Archimedes

Archimedes was both an accomplished practical inventor and brilliant theorist. He is best-known for his screw and lever machines.  According to legend, he created many useful inventions.

Background

In the realm of ancient mathematics, one name that stands out with remarkable brilliance is that of Archimedes. Hailing from the city of Syracuse in ancient Greece, Archimedes left an indelible mark on the world of mathematics, physics, and engineering. His contributions and discoveries continue to inspire and shape our understanding of these fields even today. 

Archimedes was born in 287 BCE into a family of scholars and scientists. His early education was nurtured in Alexandria, Egypt, which was a flourishing center of knowledge at the time. There, he studied under the guidance of renowned mathematicians and philosophers, acquiring a deep understanding of geometry, mechanics, and astronomy. 

Mathematical 

Archimedes was one of the greatest mathematicians of his era and his work had lasting impacts in mathematics and sciences for generations to come. He is best-known for originating many of the formulas commonly taught today in algebra and geometry courses such as pi and the right triangle.  Additionally, he made significant discoveries pertaining to circles, cylinders, paraboloids and developed an accurate system for calculating solid volumes.

Mathematically versatile, his work in geometry and mechanics made substantial contributions to both fields. He is best known for discovering fundamental plane figure theorems such as finding their centers of gravity for parallelograms, triangles, and trapeziums; discovering geometric properties of spirals; as well as determining different positions a right paraboloid of revolution takes when floating in fluid, taking into account both geometrical and hydrostatic variations.

Archimedes’ treatises are masterpieces of mathematical exposition. They demonstrate an extraordinary masterplan of attack, an expert organization of propositions, and the removal of anything not immediately relevant to the purpose of his proofs. Reading them is like standing under an enormous statue.  Your mind becomes entranced by their clarity, depth, and perfection of design and construction.

Archimedes is best-known for two fundamental axioms which form the basis for straight line calculus, the distance between points and length of line segments. His two axioms remain relevant today as the basis for modern curve calculations.

Archimedes also created a work entitled On Spirals that provided fundamental properties pertaining to spirals and explored their volumes as well as paraboloids and hyperboloids segments. He developed an accurate system for calculating volumes and diameters of spheres and cylinders while devising a number system capable of representing large numbers up to 8x 1063 in contemporary notation.

Gravity

Archimedes was not only renowned as a mathematician but also an extraordinary engineer and inventor. He created numerous innovations such as the Archimedes Screw which could raise and lower water levels as well as an amazing mirror system for reflecting sunlight to light rooms. Archimedes remains one of the greatest mathematicians who ever lived and his legacy continues to expand worldwide with each discovery after his death.

Archimedes dedicated himself tirelessly to mathematics. At times he would lose track of time while immersed in his studies, sometimes forgetting to eat! Scribbling figures onto everything available (even using olive oil to draw them on his skin!), Archimedes achieved remarkable accuracy. Although never divulging his exact methods for doing so.

He is best-known for his theorems on plane geometry, particularly those related to plane equilibriums and quadrature of circles. In his treatise On plane equilibriums he identified various plane figures such as parallelograms, triangles and trapeziums as centres of gravity in addition to solving quadrature problems.

One of his greatest contributions was developing the method to determine the circumference of a circle using area from an equilateral triangle within it and perimeter from another triangle outside; this remains one of the cornerstones of modern calculations and remains widely utilized today.

Archimedes was an accomplished astronomer who used Hipparchus’ data to develop a model of the solar system that later was refined by himself to give accurate distance estimates between planets.

Archimedes’ principle of buoyancy states that any object submerged completely or partially in water at rest will be affected by an upward force (buoyant force), equal in magnitude to the weight of liquid displaced by it, at rest. This law helps explain why objects float or sink and is the basis for determining how much weight an object will weigh when placed into liquid.

Thermodynamics

Thermodynamics is a science concerned with energy transformations.  For example, when particles’ kinetic energy is converted to thermal energy and vice versa. Furthermore, thermodynamics deals with how energy flows within systems as they change states, specifically how any lost or gained energy impacts upon overall change of entropy within them.

Understanding thermodynamics’ fundamental concepts, especially its first and second laws, is vital in order to make sense of our physical world. Luckily, these laws can be explained simply and understood easily.

One core concept is that all systems tend toward thermodynamic equilibrium. This holds even when initial temperatures or states differ significantly.  It applies equally well to systems as varied as human bodies and serves as the basis for energy conservation principles.

Archimedes’ principle dictates that when an object is submerged completely in liquid, its apparent weight loss equals that of liquid it has displaces. Furthermore, its upward force in fluid depends upon both size and shape.  As an example, a sphere floats more effortlessly than a cube which requires greater density to maintain buoyant force.

Objects may rise if their total restoring force (composed of weight of liquid displaced and meniscus weight) exceeds gravity’s downward pull, otherwise they will sink.

Bernoulli’s Principle, another basic thermodynamic principle, states that as velocity increases, pressure decreases. This principle plays an essential role in operating engines and pumps as well as moving fluids over surfaces.

Optics

Optics is the mathematical theory of light interactions between light and shaped light-absorbing or reflective surfaces, such as lenses. Optics-related theorems play an integral part in designing electromagnetic beams with subwavelength features for various uses such as imaging, radar detection, communication technology, nanoparticle trapping and manipulation, as well as surface metrology applications.

Archimedes made his mark in optics by formulating an approximation of Pi, studying sphere and cylinder, designing innovative machines such as compound pulleys and screw pumps to defend his city of Syracuse from Roman siege.

By applying the optical theorem, geometric properties of a spherically symmetric mirror or lens can be determined from its forward scattered intensity distribution in an elliptical cross section. This distribution depends on factors like curvature radius and source angle.  Additionally, backscattered radiation intensity relates directly with its geometry of absorption or reflection surface.

An example of an extreme case of optical theorem can be illustrated through an easy experiment involving an elliptical reflector and concave mirror, with caustics from a burning mirror being seen reflected back as bright filaments with cusp singularities resonant around their surfaces. Leonardo Da Vinci recorded their appearance several times in his notebooks as part of this interest area.

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