to those Who Believe that Art and Science are separate fields of human intellectual activity: Leonardo, Scientist and Artist would not agree.

The study of perspective in Art was mathematically determined, it was a scientific study of how the appearance in reality can be transported on a canvas or any bidimensional surface Rule of plant growth Paris Manuscript M Late 1490s-1500 According to Leonardo all things in nature were governed by mathematical rule. Even the growth patterns and forms of plants were conditioned by mathematical principals and geometry. On Fol 78v, Leonardo claimed that each year when the branches of plants have concluded their maturation, when added together, the sum total of their cross-section is equal to the cross-section of the trunk. In the accompanying drawing, he illustrated the principle by applying geometry to the problem. In the upper diagram, a tree is shown intersected by a series of arcs centred on the same point located on the trunk. Each of the arcs represents a “degree of ramification” at various points labelled ik, gh, ef, cd and ab. The diagram illustrates Leonardo’s theory that the cross-sectional areas of the branches will all be “equal among themselves as long as the tree is not deformed”. In the lower diagram, the branches of the tree are enclosed within an “analytical circle” in order to illustrate how the “ramifications proceed radially from a point at the centre of the crown of the tree”, marked m. In Leonardo's wordsThe elements of mathematics, that is to say number and measure, termed arithmetic and geometry, discourse with supreme truth on discontinuous and continuous quantities. Here no one argues that twice three makes more or less than six, nor that a triangle has angles smaller than two right angles, but with eternal silence, every dissension is destroyed, and in tranquility these sciences are relished by their devotees. The smallest of Leonardo’s surviving manuscripts, Manuscript M focuses on the study of Euclid’s geometry, ballistics and botany, with important theories regarding the growth of trees. This notebook also provides evidence of Leonardo’s contacts with Donato Bramante, who became the most important high Renaissance architect, and designer of the new St. Peter’s Basilica in Rome. The manuscript consists of 48 folios in their original binding. Medium Pen and ink on paper Size 10 x 7 cm Location Bibliothèque de l’Institut de France Geometric puzzles Paris Manuscript E 1513-14 Leonardo relished the challenge of geometric puzzles. Sheet 24v of Manuscript E is one of many devoted to the ancient Archimedean conundrum of “squaring the circle”, that is how to construct a square equal in area to a given circle. For this method, Leonardo used a sphere instead of a circle. He claims that a calculation of the area one eighth of the sphere will provide the solution, and so he proceeds to divide it into eight equal parts. He then takes one part and divides it into equal sections, noting how the curvature of the two sides becomes straightened, rather mysteriously “by motion over a plane location”. The next diagram shows the section transformed into a quarter circle divided into equal radiant segments. Finally, these are re-arranged to form a rectangle, but not before their curved edges are “made straight by motion”. Leonardo never did solve the problem. In fact it was impossible. The effort that he expended on it now seems rather poignant. However, his understanding of geometry gained by this type of study lead to a deeper understanding of organic structures such as the valves of the heart. In Leonardo's wordsThe night of St. Andrew’s day, I came to the end of the squaring of the circle and it was the end of the night and of the paper on which I was writing. It was concluded at the end of the hour. Painter's perspective Annunciation 1472-74 By Leonardo’s time, artist’s perspective had become an established tool in Florentine workshops. This was a system that created the illusion of space based on the principle that the scale of all elements in a painting was governed by their position in space. The Annunciation indicates that Leonardo clearly knew about artist’s perspective by the early 1470s. The tiled floor pattern on the right is diagonally orientated, indicating that the lateral point system has been applied. This was based on two vanishing points placed on the horizon line, and had long been in use by Florentine artists (one roughly where the distant town is seen and the other on the upper right edge of the picture). Leonardo’s perspective has a distorting effect due to the sudden transition from near to distant space. He wanted to make the figures as large as possible, (the Virgin’s palace is huge), while simultaneously including a panoramic view into the landscape. As such, the painting can be seen as the product of the ambitions of a young artist, albeit one who was still learning the subtleties of his craft. In Leonardo's wordsPerspective is nothing else than the seeing of an object behind a sheet of glass…these things approach the point of the eye in pyramids, and these pyramids are cut by the said glass. Paris Manuscript A c1492 Trail Catalogue No problem was too great for Leonardo! On sheet 21r, he describes an imaginative method for finding “the true height of the sun”. First, we must find two mountains within sight of each other, “as far apart as possible and of the same height”. We must then position ourselves on the summit of one and a friend ‘who is diligent’ on the other, and both build wooden huts with small apertures through which the sun may shine. When the sun makes a perpendicular angle in the first hut, a ready-prepared fire must be lit. This will make a large cloud of smoke that will alert our friend, who will then mark the point where the sun enters their hut. The result is the creation of a right-angle triangle as illustrated in the drawing, from which the result in theory would be calculable by simple trigonometry. Giovanni Cassini made the first scientific measurement of the earth-sun distance in 1672 by calculating the distance of Mars. It is worth noting that he did so by observing Mars from two places at the same time. In Leonardo's wordsThere is not to be seen in the universe a body of greater magnitude and power than the sun; and its light illuminates all the celestial bodies distributed through the universe; and the life forces descend from it, because the heat which is in living animals comes from life forces and no other heat is there in the universe, as will be shown in the fourth book. Transformation Engraving of a dodecahedron after Leonardo 1509 Luca Pacioli’s treatise on proportions, De Divina Proportione, was originally produced in two manuscripts for Ludovico Sforza and Galeazzo Sanseverino in 1498. The printed version that appeared in 1509 contained engravings of the platonic solids based on drawings by Leonardo. For the first time, Leonardo’s work was published. The platonic solids are three-dimensional shapes made up of polygonal faces, called polyhedra (cubes, prisms and pyramids are all polyhedra). There are five key solids - the tetrahedron (four sides), the cube (six sides), the octahedron (eight sides), the dodechahedron (twelve sides) and the icosahedron (twenty sides). By truncating these forms or adding “stellations”, new forms known as derivatives can be created, in a process of transformation that fascinated Leonardo. Leonardo drew the platonic solids and some of their derivatives not only in solid but also in skeletal form. This technique was highly innovative and beautifully effective as a means of conveying the three-dimensionality of the polyhedra, which are notoriously difficult to draw. Only the more complex polyhedra shown in the engravings of the printed version of Pacioli’s treatise, one of which is shown here, are based on Leonardo’s drawings. In Leonardo's wordsThe other proof which Plato gave to those of Delos is not geometry, because you proceed by the instrument of compasses and ruler, and experience shows it to us. But this is an occupation of the mind and as a consequence, geometry Leonardo drew a series of geometrical bodies for the manuscript versions of Luca Pacioli’s treatise on proportions. Two of the original manuscripts have survived. The copy in the Biblioteca Ambrosiana, Milan, is dated 1498 and is of superior quality. It is dedicated to Galeazzo Sanseverino, commander of Ludovico Sforza’s army. The other, which is now in the Bibliothèque Publique et Universitaire, Geneva, was dedicated to Ludovico Sforza himself. It appears to be a compilation of notes and illustrations of lesser quality, probably compiled after the departure of Leonardo from Milan. Luca’s treatise was published in Venice in 1509. Each of the regular solids together with some of their derivatives are illustrated in solid and skeletal form. The engravings of the more complex forms are based on Leonardo’s drawings. The body illustrated in this engraving is a dodecahedron (twelve-sided). It has been truncated to produce triangular faces, and then stellated with equilateral pyramids added on each of its faces. The engraving shows the skeletal form drawn in perspective. It is possible that Leonardo drew such complex geometric bodies with the aid of a drawing frame or “perspective window” as illustrated in the Codex Atlanticus for the depiction of an armillary sphere. Medium Engraving Size 29 x 45 cm Location Bibliothèque Nationale de France