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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
About the author
Florian Cajori
194 books7 followersFlorian Cajori was a Swiss-American historian of mathematics.
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Displaying 1 - 4 of 4 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.
May 20, 2026
Book Review: The Architecture of Scientific Language—A Review of Florian Cajori’s A History of Mathematical Notations
Rating: ★★★★☆ (4.0 out of 5 stars)
Full review available here: https://prairiefoxreads.blogspot.com/...
Bibliographic Information
Title: A History of Mathematical Notations
Author: Florian Cajori
Edition: Single-volume Dover edition (combining the original 1928 and 1929 two-volume set)
Publication Date: December 14, 2011 (First published January 1, 1974)
Publisher: Dover Publications
Page Count: 820 pages, Paperback
ISBN: 9780486677668 (ASIN: 0486677664)
Genre: Mathematics / History of Science / Nonfiction
Target Audience: Academic historians, mathematicians, cryptographers, and historians of science.
Disclaimer: I was provided a copy of this book from the publisher for review, but that has not affected the content or impartiality of this review.
I. Introduction: Purpose and Thesis
In my professional life managing government employees, as well as navigating the intersections of public health policy, epidemiological modeling, and intelligence community tradecraft, I am acutely aware of the power of a shared language. In both intelligence and science, the signal must be isolated from the noise; standardization of communication is not merely a bureaucratic preference, but a matter of operational survival.
Florian Cajori’s monumental A History of Mathematical Notations is fundamentally a study of how humanity achieved this standardization in the realm of mathematics. My central thesis in reviewing this classic work is that Cajori’s text operates as an evolutionary history of human cognition. It demonstrates that mathematical notation is not a static monolith handed down by antiquity, but a dynamic, highly contested linguistic ecosystem. Evaluated on thematic depth, historical rigor, and structural architecture, Cajori’s work remains the foundational text of its sub-discipline.
“A work that not only tells a story but reframes how we talk about its themes.”
II. Publication and Context
Originally published in a two-volume edition in 1928 and 1929, this text emerged during a historical moment of intense scientific standardization between the two World Wars. Florian Cajori (1859–1930) was a pioneering historian of mathematics, and his credentials as a professor at Tulane, Colorado College, and later UC Berkeley lend this work unassailable academic authority.
When placed in a comparative lens against his contemporaries, Cajori was a pragmatist. While others focused on the discoveries of mathematics, Cajori focused on the mechanics of transmission. In my own academic background in public health and science, we often study the spread of pathogens; Cajori studies the spread of symbols, tracking their vectors across European borders, their adoption, and their eventual mutation or extinction.
III. Summary of the Work
A History of Mathematical Notations is an encyclopedic, spoiler-free (by nature of its genre) tracing of mathematical symbols from their earliest origins to the early 20th century. The Dover edition conveniently binds both original volumes into one massive 820-page tome.
Part I covers elementary mathematics, tracing the numerals (Babylonian, Egyptian, Hindu-Arabic) and basic operational symbols (addition, subtraction, algebraic variables).
Part II addresses advanced mathematics, diving into the notations of calculus, trigonometry, and advanced geometry.
Cajori’s stated goal is to document the first appearance, the competition, the spread, and the survival or decline of mathematical symbols. He approaches this through exhaustive archival research, presenting a Darwinian survival-of-the-fittest narrative for algebraic and geometric shorthand.
IV. Analysis and Evaluation
Argument, Evidence, and Method
Cajori’s method of analysis relies heavily on historical framing and close readings of primary mathematical texts. His argument is built on an extraordinary foundation of empirical evidence, utilizing original manuscripts to pinpoint exact moments of symbolic inception. For instance, his meticulous tracing of the equals sign (==) introduced by Robert Recorde in 1557 demonstrates his reliance on primary documentation, highlighting how Recorde chose parallel lines because “noe 2 thynges can be moare equalle” (p. 164).
Themes and Ideas: The Darwinism of Symbols
A major motif is the fierce competition between competing notations. From an intelligence tradecraft perspective, evaluating how codes and ciphers become standard operational practice is fascinating. Cajori illustrates how nationalistic pride often hindered scientific progress—most notably in the bitter rivalry between the Newtonian dot notation and the Leibnizian \int∫ and dd notation for calculus (p. 612). Cajori masterfully demonstrates how the superior adaptability of Leibniz’s notation ultimately won the continent, illustrating that elegant and economical, it proves that restraint can illuminate complexity rather than obscure it.
Style, Craft, and Pace
The prose is unmistakably of its era—dense, formal, and unflinchingly academic. Yet, as a literary aficionado who finds solace in the quiet complexity of reading in a greenhouse surrounded by dormant orchids and sleeping cats, I found a distinct rhythm to his cataloging.
“Rich, precise prose that rewards patient attention and rewards fresh interpretation.”
It is not a swift read. The pacing is deliberate, requiring the reader to slow down and parse the visual architecture of the symbols printed on the page.
Representation and Limitations
If the book falters, it is in its representation and cultural scope. Written in 1929, the text exhibits the Eurocentric biases of its time. While Cajori touches upon Hindu-Arabic numerals, the vast contributions of non-Western mathematical traditions (such as those from the Islamic Golden Age, classical India, or pre-Columbian Americas) are heavily subordinated to the European narrative. Furthermore, the sheer volume of material (820 pages) can be overwhelmingly granular. Modern readers might find the lack of a unifying narrative arc challenging.
V. Contextual Analysis and Comparisons
In the realm of scientific history, this book dialogues directly with works like Carl Boyer’s A History of Mathematics. However, for readers looking for a more contemporary, narrative-driven alternative, Joseph Mazur’s Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers serves as an excellent companion. Where Mazur provides accessibility, Cajori provides exhaustive, unrelenting depth.
A History of Mathematical Notations pairs accessibility with ambition, inviting broader readership without compromising depth—though it firmly remains a reference text rather than a casual weekend read.
VI. Suitability and Audience Guidance
Reading Level & Audience: Graduate-level. Best suited for specialists, historians, and executives in scientific fields who appreciate the minutiae of structural history.
Content Warnings: None, aside from the potential for extreme eye strain for those unaccustomed to reading dense, early 20th-century mathematical typography.
Format Options: The Dover paperback is incredibly cost-effective, though its bulk (820 pages) makes it heavy. The layout includes numerous reproductions of original texts, which are historically fascinating but sometimes visually crowded.
As a middle-aged mother of four, finding uninterrupted time to digest a text of this magnitude requires discipline, but it is a book that invites rereading, revealing new layers with each visit.
VII. Conclusion and Verdict
Florian Cajori’s A History of Mathematical Notations remains the undisputed heavyweight champion of its specific niche. Its value lies in its comprehensive preservation of the structural evolution of scientific language.
Verdict: Highly recommended for academic libraries, historians of science, and professionals in quantitative or cryptographic fields. It is not recommended for the casual reader seeking a light narrative history of math.
Stakes and Implications:
Why does this book matter? In an era where algorithms dictate everything from public health policy to national security, understanding the foundation of the language that writes these algorithms is vital. Cajori leaves us with a profound respect for the cognitive leaps our ancestors made to communicate complex abstractions. It is an invitation to linger, reflect, and revisit—a testament to enduring relevance.
VIII. Supplementary Elements
Buyer’s Guide: What to Read Next
If this review piqued your interest but you are looking for varied approaches to the history of science and information, consider these companions:
For a more accessible narrative on math symbols: Enlightening Symbols by Joseph Mazur.
For the intelligence & tradecraft perspective on data/symbols: The Information: A History, a Theory, a Flood by James Gleick.
For public health/management structural thinking: The Ghost Map by Steven Johnson (exploring how data visualization and notation changed public health forever).
Discussion Prompts for Academic or Professional Settings
Management & Adoption: How does Cajori’s description of the “competition” between symbols reflect modern struggles in adopting enterprise-wide software or protocols in large government or corporate teams?
Tradecraft: In what ways does mathematical notation function similarly to cryptography, acting as both an illuminator of truth to the initiated and a barrier to the uninitiated?
Historical Bias: How might a modern historian rewrite Part I of this book to better include the global contributions to mathematical notation?
Rating: ★★★★ 4.0 / 5
- Prairie Fox 🦊📖
Rating: ★★★★☆ (4.0 out of 5 stars)
Full review available here: https://prairiefoxreads.blogspot.com/...
Bibliographic Information
Title: A History of Mathematical Notations
Author: Florian Cajori
Edition: Single-volume Dover edition (combining the original 1928 and 1929 two-volume set)
Publication Date: December 14, 2011 (First published January 1, 1974)
Publisher: Dover Publications
Page Count: 820 pages, Paperback
ISBN: 9780486677668 (ASIN: 0486677664)
Genre: Mathematics / History of Science / Nonfiction
Target Audience: Academic historians, mathematicians, cryptographers, and historians of science.
Disclaimer: I was provided a copy of this book from the publisher for review, but that has not affected the content or impartiality of this review.
I. Introduction: Purpose and Thesis
In my professional life managing government employees, as well as navigating the intersections of public health policy, epidemiological modeling, and intelligence community tradecraft, I am acutely aware of the power of a shared language. In both intelligence and science, the signal must be isolated from the noise; standardization of communication is not merely a bureaucratic preference, but a matter of operational survival.
Florian Cajori’s monumental A History of Mathematical Notations is fundamentally a study of how humanity achieved this standardization in the realm of mathematics. My central thesis in reviewing this classic work is that Cajori’s text operates as an evolutionary history of human cognition. It demonstrates that mathematical notation is not a static monolith handed down by antiquity, but a dynamic, highly contested linguistic ecosystem. Evaluated on thematic depth, historical rigor, and structural architecture, Cajori’s work remains the foundational text of its sub-discipline.
“A work that not only tells a story but reframes how we talk about its themes.”
II. Publication and Context
Originally published in a two-volume edition in 1928 and 1929, this text emerged during a historical moment of intense scientific standardization between the two World Wars. Florian Cajori (1859–1930) was a pioneering historian of mathematics, and his credentials as a professor at Tulane, Colorado College, and later UC Berkeley lend this work unassailable academic authority.
When placed in a comparative lens against his contemporaries, Cajori was a pragmatist. While others focused on the discoveries of mathematics, Cajori focused on the mechanics of transmission. In my own academic background in public health and science, we often study the spread of pathogens; Cajori studies the spread of symbols, tracking their vectors across European borders, their adoption, and their eventual mutation or extinction.
III. Summary of the Work
A History of Mathematical Notations is an encyclopedic, spoiler-free (by nature of its genre) tracing of mathematical symbols from their earliest origins to the early 20th century. The Dover edition conveniently binds both original volumes into one massive 820-page tome.
Part I covers elementary mathematics, tracing the numerals (Babylonian, Egyptian, Hindu-Arabic) and basic operational symbols (addition, subtraction, algebraic variables).
Part II addresses advanced mathematics, diving into the notations of calculus, trigonometry, and advanced geometry.
Cajori’s stated goal is to document the first appearance, the competition, the spread, and the survival or decline of mathematical symbols. He approaches this through exhaustive archival research, presenting a Darwinian survival-of-the-fittest narrative for algebraic and geometric shorthand.
IV. Analysis and Evaluation
Argument, Evidence, and Method
Cajori’s method of analysis relies heavily on historical framing and close readings of primary mathematical texts. His argument is built on an extraordinary foundation of empirical evidence, utilizing original manuscripts to pinpoint exact moments of symbolic inception. For instance, his meticulous tracing of the equals sign (==) introduced by Robert Recorde in 1557 demonstrates his reliance on primary documentation, highlighting how Recorde chose parallel lines because “noe 2 thynges can be moare equalle” (p. 164).
Themes and Ideas: The Darwinism of Symbols
A major motif is the fierce competition between competing notations. From an intelligence tradecraft perspective, evaluating how codes and ciphers become standard operational practice is fascinating. Cajori illustrates how nationalistic pride often hindered scientific progress—most notably in the bitter rivalry between the Newtonian dot notation and the Leibnizian \int∫ and dd notation for calculus (p. 612). Cajori masterfully demonstrates how the superior adaptability of Leibniz’s notation ultimately won the continent, illustrating that elegant and economical, it proves that restraint can illuminate complexity rather than obscure it.
Style, Craft, and Pace
The prose is unmistakably of its era—dense, formal, and unflinchingly academic. Yet, as a literary aficionado who finds solace in the quiet complexity of reading in a greenhouse surrounded by dormant orchids and sleeping cats, I found a distinct rhythm to his cataloging.
“Rich, precise prose that rewards patient attention and rewards fresh interpretation.”
It is not a swift read. The pacing is deliberate, requiring the reader to slow down and parse the visual architecture of the symbols printed on the page.
Representation and Limitations
If the book falters, it is in its representation and cultural scope. Written in 1929, the text exhibits the Eurocentric biases of its time. While Cajori touches upon Hindu-Arabic numerals, the vast contributions of non-Western mathematical traditions (such as those from the Islamic Golden Age, classical India, or pre-Columbian Americas) are heavily subordinated to the European narrative. Furthermore, the sheer volume of material (820 pages) can be overwhelmingly granular. Modern readers might find the lack of a unifying narrative arc challenging.
V. Contextual Analysis and Comparisons
In the realm of scientific history, this book dialogues directly with works like Carl Boyer’s A History of Mathematics. However, for readers looking for a more contemporary, narrative-driven alternative, Joseph Mazur’s Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers serves as an excellent companion. Where Mazur provides accessibility, Cajori provides exhaustive, unrelenting depth.
A History of Mathematical Notations pairs accessibility with ambition, inviting broader readership without compromising depth—though it firmly remains a reference text rather than a casual weekend read.
VI. Suitability and Audience Guidance
Reading Level & Audience: Graduate-level. Best suited for specialists, historians, and executives in scientific fields who appreciate the minutiae of structural history.
Content Warnings: None, aside from the potential for extreme eye strain for those unaccustomed to reading dense, early 20th-century mathematical typography.
Format Options: The Dover paperback is incredibly cost-effective, though its bulk (820 pages) makes it heavy. The layout includes numerous reproductions of original texts, which are historically fascinating but sometimes visually crowded.
As a middle-aged mother of four, finding uninterrupted time to digest a text of this magnitude requires discipline, but it is a book that invites rereading, revealing new layers with each visit.
VII. Conclusion and Verdict
Florian Cajori’s A History of Mathematical Notations remains the undisputed heavyweight champion of its specific niche. Its value lies in its comprehensive preservation of the structural evolution of scientific language.
Verdict: Highly recommended for academic libraries, historians of science, and professionals in quantitative or cryptographic fields. It is not recommended for the casual reader seeking a light narrative history of math.
Stakes and Implications:
Why does this book matter? In an era where algorithms dictate everything from public health policy to national security, understanding the foundation of the language that writes these algorithms is vital. Cajori leaves us with a profound respect for the cognitive leaps our ancestors made to communicate complex abstractions. It is an invitation to linger, reflect, and revisit—a testament to enduring relevance.
VIII. Supplementary Elements
Buyer’s Guide: What to Read Next
If this review piqued your interest but you are looking for varied approaches to the history of science and information, consider these companions:
For a more accessible narrative on math symbols: Enlightening Symbols by Joseph Mazur.
For the intelligence & tradecraft perspective on data/symbols: The Information: A History, a Theory, a Flood by James Gleick.
For public health/management structural thinking: The Ghost Map by Steven Johnson (exploring how data visualization and notation changed public health forever).
Discussion Prompts for Academic or Professional Settings
Management & Adoption: How does Cajori’s description of the “competition” between symbols reflect modern struggles in adopting enterprise-wide software or protocols in large government or corporate teams?
Tradecraft: In what ways does mathematical notation function similarly to cryptography, acting as both an illuminator of truth to the initiated and a barrier to the uninitiated?
Historical Bias: How might a modern historian rewrite Part I of this book to better include the global contributions to mathematical notation?
Rating: ★★★★ 4.0 / 5
- Prairie Fox 🦊📖
June 3, 2026
Reading a historical survey book written almost exactly a century ago is retro².
There's the obvious layer that's in the survey title: a deep excursion into, well, the history of mathematical notation. That by itself is nice, pleasant and instructive: for instance, I didn't know how Greeks wrote numbers, and neither did I know how long it took to settle with the notation in even basic school arithmetic. And it's impressive how the giants on whose shoulders we now stand achieved so much with tools so weak and notation so ugly. Makes me appreciate the tools we have now, since what previously took a mathematical genius is now obvious. Oh, and by the way, Leibniz almost inventing Gödel numbering 250 years before Gödel is a hell of a feat!
Then it's also pleasant in a weird way to trace the variations in the notation I've myself noticed, be it dx/dt vs dotted x for derivatives (and the story is deeper than "oh, physicists just like to use dotted symbols for time-derivative"), or why I myself sometimes write "sin^{-1}" and sometimes "arcsin", or some mild confusion I vaguely remember from my childhood and preschool years about why division is sometimes ":" and sometimes "/" and sometimes "÷". This book answers the implicit question I was carrying since then (and forgot I was carrying it) of why people couldn't just agree on one symbol?
And the book well shows why: because different people arrived at the same concepts, and then there were linguistic, geographic, and school-of-thought kinds of rivalry. Or even how cheap is it to print one symbol vs the other. Human factor, plain and simple.
But that's just the obvious, surface layer. A deeper layer is how the line of thought is continuous, really, and how the names I've only seen in textbooks were actually seen by people a century ago.
What I mean is, this book being written in 1927 makes it really easy to do a certain mental exercise: you see a reference to somebody from 1850? Just find some name from 1950's and try to correlate the specter of associations, the "remoteness", of the two. So, this survey references something from Kronecker (whom I've only seen in a textbook) from 1897? That's almost like referencing a Conor McBride's paper from 1997. Weierstrass from 1860's? Well, that's like, dunno, Hamming or von Neumann. All those names I've met in my university courses — they were actually alive once. That's not news, of course, but such a high concentration makes it felt.
And the university textbooks compress and permute the history. In an analysis book, Rolle's theorem may be a few paragraphs away from Weierstrass' ones, but Rolle is late XVII century, while Weierstrass is XIX. This survey unfolds these permutations and makes one really appreciate that math, well, did not evolve overnight.
Then there's a layer of what's missing or barely paid attention to. We know Babbage as that computer guy, but that is not mentioned at all. Babbage is just one of the myriad of names of mathematicians doing mostly continuous math. And "computer" itself is a person who computes, not a device. And an "algorithm" is something between a theory and notation rather than what we call an algorithm today.
The focus of the contemporary developments is also telling. Cantor's theory is a theory of aggregates — there are no sets. Whitehead and Russell are there, but ZF is not, really. Leibniz-Gödel numbering — well, it was four more years until Gödel's paper, and maybe a couple of decades until its importance is recognized. General relativity is mentioned only in passing, like "oh, by the way, there's this guy, Einstein, who's using tensors for his theory of relativity". And, speaking of applications, quantum mechanics isn't there at all.
And, in general, I'm into math foundations, so seeing the evolution of the logical foundations even if via notation lens strikes something.
And, while tangential, I can't help but notice how the First World War is referred to as the "Great War". Oh, little did they know!
There's the obvious layer that's in the survey title: a deep excursion into, well, the history of mathematical notation. That by itself is nice, pleasant and instructive: for instance, I didn't know how Greeks wrote numbers, and neither did I know how long it took to settle with the notation in even basic school arithmetic. And it's impressive how the giants on whose shoulders we now stand achieved so much with tools so weak and notation so ugly. Makes me appreciate the tools we have now, since what previously took a mathematical genius is now obvious. Oh, and by the way, Leibniz almost inventing Gödel numbering 250 years before Gödel is a hell of a feat!
Then it's also pleasant in a weird way to trace the variations in the notation I've myself noticed, be it dx/dt vs dotted x for derivatives (and the story is deeper than "oh, physicists just like to use dotted symbols for time-derivative"), or why I myself sometimes write "sin^{-1}" and sometimes "arcsin", or some mild confusion I vaguely remember from my childhood and preschool years about why division is sometimes ":" and sometimes "/" and sometimes "÷". This book answers the implicit question I was carrying since then (and forgot I was carrying it) of why people couldn't just agree on one symbol?
And the book well shows why: because different people arrived at the same concepts, and then there were linguistic, geographic, and school-of-thought kinds of rivalry. Or even how cheap is it to print one symbol vs the other. Human factor, plain and simple.
But that's just the obvious, surface layer. A deeper layer is how the line of thought is continuous, really, and how the names I've only seen in textbooks were actually seen by people a century ago.
What I mean is, this book being written in 1927 makes it really easy to do a certain mental exercise: you see a reference to somebody from 1850? Just find some name from 1950's and try to correlate the specter of associations, the "remoteness", of the two. So, this survey references something from Kronecker (whom I've only seen in a textbook) from 1897? That's almost like referencing a Conor McBride's paper from 1997. Weierstrass from 1860's? Well, that's like, dunno, Hamming or von Neumann. All those names I've met in my university courses — they were actually alive once. That's not news, of course, but such a high concentration makes it felt.
And the university textbooks compress and permute the history. In an analysis book, Rolle's theorem may be a few paragraphs away from Weierstrass' ones, but Rolle is late XVII century, while Weierstrass is XIX. This survey unfolds these permutations and makes one really appreciate that math, well, did not evolve overnight.
Then there's a layer of what's missing or barely paid attention to. We know Babbage as that computer guy, but that is not mentioned at all. Babbage is just one of the myriad of names of mathematicians doing mostly continuous math. And "computer" itself is a person who computes, not a device. And an "algorithm" is something between a theory and notation rather than what we call an algorithm today.
The focus of the contemporary developments is also telling. Cantor's theory is a theory of aggregates — there are no sets. Whitehead and Russell are there, but ZF is not, really. Leibniz-Gödel numbering — well, it was four more years until Gödel's paper, and maybe a couple of decades until its importance is recognized. General relativity is mentioned only in passing, like "oh, by the way, there's this guy, Einstein, who's using tensors for his theory of relativity". And, speaking of applications, quantum mechanics isn't there at all.
And, in general, I'm into math foundations, so seeing the evolution of the logical foundations even if via notation lens strikes something.
And, while tangential, I can't help but notice how the First World War is referred to as the "Great War". Oh, little did they know!
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.
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