Home  ›  Write the Equation of the Sphere in Standard Form. X2 + Y2 + Z2 + 2x âë†â€™ 2y âë†â€™ 6z = 25

Write the Equation of the Sphere in Standard Form. X2 + Y2 + Z2 + 2x âë†â€™ 2y âë†â€™ 6z = 25

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  1. Abu Al-Hasan, The Arithmetic of Al-Uqlidisi, translated by A. Saidan as The Arithmetic, D. Reidel, Dordrecht, 1978. Al Uqlidisi (the Arabic for the Euclidean) describes decimal notation, explains the algorithms for the four operations, compares the notation to sexagesimal, and explains that the latter are more suitable for scientific calculations and the former for business and everyday use.
    The use of comma's and points still remains a nuisance in understanding numbers. In the English speaking world 1,000 means a thousand in many other languages (such as Spanish) it means one, on the other hand 1.000 is a thousand in some languages and only 1 in the English speaking world!
  2. Aczel A., The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity, Four Walls Eight Windows, 2000. Contains some engaging historical accounts of mathematical mysteries, and paradoxes, and its theological dimension!
  3. Albree J., D. Arney, and V. Rickey, Station Favorable to the Pursuits of Science: Primary Materials in the History of Mathematics at the United States Military Academy, American Mathematical Society, 1999. A major part of this book is an annotated catalog of the more than 1300 works published between 1496 and 1915 found in the West Point library.
  4. Alperin R., A mathematical theory of origami construction and numbers, New York Journal of Mathematics, 16(1), 119-134, 2000.
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  9. Arsham H., A critical panoramic view of basic mathematical concepts, Teaching Mathematics and Its Applications, 17(2), 69-72, 1998.
  10. Arsham H., Computational geometry and linear programs, International Journal of Modelling and Algorithms, 1(3), 251-266, 1999.
  11. Arsham H., Links among a linear system of equations, matrix Inversion, and linear program solver routines, Journal of Mathematical Education in Science & Technology, 29(5), 764-769, 1998.
  12. Arsham H., The Art and Science of Mathematical Sin, Decision Line, 28(3), 3, 1997.
  13. Arsham H., BBC Interview: Even Numbers and Zero
    Asking the following relevant questions:
    1. Why did he feel the need to mark zero out as an even number?
    2. Do lots of people not know that zero is an even number?
    3. When was it even decided that zero is even?
  14. Artmann B., Euclid: The Creation of Mathematics, Springer Verlag, 1999.
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  18. Bashmakova I., and G. Smirnova, The Beginnings and Evolution of Algebra, Mathematical Assn. of Amer., 2000. It gives a good description of the evolution of algebra from the ancients to the end of the 19th century.
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  23. Bell E., Men of Mathematics, Touchstone Books, 1986, also Econo-Clad Books, 1999. It contains some women of mathematics too. It is a kind of inspirational literature containing a certain amount of fiction.
  24. Berggren J., Episodes in the Mathematics of Medieval Islam, Spinger-Verlag, New York, 1986.
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  27. Berlinski D., NEWTON'S GIFT: How Sir Isaac Newton Unlocked the System of the World, Free Press, 2000. It is mainly biographical in nature, but does include enough mathematics to keep your interest as well.
  28. Bernard K., Genealogical Mathematics, Vancouver, 1990. Translation of: Genealoska matematika. It is among others a Pythagorean genealogy of numbers.
  29. Berggren J. L., Episodes in the Mathematics of Medieval Islam, Springer-Verlag New York, 1986. In contains (p. 102) a good discussion on origin of the world Algebra. The word "algebra" is derived from the first word of the Arabic "al-jabr wa-l'muqabala". Al-jabr and al-muqabala are the names of basic algebraic manipulations. al-jabr means "restoring", that is, e.g., taking a subtracted quantity from one side of the equation and placing it on the other side, where it is made positive. al-muqabala is "balancing", that is, "replacing two terms of the same type, but on different sides of an equation, by their difference on the side of the larger. What makes the solution of a problem an algebraic solution is the method, not necessarily the use of notation. The book also contains a lengthy discussion of spherical trigonometry.
  30. Berggren L., J. Borwein, and P. Borwein, Pi: A Source Book, Springer-Verlag, 2000. Contains the most of the literature on the subject.
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  33. Bos H., H. Mehrtens, and I. Schneider, (eds), Social History of Nineteenth-century Mathematics, Boston, Birkhaeuser, 1981.
  34. Boyer C., History of Analytic Geometry, Scholar's Bookshelf, Princeton Junction, N.J, 1988. The author credits Newton for bringing in rectangular with four quadrants in the two dimensional Cartesian system of coordinates.
  35. Boyer C., and U. Merzbach, A History of Mathematics, John Wiley & Sons, 1991. Among other discoveries, it claims that "It is quite possible that zero originated in the Greek world, perhaps at Alexandria, and that it was transmitted to India after the decimal position system had been established in India."
  36. Brackett J., Children's conceptualizations of infinity: The association of mathematical context and middle-grade students' responses to tasks involving infinity, Journal of Interdisciplinary Mathematics, 1(1), 1-31, 1998.
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  38. Brann E., The Ways of Naysaying: No, Not, Nothing, and Nonbeing, Roman & Littlefield Pub., 2001. The author mounts an inquiry into what it means to say something is not what it claims to be or is not there or is nonexistent or is affected by nonbeing.
  39. Brann E., Plato's Sophist: The Professor of Wisdom, Focus Pub., 1996. A very good reading for understanding the concept of "nothingness" in the Sophist world view.
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  48. Burkert W., Lore and Science in Ancient Pythagoreanism, Harvard University Press, 1972. It has put an end to over 2000-years-old legend of a Pythagoras as a mathematician. It is a large book for specialist with lots of Greek citation without translation. Read also C. Huffman, "The Pythagorean Tradition" in The Cambridge Companion to Early Greek Philosophy, edited by A. Long, 1999, pp. 66-87.
  49. Burton D., History of Mathematics: An Introduction, McGraw Hill, 1997.
  50. Butterworth B., The Mathematical Brain, Macmillan, London, UK., 1999. It contains some helpful materials relevant to the so-called "dyslexia" when some children approach mathematical concepts.
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  52. Cabillon J. G., History of Mathematics, a hot discussion group.
  53. Cajori F., A History of Mathematical Notations, Chicago, Open Court, 1974, 2 vols. Also in Dover Publications, 1993. A good source for the history of the mathematical notations.
  54. Cajori F., A History of Mathematics, Chelsea Pub Co., 1999. Covers the period from antiquity to the close of World War I.
  55. Calinger R. J. Brown, and T. West, A Contextual History of Mathematics, Prentice Hall, 1999. It provide a good argument on the distinction between the words "abbacus" and "abacus", the latter referring to the counting board. The 'abbacus' is not counting board but the decimal numerals system, while mentioning that Italian teachers of the new commercial mathematics were called "Maestri d'Abbaco". pp. 367-368.
  56. Calinger R., (ed.), Classics of Mathematics, Prentice-Hall, Englewood Cliffs, 1995. Contains among other interesting topics, several proofs for the irrationality of square root of 2.
  57. Calkin N., and H. Wilf, Recounting the Rationals, American Mathematical Monthly, 107, 360-363, 2000.
  58. Capaldi N., The Art of Deception: An Introduction to Critical Thinking, Prometheus Books, 1987.
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  60. Chabert J-L., et al., A History of Algorithms: From the Pebble to the Microchip, Springer-Verlag, Berlin 1999
  61. Clawson C., Mathematical Sorcery: Revealing the Secrets of Numbers, Plenum Press, 1999.
  62. Clawson C., Mathematical Mysteries: The Beauty and Magic of Numbers, Perseus Books, 2000.
  63. Clawson C., The Mathematical Traveler: Exploring the Grand History of Numbers, Perseus Books, 1994.
  64. Cohen I. B., Revolution in Science, Harvard Univ Pr., 1987. Contains his well accepted essential criteria for scientific investigations, including mathematics and its revolution.
  65. Conant L., The Number Concept: Its Origin and Development, New York, MacMillan and Co., 1896. It has a short note (page 80) on the Hottentots' a group of Khoisan-speaking pastoral peoples of southern Africa, legend that their language had no words for numbers greater than three.
  66. Conway J., On Numbers and Games, AK Peters, 2000. Introduces a new class of numbers, called surreal numbers, which include both real numbers and ordinal numbers; these surreal numbers are applied in the author's mathematical analysis of game strategies.
  67. Conway J., and R. Guy, The Book of Numbers, Springer Verlag, 1997.
  68. Cooke R., The History of Mathematics: A Brief Course, Wiley, 1997.
  69. Coolidge J., The Mathematics of Great Amateurs, Dover, 1963.
  70. Corfield D., Towards a Philosophy of Real Mathematics, Cambridge University Press, 2003.
  71. Coxeter H., The Beauty of Geometry, Dover Publications, 1999.
  72. Creative Learning Exchange, The, Encourages the use of system dynamics in K-12 education.
  73. Crowe M., A History of Vector Analysis: The Evolution of the Idea of a Vectorial System, Dover, 1994. States that the first attempt to represent complex numbers geometrically was made in the 18 th century.
  74. Crump T., The Anthropology of Numbers, Cambridge Univ Press, 1992.
  75. Crumpacker B., Perfect Figures: The Lore of Numbers and How We Learned to Count, Thomas Dunne Books, 2007.
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  77. Dahan-Dalmedico A., and J. Peiffer, Une Histoire des Mathematiques: Routes et Dedales, Paris, Seuil, 1986. It contains a detailed historical presentation on the concept of a function. The word "function" as we use today is the noun formed from the past participle of the deponent verb "fungor," which means to execute or fulfill a duty. Hence the French word "fonctionnaire" for a civil servant and the derogatory English word "functionary" to mean a bureaucrat who merely takes orders. The earliest occurrence of the word is due to Leibniz in an article in the 1694.
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  81. Dauben J., et al., (Eds.), History of Mathematics: States of the Art, Academic Press, 1996. It is cited in Klaus Barner's preprint "Diophant und die negativen Zahlen", where he tries to credit Diophantus with the invention of negative numbers.
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  89. de Spinadel V. , From the Golden Mean to Chaos, Argentina, 1998. The author introduces a new family of positive quadratic irrational numbers, the Metallic Means, among them is the Golden Mean.
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  95. Dickson L., History of The Theory of Numbers, three-volume, Chelsea Publishing Co Inc, 1919-1923. It claims about the Chinese remainder theorem. However, this claim was questioned in the U. Libbrecht book.
  96. Dieudonne J., Mathematics: The Music of Reason, New York, Springer Verlag, 1992.
  97. Dijksterhuis E., The Mechanization of the World Picture: Pythagoras to Newton, Princeton University Press, 1986. The author states (p. 228) that it was "decidedly wrong" for H. Doerrie to have planted the mistaken notion that Archimedes used method of harmonic means, he never mentioned geometric nor harmonic means in his "Measurement of the Circle."
  98. Dilke O., Reading the Past: Mathematics and Measurement, University of California Press, 1987. This small book (only 61 pages long) provides interesting information covering the Ancient Near East including Egyptian, Babylonian, Greek and Roman mathematics.
  99. Dominguez J., and J. Ferreiros, Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics, Birkhauser, 1999. It discusses the emergence and development of the notion of set theory during the period 1850-1950.
  100. Dorrie H., 100 Great Problems of Elementary Mathematics: Their History and Solution, Dover, 1965. Contains interesting classical problems such as the "straightedge and compass restriction." These great book demonstrates that "elementary" and "simple" problems are two very different things.
  101. Driver R., J. Ewing (Editor), and F. Gehring, (Eds.), Why Math?, Springer Verlag, 1995. A very relevant book for a general education mathematics course.
  102. Dunham W., The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems, and Personalities, Wiley, 1994.
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  104. Earliest Uses of Various Mathematical Symbols, by Jeff Miller.
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  108. Emmer M., (Ed.), Mathematics and Culture, Springer , 2005. Stresses the strong links between mathematics, culture and creativity in architecture, contemporary art, geometry, computer graphics, literature, theatre and even cinema.
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  130. Gillies D., Philosophy of Science in the Twentieth Century: Four Central Themes, Blackwell Pub, 1993. It traces the development during the 20 th century of four central themes: subjective, conventionalism, the nature of observation, and the demarcation between science and philosophy.
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  134. Goldstine H., A History of the Computer from Pascal to von Neumann, Springer, 1982. A very useful reference written by someone who knows the creation process of the computing technology.
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  137. Gracia L., A. Martinez, and R. Minano, Nuevas Tecnologias y Ensenanza De Las Matematicas, Editorial Síntesis, Madrid, 1989.
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  141. Grattan-Guinness I., Fontana History of the Mathematical Sciences, Fontana Press, 1997. The connection of number Pi with a circle is as old as Archimedes' era, however, this book provides the historical account of the connection with the volume of a sphere and volumes.
    It mentioned also the used of Arabic numeral system starting with Fibonacci and gradual began to take firm place, especially in Italy, whose practitioners are called "abacists". The choice of this name is unfortunate, for it did not use any kind of abacus, p. 139.
  142. Griffin N. , The Selected Letters of Bertrand Russell, Boston, Houghton Mifflin, 1992, p 208.
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  147. Harrison E., Darkness at Night: A Riddle of the Universe, Harvard University Press, Cambridge, Mass., 1987.
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  149. Hairer E., and G. Wanner, Analysis By Its History, Springer-Verlag, New York, 1996.
  150. Hans J., The Golden Mean, State Univ of New York Press., 1994. Urges readers to be aware of the intrinsic link between the aesthetic and ethical measures of goodness!
  151. Haylock D., Mathematics Explained for Primary Teachers, Sage Publications Ltd, London, 2001. Contains curriculum on numeracy strategy, and the basic skills test in numeracy for schools in UK.
  152. Heath T., A History of Greek Mathematics, Dover Publications, UK., 1981.
  153. Heilbron J., Geometry Civilized. History, Culture, and Technique, Clarendon Press, Oxford, 1998. By reading this book you get a rich sense of of many fascinating geometric ideas suitable for teachers and students, however, not for research mathematicians or historians of geometry.
  154. Hersh R., What Is Mathematics, Really?, Oxford University Press, 1999. There is a lot of very fascinating reading with clear exposition.
  155. High Performance, To increase people's capacity for thinking, learning, communicating, and acting systemically.
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  159. Hilton P., and J. Pedersen, Build Your Own Polyhedra, Addison-Wesley Pub., 1994. Contains has some nice approximations to 'constructing' mathematical shapes through paper folding.
  160. HIST-ANALYTIC, by Steve Bayne.
  161. History of Mathematics, by David Wilkins.
  162. Hodgkin L., A History of Mathematics From Mesopotamia to Modernity, Oxford University Press, 2005. This is a relatively short (280 pp.) book contains the topics written by those who teach the subject. Teaching of history is meant to make students more "cultured", making them critical, introduce them to questions, doubts and disagreements.
  163. Hoffman P., The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth, Hyperion, 1998.
  164. Hogben L., Mathematics for the Million: How To Master The Magic of Numbers, Norton & Company, 1993.
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  166. Houben G., 5000 Years of Weights, Zwolle, Netherlands, 1990. Among others, it mentions systems of weights of power of 2. The oldest known set of weights dates the year 1229 and the longest, still existing set has weights 1/8, 1/4, 1/2, 1, 2, 4, 8 ounces.
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  169. Hudson P., History by Numbers: An Introduction to Quantitative Approaches, Edward Arnold, 2000.
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  171. Ifrah G., From One to Zero: A Universal History of Numbers, Viking Penguin Inc., New York, 2000, a translation of Histoire Universelle des Chiffres, Seghers, Paris, 1981. Ifrah drew attention to number four, claiming that "Early in this century there were still peoples in Africa, Oceania, and America who could not clearly perceive or precisely express numbers greater than 4." p.6. He also provides a discussion and cites some Arabic texts as the evidence that "early Islamic mathematics relied substantially on earlier Hindu mathematics." p.361. In addition to the Menninger book, this book is also an excellent source of information on the origin and development of number symbols in ancient and medieval societies.
  172. Ifrah G. , The Universal History of Numbers: From Prehistory to the Invention of the Computer, Wiley, 1999, (Translated from the French by D. Bellos, et al.). It is a complete account of the invention and evolution of numbers the world over. A marvelous journey through humankind's grand intellectual epic including how did many cultures manage to calculate for all those centuries without a zero?
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  185. Katz V., A History of Mathematics: An Introduction, Addison Wesley Longman, 1998. For correct pronunciation of names and keywords the book contains a nice and informative "Index and Pronunciation Guide". It also contains among other interesting topics, historical proofs for the irrationality of square root of 2.
  186. Katz V., (Ed.), Using History to Teach Mathematics: An International Perspective, Mathematical Assn of Amer., 2000. Contains 26 essays from around the world on how and why an understanding of the history of mathematics is necessary for the informed teachers.
  187. Kelley J., General Topology, Springer Verlag, 1991. The author shows how decimal expansions of real numbers can be arrived at by considering the real set R as an order-complete field.
  188. Kennedy J., Arithmetic with Roman numerals, American Mathematical Magazine, 88, 29-32, 1981.
  189. Kinyon M., and G. Brummelen, (eds.), Mathematics and the Historian's Craft. Springer, 2005. It provides a perspective on mathematical developments and deal with a variety of topics.
  190. Klein F., The Arithmetizing of Mathematics, Bulletin of The American Mathematical Society, Vol. 2, 241-249, 1996, Translated by I. Madison.
  191. Klein J., Greek Mathematical Thought and the Origin of Algebra, Dover Pub., 1992. It points out the fact that the difference between arithmetic and logic is viewed concerning relationships or not. However, they distinguished between practical and theoretical logic. Also a good discussion about the fact that to the Greeks, 1 was never a number. A number was a multitude of units and 1 is a unit, not a multitude.
  192. Kline M., Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972. He pointed out that Newton was the first who used positive, negative, integer, and fractional exponents.
  193. Kline M., Why the Professor Can't Teach: Mathematics and the Dilemma of University Education, St. Martin's Press, New York, 1977.
  194. Kline M., Mathematics in Western culture, Oxford University Press, 1964. Mostly, the book deals with the cultural history of mathematics.
  195. Klinger F., Mathematics for Everyone (US editions), Maths for Everyone (UK editions), Crown Publisher, (undated). This book is translated into almost all the European languages with lots of editions, and reprints over many years.
  196. Knorr W., Textual Studies in Ancient and Medieval Geometry, Springer Verlag, 1989. Contains a good discussion and argument on whether the Greeks have any notion for fractions and what really they meant by a "ratio?"
  197. Knuth D, Algorithmic thinking and mathematical thinking, American Mathematical Monthly, 92(3), 1985, 170-181. It is a striking essay on the subject with some historical views.
  198. Koerner S., Experience and Theory, Routledge, Circa 1964. The author uses a three-valued logic to discuss the difference between perceived or measured physical magnitudes and standard mathematics, including the non-transitivity of equality in perceived or measured magnitudes.
  199. Koetsier T., and L. Bergmans, (Eds.), Mathematics and the Divine: A Historical Study, Elsevier, 2005.
  200. Körner T., The Pleasures of Counting, Cambridge University Press, 1996.
  201. Lakatos I., Proofs and Refutations: The Logic of Mathematical Discovery, Cambridge University Press, Cambridge, 1976.
  202. Lakoff G., and R. Nuñez, Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, Basic Books, 2000. Analytical thinking and understanding from the point of view of modern cognitive science and modern understanding of brain structure and function are the main topics in this wonderful reading. It contains e.g., detailed explanations on what number e means.
  203. La lettre de la Preuve, An International Newsletter on the Teaching and Learning Mathematical Proof.
  204. Lamotke, Numbers, edited, 1999, Springer-Verlag.
  205. Lancy D., (Ed.), Cross-Cultural Studies in Cognition and Mathematics, Academic Press, 1985. Deals mostly on the anthropology aspects of counting number systems.
  206. Laubenbacher R., and D. Pengelley, Teaching with Original Historical Sources in Mathematics.
  207. Laugwitz D., Bernhard Riemann, 1826-1866: Turning Points in the Conception of Mathematics, trans. Abe Shenitzer, Birkhaeuser, 1999. It concerns with the mathematics from both the operational style of Euler and the conceptual style initiated by Riemann later.
  208. Lavine Sh., Understanding the Infinite, Harvard University Press, 1994.
  209. Lecat M., Erreurs de Mathe'maticiens des Origines a' nos Jours, Bruxelles, 1935.
  210. Le Lionnais Francois, Les Nombres remarquable, Hermann, Paris, 1983.
  211. Lesh R., and H. Doerr, Symbolizing, Communicating, and Mathematizing: Key Concepts of Models and Modeling, in P. Cobb, E. Yackel, and K. McClain (Eds.), Symbolizing and Communicating in Mathematics Classrooms: Perspectives on Discourse, Tools, and Instructional Design, Lawrence Erlbaum Associates, N.J., 361-383, 2000. By definition the mathematical modeling process of reality is the mathematization of reality as we perceive it. Mathematizing could be in the forms of quantifying, graphical visualizing, tabular coordinating and/or symbols notation systems to develop mathematical descriptions and explanations that make heavy demands on modelers' representational capabilities.
  212. Libbrecht U., Chinese Mathematics in the Thirteenth Century: The Shu Chiu-Chang of China, MIT Press, 1973.
  213. Livio M., The Accelerating Universe: Infinite Expansion, the Cosmological Constant, and the Beauty of the Cosmos, Wiley, John & Sons, 2000. This book helps the reader to think, understand, draw, and evaluate mathematical patterns of order and chaos that is a part of this universe with its physical laws.
  214. Li Yan, and Du Shiran, Chinese Mathematics: A Concise History, Oxford Univ. Press, 1987.
  215. Luce R., Krantz D., Suppes P., and A. Tversky, Foundations of Measurement, Vol. 3, Academic, London, 1990.
  216. Luoma K., Mathematical Connections.
  217. MacTutor History of Mathematics archive, The. It mentioned that the Babylonians used the identity ab = [(a+b) 2 - (a-b) 2 ]/4 for multiplying two numbers a by b. All that was needed was a good size table of squares and knowledge of dividing by 2 twice.
  218. Madison B., Challenge of Numbers: People in the Mathematical Sciences, National Academy Press, 1990.
  219. Mankiewicz R., The Story of Mathematics, Casell &Co., London, 2000. The author points out the fact that the Babylonians, and Chinese did not have a symbol for zero.
  220. Mankiewicz R., and Ian Stewart, The Story of Mathematics, Princeton Univ Press, 2001. A popular illustrated cultural history of mathematics.
  221. Maor E. To Infinity and Beyond, Princeton Univ. Press 1991.
  222. Maor E. E: The Story of a Number, Princeton Univ. Press 1998.
  223. Moritz R., On Mathematics and Mathematicians, Dover, 1942.
  224. Marshak A., The Roots of Civilization: The Cognitive Beginnings of Man's First Art, Symbol and Notation, Moyer Bell, 1991. The author claims to find numerical writing and calenders on prehistoric carved bones tens of thousands of years before the usually dated advent of writing with civilization.
  225. Martin G., Geometric Constructions, Springer Verlag, 1998.
  226. Math History: Math Forum, Math Resources.
  227. Maxwell E.A., Fallacies in Mathematics, Cambridge University Press, 1959.
  228. Mayberry J., The Foundations of Mathematics in the Theory of Sets, Cambridge University Press, 2000. Contains a good treatment of the close parallel between of the ancient notion of number and the modern notion of set. The main contribution of the book is emphasis on a central fallacy, which is called "operationalist fallacy" by the author. This fallacy comes at least in two forms:

    One basic version of the operationalist fallacy consists in the conviction that the natural numbers 0, 1, 2, ... constitute the "raw data" of mathematics. Furthermore, they are simply "given" to us as a unique infinite structure, which can be characterized fully and rigorously as "the successive images of zero under repeated applications of the "successor operation". On this conception, the principles of proof by mathematical induction and definition by recursion are simply "given" along with the natural numbers themselves, so that, in particular, these two principles can be accepted as legitimate without further justification: they are, in short, "self-evident".

    Another form of the operationalist fallacy is to be found in the view of formal syntax in which the modeling process "Constructionists" employ, which are really recursively defined functions, are somehow "given" immediately as self-evidently efficacious, and so do not require validation or justification. In this sense, the operationalist fallacy thus underlies the illusion that Formalism is a coherent foundational theory by itself.

  229. Menninger K., Number Words and Number Symbols: A Cultural History of Numbers, translated by P. Broneer, Dover, New York, 1992. He is citing linguistic evidence that the Latin tres could have the same root as trans, beyond; English three to through, p.17. This book is also an excellent source of information on the origin and development of number symbols in ancient and medieval societies.There are chapters on finger counting (digital reckoning), tally sticks, knots, and the abacus (counting board).
  230. Miller C. et al., Mathematical Ideas, Addison-Wesley Pub., 2000. An expanded standard textbook in basic mathematics for everyone to enjoy, engage and get some historical sense of the subject.
  231. Mahoney M., The Mathematical Career of Pierre de Fermat, 1601-1665, Princeton Univ Pr., 1994. The author provides an insight into the mathematical genius of a hobbyist who never sought to publish his work, yet who ranked with his contemporaries Pascal and Descartes in shaping the course of modern number theory.
  232. Moritz R., Memorabilia Mathematica: The Philomath's Quotation Book, Spectrum Series of the Mathematical Association of America, 1993. It has lots of quotations from the early 1900s and before. However, it is the best mathematical quotation books.
  233. Moscovici S., Essai sur l'histoire humaine de la nature (Essay for the Human History of the Nature), Paris, Flammarion, 1968. Does not consider something as a piece of art unless it has some components of arithmetic and geometry.
  234. Muir J., Of Men and Numbers: The Story of the Great Mathematicians, Dover Pubs, 1996.
  235. Nahin P., An Imaginary Tale: The Story of the Square Root of -1, Princeton University Press, 1998. It's is a pleasant and anecdotal introduction to complex numbers, full of ideas and stories that are seldom seen elsewhere, 257 pages.
  236. Netz R., The Shaping of Deduction in Greek Mathematics: A study in cognitive history, by Reviel (Ideas in Context, 51), Cambridge University Press, 1999. The main consideration concerning the relative unpopularity of mathematics is quite simple, the author states: "Mathematics is difficult."
  237. Neugebauer O.., The Exact Sciences in Antiquity, Dover, 1969. Provides some justifications faced by the Babylonian place value notation which are due to the lack of a symbol for zero.
  238. Neugebauer O., (editor), Astronomical Cuneiform Texts : Babylonian Ephemerides of the Seleucid Period for the Motion of the Sun, the Moon, and the Planets, Springer Verlag, 1983.
    An interesting hypothesis is the connection between partitioning a circle into 360 degrees and number of days in a year. There are two main theses about the origin of the 360º system:
    The first underlines the mathematical suitability of 360 (its factors are 2, 3, 4, 5, 6, 8, 9 ,10, 12, etc) in problems related to the division of a whole in equal parts, the second points out the connection with come astronomical constants (as 365).
    The second thesis is the fact that the Babylonian had a sexagesimal system, which was used in Greek astronomy. The fact, that a year consists of little more than 360 days, seems to be secondary. The Babylonians did have a calendar with 360 days per year, plus suitable "additional days". Actually, it is supported by a clear 'semantic' link (day=degree) and by some historical facts: for example Chinese astronomy had 365 and 1/4 degrees, the Babylonian ephemerides were based on mean synodic months divided in 30 parts and the year was divided in 12 parts, etc.
    The sexagesimal system seems to have been a basis of ancient thinking. Their day measurement was the development of a 24 hour system (spherically, each hour being one half of 30 degree segments relative to 360 degrees)... hours also divided into 60 minutes, minutes into 60 seconds. Attempts to develop measurable systems of "time" added their own bit of complexity to what was already a complex and culturally variant attempt to juxtapose precision in calendar and time systems congruent with a celestial system which seemed to defy precision at the time.
    Our desire for a mathematical modeling of the universe and its processing difficulties is apparent here too. Some interesting analogous ones existed also in music, architecture, etc. These models required the fitting between small integer numbers, easy to be represented and dealt with, and complex phenomena whose numerical parameters did not exactly fit in the integer-based scheme. It is credible that the 360-system, and the 6-8-9-12 scheme in music, were the results of this conflict, being mathematically suitable and semantically justified.
  239. Neuwirth E., Musical Temperaments, Springer Verlag, 1997.
  240. Newman J. (Editor), The World of Mathematics, Simon & Schuster, 2000
  241. Nicomacus of Gerasa; Introduction to Arithmetic, Translated by M. D'ooge, University of Michigan Press, 1926. As it is often claimed, but unfortunately, Pythagorean did nothing that could be construed as a representation by the system of congruence, i.e., a number having the remainders 2, 3, 2 when divided by 3, 5, 7 respectively.
  242. Niederman D., and D. Boyum, What the Numbers Say: A Field Guide to Mastering Our Numerical World, Broadway Publisher, 2004.
  243. Nissen H., P. Damerow, and R. Englund, Archaic Bookkeeping, University of Chicago Press, 1993. Contains an Old Babylonian school text tablet.
  244. North J., The Measure of the Universe, Clarendon Press, Oxford, UK, 1965.
  245. Olver P., Classical Invariant Theory, Cambridge Univ Press, 1999. p. 40.
  246. Ore O., Number Theory and Its History, McGraw-Hill, 1948.
  247. Pannekoek A., History of Astronomy, Dover, 1989. It gives a definition of photometric magnitude for colors as: -2.5log(intensity), approximating -(100) 1/5 log(intensity), since it was discovered that five magnitudes correspond well to an intensity ratio of 100, hence the fifth root of 100. The whole process outlined by the author introduced a classification scheme to a system of measurement which is very interesting and instructive.
  248. Pappas Th, Math-A-Day: A Book of Days for Your Mathematical Year, Wide World Publishing, 1999.
  249. Parshall K., the Art of Algebra from Al-khwarizmi to Viète: A Study In the Natural Selection of Ideas, History of Science, 26(72), 129-164, 1988.
  250. Pappas Th., The Joy of Mathematics, Wide World Pub., Tetra, 1989.
  251. Pappas Th., Mathematical Footprints: Discovering Mathematical Impressions All Around Us, Wide World Publishing/Tetra, 1999.
  252. Paulos J., Innumeracy: Mathematical Illiteracy and its Consequences, Vintage Books, 1990.
  253. Paulos J., Beyond Numeracy: Ruminations of a Numbers Man, Vintage Books, 1992.
  254. Paulos J., A Mathematician Reads the Newspaper, Basic Books, 1995.
  255. Paulos J., Once Upon a Number: The Hidden Mathematical Logic of Stories, Basic Books, 1999. A bridge between science and culture.
  256. Pears I., An Instance of Fingerpost, Penguin, 1999. (A fingerpost is a directional sign, shaped like a finger, pointing the direction to go). This book is a mathematical criminal novel about a cryptanalyst trying to solve a "code," though this word was not used that way until the early 1800's. The 17th century term was "cipher."
  257. Pesic P, Labyrinth: A Search for the Hidden Meaning of Science, MIT Press, 2000. Contains several chapters concerning the relation of codebreaking to the development of modern algebra and to the decryption of nature as a central activity of modern science.
  258. Petroski H., Invention by Design: How Engineers Get from Thought to Thing, Harvard University Press, 1998.
  259. Phillips G., Two Millennia of Mathematics: From Archimedes to Gauss, Springer Verlag, 2000.
  260. Pogliani L., M. Randic, N. Trinajstic, Much ado about nothing: An introductive inquiry about zero, Journal of Mathematical Education in Science and Technology, 29(5), 729-744, 1998.
  261. Pogliani L., M. Randic, and N. Trinajsticc, About one: An inquiry about the meanings and uses of the number one, Journal of Mathematical Education in Science, 31(6), 811-824, 2000. The long struggle to recognize one as a number and its role in philosophy, religion, mathematics and science is discussed.
  262. Polya G., Mathematical Discovery, John Willey & Sons, 1962.
  263. Possehl G., The Indus Age: The Writing System, New Delhi, Oxford & IBH Publishing Co., 1996.
  264. Rademacherm H., and O. Toeplitz, The Enjoyment of Mathematics, Dover Publications, 1990
  265. Rashed R., The Development of Arabic Mathematics: Between Arithmetic and Algebra, Boston University Press, 1994.
  266. Rashed R., (ed.), Encyclopedia of the History of Arabic Science, vol. 2, London, 1996. Mentioned that Abu Arrayhan Muhammad ibn Ahmad al-Biruni (973-1048), in his 'al-Qanun al-Mas'udi' claimed that the ratio of 'the number of the circumference' to 'the number of the diameter is irrational, pp. 126-510.
  267. Rees M., Just Six Numbers: The Deep Forces that Shape the Universe, Basic Books, 1999.
  268. Regiomontanus, Johann, De Triangulis Omnimodis, 1464. It contains a systematic account of methods for solving triangles with applications to Astronomy mostly for Calenders. An English translation by Barnabas Hughes published by the University of Wisconsin Press, 1967. The original book contributed to the dissemination of Trigonometry in Europe in the 15th century.
  269. Reid C., From Zero to Infinity, The Mathematical Association of America, 1992.
  270. Richards R., Arithmetic Operations in Digital Computers, D. Van Nostrand Company, Inc., 1955.
  271. Richardson R., and E. Landis, Fundamental Conceptions of Modern Mathematics: Variables and Quantities, London, The Open Court Pub., 1916. Unfortunately, its planned second volume never published.
  272. Ritt J., Theory of Functions, Kings Crown Press, New York, 1947. The author argues that the system of unlimited decimals is an "ordered field". Moreover, the difficulty begins with equations like 0.19999... = 0.20000... between different decimals.
  273. Robson E., and J. Wimp (eds.), Against Infinity: An Anthology of Contemporary Mathematical Poetry, Primary Press, Parker Ford, PA, 1979.
  274. Robins G., and Ch. Shute , The Rhind Mathematical Papyrus: An Ancient Egyptian Text , Dover, 1987.
  275. Rota G-C., and F. Palombi, Indiscrete Thoughts, Birkhauser, Springer Verlag, 1997. The world of mathematics between 1950 and 1990.
  276. Rothstein E., Emblems of Mind: The Inner Life of Music and Mathematics, Avon Books, NY, 1995
  277. Rotman B., Signifying Nothing: The Semantics of Zero, Stanford University Press, 1994.
  278. Rotman B., Ad Infinitum-- the Ghost in Turing's Machine: Taking God out of Mathematics and Putting the Body Back In, Stanford University Press, 1994.
  279. Russell J., Inventing the Flat Earth: Columbus and Modern Historians, Praeger, Westport Conn., 1991. The question whether "people once believed that the earth is flat" is discussed at length, in this book. Visit also, The Myth of the Flat Earth.
  280. Sachs J., Aristotle's Physics, Rutgers University Press, 1995.
  281. Salem L., F. Testard, C. Salem, and J. Wuest, The Most Beautiful Mathematical Formulas, Wiley, 1997.
  282. Sanitt N., Science As a Questioning Process, Inst. of Physics Pub., 1996. The author considers the connections and interplay of various scientific disciplines as well as their influencing a man and thinking about where we are and where to go.
  283. Sarton G., The Study of the History of Mathematics and the Study of the History of Science, Dover, New York, 1957.
  284. Saski Ch., (Ed.), The Intersection of History and Mathematics, Science Networks; Vol. 15, Boston, Birkhaeuser, 1994.
  285. Schermer F., The Proof of Nothing: A Theory of Everything, PENTA Publishing, San Francisco, CA, 2000. This is a book in the spirit of the Weinberg's "Dreams of a Final Theory."
  286. Schmidt H., (ed.), Relativity Theory: A Vision of 100 Years, Kluwer Academic Publishers, 2003. Covers the general relativity from multi-perspectives.
  287. Schubring G., Conflicts Between Generalization, Rigor and Intuition: Number Concepts Underlying the Development of Analysis in 17th-19th Century. Springer, 2005. It deals with the two main concepts: negative numbers and infinitisimals.
  288. Scriba C., and P. Schreiber, 5000 Jahre Geometrie: Geschichte, Kulturen, Menschen (5000 Years of Geometry: History, Cultures, People), Springer, 2001. Provides an overview of the historical developments of geometrical conceptions and its realizations. Its Chapter 3 deals with oriental view of geometry in the contexts of cultural environments such as Japan, China, India, and the Islamic world.
  289. Seife Ch., and M. Zimet , Zero: The Biography of a Dangerous Idea, Viking Press, 2000. Good answers to questions such as Why did the Church reject the use of zero? How did mystics of all stripes get bent out of shape over it? Is it true that science as we know it depends on this mysterious round digit?, can be found in this recent book.
  290. Selin H., (ed.), Encyclopedia of the History of Science, Technology and Medicine in Non-Western Cultures, Kluwer Academic Publ., 1997.
  291. Sesiano J., The appearance of negative solutions in medieval mathematics, Arch. Hist. Exact Sci., 32(2), 105-150, 1985. Mentions that the "number line" representing positive and negative numbers can be found in the works of Wallis as well as Newton.
  292. Shields A., Klein and Bieberbach: Mathematics, Race, and Biology, Math. Intelligencer, 10(3), 7-11, 1988.
  293. Shirley J., (Ed), A Source Book For the Study of Thomas Harriot, Arno Press, 1981. Although Leibniz is generally credited as the first Western mathematician to consider the properties of binary numbers, the credit might belongs to Thomas Harriot (1560-1621). He used the binary system with the digits 0 and 1 to carry out additions, subtractions and a multiplication of 7 digit numbers.
  294. Sidebotham Th., The A to Z of Mathematics: A Basic Guide, Wiley, 2002. An antidote to math anxiety.
  295. Sierpin W., Elementary Theory of Numbers, English edition, edited by A. Schinzel, Elsevier Science, 2000.
  296. Sigler L., Leonardo Pisano Fibonacci: The Book of Squares, An Annotated Translation into Modern English, Academic Press, 1987.
  297. Platonic Realms, by B. Sidney Smith. Visit growing Encyclopedia.
  298. Smith D. E., History of Mathematics, Vols. 1, and 2., Dover, 1958. Gives many details of the history of trigonometry in both volumes.
  299. Smith L., Reasoning by Mathematical Induction in Children's Arithmetic, Pergamon Press, London, 2002. It contains research findings on the early age cognitive development based on a sample of one hundred children in their first two years at schooling.
  300. Snape Ch., and H. Scott, Puzzles, Mazes and Numbers, Cambridge Univ Pr., 1995. It contains the historical development of the topics in its title.
  301. Snell B., The Discovery of the Mind: In Greek Philosophy and Literature, Dover Pub., 1982.
  302. Srinivasiengar N, The History of Ancient Indian Mathematics, World Press, Calcutta, 1967.
  303. Steen S., Mathematical Logic with Special Reference to the Natural Numbers, Cambridge University Press, 1972.
  304. Stein Sh., Mathematics: The Man-Made Universe, Dover Pubns., 1999.
  305. Stein Sh., Strength in Numbers: Discovering the Joy and Power of Mathematics in Everyday Life, Wiley, 1999.
  306. Stewart I., From Here to Infinity, Oxford Univ. Press, 1996.
  307. Stewart I., The Magical Maze: Seeing the World Through Mathematical Eyes, John Wiley & Sons, 1999.
  308. Stewart I., Nature's Numbers: The Unreal Reality of Mathematics, Basic Books, 1997.
  309. Stewart I., and A. Stewart, Life's Other Secret : The New Mathematics of the Living World, John Wiley & Sons, New York, 1997.
  310. Stewart I., and D. Tall, The Foundations of Mathematics, Oxford Univ. Press, 1977. Contains a section "Different decimal expansions for the same real number" on pp. 33-35 which offers a slightly different view on the question of infinite decimal expansions as series or numbers. When one talks of "foundations", usually it includes historical, psychological, and logical aspects of the subject. This book does not cover all these aspects.
  311. Struik D., Source Book in Mathematics, Harvard University Press, 1967. A really nice place to begin your search for history of mathematical notations.
  312. Sutton C., "Nullius in verba" and "nihil in verbis": public understanding of the role of language in science, British Journal for the History of Science, 27(1), 55-64, 1994.
  313. Swerdlow N., The Babylonian Theory of the Planets, Princeton University Press, 1998.
  314. Synergetic, by R. Fuller. A Web site containing many interesting items including Numerology among others.
  315. Swetz F., (ed.), From Five Fingers to Infinity: A Journey Through the History of Mathematics, Open Court Pub., 1994. This is a popular book to keep as a handbook.
  316. Tall D., The anatomy of a discovery in mathematical research, Math Notebook , Center for Teaching-Learning of Mathematics, 1(2), 25-34, 1980.
  317. Tattersall J., Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999. Among other historical theory of numbers, it includes the rule that a number having the remainders 2, 3, 2 when divided by 3, 5, and 7 respectively.
  318. Taylor III, B., Introduction to Management Science, Prentice Hall, 2010. Web Module A: The Simplex solution method applied to an unbounded problem pp. A26-A27.
  319. Temple R., The Genius of China, Simon & Schuster Inc., 1986.
  320. Teresi D., Zero, The Atlantic Monthly, 280(1), 88-94, 1997. Visit also, the Discussion page on the article.
  321. Tod M., Ancient Greek Numerical Systems, Ares Publishers, Chicago, 1979.
  322. Tropfke J., Geschichte der Elementarmathematik, Berlin, Walter de Gruyter, 1980.
  323. Tymoczko T. (editor), New Directions in the Philosophy of Mathematics, Birkhauser, 1986.
  324. Van Der Waerden B., On Greek and Hindu trigonometry, Bulletin of the Belgian Mathematical Society, ser. A, 38, 397-407, 1986.
  325. Van der Waerden B., A History of Algebra, Springer-Verlag, New York, 1980.
  326. Van Der Waerden B., Geometry and Algebra in Ancient Civilizations, Springer Verlag, 1983. Points out that unlike Greeks, the Babylonians were engage in some algebraic concepts (not algorithmic methods) such as solving systems of equations: determine x and y when the product xy, and the sum x+y (or the difference x-y) is known. However, by geometric means as application of areas, not by any algebraic methods.
  327. Vilenkin N., In Search of Infinity, Provides a good discussion on the paradoxes generated by the theory of infinite sets, Springer Verlag, 1995.
  328. Urton G., The Social Life of Numbers, University of Texas Press, Austin, 1997. The author points out the fact that the inability to count beyond three in some tribes around the world, they are able to perceive the difference in numbers, by some "gestalt" form of perception.
  329. Weinberg S., Dreams of a Final Theory, Vintage Books, 1994.
  330. Wells D., The Penguin Dictionary of Curious and Interesting Geometry, Penguin Books, 1991.
  331. Wetherbee W., Historical Topics for the Mathematics Classroom, The National Council of Teachers of Mathematics, Reston, VA, 1989. Almost all of his references are old and inaccessible.
  332. White A., (Editor), Essays in Humanistic Mathematics, Mathematical Assn. of Amer., 1993.
  333. Wolfe H., Introduction to Non-Euclidean Geometry, Holt, Rinehart, and Winston Pub., 1966. The first part of the book is an historical review and then there is a development of the hyperbolic. There is a chapter on the elliptic plane and trig, and the book closes with an account of showing the consistency of the non-Euclidean geometries. This is not an easy book for a general reader.
  334. Wustholz G., (Ed.), A Panorama of Number Theory or The View from Baker's Garden, Cambridge University Press, 2002.
  335. Zangari M., Zeno, zero, and indeterminate forms: Instants in the logic of motion, Australian Journal of Philosophy, 72, 187-204, 1984. Unfortunately, the author claimed that "0/0 is not an undefined but an indeterminate form which admists of many (correct) answers." He is trying (unsuccessfully) to solve Zeno's arrows paradox which seems to me useless.
  336. Zaslavsky C., Africa Counts, Lawrence Hill, 1999. Zaslavsky, when dealing with the early counting, has pointed out that "questions of number recognition are different from questions of counting (and from telling anthropologists about it); using a small set of number words as basis for a number system is different again , pp. 32-33.
    Note also that in classic languages the first few numbers were adjective (i.e. inflected for gender, number, case): 1, 2, 3, 4 in Greek, 1, 2, 3 in Latin. In the old Russian language when following 2, 3, 4, and all their compounds the noun is in the Genitive Singular however, when following 5, 6, 7, 8, 9, and all their compounds as well as 10 and 11 the noun is in the Genitive Plural. Also when following 100 and its multiples.

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Write the Equation of the Sphere in Standard Form. X2 + Y2 + Z2 + 2x âë†â€™ 2y âë†â€™ 6z = 25

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