A brief set of slides that attempt to speculatively identify key principles and techniques present in the thought process of Archimedes. The slides also include a brief biography and succinct outline of Archimedes major works and inventions.

1. Archimedes
How to Think like Archimedes

2. Archimedes (287 – 212 BC)
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The greatest mathematician of the
ancient world
Authored several treatise on
mathematics and pioneer in the field
of calculus
Founded new scientific disciplines
including hydrostatics
Inventor
Defender of Syracuse against Roman
attack
Figure 1.

3. Biography
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Born in Syracuse in 287 BC
Son of Pheidias, the
astronomer
Reputedly related to King
Hieron of Syracuse
No known biography
survived
Plutarch comments on
Archimedes in his work,
“Parallel Lives”
Figure 2.

4. Biography
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Likely Archimedes studied in
Alexandria in his youth with the
followers of Euclid
Commenced lifelong
correspondence with the
mathematicians “Erasthosthenes”
and “Dositheus”
For example, a greeting from
Archimedes to Dositheus proceeds
the propositions in the treatise “On
the Sphere and the Cylinder”
Figure 3.

5. Mathematical Treatise
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Author of several mathematical treatise including:
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On the equilibrium of planes
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Quadrature of the Parabola
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On the Sphere and the Cylinder
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On Spirals
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On Conoids and Spheroids
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On Floating Bodies
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Measurement of a Circle
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The Sand Reckoner
Figure 4.

6. Archimedes Inventions
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The Archimedes Screw –
developed for pumping water
Reputedly invented by
Archimedes while in
Alexandria
Figure 5.
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The Archimedes Claw –
overturned Roman ships
attacking Syracuse
Figure 6.

7. Sacking of Syracuse
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Archimedes perished in the
sacking of Syracuse in 212
BC.
Killed by a Roman soldier
after supposedly saying
“Stand away, fellow, from my
diagram”.
Figure 7.
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According to Plutarch, The Roman General Marcellus “was
most of all afflicted” by Archimedes death suggesting that
Marcellus wished to spare Archimedes perhaps in order to
profit from Archimedes' military ingenuity.

8. Archimedes Palimpsest
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Discovered in 1906 by Johan
Ludvig Heiberg in
Constantinople.
The parchments of a
manuscript containing
Archimedes writings had
been reused as a prayer book
by Byzantine monks in 1229
Figure 8.
The Archimedes Palimpsest project established in 1998 to
conserve and study the palimpsest
Unique and previously unknown translations of “The Method”,
“The Stomachion” and “On Floating Bodies”

9. How to Think like Archimedes
From known parts to the unknown whole
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Archimedes used known geometry to calculate the properties of
unknown geometrical figures
He often built up new theorems using geometrical laws
discovered in earlier theorems
An example is how
Archimedes arrived at his
calculation of Pi
Using known inscribed and
circumscribed polygons to
estimate the area of the
unknown circle
Figure 9.

10. How to Think like Archimedes
Continuous Contemplation & Reflection
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Archimedes continually contemplated his works even during
mundane daily tasks
According to Plutarch, Archimedes would draw “geometrical
figures in the ashes of the fire, or, when anointing himself, in
the oil on his body”
Another example of this is
the Eureka story of
Archimedes running naked
through Syracuse after
discovering the principle of
displacement in the bath
Figure 11.

11. How to Think like Archimedes
Visualisation & Conceptual Diagrams
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Annotated diagrams were an integral part of Archimedes
theorems.
The diagrams while assisting in the explanation of the theorems
may also have aided the development of the theorem and the
conceptualising of complex geometrical propositions
The previous slide referring
to Archimedes continual
sketching of geometry
suggests it played a strong
role in his thinking process
Figure 11.

12. How to Think like Archimedes
Observation of Phenomena
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Archimedes discovered the principle of displacement while
observing the rise in the level of the water as he emersed
himself to bathe
While contemplating a problem one should therefore observe
the related phenomena in the world around us at small and large
scales
Such observations may act as catalysts for new ideas as they
occur unexpectedly at random and therefore force the
researcher to view a problem in ways they otherwise would not
have considered

13. How to Think like Archimedes
Solve by Approximation
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Archimedes employs this technique in his treatise to
approximate the calculation of an unknown area or a value.
The inscribed and circumscribed polygons cited earlier to
approximate the area of a circle is an example of this technique.
Another example is the division of a sphere into an infinite
number of discs to calculate its volume
A general principle can be extracted for problem solving as:
define the limits which contain a problem and then repeatedly
subdivide those limits until the exact properties of the problem
are defined

14. How to Think like Archimedes
Deduction
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In many of Archimedes propositions he proves that some
property A is equal to another property B by first proving, that
A is not greater than B, and A is not less than B
This is a process of iteratively proving what is false to arrive at
what is true
The researcher should therefore, as far as possible, list all
possible conclusions, and then seek to disprove each in turn in
order to arrive at a solution
An example of this is Proposition No.1 of “Measurement of a
Circle” where Archimedes proves that the area of a circle is
equal to that of a triangle, K, by first proving that the area of the
circle is not greater or less than K.

15. Image References
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Figure 1 - http://www.math.nyu.edu/~%20crorres/Archimedes/Pictures/ArchimedesPictures.html, (Accessed December 23rd 2013).
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Figure 2 - http://www.historyandcivilization.com/Maps---Tables---Ancient-Greece---the-Aegean.html, (Accessed December 23rd 2013).
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Figure 3. - http://faculty.etsu.edu/gardnerr/Geometry-History/abstract.htm, (Accessed December 23rd 2013).
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Figure 4 - Heath T.L (2002), The Works of Archimedes
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Figure 5 - http://www.math.nyu.edu/~crorres/Archimedes/Screw/ScrewEngraving.html, (Accessed December 23rd 2013).
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Figure 6 - http://www.math.nyu.edu/~crorres/Archimedes/Claw/illustrations.html, (Accessed December 23rd 2013).
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Figure 7 - http://www.math.nyu.edu/~crorres/Archimedes/Death/DeathIllus.html, (Accessed December 23rd 2013).
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Figure 8 - http://archimedespalimpsest.org/about/, (Accessed December 23rd 2013).
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Figure 9 http://www.mathworks.com/matlabcentral/fileexchange/29504-the-computation-of-pi-by-archimedes/content/html/ComputationOfPiByArchimedes.html,
(Accessed December 23rd 2013).
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Figure 10 - http://ed.ted.com/lessons/mark-salata-how-taking-a-bath-led-to-archimedes-principle#watch, (Accessed December 23rd 2013).
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Figure 11 - Heath T.L (2002), The Works of Archimedes