‘Eureka!’ presented by Gary Rubinstein
(garyrubinstein@yahoo.com)
I. Euclid III. 17, constructing a tangent to a circle
April 26, 2012
V. Geometric explanation of the 8 basic trig identities through a point outside the circle. (300 BC)
C G
C F
C
D
H
E
H D
A F B
A A
II. Euclid I. 47, proof of Pythagorean Theorem
E
C
D
C
I
C
D
H J
III. Archimedes approximates Pi (250 BC)
AO=.5
H
B
F
A
G
E
D
C
B
K
F
I
A
G
VI. Cardano (1545) solves the cubic. i is invented.
Find the exact answer to x
Let
x
 y

1
( y

1)
3 
3( y

1)
2 
6( y

1)

20

0 becomes ‘depressed cubic’
3
y

3
3 x
2 
6 x

9 y

12

20
which can be solved with Del Ferro process from 1515.

0
Bombelli ‘invents’ imaginary numbers to solve x
3 
15 x

4
VII. Descartes solves cubics and quartics with intersecting conics. (1650 AD)
x
4 
15 x
2 
10 x

24

0 p = -15.00
20
18
16
A O A O
14
12
10
8
6
4
2 Perimeter of hexagon is approximately 3.46410
Perimeter of 12-gon is approximately 3.21539
IV. Ptolemy calculates exact value of sine 18 (100 AD
AC=AE=1,

CAE

36

C
-15 -10 -5 5 10
-2
-4 VIII. Descartes finds tangent lines without Calculus
Find the slope of the line tangent to the parabola
y
 x 2 at (3,9)
14
12
10
8
A
F
E
6
4
2
-5 5

IX. Archimedes determines the formula for the volume of a sphere in ‘The Method’
H
A S
Q
O
C
M
Resources:
Online videos of this presentation -- www.garyrubinstein.com/nctm2012
My youtube channel with more videos -- www.youtube.com/nymathteacher
AIMS ‘Historical Connections in Mathematics, volumes I, II, and III’ Three activity books based on math history, mainly for middle school, but can be adapted for high school too.
Aaboe ‘Episodes from Early Mathematics’
Cardano, ‘The Great Art’ An English translation of the 1545 book, reprinted by Dover.
Dunham, ‘Journey Through Genius’ An excellent starting point that takes you through the history of math, one
‘great theorem’ at a time. Covers ancient Greek mathematics through Cantor. This is the best book on the
History of Math that I’ve ever read.
Dunham, ‘The Mathematical Universe’
Dunham, ‘Euler, Master of us all’. A detailed look at Euler’s most influential and clever proofs. This goes into much more depth than ‘Journey Through Genius’ http://aleph0.clarku.edu/~djoyce/java/elements/elements.html. A website that has the entire 13 books of
Euclid’s Elements with interactive Sketchpad-like diagrams.
Euler, ‘Elements of Algebra’ The book on Algebra that Euler wrote in 1770. You can get a used copy still. www.eulerarchive.org. A website with many translations of Euler’s papers (and ALL of them in Latin)
Heath, ‘Euclid’s Elements’ A 3 volume set of an English translation of ‘The Elements’ with commentary.
Heath, ‘The Work of Archimedes’
Katz, ‘A History of Mathematics’ a great textbook.
Katz, ‘Historical Modules for the Teaching and Learning of Mathematics’ This is a CD with over 1,000 pages of lesson plans and activities about the history of Math for about $40.
Katz, ‘Sherlock Holmes in Babylon’
Maor ‘Trigonometric Delights’ A good history of Trigonometry.
Stein, ‘Archimedes. What did he do besides cry Eureka?’ An excellent short description of some of
Archimedes’ proofs.
Smith, ‘The Geometry Of Rene Descartes’ An English translation of the landmark book.
MAA’s convergence website: http://mathdl.maa.org/mathDL/46/