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I can't believe it, my Kindle is down to 30 books. That's unheard of for me. Between the recommendations from Amazon, the selections from BookBub and Book Release Daily, my Kindle usually has at least 50 books on it. I am notorious for purchasing anything that is free or .99 and then I get suckered in. I am one of those people that hates reading a series out of order. I generally purchase them all, start at number one and don't stop until I get to the end. I have even been known to hold off starting a series until all the books are out. With that in mind, I found a great new author recently. Her name is SJ McCoy and her books are perfect for that summer beach reading. Her series is called - The Summer Lake Series.
The four books (with at least one new one on the way) start out with four main friends that grew up together - Pete, Missy, Ben and Emma in a town called Summer Lake. With each book at least one new person is added to the group but all four friends remain constants throughout the series.
The first book - Love Like You've Never Been Hurt tells the story of Jack and Emma. Jack is one of Pete's best friends and his business partner.
The second book - Work Like You Don't Need the Money tells the story of Pete and Holly. Holly is Emma's best friend from the city where Holly has her own shop.
The third book - Dance Like Nobody's Watching is the story of Dan and Missy. Dan is the brother of Jack who starts coming to town to become a mentor to Missy's son Scot. I think this is my favorite of the four books.
The fourth book - Fly Like You've Never Been Grounded is the story of Smoke and Laura. Smoke is the pilot for Jack and Pete's company and Laura is the cousin of Jack and Dan.
We have yet to hear about Ben's story (which I can't wait to read) and the next book in the lineup according to SJ McCoy's blog is Laugh Like You've Never Cried. It will be out in August and the story is about Michael - who is a friend of the original four but who was a minor character in several of the earlier books. I have my suspicions as to who he will end up with but I'll hold off for now.
And if you want to check out what author SJ McCoy envisions her guys to look out, check out her facebook page. She has them posted.
So, if you are headed to the beach, the pool or just some down time, check out SJ McCoy and her series about Summer Lake. I can't wait to hear what you think.


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Branding and Design Concept for a New York based Art and Design Foundation.
ANAKIN calender 2018 project in cooperation with FW Medien. A visual interpretation of some of the world's greatest songs. Limited to 200pcs.
côte&ciel is an homage to the productive collision of forces you'll find at the meeting point between coast and sky.
Anakin created marketing tools and material for the avantagarde fashion brand DAMIR DOMA such as Lookbooks, Websites/Stores and Invitations.
Brand identity, Website and Product Design for DANT, a fine leather accessory brand.
Branding, websites and marketing for the biggest governmental owned Accelerator Program in the world.
Website and Branding for Dubai Future Talks. DFT is Dubai's latest government initiative to build a video community around future topics.
Re-branding and Sales Material for FW Medien, a printing house specialising in premium offset printing.
Re-branding and Sales Catalog and booth dsign for the swiss based E-Bike brand Flyer.
Brand Identity, Website and eStore for Kontextür, a NY based design company.
Annual Winter Sportswear Catalog for a local Outdoor Store chain. A well balanced mix between story telling imagery and straight forward product information.
Tropical superfood bar and restaurant. Our own bar, yeah! A place to eat, dance and express yourself
Website and branding concept for the russian Haute Couture Fashion label Ulyana Sergeenko.

subset: commoncrawl


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I will be taking the LSAT but in the presentation these tips were expressed as help for all of the graduate tests. The most interesting thing that was said was that the tests are not meaningful. At all. They are however, important. Grad tests show only that you are good at that particular test. That's about all. The keys to being successful on grad tests are:
- Be aggressive, for some reason, it's rewarded on the test.
- Laugh regularly when preparing for the test. (No, I'm serious)
- Use books, the internet, and one-on-one tutoring in your preparation
- Take a course! Studying on your own is much harder because you have to find the materials, study, and teach yourself. In a course, you only have to study, they find materials and teach you.
Also, they provided a "work-out" plan for prepping for the test, which I found incredibly helpful.
Prep minimum: 2 hours a day of high quality practice
5-6 days/week
7-8 weeks, right up until the official test day
Take 4-5 practice tests, one every two weeks
While this may seem like a very intense schedule of preparation, the presenter pointed out that this schedule is equal to only ½ of a season of a high school sport!
Lastly, practice tests from Kaplan and the Princeton Review are good, but find prior released tests. You can get these by simply purchasing them from the testing company here. I'm planning to start my studying in a few weeks. Happy testing!!


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Health Insurance Enrollment
SHB's In-person Assistants (IPA's) Amal Saleh and Sundus Ali help uninsured persons to navigate through the health insurance system. Somali Health Board's In-Person Assisters (IPAs) provide impartial information to help individuals and families determine which health insurance option best fits their needs. Whether private insurance or Washington State's Apple Health coverage, this program is offered to all individuals - to enroll in the most suitable health insurance program available. Individual appointments can be made with a certified IPA who will guide you through the entire insurance enrollment process. Our IPAs fluently speak English, Somali, Swahili, and Arabic. This service is offered free of charge.
The IPA's can also help participants renew their insurance. IPA's are available over the phone and in-person. Open enrollment is currently underway.
Documents to Bring:
· Date of birth (for all applicants)
· Social security number (for all applicants)
· Tax filing status for 2014 and 2015 (if available)
· For household members that are not US citizens: Legal Permanent Resident care or other immigration documents (visas) and Social Security number
· Username and password for healthplanfinder (if you have already created an account)
Amal Saleh: firstname.lastname@example.com (206) 619.1829
Sundus Ali: email@example.com (206) 966.5311


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3D Wing Motorcycle Air Cleaner Cover
3D Wing Motorcycle Air Cleaner Cover features an innovative design and it is one of the most unique motorcycle parts we offer. This billet Motorcycle Air Cleaner Cover is machined for fitment on custom Harley Davidson motorcycles (HD), custom choppers, bobbers and most custom motorcycles. The 3D Wing Motorcycle Air Cleaner Cover will flow with your gas tank lines for a unique and bold look and will look great combined with our 3D line products:
- 3D Motorcycle Hand Controls
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- 3D Blade Custom Motorcycle Mirrors
Black air cleaner covers are also available for bikes featuring the dark look. Aircleaner kit includes filter, choke extension (only on S&S air cleaner covers) and all necessary hardware.
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You where really going to print this?! Please go a hug a tree now!
Thank you Trends Setters:)
What do you think about heart sunglasses????
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- Home
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- Farberware Pump / 8 Cup
Farberware Pump / 8 Cup
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Pump and spring will fit Farberware 8 cup models. This pump will work on the original series model 138 or the newer series FCP280. Please note that the pumps no longer use the ceramic sleeve inserts that the original series used.
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The Envy Hotel is a hotspot for partygoers on the Gold Coast who enjoy diversity with three levels all catering for different things. Step through the doors on the bottom floor and experience the brightness with eye catching neon lights at all angles, four plasma televisions, a tab and a pokies room. The second floor plays host to the lounge bar, a place for those with class to sit, relax and enjoy a beverage. The top floor is where the party starts, a place for people to really let their hair down and experience the beats the DJ's provide.
Envy Hotel
Address: Level 2 Niecon Tower, Broadbeach, Gold Coast
Phone: 07 5527 5377
Web: www.envyhotel.com.au
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Answers 1Add Yours
Answered by jill d #170087
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But the most astonishing as well as the most interesting thing to me was the condition of the colored people, a great many of whom, like myself, had escaped thither as a refuge from the hunters of men. I found many, who had not been seven years out of their chains, living in finer houses, and evidently enjoying more of the comforts of life, than the average of slaveholders in Maryland.
I found the colored people much more spirited than I had supposed they would be. I found among them a determination to protect each other from the blood-thirsty kidnapper, at all hazards.
Narrative of the Life of Frederick Douglass


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WASHINGTON (Reuters) - Treasury Secretary Timothy Geithner will meet India's finance minister in Delhi in June to strengthen economic and financial ties between the two countries, the Treasury Department said on Monday.
Geithner and Indian Finance Minister Pranab Mukherjee are expected to discuss the implementation of financial sanctions against Iran, as well as reforms for deepening India's capital markets, the U.S. Treasury said.
The June 27-28 meetings will also include discussions with India's central bank, Reserve Bank of India, and Indian business leaders.
Reporting By Rachelle Younglai; Editing by Kenneth Barry


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Reflections of the 2013 March for Life—Washington D.C.
I had not intended to be in D.C. for the 2013 March for Life . My calendar was already full with appointments that simply could be changed.
But when a friend asked and encouraged me to join her, my thoughts turned to the possibility, and suddenly I realized I had to go. The appointments were surprisingly and easily rescheduled.
And so, like many others from across the nation, I packed my bags and headed to D.C.
I have been to previous marches; my first was in 1980. I was prepared for the cold weather, slow walking, and large crowds. But this was different, it was much colder, much slower, and a vastly larger than I had ever experienced. As many as 500,000 pro-lifers—some say even as high as 650,000—came to D.C. to march in witness to the unborn and the incomprehensible loss of 55 million babies to 40 years of legalized abortion in the U.S.
While there I was interviewed and asked the question: "What needs to happen to end abortion?" You can see the video uploaded by TFP Student Action at http://www.youtube.com/watch?v=t1OJaCB9gW8&feature=em-share_video_user. My comments are at about the 2:50 mark.
I took my place in Washington D.C. March for Life again this year because 40 years after Roe v. Wade, hearts and minds need to change to acknowledge and embrace the reality that the child in the womb is fully human and fully alive. He or she is a unique and valuable member of the human family. Just as abortion was made legal, it can be made illegal—but our ultimate goal is even more critical: For abortion to become unthinkable.
Debbie Nieport
Director of Development


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Key words: cigs solar cell; cigs pv panels; lightweight solar backpack; foldable solar charger; solar paver lights
The International Energy Agency (IEA) released an annual report during the World Oil Congress that the total global energy investment fell by 12% in 2016 and it's the second consecutive decline in global energy investment, mainly due to the continued decline in oil and gas upstream investment.
The report pointed out that last year, the global total energy investment was 1.7 trillion US dollars, accounting for 2.2 percent of the gross domestic product(GDP). What can be noticed is that it's the first time when the investment of power sector surpassed the sum of oil, natural gas and coal investment. And the clean energy investment accounted for 43% of the aggregate investment.
China is the largest energy investment country, the report said, whose energy investment accounted for 21% of the global total investment. China's coal power investment declined by 25% while clean energy power generation, power grid construction and energy efficiency investment increased. America's energy investment last year is 16% of the global total investment. India's energy investment increased by 7%, ranking the world's third largest energy investment market.
According to an executive director of IEA, Bill's words,"Our analysis reveals that savvy investment decisions are quite significant for sustaining energy security and achieving environmental goals." He emphasized that the policymakers should reverse the current downturn in investment and ensure long-term supply of energy.
Sun harmonics is a Chinese manufacturer, located in Hangzhou, Zhejiang Province. It focuses on developing and producing environmental and high-efficiency CIGS thin-film solar cells and other portable device-charging products. Sun's flexible products can be universally applied to roof power generation, outdoor activities, military areas, scenery decoration, etc.
Please click here for more details about us: Solar Cells, Solar Modules, Outdoor Modules, Solar Lights Manufacturers and Suppliers China - Wholesale Price - Sun Harmonics http://www.sygdcigscell.com/


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New York
18 April 2018
Price realised
Paul-César Helleu (French, 1859-1927)
La Lettre
signed 'P. Helleu.' (lower left)
oil on canvas
23 5/8 x 29 in. (60 x 73.7 cm.)
Painted in 1880.
Henri Letellier (1867-1960), Paris.
Mme. Ruillier, before 1962.
with Whitford and Hughes, London, by 1984.
Anonymous sale; Sotheby's, London, 16 November 1994, lot 125.
Private collection, UK, by 1998.
Acquired directly from the above by the present owner.
Contact Client Service
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F. de Watrigant, Paul-César Helleu, Paris, 2014, pp. 90-91, illustrated.
Dieppe, Musée de Dieppe, Paul Helleu, 7 June-17 September 1962, no. 1.
London, Whitford and Hughes, Peintres de l'âme, 3 May-8 June 1984, no. 6, illustrated.
Montecatini Terme, Terme Tamerici, Boldini Mon Amour. La grande mostra antologica, 18 September-30 December 2008, p. 141, illustrated p. 142, as La Lettera.
The entire oeuvre of Paul-César Helleu evokes a time of elegance and luxury. The artist's lifetime corresponds to the period of prosperity between France's Second Empire and the beginning of World War I, a time when France shone in the arts, Britain extended its empire and the United States demonstrated a wealth that seemed to be endless. Helleu's life reflected his times, and after the influential aristocrat Robert de Montesquiou arranged for the young artist to execute a portrait of his cousin, the accomplished and glamorous Comtesse de Greffuhl, Helleu became one of the most sought after portraitists in Paris. Throughout his long career, Helleu enjoyed a wide circle of friends from John Singer Sargent and Giovanni Boldini in his youthful days, to Edgar Degas, Claude Monet and Jacques-Émile Blanche in his more mature years. He counted many foreign artists among his friends: James Tissot, Walter Sickert, James McNeill Whistler, Alfred Stevens and many others. Helleu moved in a truly cosmopolitan world in the time that has come to be called La Belle Époque, and he was its illustrator.
La lettre was painted in 1880 when Paul Helleu was just twenty-one at a time when he was sharing a studio in Paris with John Singer Sargent. The model is Marie Renard, a striking brunette who Helleu painted several times and who also posed for Édouard Manet and Berthe Morisot. The young girl clearly made an impression on Helleu's roommate, for Sargent began the sketches for his masterwork El jaleo, currently in the Isabella Stewart Gardner Museum in Boston (fig. 1), at about the same time Helleu was working on La lettre and he chose to use Marie Renard as the model for the Spanish dancer (N. Bemahum, Antonio Mercé, 'La Argentina'; Flamenco and the Spanish Avant-Garde, Watertown, 2000, p. 78). Marie was also the notorious model for Henri Gervex's La femme au masque of 1886, in which she posed completely nude except for her black mask (fig. 2).
At the beginning of his artistic career, the young Helleu met and was painted by the Italian artist Giovanni Boldini and the older artist greatly influenced Helleu's style and decision to become a portraitist. The modeling of the young girl's face, the sharp, quick brushstrokes and even the palette demonstrate Boldini's influence on the developing style of his protégé. However, this is not the slavish mimicry of a mediocre artist, but rather the brilliant artistic foray of a young artist striving to create his own artistic vocabulary. Elegance and beauty were the catchwords of the day in the early 1880s, and no artist embodied these two elements more eloquently than Paul-César Helleu. Even in his early work, his innate sense of the proportion, grace and tonality of a composition brings light and drama to his figures even when they are depicted performing the most mundane of activities.
Although La lettre depicts a young woman in an interior languidly reading a letter, its envelope tossed casually on the floor at her feet, it is not a portrait in the conventional sense, but rather an 'interior still life'. Helleu's painting can almost be described as a 'Whistlerian' symphony of whites (indeed Whistler was vocal in his admiration of the Helleu's interior color scheme). The artist adds richness and elegance to the simple settee by adding texture to the French blue fabric in white; the shawl draped across the chair to the right of the composition also has its delicate pattern picked out in white. Small and simple touches of white the give definition to the chair to the left and even the toes of the young woman's black shoes. The white envelope so casually discarded on the floor is almost a mirror reflection of the letter held aloft by its beautiful reader. Against the clear, light blues and whites that dominate the painting, Helleu has chosen to dress his model in an extraordinary earthy green skirt and black velvet jacket which is set off perfectly by white lace cuffs and color. The casual pose of the reader is echoed in the shawl thrown over the chair to the right.
This combination of strong brushwork, flashy reflections and exuberant tonal contrasts exhibits the panache and confidence that would make Helleu the darling of the aristocracy and win him acceptance among the ranks of the most fashionable artists of the day. Within the course of his long career, Helleu was welcomed into the highest echelons of Parisian society and has been favorably compared to Marcel Proust as having created a visual imagery to complement the writer's literary descriptions of French society at the end of the 19th century.
We are grateful to Frédérique de Watrigant and Les Amis de Paul-César Helleu for confirming the authenticity of this work, which appears in their archives as no. HU-1253 and will be included in their forthcoming online catalogue raisonné.
(fig. 1): John Singer Sargent, El Jaleo, 1882. Isabella Stewart Gardner Museum, Boston.
(fig. 2): Henri Gervex, La Femme au masque, 1886.
Pre-Lot Text
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Three paintings that chart the changing roles of women in Belle Époque Paris — all offered in New York on 18 April
Specialist Sarah Reynolds on the women whose work made waves in Victorian Britain, from Emma Sandys to Laura, Lady Alma-Tadema
Artists, patrons and critics have argued for centuries over the right way to frame a painting. Ultimately, it's a matter of personal taste, says Andrew Graham-Dixon
Christie's London doorman - shortlisted for Doorman of the Year in Mayfair & St James's 2019 Community Awards - discusses James Bond, Francis Bacon and the day he met Madonna
Explore timeless classics that define modern French elegance with over 300 YSL designs from Catherine Deneuve's personal collection


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Product Code: 1058015
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Astro Enna Surface White LED Wall Light
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The Enna LED Wall Light's diminutive appearance means you can also create eye catching visual effects when installing in lines across a wall.
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The primary purpose of the PPA is harmonisation with relevant EU Directives, which were adopted in 2014. However, it is also supposed to simplify and streamline the entire procurement procedure, including a reduction in the administrative burden, for both contracting authorities and economic operators. In the future the contracting authorities should have available a wider scale of evaluation criteria, allowing them to take into account the price-quality ratio when assessing submitted offers.
The PPA is supposed to codify the award procedure regarding public tenders and concessions. It also regulates the powers of the body responsible for review procedures (i.e. the Czech Competition Authority).
The PPA also aims to implement e-procurement in relation to all award procedures by October 2018. Although many of the contracting authorities already communicate and award public tenders through electronic means, the PPA aims to bring in this regard complete transparency, equal treatment and anti-discrimination to all procedures by setting a clear deadline applicable to all contracting authorities.
Further relevant changes that aim to significantly simplify at least some of the tender procedures or modify the current rules, and might be of interest to economic operators, include:
(i) an increase in the financial limit when conducting tenders for construction works in a simplified below-threshold regime from the current CZK 10 million to CZK 50 million (approximately EUR 1.8 million). Within the simplified regime a contracting authority may, for instance, address a few economic operators of its choosing, while publishing the tender notice on its profile only (instead of publishing it in the central information system);
(ii) the introduction of a new, so called "light regime" for public tenders in some specific areas, such as e.g. the provision of legal services. For this light regime procedure, if the general principles of procurement procedures are followed, the contracting authority may set out specific tender procedure rules that are different from the standard regimes;
(iii) bank loans will be entirely exempted (the current regulation, under which bank loans were subject to procurement procedures, were causing delays and imposed unnecessary cost burdens on contracting authorities);
(iv) the introduction of a new set of criteria on the basis of which the contracting authority is entitled to disqualify any economic operator from a tender right at the beginning of the process due to bad prior experience working with the participant (e.g. delays in performance or other breaches of contract) if the contracting authority proves that the participant in question meets any of the disqualification criteria;
(v) "self-cleaning", which complements the above disqualification criteria and allows a disqualified economic operator to overcome its disqualification by proving that it has remedied its failures and that it no longer meets the disqualification criteria;
(vi) a more stringent regime for economic operators that have the legal form of a joint-stock company or whose form is similar to a joint-stock company. To be awarded a public contract these economic operators will be obliged to have issued solely registered shares (i.e. shares registered with a depository);
(vii) more emphasis on the "most economically advantageous tenders" rather than solely on price. The contracting authority will be free to choose the evaluation criteria for the quality of the offer, e.g., environmental, social, innovative, etc. Those criteria should nonetheless relate to the subject matter of the tender. The contracting authority will even be able to set a fixed price in advance and asses the tenders according to the defined evaluation (quality) criteria.
(viii) a more detailed definition of what constitutes a non-significant change to a public contract. The lack of clarity on this issue has led to many discussions about what changes to public contracts are still acceptable (since a new tender is required in the event of a significant change); and
(ix) an effort to limit unfounded complaints being brought before the Czech Competition Authority that may delay or even thwart procurement procedures. Starting from 1 October 2016, any person wishing to submit a complaint against a tender procedure before the relevant authority is obliged to pay a fee of CZK 10,000 (approx. EUR 370). This new provision is rather controversial, as the Czech Competition Authority is obliged by law to review all relevant information about a potential violation of public procurement rules. Apart from that, economic operators will still be obliged to pay a guarantee of 1 per cent of the offer price (in any event between CZK 50,000 and CZK 10 million) if they submit a request with the Czech Competition Authority to review a decision of a public authority.


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Microsoft Academy
The polytechnic is partnership with Microsoft to establish a state of the art Microsoft Academy in the polytechnic which will serve as the best of its kind East of the Niger. The Microsoft Academy when fully operational will enhance the professional skills of both students and staff of the polytechnic. This Academy is the brain child of the rector of the polytechnic Rev. Fr. Dr. Wence Madu.


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I've also added this to a Christmas in July contest over at Sew Festive Handmade. Winner gets some very yummy Christmas fat quarters.
Sunday, July 29, 2012
christmas kaleidoscope quilt complete
My Christmas kaleidoscope quilt is complete and I really, really love this one. Since it's July, I can't really put it on the couch now and I'm sad to have to wait four months to pull it out. I followed this tutorial found at Don't Call Me Betsy and used a two-block format. Half of the prints are from Kate Spain's line The 12 Days of Christmas. The binding is also Kate Spain but there is a lot of filler fabric from Robert Kaufman and Michael Miller as well. O ended up needed a lot more fabric than I originally planned for. Solid is Kona white. This is a generous lap-sized quilt.
3 comments:
- Sew Festive said...
You should link your quilt up at http://sewfestivehandmade.blogspot.com/ for the Christmas in July linky that opened up today. There's prizes :)
- July 30, 2012 at 1:42 AM
- Christine Carey said...
That's a beautiful quilt! Great job!
- July 30, 2012 at 7:34 AM
- Vroomans' Quilts said...
Lovely fabrics for a lovely quilt. Well done.
- July 30, 2012 at 3:44 PM
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(Updated) Just before the 2 p.m. trade freeze, sources say the Wild traded pending free agent Kim Johnsson and the rights to 2009 first-round pick Nick Leddy to the Chicago Blackhawks for 23-year-old former No. 3 overall pick Cam Barker, a talented two-way defenseman who has two more years left on his contract at almost $3.1 million per.
The Blackhawks were going to have to trade Barker this summer to get under the salary cap. They'll have to make some more moves, too, I am told, and the Wild still has interest this summer in center Patrick Sharp.
This 100 percent means, I am told by sources, that the Wild is trying to re-sign Marek Zidlicky before the March 3 trade deadline -- probably way sooner. Term is the only stumbling block right now. Trading Leddy was made easier by the influx of defensemen in the system -- Marco Scandella, Tyler Cuma, being the main two, and the long-term future of Brent Burns. The Wild, I am also told, was very concerned about Leddy's development at the U.
Barker had a career-high 40 points last year.
Wild assistant GM Tommy Thompson, reached in Vienna on a scouting trip:
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Comments will be reviewed before being published.


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Black-and-White Baby Toys
14 High-Contrast Finds to Boost Your Baby's Vision
Photo 6 of 14
Hello, Animals!
This sweet and simple board book ($6) is filled with black and white illustrations dotted with vibrant color. The perfect first book to share with your baby, it's one she will love looking at over and over.
And reading to your babe encourages her language skills!
0previous images
-17more images


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Product Description
This product goes brilliantly with all fish, either from the sea, or fresh water. It is made with a smoked Rock salt base and then layered with flavors from Lime, Smoked Paprika,Orange, ginger and Garlic. It will increase flavor to all dishes it is used in. This is the product I am using in my U Tube Video on smoking trout over Embers. . this Rub can be used on all Smoked Trout as a Pre Salt Rub. it is Tangy and mellow all at the same time. My Kids love this one so it must be good. 150 grms. I use this where normal salt is used in cooking to add flavor to dishes.


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e the EXPONENTIAL - the Magic Number of GROWTH
- Gabriel Golden
- 3 years ago
- Views:
1 e the EXPONENTIAL - the Magic Number of GROWTH Keith Tognetti School of Mathematics and Applied Statistics University of Wollongong NSW 2522 Australia 2 February 998 Introduction This Module is written as a self contained introduction to e, bringing together the main theorems and important properties of this fundamental constant of natural growth processes. Not only is an axiomatic treatment given but this is complemented by some theorems that have been selected for their unexpected beauty. It assumes only an elementary understanding of integration. Some anecdotes and history are also included. It will be seen that Euler s gamma constant also comes into the main theorem and this highlights the intimate interconnection between e, the area under the curve, and the the truncated Harmonic series. x The number e is the base of Natural logarithms but it is not the base of Naperian logarithms as will be seen by glancing at the short historical note on Naperian logarithms in Appendix A.
2 e owes its importance to mathematics because we come across it in all natural growth processes. e like π is transcendental and thus it cannot be expressed as the root of a finite polynomial equation with rational coefficients (go to Appendix C for some more about irrationals). If you read nothing else at least to help you get a feel for arithmetic look at Appendix D, the article by the Noble Prize winning physicist and sometimes lovable character Richard Feynman. For a very general, exhaustive and more elementary introduction to e see Maor (References). Let us begin our description of e by writing down the first digits e = We immediately note that 828 is repeated at the beginning. This is rather exceptional as most of the important constants of mathematics that are irrational have no obvious pattern at their beginning. However it is almost certain that if we searched through the digits of most of these numbers we would come across any pattern we liked to define, as long as the pattern was not periodic. Thus for example the above list of digits will most certainly occur somewhere in the expansion of π; although be warned that it might take us a long time to come across them. Our first introduction to the essential nature of e is through a problem in commerce - Suppose that we have $ and invest it such that at the end of the first year we are given $ interest, thats right 00% interest. Now if instead of the full year we invested our money for year it would be only fair that we would get of this amount in interest, that is we would 0 0 have a total amount of $+ at the end of this first period. 0 With simple interest we would get this same amount for each of our ten periods and hence the total interest would be $ which is the same as in the first case. This is the same as saying that our money has grown from $ to $2 by an amount of $0 0. 2
3 Now if we compound our interest the following happens. At the end of the first of 0 a year we would get $( + ) as before. However at the beginning of the second period 0 we would invest this amount instead of $. It follows that at the end of the second period we would have $( + 0 )2 =.2 and so on until at the end of the year we would have $( + 0 )0 =$ Similarly at the end of the year with 00 periods we would have $( + 00 )00 = and for 000 periods $ From this we find experimentally that as the number of periods increases the total amount appears to approach $e. We note that the derivative of ( + n )n is ( + n )n {ln( + n ) ( + n) } and that this is positive and decreases to zero as n increases. So this is further evidence that ( + n )n converges as n becomes unbounded. However we require the Main Theorem to prove this conclusively. You might like to check the following figures for reinvesting on a monthly ($2.63), weekly ($2.69) and daily ($2.7) basis. Even if we reinvest at the end of a half year the amount is $2.25 which is 2.5% better than simple interest. This is one of the reasons why banks make such a profit from simple minded customers. In general if instead of an increase of $ we had an increase of $r for the year (r is called the effective annual rate of interest) we can readily show in a similar way to the above that, with an initial investment of $, the amount at the end of the year is ( + r n )n and that this expression approaches e r as n increases without bound (see A4, below). For a rather dramatic example of the absurdity of believing that any economic system can really sustain exponential growth at even a reasonable" growth rate see Appendix D. Biological Growth Models A similar exponential process occurs during the initial growth of many biological organ- 3
4 isms. This occurs whenever the rate of change is proportional to the size or the number of the organisms. In this case we find that for example if y is the length of a root or the number of cells in an organism then y = y(0)e rt, () where y(0) is the initial value, t is time and r is the intrinsic rate of increase. This relationship is what is now generally referred to as Malthusian or exponential growth. However the original growth process referred to by Malthus was actually geometric. Knowing that dex dx side we get = ex, (from proposition Q4), it is seen that if we differentiate this equation on each dy dt = y(0)rert = ry. (2) Hence dy dt r = y. (3) Thus r, which we call the intrinsic rate of increase, is the actual rate of increase dy divided dt by the total length of, or number of, organisms. We note that if y were say the number of individuals in a population then dy dt y, and hence r, is seen to be the average rate of growth per individual. It is a remarkable fact that, in a wide range of growth processes in nature and economics, this quantity is constant during the initial stages of such growth. From (3) we see that where from now on ln x means log e x. ln { } y = rt, (4) y(0) (4) may be used as a rough check to test that we have exponential growth. We simply plot the logarithm of y against t. If this is a straight line then we have exponential growth. y(0) This exponential growth is hard to deal with intuitively and in practice it is easier to use the doubling time t d defined as the interval of time required for y to double. Then if we 4
5 consider the population at any time t we will double our population at time t + t d. Hence y(t + t d ) y(t) =2. Substituting in our exponential expression () we see that this reduces to exp(rt d )=2, (that is the y(t) cancels out and the expression is thus independent of t). From this we have t d =ln 2 r. If r is measured as a % then this is equivalent to the following - the doubling time is roughly equal to 70 divided by the interest in % or td 70 r%. This hyperbolic (or reciprocal) relationship is considered to be easier to grasp than the exponential. Thus for example with r =7% the doubling time is 0yrs, and with r =0% the doubling time is 7 yrs. Other Properties and History e has all sorts of other interesting properties. Perhaps the most beautiful relationship is Euler s formula which also involves π and i the square root of. This is e iπ =. Part of the beauty of this relationship lies in its unexpected simplicity. Here it is seen that i brings together e and π and unity, three of the fundamental constants of nature, together with the two operations minus and square root. We will show in detail in Appendix A that Napier s logarithms are somewhat different to natural logarithms which use base e. However through formula AA there is a relationship but Napier was quite unaware of it so his work only implied the use of e. 5
6 It was not until Bernoulli (Johann I, 704) that the symbol e and the name exponential were first used. This arose because it was necessary to have a way of representing e x. In fact the term Naperian antilogarithm" or e was not used until 784. The calculation of logarithms by infinite series was carried out by Gregory, Wallis and Halley (of Comet fame) beginning in the 670 s. It was not until much later (742) that William Jones ( ) used exponents to calculate logarithms and allowed present day methods to be developed. However it should be emphasised that Euler was the first to introduce this concept, and the symbol e, for the base of natural logarithms in an unpublished paper of 728 (he quoted e to 27 places). The Layout of this Note The formal part of this note begins with 2 definitions firstly of a function which is the area under the curve and then a definition for the inverse of this function. x This is followed by a set of propositions (P, P2 etc) about the logarithmic function which are developed axiomatically from the definition. Then we present some properties of the exponential function e x (Q, Q2, Q3) developed axiomatically from its definition as an inverse of the exponential. Then we show that e x is equal to its derivative (Q4). This is followed by some properties of the functions a x and x a (Q5, Q6). Q7 gives a derivation for e which we used in the interest rate problem. Then we derive an expression for e x as a series (Q8). From this we can show that e is irrational (see Appendix B). Then we give an independent proof of Q7 based on a note by Barnes which has the merit that it is coupled to Euler s γ constant. At this stage we digress into Harmonic numbers. Finally in contrast to the above axiomatic development we offer some alternative proofs A, A2,... A6. These have been selected simply on the basis of their elegance - and by elegance is meant unexpected simplicity. 6
7 An Axiomatic Approach We begin with the definition LN(x) = x t dt. def It is seen that LN(x) is simply a function of x at this stage. We hope to show that it does have the properties of a logarithm (see P4 below). We also define the inverse of LN(x) as E(y) LN(x) =y Hence E(y) =LN (y) =x. def 2 We hope to show that this inverse function is actually e y (see Q3). Properties of the logarithmic function ln x We now show that we can derive what we normally consider to be the properties of the logarithmic function from the above definition, def. We present these as a series of lemmas (minor theorems) or propositions - (it follows that a dillema is a theorem produced by a stupid person). It is not until P4 that we are convinced that LN(x) is in fact ln x. P LN()=0, directly from def P2 dln(x) dx = x. This follows directly from def by differentiation under the integral. P3 The additive relation holds, LN(ab) =LN(a)+LN(b). 7
8 Proof Let z = LN(ax) then from the chain rule dz = ( ) dx ax a =. x However dln(x) dx = x from P. Hence as the functions LN(ax) and LN(x) have the same derivative they must differ only by a constant, that is LN(ax) =LN(x)+c. In particular this relationship holds for x = and thus c = LN(a), hence LN(ax) = LN(x)+ LN(a). With x = b we have proved our lemma. P3a From P3 it follows directly that xy t dt = x t dt + y t dt, a result which is rather messy to prove directly by integration (is t the only function that has this property?) P4 LN(a n )=nln(a). This follows by continued application of P3. By substituting na = y and thus considering y n it follows that ln a n = LN(a). From this it is easy to prove that a LN(a r )=rln(a), where r is rational. And hence as the irrationals are everywhere dense this result also holds for irrationals. Note at this stage we see that LN has all the properies of our familiar function log e. P5 ln x is a convex increasing function without bound. It is increasing as its derivative is always positive. It is convex as its second derivative x 2 is always negative. It is unbounded as we can always obtain values of ln x>qwhere Q is any large quantity we like to nominate. This is neatly demonstrated as follows. In 3 n = n ln 3. But ln 3 > and thus if x =3 n then ln x>n, no matter how large n. P6 ln xdx= x ln x x 8
9 This follows by integration by parts. P7 For x, it can be shown from Taylor s formula that ln( + x) =x x2 2 + x Properties of the Exponential Function e y We now define the inverse of the logarithm of x based simply on the def. Again as we do not know as yet that this is really e y so let us call it E(x) =y, where x =lny. Q E(0)= this is merely acknowledging that ln=0. Q2 E(u)E(v) = E(u + v) Let E(u) =a, E(v) =b, E(u + v) =c then u =lna, v =lnb, u + v =lnc. It thus follows that u + v =lnab is equivalent to ab = c as required. Definition We now define e as E(). def 3. Hence This is reasonable as if e is the base of our natural logarithm then we must have ln e =. Q3 E(n) =e n. This follows immediately by the continued application of Q2, with E(u) =E(v) =E(n). In a similar way to P4 we are able to then show that Q3a E(r) =e r, where r is rational. Q3b E(x) =e x, where x is real. Suppose that x is irrational. Then for example by using continued fractions we can express x as the limit of a sequence of rationals. Thus the 9
10 relationship holds for any rational as close to x as we like and thus in the limit it must hold for x irrational. Thus we can now for the first time be convinced that if x =lny, then y = e x. Q4 dex = dx ex. From P2, d ln y dy the inverse as = dx = dx. Hence as is always positive we can obtain the derivative of dy y dy dy dx = dx dy = y = e x. Note For any function y = f(x) and its inverse f (y) =x we can see that f(x) is symmetric to f (x) about the line y = x. This is equivalent to showing that the point P =(x, y) and the point Q =(y, x) are symmetric about the line y = x (which has slope ). Now the slope of the line PQ is immediately seen to be. Also OP = OQ = x 2 + y 2. Hence the result follows. Thus the graph of e x is simply the graph of ln x reflected about the line y = x. Thus for example we see that the slope of e x at x =0is the same as the slope of ln x at x =, namely unity. Q5 The functions a x and x a. Let ln e f(x) = g(x), then by (Q3b) e f(x) = e g(x) and thus f(x) =g(x). Similarly if h(x) =e ln f(x), taking logs we have ln h(x) =lnf(x), and again f(x) =h(x). Combining these we see that f(x) =lne f(x) = e ln f(x). From this we see that we can always write a as power of e or as a logarithm a = e ln a =lne a. We can now write our functions very neatly as y = a x = e x ln a and x a = e a ln x. 0
11 Also more directly from the series for e (Q8) y = e x ln 2 +x ln 2 Q6 d(ax ) = a x ln a and d(xa ) = ax a. dx dx ( Q7 e z = lim + z ) n n n If f(x) =lnx, then f (x) = = lim f(x + h) f(x), x h 0 h = lim {ln(x + h) ln(x)}, h 0 h = lim h 0 h ln( + h x ). With x = z lim ln( + hz) =z. h 0 h Consequently e z = lim h 0 exp[ln( + hz) /h ] = lim h 0 ( + hz) /h = lim n ( + z n )n, if h = n. In particular with z =we have independently shown that ( e = n lim + n) n Q8 e x =+x + x2 + x3 +...=Σ xj 2! 3! j!, for j 0. This follows directly from the Taylor series expansion about x =0for f(x) =e x. From this we have for x = e =++ 2! + 3! +...=Σ, for j 0, j!
12 which has reasonably fast convergence and so may be used to calculate e An example Suppose we seek an approximation to y =2 x for small x. From the binomial theorem Then y = (+) x =+x + x(x ) + x(x )(x 2) + x(x )(x 2)(x 4) ! 3! 4! = +x( )+x2 (...) 2 +x ln 2. Q9 With Q8 as our definition for e we can now give the Proof that e is irrational. Assume the opposite that is e = p q p q = ++ 2! q! + (q )!p = { q!+q!+ q! 2! + q! 3! +... q! q! where p and q are integers (q+)! } (q+) (q+)(q+2) But the expressions on the left hand side and in brackets are obviously integers, whereas the remainder (q +) + (q + )(q +2) +...< (q +) + (q +) = (q+) (q+) = q. Thus for our original assumption to be valid we require an integer to be equal to an integer plus a number less than one. As this is absurd our original asssumption is invalid and e must be irrational. Theorem M To show that ( e = lim + n. n n) This presentation is based on Barnes, C. (984). As a by-product it establishes the expression for Eulers constant γ, which is defined below. Alternative proofs are given in Q7, and A. Proof From P6 n (n+) ln xdx= [ ] (n +)n ln n(n +) n +. (M) 2
13 Also we know from the mean value theorem that there exists a c n somewhere in the range of integration (that is (+n) <cn< n ) where n (n+) ln xdx= ( ) n ln c n. ( + n) (M2) From (M) and (M2) from which Hence or e = [ ] (n +) n ln =lnc n, n n+ ln (n+) n n n+ c n e = =. (n+) n n n+ c n, ( + n) n /nc n. (M3) Now let a n = (nc n). Then because of the range of c n we have <a n < + n. (M4) Hence we have the remarkable result that for all positive n ( e = a n + n) n. (M5) As a n > it follows immediately from (M5) that for example ( e> + 2 =2 2) 4. Also because a n is in the above sandwich (M4), in the limit as n, a n =and thus our theorem is proved. Definition We define Euler s constant as [{ γ = lim + n } ] ln(n +). n 3
14 Corollary M γ< π2 6 =.64.. Proof From (M5) Thus taking logarithms or ( e = a n + ) n = a n(n +) n. n n n =lna n + n{ln(n +) ln n}, n =lna/n n + ln(n +) ln n. From this by substituting successive values, 2, 3,... for n, we obtain the equations Summing =lna +ln2 ln 2 =lna/2 2 +ln3 ln 2 3 =lna/3 3 +ln4 ln 3... n =lna/n n + ln(n +) ln n n ln(n +)=lna a /2 2 a /3 3...a /n n. (M6) We note that the continued product on the right hand side is increasing in n. Also from (M4) a n < + n, hence ln a a /2 2 a /3 3...a /n n =lna + 2 ln a n ln a n, 4
15 ( < ln + ) + ( 2 ln + ) ( 2 n ln + ). n Furthermore from (M4), a n >. Thus from (M5), < ( + n) n<e. Hence ln < ln ( ) n ( ) + n < ln e, or0 < ln + n < /n. Substituting this into the inequality ln a a /2 2 a /3 3...a /n n < n 2. But it can be easily shown that the infinite series is convergent. In fact it has the value π2 6 =.644 (see Knuth - would you believe that if we select two integers at random the probability that they don t have a common factor is 6 ). Thus as the continued π 2 product is bounded, γ the l.h.s. of (M6) is bounded and hence γ< Note - It can be shown numerically that γ = Euler s constant features widely in many diverse areas of mathematics such as number theory and modified Bessel functions. Also Knuth (973) (with a rather more introductory account of this in Graham, Knuth and Patashnik) gives an interesting treatment of the above series for the summation of the first n reciprocals which he defines as H n = n =Σ k, k n, where H stands for harmonic number. Hence the infinite series H is the harmonic series. It follows that for large values of n, H(n) γ approaches ln n. It should be noted that it is only for very large values that the term γ can be discarded - for example it takes more than 0 43 terms for H(n) to exceed 00. is A more robust formulation which is a reasonable approximation for smaller values of n lim H(n) =lnn + γ + n 2n 2n n s, 4 where 0 <s< and 256n 6 A Connection Between the Harmonic Number and the Area under the curve y = x, for x> can be developed from first principles as follows. By simple graphical construction 5
16 the area under this curve, for values of x between x =and x = n is less than the sum of the n rectangles constructed, firstly on the base x =to 2 and of height (= ) and then on the base x =2to 3 and of height and so on until we construct the last rectangle on the base 2 x = n to n +of height. That is ln n<h n n. Similarly the area under this curve is greater than the sum of the of the n rectangles constructed firstly on the base x =to 2 and of height and then on the base x =2to 3 and 2 of height and so on until we construct the last rectangle of height on the base x = n 3 n to n. That is ln n>h n. def. This gives us the intuitive basis for the entry into our axiomatic treatment beginning with Now we have simply shown that +lnn>h n and this gives a better upper bound, that is γ< than Corollary M. So why have we bothered with the corollary? Well it does give some other rich relationships and it connects nicely with our main theorem. So it just could be valuable at a some later stage and that is what the essence of creative mathematics is all about - stumbling upon unexpected connections with other areas. A comment : The Generalised Harmonic Number H (r) n =+ 2 r + 3 r + 4 r n r =Σ k r, k n. For any value of r>, H (r) is convergent. H is just convergent in the sense that if r is just a touch greater than the series converges. Hence we woud expect from this that H n increases very slowly with n as we have found out above. H (r) is the corresponding infinite series is the Rieman Zeta function and is usually represented by ζ(r). Now if we attempt to solve ζ(r) =H (r) =for r, it is seen that the roots must be complex. The Riemann hypothesis makes the remarkable claim that all such roots lie along the one vertical straight line in the complex plane such that their real parts are equal to 2. 6
17 Alternative Proofs With such an important constant as e one finds that there are many rich relationships amongst the various associated expressions. Consequently it would not be expected that any particular axiomatic development would present the most elegant proofs. So we include some of these. ( A To prove e = lim n + ) n n ( Let y = n lim + n) n then ln y = lim ( n ln n ln = lim n + ( ) n + n. As this is in the form 0/0 we use l Hospitals rule and differentiate ( ) n + ln y = lim 2 n n, n 2 =, and this shows that y = e as required. n ) Notice that we have assumed P2, that is that we know the derivative of a logarithm. A2 Let f be a function such that for every pair of reals x, y f(x + y) =f(x)f(y). (M). We will say that f satisfies the multiplicative property. Then we require to prove the following: if f(x) is continuous and is not identically zero, and f() = a>0, and furthermore f(x) satisfies the multiplicative property then f(x) =a x for all x. 7
18 Proof Now f(x) =f{(x )+} = f(x )f(). Hence if f()=0then f(x) =0for all x. Hence we assume the opposite. Suppose f() = a > 0. Then again f(x) = f(x )f() = f(x )a = f(x 2)a 2 = f{x (x )}a x = f()a x = a x. Note that this shows that (M) holds only when x is integer as we subtracted integers from x to obtain. To prove this for x rational say m p = , that is p terms. p p p where m and p are integers we firstly write Thus applying (M) repeatedly f() = { f ( )} /p. p Also f ( ) ( m p = f ) p p p, where we now have m terms = { f ( )} m p = {f()} m/p = a m/p or f(x) =a x, as required for x rational. It follows that as the rationals are everywhere dense that this result also holds for real functions provided that f(x) is continuous. A3 We wish to show that the solution to dy dx = y where y(0)=, has the multiplicative property of A2. That is we claim that y(x + v) =y(x)y(v). Proof Let z(x) =y(a + x). Then z (x) =z(x). Now d y(x) = zy z y dx z(x) z 2 = zy zy z 2 =0. 8
19 Hence y(x) z(x) = K, a constant, or y(x) =Kz(x) =K y(a + x). Hence as y(0)=, with x =0we have K =. From this y(a) as required. y(a + x) =y(a) y(x), We also know independently that y(x) has a unique solution and hence the multiplicative property is unique. ( A4 Assuming A, e = lim + ) t, we wish to show that t t ( e x = lim + x n. n n) We note firstly that t may be real, that is it does not have to range over the integers for A to be valid. Now ( + x ) { n ( = + x ) } n/x x = n n with t = n. The result then follows by taking the limit. x A5 Let f(x) =b x { ( + ) t } x t Then from Q6 f (x) =b x ln b and thus f (0)=lnb But from first principles Thus with x =0 f b x+h b x (x) = lim. h 0 h b h ln b = lim h 0 h. 9
20 The relationship A5 automatically resolves the apparent conflict with the expression b a x u dx = bu+ a u+, (u +) when u, say +h, and with a =, it is seen from the above limit that this approaches ln b as required when h 0 A6 Using Q8 to define e x we wish to establish the multiplicative property. Now e u e v = m=0 u m m! m=0 v m m! But it is known that (the Cauchy convolution form for the product of two series) a n b n = c n n=0 n=0 n=0 n where c n = a n k b k. k=0 Hence and by the binomial expansion e u e v = n n=0 k=0 u n k v k (n k)! k!, n (u + v) n = n!u n k k=0 v k (n k)!k!, from which e u e v = n=0 (u + v) n n! = e u+v. References Barnes, C. Eulers Constant and e", Am Math Monthly Aug 84, p428. Coolidge, J.L. The Mathematics of Great Amateurs", (Oxford, 949). 20
21 Eves, H. Great Moments in Mathematics", (Mathematical Association of America). Knuth, D.E. The Art of Computer Programming", (Section.2.7 Harmonic Numbers p73) vol (2nd edition) 973. Graham, Knuth, D. and Patashnik, O. Concrete Maths" - (Addison Wesley - 2nd Edition 994) - Concrete is short for continuous plus discrete). Maor, Eli. e: The Story of a Number" (Princeton University Press, 994). Struik, D.J. A Source Book in Mathematics ", (Harvard, 969). APPENDIX A Napier (550-67), a Scottish Laird, was one of the great amateurs of mathematics. He began to develop logarithms in 594 and worked on this task until his death. Amongst other things he was obsessed with demonstrating that the then reigning pope was the anti christ. In fact Napier believed that his now forgotten work on the Apocalypse of St John was his greatest service to mankind and this belief was bolstered by the fact that his book on this subject went through 20 editions. Newton had a similar obsession with the Apocalypse which might indicate (at least to a statistician) that if we started off budding young geniuses on this study it could lead to great mathematical breakthroughs. Briggs was the first occupant of a chair of mathematics in Britain. This was in Geometry at Gresham College in London. Later he was the first to occupy the Savillian chair of geometry at Oxford. He extended Napier s work to base 0 and went to visit him after hearing of his wonderful discovery and communicating with him. On the interminable journey by coach to Edinburgh Briggs recorded some of his thoughts in his diary - how tall a forehead must his nobleman possess in order to house the brains that have discovered so remarkable an invention and so on. His arrival in Edinburgh was very much delayed and Napier confessed to a friend Ah John, the Professor will not come". 2
22 At that very moment a knock was heard at the gate and the Professor was ushered into the nobleman s presence. For almost a quarter of an hour each man beheld the other without speaking a word. Then Briggs said My Lord, I have undertaken this long journey purposely to see your person, and to learn by what engine of wit or ingenuity you came first to think of this most excellent help in astronomy. But, my lord being by you found out, I wonder nobody found it out before when now known it appears so easy." In understanding the importance the work of Napier it should be realised that the mathematics of his day was very deficient in both concepts and notation. Not only was there no concept of logarithm but the concept of powers was only emerging and for example the symbol x n as a shorthand for x x...had not been introduced. It is of interest to note that only because of the authority of Napier s work did the decimal notation became generally accepted, thus = instead of the clumsy The description of the technique he used for calculating his logarithms" and tables of such logarithms" are best described in his second book which was first published in Latin in 69. This was translated from the Latin into English and published by Blackwood, Edinburgh in 889, with the title The Construction of the Wonderful Canon of Logarithms". Curiously enough although the term logarithm occurs in the title the text uses only the term artificial number. As we will show shortly his logarithm is not what we now know as the natural logarithm. Detailed accounts of his work are contained in Struik, D.J. (969) and Coolidge, J.L. (949). Unfortunately his presentation is almost unintelligible to a present day mathematician and although his work is one of the great moments in mathematics it is in effect an ugly dead end, similar to the dead end taken in the pursuit of the Ptolemaic system of epicycles as a description of planetary motion. The basis for his idea was probably seen in the formula sin A sin B = {cos(a B) cos(a + B)}, 2 which was certainly known in Napier s time. 22
23 Now Stifel had earlier in Arithmetica Integra" (543) recorded the first observation that the terms of the geometric progression, r, r 2,...correspond to the arithmetic progression 0,, 2,... Stifel also noted that multiplication of two terms in this g.p. yields a term whose exponent is in effect the sum of the corresponding two terms in the a.p. Similarly he implied that division was coupled with subtraction of exponents and he was even able to manipulate negative and fractional exponents. All this was recorded without any special mathematical notation. It appears that Napier did know of this work but, incredibly, failed to see the relevance of exponents on his logarithms. If he had picked it up it would most surely have accelerated the development of mathematics. Because of his interest in spherical trigonometry he based his work on the logarithms of sines however let us warn you now that his use of the terms radius and sine simply referred to intervals on a straight line (AZ and DZ in the following). In effect he said the following. Lay out two lines AZ and A Z. Measuring in the direction from Z to A, the point A corresponds to what Napier calls the sine of 90 degrees which he has approximated by 0 7 as he could not of course represent infinity. Along A Z he lays out the corresponding logarithms" of these sines, thus he has A representing 0, or in his notation, logarithm" 0 7 =0.Nowhe lays out points B, C, D, E, along AZ and, the points B, C, D, E,...he lays out along A Z. It will be shown that A B corresponds to his logarithm of AZ, A C corresponds to his logarithm of BZ and so on. Now he has a point initially at A moving along AZ, with a velocity proportional to its distance from Z and this corresponds to a point moving along A Z with constant velocity equal to the initial velocity of the other point when it is at A. Suppose that on AZ each of the intervals AB, BC, CD...are traversed in the same small interval t. Napier assumed that if t is small enough then the velocity of the point throughout the interval is constant and equal to the velocity at the beginning of the interval. From this the intervals AZ, BZ, CZ,... form a decreasing g.p. as we will now demonstrate. Let the velocity of the point throughout the interval whose initial distance is y from B be v, in the 23
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There are no store-bought salad dressings in my house. In my humble opinion, store bought salad dressings are over seasoned and/or over sweetened and often they contain inferior oils. The flavours are much too strong and overall, none of them taste fresh.
Salad dressings are easy to make and taste infinitely better than anything you can buy so skip the store-bought varieties and make your own. To get you started, I am sharing my top three oil-free salad dressing recipes.
Remember, oil free is not the same as fat-free. There is no shortage of contradictory information available of the benefits of adding oil to your salad – indeed to your diet. A recent study published in Science Daily found that pairing vegetables with fat matters. The study found that fewer nutrients were absorbed when using low fat dressing and that overall, salads need to be paired with the right type and right amount of fat-based dressing in order for vitamins and minerals to be absorbed by the body.
Other sources of research denounce the benefits of pouring any oil (even so-called healthy oil) on your salad and suggest that a salad made from homegrown or local, organic ingredients is completely healthy all on its own. These sources indicate that vegetable oils, (yes, even olive oil) can damage the health of your arteries and that you can get all of the healthy fat you need from nuts, seeds and other whole foods such as avocados. (Click on the "Why this blog?" page for more sources citing on the dangers of consuming vegetable oils).
All three dressings below contain seeds, nuts or a combination of the two, which provide a good source of monounsaturated fat. I have come across nothing in the research to suggest that the fat found necessary for vitamin and mineral absorption must come from extracted oils and the Science Daily study didn't touch on absorption of nutrients from seeds, nuts or other whole plant-food sources, such as avocados. To me, it seems logical that health benefits would be maximized from eating the whole food rather than an extracted and processed component of the food.
These are my "go-to" dressings. I have at least one of them in my fridge at all times. As well as being oil-free, none contain refined sugar, dairy or soy either, including a "Caesar" dressing that is very satisfying and tastes incredibly like the traditional variety. Each of these dressings pack a punch of flavor and take no time at all to whip up. They all last about a week in the refrigerator.
These dressings are prepared exactly the same way so I won't belabour this post by repeating the method of preparation for each recipe – I have just listed the ingredients. For the method, place the ingredients for your selected dressing in a high-speed blender and blend until smooth. That's all there is to it!
Creamy Black Sesame Seed Dressing
A hint of sweetness from maple or date syrup make this a delicious choice for both green salads or fruit salads. You can switch the black sesame seeds with poppy seeds if you like.
- 3/4 cup water
- 1/3 cup almond meal
- 2 tbsp. lemon juice
- 1 large clove garlic, peeled
- 2 tbsp. minced shallots
- 1 tbsp. maple or date syrup (homemade)
- 1 tbsp. Dijon mustard
- 1 tbsp. black sesame seeds or poppy seeds
Yields about 3/4 cup dressing.
Chipotle Dressing
This is an adaptation of an oil-free chipotle dressing recipe by anunrefinedvegan.com The chipotle in adobo sauce gives this dressing quite a kick so start with a small amount and increase to satisfy your tastebuds. The 1/2 tsp measure is perfect for me.
- 1/2 cup water
- 1/4 cup apple cider vinegar
- juice of 1 lime
- 1/3 cup almond meal
- 2 cloves garlic, peeled
- 1 tbsp. raw sunflower or pumpkin seeds
- 1 tbsp. hemp seeds
- 1 tsp. onion powder
- 1/2 tsp. chipotle in adobo sauce
- 1/2 tsp. cumin
- 1/4 tsp. chile powder
- 1/4 tsp. smoked paprika
Yields approximately 1 1/4 cups dressing.
Vegan Caesar Salad Dressing
Imagine! A creamy Caesar dressing with no eggs, dairy or fish! And no other unhealthy ingredients either! This is an all-time favourite in my family – you'll love it!
I have included a recipe for a quick almond cheese (tastes like Parmesan) for those who would miss the addition of cheese in their Caesar salad. However, I have to say, this dressing tastes every bit as good as a traditional Caesar dressing and the salad is delightful with or without the mock Parmesan.
This dressing calls for dulse flakes. Dulse is a type of red seaweed that has been harvested for its high mineral content for hundreds of years along the shores of Canada, the Atlantic coast, Ireland and Norway. Dulse flakes are added because they impart a salty and mild fishy flavour – they take the place of the anchovies called for in a traditional Caesar dressing. But don't be put off – dulse flakes are not at all overpowering and they contribute to the rich depth of flavour in this dressing.
Dulse flakes can turn the colour of the dressing a little pink. This doesn't bother me, in fact the pinkish hue is barely noticeable when the salad is tossed, but if it bothers you, rather than adding dulse flakes to the blender with everything else, stir them into the dressing at the end, after everything else is well blended. Your dressing will be speckled, but whiter.
- ¼ cup tahini
- ¼ cup fresh lemon juice
- 1 celery stalk
- 2 Tbsp. apple cider vinegar
- 2 Tbsp. hemp seeds
- 2 tsp dulse flakes, crumbled (optional)
- 1 tsp. GF Worcestershire sauce, such as Wizard's
- 3/4 tsp Celtic sea salt
- ½ tsp white pepper
- 1 tsp maple or date syrup
- 1 large clove garlic, peeled, halved
- croutons, gluten-free
Almond Cheese (mock Parmesan)
Pulse the following ingredients in a small nut grinder or coffee grinder just until combined. Sprinkle over salad before serving to boost the cheesy flavour and add texture.
- 2 tbsp. almond meal
- ¼ tsp. salt
- ¼ tsp. lemon juice
Note: Caesar salad dressing revised November 6, 2015


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Son of Republican presidential candidate cited in alleged alcohol related crash2015-04-28 20:55:47 (GMT) (Caymanmama.com - News Providers News)
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04/22/2015 // West Palm Beach, Florida, US // href='http://justicenews.visionsmartnews.com' rel='nofollow'>JusticeNewsFlash // Justice News Flash // (press release)
Kentucky – The rel="nofollow" href="http://news.yahoo.com/rand-pauls-son-cited-dui-car-crash-kentucky-215459961.html">son of Republican presidential candidate Rand Paul was involved in a car crash in Kentucky that resulted in him being cited for driving under the influence of alcohol. An AP report says that Lexington police say William H. Paul crashed into an unoccupied parked vehicle while driving a 2006 Honda Ridgeline Sunday morning.
Authorities were alerted to the accident by individuals who were nearby at the time. Sherelle Roberts, a spokesperson for the Lexington Police, said Paul was taken to the University of Kentucky hospital for minor injuries he sustained to his face. He was reportedly alone when the incident happened.
Roberts said Paul was not arrested at the time, which is the purported standard for the type of case.
Paul received a citation for operating a motor vehicle under the influence of alcohol and failing to maintain insurance.
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