Euclid
(ca. 326265 BC)
Archimedes
(ca. 287212 BC)
Apollonius of Perga
(ca. 260200 BC)
Nichomachus of Gerasa
(ca. 100)
Claudius Ptolemy
(ca. 85165)
Diophantus of Alexandria
(ca. 200284)
Pappus of Alexandria
(ca. 300350)
Proclus
(ca. 410485)
Boethius
(ca. 480524)
Thomas Bradwardine
(ca. 12901349)
Girolamo Cardano
(15011576)
Robert Recorde
(15481620)
Johann Mller of Knigsberg,
called Regiomontanus
(14361476)
Franois Vite
(15441603)
John Napier
(15501617)
Henry Briggs
(15611630)
Adriaan Vlacq
(16001667)
Ludolph van Ceulen
(15391610)
Simon Stevin
(15481620)
Thomas Hariot
(15601621)
Galileo Galilei
(15641642)
Benedetto Castelli
(15781643)
Bonaventura Cavalieri
(ca. 15981647)
Christiaan Huygens (16291695)
Ren Descartes
(15961650)
Gottfried Wilhem Leibniz
(16461716)
Sir Isaac Newton
(16421727)
Guillaume Franois Antoine l'Hospital,
Marquis de SainteMesme
(16611704)
TOP


Brown University Library possess a copy of each sixteenthcentury translation
of Euclid's Elements of Geometry into a modern language. These vernacular
editions, grouped around the first Latin edition of 1482, are displayed in chronological
sequence, from 1533 (Greek) to 1594 (Arabic). All copies are opened at Book I,
proposition 47, "Pythagoras' Theorem," which asserts: "In rightangled triangles
the square of the side opposite the right angle is equal to the sum of the squares
of the sides containing the right angle." Most of the translations provide proof
of this equation (a^{2} + b^{2} = c^{2} using a geometrical
construction known as "the bride's chair."
Other first translations into modern European languages were published in the
seventeenth and eighteenth centuries: Dutch (1602), Russian (1739), Swedish (1744),
and Danish (1745).
Latin Editions
Euclid (ca.
326ca. 265 BC)
Preclarissimus liber elementorum Euclidis perspicacissimi: in artem geometrie
incipit ...
Venice: Erhard Ratdolt, [25 May] 1482.
Euclid's Elements of Geometry has been a primary mathematics
text for more than two thousand years. It is a compilation of early Greek mathematical
knowledge, synthesized and systematically presented by Euclid in ca. 300 BC.
Books IIV are devoted to plane geometry, Book V deals with the theory of proportions,
and Book VI with the similarity of plane figures. Books VIIIX are on number
theory, Book X on commensurability and incommensurability, Books XIXII explore
three dimensional geometric objects, and Book XIII deals with the construction
of the five regular solids. Later nonEuclidian additions include, Book XIV,
which is thought to have been contrbuted by Hypsciles (ca. 200 BC), and Book
XV, which may have been added by John of Damascus, or by a 6thcentury pupil
of Isadoros of Miletos.
The first printed edition of the Elements, based upon a translation from
Arabic to Latin presumably made by Abelard of Bath in the 12th century, was
edited and annotated by Campanus of Novara. It is justly famous for its exceptionally
fine printing and for the careful placement of geometrical diagrams in the wide
outer margins.
This incunable edition is opened at the first two printed pages. In his dedication,
on the left, Erhard Ratdolt attributed the prior lack of printed mathematical
works to the difficulty occasioned by the diagrams, and adds that he had discovered
a method for printing the illustrations as easily as the letters. On the righthand
page, the text of Book I starts with 23 definitions: A point is that which
has no part. A line is a length without breadth. The extremities of a line
are points. A straight line is a line which lies evenly with the points on
itself. A surface is that which had length and breadth only ... .
ANNMARY BROWN MEMORIAL COLLECTION
Greek Editions
Euclid.
Eukleidou Stoicheion biblon ...
Basel: Johann Herwagen, 1533.
The German theologian Simon Grynäus (14931541), using a Latin
translation made from the Greek manuscript by Bartolomeo Zamberti in 1505 and
two Greek manuscripts supplied by Lazarus Bayfius and Joannes Ruellius (14741537),
produced this first edition of the complete Greek text of the Elements, in September
1533. To this volume Grynäus appended the first publication of the four
books of Proclus' Commentary on the first book of Euclid's Elements,
taken from a manuscript provided by John Claymond, President of Magdalen College,
Oxford. In a long introduction Grynäus dedicated his translation to Cuthbert
Tunstall (14741559), Bishop of Durham, and author of the first arithmetic book
printed in English (London, 1522). A later edition of that work, De arte
supputandi libri quatuor (Paris: Robert Estienne, 1538), is in our collection.
This translation was the only comprehensive Greeklanguage version of the Elements
available until the appearance of a Greek and Latin edition of Euclid's complete
works produced by David Gregory (16591708), early in the eighteenth century
Eukleidou ta sozomena = Euclidis quae supersunt omnia (Oxford: Theatro
Sheldoniano, 1703HSC). The first edition in what is now recognized as modern
Greek was not published until 1820.
HISTORY OF SCIENCE COLLECTION
Italian Editions
Euclid
Euclide Megarense ... solo introduttore delle scientie
mathematice ...
Venice: Venturino Ruffinelli, 1543.
This Italian edition in folio is the first translation of Euclid's
work into a modern language. It was edited by Niccolò Tartaglia of Brescia
(ca.14991557), an eminent mathematician and author of the definitive sixvolume
treatise on sixteenthcentury Italian mathematics, General trattato di numeri
et misure (Venice: Curtio Troiano, 155660HSC). His translation was reprinted
at Venice, in a quarto format, during 1565, 1569, 1585 and 1586. Examples of
all but the final edition, which but for a single digit on the titlepage is
a lineforline copy of its predecessor, are in our collection.
LOWNES COLLECTION OF SIGNIFICANT BOOKS IN THE HISTORY OF SCIENCE & TECHNOLOGY
German Editions
Euclid
Das sibend, acht und neünt Büch, des hochberühmbten
Mathematici Euclidis ...
Augsburg: Valentine Ottmar, 1555.
This first translation of any part of Euclid's work into German, Elements
(Books VIIIX), was edited by Johann Scheubel (14941570), professor of mathematics
at the University of Tübingen. Scheubel had earlier produced a Latin edition
of Books IVI, Sex libri priores de geometricis principii (Basel: Johann
Herwagen, 1550HSC); an arithmetic text, De numeris et diversis rationibus
seu regulis computationum opusculum (Leipzig: Michael Blum, 1545HSC), in
which he found binomial coefficients by means of a Pascal triangle a century
before Pascal made that device famous; and an algebra text, Algebrae compendiosa
facilisque descriptio (Paris: Guillaume Cavellat, 1552HSC).
HISTORY OF SCIENCE COLLECTION
Euclid
Die sechs erste Bücher Euclidis ...
Basel: Jacob Kündig, 1562.
The second appearance of Euclid's work in German was the first translation
of the Elements (Books IVI) as edited by Wilhelm Holtzman (also known
as Wilhelm Xylander, 15321576), librarian and professor of Greek and mathematics
at the University of Heidelberg. This copy is unusual because it contains the
rare errata leaf and has an inscription from Xylander on the elaborate red and
black titlepage presenting it to his colleague Jacob Curio (14971572).
Ornatissimo viro, ac insignj mathematico medicoque D. Jacobo Curionj,
amico honorando, autor observatio[num] causa dat.
A copy of Xylander's collected works, Opuscula mathematica (Heidelberg:
Matthew Harnish, 1577), also is in our collection.
HISTORY OF SCIENCE COLLECTION
French Editions
Euclid
Les six premiers livres des Elements d'Euclide ...
Paris: Jerome de Marnef and Guillaume Cavellat, 1564.
Bound With:
Les septieme huictieme et neufieme livres des Elemens
d'Euclide ...
Paris: Charles Perier, 1565.
These two first editions of Euclid's work in French, Elements
(Books IVI and VIIIX) were translated and edited by Pierre Forcadel de Beziers
(d. 1574), Mathematical Reader in Ordinary to the King in the University of
Paris. Though Euclid's enunciations are printed in full, the demonstrations
and commentaries appear to have been derived from Forcadel's own lectures. A
decade earlier he had authored the most elaborate 3volume mathematics text
produced in sixteenthcentury France, L'Arithmetique (Paris: Guillaume
Cavellat, 155657HSC).
It was not until early in the seventeenth century that the first complete French
translation of the Elements, made by Didier Dounot (15741640), was publishedLes
quinze livres des elements d'Euclide (Paris: I. A. Ioallin, 1604). A copy
of Dounot's second edition, Les elemens della geometrie d'Euclides ... reveue
& augmentée par l'autheur (Paris: Jacques le Roy, 1613), is in our
collection.
HISTORY OF SCIENCE COLLECTION
English Editions
Euclid
The elements of geometrie of the most auncient philosopher
Euclide of Megara ...
London: John Day, 1570.
This first English translation of the complete fifteen books of Euclid's
Elements was produced by Sir Henry Billingsley (d. 1606), a wealthy merchant
and later Lord Mayor (1597) and Member of Parliament for the City of London
(1603). His address of the "Translator to the Reader" notes that this folio
volume contains "manifolde additions Scholies, Annotations and Inventions ...
gathered out of the most famous and chiefe Mathematiciens, both of old time,
and in our age." Preceeding the text of Euclid is a Preface by John Dee (15271608),
an astrologer, mathematician and fellow of Trinity College, Cambridge, in which
he celebrates the glory of the "Artes Mathematicall" and defends himself against
"the folly of Idiotes and the Mallice of the Scornfull."
Each book begins with a summary statement that often includes considerable commentary
on the efforts of Billingsley's scholarly predecessors, most notably Companus
of Novarra and Bartholomeo Zamberti. A unique feature of this edition is the
inclusion of pasted flaps of paper that can be folded up to produce three dimensional
models of the propositions in Book XI, making it one of the oldest "popup"
books known.
LOWNES COLLECTION OF SIGNIFICANT BOOKS IN THE HISTORY OF SCIENCE & TECHNOLOGY
Christoph Clavius
(15371612)
Euclidis Elementorum libri XV ...
Rome: Vincenzo Accolti, 1574.
This twovolume set is not, technically speaking, a translation, but
rather the first edition of a very personal redaction compiled by the eminent
Jesuit scholar, Christoph Clavius, professor of mathematics at the Collegium
Romanum known to his contemporaries as "the Euclid of the sixteenth century."
It is an immensely valuable edition for the enormous amount of auxiliary material
and substantial commentary contained within. By Clavius' own account, to the
486 original propositions, he added 748 others of his own devising: "in universum
ergo 1234 propositiones in nostro Euclide demonstrantur." The copy on display
once belonged to Stillman Drake, an authority on the life and works of Galileo,
with whom Clavius maintained a correspondence. Clavius' scholarship exerted
an influence on practically every subsequent publication of the Elements
for the next 200 years. Four of the seven reprints of this text issued during
his lifetime are in our collection (Rome, 1589; Cologne, 1591, 1607; Frankfurt,
1607).
A fivevolume edition of Clavius' collected works in fifteen parts, Operum
mathematicorum (Mainz: Anton Hierat, 161112HSC), was issued in the final
months of his life. The first volume contains the Euclidean geometry and his
commentary on the Sphaericorum libri III of Theodosius; the second, his
treatises on geometry, arithmetic and algebra; the third comprises his complete
commentary on the Sphaera of Joannes de Sacrobosco; the fourth is devoted
to a detailed discussion of gnomics and the construction of sundials; and the
fifth contains a fundamental exposition and defense of the calendar reform accomplished
in 15771582 under the auspices of Pope Gregory XIII.
HISTORY OF SCIENCE COLLECTION
Spanish Editions
Euclid
Los seis libros primeros dela geometria de Euclides
...
Seville: Alonso de la Barrera, 1576.
This first edition of Euclid's work in Spanish, Elements (Books
IVI) was translated and edited by by Rodrigo de Zamorano (b. 1542), who identified
himself on the titlepage as an astrologer, mathematician and cosmographer to
His Majesty. It may have been modeled, in part, on the superb Latin edition
of Federigo Commandino (15091575), Elementorum Libri XV (Pesaro: Camillo
Francischini, 1572HSC). A complete Spanish translation of all the books of
the Elements was not published until 1739.
Zamorano also authored a series of almanacs and a Compendium on the art of
navigation (Seville: Juan de Leon, 1588), a copy of which is in the John
Carter Brown Library.
HISTORY OF SCIENCE COLLECTION
Arabic Editions
Euclid
Kitab tahrir usul lUqlidus, min talif Khwajah Nasir
alDin alTusi ...
Rome: Typographia Medicea, 1594.
This Arabic edition [Recension of Euclid's Elements] is allegedly
based on a translation of Books IXIII, made from original Greek sources by
the astronomer Nasir alDin alTusi in about 1248. Two versions of this text
were issued by the Medicean Press, one with a titlepage partially in Latin
[Euclidis Elementorum geometricorum libri tredecim] and the other, displayed
here, which is entirely in Arabic. Regardless of the Latin title, only Books
IXII of the Elements were included in these printings. In the mideighteenth
century a cache of the translations, amounting to more than half of the original
edition, was discovered in the Vatican Archives and subsequently destroyed.
HISTORY OF SCIENCE COLLECTION
Archimedes
(circa 287212 BC)
Opera, quae quidem extant, omnia ...
Basel: Johann Herwagen, 1544.
Archimedes is acknowledged as the greatest mathematician of Antiquity.
Among his contributions to the history of mathematics were a method for calculating
centers of gravity, an approximation of the value of , and a system of notation
for expressing very large numbers. He demonstrated theorems relating to areas
and volumes of figures bounded by curved lines and surfaces, showed how mechanical
problems could be solved through geometrical analysis and often made use of
proof by the "method of exhaustion," an early forerunner of the calculus. His
lost treatise on levers and other mechanical devices is known only by its mention
in Pappus's "Collection" and through comments on Archimedes' boast that given
a fixed fulcrum he could move the world by using only a lever and a compound
pulley. He is credited with a number of inventions such as the waterscrew,
the compound pulley, as well as catapults and other ballastic devices. By tradition,
the latter were used unsuccessfully against a Roman seige and Archimedes was
killed during the ensuing sack of Syracuse.
This first collected Greek and Latin edition of Archimedes' Works is
composed of seven books: De sphaera et cylindro (On the sphere and the
cylinder), Circuli dimensio (On the measurement of the circle ), De
conoidibus et sphaeroidibus (On conoids and spheroids ), De lineis spiralibus
(On spirals ), Planorum aequaeponderantium inventa (On the equilibrium
of planes ), De harnae numera (The sandreckoner ), and Quadrata parabolae
(On the quadrature of the parabola ). It also contains the critical comments
of Eutocius of Ascalon (early 6th Century), In eosdem Archimedis libros commentaria.
These are noteable because, in them, Eutocius mentions Greek mathematicians
whose works would not otherwise be known. Jacob of Cremona translated the texts
in the early 1450s. They were then edited by Johann Müller, known as Regiomontanus
(14361476) who corrected Jacob's translation using a manuscript owned by Johannes
Cardinal Bessarion (14031472), the Papal Legate to the Holy Roman Empire. Over
seventy years later, Thomas Gechauff, known as Venatorius (d. 1551), revised
this edition into four, alternating Greek (1, 3) and Latin (2, 4), volumes.
In one of the propositions in "On the sphere and the cylinder," the area of
a sphere is expressed as four times that of a great circle, and in another the
volume of a sphere is given as 2/3 of the volume of the circumscribed right
cylinder. The method of exhaustion is employed in "On the measurement of the
circle," in which inscribed and circumscribed regular polygons of up to 96 sides
each are used to find a value for of approximately 3 1/7. Methods for determining
volumes for specific quadratic surfaces of revolution are found in "On conoids
and spheroids," and "On the equilibrium of planes" may contain extracts from
the lost treatise on mechanics.
As one of the propositions in "On the quadrature of a parabola," Archimedes
demonstrates, by constructing an infinite series of triangles, that the area
of a parabola is 4/3 the area of a triangle with the same base and vertex, and
2/3 of the area of the circumscribed parallelogram. "On spirals" contains the
"Spiral of Archimedes," a corkscrewlike device used to drain fields and ships'
bilges by raising water to higher levels. In "The sandreckoner," Archimedes
devises a system by which numbers of great magnitude could be expressed and
illustrates his method by attempting to find the upper limit of the number of
grains of sand it would take to fill the universe.
The heliocentric theories of Aristarchus of Samos (circa 310circa 230 BC)
are mentioned in the introduction to "The sandreckoner": "His hypotheses are
that the fixed stars and the sun remain unmoved, that the earth revolves about
the sun in the circumference of a circle, the sun lying in the middle of the
orbit ... ."
HISTORY OF SCIENCE COLLECTION (GREEK)
LOWNES COLLECTION OF SIGNIFICANT BOOKS IN THE HISTORY OF SCIENCE (LATIN)
Apollonius of Perga (circa
260circa 200 BC)
Conicorum libri quattuor ...
Bologna: Alessandro Benacci, 1566.
Apollonius of Perga composed a systematic treatise on conic sections
containing about 400 propositions divided into eight books, of which seven survived
four in Greek and three in a 9thcentury Arabic translation. By the fifteenth
century only the first four books of the "Conics" had been discovered. This
influential edition, the culmination of Greek geometry, contains glosses by
Pappus of Alexandria (4th Century) and commentary by Eutocius of Ascalon (5th
Century) as well as a two texts by Serenus of Anzi (4th Century), Unus de
sectione cylindri, alter de sectione coni. All of these were translated
into Latin, from original Greek manuscripts, and edited by Federico Commandino
(15091575) of Urbino.
The first four books are a compilation of what was then known of conics and,
according to Pappus, they might contain a restatement of the lost four books
of Euclid's Conics. Book I begins by defining a cone on a circular base and
then investigates methods of generating different plane sections of it. Apollonius
names the three types of curves which result as parabola, hyperbola and ellipse.
Book II deals with the properties of the diameter and axes of the sections,
and Book III contains theorems useful for the syntheses of solid loci and for
determinations of the possibilities of solutions. Finally, in an original contribution,
Book IV deals with how many ways the conic sections can meet one another.
This volume is opened at Book I, propositions 2427.
XXIV: If a straight line, meeting a parabola or hyperbola in
one point, when produced both ways, falls outside the section, then it will
meet the diameter.
XXV: If a straight line, meeting an ellipse between the two
(conjugate) diameters and produced both ways, falls outside the section, it
will meet each of the diameters.
XXVI: If in a parabola or hyperbola a straight line is drawn
parallel to the diameter of the section, it will meet the section in one point
only.
XXVII: If a straight line cuts the diameter of a parabola, then produced both
ways, it will meet the section.
LOWNES COLLECTION OF SIGNIFICANT BOOKS IN THE HISTORY OF SCIENCE & TECHNOLOGY
Apollonius of Perga (circa
260circa 200 BC)
Conicorum libri V. VI. VII. ... Additis in calce Archimedis assumptorum
liber ...
Florence: Giuseppe Cocchini, 1661.
Early in the seventeenth century the Medicis acquired an Arabic manuscript
containing three more books of Apollonius' Conics as paraphrased by Abalphatus
Asphahtanensus and a version of Archimedes' Liber assumptorum as recorded
by Thebit BenKora. Both texts were translated into Latin by Abraham Ecchellensis
(d. 1664), edited by Giovanni Borelli (16081679), and then published almost
one century after the first appearance of Books IIV.
According to Apollonius, the work in his final four books was largely original.
Book V is particularly important for containing a discussion of the theory of
maxima and minima which leads to his proof for the construction of the evolute
curve. In Book VI he treats equal and similar conics, and Book VII deals mainly
with propositions about inequalities between various functions of conjugate
diameters. The lost Book VIII, as described in the preface to Book I, dealt
with determinate conic problems. In the eighteenth century Sir Edmund Halley
(16561742) attempted to reconstruct that book in his Apolloni Pergæi
Conicorum libri octo (Oxford: Theatro Sheldoniano, 1710HSC).
This volume is opened at Book V, propositions 910, dealing with the basic
theorems on minima in the hyperbola (9), and in the ellipse (10). The "notes"
refer back to proposition 8, on minima in the parabola.
HISTORY OF SCIENCE COLLECTION
Nichomachus of Gerasa
(circa 100)
Arithmeticae libri duo ...
Paris: Christian Wechel, 1538.
Nichomachus, a member of the school at Alexandria, was one of the
first mathematicians of note during the early period of Roman dominance. His
"Introduction to arithmetic" covers Pythagorean number theory and contains the
earliest mention of the "mense Pythagorica," a multiplication table of Greek
origin. In great part, it deals with the same subjects as the arithmetical books
of the Elements, but where Euclid represents numbers by lines, Nichomachus
uses arithmetical notation expressed in ordinary language.
The book is opened at the beginning of Book II which deals with squares, cubes,
polygonal numbers and ten different types of proportion.
HISTORY OF SCIENCE COLLECTION
Claudius Ptolemy (circa
85circa 165) Georg von Peurbach (14231461)
Johann Müller of Königsberg, called Regiomontanus
(14361476)
Epitoma in Almagestum Ptolemaei ...
Venice: Johannes Hamman for the editors, [31 August] 1496.
Ptolemy's Almagest, a name derived from the medieval Latin
form of its Arabic title, was the most important, encyclopedic, and complex
astronomical and mathematical work of antiquity. Known in Greek as the "Mathematical
Syntaxis" or the "Mathematical Collection," its thirteen books covered every
aspect of mathematical astronomy. For over thirteen hundred years the Almagest
remained the basis for all sophisticated astronomy.
In 1460 Georg von Peurbach, professor of astronomy at the university of Vienna,
was commissioned by Johann Cardinal Bessarion, Papal Legate to the Holy Roman
Empire, to make a comprehensible Latin condensation of Ptolemy's work. Ignorant
of Greek, he based his eptiome on a copy of Gerard of Cremona's 12thcentury
Latin translation of the "Syntaxis." Peurbach died just after finishing Book
VI, and the remaining seven books were completed by his former student, Johann
Müller of Königsberg, now known simply as Regiomontanus. The manuscript was
completed sometime before April 28, 1463, but it was not until 20 years after
Regiomontanus' death, that it was first printed under the joint editorship of
Caspar Grosch and Stephan Römer.
The Epitome provided easier access to Ptolemy's masterpiece, but it was
more than a mere compressed translation. It added later observations, revised
computations and offered critical commentary on obscure points and errors in
the original text. Among the latter was the observation that Ptolemy's lunar
theory required the moon's diameter to vary much more that it really did.
This book is opened to that passage on lunar theory, Book V, proposition 22,
which attracted the attention of Nicolaus Copernicus (14731543), then a young
student at Bologna, who later overthrew the terracentric Ptolemaic system with
the heliocentric theory expounded his De revolutionibus orbium coelestium
libri VI (Nuremberg: Johann Petreius, 1543LOWNES).
LOWNES COLLECTION OF SIGNIFICANT BOOKS IN THE HISTORY OF SCIENCE & TECHNOLOGY
Diophantus of Alexandria (circa
200circa 284)
Arithmeticorum libri sex, et De numeris multangulis liber unus ...
Paris: Sébastien Cramoisy for Ambrose Drouart, 1621.
Diophantus, called "the father of algebra," is best known for his
Arithmetica, a work on the solution of algebraic equations and the theory
of numbers. Only six of the original thirteen books mentioned in its introduction
have survived. They constitute a collection of 130 determinate (having only
one solution) and indeterminate problems, which are treated by algebraic equations
and by algebraic inequalities. Diophantus introduced symbolism into algebra
(his signs were abbreviations rather than algebraic notation as we know it),
named powers up to n^{6}, and worked extensively on the solution of indeterminate
equations. The method for solving the latter (such as Ax^{2} + Bx
+ C = y^{2}) is now known as "Diophantine analysis." He accepted
only positive rational solutions and believed that irrational solutions, using
negative numbers, were impossible.
The first Latin edition of the Arithmetica, translated from the Greek
and edited by Wilhelm Xylander (also known as Wilhelm Holtzman, 15321576),
was published in 1575. This first Greek and Latin edition incorporated Xylander's
translation along with additional material and commentary supplied by ClaudeGaspard
Bachet (15811638).
It was in his copy of this edition of Diophantus that Pierre de Fermat (16011665)
scribbled his famous "Last Theorem." It states that x^{n} + y^{n} = z^{n} has no nonzero
solutions for x, y, and z when n>2. He wrote: "To divide a cube into two cubes,
a fourth power, or in general any power whatever into two powers of the same
denomination above the second is impossible. I have discovered a truly remarkable
proof which this margin is too small to hold." It was not until 1995 that Andrew
Wiles proved this theorem, although using a method that Fermat would never have
recognized.
This volume is opened at Book II, proposition 8, "To divide a square number
into two other square numbers," where Fermat wrote his "Last Theorem."
HISTORY OF SCIENCE COLLECTION
Pappus of Alexandria
(circa 300350)
Mathematicarum collectionum libri V qui extant cum commentariis Federici
Commandini ...
Peasro: Hieronymous Concordia, 1588.
The last great mathematician of the Alexandrian School was Pappus, who wrote
this "Synagoge," or "Collection," probably in the first quarter of the third
century. It originally contained eight books, or perhaps as many as twelve,
but only Books IIIVIII have survived intact. They probably were written as
separate treatises and later brought together as the name suggests. The "Collection"
is an historical synopsis of greek mathematics, works of many authors are known
only in the form that Pappus preserved them, together with Pappus' commentary
and other original contributions.
Book III is divided into four parts, the last of which shows how to inscribe
the five regular polyhedra in a sphere. Book IV deals with the properties of
curves, squaring of the circle, and the trisection of an angle. In Book V, on
isoperimetry, Pappus shows that a sphere is greater in volume than any of the
regular solids whose perimeters are equal that of the sphere. Book VI is astronomical
and may be seen as an introduction to Ptolemy's Syntaxis. Book VII examines
Euclid's Porisms, and five books by Apollonius, all of which have been
lost. Book VIII, devoted to mechanics, begins by defining center of gravity,
then gives the theory of the inclined plane, and concludes with a description
of the five mechanical powers: the wheel and axle, the lever, the pulley, the
wedge, and the screw.
This first Latin edition is based upon a translation and commentary made by
Federico Commandino (15091575), and published a dozen years after his death
by his soninlaw, Valerio Spaccioli. It is opened at the first page of Book
VII.
That which is called the Domain of Analysis, my son Hermodorus, is,
taken as a whole, a special resource that was prepared, after the composition
of the Common Elements, for those who want to acquire a power in geometry
that is capable of solving problems set to them: and it is useful for this
alone. It is written by three men: Eculid the Elementarist, Apollonius of
Perga, and Aristaeus the elder, and its approach is by analysis and synthesis
... .
HISTORY OF SCIENCE COLLECTION
Proclus
(circa 410485)
In primum Euclidis elementorum librum commentariorum ...
Padua: Gratiioso Perchacino, 1560.
Proclus was taught by the philosopher Plutarch at Plato's Academy in Athens.
More of a philosopher than a mathematician, he eventually became head of the
Academy, and remained there until his death. His "Commentary of the first book
of Euclid's elements" is our principal source of information about the early
history of Greek mathematics, from Thales to Proclus, and was probably drawn
from his own lectures at the Academy. He had access to books, such as the history
of geometry by Eudemus (4th century BC), one of Aristotle's students, and the
works of Geminus of Rhodes (circa 70 BC), which are now lost, and to extracts
from books that were lost even in his own time. This work also constitutes the
first book in the philosophy of mathematics.
A Greek version of the commentary was appended to Simon Grynäus' edition
of Euclid's Elements, published at Basle in 1533 (displayed in center
case). This first Latin edition was edited with commentary by Francesco Barozzi
(floriat 15501590), who also translated the works of Heron, Pappus
and Archimedes. It is opened at Book II, chapter I, "That geometry is a part
of general mathematics and of its subject matter."
In the preceeding discourse we have examined the common characters pervading
all mathematical science, following Plato's lead and also using thoughts collected
from other sources that are relevant to the present study. It follows next
to speak of geometry itself and of the treatise on the Elements that lies
before us and for whose sake the whole of this work has been undertaken ...
.
HISTORY OF SCIENCE COLLECTION
Boethius
(circa 480524)
Arithmetica boetij ...
Augsburg: Erhard Ratdolt, [20 May] 1488.
Today Boethius is best known for his memoir on the consolation of
philosophy, De consolatione philosophiae (Nuremberg: Anton Korberger,
1486LOWNES), although for almost a thousand years, his arithmetic and geometry
texts were considered authoritative in the Western world. Their survival may
have been due both to the fact that their author died as a martyr, and to the
poor quality of mathematical achievement during the Middle Ages.
Boethius' Arithmetica, a superficial translation of Nichomachus's
"Introduction to arithmetic," did provide some Pythagorean number theory which
was absorbed into medieval instruction as part of the quadrivium: artithmetic,
geometry, astronomy, and music. This work was both edited and printed by Erhard
Ratdolt, who, six years earlier, had issued the first printed edition of Euclid's
Elements. It is opened at Book I, chapters 25.
2. Concerning the substance of numbers.
3. The definition of numbers and the division of odd and even.
4. The definition of odd and even numbers according to Pythagoras.
5. According to a more ancient method, the division of even and odd.
ANNMARY BROWN MEMORIAL COLLECTION
Thomas Bradwardine (circa
12901349)
Geometria speculativa ... cum quodam tractatu de quadratura criculi
...
Paris: Guy Marchant, [20 May] 1495.
This is the first printed mathematical work written by an Englishman.
Bradwardine, who became Archbishop of Canterbury shortly before his death, was
educated at Merton College, Oxford, where he subsequently lectured. His two
mathematical works, Geometria speculativa and Arithmetica speculativa
(also in our collection), both probably written before 1335, were first printed
at Paris in the latefifteenth century. They were edited by the Spanish mathematician,
Pedro Sanchez Ciruelo (circa 1470circa 1550), who also published one of our
editions of Joannes de Sacrobosco's Sphaera mundi (Paris: Jean Petit,
1515HSC).
Bradwardine's geometry text is compiled from the works of Euclid, Boethius and
Campanus of Novara. Its four books covered the theory of proportions and solid
geometry, as well as some topics not developed in the Elements, especially
stellar polygons, isoperimetry, and the filling of a space by touching polyhedra.
This "brief compendium of the art of geomety" is opened at the first page which
bears a striking resemblance to the edition of Euclid's Elements printed
by Ratdolt a dozen years earlier (in center case).
HISTORY OF SCIENCE COLLECTION
Girolamo Cardano
(15011576)
Artis magnae sive de regulis algebraicis liber unus ...
Nuremberg: Johann Petrius, 1545.
According to his autobiography, Cardano had a checquered career as professor
of medicine at the Universitiy of Pavia (15401560), where he is reputed to
have divided his time between studying mathematics, debauchery and mechanics.
Nevertheless he achieved great fame as a physician, rivalling that of Andreas
Vesalius (15141564), and was frequently called upon to attend the crowned heads
of Europe. In 1560 his eldest son was executed for having poisoned his wife,
and, disturbed by the dissolute life being led by his youngest son, he cut off
the youngman's ears. In 1562, hoping to escape the family scandal, he accepted
the chair in medicine at Bologna, but resigned soon thereafter. Almost a decade
later he was imprisoned by the Inquisition for the heretical act of having cast
a horoscope of Christ, thereby attributing the events in His life to the influence
of the stars. Cardano recanted, abandoned teaching and, in 1571, travelled to
Rome where he obtained a lifetime pension from Pope Pius V as astrologer to
the papal court. This proved to be his undoing, because having predicted his
own death in his 75th year, he commited suicide on September 21, 1576, allegedly
to maintain his reputation for accuracy.
Cardano wrote more than 200 works on medicine, mathematics, physics, philosophy,
religion and music. His lasting fame, however, rests on his contributions to
mathematics and the "The great art, or the rules of algebra" is his masterpiece.
Prior algebraists dealt only with positive roots of equations, he discussed
negative and even complex roots and proved that the latter would always occur
in pairs. Before Cardano, only the solution of an equation was sought, but he
also observed the relations between the roots and the coefficients of the equation
and between the succession of the signs of the terms and the signs of the roots,
making him the originator of the theory of algebraic equations.
One of the principal aim of sixteenthcentury mathematicians had been to solve
equations of the third and fouth degree. Many solutions were achieved, but the
results were closely guarded secrets. Niccolò Tartaglia (14991557) confided
his solution of the cubic equation (in the form x^{3} + mx^{2}
= n) to Cardano, who, without permission, published that solution, which became
known as "Cardano's Rule," in Chapter XI, "On a cube and things equal to a number,"
of the Artis magnae.
Scipio Ferreus, of Bologna, invented this solution nearly thirty years ago and
taught it to Antonio Maria Florido, of Venice, who by engaging in a contest
with Nicholas Tartaglia, of Brescia, afforded Nicholas an occasion to discover
the solution. He gave it to us, but kept back the demonstration, so we, having
confidence in the solution, sought the proof, reduced it to rules (which was
a very difficult thing to do) and have set it down thus.
Tartaglia responded to this breach of faith, in Book IX of his Questi et
inventioni diverse (Venice: Venturino Ruffinelli, 1546HSC), with abusive
insults and accusations of treachery. A series of challenges and Cartelli,
two of which are in our collection, were then exchanged with Cardano's student
and soninlaw, Lodovico Ferrari (15221565). On August 18, 1548, a public contest
was held between Tartaglia and Ferrari, in which each contender was to solve
equations posed by the other within a set time limit. After one day of polemics
and posturing, Tartaglia departed and the victory fell to Ferrari by default.
Of the three, Ferrari was the more accomplished mathematician, and his solutions
to 20 cases of quartic equations (in the form x^{4} + px^{2}
+ qx + r = 0) also were published by Cardano in the Artis magnae.
LOWNES COLLECTION OF SIGNIFICANT BOOKS IN THE HISTORY OF SCIENCE &TECHNOLOGY
Robert Recorde
(15481620)
The whetstone of witte ...
London: Jhon Kyngston, 1557.
Recorde virtually founded the English school of mathematical writers. Deliberately
choosing the vernacular, he wrote simple, clear English prose of a higher quality
than scientific contemporaries. He was a skilled teacher who used a dialogue
format in his texts so that his students could be taken stepbystep through
the course of instruction. His mathematical books were written in the order
in which he intended them to be studied: arithmetic, plane geometry, practical
geometry, astronomy, and algebra.
The arithmetic was the most popular of his books. First issued in 1543, this
text, and its later editions, dealt with arabic numbers, fundamental operations,
reduction, progression, "the rule of three," and counter reckoning as applied
to whole numbers and fractions. A seventeenthcentury printing, Records
Arithmetick, or, The ground of artes (London: James Fletcher, 1652,) is
in our collection. His Pathwaie to knowledge (London: Jhon Kyngston,
1574HSC) is a translation of Books IIV of Euclid's Elements, in which
the constructions are separated from the theorems and rearranged to suit Record's
teaching methods. The castle of knowledge (London: Reginald Wolfe,
1556HSC), on the construction and use of the sphere, is an elementary Ptolemaic
astronomy book that included a brief favorable mention of Copernicus.
The whetstone of witte, displayed here, is the first English treatise
on algebra. Published as a continuation of The ground of artes, this
work showed how the square root of an algebraic expression could be extracted,
and introduced the equals sign (=) into mathematical notation. In the chapter
on "The rule of equation commonly called Algebers Rule," Recorde wrote:
Nowbeit, for easie alteration of equations I will propounde a few examples,
bicause the extraction of their rootes, maie the more aptly bee wroughte. And
to avoide the tediouse repetition of these woordes : is equalle to : I will
sette as I doe often in woorke use, a paire of paralleles, or Gemowe lines of
one lengthe [OED: gemew = twin, double or parallel], thus: ====, bicause noe
.2. thynges can be more equalle.
LOWNES COLLECTION OF SIGNIFICANT BOOKS IN THE HISTORY OF SCIENCE & TECHNOLOGY
Johann Müller of Königsberg,
called Regiomontanus (14361476)
De triangulis planis et sphaericis libri quinque ...
Basel: Henrich Petri and Petrus Perna, 1561.
Regiomontanus' "On plane and spherical triangles" was his most original
contribution to the development of mathematics. Completed in 1464, it remained
in manuscript until 1533, when it was first edited and published, as De triangulis
omnimodis, by Johann Schöner (14771547). It is represented here in
a collected edition, compiled by Daniel Santbech, which includes a third edition
of Regiomantanus' De triangulis and his Compositio tabularum sinum
recto, Georg von Peurbach's Tractatus super propositiones Ptolemæi
de sinibus & chordis and Santbech's own Problematum astronomicorum et
geometricorum.
Recognition of trigonometry as a separate discipline, independent of astronomy,
begins with this first systematic treatise on triangles. It contains the earliest
statement of the cosine law for spherical triangles, stating the proportionality
of the sides of a plane triangle to the sines of the opposite angle. This fundamental
proposition appears in Book V, Theorem II.
In every spherical triangle that is constructed from the arcs of great circles,
the ratio of the versed sine of any angle to the difference of the two versed
sines, of which one is of the side subtending this angle while the other is
of the difference on the two arcs including this angle, is as the ratio of the
square of the whole right sine to the rectangular product of the sines of the
arcs placed around the mentioned angle.
Let ABC be a triangle of this sort, having two unequal sides, AC greater than
AB, and each of them less than a quadrant ... .
HISTORY OF SCIENCE COLLECTION
François Viète (15441603)
Canon mathematicus seu ad triangula cum appendicibus ...
Paris: Jean Mettayer, 1579.
Viète was a lawyer (Mary Stuart was one of his clients) and bureaucrat as well
as a mathematician, and it was while serving as counselor to the parliament of
Brittany (15731580) that he began issuing his mathematical works. He personally
subsidized the printing of all of his books, which were published in small, but
lavish, editions that often were given away to his friends and patrons. Consequently,
they were rare books even in the seventeenth century.
This first edition of the "Mathematical canon with an appendix on trigonometry,"
Viètes first published tract, is a fundamental work on trigonometry which
he intended to form the preliminary part of a major astronomical study based
upon a Ptolemaic model (the later parts were never completed). In the tables
displayed here he tabulated all six basic trigonometric functions to the nearest
minute. He urged the use of decimal rather than sexagesimal fractions and indicated
the decimal place by a comma. In this work Viète gives for the first time
the formula equivalent to sin A sin B = 1/2[cos (AB)  cos (A+B)], which
formed the starting point for Napier's discovery of logarithms.
In the earliest work on symbolic algebra, In artem analyticam isagoge
(Tours, 1591), Viète introduced the principle of solving equations by reduction
and used this method in the solution of biquadratics. He used letters to denote
known and unknown quantities in an equation and popularized use of the "+" and
" " signs normally found only in mercantile arithmetics. In our collection
this work is represented by a French translation, Introduction en l'art analytic
ou nouvelle algèbre ... par J. L. Sieur de VauLezard (Paris: Julian
Jacquin, 1630HSC) and in Viète's Opera mathematica (Leyden: Bonaventura
& Abraham Elzevir, 1646LOWNES), edited by Frans van Schooten (16151660).
LOWNES COLLECTION OF SIGNIFICANT BOOKS IN THE HISTORY OF SCIENCE & TECHNOLOGY
John Napier
(15501617)
Mirifici logarithmorum canonis descriptio ...
Edinburgh: Andrew Hart, 1615.
One of the greatest seventeenthcentury advances in computation was the invention
of logarithms by John Napier. His "Description of the wonderful table of logarithms"
is unique in the history science as being the result of original speculation
by one individual unaided by either the work of precursors or contemporaries.
Napier began work on his tables in 1594, but it was 20 years before he was ready
to publish them, in the slim volume displayed here.
His central idea was to construct two sequences of numbers so related that when
one increases in an arithmetical progression, the other decreases in a geometrical
one. The product of two numbers in the second sequence had a simple relation
to the sum of corresponding numbers in the first, and multiplication problems
therefore could be reduced to a process of addition through use of the decimal
fraction system first propounded by Stevin in 1585. Complex division then would
be accomplished by subtraction of logarithms, and the extraction of roots by
division of logarithms.
As Napier proposed in his preface, this ingenious use of the relationship between
mathematical and geometrical progressions greatly simplified elaborate calculations.
Seeing there is nothing ... that is so troublesome to mathematical practice,
nor that doth more molest and hinder calculators, than multiplications, divisions,
square and cubical extractions of great numbers, which besides the tedious expense
of time are for the most part subject to many slippery errors, I began therefore
to consider in my mind by what certain and ready art I might remove those hindrances.
The Descriptio does not contain the logarithms of equidifferent numbers,
but the sines of equidifferent arcs for every minute in the quadrant. These
are given neither as baseten logarithms nor the natural or socalled Naperian
logarithms, but rather logarihms to a base of I/e. In the conclusion of Book
II Napier promises to develop a more convenient system, as was later accomplished
by Henry Briggs in 1624.
LOWNES COLLECTION OF SIGNIFICANT BOOKS IN THE HISTORY OF SCIENCE & TECHNOLOGY
John Napier (15501617)
Rabdologiae seu Numerationis per virgulis libri II
Edinburgh: Andrew Hart, 1617.
Napier also dealt with spherical geometry, and in the first book of
Rabdology or calculation with rods in two books he announced the invention
of "Napier's Bones." These calculating rods, or bones (socalled because they
were made of ivory), inspired by the standard multiplication tables, could be
used in multiplication, division and the extractiion of roots. Throughout the
seventeenth century, and well into the eighteenth, other authors incorporated,
or offered improvements upon, them in their own arithmetic texts. Those rods
were, in essence, the first practical calculating machines
This copy is from the library of the Scottish mathematician and bibliographer,
William Rae Macdonald (18431923).
HISTORY OF SCIENCE COLLECTION
Henry Briggs (15611630)
Arithmetica logarithma sive logarithmorum chiliades triginta, pro numeris
naturali serie crescentibus ab unitate 20,000 et a 90,000 ad 100,000 ...
London: William Jones, 1624.
All modern logarithmic tables are descended from the 14place tables
of decimal logarithms found in this first edition of Arithmetica logarithma.
In 1596 Henry Briggs became the first professor of geometry in the college founded
in London by Sir Thomas Gresham, and by 1620, at the invitation of Sir Henry
Savile, he become Savilian professor at Merton College, Oxford, a post which
he held until his death 11 years later.
Briggs learned the use of logarithms from John Napier, with whom he worked on
their development during the final two years of the latter's life. The first
part of this work is a dissertation on the nature and use of logarithms which
includes a description of the roles taken by both men in developing them. In
the preface to the reader, Briggs noted that it was Napier who first proposed
that the logarithms would be more useful if they were to baseten so that log
1 = 0 and that log 10 = 1.
That these logarithms are different from those which the distinguished Baron of
Merchiston published in his Canon Mirificus should not cause you to wonder; for
as I was explaining the doctrine of logarithms to my hearers in London, publicly
at Gresham College, I remarked that in the future it would be more convenient
if 0 be kept as the logarithm of the whole sine ... and I at once wrote to the
author about it. ... But he thought that the change should be made so that 0 would
be the logarithm of unity and 10000000000 the logarithmof the whole sine: which
I could not but acknowledge to be by far the most convenient.
The second part consists of tables of thirty thousand logarithms, from 1 to 20,000
and from 90,000 to 100,000. This copy also contains an appendix carrying the
tables to 101,000 and a leaf of square roots from 1200. A "second edition"
of Arithmetica logarithma, edited by Adrian Vlacq (16001666) and containing
the intermediate seventy chiliads, was printed at Gouda in 1628.
LOWNES COLLECTION OF SIGNIFICANT BOOKS IN THE HISTORY OF SCIENCE & TECHNOLOGY
Adriaan Vlacq (16001667)
Arithmetica logarithma sive logarithmorum chiliades tentum, pro numeris
naturali serie crescentibus ab unitate ad 100000 ...
Gouda: Petrus Rammaseyn, 1628.
Ezechiel de Decker, a Dutch surveyor, and Adriaan Vlacq, a bookseller
and publisher (formerly of London and Paris), filled in the gap between 20,000
and 90,000 in Briggs' logarithmical tables. Their augmented tables from 1 to
100,000, in 10 decimal places, were immediately welcomed by mathematicians and
astronomers throughout the Continent. A French translation, Arithmetique
logarithmetique, ou, La construction et usage d'une table contenant les logarithms
de tous les nombres depuis l'unité jusque 100000 (Gouda: Petrus Rammaseyn,
1628HSC), was issued almost simultaneously by the same publisher.
This copy is from the library of the English mathematician J. W. L. (James Whitbread
Lee) Glaisher (18481929), of Trinity College, Cambridge.
HISTORY OF SCIENCE COLLECTION
Ludolph van Ceulen
(15391610)
De arithmetische en geometrische fondamenten ...
Leyden: Jan van Closter and Jacob Marcus, 1615.
This volume contains one of the most exhaustive calculations
of
made during the seventeenth century. Originally from Germany, Ludolph
van Ceulen taught fencing and mathematics in Arnhem, Breda and Delft
before moving to Leyden in 1594. Here, in 1600, he received a position
at the engineering school founded by Prince Maurice of Nassau at Leyden
and spent the final ten years of his life there, teaching arithmetic,
surveying and fortification. His most consuming mathematical passion
appears to have been a lifelong search for the value of , which he
eventually computed to thirtyfive decimal places employing a variant
of Archimedes' method using inscribed and circumscribed regular polygons
having up to 2^{62} sides.
In this first Dutch edition of Ceulen's Arithmetische en geometrische fondamenten,
published five years after his death by his widow, Adriana Symons, is computed
to thirtythree decimal places. A Latin translation, Fundamenta arithmetica
et geometrica, was edited for the scholarly community by his student, Willebrord
Snell (15801626), and also published in 1615. Ceulen's final value for , known
in Germany for generations as "Ludolph's Number," is carved on his tombstone
in the Pieterskerk in Leyden.
 3.14159265358979323846264338327950288 
HISTORY OF SCIENCE COLLECTION
Simon Stevin (15481620)
De thiende ...
Gouda: Petrus Rammaseyn, 1626.
Stevin, a bookkeeper, mathematician, engineer, QuarterMaster General of the
Netherlands and author of 11 books, made significant contributions to the fields
of trigonometry, geography, fortification and navigation. As mathematics and
science advisor to Maurice of Nassau, Prince of Orange, he devised a plan for
flooding the Lowlands in the path of an invading army by opening selected sluices
and dikes. A byproduct of that plan was his hypothesis that pressure exerted
by a liquid upon a given surface depends upon the height of the liquid and the
area of the surface; proving that hypothesis led to the foundation of the science
of hydrostatics.
The volume displayed here is the second edtion of "The tenth" (first published
in 1585), in which Stevin proposed the adoption of decimal numbers in order
to unify systems of measurement. He defined them as "a kind of arithmetic based
on the idea of progression by tens, making use of the ordinary Arabic numerals,
in which any number may be written and by which all computations that are met
in bsiness may be performed by integers alone without the aid of fractions."
He did not invent decimal fractions, but he did introduce their widespread use
to replace the cumbersome sexagesimal fractions employed by astronomers.
Stevin concluded by noting that universal introduction of decimal coinage,
weights and measures would only be a matter of time. A French translation
of the first edition, La disme, was included in L'arithmetique de
Simon Stevin de Bruges (Leyden: Christopher Plantin, 1585HSC), and, following
their Revolution, the French applied the "metric system" to their weights
and measures. The earliest English translation, Disme: the art of tenths,
or, Decimall arithmetike, teaching how to performe all computations whatsoever
by whole numbers without fractions, was printed at London in 1608.
HISTORY OF SCIENCE COLLECTION
Thomas Hariot (15601621)
Artis analyticæ praxis, ad æquationes algebraicas nova expedita,
& generali methodo resolvendas ...
London: Robert Barker and the heirs of John Bill, 1631.
After finishing his studies at Oxford in 1580, Hariot entered the
service of Sir Walter Raleigh, who attached him, as scientific advisor, to Sir
Richard Grenville's 158586 expedition to Roanoke Island. This experience was
recounted in A briefe and true report of the new found land of Virginia
(London, 1588). In 1598 Hariot left Raleigh, and with Walter Warner and Robert
Hughes entered the service of Henry Percy, the ninth Earl of Northumberland.
This posthumous work, dedicated to Percy and edited by Warner, "embodies the
inventions by which Hariot virtually gave to algebra its modern form. The important
principle was introduced by him that every equation results from the continual
multiplication of as many simple ones as there are units in the index of its
highest power, and has consequently as many roots as it has dimensions" (DNB).
Hariot recognized negative roots and complex roots in solving equations, noted
relations between coefficients and roots, and made the observation that if a,
b, and c are roots of a thirddegree equation, then the cubic is the form (xa)(xb)(xc)
= 0. He invented simplified notation for algebra, including use of "." for multiplication,
"<" for less than, ">" for greater than, and "" for inequality, that
greatly influenced the work of Viète and Wallis. Hariot's Analytical
arts applied to solving algebraic equations, which gives birth to the English
school of algebra, did not capture the critical attention that it deserved because
Warner appparently did not completely understand nor fully appreciate the depth
of his friend's work.
The volume is opened at "Exegetice numerosa," on the solution of quadratic equations.
LOWNES COLLECTION OF SIGNIFICANT BOOKS IN THE HISTORY OF SCIENCE & TECHNOLOGY
Galileo Galilei
(15641642)
Discorsi e dimostrazioni mathematiche, intorno due nuove scienzi attenenti
alla mechanica
&
i movimenti locali ...
Leyden: Elzevier Press, 1638.
This, Galileo's last work, is the first modern textbook of physics and a fundamental
work in the science of mechanics. As a consequence of his trial for heresy,
Galileo was forbidden to publish in Florence or Rome by the Congregation of
the Index. Unable to obtain a license to print this work in Venice, Galileo
had a manuscript copy smuggled out of Italy to friends in France, who eventually
brought it to the Elzeviers in Holland.
The mathematical analyses of the two new sciences dealing with the strength
of materials and kinematics in the Discorsi complement the philosophical
discussions of the Dialogo sopre i due massimi sistemi del mondo (Florence:
Landini, 1632HSC). The three interlocutors of the Dialogo, Salviati
(Galileo's spokesman), Sagredo (an intelligent layman) and Simplicio (an upholder
of tradition), are also the three protagonists of the Discorsi. The
book is divided into four dialogues, each taking aplce on a different day.
The first two cover the constitution of matter, the nature of mathematics,
the place of experiment and reason in science, the nature of sound and the
speed of light. The last two are devoted to the treatment of uniform and accelerated
motion and a discussion of parabolic trajectories. There is also an appendix
on centers of gravity, followed by an added day of discussion on the force
of percussion.
Galileo's Discourses & mathematical demonstrations concerning the two new
sciences : pertaining to mechanics & local motions, presented a mathematical
treatment of motion and inertia that replaced classical Aristotelian theories
and gave rise to the principles of modern physics. The volume is opened at
the fourth day's discussion "On the motion of projectiles."
LOWNES COLLECTION OF SIGNIFICANT BOOKS IN THE HISTORY OF SCIENCE & TECHNOLOGY
Benedetto Castelli (15781643)
Della misura dell 'acque correnti ...
Rome: Francesco Cavalli, 1639.
A Benedictine monk, mathematician, astronomer, physicist, and student of Galileo,
Castelli was named professor of mathematics at the University of Pisa in 1613.
He taught there until 1626, one of his pupils being Bonaventura Cavalieri, then
moved on to Florence and finally to Rome, where Evangelista Torricelli and Giovanni
Alfonso Borelli and Vincenzio Galilei were among his students. In 1626 Pope
Urban VIII appointed him papal consultant on hydraulics, and requested that
he study of Rome's water supply.
The first edition of Della misura dell'acque correnti appeared two years
later. This treatise on the speed of liquids in channels and the measurement
of that speed is one of the cornerstones of modern hydraulics. Castelli proposed
the first accurate and effective methods for measuring the volume of moving
water and discussed the relationship of velocity and head, in flow through an
orifice.
This second, enlarged, edition contains an appendix, Demonstrazioni geometriche,
presenting the geometric method of using cross sections of a river to measure
the volumes of water passing in a given time. It is opened at the titlepage
engraving, also used in the first edition, of a bridge crossing the Tiber with
the papal arms above the central arch.
HISTORY OF SCIENCE COLLECTION
Bonaventura Cavalieri (circa
15981647)
Geometria indivisibilibus continuorum nova quadam reaione promota ...
Bologna: Typographia de Duciis, 1635.
In 1629, Cavalieri, a Jesuati an adherent to the Rule of St. Augustine
was appointed to the chair in mathematics at the University of Bologna, a
post he occupied until his death, largely through the recommendation of Galileo,
who proclaimed him the foremost Italian mathematician of the day. His Geometria
indivisibilibus contains the first systematic exposition, as it pertains
to the principles of summation, of what we now know as the calculus. He accomplished
this by employing the concept of "indivisibles," or "infinitesimals," which
served the same purpose as "the method of exhaustions" employed by Archimedes
and other Greek mathematicians. In principle these approaches were the same
but the system of notation for indivisibles was much more concise and convenient.
Using indivisibles, which are based upon the assumption that any magnitude may
be divided into an infinite number of small quantities which can be made to bear
any required ratio one to the other, he was able to perform the equivalent of
the integration of polynomials. This allowed him to solve problems connected with
the quadrature of curves and surfaces, to determine centers of mass, and to calculate
areas and volumes of complex geometric shapes. The latter is expressed in the
"principle of Cavalieri," which concludes that two solids of equal altitudes have
the same volume if plane cross sections at equal height have the same area. Cavalieri's
work was superseded with the invention of the integral calculus by Leibniz and
Newton at the end of the seventeenth century.
Cavalieri's Directorium generale uranometricum in quo trigonometriae logarithmicae
fundamenta ac regulae demonstrantur astronomicaeque, supputationes ad solam
ferè vulgarem additionem reducuntur (Bologna: Nicolai Tebaldini, 1632)
the first work on logarithms ever to be printed in Italy also is in our
collection.
HISTORY OF SCIENCE COLLECTION
Christiaan Huygens (16291695)
Horologium oscillatorium, sive, de motu pendulorum at horologia aptato
demonstrationes geometricae ...
Paris: F. Muget, 1673.
Huygens was recognized early as a mathematician of note and a student
of what, today, would be called mathematical physics. He studied law and mathematics
at the Unversity of Leyden, the latter with Frans van Schooten (16151660),
editor of three Latin editions of Descartes' Geometry (1649, 16561661,
1683ALL HSC), and soon achieved international recognition, becoming a Fellow
of the Royal Society of London in 1664, and one of the first Pensionnaires of
the Académie Royale des Sciences in Paris in 1666. He applied his talents
to astronomy and mathematics, and to the technological problems of optics, heat
engines, and clock design, hoping to use his pendulum clock for solving the
problem of determining longitudes at sea.
In 1657, Huygens first applied Galileo's law of the pendulum to weight clocks
and the results of that work was published almost two decades later.The
pendulum clock, or, geometrical demonstrations concerning the motion of pendulums
as applied to clocks, contained some of the most advanced forms of the
calculus developed in the period before Newton and Leibniz. The clock itself
is described in the first part of the book. Four separate and highly abstract
mathematical and mechanical treatises follow. Part 2, on cycloidal motion,
contains an account of the descent of heavy bodies under their own weight
in a vacuum, and establishs the cycloid as a tautochronous curve. Part 3 defines
evolutes and involutes and illustrates methods for finding the evolutes of
the cycloid and the parabola. Part 4, on centers of oscillation, solves the
problem of the compound pendulum, and shows that centers of oscillation and
suspension are interchangeable. Part 5, an appendix, shows that centrifugal
force on a body which moves around a circle of radius r with a uniform velocity
v varies directly as v2 and inversely a r. It was the first attempt to apply
dynamics to bodies of finite size rather than just to particles and this was
later to influence Newton's theory of gravitation.
When combined with Galileo's laws of falling bodies, Huygens' theory of the
center of oscillation implies the conservation of potential energy. His concept
is based on the axiom that if the bodies of a system start moving under the
influence of gravity alone, then the center of gravity of the system can not
rise above its original position. The volume is opened at the discussion of
that point.
LOWNES COLLECTION OF SIGNIFICANT BOOKS IN THE HISTORY OF SCIENCE & TECHNOLOGY
René Descartes (15961650)
Discours de la méthode pour bien conduire sa raison, & chercher la
verité dans les sciences ...
Leyden: Jan Maire, 1637.
This fundamental work in the philosophy of science presents Descartes' concept
of the scientific method, "to search for truth in science," summed up in the
four rules enunciated in part 2: to accept nothing that is not evidently true;
to divide the difficulty into as many parts as are needed; to start with the
simplest problems; and to be so comprehensive as to leave nothing out.
The metaphysical doctrine expounded in part 4 is often summed up in a phrase
taken from the discussion on truth and thought cogito, ergo sum. Those
philosophical and metaphysical discussions which constitute the first six parts
serve as an introduction to the three concluding treatises; La dioptrique,
which includes his derivation of the law of refraction; Les météores,
which contains an explanation of numerous atmospheric phenomena, including the
rainbow; and La géométrie, which explains Descartes' application
of algebra to geometry thereby inventing analytical geometry.
La géométrie is divided into three books. The first book,
"Problems that one can construct employing only circles and straight lines,"
contains an explanation of the principles of analytical geometry. In the second
book, "On the nature of curved lines," Descartes divides curves into two classes,
geometrical and mechanical, and also deals with the theory of tangents to
curves. An historical analysis of algebra is presented in the third book,
"On the construction of solid or supersolid problems," as well as the modern
convention of using letters from the beginning of the alphabet to denote known
quantities and those at the end of the alphabet to denote unknown quantities.
This volume is opened at a passage, in the second book, containing one of the
fundamental concepts of analytic geometry.
I could here give several other ways of tracing and conceiving a series of curved
lines, each curve more complex than any preceeding one, but I think the best
way to group together all such curves and then classify them in order, is by
recognizing the fact that all the points of those curves which we may call Geometric,
that is, those which admit of precise and exact measurement, must bear a definite
relation to all the points of a straight line, and that this relation must be
expressed by means of a single equation.
LOWNES COLLECTION OF SIGNIFICANT BOOKS IN THE HISTORY OF SCIENCE & TECHNOLOGY
Gottfried Wilhem Leibniz (16461716)
"Nova methodus pro maximis et minimis, itemque tangentibus, quae nec factas
nec irrationales quantitates moratur, & singulare pro illi calculi genus,"
Acta eruditorum ...
Leipzig: Christopher Günther for J. Gross and J.F. Gleditsch, 1684,
(pages 467473).
Leibniz, a diplomat and mathematician as well as a philosopher, was
instrumental in establishing scientific academies in Berlin, St. Petersburg,
Dresden, and Vienna. In 1682 he founded the journal Acta Erutitorum,
in which he subsequently printed his discovery of the differential calculus.
Although Newton probably had discovered the calculus in 1666, Leibniz was the
first to publish his method, which employed a system of notation that was far
superior to the fluxions advocated by Newton. His first paper in the Acta
eriditorum for October, 1684, enunciated a general method for finding maxima
and minima, for drawing tangents to curves, and finding a curve whose subtangent
is constant. This account also contained the first use of the symbols dx and
dy, and the rules of differentiation, including d(uv) = udv + vdu.
The priority dispute between Newton and Leibniz is one of the more famous controversies
in the history of science. It led to a breach between English and Continental
mathematics that lasted until the nineteenth century.
HISTORY OF SCIENCE COLLECTION
Sir Isaac Newton
(16421727)
Philosophiae naturalis princupia mathematica ...
London: Printed for the Royal Societry by Joseph Streater, 1687.
Following the researches of Galileo in the study of motion and its mathematical
analysis, and the important contributions of Descartes and Huygens, scientific
discoveries in the seventeenth century culminated with the achievements of Newton
in dynamics and gravitational astronomy.
Like Gaul, "The mathematical principles of natural philosophy" is divided into
three parts. Book I contains the dynamical theory of the whole work, beginning
with the motion of massparticles in a conic orbit. Here Newton generalizes
the law of attraction: every particle of matter in the universe attracts every
other particle of matter with a force which varies directly as the product of
their masses, and inversely as the square of the distance between them. Book
II is a treatise on fluid mechanics, including the motion of bodies in a resisting
medium. It concludes by stating that the Cartesian theory of vortices was inconsistent
with the laws of motion. Book III is devoted to astronomy, including the motion
of comets and the pull of the tides, and shows how all the phenomena of the
solar system can be predicted on the basis of universal gravitation and the
theory of dynamics.
The Principia is often described as the greatest work in the history
of science. Copernicus and Galileo had shown the way, but they only described
the phenomena they observed, Newton explained the universal laws that governed
those phenomena, and provided a great synthesis of an interconnected cosmos.
It is a very difficult book to fully comprehend at first, or second, reading,
nevertheless, Newton's principles reigned supreme for next two centuries. A
second edition of the Principia was not published until 1713 (LOWNES),
and the first English translation, by Andrew Motte, did not appear until 1729
(HSC).
Edmond Halley (16561742) paid for publishing the first edition of the Principia,
because neither Newton nor the Royal society had sufficient funds, and booksellers
were unwilling to risk their own capital on a difficult scientific treatise.
Halley also edited the work and saw it through the press. Two variant issues
of the first edition were published. The first has a twoline imprint naming
only Joseph Streater and the Royal Society, while the second has a threeline
imprint that also mentions the bookseller, Samuel Smith. The latter reflects
Halley's attempt to recoup his expenses by turning over a portion of the whole
edition of about 300 copies, to Smith, for sale on the Continent.
This copy for the first edition, first issue, of the Principia is opened to
the titlepage. A copy of the threeline imprint, second issue, is also in our
collection (HSC).
LOWNES COLLECTION OF SIGNIFICANT BOOKS IN THE HISTORY OF SCIENCE & TECHNOLOGY
Guillaume François Antoine l'Hospital, Marquis
de SainteMesme (16611704)
Analyse des infiniment petits, pour l'intelligence des lignes courbes
...
Paris: Imprimerie Royale, 1696.
In 1691, when Jean Bernoulli (16671748) spent some months in Paris
teaching l'Hospital the new calculus, a complete understanding of the infinitesimal
calculus and its power was essentially limited then to Newton, and Leibniz,
and the latter's associates Jean and Jacques Bernoulli (16541705). There was
no textbook on the subject until l'Hospital produced the first treatise which
explained the principles and use of the calculus, Analyse des infiniment
petits. The book starts with a set of definitions, in which the differential
is defined as the infinitely small portion by which a variable quantity continuously
increases or decreases, followed by a series of axioms and the basic rules of
the differential calculus. The second chapter applies those rules to the determination
of a tangent to a curve in a given point, and the third deals with maximunminimum
problems. Later chapters deal with points of inflection, cusps and higherorder
differentials.
This work had wide circulation and brought differential notation, as developed
by Leibniz, into general use throughout continental Europe. It is opened at chapter
9, "Solution of some problems using methods previously discussed," which contains
the "rule of l'Hospital," a partial investigation of the limiting value of the
ratio of functions, which, for certain values of the variable, take the indeterminite
form 0:0.
HISTORY OF SCIENCE AND COLLECTION 