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Leonhard Euler

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Leonhard Euler

Leonhard Euler ( ;[1] German pronunciation:  ( ), local pronunciation:  ( ); 15 April 1707 – 18 September 1783) was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.[2] He is also renowned for his work in mechanics, fluid dynamics, optics, astronomy, and music theory.[3]

Euler is considered to be the pre-eminent mathematician of the 18th century and one of the greatest mathematicians to have ever lived. He is also one of the most prolific mathematicians; his collected works fill 60–80 quarto volumes.[4] He spent most of his adult life in St. Petersburg, Russia, and in Berlin, Prussia.

A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all."[5][6]

  • Life 1
    • Early years 1.1
    • St. Petersburg 1.2
    • Berlin 1.3
    • Eyesight deterioration 1.4
    • Return to Russia 1.5
  • Contributions to mathematics and physics 2
    • Mathematical notation 2.1
    • Analysis 2.2
    • Number theory 2.3
    • Graph theory 2.4
    • Applied mathematics 2.5
    • Physics and astronomy 2.6
    • Logic 2.7
  • Personal philosophy and religious beliefs 3
  • Commemorations 4
  • Selected bibliography 5
  • See also 6
  • References and notes 7
  • Further reading 8
  • External links 9

Early years

Old Swiss 10 Franc banknote honoring Euler

Euler was born on 15 April 1707, in Basel to Paul Euler, a pastor of the Reformed Church, and Marguerite Brucker, a pastor's daughter. He had two younger sisters named Anna Maria and Maria Magdalena. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, where Euler spent most of his childhood. Paul Euler was a friend of the Bernoulli familyJohann Bernoulli, who was then regarded as Europe's foremost mathematician, would eventually be the most important influence on young Leonhard. Euler's early formal education started in Basel, where he was sent to live with his maternal grandmother. At the age of 13 he enrolled at the University of Basel, and in 1723, received his Master of Philosophy with a dissertation that compared the philosophies of Descartes and Newton. At this time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil's incredible talent for mathematics.[7] Euler was at this point studying theology, Greek, and Hebrew at his father's urging, in order to become a pastor, but Bernoulli convinced Paul Euler that Leonhard was destined to become a great mathematician. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono.[8] At that time, he was pursuing an (ultimately unsuccessful) attempt to obtain a position at the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition; the problem that year was to find the best way to place the masts on a ship. Pierre Bouguer, a man who became known as "the father of naval architecture" won, and Euler took second place. Euler later won this annual prize twelve times.[9]

St. Petersburg

Around this time Johann Bernoulli's two sons, Daniel and Nicolas, were working at the Imperial Russian Academy of Sciences in St Petersburg. On 10 July 1726, Nicolas died of appendicitis after spending a year in Russia, and when Daniel assumed his brother's position in the mathematics/physics division, he recommended that the post in physiology that he had vacated be filled by his friend Euler. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to St Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel.[10]

1957 Soviet Union stamp commemorating the 250th birthday of Euler. The text says: 250 years from the birth of the great mathematician, academician Leonhard Euler.

Euler arrived in the Russian capital on 17 May 1727. He was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he often worked in close collaboration. Euler mastered Russian and settled into life in St Petersburg. He also took on an additional job as a medic in the Russian Navy.[11]

The Academy at St. Petersburg, established by Peter the Great, was intended to improve education in Russia and to close the scientific gap with Western Europe. As a result, it was made especially attractive to foreign scholars like Euler. The academy possessed ample financial resources and a comprehensive library drawn from the private libraries of Peter himself and of the nobility. Very few students were enrolled in the academy in order to lessen the faculty's teaching burden, and the academy emphasized research and offered to its faculty both the time and the freedom to pursue scientific questions.[9]

The Academy's benefactress, Catherine I, who had continued the progressive policies of her late husband, died on the day of Euler's arrival. The Russian nobility then gained power upon the ascension of the twelve-year-old Peter II. The nobility were suspicious of the academy's foreign scientists, and thus cut funding and caused other difficulties for Euler and his colleagues.

Conditions improved slightly upon the death of Peter II, and Euler swiftly rose through the ranks in the academy and was made professor of physics in 1731. Two years later, Daniel Bernoulli, who was fed up with the censorship and hostility he faced at St. Petersburg, left for Basel. Euler succeeded him as the head of the mathematics department.[12]

On 7 January 1734, he married Katharina Gsell (1707–1773), a daughter of

  • Weisstein, Eric W., Euler, Leonhard (1707–1783) from ScienceWorld.
  • Encyclopædia Britannica article
  • Leonhard Euler at the Mathematics Genealogy Project
  • How Euler did it contains columns explaining how Euler solved various problems
  • Euler Archive
  • Leonhard Euler – Œuvres complètes Gallica-Math
  • Euler Committee of the Swiss Academy of Sciences
  • References for Leonhard Euler
  • Euler Tercentenary 2007
  • The Euler Society
  • Euler Family Tree
  • Euler's Correspondence with Frederick the Great, King of Prussia
  • "Euler – 300th anniversary lecture", given by Robin Wilson at Gresham College, 9 May 2007 (can download as video or audio files)
  •  .
  • Euler Quartic Conjecture

External links

  • Lexikon der Naturwissenschaftler, (2000), Heidelberg: Spektrum Akademischer Verlag.
  • Bradley, Robert E.; D'Antonio, Lawrence A.; Sandifer, Charles Edward (2007). Euler at 300: An Appreciation. Mathematical Association of America.
  • Calinger, Ronald (1996). "Leonhard Euler: The First St. Petersburg Years (1727–1741)". Historia Mathematica 23 (2): 121–166.
  • Demidov, S. S. (2005). "Treatise on the differential calculus". In
  • Dunham, William (2007). The Genius of Euler: Reflections on his Life and Work. Mathematical Association of America.
  • Fraser, Craig G. "Leonhard Euler's 1744 book on the calculus of variations".  In Grattan-Guinness 2005, pp. 168–80
  • Gladyshev, Georgi P. (2007). "Leonhard Euler's methods and ideas live on in the thermodynamic hierarchical theory of biological evolution". International Journal of Applied Mathematics & Statistics (IJAMAS) 11 (N07).  Special Issue on Leonhard Paul Euler's: Mathematical Topics and Applications (M. T. A.).
  • Heimpell, Hermann, Theodor Heuss, Benno Reifenberg (editors). 1956. Die großen Deutschen, volume 2, Berlin: Ullstein Verlag.
  • Krus, D. J. (November 2001). "Is the normal distribution due to Gauss? Euler, his family of gamma functions, and their place in the history of statistics". Quality and Quantity: International Journal of Methodology 35 (4): 445–6.
  • Nahin, Paul J. (2006). Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills. Princeton University Press.
  • du Pasquier, Louis-Gustave (2008). Leonhard Euler And His Friends. Translated by John S.D. Glaus. CreateSpace.
  • Reich, Karin. Introduction' to analysis"'".  In Grattan-Guinness 2005, pp. 181–90
  • Richeson, David S. (2011). Euler's Gem: The Polyhedron Formula and the Birth of Topology. Princeton University Press.
  • Sandifer, C. Edward (2007). The Early Mathematics of Leonhard Euler. Mathematical Association of America.
  • Sandifer, C. Edward (2007). How Euler Did It. Mathematical Association of America.
  • Simmons, J. (1996). The giant book of scientists: The 100 greatest minds of all time. Sydney: The Book Company.
  • Thiele, Rüdiger (2005). "The mathematics and science of Leonhard Euler". In Kinyon, Michael; van Brummelen, Glen. Mathematics and the Historian's Craft: The Kenneth O. May Lectures. Springer. pp. 81–140.
  • "A Tribute to Leohnard Euler 1707–1783".

Further reading

  1. ^ The pronunciation is incorrect. "Euler",
  2. ^ a b Dunham 1999, p. 17
  3. ^ Saint Petersburg (1739). "Tentamen novae theoriae musicae ex certissimis harmoniae principiis dilucide expositae".
  4. ^ a b c Finkel, B. F. (1897). "Biography—Leonard Euler". The American Mathematical Monthly 4 (12): 297–302.
  5. ^ Dunham 1999, p. xiii "Lisez Euler, lisez Euler, c'est notre maître à tous."
  6. ^ The quote appeared in Gugliemo Libri's review of a recently published collection of correspondence among eighteenth-century mathematicians: Gugliemo Libri (January 1846), Book review: "Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIe siècle, … " (Mathematical and physical correspondence of some famous geometers of the eighteenth century, … ), Journal des Savants, page 51. From page 51: " … nous rappellerions que Laplace lui même, … ne cessait de répéter aux jeunes mathématiciens ces paroles mémorables que nous avons entendues de sa propre bouche : 'Lisez Euler, lisez Euler, c'est notre maître à tous.' " ( … we would recall that Laplace himself, … never ceased to repeat to young mathematicians these memorable words that we heard from his own mouth: 'Read Euler, read Euler, he is our master in everything.)
  7. ^ James, Ioan (2002). Remarkable Mathematicians: From Euler to von Neumann. Cambridge. p. 2.
  8. ^ Ian Bruce. "Euler's Dissertation De Sono : E002. Translated & Annotated". Retrieved 14 September 2011.
  9. ^ a b Calinger 1996, p. 156
  10. ^ Calinger 1996, p. 125
  11. ^
  12. ^ Calinger 1996, pp. 128–9
  13. ^ Gekker, I. R.; Euler, A. A. "Leonhard Euler's family and descendants". Bogoli︠u︡bov, Mikhaĭlov & I︠U︡shkevich 2007, p. 402.
  14. ^ Fuss, Nicolas. "Eulogy of Euler by Fuss". Retrieved 30 August 2006.
  15. ^ "E212 – Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum". Dartmouth.
  16. ^ a b c Dunham 1999, pp. xxiv–xxv
  17. ^ Euler, Leonhard. "Letters to a German Princess on Diverse Subjects of Natural Philosophy". Internet Archive, Digitzed by Google. Retrieved 15 April 2013.
  18. ^
  19. ^ Calinger 1996, pp. 154–5
  20. ^ Gekker & Euler 2007, p. 405
  21. ^ "Book of Members, 1780–2010: Chapter E". American Academy of Arts and Sciences. Retrieved 28 July 2014.
  22. ^ A. Ya. Yakovlev (1983). Leonhard Euler. M.: Prosvesheniye.
  23. ^ "Eloge de M. Leonhard Euler. Par M. Fuss". Nova Acta Academia Scientarum Imperialis Petropolitanae 1: 159–212. 1783.
  24. ^ Marquis de Condorcet. "Eulogy of Euler – Condorcet". Retrieved 30 August 2006.
  25. ^ Leonhard Euler at Find a Grave
  26. ^ Derbyshire, John (2003).
  27. ^ a b Boyer, Carl B.; Uta C. Merzbach (1991). A History of Mathematics.
  28. ^ Wolfram, Stephen. "Mathematical Notation: Past and Future". Retrieved 23 September 2014.
  29. ^ a b Wanner, Gerhard; Hairer, Ernst (March 2005). Analysis by its history (1st ed.). Springer. p. 63.
  30. ^ Feynman, Richard (June 1970). "Chapter 22: Algebra". The Feynman Lectures on Physics: Volume I. p. 10.
  31. ^ a b Wells, David (1990). "Are these the most beautiful?". Mathematical Intelligencer 12 (3): 37–41.
    Wells, David (1988). "Which is the most beautiful?". Mathematical Intelligencer 10 (4): 30–31.
    See also: Peterson, Ivars. "The Mathematical Tourist". Retrieved March 2008.
  32. ^ Dunham 1999, Ch. 3, Ch. 4
  33. ^ Dunham 1999, Ch. 1, Ch. 4
  34. ^ Caldwell, Chris. The largest known prime by year
  35. ^ a b
  36. ^ Cromwell, Peter R. (1999). Polyhedra. Cambridge University Press. pp. 189–190.
  37. ^ Gibbons, Alan (1985). Algorithmic Graph Theory. Cambridge University Press. p. 72.
  38. ^ Cauchy, A. L. (1813). "Recherche sur les polyèdres—premier mémoire".
  39. ^ L'Huillier, S.-A.-J. (1861). "Mémoire sur la polyèdrométrie". Annales de Mathématiques 3: 169–189.
  40. ^ Calinger 1996, pp. 144–5
  41. ^ Youschkevitch, A P (1970–1990). Dictionary of Scientific Biography. New York.
  42. ^ Home, R. W. (1988). "Leonhard Euler's 'Anti-Newtonian' Theory of Light". Annals of Science 45 (5): 521–533.
  43. ^ Euler, Leonhard (1757). "Principes généraux de l'état d'équilibre d'un ﬂuide" [General principles of the state of equilibrium of a fluid]. Académie Royale des Sciences et des Belles-Lettres de Berlin, Mémoires 11: 217–273.
  44. ^ Gautschi, Walter (2008). "Leonhard Euler: His Life, the Man, and His Work". SIAM Review 50 (1): 3–33.
  45. ^ Baron, M. E. (May 1969). "A Note on The Historical Development of Logic Diagrams". The Mathematical Gazette LIII (383): 113–125.
  46. ^ "Strategies for Reading Comprehension Venn Diagrams".
  47. ^ Calinger 1996, pp. 153–4
  48. ^ a b Euler, Leonhard (1960). Orell-Fussli, ed. "Rettung der Göttlichen Offenbahrung Gegen die Einwürfe der Freygeister". Leonhardi Euleri Opera Omnia (series 3) 12.
  49. ^ ; Gillings, R. J. (February 1954). "The So-Called Euler-Diderot Incident". The American Mathematical Monthly 61 (2): 77–80.
  50. ^ Marty, Jacques. "Quelques aspects des travaux de Diderot en Mathematiques Mixtes.".
  51. ^ Brown, B.H. (May 1942). "The Euler-Diderot Anecdote".
  52. ^
  53. ^ Gillings, R.J. (Feb 1954). "The So-Called Euler-Diderot Anecdote".
  54. ^
  55. ^ E65 — Methodus... entry at Euler Archives. Retrieved on 14 September 2011.
  56. ^ Leonhard Euler at the Mathematics Genealogy Project

References and notes

See also

A definitive collection of Euler's works, entitled Opera Omnia, has been published since 1911 by the Euler Commission of the Swiss Academy of Sciences. A complete chronological list of Euler's works is available at the following page: The Eneström Index (PDF).

  • Elements of Algebra. This elementary algebra text starts with a discussion of the nature of numbers and gives a comprehensive introduction to algebra, including formulae for solutions of polynomial equations.
  • Introductio in analysin infinitorum (1748). English translation Introduction to Analysis of the Infinite by John Blanton (Book I, ISBN 0-387-96824-5, Springer-Verlag 1988; Book II, ISBN 0-387-97132-7, Springer-Verlag 1989).
  • Two influential textbooks on calculus: Institutiones calculi differentialis (1755) and Institutionum calculi integralis (1768–1770).
  • Letters to a German Princess (1768–1772).
  • Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti (1744). The Latin title translates as a method for finding curved lines enjoying properties of maximum or minimum, or solution of isoperimetric problems in the broadest accepted sense.[55]

Euler has an extensive bibliography. His best-known books include:

The title page of Euler's Methodus inveniendi lineas curvas.

Selected bibliography

On 15 April 2013, Euler's 306th birthday was celebrated with a Google Doodle.[54]

Euler was featured on the sixth series of the Swiss 10-franc banknote and on numerous Swiss, German, and Russian postage stamps. The asteroid 2002 Euler was named in his honor. He is also commemorated by the Lutheran Church on their Calendar of Saints on 24 May—he was a devout Christian (and believer in biblical inerrancy) who wrote apologetics and argued forcefully against the prominent atheists of his time.[48]

[53][52] inspired by Euler's arguments with secular philosophers over religion, which is set during Euler's second stint at the St. Petersburg academy. The French philosopher [49] There is a famous legend

Much of what is known of Euler's religious beliefs can be deduced from his Letters to a German Princess and an earlier work, Rettung der Göttlichen Offenbahrung Gegen die Einwürfe der Freygeister (Defense of the Divine Revelation against the Objections of the Freethinkers). These works show that Euler was a devout Christian who believed the Bible to be inspired; the Rettung was primarily an argument for the divine inspiration of scripture.[48]

Euler and his friend Daniel Bernoulli were opponents of Leibniz's monadism and the philosophy of Christian Wolff. Euler insisted that knowledge is founded in part on the basis of precise quantitative laws, something that monadism and Wolffian science were unable to provide. Euler's religious leanings might also have had a bearing on his dislike of the doctrine; he went so far as to label Wolff's ideas as "heathen and atheistic".[47]

Personal philosophy and religious beliefs

An Euler diagram is a diagrammatic means of representing sets and their relationships. Euler diagrams consist of simple closed curves (usually circles) in the plane that depict sets. Each Euler curve divides the plane into two regions or "zones": the interior, which symbolically represents the elements of the set, and the exterior, which represents all elements that are not members of the set. The sizes or shapes of the curves are not important: the significance of the diagram is in how they overlap. The spatial relationships between the regions bounded by each curve (overlap, containment or neither) corresponds to set-theoretic relationships (intersection, subset and disjointness). Curves whose interior zones do not intersect represent disjoint sets. Two curves whose interior zones intersect represent sets that have common elements; the zone inside both curves represents the set of elements common to both sets (the intersection of the sets). A curve that is contained completely within the interior zone of another represents a subset of it. Euler diagrams were incorporated as part of instruction in set theory as part of the new math movement in the 1960s. Since then, they have also been adopted by other curriculum fields such as reading.[46]

Euler's Diagram

Euler is also credited with using closed curves to illustrate syllogistic reasoning (1768). These diagrams have become known as Euler diagrams.[45]

F = maximum or critical force (vertical load on column),
E = modulus of elasticity,
I = area moment of inertia,
L = unsupported length of column,
K = column effective length factor, whose value depends on the conditions of end support of the column, as follows.
For both ends pinned (hinged, free to rotate), K = 1.0.
For both ends fixed, K = 0.50.
For one end fixed and the other end pinned, K = 0.699....
For one end fixed and the other end free to move laterally, K = 2.0.
K L is the effective length of the column.

F=\frac{\pi^2 EI}{(KL)^2}

Euler is also well known in structural engineering for his formula giving the critical buckling load of an ideal strut, which depends only on its length and flexural stiffness:[44]

\begin{align} &{\partial\rho\over\partial t}+ \nabla\cdot(\rho\bold u)=0\\[1.2ex] &{\partial(\rho{\bold u})\over\partial t}+ \nabla\cdot(\bold u\otimes(\rho \bold u))+\nabla p=\bold{0}\\[1.2ex] &{\partial E\over\partial t}+ \nabla\cdot(\bold u(E+p))=0, \end{align}

In 1757 he published an important set of equations for inviscid flow, that are now known as the Euler equations.[43] In differential form, the equations are:

In addition, Euler made important contributions in optics. He disagreed with Newton's corpuscular theory of light in the Opticks, which was then the prevailing theory. His 1740s papers on optics helped ensure that the wave theory of light proposed by Christiaan Huygens would become the dominant mode of thought, at least until the development of the quantum theory of light.[42]

Euler helped develop the Euler–Bernoulli beam equation, which became a cornerstone of engineering. Aside from successfully applying his analytic tools to problems in classical mechanics, Euler also applied these techniques to celestial problems. His work in astronomy was recognized by a number of Paris Academy Prizes over the course of his career. His accomplishments include determining with great accuracy the orbits of comets and other celestial bodies, understanding the nature of comets, and calculating the parallax of the sun. His calculations also contributed to the development of accurate longitude tables.[41]

Physics and astronomy

One of Euler's more unusual interests was the application of mathematical ideas in music. In 1739 he wrote the Tentamen novae theoriae musicae, hoping to eventually incorporate musical theory as part of mathematics. This part of his work, however, did not receive wide attention and was once described as too mathematical for musicians and too musical for mathematicians.[40]

\gamma = \lim_{n \rightarrow \infty } \left( 1+ \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots + \frac{1}{n} - \ln(n) \right).

Some of Euler's greatest successes were in solving real-world problems analytically, and in describing numerous applications of the Bernoulli numbers, Fourier series, Venn diagrams, Euler numbers, the constants e and π, continued fractions and integrals. He integrated Leibniz's differential calculus with Newton's Method of Fluxions, and developed tools that made it easier to apply calculus to physical problems. He made great strides in improving the numerical approximation of integrals, inventing what are now known as the Euler approximations. The most notable of these approximations are Euler's method and the Euler–Maclaurin formula. He also facilitated the use of differential equations, in particular introducing the Euler–Mascheroni constant:

Applied mathematics

Euler also discovered the formula V − E + F = 2 relating the number of vertices, edges and faces of a convex polyhedron,[36] and hence of a planar graph. The constant in this formula is now known as the Euler characteristic for the graph (or other mathematical object), and is related to the genus of the object.[37] The study and generalization of this formula, specifically by Cauchy[38] and L'Huillier,[39] is at the origin of topology.

In 1735, Euler presented a solution to the problem known as the Seven Bridges of Königsberg.[35] The city of Königsberg, Prussia was set on the Pregel River, and included two large islands that were connected to each other and the mainland by seven bridges. The problem is to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point. It is not possible: there is no Eulerian circuit. This solution is considered to be the first theorem of graph theory, specifically of planar graph theory.[35]

Map of Königsberg in Euler's time showing the actual layout of the seven bridges, highlighting the river Pregel and the bridges.

Graph theory

Euler proved Newton's identities, Fermat's little theorem, Fermat's theorem on sums of two squares, and he made distinct contributions to Lagrange's four-square theorem. He also invented the totient function φ(n), the number of positive integers less than or equal to the integer n that are coprime to n. Using properties of this function, he generalized Fermat's little theorem to what is now known as Euler's theorem. He contributed significantly to the theory of perfect numbers, which had fascinated mathematicians since Euclid. He proved that the relationship shown between perfect numbers and Mersenne primes earlier proved by Euclid was one-to-one, a result otherwise known as the Euclid–Euler theorem. Euler also conjectured the law of quadratic reciprocity. The concept is regarded as a fundamental theorem of number theory, and his ideas paved the way for the work of Carl Friedrich Gauss.[33] By 1772 Euler had proved that 231 − 1 = 2,147,483,647 is a Mersenne prime. It may have remained the largest known prime until 1867.[34]

Euler linked the nature of prime distribution with ideas in analysis. He proved that the sum of the reciprocals of the primes diverges. In doing so, he discovered the connection between the Riemann zeta function and the prime numbers; this is known as the Euler product formula for the Riemann zeta function.

Euler's interest in number theory can be traced to the influence of Christian Goldbach, his friend in the St. Petersburg Academy. A lot of Euler's early work on number theory was based on the works of Pierre de Fermat. Euler developed some of Fermat's ideas, and disproved some of his conjectures.

Number theory

Euler also pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced a new field of study, analytic number theory. In breaking ground for this new field, Euler created the theory of hypergeometric series, q-series, hyperbolic trigonometric functions and the analytic theory of continued fractions. For example, he proved the infinitude of primes using the divergence of the harmonic series, and he used analytic methods to gain some understanding of the way prime numbers are distributed. Euler's work in this area led to the development of the prime number theorem.[32]

In addition, Euler elaborated the theory of higher transcendental functions by introducing the gamma function and introduced a new method for solving quartic equations. He also found a way to calculate integrals with complex limits, foreshadowing the development of modern complex analysis. He also invented the calculus of variations including its best-known result, the Euler–Lagrange equation.

De Moivre's formula is a direct consequence of Euler's formula.

called "the most remarkable formula in mathematics" by Richard P. Feynman, for its single uses of the notions of addition, multiplication, exponentiation, and equality, and the single uses of the important constants 0, 1, e, i and π.[30] In 1988, readers of the Mathematical Intelligencer voted it "the Most Beautiful Mathematical Formula Ever".[31] In total, Euler was responsible for three of the top five formulae in that poll.[31]

e^{i \pi} +1 = 0 \,

A special case of the above formula is known as Euler's identity,

e^{i\varphi} = \cos \varphi + i\sin \varphi.\,

Euler introduced the use of the exponential function and logarithms in analytic proofs. He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative and complex numbers, thus greatly expanding the scope of mathematical applications of logarithms.[27] He also defined the exponential function for complex numbers, and discovered its relation to the trigonometric functions. For any real number φ (taken to be radians), Euler's formula states that the complex exponential function satisfies

A geometric interpretation of Euler's formula
\sum_{n=1}^\infty {1 \over n^2} = \lim_{n \to \infty}\left(\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots + \frac{1}{n^2}\right) = \frac{\pi ^2}{6}.

Notably, Euler directly proved the power series expansions for e and the inverse tangent function. (Indirect proof via the inverse power series technique was given by Newton and Leibniz between 1670 and 1680.) His daring use of power series enabled him to solve the famous Basel problem in 1735 (he provided a more elaborate argument in 1741):[29]

e^x = \sum_{n=0}^\infty {x^n \over n!} = \lim_{n \to \infty}\left(\frac{1}{0!} + \frac{x}{1!} + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!}\right).

The development of infinitesimal calculus was at the forefront of 18th Century mathematical research, and the Bernoullis—family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus became the major focus of Euler's work. While some of Euler's proofs are not acceptable by modern standards of mathematical rigour[29] (in particular his reliance on the principle of the generality of algebra), his ideas led to many great advances. Euler is well known in analysis for his frequent use and development of power series, the expression of functions as sums of infinitely many terms, such as

Euler introduced and popularized several notational conventions through his numerous and widely circulated textbooks. Most notably, he introduced the concept of a function[2] and was the first to write f(x) to denote the function f applied to the argument x. He also introduced the modern notation for the trigonometric functions, the letter e for the base of the natural logarithm (now also known as Euler's number), the Greek letter Σ for summations and the letter i to denote the imaginary unit.[27] The use of the Greek letter π to denote the ratio of a circle's circumference to its diameter was also popularized by Euler, although it did not originate with him.[28]

Mathematical notation

Euler is the only mathematician to have two numbers named after him: the important Euler's Number in calculus, e, approximately equal to 2.71828, and the Euler-Mascheroni Constant γ (gamma) sometimes referred to as just "Euler's constant", approximately equal to 0.57721. It is not known whether γ is rational or irrational.[26]

Euler worked in almost all areas of mathematics, such as geometry, infinitesimal calculus, trigonometry, algebra, and number theory, as well as continuum physics, lunar theory and other areas of physics. He is a seminal figure in the history of mathematics; if printed, his works, many of which are of fundamental interest, would occupy between 60 and 80 quarto volumes.[4] Euler's name is associated with a large number of topics.

Contributions to mathematics and physics

Euler's grave at the Alexander Nevsky Monastery

He was buried next to Katharina at the Smolensk Lutheran Cemetery on Vasilievsky Island. In 1785, the Russian Academy of Sciences put a marble bust of Leonhard Euler on a pedestal next to the Director's seat and, in 1837, placed a headstone on Euler's grave. To commemorate the 250th anniversary of Euler's birth, the headstone was moved in 1956, together with his remains, to the 18th-century necropolis at the Alexander Nevsky Monastery.[25]

il cessa de calculer et de vivre—... he ceased to calculate and to live.[24]

In St. Petersburg on 18 September 1783, after a lunch with his family, during a conversation with a fellow academician Anders Johan Lexell, about the newly discovered planet Uranus and its orbit, Euler suffered a brain hemorrhage and died a few hours later.[22] A short obituary for the Russian Academy of Sciences was written by Jacob von Staehlin-Storcksburg and a more detailed eulogy[23] was written and delivered at a memorial meeting by Russian mathematician Nicolas Fuss, one of Euler's disciples. In the eulogy written for the French Academy by the French mathematician and philosopher Marquis de Condorcet, he commented,

The situation in Russia had improved greatly since the accession to the throne of Catherine the Great, and in 1766 Euler accepted an invitation to return to the St. Petersburg Academy and spent the rest of his life in Russia. However, his second stay in the country was marred by tragedy. A fire in St. Petersburg in 1771 cost him his home, and almost his life. In 1773, he lost his wife Katharina after 40 years of marriage. Three years after his wife's death, Euler married her half-sister, Salome Abigail Gsell (1723–1794).[20] This marriage lasted until his death. He was elected a Foreign Honorary Member of the American Academy of Arts and Sciences in 1782.[21]

Return to Russia

Euler's eyesight worsened throughout his mathematical career. Three years after suffering a near-fatal fever in 1735, he became almost blind in his right eye, but Euler rather blamed the painstaking work on cartography he performed for the St. Petersburg Academy for his condition. Euler's vision in that eye worsened throughout his stay in Germany, to the extent that Frederick referred to him as "Cyclops". Euler later developed a cataract in his left eye, which was discovered in 1766. Just a few weeks after its discovery, he was rendered almost totally blind. However, his condition appeared to have little effect on his productivity, as he compensated for it with his mental calculation skills and exquisite memory. For example, Euler could repeat the Aeneid of Virgil from beginning to end without hesitation, and for every page in the edition he could indicate which line was the first and which the last. With the aid of his scribes, Euler's productivity on many areas of study actually increased. He produced on average, one mathematical paper every week in the year 1775.[4]

Eyesight deterioration

A 1753 portrait by Emanuel Handmann. This portrayal suggests problems of the right eyelid, and possible strabismus. The left eye, which here appears healthy, was later affected by a cataract.[19]
I wanted to have a water jet in my garden: Euler calculated the force of the wheels necessary to raise the water to a reservoir, from where it should fall back through channels, finally spurting out in Sanssouci. My mill was carried out geometrically and could not raise a mouthful of water closer than fifty paces to the reservoir. Vanity of vanities! Vanity of geometry![18]

Despite Euler's immense contribution to the Academy's prestige, he was eventually forced to leave Berlin. This was partly because of a conflict of personality with Frederick, who came to regard Euler as unsophisticated, especially in comparison to the circle of philosophers the German king brought to the Academy. Voltaire was among those in Frederick's employ, and the Frenchman enjoyed a prominent position within the king's social circle. Euler, a simple religious man and a hard worker, was very conventional in his beliefs and tastes. He was in many ways the antithesis of Voltaire. Euler had limited training in rhetoric, and tended to debate matters that he knew little about, making him a frequent target of Voltaire's wit.[16] Frederick also expressed disappointment with Euler's practical engineering abilities:

In addition, Euler was asked to tutor Friederike Charlotte of Brandenburg-Schwedt, the Princess of Anhalt-Dessau and Frederick's niece. Euler wrote over 200 letters to her in the early 1760s, which were later compiled into a best-selling volume entitled Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess.[17] This work contained Euler's exposition on various subjects pertaining to physics and mathematics, as well as offering valuable insights into Euler's personality and religious beliefs. This book became more widely read than any of his mathematical works, and was published across Europe and in the United States. The popularity of the 'Letters' testifies to Euler's ability to communicate scientific matters effectively to a lay audience, a rare ability for a dedicated research scientist.[16]

Concerned about the continuing turmoil in Russia, Euler left St. Petersburg on 19 June 1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the Great of Prussia. He lived for twenty-five years in Berlin, where he wrote over 380 articles. In Berlin, he published the two works for which he would become most renowned: The Introductio in analysin infinitorum, a text on functions published in 1748, and the Institutiones calculi differentialis,[15] published in 1755 on differential calculus.[16] In 1755, he was elected a foreign member of the Royal Swedish Academy of Sciences.

Stamp of the former German Democratic Republic honoring Euler on the 200th anniversary of his death. Across the centre it shows his polyhedral formula, nowadays written as ve + f = 2.

[14]. Of their thirteen children, only five survived childhood.Neva River The young couple bought a house by the [13]

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Swiss Albums Chart

Swiss Albums Chart

The Swiss Hitparade (German: ''Schweizer Hitparade'') are Switzerland's main music sales charts. The charts are a record of the highest-selling singles and albums in various genres in Switzerland.

The Swiss charts include:

  • Singles Top 75 (released since 1968)
  • Albums Top 100 (released since late 1983)
  • Compilations Top 25
  • Airplay Top 30

Since 2010, Media Control began enlisting also the highest-selling singles and albums in Romandie, the Francophone region of Switzerland:

  • Romandie Singles Top 20
  • Romandie Albums Top 50

The weekly updated charts are released on Sundays, and are posted publicly on the preceding Wednesday mornings.

See also

External links

  • Schweizer Hitparade
  • Les charts (Romandie)
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
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Details: Category: Cycling | Published: 29 March 2012 | Created: 26 November 2014 | Hits: 5115
Over the last month of so, the topic of "Hard" in relation to training has come up a fair amount. I've encountered a  fair number of athletes that really don't know what hard is or who give up too easily. Training to be a top notch athlete is all about learning to cope with failure. One must fail over and over again and learn from the experience rather than though the hands in the air claim one just can't do it. It is part of the journey.

I have posted in the last month how I have managed to drop some 80-90lbs and be able to ride a bike faster than I could ever imagine. I have learned to ride the Donut Ride without getting dropped. Three years ago, just making it to 15th and Keele St on the Donut Ride was an achievement. I have learned to race the track in London, and I have gone from never finishing a race, to almost winning a few. I still have issues keeping up for 4-5 laps in the Morning Glory Ride, but racing a 5km circuit with a 16% grade hill in the middle will still take some time to tackle. I still can't sprint well - I need to see 1300-1400W to win races, and have issues at 1100W. In time, I will get there.

The point I'm trying to make: I have come a long way. I still have a long way to go. But the experience of remembering where I started also reminds me of a major crash along the way. I found some photos recently that bring back memories. It make me think of the whiners that give up to easily. They really don't know hard.

Mark XRay Jun 23, 2009 Mark XRay Oct 1, 2009

The first image is the XRay of my right leg after I crashed during the Midweek Crit in June 2009. I beleive I was told I broke the strongest bone in my body in two places. I spent some 8hrs in surgery and two weeks in the hospital. The second image is from Oct 2009. I still have that plate in my leg and the 10 screws that hold it in place. It took 6 months to learn to walk again. I spent time in a wheelchair and had a walker for a while.

From what I recall, I was attempting to sprint for the finish line. My garmin says I was travelling around 40km/h when I flew over the handlebars and landed some 3-5meter in front of my bike. I still don't know what it happened, but I can only assume my front wheel locked up for reasons unknown. Now, I've since checked the rules and the bike and rider must finish the race to win. As I like to say, I finished the race, but the bike didn't. Of course, also hurt the pocket book. The bike I was riding was brand new - two weeks old. The frame was trashed.

The funny thing is while I managed to recover enough to walk and get around, it was not until this year that I realized how much more time I needed to get my leg back to be strong enough to be competitive for cycling. I started with Lori at FITS back in Oct. 2011 becaused I walked to learn how to do weight training correctly. I discovered that I never really completed physiotherapy.  I will keep at it. I have a goal to be able to swat at least two times my body weight. The ideal case is three times. I can currently do one times my body weight.

So, if you are racing or out riding, and get dropped. Keep it up. I have have some major setbacks in my short racing career...and I keep pushing forward. I never let anyone tell me I can't do it. I work hard. I take my lumps. I accept my defeats and considering them learning experiences. Training is hard and it is supposed to be. Racing is harder, but if it weren't why would we do it? Just remember, that there a guy in the pack that has more metal in his leg that more people have on their bikes...and I'm still doing it. I won't get started with the number of carbon bikes I've trashed.

I always like to say: Anything easy isn't worth doing.


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Another option that we assist with is short term accommodations for business travellers, we want you to enjoy your stay in Italy with as little stress as possible.
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How are hailstones formed?

Hailstones are formed from subcooled water that freezes to form ice around initial nuclei in the lower levels of storm clouds or inside a storm cell.

It is necessary for these nuclei to be relatively small in numbers so that a sufficiently large quantity of water is available to each one to enable rapid growth.

As the clouds contain large quantities of water, the latent heat produced via phase conversion results in a significant degree of instability of the temperature layers within the cloud.

The resulting updrafts can reach 20 to 30 m/s and represent another important factor for hailstone formation, as the freezing processes always result in a continuous increase in particle mass.

Without an updraft the particles would descend due to gravity, drop out of the cloud and grow no further.

The intensity of an updraft in a cloud can vary, causing particles to pass through a cycle.

First, they are drawn upwards by the updraft, only to fall back into lower air levels where they gather more water. They are then drawn upwards again where the additional water freezes.

This cycle is repeated until the hailstone is too heavy for the updraft to carry. It is therefore possible to judge the strength of an updraft inside a storm cloud from the size of the hailstone and, conversely, to forecast the diameter of hailstones based on the speed of the updraft.

The step-by-step formation of hailstones is visible from the individual layers that make up a hailstone.

Bright, translucent layers indicate a very water-rich environment with fast freezing, whilst cloudy or opaque layers indicate a low water content. The opaque appearance is caused by the inclusion of countless tiny air pockets.

Once the hailstone is heavy enough and drops out of the updraft, its size coupled with a temperature of usually less than 0°C do not allow it to melt.

Source: Wikipedia


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I am doing an educational series on living ‘green’ and swapping chemicals out of your home. This will include using Essential Oils and Essential Oils Infused products that you can make or get from Young Living.

I am very passionate about living a chemical free life mainly because of my husband’s cancer history and my own autoimmune disorder which had made me highly sensitive to non natural stuff..

Did you know that 26 sec after exposure, chemicals are found in measurable amounts in the human body? The average woman applies 300 chemicals to her body in a day – 80 before breakfast  mostly from personal / skin care products like shampoo’s, lotion etc. The top 10 most dangerous chemicals on our home: air fresheners like plug in’s, candles, chemical cleaning supplies, drain and oven cleaners, dishwashing soaps, beauty supplies and personal care products like shampoos, deodorant. One of the tops pollutants in the family home is laundry soap and fabric softener. It sits on your skin and outgases in your closet all night long.

So, what happens when your body is chemically overloaded? You may see it in something as catastrophic as cancer. But most of us feel it on other ways : lethargy, inability to focus, tiredness, sleep trouble, chronic inflammation, pain, skin issues, stress, fear, hot flashes. If you feel any of these, it is time to kick the chemicals to the curb. It is time to take this seriously and clean out the pollutants.

The point of all of this is not to monger fear. It is education and taking baby steps. No one is watching your home but you. You are the gatekeeper.

All right, where do we start. Start slow, start small, start reading labels. Ditch candles, plug-ins, airfresheners, dryer sheets etc. Take it one month at a time and swap out the things in your home. Simple tip with food for instance- read labels and ingredients. If you can’t pronounce it, don’t eat it.

Join me and learn alongside so many of us committed to this way of living – the age old saying “Health is Wealth” and remember that “For every oil that you use, that is a chemical you are not using”

Let’s start with the basics that we use every single day to clean our homes – Cleaning Supplies. We are going to focus on oils and products that we can use to clean in this series

Did you know that we can use essential oils effectively to clean ANYTHING.

Take this simple Tile and Grout cleaning recipe. Lemon is  a versatile oil which has so many uses.
Lemon EO has highly cleansing properties.

Thieves Household Cleaner :

This one item has replaced all the cleaning supplies in my home. I really can’t live without it. This is the main reason I don’t worry about my kids picking things off the floor or making sure the cabinet under the sink is locked.Try these simple recipes. #theywork

In addition, you can also add 5 drops of Pine EO to the mixture.

Or these other simple recipes

The amazing thing is that it is super cost effective in comparison. You just need a capful of Thieves Household Cleaner for a 32 oz spray bottle, costing about $1.50 a bottle. Not even the cleaners at the organic section in your grocery store can beat that along with quality. No more separate cleaners for the window, the floors, the toilet, kitchen. Just ONE item. If you would like to try a sample, contact me.

Laundry Detergent

Like mentioned above, it is one of the tops pollutants. I would highly recommend getting the Thieves Laundry Soap. It is so highly concentrated that a little goes a long way. I only use 1/4 to 1/2 a capful per load thereby stretching the usage.

If you are using dryer sheets, it is time to throw them out. They tend to be hormone / endocrine disruptors and it is better to use dryer balls instead. Add a drop of Purification or Lavender to your dryer ball and see how wonderfully smelling your clothes come out.

Hand Soaps

Did you know that the FDA has now banned Triclosan, an ingredient often found in antibaterial soaps etc. We use soaps all the time and our skin absorbs all that is in the soap. This is why I highly recommend Thieves Hand Foaming Soap which is devoid of nasty chemicals.

Dish Soap

Dish soaps often leave residue on the dishes which we then put food on. Again, Thieves has come to the rescue. I love the Thieves Dishwashing soap. It is so concentrated that a tiny bit mixed with warm water goes a long way.

Hand Sprays

Want a versatile spray that you can use anywhere specially while traveling – from cleaning hands to spraying toilet covers, try this recipe

Get the Thieves Hand Spray or the Waterless Purifier for convenience. As a mama whose 3.5 year old is touching everything, I never leave home without this

Other oils that have cleansing properties are Purification, Pine, Orange, Tea Tree in addition to Thieves and Lemon EO.

I could not help share this funny graphic. I experienced it first hand when I found my preschooler with his head inside the toilet, doing what, I don’t know. I thought this graphic was funny when I first saw it, but experiencing it first hand = not so funny

How Do You Choose?

I used to buy my Lavender for $4 a bottle online or at bulk food stores. But they never had any significant impact on my body. In the US, there is no rating system for EO’s. The closest we get is an FDA requirement that ” In order for a bottle to be labeled as “pure or therapuetic grade”, it must contain at least 5% Essential Oil.

All oils fall into 4 categories
Grade A – Therapuetic, organically grown plants, distilled at low temperatures.
Grade B– Food grade, may contain pesticides, synthetic, fertilizers, carrier oils.
Grade C– Perfume oils often containing adulaterating chemicals like Hexane to gain higher yield per harvest.
Grade D – Floral Water, aromatic only and is ually a by product of Grade A distillation. After all the oil is pulled out, leftover trash water is sold to companies which will fill 5 % of the bottle with this, add carrier oil and label it pure.

Grade A is the only true pure oil, Grade D is like walking to your fridge, taking a cup of Orange juice, diluting it 95% and then drinking it.

That is why you only want Grade A – check to see if the company grows their own plants, owns their own fields, and controls their entire process from farm to sealed bottles. Pesticides, pollution, previously farmed land all affect the quality of the oil.

Young Living’s oils are Grade A. Why would you go an extra mile of using an oil to get away from a chemical – and then use an oil laden with chemicals? It makes no sense. Their Seed to Seal process makes Young Living stand out. It means each plant is hand weeded, no pesticides, no chemicals, plants are harvested at the peak, go through vigorous testing process and then go from the farm to your home. No other company I know does that.

I have done extensive research on different EO companies out there. Because I am using it on my precious children, I will not compromise on the quality. This is the main reason I chose Young Living to build my wellness arsenal.

Want to learn more? Come to one of my classes

If you are ready to get started and experience all that Young Living has to offer in terms of “wellness, purpose and abundance” in your life, then Congratulations and here is the link to get started with Young Living. Looking forward to you being a part of my team so you can access many resources including education and support.

As always, I am here in service. If there is anything I can do for any of you, please do let me know.

With love, blessings and gratitude

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Ridley’s Harmonica

Ridley's LogoSuper Honky Tonk Sound

We’ve seen quite a few Ridley’s harmonicas arriving in lessons recently, so we thought we’d give one a closer inspection. They’re manufactured in China for Wild & Wolf Ltd, of Bath UK and Lynchburg USA. Wild & Wolf specialise in household goods that have vintage appeal. Their packaging has the feel of 1950’s and 1960’s marketing, and their range of goods is chosen for simple value and durability. No batteries or apps!

Ridley's 6iPart of the Wild & Wolf range includes Ridley’s House of Novelties, a selection of toys and children’s gifts packaged the way parents and grandparents will remember. Representing good old-fashioned fun, a 10 hole harmonica has been included in the mix and, depending on where you shop, you’ll normally find this available for anything between £4.00 and £6.00 a unit. Ours cost £5.95 on Amazon UK, plus £2.65 shipping. A total outlay of £8.60. It goes for less.

Toy or instrument?
Ridley's 5Ridley’s harmonica is marketed alongside items like yo-yo’s, balloon modelling kits, pick-up-sticks and practical joke packs. Which prompts the question, is this harmonica a toy or a musical instrument? We found online reviews that favour both views. Some buyers saw it as a source of entertainment. Here’s an example: Cheap as chips and does what it says on the box. Sturdy, well made, durable – it’s a harmonica!

But what if you’re looking for something to use in music lessons or to play more seriously? Another reviewer wrote: The one I received could not be considered anything but a toy. It was hopelessly out of tune, and virtually useless. Swan harmonicas, at a similar price, Ridley's 8are better quality, and far more usable as a serious instrument. I would say, as a bit of fun for a child these are fair enough, but steer clear if you have any sort of ambition to play harmonica seriously.

The harmonica is certainly well built and will take a knock or two. It comprises a standard injection-moulded, black ABS plastic comb. This has recesses top and bottom, within which the standard metal reed plates sit; a good arrangement for promoting air-tightness in a harmonica. As is the fact that the reed plates themselves are held in place by no less than seven mini-screws. The moulded steel cover plates are have just one screw at each end, which is fine and are also recessed within the comb unit. They carry standard hole numbering on the top plate, no branding, and open backs on both for sound projection when everything is assembled.

The proof of the tin sandwich
When we used the harmonica, the melody notes in holes 4, 5, 6 and 7 seemed to play quite well and sounded bright. However, to the ear and with a digital tuner, 4D and 6B were slightly flat and not as satisfying as the other notes. Here’s what we heard..

Listen to

Ridley's 9The biggest problem came when we tried to play chords; one of the first things newcomers will do. The result was shambolic! Which is probably why the cardboard packaging motif states super honky tonk sound. We’ve never known the term honky tonk being applied to a harmonica. Honky tonk piano certainly; this has an endearing, out-of-tune jangle. What we found was neither super, nor honky tonk. Here’s what we heard..

Listen to

From the sound file above, you can hear the chordal notes wavering, or beating as we call it. Especially at the lower end. This effect tells us the harmonica reeds are not in tune with each other, and not good for chord playing. Which, is probably why the term honky tonk is used on the packaging. This is Ridley's 4puzzling because, judging by the standard scraping on the reeds, some form of tuning has evidently taken place.

To highlight the problem more clearly, here’s what happens when we play two notes an octave apart. This technique is called octaving and it is the quickest way to check the tuning on any harmonica. We’re effectively picking out twin notes from the chords above, to see if they play sympathetically. The beating effect, becomes very apparent and is unappealing..

Listen to

Another problem is that despite its favourable design features, the harmonica has an inherent leakiness. The feel was somewhat spongy, which makes hard going for beginners. As seasoned players, when it came to bend notes, we had to work hard, especially in 3D. We also found it impossible to play the last two blow bends 10B’ and 10B” as you’ll hear..

Listen to

So what do you get for your money?
Instruction SheetThe harmonica comes in a nice sturdy case, with a small cleaning cloth. The starter information-cum-first tune sheet explains how to cup the harmonica with both hands, and how to play blow and draw notes.

The tunes chosen are Three Blind Mice and Home On The Range, two well know melodies. The triangle/circle tab for Three Blind Mice is missing some crucial repeat notes however, and doesn’t match the alliteration of the lyric. Home On The Range also has an unconventional ending. And while this may appear a little over-critical, the lack of attention to detail mirrors the poor quality control that dogs the harmonica generally.

Music SheetOur conclusion
As with all things in life, you  get what you pay for. This is not a harmonica for the serious learner, although it is fun. The packaging says the harmonica is suitable for children 6 years and over. There’s nothing to say children under 6 years couldn’t knock out a tune given the chance, or at least enjoy a few hours’ safe amusement. If Wild & Wolf tidied up the tuning and corrected the information sheet this could be a decent starter instrument. Until then, it remains a toy.


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Monthly Archives: June 2016

Industrial Design – NeoTokyo Police Car

Neo Tokyo Police Car Industrial Design Concept by Hilmyworks
Tried out the Industrial Design Challenge of the Week at The challenge is to design a police car made to track down illegal Internet comm sites.

Base idea was to have domed detector that can survey even tall buildings and pinpoint precise locations. Assuming futuristic Tokyo would be hard to navigate – with high-rises and complicated streets – drones will do the heavy work. The car will have a dedicated comms personnel surveying the site and managing the drones, while two police personnel will be able to go onsite and do arrests if needed.

Continue reading


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Our many years' activity and development on the demanding industrial market have given us a broad and diverse knowledge and a lot of experience.

production processes

Tool making services and stamping products of high-quality and for competitive prices are produced in our company in environmentally friendly production processes.

Our main advantage by all means is short delivery terms and high-quality products. Such working policy provides us a high level of flexibility, high productivity and satisfaction of our loyal customers and business partners.

Our vision is to be the most adaptable and effective partner to customers in the process of developing a product from the scheme to its functional use.Our partners also appreciate our innovativeness at searching for technological solutions.

Stamping tools

EDM Sevices

Tool making services

The company has existed since 1980 when the trade was started.

In the very beginning there were only few employees. With the development and growth of the company the number of workers began to increase, too.

Nowadays this modern and technologically well equipped company employs on average 18 to 20 workers.


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So who am I, owner of Homegrown Peppers?  And why am I writing about growing peppers?

The short answer is that I wanted something different than what was available at the grocery store.

The longer answer is below.  :)

My Name is Gail and I Love Peppers of All Kinds!

I’m Gail Nettles, the owner of this site, chief writer and pepper-eater.  Here’s how I got started on my pepper-growing journey.

Let’s see, about 16 years ago, my husband (David) and I moved into this house.  We have a bit of land, so I figured it was time to grow myself a garden.  Not only for the great-tasting produce, but also I wanted to grow my veggies as organically as possible.

So I read and read and read more about growing vegetables.  I noticed in my reading that peppers love a warm climate (South Florida is indeed warm), they caught my interest.  So I started perusing the seed catalogs and haunting the garden shops for different varieties.

I never realized there were soooo many different kinds of peppers!  I mean, I was used to the ordinary — bell peppers, jalepenos, the occasional banana pepper.  But wow, there were so many others out there to explore!  Sweet, mild, hot — they came in all shapes, sizes and colors.  Some were absolutely huge, and others were the size and shape of small marbles.

Some were even quite ornamental, with beautiful foliage and multi-color peppers dotting the plant.

Along the way, as I shared some of my bounty, friends were asking me how they could grow their own peppers.  After trying to tell them a few times, I figured that I may as well share my knowledge with everyone.

So, welcome to Homegrown Peppers!  Glad you could stop by, and I hope you enjoy your stay here.  Come back often, send friends, link to me…just get the word out on how easy it really is to grow your very own peppers!

6 Responses to About

  • Larry says:

    Hi….just wanted to say thanks for putting such a good and useful site on the web! My wife and I decided to get into pepper growing this year, on a whim We knew NOTHING when we started; we didn’t even know what the seeds looked like. Now we’ve got grow lights and heat mats and raised beds and ph-tested soil mixes, and all that stuff. We’ve managed to get around 70 seedlings going in eight varieties and they’re growing fast; we’ll be transplanting in a few weeks. It’s hard to find well-organized info on pepper growing, and your site is very helpful. Thanks!

  • Gail says:

    Hi Larry, and thanks for stopping by! I’m really glad that you found this site useful.

    Hey, good luck with growing your peppers! It sounds exciting, growing so many (you have me beat, I’m only growing about half your total this year). Let us know the varieties and stop by again later and post on how your plants are growing (not to mention producing). 🙂

    May your peppers prosper!

  • Larry says:

    Hi Gail, here’s what we’re growing this year:

    – Jalapeno Early – lots and lots of them, since we eat them in almost every meal!
    – Biker Billy (a Burpee-distributed Jal that’s supposed to be extra hot and extra large)
    – Holland Red Chili (like cayenne but milder heat, popular in Indonesia and Holland, oddly enough)
    – Serrano
    – Orange habanero (I picked the seeds out of a supermarket pepper; I hope they fruit)
    – Aji omnicolor (more of an ornamental than an eating pepper, I’m told)
    – Godfather, a Burpee Italian sweet pepper hybrid
    – Mariachi, which I think is a Santa Fe hybrid

    It does seem, as you say, like the hotter they are, the slower they grow. My sweet pepper and the milder chilis are taking over the whole growing space, while the habs, aji’s, and biker billys struggle to keep up. It’s actually kind of frustrating, waiting on the hot ones to get moving. You wonder what they’re waiting for. It also seems like the hottest ones have kind of an aversion to water that the mild ones don’t.

    An interesting side-note to this every-expanding project is that we spent weeks researching and debating the soil alternatives. We are putting in 65 square feet of brand new raised beds next week to grow the peppers (and the usual other suspects, too: tomatoes, herbs, lettuces, etc) but with a definite bias toward peppers. We are so new to this hobby and it got frustrating trying to get our heads around the soil options: whether to go peat moss – compost – vermiculite (i.e. Square Foot Gardener style), or 30 bags of Miracle Grow, or 100 % compost, or…or….or. As hard as we tried, and we tried hard, we could not find a real “consensus” strategy representing what most people do, other than that most do not use 30 bags of Miracle Grow. We heard a lot of “It just depends.” It left us with the predictable question, “Depends on what?” But nobody gave us a good answer to that question. In the end we decided to hedge our bets by putting in a little of everything. So we’re going with 2/3 Sta Green vegetable planting mix and 1/3 organic compost from two different sources, then a big big bag of Pro Mix BX, and finally a few handfuls of vermiculite thrown in. That, I think, is the best we can do at our current level of ignorance about soil.

    I read your comments on the liquified worm poop yesterday, and we found a few bottles of it today at a store. I think it might be just what we need. Over the next two weeks, I’m going to run some experiments on a few of my more advanced seedlings to see if the worm poop does anything for them versus the untreated ones. A local commercial grower told us to just use TomatoTone and nothing else for the tomatoes and peppers, but I think we’ll supplement with the worm poop to give it some extra kick.

    The funny thing is, we’ve now spent around $2,000 to, hopefully, produce peppers that you could buy for $0.30 at the grocery store (mostly for the raised beds). I’m hoping the peppers come out so totally mind-blowing that the economics and effort of it all suddenly makes sense, once we have the fruit in our hands.

    So I’ll continue to run those worm poop experiements and post when I have some results. And, of course, I would love to come back later to brag (hopefully) about how our plants are growing and producing!

    Thanks again,

  • Gail says:

    Wow! And I know what you mean, about spending lots of money for veggies you can buy for a few dollars at the grocery store. Still, the work you do this year with the raised beds will serve you for years to come, so if you take the long look, the cost really does become more reasonable.

    You might want to check out my http://www.planning-a-vegetable-garden.com site for more information on raised beds and soil. I think the only other thing I would add to the raised beds is some kelp meal and a soil activator — I did that in my original raised bed and the plants have taken off!

    (Now for some reason I am building more raised beds…can’t leave well enough alone.) 😉

    And one more thing — I’ve done the same as you, picked seeds out of grocery store veggies. Sometimes it works, sometimes it doesn’t, but it’s always interesting!

  • Lawrence J Handley says:

    Hi, Gail,

    I am a complete newbie – fell in love with Padron peppers in Portugal so am trying to grow some. Started with the seed tray with three seeds per peat cylinder. Maybe 12 of 72 have sprouted. Would you expect all 72 to sprout? How long should I wait before going to the next step?

  • Gail says:

    Hi Lawrence,

    I don’t generally expect all seeds to germinate, so I usually plant more seeds than I need. That being said, 12 our of 72 is kind of low — unless it’s been less than 3 weeks. I have planted some peppers that required 4 weeks to sprout. Usually they sprout within about 10 days.

    Keep in mind several things effect germination — soil that is too damp or dry, environment too cool, seeds planted too deep — things of that nature. It’s also possible that not all the seeds were viable.

    Peat cylinders need some careful observation, as they can dry out pretty easily — which of course the seeds don’t like.

    Best wishes!

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Wednesday, January 07, 2009

Brace yourself before reading an ingredient statement

Ya'll, this is scary. The donuts that were bought (by another member of this household thankyouverymuch) the other day have three grams of sugar LESS than the high-fiber, low-fat oatmeal bars I've been eating in my quest to not each so much junk.

And on those so-called healthy bars, the second item in the ingredient statement is high-fructose corn syrup.

And high-fructose corn syrup made the list of Dr. Oz's five food ingredients to avoid. Nice.

I guess they have to get flavor from something with all that fiber and no fat in it.

And not to ruin anyone's lunch, but if you want to continue enjoying hot dogs, cut the nutrition label off the package and throw it away. Or you could accidentally see that 110 of the 130 calories in each dog is fat. And that the ingredient statement begins with "...mechanically separated..."

I suppose it's all about moderation. But truly, I'm finding that the bigger the claim on the package, the more you need to turn it over and read the true story in the ingredient statement.

(And maybe you can have donuts guilt-free!)

~ Straight Shooter ~ said...

Holy Crap! Amen sister!
I am about to just give up with the whole rigamaroll.
Big Shooter cannot have ANYTHING with fiber. Cheese, and any kind of oil, or spice goes straight through.
Boy Child is allergic to wheat, corn, peanuts and soy.
What is left, I ask?
Hey that just gave me a post didn't it.
And have a fabulous day!

Anonymous said...

Yeah, hot dogs are scary, but taste so good. I don't buy them too often, but when I do, I at least get good ones. No red water!

Anonymous said...

If we all saw how they made hotdogs, none of us would ever eat them again. Personally, I watched a video and haven't eaten one since.

Also, I bought some low-fat fruit-on-the-bottom yogurt the other day - 23 grams of sugar.