The World of Computer Software/World_Of_Computer_Software-02-387-Vol-3of3.iso/m/math.zip/MATH.TXT

Text File | 1993-03-04 | 28KB | 618 lines

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           *              MATHEMATICS               *
           *  Ancient Science and Its Modern Fates  *
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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 atlases--was 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.

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       *     GREEK MATHEMATICS AND ITS MODERN HEIRS     *
       *  Classical Roots of the Scientific Revolution  *
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                     Objects math01 - math05

For over a thousand years--from 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 Greek-speaking 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 science--which, 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.

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                        Euclid, Elements
                            In Greek
                          Ninth century

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.

Vat. gr. 190, vol. 1                    fols. 38 verso - 39 recto
math01                                                      NS.01

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                        Archimedes, Works
                            In Latin
                Translated by Jacobus Cremonensis
                            ca. 1458

In the early 1450's, 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.  Here the beginning of Archimedes' "On Conoids and
Spheroids" displays highly ornate, and rather curious,
illumination.

Urb. lat. 261                            fol. 44 verso - 45 recto
math02                                                      NS.17

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    Piero della Francesca, De quinque corporibus regularibus
                            In Latin
                              1480s

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).

Urb. lat. 632                           fols. 40 verso - 41 recto
math03                                                      NS.18

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                         Euclid, Optics
                            In Latin
                              1458

Euclid's "Optics" is the earliest surviving work on geometrical
optics.  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.

Urb. lat. 1329                                       fol. 1 recto
math04                                                      NS.19

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                        Archimedes, Works
                            In Latin
                Translated by William of Moerbeke
                            ca. 1270

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,
made in 1269 at the papal court in Viterbo from two of the best
Greek manuscripts, 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.

Ottob. lat. 1850                        fols. 36 verso - 37 recto
math05                                                      NS.52

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              Astronomical-Mathematical Collection
                            In Greek
                          Tenth century

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.

Vat. gr. 204                                       fol. 116 recto
math06                                                      NS.02

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                       Apollonius, Conics
                            In Greek
                              1536

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 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
tangents.

Vat. gr. 205                                            pp. 78-79
math07a                                                     NS.03

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.

math07b                                                      p. 4

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                       Pappus, Collection
                            In Greek
                          Tenth century

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.

Vat. gr. 218                              fols. 39 verso-40 recto
math08a                                                     NS.05

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.

math08b                                 fols. 110 verso-111 recto

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                        Ptolemy, Almagest
                            In Greek
                          Ninth century

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 astronomy--spherical 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.

Vat. gr. 1594                             fols. 73 verso-74 recto
math09a                                                     NS.07

These pages show Book IV Chapter 2, on Hipparchus's examination
of Babylonian cycles for the motion of the moon.

math09b                                   fols. 79 verso-80 recto

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

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.

Vat. lat. 2056                            fols. 45 verso-46 recto
math10a                                                     NS.09

Shown here is Book XII Chapters 8-9, 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.

math10b                                   fols. 87 verso-88 recto

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

The most influential medieval Latin translation of the "Almagest"
was made from the Arabic in Spain in 1175 by Gerard of Cremona,
the most prolific of all medieval translators from Arabic into
Latin.

Vat. lat. 2057                            fols. 70 verso-71 recto
math11a                                                     NS.10

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.  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.

math11b                               fols. 146 verso - 147 recto

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            *         PTOLEMY'S GEOGRAPHY          *
            *  The Science of the Earth's Surface  *
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                     Objects math12 - math15

Ptolemy, who gave Greeek astronomy its final form in the second
century A.D., did the same--and more--for 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 readers--found inspiration in Ptolemy's
exaggerated value for the size of Asia for his own fateful
journey to the west.

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                      Ptolemy, Handy Tables
                            In Greek
                          Ninth century

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 well-known 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.

Vat. gr. 1291                                        fol. 2 verso
math12a                                                     NS.06

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.

math12b                                             fol. 46 recto

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           Nicholas Germanus, Almanac for Pope Paul II
                            In Latin
                            ca. 1465

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.

Vat. lat. 3107                          fols. 46 verso - 47 recto
math13                                                      NS.12

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                       Ptolemy, Geography
                            In Latin
           Translated by Jacopo d'Angelo da Scarperia
                            ca. 1470

The text of Ptolemy's "Geography" was translated into Latin by
1406-09 by Jacopo d'Angelo da Scarperia and dedicated
successively to Popes Gregory XII and Alexander V.  Maps 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 opening displayed here, from a splendid
pair of related manuscripts of text and maps, shows the
coordinates, longitude and latitude, for locations in Greece.

Vat. lat. 3810                          fols. 31 verso - 32 recto
math14                                                      NS.50

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                       Ptolemy, Geography
                    Maps by Nicholas Germanus
                            In Latin
                            ca. 1470

This map of Greece and the Aegean, very rich in detail and
elegant in execution, corresponds to the coordinates in the
preceding manuscript [Vat. lat. 3810].  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.

Vat. lat. 3811                          fols. 29 verso - 30 recto
math15                                                      NS.51

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             *           GREEK ASTRONOMY           *
             *  The Revival of an Ancient Science  *
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                     Objects math16 - math20

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 fifteenth-century 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 full-scale "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
fifteenth-century Italy.

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                Byzantine Astronomical Collection
                            In Greek
                           Before 1308

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 (shown also in Vat.
ar. 319).  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.

Vat. gr. 211                                       fol. 117 recto
math16                                                      NS.04

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                        Ptolemy, Almagest
                            In Latin
                 Translated by George Trebizond
                            ca. 1481

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
George'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.  The opening
shows Book VI Chapter 7, on the computation of the duration of
solar and lunar eclipses.

Vat. lat. 2055                        fols. 101 verso - 102 recto
math17                                                      NS.08

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          George Trebizond, Commentary on the Almagest
                            In Latin
                            ca. 1482

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.  The opening contains 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.

Vat. lat. 2058                        fols. 170 verso - 171 recto
math18                                                      NS.11

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                 Nasir ad-Din at-Tusi, Tadhkira
                            In Arabic
                       Fourteenth century

Nasir ad-Din at-Tusi 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). Shown here is his ingenious device for generating
rectilinear motion along the diameter of the outer circle from
two circular motions.

Vat. ar. 319                            fols. 29 recto - 28 verso
math19                                                      NS.15

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           Georg Peurbach and Johannes Regiomontanus,
                     Epitome of the Almagest
                            In Latin
                     Late fifteenth century

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.  On display is the
opening which 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.

Barb. lat. 156                          fols. 53 verso - 54 recto
math20                                                      NS.20

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                       Ptolemy, Geography
                            In Greek
                        Fifteenth century

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 twenty-six large regional maps,
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, particularly in the
eastward extension of Scotland and the west to east slope of
Italy.

Urb. gr. 83                           fols. 112 verso - 113 recto
math21                                                      NS.16

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                *  Object Omitted from Exhibit  *
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                          Object math22


                        Euclid, Elements
                            In Greek
                            Parchment
                          Ninth century

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 parallelpipedal solids.  The figures are excellent
early representations of three dimensional objects in a plane.

Vat. gr. 190, vol. 2                   fol. 207 verso - 208 recto
math22                                                      NS.61

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                        *****  END  *****

Note:  This file has been edited for use on computer networks.
This editing required the removal of diacritics, underlining, and
fonts such as italics and bold.

kde 12/92