Trigonometry is an area of mathematics used for determining geometric quantities. Its name, first published in 1595 by B. Pitiscus, means "the study of trigons (triangles)" in Latin. Ancient Greek Mathematicians first used trigonometric functions with the chords of a circle. The first to publish these chords in 140 BC was Hipparchus, who is now called the founder of trigonometry. In AD 100, Menelaus, another Greek mathematician, published six lost books of tables of chords. Ptolemy, a Babylonian, also wrote a book of chords. Using chords, Ptolemy knew that
sin (x + y) = sin x * cos y + cos x * sin y
and
a / sin A = b / sin B = c / sin C.
Sine first appeared in the work of Aryabhata, a Hindu. He used the word jya for sine. He also published the first sine tables. Brahmagupta, in 628, also published a table of sines for any angle. Jya became jiba in translation and jiba became jaib in later writings. Jaib means fold in Arabic. This was translated into sinus, or fold in Latin. In 1533, Regiomontanus’ published De triangulis omnimodis which dealt with planar trigonometry and inverses. Later, Rheticus published Copernicus’ book dealing with Trigonometry in Astronomy in 1542. Edmund Gunter first used the abbreviation sin in 1624. Sin was first used in a book in 1634. Other variances still were popular. Other variances for cosine and tangent were also still very popular, especially among different languages. Although sine, cosine, and tangent were used very much by astronomers and surveyors, the functions secant and cosecant were of little use to these practical minded mathematicians.
Trigonometry has been used throughout modern and ancient history dealing with practical applications, such as surveying. Modernly, it has incorporated many other ideas instead of just triangles. The six trigonometric functions now known are
sine (sin) = opposite side/ hypotenuse
cosine (cos) = adjacent side/ hypotenuse
tangent (tan) = opposite side/ adjacent
secant (sec) = hypotenuse/ adjacent side
cosecant (cosec) = hypotenuse/ opposite side
cotangent (cot) = adjacent side/ opposite side
Three of these ratios are the reverse of other ratios. All of these ratios are used widely to determine sides and angles of triangles. Without their discovery, surveyors and other practical mathematicians would not be able to efficiently determine relationships
Monday, October 27, 2008
Pi: It Will Blow Your Mind
The history of Pi encompasses many centuries. An early approximation, from the Babylonians, is 3.14159… Another is from the Egyptians, 3 1/8 or 3.125, which could be concluded to indicate that they believed a circle with diameter nine has the same area as a square of diameter 8. The Bible was content to use 3, as demonstrated by I Kings 7:23:
Also, he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about.
Archimedes of Syracuse, 250 BC, computed pi to be
pi = 256/81 = 3.1604…
The astronomer Ptolemy, of Alexandria AD 150, used
3.1408 < pi < 3.1428
Also, in China in the fifth century, Tsu Chung-Chih calculate pi correctly to seven digits. Later in 1430, Al-Kashi of Samarkand computed pi to 14 places. Machin invented the formula
pi/4 = [(1/2) - (1/(3 * 2³)) + (1/(5 * 2^5)) - (1/(7 * 2^7)) … ] + [(1/(3 * 3³)) +
(1/(5 * 3^5)) - (1/(7 * 3^7)) … ]
Mathematicians used this formula to compute 707 digits in 1873. Later, they found the error after the 527-th decimal place. Newton developed a formula of his own:
pi/4 = 4 * tan^-1 * (1/5) - tan^-1 * (1/239)
Newton published 15 digits derived from this formula and later said
"I am ashamed to tell you how many figures I carried these computations, having no other business at the time."
Euler, in the 1700s, discovered many more formulas including
pi = (3 * square root of pi) / 4 + 24 * [(1 / (3 * 2^3)) - (1 / (5 * 2^5)) …}
and the simpler
pi² / 6 = 1 + 1 / 2² + 1 / 3² + 1 / 4² + 1 / 5² + …
and
pi^4 / 90 = 1 + 1 / 2^4 + 1 / 3^4 + 1 / 4^4 + 1 /5^4 + …
In 1897, Edwin J. Goodman, MD introduced and got passed a bill into the Indiana State House of Representatives which stated that
"the ratio of the diameter and circumference is as five-fourths to four;"
Other fallacies continued, as shown by a mathematician named Buffon’s needle experiment. He dropped a needle of k < 1 on a uniform grid of parallel lines. The probability of the needle falling on a line was 2 * k / pi. He and many other got the incredible answer of
pi = 355/113 = 3.1415929.
A mathematician named Gridgeman once did a completely facetious experiment in which he threw a needle of length 0.7857 and gave the value of pi as
2 x 0.7857 / pi = ½
Also incredibly he received the answer of 3.1428 = pi!
All this leads one to wonder what their fascination is with pi. There are no practical applications for pi at an industrial level. Simple approximations are suitable for engineering. So why do they insist on calculating pi to 6,442,450,938 decimals? One might argue science for the sake of science.
Pi has baffled and intrigued scientists and mathematicians for thousands of years. Perhaps they believe there is something about pi that would ease all other types of calculations. Perhaps they believe pi holds the key to future computation in areas of mathematics yet unexplored. Whatever it is, many more surprises and discoveries are to be made.
Also, he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about.
Archimedes of Syracuse, 250 BC, computed pi to be
pi = 256/81 = 3.1604…
The astronomer Ptolemy, of Alexandria AD 150, used
3.1408 < pi < 3.1428
Also, in China in the fifth century, Tsu Chung-Chih calculate pi correctly to seven digits. Later in 1430, Al-Kashi of Samarkand computed pi to 14 places. Machin invented the formula
pi/4 = [(1/2) - (1/(3 * 2³)) + (1/(5 * 2^5)) - (1/(7 * 2^7)) … ] + [(1/(3 * 3³)) +
(1/(5 * 3^5)) - (1/(7 * 3^7)) … ]
Mathematicians used this formula to compute 707 digits in 1873. Later, they found the error after the 527-th decimal place. Newton developed a formula of his own:
pi/4 = 4 * tan^-1 * (1/5) - tan^-1 * (1/239)
Newton published 15 digits derived from this formula and later said
"I am ashamed to tell you how many figures I carried these computations, having no other business at the time."
Euler, in the 1700s, discovered many more formulas including
pi = (3 * square root of pi) / 4 + 24 * [(1 / (3 * 2^3)) - (1 / (5 * 2^5)) …}
and the simpler
pi² / 6 = 1 + 1 / 2² + 1 / 3² + 1 / 4² + 1 / 5² + …
and
pi^4 / 90 = 1 + 1 / 2^4 + 1 / 3^4 + 1 / 4^4 + 1 /5^4 + …
In 1897, Edwin J. Goodman, MD introduced and got passed a bill into the Indiana State House of Representatives which stated that
"the ratio of the diameter and circumference is as five-fourths to four;"
Other fallacies continued, as shown by a mathematician named Buffon’s needle experiment. He dropped a needle of k < 1 on a uniform grid of parallel lines. The probability of the needle falling on a line was 2 * k / pi. He and many other got the incredible answer of
pi = 355/113 = 3.1415929.
A mathematician named Gridgeman once did a completely facetious experiment in which he threw a needle of length 0.7857 and gave the value of pi as
2 x 0.7857 / pi = ½
Also incredibly he received the answer of 3.1428 = pi!
All this leads one to wonder what their fascination is with pi. There are no practical applications for pi at an industrial level. Simple approximations are suitable for engineering. So why do they insist on calculating pi to 6,442,450,938 decimals? One might argue science for the sake of science.
Pi has baffled and intrigued scientists and mathematicians for thousands of years. Perhaps they believe there is something about pi that would ease all other types of calculations. Perhaps they believe pi holds the key to future computation in areas of mathematics yet unexplored. Whatever it is, many more surprises and discoveries are to be made.
The Most Famous Teacher
Euclid, who lived around 300 BC in Alexandria, first stated his five postulates in his book The Elements that forms the base for all of his later theorems. His first postulates were
1. To draw a straight line from any point to any other.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any center and any distance.
4. That all right angles are equal to each other.
5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, if produced indefinitely, meets on that side on which are the angles less than the two right angles.
It has been stated than Euclid was not satisfied with the last postulate, and even tried to derive it from the other four. His first 28 theorems in The Elements are not proven with the last postulate—it is not used until needed. Many other authors have written of Euclid’s and others’ attempts at deducing the fifth postulate, as Proclus stated in his commentary. He mentions when another author, Ptolemy, who produces a false "proof" of the fifth postulate. He gives an alternative postulate that is the same in principle to Euclid’s fifth postulate.
Playfair’s Axiom: Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.
This postulate became Playfair’s after an Englishman named John Playfair wrote commentaries on Euclid in 1795. Another "proof" was written in 1663 by Wallis who thought he had proved the fifth postulate. He actually restated equivalently the postulate as
To each triangle, there exists a similar triangle of arbitrary magnitude.
Several other attempts to prove or disprove the fifth postulate have followed, notably that of Girolamo Saccheri. He assumed the fifth postulate to be false and attempted to derive a contradiction. Another mathematician, Gauss, started working on the postulate as early as 1792 while 15 years old. In 1813, after making little progress, he wrote
In the theory of parallels we are even now not further than Euclid. This is a shameful part of mathematics…
In 1817, Gauss stated that the fifth postulate was independent from the other postulates, and therefore needed no proof from the others, and begun to work on a different geometry in which multiple lines can be parallel to another line through a given point. In fact, the fifth postulate has been called "the one sentence in the history of science that has given rise to more publication than any other."
Euclid deduced many theorems and other conjectures from his five original postulates. Many porisms, now called corollaries, and many lemmas, or something assumed in the proof of a theorem, were used. Furthermore, many propositions in the later books were based on previous theorems proven true.
Book - Previous books or propositions upon which it depends
I - (independent)
II - I
III - I; II.5,6
IV - I; II.11; III
V - (independent)
VI - I; III.27,31; V
VII - (independent)
VIII - V.Def.; VII
IX - II.3,4; VII; VIII
X - I.44,47; II; III.31; V; VI; VII.4,11,26; IX.1,24,26
XI - I; III.31; IV.1; V; VI
XII - I; III; IV.6,7; V; VI; X.1; XI
XIII - I; II.4; III; IV; V; VI; X; XI
This leads to the conclusion that if one of the early theorems were subsequently proven false, many of the latter theorems may not be true either.
Many things now considered essential to geometry are omitted from The Elements, such as the formulas for the areas of figures. Euclid left calculations totally out of The Elements, as well as principles that can not be expressed with straight-edge and compass alone. In Book I of The Elements Euclid defines many of the terms used commonly in modern geometry, such as point, line, and figure.
Many things used in Euclid’s proofs are not proven. However, they are not postulates, either. They were called common notions by Euclid, and today we call them axioms. Some of Euclid’s axioms are
1. Things that are equal to the same thing are also equal to one another.
2. If equals be added to equals, the wholes are equal.
3. If equals be subtracted from equals, the remainders are equal.
4. Things that coincide with one another are equal to one another.
5. The whole is greater than the part.
This would suggest than many mathematicians believed that the postulates were not enough to solve all problems, and have asserted these what he considered "common" sense. Some other axioms added by later authors are
6. Two lines do not enclose a space.
7. If equals be added to unequals, the wholes are unequal.
8. If equals be subtracted from unequals, the remainders are unequal.
9. Doubles of the same thing are equal to one another.
10. Halves of the same thing are equal to one another.
However, all of these axioms can be derived from Euclid’s axioms and theorems and therefore are obsolete. Also note that axiom six should in fact be a postulate because it deals with geometric figures. Other propositions proved in The Elements include S-A-S, S-S-S, A-S-A, A-A-S, that the sum of a triangle’s angles equals that of two right angles, and the Pythagorean Theorem. The proof of the latter is believed to be originally from Euclid and is called the Bride’s Chair
1. To draw a straight line from any point to any other.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any center and any distance.
4. That all right angles are equal to each other.
5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, if produced indefinitely, meets on that side on which are the angles less than the two right angles.
It has been stated than Euclid was not satisfied with the last postulate, and even tried to derive it from the other four. His first 28 theorems in The Elements are not proven with the last postulate—it is not used until needed. Many other authors have written of Euclid’s and others’ attempts at deducing the fifth postulate, as Proclus stated in his commentary. He mentions when another author, Ptolemy, who produces a false "proof" of the fifth postulate. He gives an alternative postulate that is the same in principle to Euclid’s fifth postulate.
Playfair’s Axiom: Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.
This postulate became Playfair’s after an Englishman named John Playfair wrote commentaries on Euclid in 1795. Another "proof" was written in 1663 by Wallis who thought he had proved the fifth postulate. He actually restated equivalently the postulate as
To each triangle, there exists a similar triangle of arbitrary magnitude.
Several other attempts to prove or disprove the fifth postulate have followed, notably that of Girolamo Saccheri. He assumed the fifth postulate to be false and attempted to derive a contradiction. Another mathematician, Gauss, started working on the postulate as early as 1792 while 15 years old. In 1813, after making little progress, he wrote
In the theory of parallels we are even now not further than Euclid. This is a shameful part of mathematics…
In 1817, Gauss stated that the fifth postulate was independent from the other postulates, and therefore needed no proof from the others, and begun to work on a different geometry in which multiple lines can be parallel to another line through a given point. In fact, the fifth postulate has been called "the one sentence in the history of science that has given rise to more publication than any other."
Euclid deduced many theorems and other conjectures from his five original postulates. Many porisms, now called corollaries, and many lemmas, or something assumed in the proof of a theorem, were used. Furthermore, many propositions in the later books were based on previous theorems proven true.
Book - Previous books or propositions upon which it depends
I - (independent)
II - I
III - I; II.5,6
IV - I; II.11; III
V - (independent)
VI - I; III.27,31; V
VII - (independent)
VIII - V.Def.; VII
IX - II.3,4; VII; VIII
X - I.44,47; II; III.31; V; VI; VII.4,11,26; IX.1,24,26
XI - I; III.31; IV.1; V; VI
XII - I; III; IV.6,7; V; VI; X.1; XI
XIII - I; II.4; III; IV; V; VI; X; XI
This leads to the conclusion that if one of the early theorems were subsequently proven false, many of the latter theorems may not be true either.
Many things now considered essential to geometry are omitted from The Elements, such as the formulas for the areas of figures. Euclid left calculations totally out of The Elements, as well as principles that can not be expressed with straight-edge and compass alone. In Book I of The Elements Euclid defines many of the terms used commonly in modern geometry, such as point, line, and figure.
Many things used in Euclid’s proofs are not proven. However, they are not postulates, either. They were called common notions by Euclid, and today we call them axioms. Some of Euclid’s axioms are
1. Things that are equal to the same thing are also equal to one another.
2. If equals be added to equals, the wholes are equal.
3. If equals be subtracted from equals, the remainders are equal.
4. Things that coincide with one another are equal to one another.
5. The whole is greater than the part.
This would suggest than many mathematicians believed that the postulates were not enough to solve all problems, and have asserted these what he considered "common" sense. Some other axioms added by later authors are
6. Two lines do not enclose a space.
7. If equals be added to unequals, the wholes are unequal.
8. If equals be subtracted from unequals, the remainders are unequal.
9. Doubles of the same thing are equal to one another.
10. Halves of the same thing are equal to one another.
However, all of these axioms can be derived from Euclid’s axioms and theorems and therefore are obsolete. Also note that axiom six should in fact be a postulate because it deals with geometric figures. Other propositions proved in The Elements include S-A-S, S-S-S, A-S-A, A-A-S, that the sum of a triangle’s angles equals that of two right angles, and the Pythagorean Theorem. The proof of the latter is believed to be originally from Euclid and is called the Bride’s Chair
The First Mathematicians
Introduction
The first mathematics can be traced to the ancient country of Babylon and to Egypt during the 3rd millennium BC. A number system with a base of 60 had developed in Babylon over time. Large numbers and fractions could be represented and formed the basis of advanced mathematical evolution. From at least 1700 BC, Pythagorean triples were studied. The study of linear and quadratic equations led to form of primitive numerical algebra. Meanwhile, similar figures, areas, and volumes were studied as well as the primitive values for pi obtained. The Greeks inherited the Babylonian principles and developed mathematics from 450 BC. They discovered that all real numbers could not accurately express all values, such as relationships between sides. Irrational numbers were born. The Greeks progressed rapidly in mathematics from 300 BC. Progress also sped in the Islamic countries of Syria, India, and Iran. Their work had a different focus from that of the Greeks, but all Greek principles held true. This basis was later brought to Europe and developed further there.
Babylonians: Writing and Base 60 System
The Babylonian system of writing was called cuneiform and was based on a series of straight lined symbols. These symbols were wet and baked in the hot sun to preserve. Curved lines could not be drawn. These cuneiform symbols led to many tables used to aid calculation. As stated previously, they used a base 60 system, which has ten proper divisors, instead of our current system, base 10 with only two proper divisors. In this respect, their system may have been more advanced since many more numbers have a finite form. Two examples of these tables are the tables found at Senkerah on the Euphrates River in 1854, which date from 2000 BC. This table was used to figure the squares of numbers to 59 and cubes of numbers up to 32. However, a drawback of this system is the lack of a proper 0. Also, context was required to determine if 1 meant 1, 61, or 361, etc.
Babylonians: Multiplication and Division
The Babylonians used the fact that
ab = ((a + b)² - a ² - b²)/2
and therefore
ab = (a + b)² / 4 - (a - b)² / 4
to make multiplication possible. If the user of this formula wished to multiply two numbers, all he would need is a chart of squares. However, their process of division was a more difficult task. They used the fact that
a.b = a.(1 / b)
to solve division problems. If the user of this formula wished to divide number, all he would need is a table of reciprocals. Consider the following reciprocal chart translated into Arabic numbers.
2 30
3 20
4 15
5 12
6 10
8 7 30
9 6 40
10 6
12 5
15 4
16 3 45
18 3 20
20 3
24 2 30
25 2 24
27 2 13 20
Not all reciprocals were possible because certain numbers have no base 60 finite fractions. To compute 1/13, for instance, the Babylonians could write
1/13 = 7/91 = 7.(1/91) = approximately 7.(1/90)
Babylonians: Pythagorean Triples and Problems to Solve Them
Another Babylonian tablet, dated between 1900 BC and 1600 BC contains certain Pythagorean triples where
a² + b² = c²
Many believe that it is the oldest number theory document ever recorded. Another Babylonian tablet contains the problem
4 is the length and 5 the diagonal. What is the breadth? Its size is not known. 4 times 4 is 16. 5 times 5 is 25. You take 16 from 25 and there remains 9. What times shall I take in order to get 9? 3 times 3 is 9. 3 is the breadth.
This is a good example of using Pythagorean triples to solve certain problems by the Babylonians.
Egyptians and Romans: Number System
The Roman and Egyptian systems did not make Arithmetic calculations easy. Multiplication of Roman numerals is nearly impossible and exceedingly complex. Unlike the Babylonians, the Egyptians did not develop fully their understanding of mathematics. Instead, they concerned themselves with practical applications of mathematics.
Egyptians and Romans: Multiplication
In 1650 BC, the scribe Ahmes wrote the Rhind Papyrus, named for its Scottish Egyptologist author A. Henry Rhind. The scroll is 6 meters long and 1/3 of a meter wide. The scribe Ahmes was copying a document predated 200 years before him. Consider, for instance, the multiplication of 41 and 59.
41 59
1 59
2 118
4 236
8 472
16 944
32 1888
64 3776
Since 41 falls between 32 and 64, they carry out the following simple subtraction problem
41 - 32 = 9, 9 - 8 = 1, 1 - 1 = 0
Hence,
41 = 32 + 8 + 1
Then add the corresponding totals
59
1 59 X
2 118
4 236
8 472 X
16 944
32 1888 X
2419
Hence the answer, 2419. If the factors were reversed and the factors of 41 used, then the same answer could be reached. These are two good examples of different historical cultures solving the same types of problems totally differently before modern mathematics.
The first mathematics can be traced to the ancient country of Babylon and to Egypt during the 3rd millennium BC. A number system with a base of 60 had developed in Babylon over time. Large numbers and fractions could be represented and formed the basis of advanced mathematical evolution. From at least 1700 BC, Pythagorean triples were studied. The study of linear and quadratic equations led to form of primitive numerical algebra. Meanwhile, similar figures, areas, and volumes were studied as well as the primitive values for pi obtained. The Greeks inherited the Babylonian principles and developed mathematics from 450 BC. They discovered that all real numbers could not accurately express all values, such as relationships between sides. Irrational numbers were born. The Greeks progressed rapidly in mathematics from 300 BC. Progress also sped in the Islamic countries of Syria, India, and Iran. Their work had a different focus from that of the Greeks, but all Greek principles held true. This basis was later brought to Europe and developed further there.
Babylonians: Writing and Base 60 System
The Babylonian system of writing was called cuneiform and was based on a series of straight lined symbols. These symbols were wet and baked in the hot sun to preserve. Curved lines could not be drawn. These cuneiform symbols led to many tables used to aid calculation. As stated previously, they used a base 60 system, which has ten proper divisors, instead of our current system, base 10 with only two proper divisors. In this respect, their system may have been more advanced since many more numbers have a finite form. Two examples of these tables are the tables found at Senkerah on the Euphrates River in 1854, which date from 2000 BC. This table was used to figure the squares of numbers to 59 and cubes of numbers up to 32. However, a drawback of this system is the lack of a proper 0. Also, context was required to determine if 1 meant 1, 61, or 361, etc.
Babylonians: Multiplication and Division
The Babylonians used the fact that
ab = ((a + b)² - a ² - b²)/2
and therefore
ab = (a + b)² / 4 - (a - b)² / 4
to make multiplication possible. If the user of this formula wished to multiply two numbers, all he would need is a chart of squares. However, their process of division was a more difficult task. They used the fact that
a.b = a.(1 / b)
to solve division problems. If the user of this formula wished to divide number, all he would need is a table of reciprocals. Consider the following reciprocal chart translated into Arabic numbers.
2 30
3 20
4 15
5 12
6 10
8 7 30
9 6 40
10 6
12 5
15 4
16 3 45
18 3 20
20 3
24 2 30
25 2 24
27 2 13 20
Not all reciprocals were possible because certain numbers have no base 60 finite fractions. To compute 1/13, for instance, the Babylonians could write
1/13 = 7/91 = 7.(1/91) = approximately 7.(1/90)
Babylonians: Pythagorean Triples and Problems to Solve Them
Another Babylonian tablet, dated between 1900 BC and 1600 BC contains certain Pythagorean triples where
a² + b² = c²
Many believe that it is the oldest number theory document ever recorded. Another Babylonian tablet contains the problem
4 is the length and 5 the diagonal. What is the breadth? Its size is not known. 4 times 4 is 16. 5 times 5 is 25. You take 16 from 25 and there remains 9. What times shall I take in order to get 9? 3 times 3 is 9. 3 is the breadth.
This is a good example of using Pythagorean triples to solve certain problems by the Babylonians.
Egyptians and Romans: Number System
The Roman and Egyptian systems did not make Arithmetic calculations easy. Multiplication of Roman numerals is nearly impossible and exceedingly complex. Unlike the Babylonians, the Egyptians did not develop fully their understanding of mathematics. Instead, they concerned themselves with practical applications of mathematics.
Egyptians and Romans: Multiplication
In 1650 BC, the scribe Ahmes wrote the Rhind Papyrus, named for its Scottish Egyptologist author A. Henry Rhind. The scroll is 6 meters long and 1/3 of a meter wide. The scribe Ahmes was copying a document predated 200 years before him. Consider, for instance, the multiplication of 41 and 59.
41 59
1 59
2 118
4 236
8 472
16 944
32 1888
64 3776
Since 41 falls between 32 and 64, they carry out the following simple subtraction problem
41 - 32 = 9, 9 - 8 = 1, 1 - 1 = 0
Hence,
41 = 32 + 8 + 1
Then add the corresponding totals
59
1 59 X
2 118
4 236
8 472 X
16 944
32 1888 X
2419
Hence the answer, 2419. If the factors were reversed and the factors of 41 used, then the same answer could be reached. These are two good examples of different historical cultures solving the same types of problems totally differently before modern mathematics.
The British Society for the History of Mathematics
Web resources on the history of mathematics
Selected and annotated by June Barrow-Green (j.e.barrow-green@open.ac.uk).
The intention of this page is to give some indication of the kind of material that is available on the web, with a few examples in each case: not to list it exhaustively. Many more sites can be found through viewing the pages of Web Resources listed below.
Certain other resources are also maintained on this site: David Singmaster's Mathematical Gazetteer of the British Isles; a list of the items bequeathed by John Fauvel and held by the Open University Library on behalf of the Society (by author or by classmark); a list of books offered for sale to members from the collection of the late Neil Bibby (html or Excel spreadsheet).
Contents
* General sites
* Web resources
* Biographies
* Regional mathematics
* Museums with mathematics exhibits
* Special exhibits
* Books and articles on-line
* Student presentations
* Bibliography
* Societies
* Journals
* Philosophy of Mathematics
* History of Statistics
* History of Computing
* Education
* Miscellaneous
Some of these sites are specifically devoted to history of mathematics while others are part of larger sites. Sites which contain, or have links which contain, images and are slow to download, or which are interactive and require a specially enabled browser, have been marked with an asterisk (*). A 'hypertext' site is one which contains pages in which there are links to other pages explicitly incorporated within the text itself. A 'hypermedia' site is one which incorporates the opportunity to access additional media, such as music or animation.
Return to the top.
General Sites
All large sites have a gateway page which gives an indication of the type of resources that are available on other pages of the site. The following are the addresses of the gateways to three of the best known of the general sites on history of mathematics. (Some of the pages on these sites are also included in other sections.)
David Joyce's History of Mathematics Home Page
http://aleph0.clarku.edu/~djoyce/mathhist/
This is the starting point to a wealth of resources provided by David Joyce of Clark University, USA. There are pages on regional mathematics, subjects, books, journals, bibliography, history of mathematics texts etc, as well as an excellent list of Web Resources clearly categorised (see below), a very extensive chronology, and timelines. A highly recommended site.
The Math Forum Internet Resource Collection
http://mathforum.org/
http://mathforum.org/library/topics/history/
This site is part of the The Math Forum, an on-line mathematics education community centre, and provides an extensive list of annotated links to other sites. The sites are ordered alphabetically and the collection can be viewed in outline or annotated form. There is a well designed search engine which allows for a variety of searches, i.e. keywords, categories and dates.
St Andrews MacTutor History of Mathematics
http://www-history.mcs.st-and.ac.uk/history/
A collection of biographies of mathematicians, and a variety of resources on the developments of various branches of mathematics. The site includes an interactive (Java) famous curves index, pages on mathematical societies, medals, and honours, and birthplace maps. An extremely rich and extensive site with some excellent pages although the quality is not always consistent. In particular, the biographies should be viewed with care. Overall, though, a good place to start.
Trinity College, Dublin, History of Mathematics archive
http://www.maths.tcd.ie/pub/HistMath/HistMath.html
This site, which was created and is maintained by David Wilkins, includes biographies of some seventeenth and eighteenth century mathematicians, material on Berkeley, Newton, Hamilton, Boole, Riemann and Cantor, and an extensive directory of history of mathematics websites (see below). Another good place to start.
Convergence
http://mathdl.maa.org/convergence/1/
Convergence is an online magazine on the history of mathematics and its use in teaching, published by the Mathematical Association of America. It includes articles on the history of mathematics, material on the history of mathematics that can easily be used in teaching, reviews of books, websites, and teaching materials relevant to the history of mathematics, and more.
Selected and annotated by June Barrow-Green (j.e.barrow-green@open.ac.uk).
The intention of this page is to give some indication of the kind of material that is available on the web, with a few examples in each case: not to list it exhaustively. Many more sites can be found through viewing the pages of Web Resources listed below.
Certain other resources are also maintained on this site: David Singmaster's Mathematical Gazetteer of the British Isles; a list of the items bequeathed by John Fauvel and held by the Open University Library on behalf of the Society (by author or by classmark); a list of books offered for sale to members from the collection of the late Neil Bibby (html or Excel spreadsheet).
Contents
* General sites
* Web resources
* Biographies
* Regional mathematics
* Museums with mathematics exhibits
* Special exhibits
* Books and articles on-line
* Student presentations
* Bibliography
* Societies
* Journals
* Philosophy of Mathematics
* History of Statistics
* History of Computing
* Education
* Miscellaneous
Some of these sites are specifically devoted to history of mathematics while others are part of larger sites. Sites which contain, or have links which contain, images and are slow to download, or which are interactive and require a specially enabled browser, have been marked with an asterisk (*). A 'hypertext' site is one which contains pages in which there are links to other pages explicitly incorporated within the text itself. A 'hypermedia' site is one which incorporates the opportunity to access additional media, such as music or animation.
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General Sites
All large sites have a gateway page which gives an indication of the type of resources that are available on other pages of the site. The following are the addresses of the gateways to three of the best known of the general sites on history of mathematics. (Some of the pages on these sites are also included in other sections.)
David Joyce's History of Mathematics Home Page
http://aleph0.clarku.edu/~djoyce/mathhist/
This is the starting point to a wealth of resources provided by David Joyce of Clark University, USA. There are pages on regional mathematics, subjects, books, journals, bibliography, history of mathematics texts etc, as well as an excellent list of Web Resources clearly categorised (see below), a very extensive chronology, and timelines. A highly recommended site.
The Math Forum Internet Resource Collection
http://mathforum.org/
http://mathforum.org/library/topics/history/
This site is part of the The Math Forum, an on-line mathematics education community centre, and provides an extensive list of annotated links to other sites. The sites are ordered alphabetically and the collection can be viewed in outline or annotated form. There is a well designed search engine which allows for a variety of searches, i.e. keywords, categories and dates.
St Andrews MacTutor History of Mathematics
http://www-history.mcs.st-and.ac.uk/history/
A collection of biographies of mathematicians, and a variety of resources on the developments of various branches of mathematics. The site includes an interactive (Java) famous curves index, pages on mathematical societies, medals, and honours, and birthplace maps. An extremely rich and extensive site with some excellent pages although the quality is not always consistent. In particular, the biographies should be viewed with care. Overall, though, a good place to start.
Trinity College, Dublin, History of Mathematics archive
http://www.maths.tcd.ie/pub/HistMath/HistMath.html
This site, which was created and is maintained by David Wilkins, includes biographies of some seventeenth and eighteenth century mathematicians, material on Berkeley, Newton, Hamilton, Boole, Riemann and Cantor, and an extensive directory of history of mathematics websites (see below). Another good place to start.
Convergence
http://mathdl.maa.org/convergence/1/
Convergence is an online magazine on the history of mathematics and its use in teaching, published by the Mathematical Association of America. It includes articles on the history of mathematics, material on the history of mathematics that can easily be used in teaching, reviews of books, websites, and teaching materials relevant to the history of mathematics, and more.
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