**Pappus **of Alexandria (ca 300) Egypt, Greece.

Very little is known of Pappus’ life. Very little is known of what his actual contributions were. We do know that he recorded in one of his commentaries on the Almagest that he observed a solar eclipse on October 18, 320.

He wrote *The Collection or The Synagogue*, a treatise on geometry, which included everything of interest to him. Whatever explanations or supplements to the works of the great geometers seemed to him necessary, he formulated them as lemmas.

*Summary of Contents:*

Book I and first 13 propositions of Book II are missing. Book II was concerned with very large numbers – powers of myriads.

Book III begins with a summary of finding two mean proportionals (a:x = x:y = a:y) between two straight lines. He also defines plane problems, solid problems, and linear problems. Pappus

distinguishes (1) plane problems, solvable with straight edge and compass

distinguishes (2) solid problems, requiring the conics for solution, e.g. solving certain cubics.

Distinguishes (3) linear problems, problems invoking spirals, quadratrices, and other higher curves

gives a constructive theory of means. That is, given any two of the numbers and the type of mean (arithmetic, geometric, or harmonic), he constructs the third.

Describes the solution of the three famous problems of antiquity, asserts these are not plane problems 19 century.

*Inscribes the five regular solids in the sphere.*Book IV covers an extention of theorem of Pythagorus for parallelograms constructed on the legs of any triangle. Also in Book IV is material on the Archimedian spiral, including methods of finding area of one turn — differs from Archimedes. He also constructs the conchoid of Nicomedes. In addition, he constructs the quadratix in two different ways, (1) using a cylindrical helix, and (2) using a right cylinder, the base of which is an Archimedian spiral.

Book V reproduces the work of Zeodorus on isoperimetric figures. Here we see in the introduction his comments on the sagacity of bees.

Book VI determines the center of an ellipse as a perspective of a circle. It is also astronomical in nature. It has been called the “*Little Astronomy”*. A list of the books forming the `treasury’ is included, together with a short description of their contents.

As an independent contribution Pappus formulated the volume of a solid of revolution, the result we now call the **The Pappus – Guldin Theorem**.

Theorem. Look here: http://www.cut-the-knot.org/pythagoras/Pappus.shtml

By **Wikipedia**:

-> 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.

A= sd

->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.

V= Ad

Volume of revolution = (area bounded by the curve)(distance traveled by the center of gravity)