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A History of Mathematical Notations
This classic study notes the first appearance of a mathematical symbol and its origin, the competition it encountered, its spread among writers in different countries, its rise to popularity, its eventual decline or ultimate survival. The author's coverage of obsolete notations — and what we can learn from them — is as comprehensive as those which have survived and still enjoy favor. Originally published in 1929 in a two-volume edition, this monumental work is presented here in one volume.
820 pages, Paperback
First published January 1, 1974
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About the author
Florian Cajori
187 books7 followersFlorian Cajori was a Swiss-American historian of mathematics.
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Displaying 1 - 2 of 2 reviews
September 3, 2022
This was a hard book to read, and I mean that literally. The ebook version of A History of Mathematical Notations overwhelmed my eReader software. Published in 1928, it is 902 pages long and almost every page is filled with hand drawn images which get rendered as inline graphics and place a great burden on a reading app. The copy I checked out from my library was in the Hoopla format, and my trusty iPad was not up to the task. It would take six or seven seconds to flip the page, and after half a couple pages the app would stall and crash. Fortunately, I was able to borrow an iPad Pro, and its faster processor and additional memory were able to render the book more or less successfully. It still crashed occasionally, but by assiduously bookmarking my progress I was able to get through it.
The subject of mathematical notation is, admittedly, of interest to few people, but there is some intriguing history in the stories of how we came to use the symbols that seem so self-evident to us today. Before they existed mathematical operations had to be written out in text. For instance, prior to the idea of π as a discrete symbolic value, there was an understanding of what it represented. “Archimedes says that the length of a circle amounts to three diameters and a part of one, the size of which lies between one-seventh and ten seventy-firsts.”
Each symbol we use today was once only one of many, and it is fascinating to see other interpretations and how they expressed numerical concepts. Algebraic notation existed in China but it was expressed in an elegant though limiting form that was bound up in astronomical interpretations. “The fact that Chinese algebra reached a standstill after the thirteenth century may be largely due to its inelastic and faulty notation.”
Consider our symbol for the square root, originally called radix, Latin for “root” in the sense of a plant root. It is actually two symbols joined together, and the first person to present it in its modern form was French philosopher, mathematician, and scientist René Descartes in his 1637 work La géométrie. The sign on the left , √, was invented in Germany in the 1500s. It was Descartes who added the horizontal bar over the top, called a vinculum, which also has a number of other mathematical uses, such as separating a column of numbers from their sum.
The shapes of our numerals had a long and complicated development before they reached their modern forms. As the author states:
This book contains entire tables dedicated to showing examples of specific numerals in odd and creative shapes from the days of manuscripts and the early years of printing. And while there is probably more standardization today, when the book was written in the 1920s there was still considerable variation to be found. “An outstanding fact is that during the past one thousand years no uniformity in the shapes of numerals has been reached. An American is sometimes puzzled by the shape of the number 5 written in France. A European traveler in Turkey would find that what in Europe is a 0 is in Turkey a 5.”
There is an interesting discussion of the notation of calculus. The author simply relates the facts as they are known regarding the first uses of the symbology, and does not discuss the epic and rather sordid story of the controversy between Leibniz and Newton over who should receive credit. Newton, his followers, and Newton himself writing as a proxy under other names, claimed to have invented calculus decades before Leibniz, but he used different symbology; both the integral symbol and the dx/dy notation derived from Leibniz. “The first appearance of dx in print was in an article which Leibniz contributed to the Acta eruditorum, in 1684. Therein occur the expressions “dw ad dx,” “dx and dy,” and also “dx:dy,” but not the form dx/dy.” Similarly, “In a manuscript dated October 29, 1675...Leibnitz introduces the symbol ∫L for omn. L, that is the sum of the l’s….It was the long form of the letter s, frequently used at the time of Leibniz.” He used it as an abbreviated form of summa, which he wrote with the long s, as “∫umma”
The book is full of these sorts of interesting observations, and following are some which I found illuminating:
- Hinck’s [conjecture about the meaning of a cuneiform tablet] was confirmed by the decipherment of tablets found at Senkereh, near Babylon, in 1854...One table was found to contain a table of square numbers from 12 to 602, a second one a table for cube numbers from 13 to 323. The tablets were probably written between 2300 and 1600 B.C.
- The earliest thorough and systematic application of a symbol for zero and the principle of position was made by the Maya of Central America, about the beginning of the Christian Era.
- [In one cuneiform tablet] “the first number of every odd line can be expressed by a fraction which has 12,960,000 as its numerator and the closing number of the corresponding even line as its denominator...The question arises, what is the meaning of all this? What in particular is the meaning of the number 12,960,000 (= 60^4 or 3600^2) which underlies all the mathematical texts here treated? [It was believed that] this ‘geometrical number’ (12,960,000) which Plato [in The Republic viii. 546B-D] calls the ‘lord of better and worse births,’ is the arithmetical expression of a great law controlling the Universe.”
-The process of multiplication or division known to the Egyptians as wshtp, “to incline the head.”
- The modern algebraic signs + and – came into use in Germany late in the fifteenth century. They are first found in manuscripts. The view that our + sign descended from one of the florescent forms for et in Latin manuscripts finds further support from works on paleography. J.L. Walthier enumerates one hundred and two different abbreviations found in Latin manuscripts for the word et; one of these, from a manuscript dated 1417, looks very much like the modern +.
- In Italy the symbols p̃ and m̃ served as convenient abbreviations for “plus” and “minus” at the end of the fifteenth century and during the sixteenth.
- While the Hindu-Arabic numerals became generally known in Europe about 1275, the Roman numerals continued to hold a commanding place. For example, the fourteenth-century banking-house of Peruzzi in Florence – Campagnia Peruzzi – did not use Arabic numerals in their account-books.
- The earliest authentic document unmistakably containing the numerals with the zero in India belongs to the year 876 A.D. The earliest Arabic manuscripts containing the numerals are of 874 and 888 A.D.
- The oldest definitely dated European manuscript known to contain the Hundu-Arabic numerals is the Codex Vigilanus, written in the Albeda Cloister in Span in 976 A.D.
- [Arabic] numerals are contained in a Vatican manuscript of 1077, on a Sicilian coin of 1138, in a Regensburg (Bavaria) chronicle of 1197. The earliest coins outside of Italy that are dated in Arabic numerals are as follows: Swiss 1424, Austrian 1484, French 1485, German 1489, Scotch 1539, English 1551.
- Recently the variations in form of our numerals have been summarized as follows: ‘The form of the numerals 1, 6, 8 and 9 has not varied much among the [medieval] Arabs nor the Christians of the Occident; the numerals of the Arabs of the Occident for 2, 3 and 5 have forms offering some analogy to ours (the 3 and 5 are originally reversed, as well among the Christians as among the Arabs of the Occident); but the form of 4 and that of 7 have greatly modified themselves.
- the Arabic numeral 5 appears upside down in some Spanish books and manuscripts as late as the eighteenth and nineteenth centuries.
- [Raphael Bombelli, 1526-1572] expressed square root by R.q., cube root by R.c., fourth root by RR.q., fifth root by R.p.r., sixth root by R.q.c., seventh root by R.s.r.
- Copernicus died in 1543. His De revolutionibus orbium coelestium (1566; 1st ed., 1543) shows that the exposition is devoid of algebraic symbols and is almost wholly rhetorical. We find a curious mixture of modes of expressing numbers: Roman numerals, Hindu-Arabic numerals, and numbers written out in words.
- A page in [German mathematician Christopher Clavius’] Algebra (Rome, 1608)...shows one of the very earliest uses of round parentheses to express aggregation of terms.
- [English mathematician William Oughtred (1574-1660)] was the first to use x [the Saint Andrew’s cross] as the sign of multiplication of two numbers, as a x b. The cross appears in Oughtred’s Clavis mathematicae in 1631 and, in the form of the letter X, in E. Wright’s edition of Napier’s Descriptio (1618).
- The dot was introduced as a symbol for multiplication by G.W. Leibniz. On July 29, 1698, he wrote in a letter to John Bernoulli: “I do not like x as a symbol for multiplication, as it is easily confounded with x;….often I simply relate two quantities by an interposed dot and indicate multiplication by ZC·LM.”
- In 1659 the Swiss Johan Heinrich Rahn published an algebra in which he introduced ÷ as a sign of division. Many writers before him had used ÷ as a symbol for subtraction.
- The sign ÷ as a symbol for division was adopted by John Wallis and other English writers. It came to be adopted regularly in Great Britain and the United States, but not on the European Continent.
- In the printed books before [Robert] Recorde, equality was usually expressed rhetorically by such words as aequales, aequantur, esgale, faciunt, ghelijck, or gleich, and sometimes by the abbreviated form aeq….Recorde’s =, after its debut in 1557, did not appear again in print until 1618, or sixty-one years later.
- In 1571, a German writer, Wilhelm Holzmann, better known under the name of Xylander, brought out an edition of Diophantus’ Arithmetica in which two parallel lines || were used for equality. He gives no clue to the origin of the symbol. Moritz Cantor suggests that perhaps the Greek word íσoi (“equal”) was abbreviated in the manuscript used by Xylander, by writing of only the two letters ii. Weight is given to this suggestion in a Parisian manuscript on Diophantus where a single i denoted equality.
- If Bartholomaeus Pitiscus of Heidelberg made use of the decimal point, he was probably the first to do so. Recent writers on the history of mathematics are divided on the question as to whether or not Pitiscus used the decimal point, the majority of them stating that he did use it.
- [The symbol] √ originated in Germany. Euler guessed that it was a deformed letter r, the first letter of radix. This opinion was held generally until recently. The more careful study of German manuscript algebras and the first printed algebras has convinced Germans that the old explanation is hardly tenable; they have accepted the a priori much less probably explanation of the evolution of the symbol from a dot.
- Engström...inclines to the view that the predominance of x over y and z [to represent an unknown] is due to typographical reasons, type for x being more plentiful because of the more frequent occurrence of the letters y and z in the French and Latin languages.
- An interesting feature in our survey is the vitality exhibited by the notation dy/dx for derivatives. Typographically not specially desirable, the symbol nevertheless commands at the present time a wider adoption than any of its many rivals...For integration ∫ has practically no rival. It easily adapted itself to the need of marking the limits of integration in definite integrals
- The modern notation for pi was introduced in 1706. It was that year that William Jones made himself noted, without being aware that he was doing anything noteworthy, through his designation of the ratio of the length of a circle to its diameter by the letter π. He took this step without ostentation. No lengthy introduction prepares the reader for the bringing upon the state of mathematical history this distinguished visitor from the field of Greek letters. It simply came, unheralded.
- The introduction of the letter e to represent the base of the natural system of logarithms is due to Euler. According to G. Engström, it occurs in a manuscript written in 1727 or 1728, but which was not published until 1862. Euler used e again in 1736 in his Mechanica.
- The notation n! Is used after Kramp in Gergonne’s Annales by J.B. Durrande in 1816, and by F. Sarrus in 1819. Durrande remarks, “There is ground for surprise that a notation so simple and consequently so useful has not yet been universally adopted.” It found wide adoption in Germany, where it is read “n-Fakultät.” Some texts in the English language suggest the reading “n-admiration” (the exclamation point being a note of admiration), but most texts prefer “factorial n” or “n-factorial.”
- There has been a real need in analysis for a convenient symbolism for “absolute value” of a given number, or “absolute number,” and the two vertical bars introduced in 1841 by Weierstrass, as in |z|, have met with wide adoption.
The subject of mathematical notation is, admittedly, of interest to few people, but there is some intriguing history in the stories of how we came to use the symbols that seem so self-evident to us today. Before they existed mathematical operations had to be written out in text. For instance, prior to the idea of π as a discrete symbolic value, there was an understanding of what it represented. “Archimedes says that the length of a circle amounts to three diameters and a part of one, the size of which lies between one-seventh and ten seventy-firsts.”
Each symbol we use today was once only one of many, and it is fascinating to see other interpretations and how they expressed numerical concepts. Algebraic notation existed in China but it was expressed in an elegant though limiting form that was bound up in astronomical interpretations. “The fact that Chinese algebra reached a standstill after the thirteenth century may be largely due to its inelastic and faulty notation.”
Consider our symbol for the square root, originally called radix, Latin for “root” in the sense of a plant root. It is actually two symbols joined together, and the first person to present it in its modern form was French philosopher, mathematician, and scientist René Descartes in his 1637 work La géométrie. The sign on the left , √, was invented in Germany in the 1500s. It was Descartes who added the horizontal bar over the top, called a vinculum, which also has a number of other mathematical uses, such as separating a column of numbers from their sum.
The shapes of our numerals had a long and complicated development before they reached their modern forms. As the author states:
It is impossible to reproduce here all the forms of our numerals which have been collected from sources antedating 1500 or 1510 AD. G.F. Hill, of the British Museum, has devoted a whole book of 125 pages to the early numerals of Europe alone. Yet even Hill feels constrained to remark: ‘What is now offered, in the shape of just 1,000 classified examples, is nothing more than a vindemiatio prima.’
This book contains entire tables dedicated to showing examples of specific numerals in odd and creative shapes from the days of manuscripts and the early years of printing. And while there is probably more standardization today, when the book was written in the 1920s there was still considerable variation to be found. “An outstanding fact is that during the past one thousand years no uniformity in the shapes of numerals has been reached. An American is sometimes puzzled by the shape of the number 5 written in France. A European traveler in Turkey would find that what in Europe is a 0 is in Turkey a 5.”
There is an interesting discussion of the notation of calculus. The author simply relates the facts as they are known regarding the first uses of the symbology, and does not discuss the epic and rather sordid story of the controversy between Leibniz and Newton over who should receive credit. Newton, his followers, and Newton himself writing as a proxy under other names, claimed to have invented calculus decades before Leibniz, but he used different symbology; both the integral symbol and the dx/dy notation derived from Leibniz. “The first appearance of dx in print was in an article which Leibniz contributed to the Acta eruditorum, in 1684. Therein occur the expressions “dw ad dx,” “dx and dy,” and also “dx:dy,” but not the form dx/dy.” Similarly, “In a manuscript dated October 29, 1675...Leibnitz introduces the symbol ∫L for omn. L, that is the sum of the l’s….It was the long form of the letter s, frequently used at the time of Leibniz.” He used it as an abbreviated form of summa, which he wrote with the long s, as “∫umma”
The book is full of these sorts of interesting observations, and following are some which I found illuminating:
- Hinck’s [conjecture about the meaning of a cuneiform tablet] was confirmed by the decipherment of tablets found at Senkereh, near Babylon, in 1854...One table was found to contain a table of square numbers from 12 to 602, a second one a table for cube numbers from 13 to 323. The tablets were probably written between 2300 and 1600 B.C.
- The earliest thorough and systematic application of a symbol for zero and the principle of position was made by the Maya of Central America, about the beginning of the Christian Era.
- [In one cuneiform tablet] “the first number of every odd line can be expressed by a fraction which has 12,960,000 as its numerator and the closing number of the corresponding even line as its denominator...The question arises, what is the meaning of all this? What in particular is the meaning of the number 12,960,000 (= 60^4 or 3600^2) which underlies all the mathematical texts here treated? [It was believed that] this ‘geometrical number’ (12,960,000) which Plato [in The Republic viii. 546B-D] calls the ‘lord of better and worse births,’ is the arithmetical expression of a great law controlling the Universe.”
-The process of multiplication or division known to the Egyptians as wshtp, “to incline the head.”
- The modern algebraic signs + and – came into use in Germany late in the fifteenth century. They are first found in manuscripts. The view that our + sign descended from one of the florescent forms for et in Latin manuscripts finds further support from works on paleography. J.L. Walthier enumerates one hundred and two different abbreviations found in Latin manuscripts for the word et; one of these, from a manuscript dated 1417, looks very much like the modern +.
- In Italy the symbols p̃ and m̃ served as convenient abbreviations for “plus” and “minus” at the end of the fifteenth century and during the sixteenth.
- While the Hindu-Arabic numerals became generally known in Europe about 1275, the Roman numerals continued to hold a commanding place. For example, the fourteenth-century banking-house of Peruzzi in Florence – Campagnia Peruzzi – did not use Arabic numerals in their account-books.
- The earliest authentic document unmistakably containing the numerals with the zero in India belongs to the year 876 A.D. The earliest Arabic manuscripts containing the numerals are of 874 and 888 A.D.
- The oldest definitely dated European manuscript known to contain the Hundu-Arabic numerals is the Codex Vigilanus, written in the Albeda Cloister in Span in 976 A.D.
- [Arabic] numerals are contained in a Vatican manuscript of 1077, on a Sicilian coin of 1138, in a Regensburg (Bavaria) chronicle of 1197. The earliest coins outside of Italy that are dated in Arabic numerals are as follows: Swiss 1424, Austrian 1484, French 1485, German 1489, Scotch 1539, English 1551.
- Recently the variations in form of our numerals have been summarized as follows: ‘The form of the numerals 1, 6, 8 and 9 has not varied much among the [medieval] Arabs nor the Christians of the Occident; the numerals of the Arabs of the Occident for 2, 3 and 5 have forms offering some analogy to ours (the 3 and 5 are originally reversed, as well among the Christians as among the Arabs of the Occident); but the form of 4 and that of 7 have greatly modified themselves.
- the Arabic numeral 5 appears upside down in some Spanish books and manuscripts as late as the eighteenth and nineteenth centuries.
- [Raphael Bombelli, 1526-1572] expressed square root by R.q., cube root by R.c., fourth root by RR.q., fifth root by R.p.r., sixth root by R.q.c., seventh root by R.s.r.
- Copernicus died in 1543. His De revolutionibus orbium coelestium (1566; 1st ed., 1543) shows that the exposition is devoid of algebraic symbols and is almost wholly rhetorical. We find a curious mixture of modes of expressing numbers: Roman numerals, Hindu-Arabic numerals, and numbers written out in words.
- A page in [German mathematician Christopher Clavius’] Algebra (Rome, 1608)...shows one of the very earliest uses of round parentheses to express aggregation of terms.
- [English mathematician William Oughtred (1574-1660)] was the first to use x [the Saint Andrew’s cross] as the sign of multiplication of two numbers, as a x b. The cross appears in Oughtred’s Clavis mathematicae in 1631 and, in the form of the letter X, in E. Wright’s edition of Napier’s Descriptio (1618).
- The dot was introduced as a symbol for multiplication by G.W. Leibniz. On July 29, 1698, he wrote in a letter to John Bernoulli: “I do not like x as a symbol for multiplication, as it is easily confounded with x;….often I simply relate two quantities by an interposed dot and indicate multiplication by ZC·LM.”
- In 1659 the Swiss Johan Heinrich Rahn published an algebra in which he introduced ÷ as a sign of division. Many writers before him had used ÷ as a symbol for subtraction.
- The sign ÷ as a symbol for division was adopted by John Wallis and other English writers. It came to be adopted regularly in Great Britain and the United States, but not on the European Continent.
- In the printed books before [Robert] Recorde, equality was usually expressed rhetorically by such words as aequales, aequantur, esgale, faciunt, ghelijck, or gleich, and sometimes by the abbreviated form aeq….Recorde’s =, after its debut in 1557, did not appear again in print until 1618, or sixty-one years later.
- In 1571, a German writer, Wilhelm Holzmann, better known under the name of Xylander, brought out an edition of Diophantus’ Arithmetica in which two parallel lines || were used for equality. He gives no clue to the origin of the symbol. Moritz Cantor suggests that perhaps the Greek word íσoi (“equal”) was abbreviated in the manuscript used by Xylander, by writing of only the two letters ii. Weight is given to this suggestion in a Parisian manuscript on Diophantus where a single i denoted equality.
- If Bartholomaeus Pitiscus of Heidelberg made use of the decimal point, he was probably the first to do so. Recent writers on the history of mathematics are divided on the question as to whether or not Pitiscus used the decimal point, the majority of them stating that he did use it.
- [The symbol] √ originated in Germany. Euler guessed that it was a deformed letter r, the first letter of radix. This opinion was held generally until recently. The more careful study of German manuscript algebras and the first printed algebras has convinced Germans that the old explanation is hardly tenable; they have accepted the a priori much less probably explanation of the evolution of the symbol from a dot.
- Engström...inclines to the view that the predominance of x over y and z [to represent an unknown] is due to typographical reasons, type for x being more plentiful because of the more frequent occurrence of the letters y and z in the French and Latin languages.
- An interesting feature in our survey is the vitality exhibited by the notation dy/dx for derivatives. Typographically not specially desirable, the symbol nevertheless commands at the present time a wider adoption than any of its many rivals...For integration ∫ has practically no rival. It easily adapted itself to the need of marking the limits of integration in definite integrals
- The modern notation for pi was introduced in 1706. It was that year that William Jones made himself noted, without being aware that he was doing anything noteworthy, through his designation of the ratio of the length of a circle to its diameter by the letter π. He took this step without ostentation. No lengthy introduction prepares the reader for the bringing upon the state of mathematical history this distinguished visitor from the field of Greek letters. It simply came, unheralded.
- The introduction of the letter e to represent the base of the natural system of logarithms is due to Euler. According to G. Engström, it occurs in a manuscript written in 1727 or 1728, but which was not published until 1862. Euler used e again in 1736 in his Mechanica.
- The notation n! Is used after Kramp in Gergonne’s Annales by J.B. Durrande in 1816, and by F. Sarrus in 1819. Durrande remarks, “There is ground for surprise that a notation so simple and consequently so useful has not yet been universally adopted.” It found wide adoption in Germany, where it is read “n-Fakultät.” Some texts in the English language suggest the reading “n-admiration” (the exclamation point being a note of admiration), but most texts prefer “factorial n” or “n-factorial.”
- There has been a real need in analysis for a convenient symbolism for “absolute value” of a given number, or “absolute number,” and the two vertical bars introduced in 1841 by Weierstrass, as in |z|, have met with wide adoption.
October 23, 2024
Definitely dense and thorough. Not to be read front-to-back, better to pick and choose the topics you’re interested in.
Displaying 1 - 2 of 2 reviews