Mathematical Words: Origins and Sources

Notes for Earliest Known Uses of Some of the Words of Mathematics


The Earliest Known Uses of Some of the Words of Mathematics pages provide information on the origins of many of the English mathematical words in use today. The story of a particular word may be very complicated. There has been an English word ALGEBRA for nearly 500 years and the word can be traced back another 700 years to an expression in Arabic. The word has been around for a long time, travelled through several languages and experienced great shifts in meaning. There are patterns in the history of particular words and there are also procedures for uncovering that history. These notes describe in general terms the origins of the modern vocabulary of mathematics and the sources of information on which the Earliest Uses entries are based.


John Aldrich University of Southampton, Southampton, UK. (home) June 2009. Latest changes April 2014. I am grateful to Lianghuo Fan, Ben Fortson and Jeff Miller for suggestions.



Words and their Origins

Sources of information

Word-formation today

Works on words

Words:  historical overview

Other resources

Mathematics in English

Personal communications

Provenance:    Old English

Researching: bootstrap






General references






Words and their Origins


In the Origins section of these notes


·        There is a general discussion of word formation followed by an example—the creation of a modern English term.


·        There is an overview of the origins of the modern English lexicon and a sketch of the development of mathematical writing in English.


·        The contributions of Old English and of Latin, French, Greek and German to the English mathematical lexicon are described in more detail.


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Word-formation in English today



Do’s and Don’ts

Creating terms

Bootstrap as an example

Bootstrap story

Bootstrap in translation




All forms of human activity tend to give rise to a specialised vocabulary which participants use in conjunction with the general vocabulary of whatever language they happen to speak. The special language of mathematics contains technical terms and symbols—for the latter see Earliest Uses of Various Mathematical Symbols—and there is also an unofficial language of slang expressions. 


Mathematics does not have millions of technical terms like chemistry or biology but even so the number of terms in use is very great. General purpose mathematical dictionaries—even large ones—usually cover only a small fraction of them. The ISI Multilingual Glossary of Statistical Terms has around 5000 entries and it does not include all the statistical terms in use. (There are around 500 statistical terms in Earliest Uses.)


In some sciences there are central bodies responsible for creating or for approving terms, e.g. the IUPAC, as in some countries there are bodies responsible for the condition of the language, e.g. the Académie française. In mathematics word-formation is an individual initiative and it is up to other individuals whether they adopt the new word. Getting a new term established can be quite a process. In E Pluribus Boojum the physicist David Mermin recalls the 5 years he spent on the term boojum.


Word-formation may be open to all but only a few are successful. Much of our language of geometry comes from a single known source, Euclid's Elements. In the C19 J. J. Sylvester wrote, “Perhaps I may without immodesty lay claim to the appellation of Mathematical Adam, as I believe that I have given more names (passed into general circulation) of the creatures of mathematical reason than all the other mathematicians of the age combined.” (quoted in Parshall). Karl Pearson, Ronald Fisher and John Tukey were responsible for many statistical terms, as a search for their names on Earliest Uses will show. John Conway is another word-smith with a taste for the flamboyant.  


Earliest Uses records the births of living terms but terms also die. Old literature and old dictionaries are full of words no longer used. Some dead words are remembered because they once mattered—a good example is FLUXION—or for their association with great names—see the senses of MODULUS associated with De Moivre and Legendre. Some are remembered for their curiosity value like RADIOGRAM, an example of a still-birth—a word announced but never subsequently used. Conway’s UNLESSS may be another example.



Do’s but mostly don’ts in word-formation

Word-formation in mathematics appears not to have been studied systematically either by mathematicians or linguists. Words do not make themselves and the process of word-formation is often quite unpredictable. There may be a gap for a word and the gap stays unfilled: the word RADIUS filled a gap but mathematicians managed for centuries without a word additional to DIAMETER. Once it is decided to fill a gap there is usually a choice of term: several acceptable words may suggest themselves and it may seem a matter of chance which is adopted; RANDOM VARIABLE was a case where chance did decide the issue. Unacceptable words will also suggest themselves. Once the choice is made there is a powerful tendency to stick with it—even if the term is unacceptable!


Guides on how to write mathematics sometimes have advice on how words should be made—more often how they should not be made. The advice—sometimes contradictory—usually reflects dissatisfaction with the existing language. 


·        Do not introduce new terms is the first principle of Halmos (p. 40)

(1) Avoid technical terms, and especially the creation of new ones, wherever possible. (2) Think hard about the new ones you must create; consult Roget; and make them as appropriate as possible.


One of Sylvester’s friends told him, “terminology … is your strength as well as your weakness. You have too much of propensity to create new words. It would be well for you to forget about Greek.” (quoted in Parshall). Over appropriateness opinions often differ. Some now familiar terms originally seemed totally wrong: see the objections to the terms MATRIX and STATISTIC.


·       Do not try to undo past mistakes warns Boas (p. 728)

If you think you can invent better words than those that are currently in use, you are undoubtedly right. However, you are rather unlikely to get many people except your own students to accept your terminology; and it is unkind to make it hard for your students to understand anyone else's writing.


There is a saying “possession is nine tenths of law.”


·       You must try to undo past mistakes says Steenrod (p. 6), if you can

An author of a research monograph that is first in its area has the opportunity and the obligation to replace poor by good terminology. … The name that a research worker attaches to a new concept is usually chosen before the scope and thrust of the concept is fully understood, so his choice may be an unhappy one.

Naturally opinions differ over whether the moment has passed.


·       Boas (p. 728) complains about ambiguity (more politely, polysemy)


if you must create new words, you can at least take the trouble to verify that they are not already in use with different meanings. It has not helped communication that DISTRIBUTION now means different things in probability and in functional analysis.

There are so many different mathematical communities that polysemy is probably the rule not the exception: see CHARACTERISTIC FUNCTION, MODULUS, NORM, etc.


·       The existence of different communities and different histories also explains the proliferation of synonyms, different names for the same thing: see e.g. MODULUS and ABSOLUTE VALUE, CHARACTERISTIC FUNCTION and INDICATOR FUNCTION, and the alternatives to the EIGEN terms.



Creating mathematical terms


Given that an existing technical term cannot be adapted or a collection of existing terms cannot be combined into a new phrase, how is a new term to be created?

Mathematical words are created in much the same way and undergo the same processes of change as words in ‘ordinary’ language. Algeo provides a taxonomy of word making. In Words, Words, Words Crystal traces the word “nice” back through two thousand years of evolution. Over the same period the word ANALYSIS has undergone similarly dramatic changes.


Algeo & Pyles ch. 10-12 describe the processes by which words are formed in ‘ordinary’ English. The same processes are involved in forming ordinary words in all languages—see Durkin—and in forming technical terms:

  • Almost all words are formed from existing words, i.e. the new word has an etymon. An instance (unique?) of a mathematical word without etyma is GOOGOL. The OED gives a few other words without a root: some, like the science fiction word dalek, are intended to sound alien. That’s not the usual intention with mathematical terms—whatever students may think,


  • Some mathematical words are existing non-technical words whose meaning has undergone some kind of metaphorical extension. See e.g. the entries CHAIN, WINDOW and FUZZY and the discussion of bootstrap below. Hersh discusses the relationship between mathematical language and ordinary languages. This is an issue that particularly worries teachers of mathematics.


  • Other terms come from ordinary, non-technical language by specialising the meaning of an existing word: see e.g. ESTIMATION and SET. In AMENABLE there is an element of punning.


  • The stock of ordinary non-technical words does not have to be limited to the ‘home’ language and these last two processes can be applied to words from foreign languages especially when these languages enjoy high prestige. However borrowing whole words is much less common than borrowing roots.


  • Many terms are constructed by modifying or combining existing words or roots of existing words, e.g., IDEMPOTENT, EIGENSTATE and KURTOSIS. The elements of these particular technical English words are non-mathematical words in Latin/English, German/English and classical Greek respectively. When these particular words were formed potent and state were fully domesticated in English but they had come into the language at an earlier date.



  • There are more artful ways of making new terms from old, such as the sly back-formation of statistic out of STATISTICS or the abbreviation of binary digit to BIT and directed graph to DIGRAPH.




  • Many words are borrowed from other languages, entering with the idea the word expresses.  See below.


One peculiarity of this specialised language is the number of technical terms that are borrowed from the general language with some shading or alteration in meaning. In their Introduction to Topology T. W. Gamelin & R. E. Greene (1983, p. 80) state the theorem, “A compact Hausdorff space is normal.” All the words—apart from Hausdorff—are everyday words found in the smallest English dictionary, yet only two words have their everyday meaning, “a” and “is”. The rest are technical terms: the everyday words COMPACT  and NORMAL have been given mathematical meanings, while HAUSDORFF SPACE is an expression concocted by mathematicians.


Evidently there is little system in mathematical naming. The entry on PROBABILITY DISTRIBUTIONS, NAMES FOR shows how several principles can be at work in the same field. Schwartzman shows how 1500 or so mathematical words have been formed out of components from ordinary language. Occasionally there is movement in the other direction, when a mathematical term is used figuratively: ECCENTRIC, PARALLEL and cipher (see ZERO) are instances and so is the fashionable pre-fix cyber- from  CYBERNETICS. See also Charles Wells on Names.


Mathematicians use technical terms like those discussed above when they are doing mathematics and they also use ordinary language words in their ordinary senses. In addition there are terms used in discussing mathematics such as HARD AND SOFT MATHEMATICS, HANDWAVING and PATHOLOGICAL which are part of the occupational slang of mathematics.



  • Karen Hunger Parshall James Joseph Sylvester, Johns Hopkins. 2006. Amazon. (See index entry “Sylvester, James Joseph, neologism and.”
  • Reuben Hersh Math Lingo vs. Plain English: Double Entendre, American Mathematical Monthly, (1997), 48-51. Excerpt and Follow-up.
  • Roget’s Thesaurus, the Wikipedia artcle on the famous reference book.
  • Norman E. Steenrod, Paul R. Halmos, Menahem M. Schiffer, and Jean A. Dieudonné. How to Write Mathematics. American Mathematical Society, Providence, RI, USA, 1973. Amazon
  • P. R. Halmos  How to write mathematics, L’Enseignement Mathématique, Vol.16 (1970).
  • Donald E. Knuth, Tracy Larrabee, and Paul M. Roberts Mathematical Writing.
  • R. P. Boas Can We Make Mathematics Intelligible? American Mathematical Monthly, (1981), 727-731.
  • Nicholas J. Higham Handbook of Writing for the Mathematical Sciences, SIAM, 1998 Second Edition. homepage. The large bibliography is available on-line.
  • Bernard Ycart How to Write Mathematics has useful links.
  • Charles Wells The Language of Math.
  • Wikipedia Mathematical jargon discusses some of the occupational slang used by mathematicians.

The following do not treat mathematical words yet they offer insights into the way mathematical words are formed and how they change. 

  • Richard Nordquist Introduction to Etymology: Word Histories.
  • David Crystal Words, Words, Words, Oxford University Press. 2006. Amazon
  • Ronald Carter Vocabulary, second edition Routledge 1998.   Amazon
  • John Algeo The Taxonomy of Word Making, Word, 29, (1978), 122-31.
  • John Algeo Fifty Years among New Words, Cambridge 1993. Amazon.


The use of words in mathematics is taken very seriously by those involved in teaching mathematics. There are societies and the journals include the British Society for Research into Learning Mathematics and Journal of Mathematical Behavior.


































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An example: bootstrap


Bootstrap introduction

Bootstrap story

Bootstrap into other languages


As an example of word-formation consider a modern term coined in English, BOOTSTRAP. For this term information is plentiful and the origins story is straightforward. Sources of information are described below.


BOOTSTRAP is a recent (1979) but already well-established term in statistics. The word had been in ordinary non-mathematical English since the C18 meaning “a strap sewn on to a boot to help in pulling it on” (OED). The technical statistical meaning (a method for making inferences based on simulation from a process derived from the sample itself) draws on the expression “to pull oneself up by one’s bootstraps.” If the idiom had been, “to pull oneself up by one’s (neck) tie,” we might have the term tie.    


Bootstrap is an unusually imaginative term but it is carefully chosen term and truer to Halmos’s advice, “think hard,” than most. It would also satisfy Boas for the term was not already in use with a different meaning. The expression to BOOT a computer probably has the same root but the meanings are too dissimilar to cause confusion.


Bootstrap illustrates some points of general interest, including:

  • It is a new term for a new technique and as such is a typical addition to the lexicon. It is easier to live with an inadequate old term than overturn an established usage.
  • The suggestion of flippancy, or, at least, informality makes the term very much of its time; cf. the contemporary coinages FUZZY and HAIRY BALL THEOREM. In earlier days dignity was expected: Karl Pearson’s (1857-1936) speciality was the Greek-based neologism like HISTOGRAM and HETEROSCEDASTICITY; Ronald Fisher (1890-1962) liked to take existing English words (usually with Latin roots) and stretch their meaning, such as VARIANCE and SUFFICIENT.  Bootstrap is more in the style of John Tukey (1915-2000) who liked to make technical terms from everyday words, e.g. JACKKNIFE and WINDOW.
  • Bootstrap arrived as a singleton but there are also mass creations. A family of new words may be needed, or an old terminology may need reform. Pearson’s term KURTOSIS was part of a successful mass creation, while RADIOGRAM belonged to an abortive one.

We cannot usually hear the voice of the namers of the remote past but we can of more recent ones. Some have even described the process of finding the right word, especially when their choice is not an obvious one: see the entries FRACTAL, FUZZY, GYROID, PARAMETER and THRACKLE.


Naming is not just the business of the original namer. Others have to adopt the term and keep it going. Every time a word is used there is an opportunity to re-name it. There is a powerful tendency to respect the innovation—provided the concept is useful—and thus Sylvester’s term MATRIX survived the criticism of Lewis Carroll who thought block would be better. Yet terms do get replaced.  When a term is translated into another language there is an opportunity to deviate more or less from the original author’s intentions: EXHAUSTIVE ESTIMATION an English translation of a French translation of an English original presented two opportunities for creative translation.



Bootstrap story



The origins story of bootstrap is exceptionally tidy.


  • The first use can be dated to a specific publication by a specific author which announced the idea and with it the term. 



More often the origins story is complex and/or only partially known


  •  Bootstrap was not a new word but the new meaning was clearly announced. Sometimes terms slip into use or existing terms slip into new meanings.


  • The word bootstrap (in this new sense) arrived with the idea which was ‘fully formed.’  Sometimes the creator author tries out a number of words before settling on the ‘right’ one; see e.g. SET and MAXIMUM LIKELIHOOD. It is often not clear whether the tryouts have exactly the same meaning as the final term.


  • The same idea can arrive in different minds and be expressed in different words. See the pairs DIFFERENTIAL and FLUXION or SEMI-INVARIANT and CUMULANT. To what extent these terms represent the same idea is another difficult matter. There were no parallel creations of bootstrap.


  • There can also simultaneous coinings or re-coinings, where the same term is applied to the same thing by more than one person independently: see e.g. the term NORMAL. For ‘headline’ words the phenomenon is unusual but with derived terms it is probably the norm.


  • With old words our first sight of them is not necessarily a “first use.” What we see depends on the thoroughness of the search and ultimately on the documents that survive. We first see naught and cipher (see under ZERO) when a non-mathematical author makes a (non-mathematical) point by means of a mathematical expression familiar to his readers—a use that cannot have been a first use. Much of our knowledge of the vocabulary of Middle English (C14) comes from the writing of Geoffrey Chaucer: see e.g. ARGUMENT, DEGREE and EPICYCLE. Sometimes Chaucer was writing science but more often he was writing about people doing science or mathematics and it is not clear whether he was using their words or making up his own.


In its short history bootstrap has not been subject to significant semantic drift.


  • Some words like TRIANGLE have preserved their original meaning across centuries.



Of course further investigation of bootstrap might uncover earlier uses in pre-publication writing and in recorded conversations and possibly even show that other terms were tried and discarded.


As well as going back to before the date of publication it is possible to go forwards and notice that the term is not used in exactly the same way now as it was in 1979. A historian writing in 2020 (or even now) would be interested in both the backward and forward stories.  




Bootstrap into other languages

As translation into English has been so important building up the English mathematical lexicon (see below) it is interesting to see how bootstrap has gone from English into other languages.


It is a very tricky word and an obvious expedient—in the short run at least—is to take the word or the phrase bootstrap method into the language: ALGEBRA is a remnant of such a phrase. English has often operated in this way, borrowing wholesale from Latin, French and Greek.


The translations of bootstrap method given in the ISI Multilingual Glossary illustrate a variety of more permanent arrangements.

· Literally translate both words: the Afrikaans skoenlusmetode. OPEN SET came into English after this fashion.

· Adopt bootstrap as a loan word and translate method: the Italian metode bootstrap. EIGENVALUE is an English word formed in this way.

·       Convey something of the nature of the method using ‘local’ words: the Spanish método autosuficiente.

·       Convey something of the allusion to impossible self-locomotion in the expression “to pull oneself up by one’s bootstraps”: the French la méthode de Cyrano (Accromath Été-Automne 2008 p. 5) or the Chinese . In the translations the essence of the original joke—you can't pull yourself up by your own bootstraps—is retained but the accidentals are lost.

·       A language may support more than one translation. French also has méthode de bootstrap.

·       No language appears to have taken the EPONYMY option and called the method after its inventor, méthode d’Efron, say. In this particular case one of the main disadvantages of eponymy—the conflict between different local traditions over who should be honoured—is avoided: for such conflict see the entries PASCAL’S TRIANGLE GREGORY'S SERIES or CAUCHY-SCHWARZ INEQUALITY.


English writers of the Late Middle Ages and Renaissance had the same options when they translated Latin words. See Graddol (176).


Because of the idiom in which it is embedded, bootstrap is an especially tricky word but translation is always tricky.


  • Words taken from ordinary language can rarely be translated automatically. The English word SET and its French and German ‘equivalents’ all have several other meanings and the bundles of meanings vary from language to language; the words are not generally interchangeable.   


  • It is different with the specially constructed single-meaning technical term, such as ENTROPY. The translations show the algorithm in action: take the Greek root and apply the local affix.


Of course the translator is just one more person who makes choices: choices were made by the person who originally coined the term and the people who sanctioned its use by adopting it. Occasionally something is gained in translation: the most famous example is the mis-translation WITCH OF AGNESI. For another famous mistranslation see SINE.


Words have come into English as singletons and also when an entire work—a book or an article—has been translated. Billingsley’s C16 translation from the Greek of Euclid below may have brought more words into English mathematics than any other single work. It is mentioned in over 400 entries in the OED although not all of these involve mathematical words and not all are first uses. Some other translations appear repeatedly in Earliest Uses. They make a curious collection: De Proprietatibus Rerum by Bartholomæus Anglicus a C13 Latin encyclopaedia translated by John Trevisa at the end of the C14 brought many Latin terms into English—see ANGLE, DIAMETER, POINT and TRIANGLE. Lacroix’s Traité du calcul différentiel et integral written at the end of the C18 and translated in the early C19 brought many Continental calculus terms into English—see DIFFERENTIAL CALCULUS, VARIABLE and POLAR COORDINATES. Many entire works have been translated into English but most often they have made their impact on the language before the translation has appeared; a C20 example is Bourbaki’s Eléments de mathématique and the terms INJECTION, SURJECTION & BIJECTION.





The problems of finding the right word for topological concepts in different languages is discussed by 

  • B. Barton, Lichtenberk, F. & Reilly, I. The Language of Topology: A Turkish Case Study, Applied General Topology, 6(2), (2005), 107-117. pdf

Translation is of such cultural and commercial significance that it has given rise to an academic subject, Translation Studies. The following books give an idea of the range of the subject, although neither discusses mathematics. The textbook by Baker discusses the choice of word as “equivalence at the word level.”

  • Daniel Weissbort, Astradur Eysteinsson Translation - Theory and Practice. A Historical Reader. Oxford University Press. 2006. Amazon.
  • Mona Baker In Other Words: A Coursebook on Translation, Routledge. 1992. Amazon


Modern translators of old texts face different and very difficult problems. Three hundred years ago when Bernoulli wrote his Ars Conjectandi the (i.e. our) language of PROBABILITY was in its early stages of formation. Sylla prefaces her translation by describing the problem of choosing the right words.

  • Edith Dudley Sylla (Translator) Jacob Bernoulli The Art of Conjecturing, together with Letter to a Friend on Sets in Court Tennis, Johns Hopkins University Press. 2006. Amazon.




English Mathematical words


English as a language of mathematics presents a paradox.


  • Today more mathematics is written in English than in any other language.


  • Yet the most common English mathematical terms are adaptations of terms coined in other languages. For examples see the table below.


The explanation is that


  • Judged by the long history of mathematics, the emergence of English in the role of the international mathematical language is very recent.


  • Until recently native English speakers constituted a peripheral group in the mathematics of Western Europe with the centre(s) elsewhere. Western Europe itself has been a significant centre for only about 800 years.


The historical development of a specialised branch of language, such as mathematical language, belongs to the history of mathematics and to the history of language. There are many works on these subjects but very few on their intersection. For the history of mathematics I refer to Katz; for the history of English I refer to Algeo & Pyles, Crystal and Graddol. The Wikipedia: History of the English language is a useful summary.


Works on the history of mathematical words in English are described below. The history of mathematics in the English-speaking lands is relevant and so is the history of mathematics education there. There appear to be no specialised studies of the former, for the latter I refer to Howson. The MacTutor series on the history of mathematical education is also useful.  


The following table gives some information on the origins of a few of the words in use today.


  • The table gives the date of the first appearance in English of the terms—with something like their modern meaning. For many terms the dating is debateable. The dates given are very rough: 1950 = “around 1950”. The subject classifications too are only a rough guide and clearly do not cover the whole of mathematics. The table should not be taken too literally.


  • The table shows the language from which the term came into English and the date of the original coining when it falls in an earlier period.


A table like this would work for other Western European languages but for Chinese, say, a complete reconstruction would be required.



When and whence for some English mathematical words

























Latin 1100














Latin 1100

Greek 300BC




Latin 1100





Latin 1100

Greek 300BC









Greek 300BC





Latin 1100

Arabic  800














Greek 300BC



Latin  1100 Arabic 800

Sanskrit 400




Latin  1100

Arabic 800



Greek 300 BC



Greek 300BC



Latin  1100 Arabic 800

Sanskrit 500












Latin 1200



Latin 1600














French  1637


























 Latin ?

Greek 300 BC


































 French 1810








Latin 1800










French 1815












French 1800























 French 1822



























French 1859



























German 1900







German 1900



German 1890

















French 1892



German 1907



German 1914

























































From this table of English words it is striking


  • how late many of the terms were in appearing


  • how many of them originated in other languages


  • how long it took for words to be translated


  • how over the centuries this interval diminished


  • how recently more words originated in English.


The colour coding indicates the significant other language of the age.


Simplifying heroically


·        From 1300 to the late C18 Latin was the important language of Western European mathematics. In the beginning the main centre was in Italy but later the action moved north and west. See Katz Parts II and III.


  • By 1700 publication in the local vernacular is becoming common. Behind the C17 term EXPECTATION is a story of a Dutch text first going into Latin. The story would be repeated for other ‘minor’ languages, including English and German. In 1905 the Danish mathematician Jensen described CONVEX FUNCTIONS and JENSEN'S INEQUALITY in Danish and then in 1906 in French.


·        From the late C18 to the late C19 French rivalled Latin. In the beginning most of the French literature was produced in France but later French was adopted as an international language. See Katz Part IV.


·        From the late C19 to the Second World War German rivalled French—both because so much mathematics was done in Germany and because German was adopted as an international language. However when Poland was reconstuted after the First World War Polish mathematicians chose French as their international language: see the journal Fundamenta Mathematicae. Latin melted away. See Katz Part IV.


·        Since the Second World War English has been the most important single language. There is a large body of native English speakers doing mathematics and English is more dominant internationally than French and German were in the past. It is less universal than Latin was in the Middle Ages but the universe of Latin was limited to Western Europe. See Katz Part IV. 



The Latin, French, German and English have been at different times international or (in a smaller world) world languages. They figure with other languages in Ostler’s history of 3000 years of world languages.


To Contents  To English mathematical words




Mathematics in English: behind the table  


Before 1300


1700 to the present


Before 1300—off the table


In 1300 three languages were in use in England—French, Latin and English (in the form of Early English). There was not much demand for English mathematical words.


  • The known story of English begins around 400 AD when parts of England were occupied by settlers from the regions that are now northwest Germany and the Northern Netherlands.


·        Old English or Anglo-Saxon was a diverse group of dialects, reflecting the varied origins of these invaders.


·        The conversion to Christianity around 700 made Latin the language of religion.


·        The Christian missionaries introduced the Latin alphabet (plus a few additional letters, such as þ (thorn) that have now disappeared) and a literary tradition developed with Old English works written in the new alphabet.


·        Viking settlers whose language was Old Norse (Wikipedia Viking) English also left their mark on the language.


·        Norman French became the language of government after the conquest of 1066.


  • The first named native English speakers in the annals of mathematics appear in the C12—writing in Latin, see below.


  • Mathematics was taught in the universities which had appeared in the C13. Across Europe Latin was the language of the universities.


For this period see the MacTutor pages The teaching of mathematics in The Dark Ages and The teaching of mathematics in Britain in the Mediaeval Ages.





By the middle of the C14 French was no longer an official language in England but it remained important because ideas from the rest of the world generally came through France. By the C14 English was the language of lowland Scotland. With the Reformation of the C16 Latin ceased to be the language of religion. Scholars wrote in Latin but a market developed for mathematical works in English—especially after the introduction of printing. Many words were borrowed from Greek and Latin. Towards the end of the period a brilliant group of mathematicians appeared of whom the best known is Newton.


  • By the early C14 manuals of arithmetic written in English were appearing. Surviving documents (manuscripts) are scarce: one of these, The crafte of nombrynge, is described in the entry ADDITION; see also DIGIT and DIVISION.


  • The language of the C14 is called Middle English: see Crystal. See the entries on ANGLE and TRIANGLE for samples of English text from around 1400. There are some letters no longer used, e.g. þ (thorn), and the spelling is somewhat different but with some imagination the texts can be understood.


  • Following the introduction of printing (in the late C15) there was a spectacular expansion in mathematical works in English. The C16 author Robert Recorde favoured the creation of an English mathematical vocabulary but the new English words of the C16 were mainly classical loans like HEXAGON or EQUILATERAL; see Howson and Greek below. Recorde’s writings were the first English writings to have an impact abroad; he is remembered for a symbol—the equals sign (see Signs of Relation and (Katz 355).


·        Works of high science written for publication were still written in Latin. The most famous, Newton’s Philosophiae Naturalis Principia Mathematica, appeared in 1687; an English translation followed in 1729. The scholars wrote in Latin and so the first English use of a term was often due, not to a scholar, but to a ‘tradesman’—see the entries SINE and COSINE. However in the late C17 English mathematicians were writing to each other in English and publishing translations of their Latin works: for instances, see the entries SERIES and COMBINATION.


See the MacTutor pages The teaching of mathematics in the Renaissance and The teaching of mathematics in Britain in the Seventeenth Century. 


1700 to the present


In this period—for the first time—important original work appears in English. Mathematics in English is not the same as mathematics done by the English for Scottish and Irish mathematicians also wrote in English and the British Empire took English all over the world. In the C20 the centre of gravity of mathematics in English shifted to the United States of America. In the early part of this period the most important mathematical work in the European tradition was being done on the Continent of Europe. Before the C20 the only mathematicians who wrote in English lived in English-speaking countries. Today English is used by anybody anywhere.


  • Since the mid-C18 Anglophone mathematicians have published their main scientific work in English. However the period 1750-1850 was not noted for original work in English.


  • In the C19 the reputation of mathematics in English was founded on applied mathematics/mathematical physics. The leading applied mathematicians were not English but Scottish or Irish: for examples of their activity see the entries NABLA STOKES’ THEOREM and VECTOR. Benjamin Peirce’s term IDEMPOTENT was an early American contribution to the language of pure mathematics.


  • In the period 1750-1950 most of the new terms in pure mathematics were created in French and later in German. In the early C19 English writers took up calculus in the Continental manner; see the entry DIFFERENTIAL CALCULUS, DERIVATIVE. At the end of the century writers of the first English books on COMPLEX ANALYSIS or GROUP THEORY had to invent an English vocabulary. There is an appendix in Burnside’s Theory of Groups of Finite Order (1897) here showing how his terms correspond to the French and German originals and Harkness & Morley’s A Treatise on the Theory of Functions (1893) has a multi-lingual glossary (p. 501).


  • As the table shows, as we approach the present day, the time elapsing between the original coinage and the English translation gets shorter and the proportion of terms originally coined in English grows.


  • Modern history is reflected in three of the statistical terms coined in English “around 1950.” (EMPIRICAL) BAYES and ADMISSIBLE were coined in the United States, the first by a native speaker of English and the second by a refugee from the German-speaking world. AUTOREGRESSION was coined in Sweden by a Swede writing in English and it illustrates the rise of English as an international language.


I have been referring to English mathematical words without questioning the notion of Englishness.  Linguists are familiar with the varieties of English around the world and their differences in vocabulary, spelling, pronunciation, etc.; see ch. 9 of Algeo and Pyles. Differences in the mathematical languages of the varieties of English seem trivial.


American English and British English are the two most important national forms and Earliest Uses entries indicate some differences. Billion once had a different meaning in the two varieties of English (1012 in British English and 109 in American English) but now British English generally conforms to American; see MILLION. Different abbreviations for MATHEMATICS continue to be used, viz. Math and Maths. Modern university mathematics is very international but school mathematics is less so and differences in usage may persist: see the entry TRAPEZIUM and TRAPEZOID. When everyday words are given a mathematical sense some of the zest is lost if the word is not an everyday one in the other variety; in British English the word JACKKNIFE is not an everyday word and has currency only as a technical term in Statistics. A difference in pronunciation mathematicians will notice is that of the Greek letters Beta, Zeta, etc.



To Contents  To English mathematical words





Old English






The notes that follow describe the indebtedness of the modern English lexicon to Old English and to Latin, French, Greek and German. Of course, these four languages are not the only ones that are reflected in modern English but mathematical terms originating in, say, Arabic (e.g. ALGEBRA, ALGORITHM, RADIX (ROOT) and SURD), Sanskrit (e.g. SINE and ZERO), Italian (e.g. ABSOLUTE DIFFERENTIAL CALCULUS) or Russian (e.g. STOCHASTIC PROCESS) have usually come into English through Latin, French or German rather than directly. The small number of loan-words from a particular language may bear little relation to the extent of mathematics developed in that language; see e.g. Wikipedia: Mathematics in Medieval Islam and Wikipedia: Indian Mathematics. 


The order in which the languages are presented may seem strange but it reflects the rough order in which mathematical words from those languages first came (directly) into English.



There appear to be no works specialising in mathematical terms but several treat the foreign origins of English words in general use: at the scholarly end there is Serjeantson and at the popular end there is Metcalf. Lurquin works in the opposite direction from the source language(s). Shipley and Watkins consider the place of English in the Indo-European language family. The Wikipedia articles show how English has borrowed extensively from Arabic, Russian and Italian but not many mathematical words are included.


  • Mary S. Serjeantson A History of Foreign Words in English, Kegan Paul 1935 Amazon.


  • Allan Metcalf The World in So Many Words: A Country-by-Country Tour of Words That Have Shaped Our Language, Houghton Mifflin 1999. Amazon.
  • G. Lurquin Elsevier's Dictionary of Greek and Latin Word Constituents: Greek and Latin affixes, words and roots used in English, French, German, Dutch, Italian and Spanish, Elsevier 1998. Amazon.
  • Joseph Twadell Shipley The Origins of English Words: A Discursive Dictionary of Indo-European Roots, Johns Hopkins 2001. Amazon
  • Calvert Watkins The American Heritage Dictionary of Indo-European Roots, 2nd ed., Boston,Houghton Mifflin, 2000. Amazon.
  • List of Arabic loanwords in English Wikipedia.
  • List of English words of Russian origin Wikipedia.
  • List of English words of Italian origin Wikipedia.
  • Wikipedia Dizionario matematico inglese-italiano and dizionario italiano-inglese.



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The contribution of Old English       


The English, Latin, French, Greek and German languages belong to the family of Indo-European languages (Wikipedia) and share a common ancestry. Old English is a Germanic language; specifically it belongs to the Anglo-Frisian branch of the West Germanic line. See Crystal and Algeo & Pyles.


Only a minority of the words of Modern English are derived from Old English. Crystal p. divides the vocabulary of everyday modern English so.


  • Old English provides the core of the most basic words, not only the grammatical words—articles, prepositions and pronouns—but such content words as mother and brother,


  • 50-70% of the common words come from French or Latin.


The overlay of French and Latin words is so heavy that modern English sometimes has three ways of expressing the same basic idea: Crystal (48 & 124) illustrates the possibility with the triples rise/mount/ascend and kingly/royal/regal. “The Old English word is usually the more popular one, with the French more literary, and the Latin more learned.” 


There was a literary tradition in Old English—the Anglo-Saxon Chronicle of the C9 has been described as the first continuous history of any Western people in their own language. But there was no mathematical literature and therefore no specialised mathematical vocabulary for Latin and French words to augment or to displace.


A form of the popular/learned division applies to mathematical words. Very few of Schwartzman’s 1500 mathematical words are labelled “native English.” There is a rare run of such words: on, one, onto and open. These words illustrate some general points about the Old English presence in the modern language.


  • These are among the most frequently used words in the language and are learnt well before a child goes to school and ‘learns mathematics’.


  • The ‘school’ words for discussing numbers and space, including NUMBER and SPACE, are more recent additions to English and are usually adaptations of foreign words—from Latin or French.


  • One and the other Old English number words have survived. But they were not used as components in the words created as equivalents to the classical Greek terms when the latter were put into English in the C16. Instead the Greek or Latin forms of the numbers were adopted: thus TRIANGLE, QUADRILATERAL and PENTAGON. Well into the C19 the classical languages also provided the ingredients when words were sought for new ‘numerical’ concepts–see e.g. QUATERNION and OCTONION.


  • More recently the full resources of the language have been used in forming mathematical terms and the Old English words have figured in such constructions as OPEN SET and ONTO FUNCTION from the C20.  The first is a translation from the similarly colloquial expression in French.


Behind Schwartzman’s label “native English” is often a complicated story in which an ancestor word has been greatly modified. The OED entry for “one” relates how the word “an” used by writers of some version of Old English in the C9 was transformed into the word “one” which became the standard but not until the C16: Harper gives the basic story. Harper also treats the other number words.







































































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The contribution of Latin       


Latin (Wikipedia), as the language of the Roman Empire, had a profound influence on all the languages of Western Europe. It had most influence on the languages that are its direct descendants, the Romance languages (Wikipedia). One of these, French, had a very strong influence on English.


In all parts of Western Europe Latin was used in conjunction with the local language for religious, political and scholarly purposes. For aspects of medieval learning see QUADRIVIUM and LOGIC.  


If we count by years, Latin has been the most important language of mathematics in the European tradition. Yet as a language of mathematics, Latin presents a paradox:


  • The native speakers of Latin, the classical Romans and the people of the Western Empire, are notorious as non-contributors to mathematics. Katz gives them one page in his account of the “Final Chapters of Greek Mathematics.”


  • Yet for more than 1000 years non-native speakers published their mathematics in Latin and in that time Latin was the language of Western mathematics.


Latin only became a dead language in mathematics at the end of the C19. Before then new works were published in Latin and new words were created or old ones given new meanings.


·        In the time of the Roman Empire some Greek mathematical terms were taken into Latin. 


·        Later Latin translations of ancient Greek and contemporary Arabic works made those works became known in Western Europe. The first Englishmen recorded in histories of mathematics were translators of Arabic into Latin: Adelard of Bath (1075-1164) and Robert of Chester (fl. 1144–1150). For traces of their activity see the entries ALGEBRA , ALGORITHM FRACTION and SINE. (Katz 244 & 290-1)


·        Original work came later and it too was written in Latin. The Italian mathematicians of the medieval and early Renaissance periods, Fibonacci, Tartlaglia and Cardano, all wrote in Latin. One English contributor was Thomas Bradwardine (ca. 1295-1349); see the entry RATIO (Katz, 315).



  • The Italians and the French were the first to make the break with Latin: Galileo’s Dialogo sopra i due massimi sistemi del mondo appeared in 1632 and Descartes La Géométrie in 1637.


  • In England Latin long continued to be the language of learning. The Artis logicae compendium a treatise on logic by Henry Aldrich was published in 1691 and was the main text in England for 150 years. It was translated into English only at the beginning of the C19.



  • Until well into the C19 German authors published in Latin.   A glance at the dates of works with titles beginning with De (= “On”) on DML shows how long Latin lasted as an important medium.




The death of Latin


By the end of the C19 Latin had ceased to be a living language for mathematicians and ceased to be the international language for mathematics.


  • At the beginning of the C20 English French and German were the accepted international languages. The Proceedings of the Imperial Academy of Japan founded in 1912 took articles in these languages (and Japanese). The international journal Acta Mathematica (published in Sweden) took articles in French and German. The journal Mathematische Annalen published articles in English and French as well as in German. Italian work on TENSORS reached a German audience through the medium of French.


  • Some mathematicians looked to artificial languages (Esperanto was the best known) to succeed Latin. One of these was Peano who also tried to revive Latin in the form of a simplified Latino sine flexione.


  • A century after the demise of the old Latin, English rather than an invented language has become the new Latin.


Gray describes the linguistic situation at the turn of the C20.




Latin mathematics and English words


The table above illustrates how from ADDITION and ARGUMENT in the C13 to MODULUS in the C19 large numbers of Latin words passed into English.


The author expressing new ideas in Latin or translating from Greek or Arabic had the usual options


  • Use an existing word and stretch its meaning as Gauss did with residuum; see RESIDUE in number theory.


  • Create a new Latin word; as ALGEBRA came from Arabic.



In the early centuries the words were created because the author was writing Latin text; later when the mathematician was writing in his own language he might construct a word from Latin roots.


Mathematicians writing in their own language often chose Latin roots for the words they invented. Examples from English-speaking writers include QUATERNION and IDEMPOTENT. The the C20 graph theory terms ARBORESCENCE and ARBORICITY both have Latin roots but the components were long established in English and the first word was already in English. 



Latin references


The first two references are introductory surveys of Latin and its role in Western history. The next one is an on-line resource for beginning students of Latin and Greek.

  • Tore Janson A Natural History of Latin, Oxford University Press 2004.  Amazon
  • Nicholas Ostler Ad Infinitum: A Biography of Latin, Harper 2007. Amazon
  • N. S. Gill Etymology of Geometry Terms
  • Barnabas Hughes “Mathematics and Geometry” pp. 348-54 of F. A. C. Mantello, A. G. Rigg (eds.) Medieval Latin: an introduction and bibliographical guide, Catholic University of America Press, 1996. Google Books.
  • Ostler Empires of the Word chapters 7 and 8.
  • Jeremy J. Gray (2002) Languages for mathematics and the language of mathematics in a world of nations, pp. 201-228 of K. Parshall & A. Rice (eds.) Mathematics unbound: the evolution of an international mathematical community, 1800-1945 American and London Mathematical Societies.






The contribution of Greek      


Before the Renaissance Greek mathematical words came into English through Latin (e.g. GEOMETRY) but since the C16 English writers have had direct contact with Greek and many English words have been taken from the Greek (e.g. ISOSCELES). The modern mathematical lexicon not only has the old Greek words but a large number of newly-minted words based on Greek roots (e.g. TOPOLOGY). 


Greek (Wikipedia) occupied a very different place in Western European mathematics from that of Latin, the other language from classical antiquity.


·        There was a very important body of Greek mathematics produced between 300 BC and 300 AD by Euclid, Archimedes, Appollonius, Diophantus amongst others.  See Katz Part One.


·        In the Eastern Mediterranean Greek served as a lingua franca long after the decline of Greece as a political force but it never became a working language in Western Europe and no new mathematics was written in Greek


·        In the West Greek was a language to be translated from. 


English mathematical language reflects the influence of the Greek language in two distinct ways.


  • Old words, i.e. translations of the terms used in the classics.  By the C17 the vocabulary of the Greek classics had been taken into English and the other modern European languages either via Latin or directly from the Greek.  English was left with a large number of loan words formed by transliteration.


  • New words. Since the C18 writers in English (and other European languages) have been creating new words in their languages from Greek elements.



Old words


To English speakers Greek loans like CONVEX and ISOSCELES appear difficult and complicated but they were just built out of everyday components, just as STEM AND LEAF DISPLAY is in modern English.  


Many Greek words came into English not directly but via Latin: e.g. the English words MATHEMATICS and ARITHMETIC derive from Greek loans to Latin. The Greek words were simply lifted into Latin, transforming letter by letter.


The translation into English of Euclid's Elements, the most influential work of Greek mathematics, brought many more Greek words into English. In the Renaissance translations of were made in Italian, French, German, Spanish and English. (Katz 385). These were printed books for the technology of printing had been introduced some time before.


The English translation of 1570 was the work not of a scholar but of a merchant Sir Henry Billingsley. The Elements had a unique status amongst mathematical works and Billingsley was not writing primarily for scholars. For them a more useful language was Latin: thus in 1706-10 when Edmond Halley translated the more esoteric Conics of Appollonius for the learned world he translated it into Latin. Heath’s edition of the Elements (chapter VIII of vol. 1) surveys the early editions published in England: there were editions of the Greek text with Latin translation in 1620 and 1703, Latin translation for use in schools in 1655 with an English translation following in 1660.


A typical Billingsley effort is POLYHEDRON from πολíεδρον. This is a loan word like bootstrap in Italian though, because English and Italian share an alphabet, there is no need for transliteration. Of course, bootstrap is a unit in Italian while polyhedron in English breaks into two, poly- and –hedron, and belongs to two large families of words.


Billingsley’s translation strategy was the opposite of his contemporary Robert Recorde who took Greek compounds and translated their components into English and re-combined them; see above.  




New words


After the period of translation was over Greek had a continuing influence on the language of mathematics through the construction of new words. Latin enjoyed a similar role. The strategy was to create new words from classical elements rather than stretch the meaning of existing Greek and Latin words. 


  • The C19 German physicist Rudolf Clausius made it a principle to construct terms from classical roots: there is a statement of his philosophy in the entry for ENTROPY; In the case of this word the strategy was very successful as the translations show. There were many other German creations including TOPOLOGY and HOMOTOPY.


  • The same process was at work in all the European languages: cf. the entries HOLOMORPHIC (from a French coinage) and HETEROSCEDASICITY (an English one).


·        Some Greek elements, such as tele and bio, are very well domesticated in English while others, such as platy and lepto seem more exotic. For an amusing mnemonic for the latter as qualifiers of kurtosis see the entry KURTOSIS. Greek and Latin elements dominate Quinion’s dictionary of suffixes.


  • In the history of English there have been times when there has been a reaction against creating words derived from the classical languages—a wish to return to the roots of English. Perhaps mathematical terms have not been immune to such trends.


In mathematics the fashion for Greek neologisms peaked in the C19 and has faded as familiarity with the Greek language has declined: there are some more recent coinages, including CYBERNETICS, GYROID  and PLETHYSM. 


Greek references            




































































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The contribution of French


Of the modern languages French (Wikipedia) has had the strongest influence on the English language.


·        French influenced the English language even before the Norman Conquest of 1066 but the replacement of the ruling elite led to a mass importation of French words. See above.


·        French has always been the language through which ideas from Continental Europe entered Britain.


  • Besides this special influence French had a general influence on European culture in the C17-C19. At the end of the C18 French was so much the international language for European intellectuals that in 1782 the Berlin Academy offered a prize for the best essay on “Qu’est-ce qui a rendu la langue françoise universelle?” See Rickard (p. 121); also see Ostler.


  • The influence was particularly strong in mathematics. In the course of the C17 French mathematicians stopped writing in Latin and wrote in French. Among those who lived in the age of transition were Viète, Desargues, Descartes,  Fermat and Pascal. In the late C18 and early C19 European mathematics was dominated by mathematicians who wrote in French—Lagrange, Laplace, Cauchy, etc.  See Katz Part 4.


The influence of English on French has been much weaker. Wise writing in 1997 reported that in a recent edition of one of the non-technical French dictionaries only 350 of the 45,000 entries were of English origin!



French mathematics and English words


The table above contains a sample of English words that are derived from French and a glance at a French-English glossary like MathVoc shows just how similar the basic vocabulary is. For the older terms the traffic was from French to English while lately the traffic has been in the opposite direction. Dictionaries, however, give greater weight to the past. While everyday English contains many ‘undigested’ French words and expressions, such as concierge and joie de vivre, such French stand-outs are rare in mathematical language. Consider the three terms INJECTION, SURJECTION and BIJECTION which have been in English and French mathematics for around 50 years. Of these surjection is a stand-out and clearly of French origin for, unlike in- and bi-, sur- is not a productive prefix in English, i.e. it is not used for making new words. Many similar loans, such as SURFACE have been around for so long that they no longer look French.


Many everyday French words and everyday English words are twins, either through borrowing or because both words have a common origin in a third language, most often Latin.


From many instances, consider imaginaire/imaginary (both from Latin), dérivative/derivative (from French to English), matrice/matrix (from French to English or both from Latin), mesure/measure (from French to English), estimation/estimation (from French to English) and appartement/apartment (from French to English); details are in the OED and Le Robert (below).


The words are also mathematical twins and they illustrate a widely-applied principle of parallel extension: when an existing word is used to express a new idea, use the twin to express the idea in the other language. Of course the new direction of travel is not determined by the old and as mathematical terms, matrix and estimation went from English to French. See IMAGINARY, DERIVATIVE, MATRIX, MEASURE, ESTIMATION and APARTMENT.


In the centuries when most of the mathematical literature was in Latin a word could be taken into English directly from Latin or from a French translation of the Latin original. Examples of the second route are DERIVATIVE and DETERMINANT. It is not always possible to say which route was followed.

  • The English word DETERMINANT and the French word déterminant appear to have come independently from the Latin and neither originally had a mathematical meaning. When the French term was given a mathematical meaning English mathematicians extended the English word in the same way.  See below for more on this word.
  • The everyday English word RESIDUE came from the French word résidu (ultimately from Latin) in the C14. When the French word was given a mathematical meaning in complex analysis in the C19, the English word was given a parallel stretch.
  • RESIDUE is also a term in number theory in which sense it originated with Gauss writing residuum in Latin. Whether the English word came directly from the Latin or via the French is not known.
  • When a new technical term was created in French, especially one with a classical root such as holomorphique, it went smoothly into English as HOLOMORPHIC because the same move had been made so often before.  

Of course plenty of French words have not been borrowed. When the everyday French word faisceau (of Latin origin and related to the English word fascism) acquired a technical meaning it went into English as SHEAF an everyday word descended from Old English. The word immeuble went into English as building also derived from Old English: see APARTMENT, BUILDING & CHAMBER.


French references



















































































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The contribution of German                  


Modern German and modern English are related languages with common ancestors—see Waterman or Wikipedia: German language—but English speakers find many more familiar words in French—or even in Spanish—than in German: compare MathVoc with Manolov: Math dictionary. Pfeffer & Cannon find a total of around 5000 German loan words in the English language and around 60 in mathematics. Most of the mathematical borrowings took place in the C19 or early C20. This was also the period in which German culture exercised its greatest influence; for a survey see Watson.


Keller (pp. 485-92) reports that in book production German books first outnumbered books in Latin in 1681 and in 1781 one eighth of the books were still in Latin. Later French had such prestige that ten per cent of all books published in Germany between 1750 and 1780 were in French. In 1750 Voltaire wrote of the use of French in the Prussian court “On ne parle que notre langue. L’allemand est pour les soldats et pour les chevaux…” 


In the C17 Leibniz published his mathematical works in Latin, as was the custom throughout Western Europe. At the beginning of the C19 Gauss still published in Latin but at the end Hilbert published only in German; cf. the entries under GAUSS with those under HILBERT. In the course of the century German became not only an acceptable written medium for German-speaking mathematicians but a major international language. The importance of German increased in the C20 until mathematics in Germany almost collapsed with the rise of the Third Reich. Part Four of Katz covers this period; see also Siegmund-Schultze.



The development can be traced as follows


·        At the beginning of the C19 German mathematicians were still publishing in Latin to reach an international audience.


·        Later in the century German mathematicians were publishing in German but the language was still not widely known. Thus in the 1880s the international journal Acta Mathematica re-issued Cantor’s papers in French translation; see SET and SET THEORY.


·        In the C19 Russian mathematicians used French as their international language: see the CHEBYSHEV entries. In the early C20 there was a movement towards German with important work on probability and topology appearing in German periodicals—see the entries LAW OF THE ITERATED LOGARITHM and METRIZABLE. In the late 1930s mathematicians came under pressure to publish in Soviet journals and in Russian.


·        In the course of the C19 American mathematics changed from being a satellite of British mathematics to being a satellite of German mathematics. The Harvard professor W. F. Osgood even wrote books in German—e.g. Lehrbuch der Funktionentheorie (1907)—an unusual decision but an intelligible one. See Parshall & Rowe.


·        The German-writing refugees who settled in the United States in the 1930s stopped writing in German and wrote in English without adding new German loan words to the English lexicon. Occasionally immigrants replaced German loan-words with English words, see CONVOLUTION and Wintner’s role in its introduction. 




German mathematics and English words


Given the importance of German as source of mathematical ideas and the historical connections between the English and German languages it is surprising how few German words—or at least words with German roots—are in the English mathematical lexicon. The situation in subjects such as chemistry and mineralogy is different: Pfeffer & Cannon’s 60 for mathematics can be compared with 857 for mineralogy. (The exact numbers are unconvincing but the relative magnitudes are more plausible.)


Some German mathematical words look very different from their English counterparts but nevertheless are related to English words.  The German words Mittel = MEAN and Raum = SPACE are related to the English middle and room but the technical English words come from French and ultimately Latin. German writers were more willing than English to produce new words from their language’s own resources: compare the English TRIANGLE from the Latin with the German Dreieck based on the German words for three and corner.


It has been usual for German terms to be translated into English for the English language is at least as rich in resources as the German. The usual pattern is that of topologischer Raum to TOPOLOGICAL SPACE, inneres Produkt to INNER PRODUCT or transfinit to TRANSFINITE where the original German components are replaced by their English equivalents.


However some German words have not been translated but rather carried into English.


·        One way for a German mathematical word to pass into English is by being identical to an English word with the same non-technical meaning; cf. the discussion of French-English twins above. RING and IDEAL are such words and SIGNATURE (from Signatur) is nearly so. The first is in Old English and old German while the latter two are derived from Latin in both languages. See Online Etymology Dictionary.


·        German neologisms were more likely to be borrowed if they had classical roots: thus Topologie became TOPOLOGY, Entropie became ENTROPY, Homotopie HOMOTOPY, Statistik STATISTICS and Combinatorik COMBINATORICS. (The same was true in the opposite direction: the English words MATRIX and QUATERNION went into German unchanged. It is impossible to tell from the appearance of the words in which language they originated.)


·        The outstanding Germanism is the eigen prefix—see EIGENVALUE—a reminder of the importance of German science at its peak in the 1920s. Ironically eigen- was an unnecessary borrowing in that there were already English terms, latent and characteristic, that could do the job.


·        Sometimes it has seemed difficult to find, or to create, an English equivalent. The entries for CONVOLUTION (Faltung), DECISION PROBLEM (Entscheidungsproblem), SQUAREFREE (quadratfrei) and TRACE (Spur) show a German loan in use for a few years or even a few decades before being replaced by an English term.


·        One loanword still in use after nearly a century is SCHLICHT although it is giving way to the French borrowing UNIVALENT which is both a more accurate descriptive term and an established English word in Physics/Chemistry.


·        Occasionally what was originally a description in German becomes a name in English (cf. Gauss’s Latin expression THEOREMA EGREGIUM). The terms FREIHEITSSATZ, HAUPTSATZ, HAUPTVERMUTUNG and HILBERT'S NULLSTELLENSATZ have this status and, so it seems, does Entscheidungsproblem for the orginal DECISION PROBLEM posed by Hilbert. In German the term Hauptsatz (fundamental theorem) can be applied to new theorems but in English it is tied to one result from the 1930s.     


·        Abbreviations such as T1 (from Trennungsaxiom 1) or Gδ (from Gebiet and Durchschnitt) appear to be more transportable than words. Going the other way, the English abbreviation ANOVA (see the entry VARIANCE) is used in German as an abbreviation for the term Varianzanalyse.  


·        The widely-used abbreviation, Z for the integers, is from the German word Zahlen, although the letter was chosen by the Bourbaki group of French mathematicians. See Earliest Uses of Symbols of Number Theory.


·        Until well into the C20 many German texts including newspapers were printed in black letter or Gothic script. These characters were used in German mathematics and they have passed into English mathematics.


·        The practice in German is also to translate but German mathematics has its modern English loans: thus BOOTSTRAP (methods) has become become Bootstrap-Methoden and JACKKNIFE  (method) has become Jackknife-Verfahren. 



German references




Sources of information


Works on words

Other works

Personal communications



Earliest Uses is not a transcription of a paper original or masterwork—no such work exists. The following notes describe how the entries are constructed.    

The entries draw on secondary sources and on original research. Some of the secondary sources are described below—both specialist works and others. Of course these works and even those listed in the Earliest Uses References make up only a very small proportion of all the sources used; many more specialised references appear in the entries for individual words. Finally there are some examples illustrating the use of these resources in researching individual words.



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Works on the history of words and symbols

Some of the information on the Words and Symbols pages comes from specialist works on the history of words and symbols. Such works fall into several categories


  • H. A. David’s “First (?) Occurrence of Common Terms in Statistics and Probability” appears to be the only systematic work on the origins of (some category of) mathematical terminology; the first articles appeared in American Statistician 1995 and -8 and the latest version is available here.  As well as being an important source of information on statistics and probability, “First (?) Occurrence” has been a model for Earliest Uses in a wider sense. However its journal article format and the space restrictions this imposes have meant that most entries give only a bare reference to the first use.


  • Florian Cajori’s A History of Mathematical Notations (originally in 2 volumes 1928/9) Amazon is another unique work. It has provided the nucleus for the entries on the Symbols pages. There is no corresponding work for mathematical words, although Cajori is useful for them too. The standard of scholarship is very high but the work’s great limitation is that it reflects the mathematical universe of the early C20. 


  • Steven Schwartzman’s The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English (1994) Amazon gives the meaning of over 1500 words and investigates their roots. Its limitation is that, unlike Cajori, it usually gives no details of who, where, when. Thus it is more of a complement to Earliest Uses rather than a source for it. Words of Mathematics was reviewed by Henry J. Ricardo in American Mathematical Monthly, 102, (June), (1995) 563-565 and by Holger Becker in English Today, 15, (1) (1999), 55-6 here.



  • Anthony Lo Bello’s Origins of Mathematical Words: A Comprehensive Dictionary of Latin, Greek, and Arabic Roots (2013) Amazon contains essays of varying lengths and no fixed pattern on a large number of mathematical words. It is an entertaining and very polemical dictionary for Lo Bello is as concerned to judge usage as to record it: thus we learn of INJECTION, SURJECTION and BIJECTION that the first two are “formed on good analogy” while the third is “low, very low” while FACTORIAL with an i “should really mean having to do with a factory.” There is a review by Benjamin Wardhaugh in the LMS Newsletter April 2014.


  • Bertrand Hauchecorne’s Les Mots & les Maths: Dictionnaire historique et étymologique du vocabulaire mathématique (2003) Amazon is like a French Schwarzmann. It covers fewer words but it gives some indication of who, where, when though not in the depth of Earliest Uses.



  • English dictionaries. English mathematical words are English words and the greatest single work on the words of the English language is the Oxford English Dictionary (Wikipedia). The OED, which contains more than half a million words, descends from John Murray’s New English Dictionary on Historical Principles; Founded Mainly on the Materials Collected by the Philological Society (first volume, 1884). Each entry gives the meaning of the word, its etymology and quotations illustrating how the word is used—including early as well as typical uses. For the history of mathematical words the main limitations of this and the many smaller dictionaries on the same pattern are limited coverage and a mission that leads them to concentrate on the English origins of words. The OED has a unique problem—its age for some entries have not been revised since they were first published more than a century ago. There are numerous dictionaries online at 



  • Etymological Dictionaries. These are potentially more useful than general purpose dictionaries, although they generally contain few technical mathematical words. Douglas Harper’s Online Etymology Dictionary lists the major English etymological dictionaries and itself illustrates the information such works can provide. There are similar dictionaries for other languages such as the Dictionnaire historique de la langue française. Other web resources on etymology include an email discussion list run by the American Dialect Society and a site by Michael Quinion.  


  • Dictionaries of Mathematical Terms sometimes have information about the origins of terms though this is not usually considered essential to their function. Thus the Oxford Dictionary of Statistical Terms (Y. Dodge (ed.)) Amazon has a lot of historical information without being systematically historical. See here for Eric W. Weisstein’s annotated list of Dictionaries of Mathematics.


  • There are also articles dispersed across mathematics, history and education journals. Thus Derek Bissell’s “Statisticians Have a Word for It,” Teaching Statistics, 18 (3), (1996), 87-89 and M. E. McIntosh’s “Word Roots in Geometry” Mathematics Teacher, 87(7), (1994), 510-515 are designed for teachers interested in the history of words.






























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Works produced for other purposes


Most of the information on the Words and Symbols pages is gleaned from works that are not primarily about the history of words and symbols. Among such resources are the following.


  • Histories, both general histories of mathematics, which describe the origins of the most important concepts and the words used to express them and monographs and journal articles which do the same in more detail for more specialised topics. David Joyce has lists of general books, of specialised books and of journals. Of the journals Archive for the History of Exact Sciences and Historia Mathematica are probably the most useful. The entries on EIGENVALUE and ST. PETERSBURG PARADOX illustrate the use of journal articles. Of the general histories Smith’s (especially volume II) pays most attention to language although it treats only elementary mathematics.  



  • Biographies of the creators of terms and of the people after whom terms are named often have useful information: see for instance the use of Dauben’s biography of Cantor in the entry for SET. Besides works on individuals there are collective works such as the Dictionary of Scientific Biography and the biographies in The MacTutor History of Mathematics archive.



Histories, treatises and biographies tend to document the original innovation regardless of the language in which it was first expressed and so they complement the English dictionaries with their emphasis on English usage. Histories generally only attend to the englishing of terms when it is associated with important mathematical developments. Thus most of the translation activity of the C14 to C17 goes unnoticed in the general histories.  






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Personal communications

Many of the entries on the Earliest Uses pages are based on personal communications by which individuals share their expert knowledge. (See Invitation) Questions about origins are regularly raised in internet discussion groups such as Discussion: math-history-list and Philomathes Discussion Group. The very useful Historia-Matematica seems to be inactive, although its archive is preserved.
























































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Researching bootstrap, determinant and point

To illustrate the use of these resources in researching words consider three examples from different periods.


BOOTSTRAP an English mathematical word for 30 years

We saw this word above.


Sources of information on the history of bootstrap

  • It is a very recent word and many people alive today know its history because they have lived it.
  • The word is too new to be covered in histories or in the OED although of course the original sense is given there.
  • There are already treatises and encyclopedia articles on bootstrapping and these usually discuss the origin of the term.
  • There is an entry in “First (?) Occurrence”.
  • The original article is reviewed in Mathematical Reviews.
  • A JSTOR search will find the first use, i.e. the earliest appearance on JSTOR is the first use.


JSTOR is so useful in researching some words and so useless for others that it deserves some comment. (There are several JSTOR packages and the specific comments apply to the combined collections Arts & Sciences I and Arts & Sciences II).


  • JSTOR’s coverage of statistics is very good—it has complete runs of around 20 of the most important journals. Its mathematics category (disjoint) has around 40 journals!


  • JSTOR contains only material in the English language but as much of the modern international statistical vocabulary was created in English (from the late C19 onwards) the limitation is not catastrophic. In the neighbouring field of probability it is catastrophic for English only became an important language after the Second World War.


  • JSTOR contains only material published in journals. The limitation is not so bad for recent terms (coined in the last 50 years, say) because they are likely to have first appeared in journal articles. In earlier days terms were often introduced in books. For instance, R. A. Fisher’s book Statistical Methods for Research Workers (1925) has an important place in the history of statistical terminology: see e.g.  STUDENT’S t DISTRIBUTION and LATIN SQUARE. 

Most of the mathematical words in use are not new creations and have to be researched in other ways. Researching a modern—but not new—term like HOMEOMORPHISM or EIGENVALUE is more like researching the old words discussed below.


DETERMINANT an English mathematical word for 150 years

There has been an English word determinant for around 400 years but its use as an algebraic term goes back about 150 years. In that sense it is a translation of the French word déterminant used by Cauchy which was itself inspired by the Latin word determinant used by Gauss. The Gauss-Cauchy terms were late arrivals in the sense that there was already a tradition of studying determinants that was more than 100 years old.


Sources of information on the history of determinant

·        Modern textbooks of algebra do not usually spend much time on the origins of the terms used.

·        The OED documents the appearance of the English word both in its pre-mathematical sense and in its mathematical sense.

·        General histories of mathematics like Katz discuss determinants without paying much attention to terminology.

·        The history of determinants is told in great detail in the volumes by Muir and these are the most authoritative sources for usage.


POINT an English mathematical word for 600 years

Although the word was already in the language (Harper), point began its mathematical career in English around 600 years ago. It was used to express an idea that went back two millennia.


Sources of information on the history of point

·        Modern textbooks of geometry do not usually detail the origins of the terms used.

·        Histories of mathematics like Katz discuss the treatment of the concept of point in Euclid’s Elements without discussing the language Euclid used.

·        The notes in Heath’s edition of Euclid’s Elements discuss the word(s) used by Euclid and his predecessors and the translators of the Elements.

·        The OED documents the first appearance of the word in English.














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General References


  • John Algeo and Thomas Pyles The Origins and Development of the English Language, 5th edition, Heinle. 2004.  Amazon
  • David Crystal The Cambridge Encyclopedia of the English Language, Cambridge University Press. 1995.  Amazon
  • Philip Durkin The Oxford Guide to Etymology, Oxford University Press, 2009. Amazon
  • David Graddol, Dick Leith, Joan Swann (editors) English: History, Diversity and Change, London: Routledge in association with the Open University, 1996.
  • Geoffrey Howson A History of Mathematics Education in England, Cambridge University Press 1982.