Archimedes made several important mathematical discoveries, including determining that the surface area of a sphere is 2/3 that of a cylinder with the same diameter. He also invented war machines, including a claw that could lift ships out of the water. Taking a bath one day, Archimedes noticed that the water level rose as he got in, inspiring his principle of buoyancy, which he ran through the streets naked announcing his discovery.

2. One of the many great
mathematical discoveries of
Archimedes was the
relationship between the
surface area of a cylinder
and a sphere. Archimedes
discovered that a sphere
that has the same diameter
as the height and width of
the cylinder is 2/3 of the
surface area of the cylinder

3. Archimedes’ first war
invention was a claw that was
said to be able to lift ships out
of the water and then smash
them. From Pappus we have
learned that in connection with
his discovery of the solution to
the problem of moving a given
weight by a given force, that
Archimedes upon applying the
law of the lever is to have said,
“Give me a place to stand on,
and I can move the earth.”

5. Archimedes taking his bath on day, he
noticed that that the level of the water in
the tub rose as he got in, and he had the
sudden inspiration that he could use this
effect to determine the volume (and
therefore the density) of the crown. In
his excitement, he apparently rushed out
of the bath and ran naked through the
streets shouting, "Eureka! Eureka!" (“I
found it! I found it!”). This gave rise to
what has become known as Archimedes’
Principle: an object is immersed in a
fluid is buoyed up by a force equal to the
weight of the fluid displaced by the
object.

6. The circumference of a circle is the actual
length around the circle which is equal to
360°. Pi (p) is the number needed to
compute the circumference of the circle.
p is equal to 3.14.
Pi is greek and has been around for over
2000 years!
In circles the AREA is equal to 3.14 (p) times
the radius (r) to the power of 2.
Thus the formula looks like:
A= pr2.
In circles the circumference is 3.14 (p) times
the Diameter.
Thus the formula looks like:
2pr or pd.
Lines in Circles.
AB = Diameter,
OC = Radius,
ED = Chord,
FG = Tangent,
EHD = Arc,
ADB = Semicircle,
OCB = Sector,
COB = Central Angle.

7. Plate portraying soldiers
using a compass to measure
the barrel of a cannon. Jim
Bennet, Stephen Johnston
(edited by), The Geometry of
War (1500-1750), Oxford,
1996, p. 15.
Using instruments was
indispensable in the military
field, where the technology
of firearms called for
increasingly more precise
mathematical knowledge.al

8. This 19th century model is based on a
drawing made by Galileo's (1564-1642)
friend and biographer Viviani (1622-
1703) of a pendulum clock, which
Galileo designed just before his death
and which was partly constructed by
his son Vincenzio in 1649. It represents
the first known attempt to apply a
pendulum to control the rate of a
clock. He recognised the potential of
using a pendulum to control a clock but
died before his work could be
completed.

9. Box for mathematical
instruments (17th ca.),
Florence, Istituto e Museo di
Storia della Scienza .
This box contains a set of
mathematical instruments
dating from the 17th century,
coming from the Medicean
collections. The interior,
divided into nineteen
compartments, now holds
thirteen pieces, all made of
brass: various instruments for
drawing, a pair of knives and
a proportional compass.

10. Mordente’s compass (1591),
Florence, Istituto e Museo di
Storia della Scienza.
Invented by Fabrizio
Mordente (1532 – c. 1608)
to measure the smallest
fraction of a degree, this
particular proportional
compass with eight points is
distinguished by the
presence of sliding cursors.
Based on their positions, it
was possible to establish the
proportions between lines,
geometric figures and solid
bodies.

11. Geometric and military
compass of Galileo Galilei
(c. 1606), Florence,
Istituto e Museo di Storia
della Scienza
The Istituto e Museo di
Storia della Scienza of
Florence possesses one of
the very rare surviving
examples of Galileo’s
compass, probably the
one donated by the Pisan
scientist to Cosimo II along
with a printed copy of the
Operazioni del compasso
geometrico et militare.

12. 3-Dimensional figures
Leonardo da Vinci kept also busy with complex
3-dimesional geometric figures. Leonardo drew these
in all their variants. In his period in Florence he was
already introduced to the perspective geometry . The
abstract perfection from these complex figures must
have charmed and fascinated him.

14. The rhombicuboctahedron, as
published in De divina proportione.

16. Tycho Instrument – Sextant

19. A calculating machine in
the 17th century.
In 1888, William S. Burroughs
patented his first calculator. Like
the Comptometer, it was really an
adder-subtracter, but it could
multiply and divide via repeated
additions and subtractions

20. 1876 Centennial
Expo Geo B Grants
calculating
machine.
In 1851, J. W. Nystroms
calculating machine.

21. The Arithmetical machine (pictured) is one of the
first mechanical calculating devices known to
exist.

23. Magnetic compass
A Geometrical
Instrument used in
measuring angles.

24. Earliest surviving
Astrolabes…

25. Ancient Water
Clock in Quanat
of Gonabad,
500 years ago.

28. The World’s oldest known
measuring device, The Lebombo
Bone (35,000 BC) with 29 lines,
Africa.

30. Water Clock Sun Dial

31. A Sun Dial, Dublin in 1742