Mathematics of the Past

February 24, 2012

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Mathematics of the Past

by Garry Kasparov

Garry Kasparov won the title of the Chess World Champion in 1985 – at the age of 22. He was the world No.1 ranked player for 255 months, by far the most of all-time and nearly three times as long as his closest rival, Anatoly Karpov.In 1997, during a historical chess challenge that made headlines all over the world, he defeated IBM’s Deep Blue supercomputer. Read more about the author at the end of this article.

Since my early childhood, I have been inspired and excited by ancient and medieval history. I also have a good memory, which allows me to remember historical events, dates, names, and related details. So, after reading many history books, I analyzed and compared the information and, little by little, I began to feel that there was something wrong with the dates of antiquity. There were too many discrepancies and contradictions that could not be explained within the framework of traditional chronology. For example, let’s examine what we know of ancient Rome.

The monumental work The Decline and Fall of the Roman Empire, written by English historian and scholar Edward Gibbon (1737-1794), is a great source of detailed information on the history of the Roman Empire. Before commenting on this book, let me remark that I cannot imagine how – with their vast territories – the Romans did not use geographical maps, how they conducted trade without a banking system, and how the Roman army, on which the Empire rested, was unable to improve its weapons and military tactics during nine centuries of wars. With the use of simple mathematics, it is possible to discover in ancient history several such dramatic contradictions, which historians don’t seem to consider. Let us analyze some numbers.

E. Gibbon gives a very precise description of a Roman legion, which “ … was divided into 10 cohorts … The first cohort, … was formed of 1 105 soldiers … The remaining 9 cohorts consisted each of 555 soldiers, … The whole body of legionary infantry amounted to 6,100 men.”
He also writes, “The cavalry, without which the force of the legion would have remained imperfect, was divided into 10 troops or squadrons; the first, as the companion of the first cohort, consisted of a 132 men; while each of the other 9 amounted only to 66. The entire establishment formed a regiment … of 726 horses, naturally connected with its respected legion …”
Finally, he gives an exact estimate of a Roman legion: “We may compute, however, that the legion, which was itself a body of 6,831 Romans, might, with its attendant auxiliaries, amount to about 12,500 men. The peace establishment of Hadrian and his successors was composed of no less than 30 of these formidable brigades; and most probably formed a standing force of 375 000.”

This enormous military force of 375,000 men, maintained during a time of peace, was larger than the Napoleonic army in the 1800s. After 1800, Napoleon routinely maneuvered armies of 250 000.  [ See Wikipedia or Encyclopedia Britannica online at http://www.britannica.com/ ]

Let me point out that according to the Encyclopedia Britannica6 “Battles on the Continent in the mid-18th century typically involved armies of about 60,000 to 70,000 troops.” Of course, an army needed weapons, equipment, supplies, etc.

Again, E. Gibbon gives us a lot of details: “Besides their arms, which the legionaries scarcely considered as an encumbrance, they were laden with their kitchen furniture, the instruments of fortifications, and the provisions of many days. Under this weight, which would oppress the delicacy of a modern soldier, they were trained by a regular step to advance, in about six hours, nearly twenty miles. On the appearance of an enemy, they threw aside their baggage, and by easy and rapid evolutions converted the column of march into an order of battle.”

This description of the physical fitness of an average Roman soldier is extraordinary. It brings us to the very strange conclusion that, at some point, the human race retrogressed in its ability to cope with physical problems. Is it possible that there was a gradual decline of the human race, with hundreds of thousands of Schwarzenegger-like athletes of Roman times evolving into medieval knights with relatively weak bodies (like today’s teenage boys), whose little suits of armuor are today proudly displayed in museums? Is there a reasonable biological or genetic explanation to this dramatic change affecting the human race over such a short period of time?

In order to supply such an army with weapons, a whole industry would have been needed. In his work, E. Gibbon explicitly mentions iron (or even steel) weapons: “Besides a lighter spear, the legionary soldier grasped in his right hand the formidable pilum …, whose utmost length was about six feet, and which was terminated by a massy triangular point of steel of eighteen inches.” In another place, he indicates “The use of lances and of iron maces …”

It is believed that the extraction of iron from ores was very common in the Roman Empire. However, to smelt pure iron, a temperature of 1, 539oC is required, which couldn’t be achieved by burning wood or coal without the blowing or the blast furnaces invented more than a 1,000 years later. Even in the 15th century, the iron produced was of quite poor quality because large amounts of carbon had to be absorbed to lower the melting temperature to 1,150oC. There is also the question of sufficient resources – the blast furnaces used in the mid-16th century required large amounts of wood to produce charcoal, an expensive and unclean process that led to the eventual deforestation of Europe. How could ancient Rome have sustained a production of quality iron on the scale necessary to supply thousands of tonnes of arms and equipment to its vast army?

Just by estimating the size of the army, we can conclude that the population of the Eastern and Western Roman Empire in the second century AD was at least 20 million people, but it could have been as high as 40 or even 50 million. According to E.Gibbon, “Ancient Italy … contained eleven hundred and ninety seven cities.” The city of Rome had more than a half-million inhabitants, and there were other great cities in the Empire. All of these cities were connected by a network of paved public highways, their combined lengths more than 4,000 miles! This could only be possible in a technologically advanced society. According to J.C. Russell, in the 4th century, the population of Western Roman Empire was 22 million (including 750,000 people in England and five million in France), while the population of the Eastern Roman Empire was 34 million.

It is not hard to determine that there is a serious problem with these numbers. In England, a population of four million in the 15th century grew to 62 million in the 20th century. Similarly, in France, a population of about 20 million in the 17th century (during the reign of Louis XIV), grew to 60 million in the 20th century … and this growth occurred despite losses due to several atrocious wars. We know from historical records that during the Napoleonic wars alone, about three million people perished, most of them young men. But there was also the French Revolution, the wars of the 18th century in which France suffered heavy losses, and the slaughter of World War I. By assuming a constant population growth rate, it is easy to estimate that the population of England doubled every 120 years, while the population of France doubled every 190 years.

Graphs showing the hypothetical growth of these two functions are provided in Figure 1. According to this model, in the 4th and 5th centuries, at the breakdown of the Roman Empire, the (hypothetical) population of England would have been 10,000 to 15,000, while the population of France would have been 170,000 to 250,000. However, according to estimates based on historical documents, these numbers should be in the millions.

It seems that starting with the 5th century, there were periods during which the population of Europe stagnated or decreased. Attempts at logical explanations, such as poor hygiene, epidemics, and short lifespan, can hardly withstand criticism. In fact, from the 5th century until the 18th century, there was no significant improvement in sanitary conditions in Western Europe, there were many epidemics, and hygiene was poor. Also, the introduction of firearms in the 15th century resulted in more war casualties. According to UNESCO demographic resources, an increase of 0.2 per cent per annum is required to assure the sustainable growth of a human population, while an increase of 0.02 per cent per annum is described as a demographic disaster. There is no evidence that such a disaster has ever happened to the human race. Therefore, there is no reason to assume that the growth rate in ancient times differed significantly from the growth rate in later epochs.

These discrepancies lead me to suspect that there is a gap between the historical dates attributed to the Roman Empire and those suggested by the above computations. But there are more inconsistencies in the historical record of humankind. As I have already noted, there are similar gaps of several centuries in technological and scientific development. Notice that knowledge and technology traditionally associated with the ancient world presumably disappears during the Dark Ages, only to resurface in the 15th century during the early Renaissance. The history of mathematics provides one such example. By chronologically and logically ordering major mathematical achievements, beginning with arithmetic and Greek geometry and finishing with the invention of calculus by I. Newton (1643-1727) and G.W. Leibnitz (1646-1716), we see a thousand-year gap separating antiquity from the new era. Is this only a coincidence? But what about astronomy, chemistry (alchemy), medicine, biology, and physics? There are too many inconsistencies and unexplained riddles in ancient history. Today, we are unable to build simple objects made in ancient times in the way they were originally created – this in a time when technology has produced the space shuttle and science is on the brink of cloning the human body! It is preposterous to blame all of the lost secrets of the past on the fire that destroyed the Library of Alexandria, as some have suggested.

It is unfortunate that each time a paradox of history unfolds, we are left without satisfactory answers and are persuaded to believe that we have lost the ancient knowledge. Instead of disregarding the facts that disagree with the traditional interpretation, we should accept them and put the theory under rigorous scientific scrutiny. Explanations of these paradoxes and contradictions should not be left only to historians. These are scientific and multidisciplinary problems and, in my opinion, history – as a single natural science – is unable to solve them alone.

I think that the chronology of technological and scientific development should be carefully investigated. The too numerous claims of technological wonders in antiquity turn history into science friction (e.g., the production of monolithic stone blocks in Egypt, the precise astronomical calculations obtained without mechanical clocks, the glass objects and mirrors made 5,000 years ago, and so on). It is unfortunate that historians reject scientific incursion into their domain. For instance, the most reasonable explanation of Egyptian pyramid-building technology, presented by French chemist Joseph Davidovits (the creator of the geopolymer technology), was rejected by Egyptologists, who refused to provide him with samples of pyramid material.

I came across several books written by two mathematicians from Moscow State University: academician A.T. Fomenko and G.V. Nosovskij. The books described the work of a group of professional mathematicians, led by Fomenko, who had considered the issues of ancient and medieval chronology for more than 20 years with fascinating results. Using modern mathematical and statistical methods, as well as precise astronomical computations, they discovered that ancient history was artificially extended by more than 1,000 years. For reasons beyond my understanding, historians are still ignoring their work.

But let us return to mathematics and to ancient Rome. The Roman numeral system discouraged serious calculations. How could the ancient Romans build elaborate structures such as temples, bridges, and aqueducts without precise and elaborate calculations? The most important deficiency of Roman numerals is that they are completely unsuitable even for performing a simple operation like addition, not to mention multiplication, which presents substantial difficulties (see Figure 2). In early European universities, algorithms for multiplication and division using Roman numerals were doctoral research topics. It is absolutely impossible to use clumsy Roman numbers in multi-stage calculations. The Roman system had no numeral “zero.” Even the simplest decimal operations with numbers cannot be expressed in Roman numerals. N.P.

Just try to add Roman numerals: MCDXXV+ MCMLXV
or multiply : DCLIII by CXCIX

Try to write a multiplication table in Roman numerals. What about fractions and operations with fractions?

Despite all these deficiencies, Roman numerals supposedly remained the predominant representation of numbers in European culture until the 14th century. How did the ancient Romans succeed in their calculations and complicated astronomical computations? It is believed that in the 3rd century, the Greek mathematician Diophantus was able to find positive and rational solutions to the following system of equations, called Diophantic today 

x1 3 + x2 = y3
x1 + x2 = y

According to historians, at the time of Diophantus, only one symbol was used for an unknown, a symbol for “plus” did not exist, neither was there a symbol for “zero.” How could Diophantic equations be solved using Greek letters or Roman numerals (see Figure 2)? Can these solutions be reproduced? Are we dealing here with another secret of ancient history that we are not supposed to question? Let us point out that even Leonardo da Vinci, at the beginning of the 16th century, had troubles with fractional powers. It is also interesting that in all of da Vinci’s works, there is no trace of “zero” and that he was using  22/7 as the approximation of p – probably it was the best approximation of p available at that time.

It is also interesting to look at the invention of the logarithm. The logarithm of a number x (to the base 10) expresses simply the number of digits in the decimal representation of x, so it is clearly connected to the idea of the positional numbering system. Obviously, Roman numerals could not have led to the invention of logarithms.

Knowledge of our history timeline is important, and not only for historians. If indeed the dates of antiquity are incorrect, there could be profound implications for our beliefs about the past, and also for science. Historical knowledge is important to better understand our present situation and the changes that take place around us. Important issues such as global warming and environmental changes depend on available historical data. Astronomical records could have a completely different meaning if the described events took place at times other than those provided by traditional chronology. I trust that the younger generation will have no fear of “untouchable” historical dogma and will use contemporary knowledge to challenge questionable theories. For sure, it is an exciting opportunity to reverse the subordinate role science plays to history, and to create completely new areas of scientific research.

REFERENCES :

  1. E. Gibbon. The Decline and Fall of the Roman Empire. Peter Fenelon Collier & Son, vol. 1, New York, 1899. This book is also available online at:  http://www.ccel.org/g/gibbon/decline/
  2. I. Davidenko and Y. Kesler. Book of Civilization, (with preface by Garry Kasparov). EkoPress-2000, Moscow, 2001.
  3. J. Davidovits and M. Morris. The Pyramids: An Enigma Solved. New York: Hippocrene Books, 1988 (4th printing). Later by Dorset Press, New York, 1989, 1990.
  4. A. T. Fomenko. Empirico-Statistical Analysis of Narrative Material and its Applications to Historical Dating. Volume 1: The Development of the Statistical Tools, and Volume 2: The Analysis of Ancient and Medieval Records. Kluwer Academic Publishers, 1994, The Netherlands.
  5. A. T. Fomenko , V.V. Kalashnikov and G.V. Nosovskij. Geometrical and Statistical Methods of Analysis of Star Configurations: Dating Ptolemy’s Almagest. CRC Press, 1993, USA.
  6. J. C. Russell. Late Ancient and Medieval Population. American Philosophical Society. 152 p., (Transactions of the American Philosophical Society 48 pt. 3), Philadelphia, 1958.
  7. J.E. Dayton. Minerals, Metals, Glazing and Man. Harrap, London, 1978. ISBN: 0245528075.
  8. The Notebooks of Leonardo da Vinci, 2nd ed., 2 vol. (1955, reissued 1977); and Jean Paul Richter (compiler and ed.). Original kept at Institut de France, Paris.
  9. Leonardo da Vinci. Codex Atlanticus. Kept in Biblioteca Ambrosiana in Milan, Italy.

Copyright © 2002-2012 by Garry Kasparov. All Rights Reserved
Reprinted with Permission of New Tradition Sociological Society
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About the Author

Kasparov, Garry Kimovich (1963- ), chess player and longtime world champion, who competes for Russia. At the age of 22 he became the youngest world chess champion in history. Born Garri Weinstein in Baku, Azerbaijan, in what was then the Union of Soviet Socialist Republics (USSR), he learned chess from his father, who died when Garri was seven years old. He subsequently adopted his mother’s maiden name, and his first name is now widely spelled as “Garry.”

At the age of 12 Kasparov won the Azerbaijan championship and the USSR junior championship, and at the age of 16 he won the world junior championship. In 1980, at the age of 17, he earned the International Grandmaster title. Two years later Kasparov became a candidate for the world championship, and in 1984 he earned the right to challenge the world champion, Russian Anatoly Karpov. Their first match was stopped by Florencio Campomanes of the Philippines, president of the Fédération Internationale des Échecs (FIDE), after it had lasted six months without a deciding result. In 1985 Kasparov won a match against Karpov and became the world champion. He defended his title by beating Karpov in 1986, then tied a match with him in 1987 (FIDE rules permit a champion to keep the title if the match ends in a tie). Kasparov beat Karpov again in 1990 and retained his championship.

When Karpov failed to qualify to challenge Kasparov for the world championship in 1993, Kasparov and British challenger Nigel Short broke away from the FIDE and held a championship match under the governance of the Professional Chess Association (PCA). Spurned by Kasparov, the FIDE sanctioned a championship match between Karpov and Dutch grandmaster Jan Timman. Kasparov and Karpov won their respective matches, and both claimed the title of world champion. In 1995 Kasparov retained his PCA title by defeating Indian challenger Viswanathan Anand, though the association fell apart soon afterward.

In 1996 Kasparov competed against an International Business Machines (IBM) computer named Deep Blue, the first time a world champion had competed against a computer under standard match conditions. Deep Blue, operated by a team of IBM programmers, was capable of processing millions of chess positions per second. Applying this massive computational power, a technique of artificial intelligence known as brute force, Deep Blue won the first game of the match to become the first computer to defeat a world champion under regulation time controls. Kasparov subsequently defeated Deep Blue by a score of 4 games to 2 to win the match. A year later, however, Kasparov accepted a rematch against an enhanced version of Deep Blue, capable of processing 200 million chess positions per second. Although Kasparov won the first game, he was defeated in the six-game series 3.5 games to 2.5 games. It was the first time an international grandmaster lost a series to a computer.

Kasparov continued to hold the “Classical” World Chess Championship until his defeat by Vladimir Kramnik in 2000. He is also widely known for being the first world chess champion to lose a match to a computer under standard time controls, when he lost to Deep Blue in 1997.

Kasparov’s ratings achievements include being rated world No. 1 according to Elo rating almost continuously from 1986 until his retirement in 2005 and holding the all-time highest rating of 2851. He was the world No. 1 ranked player for 255 months, by far the most of all-time and nearly three times as long as his closest rival, Anatoly Karpov. He also holds records for consecutive tournament victories and Chess Oscars.

Kasparov announced his retirement from professional chess on 10 March 2005, to devote his time to politics and writing. He formed the United Civil Front movement, and joined as a member of The Other Russia, a coalition opposing the administration of Vladimir Putin. He was considered to become a candidate for the 2008 Russian presidential race, but later withdrew. Widely regarded in the West as a symbol of opposition to Putin, Kasparov’s support in Russia is low.    More about G. Kasparov >>

Click to order: Game Over: Kasparov & The Machine [DVD] [2004]

{ 2 comments… read them below or add one }

nirmalendu das February 27, 2012 at 11:24 pm

Present Mathematics to find the distance of Planets, Stars, Galaxies etc.

INTRODUCTION OF A NEW UNIT TO MEASURE THE HEAVENLY BODIES
( This may term as magic unit )
NIRMALENDU DAS*

There are few units to measure the distances of planets, stars, galaxies etc. The new unit has been drawn by using the ratio of atom & electron with the value of Pi & Avogadro’s number which has taken as centimeter (C.G.S. unit) in the form of 2?2 NA R & it shows the radius of the universe. The arrangement of this formation indifferent way brings good relation of distance between satellites, planets, stars, galaxies, quasars etc. The calculated results are almost equal to the observed distance made by the investigators time to time.

INTRODUCTION
Astronomical Units
The well known astronomical units are normally used to measure the distance of planets, stars, galaxies etc at the present scientific world. These units are given below:
1. One astronomical unit = 1.49597892 x 10^13 cm
2. One light year = 9.4605 x 10^17 cm
3. One parsec = 3.086 x 10^18 cm

DISCUSSION:
One parsec (one parallax – second or parsec or pc) is 2.062863 x 10^5 & 3.2619840 (Pi ??=3.141592654) times larger than one astronomical unit and one light year respectively.
Now, the ratio of 3.261984039 = 1.038321768 ? = 1.03642035
( ? =) 3.141592654
Again we see that the ratio of one parsec & one astronomical unit = 207674.5476
But, 207674.5476 = 455.7132296 ———————— ( a )
? ¼ x mu / me = 455.7221325 — ( b )
( The symbol have their usual meaning )
The ratio R = 1822.8885 which is known to us as the mass of atom (mu = 1.6605402 x 10 ^- 24 gm) is divided by the mass of an electron (me = 9.1093897 x 10^ – 28 gm) & one forth of this value is 455.7221325 & tallies with the value 207674.5476 = 455.7132296. It brings an interesting relation between ( a ) & ( b ). From this view we can assume that during the creation of celestial’s bodies, it was set right in proper place in the universe as a distance from the source with taking one of the relations between the atom & electron The mass of atom was estimated as unified mass unit 2.
INTRODUCTION : The International Unions of Pure and Applied Physics ( IUPAP ) on September, 8,1960 adopted a new mass scale based on C12 to replace the old scale , based on O16. In Kilograms, the unified mass unit is 1u = 10^ -3 / NA = 1.66043 x 10 ^-27 Kg. Where NA is the new Avogadro number (6.0221367 x 10^23). But the accurate value of Avogadro’s Number3. (NA) is 6.0221367 x 10^23 . We can use this unit for more accuracy.

The above numbers like ?, NA , R has no units. If we consider the value of Avogadro No. as a length in centimeter, the length for calculating the distance of planets, stars etc, then we will able to measure the distance of celestials bodies and this will prove that this new unit is more active, sensitive and useful.

The 6.0221367×10^23cm is 6.365558586×10^5 times larger than one light year.

For example
The cosmological constant4 is denoted by the capital Greek letter Lambda ( ^ ) and it is bounded empirically by ( ^ ) < 3x 10 – 52 m^ – 2

If we consider the value of Avo. No as a length, then, 1/ NA = 2.757393712 x10 ^- 48 cm ^-2 = 2.757393712 x10 ^- 52 m^ -2 . Now, 1/ NA = (^) = 2.757393712 x 10^ – 52 m ^–2 may treated as a correct value & almost equal to the empirical value 3x 10^ – 52 m ^- 2
Therefore, Avo. No. as a length is meaningful and it can use in the case of calculating the distance of planet, star, galaxy, quasar etc.

2) Radius of the universe5
INTRODUCTION : The experiment was done in laboratory & found spectrum of binary star of Germanium at different times of the binary star d -Germanium. During measurement on RECESSIONAL RED SHIFT, put the age of the universe as 10^12 years and 2 x10^28 cm as its radius. However, the real significance of these figure is not clear at present, although several cosmological models have been proposed 6 .
We can get the above radius of the universe by this system of : 2 ?^2 NA R = 2.1669 x 10^28 cm .
It proves that, we can use ? , NA , R to calculate the distances of planets, stars in different way & given bellow :
Determination of different distance of planets of our solar system :
The value of 2 ?^2 NA R without length concern is 2.1669 x 1028 number only, we can use this number as ? 2 ?^2 NA R = 1.47204 x 10^14 and when this number will be treated as 1.47204x 10^14 cm to measure the distance as a length. Then we observed that (This 2 ?^2 NA R may term as magic unit at the time of use to determine the distance)
1) 1x ? 2 ?^2 NA R = 1.47204 x 10^14 cm = distance of Saturn
2) 2x ? 2 ?^2 NA R = 1.47204 x 10^14 cm = 2.944 x10^14 cm = distance of Uranus.
3) 3x ? 2 ?^2 NA R = 1.47204 x 10^14 cm = 4.416×10^14 cm = distance of Neptune.
4) 4x ? 2 ?^2 NA R = 1.47204 x 10^14 cm = 5.888×10^14 cm = distance of Pluto.

5) ?NA / 2 ?^2 = distance of Moon from the Earth.

Let, 1.47204 x 10^14 cm = P distance. So,
6) P / 2 = Distance of Jupiter
7) P / 2 Pi = Distance of = Mars
8) P / Pi ^2 = Distance of Earth
9) P / 4 Pi = Distance of Venus
10) P / 8 Pi = Distance of Mercury.

The above calculated distances almost near value of observed distance of planets. But the distance of planets VOKAN11– X1 and VOKAN11 – X2 are not known in our solar system after the planet Pluto. So we can calculate the distance of these two planets using the above process as e) Distance of VOKAN –X1 = 5 x ? 2 ?^2 NA R = 7.3602 x 10^14 cm ( The distance of this planet is not known )
f) Distance of VOKAN –X2 = 6 x ? 2 ?^2 NA R = 8.8322 x 10^14 cm ( The distance of this planet is not known )
The above process is multiplying process of ? 2 ?^2 NA R with numbers.
This proves that we can use 2 ?^2 NA R to measure the distance of planets, stars etc to arrange this relation in different way in various fields.
and ?^2 (9.869604401), 2? (6.28318530) & 2, then we will get the distance of Mercury, Venus, Earth, Mars, Jupiter respectively. In the case of stars, galaxies, quasars, to calculate the distance, it require to arrange to dress the equation 2?^2 NA R in the form of : ( few examples are given bellow )
( A ) 2?^2 NAR / ?2 =2 NA R =2.1955×10^27 cm = distance of a quasar, ( Ref. Value of quasar – 3C273 is 1.892 x10^27cm )
( B ) NA / 2 ? = 9.5845 x10^22 cm = distance of our galaxy, ( Ref. Value of galaxy is 9.46 x 10^22 cm ( C ) NA / 2 ?^2 = 3.0508 x10^22 cm = distance as position of the sun in galaxy.( Ref. Value is 2.833 x 10^22 cm )

Nirmalendu Das, Email: [email protected]

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Richard Wallace February 26, 2012 at 2:22 pm

Great article. I hope Gary Kasparov expands this research. My pet peeve is archaeologists who explain those things they do not understand by using words such as “Ritual” or “Sacrificial” (and they have many more such words) to explain their own ignorance of what they have in front of them. I would like to see math applied to the devastation of the Irish population between the years of 1841 and 1851. I am not mathematical and I have trouble plotting such a graph. I would like to see a mathematical explanation for why in 1841 the population was 7.9 million, while in 1851 the population was roughly 5 million. We know the potato developed blight, but it did so all over Europe. Why, only in Ireland, did the population drop 2.9 million over a period of 20 years, while the reproduction rate of the Irish was one of the highest in the world? What would the math reveal if the birth rate, left unmolested, would have added a significant number to the population? What would the population have been if the populace was left unmolested? And what would the “disappeared” number be if normal population growth were added into the final formula?

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