Read about Mathematical Treasures in the December 2012/January 2013 issue of

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

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 ofhastasequal to half that difference.152: There is a hole at the foot of a pillar ninehastashigh, 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.

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.

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:

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

This is the title page of

See more pages from Thomas Digges'

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

These pages are from the brief work

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,

Van Heuraet's Rectification of Curves

This pages is from the

Zhoubi suanjing

Christopher Clavius's edition of Euclid's Elements

This image is page 263 of

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

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

More pages from Francisco Feliciano's Libro di Arithmetica

This is the title page of

Johann Boschenstein's Rechenbuch

Pages 202-203 (click to enlarge) of Christian Wolff's

Christian Wolff's

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.

This diagram from Michael Stifel's

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.

Michael Stifel's

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.

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.

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

More pages: Gerolamo Cardano's Practica Arithmetice

This is the title page of

More pages: John Ward's Compendium of Algebra

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

Gemma Frisius's Arithmeticae Methodus Facilis

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

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

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

German protractor

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

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.

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

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

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 (1494-1555) was a French mathematician and astronomer who served as the Chair of Mathematics at the CollÃ©ge 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

Oronce Fine's Le Sphere du Monde

In the year 1794, AndrÃ© Marie Legrendre (1752 - 1833) published his

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

Legendre's Elements of Geometry

Leonhard Zubler wrote

The image above is page 23 of Zubler’s

Leonhard Zubler's Nova Instrumentum Geometricum

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

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.

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