Math Words, pg 8

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Abundant Abundant numbers are numbers whose proper divisors total to more than the number. Twelve is the smallest abundant number because the sum of its proper divisors, 1+ 2+ 3+ 4+ 6 = 16 which is greater than 12. The word is from the union of the Latin roots ab, away, and undare, to flow. Literally then, the word means overflowing, and refers to a general excess of something.
The ancient Greeks often applied mystical properties to numbers and used them to predict futures and personal fortunes. They worked with Perfect, abundant, and deficient numbers as early as 100 years before the Christian Era. The first known use of "abundant number" in English was by Robert Recorde in Whetstone of Witte during the Sixteenth Century.

Addend The word we now use for the numbers to be added together is shortened from the word addendum. Originally the word was applied only to the values after the first. For example in 4 + 3 the three was considered an addend but not the four, since only the three was being "added". The Latin phrase was numeri addenda. By the 1600's the word seems to have been applied equally to any of the numbers to be added. According to the Oxford English Dictionary, the word was first used in English in a 1696 book by Samuel Jeake "Place the Addends in rank and file one directly under another".

Amplitude is used in mathematics as a term for size or magnitude. The origin is the same Indo-European ple root that gives us plus and complement. The Latin source was amplus for wide. Today the word is used to describe the distance a periodic function varies from its central value, and the magnitude of a complex number.

Arc It is uncertain if the early root of arc referred to a bow, an arrow, or the two as a unit. German and old English derivatives lead to arrow, but the Latin root arcus was for a bow. The use as a term for the arc of a circle seems reasonably to have come from the shape of the bent bow. Other common words from the same source include arch and arcade (which was originally a series of arches, not a place for video games).

AreaThe general term for a measure of two dimensional space comes unchanged from the Latin. The more common meaning of area applies, as it did 2000 years ago, to an flat expanse of open unoccupied land. The French shortened the word to "are" as a name for a measure of land equal to one hundred squre meters.

Argand Diagram Argand Diagrams, the method of drawing complex numbers as vectors on a coordinate plane, are named for Jean R. Argand (1768-1822), an amateur mathematician who described them in a paper in 1806. A similar method, although less complete, had been suggested as early as 120 years before by John Wallis, and developed extensively by Casper Wessel(1745-1818), a Norwegian surveyor. (Actually, at the time Wessel lived, the area where he was born was a part of Denmark. Norway became an independent government in 1905 after years of domination by Denmark and Sweden.) It may be that even then, the method was unknown to Gauss and he had to rediscover it for himself in 1831 although it has been suggested that Gauss may have discovered the idea as early as Wessel. Some parts of his Demonstratio Nova would seem almost miraculously derived without a knowledge of the ideas of the geometry of complex numbers.

Wessel's paper was published in Danish, and was not circulated in the languages more common to mathematics at that time. It was not until 1895 that his paper came to the attention of the mathematical community, long after the name Argand Diagram had stuck. Incredibly, there were at least three more individuals who may have independently discovered and written on the same idea; Abbe Bruee, C. V. Mourney, and John Warren.

Argand's Book, Essai sur une maniere de representer les quantities imaginaires dans les constructions geometriques, might have suffered the same fate as Wessel except for an unusual chain of events. I give here the version as presented by Michael Crowe in his A History of Vector Analysis

In 1813 J. F. Francais published a short memoir in volume IV of Gergonne's Annales de mathematiques in which Francais presented the geometrical representation of complex numbers. At the conclusion of his paper Francais stated that the fundamental ideas in his paper were not his own, he had found them in a letter written by Legendre to his (Francis') brother who had died. In this letter Legendre discussed the ideas of an unnamed mathematician. Francis added that he hoped this mathematician would make himself known and publish his results.

The unnamed mathematician had in fact already published his ieas, for Legendre's friend was Jean Robert Argand. Hearing of Francais' paper, Argand immediately sent a communication to Gergonne in which he identified himself as the mathematician in Legendre's letter, called attention to his book, summarized its contents, and finally presented an (unsuccessful) attempt to extend his system to three dimensions.

Even with so much interest and attention to the geometry of complex numbers, it was not until Gauss published a short work on the ideas that they became popular.

Translations of both Wallis' and Wessel's papers on the imaginaries can be found in A Sourcebook of Mathematics by David Eugene Smith.

Argument Although most current meanings of argument indicate a disagreement, the Latin root arguere was closer to such present meanings as declare or prove. Perhaps today's meanings come from the continued practice of opposing sides alternately declaring their position in an attempt to prove their view. Whatever the source of its common meanings, today when mathematicians talk about the argument of a function, they are declaring a value for the independent variable of the function. In Sin(2), 2 is the argument of the function.

Catenary A catenary curve is the shape that a perfectly uniform rope would form when suspended between two points. The word is from the Latin catena for chain. The name was applied by Christen Huygens while studying the form of suspended chains. Galileo thought the shape would be a parabola. As can be seen from the image below, near the vertex a parabola and a catenary look very similar. When x is slightly greater than three however, the catenary begins to rapidly outgrow the value of the parabola.
The two shapes are related by another relationship. If a parabola is rolled along a straight line, the focus of the parabola will move along a catenary curve. In the figure the parabola y=x2 + 1 is on the inside and the catanary whose equation is y= (ex+e-x)/2, is on the outside.

For students of American History, it may be interesting that the first use of "catenary", rather than the longer, more formal "catenaria", may have been in a letter from Thomas Jefferson to Thomas Paine. Jeff Miller's wonderful web-site on the first use of mathematical words has

In a letter to Thomas Jefferson dated Sept. 15, 1788, Thomas Paine, discussing the design of a bridge, used the term catenarian arch:
Whether I shall set off a catenarian Arch or an Arch of a Circle I have not yet determined, but I mean to set off both and take my choice. There is one objection against a Catenarian Arch, which is, that the Iron tubes being all cast in one form will not exactly fit every part of it. An Arch of a Circle may be sett off to any extent by calculating the Ordinates, at equal distances on the diameter. In this case, the Radius will always be the Hypothenuse, the portion of the diameter be the Base, and the Ordinate the perpendicular or the Ordinate may be found by Trigonometry in which the Base, the Hypothenuse and right angle will be always given.
In a reply to Paine dated Dec. 23, 1788, Thomas Jefferson used the word catenary:
You hesitate between the catenary, and portion of a circle. I have lately received from Italy a treatise on the equilibrium of arches by the Abb� Mascheroni. It appears to be a very scientifical work. I have not yet had time to engage in it, but I find that the conclusions of his demonstrations are that 'every part of the Catenary is in perfect equilibrium.'
The earliest citation for catenary in the OED2 is from the above letter.

The shape is sometimes called a chainette and a funicular curve from the Latin "funiculus" for a cord or rope.

One day in a fit of curiosity, I went searching for more information about Thomas Paine the Bridge Builder. I posted the results on this blog.

Circumference The Latin roots of this word unite circus, the root of circle, with ferre which meant, "to carry", thus literally "to carry around".
The word is the Latin translation of the Greek word periphereia with the same meaning. The word was used before Euclid and seems to have been common in mathematics. According to Jeff Miller's excellent web page on the first use of some math words, "..... Periphereia was used by Heraclitus (who wrote) The beginning and end join on the circumference of the circle (kuklou periphereias)". Perhaps its long history accounts for the fact that the Greek form of the word also persisted in Latin as a borrowed word and made its way into English in such words as peripheral.

Complex Numbers The use of the word complex for a number of the form a+bi, where i= was introduced by Karl F. Gauss around 1830. Gauss wanted to distinguish between the imaginary part bi and the sum of the real and imaginary part and used the phrase "numeros integros complexos". The first use of the term in English seems to have happened about 25 years later in a paper by Sir William R Hamilton.

Cosine, Cotangent, Cosecant The cosine function was created to compute the sine of the complimentary angle, and hence its name. The Hindu mathematicain Aryabhata (475-550 AD) called it kotijiva (see Sine for relation of jiva).
Edmund Gunter created the terms "co.sinus" and "co.tangens", one of which was quickly modified to "cosinus" by John Newton around 1660. By 1675 Sir Jonas Moore had abbreviated this down to "cos". [Jonas Moore played an important part in drainging the fens of East Anglia as the chief surveyor of the Dutch dredging expert Cornelius Vermeyden during the 1650's] Other abbreviations have been attempted. William Oughtred used "t" for tangent and "t co" for cotangent while John Wallis used a capital and a lowercase t. Richard Norwood used "tc" (tangent compliment) as his symbol for contangent and "sc" for cosine. Even today some European texts use "tg" and "ctg".

Jeff Miller's web site on "Earliest Known Uses of Some of the Words of Mathematics " credits cosecant to Rheticus, "The Latin cosecans appears in Opus Palatinum de triangulis ("The Palatine Work on Triangles"), which was written by Georg Joachim von Lauchen Rheticus (1514-1574). This treatise was published after his death by his pupil Valentin Otto in 1596. According to Ball (page 243) and Smith (vol. 2, page 622), the term seems to have been first used by Rheticus."

Denominator The bottom number in a rational fraction serves to name the type of fraction being counted in the numerator (see numerator). The root nomen for name, carries this meaning. Nominate, noun, nomenclature, and of course, name are all derived from this same root.
The word, denominans, appears in the famous math book Liber Abbaci by Leonardo de Pisa which brought the Arabic numerals to the Western World. [See the historical note below under numerator]

DodecahedronA dodecahedron is one of the five Platonic solids. It is formed by twelve faces that is each a regular pentagon. The name is derived from the Greek root for the number twelve duodeka, literally ten and two. The figure is produced by WinGeom, a wonderful freeware program, provided by Richard Parris of Phillips Exeter Academy. The Platonic Solids page above also has a link to a site where you can download Nets of a dodecahedron to cut and fold to make your own models.

A twelve sided plane figure is called a dodecagon. It is known that the area of a regular dodecagon inscribed in a unit circle has an area of 3 square units. This is called Kurschak's theorem after the Hungarian mathematician Jozsef Kurschak. This link shows a really nice animated disection proof of the theorem by Antonio Gutierrez.

Fibonacci's Sequence This sequence is named for Leonardo of Pisa (1175-about 1250). Now he is also called Leonardo Fibonacci, which seems to have been a contraction of Filius Bonacci, son of Bonacci, although the name was never applied during his lifetime. He was one of the three men who are most responsible for introducing the Arabic numerals and methods of algebra to the Western World. In his classic book Liber abaci he poses, and solves the famous rabbit problem which produces the now famous sequence, {0, 1, 1, 2, 3, 5, 8, 13, 21...} in which each value is the sum of the two previous values.

Although it is almost certain that he knew, Fibonacci never wrote that each term was found by adding the two previous terms. The first record of such a statement occurred almost 400 years after Fibonacci by Kepler. Almost another hundred years would pass before R. Simson, for whom the Simson line is mis-named, recognized that each term was the convergent of the continued fraction

. Since all the partial quotients are one, this is the simplest, and therefore slowest to converge, of all the infinite continued fractions. Yet another 150 years would pass before the explicit formula for fn would be found by J. Binet. Letting the golden ratio be represented by he was able to express the Nth term of the series as .

It is not clear when we began to call it the Fibonacci sequence. As late as the mid-1800's French Mathematician Gabrielle Lame' proved that the number of n-digit Fibonacci numbers is at least 4 and at most 5; but Lame' did not use the name "Fibonacci numbers". In fact, they were often called Lame's numbers because of his proof. Lame also contributed to the story of Fermat's Last Theorem in 1839 when he confimed that there could be no integer values for which x7 + y7 = z7

According to Paul J. Nahin, author of AN IMAGINARY TALE, the name Fibonacci was not common until centuries after Leonardo's death, and during his lifetime he was called Bigollo, a slang term for a loafer drawn from the word bighellone. Julio Gonzalez Cabillon has written, "The name 'Fibonacci' most probably originated with the historian of mathematics Guillaume Libri (1803-1869)."

In September of 2001, Heinz Lueneburg posted a note that seemed to suggest that there may have been earlier uses than Julio suggested. He writes (with some editing by me):

I was in Rome and checked the Boncompagni paper
I quoted in my posting of August 28. The paper starts with the diskussion of what is known of persons of the Bonacci family other than Leonardo: One Matteo Bonacci is known because he is mentioned as a witness of the treaty Pisa and Genova signed on February 13, 1188.
Then he lists the names of authors who use the name "Leonardo Pisano".
Then he lists the names of authors who use the name "Fibonacci".
John Leslie 1820
Cossali 1797-99
Giovanni Gabriello Grimaldi 1790-1792
Libri 1838-1841
Chasles 1837
Nicollet 1811-1818
S. Ersch & I. G. Gruber 1818 and subsequent years
August de Morgan 1847. He also uses Bonacci.

Then he lists the names of authors who explain "Fibinacci = filio Bonacci"
Flaminio dal Borgo 1765
Tiraboschi 1822-1828
Ranieri Tempesti 1787
Giovanni Andres 1808-1817
Grimaldi 1790-1792
Libri 1838-1841

Then his arguments for Fibonacci = de filiis Bonacci follow.
Then he discusses the sobriquet Bigollone, Bigollo, Bigoloso.
Finally, in the major part of the paper, he discusses the various manuscripts of the various works of Fibonacci still in existence.
The question "who gets the credit?" is still open.
Regards, Heinz Lueneburg

Those interested in learning more about the sequence may find a good reference at the Dr. Math Faq page. And there is a large body of information related to the Fibonacci sequence and the Golden Ratio at this page from England. The page is also the source of the photo at right of the statue to Fibonacci in the cemetary at the Duomo (Cathedral) in Pisa.

A nice biorgraphy of Leonardo of Pisa can be found at a page by Clark Kimberling. The page shows an earlier picture of the same statue prior to its restoration when it stood on the street named for Fibonacci. My thanks to Gian Marco Rinaldi who corrected a previous mistake about the statue. He also introduced me to a web page where Alberto Rodr�guez Santos maintains a site which has a history of the many moves of the statue around the city over the years. On my last day of my first visit to Pisa I missed a turn and came upon the Via Fibonacci quite by accident.

Imaginary Numbers The word imaginary was first applied to the square root of a negative number by Rene Descartes around 1635. Descartes wrote that although one can imagine every Nth degree equation had N roots, there were no real numbers for some of these imagined roots. Around 1685 the English mathematician, John Wallis wrote, "We have before had occasion to make mention of Negative Squares, and Imaginary Roots". Some mathematicians have suggested the name be changed to avoid the stigma that it seems to create in young students; "Why do we have to learn them if they aren't even real?", but the weight of unfortunate history seems too much to overcome. Prior to Descartes ill-fated use of imaginary, the square roots of negatives had been called sophisticated or subtle.

The first person ever to write about employing the square roots of a negative number was Girolamo Cardano (1501-1576). In his Ars Magna (great arts) he posed the problem of dividing ten into two parts whose product was forty. After pointing out that there could be no solution, he proceeded to solve the two equations x+y=10 and xy=40 to get the two solutions 5+ (-15) and 5- (-15). He then multiplied the two answer to show that their product was indeed forty, and concluded that the process was "as subtle as it is useless." You can see the cover sheet of Cardano's masterpiece here, and a partial translation here.

Cardano also left a seed to inspire future work in the mystery of roots of negative numbers. Cardano had published a method of finding solutions to certain cubic equations of the form x3 = ax + b. His solution required finding the square root of (b/2)2 - (a/3)3. For equations in which the value was negative, his method seemed not to work, even in cases that had a known real solution like x 3 = 15x + 4 with a known solution of 4.

About thirty years later Rafael Bombelli found a way to produce an answer of four using Cardano's method and developed a set of operations for these roots of negative numbers. By the eighteenth century the imaginary numbers were being used widely in applied mathematics and then in the nineteenth century mathematicians set about formalizing the imaginary numbers so that they were mathematically "real".

The St Andrews University website on math history has a longer article about the development of the solutions for quadratic, cubic and quartic equations that includes Cardano's contribution.

At Jeff Miller's web site on the first use of math symbols I found the following discussion on the origin of the symbol "i" for the square root of negative one:

i for the imaginary unit was first used by Leonhard Euler (1707-1783) in a memoir presented in 1777 but not published until 1794 in his "Institutionum calculi integralis."
On May 5, 1777, Euler addressed to the 'Academiae' the paper "De Formulis Differentialibus Angularibus maxime irrationalibus quas tamen per logarithmos et arcus circulares integrare licet," which was published posthumously in his "Institutionum calculi integralis," second ed., vol. 4, pp. 183-194, Impensis Academiae Imperialis Scientiarum, Petropoli, 1794.
Quoniam mihi quidem alia adhuc via non patet istud praestandi nisi per imaginaria procedendo, formulam littera i in posterum designabo, ita ut sit ii = -1 ideoque 1/i = -i.
According to Cajori, the next appearance of i in print is by Gauss in 1801 in the Disquisitiones Arithmeticae. Carl Boyer believes that Gauss' adoption of i made it the standard. By 1821, when Cauchy published Cours d'Analyse, the use of i was rather standard, and Cauchy defines i as "as if was a real quantity whose square is equal to -1."
Throughout his Introductio, Euler consistently writes , denoting by i the "numerus infinite magnus" [namely, an infinitely large number]. Nonetheless, there are very few occasions where Euler chose i with a different meaning. Thus, chapter XXI (volume 2) of Euler's Introductio contains the first appearance of i as quantitas imaginaria:
Cum enim numerorum negativorum Logarithmi sint imaginarii (...) erit log(-n) quantitas imaginaria, quae sit = i. The citation above is from "Introductio in analysin infinitorum," Lausannae, Apud Marcum-Michaelem Bousquet & socios, M.DCC.XLVIII (1748). Please note that, in this fascinating passage about logarithms, Euler does not introduce the symbol i such that i^2 = -1.
[This entry was contributed by Julio Gonz�lez Cabill�n.]

Integer The base root comes from the proto Indo-European root tag for touch. The root evolved into a nasalized tang and became the root for many "touch" words such as tangent and contaminate.
In integer it means "untouched" and that was the original Latin base meaning. It also carried peripheral meanings associated with being untouched, such as virtuous, pure, and whole or complete. These last terms brought it into mathematics.
It appears, according to the Oxford English Dictionary, that the word was first used as a noun in English by T. Digges in a 1571 in Pantometria, a book on geometry.

I recently found a note on a statistics history web page that said that Fibonacci used the term sano from the Latin for Healthy or whole for integer values. It also had, "According to Heinz Lueneburg, the term numero sano 'was used extensively by Luca Pacioli in his Summa'. Before Pacioli, it was already used by Piero della Francesca in his Trattato d'abaco. I also find it in the second edition of Pietro Cataneo's Le pratiche delle due prime matematiche of 1567. I haven't seen the first edition. Counting also Fibonacci's Latin numerus sanus, the word sano was used for at least 350 years to denote an integral (untouched, virginal) number. Besides the words sanus, sano, the words integer, intero, intiero were also used during that time.'

Isometry Isometry is from the Greek roots iso for same, and metros for measure, and literally interpret to same measure. Two systems are isometries if they preserve measures. In geography two points on the Earth are isometric if they are the same distance above sea-level.

Isomorphic Isomorphic is another word from the Greek root iso for same. The second root morphos for shape. Two systems are isomorphic if the image of A operate B is the same as the image of A operated on the image of B, or more mathematically, (A*B)'=A'*B'.

Jerk The origin of the common English word jerk is unknown. The mathematical origin seems to have been in 1955 by J. S. Beggs. The mathematical meaning of jerk is the instantaneous rate of change of the acceleration, . This is also expressed as the third derivative of position with respect to time.

Limit The Latin word for the fence or path between two pieces of land was limes. The idea of a point or boundary beyond which something does not extend is the basis for both the common English meaning of limit, and the mathematical meaning. Isaac Newton first used limit in a mathematical sense around 1725.

Locus The set of all points are "places" that meet a given set of geometric conditions is called the locus of those conditions. For example, a mathematician might say, "The locus of all points equidistant from a given point is a circle." The ancient Greek geometers would have used the word topos which was the Greek word for place. When these works were translated into Latin, the word was replaced with the Latin word for place, which was locus. That word persisted in mathematics to the present.

Mantissa Mantissa is probably one of those math words that calculators will render obsolete. In the times before calculators, a mathematician who needed to work with logarithms would look up the decimal part, the mantissa, in a table of logarithms. The whole number part of a logarithm is just a measure of magnitude, for example, log(2.3), log(23), and log(230) all have the same decimal part or mantissa.
The word seems to have come to us from the ancient Etruscans through the Latin language where it meant some small addition or something of minor value. Over time it came to mean an appendix. John Wallis used the word "appendage" in his English works, but then in the Latin translation applied the term "mantissa". As with some other words, it was not applied widely until it was used by Euler. The OED suggests that Wallis was the first to use the word in a mathematical sense, "but appears in fact to have originated with J. Wallis in his Opera Math. (1693) II. x. 41, where the Latin word is used for the digits of a decimal number following the decimal point (in the English edition of his Algebra (1685) Wallis used �appendage�); the Latin word was used in relation spec. to the decimal part of a logarithm by L. Euler in his Introductio in Analysin Infinitorum (1748) I. vi. 83.]" At one time Gauss suggested using the word for the decimal part of any mixed number.

The word is also sometimes used for the decimal number multiplier in floating point arithmetic. The number 63, 400 may be written as .634 x 105, and in this case the decimal number .634 is also sometimes called the mantissa


The first mathematical use of the word matrix was around 1850 by Sylvester. Sylvester saw a matrix as a way of obtaining determinants, but did not fully realize their potential. Within a year of his first use of the term he introduced the idea to Cayley who was the first to publish the inverse of a matrix and treat them as purely abstract mathematical forms. Sylvester was also the first to use the word "minor" (see same below) for a smaller matrix. I came across the following anecdote about Sylvester and didn't know where else to put it:

Sylvester, incidentally, spent a very brief period teaching at the University of Virginia. It ended after an enraged Sylvester hit a student reading a newspaper with a stick and fled the country believing he had killed him.

The use of mathematical arrays to solve problems predates the application of the name by about 2000 years. Around 200 BC in the Chinese text Juizhang Suanshu (Nine Chapters on the Mathematical Arts) the author solves a system of three equations in three unknowns by placing the coefficients on a counting board and solving by a process that we today call Gaussian Elimination. Elimination would not become well known in the West until the beginning of the Nineteenth Century. By the early 1500's, Cardan states a rule, regula de modo, for solving two equations in two unknowns by much the same method as Creamer's rule.

The word matrix comes from the same Latin root that gives us mother, and was used to refer to the womb, and pregnant animals. It became generalized to mean any situation or substance that contributes to the origin of something.


Measurement, exact or approximate, stretches back to man's earliest mathematical efforts, and the Indo-European root of measure me shows up today in words that at first thought often seem unrelated to measuring. The Greeks extended the word to metron from which we get meter, metric. The Latin variant of this took the forms of med or men, and gave us words like moon, month, menopause and semester (semi-meter). The old German variation had an "l" ending and gives us meal, which originally meant "an appointed time". The time meaning of meal still shows up in the word piecemeal, literally a piece at a time.

The word measure has often been used in much the same way we now use the word factor. In his Universal Arithmetick Newton distinguishes three kinds of numbers, "integer, fracted, and surd", and defines an integer as "what is measured by Unity." Frederick Emerson's North American Arithmetic(1850)says "One number is said to MEASURE another, when it divides it without leaving any remainder." (pg 18) Later it states," A number which divides two or more numbers without a remainder is called their COMMON MEASURE."

Minor The word minor, for small or lesser, comes to us from the Greek root meion for small. This is the same root that gives us such words as minus, minimum, and minister (a servant or one of low standing, a "magister" was a master).
The word was introduced to mathematics by J. J. Sylvester in 1850 to describe a smaller matrix made by eliminating one row and one column from a larger matrix. The image shows a 3x3 matrix at the top, and below it are two of its minors.

Numerator The top number in a rational fraction serves to tell how many of the fractional parts their are. It is drawn from the Latin "enumerate", to count out. I had a note that said that both numerator and denominator were used in Liber Abbaci(1202) by Leonardo de Pisa, however, in the notes of his English translation, L.E. Sigler states, "The modern terms numerator and denominator are used respectively to translate the Latin terms denominatus and denominans used by Leonardo."
The note refers to a statement in the web page of Jeff Miller, a portion of which is shown below:The words NUMERATOR and DENOMINATOR are found in 1202 in Liber abbaci by Leonardo of Pisa:
Cum super quemlibet numerum quedam uirgula protracta fuerit, et super ipsam quilibet alius numerus descriptus fuerit, superior numerus partem uel partes inferioris numeri affirmat; nam inferior denominatus, et superior denominans appellatur. Vt si super binarium protracta fuerit uirgula, et super ipsam unitas descripta sit ipsa unitas unam partem de duabus partibus unius integri affirmat, hoc est medietatem sic 1/2 [When above any number a line is drawn, and above that is written any other number, the superior number stands for the part or parts of the inferior number; the inferior is called the denominator, the superior the numerator. Thus, if above the two a line is drawn, and above that unity is written, this unity stands for one part of two parts of an integer, i. e., for a half, thus 1/2].
Numerator and denominator are found in Algorismus proportionum by Nicole Oresme (ca. 1323-1382). The work is in Latin but the words are spelled as they are in English, and are defined as "the number above the line" and "the number below the line" (Cajori vol. 1, page 91).

Per cent The term for the parts out of each hundred comes to us almost unchanged from its Latin roots per for each, and centum for hundred. In fact, "per centum" is still found in many dictionaries as an antiquated term for percent. The symbol, %, is of uncertain origin. One source suggests it may have been used as early as 1425.

Math historian D. E. Smith traced the development of the % symbol from a manuscript in 1425 from an unknown writer. Prior to this date, he suggests it was common for Italian writers to use �per 100� or �per c� (c was the Roman Numeral for 100) for percent. It was also common at that time to use a raised circle directly above a numeral or as a superscript to indicate primary and secondary parts�. 1o, 2o, 3o, etc. �. . In the 1425 manuscript the author used a script p with a circle above the letter c, which was written with a long upper curve to represent �per c�. The �per� disappeared over time from the written symbol, and the circle over c became a circle over a circle, perhaps through careless writing and recopying of the symbol. By 1650 the horizontal vinculum was often replaced with a slanted solidus to produce 0/0. Writing quickly the first zero often ran into the bar making a version of the common percent symbol, % .

Around 1900, apparently using the idea that since the two zeros present in % represented 100ths, a symbol like 0/00, for per mille could represent thousandths. It appears not to have lasted.

Apparently the last statement is not true. The per mille symbol seems to have lasted in some particular contexts. Early in 2003 while researching for a new textbook, Rex Boggs posted a copy of the previous information to a newsgroup and quickly received the following correction from Victor Steinbok

Not true. The "per mille" or "promille" is common notation to describe salinity of sea water. I imagine there are other *applied* areas where it might have escaped attention of mathematicians. I would look in old (30-40 years ago) environmental reports. I've even seen 0/000 and 0/0000 used a couple of times. Some old typewriters (particularly, I believe, those of German, Scandinavian and Eastern European origin) even used to have a 0/00 key--actually the shifted position of 0/0 key. Perhaps someone can find one and take a picture for Pat's collection."
Rex Boggs then added, "I even found the symbol in Windows! Select Insert, Symbol, normal text. Find the General Punctuation subset, and there it is." To which I could only add "OOOPS", and Thanks to Victor and Rex. And here it is

Pick's Thm. This is a theorem for finding the area of a polygon if all the vertices lie on lattice points (points whose x and y coordinates are both integers). The formula is named for Georg Pick and was presented by him in 1899. The formula states that if N is the number of lattice points inside the polygon, and B is the number of lattice points on the boundary, then the area is given by Area = N + B/2 -1. In the figure below N=31 and B=15, so the area should equal 37.5 square units.
You can find more about the theorem and links to some of its extensions at Drexel University where I also found that the original theorem was published in "Geometrisches zur Zahlenlehre" Sitzungber. Lotos, Naturwissen Zeitschrift Prague, Volume 19 (1899) pages 311-319. They also have the image shown here.

Power The word power comes from the French poeir and perhaps the earlier Latin word potere from which we get potent. Both words refer to ability or being able. In mathematics, power refers to the number arrived at by raising a number to an exponent. In the mathematical expression 3^2=9, three is the base, two is the exponent, and nine is the power. Students often refer to the exponent as the power, but this is not historically correct, although it has become so common, even among many teachers, that some dictionaries refer to the power as the exponent. From Jeff Miller's web page on the first use of some math words I found that "POWER appears in English in 1570 in Sir Henry Billingsley's translation of Euclid's Elements: 'The power of a line, is the square of the same line.' ".

Proportion The word proportion comes from the Latin proportio, a translation from the Greek word for analogy . The roots pro, for, and portio, share, literally combine to mean "for (his or its) share". The portio root is an extension of the earlier Latin root pars and they are preserved in words such as portion, parity, compare, peer, pair, and the golfer's favorite, par.

One of the earliest symbols for a geometric proportion has lasted, with minor modification, to the present day. William Oughtred developed the symbol a.b::c.d in his Clavis mathematicae in 1631. Since the "=" sign had become common by then, one wonders that he did not use it, or the ae symbol for equality, in place of the four dots. The use of a single dot to signify proportions became confusing with the increasing use of the decimal point, and Wing and others modified the symbol to a:b::c:d. With an equals sign replacing the four dots, the symbol continues to this day.

The four values in the proportion are called the terms. The first and fourth term, a and d in the previous example, are called the extremes of the proportion. The two inner terms, b and c, were called the means. All solutions to proportions are essentially founded in one rule, the product of the means is equal to the product of the extremes. When proportions are written as an equivalance between two fractions, and they frequently are, the two numerators (a and c) are often called the antecedents[literally, that which comes first from the Latin, antecedens]. The other two terms are called the consequents[that which occurs as the result of the antecedent; frequently the terms suggested a cause and effect relationship].

A Mean proportional between two numbers is a number b such that a:b = b:c. The Mean proportional is also called the geometric mean of the two numbers. In the same proportional ordering, a:b = b:c, c is called the third proportional to a and b. Students in the 1800's were often instructed to set up proportions so that the unknown came last, a:b=c:x, and the unknown x was called the fourth proportional. The fourth proportional was solved by using the Rule of Three

Quotient "How many times?" is the question in a division problem. How many times can you make a group of that many from this? And the name for that answer comes from the Latin root quot for that very question, how many. The word seems to have made its way into English in the fourteenth century. The OED credits this to the English mathematician, John of Holywood, also known as Sacrobosco; " c1430 Art Nombryng (E.E.T.S.) 12 Above at most sette a cifre in ordre of the nombre quocient".

Quote, quota, and quorum all come from the same root.

Calculus students have to learn The Quotient Rule in first semester calculus, and frequently get the order backward, causing an error in the resulting sign of the derivative. Many learn the following song, credited to Edward Newman:

If the quotient rule you wish to know,
It's low-de-high less high-de-low
Then draw the line and down below
Denominator squared will go.
Somewhere I picked up the following simple variant, which I teach to my students: "It's bottom-de-top minus top-de-bottom; all over bottom-bottom."


The word Radian is believed to be a made up word. Some suggest it may have been intended as an abbreviation for "RADIus ANgle".
Here is a quote from Cajori's History of Mathematical Notations, vol 2 (1929) as provided by Julio Cabellion to the Historia-Matematica Newsgroup:

"An isolated matter of interest is the origin of the term 'radian', used with trigonometric functions. It first appeared in print on June 5, 1873, in examination questions set by James Thomson at Queen's College, Belfast. James Thomson was a brother of Lord Kelvin. He used the term as early as 1871, while in 1869 Thomas Muir, then of St. Andrew's University, hesitated between 'rad', 'radial' and 'radian'. In 1874, T. Muir adopted 'radian' after a consultation with James Thomson. (+)" (+) _Nature_, Vol. 83, pp. 156, 217, 459, 460.

The concept of a radian measure, as opposed to the degree of an angle, should probably be credited to Roger Cotes. According to a recent post to a math history newsgroup by Bob Stein; "He then calculated this as approximately 57.295 degrees. He had the radian in everything but name, and he recognized its naturalness as a unit of angular measure." In the same string, Don Cook added that "Joe Zund and I did a search for who was the first person to discover that the derivative of the sine was the cosine. It was Cotes. "

The picture below shows an angle with one radian measure which is approximately the same as 57 degrees.

I came across the idea of using a symbol for radian measure in an old thread on the Math History groupOn Tue, 21 Nov 1995 C.J.Earle wrote:
In some textbooks in use in England (and elsewhere possibly) values which are expressed in radian measure are denoted by the use of a superscript c after the value. [inserted note... such as writing 1.2c for 1.2 radians] Rather like the way that the degree symbol is used.
As radian measure is a ratio and therefore not expressed in any unit based system this seems technically incorrect.
Does anyone know the origins of this usage and how widely it is found?
John Earle


The earliest discussion I know of regarding symbols for radian measurement can be found in Florian Cajaori's "A History of Mathematical Notations," volume 2, pp. 147-148. The little c is there together with other symbols, and the usual sources for Cajori's remarks.
Barnabas Hughes

The "c" you speak of stands for "circular measure". So it isn't really the name of a unit, and doesn't raise the "correctness" problem you allude to.
John Conway

Tetrahedron A tetrahedron is the most simple of three space shapes since it consists of only four vertices (see figure below). The Greek tetra stands for four, and can still be found in some science words such as tetrachloride or tetravalent. The hedra is from the Greek for base, or seat. See polyhedron.

Velocity In mathematics, velocity is the rate of change of position with respect to time. In more general terms it can be thought of as the speed with which an object moves along its path. The ancient origin of the word is from a Indo-European root for healthy or strong, and the more modern usage comes from the Latin velox for fast. The French adapted this to velocite which, with minor changes, produced the word we use today. Related words in use today include vigil, vigor, and vegetable. From the same root through the German dialects we get watch (as in to stand watch), and awake.