De quinque corporibus regularibus

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''De quinque corporibus regularibus'' (sometimes called ''Libellus de quinque corporibus regularibus'') is a book on the
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
of
polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...
written in the 1480s or early 1490s by Italian painter and mathematician
Piero della Francesca Piero della Francesca (, also , ; – 12 October 1492), originally named Piero di Benedetto, was an Italian painter of the Early Renaissance. To contemporaries he was also known as a mathematician and geometer. Nowadays Piero della Francesca ...
. It is a
manuscript A manuscript (abbreviated MS for singular and MSS for plural) was, traditionally, any document written by hand – or, once practical typewriters became available, typewritten – as opposed to mechanically printed or reproduced in ...
, in the Latin language; its title means '' he little bookon the five regular solids''. It is one of three books known to have been written by della Francesca. Along with the Platonic solids, ''De quinque corporibus regularibus'' includes descriptions of five of the thirteen
Archimedean solid In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...
s, and of several other irregular polyhedra coming from architectural applications. It was the first of what would become many books connecting mathematics to art through the construction and perspective drawing of polyhedra, including
Luca Pacioli Fra Luca Bartolomeo de Pacioli (sometimes ''Paccioli'' or ''Paciolo''; 1447 – 19 June 1517) was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and an early contributor to the field now known as accounting ...
's 1509 ''
Divina proportione ''Divina proportione'' (15th century Italian for ''Divine proportion''), later also called ''De divina proportione'' (converting the Italian title into a Latin one) is a book on mathematics written by Luca Pacioli and illustrated by Leonardo da V ...
'' (which incorporated without credit an Italian translation of della Francesca's work). Lost for many years, ''De quinque corporibus regularibus'' was rediscovered in the 19th century in the
Vatican Library The Vatican Apostolic Library ( la, Bibliotheca Apostolica Vaticana, it, Biblioteca Apostolica Vaticana), more commonly known as the Vatican Library or informally as the Vat, is the library of the Holy See, located in Vatican City. Formally es ...
and the Vatican copy has since been republished in facsimile.

Background

The five Platonic solids (the regular
tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...
,
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the onl ...
,
octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ...
,
dodecahedron In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentago ...
, and
icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
) were known to della Francesca through two classical sources: '' Timaeus'', in which
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
theorizes that four of them correspond to the
classical elements Classical elements typically refer to earth, water, air, fire, and (later) aether which were proposed to explain the nature and complexity of all matter in terms of simpler substances. Ancient cultures in Greece, Tibet, and India had s ...
making up the world (with the fifth, the dodecahedron, corresponding to the heavens), and the ''
Elements Element or elements may refer to: Science * Chemical element, a pure substance of one type of atom * Heating element, a device that generates heat by electrical resistance * Orbital elements, parameters required to identify a specific orbit of o ...
'' of
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
, in which the Platonic solids are constructed as mathematical objects. Two apocryphal books of the ''Elements'' concerning the metric properties of the Platonic solids, sometimes called ''pseudo-Euclid'', were also commonly considered to be part of the ''Elements'' in the time of della Francesca. It is the material from the ''Elements'' and pseudo-Euclid, rather than from ''Timaeus'', that forms della Francesca's main inspiration. The thirteen
Archimedean solid In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...
s, convex polyhedra in which the vertices but not the faces are symmetric to each other, were classified by
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
in a book that has long been lost. Archimedes' classification was later briefly described by
Pappus of Alexandria Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem ...
in terms of how many faces of each kind these polyhedra have. Della Francesca had previously studied and copied the works of Archimedes, and includes citations to Archimedes in ''De quinque corporibus regularibus''. But although he describes six of the Archimedean solids in his books (five in ''De quinque corporibus regularibus''), this appears to be an independent rediscovery; he does not credit Archimedes for these shapes and there is no evidence that he knew of Archimedes' work on them. Similarly, although both Archimedes and Della Francesca found formulas for the volume of a cloister vault (see below), their work on this appears to be independent, as Archimedes' volume formula remained unknown until the early 20th century. ''De quinque corporibus regularibus'' is one of three books known to have been written by della Francesca. The other two, ''De prospectiva pingendi'' and ''Trattato d'abaco'', concern
perspective drawing Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation, ...
and arithmetic in the tradition of
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western ...
's ''
Liber Abaci ''Liber Abaci'' (also spelled as ''Liber Abbaci''; "The Book of Calculation") is a historic 1202 Latin manuscript on arithmetic by Leonardo of Pisa, posthumously known as Fibonacci. ''Liber Abaci'' was among the first Western books to describ ...
'', respectively. The other mathematical book, ''Trattato d'abaco'', was part of a long line of abbacist works, teaching arithmetic, accounting, and basic geometrical calculations through many practical exercises, beginning with the work of
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western ...
in his book ''
Liber Abaci ''Liber Abaci'' (also spelled as ''Liber Abbaci''; "The Book of Calculation") is a historic 1202 Latin manuscript on arithmetic by Leonardo of Pisa, posthumously known as Fibonacci. ''Liber Abaci'' was among the first Western books to describ ...
'' (1202). Although the early parts of ''De quinque corporibus regularibus'' also borrow from this line of work, and overlap extensively with ''Trattato d'abaco'', Fibonacci and his followers had previously applied their calculation methods only in two-dimensional geometry. The later parts of ''De quinque corporibus regularibus'' are more original in their application of arithmetic to the geometry of three-dimensional shapes.

Contents

After its dedication, the title page of ''De quinque corporibus regularibus'' begins ''Petri pictoris Burgensis De quinque corporibus regularibus''. The first three words mean "Of Peter the painter, from Borgo", and refer to the book's author, Piero della Francesca (from Borgo Santo Sepolcro); the title proper begins after that. A decorative
initial In a written or published work, an initial capital, also referred to as a drop capital or simply an initial cap, initial, initcapital, initcap or init or a drop cap or drop, is a letter at the beginning of a word, a chapter, or a paragraph that ...
begins the text of the book. The first of the book's four parts concerns problems in plane geometry, primarily concerning the measurement of
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two t ...
s, such as calculating their
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open ...
,
perimeter A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several prac ...
, or side length, given a different one of these quantities. The second part concerns the
circumscribed sphere In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the term ''circumcircl ...
s of the Platonic solids, and asks similar questions on lengths, areas, or volumes of these solids relative to the measurements of the sphere that surrounds them. It also includes the (very likely novel) derivation for the height of an irregular tetrahedron, given its side lengths, equivalent (using the standard formula relating height and volume of tetrahedra) to a form of
Heron's formula In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, the area is, :A = \sqrt. It is named after first-centur ...
for tetrahedra. The third part includes additional exercises on circumscribed spheres, and then considers pairs of Platonic solids inscribed one within another, again focusing on their relative measurements. This part is inspired most directly by the 15th (apocryphal) book of the ''Elements'', which constructs certain inscribed pairs of polyhedral figures (for instance, a regular tetrahedron inscribed within a cube and sharing its four vertices with the four of the cube). ''De quinque corporibus regularibus'' aims to arithmetize these constructions, making it possible to calculate the measurements for one polyhedron given measurements of the other. The fourth and final part of the book concerns other shapes than the Platonic solids. These include six
Archimedean solid In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...
s: the
truncated tetrahedron In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedr ...
(which appears also in an exercise in his ''Trattato d'abaco''), and the truncations of the other four Platonic solids. The
cuboctahedron A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...
, another Archimedean solid, is described in the ''Trattato'' but not in ''De quinque corporibus regularibus''; since ''De quinque corporibus regularibus'' appears to be a later work than the ''Trattato'', this omission appears to be deliberate, and a sign that della Francesca was not aiming for a complete listing of these polyhedra. The fourth part of ''De quinque corporibus regularibus'' also includes domed shapes like the domes of the
Pantheon, Rome The Pantheon (, ; la, Pantheum,Although the spelling ''Pantheon'' is standard in English, only ''Pantheum'' is found in classical Latin; see, for example, Pliny, '' Natural History'36.38 "Agrippas Pantheum decoravit Diogenes Atheniensis". Se ...
or the (at the time newly constructed)
Santa Maria presso San Satiro Santa Maria presso San Satiro (Saint Mary near Saint Satyrus) is a church in Milan. The Italian Renaissance structure (1476-1482) houses the early medieval shrine to Satyrus, brother of Saint Ambrose. The church is known for its false apse, an ...
in
Milan Milan ( , , Lombard: ; it, Milano ) is a city in northern Italy, capital of Lombardy, and the second-most populous city proper in Italy after Rome. The city proper has a population of about 1.4 million, while its metropolitan city ...
formed from a ring of triangles surrounded by concentric rings of irregular quadrilaterals, and other shapes arising in architectural applications. The result that calls della Francesca's "most sophisticated" is the derivation of the volume of a
Steinmetz solid In geometry, a Steinmetz solid is the solid body obtained as the intersection of two or three cylinders of equal radius at right angles. Each of the curves of the intersection of two cylinders is an ellipse. The intersection of two cylinders i ...
(the intersection of two cylinders, the shape of a cloister vault), which della Francesca had illustrated in his book on perspective. Despite its curves, this shape has a simple but non-obvious formula for its volume, 2/3 of the volume of its enclosing cube. This result was known to both Archimedes and, in ancient China,
Zu Chongzhi Zu Chongzhi (; 429–500 AD), courtesy name Wenyuan (), was a Chinese astronomer, mathematician, politician, inventor, and writer during the Liu Song and Southern Qi dynasties. He was most notable for calculating pi as between 3.1415926 and 3 ...
, but della Francesca was unaware of either prior discovery. ''De quinque corporibus regularibus'' is illustrated in a variety of styles by della Francesca, not all of which are in correct mathematical perspective. It includes many exercises, roughly half of which overlap with the geometric parts of della Francesca's ''Trattato d'abaco'', translated from the Italian of the ''Trattato'' to the Latin of the ''De quinque corporibus regularibus''.

Dissemination

Della Francesca dedicated ''De quinque corporibus regularibus'' to Guidobaldo da Montefeltro, the
Duke of Urbino The Duchy of Urbino was an independent duchy in early modern central Italy, corresponding to the northern half of the modern region of Marche. It was directly annexed by the Papal States in 1625. It was bordered by the Adriatic Sea in the eas ...
. Although the book is not dated, this dedication narrows the date of its completion to the range from 1482 when Guidobaldo, ten years old, became duke, until 1492 when Della Francesca died. However, della Francesca likely wrote his book first in Italian, before translating it into Latin either himself or with the assistance of a friend, Matteo dal Borgo, suggests that della Francesca did not know Latin and would have needed dal Borgo's assistance, but this is contradicted by the later discovery by of a Latin manuscript of the works of Archimedes, copied by della Francesca. so its original draft may have been from before Guidobaldo's accession. In any case, the book was added to the library of the duke. It was kept there together with della Francesca's book on perspective, which he had dedicated to the previous duke. In what has been called "probably the first full-blown case of plagiarism in the history of mathematics",
Luca Pacioli Fra Luca Bartolomeo de Pacioli (sometimes ''Paccioli'' or ''Paciolo''; 1447 – 19 June 1517) was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and an early contributor to the field now known as accounting ...
copied exercises from ''Trattato d'abaco'' into his 1494 book ''
Summa de arithmetica ' (''Summary of arithmetic, geometry, proportions and proportionality'') is a book on mathematics written by Luca Pacioli and first published in 1494. It contains a comprehensive summary of Renaissance mathematics, including practical arithmeti ...
'', and then, in his 1509 book ''
Divina proportione ''Divina proportione'' (15th century Italian for ''Divine proportion''), later also called ''De divina proportione'' (converting the Italian title into a Latin one) is a book on mathematics written by Luca Pacioli and illustrated by Leonardo da V ...
'', incorporated a translation of the entire book ''De quinque corporibus regularibus'' into Italian, without crediting della Francesca for any of this material. It is through Pacioli that much of della Francesca's work became widely known. Although
Giorgio Vasari Giorgio Vasari (, also , ; 30 July 1511 – 27 June 1574) was an Italian Renaissance Master, who worked as a painter, architect, engineer, writer, and historian, who is best known for his work '' The Lives of the Most Excellent Painters, Sculp ...
denounced Pacioli for
plagiarism Plagiarism is the fraudulent representation of another person's language, thoughts, ideas, or expressions as one's own original work.From the 1995 '' Random House Compact Unabridged Dictionary'': use or close imitation of the language and though ...
in his 1568 book, ''
Lives of the Most Excellent Painters, Sculptors, and Architects ''The Lives of the Most Excellent Painters, Sculptors, and Architects'' ( it, Le vite de' più eccellenti pittori, scultori, e architettori), often simply known as ''The Lives'' ( it, Le Vite), is a series of artist biographies written by 16th-ce ...
'', he did not provide sufficient detail to verify these claims. Della Francesca's original work became lost until, in 1851 and again in 1880, it was rediscovered in the Urbino collection of the
Vatican Library The Vatican Apostolic Library ( la, Bibliotheca Apostolica Vaticana, it, Biblioteca Apostolica Vaticana), more commonly known as the Vatican Library or informally as the Vat, is the library of the Holy See, located in Vatican City. Formally es ...
by Scottish antiquary James Dennistoun and German art historian , respectively, allowing the accuracy of Vasari's accusations to be verified. Subsequent works to study the regular solids and their perspectives in similar ways, based on the work of della Francesca and its transmission by Pacioli, include
Albrecht Dürer Albrecht Dürer (; ; hu, Ajtósi Adalbert; 21 May 1471 – 6 April 1528),Müller, Peter O. (1993) ''Substantiv-Derivation in Den Schriften Albrecht Dürers'', Walter de Gruyter. . sometimes spelled in English as Durer (without an umlaut) or Due ...
's ''Underweysung der Messung'' (1525), which focuses on techniques for both the perspective drawing of regular and irregular polyhedra as well as for their construction as physical models, and Wenzel Jamnitzer's '' Perspectiva corporum regularium'' (1568), which presents images of many polyhedra derived from the regular polyhedra, but without mathematical analysis. Although a book with the same title was recorded to exist in the 16th century in the private library of
John Dee John Dee (13 July 1527 – 1608 or 1609) was an English mathematician, astronomer, astrologer, teacher, occultist, and alchemist. He was the court astronomer for, and advisor to, Elizabeth I, and spent much of his time on alchemy, divination, ...
, the Vatican copy of ''De quinque corporibus regularibus'' (Vatican Codex Urbinas 632) is the only extant copy known. An 1895 catalog of the Vatican collection lists it between volumes of Euclid and Archimedes. Reproductions of it have been published by the
Accademia dei Lincei The Accademia dei Lincei (; literally the "Academy of the Lynx-Eyed", but anglicised as the Lincean Academy) is one of the oldest and most prestigious European scientific institutions, located at the Palazzo Corsini on the Via della Lungara in Ro ...
in 1916, and by Giunti in 1995.