Prove Pappus's centroid theorems without calculus

[Happiness]

Pappus's centroid theorems were discovered 17 centuries ago, when calculus wasn't invented yet. How are these theorems proved without using calculus?

"The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on the same plane is equal to the product of the arc length s of C and the distance d traveled by its geometric centroid."

The centroid of an object is its center of mass supposing its density is uniform.

"The second theorem states that the volume V of a solid of revolution generated by rotating a plane figure F about an external axis is equal to the product of the area A of F and the distance d traveled by its geometric centroid."

Quotes from https://en.wikipedia.org/wiki/Pappus's_centroid_theorem

 

[a1call]

Pappus's centroid theorems were discovered 17 centuries ago, when calculus wasn't invented yet. How are these theorems proved without using calculus?

"The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on the same plane is equal to the product of the arc length s of C and the distance d traveled by its geometric centroid."

The centroid of an object is its center of mass supposing its density is uniform.

"The second theorem states that the volume V of a solid of revolution generated by rotating a plane figure F about an external axis is equal to the product of the area A of F and the distance d traveled by its geometric centroid."

Quotes from https://en.wikipedia.org/wiki/Pappus's_centroid_theorem

Interesting post.
What does it refer to in its geometric centroid?
If I am not mistaking a centroid is a property of surfaces and solids only and not that of a 2-D open arc which can be the object of revolution here.
One thing to keep in mind is that mathematical proofs have not always been as strict as they are today.
Another is that an arc can-be/has-been defined by sections of a circle only and not a general curve, which would simplify the proof and would not be based on a centroid.
a simpler calculation would be derived from the fact that like a flat triangle, the area of a spherical triangle is equal to
1/2 x base-arc x height-arc
The height-arc would be the arc passing through the none base corner and perpendicular to the base arc.
It would not be too difficult to derive areas of more complex rotations from the above fact.

 

[a1call]

ETA In my post above arcs in calculating the area refer to arc lengths and not angles

 

[Happiness]

Interesting post.
What does it refer to in its geometric centroid?
If I am not mistaking a centroid is a property of surfaces and solids only and not that of a 2-D open arc which can be the object of revolution here.

The geometric centroid of an arc is the average distance the arc is from the axis of rotation.