A History Of Mathematical Notations
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''A History of Mathematical Notations'' is a book on the
history of mathematics The history of mathematics deals with the origin of discoveries in mathematics and the History of mathematical notation, mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples ...
and of
mathematical notation Mathematical notation consists of using glossary of mathematical symbols, symbols for representing operation (mathematics), operations, unspecified numbers, relation (mathematics), relations, and any other mathematical objects and assembling ...
. It was written by Swiss-American historian of mathematics
Florian Cajori Florian Cajori (February 28, 1859 – August 14 or 15, 1930) was a Swiss-American historian of mathematics. Biography Florian Cajori was born in Zillis, Switzerland, as the son of Georg Cajori and Catherine Camenisch. He attended schools firs ...
(1859–1930), and originally published as a two-volume set by the
Open Court Publishing Company The Open Court Publishing Company is a publisher with offices in Chicago and LaSalle, Illinois. It is part of the Carus Publishing Company of Peru, Illinois. History Open Court was founded in 1887 by Edward C. Hegeler of the Matthiessen-Hege ...
in 1928 and 1929, with the subtitles ''Volume I: Notations in Elementary Mathematics'' (1928) and ''Volume II: Notations Mainly in Higher Mathematics'' (1929). Although Open Court republished it in a second edition in 1974, it was unchanged from the first edition. In 1993, it was published as an 820-page single volume edition by
Dover Publications Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, book ...
, with its original pagination unchanged. The Basic Library List Committee of the
Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university A university () is an educational institution, institution of tertiary edu ...
has listed this book as essential for inclusion in undergraduate mathematics libraries. It was already described as long-awaited at the time of its publication, and by 2013, when the Dover edition was reviewed by Fernando Q. Gouvêa, he wrote that it was "one of those books so well known that it doesn’t need a review". However, some of its claims on the history of the notations it describes have been subsumed by more recent research, and its coverage of modern mathematics is limited, so it should be used with care as a reference.


Topics

The first volume of the book concerns elementary mathematics. It has 400 pages of material on
arithmetic Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...
. This includes the history of notation for numbers from many ancient cultures, arranged by culture, with the
Hindu–Arabic numeral system The Hindu–Arabic numeral system (also known as the Indo-Arabic numeral system, Hindu numeral system, and Arabic numeral system) is a positional notation, positional Decimal, base-ten numeral system for representing integers; its extension t ...
treated separately. Following this, it covers notation for arithmetic operations, arranged separately by operation and by the mathematicians who used those notations (although not in strict chronological order). The first volume concludes with 30 pages on
elementary geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, including also the struggle between symbolists and rhetoricians in the 18th and 19th centuries on whether to express mathematics in notation or words, respectively. The second volume is divided more evenly into four parts. The first part, on arithmetic and algebra, also includes
mathematical constant A mathematical constant is a number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an Letter (alphabet), alphabet letter), or by mathematicians' names to facilitate using it across multiple mathem ...
s and
Special functions Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by ...
that would nowadays be considered part of
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
, as well as notations for
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s and other topics in
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
, and even the history of the
dollar sign The dollar sign, also known as the peso sign, is a currency symbol consisting of a Letter case, capital crossed with one or two vertical strokes ( or depending on typeface), used to indicate the unit of various currency, currencies around ...
. The second part is entitled "modern analysis", but its topics are primarily
trigonometry Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The fiel ...
,
calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...
, and
mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
, including the conflicting calculus notations of
Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
and
Gottfried Wilhelm Leibniz Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to ...
. The third part concerns geometry, while the fourth concerns scholarship in the history of mathematics as well as the movement for international standardization.


Audience and reception

This book is mainly a reference work and sourcebook, containing excerpts from many texts illustrating their use of notation. Among reviewers from the time of the work's original publication, George Sarton took as the main lesson from the book "the slowness and timidity of human advance", while some other reviewers took the different view that the confusing multiplicity of notations documented by the book should lead to a greater push for standardization. Although praising the book's "richness of explanation" and "familiarity with the ground", Lao Genevra Simons expressed a wish that Cajori had access to a greater number of original sources, and pointed to some historical inaccuracies in the work. Sarton concluded, accurately, that the book "will remain a standard work for many years to come". Although one reviewer found the treatment of dollar signs appropriate for an American book, the reviewer G. Feigl disagreed, regarding this subject as off-topic. In 1974, and echoing Feigl, reviewer complained that the book's coverage of mathematics from after the beginning of the 19th century was inadequate. In a review published in 2013, Fernando Q. Gouvêa wrote that the book remained useful, especially for its photographic reproductions of samples of old notation. He added that it was still the only comprehensive text in this area, although other works cover more specialized subtopics. However, Gouvêa wrote that modern scholarship on the numbering systems of past civilizations and on the first uses of some symbols has changed since Cajori's work, so such claims need to be checked against more recent publications instead of taking Cajori's word for them. In the case of ancient number systems, Gouvêa recommends instead '' Numerical Notation: A Comparative History'' by Stephen Chrisomalis (Cambridge University Press, 2010).


References


External links


''A History of Mathematical Notations, Vol. I''
an
''A History of Mathematical Notations, Vol. II''
on the Internet Archive {{DEFAULTSORT:History of Mathematical Notations, A 1928 non-fiction books 1929 non-fiction books Books about the history of mathematics Mathematical notation Open Court Publishing Company books