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- Mathematics In Europe - 2012: P.-S. Laplace wrote the "Théorie analytique des probabilités" in 1812

to be mere curiosities and in fact their promotion led by the middle of the nineteenth century to a temporary rejection of probability theory by the majority of mathematicians These concerned the believability of eye witness testimony in court the probability of just verdicts and the accuracy of elections and voting procedures in the light of probability and the integration of psychological points of view Altogether for Laplace and his followers considerations of probability are motivated by nothing other than incomplete knowledge of something that has to be judged or decided Laplace s Théorie analytique was preceded by numerous publications on separate topics beginning in 1774 with the Memoir on recurrent series and their application to the theory of games of chance and the Memoir on the probability of causes of events Finally in 1814 to the second edition of the Théorie analytique was added a preface consisting of an edited version of a popular that is without formulas lecture a philosophical essay on probabilities that he had given in 1795 at the École normale We quote from the final sentences One sees in this essay that the theory of probabilities is basically only common sense reduced to a calculus It makes one estimate accurately what right minded people feel by a sort of instinct often without being able to give a reason for it It leaves nothing arbitrary in the choice of opinions and of making up one s mind every time one is able by this means to determine the most advantageous choice if one observes also that even in matters which cannot be handled by the calculus it gives the best rough estimates to guide us in our judgements and that it teaches us to guard ourselves from the illusions which often mislead us one will see

Original URL path: http://mathematics-in-europe.eu/tr/anasayfa/47-information/math-calendar/753-2012-p-s-laplace-wrote-the-theorie-analytique-des-probabilites-in-1812 (2013-11-18)

Open archived version from archive - Mathematics In Europe - Postage stamps and the history of mathematics

The European Mathematical Society Our Sponsor Munich RE Postage stamps and the history of mathematics Ayrıntılar Kategori Strick Here we present translations of articles that have been written by Heinz Klaus Strick Germany translations David Kramer They appeared originally in German in 2009 and 2011 in the German science journal Spektrum der Wissenschaft They are republished here with the kind permission of the author and Spektrum der Wissenschaft Abel d

Original URL path: http://mathematics-in-europe.eu/tr/anasayfa/76-enjoy-maths/strick/598-postage-stamps-and-the-history-of-mathematics (2013-11-18)

Open archived version from archive - Mathematics In Europe - The Reuleaux Stones

lid rescuing There is of course a simple way to avoid the sewer lid accident using only circular lids Opposite to the shapes mentioned above the circle has constant diameter whatever its orientation its width maximum distance between two different points is constant So there is no way that a circular lid can fall through a circular opening of the same diameter in fact slightly smaller because of the inner edge it will always become stuck This would be the right answer for a Microsoft interviewer To a mathematician however it suggests a new question is the circle the only curve with the property of having constant width Maybe it seems counterintuitive but there exists an infinite number of curves with the constant width property The first one was introduced by the German mechanical engineer Franz Reuleaux 1829 1905 a professor in Berlin in the context of machine engineering Constructed from an equilateral triangle it became known as the Reuleaux triangle fig 1 although it is obviously not a triangle in the mathematical sense Reuleaux called it a curved triangle Figur 1 The Reuleaux triangle Here s the construction of the Reuleaux triangle We start off with an equilateral triangle with side L With center at each vertex we sequentially trace an arc of circle of radius L joining the two opposite vertices These three arcs of circle compose the boundary of the Reuleaux triangle Now given an arbitrary point P at this boundary let us determine the width at P that is the maximum distance between P and any other point of this curve Since P is on an arc of circle of radius L centered at the farthest vertex it follows that the width at P is L But P is an arbitrary point on the Reuleaux triangle Thus the Reuleaux triangle has constant width L It is easy to see that it is the constant width property which is at stake when trying to avoid the sewer lid accident In fact some cities use Reuleaux shaped sewer lids see fig 2 Fig 2 Reulaeux shaped sewer lids Mathematically approved More generally we may apply Reuleaux s construction to any odd sided regular polygon obtaining the so called Reuleaux pentagons Reuleaux heptagons and so on see fig 3 an infinite family of constant width curves What about if we start with even sided polygons Well in that case we still end up with a constant width curve but in the even side case we always end up with a circle so there s nothing new to learn here Fig 3 The first four Reuleaux polygons Reuleaux polygons are much more than a simple curiosity British people touch them literally every day several coins in the British currency are Reuleaux heptagons fig 4 Given the wish for aesthetic originality in designing a non round coin why choose a Reuleaux shape Because the constant width property ensures that it will always roll smoothly not becoming stuck for instance in vending

Original URL path: http://mathematics-in-europe.eu/tr/anasayfa/13-frontpage/popular-articles/748-the-reuleaux-stones (2013-11-18)

Open archived version from archive - Mathematics In Europe - The proposals of May 2012 for the "top five inspirations of the month"

org millennium on the millennium problems has links to an online video of a lecture by Fields Medal winner Tim Gowers on The importance of mathematics Aimed at the general public it assumes little mathematical background and is also on YouTube at http www youtube com watch v BsIJN4YMZZo in 8 parts and there is a transcript at http www dpmms cam ac uk wtg10 importance pdf In the Rare Book Room website http rarebookroom org are several mathematical works including versions of Euclid s Elements Newton s Principia Mathematica and Napier s logarithms as well as astronomical works of Copernicus Kepler and Galileo There are also a number of other versions of Euclid s Elements the earliest surviving edition dating from AD 888 can be found on http www claymath org library historical euclid and a dynamic version of it using a Java applet is on http aleph0 clarku edu djoyce java elements elements html A good Archimedes site containing some original sources is http www cs drexel edu crorres Archimedes contents html For the famous Archimedes palimpsest described in Reviel Netz and William Noel s prizewinning book The Archimedes codex look at http www archimedespalimpsest org and especially the links to The Palimpsest and Conservation For Newton there are several useful websites The Newton project has biographical information and many of his writings http www newtonproject sussex ac uk prism php id 1 The following is taken from their website Welcome to the Newton Project Although Sir Isaac Newton 1642 1727 is best known for his theory of universal gravitation and discovery of calculus his interests were much broader than is usually appreciated In addition to his celebrated natural philosophical writings and mathematical works Newton also wrote many theological texts and alchemical tracts We already have texts and

Original URL path: http://mathematics-in-europe.eu/tr/77-enjoy-maths/topfive/680-the-proposals-of-may-2012-for-the-top-five-inspirations-of-the-month (2013-11-18)

Open archived version from archive - Mathematics In Europe - Magical Mathematics: A Quick Arithmetic Trick

follows You ask a spectator to choose which of two numbers say 6 and 11 he she prefers but not to tell you his her choice Then tell him her to multiply the number he she prefers by say 3 and the other by say 2 add the two products and tell you the result You immediately know which number 6 or 11 was the sprectator s favourite How You may have noticed the many say s in the preceding paragraph This is because both for the two numbers among which the spectator has a choice you can choose any two numbers as long as one is even and the other is odd as you can for the two multipliers 3 and 2 in our example The only important thing is that you instruct that the spectator multiplies the chosen number by the odd multiplier and the other by the even multiplier and then to add the products The secret is in the arithmetic of even and odd numbers A product of an even number with any other number is even and the product of two odd numbers is odd Also the sum of two numbers of different parity is

Original URL path: http://mathematics-in-europe.eu/tr/anasayfa/78-enjoy-maths/recreational-mathematics/1028-magical-mathematics-a-quick-arithmetic-trick (2013-11-18)

Open archived version from archive - Mathematics In Europe - How many candies do you have?

it s time to do the trick First you instruct Alexander to take any number of candies from the bag You must turn away so that at no moment you see how many candies who of the two volunteers has Now Alexander is to show Barbara his candies and she should count them and then she should take 6 times as many from the bag For example if Alexander took 8 candies she should take 48 Now tell Alexander to give Barbara 2 more from his candies in the described example this leaves Alexander with 6 and Barbara with 50 candies He may be grumbling now because Barbara has so many candies but the next step will make Barbara grumble tell her to give to Alexander from her candies 6 times as many as he still has In our example this would mean Barbara giving 6 6 36 candies to Alexander Finally its your turn to make a dramatical announcement that you know how many candies Barbara still has the number is 14 Of course if you chose some other two numbers than 6 for the two multiplication instructions and 2 for the one subtraction the number of candies Barbara will end up is the corresponding number you have calculated in the beginning Finally think about the maths The maths behind this trick is as often simple algebra the instructions are designed so that the number Barbara ends up with does not depend on the number of candies Alexander chooses in the beginning If you closely follow the described procedure you should be able to work it out but in case you prefer to have it all spelled out here it is Denote your first number the one used for multiplication by m and the other by s consequently the

Original URL path: http://mathematics-in-europe.eu/tr/78-enjoy-maths/recreational-mathematics/749-how-many-candies-do-you-have (2013-11-18)

Open archived version from archive - Mathematics In Europe - The Three Ducks Trick

duck but not name it Then the mathemagician turns his back to the table and gives the following instruction First switch the two ducks you haven t chosen and do not tell me the positions involved After that you can switch any two ducks as many times as you want but for all these other switches you tell me the positions involved After that the mathemagician turns around and can instanteneously tell which duck the participator has chosen in the beginning Say the participator selected the yellow duck Then in the first switch he would change the positions of the pink and blue duck After that he continues switching but with naming the positions say he switches now the pink and yellow duck In this case he says that he has switched positions 1 and 2 Then he decides to switch the pink and yellow duck again saying the switch involved positions 1 and 2 When switching the pink and blue duck he says that he switches 2 and 3 Say that the participator decides to stop here The mathematician turns around and says You chose the yellow duck How it works The principle of this trick is simple logic All the mathemagician has to do is to choose a duck himself in the beginning When the participator starts announcing the position changes it is not hard to follow the path of his duck Say the mathemagician chose the blue duck in the beginning and remembers its position 2 When participator says that he has changed 1 and 2 the mathemagician supposes the blue duck is now at position 1 In the next switch above the same positions are involved so the blue duck should be back to 2 I the final switch 2 and 3 are involved so in

Original URL path: http://mathematics-in-europe.eu/tr/78-enjoy-maths/recreational-mathematics/731-the-three-ducks-trick (2013-11-18)

Open archived version from archive - Mathematics In Europe - The clairvoyant (mathe)magician

1 and jacks queens and kings have value 10 Is this O K with you Yes Fine Now I return the pack of cards to you Four cards are left on the table Well you ask Well I say now deal from the pack onto each of the face up cards as many cards from the pack as is the difference between 10 and the card value Face up you ask I don t care just take care to count correctly is my answer So you deal 7 cards on the three of hearts one card on the nine of hearts 6 cards on the four of diamonds and no cards on the king of clubs And you still have some cards left in your hand Onto each card as many cards are dealt as is the value of the difference between 10 and the card value Now I will concentrate is my line and you see me closing my eyes putting my thinking magician s hat on Yes I have it You have what you ask I think I am able to see the twenty sixth card in the pack in your hand Please find it Holding the pack face up or face down Down if you please And you do as you have been told to do And I say I see something red and a number with two digits and a man with a hat he looks somehow sleepy It s Robert Mitchum i e the 10 of diamonds isn t it And you are surprised or at least you act as if you were because I am right The performer knows the 26th card if the cards have been chosen as shown in the previous pictures Now dear reader you ask yourself what has happened First

Original URL path: http://mathematics-in-europe.eu/tr/78-enjoy-maths/recreational-mathematics/690-the-clairvoyant-mathe-magician (2013-11-18)

Open archived version from archive