Archimedes' Principle?
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1. The bouyant force of the water on the object is the difference in the reading, i.e. BF=(.0554-.0370)g\n                                   =0.0184gN\n\nBased on Archimedes' Principle, BF =Volume*Density of water displaced.\n \n0.018g=Volume*1000g\nVolume=.018/1000\n            =1.8*10^-5m3\n\n2. Density of Object=Mass/Volume\n                                =.0554/1.8*10^-5\n                                =3078kg/m3
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You might find this paper useful\nhttp://arxiv.org/abs/math-ph/0106027
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A sicily born mathematician/physicist.\n\nArchimedes (Greek: Αρχιμήδης ) (c. 287 BC–212 BC) was an ancient Sicilian mathematician, physicist, engineer, astronomer and philosopher born in the seaport colony of Syracuse. He is considered by some historians of mathematics to be one of the greatest mathematicians in antiquity; Carl Friedrich Gauss considered him one of the two greatest ever.\n\nWell known for proposing Archimedes principle related to relative density of object and floatation. When he found the truth about relative density.. it is said that, he ran out of the bath yelling at the top of his voice -- "Eureka, eurteka" -- meaning I found it i found it.
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Pythagorean Theorem\n \nLet's build up squares on the sides of a right triangle. Pythagoras' Theorem then claims that the sum of (the areas of) two small squares equals (the area of) the big one.\n\nIn algebraic terms, a2 + b2 = c2 where c is the hypotenuse while a and b are the legs of the triangle.\n\nThe theorem is of fundamental importance in the Euclidean Geometry where it serves as a basis for the definition of distance between two points. It's so basic and well known that, I believe, anyone who took geometry classes in high school couldn't fail to remember it long after other math notions got thoroughly forgotten.\n\nBelow is a collection of 69 approaches to proving the theorem. Some of the proofs are accompanied by Java illustrations, but most have been written in plain HTML.\n\nRemark\nThe statement of the Theorem was discovered on a Babylonian tablet circa 1900-1600 B.C. Whether Pythagoras (c.560-c.480 B.C.) or someone else from his School was the first to discover its proof can't be claimed with any degree of credibility. Euclid's (c 300 B.C.) Elements furnish the first and, later, the standard reference in Geometry. In fact Euclid supplied two very different proofs: the Proposition I.47 (First Book, Proposition 47) and VI.31. The Theorem is reversible which means that a triangle whose sides satisfy a2 + b2 = c2 is necessarily right angled. Euclid was the first (I.48) to mention and prove this fact.\n\nW. Dunham [Mathematical Universe] cites a book The Pythagorean Proposition by an early 20th century professor Elisha Scott Loomis. The book is a collection of 367 proofs of the Pythagorean Theorem and has been republished by NCTM in 1968.\n\nIn all likelihood, Loomis drew inspiration from a series of short articles in The American Mathematical Monthly published by B. F. Yanney and J. A. Calderhead in 1896-1899. Counting possible variations in calculations derived from the same geometric configurations, the potential number of proofs there grew into thousands. For example, the authors counted 45 proofs based on the diagram of proof #6 and virtually as many based on the diagram of #19 below. I'll give an example of their approach in proof #56. (In all, there were 100 "shorthand" proofs.)\n\nPythagorean Theorem generalizes to spaces of higher dimensions. Some of the generalizations are far from obvious.\n\nLarry Hoehn came up with a plane generalization which is related to the law of cosines but is shorter and looks nicer.\n\nThe Theorem whose formulation leads to the notion of Euclidean distance and Euclidean and Hilbert spaces, plays an important role in Mathematics as a whole. I began collecting math facts whose proof may be based on the Pythagorean Theorem.\n\nWherever all three sides of a right triangle are integers, their lengths form a Pythagorean triple (or Pythagorean numbers). There is a general formula for obtaining all such numbers.\n\nMy first math droodle was also related to the Pythagorean theorem. Unlike a proof without words, a droodle may suggest a statement, not just a proof.\n\nSeveral false proofs of the theorem have also been published. I shall collect some in a separate page. It is better to learn from mistakes of others than to commit one's own.\n\nIt is known that the Pythagorean Theorem is Equivalent to Parallel Postulate.\n\nThe Pythagorean configuration is known under many names, the Bride's Chair being probably the most popular. Besides the statement of the Pythagorean theorem, Bride's chair has many interesting properties, many quite elementary.\n\nProfessor Edsger W. Dijkstra found an absolutely stunning generalization of the Pythagorean theorem. If, in a triangle, angles a, b, g lie opposite the sides of length a, b, c, then\n\n(EWD) sign(a + b - g) = sign(a2 + b2 - c2),  \n\nwhere sign(t) is the signum function:\n\n  sign(t) = -1, for t < 0, \nsign(0) =  0, \nsign(t) =  1, for t > 0. \n \n\nThe theorem this page is devoted to is trea
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I assume this is an analysis of Archimedes' famous quote "Give me a place to stand, and I will move the Earth."  Unfortunately, while this is mathematically sound, it is practically impossible!\n\nLet me explain.  Say that the pole has a length L.  We'll place the Earth at one end, a person at the other end, and a fulcrum at a distance D from the Earth.  If the Earth exerts a force F_E and the person exerts a force F_0, then we have the equation\n\nF_E * D = F_0 * (L-D)\n\nThis was proved by Archimedes, of course.  We know that an object of mass M exerts a gravitational force of F = M g, where g = 9.81 m/s^2.  The Earth has a mass M_E = 5.9742 x 10^24 kg, and an average person has a mass M_0 = 70 kg.  This gives the the equation\n\n5.9742 x 10^24 * D = 70 * (L-D)\n\nso that L/D = 8.53457 x 10^22.  The length of the pole L depends on the location D where the fulcrum is placed.\n\nUnfortunately, this ratio is impractically large: If we place the fulcrum at a distance of D = 2 micrometers (which is slightly smaller than a strand of spider web), then the pole must be a length of L = 16.3 light-years.  This is farther in distance than the fartest known star!
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Galileo's principle of relativity essentially states that, regardless of an observer's position or velocity in the universe, all physical laws will appear constant. From this principle, it follows that an observer cannot determine either his absolute velocity or direction of travel in space.\n\nhttp://farside.ph.utexas.edu/teaching/jk1/lectures/node7.html\n\nhttp://en.wikipedia.org/wiki/Principle_of_relativity
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You find it in the normal coordinates way; i.e.\n((x1+x2)/2, (y1+y2)/2)\n\nThe curvature of our Earth does not affect it as the coordinates we are using is 2-dimensional.
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Archimedes. You have to take into account the shape of the hull. The floatline will be in the line where the mass of water displaced ( this is the volume of the boat under the water x water density ) equals the mass of the boat.
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yes.\nthe principle of relativity which was discovered as far back as the 17th century essentially states that motion and position of an object are  RELATIVE to the frame of referance of the observer.For example,from the earth's POINT OF VIEW, the sun moves.(which is why we say that the sun rises in the east & sets in the west).\nBut from the sun's frame of referance,it is the earth that moves around it. this is an obvious & established fact and is known as principle of relativity.\nwhen Einsten came around, he proposed the special theory of relativity which predicts stronger & surprising results.For eg,all along,the principal of relativity assumes that the length of an object does not change when it's in relative motion.it also assumes that the time passed on a moving frame is the same as that of a stationary frame.these assumptions were proved wrong using  special relativity.\nSo there is no reason to 'mix' up the principal of relativity & special relativity.for a fuller treatment,you may visit the folowing webpage:\nhttp://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Special_relativity.html
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You **did** say that the Pythagorean Theorem was a given, right? I think that your INTENT is to prove the converse of the Pythagorean Theorem, right?\n\nLet x, y, and z be positive integers such that\nx^2 + y^2 = z^2.\nClearly z > x and z > y (otherwise either x^2 or y^2 must be less than zero; prove this for yourself if you must).\n\nLet x and y be the sides of a right triangle that form the right angle.\nThen, by the Pythagorean Theorem the length of the hypotenuse h is\nh = √(x^2 + y^2)\nh = √(z^2) . . . . . . . . (given equation above)\nh = z . . . . . . . . . . . (given z is positive) \n        \nTherefore, by side-side-side congruence, any triangle whose sides are of lengths, x, y, and z, where x^2 + y^2 = z^2, must be a right triangle with hypotenuse z.\n\n\nASIDE TO HOOD...: You are using the Pythagorean Theorem to prove the Pythagorean Theorem. Although the questioner's intent is not perfectly clear, I don't think the intent was to prove "if x, then x". If so there are easier ways to do so.
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If you are referring to an area of a circle then Archimedes did that a while ago by method of exhoustion. http://people.bath.ac.uk/slt20/archimedesmethod.html \n\nHowever I think you are looking for a derivation in Cartesian coordinates.\n1.Determine what is the relationship of a radius to the x and y. \nFrom Pythagorean theorem x^2 + y^2 = r^2, where radius r is the hypotenuse. \n2.The equation above is good but not general enough since it does not incorporates shifts in x and y axis.  The x-a shifts to the right and x + a shifts the circle to the left. Similarly for y. then we have a general equation\n\n(x-a)^2 + (y-b)^2=r^2  \n\n\nI hope that helps.
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The Pythagorean Theorem definitely: a^2+b^2=c^2 \n\nPoint slope form: y - y1 = m(x-x1) \nStandard y equation: y=mx+b\n...Sorry, cant think of any more. \nSee, the rest of the equations that i could give you are not the ones EVERYONE should know :)
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The water supplies lift force. Archimedes.
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This is tricky without a particular formula to derive. The master of this was Richard Feynmann. Try his Six Easy Pieces. His lectures on the motion of the planets and his autobiographies are fun easy reading.\n"Surely You're Joking Mr Feynmann"  is the first.\nThe Physics Lecture series are classic physics books.
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Is that the same principle as two negatives making a positive?\nThen the answer would be yes.
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Well, I think I found something; it says that the rule is part of the Newton-Cotes formulas (Isaac Newton and Roger Cotes).\n\nThis is the page where I found the info:\nhttp://www.absoluteastronomy.com/ref/trapezium_rule
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The Pythagorean Theorem says that if you have a right triangle, then a squared + b squared = c squared.  The converse says that if you have a triangle in which a squared + b squared = c squared, then your triangle must be a right triangle.