Special Collections- Vatican Film Library : Exact Sciences

RR, Plate 101: Euclid's Elements, written about 300 B.C., a comprehensive treatise on geometry, proportions, and the theory of numbers, is the most long-lived 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.

RR, Plate 102: This plate from the same manuscript of Euclid's Elements as Vat. gr. 190, vol. 1, shows Book XI Propositions 31-33 on the volumes of parallelepipedal solids. The figures are excellent early representations of three dimensional objects in a plane.

RR, Plate 103: 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.

RR, Plate 104: This is the holograph of Moerbeke's translation of 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.

RR, Plate 105: 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. The pages displayed here show the beginning of Archimedes' On Conoids and Spheroids with highly ornate, and rather curious, illumination.

RR, Plate 106: Francesca wrote this treatise 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 in a cube, and of a cube in an octahedron.

RR, Plate 107: Apollonius's Conics on conic sections, the ellipse, parabola, and hyperbola, is the most complex and difficult single work of all Greek mathematics. This magnificent copy, probably the most elegant of all Greek mathematical manuscripts, was made in 1536 for Pope Paul III. This page from the same manuscript of Apollonius's Conics shows Book I Propositions 4-6, with figures of the formation of the conic sections by a plane cutting a cone.

RR, Plate 108: The pages on display show the particularly elaborate figures illustrating Propositions 2-4 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.

RR, Plate 109: 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. These pages show Book VI Propositions 53, an extension of Euclid, Optics 35-36, showing that a circle viewed from outside its plane will appear as an ellipse with its center removed from the center of the circle.

RR, Plate 122: Ptolemy's Geography contains instructions for drawing maps of the entire inhabited world and particular regions, along with the longitudes and latitudes of about eight thousand locations in Europe, Africa, and Asia. There are two versions, the A recension and the B recension, displayed here, with sixty-four 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.

RR, Plate 123: By the middle of the century increasingly opulent manuscripts of the Geography 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.

RR, Plate 124: 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.

RR, Plate 110: 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. 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.

RR, Plate 111: The Almagest is a comprehensive treatise on all aspects of mathematical astronomy--spherical astronomy, solar, lunar, and planetary theory, eclipses, and the fixed stars. 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. These pages show Book IV Chapter 2, on Hipparchus's examination of Babylonian cycles for the motion of the moon.

RR, Plate 112: 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. These pages show Book X Chapters 6-7, Ptolemy's description of his kinematic model for the motion of the superior planets--Mars, Jupiter, and Saturn.

RR, Plate 113: In about 1160 a very literal translation of the Almagest was made directly from the Greek by an unknown translator in Sicily. In the early fifteenth century this manuscript, the only known complete copy, came into the hands of the great book collector Coluccio Salutati. Shown here is Book XII Chapters 8-9, the table of stations of the planetswritten entirely in Roman numerals, and the method of computing a table of the greatest elongations of Mercury and Venus from the sun.

RR, Plate 114: 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.

RR, Plate 115: George Trebizond also wrote a commentary as long as the original text. The commentary was severely criticized which resulted in a falling out with Pope Nicholas V. This opulent manuscript was dedicated to Pope Sixtus IV 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.

RR, Plate 116: 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.

RR, Plate 117: Nasir ad-Din at-Tusi was among the first of several Arabic astronomers 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.

RR, Plate 118: This short treatise, translated by Gregory Chioniades, describes Tusi's lunar theory, with illustrations such as this figure of the motion of the moon.

RR, Plate 119: Ptolemy's Handy Tables (4th century) became 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 well-known for its illumination drawn elegantly in white against the dark blue of the night sky, showing the northern part of the zodiac.

RR, Plate 120: 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.

RR, Plate 121: This uncommonly beautiful example of an almanac, computed for the years 1466 to 1484, was prepared for Paul II. 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.