Term Paper on "Historic Mathematicians"

Term Paper 6 pages (2172 words) Sources: 1+

[EXCERPT] . . . .

Historic Mathematicians

Born on January 29, 1700 Daniel Bernoulli was a famous Swiss Mathematician. His father -- Johan Bernoulli was the head of mathematics at Groningen University in the Netherlands. His father planned his future so that Daniel would become a merchant. but, Daniel never wanted to become a merchant, as he his favorite was calculus. His father could not admit this and began to thrust him to become a doctor. His father unwillingly taught him mathematics while thrusting his studies of medicine especially. While studying medicine he read a book by English physician named William Harvey. This book was about the hearts of animals. Daniel was fascinated by Harvey's work because it mixed his love of mathematics and fluids while at the same time expanding his medical knowledge, which satisfied his father. Daniel completed his medical studies at the age of 21. He submitted his applications for two teaching positions, at a university, but was refused. Daniel taught at the Emperial Academy in St. Petersburg in Russia at the age of 25. (Daniel Bernoulli: Personal Life and Significant Contributions)

Daniel carried out many researches in the flow of fluids. Particularly he wanted to know about blood flow and pressure. During his research, Daniel came across a new method to measure blood pressure by piercing an artery with a glass pipe. This system was used for nearly 200 years after its invention. This way of finding pressure is even now used to find the air speed of aircraft. Furthermore, Bernoulli discovered the fluid equation, which is even now one of the most significant discoveries in physics, as it governs why airplanes fly. Daniel carried out his research on fluids till he published a famous work called Hydrodynamics. There was a dispute between Daniel and his father about who made the discoveries, after the release of this book. This dispute turned his concentration towards medicine. He lived in Switzerland until his death at the age of 82, on March 17, 1782. (Daniel Bernoulli: Personal Life and Significant Contributions)

Bernoulli's Contributions to Mathematics:

Daniel worked on mathematics and his first mathematical work "Mathematical exercises" was published in 1724 in Venice with the help of Goldbach. This had four different parts of four different topics. The first part was a game of faro and was about probability. Second part was about the flow of water from a hole in a container and discussed Newton's theories. His medical effort on the flow of blood and blood pressure also attracted him in fluid flow. The third part was on the Riccati differential equation and the last part was a geometrical query of figures bordered by two arcs of a circle. Daniel did a research on vibrating systems in St. Petersburg. He made one of his most illustrious discoveries in St. Petersburg, when he described the simple nodes and the frequencies of oscillation of a system. He proved that the oscillations of strings of musical instruments are made up of a vast number of harmonic vibrations all placed over on the string. Probability and political economy was another famous work of Daniel while he was at St. Petersburg. (Daniel Bernoulli www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Bernoulli_Daniel.html": (www-groups.dcs.st-and.ac.uk)

But the most significant work of Bernoulli when he was at St. Petersburg was his work on hydrodynamics. His book was named "Hydrodynamica" and before he left St. Petersburg, he left a rough copy of the book with the publisher. Though the book was not published till 1729, he modified it substantially between 1734 and 1738, but the matter was not changed. For the first time this book had correct analysis about water flowing from a hole in a container. This was on the basis of the principle of conservation of energy, which he had learnt from his father in 1720. One notable discovery is mentioned in Chapter 10 of Hydrodynamica where Daniel talks the basis for the kinetic theory of gases. He was capable of providing the basic laws for the theory of gases and gave, though not in full detail, the equation of state discovered by Van der Waals a century later. Another significant feature of Daniel Bernoulli's work that was essential in the improvement of mathematical physics was his recognition of many of Newton's theories and his utilization of these together with the tolls coming from the more powerful calculus of Leibniz. He also analyzed the oscillation of bodies in resisting medium using Newton's methods. He has also presented a good work on the theory of oscillations and has given details about the oscillations of air in organ pipes. (Daniel Bernoulli www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Bernoulli_Daniel.html": (www-groups.dcs.st-and.ac.uk)

Bernoulli had a clever assistant, Leonard Euler for the period 1727 to 1733. They concentrated on the mechanics of supple and elastic bodies and got the equilibrium curves for these bodies. But his prominent discovery was when he described the simple nodes and the frequencies of oscillation of a system. Both Bernoulli and Euler aimed at discovering more about the flow of fluids. They wanted to know about the connection between the speed at which blood flows and its pressure. To study this, Daniel tested by piercing the wall of a pipe with a small open ended straw and found that the height to which the fluid rose up the straw was connected to fluid's pressure in the pipe. Soon doctors all over Europe were finding patient's blood pressure by attaching point-ended glass tubes straight into their arteries. (Daniel Bernoulli (1700-1782))

Bernoulli's way of measuring pressure is even now used in present aircraft to find the speed of the air passing the plane; that is its air speed.

Daniel Bernoulli now focused on his earlier work on Conservation of Energy. It was known that a moving body swaps its kinetic energy for potential energy when it gains height. Daniel recognized that likewise, a moving fluid swaps its kinetic energy for pressure. Mathematically this law is now written: 1/2*rho*u^2 + p = constant, where p is pressure, "rho" is the density of the fluid and u is its velocity. This is oppressed by the wing of an airplane, which is planned in such a way to form an area of fast flowing air above its surface. The pressure of this area is lesser and so the wing is sucked upwards. (Daniel Bernoulli (1700-1782))

Effect of Bernoulli's work on today's world:

Daniel Bernoulli used his analytical skills to problems spread across a broad range of scientific disciplines. He made important contributions to the study of fluid flow. In spite his distinction, one of his most significant contributions remained ignored for more than a century. Bernoulli imagined gases as having a vast number of particles in fast, independent motion. The impact of the particles with one another and with the walls of the vessel having the gas was considered to be completely elastic. He proposed that the speeds of the particles were related to temperature. This form for the gas entails that gas pressure is the result of the force on the vessel wall exercised by the gas particles crashing with the walls. The model guided Bernoulli to a mathematical treatment that gave an explanation of gas behavior in agreement with experiment. For instance, it led to Boyle's law. (the Classical Atom)

In addition, it explained why the pressure in a vessel goes up on heating: As temperature rises, the speeds of the gas particles rise. Thus they clash more often with the wall and exercise greater force on the vessel wall when they do so. Bernoulli's works are a set of connected descriptions, and mathematical expression that conveys the contents of those descriptions. The main metaphorical elements are: Atoms are in continuous, frenzied motion; Clashes with other atoms and with the vessel walls are supple; the speeds of atoms rises with increasing temperature; the average kinetic energy which means the energy of motion of a gas atom is dependent only on the gas temperature, not on the mass of the atom; Pressure is due to the collisions of gas atoms with the vessel walls. (the Classical Atom)

Aerodynamics is a subdivision of fluid mechanics that deals with the motion of air and other gaseous fluids, and with the forces acting on bodies in motion relative to such fluids. Some of the examples of aerodynamic actions are: the movement of an aircraft through the air, the wind forces applied on a structure and the working of a windmill. Daniel Bernoulli's principle is the main law dictating the motion of fluids, which links an increase in flow velocity to a decrease in pressure. For instance, for the same quantity of air at the entry to the venturi tube below to flow through the restriction in the middle, the air must accelerate. On the basis of Newton's theory that energy cannot be formed or shattered, it can only be transmitted, this increased speed must have an equivalent decrease in pressure, if the same quantity of air is to pass through the tube. As the air comes out of the constriction, it decelerates and… READ MORE

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