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Convergence

A woman, the Muse of Geometry, instructs a group of stonemasons. Welcome to Convergence, MAA's refereed online journal where history, mathematics, and teaching meet! Since 2004, Convergence has offered a wealth of resources to help instructors enrich their students’ learning experiences by teaching mathematics using its history. Make Convergence your source for math history and its use in teaching!

Mathematical Treasures

Completing the square in a manuscript copy of Al-Khwarizmi's 9th century algebra text

Finding heights in Galileo's Geometrical Compass (1640)

Title page of Galileo's Geometrical Compass (1640)

Spiral from Albrecht Durer's geometry book (1538)

Ellipse from Albrecht Durer's geometry book (1538)

Title page of Cardano's Ars Magna (1545)

From a 1650 copy of the 12th century Lilavati of Bhaskara II

Title page of Simon Jacob's Rechenbuch (1565 edition)

Finger counting in Luca Pacioli's Summa de arithmetica (1494)

"Allegory of Arithmetic" from Gregor Reisch's Margarita Philosophica (1503)

English tally stick from 1296 is marked William de Costello, Sheriff of London.

This large English tally stick was used as a receipt.

Notched wooden English tally sticks served as receipts.

German brass protractor with Baroque decoration from about 1700
 

Nested Austrian weights are elaborately decorated and marked 1787.

This Italian astrolabe was constructed by Bernard Sabeus of Padua in 1558.

Da Vinci's stellated dodecahedron in Pacioli's Divina proportione (1509)

Completing the square in a manuscript copy of Al-Khwarizmi's 9th century algebra text

Thomas Digges used a quadrant to measure the height of a tower in his Pantometria (1571).

Peter Apian's A Geographical Introduction (1534) applied geometry and trigonometry.

Peter Apian's device for computing sines in his Instrumentum sinuum (1534)

Peacock and snake problem from 1650 copy of the 12th century Lilavati of Bhaskara II

Kepler's sketch of the apparent retrograde motion of Mars in Astronomia Nova (1609)

Kepler compared planetary motion per Ptolemy, Copernicus, Brahe in Astronomia Nova (1609).

Title page of Johannes Kepler's Astronomia Nova (1609) based on Brahe's data

Notched wooden tally sticks used to record financial transactions in England (1296)

Late 19th century Korean wooden computing rods or sangi

This 15th century Italian compass and ruler converted Roman units to other units in use at the time.

Italian armillary sphere (1550): mechanical model of universe showing circular orbits of planets

Featured Items

Historically Speaking: 4. George Washington and Mathematics Education

An early American mathematics textbook was brought to the attention of the first President of the United States.
Quotations in Context: Fermat

The origin and context of a quotation about mathematics by Pierre de Fermat.
Quotations in Context: Germain

The origin and context of a quotation about mathematics by Sophie Germain.
Quotations in Context: Bowditch

The origin and context of a quotation about mathematics by Nathaniel Bowditch.
Geometría Maya en la Sala de Clases

Cómo los mayas clásicos probablemente usaban cuerdas anudadas para formar formas geométricas deseadas en el arte y la arquitectura.
A Compact Introduction to a Generalized Extreme Value Theorem: A Mini-Primary Source Project for Topology Students

A classroom-ready project for topology students, based on a 1904 paper by Maurice Fréchet.
HOM SIGMAA 2024 Student Paper Contest Winners

Download the winning papers from the 2024 contest.
Solving First-Order Linear Differential Equations: Three Mini-Primary Source Projects for Differential Equations Students

A set of three mini-Primary Source Projects based on works by Leibniz, Johann Bernoulli and Euler, for use in Differential Equations.
Historically Speaking: 3. “Large” Roman Numerals

Phillip S. Jones offers “a few comments on Roman numerals, especially devices for writing large numbers with them.” Convergence co-founder Victor J. Katz responds to the column, especially the author's claim that the Romans “contributed little to the development of real mathematics.”
Three Hundred Years of Helping Others: Maria Gaetana Agnesi on the Product Rule – A Mini-Primary Source Project for Calculus 1 Students

A student project based on the original writings of Maria Gaetana Agnesi.
Pitfalls and Potential Solutions to Your Primary Source Problems

Suggestions for dealing with difficulties that can arise when an instructor brings primary sources into the mathematics classroom.
Primary Source Projects and Reading Apprenticeship in Mathematics History

A classroom testimonial on the effect of combining primary source-based materials designed to teach mathematics with a framework designed to engage students as readers.

 

 
Lagrange’s Work on Wilson’s Theorem: Three Mini-Primary Source Projects for Number Theory Students

A set of three student projects for number theory based on Lagrange’s original paper on Wilson’s Theorem.
Keys to Mathematical Treasure Chests: Classroom Slide Rules

A discussion of slide rules used in classrooms for whole-class demonstration and by individual students, with pointers to online collections databases containing more information.