- History of Mathematics, Mathematics, Mesopotamian mathematics, Leonhard Euler, History and pedagogy of mathematics, Stanislaw Lesniewski, and 19 moreMereology, Jan Lukasiewicz, Evangelista Torricelli, XVII century Science, 18th Century Land Surveying, Department of the Geographer to the United States, 1777-1783, Land Surveying, History of calculus, History of Logic in Poland, Lvov-Warsaw School, West point history, Protothetic, George Washington, History of Logic, Alfred Tarski, Formal Ontology, History of Mathematics (History), Surveying, and Historical Scientific Instrumentsedit
This is a draft of a paper which I will submit to the CSHPM proceeding in mid August. All comments are most welcome. My email is fred.rickey@me.com. This paper deals with a technique for deciding how many ways an integer is a sum of two... more
This is a draft of a paper which I will submit to the CSHPM proceeding in mid August. All comments are most welcome. My email is fred.rickey@me.com.
This paper deals with a technique for deciding how many ways an integer is a sum of two squares and whether it is prime or rnot.
This paper deals with a technique for deciding how many ways an integer is a sum of two squares and whether it is prime or rnot.
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How can you decide if a number is the sum of two squares? Euler begins with the dumbest possible algorithm you can think of: Take the number, subtract a square, and check if the remainder is a square. If not, repeat, repeat, repeat. But... more
How can you decide if a number is the sum of two squares? Euler begins with the dumbest possible algorithm you can think of: Take the number, subtract a square, and check if the remainder is a square. If not, repeat, repeat, repeat. But Euler, being Euler, finds a way of converting all those subtractions into additions. Then he does several things to speed up the computation even more (but, sadly, does not explain himself very well). He applies this to 1,000,009, and — in less than a page — finds that there are two ways to express this as a sum of squares. Hence, by earlier work in E228, it is not a prime. Amusingly, when he later described how to prepare a table of primes "ad milionem et ultra" (E467), he includes this number as prime. So then feels obliged to write another paper, E699, using another refinement of his method, to show that 1,000,009 is not prime.
This paper sprang from a much larger effort to better understand the mathematics education of the young George Washington – an education that let him eventually to trigonometry and logarithms. He preserved his ciphering papers throughout... more
This paper sprang from a much larger effort to better understand the mathematics education of the young George Washington – an education that let him eventually to trigonometry and logarithms. He preserved his ciphering papers throughout his life and they were passed down with the balance of his papers. Most of them are preserved today at the Library of Congress. The fate of some of these whose provenance is lost will be discussed below. This paper is a spin-off of the larger study of Washington and mathematics; it is the joint effort of Theodore J Crackel, Professors V. Frederick Rickey, and Joel Silverberg.
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This is my first historical paper.
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Abstract. You will probably remember from your grade school education that George Washington spent several of his youthful years as a professional surveyor. But how much mathematics did he know and how did he use it as a surveyor? Thanks... more
Abstract. You will probably remember from your grade school education that George Washington spent several of his youthful years as a professional surveyor. But how much mathematics did he know and how did he use it as a surveyor? Thanks to two “cyphering books” he compiled as a teenager, we are able to show what he learned of trigonometry and surveying. The combined use of these subjects is perplexing to the modern reader, so we shall illustrate and explain the methods he used. Finally, in contrast to what one would expect, we will argue that he did not use trigonometry in surveying.
===========================Vous vous souvenez probablement de vos ann ́ees `a l’ ́ecole ́el ́ementaire que Georges Washington a ́et ́e pendant plusieurs ann ́ees de sa jeunesse arpenteur professionnel. On peut alors se demander quelle ́etait sa connaissance des math ́ematiques et comment il s’en servait `a titre d’arpenteur. Grˆace `a deux «cahiers math ́ematiques» qu’il a compil ́es `a l’adolescence, nous sommes en mesure de montrer exactement ce qu’il connaissait de la trigonom ́etrie et de l’arpentage. L’emploi combin ́e de ces sujets est assez d ́eroutant pour le lecteur moderne. Nous allons ici illustrer et expliquer les m ́ethodes qu’il a utilis ́ees. Enfin, contrairement `a ce qu’on pourrait croire, nous montrerons qu’il n’a pas vraiment utilis ́e la trigonom ́etrie dans ses exemples concrets d’arpentage.
===========================Vous vous souvenez probablement de vos ann ́ees `a l’ ́ecole ́el ́ementaire que Georges Washington a ́et ́e pendant plusieurs ann ́ees de sa jeunesse arpenteur professionnel. On peut alors se demander quelle ́etait sa connaissance des math ́ematiques et comment il s’en servait `a titre d’arpenteur. Grˆace `a deux «cahiers math ́ematiques» qu’il a compil ́es `a l’adolescence, nous sommes en mesure de montrer exactement ce qu’il connaissait de la trigonom ́etrie et de l’arpentage. L’emploi combin ́e de ces sujets est assez d ́eroutant pour le lecteur moderne. Nous allons ici illustrer et expliquer les m ́ethodes qu’il a utilis ́ees. Enfin, contrairement `a ce qu’on pourrait croire, nous montrerons qu’il n’a pas vraiment utilis ́e la trigonom ́etrie dans ses exemples concrets d’arpentage.
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Abstract. Soon after we began the study of George Washington’s cyphering manuscript we realized that some of the pages were missing. To understand how this happened, we shall first discuss the provenance of the cyphering books. Then we... more
Abstract. Soon after we began the study of George Washington’s cyphering manuscript we realized that some of the pages were missing. To understand how this happened, we shall first discuss the provenance of the cyphering books. Then we present some “missing” pages that we have located, provide evidence that there are still more missing pages, and describe the detective work involved in situating these pages in the manuscript.======================
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We are gathered tonight to celebrate the centenary of the Mathematical Association of America. It was founded as a home for the American Mathematical Monthly which was founded by Finkel and started publication in 1894. We are here in Ada... more
We are gathered tonight to celebrate the centenary of the Mathematical Association of America. It was founded as a home for the American Mathematical Monthly which was founded by Finkel and started publication in 1894. We are here in Ada because Finkel earned his undergraduate degree here at Ohio Northern University. We will discuss Finkel's life, education, long involvement with the Monthly, and his impact on American Mathematics.
"B. F. Finkel, the Monthly, and the MAA" is the written version of this talk. It is available on this web page: https://usma.academia.edu/FredRickey
"B. F. Finkel, the Monthly, and the MAA" is the written version of this talk. It is available on this web page: https://usma.academia.edu/FredRickey
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Fortunately Newton and Leibniz had different conceptions of the integral, for they provide us with two ways to convince students of the intuitive truth of the Fundamental Theorem of Calculus. This is but one of the many ways that we can... more
Fortunately Newton and Leibniz had different conceptions of
the integral, for they provide us with two ways to convince students of the intuitive truth of the Fundamental Theorem of Calculus. This is but one of the many ways that we can use the history of mathematics to help students to gain a better understanding of mathematics and how it is done. Today we shall present a plethora of proven examples that you can use in your classroom.
the integral, for they provide us with two ways to convince students of the intuitive truth of the Fundamental Theorem of Calculus. This is but one of the many ways that we can use the history of mathematics to help students to gain a better understanding of mathematics and how it is done. Today we shall present a plethora of proven examples that you can use in your classroom.
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After the surveyor travels out to the field with his circumferentor, hires some men to drag the chain through the woods, and records bearings and distances in his field book, he retires to a place where he can do the careful work of... more
After the surveyor travels out to the field with his circumferentor, hires some men to drag the chain through the woods, and records bearings and distances in his field book, he retires to a place where he can do the careful work of drawing a plat. For this task he needs some tools: paper, a quill pen, some bread to do erasures, and a case of mathematical instruments containing a plain scale, a pair of dividers, a pair of compasses, and a sector or Gunter's rule. We will explain how a surveyor could create the essential (and costly) tools for drawing a plat: a plain scale and a scale of chords. We will see how both a Gunter scale and a sector were used to find inaccessible distances and angles, and how to calculate the areas of plots without the need for either algebra or trigonometry. Lastly, we will set you to work replicating some of the plats that twenty-year-old George Washington drew. We will supply paper Gunters, but bring your own dividers and compass (these are instruments of math instruction so the TSA won't let us supply them)
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"Just as Grant Wood portrayed Parson Weems pulling the curtain back on the life of George Washington, we shall illuminate Washington's mathematical education. We are blessed with 179 pages of handwritten manuscript in Washington's... more
"Just as Grant Wood portrayed Parson Weems pulling the curtain back on the life of George Washington, we shall illuminate Washington's mathematical education. We are blessed with 179 pages of handwritten manuscript in Washington's youthful hand and while this material has frequently been mentioned by scholars, it has never been analyzed; we shall present an abundance of detail. Little is known about Washington's youth, so these papers provide a way of learning about his education.
The individuals and organizations that have controlled these papers have organized and reorganized them into disorder. Is it reasonable for a thirteen-year-old --- living in plantation Virginia where there were few schools --- to begin his mathematical education with the study of formal geometry? Could he have learned surveying before studying arithmetic and trigonometry? Using physical evidence, handwriting analysis, and mathematical context, we shall present our conjectured order of the pages of the manuscript.
This is a case study in mid-eighteenth century mathematical education in the American Colonies. We shall contrast the surprising depth of his theoretical education --- including logarithms and trigonometry --- with his practical use of mathematics as a field surveyor.
After serving two terms as President, Washington took pains to preserve his papers for he believed that someday they might be "of interest.'' You will be the judge of how interesting these papers are."
The individuals and organizations that have controlled these papers have organized and reorganized them into disorder. Is it reasonable for a thirteen-year-old --- living in plantation Virginia where there were few schools --- to begin his mathematical education with the study of formal geometry? Could he have learned surveying before studying arithmetic and trigonometry? Using physical evidence, handwriting analysis, and mathematical context, we shall present our conjectured order of the pages of the manuscript.
This is a case study in mid-eighteenth century mathematical education in the American Colonies. We shall contrast the surprising depth of his theoretical education --- including logarithms and trigonometry --- with his practical use of mathematics as a field surveyor.
After serving two terms as President, Washington took pains to preserve his papers for he believed that someday they might be "of interest.'' You will be the judge of how interesting these papers are."
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There are many interesting things in the cyphering books that George Washington compiled as a teenager: His study of decimal arithmetic is straightforward, but understanding some of the errors he made can be a fun. Some things are hard to... more
There are many interesting things in the cyphering books that George Washington compiled as a teenager: His study of decimal arithmetic is straightforward, but understanding some of the errors he made can be a fun. Some things are hard to understand for time has passed them by. For example, his pre-Eulerian trigonometry is a mystery today, so we shall elucidate it. But some of his work shows that his sources knew some calculus; a formula for gauging casks relies on Simpson's rule. We shall present one page of his cyphering books which the Library of Congress neglected to digitize and another that ended up at Dartmouth.
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This is joint work with Fabio Zanasi, who deserves most of the credit for this work. This paper shows that there are developments of Lesniewski's logic with arbitrarily many creative definitions.
