The Printing of Important Mathematics Texts Leads the Way to the Scientific Revolution

Overview

By the late fifteenth century the scholars of Europe were poised to fully reclaim the classical mathematical heritage that was nurtured and expanded by Islamic scholars and reintroduced into the West in the thirteenth century. Crucial in the growth of the mathematics of the period was the publication of mathematical texts. These texts helped pave the way for the growth of commerce and the onset of the Scientific Revolution.

Background

Many changes in economics, culture, and education were taking place in Europe in the fourteenth and fifteenth centuries. These changes had an effect on the mathematics of that time. These changes can be traced back even further to the Middle Ages. National boundaries became stabilized. In order for these new nations to expand economically and culturally, both secular and religious leaders initiated the founding of universities in major cities. The Crusades, while largely an exercise in futility, did manage to open trade routes to the Byzantine and Islamic peoples. Both of these events led to the recovery of classical learning. At first, scholars set to work on translating Greek and Arabic texts into Latin. Next, some scholars, notably Leonardo of Pisa (a.k.a. Fibonacci, 1170-1240) and Jordanus de Nemore (fl. 1200s), contributed original mathematics. New mathematical techniques were necessary to accommodate the new business of the trading companies of Italy. In order to succeed, these new kinds of merchants needed to learn this new mathematics, so some mathematicians soon found themselves in demand as teachers and textbook authors. All of this work was more widely distributed with the invention of the printing press by Johannes Gutenberg (c. 1398-1468) around 1450, and the subsequent rapid growth of printing with moveable type.

What are some of the influential books printed during this period? The first printed arithmetic text was published in 1478. This work had no title and was by an unknown author. It is now commonly referred to as the Treviso Arithmetic because of its place of publication—Treviso, north of Venice in Italy. The text is meant to be a training manual for young students seeking to enter the new trading businesses in Italy. In it one finds how to write and compute with numbers (only the four basic operations) and how to apply these techniques to questions of partnerships and trading. For example, one problem tells of three partners who invest unequal amounts in a partnership. If the partnership earns a profit, one must solve the problem of how the profit should be fairly divided. The Treviso Arithmetic itself was not very influential, except for the fact that its publication paved the way for an incredible amount of printed mathematical texts in the years following.

The first significant work in mathematics to be printed was the Elements of Euclid (c. 330-c. 260 b.c.), printed in Venice in 1482. All mathematics of the time relied heavily on the Elements. Indeed the Elements was a foundational work for all mathematics well into the eighteenth and nineteenth centuries. It is natural, then, that this work should be among the first printed. Euclid's work had been translated before by Islamic mathematicians and in turn these Arabic versions were translated into Latin, notably by the English monk Adelard of Bath (1075-1164) and Gerard of Cremona (1114-1187). Adelard translated the Elements from the Arabic version of al-Hajjai (c. 786-833). To Adelard's translation the Italian Johannes Campanus of Novara (1220-1296) added proofs for the propositions and other supplementary material. This became the standard translation of the Elements and was the one published in 1482. The fact that this version was the one that was printed illustrates its importance during this period.

Printed versions of other translations of the Elements later appeared on the scene. Interestingly, one of the defenders of the Campanus version was the Italian friar Luca Pacioli (1445-1517), who published corrections and commentary to the Campanus version. However, it is Pacioli's work Summa de Arithmetica, geometria, proportioni et proportionalita, finished in 1487 and first printed in 1494, that is the next significant publication under consideration.

The Summa is regarded as the first printed work on algebra. Again the honor does not go to an original piece of work. Concerned with the lack of teaching materials, Pacioli gathered mathematical materials from various sources and published them in a large comprehensive text. Very little in the book is the original work of Pacioli; in fact, much of the material can be found in the Liber abbaci of Leonardo of Pisa, a very influential work of its time. However, given the book's scope and the fact that is was one of the first texts printed, it became widely circulated and hence very influential. Its circulation and influence was also extended due to the fact that the book was printed in the vernacular Italian and not Latin.

The arithmetic in the Summa is standard for the time, including many techniques for the four arithmetical operations and the extraction of square roots. There are also problems in commercial arithmetic and an important study of double entry bookkeeping. The Summa also deals with algebra up to and including the solution of quadratic equations. The algebra here is verbal with no symbols used. Pacioli is one of the first to write on the so-called problem of the points for determining the division of the stakes in a game of chance if the game is stopped before it is finished. The study of this problem later led Blaise Pascal (1623-1662) and Pierre de Fermat (1601-1665) to the invention of the science of probability. The geometry in the Summa is basic, being a rehash of portions of Euclid along with some work on using algebra to solve geometric problems, which was first done by Regiomontanus (born Johann Müller, 1436-1476).

Regiomontanus' greatest work is De Triangulis Omnimodis (On Triangles of Every Kind) finished in 1464 (so Pacioli certainly had access to it) but not printed until 1533. Regiomontanus was one of the scholars involved in translating classical works, having translated Ptolemy's (100-170) Almagest from Greek to Latin. Regiomontanus was also a leader in using the new printing technology to publish mathematical works.

De Triangulis Omnimodis is the first European trigonometry text, even though it does rely heavily on earlier Islamic texts. Trigonometry was important to kings and merchants alike—for astrological predictions and making calendars and for astronomy and navigation. Trigonometry at the time was primarily concerned with the sine of an arc of a circle, which is defined to be half of the chord of double the arc on a circle. Regiomontanus' work discussed this function and also made use of the sine of the complement of the arc, the precursor of the modern cosine function. These are the only two trigonometric functions used in De Triangulis Omnimodis.

Regiomontanus's work is made up of five books and is a compilation of the trigonometry known in Europe at the time. The first book is concerned with lengths and ratios and is based on Euclid's Elements. In fact the entire work is written in the style of the Elements, with theorems proven based on axioms and results from Euclid's work. However, Regiomontanus also includes many examples. In this book Regiomontanus shows how to solve triangles; that is, given various combinations of sides and/or angles, to find the other sides and angles. For example, Theorem 28 of Book I states, "When the ratio of two sides of a triangle is given, its angles can be ascertained." Theorem 1 of Book II of De Triangulis Omnimodis is a perfectly modern statement of the law of sines for plane triangles. The remainder of Book II uses the law of sines to solve other triangles. This is standard material in a modern course on triangle trigonometry. To solve these triangles, Regiomontanus used his sine table, which was based on a circle of radius 60,000. The sines for other radii could be determined from this using the properties of similar triangles.

In De Triangulis Omnimodis Regiomontanus was also the first European to make use of algebra when solving triangles, solving a quadratic equation to find the remaining sides of a triangle given the lengths of the base, the altitude to the base, and the ratio of the unknown sides. This can be accomplished using the Pythagorean theorem and the given ratio. Book III of De Triangulis Omnimodis deals with spherical geometry, and correspondingly Book IV deals with spherical trigonometry. Naturally this material is of great use in astronomy. In these books Regiomontanus shows how to solve spherical triangles, whose sides are arcs of great circles on a sphere. Included is the spherical law of sines and the first use of the law of cosines for spherical triangles.

Because of their trading companies and banks, the Italians led the way in arithmetic and algebra related to commerce, followed in turn by the Germans with their trading towns on the Baltic Sea and the support of German banking tycoons. It is German texts that greatly influenced the important texts written in English.

The first English text author of any importance is Robert Recorde (1510-1558). Recorde is responsible for the first series of mathematics texts in England written in English, the first of which is The Ground of Arts (1543), an arithmetic text with commercial applications along the lines of Pacioli's Summa. Recorde's works are written in dialogue form, with an engaging style in which all steps are carefully explained. This might explain the fact that The Ground of Arts remained in print for over 24 editions over a period of 150 years. Recorde also wrote less successful companion texts on geometry (The Pathway to Knowledge, 1551), algebra (The Whetstone of Witte, 1557), and astronomy (The Castle of Knowledge, 1556).

The algebra texts of the time concentrated on solving equations no higher than second degree. Indeed, Pacioli deemed that the solution of the general cubic equation by a formula analogous to the quadratic formula could not be found. Pacioli was proven wrong by Scipione del Ferro (1465-1526) and Niccolò Fontana, better known as Tartaglia (1499-1557). The publication of the solution of the general cubic equation occurred in 1545 with the publication of Ars Magna, sive de Regulis Algebraicis (The Great Art, or On the Rules of Algebra) by Girolamo Cardano (1501-1576).

The Ars Magna is a book on solving algebraic equations, including cubic and quartic equations. The major breakthrough in the work is the first publication of the solution of cubic and quartic equations by radicals, that is, with a formula similar to the quadratic formula. Cardano also shows his awareness of imaginary numbers in Ars Magna because they appear in the formula for the solution of the cubic equation. Our modern algebraic symbolism is not present in the work, nor is the use of general coefficients. Cardano lists many different cases of cubic equations corresponding to the various ways necessary to write the equation with all positive coefficients. Cardano also supplies geometric proofs of the validity of his results. While this renders the work quite difficult for a modern reader, it was a revelation to the readers of the time. Until the invention of a suitable algebraic notation, the Ars Magna was the best that any mathematician could do on the subject.

By the time of René Descartes (1596-1650), suitable algebraic notation and the use of general coefficients had both been developed. Descartes uses these to good effect in La Géométrie, an appendix to his 1637 work Discours de la méthode pour bien conduire sa raison et chercher la vérité dans les sciences (A Discourse on the Method of Rightly Conducting the Reason and Seeking Truth in the Sciences). La Géométrie is a ground-breaking work, one of the greatest scientific works of the Renaissance. However, its influence was not initially felt. Descartes wrote in French rather than Latin, and while writing in the vernacular was essential for works that wished to popularize mathematics (see Pacioli and Recorde), for works that were on the cutting edge, Latin was the language of the learned. Also, Descartes was not very forthcoming with details and explanations in his work. The profound influence that La Géométrie exerted on mathematics is due to Franz van Schooten (1615-1660) and his publication of Geometria a Renato Des Cartes in 1649. Analytic geometry is truly born in this work, and from there it is a short step to the invention of the calculus and a whole new era of mathematics. In his Latin edition van Schooten added commentary, figures, alternate techniques, and a logical ordering of the material. He also added the work of other mathematicians, notably Sluse, van Heuraet, and de Witt. It is important to realize that Descartes's work accounts for approximately 100 pages, while the additional work of van Schooten and the others accounts for nearly 900 pages. This and later editions were the primary source of analytic geometry for mathematicians until the middle of the eighteenth century.

Impact

All of these works exerted significant influence. They became the standard textbooks of the time in the universities and training schools. Important pioneers in mathematics and the sciences studied from these works. Isaac Newton (1642-1727) mastered Euclid's Elements, although it is unclear what version he used. However it is known that Newton studied from van Schooten's Geometria a Renato Des Cartes, as did Christiaan Huygens (1629-1665), Gottfried Leibniz (1646-1716), John Wallis (1616-1703), Jakob and Johann Bernoulli (1654-1705 and 1667-1758, respectively), and Leonhard Euler (1707-1783). Pacioli was well versed in Campanus' version of Euclid, as was every other mathematician of the time, including John Dee (1527-1608), the editor of the first English version of the Elements.

Pacioli's Summa was studied by the sixteenth-century Italian mathematicians; it was the common starting point from which they advanced the study of algebra. Scipione del Ferro and Cardano were known to be familiar with the Summa. It also helped popularize the use of abbreviations for mathematical terms, which led to the invention of algebraic symbolism, which in turn was crucial to the growth of mathematics. The Summa also popularized using the methods of algebra to solve geometric problems, which were initiated in the West in Regiomontanus' De Triangulis Omnimodis. This started mathematicians down the road towards analytic geometry, a road that ended with van Schooten's Geometria a Renato Des Cartes.

After Regiomontanus' work, many other trigonometry works were published, and European trigonometry began to surpass the trigonometry of the Islamic mathematicians. In addition, trigonometry supplied the tools necessary for the advances in astronomy soon to follow, culminating with the overthrow of the Ptolemaic system of the world in favor of the Copernican system.

Research in algebra was stimulated by the Ars Magna. For the next 200 years mathematicians, inspired by the discovery of solutions in radicals to the cubic and quartic equations, sought to generalize those results to higher degree equations. Although we now know that they would search in vain (general equations of degree five and higher cannot be solved using radicals only), a great deal of important mathematics in the fields of higher algebra, which includes the theory of groups, rings, and fields, was discovered along the way. In addition, a new kind of number was born of the solution of the cubic. Mathematicians turned to the study of imaginary numbers to better understand the solutions given by Cardano's cubic formula.

Cardano's book also showed that analytic methods could be quite powerful. Problems could be solved without having to resort to geometry, which up to this time was the foundation for all of mathematics. The Ars Magna exerted such influence that it could be said that too much emphasis was placed on algebraic methods, perhaps postponing the marriage of algebra and geometry later found in van Schooten's Latin Géométrie and in the development of the calculus.

GARY S. STOUDT