Until recently, historians of the Scientific Revolution of the 16th and 17th centuries treated it as a kind of rebellion against the authority of ancient books and humanist scholarship. In fact, however, it began with the revival of several tremendously important and formidably difficult works of Greek science. The mathematics and astronomy of the Greeks had been known in medieval western Europe only through often imperfect translations, some of them made from Arabic intermediary texts rather than the Greek originals. The papal curia became a center for the recovery of the original Greek manuscripts, often very old and remarkably elegant, and the production of new translations of these works. Ptolemy's "Geography"the book which inspired Columbus to attempt his voyage, and remains the model of all systematic atlaseswas dedicated to Popes Gregory XII and Alexander V by its first translator, the apostolic secretary Jacopo Angeli. Illustrated texts of this elegant atlas found readers everywhere in Europe. Nicholas V supported translations of the greatest of Greek mathematicians, Archimedes, and the greatest of Greek astronomers, Ptolemy. Cardinal Bessarion collected a vast range of Greek texts (which eventually wound up in Venice, as the nucleus of another great Renaissance library). A scholar whom he helped in many ways, Joannes Regiomontanus, became the first western European in centuries really to master Ptolemy's astronomy, which had been preserved and improved in the Islamic world. His work done in and for the curia laid the essential foundations on which Copernicus and other innovators built a new astronomy in the sixteenth century, using the Greek texts as their basic source of data and methods. Scholarship supported science in this world where faith and science were not yet seen as two, irreconcilable cultures.
Greek Mathematics and its Modern Heirs
For over a thousand yearsfrom the fifth century B.C. to the fifth century A.D.Greek mathematicians maintained a splendid tradition of work in the exact sciences: mathematics, astronomy, and related fields. Though the early synthesis of Euclid and some of the supremely brilliant works of Archimedes were known in the medieval west, this tradition really survived elsewhere. In Byzantium, the capital of the Greekspeaking Eastern empire, the original Greek texts were copied and preserved. In the Islamic world, in locales that ranged from Spain to Persia, the texts were studied in Arabic translations and fundamental new work was done. The Vatican Library has one of the richest collections in the world of the products of this tradition, in all its languages and forms. Both the manuscripts that the Vatican collected and the work done on them in Rome proved vital to the recovery of ancient sciencewhich, in turn, laid the foundation for the Scientific Revolution of the 16th and 17th centuries. In the Roman Renaissance, science and humanistic scholarship were not only not enemies; they were natural allies.
Euclid, Elements
Euclid's Elements, written about 300 B.C., a comprehensive treatise on geometry, proportions, and the theory of numbers, is the most longlived of all mathematical works. This manuscript preserves an early version of the text. Shown here is Book I Proposition 47, the Pythagorean Theorem: the square on the hypotenuse of a right triangle is equal to the sum of the squares on the sides. This is a famous and important theorem that receives many notes in the manuscript.
Euclid, Elements. In Greek. Parchment. Ninth century
Bookmark this item: //www.loc.gov/exhibits/vatican/math.html#obj1
Archimedes, Works
In the early 1450s, Pope Nicholas V commissioned Jacobus de Sancto Cassiano Cremonensis to make a new translation of Archimedes with the commentaries of Eutocius. This became the standard version and was finally printed in 1544. This early and very elegant manuscript may have been in the possession of Piero della Francesca before coming to the library of the Duke of Urbino. The pages displayed here show the beginning of Archimedes' On Conoids and Spheroids with highly ornate, and rather curious, illumination.
Archimedes, Works. In Latin. Translated by Jacobus Cremonensis. ca. 1458
Bookmark this item: //www.loc.gov/exhibits/vatican/math.html#obj2
Piero della Francesca, De quinque corporibus regularibus
The early Renaissance artist Piero della Francesca developed a mathematically rigorous system of perspective on which he wrote the treatise De prospectiva pingendi. His interest in mathematics increased as he grew older and late in his life he wrote two other treatises, a Trattato d'abaco, on algebra and the measurement of polygons and polyhedra (solids), and "De quinque corporibus regularibus," on the five regular polyhedra, which survives only in this unique manuscript from the library of the Duke of Urbino. The figures are said to be by Piero himself. Shown here are the inscriptions of an icosahedron (a solid composed of twenty equilateral triangular faces) in a cube, and of a cube in an octahedron (a solid of eight equilateral triangular faces).
Piero della Francesca, De quinque corporibus regularibus. In Latin. Parchment. 1480s
Bookmark this item: //www.loc.gov/exhibits/vatican/math.html#obj3
Euclid, Optics
Euclid's Optics is the earliest surviving work on geometrical optics, and is generally found in Greek manuscripts along with elementary works on spherical astronomy. There were a number of medieval Latin translations, which became of new importance in the fifteenth century for the theory of linear perspective. This technique is beautifully illustrated here in the miniature of a street scene in this elegant manuscript from the library of the Duke of Urbino. It may once have been in the possession of Piero della Francesca, who wrote one of the principal treatises on perspective in painting.
Euclid, Optics. In Latin. Parchment. 1458
Bookmark this item: //www.loc.gov/exhibits/vatican/math.html#obj4
Archimedes, Works
William of Moerbeke was the most prolific medieval translator of philosophical, medical, and scientific texts from Greek into Latin. This is the holograph of his translation of the greatest Greek mathematician, Archimedes, with the commentaries of Eutocius. The translations were made in 1269 at the papal court in Viterbo from two of the best Greek manuscripts of Archimedes, both of which have since disappeared. Shown here is a part of Eutocius's commentary on Archimedes' On the Sphere and the Cylinder, in which he reviews solutions to the classical problem of the duplication of the cube, i.e. how to construct a cube twice the volume of a given cube.
Archimedes, Works. In Latin. Translated by William of Moerbeke. ca. 1270
Bookmark this item: //www.loc.gov/exhibits/vatican/math.html#obj5
AstronomicalMathematical Collection
This is the oldest and best manuscript of a collection of early Greek astronomical works, mostly elementary, by Autolycus, Euclid, Aristarchus, Hypsicles, and Theodosius, as well as mathematical works. The most interesting, really curious, of these is Aristarchus's On the Distances and Sizes of the Sun and Moon, in which he shows that the sun is between 18 and 20 times the distance of the moon. Shown here is Proposition 13, with many scholia, concerned with the ratio to the diameters of the moon and sun of the line subtending the arc dividing the light and dark portions of the moon in a lunar eclipse.
AstronomicalMathematical Collection. In Greek. Parchment. Tenth century
Bookmark this item: //www.loc.gov/exhibits/vatican/math.html#obj6
Apollonius, Conics
Apollonius's Conics, written about 200 B.C., on conic sections, the ellipse, parabola, and hyperbola, is the most complex and difficult single work of all Greek mathematics and was all but unknown in the west until the fifteenth century. This magnificent copy, probably the most elegant of all Greek mathematical manuscripts, was made in 1536 for Pope Paul III. The pages on display show the particularly elaborate figures illustrating Propositions 24 of Book III on the equality of areas of triangles and quadrilaterals formed by tangents and diameters of conics, and by tangents and lines parallel to the tangents.
1 of 2

Apollonius, Conics. In Greek. Paper. 1536

Apollonius, Conics [image not available at this time]. In Greek. Paper. 1536. This page from the same manuscript of Apollonius's Conics shows Book I Propositions 46, with figures of the formation of the conic sections by a plane cutting a cone.
Bookmark this item: //www.loc.gov/exhibits/vatican/math.html#obj7
Pappus, Collection
Pappus's Collection, consisting of supplements to earlier treatises on geometry, astronomy, and mechanics, dates from the late third century A.D. and is the last important work of Greek mathematics. This manuscript reached the papal library in the thirteenth century, and is the archetype of all later copies, of which none is earlier than the sixteenth century.
1 of 2

Pappus, Collection. In Greek. Parchment. Tenth century

Pappus, Collection [image not available at this time]. In Greek. Parchment. Tenth century. These pages show Book VI Propositions 53, an extension of Euclid, Optics 3536, showing that a circle viewed from outside its plane will appear as an ellipse with its center removed from the center of the circle.
Bookmark this item: //www.loc.gov/exhibits/vatican/math.html#obj8
Ptolemy, Almagest
Claudius Ptolemy, who lived in the second century A.D., did work of enormous importance in astronomy and geography in which the Vatican Library has particularly rich holdings. The Almagest, written about A.D. 150, is a comprehensive treatise on all aspects of mathematical astronomyspherical astronomy, solar, lunar, and planetary theory, eclipses, and the fixed stars. It made all of its predecessors obsolete and remained the definitive treatise on its subject for nearly fifteen hundred years. This, the most elegant of all manuscripts of the Almagest, is one of the oldest and best witnesses to the text, and is very rich in notes.
1 of 2

Ptolemy, Almagest. In Greek. Parchment. Ninth century

Ptolemy, Almagest [image not available at this time]. In Greek. Parchment. Ninth century. These pages show Book IV Chapter 2, on Hipparchus's examination of Babylonian cycles for the motion of the moon.
Bookmark this item: //www.loc.gov/exhibits/vatican/math.html#obj9
Ptolemy, Almagest
In about 1160 a very literal translation of the Almagest was made directly from the Greek by an unknown translator in Sicily. The version had little circulation, but in the early fifteenth century this manuscript, the only known complete copy, came into the hands of the great Florentine book collector Coluccio Salutati.
1 of 2

Ptolemy, Almagest. In Latin. Salernitan Translation. Parchment. Late thirteenth or early fourteenth century

Ptolemy, Almagest [image not available at this time]. In Latin. Parchment. Salernitan Translation. Late thirteenth or early fourteenth century. Shown here is Book XII Chapters 89, the table of stations of the planets (the place on the epicycle where the planet appears stationary) written entirely in Roman numerals, and the method of computing a table of the greatest elongations of Mercury and Venus from the sun.
Bookmark this item: //www.loc.gov/exhibits/vatican/math.html#obj10
Ptolemy, Almagest
The most important medieval Latin translation of the Almagest, which is found in many manuscripts, was made from the Arabic in Spain in 1175 by Gerard of Cremona, the most prolific of all medieval translators from Arabic into Latin.
1 of 2

Ptolemy, Almagest. In Latin. Translated by Gerard of Cremona. Parchment. Thirteenth century

Ptolemy, Almagest [image not available at this time]. In Latin. Translated by Gerard of Cremona. Parchment. Thirteenth century. These pages show Book X Chapters 67, Ptolemy's description of his kinematic model for the motion of the superior planetsMars, Jupiter, and Saturn. The separation of the center of uniform motion from the center of uniform distance of the center of the epicycle is explained, as well as the beginning of the derivation of the elements of the model for Mars, through a lengthy iterative computation. The earth is at rest at (e) and the planets move uniformly with respect to a point (r) which is separated from the center of their spheres, (d). This device closely approximated the elliptical orbit in which planets actually move.
Bookmark this item: //www.loc.gov/exhibits/vatican/math.html#obj11
Ptolemy's Geography
Ptolemy, who gave Greeek astronomy its final form in the second century A.D., did the sameand morefor geography and cartography. His massive work on the subject, which summed up and criticized the work of earlier writers, offered instruction in laying out maps by three different methods of projection, provided coordinates for some eight thousand places, and treated such basic concepts as geographical latitude and longitude. In Byzantium, in the thirteenth century, Ptolemic maps were reconstructed and attached to Greek manuscripts of the text. And in the fifteenth century, a Latin translation of this text, with maps, proved a sensation in the world of the book. A best seller both in the age of luxurious manuscripts and in that of print, Ptolemy's Geography became immensely influential. Columbus one of its many readersfound inspiration in Ptolemy's exaggerated value for the size of Asia for his own fateful journey to the west.
Ptolemy, Handy Tables
Ptolemy's Handy Tables, intended for practical computation, were edited by Theon of Alexandria in the fourth century A.D. and became, with various modifications, the basis of later astronomical tables in Greek, Arabic, and Latin. The Handy Tables allow the calculation of solar, lunar, and planetary positions and eclipses of the sun and moon far more rapidly than the tables included in the Almagest. This early and elegant uncial manuscript is wellknown for its illumination, which appears to descend from a prototype in late antiquity as can clearly be seen in this map of the constellations, drawn elegantly in white against the dark blue of the night sky, showing the northern part of the zodiac.
1 of 2

Ptolemy, Handy Tables. In Greek. Parchment. Ninth century

Ptolemy, Handy Tables [image not available at this time]. In Greek. Parchment. Ninth century. This is another illustration from the same manuscript of the Handy Tables. In this table for the latitude of the moon, figures of distinctly classical appearance grace the tops of the columns, evidently a copy of a prototype from late antiquity.
Bookmark this item: //www.loc.gov/exhibits/vatican/math.html#obj12
Nicholas Germanus, Almanac for Pope Paul II
By the fifteenth century it had become common to compute annual ephemerides or almanacs giving daily positions of the sun, moon, and planets (for the casting of horoscopes) and eclipses of the sun and moon. Most were utilitarian but this uncommonly beautiful example, computed for the years 1466 to 1484, was prepared for Paul II by Nicholas Germanus, best known for his maps for Ptolemy's Geography. The entries for the months of April and May of 1473, shown here, illustrate in the margins a partial solar eclipse on April 26 and a partial lunar eclipse on May 11.
Nicholas Germanus, Almanac for Pope Paul II. In Latin. Parchment. ca. 1465
Bookmark this item: //www.loc.gov/exhibits/vatican/math.html#obj13
Ptolemy, Geography
The text of Ptolemy's Geography was translated into Latin by 140609 by Jacopo Angeli da Scarperia and dedicated successively to Popes Gregory XII and Alexander V. Maps based on this translation followed independently within less than twenty years. By the middle of the century, increasingly opulent manuscripts of the Geography, mostly from Florence, had become fashionable as conspicuous displays of wealth; and travellers and explorers as well as scholars read them. The pages displayed here, from a splendid pair of related manuscripts of text and maps, shows the coordinates, longitude and latitude, for locations in Greece.
Ptolemy, Geography. In Latin. Parchment. Translated by Jacopo d'Angelo da Scarperia. ca. 1470
Bookmark this item: //www.loc.gov/exhibits/vatican/math.html#obj14
Ptolemy, Geography
This map of Greece and the Aegean, very rich in detail and elegant in execution, corresponds to the coordinates in the preceding manuscript. The trapezoidal projection, reducing the distortion of longitudinal distances in a rectangular projection by having the meridians converge toward the pole, was the invention of Nicholas Germanus, who dedicated editions of the Geography to Borso d'Este of Ferrara and Pope Paul II. Nicholas personally supervised the preparation of a number of fine copies, perhaps this one among them, and his maps and projections continued to appear in the most important of the early printed editions.
Ptolemy, Geography. Map by Nicholas Germanus. In Latin. Parchment. ca. 1470
Bookmark this item: //www.loc.gov/exhibits/vatican/math.html#obj15
Greek Astronomy
One of the most powerful creations of Greek science was the mathematical astronomy created by Hipparchus in the second century B.C. and given final form by Ptolemy in the second century A.D. Ptolemy's work was known in the Middle Ages through imperfect Latin versions. In fifteenthcentury Italy, however, it was brought back to life. George Trebizond, a Cretan emigre in the curia, produced a new translation and commentary. These proved imperfect and aroused much heated criticism. But a German astronomer, Johannes Regiomontanus, a protege of the brilliant Greek churchman Cardinal Bessarion, came to Italy with his patron, learned Greek, and produced a fullscale "Epitome" of Ptolemy's work from which most astronomers learned their art for the next century and more. Copernicus was only one of the celebrities of the Scientific Revolution whose work rested in large part on the study of ancient science carried out in fifteenthcentury Italy.
Byzantine Astronomical Collection
In the thirteenth and fourteenth centuries, a number of recent Arabic and Persian astronomical works were translated into Greek by scholars who traveled to Persia under the Ilkhanid Empire. One short and confused treatise, translated by Gregory Chioniades, describes Tusi's lunar theory, illustrated, not altogether correctly, in this figure along with Tusi's device for producing rectilinear from circular motions. A part of the planetary and lunar theory of the astronomers of Maragha was later utilized by Copernicus, though scholars do not know how he gained access to this material.
Byzantine Astronomical Collection. In Greek. Paper. Before 1308
Bookmark this item: //www.loc.gov/exhibits/vatican/math.html#obj16
Ptolemy, Almagest
George Trebizond, one of the notable Greek scholars who came to Italy in the early fifteenth century, made a new translation of the Almagest from the Greek for Pope Nicholas V between March and December of 1451. Due to a dispute about the quality of Trebizond's commentary on the text, the translation was never dedicated to Nicholas. This very elaborate manuscript of the translation, with the figures drawn in several colors, was dedicated to Pope Sixtus IV by George's son Andreas. These pages show Book VI Chapter 7, on the computation of the duration of solar and lunar eclipses.
Ptolemy, Almagest. In Latin. Translated by George Trebizond. Parchment. ca 1481
Bookmark this item: //www.loc.gov/exhibits/vatican/math.html#obj17
George Trebizond, Commentary on the Almagest
During the same nine months that George Trebizond made his translation of the Almagest, he also wrote a commentary as long as the original text. The commentary was severely criticized, however, which resulted in a falling out with Pope Nicholas V. This opulent manuscript was dedicated to Pope Sixtus IV by George's son Andreas along with Vat. lat. 2055 of the translation. These pages contain a large figure of the model for the planet Mercury, shown at its least distance from the earth, with a list of Mercury's parameters and distances, and then the beginning of the treatment of Venus in Book X.
George Trebizond, Commentary on the Almagest. In Latin. Parchment. ca. 1482
Bookmark this item: //www.loc.gov/exhibits/vatican/math.html#obj18
Nasir adDin atTusi, Tadhkira
Nasir adDin atTusi was among the first of several Arabic astronomers of the late thirteenth century at the observatory of Maragha in Persia who modified Ptolemy's models based on mechanical principles, in order to preserve the uniform rotation of spheres. This early Arabic manuscript contains his principal work on the subject, the Tadhkira fi'ilm al'haya (Memoir on Astronomy). The figure shown here is his ingenious device for generating rectilinear motion along the diameter of the outer circle from two circular motions.
Nasir adDin atTusi, Tadhkira. In Arabic. Paper. Fourteenth century
Bookmark this item: //www.loc.gov/exhibits/vatican/math.html#obj19
Georg Peurbach and Johannes Regiomontanus, Epitome of the Almagest
The Epitome of the Almagest was written between 1460 and 1463 by Georg Peurbach and Johannes Regiomontanus at the suggestion of Cardinal Bessarion. It gave Europeans the first sophisticated understanding of Ptolemy's astronomy, and was studied by every competent astronomer of the sixteenth century. The illustration here shows the distance of the sun from the earth as 1210 terrestrial radii (about 4,800,000 miles), which is too small by a factor of twenty, but gives a solar parallax (the maximum displacement due to observing the sun from the surface rather than from the center of the earth) of less than 3 minutes, still well below the limit of observational accuracy.
Georg Peurbach and Johannes Regiomontanus, Epitome of the Almagest. In Latin. Paper. Late fifteenth century
Bookmark this item: //www.loc.gov/exhibits/vatican/math.html#obj20
Ptolemy, Geography
Ptolemy's Geography contains instructions for drawing maps of the entire oikoumene (inhabited world) and particular regions, along with the longitudes and latitudes of about eight thousand locations in Europe, Africa, and Asia. The maps in manuscripts of the Geography, however, date only from about 1300, after the text was rediscovered by Maximus Planudes. There are two versions, the A recension with twentysix large regional maps, and the B recension, displayed here, with sixtyfour smaller regional maps and four large additional maps. Shown here is the additional map of Europe which reveals Ptolemy's systematic exaggeration of west to east distances, particularly in the eastward extension of Scotland and the west to east slope of Italy.
Ptolemy, Geography. In Greek. Paper. Fifteenth century
Bookmark this item: //www.loc.gov/exhibits/vatican/math.html#obj21
Euclid, Elements
This plate from the same manuscript of Euclid's Elements as Vat. gr. 190, vol. 1, shows Book XI Propositions 3133 on the volumes of parallelpipedal solids. The figures are excellent early representations of three dimensional objects in a plane.
Euclid, Elements. In Greek. Parchment. Ninth century
Bookmark this item: //www.loc.gov/exhibits/vatican/math.html#obj22