Index to the *Convergence* Collection of Mathematical Treasures

Index to the Collection of Mathematical Treasures from the David Eugene Smith and George Arthur Plimpton collections (alphabetical by author)

During the first half of the twentieth century, David Eugene Smith (1860-1944) was a moving force in the world of mathematics education. As the chairman of the mathematics education department at Columbia University’s Teachers College, Smith led the way in teaching reforms attuned to the Progressive Education Movement. He firmly believed that the teaching of mathematics should be closely associated with the history of the subject. As an historian of mathematics, he wrote and lectured widely on the subject and also collected historical mathematical materials: texts, documents and artifacts. Smith befriended the wealthy New York book publisher and bibliophile, George Arthur Plimpton (1855-1936). While Plimpton was an avid collector of materials from the liberal arts that comprised “the tools of learning” for western civilization, under Smith’s influence Plimpton greatly enriched his collection with mathematical manuscripts and many early Renaissance texts on arithmetic. When Plimpton died in 1936, he bequeathed his collection to Columbia University. Similarly, beginning in 1931, David Eugene Smith began donating his extensive collection of mathematical memorabilia: historical texts; correspondence; portraits of famous mathematicians; signatures and concrete artifacts to the Columbia University Library.

Today, these two collections exist as rich resources for understanding the development of mathematics and the lives and work of many of the persons responsible for its advance. These archives are available to researchers through the Rare Book and Manuscript Collection at Columbia University. The Mathematical Association of America, in cooperation with the Columbia University Libraries, is pleased to display a selection of items, *Mathematical Treasures*, from these two separate collections. The editors of *Convergence* would like to particularly thank Dr. Michael Ryan, Director of Rare Books and Manuscripts, and Jennifer Lee, Librarian for Public Service and Programs, for their assistance in making this display possible.

The next page (Page 2) contains an index to the collection (alphabetical by author). Each item is posted in this article at the standard web resolution of 72dpi. But if you right-click on the name of the item, found in the first sentence of the item description, you can download the item as a tif file at a resolution of approximately 150-200dpi (3-5mb) as well. That version should be suitable for most purposes in a classroom setting. If you want a version in even higher resolution, please contact the second author, Victor Katz. (Note that if you just click on the name of the item, you will get the tif file on your screen, but you may not be able to save it to your computer.)

These images may be used for instructional / educational purposes; for all other uses, please contact the Columbia University Libraries to obtain permission.

Index to the Collection of Mathematical Treasures from the David Eugene Smith and George Arthur Plimpton collections (alphabetical by author)

Index to the *Convergence* Collection of Mathematical Treasures

Index to the *Convergence* Collection of Mathematical Treasures

Index to the Collection of Mathematical Treasures from the David Eugene Smith and George Arthur Plimpton collections (alphabetical by author)

During the first half of the twentieth century, David Eugene Smith (1860-1944) was a moving force in the world of mathematics education. As the chairman of the mathematics education department at Columbia University’s Teachers College, Smith led the way in teaching reforms attuned to the Progressive Education Movement. He firmly believed that the teaching of mathematics should be closely associated with the history of the subject. As an historian of mathematics, he wrote and lectured widely on the subject and also collected historical mathematical materials: texts, documents and artifacts. Smith befriended the wealthy New York book publisher and bibliophile, George Arthur Plimpton (1855-1936). While Plimpton was an avid collector of materials from the liberal arts that comprised “the tools of learning” for western civilization, under Smith’s influence Plimpton greatly enriched his collection with mathematical manuscripts and many early Renaissance texts on arithmetic. When Plimpton died in 1936, he bequeathed his collection to Columbia University. Similarly, beginning in 1931, David Eugene Smith began donating his extensive collection of mathematical memorabilia: historical texts; correspondence; portraits of famous mathematicians; signatures and concrete artifacts to the Columbia University Library.

Today, these two collections exist as rich resources for understanding the development of mathematics and the lives and work of many of the persons responsible for its advance. These archives are available to researchers through the Rare Book and Manuscript Collection at Columbia University. The Mathematical Association of America, in cooperation with the Columbia University Libraries, is pleased to display a selection of items, *Mathematical Treasures*, from these two separate collections. The editors of *Convergence* would like to particularly thank Dr. Michael Ryan, Director of Rare Books and Manuscripts, and Jennifer Lee, Librarian for Public Service and Programs, for their assistance in making this display possible.

The next page (Page 2) contains an index to the collection (alphabetical by author). Each item is posted in this article at the standard web resolution of 72dpi. But if you right-click on the name of the item, found in the first sentence of the item description, you can download the item as a tif file at a resolution of approximately 150-200dpi (3-5mb) as well. That version should be suitable for most purposes in a classroom setting. If you want a version in even higher resolution, please contact the second author, Victor Katz. (Note that if you just click on the name of the item, you will get the tif file on your screen, but you may not be able to save it to your computer.)

These images may be used for instructional / educational purposes; for all other uses, please contact the Columbia University Libraries to obtain permission.

Index to the *Convergence* Collection of Mathematical Treasures

Selected pages from the following documents from the George Arthur Plimpton and David Eugene Smith Collections, Columbia University Libraries, are available here. Click on the name of the document to go to the illustration, where more information is provided. An index that includes not only these "mathematical treasures," but also hundreds of others from a wide array of institutions, museums, and individuals, is available in *Convergence* as A Collection of Mathematical Treasures - Index.

Author |
Title |
Date |

Agnesi, Maria Gaetana | Instituzioni analitiche (8 images) | 1748 |

Alberti, Giuseppe | Instruzione per l'Ingenero | 1774 |

Al-Khwarizmi, Muhammad | Algebra | (Original dates to ~825) |

Al-Tusi, Nasir al-Din | Commentary on Euclid | (13th century original) |

Anonymous | Antichissimo di algorismo (2 images) | 14th century |

Anonymous | Austrian measuring rod | 1732 |

Anonymous | Austrian weights | 18th century |

Anonymous | English gauger's scale | 18th century |

Anonymous | English tally sticks (5 images) | 1296 |

Anonymous | German protractor | 1700 |

Anonymous | Italian armillary sphere | 1550 |

Anonymous | Italian compass | 15th century |

Anonymous | Jiuzhang suanshu | 1603 (original from c. 250 CE) |

Anonymous | Korean sangi rods (2 images) | 19th century |

Anonymous | Plimpton 322 | c. 1850 BCE |

Anonymous | Treviso Arithmetic | 1478 |

Anonymous | Zhoubi suanjing (2 images) | 1603 (original from 100 BCE) |

Apianus, Petrus | Instrumentum sinuum | 1534 |

Apianus, Petrus | Introductio geographica | 1534 |

Barozzi, Francesco | Procli Diadochi (2 images) | 1560 |

Barrow, Isaac | Geometrical Lectures | 1735 edition |

Barrow, Isaac | Archimedes, Apollonius | 1675 |

Barrow, Isaac | Euclid's Elements (2 images) | 1665 |

Benedetto da Firenze | Trattato d'arismetriche | c. 1460 |

Bernoulli, Jakob | Ars conjectandi | 1713 |

Bettini, Mario | Aerarium philosophiae mathematicae (5 images) | 1648 |

Bhaskara | Lilavati (2 images) | 1650 |

Billingsley, Henry | Euclid's Elements (2 images) | 1570 |

Boeschenstein, Johann | Ain neu geordnet Rechenbiechlin (3 images) | 1514 |

Boethius, Anicius Manlius | Arithmetic (2 images) | 1294 manuscript |

Boethius, Anicius Manlius | Liber circuli | 1503 |

Bossut, Charles | Traite elementaire de geometrie (6 images) | 1775 |

Bowditch, Nathaniel | Celestial Mechanics (5 images) | 1818/1829 |

Brahe, Tycho | Astronomiae instauratae (2 images) | 1602 |

Byrne, Oliver | Elements of Euclid | 1847 |

Cardano, Gerolamo | Ars magna | 1545 |

Cardano, Gerolamo | Practica arithmetice (5 images) | 1539 |

Clavius, Christopher S.J. | Opera mathematica (2 images) | 1612 |

Clavius, Christopher S.J. | Epitome arithmetica practica (2 images) | 1584 |

Clavius, Christopher S.J. | Edition of Euclid (2 images) | 1574 |

de Witt, Jan | Elements of Curves (5 images) | 1683 |

Desargues, Girard | Universal Way of Dyaling | 1659 |

Descartes, René | Discourse on Method | 1637 |

Digges, Leonard | Tectonicon (4 images) | 1556/1634 |

Digges, Thomas | Pantometria (3 images) | 1571 |

Dürer, Albrecht | Treatise on Mensuration (4 images) | 1538 |

Emerson, William | Treatise of Algebra (5 images) | 1764 |

Euclid | Elements: Late 13th century manuscript (5 images) | 1294 |

Euclid | Elements: Nasir al-Din al-Tusi commentary | (13th century original) |

Euclid | Elements: 14th century manuscript (3 images) | 1390 |

Euclid | Elements: Henry Billingsley edition (2 images) | 1570 |

Euclid | Elements: Christopher Clavius edition (2 images) | 1574 |

Euclid | Elements: Isaac Barrow ediiton (2 images) | 1665 |

Euclid | Elements: Oliver Byrne's edition (3 images) | 1847 |

Euler, Leonhard | Algebra (in French) | 1795 |

Euler, Leonhard | Calculus of Variations | 1744 |

Euler, Leonhard | Integral Calculus (3 images) | 1768 |

Euler, Leonhard | Analysis of the Infinite (4 images) | 1748 |

Eyre, John | The Exact Surveyor (2 images) | 1654 |

Faulhaber, Johann | Academia algebrae (2 images) | 1631 |

Feliciano, Francisco | Libro di Arithmetica i Geometria (3 images) | 1536 |

Fine, Oronce | Geometry | 1551 |

Fine, Oronce | Le sphere du monde (3 images) | 1551 |

Frisius, Gemma | Arithmeticae methodus facilis (3 images) | 1540 |

Galilei, Galileo | Siderius nuncius (2 images) | 1610 |

Galilei, Galileo | Geometrical Compass (2 images) | 1640 |

Ghaligai, Francesco | Practica d'Arithmetica (3 images) | 1552 |

Heuraet, Hendrick van | Rectification of Curves (3 images) | 1659 |

Honoratus | Opus arithmetica (2 images) | c. 1560 |

Huygens, Christiaan | De circuli magnitude inventa | 1654 |

Jacob, Simon | Rechenbuch (7 images) | 1565/1599 |

Jordanus de Nemore | Arithmetica | (13th century original) |

Kepler, Johann | Rudolphine Tables | 1627 |

Kepler, Johann | Uralten messekunst Archimedes | 1616 |

Kepler, Johann | Epitome of Copernican Astronomy (2 images) | 1619 |

Kepler, Johann | Chilias logarithmoria | 1624 |

Khayyam, Omar | Algebra | (Original dates to ~1100) |

Köbel, Jacob | Geometry (4 images) | 1608 |

Lacroix, Sylvestre | Calculus of Probability (2 images) | 1816 |

Lagrange, Louis | Analytical Mechanics (2 images) | 1788 |

Laplace, Pierre-Simon | Celestial Mechanics (5 images) | 1799/1818/1829 |

Maseres, Francis | Doctrine of Permutations and Combinations (3 images) | 1795 |

Moore, Jonas | Arithmetick in Four Books (5 images) | 1650/1688 |

Newton, Isaac | Collected Works (5 images) | 1779 |

Pacioli, Luca | Divine Proportion (2 images) | 1509 |

Pacioli, Luca | Summa (3 images) | 1494 |

Pitiscus, Bartholomew | Trigonometry (2 images) | 1600 |

Qadi Zada al-Rumi | Geometry | 16th century (original 1412) |

Ramus, Peter | The Way to Geometry (2 images) | 1636 |

Recorde, Robert | Pathway to Knowledge | 1551 |

Recorde, Robert | Grounde of Artes (4 images) | 1543 |

Recorde, Robert | Whetstone of Witte (6 images) | 1557 |

Regiomontanus | On Triangles (2 images) | 1533 |

Reisch, Gregor | Margarita philosophica (6 images) | 1503 |

Riese, Adam | Rechenbuch (2 images) | 1550 |

Sabeus, Bernard | Italian astrolabe | 1558 |

Sault, Richard | New Treatise of Algebra (5 images) | 1694 |

Scheubel, Johann | De numeris (2 images) | 1545 |

Schooten, Frans van | Exercitationes mathematicae (7 images) | 1657 |

Seki Kowa | Essentials of Mathematics (2 images) | 1712 |

Stevin, Simon | Oeuvres Mathematiques (5 images) | 1634 |

Stifel, Michael | Arithmetica integra (5 images) | 1544 |

Tartaglia, Niccolo | Nova scientia (2 images) | 1537 |

Tartaglia, Nicolo | Treatise on Number & Measure (2 images) | 1556 |

van Heuraet, Hendrick | Rectification of Curves (3 images) | 1659 |

van Schooten, Frans | Exercitationes mathematicae (7 images) | 1657 |

Wallis, John | Treatise of Algebra (4 images) | 1685 |

Ward, John | Compendium of Algebra (3 images) | 1724 |

Witt, Jan de | Elements of Curves (5 images) | 1683 |

Wolff, Christian | Treatise of Algebra (5 images) | 1765 |

Zubler, Leonhard | Nova instrumentum geometricum (5 images) | 1607 |

These images may be used for instructional and educational purposes; for all other uses, please obtain permission from the Columbia University Libraries.

Index to the *Convergence* Collection of Mathematical Treasures

Plimpton 322, an Old Babylonian tablet from Larsa, has four columns of numbers, two of which, most experts believe, contain, in each of the fifteen rows, two of the three numbers in a Pythagorean triple. This tablet was first analyzed by Otto Neugebauer and Abraham Sachs in their 1945 book, *Mathematical Cuneiform Texts* (New Haven, American Oriental Society)*. * There have been numerous discussions of this tablet since that time. In particular, two articles, "Sherlock Holmes in Babylon," (1980) by R. Creighton Buck, and "Words and Pictures: New Light on Plimpton 322," (2002) by Eleanor Robson are included in Marlow Anderson, Victor Katz, & Robin Wilson, eds., *Sherlock Holmes in Babylon and Other Tales of Mathematical History* (Washington: Mathematical Association of America, 2004), pp. 5-26. Further references to the literature are included in those two articles. More recently, Jöran Friberg, in *A Remarkable Collection of Babylonian Mathematics Texts* (New York: Springer, 2007) (pp. 433-452) has challenged the interpretation of the numbers on the tablet as parts of Pythagorean triples.

This is the title page from the *Ars Magna*, by Gerolamo Cardano, published in 1545. The page contains an engraving of a portrait of the author. This is the text in which the algebraic solution of cubic equations was first printed. Although the initial discovery of the solution of one type of cubic equation was due to Scipione del Ferro, and solution methods for at least three types were worked out by Niccolò Tartaglia, it was Cardano, along with his student Lodovico Ferrari, who worked out the details for thirteen cases and then published them in this book. The book also contains Ferrari's basic method of solution for quartic equations, as well as much else.

The title page may be translated as follows:

Book one of The Great Art, or the Rules of Algebra, by Gerolamo Cardano, most distinguished mathematician, philosopher, and physician, which is the tenth in order of the whole work on arithmetic, which is titled the perfect work.

You have in this book, diligent reader, the rules of algebra (in Italian, called rules of the coss), so abounding in new discoveries and demonstrations by the author, more than seventy, that earlier works now amount to little (or, in the vernacular, are washed out). It disentangles the knots not only where one term is equal to another or two to one, but also where two are equal to two or three to one. It is a pleasure, therefore, to publish this book separately so that, this most abstruse and unsurpassed treasury of all of arithmetic being brought to light, and as, in a theater exposed to the sight of all, its readers may be encouraged and will all the more readily embrace and with less aversion study thoroughly the remaining books of this perfect work, which will be published volume by volume.

(Translation adapted from that of T. Richard Witmer, *The Great Art,* Cambridge: MIT Press, 1968.)

This is a page from al-Khwārizmī's algebra text, *Kitāb al-jabr wa'l-muqābala*, written in about 825, the first extant algebra text, by Muḥammad ibn Mūsā al-Khwārizmī. This copy itself is undated, however. It corresponds to page 15 in the translation by Frederic Rosen: *The Algebra of Muhammed ben Musa* (London: Oriental Translation Fund, 1831), which is also available in a reprinting in the series on Islamic Mathematics and Astronomy, from the Institute for the History of Arabic-Islamic Science at the Johann Wolfgang Goethe University, Frankfurt am Main. On this page is al-Khwārizmī's proof of the rule for solving a quadratic equation of the form "squares plus roots equal numbers" (\(x^2 + bx = c\)). The central square in the diagram represents the square on the unknown. The four rectangles on the four sides of the square each have width \(\frac{b}{4}.\) Thus the area of the central square plus the four rectangles is \(c.\) The square is then completed by adding the four corner squares, each of side \(\frac{b}{4}.\) Thus, the area of the large square is, in modern notation, \[x^2 + bx + {\frac{b^2}{4}} = {\left(x + {\frac{b}{2}}\right)}^2,\] and this is in turn equal to \(c + {\frac{b^2}{4}}.\) The solution to the equation is then evident.

This is a page from a manuscript of the *Lilavati* of Bhaskara II (1114-1185). This manuscript dates from 1650. The rule for the problem illustrated here is in verse 151, while the problem itself is in verse 152:

151: The square of the pillar is divided by the distance between the snake and its hole; the result is subtracted from the distance between the snake and its hole. The place of meeting of the snake and the peacock is separated from the hole by a number of

hastasequal to half that difference.152: There is a hole at the foot of a pillar nine

hastashigh, and a pet peacock standing on top of it. Seeing a snake returning to the hole at a distance from the pillar equal to three times its height, the peacock descends upon it slantwise. Say quickly, at how manyhastasfrom the hole does the meeting of their two paths occur? (It is assumed here that the speed of the peacock and the snake are equal.)

These verses and much else from the *Lilavati* may be found in Kim Plofker, "Mathematics in India", in Victor Katz, ed., The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook * * (Princeton: Princeton University Press, 2007), pp. 385-514.

This page from the *Lilavati* gives another illustration of the Pythagorean Theorem.

This is the Tree of Proportions and Proportionality from the *De* *Divina Proportione* of Luca Pacioli (1445 - 1509), published in 1509. Some of the terms in Pacioli's tree are familiar today and some are taken from the study of proportions by Nicomachus in his *Arithmetic,* but the meanings of some other terms are not generally known. It would be a worthwhile project to go through the tree and decipher the meanings of all of the terms in it.

One image from *De* *Divina Proportione* of a stellated dodecahedron. This was a woodcut print of an illustration by Leonardo Da Vinci.

- Another "mathematical treasure" in
*Convergence*provides additional information about Pacioli’s*Divina proportione,*along with two images from the copy housed at the University of Oklahoma Library. - Another article in
*Convergence*displays illustrations of the Platonic solids, prepared for*De Divina Proportione*by Leonardo da Vinci.

This is the first page of Pathway to Knowledge by Robert Recorde (1510-1558). The book is essentially a version of Euclid's *Elements*, with commentary by Recorde to make it easier to understand. On this page are Euclid's first three postulates, that one can draw a line between any two points, that one can extend a straight line, and that one can draw a circle with a given center and a given radius.

This is a page from the Treviso Arithmetic (1478), the earliest known example of a printed book on arithmetic. The work has no title, and no author's name is found anywhere in the book. It was printed in Treviso, a town about 26 km northwest of Venice. In many book catalogues, it is called the *Arte del Abbaco* (*Art of Calculation*), because it demonstrates not only how to use the Hindu-Arabic decimal place-value system, but how to solve numerous practical problems, mostly using the rule of three. The problem that begins in the middle of the left-hand page of this illustration is the following:

Two merchants, Sebastiano and Jacomo, enter into partnership. Sebastiano put in 350 ducats on the first day of January, 1472; Jacomo put in 500 ducats, 14 grossi on the first day of July, 1472. On the first day of January, 1474 they find that they have gained 622 ducats. Required is the share of each.

The first calculation for this problem, toward the bottom of the page, converts both amounts of money into grossi, given that there are 24 grossi in a ducat. The next instruction is to multiply each partner's amount by the length of time it was invested; so on the next page, Sebastiano's amount is multiplied by 24 and Jacomo's by 18.

A complete translation and analysis of the *Treviso Arithmetic* can be found in Frank Swetz, *Capitalism and Arithmetic: The New Math of the 15th Century* (La Salle, IL: Open Court, 1987). For further information on the *Treviso Arithmetic*, click here.

This is the title page of *On Triangles* by Regiomontanus (Johannes Müller) (1436-1476). Although the work was written in 1464, it was not published until 1533. The page can be translated as follows:

Five Books on Triangles of Every Kind, by the most learned man and extraordinary professor of the mathematical disciplines, Johannes Regiomontanus: In which is explained all things necessary for one who wishes to reach perfection in his knowledge of the astronomical sciences. Since these matters have never been developed elsewhere before this time, one will aspire in vain to learn these things without these ideas.

An appendix contains the works of Nicolas of Cusa on the quadrature of the circle and the measuring of straight and curved lines, together with their never before published refutation by Johannes Regiomontanus.

All of these have been thoroughly edited with singular fidelity and diligence at the publishing house of John Petreus, in Nürnberg, A.D. 1533.

A complete translation and analysis of *On Triangles* is available in Barnabas Hughes, editor and translator, *Regiomontanus on Triangles* (Madison: University of Wisconsin Press, 1967).

This page contains theorems 12 (XII) and 13 (XIII) of Book II of *On Triangles* (*De triangulis*):

**Theorem 12:** If the perpendicular is given and the base and the ratio of the sides are known, then each side is known.

**Theorem 13:** If each of the segments of the base is known and also the ratio of the sides, then each side and the perpendicular may be found.

There is a calculation in the margin by an owner of this book. It would be interesting to work out the details.

This is the title page to the English version of Isaac Barrow's *Geometrical Lectures*, which were originally given in his position as Lucasian Professor of Mathematics at Cambridge University. These lectures contain one of the earliest statements and proofs of what is today known as the fundamental theorem of calculus.

This is the title page of Jacob Bernoulli's *Ars Conjectandi* (The Art of Conjecturing), published posthumously in 1713 by his nephew Nicolaus I Bernoulli. The *Ars Conjectandi *is famous for Bernoulli's statement of a general rule for summing integral powers as well as for the first statement and proof of the Law of Large Numbers. The first parts of the book contain an extensive treatment of the theory of probability, mostly based on earlier work of Huygens and others. The final section is entitled "The Use and Application of the Preceding Doctrine in Civil, Moral, and Economic Matters." It is here that the Law of Large Numbers is stated, but despite the imposing title of the section, it is quite short. Evidently, Bernoulli found it difficult actually to apply probability to "civil, moral, and economic" matters. Note that the book as published includes a treatise on infinite series and a letter Bernoulli wrote to a friend about probabilities in the game of tennis.

A complete translation of the *Ars Conjectandi* is available as Jacob Bernoulli, *The Art of Conjecturing, together with Letter to a Friend on Sets in Court Tennis*, translated by Edith Dudley Sylla (Baltimore: Johns Hopkins University Press, 2006).

This is the title page to René Descartes' *Discourse on Method* or *Treatise on Method*, with its three famous essays on Dioptrics (or Optics), Meteorology, and Geometry. The third essay contains Descartes' treatment of analytic geometry as well as his theory of equations and his rule for finding the normal to a given curve.

A complete translation of the *Treatise* with the essays is in René Descartes, *Discourse on Method, Optics, Geometry, and Meteorology*, translated by Paul Olscamp (Indianapolis, Bobbs-Merrill Co., 1961).

This is the opening page of the section of the geometrical work of Oronce Fine (1494-1555) dealing with Euclid's work on circles. This is part of Fine's major work entitled *Protomathesis* (1530), which covered arithmetic, astronomy and astronomical instruments, as well as geometry. The pages displayed give a somewhat theoretical treatment of geometry, but Fine then goes on to practical matters dealing with measuring length, height, surface area, and volume.

This is the title page of the French edition of the *Algebra* of Leonhard Euler (1707-1783). The text was originally published in German in 1767 in St. Petersburg and then translated into French in 1795. An English version appeared in 1840 and was reprinted in 1984 by Springer-Verlag.

This is the title page of the "New and Fully Revised" Rechenbuch of Simon Jacob (d. 1564), one of the best-known Rechenmeisters of the sixteenth century. The book was first published in 1560, but this illustration is from the 1565 edition. Other editions were printed up to 1600. The book was a commercial arithmetic text, in which the basic laws of arithmetic were applied to many practical problems. A wide variety of mathematical instruments are illustrated on this page.

This is the title page of the 1599 edition of Jacob's first book, the Rechenbuchlein, originally published in 1557. It also went through several editions.

This illustration and those that follow are all from the 1565 book. On this page, Jacob has sketched the decomposition of a solid as an illustration of his method of calculating cube roots.

On the left-hand page there is part of the computation of the fourth root of 76656. Note that the answer is not a whole number. On the right-hand page, Jacob begins his description of the general procedure for calculating higher roots.

Since the general method of calculating roots involves what we call the Pascal triangle, Jacob gives this triangle through row 12 on page 104.

Jacob continues his description of the method of finding roots. On the right-hand page, he explicitly states the binomial theorem in several cases.

On the right-hand page, Jacob illustrates the calculation of the fifth root of 119643398733624. The answer is 654.

This is the title page of the geometry book of Jacob Köbel (1460 - 1533). Köbel was not only a Rechenmeister (reckoning master, or arithmetic teacher) but was also the town clerk in Oppenheim, a German town near Mainz. Köbel wrote the geometry book toward the end of his life, but it was reprinted many times after his death, including this 1608 edition. The book dealt primarily with measurement, showing its readers how to use various instruments to measure fields, determine heights of buildings, and perform various other necessary tasks.

Three other pages from this book follow:

In this image, Köbel demonstrates how to determine a "rood": one takes sixteen men leaving church on a Sunday and lines up their feet. The total length of the sixteen feet is a "rood". For more information, see the article by Peter Ransom, entitled "The Right and Lawful Rood."

Measuring the height of a tower, when one can measure the distance to the tower. To determine the angle, the surveyor is employing reflection on a smooth surface or mirror (here a polished plate is being used). This technique is based on the principle of "angle of incidence equals angle of reflection."

Measuring the height of a tower using quadrants, when one cannot measure the distance to the tower.

This is the title page of the *Margarita philosophica* (*Pearl of Wisdom*) of Gregor Reisch (1467 - 1525). The first edition was published in 1503. This work was used as a university textbook in the early sixteenth century. Among its twelve chapters are seven dealing with the seven liberal arts commonly taught at the universities: the trivium of logic, rhetoric, grammar and the quadrivium of arithmetic, music, geometry, and astronomy. There are also several chapters on more advanced topics. Most of the chapters have engravings dealing with the chapter's subject matter, some of which are included below.

This is the engraving depicting Grammar, often called the Tower of Learning. Reisch depicts Grammar, the foundation of all learning, as Nicostrata (legendary inventor of the alphabet) holding a hornbook and key and introducing a child into a tower of learning with six levels. Toward the top of the tower are portraits representing the subjects of the trivium and the quadrivium.

This is the classic engraving of Arithmetica (or the Allegory of Arithmetic) supervising a contest between Boethius, representing written calculation using Hindu-Arabic numbers, and Pythagoras, represented as using a counting board.

Here is an illustration from the geometry chapter (Allegory of Geometry) illustrating various geometric figures as well as geometric instruments.

At the beginning of the astronomy chapter is the Allegory of Astronomy, that is, the image of Astronomia instructing Ptolemy on measuring the heavens by the use of the quadrant.

Also, in the astronomy chapter, there is an image of the Ptolemaic (and also the medieval) version of the geocentric universe. Note that man is in the center of the universe.

*Editor's note: *This article was posted May 14, 2008.

This is the title page of the first textbook in the calculus of variations, the *Method of Finding Curved Lines that Show some Property of Maximum or Minimum*, by Leonhard Euler (1707-1783). The book was published in 1744.

This is a page from the Rechenbuch of Adam Riese (1492 - 1559), one of the greatest reckoning masters of the 16th century. The complete title of the book is *Rechnung nach der lenge/auff den Linien und* Feder. This page (78) deals with the gauging of barrels, that is, with measuring the contents of the barrel.

This page from the Rechenbuch (p. 6) has a table showing numbers both in German words, in Roman numerals, and in the Hindu-Arabic numerals.

This is a page from a manuscript of the *Algebra* (*Maqalah fi al-jabra wa-al muqabalah*) of Omar Khayyam (1048-1131). This work is known for its solution of the various cases of the cubic equation by finding the intersections of appropriately chosen conic sections. On this page, Omar is discussing the case "a cube, sides and numbers are equal to squares", or, in modern notation, x^{3} + cx + d = bx^{2}. The two conics whose intersection provides the solution are a circle and a hyperbola. In the case illustrated, these curves intersect twice, thus providing two (positive) solutions of the given cubic equation. Khayyam even provides a problem which leads to this case: Divide ten into two parts so that the sum of the squares of the parts together with the quotient of the division of the greater by the smaller be seventy-two. For more details, see pp. 90ff of Daoud Kasir, *The Algebra of Omar Khayyam* (New York: Teachers College Press, 1931) or pp. 144ff of R. Rashed and B. Vahabzadeh, *Omar Khayyam, the Mathematician* (New York: Bibliotheca Persica Press, 2000).

This particular manuscript was copied in the thirteenth century in Lahore, India. Among the other fourteen works contained in the volume are two by Sharaf al-Din al-Tusi (1135-1213) on determining vertical heights of objects and a treatise by ibn al-Haytham (965-1039) on the astrolabe.

This is the title page of Robert Recorde's *The Grounde of Artes *(1543), which deals with the elements of arithmetic.

On this reverse of the title page, there is a poem by John Dee (1527-1609) dealing with arithmetic.

On this page there is an outline of the first dialogue in this book. (Recorde wrote his book in the form of a dialogue between student and master.) This dialogue deals with some of the elements of arithmetic, including the basic operations and the use of the rule of three (or the Golden rule).

A page dealing with various "rules of practice" in arithmetic.

This is the title page of the *Treatise of Algebra* (1685), by John Wallis (1616-1703). This is probably the first attempt at a history of the subject of algebra, presented in the context of a text on the subject. There is a discussion of Wallis's text in Jacqueline Stedall, A Discourse Concerning Algebra: English Algebra to 1685 (Oxford: Oxford University Press, 2002).

A portrait of John Wallis from this text.

Among the most famous parts of this treatise is Wallis's discussion of the work of Thomas Harriot, especially his contention that René Descartes plagiarized Harriot's symbolization procedure in algebra. This discussion is summarized on the initial pages (3, 4, and 5) of Wallis's preface. After giving a list of Harriot's discoveries in algebra, Wallis notes that there is "scarce anything in (pure) algebra in Descartes which was not before in Harriot." Most historians did not believe Wallis, because Harriot's published work did not include a lot of what Wallis stated. But since the recent discoveries of Harriot's algebra manuscripts (newly published by Jacqueline Stedall), there is certainly some reason to believe that Wallis was correct. There is certainly some similarity between Harriot's manuscripts and Descartes' algebraic work in his *Geometry*.

The *Arithmetic* of Boethius (480-524) dates from the early sixth century. This page is from a mansucript (Plimpton MS 165) that dates from approximately 1294, written on vellum. This page (folio 13) lists the powers of 2, 3, and 4 (at the bottom) and also has some other tables representing multiplication by some of these powers. Note that the forms of the figures are not always identical to the modern form.

This manuscript page (folio 15) contains illustrations of square and pentagonal numbers.

The four images below are of the same manuscript (Plimpton MS 165) of Boethius's *Arithmetic* as those above. They were added to this article in 2018. While the two images above were prepared specifically for the Mathematical Association of America (MAA) by the Rare Book and Manuscript Library of the Columbia University Libraries, those below are presented courtesy of the Columbia University Libraries via *Digital Scriptorium.*

Boethian arithmetics were quite popular during the Early Middle Ages. These were copies of the *Arithmetic* of Boethius, who lived from about 480 to 524 in what is now Italy. This illustrated French manuscript from 1294 (Plimpton MS 165) attests to its scribe’s craftsmanship. We begin with an image of the page of figurate numbers shown just above (folio 15) and its facing page (folio 14v). The facing page, not surprisingly, discusses and illustrates triangular numbers.

A “Table of Multiplication Facts” is rendered in red and green ink in folio 9, below.

The scribe's beautiful work is shown especially well in folio 12, below.

Much of the textual discussion concerns number properties, also demonstrated by the use of diagrams.

*The images above have been obtained through the kind cooperation of the Rare Book and Manuscript Library of the Columbia University Libraries. These and more images may be accessed via Digital Scriptorium, a digital collection of medieval and early Renaissance manuscripts made available by a consortium of cooperating university libraries headed by the University of California, Berkeley.*

This is the title page of the Siderius Nuncius (Starry Messenger) of Galileo (1564-1642), published in 1610. The page reads, "The Starry Messenger, Revealing great, unusual, and remarkable spectacles, opening these to the consideration of every man, and especially of philosophers and astronomers; as observed by Galileo Galilei, Gentleman of Florence, Professor of Mathematics in the University of Padua, with the aid of a spyglass, lately invented by him, In the surface of the Moon, in innumerable Fixed Stars, in Nebulae, and above all in four planets swiftly revolving about Jupiter at differing distances and periods, and known to no one before the author recently perceived them and decided that they should be named The Medicean Stars." This book was a report on Galileo's first investigations with a telescope, although the telescope was certainly not "invented" by him. The discussion of the moons of Jupiter was influential in gaining acceptance for the Copernican theory of the solar system.

This page gives Galileo's initial sketches of the surface of the moon, with various craters, and the line between darkness and light clearly visible.

Jordanus de Nemore (1225-1260) was a German scholar who wrote several books on arithmetic and geometry. In the ten books of his *De elementis arismetice artis,* he attempted to write a comprehensive work on arithmetic similar to Euclid’s *Elements *in geometry. The images below are from an early printed edition of the *Arithmetica* of Jordanus de Nemore, who would have handwritten the original.

A discussion on the operation of addition employs diagrams that look a bit like “number lines” along with operational “jumps” to demonstrate the additive process. The first diagram illustrates that \(2\times(2+4)=4+8=12.\)

This is page 7. The page contains theorems 14 through 19 of Book I. Rough translations of the statements of the theorems follow.

14. If a number is divided into two parts and the whole number is multiplied by one of the parts, the result is that number multiplied by itself added to the product of the two parts.

15. If a number is divided into two parts and the whole is multiplied by itself, the result is the sum of each part multiplied by itself and twice one part multiplied by the other.

16. If a number is divided into two parts, then the sum of the whole multiplied by itself and one part multiplied by itself is twice the whole multiplied by that part and the second part multiplied by itself.

17. If a number is divided into two parts, then the whole multiplied by itself is the same as four times the product of one part multiplied by the other and the difference between the two parts multiplied by itself.

18. If a number is divided into two parts, then the smaller multiplied by itself and the whole number multiplied by the difference of the two parts is the same as the larger multiplied by itself.

19. If a number is divided into two equal parts and also into two unequal parts, then one of the equal parts multiplied by itself is the same as the product of the unequal parts and the product of the differences between the unequal parts and the equal parts.

Note that illustrations of each theorem are given by diagrams with numbers in the left margin. Thus, the first diagram illustrates that \(2\times(2+4)=4+8=12.\)

An introductory page from the same copy of the work follows:

*These images appear courtesy of the Columbia University Libraries, where the book featured above is part of the Plimpton and Smith Collections.*

See images from a manuscript copy of Jordanus de Nemore's *De triangulis* here in *Convergence*.

This is the title page of the 1640 printing of Galileo's *Operation of the Geometrical and Military Compass*, originally published privately in 1606. Galileo had invented a version of this geometrical compass a few years earlier and evidently gave copies of this manual to those who bought the compass. The compass had many uses, from performing square root calculations to determining ranges of cannons to solving surveying problems. Click here for more information on the compass.

On this page, Galileo demonstrates how to find the height of a distant object by using the compass twice to sight the object at different distances.

This is the title page of the *Summa de arithmetica, geometria, proportioni et proportionalita*, published by Luca Pacioli (1445-1509) in 1494. This was the most comprehensive mathematical text of the time and one of the earliest printed mathematical works. It contained not only practical arithmetic, but also algebra, practical geometry and the first published treatment of double-entry bookkeeping.

This page is the reverse of the title page. A translation of this page would be valuable.

Above: On this page (f. 36v) Pacioli illustrates one of the methods of finger counting prevalent at his time in Italy.

*Editor's note:* The images above, from Columbia University Library's David Eugene Smith and George Arthur Plimpton collections, were posted on 5/14/2008. The images below, obtained through the courtesy of the Dibner Library of Science and Technology, Smithsonian Libraries, were added on 5/14/2017.

The opening page of text in the *Summa* discusses the properties of numbers:

Pacioli considered constructions involving the geometrical inscription of various figures within a given circle:

For additional images from Pacioli’s *Summa,* see the *Convergence* article, Mathematical Treasure: Luca Pacioli's *Summa*, by Cynthia J. Huffman. The *Summa* can be viewed in its entirety on Google Books.

Index to the Collection of Mathematical Treasures from the David Eugene Smith and George Arthur Plimpton collections, Columbia University Library

Index to the *Convergence* Collection of Mathematical Treasures

This is the page (30) illustrating the construction of an ellipse, from the Latin translation (1538) of the *Treatise on Mensuration* of Albrecht Dürer (1471-1528). This book was originally written in German and published in 1525 and was designed to teach German artists the geometrical ideas on which perspective in painting was based. On this page, Dürer shows how to construct an ellipse from its definition as a section of a right circular cone. Unfortunately, Dürer did not get the curve quite right, presumably because he believed that it was wider at one end than the other. (For more information on this, see Roger Herz-Fischler, "Dürer's Paradox or Why an Ellipse is Not Egg-Shaped," *Mathematics Magazine* 63 (1990), 75-85.)

This page (33) illustrates Dürer's construction of a parabola, again by considering it as a section of a cone.

T

This page (92) illustrates the drawing of a spiral in space.

This woodcut print on page 185, called "The Designer of the Lute", illustrates how one uses projection to represent a solid object on a two-dimensional canvas.

This is the title page of the *Nova Scientia* (1537) of Nicolo Tartaglia (1499-1557). In this work, Tartaglia discussed the mathematics of artillery and developed methods for determining the range of a cannon. The caption below the illustration reads,

The Mathematical sciences speak: Who wishes to know the various causes of things, learn about us. The way is open to all.

The illustration itself depicts a walled compound, the compound of knowledge. The high wall keeps out the man who attempts to scale it and enter improperly. Entrance into the compound is through a single door opened by Euclid. In the first courtyard, a crowd comprised of Tartaglia and the muses of the seven liberal arts watch a demonstration of Tartaglia's new knowlege, a theory of trajectories. Beyond the first courtyard is a second smaller, more exclusive and highly elevated one. Its entrance is manned by Aristotle and Plato. Plato holds a banner proclaiming,

No one can enter who does not know geometry.

Enthroned at the rear of this compound, in the highest position of all, is philosophy.

Above: On this page (folio 29v), we see a method of determining the height of a distant object using a quadrant, or set-square.

*Editor's note:* The images above, from Columbia University Library's David Eugene Smith and George Arthur Plimpton collections, were posted on 5/14/2008. The images below, obtained through the courtesy of the Dibner Library of Science and Technology, Smithsonian Libraries, were added on 5/14/2017.

Tartaglia established that the trajectories of projectiles follow a (somewhat) parabolic path:

The range of a cannon is determined by the elevation of the barrel. Tartaglia designed a set-square, or quadrant, that would help gauge cannon barrel elevation:

Index to the Collection of Mathematical Treasures from the David Eugene Smith and George Arthur Plimpton collections, Columbia University Library

Index to the *Convergence* Collection of Mathematical Treasures

This is the title page to the 1811 edition of the *Descriptive Geometry *of Gaspard Monge (1746-1818). This book deals with methods for representing three-dimensional objects in two dimensions. It was written to accompany Monge's courses at the Ècole Polytechnique in Paris.

A page from Monge's book (plate 14) illustrating projections obtained by cutting a cone with an oblique plane.

This is the title page of the Latin edition of the works of Archimedes, the first four books of Apollonius's *Conics*, and Theodosius's *Spherica*, by Isaac Barrow (1630-1677). It was published in 1675.

Gottfried Wilhelm Leibniz carried on an active correspondence within the intellectual community of his time. In particular, two of his main correspondents were the brothers Jacob and Johann Bernoulli. Johann began corresponding with Leibniz in 1693. Depicted is the title page of the 1745 edition of the *Leibniz Bernoulli Correspondence*, vol.1. This book contains their correspondence for the years 1694 - 1699.

This is from a letter of June, 1695 in which Leibniz is discussing differentiation with Johann Bernoulli. Notice that Leibniz sometimes uses negative powers on the "*d*" to represent anti-differentiation.

In this letter from December, 1696 from Leibniz to Bernoulli there is a discussion of integration by parts applied to functions having powers of *x* and powers of the logarithm.

This is the first page of *De Circuli Magnitudine Inventa* (*On Finding the Magnitude of the Circle*) (1654) by Christiaan Huygens (1629-1695). In this work, Huygens considered various ways to determine the accurate measurement of a circle.

This is the title page of* **Exercitationes mathematicae libri quinque *(*Five Books of Mathematical Exercises*) (1657) by Frans van Schooten (1615-1660)*. *In particular, according to the inscription, this copy was originally owned by Johann Hudde (1628-1704), a student of van Schooten. The five books collected in this work contained a variety of mathematical material, including sections on basic arithmetic and geometry, simple geometric constructions, an attempted reconstruction of a work of Apollonius, constructions of conic sections, and some combinatorial techniques enabling him to find amicable numbers. Van Schooten also included as an appendix the first text on probability, written by Christiaan Huygens (1629-1695), again one of his students. This collection was one of the books read by Isaac Newton while he was a student at Cambridge University and helped to introduce him to modern mathematics.

This is page 167 in the section on geometrical constructions. This particular construction shows how to find the distance across a river.

On p. 223, van Schooten begins his section on reconstruction of the Plane Loci of Apollonius. Interestingly, although van Schooten only knew of the Plane Loci from various Greek references, more recently the actual work was discovered in an Arabic version.

On p. 224, van Schooten shows how, given points A and B, to construct two line segments AG and BG that meet at a given angle.

On page 327, van Schooten demonstrates the use of linkages to construct ellipses.

On p. 533 is the final proposition in Huygens' *Treatise on Reckoning in Games of Chance*. This particular problem asks to determine the ratio of "chances" in a game where two people play alternately with two dice, on the condition that the first one wins when he throws a 7, while the second one wins if he throws a 6, assuming that the second player throws first. Huygens' solution is that the ratio is 31:30.

At the bottom of p. 533 and on p. 534 are five exercises that conclude Huygens' treatise. For example, in problem 2, Huygens sets up the situation where there are 12 balls in an urn, of which 4 are white and 8 are black. There are three blindfolded players A, B, and C who draw a ball out in turn with the winner being the first one to draw out a white ball. Huygens asks the ratio of the chances of the three players. (As stated, the problem is somewhat ambiguous, but it is probably easiest to assume that if a black ball is drawn, it is put back into the urn.)

Here is the title page of Part I of the *General Trattato di Numeri* (*General Treatise on Number and Measure*) (1556) of Nicolo Tartaglia (1500-1557). This is an extensive work on elementary mathematics that was popular in Italy for several decades after its publication.

Here is folio 29 of the *General Treatise*. Here Tartaglia is showing how to determine the area of an irregular curved shape.

This is the title page of what would be a five-volume edition of all works of Isaac Newton (1642-1727) known at that time. Note that this five-volume work (1779-1785), edited by Samuel Horsley (1733-1806), is much shorter than the more recent eight-volume collection, *The Mathematical Papers of Isaac Newton* (1967-1981), edited by Derek Thomas Whiteside (1932-2008).

This is the beginning of Newton's first letter to Gottfried Wilhelm Leibniz (1646-1716), the "*Epistola prior,*" sent through Henry Oldenburg (*ca.* 1618-1677) on June 13, 1676. This letter (from vol. 1, p. 285) contains some of Newton's work on the calculus, including his first statement of the binomial theorem.

This page (vol. 2, p. 359) is from Newton's *Principia Mathematica. * It is the discussion of Proposition 29, Problem 6 from Book 2, Section 6 of that work.

This page, from vol. 3, p. 420, illustrates refraction of light through triangular prisms.

This diagram, Table IV from volume 4, shows how refraction of light is the cause of the rainbow.

Volume 5, containing papers on history and theology, was published in 1785. It is available via GoogleBooks.

These two pages are from the *Zhoubi suanjing* (*Arithmetical Classic of the Gnomon and the Circular Paths of Heaven*), a Chinese book on astronomy and mathematics dated to approximately 100 BCE. These images are from a Ming dynasty copy printed in 1603. These diagrams were added to the original text at some point in an attempt to illustrate a dissection proof of the "Pythagorean Theorem," known by the Chinese as the *Gougu* theorem. A complete English translation and analysis of the *Zhoubi suanjing* is given by Christopher Cullen in his *Astronomy and Mathematics in Ancient China: the Zhou bi suan jing* (Cambridge University Press, 1995). See, in particular, Appendix 1.

On these pages, the diagram on the right is usually called the "hypotenuse diagram" and illustrates the proof of the *Gougu* (or Pythagorean) theorem in the 3-4-5 case. The diagram on the left shows how a square of side 3 fits into a square of side 5.

This diagram illustrates a square of side 4 fitting into a square of side 5.

This page is from a sixteenth century Ming dynasty edition of the *Jiuzhang suanshu* (*Nine Chapters on the Mathematical Art*). The work was originally written around the beginning of our era, but the extant copies we have today all stem from an edition and commentary prepared by Liu Hui in the third century. This illustration explains Liu's exhaustion method for determining pi. He obtained a value of 3.14024. A successor, astronomer-mathematician Zu Chongzhi (429-501) extended the method further and obtained a lower bound of 3.1415926 for pi and an upper bound of 3.1415927.

For more details on Liu Hui's calculation, see: Victor Katz, ed., *The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook* (Princeton University Press, 2007), pp. 235-240.

This is the title page of the *Tectonicon* by Leonard Digges (1520-1559), a 1634 edition of the work originally published in 1556. Digges was an English astronomer, mathematician, and instrument maker, credited with the invention of the reflecting telescope and the theodolite. His book discusses methods of measurement.

At the beginning of the book, Digges discusses basic ideas of arithmetic, preliminary to dealing with measurement.

On page 4, basic ideas of measurement are discussed.

And on page 5, we have definitions of various parts of the circle. See also images from *Stratioticos,* by Leonard Digges, and from *Pantometria,* by Leonard Digges and his son, Thomas Digges.

This is the title page of *A Geometrical Practise Named Pantometria*, a guide to applied geometry published by Thomas Digges (1546-1595) in 1571. *Pantometria* was completed by Thomas from a manuscript left by his father, Leonard Digges, who died when Thomas was 13 years old. After his father's death, Thomas became the ward of John Dee (1527-1609), sometime scientific advisor to Queen Elizabeth I. Thomas was greatly influenced by Dee, and remained friends with him throughout his life. Thomas Digges became a recognized astronomer and the leader of the English Copernicans.

This diagram shows how to measure the height of a tower using a quadrant.

This diagram shows how to use a plane table to take sightings on a fortified city during a siege.

Both Leonard and Thomas Digges, father and son, wrote about military applications of mathematics. See also images from *Tectonicon* and from *Stratioticos,* by Leonard Digges.

This is a page from the *Sharḥ ashkāl al-taʾsīs* (*Commentary on the Fundamental Theorems,* 1412-13) of Qāḍī Zāda Mūsā ibn Muḥammad, also known as al-Rūmī ("the Roman" or "the Byzantine") because he was born in Asia Minor in about 1364. He died in 1436. Qāḍī Zāda's book was a commentary on the *Ashkāl al-taʾsīs (Fundamental Theorems),* written by al-Samarḳandī (1250-1310) and containing a discussion of 35 of Euclid's propositions. The book shown in the image is a later copy of Qāḍī Zāda's work, probably written in the sixteenth century. At the top of the page is a discussion of Euclid's Proposition I-5, the "Bridge of Asses" proposition that the base angles of an isosceles triangle are equal. At the bottom, there is a discussion of Proposition I-6, the converse of I-5. Qāḍī Zāda was an astronomer and mathematician in the court of Ulugh Beg (1393-1449) in Samarkand. He and his colleagues compiled the first complete star catalogue since the time of Ptolemy.

*Posted 5/14/2008 (or earlier); updated 5/23/17*

See pages from two other copies of Qāḍī Zāda’s commentary on al-Samarḳandī’s *The Fundamental Theorems* (or *The Forms of Foundation,* based on Euclid’s *Elements*) in Mathematical Treasure: Commentary on *The Fundamental Theorems* and Mathematical Treasure: Qadi Zada’s Commentary on *The **Forms of Foundation.*

An example of galley division from an unpublished 16th century manuscript, *Opus Arithmetica D. Honorati veneti monachj coenobij S. Lauretij*. Honoratus was a Venetian monk, and the manuscript was written in the second half of the 16th century. But the manuscript was copied by a pupil, probably also a monk, who also did the illustrations. His computation “explodes” against a skyline of Venice. An observer raises his hands in amazement at the result. The computation divides 965347653446 by 6543218 and obtains as the answer 147534. This document is Plimpton MS 227, f. 14v.

Another illustration from *Opus Arithmetica* depicts the compound operation necessary to solve a given problem. The multiplication of 16299 by 613 resulting in the product 9991287 can be discerned in the central configuration. In the lower left corner is another galley division. This page is f. 56v.

This image is f. 5 from a late 14th century manuscript containing the first five books of Euclid's *Elements *in Latin translation. The manuscript probably comes from England, but the scribe is unknown. This page contains four propositions from Book I with their diagrams. First is proposition 26 (at the top), which is the AAS congruence theorem. Next is proposition 27, stating that if a line falling on two straight lines makes the alternate angles equal, then the two lines are parallel. Proposition 28 states that if such a lines an exterior angle equal to the opposite interior angle or makes the interior angles on one side equal to two right angles, then the two lines are parallel. And finally, proposition 29 is the converse to both propositions 27 and 28 and is the first proposition in the *Elements* requiring the famous parallel postulate.

This page, f. 8, contains propositions 46 and 47 of Book I. Proposition 46 demonstrates how to construct a square on a give straight line, while proposition 47 is the Pythagorean Theorem. Note that the scribe has two versions, neither very neat, of the famous diagram illustrating Euclid's proof of this theorem.

This page is f. 10, and contains three results from Book II, often characterized as results in geometric algebra. Proposition 7 states that if a straight line is cut at random, then the sum of the square on the whole and that on one of the segments is equal to twice the rectangle, one of whose sides is the given line and the other the given segment, together with the square on the second segment. Proposition 8 states that if a straight line is cut at random, then the sum of four times the rectangle whose sides are the whole and one of the segments and the square on the other segment is equal to the square on the whole line together with the original segment. Finally, proposition 9 states that if a straight line is cut into equal and unequal segments, then the sum of the squares on the two unequal segments is equal to twice the square on half the original line plus twice the square on the segment between the points of section. All of these propositions can be translated into algebraic results and easily checked, but Euclid treats these as geometric propositions and draws diagrams (in the first two propositions) confirming the proposed equality of areas. He gives a somewhat different proof of proposition 9, using the Pythagorean Theorem, but the scribe, in the diagram at the bottom, seems to have drawn the squares of each application of the Pythagorean Theorem in the proof and used these to demonstrate explicitly the equality of areas.

These pages are from Plimpton MS 165. This is a manuscript from c. 1294 that contains a copy of Boethius's *Arithmetic* followed by Euclid's *Elements*. (Pages from the Boethius are elsewhere in this document.) This page contains I-46 (On a given straight line to describe a square), I-47 (the Pythagorean Theorem) and I-48 (the converse of the Pythagorean Theorem). At the bottom of the page are the opening definitions of Book II. The diagrams, however, are not entirely in line with the propositions to which they refer. For example, the last diagram is one accompanying proposition II-1.

This page contains several propositions from Book III, dealing with properties of circles (propositions III-28, 29, 30, and 31). Note that the circles are all drawn with a compass and the straight lines with straightedges.

This is the first page of Book V, with the fancy letter at the top of the page. The text contains the various definitions of Book V dealing with ratio and proportion. The illustration in the right margin is of definition 5, the famous definition of equal ratio.

This page is the beginning of Book VI. Note the fancy initial letter desinating the beginning of a book. The two diagrams toward the top illustrate the definitions of similar figures. The text also includes propositions 1 and 2 of the book. Proposition 1 states that "triangles and parallelograms which are under the same height are to one another as their bases," while proposition 2 is "if a straight line is drawn parallel to one of the sides of a triangle, then it will cut the sides of the triangle proportionally, and conversely." The proof of proposition 1 is the only one in Book VI that makes explicit use of Euclid's definition 5 in Book V giving the definition of the equality of ratios.

On this page appears proposition 28 of Book VI (even though the number in the margin says "26"). The proposition, actually a construction, states, "To a given straight line to apply a parallelogram equal to a given rectilinear figure and deficient by a parallelogrammic figure similar to a given one; thus the given rectilineal figure must not be greater than the parallelogram described on the half of the straight line and similar to the defect." Note that in the accompanying diagram, the parallelogram is a rectangle and the given figure to which the defect is similar (labeled **d**) is a square. In this form, the problem can be translated (and was translated by Islamic mathematicians) into the question of solving a quadratic equation. Namely, if the given line segment has length *b*, if the given rectilinear figure (here a triangle) has area *c*, and if the length of the right hand segment of the line segment is labeled *x*, then the problem can be translated into the quadratic equation *x*(*b - x*) = *c* or *bx - x*^{2} = *c*.

Benedetto da Firenze (1429 – 1479) was a respected Florentine *maestro d’abaco*. Here, on page 114 of his unpublished manuscript *Trattato d’arismetricha* (ca. 1460), a work on mercantile arithmetic, is a discussion of *regula del chataina,* the chain rule, used to compute exchange rates. This is not the chain rule you know from calculus, of course!

The two images below are of the same manuscript (Plimpton MS 193) of Boethius's *Arithmetic* as that above. They were added to this article in 2018. While the image above was prepared specifically for the Mathematical Association of America (MAA) by the Rare Book and Manuscript Library of the Columbia University Libraries, those below are presented courtesy of the Columbia University Libraries via *Digital Scriptorium.*

Benedetto da Firenze, or Benedict of Florence, was a noted mathematician of his city, known especially for his mercantile arithmetic, *Trattato* *d’arismetricha* (ca. 1460, Plimpton MS 193). Although highly valued for its content, the manuscript was never printed. It contains several classical problems of the times: grains of wheat on a chessboard, the hare and the hound, jealous husband, and the testament of the dying man.

The *Arithmetic* also considered geometry problems.

*The images above have been obtained through the kind cooperation of the Rare Book and Manuscript Library of the Columbia University Libraries. These and more images may be accessed via Digital Scriptorium, a digital collection of medieval and early Renaissance manuscripts made available by a consortium of cooperating university libraries headed by the University of California, Berkeley. *

Nasīr al-dīn al-Ṭūsī (1201 – 1274) was a Persian astronomer and mathematician. He is noted for writing the first major work on pure trigonometry as well as for his commentaries on Greek works. This is a page from a later Arabic edition of his commentary on Euclid’s *Elements,* a page dealing with Euclid's method of exhaustion.

This is the title page of Seki Kowa’s (1642-1708) *Katsuyo sampo* [*Essentials of Mathematics,* 4 vols.], published posthumously in 1712. It is a collection of Seki’s discoveries including:

- an accurate value of \(\pi\) to 10 decimal places and the computation of the volume of a sphere using
*enri,*which was a “method of circles”; - “Newton’s” interpolation formula;
- properties of spirals and ellipses;
- derivation of “Bernoulli” numbers; and
- the “Pappus-Guldin” theorem.

These are two pages in Volume I showing the table employed to obtain “Bernoulli” numbers. They are given in the right column and were obtained by a technique called *Ruisai Shosa-ho*.

This is the title page of Oliver Byrne’s 1847 edition of *Euclid's Elements*, the first six books. Using color and pictures, Byrne attempted to teach Euclid's geometry by minimizing textual discussion, including labels. This innovative approach stressed visualization. The picture on this page illustrates Euclid’s Proposition 47 of Book I, the Pythagorean Theorem. Oliver Byrne (1810-1880) was trained as a civil engineer but was also employed as a surveyor and a sometime professor of mathematics. Augustus DeMorgan described him as “eccentric and a minor mathematician.” However, his *Elements of Euclid* was proudly displayed at The Great Exhibition of 1851 and credited with being “one of the oddest and most beautiful books of the century.”

Page 45 of Byrne’s *Euclid* presenting Proposition 44 of Book I.

Page 117 of Byrne’s *Euclid* discusses Proposition 32 of Book III.

A complete copy of Byrne's *The First Six Books of the Elements of Euclid* can be viewed at Bill Casselman's website at the University of British Columbia, Canada.

For the most complete biography of Oliver Byrne to date and ideas for how to use Byrne's 'Euclid in Colours' in the classroom, see the *Convergence* article, "Oliver Byrne: The Matisse of Mathematics."

Frontispiece for Kepler's *Rudolphine Tables.* The image, a monument for astronomy, pays homage to the great astronomers of the past: Hipparchus, Ptolemy, Copernicus and Tycho Brahe. Copernicus and Brahe appear to be debating an issue while Hipparchus ponders and Ptolemy computes.

This is the title page of Johannes Kepler's *Rudolphine Tables* (1627), the most accurate and comprehensive star catalogue and planetary tables published up until that time. It contained the positions of over 1000 stars and directions for locating the planets within our solar system. Kepler finished the work in 1623 and dedicated it to his patron, the Emperor Rudolf II, but actually published it in 1627. It is the first scientific work which extensively employs a new concept of logarithms. The table's findings support Kepler's laws and the theory of a heliocentric astronomy.

Page 78 of the *Tables* discusses the computation of planetary orbits.

For additional images from this work, see also Mathematical Treasure: Kepler's *Rudolphine Tables* and defense of Brahe.

Title page of Tycho Brahe's *Astronomiae instauratae mechanica*. Brahe was the foremost astronomer of his time, particularly noted for his accurate celestial measurements. His angular observations were consistently correct within one minute of arc. This level of accuracy was due in large part to the excellence of his self designed instruments, most of which are described in this book. For a complete translation of his table of contents and listing of instruments see http://www.kb.dk/en/nb/tema/webudstillinger/brahe_mechanica/.

An illustration from Brahe’s *instauratae* showing his great mural quadrant. This functioning quadrant was actually painted on the wall of his observatory, Uraniborg at Hven.

Petrus Apianus (Peter Apian) was a German humanist known for his work in mathematics, astronomy, and cartography. This is the title page of his *Instrumentum primi mobilis* (1534), which discusses the use of measuring instruments and presents new developments in the compiling of trigonometric tables. This book is also noteworthy in that it contains the first European printing of the Latin translation by Gerard of Cremona of the revised edition of Ptolemy's *Almagest *by Jabir ibn Aflah (fl. 12th century)*.* Jabir is commonly known in the West by the Latinized version of his name, "Geber".

This page of *Instrumentum* directly precedes a listing of sine values. Apian illustrated and explained the nomogram employed to graphically determine his sine and versed sine values (versed sine of arc ø = 1 - cos ø).

Title page of Petrus Apianus’s *A Geographical Introduction* (1534). In this book, Apian reviewed the theories of Vernerus, or Johannes Werner (1468-1522), a Nuremburg priest and mathematician who devised a method of using lunar observations to find longitude, and explained applications of trigonometry (specifically, sines and chords) in geography.

*Convergence*'s Mathematical Treasures also include images from Apian's 1533 Folium populi: Instrumentum and from his 1545 *Cosmographia,* the latter extended from his original 1524 edition by his student Gemma Frisius.

Index to Mathematical Treasures

The title page of Leonard Euler’s *Integral Calculus*, vol. 1 (1768). The complete work in three volumes appeared in the interval 1768 – 1770. This was the first complete textbook published on the integral calculus. The entire *Integral Calculus* is available at the Euler Archive.

Euler’s discussion on page 121 of volume 1 of his *Integral Calculus* concerning the integration of logarithmic and exponential functions. Note that Euler used *lx* to represent what we write as ln(*x*)*.*

An excerpt, pp. 182–183, from volume 2 of Euler’s *Integral Calculus*, demonstrating a series solution for a differential equation.

Christopher Clavius S. J. (1537 - 1612) was a German mathematician and astronomer. Renowned as a teacher and writer of textbooks, Clavius was particularly active in the reform of the Gregorian calendar. This is the title page of his collected works, *Opera Mathematica*,* *(1612), five volumes. The first volume contains his commentaries on Euclid and the “Spheric” of Theodosius. The second volume contains his works on algebra and geometry and the third volume contains a commentary on Sarobosco’s “Sphaera” An account of the construction of sundials is given in the fourth volume, while the last volume contains information on calendar reform.

The English translation of this page is as follows:

Christopher Clavius of Bamberg and the Society of Jesus Mathematical Works in five volumes, now once more corrected by the author and revised in many places. Dedicated to the Most Reverend and Illustrious Prince and Lord, Dom John Geoffrey, Bishop of Bamberg at Mainz printed by Reinhard Eltz, at the expense of Anthony Heirat, with leave and permission of His Majesty the Holy Emperor. In the year of Our Lord1612. God has given me the knowledge of the progress of the years and of the constellations of the stars. Wisdom 7,19.

Index of problem demonstrations from volume 1 of Clavius’ *Opera.*

Plate VII from Giuseppe Alberti’s *Instruzioni pratiche per l’ingenero civile* (1774) [*Practical Instructions for Civil Engineers*]. Alberti (1712 - 1768) was an Italian engineer and architect. This illustration on page 298 explains the triangulation method of land measurement employing a sighting staff or surveyor’s cross. The instrument shown contains a compass for marking bearings.

Index to Mathematical Treasures

This is the title page of the *Academia Algebrae* by Johann Faulhaber (1580 - 1635). Faulhaber was a German cossist, who evidently had influence on both Johann Kepler and René Descartes. In this work, he exhibited formulas (in the German cossist notation) for sums from 1 to *n* of the *k*th powers of the positive integers for *k* = 13, 14, 15, 16, and 17. He had earlier exhibited the formulas for smaller values of *k*. Unfortunately, he left little indication as to how he had developed these formulas.

On this double page, signature B, f. i (verso) and ii (recto) of the *Academia Algebrae,* Faulhaber gave the formula for the sum of the 13th powers of the integers, with some steps leading to this formula. The final formula appears in the third from the last line of the paragraph with all the algebraic notation. In translation to modern notation, it reads that the sum of the 13th powers from 1 to *n* is equal to (30*n*^{14} + 210*n*^{13} + 455*n*^{12} – 1001*n*^{10} + 2145*n*^{8} – 3003*n*^{6} + 2275*n*^{4} –^{ }691*n*^{2})/420. For more information on Faulhaber and sums of powers in general, consult the article in this magazine by Janet Beery, "Sums of Powers of Positive Integers."^{ }

This is the title page of the *Whetstone of Witte* (1557) by Robert Recorde (1510-1558). Recorde explains in this poem the reason for the name of what is essentially an algebra text, one of the earliest in England.

On this page (Sig. S, f. i v & f. 2 r), Recorde explains the notation for a unknown and its various powers. Note that the owner of this particular copy wrote notes to help him understand the various names and abbreviations for the powers.

Recorde explains subtraction of polynomials by use of a poem (Sig. X, f. ii r).

On this page (Sig. Ff, f. i r), we see Recorde introducing, for the first time, the "equal" sign. He explains that he picked two parallel lines to represent this concept "because no two things can be more equal." He then gives various examples of the use of the equal sign in algebraic equations.

On these pages (Sig. Ii, f. iv r & v and Sig. Kk, f. i r) is Recorde's attempt to design a real problem whose solution requires a quadratic equation. This problem is entitled a "question of jorneying" and requires knowledge of the formula for the sum of an arithmetic progression.

Christopher Clavius S. J. (1537 - 1612) was highly respected in his time as a mathematical educator and curricular reformer. His textbooks were valued and widely used. This is the title page of the 1584 edition of his *Practical Arithmetic*, first published in 1583 in Rome. Among its notable users were René Descartes and Gottfried Leibniz. The missionary, Matteo Ricci S.J. (1552-1610) would eventually adapt and translate this work for the Chinese. It was published in China in1613 after Ricci’s death, and introduced Western arithmetic to the Celestial Empire.

Here are pages 76 and 77 of Clavius’ *Arithmetic *in which the author demonstrates shortcuts in using galley division and accommodating fractional remainders. On page 76, in obtaining the quotient of 6709456 and 2808, the division by 2808 is undertaken with the first four digits of the dividend. A remainder is obtained, 913, indicating the division process must continue. The large “X” at the side of the computation marks a “casting out of nines” was used to check the work. On page 77, the quotient of 13946007693 and 38000000 is sought. Here the abbreviated division process works as 3800, the “shortened” divisor, is an integral factor of the “shortened” dividend, 1394600. The “cut off tail” of the dividend, 7693, becomes the numerator of the remainder.

This is the title page of the brief work *On the Transformation of Curves into Straight Lines*, by Hendrick van Heuraet (1634 - 1660), published in the 1659 Latin edition of Descartes's *Geometry*, edited by van Schooten. Although van Heuraet was not the first to accomplish a rectification, a task that Descartes had said could not be done, this is the first publication of a general procedure, a procedure very close to our standard calculus procedure for finding the length of a curve.

On these two pages, van Heuraet describes his general procedure for rectification, one which tranforms the length into an integral, that is, the area under a curve. He then illustrates the procedure by calculating the length of the semi-cubical parabola, *y*^{2} = *x*^{3}/*a. * (We can take *a* = 1 for simplicity.) Note that since the procedure for finding arc length involved first finding *dy*/*dx* (or the tangent to the curve)*,* van Heuraet accomplishes this by using Descartes's normal method and Hudde's rule for finding a double root. Note also that van Heuratet uses Descartes's symbol for "equal" rather than our modern equal sign.

On this page, van Heuraet completes his calculation, noting that the answer is found by determining the area under the parabola. He further notes that he could also determine the lengths of the curves *y*^{4} = *x*^{5}, *y*^{6} = *x*^{7}, and so on. Finally, he shows that to find the length of a parabola he needs to be able to find the area under a hyperbola.

This is the frontispiece of *Procli Diadochi* by Francesco Barozzi, published in Venice, 1560. Barozzi (1537 - 1604) was a Venetian nobleman, mathematician, astronomer and humanist. A correspondent of Christopher Clavius, he was well known in the Italian mathematical community of the time. He was a translator of and commentator on ancient mathematical classics and was particularly active in the 16th century movement to revive an interest in Euclidean geometry. His book is a translation of and commentary on the edition of Euclid's *Elements* by Proclus or Proclus Diadochus (411 - 485), a manuscript known simply as *Procli Diadochi*. The portrait depicts Barozzi.

A discussion of parallel lines from *Procli Diadochi*, pages 214 - 215. Barozzi was noted for devising fourteen different methods for constructing a set of parallel lines.

The title page of Joseph Lagrange’s *Mécanique analytique* published in 1788. This was Lagrange’s definitive work. It's summarized all the accomplishments in the field of mechanics from the time of Isaac Newton forward. In his discussions, Lagrange employed applications of the calculus of variations and differential equations, transforming mechanics into a branch of mathematical analysis. Admired for its clarity and structure, this book has been described as a great mathematical classic and a “scientific poem”.

A discussion from Langrange’s *Mechanics, *p. 58-59, on the resolution of three forces, acting at a distance on a fixed point.

The title page of *Universal Geometry,* by Marin Mersenne (1588 - 1648), published in Paris in 1644. A compendium of material on geometry, mechanics and optics, it considers the theories of Euclid, Archimedes, Ramus, Theodosius, and Apollonius and includes a treatise on optics by the contemporary natural philosopher Thomas Hobbes (1588 - 1679).

This is the title page of Leonard Euler’s *Introductio in Analysis Infinitorum* vol.I, published in 1748. This book is considered by some mathematical historians to be one of the most influential mathematical texts of all time. It serves as an introduction to Euler’s later series of texts on the calculus: *Differential Calculus* (1755) and *Integral Calculus*, completed in 1770. Among its many contributions the *Analysis*: defines “function”; presents methods of transforming and representing functions; establishes the category of even and odd functions, defines the trigonometric functions in a modern manner; presents a proof (not complete) for the Fundamental Theorem of Algebra; popularized the use of the symbol “*e*” for the number 2.71828…and “π” for the number 3.1417... and established the relationship cos *θ* + *i* sin *θ* = *e ^{iθ}*. A complete English translation of this work is available: Leonard Euler

The frontispiece of Euler’s *Analysis* reflects the romantic era of his time and shows two women contemplating a mathematical problem while a winged muse hovers above. The engraving is entitled "Analysis of the infinitely small."

Image of page 46 of the *Analysis,* the beginning Chapter IV, “On the development of functions in infinite series”. Here Euler makes the argument that many functions can be simplified for computation by converting them into power series: *A + Bz + Cz*^{2 +} *Dz*^{3} +...

This image of page 47 continues the discussion of series conversion of a function. The example of the rational function 1/(α + βz) is given. By repeated division the function is transformed into a power series and Euler then proceeds to demonstrate techniques for determining the unknown coefficients *A*, *B*, *C*,… .

This is the title page of *Calculus of Probability*, published in 1816. It was written by Sylvestre Lacroix (1765 - 1843). Lacroix taught at the Ecόle Polytechnique, where he assisted Monge in producing his book on descriptive geometry. He later became a Professor of Mathematics at Collége de France. Lacroix produced textbooks based on his lecture notes. These books were well received and their style and form were emulated by other European textbook writers of this period.

Here, on pages 122-123 of the *Calculus of Probability*, Lacroix discusses the “St. Petersburg Paradox” first proposed by Nicolaus Bernoulli in 1713. The problem is as follows: “A fair coin will be tossed until heads appears; if the first head appears on the* n*th toss, then the payoff is 2^{n }ducats. How much should one pay to play this game?”

Here is the title page of the original Italian version of the *Instituzioni analitche ad uso della gioventu italiana* (*Foundations of Analysis for the Use of Italian Youth*) of Maria Agnesi (1718-1799). The text was one of the earliest treatments of calculus written on the European continent. Because Agnesi originally wrote this to instruct her younger brothers in analysis, she explained concepts very clearly and gave numerous examples. (For more on Agnesi, see C Truesdell, Maria Gaetana Agnesi, *Archive for History of Exact Science* **40** (1989), 113-142, C Truesdell, Correction and Additions for Maria Gaetana Agnesi, *Archive for History of Exact Science* **43** (1991), 385-386, M Mazzotti, *The World of Maria Gaetana Agnesi, Mathematician of God* (Baltimore, 2007), and A. Cupillari, *A Biography Of Maria Gaetana Agnesi, An Eighteenth-Century Woman Mathematician: With Translations.*)

Among Agnesi's examples was a description of a curve which she called *la Versiera*. We give here (pp. 380-381) her geometric description of the curve and her derivation of its analytic formula. The figure is given below.

In the 1801 English translation of Agnesi's book by John Colson, the translator translated her name for the curve as "Witch", because the Italian word *versiera*, which actually is derived from the Latin meaning "to turn", could be considered an abbreviation for *avversiera*, meaning "wife of the devil." Unfortunately, this term has stuck, and the curve is usually referred to in English as the "witch of Agnesi." It is, in fact, the item in her book for which Agnesi is most famous. Here is page 222 of the English edition and below is the continuation on p. 223 in Colson's words.

There was, naturally, much more in the *Analytical Institutions, *including what some regard as the best treatment of calculus written in the first half of the eighteenth century.* *The following three pages deal with some procedures of integration.

On page 656, Agnesi shows how to use algebraic substitution to deal with integrals of square roots of quadratic expressions in *x*. This is different from our modern method of trigonometric substitution.

On pages 656-657, Agnesi continues to find substitutions in more complicated situations.

Finally, on pages 658-659, Agnesi completes this discussion and gets some complicated integrals involving logarithms. It is an interesting exercise to compare her methods with modern methods of integration.

Johannes Kepler ( 1571—1630 ) was concerned that Austrian wine merchants were cheating their customers by gauging the volume of their barrels incorrectly. To correct the situation, he undertook a study of the volume of wine barrels. He published his findings, *Nova Stereometria Doliorum vinarorum*, in 1615. Forsaking classical techniques of volume calculation, Kepler produced solids of revolution, dissected them into an infinite number of circular laminae and obtained a volume summation. He applied this technique to consider solids other than wine barrels; in total studying the volumes of 92 different solids. Written in Latin this work was scholarly and had a limited audience. In order to increase his financial returns in 1616, he published a popular German language version of his work, *Ausszag aus der Uralten Messekunst Archimedes*. The page images are from the *Messekunst*. Page 27 contains a discussion on the volume of a torus. Page 28 returns to a consideration of the volume of wine barrels.

In 1570 Sir Henry Billingsley (d. 1606) published the first English-language edition of Euclid's *Elements** *as* The elements of geometrie of the most ancient philosopher Euclide of* *Megara* [*sic*]. This is the title page. Billingsley committed an error common in the Renaissance of confusing the mathematician, Euclid of Alexandria, with the philosopher, Euclid of Megara. For many years a controversy raged as to who was the real author of this work: Billingsley or, the more well known John Dee (1527–1608), mystic and scientific advisor to Queen Elizabeth I. At the end of the nineteenth century, the copy of Theon’s Euclid used by Billingsley in his translation was discovered. The numerous marginal notes and comments left no doubt that Henry Billingsley was indeed the translator and major commentator of the disputed work. Further, his conjectures affirmed the fact that he was a talented mathematician. Above: The title page design is an allegorical woodcut print on the applications of geometry. Billingsley’s annotated copy of Theon’s Euclid now resides in the Princeton University Library.

Billingsley was a rich merchant and served as Lord Mayor of London. He was also a mathematics graduate of Cambridge University and versed in Greek. His edition of the *Elements* was an *opus magnus*. Billingsley translated the thirteen books of Euclid from the Greek edition of Theon of Alexandria (ca. 390), added three additional books attributed to Euclid, and included notes from various ancient and modern commentators. The finished work was over a thousand pages long, included unique “pop up” models of geometric solids, and contained a preface written by Dee. The image above shows Billingsley’s preface note to the reader advocating the study of mathematics.

Above: Folio 314 of Billingsley's *Elements*. This page contains three pop up models of pyramids. These pop-up models occur throughout Book XI on solid geometry and were hand-glued into each copy of the work.

For images from Billingsley's Greek and Latin source material, see Mathematical Treasure: Billingsley's Sources for the First English Euclid's Elements here in *Convergence*.

This is the title page of Nathaniel Bowditch’s (1773 - 1838) English language translation of Volume I of Laplace's *Celestial Mechanics*. Pierre-Simon de Laplace (1749 - 1827) produced his monumental work: *Mécanique Céleste* in five volumes during the years 1799 to 1825. Bowditch’s translation was one of the first translations of a major European mathematical work in the new United States of America. He completed the translation in 1818 but delayed publication until 1829 due to a lack of funds for its printing.

Pages 230 and 231 of *Celestial Mechanics* discuss the attraction of a body on a point in space.

Illustration from the 1503 edition of Boethius’s *Liber Circuli*. Folio 86 discusses the quadrature of a circle.

This is the title page of Christopher Clavius’ ( 1538--1612) *Elements* published in Rome in 1574. Note that Clavius indicates his volume contains 15 books of Euclid. Many medieval authors erroneously attributed two extra books to Euclid's *Elements.* Book XIV extends Euclid discussion in book XIII on the comparison of the regular solids inscribed in a sphere. This work is now believed to have been composed by Hypsicles of Alexandria (ca.190 BCE—ca 120 BCE). Book XV also deals with the properties of regular solids and is believed to have been compiled by Isidore of Miletus (fl.ca. 532), who was the architect responsible for the Cathedral of Holy Wisdom in Constantinople, later to become the Hagia Sophia.

Below are Euclid’s propositions I-46 and I-47 as given in Clavius’ *Elements*. Proposition 47 is the Pythagorean Theorem.

This is the title page of Johannes Kepler’s (1571—1630) *Chilias Logarithmoria*, 1624, logarithmic tables he constructed by geometric procedures. Kepler also gave a proof of why logarithms worked and then used them extensively in his calculations of the Rudolphine astronomical tables.

Girard Desargues (1591—1650) was a French mathematician and engineer best known for his contributions to projective geometry. He published materials on many technical and engineering topics. In 1640, he published a tract on dialing-constructing sun dials. This is the title page of the 1659 English language translation of this work.

Title page of Leonhard Zubler's *Nova Instrumentum Geometricum*, 1607. Zubler (d. 1611) was a Swiss goldsmith and instrument maker. He is credited with introducing the use of the plane table into modern surveying. This book demonstrates and promotes the use of his instruments in techniques of triangulation. Many of the situations depicted concern warfare. Note the two surveyors flanking the title page, proudly holding their measuring instruments.

This illustration on page 11 of Zubler’s text shows a technique of multiple sightings used for topographical survey. The double armed sighting instrument allowing for the measure of angles was developed by the author.

Page 17 of Zubler's book illustrates the use of the plane table in a multi-sighting land survey.

Page 23 of Zubler’s *Instrumentum* demonstrates a triangulation technique for obtaining the distance to a fortress.

Illustration on page 49 of Zubler's book demonstrates a sighting technique used to determine distance between objects below the observer’s line of sight.

In the year 1794, André Marie Legrendre (1752 - 1833) published his *Eléments de* *géométrie.* In its preface, Legrendre says he tried to produce a geometry that will testify to the *l’esprit* of Euclid. The book became an immediate success in Europe and eventually went through 20 additions. The first American translation appeared in 1819, a work by John Farrar (1779 – 1853), Hollis Professor of Mathematics and Natural Philosophy (Science) at Harvard. This is the title page of Farrar’s translation of Legendre's *Elements*, the second edition (1825). Farrar went on to translate five French mathematical classics of this time. The style and format of these books transformed American mathematics teaching, and they became models for the new mathematical textbooks employed in the U.S military academies.

Pages 106 and 107 of Legendre’s *Elements of Geometry* discuss the construction and properties of planes. Note the use of the symbol for line segment. This is one of the first applications of this symbol in an American textbook.

Oronce Fine (1494-1555) was a French mathematician and astronomer who served as the Chair of Mathematics at the Collége Royal from 1531 until the time of his death. He revised the classical works of great masters such as Ptolemy, Aristotle, and Sacrobosco; compiled encyclopedic texts on mathematics; and developed astronomical measuring instruments. This illustration is from page 2 of the 1551 edition of his *Le sphere du monde*. It shows the interaction between the Four Elements and the Four Humours. It was preceded in 1549 by a royal manuscript edition. Harvard University's Houghton Library has digitized its 1549 manuscript edition of *Le sphere du monde: proprement dicte Cosmographie. *See images from this 1549 manuscript in *Convergence.*

This illustration on page 45 of *Le sphere* shows the division of the earth into its various zones with the boundaries of the zones including the Tropic of Cancer, the Tropic of Capricorn, and the Arctic and Antarctic circles.

Illustrations from Book V of *Le sphere* demonstrating Fine’s “heart-shaped” projection of the spherical earth onto a flat surface.

See images from Fine's 1549 illustrated presentation manuscript of his *Le sphere du monde* for King Henry II of France.

This is the title page of John Eyre’s, *The Exact Surveyor* published in London in 1654. It describes procedures for the measurement of land, using the instruments of plane table, theodolite, and circumferentor.

Pages 154 and 155 of *The Exact Surveyor* discuss procedures for partitioning plots of land.

Bartholomew Pitiscus (1561-1613) was a mathematician and the Court Chaplain for Frederick IV, elector of the Palatine. In 1595, Pitiscus published a comprehensive trigonometry text, *Trigonometria: sive de solutione tractus brevis et perspicius*. This was the first time the word “trigonometry” was used to describe the study of the properties of triangles. This image is the title page of Pitiscus’s revised edition of his *Trigonometria* from 1600. As before, it considers the properties of spherical and plane triangles, particularly in the applied areas of geodesy, altimetry, geography, gnomonetry, and astronomy.

Pages 90 and 91 of Pitiscus’s *Trigonometria* discuss an illustrative example.

This is the title page of *The Doctrine of Permutations and Combinations, being an* *essential and fundamental part of the Doctrine of Chance,* compiled and published by Francis Maseres, in London in 1795. Maseres (1731 -- 1824) was a London solicitor and minor mathematician. In this book, he collects some of the writings of leading mathematicians of his time dealing with permutations and chance. Its contents include: an English language translation of the first three chapters of the second part of Jacob Bernoulli's *Ars Conjectandi;* Isaac Newton’s discussions of the binomial theorem and comments by John Wallis, Thomas Simpson, Charles Hutton, Nicholas Saunderson, Gottfried Leibniz and others.

Page 282 of the *Doctrine of Permutations* introduces the section by Wallis on counting techniques.

Page 283 of the *Doctrine of Permutations and Combinations* contains John Wallis’ discussion of techniques of counting.

See images of Maseres’ *Doctrine of Permutations* from the University of Pennsylvania Libraries.

This is the title page of Isaac Barrow’s (1630-1677) *Euclide's Elements,* 1660 edition. Barrow was self-taught in geometry. He originally published this book in 1655 as a simplified version of the *Elements*. It became very popular, and for the next half century was the standard English language text on the subject.

Below are pages 10 and 11 of Barrow’s *Euclide's Elements**.* The illustration on page 11 presents Euclid's proof of Proposition I.5 (The base angles of an isosceles triangle are equal). This is the Pons Asinorum [Bridge of Asses] of medieval geometry. If one could prove this proposition, he or she was considered a competent mathematician.

For a text showing the original Latin version of this work, please see Mathematical Treasures: Isaac Barrow’s Euclidean Geometry.

Johannes Kepler's (1571-1630) most influential work in introducing the heliocentric theory to a broad audience was his *Epitome Astronomia Copernicanae* [*Epitome of* *Copernican Astronomy*], published in the years 1618-1621. This work comprises three volumes and contains seven books: Volume I contains three books, Volume II contains Book Four, and Volume III contains the remaining three books. Here is the title page from Volume I of the 1635 edition. The first three books are devoted to the “Doctrine of the Sphere.”

Pages 278 and 279 of Kepler’s *Epitome*. The diagram on page 279 illustrates the rotation of the Earth about the Sun. Note the text above the diagram telling the reader, “Solis S immobile”—the Sun does not move.

This is the title page of Mario Bettini’s *Aerarium Philosophiae Mathematicae* [*Treasury of Mathematics*], published in 1648. Bettini (1582 - 1657) was a Jesuit, a mathematician and a well respected astronomer. This book is an encyclopedic collection of mathematical curiosities.

The frontispiece of Bettini’s book. The picture is a pun on the title, *The Treasury of* *Mathematics*. The picture shows a Prince providing a Jesuit with a chest of money. The Jesuits are thus paid to provide the common people with knowledge of mathematics.

Pages 48 and 49 of Bettini’s *Treasury* discuss the generation of a spiral as a path around the surface of a cylinder. The diagram on page 49 shows that if a spiral is unrolled from its cylinder it will revert into the hypotenuse of a right triangle.

Pages 74 and 75 of Bettini’s book illustrate the erroneous use of simple geometry in determining the diameter of the sun.

A fold-out plate from Bettini’s book provides views of various solids and nets for the construction of polyhedra.

Pierre de la Ramée (1515 - 1572) was a French scholar, philosopher, and mathematician who used the Latin pen name Petrus (or Peter) Ramus. He was a fervent anti-scholastic. As the result of his campaign against scholasticism, he was forbidden by Francis I to publish in philosophy. He turned his attention to mathematics and the writing of school textbooks. In 1569, he published an edition of Euclid's *Elements**.* This is the title page of *The Way to Geometry*, an English language translation of his work. This translation was done from the Latin into English by William Bedwell and published in 1636. It is obvious that this book is intended for the common craftsman.

Pages 16 and 17 of *The Way to Geometry *display a series of definitions of simple geometric concepts and terms.

This is the title page of the *Arithmeticorum libri duo *(*Two Books of Arithmetic*) (1575) by Franciscus Maurolycus (1494-1575). Maurolycus was an Italian mathematician and astronomer, who wrote numerous mathematical works. This particular book, completed several years earlier but only published in the year of his death, contains much material on what we would call number theory. Some of Maurolycus's proofs in the book are essentially by the method of mathematical induction, although Maurolycus does not identify the method specifically.

In this introductory material, Maurolycus defines the various types of figurate numbers - triangles, squares, pentagons, hexagons, and so on. He gives a table of the first ten instances of each type of number. For example, the first ten pentagonal numbers are 1, 5, 12, 22, 35, 51, 70, 92, 117, and 145.

On p. 7 of the *Arithmeticorum libri duo*, Maurolycus states in proposition 16 that the sum of a certain quantity of odd numbers beginning with 1 is the square of the quantity, or, in modern language, that the sum of the first *n* odd numbers is *n*^{2}. It is often stated that Maurolycus's proof is by mathematical induction, based on his use of the two previous propositions. Proposition 14 states that if you add to the square of a particular number the odd number indexed by the next number, the result will be the square of the next number. Proposition 15 states if you add to the square of a particular number twice that number plus one, you will get the square of the next number. The reader should consider how Maurolycus's results relate to the principle of mathematical induction.

On p. 10 of his book, Maurolycus proves that every perfect number is hexagonal and also triangular.

On p. 27 of his book, Maurolycus shows how cubes are generated from sums of consecutive odd numbers. That is, 1 = 1^{3}, 3 + 5 = 8 = 2^{3}, 7 + 9 + 11 = 27 = 3^{3}, and so on.

On p. 70, Maurolycus shows how sums of consecutive odd numbers give fourth powers: 1 = 1^{4}, 1 + 3 + 5 + 7 = 16 = 2^{4}, 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81 = 3^{4}, and so on.

This is the title page of *Moore's Arithmetick in Four Books*, by Jonas Moore (1627 - 1679). Moore served many years as a surveyor, and his work was necessary in the draining of the Fens, a marshland in eastern England near Cambridge. His *Arithmetick *first appeared in 1650, but this is a copy of the third edition, published nine years after his death. Moore's portrait is on the left. Note that, according to the title page, it is in algebra that "all difficult questions receive their analytical laws and resolutions."

On pp. 274-275, there is a mixture problem dealing with the composition of an ointment. Notice the ingredients.

Moore gives on p. 407 various Renaissance and 17th century names and notations for the powers of an unknown in a comprehensive table. Note that columns 7, 8, and 9 give Viète's, Harriot's, and Descartes' notation for powers, respectively. (The actual powers are below the central line; above the reciprocals of the powers.)

On pp. 454 - 457, Moore gives a careful treatment, with examples, of the procedure for turning a word problem into an algebraic equation. But he always aims at generality, if possible. Interestingly, Moore gives on p. 454 both Viète's and Descartes' procedure for choosing letters for knowns and unknowns.

Note Moore's final statement, which still applies today: "And note that of all the parts of equation, this only of the invention is most difficult." Students will probably agree with that statement.

This is the title page of the *De Numeris et Diversis Rationibus* (1545) of Johann Scheubel (1494-1570). Scheubel was a German cossist, who wrote textbooks in arithmetic, algebra, and geometry. In general, his works were written in Latin and were theoretical rather than practical. Thus, they appealed to the academically-minded rather than the merchants for whom other cossists were writing.

On this page (sig. b, f. 2, r), Scheubel displayed his version of the Pascal triangle. Note the typographical errors toward the bottom. Scheubel noted that the triangle would aid in finding roots of numbers. He spent many pages in the text working through the procedures using the triangle that would enable one to extract higher roots.

This is the title page of the *Elements of Linear Curves *by Jan de Witt (1625 - 1672). De Witt was a student of Frans van Schooten, who published this work in his 1661 edition of Descartes' *Geometry*. (This copy is from the 1683 edition. And, in fact, de Witt himself probably finished the work by 1646.) The first of the two books of this treatise was devoted to developing the properties of the conic sections using synthetic methods based on the work of Apollonius. But in the second book, de Witt produced a complete algebraic treatment of the conics, beginning with equations in two variables, based on the work of Fermat and Descartes.

On pages 250-251, de Witt shows that the equations *yy = ax + bb* and *xx = ay + bb* determine parabolas. The proof is based on the "nature of a parabola." Note that de Witt still uses Descartes' symbol for "=" as well as his homogeneity rules in making sure that each side of the equation is two-dimensional. Note also that the diagrams show that de Witt did not stick to perpendicular axes and that he only considered what we would call the first quadrant. That is, all of the values displayed were positive.

Here, on p. 263, de Witt shows how to rotate the axes to turn a complicated second degree equation in two variables into the standard one displayed earlier. Unlike in modern treatises, de Witt does not use trigonometry, but gives the equations of the new axes in terms of the old ones. That is, he uses a transformation of coordinates based on the form of the given equation.

On pp. 276-277, de Witt shows that certain equations in two variables determine a hyperbola, in particular, the equation *lyy/g = xx - ff *and its analog with the variables reversed. Again, his proof requires the "nature of a hyperbola", and, of course, this "nature" comes from de Witt's understanding of the work of Apollonius.

On p. 283, de Witt shows how to make the appropriate transformation of coordinates to change a complicated quadratic function of two variables into the recognizable form of a hyperbola. As before, he does not use trigonometry.

This is the title page of the 1536 edition of the *Libro di Arithmetica i Geometria* of Francesco Feliciano (first half of 16th century). Not much is known about Feliciano, except that he was born in Lazisa, near Verona and was still living in 1563. This book is basically a revision of Feliciano's earlier *Libro de Abaco*, which appeared in 1517. The book contains much commercial arithmetic, but also a treatment of roots, the rule of false position, some algebra, and a section on practical geometry. The book had a good deal of influence on the teaching of elementary mathematics, appearing in numerous editions including one in 1669, 143 years after the original edition.

On sig. S, f. 3r, Feliciano shows how to calculate the circumference and area of a circle. Note that he approximates pi by 22/7 to calculate the circumference and area of a circle of diameter 14.

Here, on sig C, f. 4v and sig. D, f. 1r, Feliciano illustrates the method of multiplication "per crocetta", one of several methods that were in use in the Italian Renaissance.

This is the title page of *Ain neu geordnet Rechenbiechlin* (1514) by Johann Böschenstein (1472-1540). Böschenstein was best known as a professor of Hebrew in several German universities. In fact, Martin Luther studied Hebrew with him at one time. This *rechenbuch* (reckoning book, or arithmetic text) introduced students to the basic principles of arithmetic, with application to various business problems. The engraving on the title page shows someone working on an arithmetic problem.

Here, on sig. A, f. vi, Böschenstein presents eight examples of adding fractions and mixed numbers.

On sig. B, f. iii, Böschenstein presents the rules for dividing fractions, along with eight examples.

This is the title page of the *Treatise of Algebra* by Christian Wolff (1679-1754). Wolff was a student of Leibniz and is most famous for his work in philosophy. His school of philosophy, in fact, was the most prominent in Germany prior to Kant. This book was originally written in Latin in 1713. It first appeared in English in 1739, though this copy is of the second edition of 1765.

On pages 140-141 of his book, Wolff presents a derivation of the formula for the sums of the integers as well as the sums of the squares and cubes of the integers, a derivation which is generalizable to higher powers as well.

On pages 142-143, Wolff generalizes his earlier results and shows how to derive formulas for the sums of higher powers of the integers.

Here, on page 144, Wolff completes the derivation with the complete formula for the sums of third and fourth powers.

On pages 202-203, Wolff discusses some elements of the theory of equations. Note that he mentions Descartes' rule of signs, without attribution to Descartes. In fact, he attributes it to Thomas Harriot and claims further that no one had yet proven it. The first published proof of the result was due to Jean Paul de Gua de Malves (1713 - 1785), who gave two proofs in 1741 in a paper in the *Memoires *of the Paris Academy.

This is the title page of the *Arithmetica Integra* (1544), by Michael Stifel (1487-1567), one of the best-known German cossists of the sixteenth century. Stifel's work covered the basics of algebra, using the German symbols for powers of the unknown and also considering negative exponents for one of the first times in a European book. He also presented the Pascal triangle as a tool for finding roots of numbers and was one of the first to present one combined form of the algorithm for solving quadratic equations.

Page 227 gives Stifel's introduction to the rules of algebra.

The diagram here on p. 255 represents the solution to the pair of simultaneous equations

x^{2} + y^{2} - (x + y) = 78, xy + (x + y) = 39.

Here, the two unknowns are represented by AC and BC, while the sum AB is called "B" by Stifel. Also, the script z is Stifel's notation for the square of the (first) unknown, namely x^{2}. Note that therefore the smaller square (on the upper right) is labeled with the script z, the two rectangles are labeled 39 - 1B (since their areas are each xy, which is equal to 30 - (x + y)), and the larger square, which is equal to y^{2}, is labeled 78 + B - z, that is 78 + (x + y) - x^{2}. Stifel completes the problem as follows: The sum of the areas of all four regions of the diagram is equal to 156 - B, and this equals B^{2}. It follows that B = 12. Therefore the larger square has area 90 - x^{2}, and the two rectangles each have area 27. But either of those rectangles is the mean proportional between the larger square and the smaller square. Therefore, (90 - x^{2}):27 = 27:x^{2}. It follows that 90x^{2} - x^{4} = 729. So x^{2} = 9 and x = 3. Then y = 9 and the problem is solved.

On page 312 (front and back), Stifel discusses the following problem, which he has taken from a work of Cardano:

Find two numbers such that their difference multiplied by the difference of their squares is 792, while their sum multiplied by the sum of their squares is 5720.

We can follow how Stifel makes use of two unknowns in this problem.

Here is the reverse of page 312.

This is the title page of *A New Treatise of Algebra* by Richard Sault (d. 1702). Not much is known about Sault, except that he ran a mathematical school in London in the 1690s near the Royal Exchange and was an editor of and contributor to the *Athenian Mercury*, a literary journal that was published between 1690 and 1697. The *Treatise of Algebra *was published as an appendix to William Leybourne's *Pleasure with Profit*, and included a chapter by Joseph Raphson on converging series.

On page 1, Sault describes algebra as the "art of reasoning with unknown quantities, in order to discover their habitude or relation to such as are known."

On page 19, Sault describes in some detail, with an example, how to convert a word problem into algebraic notation. Note that he generalizes his problem by using arbitrary constants, instead of just the given numbers.

On page 20, Sault continues his description of solving equations, giving more examples and then various questions for practice.

On page 33, Sault introduces quadratic equations, showing how the solution procedure for these equations enables one to solve a problem that Clavius was unable to solve in his own treatise on algebra.

This is the title page of the *Oeuvres Mathematiques* of Simon Stevin (1548-1620), edited by Albert Girard (1595 - 1632) and published in 1634.

This is the first page of Stevin's *Arithmetique,* originally published in 1585, in which he gives several definitions. In particular, he argues, contrary to Euclid, that unity is a number and that "number is that which explains the quantity of each thing." Thus, "number" is not only a collection of units and, in essence, Euclid's distinction between "number" and "magnitude" (coming from Aristotle) no longer makes sense.

On this page, Stevin shows how to solve various types of quadratic equations. In his notation, the circle around a given number designates the unknown raised to that power.

On page 103, Stevin shows how to translate some problems from Diophantus's *Arithmetica* into his own algebraic notation and then how to solve them.

On page 209 of the *Oeuvres,* we find the introduction to Stevin's *La Disme,* his work of 1585 explaining how to use decimal fractions. Again, he uses circles around digits, but these now stand for the appropriate decimal place.

This is the title page of the *Treatise of Algebra* (1764)* *by William Emerson (1701 - 1782). Emerson was a prolific writer on mathematical subjects, including works on fluxions, trigonometry, and mechanics.

On pages 212-213, Emerson discusses the question of the value of a fraction x/y assuming both numerator and denominator vanish. Emerson treats this as an algebra problem, although, since he had written on the topic of fluxions, he understood that calculation of the values of such fractions was based on some notion of limit, even if that was only imperfectly understood.

On pages 214-215, Emerson continues the discussion of the values of fractions. Note that example 5 is taken directly from *l'Hospital's Analyse des Infiniment Petits*. In fact, the example was originally given to L'Hospital by Jacob Bernoulli and is the first example in L'Hospital's text of what we now know l'Hospital's rule.

On pages 336-337, we see how Emerson explained the solution of some algebra problems, showing his readers the step by step approach to solving these.

On pages 338-339, Emerson presents the general problem of oxen eating grass, where the grass continues to grow. A specific example of this problem was given by Isaac Newton in his *Arithmetica Universalis*, first published in 1707.

This is the title page of the *Practica Arithmetice *of Gerolamo Cardano (1501-1576), published in 1539. It was a comprehensive work on arithmetical questions, with numerous practical problems and even some elementary algebra and geometry.

In paragraph 12, Cardano details the side length of regular polygons inscribed in a circle of diameter 10000. Thus the side of an equilateral triangle is 8860, of a square is 7071, and of a hexagon is 5000.

In paragraph 63, Cardano solves a classical problem coming from Leonardo of Pisa: To find a number that when divided by each of 2, 3, 4, 5, and 6 falls short by 1 of exactly dividing, but is divisible by 7. The answer is 119.

In paragraph 97, Cardano solves another problem coming from Leonardo: If of four men, the first, second, and third together have 34 coins; the first, second, and fourth have 73; the first, third, and fourth have 72; and the second, third, and fourth have 88; then how many coins does each have? Cardano simply adds the four numbers together, noting that this sum, 267, is equal to three times the total. Therefore, the total number of coins is 89 and it is straightforward to calculate the number held by each man.

In paragraph 122, Cardano solves a cubic equation without using any formula.

This is the title page of the 1552 edition of the *Practica d'Arithmetica* of Francesco Ghaligai (d. 1536). The book was originall published in 1521, but this printing, like several other printings, is identical with the original. Its intended audience was merchants, so there are many practical problems dealing with issues of trade. In the sections on algebra, Ghaligai introduces his own notation.

On pages 71 and 72, Ghaligai proposes a new notation for powers of the unknown. Notice on the left hand page that his notation for the second power (censo) is just a square, but the other notations never caught on with other authors. On the right hand page, Ghaligai illustrates the notation by calculating the powers of 2 up to the fifteenth power.

This page illustrates a typical algebra problem and makes some use of the "square" notation for censo.

This is the title page of the *Traité élémentaire de géométrie et de la maniere d'appliquer l'algébre a la géométrie* (1775), written by Charles Bossut (1730-1814). This work was one of many texts written by Bossut in connection with his teaching at several different institutions in France. It was one of the first books to give a detailed explanation of analytic geometry.

On page 397 and the following pages, Bossut discusses the equation of the circle. Note that he does not assume that the two coordinate axes are perpendicular.

On pages 398-399, Bossut carefully works out the equation of the circle, displaying it as equation (A) on p. 399. Note that because his axes are not assumed to be perpendicular, the equation has an xy term.

On page 400, Bossut shows various simplifications of the equation of the circle. He derives equation (B) in the case where the two axes are perpendicular. In the next two equations, the position of the origin is simplified.

On pages 424-425, Bossut discusses certain properties of tangents to parabolas and deals with them algebraicially.

On pages 426-427, Bossut proves the familiar property of a paraboloa, that the line from the focus to a point on the parabola makes the same angle with the tangent as the line from that point parallel to the axis.

This is the title page of *A Compendium of Algebra* (1724), written by John Ward, an English mathematicians about whom very little is known. He was born in 1648 and died sometime around 1730. It is known that he taught mathematics in Chester and is famous for another mathematics work, the *Young Mathematician's Guide*, first published in 1707. That work was imported in large quantities to New England and was used for a time as a textbook at Harvard University. It contains a very interesting method of calculating pi.

On pages 38-39, Ward discusses what happens when the number of equations is more than, equal to, or less than the number of unknowns.

Pages 52-53 show two typical problems from the book, each with a detailed description of the steps in the left hand column of the solution of the problem.

This is the title page of the *Arithmeticae Practicae Methodus Facilis *(1540), by Gemma Frisius (originally Regnier Gemma) (1508-1555). Gemma Frisius was best known for his work in astronomy and map-making; he worked closely with Gerardus Mercator in making an early globe. He also suggested a method for determining longitude at sea.

On this page is an example of the use of double false position to solve a problem in two unknowns.

Here, Gemma describes a method of finding square roots.

Two illustrations follow from the fourteenth century Italian codex, *Antichissimo di Algorismo*. This is one of many algorisms written at this time. They were arithmetics designed to introduce the Hindu-Arabic numerals, their operational algorithms and demonstrate their use in problem solving. The majority of the problems considered in this codex are commercial in nature. A few might be categorized as “recreational problems.” These two are of this nature and have remained favorites over the years. A special feature of this codex is that it contains 42 illustrations, many of which supplement problems. These two illustrations are of this kind.

The picture found on folio 59 depicts a situation where two men must divide an item equally. The two companions have eight ounces of balm in an eight ounce vase. At their disposal they have two other vases, one holding three ounces, and the other five ounces. How can they accomplish the equal division?

The illustration on folio 60 presents the situation where three couples wish to cross a stream. The small boat they have will only accommodate two persons at a time. How can they all get to the other shore if no man is to cross with another’s wife? This is a variation of the puzzle-type “River Crossing Problem” that has been posed over the centuries in many guises.

Brass protractor from about 1700 of German manufacture. Its base plate contains some Baroque decoration. Note its similarity to a present day student protractor.

Nest of Austrian weights of the 18^{th} century. Selected by D.E.Smith for his collection to illustrate the ancient, “Problem of Weights”. One example of this problem is given by Claude Bachet as:

*What is the least number of weights that can be used on a scale pan to weigh any in**tegral number of pounds from *1* to *40* inclusive, if the weights can be placed in ei**ther of the scale pans?*

This particular set of weights is elaborately decorated and is one of the best specimens of the weight maker’s art of the period. It bears at least ten official seals, one of which contains the date 1787.* *

An English gauger’s scale, probably 18th century. The units measured on it are: Hogshead, kilderkin, barrel, etc. It may have been the possession of a tavern keeper employed to measure his wares. Gauging, or measuring the liquid volume in a cask or barrel was an important mathematical activity from the Middle Ages through the 19th century. Johannes Kepler devoted a book to the subject.

Austrian measuring rod for an ell. The rod bears the date 1732. The ell as a unit of measure evolved from the ancient cubit, the distance from the tip of a man’s middle finger to the “point” of his elbow. At this time, the ell was still used as a measure in the tailoring business. Its exact length varied considerably among European cities. This one is approximately 26” long, more exactly 65.8cm. This rod is particularly interesting due to its curious indication of fractional divisions.

Notched pieces of wood or bone were used by many ancient peoples to record numbers. The most common type of these “tally sticks” was made of wood. Tally sticks served as records or receipts for financial transactions such as the payment of taxes, debts and fines. From the 12th century onward tally sticks were officially employed by the Exchequer of England to collect the King’s taxes. Local sheriffs were given the task of actually collecting the taxes. The depth and series of notches on these sticks represented the value of the transaction. In recording a debt, a wooden stick was often split horizontally into two parts: the lender receiving one part, the *stock*; and the debtor, the other part, the *foil*. This box contains sticks that date from the year 1296 and were found in the Chapel of the Pyx, Westminster Abbey in 1808. England abolished the use of tally sticks in 1826. The accumulation of tally sticks in the Office of the Exchequer were burned in 1834 resulting in a fire that destroyed the Parliament Building.

The box is opened and the tally sticks removed.

Close-up of three smaller sticks from the box showing notches.

Close-up of large stick revealing notches.

Second view of larger stick showing the name of the of King’s agent, William de Costello, Sheriff of London in 1296.

Index of Mathematical Treasures

This set of late 19th century *sangi**,* wooden computing rods, originated in Korea. They are contained in their cloth carrying case. Sangi were also used in Japan up until about 1700. These computing rods or sticks, and their resulting numeration system, were originally derived from *suanzi*, rods used in China from ancient times through the Yuan Dynasty (1271-1368). The Chinese rods were replaced by the *suanpan*, or bead abacus, which was then adapted with variations throughout Asia.

Another view of the set.

This Italian compass of the 15th century displays a high level of workmanship. Its scales allow for conversion of Roman linear measuring units to other contemporary systems. The compass was more than a mere measuring instrument, it was a computing device.

An armillary sphere is a mechanical model of the universe. The metal bands within the spheres represented the circular orbits of the planets revolving around a central Earth or the sun, depending on the particular scientific theory depicted; pre or post Copernican. When devised, they were among the most complex mechanical devices of their time. Renaissance personages frequently had themselves portrayed in paintings standing next to an armillary sphere indicating their association with wisdom and knowledge. The sphere shown is dated 1550 and is probably of Italian origin. The wide, foremost, band contains the divisions of the zodiac. Its wooden stand was constructed at a later date. This sphere was probably ornamental, decorating the house of a rich merchant or aristocrat, attesting to the fact that he was aware of the importance of science.

This Western astrolabe was constructed by Bernard Sabeus in 1558. Sabeus was a craftsman who is known to have worked in Padua during the years 1552 - 1559. The artisan’s skill that had been previously used to decorate objects of warfare such as swords and suits of armor was now directed at embellishing the new objects of status and power, scientific instruments. However, in creating these new instruments, high levels of precision and mechanical ability were also required. An astrolabe was an instrument used to measure plane angles associated with navigational, terrestrial and astronomical sightings. This astrolabe exhibits a high level of workmanship.

Index to the Collection of Mathematical Treasures from the David Eugene Smith and George Arthur Plimpton collections

The Plimpton library was formally presented to Columbia University in 1936 shortly before the donor's death. The collection of more than sixteen thousand volumes was assembled by George Arthur Plimpton (1855-1936), who served as a board member of the textbook publisher Ginn & Company, to show the development of "our tools of learning." He stated his notable purpose in the preface to his *The Education of Shakespeare* as "the privilege to get together the manuscripts and books which are more or less responsible for our present civilization, because they are the books from which the youth of many centuries have received their education." In general, the Plimpton Library may be described as an assemblage of notable treatises on the liberal arts, particularly grammar, rhetoric, arithmetic, algebra, geometry, geography, astronomy and handwriting. Represented in the Library are the forms of knowledge from the most rudimentary, the hornbook, to the most sublime heights reached in the writings of Aristotle, Donatus, Cicero, Boethius, Euclid, Ptolemy, Pliny and Petrus Lombardus. It is hardly surprising that one of the earliest items in the collection may be the most remarkable, a cuneiform clay tablet on which is written in Old Babylonian (1900-1600 BCE) script a mathematical listing of Pythagorean triples.

The 317 medieval and renaissance manuscripts collected by George Plimpton form the largest such group in the Rare Book and Manuscript Library. Among these are manuscripts of texts by classical authors, antiphonals, early grammars, mathematical and philosophical treatises, and the writings of the church fathers. There is also a complete copy manuscript of Chaucer’s *A* *Treatise on the Astrolabe*, written in England in the fifteenth century.

Association books are also found in the Plimpton library, for the collector was interested in why and how books were used, and who used them. Volumes once owned by notable humanists and scientists include: Sir Thomas More's copy of *Euclid's Elements*, 1516; Isaac Newton's copy of Vincent Wing's *Harmonium coeleste*, 1651. Among the nearly six hundred copybooks and handwriting manuals is one of the two recorded copies of the first edition of *A Boke Containing Divers Sortes of Hands*, 1570, by John De Beauchesne and John Baildon, the first handwriting book published in England. Plimpton's library does, indeed, represent the labor of a collector and the experience of a career, the blending of a vocation as publisher and an avocation as bibliophile and author.

David Eugene Smith (1860-1944), noted mathematics educator at Columbia University’s Teachers College, befriended Plimpton and encouraged him to greatly enrich his collection with mathematical manuscripts and many early Renaissance texts on arithmetic. Today, the collections of both men exist as rich resources for understanding the development of mathematics and the lives and work of many of the persons responsible for its advance. These archives are available to researchers through the Rare Book and Manuscript Collection at Columbia University. The Mathematical Association of America, in cooperation with the Columbia University Libraries, is pleased to display a selection of items, *Mathematical Treasures from the Plimpton and Smith Collections,* from these two separate collections. The editors of *Convergence* would like to particularly thank Dr. Michael Ryan, Director of Rare Books and Manuscripts, and Jennifer Lee, Librarian for Public Service and Programs, for their assistance in making this display possible.

A list of images in *Convergence* from the Plimpton and Smith collections can be found in our Index to the Collection of Mathematical Treasures from the David Eugene Smith and George Arthur Plimpton collections (alphabetical by author). Each item from the Plimpton and Smith collections is posted here in *Convergence* at the standard web resolution of 72dpi. But if you right-click on the name of the item, found in the first sentence of the item description, you can download the item as a tif file at a resolution of approximately 150-200dpi (3-5mb) as well. That version should be suitable for most purposes in a classroom setting. If you want a version in even higher resolution, please contact the second author, Victor Katz. (Note that if you just click on the name of the item, you will get the tif file on your screen, but you may not be able to save it to your computer.)

These images may be used for instructional and educational purposes; for all other uses, please obtain permission from the Columbia University Libraries.

Index to the *Convergence* Collection of Mathematical Treasures

Index to the Collection of Mathematical Treasures from the David Eugene Smith and George Arthur Plimpton collections

David Eugene Smith (1860-1944) was a driving force and innovator in mathematics education. As the chairman of the mathematics education department at Columbia University’s Teachers College, Smith led the way in teaching reforms attuned to the Progressive Education Movement. He firmly believed that the teaching of mathematics should be closely associated with the history of the subject. As an historian of mathematics, he wrote and lectured widely on the subject and also collected historical mathematical materials: texts, documents and artifacts.

When Smith began giving his collection to the Columbia University Libraries in 1931, it included 12,000 printed books on the history of mathematics, ranging from the 15th through the 20th century. It also included 35 boxes of historical documents relating to mathematics; 140 boxes of his own professional papers; 350 volumes of western European manuscripts dating from the 15th to the early 20th century; 670 volumes of Oriental (primarily Arabic and Persian) manuscripts dating from the 8th to the early 20th century; 88 volumes of Chinese manuscripts; 363 volumes of Japanese manuscripts; 3,000 prints and portraits of mathematicians; and some 300 mathematical instruments and related objects.

Today, the collections of Smith and his friend, George Arthur Plimpton, exist as rich resources for understanding the development of mathematics and the lives and work of many of the persons responsible for its advance. These archives are available to researchers through the Rare Book and Manuscript Collection at Columbia University. The Mathematical Association of America, in cooperation with the Columbia University Libraries, is pleased to display a selection of items, *Mathematical Treasures from the Plimpton and Smith Collections,* from these two separate collections. The editors of *Convergence* would like to particularly thank Dr. Michael Ryan, Director of Rare Books and Manuscripts, and Jennifer Lee, Librarian for Public Service and Programs, for their assistance in making this display possible.

A list of images in *Convergence* from the Plimpton and Smith collections can be found in our Index to the Collection of Mathematical Treasures from the David Eugene Smith and George Arthur Plimpton collections. Each item from the Plimpton and Smith collections is posted here in *Convergence* at the standard web resolution of 72dpi. But if you right-click on the name of the item, found in the first sentence of the item description, you can download the item as a tif file at a resolution of approximately 150-200dpi (3-5mb) as well. That version should be suitable for most purposes in a classroom setting. If you want a version in even higher resolution, please contact the second author, Victor Katz. (Note that if you just click on the name of the item, you will get the tif file on your screen, but you may not be able to save it to your computer.)

These images may be used for instructional and educational purposes; for all other uses, please obtain permission from the Columbia University Libraries.

Index to the *Convergence* Collection of Mathematical Treasures