Graphs and Graphing
The use of graphs to display information became a common occurrence in the nineteenth and twentieth centuries. Newspapers, magazines, science journals, financial reports, political surveys, and, of course, the sports page, all routinely contain tables, charts, and graphs to summarize and explain current events, scientific studies, or statistical data. It was actually Florence Nightingale who became one of the first innovators in the tabulation and graphical display of descriptive statistics in her attempts to reform hospital sanitation methods. But to most people, graphs and graphing still belong primarily to the domain of high school algebra and other fields of mathematics.
One of the first graphical representations of a function occurred around the middle of the 14th century. At that time Nicole d'Oresme gave a geometrical verification of a theorem, discovered at Merton College of Oxford and hence known as the Merton rule, concerning the distance covered by a body moving with uniform acceleration. Oresme's novel idea was to draw a time-velocity graph for a body moving with uniform acceleration. He marked instants of time along a horizontal line and at each such mark he then drew a perpendicular line segment (which he called the latitude) whose length represented the velocity at that time. The end points of the latitudes lie along a straight line which we would now call the graph of the velocity as a function of time. Oresme also observed that the distances covered by the body corresponded to areas of right triangles formed by the velocity graph.
The graphical representation of functions became known as the latitude of forms. It was not studied in a systematic way, however, until the development of analytic geometry by the French philosopher and mathematician René Descartes in 1637. The introduction of a coordinate system allowed a point to be specified as a pair of numbers called its coordinates. A curve could then be graphed by plotting the points whose coordinates satisfied the equation of the curve. To Descartes, the acceptable geometric curves were those that could be expressed by a unique algebraic equation in two variables.
In 1671 Issac Newton made extensive use of the new coordinate geometry to sketch the graphs of curves from their equations. In The Method of Fluxions and Infinite Series he also introduced new coordinate systems, including the polar coordinate system in which points are graphed based on their distance from a fixed reference point and the angle with a fixed reference line. This work was not published, however, until 45 years after Jakob Bernoulli had devised essentially the same method for graphing special curves in polar coordinates. The main development for graphing surfaces occurred in the eighteenth century. Although Philippe de La Hire (1640-1718) had represented a surface by specifying points with three coordinates based upon a rectangular coordinate system, it was Johann Bernoulli in 1715 who introduced the three coordinate planes that we use today for graphing in three dimensions.
Modern technology has greatly increased our ability to generate accurate graphs quickly. Graphing calculators and computer programs use essentially the same technique as that developed by the 17th century mathematicians to draw the graphs of functions. A finite number of points lying on the graph are plotted and adjacent data points are connected by line segments to achieve the appearance of a smooth curve. The more points that are used, the more accurate will be the graph. Undersampling occurs when too few data points are used and important features of the graph are missed, much as a photograph may miss details if the resolution is too low. Advanced computer algebra systems often use an adaptive sampling strategy in which more points are used in regions where the graph appears to wiggle up and down over a short interval. The ability of a computer to generate graphical solutions to a differential equation has allowed mathematicians and scientists to study the qualitative behavior of these equations without having to solve them analytically. Computer graphics have also provided the opportunity to draw the graph of a surface or other three-dimensional object and actually rotate the object in real time to examine it from many different directions, even from the inside. This has yielded important applications to many areas such as medical diagnostics and aeronautical design.
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