Part 4 (1/2)
Figure 48
Figure 49 Like Fibonacci before him, Piero wrote the Treatise on the Abacus Treatise on the Abacus mainly to provide the merchants of his day with arithmetic recipes and geometrical rules. In a commercial world that had neither a unique system of weights and measures nor even agreed-upon shapes or sizes of containers, the ability to calculate volumes of figures was an absolute must. However, Piero's mathematical curiosity carried him well beyond the subjects that had simple everyday applications. Accordingly, we find in his books ”useless” problems, such as calculating the side of an octahedron inscribed inside a cube or calculating the diameter of the five small circles inscribed inside a circle of a known diameter ( mainly to provide the merchants of his day with arithmetic recipes and geometrical rules. In a commercial world that had neither a unique system of weights and measures nor even agreed-upon shapes or sizes of containers, the ability to calculate volumes of figures was an absolute must. However, Piero's mathematical curiosity carried him well beyond the subjects that had simple everyday applications. Accordingly, we find in his books ”useless” problems, such as calculating the side of an octahedron inscribed inside a cube or calculating the diameter of the five small circles inscribed inside a circle of a known diameter (Figure 49). The solution of the latter problem involves a pentagon and, therefore, the Golden Ratio.
Much of Piero's algebraic work was incorporated into a book published by Luca Pacioli (14451517), ent.i.tled Summa de arithmetica, geometria, proportioni et proportionalita Summa de arithmetica, geometria, proportioni et proportionalita (The collected knowledge of arithmetic, geometry, proportion and proportionality). Most of Piero's work on solids, which appeared in Latin, was translated into Italian by the same Luca Pacioli and again incorporated (or, many less tactfully say, simply plagiarized) into his famous book on the Golden Ratio: (The collected knowledge of arithmetic, geometry, proportion and proportionality). Most of Piero's work on solids, which appeared in Latin, was translated into Italian by the same Luca Pacioli and again incorporated (or, many less tactfully say, simply plagiarized) into his famous book on the Golden Ratio: Divina Proportione Divina Proportione (The divine proportion). (The divine proportion).
Who was this highly controversial mathematician Luca Pacioli? Was he the greatest mathematical plagiarist of all times or rather a great communicator of mathematics?
UNSUNG HERO OF THE RENAISSANCE?.
Luca Pacioli was born in 1445 in Borgo San Sepolcro (the same Tuscan town in which Piero della Francesca was born and where he had his workshop). In fact, Pacioli had his early education in Piero's workshop. However, unlike other students who displayed skill in the art of painting, and some, like Pietro Perugino, who were destined to become great painters themselves, he showed greater promise in mathematics. Piero and Pacioli were closely a.s.sociated later in life, as manifested by the fact that Piero included a portrait of Pacioli, as St. Peter Martyr, in a painting of ”Madonna and Child with Saints and Angels.” Pacioli moved to Venice at a relatively young age and became the tutor of the three sons of a wealthy merchant. In Venice he continued his mathematical education (under the mathematician Domenico Bragadino) and wrote his first textbook on arithmetic.
In the 1470s, Pacioli studied theology and was ordained as a Franciscan friar. Consequently, he is customarily referred to as Fra Luca Pacioli. In the following years, he traveled extensively, teaching mathematics at the universities of Perugia, Zara, Naples, and Rome. During this period he may have also tutored for some time Guidobaldo of Montefeltro, who was to become the Duke of Urbino in 1482.
In what may be the best portrait of a mathematician ever produced, Jacopo de' Barbari (14401515) depicts Luca Pacioli giving a lesson in
Figure 50 geometry to a pupil (Figure 50; the painting is currently in the Galleria n.a.z.ionale di Capodimonte in Naples). One of the Platonic solids, a dodecahedron, is seen on the right resting on top of Pacioli's book Summa. Summa. Pacioli himself, dressed in his friar robes and almost resembling a geometrical solid, is shown copying a diagram from volume XIII of Euclid's Pacioli himself, dressed in his friar robes and almost resembling a geometrical solid, is shown copying a diagram from volume XIII of Euclid's Elements. Elements. A transparent polyhedron known as a rhombicuboctahedron (one of the Archimedean Solids, with twenty-six faces of which eighteen are squares and eight equilateral triangles), half filled with water and hanging in mid-air, symbolizes the purity and timelessness of mathematics. The artist has captured the reflections and refractions from this gla.s.s polyhedron with extraordinary skill. The ident.i.ty of the second person in the painting has been the subject of some debate. One of the suggestions is that the student is Duke Guidobaldo. British mathematician Nick MacKinnon raised an interesting possibility in 1993. In a well-researched article ent.i.tled ”The Portrait of Fra Luca Pacioli,” which appeared in the A transparent polyhedron known as a rhombicuboctahedron (one of the Archimedean Solids, with twenty-six faces of which eighteen are squares and eight equilateral triangles), half filled with water and hanging in mid-air, symbolizes the purity and timelessness of mathematics. The artist has captured the reflections and refractions from this gla.s.s polyhedron with extraordinary skill. The ident.i.ty of the second person in the painting has been the subject of some debate. One of the suggestions is that the student is Duke Guidobaldo. British mathematician Nick MacKinnon raised an interesting possibility in 1993. In a well-researched article ent.i.tled ”The Portrait of Fra Luca Pacioli,” which appeared in the Mathematical Gazette Mathematical Gazette, MacKinnon suggests that the figure is that of the famous German painter Albrecht Durer, who had great interest in geometry and perspective (and to whose relations.h.i.+p with Pacioli we shall return later in this chapter). The face of the student does in fact bear a striking resemblance to Durer s self-portrait.
Pacioli returned to Borgo San Sepolcro in 1489, after having been granted some special privileges by the Pope, only to encounter jealousy from the existing religious establishment. For about two years he was even banned from teaching. In 1494, Pacioli went to Venice to publish his Summa Summa, which he dedicated to Duke Guidobaldo. Encyclopedic in nature and scope (some 600 pages), the Summa Summa compiled the mathematical knowledge of the time in arithmetic, algebra, geometry, and trigonometry. In this book, Pacioli borrows freely (usually with an appropriate acknowledgment) problems on the icosahedron and dodecahedron from Piero's compiled the mathematical knowledge of the time in arithmetic, algebra, geometry, and trigonometry. In this book, Pacioli borrows freely (usually with an appropriate acknowledgment) problems on the icosahedron and dodecahedron from Piero's Trattato Trattato and problems in algebra and geometry from Fibonacci and others. Identifying Fibonacci as his main source, Pacioli states that when no other is quoted, the work belongs to Leonardus Pisa.n.u.s. An interesting part of the and problems in algebra and geometry from Fibonacci and others. Identifying Fibonacci as his main source, Pacioli states that when no other is quoted, the work belongs to Leonardus Pisa.n.u.s. An interesting part of the Summa Summa is on double-entry accounting, a method of record keeping that lets you track where money comes from and where it goes. While Pacioli did not invent this system but merely summarized the practices of Venetian merchants during the Renaissance, this is considered to be the first published book on accounting. Pacioli's desire to ”give the trader without delay information as to his a.s.sets and liabilities” thus gained him the t.i.tle ”Father of Accounting,” and accountants from all over the world celebrated in 1994 (in Sansepolcro, as the town is now known) the 500 is on double-entry accounting, a method of record keeping that lets you track where money comes from and where it goes. While Pacioli did not invent this system but merely summarized the practices of Venetian merchants during the Renaissance, this is considered to be the first published book on accounting. Pacioli's desire to ”give the trader without delay information as to his a.s.sets and liabilities” thus gained him the t.i.tle ”Father of Accounting,” and accountants from all over the world celebrated in 1994 (in Sansepolcro, as the town is now known) the 500th anniversary of the anniversary of the Summa. Summa.
In 1480, Ludovico Sforza became effectively the Duke of Milan. In fact, he was only the regent of the real seven-year-old duke, following an episode of political intrigue and murder. Determined to make his court a home for scholars and artists, Ludovico invited Leonardo da Vinci in 1482 as a ”painter and engineer of the duke.” Leonardo had considerable interest in geometry, especially for its practical applications in mechanics. In his words: ”Mechanics is the paradise of the mathematical sciences, because by means of it one comes to the fruits of mathematics.” Consequently, Leonardo was probably the one who induced the duke to invite Pacioli to join the court, as a teacher of mathematics, in 1496. Undoubtedly, Leonardo learned some of his geometry from Pacioli, while he infused in the latter a greater appreciation for art.
During his stay in Milan, Pacioli completed work on his three-volume treatise Divina Proportione Divina Proportione (The divine proportion), which was eventually published in Venice in 1509. The first volume, (The divine proportion), which was eventually published in Venice in 1509. The first volume, Compendio de Divina Proportione Compendio de Divina Proportione (Compendium of the divine proportion), contains a detailed summary of the properties of the Golden Ratio (which Pacioli refers to as the ”Divine Proportion”) and a study of Platonic solids and other polyhedra. On the first page of (Compendium of the divine proportion), contains a detailed summary of the properties of the Golden Ratio (which Pacioli refers to as the ”Divine Proportion”) and a study of Platonic solids and other polyhedra. On the first page of The Divine Proportion The Divine Proportion Pacioli declares somewhat grandiloquently that this is: ”A work necessary for all the clear-sighted and inquiring human minds, in which everyone who loves to study philosophy, perspective, painting, sculpture, architecture, music and other mathematical disciplines will find a very delicate, subtle and admirable teaching and will delight in diverse questions touching upon a very secret science.” Pacioli declares somewhat grandiloquently that this is: ”A work necessary for all the clear-sighted and inquiring human minds, in which everyone who loves to study philosophy, perspective, painting, sculpture, architecture, music and other mathematical disciplines will find a very delicate, subtle and admirable teaching and will delight in diverse questions touching upon a very secret science.”
Pacioli dedicated the first volume of The Divine Proportion The Divine Proportion to Ludovico Sforza, and in the fifth chapter he lists five reasons why he believes that the appropriate name for the Golden Ratio should be to Ludovico Sforza, and in the fifth chapter he lists five reasons why he believes that the appropriate name for the Golden Ratio should be The Divine Proportion. The Divine Proportion.
1. ”That it is one only and not more.” Pacioli compares the unique value of the Golden Ratio to the fact that unity ”is the supreme epithet of G.o.d himself.”
2. Pacioli finds a similarity between the fact that the definition of the Golden Ratio involves precisely three lengths (AC, CB (AC, CB, and AB AB in in Figure 24 Figure 24) and the existence of a Holy Trinity, of Father, Son, and Holy Ghost.
3. To Pacioli, the incomprehensibility of G.o.d and the fact that the Golden Ratio is an irrational number are equivalent. In his own words: Just like G.o.d cannot be properly defined, nor can be understood through words, likewise our proportion cannot be ever designated by intelligible numbers, nor can it be expressed by any rational quant.i.ty, but always remains concealed and secret, and is called irrational by the mathematicians.”
4. Pacioli compares the omnipresence and invariability of G.o.d to the self-similarity a.s.sociated with the Golden Ratio-that its value is always the same and does not depend on the length of the line being divided or the size of the pentagon in which ratios of lengths are calculated.
5. The fifth reason reveals an even more Platonic view of existence than Plato himself expressed. Pacioli states that just as G.o.d conferred being to the entire cosmos through the fifth essence, represented by the dodecahedron, so does the Golden Ratio confer being to the dodecahedron, since one cannot construct the dodecahedron without the Golden Ratio. He adds that it is impossible to compare the other four Platonic solids (representing earth, water, air, and fire) to each other without the Golden Ratio.
In the book itself, Pacioli raves ceaselessly about the properties of the Golden Ratio. He a.n.a.lyzes in succession what he calls the thirteen different ”effects” of the ”divine proportion” and attaches to each one of these ”effects” adjectives like ”essential,” ”singular,” ”wonderful,” ”supreme,” and so on. For example, he regards the ”effect” that Golden Rectangles can be inscribed in the icosahedron (Figure 22) as ”incomprehensible.” Pacioli stops at thirteen ”effects,” concluding that, ”for the sake of salvation, the list must end,” because thirteen men were present at the table at the Last Supper.
Figure 51 There is no doubt that Pacioli had a great interest in the arts and that his intention in The Divine Proportion The Divine Proportion was partly to perfect their mathematical basis. His opening statement on the book's first page expresses his desire to reveal to artists, through the Golden Ratio, the ”secret” of harmonic forms. To ensure its attractiveness, Pacioli secured for was partly to perfect their mathematical basis. His opening statement on the book's first page expresses his desire to reveal to artists, through the Golden Ratio, the ”secret” of harmonic forms. To ensure its attractiveness, Pacioli secured for The Divine Proportion The Divine Proportion the services of the dream ill.u.s.trator of any author-Leonardo da Vinci himself provided sixty ill.u.s.trations of solids, depicted in both skeletal ( the services of the dream ill.u.s.trator of any author-Leonardo da Vinci himself provided sixty ill.u.s.trations of solids, depicted in both skeletal (Figure 51) and solid forms (Figure 52). Pacioli was quick to express his grat.i.tude; he wrote about Leonardo's contribution: ”the most excellent painter in perspective, architect, musician, the man endowed with all virtues, Leonardo da Vinci, who deduced and elaborated a series of diagrams of regular solids.” The text itself, however, falls somewhat short of its declared high goals.
While the book starts with a sensational flourish, it continues with a rather conventional set of mathematical formulae loosely wrapped up in philosophical definitions.
Figure 52 The second book in the Divina Proportione Divina Proportione is a treatise on proportion and its application to architecture and the structure of the human body. Pacioli's treatment was largely based on the work of the eclectic Roman architect Marcus Vitruvius Pollio (ca. 7025 is a treatise on proportion and its application to architecture and the structure of the human body. Pacioli's treatment was largely based on the work of the eclectic Roman architect Marcus Vitruvius Pollio (ca. 7025 B.C. B.C.). Vitruvius wrote: ... in the human body the central point is naturally the navel. For if a man be placed flat on his back, with his hands and feet extended, and a pair of compa.s.ses centered at his navel, the fingers and toes of his two hands and feet will touch the circ.u.mference of a circle described therefrom. And just as the human body yields a circular outline, so too a square figure may be found from it. For if we measure the distance from the soles of the feet to the top of the head, and then apply that measure to the outstretched arms, the breadth will be found to be the same as the height, as in the case of plane surfaces which are perfectly square.
This pa.s.sage was taken by the Renaissance scholars as yet another demonstration of the link between the organic and geometrical basis of beauty, and it led to the concept of the ”Vitruvian man,” drawn beautifully by Leonardo (Figure 53; currently in the Galleria dell'Accademia, Venice). Accordingly, Pacioli's book also starts with a discussion of proportions in the human body, ”since in the human body every sort of pro portion and proportionality can be found, produced at the beck of the all-Highest through the inner mysteries of nature.” However, contrary to frequent claims in the literature, Pacioli does not insist on the Golden Ratio as determining the pro portions of all works of art. Rather, when dealing with de sign and proportion, he specifically advocates the Vitruvian system, which is based on simple (rational) ratios. Author Roger Herz-Fischler traced the fallacy of the Golden Ratio as Pacioli's canon for proportion to a false statement made in the 1799 edition of Histoire de Mathematiques Histoire de Mathematiques (History of mathematics) by the French mathematicians Jean Etienne Montucla and Jerome de Lalande. (History of mathematics) by the French mathematicians Jean Etienne Montucla and Jerome de Lalande.
Figure 53 The third volume of the Divina Divina (A short book divided into three partial tracts on the five regular bodies) is essentially an Italian word-by-word translation of Piero's (A short book divided into three partial tracts on the five regular bodies) is essentially an Italian word-by-word translation of Piero's Five Regular Solids Five Regular Solids composed in Latin. The fact that nowhere in the text does Pacioli acknowledge that he was merely the translator of the book provoked a violent denunciation from art historian Giorgio Vasari. Vasari writes about Piero della Francesca that he composed in Latin. The fact that nowhere in the text does Pacioli acknowledge that he was merely the translator of the book provoked a violent denunciation from art historian Giorgio Vasari. Vasari writes about Piero della Francesca that he ... was regarded as a great master of the problems of regular solids, both arithmetical and geometrical, but he was prevented by the blindness that overtook him in his old age, and then by death, from making known his brilliant researches and the many books he had written. The man who should have done his utmost to enhance Piero's reputation and fame, since Piero taught him all he knew, shamefully and wickedly tried to obliterate his teacher's name and to usurp for himself the honor which belonged entirely to Piero; for he published under his own name, which was Fra Luca dal Borgo [Pacioli], all the researches done by that admirable old man, who was a great painter as well as an expert in the sciences.
So, was Pacioli a plagiarist? Quite possibly, although in Summa Summa he did render homage to Piero, whom he regarded as ”the monarch of our times in painting” and one who ”is familiar to you in that copious work which he composed on the art of painting and on the force of the line in perspective.” he did render homage to Piero, whom he regarded as ”the monarch of our times in painting” and one who ”is familiar to you in that copious work which he composed on the art of painting and on the force of the line in perspective.”
R. Emmett Taylor (18891956) published in 1942 a book ent.i.tled No Royal Road: Luca Pacioli and His Times. No Royal Road: Luca Pacioli and His Times. In this book, Taylor adopts a very sympathetic att.i.tude toward Pacioli, and he argues that, on the basis of style, Pacioli may have had nothing to do with the third book of the In this book, Taylor adopts a very sympathetic att.i.tude toward Pacioli, and he argues that, on the basis of style, Pacioli may have had nothing to do with the third book of the Divina Divina and it was just appended to Pacioli's work. and it was just appended to Pacioli's work.
Be that as it may, there is no question that if not for Pacioli's printed printed books, Piero's ideas and mathematical constructions (which were not published in printed form) would not have reached the wide circulation that they eventually achieved. Furthermore, up until Pacioli's time, the Golden Ratio had been known only by rather intimidating names, such as ”extreme and mean ratio” or ”proportion having a mean and two extremes,” and the concept itself was familiar only to mathematicians. The publication of books, Piero's ideas and mathematical constructions (which were not published in printed form) would not have reached the wide circulation that they eventually achieved. Furthermore, up until Pacioli's time, the Golden Ratio had been known only by rather intimidating names, such as ”extreme and mean ratio” or ”proportion having a mean and two extremes,” and the concept itself was familiar only to mathematicians. The publication of The Divine Proportion The Divine Proportion in 1509 gave a new topical interest to the Golden Ratio. The concept could now be considered with fresh attention, because its publication in book form identified it as worthy of respect. The infusion of theological/philosophical meaning into the in 1509 gave a new topical interest to the Golden Ratio. The concept could now be considered with fresh attention, because its publication in book form identified it as worthy of respect. The infusion of theological/philosophical meaning into the name name (”Divine Proportion”) also singled out the Golden Ratio as a mathematical topic into which an increasingly eclectic group of intellectuals could delve. Finally, with Pacioli's book, the Golden Ratio started to become available to artists in theoretical treatises that were not overly mathematical, that they could actually use. (”Divine Proportion”) also singled out the Golden Ratio as a mathematical topic into which an increasingly eclectic group of intellectuals could delve. Finally, with Pacioli's book, the Golden Ratio started to become available to artists in theoretical treatises that were not overly mathematical, that they could actually use.
Leonardo da Vinci's drawings of polyhedra for The Divine Proportion The Divine Proportion, drawn (in Pacioli's words) with his ”ineffable left hand,” had their own impact. These were probably the first ill.u.s.trations of skeletal solids, which allowed for an easy visual distinction between front and back. Leonardo may have drawn the polyhedra from a series of wooden models, since records of the Council Hall in Florence indicate that a set of Pacioli's wooden models was purchased by the city for public display. In addition to the diagrams for Pacioli's book, we can find sketches of many solids scattered throughout Leonardo's notebooks. In one place he presents an approximate geometrical construction of the pentagon. This fusion of art and mathematics reaches its climax in Leonardo's Trattato della pittura Trattato della pittura (Treatise on painting; organized by Francesco Melzi, who inherited Leonardo's ma.n.u.scripts), which opens with the admonition: ”Let no one who is not a mathematician read my works”-hardly a likely statement to be found in any contemporary art handbook! (Treatise on painting; organized by Francesco Melzi, who inherited Leonardo's ma.n.u.scripts), which opens with the admonition: ”Let no one who is not a mathematician read my works”-hardly a likely statement to be found in any contemporary art handbook!
The drawings of solids in the Divina Divina have also inspired some of the have also inspired some of the intarsia intarsia constructed by Fra Giovanni da Verona around 1520. Intarsia represent a special art form, in which elaborate flat mosaics are constructed of pieces of inlaid wood. Fra Giovanni's intarsia panels include an icosahedron, which almost certainly used Leonardo's skeletal drawing as a template. constructed by Fra Giovanni da Verona around 1520. Intarsia represent a special art form, in which elaborate flat mosaics are constructed of pieces of inlaid wood. Fra Giovanni's intarsia panels include an icosahedron, which almost certainly used Leonardo's skeletal drawing as a template.
The lives of Leonardo and Pacioli continued to be somewhat intertwined even after the completion of The Divine Proportion. The Divine Proportion. In October of 1499 the two men fled Milan, after the French army, led by King Louis XII, captured that city. After spending brief periods of time in Mantua and Venice, both settled for some time in Florence. During the period of their friends.h.i.+p, Pacioli's name became a.s.sociated with two other major mathematical works-a translation into Latin of Euclid's In October of 1499 the two men fled Milan, after the French army, led by King Louis XII, captured that city. After spending brief periods of time in Mantua and Venice, both settled for some time in Florence. During the period of their friends.h.i.+p, Pacioli's name became a.s.sociated with two other major mathematical works-a translation into Latin of Euclid's Elements Elements and an unpublished work on recreational mathematics. Pacioli's translation of the and an unpublished work on recreational mathematics. Pacioli's translation of the Elements Elements was an annotated version, based on an earlier translation by Campa.n.u.s of Novara (12201296), which appeared in printed form in Venice in 1482 (and which was the first was an annotated version, based on an earlier translation by Campa.n.u.s of Novara (12201296), which appeared in printed form in Venice in 1482 (and which was the first printed printed version). Pacioli did not manage to publish his compilation of problems in recreational mathematics and proverbs version). Pacioli did not manage to publish his compilation of problems in recreational mathematics and proverbs De Viribus Quant.i.tatis De Viribus Quant.i.tatis (The powers of numbers) before his death in 1517. This work was a collaborative project between Pacioli and Leonardo, and Leonardo's own notes contain many of the problems in (The powers of numbers) before his death in 1517. This work was a collaborative project between Pacioli and Leonardo, and Leonardo's own notes contain many of the problems in De Viribus. De Viribus.
Fra Luca Pacioli certainly cannot be remembered for originality, but his influence on the development of mathematics in general, and on the history of the Golden Ratio in particular, cannot be denied.
MELANCHOLY.
Another major Renaissance figure who entertained an intriguing combination of artistic and mathematical interests is the German painter Albrecht Durer.
Durer is considered by many to be the greatest German artist of the Renaissance. He was born on May 21, 1471, in the Imperial Free City of Nurnberg, to a hardworking jeweler. At age nineteen, he already demonstrated talents and ability as a painter and woodcut designer that surpa.s.sed those of his teacher, the leading painter and book ill.u.s.trator in Nurnberg, Michael Wolgemut. Durer therefore embarked on four years of travel, during which he became convinced that mathematics, ”the most precise, logical, and graphically constructive of the sciences,” has to be an important ingredient of art.
Consequently, after a short stay in Nurnberg, during which he married Agnes Frey, the daughter of a successful craftsman, he left again for Italy, with the goal of expanding both his artistic and mathematical horizons. His visit to Venice in 14941495 seems to have accomplished precisely that. Durer's meeting with the founder of the Venetian School of painting, Giovanni Bellini (ca. 14261516), left a great impression on the young artist, and his admiration for Bellini persisted throughout his life. At the same time, Durer's encounter with Jacopo de' Barbari, who painted the wonderful portrait of Luca Pacioli (Figure 50), acquainted him with Pacioli's mathematical work and its relevance for art. In particular, de' Barbari showed Durer two figures, male and female, that were constructed by geometrical methods, and the experience motivated Durer to investigate human movement and proportions. Durer probably met with Pacioli himself in Bologna, during a second visit to Italy in 1505 to 1507. In a letter from that period, he describes his visit to Bologna as being ”for art's sake, for there is one there who will instruct me in the secret art of perspective.” The mysterious ”one” in Bologna has been interpreted by many as referring to Pacioli, although other names, such as those of the outstanding architect Donato di Angelo Bramante (14441514) and the architectural theorist Sebastiano Serlio (14751554), have also been suggested. During the same Italian trip Durer also met again with Jacopo de' Barbari. This second visit, though, was marked by Durer's somewhat paranoiac nervousness about harm that might be done to him by artists envious of his fame. For example, he refused invitations to dinner for fear that someone might try to poison him.
Starting in 1495, Durer showed a serious interest in mathematics. He spent much time studying the Elements Elements (a Latin translation of which he obtained in Venice, although he spoke little Latin), Pacioli's works on mathematics and art, and the important works on architecture, proportion, and perspective by the Roman architect Vitruvius and by the Italian architect and theorist Leon Baptista Alberti (14041472). (a Latin translation of which he obtained in Venice, although he spoke little Latin), Pacioli's works on mathematics and art, and the important works on architecture, proportion, and perspective by the Roman architect Vitruvius and by the Italian architect and theorist Leon Baptista Alberti (14041472).
Durer's contributions to the history of the Golden Ratio come both in the form of written work and through his art. His major treatise, Unterweisung der Messung mit dem Zirkel und Richtscheit Unterweisung der Messung mit dem Zirkel und Richtscheit (Treatise on measurement with compa.s.s and ruler), was published in 1525 and was one of the first books on mathematics published in German. In it Durer complains that too many artists are ignorant of geometry, ”without which no one can either be or become an absolute artist.” The first of the four books of the (Treatise on measurement with compa.s.s and ruler), was published in 1525 and was one of the first books on mathematics published in German. In it Durer complains that too many artists are ignorant of geometry, ”without which no one can either be or become an absolute artist.” The first of the four books of the Treatise Treatise gives detailed descriptions of the construction of various curves, including the logarithmic (or equiangular) spiral, which is, as we have seen, closely related to the Golden Ratio. The second book contains precise and approximate methods for the construction of many polygons, including two constructions of the pentagon (one exact and one approximate). The Platonic solids, as well as other solids, some of Durer's own invention, together with the theory of perspective and of shadows, are discussed in the fourth book. Durer's book was not intended to be used as a textbook of geometry-for example, he gives only one example of a proof. Rather, Durer always starts with a practical application and then continues with an exposition of the very basic theoretical aspects. The book contains some of the earliest presentations of nets of polyhedra. These are plane sheets on which the surfaces of the polyhedra are drawn in such a way that the figures can be cut out (as single pieces) and folded to form the three-dimensional solids. gives detailed descriptions of the construction of various curves, including the logarithmic (or equiangular) spiral, which is, as we have seen, closely related to the Golden Ratio. The second book contains precise and approximate methods for the construction of many polygons, including two constructions of the pentagon (one exact and one approximate). The Platonic solids, as well as other solids, some of Durer's own invention, together with the theory of perspective and of shadows, are discussed in the fourth book. Durer's book was not intended to be used as a textbook of geometry-for example, he gives only one example of a proof. Rather, Durer always starts with a practical application and then continues with an exposition of the very basic theoretical aspects. The book contains some of the earliest presentations of nets of polyhedra. These are plane sheets on which the surfaces of the polyhedra are drawn in such a way that the figures can be cut out (as single pieces) and folded to form the three-dimensional solids.