Len Berggren


W ELCOME  and thanks for visiting my website! My academic activities center on the history of mathematics, so included here is information related to my work in this area. My research interests in the history of mathematics include ancient Greece and medieval Islam, as well as the history of such scientific instruments as the sundial and the astrolabe. Sundials are a particular interest of mine so they have their own section here.

I retired from the Department of Mathematics at SFU in 2006 and am no longer taking students. The courses that I taught there include calculus, geometry, linear algebra, history of mathematics, and mathematics in science and civilization.

History of Math

Books Published

Episodes in the Mathematics of Medieval Islam (2nd Edition)

Episodes in the Mathematics of Medieval Islam (2nd Edition)

J.L. Berggren. Berlin, Heidelberg, New York: Springer, 2016.

This book, due to be published in 2017, contains considerable additional material on each of the topics covered in the first edition. It also contains a new chapter on number theory, magic squares and permutations and combinations in medieval Islam. The coverage of mathematics in North Africa and medieval Spain has been greatly expanded over that in the first edition. The book is illustrated with diagrams and color plates.

Sourcebook in the Mathematics of Medieval Europe and North Africa

Sourcebook in the Mathematics of Medieval Europe and North Africa

Editor: Victor J. Katz; Contributors: Menso Folkerts, Barnabas Hughes, Roi Wagner & J.L. Berggren. Princeton University Press, 2016.

In some ways a complement to the Sourcebook listed below, this work makes available translations of texts dealing with arithmetic, algebra, geometry, trigonometry and other parts of mathematics from western Christian, Jewish and Muslim societies and shows how important mathematics developed in, and passed among, these three cultures.
“This book provides ... English translations of key mathematical texts from medieval Western Europe and North Africa, all originally written in one of the three scientific languages of that time: Latin, Hebrew, and Arabic. Until now, there has been no sourcebook that deals with Hebrew mathematics as such, or the mathematics of the Muslim West. This volume fills that gap and brings the field an important step forward.”
– Pascal Crozet, Université Paris Diderot

The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook

The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook

J.L. Berggren. Section "Mathematics in Medieval Islam" in The Mathematics of Egypt, Mesopotamia, China, India and Islam: A Sourcebook. (V. Katz, ed.) Princeton, NJ: Princeton University Press, 2007, pp. 515 - 675.

This Sourcebook presents English translations of original mathematical texts from the five ancient societies listed in the title. Almost everyone knows the names of Euclid and Archimedes, but few could name one mathematician from China, or recognized the names Brahmagupta or al-Khwarizmi. This book goes some way to set the historical record of mathematical achievement straight and shows that, in the words of the 7th-century Bishop, Severus Sebokht, “there are others who know something, not only Greeks but men of another language!”
“[This book] is notable for the wide variety of sources, which will challenge preconceptions everywhere, as well as for the clarity and force of the introductions to the mathematical cultures on display. The authors are leading authorities in their subjects and the scholarship is of the highest order.”
– Jeremy Gray, The Open University

Geminos's Introduction to the Phenomena: A Translation and Study of a Hellenistic Survey of Astronomy

Geminos's Introduction to the Phenomena: A Translation and Study of a Hellenistic Survey of Astronomy

J.C. Evans and J.L. Berggren. Princeton University Press, 2006.

Geminos’s Introduction to the Phenomena is a textbook written for the beginning student of astronomy by a talented Greek writer in the first century BC. Among the topics treated are the zodiac, the constellations, the celestial sphere, the succession of lunar phases, the length of the lunar month, eclipses and the motions of the five planets visible to the naked eye. This first English translation and study of Geminos’s work is illustrated with numerous diagrams and photographs related to the topics that Geminos discusses.
“Everything about this book is excellent. The translation is a model of clarity and the notes to each section explain the text in detail, both technically and historically ... Evans and Berggren have done such an excellent job in every way, their knowledge of the subject is so complete, that I can only admire their work and recommend it as exemplary.”
- Noel Swerdlow, University of Chicago

Episodes in the Mathematics of Medieval Islam

Episodes in the Mathematics of Medieval Islam (1st Edition)

J.L. Berggren. Berlin, Heidelberg, New York: Springer, 1986 (softcover edition 2003). A revised version of this work has been published in German, in 2011, as Mathematik im mittelalterlichen Islam (Springer).

This book presents specific examples of mathematics translated from Arabic texts in their historical context. The reader will learn how mathematicians and astronomers in the medieval Islamic world transformed the arithmetic, algebra and trigonometry of the ancient world to create much of the mathematics taught in schools today. The reader will also see some beautiful geometric results and learn of how the first analogue computer – the astrolabe – was used to solve important problems in astronomy.

The book is illustrated with numerous diagrams and photographs.
“This book is, in spite of the author’s more modest claims, an introductory survey of the main developments in those disciplines which were particularly important in medieval Islamic mathematics ... No knowledge of mathematics ... beyond normal high-school level is presupposed and everything beyond that ... is explained carefully and clearly.”
- Jens Høyrup, Mathematical Reviews

From Ancient Omens to Statistical Mechanics: Essays on the Exact Sciences Presented to Asger Aaboe

From Ancient Omens to Statistical Mechanics: Essays on the Exact Sciences Presented to Asger Aaboe

J.L. Berggren and B.R. Goldstein, Editors. Acta Historica Scientiarum Naturalium et Medicinalium. Vol. 39. Copenhagen: University Library, 1987.

Out of Print.

Euclid's Phaenomena: A Translation and Study of a Hellenistic Work in Spherical Astronomy

Euclid's Phaenomena: A Translation and Study of a Hellenistic Work in Spherical Astronomy

J.L. Berggren and R.S.D. Thomas. Sources and Studies in Hellenistic Exact Sciences New York: Garland Publishing, 1996.
Reprinted in 2006 in a softcover edition by the American Mathematical Society and the London Mathematical Society as Vol. 29 in the series History of Mathematics: Sources.

Most are unaware that Euclid, famous for his Geometry, also wrote what is arguably the first book on mathematical astronomy – ironically dealing with the non-euclidean geometry of circles on the sphere. With no numerical computations or formulas, and using only the geometry of circles on the sphere, he made a comparative study of the length of daylight for different days at different places on the earth’s surface. A historical and mathematical commentary by the authors makes it possible to understand Euclid’s arguments without any prior knowledge of astronomy or advanced mathematics. This is the first English translation of this historic work.
“The edition and translation are beautifully done ... The notes and introduction are very helpful. Anyone interested in the history of Greek science will want to have a copy. So will anyone who is curious about the history of how the Greeks deployed mathematical models to understand astronomical phenomena.”
- Fernando Q. Gouvêa, Mathematical Association of America Reviews

Ptolemy's Geography

Ptolemy's Geography, An Annotated Translation of the Theoretical Chapters

J.L. Berggren and Alexander Jones. Princeton University Press (Hardcover edition: 2000; softcover edition 2002.)

Ptolemy’s Geography is the only work on cartography to have survived from the ancient world. This book, which introduced the practice of using latitude and longitude as a basis for map-making, is one of the most important works ever written on that topic, and its methods inspired the cartographers of the Renaissance. The present volume provides the only reliable English translation of the parts of Ptolemy’s work dealing directly with creating maps. It also explains the mathematics and the historical context of the Geography and is illustrated with numerous diagrams, plates and maps.
“Berggren and Jones have made a difficult, detailed and challenging text far more accessible and have produced a work that will be the standard for many years to come.”
– Liba Taub, Cambridge University

Pi: A Source Book

Pi: A Source Book (3rd Edition)

J.L. Berggren, J. Borwein and P. Borwein. New York: Springer, 2004.

INTRO VIDEO  (57 min)
This book provides English translations of important sources from all over the world concerning what is perhaps the most famous number known, 3.1415... expressing the ratio of the circumference of a circle to its diameter. With selections from mathematical works, amusing stories and even a song this book carries the story of pi from ancient Egypt up to the present.
“...admirably designed to cater for a broad spectrum of tastes: professional mathematicians , , ,, historians of mathematics, teachers at all levels searching out material for individual talks and student projects, and amateurs who will find much to amuse and inform them in this leafy tome. The authors are to be congratulated...”
- Roger Webster, Bulletin of the London Mathematical Society

Refereed Journal Articles and Conference Proceedings

Invited Articles


Encyclopedia Articles

Current Scholarly Projects

I am engaged in ongoing research centered on the translation and study of ancient Greek and medieval Arabic mathematical texts dealing with geometry, mathematical geography and astronomy. This includes studies of the mathematical ideas in these texts and their historical, scientific and social contexts.

Some of these researches will be presented in the paper “Al-Bīrīnī’s Mappings of the Heavens and the Earth” which I will present at the International Congress for the History of Science and Technology” in Rio de Janeiro in July 2017.

Recent Lectures


equatorial sundial in front of the Trottier Family Observatory at SFU

Sundial Publications

The following articles were published in The Compendium, which is the official publication of the North American Sundial Society.

Sundial Lectures

My published lectures on this topic may be accessed in the previous section. My unpublished lectures include:

Sundial Examples

Dr. Berggren demonstrating how to read time using the analemmatical sundial at SFU.
Dr. Berggren demonstrating how to read time using the analemmatical sundial at SFU.
Dr. Berggren with equatorial sundial in front of the Trottier Family Observatory at SFU.
Dr. Berggren with equatorial sundial in front of the Trottier Family Observatory at SFU.

Media Coverage

The Sky's the Limit

The Sky's the Limit at SFU

Written by Wanda Chow. Photo by Mario Bartel. Excerpt from Burnaby Now, April 23, 2015

...a nod to decidedly older technology [is] a sundial, designed by an SFU math professor and built by students from BCIT's mechanical technology program. It shows local clock time as well as sun time, which does not recognize time zones or daylight savings time changeovers. [Source Article]

Telling Time the Old Fashioned Way

Telling Time the Old Fashioned Way

By Diane Luckow. Reprinted from SFU News, Oct 30, 2003, Vol. 28, No. 5

An analemmatical sundial - the first [split analemmatical dial] in Canada – can now be found at the northeastern end of "C" parking lot, painted on the pavement. An ancient form of telling time, the first analemmatical sundial appeared in France in the early 17th century at the church of Brou.

SFU’s sundial features two analemmas (the large figure eight’s painted on the pavement). The analemma on the west is used for morning hours, the other for afternoon hours.

To tell clock time select the appropriate analemma for morning or afternoon and find the date position on its curved boundary. The first of each month is shown as a white dot; thus Sept. 15 is half way between Sept. 1 and Oct. 1.

The hours are shown on the dial beside the large yellow circles, labelled in blue for Pacific standard time and red for Pacific daylight savings (PST) time – one hour later of PST.

Someone telling time on a sunny day must face away from the sun on the coloured analemma with the date centered between their feet. The approximate mean time will be shown by the person’s shadow on the dial.

“It’s quite accurate,” says mathematics professor Len Berggren, who collaborated on the dial with North Vancouver retired engineer and sundial enthusiast Brian Albinson.  “The average absolute error in the morning is 23 seconds and in the afternoon, 54 seconds.”

CBC Radio, Ideas Program, The Path of Knowledge

CBC Radio, Ideas Program, The Path of Knowledge

November 9, 2004: Part 1

November 16, 2004: Part 2

Commanded by the Qur’an to seek knowledge and examine nature for signs of the Creator, the Islamic world was synonymous with learning and science for five-hundred years. In the twenty-first century, the relationship between science and religion generates much debate among Muslims. Chris Tenove asks if there is there a contradiction between Islam and modern science.

Ptolemy Translated to English

Ptolemy Translated to English

By Carol Thorbes. Reprinted from SFU News, March 22, 2001 Vol. 20, No. 6

A groundbreaking treatise on cartography that's been Greek to many map buffs for centuries is now making sense, thanks to two scholars' knowledge of the ancient language and ancient mathematics.

Len Berggren, the chair of Simon Fraser University's mathematics department, and Alexander Jones, a University of Toronto classicist, are the authors of the first complete and reliable English translation of Ptolemy's Geography.

"Up until now most of the book, which is Ptolemy's exposition of the principles of map making, has only been available in Greek, Latin and Arabic," explains Berggren, who reads Greek and Arabic. Originally written in Greek by the Alexandrian astronomer and geographer Claudius Ptolemy in the second century A.D., Ptolemy's Geography is the only surviving work on cartography from classical antiquity.

After its translation into Latin around 1410 A.D., the book became a cartographer's bible. For the next two centuries, cartographers and explorers regarded it as the best source of map-making techniques and depictions of the world's geography, including its curvature.

Berggren, a historian of ancient mathematics, says the English translation, which took him and Jones 15 years to complete, will help popularize Ptolemy's farsightedness.

"Ptolemy's Geography introduced the use of longitudinal and latitudinal coordinates to map the world," says Berggren. "His technique enabled map makers to make their own maps using only the text of the Geography. For 15 centuries, Ptolemy's Geography represented the most thorough discussion of the importance of relying on astronomical observation and applied mathematics to determine location," says Berggren.

However, Ptolemy's cartography had its flaws. Berggren and Jones point out several in their in-depth analysis of the geographer's work.

"When Christopher Columbus landed on the shores of the Americas in 1492 he mistook them for the shores of Asia because he had relied on Ptolemy's book, which seriously shortened the distance between Europe and Asia," notes Berggren.

He and Jones illustrate Ptolemy's view of the world by comparing colour reproductions of the geographer's original maps to present day maps. The two also trace the evolution of Ptolemy's maps in the hands of latter day mapmakers.

Meredith, Berggren win President's Media Award

Meredith, Berggren win President's Media Award

By Marianne Meadahl. Excerpt from SFU News, Feb. 8, 2001 Vol. 20, No. 3

Two SFU professors are being honoured for their part in making SFU a better known institution.

Lindsay Meredith, associate dean of business and Len Berggren, chair of mathematics and statistics are the recipients of the annual President's award for service in media and public relations.

The award committee unanimously chose to honour Meredith for his extensive work with the media and Berggren for his longstanding role in community relations. They'll both be recognized at a reception and at SFU's annual award ceremony Feb. 16. [Click here for section on Dr. Meredith.]

Like Meredith, Berggren is no stranger to an audience, whether the crowd is a group of stroke recovery patients or a mensa gathering. For the past three decades he's been one of the most available speakers involved with SFU's speakers' bureau and has given more than 70 talks.

An expert on the history of mathematical sciences in ancient Greece and medieval Islam, he has developed a wide range of interests – from time measurement to the politics of Cyprus – as well as a commitment to sharing them.

"The speakers' bureau is a great opportunity for the university and for those of us who are passionate about our work," says Berggren (right), who is also a regular in local schools, where he addresses student and teacher groups.

Finding time to prepare and deliver talks isn't always easy. Berggren says it's a matter of commitment. "You have to decide it's a priority," he says. "It's one way a university researcher can help the community. It's public education of your work."

Berggren maps out his thoughts in advance. "I wish I could speak more off the cuff, but I need to be well-prepared," he admits. "I'm always redoing my talks because they reflect my research interests, and those are always developing."

Berggren has also had his share of the spotlight. Called on to demystify concepts like leap year and the start of the true millennium, he faces the task of explaining difficult concepts in short time frames.

"I'm still a bit nervous," he concedes. "It's hard to choose what honestly reflects what you know without unnecessary complexity. It's something you learn as you do it."

Besides appearing on local radio and in print, Berggren's international media exposure includes the BBC in Scotland and the New York Times.

"I think journalists are appreciative of our expertise," says Berggren. "They are simply trying to tell a story. If we can make it a better one then everybody wins."



Academic Administration

Sundial Design and Construction


John Lennart Berggren

Professor Emeritus, Department of Mathematics, Simon Fraser University

Educational Background

Affiliations with Academic Institutions

Period Position Department Institution
2006 - present Professor Emeritus Mathematics Simon Fraser University
2006 - 2016 Committee Member Academic Standards Alexander College
1984 - 2006 Professor Mathematics and Statistics Simon Fraser University
1992 Visiting Scholar History of Science Harvard University
1990 - 1991 Visiting Scholar History of Science Harvard University
1973 - 1984 Associate Professor Mathematics and Statistics Simon Fraser University
Sept. 1975 - Dec. 1976 Visiting Fellow History of Medicine and Science Yale University
1963 - 1973 Assistant Professor Mathematics Simon Fraser University
Sept. 1972 - May 1973 Visiting Fellow History of Medicine and Science Yale University
Sept. 1968 - June 1969 Visiting Lecturer Mathematics University of Warwick

Selected Theses Supervised

Name Degree Title Completed
Underell, John M.A.L.S. An Enquiry into Human Creativity 2003
Sinclair, Nathalie M.Sc. Applications of Conic Sections in Ancient Mathematics 1995
Jeffries, William M.A.L.S. Extended Essays: "Claude Bernard"; "Cartesian Chalk Circle" 1995
Stanley, Daryn M.Sc. The Quadrant in Middle Ages and Renaissance 1994
Van Brummelen, Glen Ph.D. Mathematical Tables in Ptolemy's Almagest 1993
Vogt, David Ph.D. An Information Analysis of Great Plains Medicine Wheels 1990

Current Research Interests

My research centers around reading and translating ancient Greek and medieval Arabic mathematical texts dealing with geometry, mathematical cartography and astronomy, as well as studies of the mathematical ideas in these texts and their historical development.

I have also taught and done research in how the mathematical sciences have affected, and been affected by, human values and cultures including moral, religious and cognitive values.

Books Published

Please see this section.

Selected Refereed Publications

Please see this section.

Membership in the Academic Community


Other Interests


Dr. J. L. Berggren
Professor Emeritus
Department of Mathematics
Simon Fraser University
8888 University Drive
Burnaby, BC, V5A 1S6