From Euclid to Newton:
An Exhibition in Honor of the 1999 Conference of the
Mathematical Association of America
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   (ca. 326-265 BC)
   (ca. 287-212 BC)
Apollonius of Perga
   (ca. 260-200 BC)
Nichomachus of Gerasa
   (ca. 100)
Claudius Ptolemy
   (ca. 85-165)
Diophantus of Alexandria
   (ca. 200-284)
Pappus of Alexandria
   (ca. 300-350)
   (ca. 410-485)
   (ca. 480-524)
Thomas Bradwardine
   (ca. 1290-1349)
Girolamo Cardano
Robert Recorde
Johann Mller of Knigsberg,
called Regiomontanus
Franois Vite
John Napier
Henry Briggs
Adriaan Vlacq
Ludolph van Ceulen
Simon Stevin
Thomas Hariot
Galileo Galilei
Benedetto Castelli
Bonaventura Cavalieri
   (ca. 1598-1647)
Christiaan Huygens    (1629-1695)
Ren Descartes
Gottfried Wilhem Leibniz
Sir Isaac Newton
Guillaume Franois Antoine l'Hospital,
Marquis de Sainte-Mesme





Brown University Library possess a copy of each sixteenth-century 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 right-angled 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 (a2 + b2 = c2 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. 326-ca. 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 I-IV are devoted to plane geometry, Book V deals with the theory of proportions, and Book VI with the similarity of plane figures. Books VII-IX are on number theory, Book X on commensurability and incommensurability, Books XI-XII explore three dimensional geometric objects, and Book XIII deals with the construction of the five regular solids. Later non-Euclidian 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 6th-century 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 right-hand 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 ... .


Greek Editions

Eukleidou Stoicheion biblon ...
Basel: Johann Herwagen, 1533

The German theologian Simon Grynäus (1493-1541), 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 (1474-1537), 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 (1474-1559), 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 Greek-language version of the Elements available until the appearance of a Greek and Latin edition of Euclid's complete works produced by David Gregory (1659-1708), early in the eighteenth century Eukleidou ta sozomena = Euclidis quae supersunt omnia (Oxford: Theatro Sheldoniano, 1703-HSC). The first edition in what is now recognized as modern Greek was not published until 1820.


Italian Editions

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.1499-1557), an eminent mathematician and author of the definitive six-volume treatise on sixteenth-century Italian mathematics, General trattato di numeri et misure (Venice: Curtio Troiano, 1556-60-HSC). 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 title-page is a line-for-line copy of its predecessor, are in our collection.


German Editions

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 VII-IX), was edited by Johann Scheubel (1494-1570), professor of mathematics at the University of Tübingen. Scheubel had earlier produced a Latin edition of Books I-VI, Sex libri priores de geometricis principii (Basel: Johann Herwagen, 1550-HSC); an arithmetic text, De numeris et diversis rationibus seu regulis computationum opusculum (Leipzig: Michael Blum, 1545-HSC), 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, 1552-HSC).

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 I-VI) as edited by Wilhelm Holtzman (also known as Wilhelm Xylander, 1532-1576), 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 title-page presenting it to his colleague Jacob Curio (1497-1572).

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.


French Editions

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 I-VI and VII-IX) 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 3-volume mathematics text produced in sixteenth-century France, L'Arithmetique (Paris: Guillaume Cavellat, 1556-57-HSC).

It was not until early in the seventeenth century that the first complete French translation of the Elements, made by Didier Dounot (1574-1640), 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.


English Editions

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 (1527-1608), 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 "pop-up" books known.


Christoph Clavius (1537-1612)
Euclidis Elementorum libri XV ...
Rome: Vincenzo Accolti, 1574

This two-volume 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 five-volume edition of Clavius' collected works in fifteen parts, Operum mathematicorum (Mainz: Anton Hierat, 1611-12-HSC), 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 sun-dials; and the fifth contains a fundamental exposition and defense of the calendar reform accomplished in 1577-1582 under the auspices of Pope Gregory XIII.


Spanish Editions

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 I-VI) was translated and edited by by Rodrigo de Zamorano (b. 1542), who identified himself on the title-page as an astrologer, mathematician and cosmographer to His Majesty. It may have been modeled, in part, on the superb Latin edition of Federigo Commandino (1509-1575), Elementorum Libri XV (Pesaro: Camillo Francischini, 1572-HSC). 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.


Arabic Editions

Kitab tahrir usul l-Uqlidus, min talif Khwajah Nasir al-Din al-Tusi ...
Rome: Typographia Medicea, 1594.

This Arabic edition [Recension of Euclid's Elements] is allegedly based on a translation of Books I-XIII, made from original Greek sources by the astronomer Nasir al-Din al-Tusi in about 1248. Two versions of this text were issued by the Medicean Press, one with a title-page 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 I-XII of the Elements were included in these printings. In the mid-eighteenth century a cache of the translations, amounting to more than half of the original edition, was discovered in the Vatican Archives and subsequently destroyed.


Archimedes (circa 287-212 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 fore-runner 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 water-screw, 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 sand-reckoner ), 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 (1436-1476) who corrected Jacob's translation using a manuscript owned by Johannes Cardinal Bessarion (1403-1472), 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 corkscrew-like device used to drain fields and ships' bilges by raising water to higher levels. In "The sand-reckoner," 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 helio-centric theories of Aristarchus of Samos (circa 310-circa 230 BC) are mentioned in the introduction to "The sand-reckoner": "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 ... ."


Apollonius of Perga (circa 260-circa 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 9th-century 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 (1509-1575) 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 24-27.

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.


Apollonius of Perga (circa 260-circa 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 Ben-Kora. Both texts were translated into Latin by Abraham Ecchellensis (d. 1664), edited by Giovanni Borelli (1608-1679), and then published almost one century after the first appearance of Books I-IV.

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 (1656-1742) attempted to reconstruct that book in his Apolloni Pergæi Conicorum libri octo (Oxford: Theatro Sheldoniano, 1710-HSC).

This volume is opened at Book V, propositions 9-10, 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.


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.


Claudius Ptolemy (circa 85-circa 165) Georg von Peurbach (1423-1461) Johann Müller of Königsberg, called Regiomontanus (1436-1476)
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 12th-century 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 (1473-1543), then a young student at Bologna, who later overthrew the terra-centric Ptolemaic system with the helio-centric theory expounded his De revolutionibus orbium coelestium libri VI (Nuremberg: Johann Petreius, 1543-LOWNES).


Diophantus of Alexandria (circa 200-circa 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 n6, and worked extensively on the solution of indeterminate equations. The method for solving the latter (such as Ax2 + Bx + C = y2) 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, 1532-1576), was published in 1575. This first Greek and Latin edition incorporated Xylander's translation along with additional material and commentary supplied by Claude-Gaspard Bachet (1581-1638).

It was in his copy of this edition of Diophantus that Pierre de Fermat (1601-1665) scribbled his famous "Last Theorem." It states that xn + yn = zn has no non-zero 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."


Pappus of Alexandria (circa 300-350)
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 III-VIII 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 (1509-1575), and published a dozen years after his death by his son-in-law, 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 ... .

Proclus (circa 410-485)
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 1550-1590), 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 ... .

Boethius (circa 480-524)
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, 1486-LOWNES), 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 2-5.

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.


Thomas Bradwardine (circa 1290-1349)
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 late-fifteenth century. They were edited by the Spanish mathematician, Pedro Sanchez Ciruelo (circa 1470-circa 1550), who also published one of our editions of Joannes de Sacrobosco's Sphaera mundi (Paris: Jean Petit, 1515-HSC).

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).


Girolamo Cardano (1501-1576)
Artis magnae sive de regulis algebraicis liber unus ...
Nuremberg: Johann Petrius, 154

According to his autobiography, Cardano had a checquered career as professor of medicine at the Universitiy of Pavia (1540-1560), 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 (1514-1564), 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 sixteenth-century 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 (1499-1557) confided his solution of the cubic equation (in the form x3 + mx2 = 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, 1546-HSC), 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 son-in-law, Lodovico Ferrari (1522-1565). 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 x4 + px2 + qx + r = 0) also were published by Cardano in the Artis magnae.


Robert Recorde (1548-1620)
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 step-by-step 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 seventeenth-century printing, Records Arithmetick, or, The ground of artes (London: James Fletcher, 1652,) is in our collection. His Pathwaie to knowledge (London: Jhon Kyngston, 1574-HSC) is a translation of Books I-IV 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, 1556-HSC), 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.

Johann Müller of Königsberg, called Regiomontanus (1436-1476)
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 (1477-1547). 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 ... .

François Viète (1544-1603)
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 (1573-1580) 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 (A-B) - 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 Vau-Lezard (Paris: Julian Jacquin, 1630-HSC) and in Viète's Opera mathematica (Leyden: Bonaventura & Abraham Elzevir, 1646-LOWNES), edited by Frans van Schooten (1615-1660).


John Napier (1550-1617)
Mirifici logarithmorum canonis descriptio ...
Edinburgh: Andrew Hart, 1615.

One of the greatest seventeenth-century 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 base-ten logarithms nor the natural or so-called 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.


John Napier (1550-1617)
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 (so-called 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 (1843-1923).


Henry Briggs (1561-1630)
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 14-place 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 base-ten 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 1-200. A "second edition" of Arithmetica logarithma, edited by Adrian Vlacq (1600-1666) and containing the intermediate seventy chiliads, was printed at Gouda in 1628.


Adriaan Vlacq (1600-1667)
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, 1628-HSC), 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 (1848-1929), of Trinity College, Cambridge.


Ludolph van Ceulen (1539-1610)
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 life-long search for the value of , which he eventually computed to thirty-five decimal places employing a variant of Archimedes' method using inscribed and circumscribed regular polygons having up to 262 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 thirty-three decimal places. A Latin translation, Fundamenta arithmetica et geometrica, was edited for the scholarly community by his student, Willebrord Snell (1580-1626), 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 -


Simon Stevin (1548-1620)
De thiende ...
Gouda: Petrus Rammaseyn, 1626.

Stevin, a bookkeeper, mathematician, engineer, Quarter-Master 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, 1585-HSC), 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.


Thomas Hariot (1560-1621)
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 1585-86 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 third-degree equation, then the cubic is the form (x-a)(x-b)(x-c) = 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.


Galileo Galilei (1564-1642)
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, 1632-HSC). 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."


Benedetto Castelli (1578-1643)
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 title-page engraving, also used in the first edition, of a bridge crossing the Tiber with the papal arms above the central arch.


Bonaventura Cavalieri (circa 1598-1647)
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.


Christiaan Huygens (1629-1695)
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 (1615-1660), editor of three Latin editions of Descartes' Geometry (1649, 1656-1661, 1683-ALL 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.


René Descartes (1596-1650)
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.

Gottfried Wilhem Leibniz (1646-1716)
"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 467-473).

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.


Sir Isaac Newton (1642-1727)
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 mass-particles 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 (1656-1742) 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 two-line imprint naming only Joseph Streater and the Royal Society, while the second has a three-line 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 title-page. A copy of the three-line imprint, second issue, is also in our collection (HSC).


Guillaume François Antoine l'Hospital, Marquis de Sainte-Mesme (1661-1704)
Analyse des infiniment petits, pour l'intelligence des lignes courbes ...
Paris: Imprimerie Royale, 1696
In 1691, when Jean Bernoulli (1667-1748) 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 (1654-1705). There was no text-book 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 maximun-minimum problems. Later chapters deal with points of inflection, cusps and higher-order 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.



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