Mathematics

Leonardo’s admiration for mathematics was unconditional, and found expression in his writings in such statements as “No certainty exists where none of the mathematical sciences can be applied” (MS G, fol. 96v). It is therefore useful to consider the development of his mathematical thought, drawing upon the whole of his manuscripts in order to reconstruct its principal stages; such a study will also serve to illustrate the sources for much of his work.

Leonardo particularly valued the rigorous logic implicit in mathematics, whereby the mathematician could hope to attain truth with the same certitude as might a physicist dealing with experimental data. (Students of the moral and metaphysical sciences, on the other hand, had no such expectation, since they were forced to proceed from unascertainable and infinitely arguable hypotheses.) His predilection did not, however, presuppose a broad mathematical culture or any real talent for calculation. Certainly, his education was that of an artisan rather than a mathematician, consisting of reading, writing, and the practical basics of calculus and geometry, together with the considerable body of practical rules that had been accumulated by generations of craftsmen and artists.

The effects of his early education may be seen in Leonardo’s literary style (he preferred aphorisms and definitions to any prolonged organic development of ideas) and in his mathematics, which contains grave oversights not entirely due to haste. An example of the latter is his embarrassment when confronted with square and cube roots. In Codex Arundel (fol. 200r) he proposed, as an original discover), a simple method for finding all roots “both irrational and rational,” whereby the “root” is defined as a fraction of which the numerator alone is multiplied by itself one or two times to find the square or cubic figure. Thus the square root of 2 is obtained by multiplying 2/2 by 2/2 to find 4/2, or simply 2 ; while the cube root of 3 is reached by taking 3/9 x 3/9 x 3/9 = 27/9, or 3. He applied this erroneous method several times in his later work; however absurd they might be, Leonardo seems to have been prouder of his discoveries in mathematics than in any other field.

When he was in his late thirties, Leonardo began to try to fill the gaps in his education and took up the serious study of Latin and geometry, among other subjects. At the same time he began to write one or more treatises. He wished to write for scientists, although scholars of the period were not prepared to admit the knowledge possessed by an artist as science. (An illustration of this occurs in Alberti’s definition of the “principles” of geometry, in which he stated that he could not use truly scientific terms because he was addressing himself to painters.) It was only after Leonardo entered the Sforza service in Milan and became associated with philosophers and men of letters on a basis of mutual esteem that he took up his theory of the supremacy of painting. This theory was in part incorporated into the Treatise on Painting compiled by Leonardo’s disciple Francesco Melzi, who, however, omitted Leonardo’s definitions of the geometric “principles” of point, line, and area—definitions that Leonardo had drawn up with great care, since he wished them to be read by mathematicians rather than artists.

As a painter, Leonardo was, of course, concerned with proportion and also with “vividness of the actions” (this is how he translated Ficino’s definition of beauty as “actus vivacitas"). In his pursuit of the latter, Leonardo inclined toward physics, seeking motions that could be studied experimentally. Perspective, too, had an important place in the treatise on painting, but Leonardo was more concerned with aerial perspective (chiaroscuro) than with linear. MS C (1490) is rich in geometrical drawings, although limited to the depiction of the projection of rays of light; it contains no organic system of theorems.

Aside from those in MS C, the pages dedicated to geometry in other early manuscripts are few; there are nine in MS B (ca. 1489) and nineteen in MS A (1492). The prevalent matter in all of these is the division of the circumference of the circle into equal parts, a prerequisite for the construction of polygons and for various other well-known procedures. The notes set down are not connected with each other and represent elementary precepts that were common knowledge to all “engineers,” Because of these limitations, it is impossible to accept Caversazzi’s thesis concerning MS B, folios 27v and 40r, wherein Leonardo explained how to divide a circumference into equal parts by merely “opening the compass” ; Caversazzi sees this as an “exquisite geometric discovery,” one that may be applied to solve Euclidean problems of the first and second degree. But since at this time Leonardo had not yet begun to study Euclid, it is probable that he was referring to precepts known to all draftsmen.

Seven pages are devoted to geometry in Codex Forster 111 (1493-1495) and, again, these contain only elementary formulas, expressed in language more imaginative than scientific (the volume of a sphere, for example, is described as “the air enclosed within a spherical body” ). Only in Codex Forster II₁ written between 1495 and 1497, are there the first signs of a concentrated interest in geometrical problems, particularly those that were deeply to concern Leonardo in later years—lunes and the equivalence of rectilinear and curvilinear surfaces. An additional seventeen pages of this manuscript are devoted to the theory of proportion.

The two manuscripts that mark the close of Leonardo’s first sojourn in Milan-MS M and MS I-are of far greater importance. The first thirty-six pages of MS M contain translations of Euclid (“Petitioni” and “Conceptioni”) as well as derivations of propositions 1-42 (with a few omissions) of book I of the Elements, together with a group of propositions from the tenth book. Propositions 43-46 of book I appear in the first sixteen folios of MS I (which must for that reason be considered as being of a later date than MS M), as do propositions 1-4 and 6-10 of book 11, selections from book 111, and occasional references to book X. These evidences of Leonardo’s systematic study of Euclid may be related to his friendship and collaboration with Luca Pacioli.

Leonardo must have read the Elements and acquired some deeper knowledge of geometry before undertaking the splendid drawings of solid bodies with which he illustrated the first book of Pacioli’s Dirina proportione. His interest in this work is probably reflected in the pages of Codex Forster II₁ on proportion; it would also explain his study of the tenth book of the Elements, the book least read because of its difficulty and its practical limitation to the construction of regular polygons. Pacioli was thus responsible for arousing Leonardo’s enthusiasm for geometry and for introducing him to Euclid’s work; indeed, he may have helped him read Euclid, since the text would have been extremely difficult for a man as relatively unlettered as Leonardo. (It is interesting to note that MS I also contains some first principles of Latin, copied from Perotti’s grammar.) Pacioli’s influence on Leonardo was probably also indirect, through his Summa arithtietica and perhaps through his translation of the Elements.

That Leonardo had at hand the Latin text of the Elements of 1482 or 1491 can be seen through the identity of some of his drawings with those texts as well as by certain verbal correspondences. At the same time, his method was not to transcribe sentences from the text he was studying but to attempt to present geometrical ideas graphically. each page of his geometrical notes represents an aid to memorizing Euclid’s text, rather than a compendium; hence it is not always easy to trace the specific passages studied. For example, Leonardo would often begin with a sketch, a number, or occasionally a word which he must have used initially to impress upon his mind some of the intricacies of Euclid’s theses and later to recall the whole content. In MS M, folio 29v, three figures appear: a point and a line; the same point and line with an additional transversal line; and two parallel lines joined by the transversal, the original point having disappeared. The correspondence with the Euclidean thesis, together with the coincidence of similar notes on the preceding and following pages to propositions 27, 28, 29, 30, and 32, respectively, prove this to be Euclid’s book 1, proposition 31.

Notes on Euclid’s first books also appear in certain folios of the Codex Atlauticus, where they are set out in better order and in a more complete form. On folio 169r-b, for instance, the “Petitioni” and “Conceptioni” are presented symbolically, while on folio 177v-a proposition 1.7 is transformed into a series of thirteen drawings.

Leonardo continued to work with Pacioli after they both left Milan in 1499. In the Codex Atlanticus he stated that he would learn “the multiplication of roots” from his friend “master Luca” (fol. 120r-d). and he had in fact transcribed all the rules for operations with fractions from the Summa arithmetica (fol. 69, a-b). Pietro da Novellara recorded in 1503 that Leonardo was neglecting painting in favor of geometry; this activity is reflected in MS K, Codex Madrid, 11, and Codex Forster_I,as well as in many folios of the Codex Atlanticus.

Two-thirds of MS K₁(1504) are devoted to Euclid. From folio 15v to folio 48v Leonardo copied, in reverse order, almost all the marginal figures of books V and VI of the Elements. He transcribed none of the text, although the drawings are accoumpanied by unmistakable signs of his contemplation of the theory of proportion. MS K₂ contains notes similar to those in MS M and MS 1; here they refer to the whole of the second book of the Elements, to a few propositions of books I, II and III, and to nine of the first sixteen definitions of book V.

Leonardo’s interest in the theory of proportion is further evident in Codex Madrid, II, folios 46v–50, in which he summarizes and describes the treatise “De proportionibus et proportionalitatibus” that is part of Pacioli’s Summa arithmetica. A drawing on folio 78 of the same manuscript, graphically illustrating all the various kinds of proportions and proportionalities, is also taken from Pacioli’s book, while Euclid reappears in the last five pages of the manuscript proper. In the latter portion Leonardo transcribed, in an elegant handwriting, the first pages of an Italian version of the Elements, the author of which is not known (the list of Leonardo’s books cites only “Euclid, in Italian, that is, the first three books” ). Folio 85r contains the ingenious illustration of Euclid’s so-called algorithm (VII.2, X.2-3) that is repeated in more detail in Codex Atlanticus, folio 207r-b.

It is thus clear from manuscript sources that from the time that Leonardo began to collaborate with Pacioli on the Divina proportione, he concentrated on books I and II of Euclid—an indispensable base—and on books V and VI, the theory of proportion that had also been treated by Pacioli in the Summa arithmetica. The few references to book X deal with the ratios of incommensurate quantities; therefore there can be no doubt that the subject of proportions and proportionalities remained a constant center of Leonardo’s interest.

There is no evidence of a similar study of the last books of Euclid, but it must not be forgotten that about two-thirds of Leonardo’s writings have been lost. There is, moreover, no lack of practical applications of the propositions of the last books in Leonardo’s writings. But Codex Madrid, II, is of particular importance because it demonstrates that Leonardo, after conducting a modest study of Euclid from approximately 1496 to 1504, began to conduct ambitious personal research. Having copied out the first pages of the Elements as, perhaps, a model, Leonardo wrote on folios IIIr and 112r two titles indicative of the plan of his work, “The Science of Equiparation” and “On the Equality of Unequal Areas.” For this new science he wrote on folio 107v a group of “Petitioni” and “Conceptioni” based upon Euclid’s. Here Leonardo’s chief concern lay in the squaring of curvilinear surfaces, which he generally divided into “falcates” (triangles with one, two, or three curved sides) and “portions” (circular segments formed by an are and a chord, which could also be the side of a polygon inscribed in a circle). He carried out the transformation of these figures into their rectilinear equivalents by various means, some of which (such as superimposition and motion, or rotation) were mechanical.

The use of mechanical solutions, rarely accepted by Euclid, may perhaps be attributable to Leonardo’s engineering background; he would almost seem to have recognized the unorthodoxy of his procedures in applying to himself Giordano’s words: “This method is not simply geometrical but is subordinated to, and participates in, both philosophy and geometry, because the proof is obtained by means of motion, although in the end all mathematical sciences are philosophical speculations” (Codex Madrid, II,fol. 107r). Leonardo noted, however, that the squaring of curvilinear surfaces could be accomplished by more orthodox geometrical methods, provided the curved sides of the figure form parts of circles proportional among themselves (Codex Atlanticus, fol. 139v-a).

It is a short step from squaring falcates to squaring the circle, and Leonardo proposed several solutions to the latter problem. Again, some of them were mechanical and were suggested to him by studying Vitruvius. These consist in, for example, measuring the rectilinear track left by a wheel of which the width is one-quarter of the wheel’s diameter (MS G, fol. 61r; also MS E, fol. 25v, where the width of the wheel that makes “a complete revolution” is mistakenly given as being equal to the radius). The circumference is obtained by winding a thread around the wheel, then withdrawing and measuring it (MS K, fol. 80r).

Leonardo was also well acquainted with the method for the “quadratura circuli per lunulas,” which he knew from the De expetendis et fugiendis rebus of Giorgio Valla. A drawing corresponding to this method appears in MS K, folio 61r. (The identity of “Zenophont” or “Zenophonte,” a mathematician criticized by Leonardo on the previous page of the same manuscript, is probably resolvable as Antiphon,

whose method of quadrature was also criticized by Valla.)

Leonardo of course knew Archimedes’ solution to squaring the circle; Clagett has pointed out that he both praised and criticized it without ever having understood it completely. He accepted Archimedes’ first proposition, which establishes the equivalence of a circle and a rectangular triangle of which the shorter sides are equal to the radius and the circumference of the circle, respectively; he remained unsatisfied with the third proposition, which fixes the approximate ratio between the circumference (taken as an element separating two contiguous classes of polygons) and the diameter as 22:7. Leonardo tried to take this approximation beyond the ninety-six-sided polygon; it is in this attempt at extension ad infinitum that the value of his effort lies.

Valla provided Leonardo with a starting point when, in describing the procedure of squaring “by lunes,” he recommended “trapezium dissolvatur in triangula. “Leonardo copied Valla’s drawing, dividing the trapezium inscribed in a semicircle into three triangles; he extended the concept in MS K , folio 80r, in which he split the circle into sixteen triangles, rectified the circumference, then fitted the triangles together like gear teeth, eight into eight, to attain a rectangle equivalent to the circle. He took a further step in Codex Madrid, II, folio 105v, when departing from Euclid 11.2 (circles are to each other as the squares of their diameters), he took two circles, having diameters of 1 and 14, respectively, and divided the larger into 196 sectors, each equivalent to the whole of the smaller circle.

In Codex Atlanticus, folio 118v-a, Leonardo dealt with circles of the ratio 1:1,000 and found that when the larger was split into 1,000 sectors, the “portion” (the difference between the sector and the triangle) was an “imperceptible quantity similar to the mathematical point.” By logical extension, if each sector were “a millionth of its circle, it would be a straight portion” or “almost plane, and thus we would have carried out a squaring nearer the truth than Archimedes”’ (Codex Madrid, II, fol. 105v). In Codex Arundel, folio 137v, Leonardo squared the two diameters, writing “one by one is one” and “one million by one million is one million”; his disappointment in seeing that the squares did not increase suggested the variant “four million by four million is sixteen million” (that he corrected his error in Codex Atlanticus, fol. 118r-a, writing “a million millions,” reveals a poor grasp of calculation).

Leonardo was clearly proud of his discovery of the quadrature, and in Codex Madrid, II, folio 112r he recorded the exact moment that it came to him—on the night of St. Andrew’s day (30 November), 1504, as hour, light, and page drew to a close. In actual fact, however, Leonardo’s discovery amounted only to affirming the equivalence of a given circle to an infinitesimal part of another without calculating any measure; to have made the necessary calculations, he would have had to use Archimedes’ formula 22:7, which he rejected. Leonardo characterized Archimedes’ method of squaring as “ben detta e male data” (Codex Atlanticus, fol. 85r-a), that is, “well said and badly given.” It would seem necessary to describe his own solution simply as “detta,” not “data.”

The presumed discovery on St. Andrew’s night did, however, encourage Leonardo to pursue his geometrical studies in the hope of making other new breakthroughs. MS K₃, MS F, and a large number of pages in the Codex Atlauticus demonstrate his continuous and intense work in transforming sectors, “falcates,” and “portions” into rectilinear figures. Taking as given that curved lines must have equal or proportional radii, Leonardo practiced constructing a series of squares doubled in succession, in which he inscribed circles proportioned in the same way; to obtain quadruple circles, he doubled the radii. To obtain submultiple circles, he divided a square constructed on the diameter of the circle into the required number of rectangles, transforming one into a square and inscribing the submultiple circle therein. In order to double the circle, however, instead of using the diagonal of the square constructed on the diameter, he turned to arithmetic calculus to increase proportionally the measure of the radius, adopting a ratio of 1 :3/2, which is somewhat greater than the true one of 1:1/2. To obtain a series of circles doubling one another, he divided the greatest radius into equal parts, forgetting that arithmetic progression is not the same as geometric (Codex Madrid, II, fols. 117v, 132).

Alternating with the many pages in the Codex Atlanticus devoted to the solution of specific problems are others giving the fundamental rules of Leonardo’s new science of comparison. Within the same sphere falls a group of pages in Codex Madrid, II, which are concerned with transforming rectilinear figures into other equivalent figures or solids into other solids. These sections of the two collections, presumably together with pages that are now lost, are preparatory to the short treatise on stereometry contained in Codex Forster I₁ (1505) and entitled “Book on Transformation, That Is, of One Body Into Another Without Decrease or Increase of Substance.” In his study of this treatise, Marcolongo judges it to be suitable “for draftsmen more skilled in handling rule, square, and compass than in making numerical calculations” —Leonardo’s viewpoint was often that of the engineer.

One of the problems to which Leonardo chose to apply his new science was that of how to transform a parallelepiped into a cube, or how to double the volume of a cube and then insert two proportional means between two segments. In the Summa arithmetica, Pacioli had solved the problem and had described how to find the cube root of 8 by geometrical methods, although he provided no geometrical demonstrations. Leonardo, in Codex Atlanticus, folio 58r, stated that if the edge of a cube is 4, the edge of the same cube doubled will be 5 plus a fraction that is “inexpressible and easier to make than to express.” By 1504 Leonardo also possessed Valla’s book, which gave the solution of the problem, with numerous demonstrations from the ancients. Leonardo copied various figures from Valla (in Codex Arundel, fols. 78r-79v, where the last one is, however, Pacioli’s) and transcribed a vernacular translation of the part of Valla’s book referring to the demonstrations of Philoponus and Parmenius. (That, having the Latin text at hand, Leonardo felt the need to translate or to have translated the two pages that interested him confirms his difficulty in reading Latin directly-Clagett and Marcolongo have both pointed out many mistakes in the translation of this passage which, if uncorrected, distort the text.)

Leonardo repeatedly expressed his dissatisfaction with the solution reported by Valla. He particularly objected to having to make the rule swing until the compass has fixed two points of intersection; this “negotiation,” he stated (Codex Atlanticus, fol. 218v-b), seemed to him “dubious and mechanical.” Since Valla attributed this procedure to Plato, Leonardo further stated, in MS F (1508), folio 59r, that “the proof given by Plato to the inhabitants of Delo is not geometrical.” He continued to look for other, more truly “geometrical,” means to double the cube or find the cube root.

In folios 50v-59v of MS F and in the similar folios 159r-a and b of the Codex Atlanticus, Leonardo tried, in opposition to the “ancient system,” to resolve the problem by decomposing and rebuilding a cube; he further attempted to apply Pythagoras’ theorem by substituting three cubes for three squares constructed on the sides of a right triangle. Since every square is equal to half the square constructed on its diagonal, he attempted to halve each cube with a diagonal cut, on the assumption that the face of a doubled cube could be obtained through manipulating the rectangular face formed by the cut. He arranged nine cubes in the form of a parallelepiped upon which he analyzed the diagonals that he supposed to represent the square and cube roots, respectively (Codex Atlanticus, fols. 159r-b, 303r-b; Windsor Collection, fol. 19128). He forgot, however, that the progression of squares and cubes does not correspond to the natural series of numbers. Although he studied the proportions between the areas that “cover” a cube and those of the same cube doubled, he realized that he could not apply “the science of cubes based on surfaces, but based on bodies [volumes]” (Codex Arundel, fol. 203r). He was thus aware that his formulation of the problem was inaccurate.

Leonardo finally arrived at a solution. He recorded in Codex Atlanticus, folios 218v-b, 231r-b, that in order to “avoid the difficulty of the mechanical system” taught by Plato and other ancients, he had

eliminated the compass and the imprecise movement of the rule in favor of placing two sides of the rectangle in the ratio b = 2a (thereby unknowingly reestablishing the ratio used in the Greek text). Step by step, then, his procedure was to join two faces of a cube to make a rectangle, connect the upper-left angle to the center of the right square, then carry the line obtained to the upper extension of side a; from the extreme point thus reached, he drew a line that, touching the upper-right angle of the rectangle, cut the extension of its base. On this line Leonardo was able to determine the measure of the edge of the doubled cube, a measure that is to be found four times in the figure thus constructed. His results are reasonably accurate; with a = 2, Leonardo obtained a cube root for 16 as little as 0.02967 in excess of its true value

Leonardo was unable to demonstrate the truth of his “new invention” save through criticism of the “mechanical test” of the compass, which after “laborious effort” helped the ancients to discover the proportional means—whereas now, Leonardo claimed, “without effort I use it to confirm that my experiment was confirmed by the ancients “—which, he added, “could not be done before our time.” What Leonardo had discovered was that the third proportional number, used to determine the second, coincides with a line that can be more accurately constructed inside the rectangle; he thereby simplified the classical procedure without in any way discussing its scientific demonstrations, which he took for granted. He must, however, have checked his results experimentally and judged them with his unerring eye.

Other pages of the Codex Atlanticus contain summaries of problems to be solved, as, for example, folio 139r-a, which is entitled “Curvilinear Geometrical Elements.” This, together with other titles- “On Transformation,” folio 128r-a; “Book on Equation,” folio 128r-a; and “Geometrical Play” ( “De ludo geometrico” ), folios 45v-a, 174v-b, 184v-c, and 259v-a —including the previously mentioned “Science of Equiparation” —might suggest that Leonardo had actually written systematic books and treatises that are now lost. But it is known that he cherished projects that he did not complete; and it is unlikely that he composed treatises using methods and forms different from those set out in, for example, Codex Forster I_2. The titles of treatises projected but unwritten do provide some record of the stages of his mathematical work, in addition to which Leonardo has recorded some specific dates for his successes.

Leonardo’s discovery on St. Andrew’a night has already been mentioned: in the same manuscript, Codex Madrid, II, on folio 118r, he further mentioned that a certain invention had been given to him “as a gift on Christmas morning 1504.” In Windsor Collection, folio 19145, he claimed to have discovered, after prolonged research, a way of squaring the angle of two curved sides on Sunday, 30 April 1509. (Folio 128r-a of the Codex Atlanticus, the beginning of the “Book of Equation,” is concerned with the squaring of a “portion” of a single curved side and promises a second book devoted to the new procedure.)

Although the last discovery is exactly dated, Leonardo’s explanation of it is not very clear. (This is often the case in Leonardo’s work, since the manuscripts are marked by incomplete and fragmentary discussions, geometrical figures unaccompanied by any explanation of the construction, and the bare statements of something proved elsewhere.) His accomplishment is, however, apparent—he had learned how to vary a “portion” in infinite figures, decomposing and rebuilding it in many different ways. The examples drawn on the Windsor Collectionpage cited reappear, with variations, in hundreds of illustrations scattered or collected in the Codex Atlanticus. The reader can only be dismayed by their complexity and monotony.

The mathematical pages of the Codex Atlanticus (to which all folio numbers hereafter cited refer) provide an interesting insight into the late developments of Leonardo’s method. On folio 139r-a, among other places, he confirmed his proposal to vary “to infinity” one or more surfaces while maintaining the same quantity. It is clear that he received from the elaboration of his geometric equations the same pleasure that a mathematician derives from the development of algebraic ones. If, at the beginning of his research on the measure of curvilinear surfaces, his interest would seem to have been that of an engineer, his later work would seem to be marked by a disinterested passion for his subject. Indeed, the title of the last book that he planned— “Geometrical Play,” or “De ludo geometrico” —which he began on folio 45v-a and worked on up to the last years of his life, indicates this.

For example, one of Leonardo’s basic exercises consists of inscribing a square within a circle, then joining the resultant four “portions” in twos to get

“bisangoli.” The “bisangles” can then be broken up, subdivided, and distributed within the circumference to create full and empty spaces which are to each other as the inscribed square is to the four “portions” of the circumscribed circle. An important development of this is Leonardo’s substitution of a hexagon for the square, the hexagon being described as the most perfect division of the circle (fols. 111v-s and b [Figs. 3. 4]). This leaves six “portions” for subsequent operations, and Leonardo recommended their subdivision in multiples of six. (On the splendid folio 110v-a [Fig. 5], he listed and calculated the first fifty multiples of six, but through a curious oversight the product of 34 × 6 is given as 104, rather than 204: and all the following calculations are thus incorrect.)

To obtain the exact dimensions of submultiple “portions,” Leonardo devised the “square of proportionality” (fol. 107v-a), which he constructed on the side of the hexagon inscribed within the circle and then subdivided into the required number of equal rectangles. He next transformed one such rectangle into the equivalent square, the side of which is the radius of the submultiplied circle, the side of the corresponding inscribed hexagon, and the origin or chord of the minor “portions.” These were then joined in pairs (“bisangoli.”), combined into rosettes or stars, and distributed inside the circumference of the circle in a fretwork of full and empty spaces, of which the area of all full spaces is always equal to that of the six greater “portions” and the area of all the empty spaces to the area of the inscribed hexagon. The result is of undoubted aesthetic value and may be taken as the culmination of the geometrical adventure that began for Leonardo with Pacioli’s Divina proportione.

Although Leonardo remained faithful throughout his life to the idea of proportion as the fundamental structure of reality, the insufficient and inconstant rigor of his scientific thesis precludes his being considered a mathematician in the true sense of the word. His work had no influence on the history of mathematics; it was organic to his reflections as an artist and “philosopher” (as he wished to call himself and as he was ultimately called by the king of France). In his geometrical research he drew upon Archimedes and the ancients’ doctrine of lunes to develop the most neglected parts of Euclid’s work—curvilinear angles, the squaring of curvilinear areas, and the infinite variation of forms of unaltered quantity. Thus, in Leonardo’s philosophy, does nature build and infinitely very her forms, from the simplest to the most complex; and even the most complex have a rational structure that defines their beauty. It was Leonardo’s profound intuition that “Necessity... bridle and eternal rule of Nature” must have a mathematical foundation and that the infinite forms found in nature must therefore be the infinite variations of a fundamental “equation.”

Leonardo thus expressed a Pythagorean or Platonic conception of reality, common to many artists of the Renaissance. It was at the same time a revolutionary view, inasmuch as it identified form as function and further integrated into the concept of function the medieval notion of substance. Leonardo did not formulate his conception of reality in the abstract terms of universal scientific law. Rather, as the purpose of his painting was always to render the most subtle designs of natural structures, his last geometric constructions were aimed at discovering the mathematical structure of nature.

Augusto Marinoni