Archimedes has generous contributions to various field include mathematics, physics, mechanics and so on. Meanwhile, he has many famous works which represents his principal achievements. In particular，his glorious achievements in geometry area have profound impact of the development of mathematics.
The Method of Exhaustion
In "On the Sphere and the Cylinder", he made adventurous research of the method of exhaustion to prove his mathematical discoveries, which are regarded as forerunners to modern calculus. Archimedes made effective use of the concept of exhaustion proposed by Euclid, and he used the "exhaustion method" to calculate the sphere area, sphere volume, parabola, elliptical area, and so on. Generations mathematicians based on this "exhaustion method" to develop into modern calculus.
The Formula for Sphere
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| inscribed sphere in a circular cylinder |
Through more efficient use of "exhaustion method", Archimedes also pointed out the formula for the surface area (4πr^2) and enclosed volume (4/3πr^3) of a sphere in his work, On the Sphere and Cylinder. Archimedes also found that if an equilateral sphere has an inscribed sphere, the full area of the cylinder equal to the surface area of the sphere and its volume are the volume of the sphere.
The Value of Pi
Archimedes is considered the first to calculate an accurate estimation of the value of pi. In "On the Measurement of a Circle", he drew a regular polygon outside a circle and a regular polygon inside it; and gradually increased the number of sides of both the polygons till they had 96 sides each. Archimedes used the method of exhaustion to estimate the area of a circle, and he obtained that the value of pi lay between 223/71 (approximately 3.1408) and 22/7 (approximately 3.1429). In the history of mathematics, this is the first time to clearly point out pi value.
In "The Quadrature of the Parabola", Archimedes introduces some methods and skills of calculating the figure acreage enclosed by plane curve. Archimedes used the method of exhaustion to prove that the area of a parabolic segment (the region enclosed by a parabola and a line) is 4/3 that of a certain inscribed triangle.
The Area Enclosed by Parabola
Archimedean spiral
Archimedes cleared and defined the Archimedean spiral, as well as, the calculation of the spiral in "On Spirals". In the same work, Archimedes also derived geometric methods for the summation of arithmetic progression and geometrical progression. Archimedean spiral is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity. Similarly, in polar coordinates, Archimedean spiral can be described by the equation r=a+b(Theta) with real numbers a and b.
Reference List
Archimedes, Netz, R., & Eutocius, o. A. (2004). The works of Archimedes : translated into English, together with Eutocius' commentaries, with commentary, and critical edition of the diagrams. Cambridge ; New York : Cambridge University Press, 2004-.
Archimedes, & Clagett, M. (1964). Archimedes in the Middle Ages. Philadelphia : American Philosophical Society, 1964-.
Archimedes, Netz, R., & Eutocius, o. A. (2004). The works of Archimedes : translated into English, together with Eutocius' commentaries, with commentary, and critical edition of the diagrams. Cambridge ; New York : Cambridge University Press, 2004-.
Archimedes, & Clagett, M. (1964). Archimedes in the Middle Ages. Philadelphia : American Philosophical Society, 1964-.
Comments: "Archimedes also found that if an equilateral sphere has an inscribed sphere, the full area of the cylinder equal to the surface area of the sphere and its volume are the volume of the sphere." Is this really what you meant? Aren't all spheres equilateral? Also, I'm not sure this is what he found.
ReplyDelete"figure acreage" (Do you mean figure's area?)
"On Spirals". (Period goes inside quotes.)
Very nice overall, except check on those few ideas I mentioned above.