Why, in the portrait shown here, is the founding father of the Roman academy of arts, Federico Zuccari, depicted displaying some rudimentary geometric diagrams? He shows us a point, parallel and perpendicular lines, angles of 60 and 45 degrees, triangles, a circle, a semicircle and a square. Below the diagrams is a sectioned human figure with an oddly detached arm.

Credit: ACCADEMIA NAZIONALE DI SAN LUCA, ROME

He has drawn these diagrams on a board or primed canvas with the white chalk gripped in its metal holder. His right hand, emerging from its white linen cuff, is poised with the precise refinement of a Victorian lady sipping tea from the finest porcelain.

The posthumous portrait of Zuccari, which can be seen in the exhibition 'Dürer e l'Italia' (Dürer and Italy) at the Scuderie del Quirinale in Rome until 10 June, was painted by Giovanni Maria Morandi in 1695 to mark the centenary of the Accademia di San Luca. St Luke was the patron saint of painters, which is why the Italian art academy bears his name. The large creased document in the painting is a Papal bull assigning the Church of Santa Martina to the painters as the base for their devotions.

Zuccari's pens and brushes protrude through the thumbhole of his painter's palette, held by a knotted ribbon. We can easily read the Latin inscription, “AEQUA POTESTAS”, a phrase from Ars poetica by the Roman author Horace that alludes to the “equal daring” of the poet and painter in creative invention. The painter, then, is the equal of the poet and operates with a high level of papal favour. But why the geometry?

The answer lies with what had become the stock opening sections of Renaissance treatises on geometry for painters. Leon Battista Alberti wrote the first of these, De pictura, in 1435. He began with the definitions of point, line, surfaces and triangles, taking his cue from Euclid. Step by step, he built towards the three-dimensional construction of painter's perspective according to the rules of geometric optics.

This euclidian grounding became a long-lasting tradition. When the English landscape painter J. M. W. Turner was appointed professor of perspective at the Royal Academy of Arts in London in 1807, he schooled himself and his charges in the basics of Euclid's Elements. To a modern spectator familiar with Turner's evanescent landscapes suffused with veils of colour, this dedication of the geometric basics comes as something of a surprise.

Zuccari was an intellectually ambitious author on the visual arts, well versed in all the leading ideas. The sectioned figure testifies to his knowledge. It is taken from Four Books on Human Proportion by Albrecht Dürer, the German painter and print-maker. Dürer was the first to formulate systems of proportion for figures of all types: fat and thin, tall and short, stocky and attenuated, and even children.

Dürer's books on proportion and geometry have a claim to be the most influential of all artists' books on the worlds of art and science, as they pioneer several practical techniques in geometry. Zuccari has copied the figure, whose head is 1/7 of his total height, larger than the norm.

Geometry was not just useful for perspective and proportions, but was something that should be seen to be known. Emerging in the Renaissance from a predominantly artisanal tradition, artists were keen to brandish the 'science' of their art. Geometry played a crucial role in their quest for more elevated status, both intellectually and socially.

As with any profession that harbours growing pretensions, the public parade of expertise is important. They sold their skills, so there was a little point in painters having knowledge of geometry unless they were seen to have it.