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It was Dirichlet who proved the fundamental Theorem of Fourierseries: that periodic analytic functionscan always be represented as a simpletrigonometric series.

for a triangle's smallest circumscribing and largest inscribing ellipses,and for its "Malfatti circles."Among many famous and importanttheorems of classic and projective geometry,he proved that theWallace lines of a triangle lie in a 3-pointed hypocycloid,developed a formula for the partitioning of space by planes,a fact about the surface areas of tetrahedra,and proved several facts about his famousSteiner's Chain of tangential circles and his famous "Roman surface."Perhaps his three most famous theorems arethe Poncelet-Steiner Theorem (lengths constructiblewith straightedge and compass can be constructed with straightedgealone as long as the picture plane contains the centerand circumference of some circle), the Double-Element Theoremabout self-homologous elements in projective geometry,and the Isoperimetric Theorem that among solids of equalvolume the sphere will have minimum area, etc.

**CORRECTION: **It's easy to think that what scientists do in far-off laboratories and field stations has little relevance to your everyday life after all, not many of us deal with super colliders or arctic plankton on a regular basis but take another look around you. All the technologies, medical advances, and knowledge that improve our lives everyday are partly the result of scientific research. Furthermore, the choices you make when you vote in elections and support particular causes can influence the course of science. Science is deeply interwoven with our everyday lives. To see how society influences science, visit . To learn more about how scientific advances affect your life, visit

**CORRECTION: **Memorizing facts from a textbook can be boring but science is much more than the knowledge that makes its way into school books. Science is an ongoing and unfinished process of discovery. Some scientists travel all over the world for their research. Others set up experiments that no one has ever tried before. And all scientists are engaged in a thrilling quest to learn something brand new about the natural world. Some parts of scientific training or investigations may be tedious, but science itself is exciting! To see how a scientific perspective can make the world a more exciting and intriguing place, visit our side trip .

**CORRECTION: **Some students find science class difficult but this doesn't translate to not being good at science. First of all, school science can be very different from real science. The background knowledge that one learns in school is important for practicing scientists, but it is only part of the picture. Scientific research also involves creative problem-solving, communicating with others, logical reasoning, and many other skills that might or might not be a part of every science class. Second, science encompasses a remarkably broad set of activities. So maybe you don't care much for the periodic table but that doesn't mean that you wouldn't be great at observing wild chimpanzee behavior, building computer models of tectonic plate movement, or giving talks about psychology experiments at scientific meetings. Often when a student claims to "not be good at science," it really just means that he or she hasn't yet found a part of science that clicks with his or her interests and talents.

(Some even suspect that Descartes arranged the destructionof Pascal's lost .)And Descartes made numerous errors in his development ofphysics, perhaps even delaying science,with Huygens writing "in all of [Descartes'] physics,I find almost nothing to which I can subscribe as being correct."Even the historical importance of his mathematicsmay be somewhat exaggerated since others, e.g.

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He noted that the obvious packing of cannonballs gave maximumdensity (this became known as ; optimalitywas proved among regular packings by Gauss, but it wasn't until1998 that the possibility of denser packingswas disproven).

For example, the Law of the Pendulum, based on Galileo's incorrectbelief that the tautochrone was the circle, conflictedwith his own observations.)Despite his extreme importance to mathematical physics,Galileo doesn't usually appear onlists of greatest .

Bell wrote "it can be argued that Fermat wasat least Newton's equal as a pure mathematician."Fermat's most famous discoveries in number theoryinclude the ubiquitously-used ;the case ofhis conjectured (he may haveproved the case as well);and (that any prime (4n+1) can be represented as the sum of twosquares in exactly one way) which may be considered themost difficult theorem of arithmetic which had beenproved up to that date.

of all time."His book studiedmultivariate polynomials and is consideredthe best mathematics in ancient China and describes methods notrediscovered for centuries; for exampleZhu anticipated the Sylvester matrix method for solving simultaneouspolynomial equations.

Referring to this system,Gauss was later to exclaim "To what heights would sciencenow be raised if Archimedes had made that discovery!"Some histories describe him as bringing Islamic mathematicsto Europe, but in Fibonacci's own preface to ,he specifically credits the Hindus:

...

(Muhammed the Prophet of Allah is #1.)Whatever the criteria, Newton would certainly rank firstor second on any list of physicists, or scientists in general,but some listmakers would demote him slightly on a list ofpure mathematicians:his emphasis was physics not mathematics,and the contribution of Leibniz(Newton's rival for the title )lessens the historical importance of Newton's calculus.

(Al-Biruni's contemporary Avicenna was not particularly a mathematicianbut deserves mention as an advancing scientist, as does Avicenna's discipleAbu'l-Barakat al-Baghdada, who lived about a century later.)Al-Biruni has left us what seems to be the oldest surviving mentionof the Broken Chord Theorem (if M is the midpoint of circular arc ABMC,and T the midpoint of "broken chord" ABC, then MT is perpendicular to BC).

I won't try to summarize Leibniz' contributions to philosophyand diverse other fields including biology; as justthree examples: he predicted the Earth's molten core,introduced the notion of subconscious mind,and built the first calculator that could do multiplication.

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