Mathematics around

the World
Lecture #11
Having studied in some detail the mathematics of China, India, the Islamic
world, and Europe up to about the year 1300, it is useful to compare the
mathematics known in these places at that time. We then consider the issue
of why it was that modern mathematics developed in Europe rather than
elsewhere. We can also ask the question as to what mathematical ideas were
known in other parts of the world at this time. This question is difficult to
answer in much detail, given the current state of our knowledge.
Nevertheless, we will present in the second half of this chapter a selection of
mathematical ideas that are known to have been developed in the Americas,
in sub-Saharan Africa, and in the Pacific.
 Common ides of Mathematics
In practical geometry—measuring fields, finding distances, calculating volumes—societies
in the study had similar techniques. They all calculated areas and volumes, used the
Pythagorean Theorem for right triangles, and had similar methods for determining the
height of distant towers.

For theoretical geometry, the Islamic world preserved and advanced classical Greek
geometry. They tackled questions about volumes of solids, locations of centers of gravity,
Euclid’s parallel postulate, and the separation of number and magnitude. The Greeks' idea
of proof from stated axioms was well understood.

While Europe always had Euclid's Elements, renewed interest emerged in the 14th century
due to translations in the 12th and 13th centuries. The idea of proof persisted, but there
was no new theoretical geometry. India and China, not exposed to classical Greek
geometry, had notions of proof. Chinese mathematicians used logical arguments, while
written derivations in India became more explicit from the 14th century onward.
By 1300, trigonometry was actively used in India, Islam, and Europe for studying the
heavens, although China seemed to lack it. In certain aspects of algebra, China
pioneered techniques for solving systems of linear equations and developed
root-finding methods using Pascal's triangle. India solved linear congruences using
the Euclidean algorithm and took pride in techniques for solving quadratic
indeterminate equations, known as Pell equations. Islamic mathematicians
extensively studied quadratic and cubic equations, developed numerical methods
for solving polynomial equations, and worked with the Pascal triangle.

While Islam excelled in algebra, Europe was just beginning to adopt algebraic
techniques around the 14th century, drawing heavily from Islamic work. European
algebra, like its Islamic counterpart, didn't consider negative numbers. India and
China, however, were comfortable using negative quantities in calculations. Europe
uniquely explored the mathematical concepts of motion, delving into instantaneous
velocity and developing the mean speed rule, laying the groundwork for calculus
three centuries later. In the 14th century, India also started exploring infinitesimally
small quantities in connection with calculus.
Possible transmissions of ideas
Around the 14th century, the level of mathematics in China, India, the
Islamic world, and Europe was comparable. While each culture had specific
techniques, many mathematical ideas and methods were shared among
them. The question arises whether these ideas developed independently or
through transmission. Clear lines of transmission exist, such as
trigonometry moving from Greece to India to Islam and back to Europe.
The decimal place value system also moved from China or India to
Baghdad and then to Europe. However, for other ideas, like the Pascal
triangle or methods of determining heights and distances, the transmission
is less clear. For instance, the tangent function, crucial in trigonometry,
appeared in China in the 8th century and later in Islam. It reached Europe
in the 12th century, but its path is uncertain. Similarly, the Pascal triangle
showed up in Islam and China. Possible transmission routes include
contact along the silk route or scholars traveling between the Islamic
world and Europe.
Methods like the basic sighting technique for determining
heights and distances, as well as recreational problems,
appeared in China and later in Europe. Speculation surrounds
whether these ideas traveled, potentially through the silk route
or specific trade routes like those of the Radhanites, a group of
Jewish merchants. For instance, the Jacob’s Staff, a surveying
device described in Europe by Levi ben Gerson in the 14th
century, was available in China by the 11th century. It's unclear
if this device was carried from China by Jewish merchants.
While many questions of transmission remain speculative due
to the lack of documentation, it's evident that common
mathematical ideas were adapted to meet the specific needs of
each civilization.
Mathematics in America,
Africa and Pacific
 The Mayans

We'll start in the Americas with the Mayans, known for their
mathematics due to their written language. Flourishing between the
3rd and 9th centuries in southern Mexico, Guatemala, Belize, and
Honduras, the Mayans had a priestly class studying math and
astronomy, maintaining the calendar. Unfortunately, the Spanish
conquerors destroyed many Mayan documents, and deciphering the
remaining ones, like the Dresden Codex, has been a slow process.
The Mayans still exist today, with around two and a half million
speakers of Mayan languages. Scholars understand the basics of the
Mayan calendar and numeration systems, but the methods behind
their calculations remain speculative due to the lack of records.
 The Incas
About 2000 miles south of the Mayan civilization was the Inca civilization, thriving in
what is now Peru from about 1400 to 1560. Unlike the Mayans, the Incas did not have a
written language, but they used a unique method for recording information: quipus.
Quipus were collections of colored knotted cords, serving as a means of communication
for the Inca leadership. Messages about needed items, owed taxes, or required workers
for public projects were encoded on quipus and sent through a network of runners.

Each quipu had a main cord with pendant cords attached, and sometimes a top cord. The
color, placement, knots, and spaces between knots on these cords conveyed the
recorded data. The knots were grouped and spaced to represent numbers, using a
base-10 place value system with the highest value closest to the main cord. For example,
a cord with three knots near the top and nine knots near the bottom represented the
number 39. The largest number found on a quipu is 97,357. Zeros were represented by a
wide space between knots.
The pendant cords on quipus are grouped and often have distinct colors, assumed to
represent different types of data. There is usually a top cord associated with a group
that records the sum of the numbers on the individual cords. Quipus are records, not
calculating tools; the actual calculations were likely done elsewhere, possibly with
counting boards.

The specific data recorded on a quipu is not always known, but one example is a
quipu documenting census data for seven provinces. People in each province were
classified into two groups, further divided into subgroups. Cords on the quipu
recorded the number of households in each province belonging to each subgroup,
with other cords representing sums of this data and one cord giving the grand total
for the entire region.

In modern terms, a quipu can be viewed as a type of graph known as a tree. Quipu
makers likely dealt with questions related to tree structures, such as how many
different trees could be constructed with a given number of edges. Since Inca officials
associated numbers and colors with each edge of the tree, designing these quipus to
be useful involved non-trivial questions.
 The North American Indians
In the Inca civilization, as in the Mayan, there was a professional class of
"mathematicians" who dealt with the mathematics of the culture regularly.
In other cultures to be discussed, such a class did not exist, but there
were mathematical aspects in their daily lives recognized today as
"ethnomathematics." The Anasazi, a pre-Columbian civilization in the Four
Corners area of the Southwest (600 BCE to around 1300 CE), displayed
sophisticated knowledge through the alignment of structures to important
astronomical events like solstices and the moon's 18.6-year cycle.
Although no documented records of their mathematics exist,
archaeological remnants suggest they aligned major buildings with the
cardinal directions.
For example, the Anasazi aligned roads and buildings with the four
cardinal directions, emphasizing the importance of these directions
in their religion. Casa Rinconada, a ceremonial structure in Chaco
Canyon, featured a 63-foot diameter circle aligned with cardinal
directions. Speculating on how they determined true north, one
possibility is using techniques similar to Roman surveyors, drawing
a circle centered on a pole and recording the shadow's curve
throughout the day to establish east-west and north-south lines.

Other North American Indian civilizations, like the Mississippian in

East St. Louis, Illinois, and Plains Indians with structures like the
Bighorn Medicine Wheel in Wyoming, displayed knowledge of
astronomy and geometry through carefully aligned structures and
urban planning. These examples highlight the importance of
mathematical ideas in various cultures, even when not explicitly
classified as such.
 Sub-Saharan Africa
In most African cultures of the past, especially those without written
records, it's challenging for historians to pinpoint when and how
mathematical ideas developed. Great Zimbabwe, a twelfth-century
stone complex, showcases the need for mathematics in administrative,
engineering, trade, tax, and calendar aspects. West African states like
Ghana, Mali, and Songhai likely incorporated mathematics into their
bureaucracies. The influence of Islam in West Africa might have
introduced Islamic mathematics, but direct information on
mathematics or mathematicians from this time and place is scarce.
Ethnographers' reports from the 19th and 20th centuries
provide insights into African civilizations. The Bushoong
and Tshokwe cultures in Zaire and northeastern Angola
show early awareness of graph theoretical ideas, tracing
figures continuously without lifting a finger. The Tshokwe
use this for storytelling, with dots representing humans and
detailed drawing rules for constructing curves. Geometric
patterns are another prevalent mathematical idea, seen in
cloth weaving and decorative metalwork across Africa.
Mathematical games and puzzles, like the board game wari,
omweso, and mankala, are played throughout Africa,
teaching children counting and strategy. The puzzle story
of transporting objects across a river occurs in various
African cultures, similar to the eighth-century European
puzzle known as Alcuin's Propositiones.
 The South Pacific
In most African cultures of the past, especially those without written
records, it's challenging for historians to pinpoint when and how
mathematical ideas developed. Great Zimbabwe, a twelfth-century
stone complex, showcases the need for mathematics in administrative,
engineering, trade, tax, and calendar aspects. West African states like
Ghana, Mali, and Songhai likely incorporated mathematics into their
bureaucracies. The influence of Islam in West Africa might have
introduced Islamic mathematics, but direct information on
mathematics or mathematicians from this time and place is scarce.
In the Marshall Islands of the South Pacific, stick charts are part of the
navigation tradition. These charts, some with idealized shapes, serve
to train navigators in understanding wave motion and its interaction
with land masses. They also function as maps of the archipelago,
representing essential elements for sailing between islands. In Bali,
Indonesia, there's a unique calendar based on cycles of arbitrary
lengths. The Balinese use a wooden board called the tika, marked with
symbols, to perform calendrical calculations. These examples from
ethnomathematics highlight the presence of logical thought and
pattern analysis, demonstrating that mathematics is a universal force
in societies worldwide.