Boethius's Liber Circuli

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

Boethius's Liber Circuli

Billingsley Euclid

In 1570 Sir Henry Billingsley (d.1606) published the first English-language edition of Euclid's Elements: The elements of geometrie of the most ancient philosopher Euclide of Megara [sic].

The image is 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.

Billingsley Euclid

Maria Agnesi's Analytical Institutions

Pages from 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.

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.

Maria Agnesi's Analytical Institutions

Leonhard Euler's Integral Calculus

Leonard Euler’s discussion of volume 1 of his Integral Calculus concerning the integration of logarithmic and exponential functions on page 121 of Integral Calculus, vol. 1 (1768). Note that Euler used lx to represent what we write as ln(x).

The complete Integral Calculus in three volumes appeared in the interval 1768 – 1770. This was the first complete textbook published on the integral calculus.

Leonhard Euler's Integral Calculus

Christopher Clavius's Opera Mathematica

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.

Christopher Clavius's Opera Mathematica

Giuseppe Alberti's Instruzioni pratiche per l’ingenero civile

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.

Giuseppe Alberti's Instruzioni pratiche per l’ingenero civile

Peter Apianus's trigonometry and geography

Title page of Petrus Apianus’ A Geographical Introduction (1534). In this book, he reviews the theories of Vernerus [Johannes Werner (1468-1522), a Nuremburg priest and mathematician who devised a method of using lunar observations to find longitude] and explains the applications of trigonometry (i.e. sines and chords) in geography.

Peter Apianus's trigonometry and geography

Tycho Brahe's astronomical instruments

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

Tycho Brahe's astronomical instruments

Oliver Byrne's Euclid

Page 117 of Oliver Byrne’s 1847 edition of Euclid's Elements discusses Proposition 32 of Book II.

Oliver Byrne's Euclid

Seki Kowa's Essentials of Mathematics

These are two pages from volume one of Seki Kowa’s (1642-1708) Katsuyo sampo [Essentials of Mathematics]. These pages show the table employed to obtain “Bernoulli” numbers. They are given in the right column and were obtained by a technique called Ruisai Shosa-ho.

Seki Kowa's Essentials of Mathematics

Nasir al-Din al-Tusi's Commentary on Euclid's Elements

Nasir al-Din al-Tusi (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.

Nasir al-Din al-Tusi's Commentary on Euclid's Elements

Benedetto da Firenze's Trattato d'arismetriche

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.

Benedetto da Firenze's Trattato d'arismetriche

Euclid's Elements in a 14th century manuscript

This image is 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 is f. 10, and contains three results from Book II, often characterized as results in geometric algebra.

Euclid's Elements in a 14th century manuscript

Opus Arithmetica of Honoratus

An illustration 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.

The above illustration 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.

Opus Arithmetica of Honoratus

Qadi Zada al-Rumi's Geometry

This is a page from the Geometry (1412) of Qadi Zada al-Rumi (1364-1436). Al-Rumi's book was a commentary on the Fundamental Theorems, written by al-Samarqandi (1250-1310), where he discusses twenty-five of Euclid's propositions in detail. The book shown in the image is a later copy of al-Rumi'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 I-6, the converse of I-5. Al-Rumi was an astronomer and mathematician in the court of Ulugh Beg (1393-1449) in Samarkand. He and his colleagues compiled the first complete star catlogue since the time of Ptolemy.

Qadi Zada al-Rumi's Geometry

The Grounde of Artes by Robert Recorde

A page from Robert Recorde's The Grounde of Artes (1543) dealing with various "rules of practice" in arithmetic. See more pages.

The Grounde of Artes by Robert Recorde

Omar Khayyam's Algebra

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, x³ + cx + d = bx². Read more about Omar Khayyam's Algebra.

Margarita philosophica of Gregor Reisch

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.

Margarita philosophica of Gregor Reisch

De Divina Proportione by Luca Pacioli

Image of a stellated dodecahedron from the book De Divina Proportione of Luca Pacioli (1445 - 1509), published in 1509.

De Divina Proportione by Luca Pacioli

Leonhard Euler's Calculus of Variations

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.

Leonhard Euler's Calculus of Variations

Omar Khayyam's Algebra

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.

Omar Khayyam's Algebra

A Treatise of Algebra by John Wallis

A 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.

A Treatise of Algebra by John Wallis

Boethius's Arithmetic

The Arithmetic of Boethius (480-524) dates from the early sixth century. This page is from a manuscript (Plimpton MS 165) that dates from approximately 1294, written on vellum. The above image is from page f. 13 and lists the powers of 2, 3, and 4 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.

Boethius's Arithmetic

Galileo's Siderius Nuncius

This page from Galileo's Siderius Nuncius 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's Arithmetica

This is page 7 from an early printed edition of the Arithmetica of Jordanus de Nemore (early 13th century). The page contains theorems 14 through 19 of Book I. Rough translations are available here.

Jordanus de Nemore's Arithmetica

Galileo's Geometrical Compass

This image is from the 1640 printing of Galileo's Operation of the Geometrical and Military Compass. In this image, Galileo demonstrates how to find the height of a distant object by using the compass twice to sight the object at different distances.

Galileo's Geometrical Compass

Pacioli's Summa

This is the title page of the Summa de arithmetica, geometrica, 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.

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

Pacioli's Summa

Gaspard Monge's Descriptive Geometry

A page from 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.

This page (plate 14) illustrates projections obtained by cutting a cone with an oblique plane.

Gaspard Monge's Descriptive Geometry

Niccolo Tartaglia's General Trattato di Numeri et Misure

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

Here Tartaglia is showing how to determine the area of an irregular curved shape.

Niccolo Tartaglia's General Trattato di Numeri et Misure

Leibniz - Bernoulli Correspondence

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.

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

Leibniz - Bernoulli correspondence

Albrecht Durer's Treatise on Mensuration

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.

Albrecht Durer's Treatise on Mensuration

Jacob Kobel's Geometry

Jacob Kobel's Geometry

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. Read more.

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