Landscape filled a large place in the thoughts of Leonardo. His oldest-dated drawing—an Alpine, view—bears witness to the efforts he made in that direction, even in his youth! In the Trattato he often reverts to the subject. According to him, landscapes should be so represented that the trees are half in light, half in shadow, but the best way is to paint them when the sun is hidden by clouds, so that the trees may be illuminated by the general light of the sky, and shadowed by the universal shadow of the earth. "And these," he adds, "will be most obscure in the parts nearest to the centre of the tree, and to the earth."*

His studies of the proportions and movements of the human figure were intended to complete the Trattato. For the most part these researches were carried out between the years 1489 and 1498. At this latter date, Pacioli notes the completion of Leonardo's work in the dedication to his own De divina Proportione (" Leonardo da Vinci .... havenda gia. con tutta diligentia al degno libro de pictura e movimento humani posto fine2").

Naturally enough, Leonardo made use of the labours of his Greek and Roman predecessors. But on one occasion of his taking count of antique opinions he was ill-inspired. Basing himself on Vitruvius, he adopted eight heads, or ten faces, as the normal height of the human figure (cap. 264, etc.). Now this calculation is false. Modern science has proved that the normal height equals seven and a half heads, or, at most, seven and three quarters. As for the head itself, he divided it into 248,832 (?) parts, 12 grades, subdivided into 12 "punti," 12 "aminuti," 12 " minimi," and 12 "semi-minimi."3

All these studies of proportion have come down to us, partly in the manuscripts of Leonardo himself, partly in the echoes of his ideas to be found in Pacioli's treatise, De divina Proportione.

1 Manuscript G, folio 19.

2 Leonardo commenced the book entitled De Figura umana on April 2, 1489 (Richter, vol. ii., p. 415).—Zeising gives a very short resume of Leonardo's theory of proportions in his Neue Lehre von den Proporlionen des menschlichen Korpers (Leipzig, 1854, p. 50).

3 One might be tempted to believe that the engravings of Fra Giocondo (M. Vitruvius per Jocundum, 1511), and of Cesare Cesariano (Di Lucio Vitruvio Pollione de Architectura Libri decern; Como, 1521, fol. L), were taken from Leonardo's drawing of a man standing in a circle with outstretched arms and legs. It was not so. The engravings in question proceed naturally and inevitably from the text of Vitruvius.

A few words, before going farther, on this very common-place satellite of the great Leonardo.

Luca Pacioli was born at Borgo San Sepolcro in 1450; he was therefore two years older than da Vinci. A compatriot of Piero della Francesca, he began, like him, with the study of mathematics, and pushed his admiration of his teacher and fellow townsman so far as to appropriate Piero's Tractates de quinque Corporibus.1 Entering the

Franciscan order, he lived sometimes in Rome, where he enjoyed the hospitality of L. B. Alberti, sometimes at Perugia, where from 1477 to 1480, and from 1487 to 1481, he filled the chair of mathematics in the University. He also appeared now and then at Naples, at Florence, at Padua, at Assisi, and at Urbino.2 His Summa de Arithmetic a appeared at Venice in 1494, with a dedication to Guidobaldo of Urbino.3 Here Pacioli betrays himself as the most insipid of bookmakers, as well as a gossip and general blunderer.4 His Latin is barbarous and his

1 The fact of these borrowings has been established by Hubert Janitschek in the Kunstchronik of 187S (no. 42), and by Jordan in the Jahrbuch for 1880, vol. i., p. 112-118. See also Winterberg and Uzielli (second edition, vol. i., p. 451)- ^'e must not forget, however, that Pacioli, far from concealing his indebtedness to Piero, proclaims it with enthusiasm: "E anco con quello prometto darve piena notitia de prospectiva medianti li documcnti del nostro conterraneo et contemporale di tal facolta all tempi nostri Monarca Maestro Petro de Franceschi, di la qual gia feci dignissimo compendio e per noi ben aprcso. E del suo caro quanto fratello Maestro Lorenzo Canozo da Lcndenara." (Winterberg's edition, p. 123.)

2 Uzielli, 2nd edition, vol. i., pp. 388 et seq.

3 See Narducci, Interno a due Edizioni della Summa de Arithmetica di Fra Luca Pacioli, Rome, 1863.

4 His last biographer, M. Uzielli, nevertheless credits him with having popularised the highest branches of mathematics.



Italian unworthy of a Milanese, to say nothing of a Tuscan. In spite of his mediocrity he was, however, superior to Leonardo in one point—he gave the results of his labours to the world, while the greater master jealously guarded his from the knowledge of his contemporaries.

The fact that Pacioli never refers to Leonardo in his preface, while he mentions a crowd of other living artists,1 justifies us in supposing that his acquaintance with the great painter did not begin till later. It was not, in

fact, until 1496 that he entered the service of the Sforzi. Lodovico appointed him professor of arithmetic and geometry in the University of Pavia. His pay was modest enough, for while a professor of civil law enjoyed an annual salary of 3,600 lire, he received no more than 310. From 1496 to 1499 Pacioli worked side by side with Leonardo, to whom he devotes a generous eulogium in his De Divina Proportioned After the fall of Lodovico, Pacioli quitted Milan at the same time as Leonardo. In 1500 we find him

1 I reprinted this preface in Les Archives des Arts, p. 34 et seq. In one of those now incomprehensible memoranda with which he filled his notebooks, Leonardo writes, "Learn the multiplication of roots from Maestro Luca." Richter, vol. ii., p. 433.

2 Finished in December, 1497. The dedication is dated February, 1498. The work was not published until 1509. The Divina Proportione itself is followed by "Libellus in tres partiales tractatus divisus quinque corporum regularium et dependentium, active perscrutationis, D. Petro Soderino principi perpetuo populi florentini, a M. Luca Paciolo Burgense Minoritano particularity dicatus. Feliciter incipit." (27 folios.) Next come




(Windsor Library.)

living once more at Perugia, and afterwards with da Vinci at Florence.1 Here, in 1509, he dedicated to the Gonfaloniere Soderini his Divina Proportione, which had previously borne a dedication to II Moro. In the meantime, between 1500 and 1505,2 he had been teaching at Pisa, and had, in 1508, put in an appearance at Venice. In 1510 we find him again in Perugia, after which all trace of him is lost.

The following headings will give some idea of the contents of this strange compilation. Perspective, like music, and for the same reason, forms a branch of mathematics (book 1, chapter iii). How to divide a dimension, according to the rules of proportion, into a medium part and two extreme parts (chapter viii). How the hexagon and decagon form between them a dimension susceptible of division according to the rules of proportion (chapter xvi).3

I must make some reference to the figures inserted in the text of the Divina Proportione. Setting aside the separate plates, they are all geometrical diagrams, except those of fol. 25, v°-, a man's head in profile, turned to the left, and geometrically divided. We have already said something about Leonardo's share in the production of these engravings.

We know from the evidence of Geoffroy Tory, brought to light by the Marchese d'Adda and M. Dehio,4 that the initials in Pacioli's

the plates, printed only on one side of the leaf. The first, inscribed "Divina Proportio," is the male head described below; next come twenty-three plates numbered from A to Y; and finally three plates, the first columns, the second entablatures, the third "Porta templi domini dicta speciosa. Hierosolomis." There are besides some geometrical diagrams. Note that the majority of the initials contain those interlaced ornaments so dear to Leonardo.

1 De Architecture, ed. Winterberg, p. 144.—Mariotti, Lettcre pittoriche perugitie, p. 127.

2 Fabroni, Historia Academics Pisana, vol. i., p. 392.

3 A German savant, Herr Winterberg, has had the courage to translate this chaotic work, and to expound its fundamental law, the Golden Section, a magic formula, which, it is asserted, enables the student to establish the value of any work of art by means of three propositions! This was an honour certainly undreamt of by the humble Pacioli!

4 Repertorium fur Kunstivissenschaft, i88r, p. 269-279.—"Frere Lucas Pacioli de Bourg sainct Sepulchre, de l'ordre des freres mincurs et the'ologien, qui a faict en vulgar italien un libre intitule Divina Proportione, et qui a volu figurer lesdictes lettres Attiques, n'en a point aussi parle" ne bailie" raison : et je ne m'en ebahis point, car j'ay entendu par aulcuns Italiens qu'il a desrobe" sesdictes lettres, et prinses de feu messire

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