Trapeziod

A trapezoid is geometrical figure with two parallel sides. In an isoceles trapezoid, the two non-parallel sides are equal in length. In a rectangular trapezoid, one of the non-parallel sides is perpendicular to the two parallel sides.

Ellipse

Ellipse, in geometry, closed plane curve, one of the conic sections (see Cone), formed by a plane that cuts all the elements of a right circular cone. A circle, which is formed by a plane perpendicular to the axis of the cone, is a specialized form of ellipse.

An ellipse may be defined as the locus of all points, P, the sum of whose distances, d ₁ and d ₂, from two fixed points is a constant (see Fig. 1). The two fixed points that define an ellipse are known as its foci and are labeled F and F ’ in Fig. 1. This property of an ellipse is often used for drawing the figure. If pins are placed in the drawing surface at the two foci and a length of string is tied loosely between them, a point holding the string taut will trace an ellipse as it moves.

Any ellipse is symmetrical with respect to its major axis, which is a straight line passing through the two foci and extended to meet the curve at each end. It is also symmetrical with respect to its minor axis, a line perpendicular to the major axis at the midpoint between the two foci. In a circle the two foci of the ellipse coincide, and the major and minor axes are equal.

The eccentricity of an ellipse, that is, the ratio of the distance between the foci to the length of the major axis, is always less than 1. The eccentricity of a circle is 0.

The ellipse is one of the most important curves in physical science. In astronomy, the orbits of the earth and the other planets around the sun are ellipses. It is used in engineering in the arches of some bridges and the design of gears for certain types of machinery such as punch presses.

Archimedes

Archimedes (287-212 bc), preeminent Greek mathematician and inventor, who wrote important works on plane and solid geometry, arithmetic, and mechanics.

Archimedes was born in Syracuse, Sicily, and educated in Alexandria, Egypt. In pure mathematics he anticipated many of the discoveries of modern science, such as the integral calculus, through his studies of the areas and volumes of curved solid figures and the areas of plane figures. He also proved that the volume of a sphere is two-thirds the volume of a cylinder that circumscribes the sphere.

In mechanics, Archimedes defined the principle of the lever and is credited with inventing the compound pulley. During his stay in Egypt he invented the hydraulic screw for raising water from a lower to a higher level. He is best known for discovering the law of hydrostatics, often called Archimedes' principle, which states that a body immersed in fluid loses weight equal to the weight of the amount of fluid it displaces. This discovery is said to have been made as Archimedes stepped into his bath and perceived the displaced water overflowing.

Archimedes spent the major part of his life in Sicily, in and around Syracuse. He did not hold any public office but devoted his entire lifetime to research and experiment. During the Roman conquest of Sicily, however, he placed his gifts at the disposal of the state, and several of his mechanical devices were employed in the defense of Syracuse. Among the war machines attributed to him are the catapult and—perhaps legendary—a mirror system for focusing the sun's rays on the invaders' boats and igniting them.

After the capture of Syracuse during the Second Punic War, Archimedes was killed by a Roman soldier who found him drawing a mathematical diagram in the sand. It is said that Archimedes was so absorbed in calculation that he offended the intruder merely by remarking, “Do not disturb my diagrams.” Several of his works on mathematics and mechanics survive, including Floating Bodies, The Sand Reckoner, Measurement of the Circle, Spirals, and Sphere and Cylinder. They all exhibit the rigor and imaginativeness of his mathematical thinking.

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