History of Mathematics
for PS421
Maurice OReilly
CASTeL, Dept of Mathematics,
St Patrick’s College Drumcondra
History of Mathematics for PS421
20/4/2012
The plan
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2.
3.
4.
5.
6.
7.
HoM & me
History in Maths Education
HoM Timeline & Themes
HoM Resources
HoM & Ireland
HoM Issues
Conclusion
History of Mathematics for PS421
20/4/2012
1. HoM & me
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Gerdes on ethnomathematics, c. 1992
Baghdad & Monasterboice, 1997
IMTA in Dundalk, 1996-7
Computus (Ó Cróinín, McCarthy)
Stevick on bookarts
ICMI & ICME, Monterrey, Mexico, 2008
van Maanen at MEI 3, Dublin, 2009
HPM ESU 6, Vienna, 2010
WG12, CERME7, Rzeszów, 2011
PS421, 2012
History of Mathematics for PS421
20/4/2012
Gerdes on ethnomathematics, c. 1992
History of Mathematics for PS421
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Baghdad & Monasterboice, 1997
History of Mathematics for PS421
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Where do the history of maths and the curriculum meet?
NUMBERS
ALGEBRA
PROBABILITY
GEOMETRY
CALCULUS
FUNCTIONS
IMTA in Dundalk, 1996-7
History of Mathematics for PS421
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Computus (Ó Cróinín, McCarthy)
History of Mathematics for PS421
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Stevick on Early Irish & English Bookarts
History of Mathematics for PS421
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ICMI & ICME, Monterrey, Mexico, 2008
History of Mathematics for PS421
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x² + 10x = 39
van Maanen at MEI 3, Dublin, 2009
History of Mathematics for PS421
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HPM ESU 6, Vienna, 2010
History of Mathematics for PS421
20/4/2012
WG12, CERME7, Rzeszów, 2011
History of Mathematics for PS421
20/4/2012
PS421, DCU 2012
History of Mathematics for PS421
20/4/2012
The plan
1.
2.
3.
4.
5.
6.
7.
HoM & me
History in Maths Education
HoM Timeline & Themes
HoM Resources
HoM & Ireland
HoM Issues
Conclusion
History of Mathematics for PS421
20/4/2012
2. History in Maths Education
• International setting
• International Commission on Mathematical
Instruction (ICMI)
• History and Pedagogy of Mathematics (HPM)
• ICMI Study: History in Mathematics Education
• European Summer School (ESU) on History and
Epistemology in Mathematics Education
• Congress of the European Society for Research in
Mathematics Education (CERME)
• Why history?
History of Mathematics for PS421
20/4/2012
A view of the international setting
History of Mathematics for PS421
20/4/2012
International Commission on Mathematical Instruction
(ICMI)
at ICM, Rome 1908
by Felix Klein
reconstituted/reborn 1952
first ICME, Lyon 1969
History of Mathematics for PS421
20/4/2012
International Study Group on the Relations between
the History and Pedagogy of Mathematics (HPM)
Not:
but …
History of Mathematics for PS421
20/4/2012
•formally affiliated to ICMI at ICME 3, Karlsruhe 1976
•conferences:
satellites at ICME (starting ICME 5, Adelaide 1984)
European Summer Schools (ESU, since Montpellier 1993)
working groups at ICME & CERME
•10th ICMI Study devoted to H in ME
•HPM Newsletter – online
History of Mathematics for PS421
20/4/2012
10th ICMI Study
• Napier & Briggs meeting, Edinburgh 1616
• Does HoM have a role in ME?
• Relation between how students achieve
understanding and the historical
construction of mathematical thinking
• Integration of HoM into teaching
– including use of original sources
• Use of technology, e.g. GeoGebra
• HoM on the Web, & its use for ME
History of Mathematics for PS421
20/4/2012
Examples:
• Letter from Leibniz to Riccati, c. 1715
on 1-1+1-1+ … = ½
• Cayley’s 1878 paper on groups of order 6:
a,b: a² = b³ = 1, ab = ba
• Hamilton’s perception of multiplication of
complex numbers: z1z2
History of Mathematics for PS421
20/4/2012
ESU 6, Vienna 2010
• Uffe Janqvist
on history as tool or history as goal
• Michael Fried
on how can ME accommodate HoM
• Michael Glaubitz & James Kiernan
on using original sources
• Olivier Bruneau
on ICT & pedal curves, C 17th- 19th
• Man Keung Siu
on Sino-European dialogue, 1587 …
History of Mathematics for PS421
20/4/2012
WG12, CERME7, Rzeszów, 2011
History of Mathematics for PS421
20/4/2012
Why history?
“… mathematics, as persented to students, fails on
two major grounds which you’ve pointed out and
that it’s taught in a very dry formal way without any
explanation of the origins and the motivations of
where things are coming from. That is a terrible
mistake and shouldn’t happen, but people are
human.”
Michael Atiyah @ RIA
IMS Bulletin #58 (2006), p49
Consider student activities:
Women in Mathematics Day: Ireland
UL 27th April 2010
National Committee for Commemorative Plaques in Science and Technology
The plan
1.
2.
3.
4.
5.
6.
7.
HoM & me
History in Maths Education
HoM Timeline & Themes
HoM Resources
HoM & Ireland
HoM Issues
Conclusion
History of Mathematics for PS421
20/4/2012
3. HoM Timeline & Themes
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Beginnings
Greece, India, China
Arabic & Persian mathematics
Europe to early 16th century
Solving polynomials
Calculus & physics
Rigour & Foundations
Abstraction, specialisation & new synergies
History of Mathematics for PS421
20/4/2012
Beginnings
Plimpton 322
c. 1800 BCE
Mesopotamia
Cuneiform
Sexagesimal
Pythagorean triples
History of Mathematics for PS421
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Rhind papyrus
c. 1650 BCE by the scribe Ahmes
Hieratic Egyptian script
Calculations with fractions; areas & volumes
710 = 23+130
History of Mathematics for PS421
20/4/2012
“Pythagoras’ theorem”
from Euclid
Archimedes and 
using inscribed &
circumscribed polygons
GREECE
 from length, area or volume?
History of Mathematics for PS421
20/4/2012
Brahmagupta’s
theorem
(7th cent CE)
INDIA
Number system
with place values
and zero
Negative numbers
Trigonometry
|AF|=|FD|
History of Mathematics for PS421
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Jiuzhang Suanshu
or
The Nine Chapters on
the Mathematical Art
Variety of practical
problems with solutions
“Pythagoras”
500-200 BCE
c. 100 BCE
CHINA
History of Mathematics for PS421
20/4/2012
Al-Khwarizmi’s
al-jabr wa-l-muqabala
x² + 10x = 39
Textual (rather than symbolic), but ‘justified’ by geometry.
Specific examples, yet general approach: algorithm
Omar Khayyam’s solution of x³+cx=d
1048-1131 CE
The parabola x²=-√c y
intersects the semi-circle
(having radius dc) at B & D.
x=|BE|
is the solution of the cubic.
Europe in the Middle Ages
c. 500-1500
Trivium (logic, rhetoric & grammar)
Quadrivium (arithmetic, geometry, music & astronomy)
Boethius (480-524)
Isidore of Seville (560-636) – Etymologiae
Scotii
Gathering universal knowledge
Charlemange
Bede
Calendar studies important
Translations: geometry, number systems
Greek knowledge through Arab world, extended by them
Leonardo di Pisa (Fibonacci, c. 1170-1240)
Nicole Oresme (1320-1382) – kinematics, exponentials
Solution of cubic & quartic
1540-1640
Tartaglia (1499-1557) & Cardano (1501-1576)
Ars Magna, 1545
what about Fermat, Wallis, Pascal, …
and logarithms – who thought of them?
Algebrization of geometry
La Géometrie, 1637
Calculus …
Leibniz (1646-1716) & Newton (1642-1727)
1660-1780
Johann Bernoulli (1667-1748)
& l’Hôpital (1661-1704)
The priority dispute!!
what about other Bernoullis, Taylor, …
doubting Berkeley …
also d’Alembert & Agnesi?
… becomes analysis
Differential equations; books
Leibniz’ characteristic triangle
transmutation rule
and Gregory-Madhava-Leibniz series:
Political turmoil in France
1780-1820
Lagrange (1736-1813)
Theory of equations
Monge (1746-1818)
Analytic & differential geometry)
Laplace (1749-1827)
Statistical inference
Legendre (1752-1833)
Number theory
… but maths is healthy
Is England still in the races?
number theory, congruences, fundamental
theorem of algebra, non-Euclidean and
differential geometry, etc., etc.
FTA
w=z²+z+1
replace 2 by 3, 4, …
Complex numbers?
As r=|z| increases, the
locus of w must pass
through 0, and so …
The plan
1.
2.
3.
4.
5.
6.
7.
HoM & me
History in Maths Education
HoM Timeline & Themes
HoM Resources
HoM & Ireland
HoM Issues
Conclusion
History of Mathematics for PS421
20/4/2012
4. HoM Resources
• Original sources
• HoM books
• The Web
History of Mathematics for PS421
20/4/2012
The plan
1.
2.
3.
4.
5.
6.
7.
HoM & me
History in Maths Education
HoM Timeline & Themes
HoM Resources
HoM & Ireland
HoM Issues
Conclusion
History of Mathematics for PS421
20/4/2012
5. HoM in Ireland
• Early medieval Ireland
• Irish mathematicians
• Foreign mathematicians in Ireland
History of Mathematics for PS421
20/4/2012
Mathematics in Early
Medieval Ireland
Two areas: the Computus & Geometry
Insular manuscripts
Imago Leonis
Echternach Gospels f.75v
Fr Ingram lecture, IMTA, Carlow, 2000
St Matthew
Book of Moling , f.12v
DIT, February 2008
Carpet page (Luke)
Lindisfarne f.138v
Galway Computus conference, 2008
Hamilton in Cabra
plaque
Seán Keating’s version
tea @ Dunsink
Enormous insights:
Conical refraction
General theory of dynamics
Quaternions
graffiti
Casey in Reykjavík!?
New triangle geometry of:
•Lemoine
•Brocard
•Casey
1820-1891
Alastair Wood on Stokes
George Gabriel Stokes
1819 - 1903
An Irish Mathematical
Physicist,
IMS Bulletin #35
Boole in Cork
An investigation into the Laws
of Thought, on Which are
founded the Mathematical
Theories of Logic and
Probabilities (1854)
Alicia Boole Stott
worked on polytopes:
there are six regular polytopes on four dimensions
and that they are bounded by 5, 16 or 600 tetrahedra,
8 cubes, 24 octahedra or 120 dodecahedra.
Philip Boland on Gosset
A biographical
glimpse of
WS Gosset,
IMS Newsletter #8
The plan
1.
2.
3.
4.
5.
6.
7.
HoM & me
History in Maths Education
HoM Timeline & Themes
HoM Resources
HoM & Ireland
HoM Issues
Conclusion
History of Mathematics for PS421
20/4/2012
6. HoM Issues
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Proof & mathematical truth
Notation
Eurocentrism
War & maths
Women in maths
Maths & applications
History of Mathematics for PS421
20/4/2012
Eurocentrism
• Picture removed
(awaiting artist’s
permission)
Pythagoras’ theorem
Pascal’s triangle
Gaussian elimination
The History of Europe 2
Aidan Hickey
History of Mathematics for PS421
20/4/2012
“Since researching my essay on women mathematicians
my attitude towards mathematics has been altered.
Although of course these male mathematicians achieved
great things during the 16th & 17th century, I can’t help
but think about women such as Sophie Germain and
Maria Agnesi. These women had to fight so much harder
than their male counterparts to achieve or succeed in this
area. They did not receive the same respect or education
that they should have been entitled to.”
from (female) student’s learning journal
HoM module, spring 2010
The plan
1.
2.
3.
4.
5.
6.
7.
HoM & me
History in Maths Education
HoM Timeline & Themes
HoM Resources
HoM & Ireland
HoM Issues
Conclusion
History of Mathematics for PS421
20/4/2012
“Some treatment of the history of mathematics is
very important I think and part of that history is, of
course, talking about the people and where they
came from with their contribution and it also gives
you a chance to explain the motivational origins –
the roots if you want – and to follow these things
back into the past”
Michael Atiyah @ RIA
IMS Bulletin #58 (2006), p49
See the big picture with HoM!
History of Mathematics for PS421
20/4/2012
“As from his small window
the astronaut sees all that he has sprung from”
Alphabets by Seamus Heaney
History of Mathematics for PS421
20/4/2012