In 1418, the Wool Guild , which was overseeing construction, announced a competition to find a project to erect the cupola of the Duomo of Florence, which had to rise from the octagonal tambour; a project thus far unrealised given the difficulty, especially in technical terms, that the task presented. The height and width of the structure would not allow for, in fact, the use of traditional methods of scaffolding in wood from ground level, and a centred framework.
In 1420, having presented his brickwork model, which was created with the help of Donatello and Nanni di Banco, Brunelleschi was nominated, along with Ghiberti and Battista d’Antonio, ‘superintendent of the cupola’. By 1423, however, his was the only name which appeared as ‘inventor and governor of the great cupola’.
The work was long and difficult, but Brunelleschi managed to resolve all problems with technical skill, proposing to erect the cupola without using scaffolding, having invented a new technique based on mathematical calculation. Polygonal cupolas, such as the S. Maria Del Fiore’s octagonal one, have a more complex articulation than circular cupolas, which are achieved by rotating a quarter-circle around a vertical axis. In order to create a cupola without using scaffolding, it is necessary to complete, one by one, the various rings of which it is composed.
Brunelleschi, so as to guarantee the stability of the brickwork -  which was laid out on inclined foundations - prior to closing the rings, placed bricks with their long side sticking out with respect to those resting on the conical surface; thus realising the so-called herring-bone effect, which appeared as a spiralling warp.
The model of the constructive geometry of Brunelleschi’s cupola reveals two other fundamental characteristics. The visualisation of the method of  determining the  sharp curvature of the costoloni (ribs) supporting the cupola. This was achieved by dividing into five equal parts, the diameter of the circle on which is inscribed the octagon inside the cupola, and then outlining an opening which is is equal to 4/5 of the diameter. Also evident is the inclination of the masonry, known as ‘corda blanda’ (loose-roped), because it corresponds to that of a rope which is not taut. This is due to the fact that the foundations on which the bricks lie are placed on the surface of an upside-down cone, the axis of which coincides with the axis of the cupola. The top of the cone, a bit at a time, moves up as the work goes ahead. This method of construction corresponds to that which  you can observe when you use a pencil sharpener. The polygonal pencil corresponds to the cupola, whilst the conical cavity of the sharpener represents the upturned cone. The sharpener transforms the vertical walls of the pencil into a conical surface.
The equilibrium of the cupola is achieved through the use of an underlying roof, which is self-supporting and attached to the outer shell via a scaffolding of vertical and horizontal elements.
The tribunes, which serve to cover the connecting structure of the apses, are based on the form of a circle. Their semi-circular perimeter has been hollowed out of niches, which are divided by ornate pilasters, each one of which is decorated by twin Corinthian semi-columns.
The effect is one of great movement and plasticity, accentuated by the profound chiaroscuro obtained by alternating space and matter.
The cupola is a volume which is well-defined in space, and rationally dominated, and thus represents the first example of the application of the principles of the Renaissance.
The structure of the white marble costoloni coincides with the corners of the octagonal tambour, thereby creating eight  prominent ribs, which stand out against the red tiles.
To crown his work, Brunelleschi came up with a lantern in the form of a little, circular temple. (The model was approved in 1463.) Its elegant, modern structure acted as the pivot for the whole system. In the lantern, the eight costoloni of the cupola are reflected in an equal number of rampant arches which, dividing the high windows, sustain the spire, and draw your gaze upwards.
 

Measurements and Characteristics.

The impost of the cupola, which is 33 metres high, begins 55 metres above the ground. The distance between the facing corners of the octagon is about 55 metres. The height of the lantern is greater than 22 metres. The weight of the cupola has been estimated at about 37,000 tons, whilst it has been calculated that some 4 million bricks of differing shapes and sizes have been used in its construction.

The Building Site of the Cupola.

The perfect organisation of the building site, and the availability of the most efficient tools, such as ulivelle, a wedge-shaped gadget used for making an aperture in stone blocks, and turnbuckles, and machines able to lift enormous weights, played a determining role in Brubelleschi’s achievement. Filippo invented a variety of machines for the different tasks, of which he left neither drawings or verbal descriptions. We have knowledge of these thanks to the drawings of Taccola, Francesco di Giorgio, Giuliano da Sangallo, Bonaccorso Ghilberti, and the young Leonardo. One of these machines is the winch with three speeds, which, powered by a pair of oxen, guaranteed the lifting from the ground of all the materials necessary for the construction of the cupola. Enormous in its proportions, the winch could work at three different speeds, depending upon the load that needed to be lifted.

Perspective, in art, is the system of representative conventions which seeks to recreate on the two-dimensional surface of the canvas or bas-relief, the depth of real space. Perspective is based on the elementary laws of optics, and in particular on the fact that distant objects appear smaller and less well-defined than those which are nearer. ‘Linear’ perspective graphically reproduces the effect of the reduction in scale of objects as determined by their distance. ‘Aerial’ perspective reproduces the effects of the atmosphere and light, also in relation to increasing distance, for example in the variation of colour apparent in the mountains when viewed from a distance.

The simplest intuitinve example of the principle of linear perspective is that offered by the optical illusion of railway tracks, which seem to draw ever closer until they finally join together on the horizon. In perspective drawing, the surface of the sheet of paper or canvas is called the ‘projection plane’.  The ‘horizon’ is the line that divides the projection plane, thereby individuating the height of the point of view of the ideal observer. The ‘vanishing point’, placed on the horizon, is that in which all depth lines converge. There can be more than one vanishing point, depending on the objects present in the scene depicted.

The ancient Egyptians, Greeks and Romans indicated spatial depth through a series of more or less rudimental expedients, such as the partial superimposition of figures. In Rome, the word perspectiva (from the verb perspicere, ‘to see clearly’) referred to the science of vision, and corresponded to the Greek ottica; optics. That said, notwithstanding the fact that at times they utilised the apparent convergency of parallel depth lines, Greek and Roman painters and scenographers, tied, as they were, to the experience of  real vision, never managed to create a fixed and immutable ‘point of view’ capable of  coordinating all the features of vision.

Scientific understanding of the laws of perpective is a relatively recent historical aquisition. They were first described accurately in Italy, in the 1400s. By the beginning of that century, artists had developed an intuitive understanding of perspective, but it was the Florentine architect, Filippo Brunelleschi, who, with a series of experiments between 1417 and 1420, indicated precisely the laws of linear, centric perspective. The Florentine painters Masaccio and Paolo Uccello were among the first to apply Brunelleschi’s rules of perpsective; this being a real turning pointin the birth of the artistic culture of the Renaissance.

In 1435, the architect Leon Battista Alberti wrote, in Latin, the treatise Della pittura, which was subsequently published in Italian in 1436. The treatise outlined and explained Brunelleschi’s method and fixed the theoretical bases for all future developments of perspective theory. Further theoretical clarifications and practical practical applications were proposed by Piero della Francesca and Leonardo da Vinci.

The command of the laws of aerial and linear perspective made a huge impact on the development of artistic production in the West. It was not until the 20th Century, and in particular given the experience of the avant-garde, that the notions of perspective theorised in the Renaissance, began to be questioned in the figurative arts. Moreover the search for a solution, which might be defined scientific, to the problem of the representation of depth, is something that is unique to Western culture. The problem has never been posed anywhere else in a similar way; not even by civilisations whose art enjoyed a level of great refinement and complexity, such as in the Far East or in Pre-Columbian America.