATERAL TRIANGLE IN RENAISSANCE PAINTING]

[Illustration 56:  GEOMETRICAL BASIS OF THE SISTINE CEILING PAINTINGS]

[Illustration 57:  ASSYRIAN; GREEK]

[Illustration 58:  THE GEOMETRICAL BASIS OF THE PLAN IN ARCHITECTURAL DESIGN]

[Illustration 59]

Sculpture, even more than painting, is based on geometry.  The colossi of Egypt, the bas-reliefs of Assyria, the figured pediments and metopes of the temples of Greece, the carved tombs of Revenna, the Della Robbia lunettes, the sculptured tympani of Gothic church portals, all alike lend themselves in greater or less degree to a geometrical synopsis (Illustration 57).  Whenever sculpture suffered divorce from architecture the geometrical element became less prominent, doubtless because of all the arts architecture is the most clearly and closely related to geometry.  Indeed, it may be said that architecture is geometry made visible, in the same sense that music is number made audible.  A building is an aggregation of the commonest geometrical forms:  parallelograms, prisms, pyramids and cones—­the cylinder appearing in the column, and the hemisphere in the dome.  The plans likewise of the world’s famous buildings reduced to their simplest expression are discovered to resolve themselves into a few simple geometrical figures. (Illustration 58).  This is the “bed rock” of all excellent design.