# Mathematicians: Contributions to the Mathematics of Volume and Surface Area

## Introduction

Mathematics has been in existence since antiquity and has experienced immense development in scope and changes. Notable contributions in diverse areas of mathematics date back to the pre-historic times. Some of these works, owing to their proven soundness and credibility, continue to be used to date. This paper explores contributions to the mathematics of volume and surface area by some of these mathematicians.

## Archimedes of Syracuse

Archimedes (287-212 BC), probably, the most acclaimed scientist of ancient times, made many contributions in computing volume and area among other great inventions (Weinsten, n.pag). He is credited with inventing how to determine the volume of and irregularly shaped object. A popular anecdote alleges that he made this discovery while working on an assignment to determine the purity of a golden crown for his king. He discovered that the volume of water displaced is equal to the volume of the body and therefore by dividing this volume by the weight of crown he could get its density and compare it to that of pure gold.

Archimedes extensive studies on geometry led him to come up with an approximation of the value of pi (223/71 < pi <22/7) which is very close to the present universally adopted one (Allen, n.pag). This value enable a more accurate estimation of areas and volume of geometric bodies such as spheres, Cylinders and cones. His favorite discovery was when he showed the volume of a sphere is “two-thirds the volume of the smallest cylinder that it can contain” (Weinstein, n.pag). This can be explained whereby a cylinder will contain either two cones or spheres if the radius of the sphere or cone is exactly twice the height of the cylinder (Calkins, n.pag). Archimedes used the method of exhaustion to arrive at his conclusions for example by inscribing and circumscribing polygons on a circle when working on the value of pi. He also showed ways of working out the area of geometric figures such as triangles, parallelogram and parabolas. For instance, he proved the area segment under a parabola to be ^{4}/_{3} of that of a triangle drawn inside it.

## Others

Paul Guldin (1577-1643) and Pappus of Alexandria (290-350 BC) are other great mathematicians who also showed how to determine the surface area and volume of solids of revolutions such as cone, sphere and cylinder. The Pappus-Guldinus theorem is named after them (Weinstein, n.pag).Their theorem on volume states that the “volume of a solid of revolution generated by the revolution of a Lamina about an external axis is equal to the product of the Area of the Lamina and the distance traveled by the lamina geometric centroid” (Weinstein. n.p). For the surface area, it states that the surface area of a revolution generated by the revolution of a curve about an external axis is equal to the product of the arc length of the generating curve and the distance traveled by curve’s geometric centroid (Weinstein. n.pag).

Eudoxus of Cnidus, another great mathematician is also credited with “establishing rigorous methods for finding areas and volumes of curvilinear figures” such as cones and spheres (Allen n.pag).

Boventura Cavalieri, a great Italian mathematician contribution in areas and volumes is summed up in his Cavalieri Principle which states that “the volumes of two objects are equal to the area if their corresponding cross-sections are in all cases equal (Wenstein, n.pag). According to Encyclopedia Britannica he determined areas and volumes using a “Method of Indivisibles” which was similar to the methods of integral Calculus.

## Works Cited

Allen, James D. “Greatest Mathematicians of All Time”.n.p. 2011. Web. 2011.

“Bonaventura, Cavalieri , ” *Encyclopedia Britannica, Encyclopedia Britannica Online*. Encylopedia. Britannica, 2011. Web.

Calkins, Keith G. “Review of Basic Geometry- Lesson 10”. Andrews University. 2006. Web.

Weinsten Eric W. “Cavalieri’s Principle”.MathsWorld. 2011 Web.