Enjoy Mathematical Treasures!

Read about Mathematical Treasures in the December 2012/January 2013 issue of MAA FOCUS, the newsmagazine of the Mathematical Association of America. 

Lilavati of Bhaskara

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 hastas equal to half that difference.

152:  There is a hole at the foot of a pillar nine hastas high, 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 many hastas from 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.

Lilavati of Bhaskara

Francesco Ghaligai's Practica D'Arithmetica

                   

These pages are from the 1552 edition of the Practica d'Arithmetica of Francesco Ghaligai (d. 1536). On these pages, 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.

Practica d'Arithmetica was originally 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.

Francesco Ghaligai's Practica D'Arithmetica

Austran weights

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 integral number of pounds from 1 to 40 inclusive, if the weights can be placed in either 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.

Austran weights 

Thomas Digges' Pantometria

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.

See more pages from Thomas Digges' Pantometria 

Francesco Barozzi's Procli Diadochi

This is the frontispiece of Procli Diadochi by Francesco Barozzi, published in Venice, 1560. Barozzi (1537 - 1604) was a Venetian nobleman, a 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 Proclus Diadochus’ ( 411 - 485 ) edition of Euclid's Elements. The portrait depicts Barozzi.


Francesco Barozzi's Procli Diadochi

Van Heuraet's Rectification of Curves

These pages are from 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, y² = x³/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.


Van Heuraet's Rectification of Curves

Zhoubi suanjing

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

Zhoubi suanjing

Christopher Clavius's Edition of Euclid's Elements

Euclid’s propositions I-46 and I-47 as given in 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.


Christopher Clavius's edition of Euclid's Elements

Jan de Witt's Elements of Curves

This image is page 263 of 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 this page, 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.


More pages from Jan de Witt's Elements of Curves 

Francisco Feliciano's Libro di Arithmetica

This page is from 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 this page, 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.

More pages from Francisco Feliciano's Libro di Arithmetica

Johann Boschenstein's Rechenbuch

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 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.


Johann Boschenstein's Rechenbuch

Christian Wolff's Treatise of Algebra

Pages 202-203 (click to enlarge) of Christian Wolff's Treatise of Algebra. On these pages, 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.


Christian Wolff's Treatise of Algebra


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 on 1765. Read more. 

Michael Stifel's Arithmetica Integra

This diagram from Michael Stifel's Arithmetica Integra represents the solution to the pair of simultaneous equations

x² + y² - (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 x². 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 y², is labeled 78 + B - z, that is 78 + (x + y) - x². 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 B². It follows that B = 12. Therefore the larger square has area 90 - x², 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 - x²):27 = 27:x². It follows that 90x² - x⁴ = 729. So x² = 9 and x = 3. Then y = 9 and the problem is solved.

Michael Stifel's Arithmetica Integra


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.

Richard Sault's New Treatise of Algebra

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.  TheTreatise of Algebra was published as an appendix to William Leybourne's Pleasure with Profit, and included a chapter by Joseph Raphson on converging series.

This image is page 19. Notice that 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.

Richard Sault's New Treatise of Algebra

Simon Stevin's Oeuvres Mathematiques

This is the title page of the Oeuvres Mathematiques of Simon Stevin (1548-1620), edited by Albert Girard (1595 - 1632) and published in 1634. More pages are available on MathDL.

Simon Stevin's Oeuvres Mathematiques

Gerolamo Cardano's Practica Arithmetice

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.


More pages: Gerolamo Cardano's Practica Arithmetice

John Ward's Compendium of Algebra

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 1703.  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.


More pages: John Ward's Compendium of Algebra

Gemma Frisius's Arithmeticae Methodus Facilis

An example of the use of double false position to solve a problem in two unknowns found in 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.  


Gemma Frisius's Arithmeticae Methodus Facilis

Antichissimo di Algorismo

One of two illustrations 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”. A special feature of this codex is that it contains 42 illustrations, many of which supplement problems. 

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. 


Antichissimo di Algorismo

German protractor

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


German protractor

Simon Jacob's Rechenbuch

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.  


Simon Jacob's Rechenbuch

English tally sticks

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, wooden sticks were 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. 

English tally sticks

Korean Sangi rods

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.

Korean Sangi rods

Italian armillary sphere

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.


Italian armillary sphere

Italian astrolabe

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 (click image to enlarge) exhibits a high level of workmanship.


Italian astrolabe

Oronce Fine's Le Sphere du Monde

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 Sorobosco; compiled encyclopedic texts on mathematics; and developed astronomical measuring instruments. 


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


Oronce Fine's Le Sphere du Monde 

Legendre's Elements of Geometry

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.


The image above (click to enlarge) is of pages 106 and 107 of Legendre’s Elements of Geometry and 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.

Legendre's Elements of Geometry 

Leonhard Zubler's Nova Instrumentum Geometricum

Leonhard Zubler wrote Nova Instrumentum Geometricum in 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.


The image above is page 23 of Zubler’s Instrumentum and demonstrates a triangulation technique for obtaining the distance to a fortress.


Leonhard Zubler's Nova Instrumentum Geometricum 

Desargues' Sundials

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.


Desargues' Sundials

Kepler's Logarithms

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.


Kepler's Logarithms

Johann Kepler's Uralten Messekunst Archimedes

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.

Johann Kepler's Uralten Messekunst Archimedes