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1. THE FINAL CHAPTER OF
GREEK MATHEMATICS

2. DIOPHANTUS

3. DIOPHANTUS;
Diophantus of Alexandria (c. 200 – c. 298
AD) was a Greek mathematician
renowned for his contributions to
algebra and number theory. He is often
referred to as the "Father of Algebra" due
to his groundbreaking work in the field,
particularly his book Arithmetica.

4. Diophantus' Life and Legacy
While much of Diophantus' life remains shrouded in mystery, a few
key facts are known. He lived in Alexandria, Egypt, during the
Roman era, and his work significantly influenced the development
of algebra in both the Arab world and Europe.
The most famous source of information about Diophantus' life is a
5th-century Greek anthology of number games and puzzles, which
includes a problem sometimes called his epitaph. This problem,
expressed as an algebraic equation, suggests that Diophantus lived
to be 84 years old. However, the accuracy of this information is
debated.

5. Problem : Finding Two Numbers with a
Given Sum and Difference
x + y = 20
x - y = 4
He solved this system of equations using a combination of
substitution and elimination, arriving at the solution: x = 12
and y = 8.
Diophantus' Work: Arithmetica

6. PAPPUS;
. Pappus of Alexandria (The Mathematician)
Pappus of Alexandria was a Greek mathematician who lived around
290–350 AD. He is best known for his work in geometry and is
particularly famous for the Pappus's Theorem and his influential
text, the Synagoge (or Collection), which compiled and expanded
upon the mathematical knowledge of earlier Greek mathematicians,
including Euclid and Archimedes.
His Centroid Theorem (also known as Pappus's Theorem) deals with
the surface area and volume of solids generated by rotating a plane
curve around an external axis. The formulas derived from Pappus’s
Theorem are foundational in understanding solids of revolution.

7. pappus of alexandria

8. PAPPUS ANALYSIS;
Pappus' Theorem, also known as Pappus's
centroid theorem, is a fundamental result in
geometry and calculus related to the area and
volume of solids generated by revolving
planar curves. The theorem has two key
versions: one dealing with the surface area of
a solid of revolution and the other dealing
with its volume.

9. Pappus' Theorem;
Theorem for Surface Area;
A = L * d
The first theorem states that the surface area A of a surface of revolution
obtained by rotating a plane curve C about a non-intersecting axis lying in
the same plane is equal to the product of the curve length L and the
distance d traveled by the centroid of C:

10. Pappus' Theorem for Volume;
The second theorem states that the volume V of a solid of
revolution obtained by rotating a lamina F about a non-
intersecting axis lying in the same plane is equal to the
product of the area A of the lamina F and the distance d
traveled by the centroid of F:
V = A * d

11. Pappus' Theorems find numerous applications in
various fields, including:
Engineering: Calculating the surface area and volume of
complex shapes, such as gears, propellers, and other
rotational components.
Physics: Determining the moment of inertia of objects
with rotational symmetry.
Mathematics: Proving geometric theorems and exploring
the properties of solids of revolution.

12. Generalization and Extensions;
Pappus' Theorems have been
generalized to cases where the rotating
figure is not necessarily a simple curve
or a plane figure. For instance, the
theorems can be applied to regions
bounded by multiple curves or to three-
dimensional objects.

13. Conclusion;
Pappus' Theorems provide a powerful and elegant
approach to calculating surface areas and volumes
of solids of revolution. Their simplicity and wide
applicability make them valuable tools in various
fields, from engineering and physics to mathematics
and beyond. The theorems highlight the
interconnectedness of geometry and calculus and
offer a glimpse into the ingenuity of ancient Greek
mathematicians.

14. THANK YOU