|
 |
May 04, 2003 THE RIEMANN HYPOTHESIS....I finished my book on the Riemann Hypothesis, and unfortunately I didn't think much of it. It turns out that I've already read another book by the same author, and I didn't like that book too much either, so maybe I just don't like his books. But it goes beyond that, I think, and the problem is one that infects an awful lot of popular books about math: most of them discuss things that require at least some familiarity with topics beyond basic arithmetic. In this case, for example, you really need to know what a complex number is. So the author has two choices: (a) write the book for an audience that already knows what complex numbers are, which reduces your potential readership to a Very Small Number, or (b) explain complex numbers. Most authors, including this one, choose option (b). This is annoying to me because I then have to wade through dozens of pages I don't need to read, something that has to be done with care since these pages invariably also contain details here and there that are important to other parts of the story. What's more, I have my doubts that this works anyway. First of all, I wonder just how many non-mathophiles are going to read a book like this in the first place, and of the ones who do, I wonder how many actually end up understanding the kinds of tortured analogies that are usually used to explain difficult concepts. In this case the author analogizes complex numbers to streets and cross streets in New York, and then tries to convince us that the cross streets are somehow related to the square root of -1. Fair enough, and I don't know that I could do any better, but does it actually help? Does anyone who didn't understand the concept in the first place understand it better after reading an explanation like this? I honestly don't know, but I can't help but think that these efforts are doomed. The book has plenty of scary looking equations, and I'm pretty sure that simply saying "don't be scared!" doesn't do much to broaden your audience. It might be better to simply assume that anyone interested in a topic like this is already familiar with high school algebra, accept the fact that this will reduce your audience a bit, and be done with it. POSTSCRIPT: Of course, a lot of this has to do with the skill of the author. For example, a very good book about math that I read a few years ago was The Mystery of the Aleph, by Amir Aczel, a short book about Georg Cantor and transfinite set theory. Aczel did a great job of explaining the math in an understandable way, weaving it seamlessly with interesting historical background and a biography of Cantor himself. My recollection is also that Aczel simply assumed, for example, that his readers knew what an exponent was and didn't waste time trying to teach basic algebra. It made for a much better book. Posted by Kevin Drum at May 4, 2003 03:02 PM | TrackBackComments
>Does anyone who didn't understand the concept in the first place understand it better after reading an explanation like this? I am not even sure what it means to understand complex numbers. I went through an electrical engineering education, and most of us just sort of accepted that sqrt(-1) has nice (if weird) properties, e.g. a Fourier transform can be written much more elegantly using complex numbers, or that complex eigenvalues in a linear system implies oscillatory behavior. In other words, I know how to formally manipulate complex numbers, but don't have a "gut feel" for how those formal manipulations came about in the first place. Perhaps this is the difficulty with writing about math. An idea in, say, economics, can be illustrated with a few key examples -- one can look "under the hood" of the formulas and talk about the underlying concept. With "complex number" this does not seem possible. There seems no way to look "under the hood" of the formal operations (maybe a graduate-level math education is needed for that). Posted by: Gautam Vallabha at May 4, 2003 04:08 PM | PERMALINKI think the explanation of what are to some people basic concepts is for a very specific portion of the audience: those who like math, are interested in math, but haven't actually done anything with the types of math discussed in the book for several years. You know, a brief refresher course before you can attack the rest of the book. I mean, I've been out of highschool for a relatively short amount of time compared to the general population, but since my field of study doesn't involve much higher-level math, some of the old concepts, while easily grasped back in the day, have accumulated a bit of dust in my skull. A few pages devoted to teaching a relativley concept makes a book that would be frustrating to mathematical laymen suddenly accesible. Posted by: The Mighty Reason Man at May 4, 2003 04:43 PM | PERMALINKGautam, I think the biggest difficulty is just getting people to realize that the number system they've used since kindergarten isn't adequate to represent everything. And that alone is hard because we're taught that numbers should have some sort of physical representation, which isn't possible with sqrt(-1) = i (or j for us EEs.) There simply is no real-world example of i apples. You actually do run into the same sort of problem with negative numbers. Those are a little more tangible, though, since you can liken them to ideas such as debt. I think I got around it by deciding that it's perfectly reasonable to want to take the square root of a negative value. But since the real number representation I'd known so well couldn't do that, it made sense to use a more general notion of a number. Posted by: Jon M at May 4, 2003 04:43 PM | PERMALINKChange that last sentence in that last post to "A few pages devoted to teaching a relatively basic concept can make a book that would be frustrating to a mathematical layman suddenly accessible." I need more editors around here. Posted by: The Mighty Reason Man at May 4, 2003 04:46 PM | PERMALINKI haven't read this book, but this is a pervasive and very difficult problem in semi-popular writing about mathematics---the audience is very spotty in terms of what they know. It's partly a question of critical mass---there is much more science writing in biology, physics, chemistry, etc. than in math, despite the fact that most people "take" much more math than these other fields. Another issue: even when people think they are acquainted with complex numbers, how they understand them is highly variable. Once when I gave a presentation to some high school students, I was startled to learn that although they had "studied" complex numbers, they were purely formal symbols to manipulate---they had no idea that they could be represented as points in the plane, with easy geometric interpretations for the arithmetical operations. These kinds of issues are crucial. As an intellectual enterprise, mathematics is tall and interconnected---new ideas depend on old ideas, and they build into tall scaffoldings. If someone's mental scaffold is shaky at any lower level, the whole thing is in danger of collapse. For what it's worth, I think there are ways to ameliorate the situation. First, mathematics is actually quite broad, there are a lot of different possible things to focus on when writing about it. Since there's a lot of choice, it's better to emphasize areas that are more novel to more people, so that there's not as big a mixture of background. Second, it's hard but usually possible to find ways to discard technical constructions and instead give informal explanations of almost anything, provided you understand it well enough yourself. A good informal explanation can communicate at different levels to a varied audience. In this regard, I'd like to shamelessly plug to the videos "Not Knot" and "Outside In" (with pamphlet-sized supplements that answer a lot of questions) available from the publisher, A.K. Peters http://akpeters.com/. They're both full of mathematical content at many levels, but research mathematicians as well as people totally unversed in mathematics often seem to get something from them. E.g. the Grateful Dead used parts of Not Knot in their shows for about a year. We tried very hard to find ways to communicate sophisticated ideas without getting too pedantic. Of course, as the internet and hyperlinking becomes more an everyday thing, it becomes easier to refer people somewhere outside the main narrative to learn some subject such as complex numbers. In actual math books, this is often done using appendices or extended glossaries, and can work pretty well. In a semi-popular physical book it's not what people expect to encounter, but when people start to habitually read books electronically it'll probably become the norm. Posted by: Bill Thurston at May 4, 2003 05:02 PM | PERMALINKI suspect a large part of the problem is the question of "why is this book being written"? I think there is (one day, but not quite yet) hope for books written by scientists to appeal to science-minded people of whatever ability is of interest to the author. What defeats such books right now is that they are routed through a publishing process that ruthlessly strips out anything "scary looking", meaning equations, graphs and suchlike. Future web-published material (ie material written for love, not money) has the potential to do better. Of course books written by professional authors are doomed and pretty much always will be, since their primary interest is as large an audience as possible. As for what to do today, my advice is simply don't waste your time buying what I call "baby-science" books. You're a smart guy, which means that whether it's a pop-science book about biology, chemistry, physics, math or astronomy, you probably know pretty much everything in it; and you'll just be disappointed yet again by the predictability and lack of content [much like whenever you pay to see a hollywood movie]. Do yourself a favor and buy textbooks --- either glossy 1st yr textbooks (second-hand bcs new they cost $100 a pop) to get an overview of a field --- or Dover paperbacks ($5-$25, and generally already screened as being classics in the field). Finally remember how to read math/physics text books. The first time
you read a field, you're going to get lost pretty much after the second
chapter. That's fine, persist, skimming over the parts that really mean
nothing like the proofs. The goal is to get a feel for the terminology
and notation of the field, to see what its concerns are, where it is
headed, what it does that's ultimately interesting and what it does that
is merely scaffolding, necessary to get to the good stuff but not much
interesting in and of itself. Is assuming that (in order to describe some phenomena conveniently) it makes sense after all to regard -1 as having a square root really any more outlandish than assuming that you can multiply a number by itself a FRACTIONAL number of times? Or, for that matter, that there can be such a thing as "less than zero"? Posted by: Bruce Moomaw at May 4, 2003 09:46 PM | PERMALINKMaynard: what you said, except for the stuff about price. You can find all kinds of textbooks dirt cheap at thrift stores and garage sales, especially if you're not looking for anything particular. Me, I'm a humanities kind of guy (actually, I did have an early affinity for math, but I must have burned away those particular brain cells in high school), but I've lately become interested in economics, so I picked up a copy of Baumol and Blinder's Macroeconomics: Principles and Policy (6th ed.: 1994) at the local Sally. Price: 37 cents. Good readin', with lots of pictures, even the examples weren't quite up to date, and it was, I think, subtly right-leaning (soft on Reagonamics, and nothing about stuff liken the greater tendency of poor people to spend rather than save). Posted by: David at May 4, 2003 09:52 PM | PERMALINKGautam: "In other words, I know how to formally manipulate complex numbers, but don't have a "gut feel" for how those formal manipulations came about in the first place." Did you have a gut feel for similar triangles when you studied geometry? The shape of a triangle is essentially the same thing as a complex number---many phenomena in plane geometry can be translated into complex numbers in a way that clarifies both. This is better illustrated with pictures, but a good way to think about complex numbers, especially on the "gut" level, is that they measure the shape of a triangle. The ancient Greeks thought of proportions of line segments, rather than real numbers--- essentially the same concept, just a very different mental image to imagine a ratio of line segments rather than a string of decimal digits. Complex numbers are really a 2-dimensional version of a ratio---i.e. a complex number describes the relationship of two sides of a triangle that meet at a vertex. To multiply two complex numbers, you put similar copies of the triangles together at the vertex. i = sqrt(-1) is basically the statement that if you turn left twice, you're going backwards. Complex analysis is actually a very beautiful and intuitive geometric subject, just usually not taught very geometrically. Quite a lot of it is accessible on "from scratch" --- a lot can readily be absorbed and appreciated say by an interested 4th grader. It also has surprisingly many interesting and significant applications. If you're actually interested, I think Tristan Needham's book "Visual complex analysis" takes a pretty good approach---various people have told me they liked it, but I haven't actually taught from it or more than browsed it so I'm not completely sure. Posted by: Bill Thurston at May 4, 2003 11:18 PM | PERMALINK> i = sqrt(-1) is basically the statement that if you turn left twice, you're going backwards. Thanks, Bill! This is really intriguing, and I going to take a look at the Needham book you recommended. By the way, this notion of geometric intuition seems to bear on the relation topic of whether mathematics is symbolic thinking. We seem to be biologically wired for thinking in certain ways about spatiality and causality, and "gut feel" seems to relate to how well we can map a domain into these biologically wired abilities. Posted by: Gautam Vallabha at May 5, 2003 07:43 AM | PERMALINKWhy not have an appendix where the mathematical concepts If you are interested in the physical properties of i, I would suggest the excellent book: An Imaginary Tale, by Paul J. Nahin. I was a math major for 2 years before I changed my field of study, and it wasn't until I read this book that I really understood the function of i in real-world terms. Posted by: Sebastian Holsclaw at May 5, 2003 09:49 AM | PERMALINKI would second the recommendation for Visual Complex Analysis by Needham. As a math teacher, I have found that knowing many different analogies and representations for an idea can be very help for both understanding and explaining mathematical ideas. There is no one correct way of thinking of math ideas, which is why I think (back to a previous discussion) math is more than just pushing symbols around: the connections, patterns and representations are HOW we understand the math. I would recommend the Lakoff and Nunez book, Where Mathematics Comes From, not for its repetitious style or its minor errors, but because it makes explicit how much of how we think mathematically comes from our analogies and metaphors, i.e., our representations. Cheers Posted by: Eddi Vulic at May 5, 2003 11:05 AM | PERMALINKUK reader here. I've actually skimmed the book a few times (being a cheapskate I'm waiting for the paperback) and find it quite enjoyable. Certainly preferred it to Singh's book on FLT, tho' that's probably a matter of personal taste. Some scattered thouhgts and responses: "I wonder just how many non-mathophiles are going to read a book like this in the first place, and of the ones who do, I wonder how many actually end up understanding the kinds of tortured analogies that are usually used to explain difficult concepts." Good point. On t'other hand one reason I'd suggest this book to non-math friends is that it gives an impression (to me at least) of what *doing pure maths* is like. Maybe you do well out of the broader USA system but over here most people are convinced early in life that "pure" maths is about doing lots of calculations with unfamiliar notation; the creative possibilities and the sense of inter-dependent structure don't get much of a mention, because it's so hard to communicate them in "popular" terms. The whole de Branges aspect - which is more about mathematicians as a social group than the actual operator theory - is also dealt with quite well. Regarding the question "why this book?" -- again, I think the main reason is (if you're not a mathematician) to get some idea of what some mathematicians do. "A few pages devoted to teaching a relatively basic concept can make a book that would be frustrating to a mathematical layman suddenly accessible." Amen. Same goes for any other subject really; the reader needs a way in, or a foothold of some sort. BTW Kevin, I'd be interested to know if you felt the book left you with a better (or worse) idea of the mathematics involved. P.S. I'm sure loads of people have already pointed this out, but ... Bill Thurston? Wow. Glad to see some people believe as much in communicating about maths as in advancing research. Posted by: Yemon Choi at May 7, 2003 01:11 PM | PERMALINKI search for blog like this long time.You website is very good!I will come next time!
Cialis has been an eventual success in Europe since its introduction in Early 2003.Cialis will now be available in US soon. You may buy Cialis through various registered pharmacies. levitra buy cialis cheap cialis buy levitra cheap levitra http://www.one-levitra.com/ http://www.one-cialis.com/levitra.htm/ http://www.one-cialis.com/ Posted by: cialis at March 14, 2004 04:57 PM | PERMALINKWe provide a comprehensive list of e-pharmacies to help you get the best Levitra deals. cialis online cialis buy cialis cheap cialis buy levitra cheap levitra online levitra http://www.one-cialis.com buy cialis http://www.one-cialis.com http://www.one-levitra.com http://www.one-levitra.com/cheap_levitra_one.htm Posted by: levitra at March 14, 2004 07:57 PM | PERMALINKTramadol hydrochloride is a centrally acting analgesic. The chemical name for tramadol hydrochloride is (±)cis-2- [(dimethylamino)methyl]-1-(3-methoxyphenyl) cyclohexanol hydrochloride. The molecular weight of tramadol hydrochloride is 299.8! buy tramadol online tramadol http://www.top-tramadol.com cheap tramadol Posted by: tramadol at March 15, 2004 07:35 PM | PERMALINKNice site. Cheap Propecia http://www.one-propecia.com/ is a new and effective treatment for male pattern baldness. View Online Propecia News. It is a capsule taken by mouth vs. a cream. A net increase in scalp hair count and hair regrowth was seen in over 80% of men for whom it was prescribed. Buy Propecia Now! Posted by: propecia at March 17, 2004 11:33 PM | PERMALINKAspirin most likely reduces heart attack risk by irreversibly blocking the enzyme COX-1 online fioricet, thereby impairing the ability of platelets in the blood to form clots, Dr. Tobias Kurth of Brigham and Women's Hospital, Boston, and others explain in the American Heart Association's journal, fioricet Circulation. NSAIDs buy fioricet also lock on to COX-1, but the effect is reversible. Cheap Generic Fioricet or visit this site: http://www.top-fioricet.com ! Posted by: fioricet at March 18, 2004 03:39 AM | PERMALINKCheap Soma Carisoprodol is a prescription medication that is used to relax your body, relax your muscles and help put stress and other difficulties behind you. Online Soma is now available online with a prescription. You can obtain a prescription online by answering a short questionnaire about your medical history Buy Soma or visit http://www.top-soma.com. Posted by: soma at March 18, 2004 11:49 AM | PERMALINKPhentermine Online and Adipex Online is the fastest growing online source for Buy Phentermine and Buy Adipex prescription medication that you can trust to get your order out quickly and privately. We provide info for Cheap Adipex and Cheap Phentermine. Visit this site: http://www.hot-phentermine.com/ Posted by: phentermine at March 18, 2004 12:06 PM | PERMALINKDo you know the difference between a failed interview and an amazing interview? Do you want to be able to answer even the toughest, meanest, and most low-down interview questions that you could ever be possibly asked? Do you want to go though your interview with confidence? Do you want to feel prepared, impress the interviewer, and win the job interview of your dreams? We provide you with job interview tips, visit http://www.job-interview-questions-tips.com Posted by: job interview questions at March 21, 2004 06:40 AM | PERMALINKGreat site. Order Phentermine COD online. Viagra Ultram Generic Fioricet, Fioricet most likely reduces heart attack risk by irreversibly blocking the enzyme COX-1 online fioricet, thereby impairing the ability of platelets in the blood to form clots, Dr. Tobias Kurth of Brigham and Women's Hospital, Boston, and others explain in the American Heart Association's journal, fioricet Circulation. NSAIDs buy fioricet also lock on to COX-1, but the effect is reversible. Cheap Generic Fioricet or visit this site: http://www.x-fioricet.com ! Posted by: fioricet at April 17, 2004 06:14 AM | PERMALINKOnline Ambien, ambien is a sleep inducing medication. A good night's sleep is invaluable for your mental and physical well being. If you suffer from insomnia, you may want to look into cheap Ambien (zolpidem tartrate), the most prescribed sleep medication in the U.S. Buy Ambien Now or visit this site: http://www.i-ambien.com! Posted by: ambien at April 18, 2004 09:23 AM | PERMALINKOnline Wellbutrin, wellbutrin, wellbutrin XL, wellbutrin SR is prescribed for the treatment of depression, but it is not for everyone. If you take cheap WELLBUTRIN XL, there is a risk of seizure, which is increased in patients with certain medical problems or in patients taking certain medicines. Buy Wellbutrin XL Now or visit this site: http://www.i-wellbutrin.com! Posted by: wellbutrin at April 18, 2004 09:25 AM | PERMALINKOnline Flexeril, flexerilis used to treat the pain and stiffness of muscle injuries, including strains, sprains and muscle spasms.Buy Flexeril, Cheap Flexeril Now or visit this site: http://www.online-flexeril.com! Posted by: flexeril at April 18, 2004 09:37 AM | PERMALINKOnline Xenical, Xenical is one of the most prescribed treatments for weight loss in the world. cheap Xenical is used with a reduced-calorie diet to help significantly overweight persons lose weight. It also helps to prevent regaining weight previously lost. Orlistat works by blocking some of the fat normally digested by the intestine. Buy Xenical Now or visit this site: http://www.x-xenical.com! Posted by: xenical at April 18, 2004 09:37 AM | PERMALINKCheap Soma Carisoprodol, Soma - Carisoprodol is a prescription medication that is used to relax your body, relax your muscles and help put stress and other difficulties behind you. Online Soma, Cheap Soma is now available online with a prescription. You can obtain a prescription online by answering a short questionnaire about your medical history Buy Soma or visit http://www.one-soma.com. Posted by: soma at April 18, 2004 09:38 AM | PERMALINKOnline Tramadol, Tramadol, Tramadol HCL is one of the most prescribed treatments for pain in the world. More than 55 million people have taken cheap Tramadol to relieve their back pain, shoulder pain, and other chronic conditions. By acting on parts of the brain that trigger pain, and by reducing the size of pain signals that travel throughout the body, Ultram Tramadol provides powerful pain relief in just minutes! Buy Tramadol Now or visit this site: http://www.x-tramadol.com! Posted by: tramadol at April 18, 2004 09:46 AM | PERMALINKPhentermine Online and Adipex Online is the fastest growing online source for Buy Phentermine and Buy Adipex prescription medication that you can trust to get your order out quickly and privately. We provide info for Cheap Adipex and Cheap Phentermine. Visit this site: http://www.hot-phentermine.com/ Posted by: phentermine at April 18, 2004 03:31 PM | PERMALINKPhentermine Online, Phentermine and Adipex Online is the fastest growing online source for Buy Phentermine and Buy Adipex prescription medication that you can trust to get your order out quickly and privately. We provide info for Cheap Adipex and Cheap Phentermine. Visit this site: phentermine diet pill, phentermine sale, phentermine free shipping, cheapest phentermine! http://www.x-phentermine.com/ Posted by: phentermine at April 18, 2004 03:35 PM | PERMALINKOnline Skelaxin, skelaxin, 800 mg skelaxin, 800mg skelaxin,400mg skelaxin is used to treat the pain and stiffness of muscle injuries, including strains, sprains and muscle spasms. Cheap Xenical should not be used if you have ever had an allergic reaction to carisoprodol, meprobramate or tybamate. Buy Xenical Now or visit this site: http://www.x-xenical.com! Posted by: skelaxin at April 19, 2004 03:41 AM | PERMALINKViagra, Are you looking for cheap Viagra online? Then you are on the right place. We provide one of the largest databases of online pharmacies on Web to help you buy only the cheap drugs for your problems. online Viagra buy viagra generic viagraVisit this site: http://www.x-viagra.com Posted by: viagra at April 21, 2004 02:32 PM | PERMALINKAdipex is used as an appetite suppressant. It is used in conjunction with an overall diet plan to reduce weight. online adipex buy adipex cheap adipex Visit this site: http://www.top-adipex.com Posted by: adipex at April 22, 2004 01:40 PM | PERMALINKHit it to scan your computer for spyware Posted by: free spyware scanner at April 28, 2004 02:38 PM | PERMALINKThanks for the great site. cheap Credit, debt and loan are all different words that boil down to the same thing: borrowing money from someone with a promise to pay it back, usually with interest. Personal loan allow you to borrow a fixed amount and then pay it back according to a fixed schedule. The same is for car loan, home loan, auto loan and college loan.Sometimes a loan will require collateral, which is basically property or assets that you promise to give the lender in the event that you are unable to repay the loan. Often, borrowers use personal online loan to pay for big purchases. http://www.i-loan-online.com Posted by: loan at May 12, 2004 02:15 AM | PERMALINKHere is the story: BlackJack originated in French casinos around 1700 where it was called "vingt-et-un" ("twenty-and-one") and has been played in the U.S. since the 1800's. Online BlackJack is named as such because if a player got a Black Jack of Spades and internet black jack an Ace of Spades as the first two cards (Spade being the color jack black of course), the player was additionally remunerated. http://www.888-blackjack.com/ Posted by: blackjack at May 12, 2004 01:14 PM | PERMALINKDo you need urgent cash till pay day? Now? Payday loan online is alot easier faster and less hassle then going to your bank for a personal loan or credit card.Payday on your paycheck, usually ranging from $100 to $500. What is a payday loan? Payday loans are meant to help tie you over when you are short on cash between paychecks. http://www.payday-loan-payday.com Posted by: payday at May 12, 2004 01:14 PM | PERMALINKKeno, Bingo, Slots, Poker and other all-time favorite games delivered by top-notch software at only trusted and certified casinos. A comprehensive online bingo and unique casino games portal. Select your game with the best bonuses and payouts! http://www.i-online-bingo.com Posted by: bingo at May 12, 2004 01:17 PM | PERMALINKDating and picture personal dating ads services for singles looking for love, romance, dating dates pen pals or relationships. Personals for the online dating and singles community. Post Your Free Dating, Singles, Picture Personal ads today. Visit us to chat with singles, look through personals, and read up on all the hilarious and heart-rending tribulations of dating service and romance today. http://www.dating-free-dating.com Posted by: dating at May 12, 2004 05:51 PM | PERMALINKDebt Counseling companies currently help over 1 million people to debt consolidation their unsecured loans, bills, and credit card debts into one easy payment, while saving thousands of dollars in unnecessary interest & credit fees. We can help you consolidate r debts with a consolidation plan that is just right for you! This free debt consolidation could save you 50% or more in monthly credit card payments. http://www.free-debt-consolidation-free.com Posted by: debt consolidation at May 12, 2004 05:56 PM | PERMALINKWhether you have a rigid poker strategy in mind before you hit the table or whether you play it by ear, having a strong foundation of poker knowledge is important to every player. If you know and understand something about this game (video poker, strip poker, online poker )that your opponent does not, you will play a better game. http://www.888-online-poker.com Posted by: poker at May 12, 2004 05:59 PM | PERMALINKWe offer online personals ads services for dating singles with many free services such as: anonymous dating email addresses, relationship advice for marriage, dating or singles, personal ads posting, match, and many more free online Dating Singles Personals: Personal ad Services... for all men and women. Our Dating Singles Personals site is more than those matchmaker, marriage, pen pals or photo match personal ads sites; we're much more! Want more than the strictly dating service, online personal ads and matchmaker dating singles sites? Then. http://www.dating-service-dating.com Posted by: dating at May 12, 2004 06:24 PM | PERMALINKOur best online gambling review pages will surpass all your expectations offering the best online casino sites on the web. Look no further, whether it's online sports betting, progressive poker, free games, flash downloads or fast casino downloads, we have the best online gambling reviews to meet anyone's needs. Looking for online gambling news? you'll find everything in once place including online gambling online tips, reviews and promotions. http://www.666-gambling.com Posted by: gambling at May 17, 2004 04:53 AM | PERMALINKAll Online Casino Gambling sites chosen & recommended here at Online Casino have been thoroughly tested by us for fairness, reliability, online casino game play realism, casino payout percentages, game speed, and bonuses. We have tested hundreds of casino sites ourselves in an effort to bring you the very best online casino promotions. The following is a small sampling of internet casino - only the best from Online Casino Rewards. http://www.666-casino.com Posted by: casino at May 17, 2004 04:54 AM | PERMALINKWith today’s low mortgage rates, many people are finding that owning a home is often as affordable as renting.Check home mortgage. If you have the funds for mortgage loan saved up to put together a downpayment as well as cover off various closing costs, then the ability to afford monthly mortgage online, property tax, insurance and utility payments, then home ownership is a great way to build up equity over time.http://www.i-mortgage-online.com Posted by: mortgage at May 17, 2004 06:48 AM | PERMALINKThanks for a great site. Visit our site too and shop Ambien. Posted by: Shop Ambien at May 18, 2004 07:24 AM | PERMALINKThey offer a variety of casino great online games, fabulous colorful graphics, music, other familiar If you are looking for a debt consolidation loan There is no obligation to the debt consolidation credit counseling, it is just a educational consultation of debt assistance. http://www.site-debt-consolidation.com Posted by: debt consolidation at June 1, 2004 07:28 PM | PERMALINKThey offer a variety of casino great online games, fabulous colorful graphics, music, other familiar Learn what more home poker players are learning. Low limit casino If you are looking for a debt consolidation loan There is no obligation to the debt consolidation credit counseling, it is just a educational consultation of debt assistance. http://www.own-debt-consolidation.com Posted by: debt consolidation at June 1, 2004 07:29 PM | PERMALINKLearn what more home poker players are learning. Low limit casino Learn what more home poker players are learning. Low limit casino If you are looking for a debt consolidation loan There is no obligation to the debt consolidation credit counseling, it is just a educational consultation of debt assistance. http://www.live-debt-consolidation.com Posted by: debt consolidation at June 1, 2004 07:29 PM | PERMALINKThey offer a variety of casino great online games, fabulous colorful graphics, music, other familiar We are an payday loan lender guide offering access to We are an payday loan lender guide offering access to The worlds best known Internet Casino you bet If you are looking for a debt consolidation loan There is no obligation to the debt consolidation credit counseling, it is just a educational consultation of debt assistance. http://www.fastest-debt-consolidation.com Posted by: debt consolidation at June 1, 2004 07:30 PM | PERMALINKThe worlds best known Internet Casino you bet If you are looking for a debt consolidation loan There is no obligation to the debt consolidation credit counseling, it is just a educational consultation of debt assistance. http://www.max-debt-consolidation.com Posted by: debt consolidation at June 1, 2004 07:30 PM | PERMALINKWe are an payday loan lender guide offering access to The worlds best known Internet Casino you bet Credit quality and debt-to-income-ratio affect the Credit quality and debt-to-income-ratio affect the Credit quality and debt-to-income-ratio affect the Whether you are looking for a free satellite TV system from the Credit quality and debt-to-income-ratio affect the Whether you are looking for a free satellite TV system from the But the increased number of online bingo cards was exactly what was needed to make bingo a staple at churches. http://www.i-play-bingo.com Posted by: bingo at June 1, 2004 07:31 PM | PERMALINKWhether you are looking for a free satellite TV system from the Learn what more home poker players are learning. Low limit casino But the increased number of online bingo cards was exactly what was needed to make bingo a staple at churches. http://www.i-win-bingo.com Posted by: bingo at June 1, 2004 07:31 PM | PERMALINKLearn what more home poker players are learning. Low limit casino But the increased number of online bingo cards was exactly what was needed to make bingo a staple at churches. http://www.e-play-bingo.com Posted by: bingo at June 1, 2004 07:32 PM | PERMALINKBut the increased number of online bingo cards was exactly what was needed to make bingo a staple at churches. http://www.x-bingo.com Posted by: bingo at June 1, 2004 07:32 PM | PERMALINKIf you are looking for a debt consolidation loan There is no obligation to the debt consolidation credit counseling, it is just a educational consultation of debt assistance. http://www.inet-debt-consolidation.com Posted by: debt consolidation at June 1, 2004 07:32 PM | PERMALINKThis is the online ambien to go there , big surptice men ambien is a sleep inducing medication. A good night's sleep is of great dental need for your health. Avoid insomnia, you may want to look into cheap ambien the most prescribed sleeping pill in the U.S today . Ya is true buy ambien But visit this site for more info: http://www.top-ambien.com Posted by: ambien at June 1, 2004 07:32 PM | PERMALINKa href='http://www.i-buy-ambien.com'>online ambien to go there , yes that is true ambien is a sleep inducing medication. A good night's sleep is invaluable for your mental and physical well being. If you suffer from insomnia, you may want to look into cheap ambien top (zolpidem tartrate), the most prescribed sleeping pill in the U.S. Tha is true buy ambien Now or visit this site: http://www.i-buy-ambien.com Posted by: buy ambien at June 1, 2004 07:32 PM | PERMALINKGot here Fioricet most likely reduces heart attack risk by irreversibly blocking the enzyme COX-1 online fioricet, thereby impairing the ability of platelets in the blood to form clots, Dr. Tobias Kurth of Brigham and Women's Hospital, Boston, and others explain in the American Heart Association's journal, fioricet Circulation. NSAIDs buy fioricet also lock on to COX-1, but the effect is reversible. Cheap Generic Fioricet or visit this site: http://www.x-fioricet.com Posted by: fioricet at June 1, 2004 07:32 PM | PERMALINKGo there guys Cheap Soma Carisoprodol is a prescription medication that is used to relax your body, relax your muscles soma and help put stress and other difficulties behind you. Online Soma is now available online with a prescription. You can obtain a prescription online by answering a short questionnaire about your medical history Buy Soma or visit http://www.top-soma.com Posted by: soma at June 1, 2004 07:32 PM | PERMALINKBest deal everywhere Soma - Carisoprodol is a prescription medication that is used to relax your body, relax your muscles and help put stress and other difficulties behind you. Online Soma, to do Cheap Soma is now available online with a prescription. You can obtain a prescription online by answering a short questionnaire about your medical history Buy Soma or visit http://www.one-soma.com Posted by: soma at June 1, 2004 07:33 PM | PERMALINKBest before buy Online Ambien, yeaj hthe buy ambien is a sleep inducing medication. A good night's sleep is invaluable for your mental and physical well being. If you suffer from insomnia, you may want to look into cheap Ambien (zolpidem tartrate), the most prescribed sleep medication in the U.S. Buy Ambien Now or visit this site: http://www.i-ambien.com Posted by: ambien at June 1, 2004 07:33 PM | PERMALINKThey offer a variety of affiliate software great online games, fabulous colorful graphics, music, other familiar Best Cialis and Online Cialis has been an eventual success in Europe since its introduction in Early 2003. Cialis will now be available in US soon. You may buy cialis through various registered pharmacies. cheap cialis http://www.top-cialis.com/ Posted by: cialis at June 1, 2004 07:33 PM | PERMALINKBest Cialis and Online Cialis has been an eventual success in Europe since its introduction in Early 2003. Cialis will now be available in US soon. You may buy cialis through various registered pharmacies. cheap cialis http://www.new-cialis.com/ Posted by: cialis at June 1, 2004 07:33 PM | PERMALINKBest Cialis and Online Cialis has been an eventual success in Europe since its introduction in Early 2003. Cialis will now be available in US soon. You may buy cialis through various registered pharmacies. cheap cialis http://www.hot-cialis.com/ Posted by: cialis at June 1, 2004 07:33 PM | PERMALINKWe provide a comprehensive list of e-pharmacies to help you get the best Levitra that is true Cialis Viagra deals. Cheap Levitra, in all clinical trials, has proven to be extremely successful.Each Levitra pill may work in as quickly as 16 minutes and may work for up to 24 hours, far surpassing the length of Viagra's effects which is an average of 4 hours. Buy Levitra http://www.hot-levitra.com Posted by: levitra at June 1, 2004 07:33 PM | PERMALINKBet the dumb out oif up online black jack thus we can get rid of it. Posted by: online blackjack at June 3, 2004 05:51 PM | PERMALINKFuck you all, fuck my siter and all online poker this is the case. Posted by: online poker at June 3, 2004 05:51 PM | PERMALINKGreat Poker texas holdem poker thar is trye http://texas-holdem.666-casino.com/texas-holdem.htm Posted by: texas holdem at June 3, 2004 05:51 PM | PERMALINKWhether you have a rigid online poker strategy in mind before you hit the table or whether you play it. Posted by: online poker at June 3, 2004 05:51 PM | PERMALINKGambling sites chosen recommended here at Online Casino have been thoroughly tested by us for fairness, reliability! Posted by: online casino at June 3, 2004 05:51 PM | PERMALINKA comprehensive online bingo and unique casino games portal. Posted by: online bingo at June 3, 2004 05:51 PM | PERMALINKGod the fuck online casino are you sure. Posted by: online casino at June 3, 2004 05:51 PM | PERMALINKGreat Blog online poker Great Blog. I love it Why you are not so bad The poker I have ever played God damn it, you are such an online casino Ohhh this is strange fucker online casino gambling sounds. Go there and try this Drugs are wrong opportunity, be aware of that Try this fun men online poker you think. news all aroung the world online poker try it for free that right. Posted by: online poker at June 3, 2004 05:53 PM | PERMALINK
They offer a variety of But the increased number of online bingo cards was exactly what was needed to make. Posted by: online bingo at June 3, 2004 05:53 PM | PERMALINKLearn this from me, never online poker play it you think. Posted by: online poker at June 3, 2004 05:53 PM | PERMALINKTo do this go there and see of that Try this and feel it online bingo not to be missed. Posted by: online bingo at June 3, 2004 05:53 PM | PERMALINKBack to track online blackjack casino games on the Internet. Posted by: online black jack at June 3, 2004 05:53 PM | PERMALINKPlay at the best dumb online bingo not hte rest. Posted by: online bingo at June 3, 2004 05:54 PM | PERMALINKSurely not a chance for you but online poker nevertheless try this. Posted by: online poker at June 3, 2004 05:54 PM | PERMALINKGreat Poker Rooms I suggest too online texas holdem poker thar is trye http://online-texas-holdem.666-casino.com/online-texas-holdem.htm Posted by: online texas holdem at June 3, 2004 05:54 PM | PERMALINKGot here and seen this right online poker By the way ever played the casino played why play it as you can online bingo feel free with this opportunity. Posted by: online bingo at June 3, 2004 05:54 PM | PERMALINKOnline, trust our 24hr live support security, and Good site debt consolidation loan that is right. The http://www.all-debt-consolidation.org Posted by: debt consolidation at June 3, 2004 05:54 PM | PERMALINKI thoroughly enjoyed reading everyones opinion on these matters. Posted by: Online Casino Games at June 15, 2004 06:40 PM | PERMALINK
What is a Pocket Bike? Pocket Bikes are designed in class and style
of a mini-motorcycle? Its well-built aluminum frame gives it a sleek
look of a racing style bike. Top speeds range from between 34-45 mph,
depending on the weight and height of the rider. Pocket Bikes are fully
equipped with either a 49cc or a 47cc two stroke engine with a
centrifugal clutch. Pocket Bikes comes with front and rear disc brakes. Pocket bikes Pocket bikes Airline Tickets bikini girls Free Reality Sex Reviewed - All Reality Pass - Mr Chews Asian Beaver - Pump that Ass - Texas Dildo Massacre - Tinys Black Adventures - The Big Swallow - Barefoot Confidential - Vanilla Teens Black Cream - Horny Spanish Files - Lesbian Adult Movie Matrix - Teen Adult Movie Matrix - Chloe Jones - Gina Lynn - Kendra Jade - Krystal Steal - Tera Patrick - Glory Hole - Blacks on Blondes - Exploited Black Teens - Pimp 4 a Day - Milf Riders - Black Dicks White Chicks - Fucking Machines - Hogtied - Suck Me Bitch - Xxx Proposal - College Fuck Tour Posted by: Karasxxx at June 21, 2004 06:58 AM | PERMALINKYou can also check some information dedicated to lipitor Cialis helps to overcome the main problem of embarra-ssment and false
stigma attached to e-rectile dysfunction (ED). No more anxiety when it
comes to performing in the bed.
You may find it interesting to check the sites dedicated to http://www.melincs.org/ phentermine http://worldwide-deals.net/ phentermine http://phentermine-today.com/ phentermine http://phentermine-pharmacy-deals.com/ phentermine http://www.phentermine-experts.com/ phentermine http://www.phentermine-375-deals.com/ phentermine http://www.phentermine-use.com/ phentermine http://www.online-deals4u.com/ phentermine http://www.worldwide-90.com/ phentermine http://www.phentermine-online-sale.net/ phentermine http://www.phentermine375.net/ phentermine http://www.bestbuys-online.net/ phentermine http://www.bestbuys-online-24x7.net/ phentermine http://www.kmag.biz/ online casino http://www.sydney-harbour.info/ online casino http://www.online-casino-4u.com/ online casino http://www.onlinecasino-4u.com/ online casino http://www.online-casino-555.com/ online casino http://www.worldwide-21.com/ online casino http://www.online-deals21.com/ online casino http://www.bestbuys-win.com/ online casino http://www.onlinecasino777.net/ online casino http://www.online-casino-winner.net/ online casino http://www.hokitika.info/ online poker http://www.online-poker-4u.com/ online poker http://www.onlinepoker-4u.com/ online poker http://www.online-poker-555.com/ online poker http://www.worldwide-em.com/ online poker http://www.online-deals99.com/ online poker http://www.bestbuys-winner.com/ online poker http://www.onlinepoker777.net/ online poker http://www.online-poker-winner.net/ online poker http://www.onlinepokersite.net/ online poker http://www.mrspike.biz/ viagra http://www.viagra-experts.com/ viagra http://www.viagra-100-deals.com/ viagra http://www.viagra-use.com/ viagra http://www.online-top-deals.com/ viagra http://www.worldwide-50.com/ viagra http://www.viagra-online-sales.net/ viagra http://www.viagra24x7.net/ viagra http://www.buyviagra50.net/ viagra http://www.viagrapills100.com/ viagra ... Thanks!!! Posted by: online casino at July 28, 2004 07:37 AM | PERMALINKI do like your site very much, I am learning how to build a site like
yours. The following sites are my sites, pls give me some suggestions: I do like your site very much, I am learning how to build a site like
yours. The following sites are my sites, pls give me some suggestions: I do like your site very much, I am learning how to build a site like
yours. The following sites are my sites, pls give me some suggestions: I do like your site very much, I am learning how to build a site like
yours. The following sites are my sites, pls give me some suggestions: I do like your site very much, I am learning how to build a site like
yours. The following sites are my sites, pls give me some suggestions: I do like your site very much, I am learning how to build a site like
yours. The following sites are my sites, pls give me some suggestions: I do like your site very much, I am learning how to build a site like
yours. The following sites are my sites, pls give me some suggestions: I do like your site very much, I am learning how to build a site like
yours. The following sites are my sites, pls give me some suggestions: HGH, Human Growth Hormone dvd copy, copy dvdhttp://www.copiesanything.net/DVD-Copy.html Posted by: dvd copy at August 10, 2004 11:12 AM | PERMALINKgas scootershttp://gas.scooters-web.com/gas_scooters.html 3841 check out the hot blackjack at http://www.blackjack-p.com here you can play blackjack online all you want! So everyone ~SMURKLE~ Posted by: play blackjack at August 23, 2004 12:56 PM | PERMALINK6170 Herie http://blaja.web-cialis.com is online for all your black jack needs. We also have your blackjack needs met as well ;-) Posted by: blackjack at August 25, 2004 06:02 AM | PERMALINK7773 check out http://texhold.levitra-i.com for texas hold em online action boodrow Posted by: online texas hold em at August 26, 2004 01:24 PM | PERMALINKbreast enlargement http://breast-enlargement.never2big.com/breast-enlargement.html |
|
|
Powered by Movable Type 2.63 
|
||||
