FYI May 26, 2017

May 26th National Blueberry Cheesecake Day!

On this day:

1783 – A Great Jubilee Day held at North Stratford, Connecticut, celebrated the end of fighting in American Revolution.
A Great Jubilee Day, first held on Monday May 26, 1783 in North Stratford, now Trumbull, Connecticut, commemorated the end of fighting in the American Revolutionary War.[1] This celebration included feasting, prayer, speeches, toasts, and two companies of the North Stratford militia performing maneuvers with cannon discharges and was one of the first documented celebrations following the War for Independence and continued as Decoration Day and today as Memorial Day with prayer services and a parade.[2]

Reverend Beebe’s personal reflections
Monday the 26th day of May 1783 the inhabitants of North Stratford set apart as a day of public rejoicing for the late publication of peace. At one o’clock, PM, the people being convened at the meeting house, public worship was opened by singing. The Reverend James Beebe said a prayer well adapted and suitable for the occasion.[3][4] They all sang a Psalm. Mr. David Lewis Beebe, a student at Yale College, made an oration with great propriety.[5] The congregation then sung an anthem. The Reverend Beebe, then requested the Ladies to take their seats prepared on an eminence for their reception when they walked in procession, and upwards of 300 being seated the committee who were appointed to wait on them supplied their table with necessaries for refreshments. In the meantime the two companies of militia being drawn up performed many maneuvers, and firing by platoons, general volleys and street firing, and the artillery discharging their cannon between each volley with much regularity and accuracy. After which a stage was prepared in the center and the following toasts were given:

1st. The United States in Congress Assembled.
2d. General Washington and the brave Officers and soldiers of his command.
3d. Our Faithful and Illustrious Allies.
4th. The Friendly Powers of Europe.
5th The Governor and Company of the State of Connecticut.
6th. May the present peace prove a glorious one and last forever.
7th. May tyranny and despotism sink, and rise no more.
8th. May the late war prove an admonition to Great Britain, and the present peace teach its inhabitants their true interests.
9th. The Navy of the United States of America.
10th. May the Union of these States be perpetual and uninterrupted.
11th. May our Trade and Navigation Extend to both Indies and the Balance be found in our favour.
12th. May the American Flag always be a scourge to tyrants.
13th. May the Virtuous Daughters of America bestow their favors only on those who have Courage to defend them.
14th. May Vermont be received into the Federal Union and the Green Mountain Boys flourish.

At the end of each toast a cannon was discharged. The whole was conducted with the greatest decency and every mind seemed to show satisfaction.

North Stratford Militia
The Connecticut general assembly named Robert Hawley the Ensign of the North Stratford Train Band or Company of the 4th regiment of the Connecticut Colony militia in October 1765.[7] He was promoted to Lieutenant in October 1769 and ultimately to Captain in May 1773.[8] At a special meeting assembled in North Stratford on November 10, 1777 he was appointed to a committee to provide immediately all those necessaries for the Continental soldiers.[9] On March 12, 1778, the parish of North Stratford made donations of provisions for those residents serving in the southern army stationed at Valley Forge, Pennsylvania under the command of General George Washington. Mr. Stephen Middlebrook donated the sum of seven pounds (money), three shillings and ten pence to transport the almost two hundred pounds of provisions.[10] George Washington called Connecticut the Provision State because of supplies contributed to his army by Governor Jonathan Trumbull the only Colonial Governor to support the cause of America’s Independence from Great Britain.[11]


Born on this day:

1667 – Abraham de Moivre, French-English mathematician and theorist (d. 1754)
Abraham de Moivre (French pronunciation: ​[abʁaam də mwavʁ]; 26 May 1667 – 27 November 1754) was a French mathematician known for de Moivre’s formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He was a friend of Isaac Newton, Edmond Halley, and James Stirling. Even though he faced religious persecution he remained a “steadfast Christian” throughout his life.[1] Among his fellow Huguenot exiles in England, he was a colleague of the editor and translator Pierre des Maizeaux.

De Moivre wrote a book on probability theory, The Doctrine of Chances, said to have been prized by gamblers. De Moivre first discovered Binet’s formula, the closed-form expression for Fibonacci numbers linking the nth power of the golden ratio φ to the nth Fibonacci number. He also was the first to postulate the central limit theorem, a cornerstone of probability theory.

Scrolled down:

See also: de Moivre–Laplace theorem

De Moivre pioneered the development of analytic geometry and the theory of probability by expanding upon the work of his predecessors, particularly Christiaan Huygens and several members of the Bernoulli family. He also produced the second textbook on probability theory, The Doctrine of Chances: a method of calculating the probabilities of events in play. (The first book about games of chance, Liber de ludo aleae (On Casting the Die), was written by Girolamo Cardano in the 1560s, but it was not published until 1663.) This book came out in four editions, 1711 in Latin, and in English in 1718, 1738, and 1756. In the later editions of his book, de Moivre included his unpublished result of 1733, which is the first statement of an approximation to the binomial distribution in terms of what we now call the normal or Gaussian function.[9] This was the first method of finding the probability of the occurrence of an error of a given size when that error is expressed in terms of the variability of the distribution as a unit, and the first identification of the calculation of probable error. In addition, he applied these theories to gambling problems and actuarial tables.

An expression commonly found in probability is n! but before the days of calculators calculating n! for a large n was time consuming. In 1733 de Moivre proposed the formula for estimating a factorial as n! = cnn+1/2e−n. He obtained an approximate expression for the constant c but it was James Stirling who found that c was √(2π) .[10]

De Moivre also published an article called “Annuities upon Lives” in which he revealed the normal distribution of the mortality rate over a person’s age. From this he produced a simple formula for approximating the revenue produced by annual payments based on a person’s age. This is similar to the types of formulas used by insurance companies today.

Priority regarding the Poisson distribution
Some results on the Poisson distribution were first introduced by de Moivre in De Mensura Sortis seu; de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus in Philosophical Transactions of the Royal Society, p. 219.[11] As a result, some authors have argued that the Poisson distribution should bear the name of de Moivre.[12][13]
De Moivre’s formula

In 1707 de Moivre derived:

cos ⁡ x = 1 2 ( cos ⁡ ( n x ) + i sin ⁡ ( n x ) ) 1 / n + 1 2 ( cos ⁡ ( n x ) − i sin ⁡ ( n x ) ) 1 / n {\displaystyle \cos x={\tfrac {1}{2}}(\cos(nx)+i\sin(nx))^{1/n}+{\tfrac {1}{2}}(\cos(nx)-i\sin(nx))^{1/n}} \cos x={\tfrac {1}{2}}(\cos(nx)+i\sin(nx))^{{1/n}}+{\tfrac {1}{2}}(\cos(nx)-i\sin(nx))^{{1/n}}

which he was able to prove for all positive integers n.[14] In 1722 he suggested it in the more well known form of de Moivre’s Formula:

( cos ⁡ x + i sin ⁡ x ) n = cos ⁡ ( n x ) + i sin ⁡ ( n x ) . {\displaystyle (\cos x+i\sin x)^{n}=\cos(nx)+i\sin(nx).\,} (\cos x+i\sin x)^{n}=\cos(nx)+i\sin(nx).\,

In 1749 Euler proved this formula for any real n using Euler’s formula, which makes the proof quite straightforward. This formula is important because it relates complex numbers and trigonometry. Additionally, this formula allows the derivation of useful expressions for cos(nx) and sin(nx) in terms of cos(x) and sin(x).

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