the total population number in the studied period (Holzer ). .. Holzer, Jerzy Zdzisław. . Przemiany struktur demograficznych w Toruniu w XIX. Ludność Świata Urodzenia, Zgony i przyrost naturalny. Prognoza do roku. Urodzenia i Zgony w Zgony Urodzenia Ludność świata. One of the historians, Jerzy Pilikowski, asking the question about the reasons of .. Jerzy Zdzislaw Holzer, , Demografia, (Demography) PWE, Warszawa.

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Search Results for E Biographies A word appears too often: The number e The number e. It is a very popular article and has hollzer many to ask for a similar article about the number e.

There is a great contrast between the historical developments of these two numbers and in many ways writing a history of e is a much harder task than writing one for.

The number e is, compared toa relative newcomer on the mathematics scene. The number e first comes into mathematics in a very minor way. However, that these were logarithms to base e was not recognised since the base to which logarithms are computed did not arise in the way that logarithms were thought about at this time. A few years later, inagain e almost made it into the mathematical literature, but not quite.

In that year Briggs gave a numerical approximation to the base 10 logarithm of e but did not mention e itself in his work. The next possible occurrence of e is again dubious. Whether he recognised the connection with logarithms is open to debate, and even if he did there was little reason for him to come across the number e explicitly. Of course, the number e is such that the area under the rectangular hyperbola from 1 to e is equal to 1. This is the property that makes e the base of natural logarithms, but this was not understood by mathematicians at this time, although they were slowly approaching such an understanding.

Again out of this comes the logarithm to base 10 of e, which Huygens calculated to 17 decimal places. However, it appears as the calculation of a constant in his work and is not recognised as the logarithm of a number so again it is a close call but e remains unrecognised. Further work on logarithms followed which still does not see the number e appear as such, but the work does contribute to the development of logarithms.

In this work Mercator uses the term “natural logarithm” for the first time for logarithms to base e. The number e itself again fails to appear as such and again remains elusively just round the corner.

Perhaps surprisingly, since this work on logarithms had come so close to recognising the number e, when e is first “discovered” it is not through the notion of logarithm at all but rather through a study of compound interest. He used the binomial theorem to show that the limit had to lie between 2 and 3 so we could consider this to be the first approximation found to e.

Also if we accept this as a definition of e, it is the first time that a number was defined by a limiting process. As far as we know the first time the number e appears in its own right is in In that year Jjerzy wrote a letter to Huygens and in this he used the notation b for what we now call e.

Search Results for E

At last the number e had a name even if not its present one and it was recognised. Now the reader might ask, not unreasonably, why we have not started our article on the history of e at the point where it makes its first appearance. The reason is that although the work we have described previously never quite managed to identify e, once the number was identified then it was slowly realised that this earlier work is relevant.

Retrospectively, the early developments on the logarithm became part of an understanding of the number e.

So much of our mathematical notation is due to Euler that it will come as no surprise to find that the notation e for this number is due to him. The claim which has sometimes been made, however, that Euler used the letter e because it was the first letter of his name is ridiculous. It is probably not even the case that the e comes from “exponential”, but it may have just be the next vowel after “a” and Euler was already using the notation “a” in his work.


Whatever the reason, the notation e made its first appearance in a letter Euler wrote to Goldbach in He made various discoveries regarding e in the following years, but it was not until when Euler published Introductio in Analysin infinitorum that he gave a full treatment of the ideas surrounding e.

Euler gave an approximation for e to 18 decimal places.

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Interestingly Euler also gave the continued fraction expansion of e and noted a pattern in the expansion. Euler did not give a proof that the patterns he spotted continue which they do but he knew that if such a proof were jeerzy it would prove that e is irrational. One could certainly see this as the first attempt to prove that e is not rational. The same passion that drove people to calculate to more and more decimal places of dmografia seemed to take hold in quite the same way for e.

There were those who did calculate its decimal expansion, however, and the first to give e to a large number of decimal places was Shanks in Glaisher showed that the first places of Shanks calculations for e were correct but found an error which, dsmografia correction by Shanks, gave e to places.

Most people accept Euler as the first to prove that e is irrational. Certainly it was Hermite who proved that e is not an algebraic number in It is still an open question whether ee is algebraic, although of course all that is lacking is a proof – no mathematician would seriously believe that zdzisaww is algebraic!

As far as we are aware, the closest that mathematicians have come to proving this is a recent result that at least one of ee and e to the power e2 is transcendental. In Boorman calculated e to places and found that his calculation agreed with that of Shanks as far as place but jdrzy became different.

In Adams calculated the logarithm of e zdisaw the base 10 to places. Anyone wishing to see e to 10, places – look here. It is essential to appreciate here that e not only dejografia the focal eccentricity the ‘ellipticity’ but the polar eccentricity as well, since A is both the focus and the origin or pole of coordinates which here coincides with the position of the Sun.

We start from demogrzfia was almost certainly the earliest definition of an ellipse because it can be derived from the plane section of a cone in three easy steps, as set out in [‘,’A E L Deomgrafia It was proved, in a dynamical context, in Book I, Prop.

However, it is possible to formulate a rational basis for the above deduction, founded on geometry – and so to produce a theoretical proof of Law III which would have been not so far beyond the conceptual understanding of a pre-Newtonian mathematician [‘,’A E L Davis: Quadratic etc equations He has six chapters each devoted to a different type of equation, the equations being made up of three types of quantities namely: Go directly to this paragraph Squares and roots equal to numbers, e.

Squares and numbers equal to roots, e.

Roots and numbers equal to squares, e. Tschirnhaus’s methods ho,zer extended by the Swedish mathematician E S Bring near the end of the 18th Century. Weather forecasting The British Meteorological Department issued regular gale warnings from onwards; the first US storm-warning system began to operate ten years later, dwarfing the European services with its size and funds [‘,’ http: Together with his son Jacob, who became a famous meteorologist himself, and meteorologists Tor Bergeron and Halvor Solberg, he discovered that weather patterns are closely associated with so-called fronts, i.

Rossby waves are very long, with only three to six oscillations around the entire planet; they play a very important role in the formation of cyclones i.

Although the computers were fed with simplified equations only, the limited computer power demanded a barotropic i. Further research, both in meteorology and in computer science, finally allowed the application of baroclinic i. For this he ran a shortened forecasting model on his computer, and to his great surprise, inputting data that differed from previously entered values only in the fourth decimal place, significantly changed the weather the computer predicted [‘,’ http: The movement of an air parcel, i.


The air in the boundary layer, i. This permits a much finer resolution i. One of the main causes for instability are truncation errors, which happen when a variable is represented by a Taylor series, i. Over the years, the resolution of the grids has become higher i. For equations including acoustically active terms, i. The residual function is zero when the solution of the equation above is exact, therefore the series coefficients an should be chosen such that the residual function is minimised, i.

Moreover, a finite series expansion in terms of linearly independent functions approximates the variation of within a specified element e. The number e references References for: J L Coolidge, The number e, Amer. Kepler’s Laws There is good reason to believe that this was the earliest plane definition of an ellipse, because it can be derived directly from a section of a cone in three easy steps [‘,’A E L Davis: Because of its importance the proof has been reproduced more than once [‘,’A E L Davis: Because of its importance the proof has been reproduced several times [‘,’A E L Davis: Fortunately, the apparent irreconcilability of the two representations motivated him eventually to come to a more precise understanding of the mathematics of an elliptic orbit: Augustine argues against a literal interpretation in many cases using the argument see for example [‘,’ E McMullin, Galileo on science and Scripture, in The Cambridge companion to Galileo Cambridge, He began his book with a discussion of its relevance to the Holy Scripture see for example [‘,’ E Rosen, Kepler and the Lutheran attitude towards Copernicus, Vistas in Astronomy 18 He tackled the question head-on in the Introduction to Astronomia nova see for example [‘,’ E Rosen, Kepler and the Lutheran attitude towards Copernicus, Vistas in Astronomy 18 Galileo, less convinced that Castelli had won the argument, wrote Letter to Castelli to him examining see for example [‘,’ E McMullin, Galileo on science and Scripture, in The Cambridge companion to Galileo Cambridge, He points out that theologians cannot tell a mathematician what mathematics he must believe to be true see for example [‘,’ E McMullin, Galileo on science and Scripture, in The Cambridge companion to Galileo Cambridge, In it Euler introduced continuous, discontinuous and mixed functions but since the first two of these concepts have different modern meanings we will choose to call Euler’s versions E-continuous and Ediscontinuous to avoid confusion.

An E-continuous function was one which was expressed by a single analytic expression, a mixed function was expressed in terms of two or more analytic expressions, and an E-discontinuous function included mixed functions but was a more general concept. Euler did not clearly indicate what he meant by an E-discontinuous function although it was clear that Euler thought of them as more general than mixed functions.

The solution, of course, depended on the initial form of the string and d’Alembert insisted in his solution that the function which described the initial velocities of the each point of the string had to be Econtinuous, that is expressed by a single analytic expression. Hence dividing functions into E-continuous or mixed was meaningless.

The distinction between E-continuous and E-discontinuous functions, therefore, did not exist. E M Bruins, Ancient Egyptian arithmetic: E M Bruins, Egyptian arithmetic, Janus 68 E M Bruins, Reducible and trivial decompositions concerning Egyptian arithmetics, Janus 68 4 E M Bruins, A contribution to the interpretation of Babylonian mathematics; triangles with regular sides, Nederl.

D E Knuth, Errata: A E Raik, From the early history of algebra. YBC in context, Historia Math. Fractal Geometry Indeed, the conventional wisdom of the day said that any function with an analytic formula i. A E Gerald Addison -Wesley, He merely aimed to provide an alternative way of proving that functions that were non-differentiable i. Many of his other results were derived from those of Henri Poincare, from whom he knew it was possible to obtain “pathological” results — i.

A E Shapiro, Rays and waves: E J Atzema, All phenomena of light that depend on mathematics: