History Of Vodka

Vodka is a drink which originated in Eastern Europe, the name stemming from the Russian word 'voda' meaning water or, as the Poles would say 'woda'. 

The first documented production of vodka in Russia was at the end of the 9th century, but the first known distillery at, Khylnovsk, was about two hundred years later as reported in the Vyatka Chronicle of 1174. Poland lays claim to having distilled vodka even earlier in the 8th century, but as this was a distillation of wine it might be more appropriate to consider it a crude brandy. The first identifiable Polish vodkas appeared in the 11th century when they were called 'gorzalka', originally used as medicines. 

Medicine & Gunpowder

During the Middle Ages, distilled liquor was used mainly for medicinal purposes, as well as being an ingredient in the production of gunpowder. In the 14th century a British emissary to Moscow first described vodka as the Russian national drink and in the mid-16th century it was established as the national drink in Poland and Finland. We learn from the Novgorod Chronicles of 1533 that in Russia also, vodka was used frequently as a medicine (zhiznennia voda meaning 'water of life'). 

In these ancient times Russia produced several kinds of 'vodka' or 'hot wine' as it was then called. There was 'plain wine' (standard), 'good wine' (improved) and 'boyar wine' (high quality). In addition stronger types existed, distilled two ('double wine') or more times. 

Since early production methods were crude, vodka often contained impurities, so to mask these the distillers flavoured their spirits with fruit, herbs or spices. 

The mid - 15th century saw the first appearance of pot distillation in Russia. Prior to that, seasoning, ageing and freezing were all used to remove impurities, as was precipitiation using isinglass ('karluk') from the air bladders of sturgeons. Distillation became the first step in producing vodka, with the product being improved by precipitation using isinglass, milk or egg white. 

Around this time (1450) vodka started to be produced in large quantities and the first recorded exports of Russian vodka were to Sweden in 1505. Polish 'woda' exports started a century later, from major production centres in Posnan and Krakow. 

From acorns to melon

In 1716, owning distilleries became the exclusive right of the nobility, who were granted further special rights in 1751. In the following 50 or so years there was a proliferation of types of aromatised vodka, but no attempt was made to standardise the basic product. Types produced included; absinthe, acorn, anisette, birch, calamus root, calendula, cherry, chicory, dill, ginger hazelnut, horseradish, juniper, lemon, mastic, mint, mountain ash, oak, pepper, peppermint, raspberry, sage, sorrel, wort and water melon! A typical production process was to distil alcohol twice, dilute it with milk and distil it again, adding water to bring it to the required strength and then flavouring it, prior to a fourth and final distillation. It was not a cheap product and it still had not attained really large-scale production. It did not seek to compete commercially with the major producers in Lithuania, Poland and Prussia. 

In the 18th century a professor in St. Petersburg discovered a method of purifying alcohol using charcoal filtration. Felt and river sand had already been used for some time in Russia for filtration. 

Vodka marches accross Europe

The spread of awareness of vodka continued throughout the 19th century, helped by the presence in many parts of Europe of Russian soldiers involved in the Napoleonic Wars. Increasing popularity led to escalating demand and to meet this demand, lower grade products were produced based largely on distilled potato mash. 

Earlier attempts to control production by reducing the number of distilleries from 5,000 to 2,050 between the years 1860 and 1890 having failed, a law was enacted in 1894 to make the production and distribution of vodka in Russia a state monopoly. This was both for fiscal reasons and to control the epidemic of drunkenness which the availability of the cheap, mass-produced 'vodkas' imported and home-produced, had brought about. 

It is only at the end of the 19th century, with all state distilleries adopting a standard production technique and hence a guarantee of quality, that the name vodka was officially and formally recognised. 

After the Russian Revolution, the Bolsheviks confiscated all private distilleries in Moscow. As a result, a number of Russian vodka-makers emigrated, taking their skills and recipes with them. One such exile revived his brand in Paris, using the French version of his family name - Smirnoff. Thence, having met a Russian émigré from the USA, they set up the first vodka distillery there in 1934. This was subsequently sold to a US drinks company. From this small start, vodka began in the 1940s to achieve its wide popularity in the Western World.

Source: http://www.ginvodka.org/history/vodkaHistory.asp

Definition Of Limit

(ε, δ)-definition of limit

In calculus, the (ε, δ)-definition of limit ("epsilon-delta definition of limit") is a formalization of the notion of limit. It was first given byBernard Bolzano in 1817. Augustin-Louis Cauchy never gave an () definition of limit in his Cours d'Analyse, but occasionally used arguments in proofs. The definitive modern statement was ultimately provided by Karl Weierstrass.

History

Isaac Newton was aware, in the context of the derivative concept, that the limit of the ratio of evanescent quantities was not itself a ratio, as when he wrote:

Those ultimate ratios ... are not actually ratios of ultimate quantities, but limits ... which they can approach so closely that their difference is less than any given quantity...

Occasionally Newton explained limits in terms similar to the epsilon-delta definition. Augustin-Louis Cauchy gave a definition of limit in terms of a more primitive notion he called a variable quantity. He never gave an epsilon-delta definition of limit (Grabiner 1981). Some of Cauchy's proofs contain indications of the epsilon-delta method. Whether or not his foundational approach can be considered a harbinger of Weierstrass's is a subject of scholarly dispute. Grabiner feels that it is, while Schubring (2005) disagrees. Nakane concludes that Cauchy and Weierstrass gave the same name to different notions of limit.

Informal Statement

Let f be a function. To say that

means that f(x) can be made as close as desired to L by making the independent variable x close enough, but not equal, to the value c.

How close is "close enough to c" depends on how close one wants to make f(x) to L. It also of course depends on which function f is and on which number c is. Therefore let the positive number ε (epsilon) be how close one wishes to make f(x) to L; strictly one wants the distance to be less than ε. Further, if the positive number δ is how close one will make x to c, and if the distance from x to c is less than δ (but not zero), then the distance from f(x) to L will be less than ε. Therefore δ depends on ε. The limit statement means that no matter how small ε is made, δ can be made small enough.

The letters ε and δ can be understood as "error" and "distance", and in fact Cauchy used ε as an abbreviation for "error" in some of his work. In these terms, the error (ε) in the measurement of the value at the limit can be made as small as desired by reducing the distance (δ) to the limit point.

This definition also works for functions with more than one argument. For such functions, δ can be understood as the radius of a circle or a sphere or some higher-dimensional analogy centered at the point where the existence of a limit is being proven, in the domain of the function and, for which, every point inside maps to a function value less than εaway from the value of the function at the limit point.

Precise Statement

The  definition of the limit of a function is as follows:

Let  be a function defined on a subset , let  be a limit point of , and let  be a real number. Then

the function  has a limit  at 

is defined to mean

for all , there exists a  such that for all  in  that satisfy , the inequality  holds.

Symbolically:

Example
PROBLEM 1 : Prove that 
Begin by letting  be given. Find  so that if  , then , i.e.,  , i.e.,  . But this trivial inequality is always true, no matter what value is chosen for . For example,  will work. Thus, if  , then it follows that  . This completes the proof.


PROBLEM 2 : Prove that 
Begin by letting  be given. Find  (which depends on  ) so that if  , then . Begin with  and ``solve for"  . Then,

 iff 

iff 

iff 

iff 

iff 

iff  .

Now choose  . Thus, if  , it follows that . This completes the proof.


PROBLEM 3 : Prove that 
Begin by letting  be given. Find  (which depends on  ) so that if  , then . Begin with  and ``solve for" | x - 2 | . Then,

 iff 

iff 

iff  .

We will now ``replace" the term |3x+5| with an appropriate constant and keep the term |x-2| , since this is the term we wish to ``solve for". To do this, we will arbitrarily assume that  (This is a valid assumption to make since, in general, once we find a  that works, all smaller values of  also work.) . Then  implies that -1 < x-2 < 1 and 1 < x < 3 so that 8 < |3x+5| < 14 (Make sure that you understand this step before proceeding.). It follows that (Always make this ``replacement" between your last expression on the left and . This guarantees the logic of the proof.)

iff 

iff  .

Now choose  (This guarantees that both assumptions made about  in the course of this proof are taken into account simultaneously.). Thus, if  , it follows that . This completes the proof.

Source:
https://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit
https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/preciselimdirectory/PreciseLimit.html