Archimedes

How to Think like Archimedes

Archimedes (287 – 212 BC)

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.

Biography

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.

Biography

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.

Mathematical Treatise Author of several mathematical treatise including:

On the equilibrium of planes

Quadrature of the Parabola

On the Sphere and the Cylinder

On Spirals

On Conoids and Spheroids

On Floating Bodies

Measurement of a Circle

The Sand ReckonerFigure 4.

Archimedes Inventions The Archimedes Screw –

developed for pumping water

Reputedly invented by Archimedes while in Alexandria

The Archimedes Claw – overturned Roman ships attacking Syracuse

Figure 5. Figure 6.

Sacking of Syracuse

Archimedes perished in the sacking of Syracuse in 212 BC.

Killed by a Roman soldier after supposedly saying “Stand away, fellow, from my diagram”.

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.

Figure 7.

Archimedes Palimpsest 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

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”

Figure 8.

How to Think like Archimedes

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

From known parts to the unknown whole 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

Figure 9.

How to Think like Archimedes

Another example of this is the Eureka story of Archimedes running naked through Syracuse after discovering the principle of displacement in the bath

Continuous Contemplation & Reflection 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”

Figure 11.

How to Think like Archimedes

Visualisation & Conceptual Diagrams 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.

How to Think like Archimedes

Observation of Phenomena 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

How to Think like ArchimedesSolve by Approximation 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

How to Think like ArchimedesDeduction 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.

Image References Figure 1 - http://www.math.nyu.edu/~%20crorres/Archimedes/Pictures/ArchimedesPictures.html, (Accessed December 23rd 2013).

Figure 2 - http://www.historyandcivilization.com/Maps---Tables---Ancient-Greece---the-Aegean.html, (Accessed December 23rd 2013).

Figure 3. - http://faculty.etsu.edu/gardnerr/Geometry-History/abstract.htm, (Accessed December 23rd 2013).

Figure 4 - Heath T.L (2002), The Works of Archimedes

Figure 5 - http://www.math.nyu.edu/~crorres/Archimedes/Screw/ScrewEngraving.html, (Accessed December 23rd 2013).

Figure 6 - http://www.math.nyu.edu/~crorres/Archimedes/Claw/illustrations.html, (Accessed December 23rd 2013).

Figure 7 - http://www.math.nyu.edu/~crorres/Archimedes/Death/DeathIllus.html, (Accessed December 23rd 2013).

Figure 8 - http://archimedespalimpsest.org/about/, (Accessed December 23rd 2013).

Figure 9 - http://www.mathworks.com/matlabcentral/fileexchange/29504-the-computation-of-pi-by-archimedes/content/html/ComputationOfPiByArchimedes.html, (Accessed December 23rd 2013).

Figure 10 - http://ed.ted.com/lessons/mark-salata-how-taking-a-bath-led-to-archimedes-principle#watch, (Accessed December 23rd 2013).

Figure 11 - Heath T.L (2002), The Works of Archimedes