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Mathematics & Economics
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Topic:

Ancient Mathematics: Probability Theory During the Seventeenth Century

Essay Instructions:

HPS / MAT 390    Essay Guidelines and Topics

 

The essay should be around 2000 words and must be submitted online by 5 pm on December 4.  There will be a deduction of 5% per business day assessed for the submission of essays after December 4.

 

Bibliographical References and Essay Guidelines

 

    These books are available in Gerstein Library, as well in some of the branch libraries.

 

J. W. Dauben The History of Mathematics from Antiquity to the Present. A Selective Bibliography (1985).

 

K. O. May Bibliography and Research Manual of the History of Mathematics (1973).

 

C. C. Gillispie Dictionary of Scientific Biography 16 volumes (1970-).

 

One may also consult the references at the end of chapters in the standard histories of mathematics by Katz, Kline, Boyer, Eves, Struik, Burton, Smith and so on. The MacTutor website for the history of mathematics contains references to the historical literature that are useful. The weekly readings may also provide the basis for your research on an essay topic.

 

    The essay is a major component of the grade for the course. The essay should consist of your own work, and may be run through a standard database to verify that there has been no copying from some source. In particular, do not cut and paste text from websites or Wikipedia articles into your essay. Follow standard conventions for citing and quoting from sources. References and citations should be done according to APA style. You should choose a fairly specific subject, and develop it in an historical and critical way.

 

Please read the document “How not to plagiarize” which is contained in this module.

 

Evaluation scheme:

 

Arguments (50%): The essay presents a concise, well-stated, interesting and
non-trivial thesis; and it is argued for persuasively. The student engages
with historical sources (primary and/or secondary).

Style: "proper essay" (20%): The essay has a clear introduction, body,
conclusion, transitions, thesis statement etc.

Style: writing ability (10%): Clarity, sentences, paragraphs, foot/end
notes, formal (academic) style, etc.
Sources (10%): Uses good sources (number, quality, level) and proper
bibliographic style (clear, consistent)

Overall effort (10%): General impression - was a lot of work put in, or was
it written at the last minute?

 

Essay Topics

 

    The list below is intended as a general guide in choosing a topic. The essay itself should be fairly specific in developing some theme or exploring some issue that arises in these or any other subject areas.   Avoid general descriptive overviews or reports on the literature. Although the essay may contain a synthesis of factual material, it should be focussed, analytical and issue-oriented.

 

 

 

Decomposition of unit fractions in Egyptian mathematics

 

Babylonian mathematical astronomy

 

The role of the "crisis" of incommensurables in the development of pre-Euclidean Greek mathematics

 

Geometric algebra in Euclid’s Elements

 

The method of exhaustion in Euclid and Archimedes

 

The place of construction in Greek geometry

 

Numerical methods in Ptolemy’s Almagest

 

The role of mathematics in the development of Greek astronomy

 

Contributions of the Arabs to algebra and arithmetic

 

Mathematical astronomy in Islamic science

 

Trigonometry and Islamic mathematics

 

Foundations of geometry in Islamic mathematics

 

Indian work on infinite series

 

The reception and transmission of Euclid's Elements in Medieval Europe

 

Proportion theory in the Middle Ages

 

Mathematical dynamics in the Middle Ages

 

Oresme and the latitude of forms

 

The handling of imaginary numbers by Cardano and Bombelli

 

Viète and the invention of the analytic art 

 

Mathematics in Copernicus’s De Revolutionibus

 

Linear perspective in art and the origins of projective geometry

 

The construction of curves in Descartes' Géométrie

 

Number theory in the seventeenth century

 

Theory of probability in the seventeenth century

 

Concepts of the continuum in Medieval mahematics

 

Kepler’s derivation of the elliptical orbit

 

Napier and the invention of logarithms

 

The history of the concept of analysis from Pappus to Descartes

 

Method of indivisibles in 17th-century mathematics

 

Tangent methods in the pre-calculus period

 

Transcendental curves in 17th-century mathematics

 

Optimization problems in seventeenth-century mathematics

 

Mathematical dynamics and the invention of calculus

 

The Newton-Leibniz priority dispute

 

Essay Sample Content Preview:

Ancient Mathematics: Probability Theory During the Seventeenth Century
Name
Institutional Affiliation
Ancient Mathematics: Probability Theory During the Seventeenth Century
Introduction
Participation in gaming and gambling is as old as humankind and the idea of probability is thought to be as old as humanity. However, the conception that it was possible to predict outcomes to some degree of accuracy was never conceivable until the 16th and the 17th centuries. The idea was prompted by the need to achieve a predictable balance between risks and gains to make profits from gambling. Statistics, a field closely related to probability began in the 1600s following John Grant's work on “Bills of Mortality” in 1662 (Burton, 1999). In this manual, Graunt illustrated the most common causes of death and age expectancy in London. Probability was initially inspired through gambling, and the original work on the theory was done by Girolamo Cardano (1501-1576), who specialized in physics and mathematics (Hald, 2003). Cardano in his manual “Liber de Ludo Aleae” discusses several basic constructs of probability theory complete with a systematic examination of gambling challenges. However, his work had no significant impact on the growth of the theory since it never featured in print format until 1663, which made it receive little attention. Later years saw Chevalier de Mere's invention of a gambling system in 1654, which made him convinced that it would help in making money. De Mere decided to bet an amount that he could roll a twelfth in 24 rolls of the two dice. As de Mere started losing his money, he requested Blaise Pascal to investigate his gambling approach (Grinstead & Snell, 1997). Pascal could later discover that de Mere’s system was vulnerable to losing almost 51% each time it is applied. Pascal could, later on, become intrigued with probability and started studying more probabilistic challenges, which he shared with Pierre de Fermat, another famous mathematician and they established the foundation of probability theory (Dale 1999). This paper explores the advancement of probability theory and illustrates how the theory emerged from the need to solve fundamental challenges in predicting outcomes and gambling for profits.
The theory of probability is largely concerned with establishing the connection between the frequency of event occurrence (Grinstead & Snell, 1997). For instance, in flipping a coin, it will be curious to know the times the heads will show up if that coin was to be flipped a hundred times. Answering this problem requires both empirical and theoretical approaches and the example of flipping a coin illustrates the differences that manifest in the two ways of solving probability problems. While using the theoretical approach, it can be reasoned that in each flip, there are only two chances, that of a tail and ahead. By assuming that each of the two events is likely to be equal, then the probability that the heads will appear is only 0.5 or half.
On the other hand, the empirical approach avoids such assumptions of having equal likelihoods. By performing a real flipping experiment, it is possible to count the number of tails and heads each time the coin is tossed. In this ca...
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