Origins of some Math terms

Math Words Alphabetical Index

Abscissa is the formal term for the x-coordinate of a point on a coordinate graph. The abscissa of the point (3,5) is three. The word is a conjunction of ab(from) + scindere (tear). Literally then, abscissa is a line that has been cut or torn from another line. The main root is closely related to the Latin root from which we get the word scissors. I have a note that credits Leibniz with the orinin of the term in 1692, but in 2006 I received a note from Professor Barney Hughes that, "Fibonacci used the word in our meaning several times in his book, De practica geometrie. " .

I read a blog (ok, I read it because it mentioned me..in fact, misquoted me, but there was still some new stuff there, so : They quoted Jeff Miller's site (always a good bet to be very accurate) as saying that the first use was in 1659 in Miscellaneum Hyperbolicum, et Parabolicum by Stephano degli Angeli. The site also quoted some translation of the usage by Fibonacci, and the usage seemed distinctly different than the present usage.. here is one for you to decide. "Not to be overlooked is to show how to find the square on line eb called the residue, recisum, or abscissa. It is the difference between two lines commensurable only in their squares, such as between lines ae and ab. For example, let ae be the root of the rational number 720 and ab the number 10. Because line ae was divided into two parts at point b, the squares on lines ae and ab equal twice the product of ab by ae and the square on line eb, as was shown above. Therefore subtract twice the product of ab and ae from the squares on lines ae and ab; that is, subtract 20 times the root of 720 from 820. Now 20 roots of 720 equal the root of 288000, the number arising from the product of 400 the square of 20 and 720. The residue then is 820 less the root of 288000"

Absolute Value The word absolute is from a variant of absolve and has a meaning related to free from restriction or condition. The first use of "absolute value" in English seems to have been to apply to real values. Jeff Miller's

website on the Earliest Known Uses of Some of the Words of Mathematics says," Absolute value is found in English in 1850 in The elements of analytical geometry; comprehending the doctrine of the conic sections, and the general theory of curves and surfaces of the second order by John Radford Young (1799-1885): "we have AF the positive value of x equal to BA - BF, and for the negative value, BF must exceed BA, that is, F must be on the other side of A, as at F', hence making AF' equal to the absolute value of the negative root of the equation" [University of Michigan Digital Library]." [See the page here] In 1876 Karl Weierstrass applied the term to magnitude of complex numbers. From Miller's site again we find "Absolute value was coined in German as absoluten Betrag by Karl Weierstrass (1815-1897), who wrote: Ich bezeichne den absoluten Betrag einer complex Groesse x mit |x|. [I denote the absolute value of complex number x by |x|.]"

In "The Words of Mathematics", Steven Schwartzman suggests that the use of the word for real values only became common in the middle of the 20th century. This may be true, but the use for signed numbers also appears in 1889 by Wentworth according to Miller; "In 1889, Elements of Algebra by G. A. Wentworth has: 'Every algebraic number, as +4 or -4, consists of a sign + or - and the absolute value of the number; in this case 4.' " (above). In the 1893 edition of the same book he uses the term again, as shown below, without any symbol.

The revision of Hall and Knight's Algebra, for Colleges and Schools {"Revised and Enlarged for the use of American Schools"} by F. L. Sevenoak in 1905 also uses the term without a sign. By 1934, the word is still used without symbol in Walter W. Hart's Progressive First Algebra,(pg 78), but in the 1939 edition of College Algebra by Rosenbach

and Whitman, the symbol is used as shown below

The symbol for absolute value is usually a pair of vertical lines containing the number, as created by Weierstrass in 1876 (see above). |3| is read as "The absolute value of three". The absolute value of a real number is its distance from zero, so |3| = |-3| = 3. In words that says that the absolute value of three is equal to the absolute value of -3 , and that both have a value of three.

For complex numbers the absolute value is also called magnitude or length of the complex number. Complex numbers are sometimes drawn as a vector using an Argand Diagram, and the length of the vector Z=a+bi is |a+bi|. Stated another

way, the value of |a+bi|=

A symbol for the Absolute Difference of two numbers, or the absolute value of the difference was created by Oughtred around 1630. Miller writes, "The tilde was introduced for this purpose by William Oughtred (1574-1660) in the Clavis Mathematicae (Key to Mathematics), composed about 1628 and published in London in 1631, according to Smith, who shows a reversed tilde (Smith 1958, page 394)." I have seen this symbol used in an American text as late as 1893 when Irving Stringham used it in his list of symbols in Uniplaner Algebra. The symbol seems no longer to be common in basic maths classes in the US or in England today. After posting a request for information to the Historia Matematica discussion group about the use of the tilde to indicate absolute difference in England I received the following update from Herbert Prinz:

"In modern English texts on navigation, nautical astronomy or its history, the tilde is frequently used to express the function | a - b |, where |x| stands for absolute value. E. g. Cotter, The Complete Nautical Astronomer, 1969. I am not sure when this practice started. In older texts on the same subject, say, Moore, The Practical Navigator, 1800, one does not find the tilde used in this way. For one, because

instructions were given mostly verbally without the use of any symbols at all. And second, the distinction from '-' was unnecessary, as it was always understood, if not explicitly stated, that one must subtract the smaller

In England the absolute value is often referred to as the modulus function, and the two bars that make up the symbol are sometimes called "modulus signs" according to a note posted by Vicky Neale on the Ask NRich math site. The term modulus is used both in America and England to represent the magnitude or length of a complex number. The term is also used in a number of other specialty ways in mathematics, the best known being the "congruence modulus". The modulus of a congruence, often shortened to "mod" is the base value with which the congruence is computed. We say A is Congruent to B modulus C, if A divided by C and B divided by C have the same remainder. C is called the modulus of congruence. It would be written AB [mod C]

Modulus comes almost unchanged from the Latin from the diminutive of modus (measure or amount), modulus for a small measure. Vicky also pointed out that at one time the term was used for, "A unit of payment used at Trinity College.... Fellows received some number of moduli". Ms Neale also said she was unfamiliar with the use of the ~ for absolute difference.

It was Gauss, Disquisitiones arithmeticae in 1801, who introduced the term modulus of congruence, and the abreviation, "mod". Cajori credits Jean Argand for the first use of modulus for the length of a vector in 1814. I am not sure when the British public schools started to use the term for the absolute value of a number, and would love to know if someone has old books with these terms (or others for the same idea).

Acute is from the Latin word acus for needle, with derivatives generalizing to anything pointed or sharp. The root persists in the words acid (sharp taste), acupuncture (to treat with needles) and acumen (mentally sharp). An acute angle then, is one which is sharp or pointed. In mathematics we define an acute angle as one which has a measure of less than 90o.

The first use of the term in Enlish was in Henry Billingsley's translation of the Elements of Euclid. "An acute angle is that, which is lesse then a right angle"; "an obtuse angle is that which is greater then a right angle" .

Algebra comes from a book written in Arabic that revolutionized how mathematics was done in western cultures. "Al-jebr w'al-mugabalah" written by Abu Ja'far Ben Musa (about 825 AD) who was also known as al-Khowarizmi. He is as famous among Arabs as Euclid and Aristotle are to the Western World. He was probably the greatest living mathematician of his period. The phrase Al-jebr at the start of the title became the word Algebra in western languages. The phrase loosly translated means "the reunion of broken parts". Later, in medieval Europe, "algebrista" was became a term for the person who set bones (the reunion of broken parts) and since it was the barbers who did the bonesetting and blood-letting, they were called an "algebrista".

Abu Ja'far Ben Musa is often mistakenly listed as an Arab mathematician, but was in fact Persian, and Khowarizmi refers to the area which was his home. Modern scholars believe he was born near the Aral sea in what is now Turkestan. The literal translation of his name means "father of Jafar and Son of Musa, from Khowarizmi."

The first use of the word "algebra" in English was by the Welsh mathematician and textbook writer, Robert Recorde in his Pathway of Knowledge written about 1550.

Abu Ja'far Ben Musa was also the source of the word algorithm (see below). His book, above, also includes the first use of what we would today call the quadratic formula; although his description was verbal and not in modern mathematical notation.

Algorithm, as it is used in mathematics means a systematic procedure to solve a problem. The word is derived from the name of the Persian mathematician, al-Khowarazmi (See algebra). The first use of the word I am aware of was by G W Liebniz in the late 1600.

Julio Gonzalez Cabillon posted the notes below in summarizing earlier posts at Historia Matematica

I checked out the Latin of this Leibniz' first published account of the calculus [_Acta Eruditorum_, vol. 3, pp. 467-473, October 1684], and I certainly find the word "Algorithmo". [*] "Nova Methodvs pro maximis et minimis, itemque tangentibus, quae nec fractas, nec irrationales quantitates moratur, & singulare pro illis calculi genus, per G.G.L." (Leibniz's initials in Latin): On page 469, Leibniz states: "Ex cognito hoc velut *Algorithmo*, ut ita dicam, calculi hujus, quem voco *differentialem*, omnes aliae aequationes differentiales inveniri possunt per calculum communem, maximae que & minimae, item que tangentes haberi, ita ut opus non sit tolli fractas aut irrationales, aut alia vincula, quod tamen faciendum fuit secundum Methodos hactenus editas." A few comments: 1. Both terms "Algorithmo" and "differentialem" are italicised in the original. This must be emphasized, since either Smith or his editor overlooked this 'petit' detail in "A Source Book in Mathematics". 2. Please note that in the quoted passage, Leibniz employs "algorithm" (in the sense of a systematic technique for solving a problem) with a meaning that may suggest a new term -- the context, and the italics conveys that possibility. 3. Apparently, the first English translation of Leibniz' "Nova Methodvs pro maximis et minimis..." was carried out by Joseph Raphson in "The Theory of Fluxions, Shewing in a compedious manner The first Rise of, and various Improvements made in that Incomparable Method", London, 1715. Thereby, most probably the earliest printed appearance, in English, of the term ALGORITHM (in the sense of a systematic technique for solving a problem) is in that treatise.

Algorithm remained a little known and little used term in

western mathematics until the Russian mathematician Andrei Markov (1903-????) introduced it. The term became very popular in the areas of math focused on computing and computation.

Analogy The word analogy comes from the early Greek roots ana + logos . Logos was the early Greek root for lots of related mental constructions such as word, speech, logic, and reason. An analogy refers to things that share a similar relation. Originally it was more of a mathematical term interchangeable with ratio or proportion; as in "2,4,8 is analogous to 3,6,12". Later this idea of similar relations was extended to things that shared a logical relationship. Analog clocks and computers are so named because they operate off mechanical objects (gears and pulleys) that transform motions in proportional movements.

Angle comes from the Latin root angulus, a sharp bend. As with many g sounds the transfer from Latin to the German and English languages switched to a k spelling. The word ankle is from the same root. An angle is formed by two rays with a common endpoint.

The word Angles for the Germanic tribe that invaded England in the 5th century, and from which words like Anglo-Saxon and English are derived, was also drawn from the same root. "The Angles, says the OED, are the people of Angul "a district of Holstein, so called from its shape"; it goes on to say that Angul is the same word as the Old English, Old Saxon and Old High German angul, a fish-hook - which gives us the English word angling." [granthutchison, post on Agora]

Apothem The distance from the center of a regular polygon to the sides, the apothem, comes from the Greek term "to set off", as in to set apart. The word is frequently pronounced "a poth' em' with the accent on the second syllable, but the traditional, and dictionary pronunciation is with the accent on the first syllable, "ap' e thum" as in apogee, which

shares the ap root, and means off from the Earth (gee from geos). Apothem appears to be of modern origin despite its ancient name, and seems to have first appeared in English in the mid 1800's

According to Jeff Miller's website on the first use of math terms:

APOTHEM is found in 1828 in Elements of Geometry and Trigonometry (1832) by David Brewster (a translation of Legendre): The radius OT of the inscribed circle is nothing else than the perpendicular let fall from the centre on one of the sides: it is sometimes named the apothem of the polygon.

Are An are is a unit of measure for area equal to 100 square meters. The word, and the unit of measure, seems to have been created by the French and derived from the Latin word area with its current meaning. The are is seldom used today, but its derivative form, the hectare, is still a common unit of land measure in some countries.

Arithmetic was the Greek word for number, and is closely related to the root of reckon, which is becoming an obsolete term for count (except in some parts of the western and southern US where they "reckon" almost anything). . . . (that was a joke folks). In the middle ages the best mathematicians of Germany were called Reichenmeister and their arithmetic texts were reichenbucher The beginning of the word is drawn from the Indo-European root ar which is related to "fitting together" and gives us words like army, and art. Order, adorn, and rate all come from variants of the same root.

The first arithmetic book published in North America was Sumario compendioso de las quentas de plata y oro que in los

reynos del Piru son necessarias a los mercaderes y todo

genero de tratantes Los algunas reglas tocantes al

Arithmetica. The title translates to "Comprehensive Summary of the counting of silver and gold, which, in the kingdoms of Peru, are necessary for merchants and all kinds of traders". The author was Brother Juan Diez, a priest who

arrived in Mexico with Cortez in 1519. The following is clipped from an article, THE SUMARIO COMPENDIOSO: THE NEW WORLD'S FIRST MATHEMATICS BOOK , in the Mathematics Teacher in February of 2001 by Shirley Gray and C. Edward Sandifer.

The author, Brother Juan Diez, arrived in Mexico with Cortez in 1519. In 1536, a printing press was set up in Mexico City, and the following year, it went into operation and was used for printing religious books. In 1556, the Sumario became the first book that was not a religious book, and the twenty-fifth book of any kind, to be published in the New World. The publication date of 1556 is remarkable. It was long before any settlement in Jamestown (1607), Plymouth Colony (1620), or Quebec City (1608). The New World's first mathematics book in English was not published until 1703. A Dutch mathematics book was published in 1730; a German book, in 1742; a French book, in 1775; a Portuguese one, in 1813; a Hawaiian one, in 1833; and a mathematics book in Choctaw in 1835. Of all the colonial mathematics books, the ones in Spanish are the most interesting because they were mostly written in America for use by people living in America. Books from the other colonies were mostly American editions of European books or else were closely based on European editions.

According to Bruce Burdick of Roger Williams University, "The New World's first printed arithmetic (as opposed to a book, like the Sumario Compendioso, that contains arithmetic or algebra but whose main theme was something else) was the Arte para Aprender by Pedro de Paz (Mexico, 1623)." For those seeking more information about early Spanish math books in the Americas, Professor Sandifer has an article about "mathematics books published in the Spanish American colonies before 1700" and another article on the Breve Arithmetica on his web page.

Associative The root of the word associative, is the Greek root for our word social, soci. The first use of the word in the sense of a mathematical property was probably by W R Hamilton around 1850.

Association in mathematics refers to changing the grouping of objects to be operated upon first. Since addition and multiplication are binary operations (they work with two numbers at a time) if we wish to add three numbers we have to choose which two to add first. The associative property of addition says that in adding 2+4+7, the same result will occur if we add 2+4 first, and then add the result to 7 as would occur if we added 4+7 first, and then added 2 + (the answer to 4+7). Formally the distributive property of addition is written (A+B)+C= A+(B+C). There is an identical property for multiplication.

Asymptote The asymptote of a function as it is now used is a much narrower definition than the original Greek meaning. The word joins the roots a (not), with sum (together) + piptein (to fall) and literally means "not falling together", or not meeting. The word is believed to have been known to Apollonius of Perga before 200 BC. Originally it was used for any two curves that did not intersect. Proclus writes about both asymptotic lines, and symptotic lines (those that do cross). Now symptotic is almost never heard, and asymptote is used primarily for straight lines that serve as a limiting barrier for some curve as one of its parameters approaches infinity (+/-).

The ~ symbol is often used to indicate that one function is asymptotic to another. One might write f(x)~ g(x) if the ratio of f(x) and g(x) approach 1 as x -> infinity.

The pet base of the root piptein gives us words like petition, petal, petite and propitious.

Average The meaning of average, as it is used in math today, comes from a commercial practice of the shipping age. The root, aver, means to declare, and the shippers of goods would declare the value of their goods. When the goods were sold, a deduction was made from each persons share, based on their declared value, for a portion of the loss, their AVERAGE.

In response to a question about the use of the x-bar

symbol, , for averge(mean) value of a sample, John Harper of Victoria University in New Zealand sent a response including the following information:

> R.A. Fisher used that notation, in "On an absolute criterion for fitting frequency curves Messenger of Mathematics", v. 41: 155-160 (1912) on p.157. (Univ of Adelaide has put Fisher's collected works on the Web) I don't know if he was the first. John Harper, School of Mathematical and Computing Sciences, Victoria University, PO Box 600, Wellington, New Zealand

Jeff Miller's web page provides some additional material:

for the sample mean. This usage derives from the practice of applied mathematicians of representing any kind of average by a bar. J. Clerk Maxwell's "On the Dynamical Theory of Gases (Philosophical Transactions of the Royal Society, 157, (1867) p. 64) uses v-bar for the "mean velocity" of molecules while W. Thomson & P. G. Tait's

Treatise on Natural Philosophy (1879) uses for the centre of inertia, wx / x. Karl Pearson, the leading statistician of the early 20th century, was from such a physics background. Pearson and his contemporaries used the bar for sample averages and for expected values but eventually E

replaced it in the latter role. The survival of for the sample mean is probably due to the influential example of R. A. Fisher who used it in all his works from the first, " On an Absolute Criterion for Fitting Frequency Curves," (1912).

Billion seems to have been a French creation, and was originally bi-million. The term originally meant 10^12 or one million millions, and still has this meaning in many countries today. In the US and some other countries it is used for 10^9 or one thousand million. The table below compares the names as used in the US and in Germany: Value -----German name--------US name 10^6 ----- Million ---------- Million 10^9 ------ Millard------------Billion

10^12 ----- Billion -----------Trillion 10^15------ Billiarde -------- Quadrillion

Cajori attributes the first publication of the words above million to Nicholas Chuquet. Here is a quote from his A History of Elementary Mathematics with Hints on Methods of Teaching:

Their origin dates back almost to the time when the word million was first used. So far as known, they first occur in a manuscript work on arithmetic by that gifted French physician of Lyons, Nicolas Chuquet He employs the words byllion, tryllion, quadrillion, quyllion, sixlion, septyllion, octyllion, nonyllion, "et ainsi des aultres se plus oultre on voulait proceder" to denote the second, third, etc. powers of a million, i.e. (1,000,000)2, (1,000,OO0)3, etc. Evidently Chuquet had solved the difficult question of numeration. The new words used by him appear in 1520 in the printed work of La Roche. Thus the great honour of having simplified numeration of large numbers appears to belong to the French. In England and Germany the new nomenclature was not introduced until about a century and a half later. In England the words billion, trillion, etc., were new when Locke wrote, about 1687. In Germany these new terms appear for the first time in 1681 in a work by Heckenberg of Hanover, but they did not come into general use before the eighteenth century. About the middle of the seventeenth century it became the custom in France to divide numbers into periods of three digits, instead of six, and to assign to the word billion, in place of the old meaning, (1000,000)2 or 1012, the new meaning of 109

In The Book of Numbers by John Conway and Richard Guy (pp. 14-15) they write

These arithmeticians [Chuquet and de la Roche] used "illion" after the prefixes b, tr, quadr, quint, sext, sept, oct and non to denote the 2nd, 3rd, 4th, 5th, 6th, 7th, 8th and 9th powers of a million. But around the middle of the 17th century, some other French arithmeticians used them instead for the 3rd, 4th, 5th, 6th, 7th, 8th, 9th and 10th powers of a thousand. Although condemned by the greatest lexicographers as "erroneous" (Litr'e) and "an entire perversion of the original nomenclature of Chuquet and de la Roche" (Murray), the newer usage is now standard in the U.S., although the older one survives in Britain and is still standard in the

continental countries (but the French spelling is nowadays "llon" rather than "llion". Because of continued conflict with England for the first fifty years of the new United States existance, it was much more willing to base the foundation for its numeration system on the method of the French, who had supported them in their revolution. In spite of this, "In many textbooks prior to the War of 1812 (eg. those by Consider and John Stery 1790, John Vinall 1792, and Johann Ritter 1807) if any numbers higher than 999,999,999 were discussed, the British system was used." [for example 1,000,000,000 was one-thousand million rather than one-billion ] {from Karen D. Michalowicz and Arthur C Howard in "Pedagogy in Text", from the NCTM's A History of School Mathemaitics}

Cardinal numbers are numbers that express amounts, as opposed to ordinal numbers, which express order or rank. The term is from the Latin, cardin, for stem or hinge. Cardinal today means most important or principal, with other things depending (hinging) on it. The first use appears to have been by R Percival in 1591,

Cardioid The path of a point on a circle as it rolls around another circle of the same size is sort of heart shaped and thus the term is from the Greek root for heart, kardia.

Here is a note on the origin of the term from a post by Julio Gonzalez Cabillon: CARDIOID was first used by Giovanni Francesco Mauro Melchior Salvemini de Castillon in "De curva cardiode" in the Philosophical Transactions of the Royal Society (1741). Giovanni Castillon was born on January 15, 1708, in Castiglione (hence his name), and died on October 11, 1791, in Berlin. I've taken his dates from Poggendorff's _Bibl.-lit. Handwoerterbuch_.

A nice animation of the generation of a cardiod by a circle rolling upon it is given at the Mathworld page.

The cardiod is a degenerate form of a limacon. The Polar equation of a limacon is r = b + a Cos(t). If b is smaller than a then the limacon will have an internal loop. If b is larger than a, but smaller than 2a, then the limacon will have a concave "dimple". if b is greater than 2a then the limacon is convex. When b=a, the shape is a cardiod.

At Jeff Miller's web site on the first use of math words I found,

The term LIMAON was coined in 1650 by Gilles Persone de Roberval (1602-1675) (Encyclopaedia Britannica, article: "Geometry"). It is sometimes called Pascal's limaon, for tienne Pascal (1588?-1651), the first person to study it. Boyer (page 395) writes that "on the suggestion of Roberval" the curve is named for Pascal.

Center The word center comes to us from a Greek root, kentrus, for a spur or sharp pointed object. The relation to the center of a circle seems obvious. A sharp point was made at a center to fix the spot, and a more dull object was dragged around the center to form the circle.

Century Although now used almost exclusively for a period of one hundred years, century was originally the Latin term for any collection of one hundred items. In the Roman army a company consisted of one hundred men, and each was called a centurion.

Cevian A word created by French geometers around the end of the 19th century to honor the Italian Giovanni Ceva (1650?-1735). A cevian is a line segment from a vertex of a triangle to a point on the opposite side. The median, altitude, and angle bisector are all examples of cevians. The perpendicular bisector, in most cases, is not a cevian because it does not pass through a vertex of the triangle. Note that a cevian, may cut the opposite side outside the triangle.

Julio Gonzalez Cabillon wrote, "the French word CEVIENNE, which was proposed by Professor A. Poulain (Faculte catholique d'Angers, France) in 1888. Naturally, he derived the word from the surname of the Italian mathematician Giovanni Ceva (1647?-1734)."

Chaos Although the ideas of chaos theory as we know it today have been actively studied at some level for most of the 20th century, the word as a mathematical term dates only from an article in American Mathematical Monthly in 1975, "Period Three Implies Chaos". The Greek root khaox was for an empty space. This meaning still persists in archaic usage where it refers to a canyon or abyss. The evolution of the word to mean disorder seems to come from reference to the time before God created the universe. The empty space was with out order and the creation filled the emptiness and created order.

A more common form of the word exists today, but few people are aware of the connection. At the start of the 17th century, a Flemish scientist named Jan Baptist van Helmont was studying the bubbles that rise when fruit juice was allowed to stand. These strange vapors, without shape or form, reminded him of the Greek idea of Chaos, so he called

them by the Germanic (Flemish is a dialect of German) spelling of chaos, gas.

The physical objects formed out of the void were called the cosmos, the Greeks word for orderly or well formed. Today we often hear people refer to the Universe as the cosmos. When Robert Milliken, the American physicist, sought a term for the radiation that seemed to be coming from everywhere in the universe (the cosmos) he suggested the name Cosmic Rays . Today the word cosmos also remains as the root of words like cosmopolitan and cosmetics.

Chi Square The statistical test, and the name for it are both credited to Karl Pearson around the year 1900. The actual distribution now called the Chi-Square distribution was discovered earlier by Helment

The Chi_Square test is often used to assess the "goodness of fit" between an obtained set of frequencies in a random sample and what is expected under a given statistical hypothesis.

The distribution is named for the letter Chi,, the 22nd letter of the Greek alphabet.

Chord The Greek root of the chord, chorde, means gut or string. The musical use of the term comes from a contraction of accord, two strings played together.

Circle The Latin root of the word circle is circus. The traditional shape of the large roofless enclosures in which the famous Roman Chariot races were run was circular or oblong, and thus the word came to described this shape as well.

Congruent The Latin word congruere meant "coming together" or "working together". I learned from Glen Woodburn recently that, "Actually, gruere comes from the latin word grui which means to be in harmony with. So congruent translates to mean together in harmony with." Whether applied to a geometric shape, or a military unit, it meant that all the parts fit together. According to a message from Nathan Sidoli, in Euclid's Elements the "word that Heath translates as "coincides" is *efarmo^zein* - to fit exactly"

. Nathan refers to Common Notion 4 in Book one, which Heath translates as "Things which coincide with one another, are equal to one another."

During the 16th century translations of Euclid into Latin began to use the Latin term for Common Notion 4. In a note to the Math Hisotry list J. Cabilon wrote that "Christoph Clavius (1537?-1612) wrote: '...Hinc enim fit, ut aequalitas angulorum ejusdem generis requirat eandem inclinationem linearum, ita ut lineae unius conveniant omnino lineis alterius, si unus alteri superponatur. Ea enim aequalia sunt, quae sibi mutuo congruunt.' (vol. I, p. 363)"[emphasis added]

At Jeff Miller's web site there are several notes on the development of the term congruence. In particular he says that, "In English, writers commonly refer to geometric figures as equal as recently as the nineteenth century. In 1828, Elements of Geometry and Trigonometry (1832) by David Brewster (a translation of Legendre) has: Two triangles are equal, when an angle and the two sides which contain it, in the one, are respectively equal to an angle and the two sides which contain it, in the other."

The modern symbol for congruence common to most US high school texts, which combines the tilde ~ above an equal

sign, =, was first used by many writers for similarity as well. It is sometimes used with the wave inverted also.

Leibniz used a tilde with a single underline as a unique symbol for congruence, but so many symbols were in use that it did not catch on. According to Cajori, the use of the modern symbol for congruence became the accepted practice around the beginning of the 20th century. He suggests the first use was by G. A. Hill and George B Halstead. The symbol is still not universally accepted and was not used in England at the time of his writing because of confusion with the tilde symbols use for difference.

I recently (Feb, 2004) posted a request for information to the Historia Matematica discussion group and received the following update on the use of the ~ symbol in England, and the related question of a symbol for congruence. According to a post from Herbert Prinz,

"In modern English texts on navigation, nautical astronomy or its history, the tilde is frequently used to express the function | a - b |, where |x| stands for absolute value. E. g. Cotter, The Complete Nautical Astronomer, 1969. I am not sure when this practice started. In older texts on the same subject, say, Moore, The Practical Navigator, 1800, one does not find the tilde used in this way. For one, because instructions were given mostly verbally without the use of any symbols at all. And second, the distinction from '-' was unnecessary, as it was always understood, if not explicitly stated, that one must subtract the smaller from the larger value." .

Tony Mann pointed out that in England the symbol was, "commonly used for 'is isomorphic to', and is used colloquially for 'is essentially equivalent to'." John Harper of Victoria University in New Zealand added that, "Geometric congruence was indicated by 3 parallel equal lines: , an equals sign with a straight underline. It still is, according to Borowski & Borwein "Collins Dictionary of Mathematics" (HarperCollins, Great Britain 1989) who give ~ on top of = only for approximate equality." Cajori credits the creation of the symbol for geometric congruence to Reimann and was used by Bolyai.

Gauss used the term congruent in modular arithmetic to refer to numbers which had the same remainder upon divison, for example 12 7 mod 5 since each has a remainder of two when divided by five.

Congruent Numbers was a brand new term to me when I read a neat blog at Bit-Player about it after a recent news release from AIM (American Institute of Mathematics) announced that all the congruent numbers up to 1 trillion have been enumerated. Well, job done I guess. The blog is so well written that I am not about to try to replicate all that

good work, go read it. If you want more, here is a link from the AIM on the topic.

Conjugate is the union of the common Latin prefix com (together) and the root juge (yoke) and means to bind together in a pair. Mathematically it is often used for things that are opposites in some way, as in the complex conjugates. The same word in grammar refers to words of a common origin and related meaning, and in biology to an act of sexual union, for which the more common term is conjugal relations.

Converse is from the Latin roots com(great or intense) + vertere (to turn). The literal meaning is "to turn away". The verb converse (as in conversation), which has the same spelling, is from a completely different root.

Dean The term now used for the head of a department or faculty at a college is derived from the Latin deaconus which meant "chief of ten". The similar sounding deacon, for a church leader, is not related and comes from the Greek root diakonos for a servant. According to John Conway, the literal meaning is "one who raises the dust".

The Online Etymology Dictionary indicates that doyen comes from the same term... "doyen 1422, from M.Fr. doyen "commander of ten," from O.Fr. deien (see dean).

Diagonal comes from the Greek roots dia( to pass through or join) + gonus [angle] and describes the line segment which passes from the vertex of one angle to another in a polygon. The word diagonal was probably first used in a geometric sense by Heron of Alexandria.

Diagram joins the roots dia(to pass through or join) with gram (written or drawn and earlier carved). It literally means "that which is marked out", as by the crossing of two lines. This leads me to wonder how old the expression, "X marks the spot", could be.

Divide shares its major root with the word widow. The root vidua refers to a separation. In widow the meaning is obvious, one who is separated from the spouse. A similar version of the word was often meant to describe the feeling of bereavement that a widow would feel. The prefix, di, of divide is a contraction of dis, a two based word meaning apart or away, as in the process of division in which equal parts are separated from each other. Notice that the vi part of vidua is also derived from a two word, and is the same root as in vigesimal (two tens), for things related to twenty. An individual is one who can not be divided.

In a division problem such as 24 / 6 = 4 the number being divided, in this case the 24, is called the dividend and the number that is being used to divide it, the 6, is called the divisor. The four is called the quotient. If the quotient is not a factor of the dividend, then some quantity will remain after division. This quantity is usually called the remainder, although residue sometimes is used. The Treviso Arithmetic uses the word lauanzo for remainder. In Frank Swetz's book, Capitalism and Arithmetic he gives, "The term lauanzo apparently evolved from l'avenzo, meaning a surplus, or in a business context, a profit." Swetz also points out that in the 15th Century the term partition (partire in Latin) was synonymous with the word divisision

In today's schools almost every grade school student learns to divide, so students may be surprised to learn that in the 16th century schools Division was only taught in the University. One of the first arithmetics for the general public that treated the subject of division was Rechenung nach der lenge, auff den Linihen vnd Feder by Adam Riese. Here is how the Math History page at St Andrews University in Scotland described it,

"It was published in 1550 and was a textbook written for everyone, not just for scientists and engineers. The book contains addition, subtraction, multiplication and, very surprisingly for that period, also division. At that time division could only be learnt at the University of Altdorf (near Nrnberg) and even most scientists did not know how to divide; so it is astonishing that Ries explained it in a textbook designed for everyone to use." I think it is even more astonishing that the sitution described still existed in 1550 in Germany. Perhaps the earliest "arithmetic" to provide instruction in the local vernacular of the common people was the 1478 "Treviso Arithmetic", so named because it was printed in the city of Treviso (the author is unknown) just north of Venice. Frank J Swetz writes about the situation in Capitalism and Arithmetic (pg 10): From the fourteenth century on, merchants from the north travelled to Italy, particularly to Venice, to learn the arte de mercadanta, the mercantile art, of the Italians. Sons of German businessman flocked to Venice to study...

Early algorithms for division: By the middle ages there seem to have been five approaches to the process of division.

The first was called the Galley, galea, or Scratch method. This method was efficient in a period of expensive paper and quill pens since it required less figures than other methods. Even the modern long division method requires more figures. The name Galley was used because the resulting pattern after the division left a picture that seemed to remind the early reckoning masters of the shape of a ship at sail. The term scratch has to do with the crossing out of values to be replaced with new ones in the process. The ease with which this could be done on a sand board or counting board made it a popular approach in the cultures of the East, and the method is believed to come from the early Hindu or Chinese. For example, Cajori writes, "It will be remembered that the scratch method did not spring into existence in the form taught by the writers of the sixteenth century. On the contrary, it is simply the graphical representation of the method employed by the Hindus, who calculated with a coarse pencil on a small dust-covered tablet. The erasing of a figure by the Hindus is here represented by the scratching of a figure." He also comments on the popularity of this method, " For a long time the galley or scratch method was used almost to the entire exclusion of the other methods. As late as the seventeenth

century it was preferred to the one now in vogue. It was adopted in Spain, Germany, and England. It is found in the works of Tonstall, Kecorde, Stifel, Stevin, Wallis, Napier, and Oughtred. Not until the beginning of the eighteenth century was it superseded in England. "

Here is an image comparing how the galley method works shown beside the current US Model for long division, which the Italians called a danda. The page the image is from has a nice step by step illustration of the process. I have recently (2005) acquired a German student "copy book" from 1804 which seems to show the Galley division method and the student's illustration of the ship around the work. (below right)

A second method that was sometimes taught was the process of repeated subtraction. The image below shows an example from a popular Arithmetic in the US by Charles Davies, published in 1833. I have seen this method in an English textbook as late as 1961 (Public School Arithmetic by Baker and Bourne). It also appears in a 1932 US publication of Practical Arithmetic, by George H. Van Tuyl, and perhaps in others

A third method was called per repiego by parts, which I have seen in books into the 20th century. In this method a division was accomplished by breaking the divisor into its factors, and then dividing the dividend by one of the factors, and sequentially dividing the resulting quotient by each remaining factor in turn to get a final quotient. The problem below is modeled on a problem in the 1919 copyright A School Arithmetic, by Hall and Stevens.

divide 92467 by 168 or 4 x 6 x 7 4|92467 6|23116 . groups of four and 3 units over 7| 3852 .. groups of 24 (4x6) and 4 foursover ___550 groups of 168 and 2 twenty-fours over The complete remainder is 2 (24) + 4(4) + 3 = 67 This method was presented in Liber Abaci, by Fibonacci in 1202. After introducing how to divide by numbers of one digit, and then larger primes, he develops a set of "Composition Rules" for numbers with more than one digit. A

composed fraction might look like . Fibonacci used the Arabic method of writing fractions from right to left, and this composed fraction would be read as 4/5 + 2/25 + 1/75; or in modern notation, 67/75 with each part of the numerator being read over the product of all the denominators below or to the right. The "composition" of 75 would be a fraction with 1 0 0 and 3 5 5 in the denominator, the fraction 1/75. When he divides 749 by 75, he first uses only the first denominator, 3. The quotient of 749 by three is 249 with a remainder of 2. The 2 is placed as a numerator over the three, and the 249 is divided by the second number in the denominator (a five). 249 divided by 5 gives 49 with a remainder of four. This remainder, 4, is placed as a second number in the numerator over the five in the denominator. Now the 49 is divided by

the final number in the denominator (another five) and the quotient is 9 with another remainder of four. This four is placed over the final five and the nine is placed to the right as the quotient. Fibonacci then gives the answer of

749 divided by 75 as which would be 9 and 4/5 + 4/25 + 2/75 or 9 74/75.

A fourth method, which is similar to what we would now called short division except that the student used a table of division or multiplication facts. The method was called per colona, by the column, or per tavoletta by the table, in reference to the table of facts used. An example of this method is shown below from another popular American arithmetic by Nicholas Pike, from 1826. The use of tables to aid in multiplication and division were a common practice from the 1400s up to the early 20th century.

The fifth is the true ancestor of the method most used for long division in schools today, and was called a danda, "by giving". In his Capitalism and Arithmetic, Frank J Swetz gives The rationale for this term was explained by Cataneo (1546), who noted that during the division process, after each subtraction of partial products, another figure from the dividend is given to the remainder. He also says that the first appearance in print of this method was in an arithmetic book by Calandri in 1491. The method was frequently called the Italian method even into the 20th century (Public School Arithmetic, by Baker and Bourne, 1961) although sometimes the term Italian method was used to describe a form of long division in which the partial products are omitted by doing the multiplication and subtraction in one step. The image below shows a typical long division problem with the partial products crossed out and the resulting "Italian method" on the right.

The early uses of this method tend to have the divisor on one side of the dividend, and the quotient on the other as the work is finished, as shown in the image below taken from the 1822 The Common School Arithmetic : prepared for the use of academies and common schools in the United States by Charles Davies. Swetz suggests that it remained on the right by custom after the galley method gave way to the Italian method in the 17th century. It was only the advent of decimal division, he says, and the greater need for alignment of decimal places, that the quotient was moved to above the number to be divided.

In a recent Greasham College lecture by Robin Wilson at Barnard's Inn Hall in London, he credited the invention of the modern long division process to Briggs, "The first Gresham Professor of Geometry, in early 1597, was Henry Briggs, who invented the method of long division that we all learnt at school."

I recently found a site called The Algorithm Collection Project. where the authors have tried to collect the long division process as used by different cultures around the world. Very few of the ones I saw actually put the quotient on top as American students are usually taught. In one interesting note, a respondent from Norway showed one method, then explained that s/he had been taught another way, and then demonstrates the common American algorithm, but adds a note that says, but no one is using this algorithm in Norway anymore. I might point out that the colon, ":" seems to be the division symbol of choice if this sample can be generalized as it was used in Norway, Germany, Italy, and Denmark. The Spanish example uses the obelisk, and the other three use a modification of the "a danda" long division process. The method labled "Catalan" is like the "Italian Method" shown above where the partial products are omitted. (More about division symbols at Symbols of Division

Dozen The word dozen is a contraction of the Latin Duodecim (two + ten). This root also appears in dodecagon (from duodecagon) and duodenum, the first part of the intestine

that is about twelve inches long. Some math and language historians think that a dozen is one of the earliest primitive groupings, perhaps because there are approximately a dozen cycles of the moon in a cycle of the sun. It appears to be the basis of many larger values that were developed by many cultures. A shock was 60, or five dozen (a dozen for each finger on one hand) and many cultures had a "great hundred" [see hundred] of 120 or ten dozen (a dozen for each finger on both hands). The Romans used a fraction system based on 12 and the smallest part, an uncil became our word for an ounce. Charlemagne established a monetary system that had a base of twelve and twenty and the remnants persist in many places. In English money today 100 pence equals a Pound, but only a few short years ago a Pound was divided into 20 shillings of 12 pence each.

First is a native English word from the Old English fyrst which was a variant of fore (front)

Fraction comes from the Latin word frangere, to break. A fraction, then, originally represented the broken portion of some whole. The first known use of the word in English is by Geoffrey Chaucer in 1391 in the work, A Treatise on the Astrolabe.

By the middle of the 19th Century fraction was used to describe parts larger than the whole as well. In the 1876 edition of Davies' Practical Arithmetic he lists as Article 114. "There are six kinds of fractions:" He then goes on to define

"1. A Proper Fraction is one whose numerator is less than the denominator" "2. An Improper Fraction is one whose numerator is equal to, or exceeds the denominator." "3. A Simple Fraction is one whose numerator and denominator are both whole numbers." (Note this is not necessarily what modern teachers would call in "simplest form", for example 8/4 is a simple fraction) "4. A Compound Fraction is a fraction of a fraction or several fractions connected by the word of or x. The

following are compound fractions: 1/2 of 1/4, 1/3 of 1/3 of 1/3, 1/7 x 1/3 x 4." "5. A Mixed Number is a number expressed by an integer and a fraction." "6. A Complex Fraction is one whose numerator or denominator is fractional; or, in which both are fractional," In the Fourth Yearbook of the NCTM in 1929 one of the curriculum changes listed for the State of New York included in the list for the 1910 syllabus, "Fractions, including complex fractions of the 'apartment house' type." (page 161) I assume the "mixed number over a mixed number" is the type of problem referred to, but am still trying to find confirmation of this.

Many modern elementary teachers get upset by the use of the term "reduce a fraction". I think this is mostly because they are not familiar with the origin of the term and only understand the word "reduce" to mean "make smaller", which is certainly one of the most common definitions of the word in modern dictionaries. I hope the the following will make them more understanding of those of us who are VERY old, and still remember when the term had a broader meaning.

According to the OED, the first use of the term in the sense of reducing a fraction was in 1579 in a book by Thomas Digges. Reduction is defined in the 1850 edition of Frederick Emerson's North American Arithmetic, Part Third, for Advanced Scholars as "the operation of changing any quantity from its number in one denomination to its number in another denomination."(pg 29, see image here) On the following page it asks the student to "reduce 7 bushels and 6 quarts to pints.". Later in the section on fractions it defines, "Reduction of fractions consists in changing them from one form to another, without altering their value." This broader language is preserved in most later texts for the next seventy or so years. It is defined in Milne's Progressive Arithmetic (1906, William J Milne) thusly, "The process of changing the form of any number without changing its value is called reduction." An almost identical definition appears in Davies and Peck's 1877 Complete Arithmetic, Theoretical and Practical(page 84, art. 66). All the books include reduction of fractions to higher terms as well as lower terms, and reduction of "decimals to common fractions".

In the Late 1930's and 40's arithmetic textbooks seemed to have totally omitted the broader definition, and treat

reduce as a vade mecam for fractions in "lowest terms" or "simplest terms". In Learning Arithmetic (6) by Lennes, Rogers and Traver, (1942) the term reduction appears in the index only as a subheading under "fractions". The first occurance in the text, on page 36, without prior definition introduces students to a set of problems with the directions, "Reduce the fractions below to simplest forms". In Making Sure of Arithmetic by Silver Burdett (1955) the word "reduce" does not appear in the index at all, but on page 8 it contains, "When the two terms of a fraction are divided by the same number until there is no number by which both terms can be divided evenly, the fraction is reduced to lowest terms." [emphasis is from text]. By 1964, The Universal Encyclopedia of Mathematics by Simon and Schuster contains "A fraction is reduced, or cancelled, by dividing numerator and denominator by the same number." (pg 364) Later on the same page they note, "a fraction cannot be reduced if numerator and denominator are mutually prime" indicating that when they said "the same number" in the first statement, they meant a positive integer. This definition leads to "reduction" of fractions as making the numerator and denominator both smaller.

The roots of the word reduce are from the Latin re for back or again, and dicere which means "to lead". The latter root is also found in the word educare which is literally, to lead out, and is the source of our modern English word, educate.

Frustum (sometimes spelled frustrum) is from the Latin and means "a piece broken off". Mathematically it usually refers to a part of a solid cut off between two parallel planes, as opposed to truncated. The Indo-European root of frustum is bhreus and is related to cutting, crushing, or pounding. Related words from the same root are fragment, bruise, and possibly brush (from a bundle of cut twigs).

Jeff Miller's web site on the earliest use of math words includes a note on how frequently the term is misspelled as "frustrum".

"This word is commonly misspelled as "frustrum" in, for example, Samuel Johnson's abridged 1843 Edition of his

dictionary. The word is spelled correctly in the "Frustum" entry and the "Hydrography" entry in the 1857 Mathematical Dictionary and Cyclopedia of Mathematical Science, but it is misspelled in the entry "Altitude of a Frustrum." The word is misspelled in the 1962 Crescent Dictionary of Mathematics and remains misspelled in the 1989 Webster's New World Dictionary of Mathematics, which is a revision of the Crescent dictionary. The word is also misspelled in at least three places in The History of Mathematics: An Introduction (1988) by David M. Burton. This has become so common it may almost be considered an alternative spelling. He also has, of course, a notation on the first use of the term in English, "FRUSTUM first appears in English in 1658 in The Garden of Cyrus or the Quincuncial Lozenge, or Net-work Plantations of the Ancients ... Considered by Sir Thomas Brown: "In the parts thereof [plants] we finde..frustums of Archimedes" (OED2)."

Geometry is derived from the conjunction of the Greek word for the Earth, Geos, and the term for "to measure", metros. Literally then, Geometry means "to measure the Earth". According to the Greek legends the first things created out of the "chaos" were the earth, Gaia and the sky, Ouranos (which would become Uranus in Latin). The Greek word Gaia was a name not only for the earth, but for the goddess of the Earth. Although the Greeks and the Latins pronounced the word Gay' yuh it came into English pronounced more like Jee' uh from which we get the many Geos rooted words such as geography and geology.

I recently read a post by G.L.Narasimham to a geometry discussion list that pointed out that the common term in some Indian dialects, was very closly related

A commercial (advertisement for onion soup spice, CBR Masala ) in Hindu newspaper in India a couple of years ago mentioned origin of the word Geometry.As it is intersting, I reproduce it below, not verbatim: --- Geometry ( Jyo-Miti) Kalpasutra is an important source of Vedic mathematics. The sections called sulba-sutras deal with measurement and construction of such Yagna-vedis (platform for religious rites) that involved geometrical propositions and problems

related to geometrical figures. The word sulba means a cord, rope, filament or string and the word root means 'measuring'. It is interesting to note that among the Egyptians, geometry of surveying was considered to be the science of rope structures (harpendona 'ptae) !! .. They thus appear to be the Egyptian counterparts of Indian sulbavids. Some 50 years ago text books on geometry in Tamil language were entitled 'Jyomiti Ganitam' The word geometry owes its origin to India, Jya=Earth. Miti = Measure. Hence 'Jyamit' meaning measurement of earth or figures drawn on earth, gave rise to the present term geometry. Courtesy Dr. V.S. Narasimhan, Chennai. 'A Concise History of Science of India', Indian National Science Academy, pp 139,149. -------

Just as the study of the Earth, Geology, recalls the ancient Greek goddess of the earth, the term Uranology is the study of the sky, but the more common term today is astronomy. But do not despair for the lost memory of the Greek god of the sky, for he is preserved in the name of the planet discovered in 1781 by William Herschel. Herschel thought he had discovered a comet and published the result for the Royal Society. Although Herschel wanted to name the planet for King George III, remembered as the bad guy in the American revolution by U.S. History; Johann Bode (see Bode's Law) came up with the suggestion for the name that stuck, Uranus. Hershcell continued to call the planet Georgium Sidus, the Georgian Star after his royal patron.

Uranus, the sky god, is also remembered through a discovery a few years later (1798) by chemist Martin Klaproth. It was a tradition of chemists to name metals after planets, so Klaproth named his new metal after the new planet, calling it Uranium. Strangely, he later discovered another new metal and decided to name it after the earth, but instead of using the Greek goddess of earth, he chose the Roman equivalent and called his new metal, tellurium. The Roman goddess of the earth was known by two names. The first was terra, which gives us words like terrestrial, and its better known opposite, extra-terrestrial. The other was tellus, which is almost non-existent today, except in the name of Klaproth's metallic discovery.

Googol A number invented by Milton Sirotta, the eight year old nephew of Dr Edward Kasner, when asked to think of a name for a 1 followed by 100 zeros. 10^100 is an incredibly large number. The largest reasonable estimates for the number of particles in the universe is only about 10^85. A googol is a million times a billion times this much.

I've been asked so many times that I finally tracked down the answer to "Whatever happened to Milton Sirotta?" From his obituary, it seems he died in 1980, I also found a note that said, "Edwin(his brother) and Milton worked for most of their lives in their father's factory in Brooklyn, NY, pulverizing apricot pits into an abrasive used for industrial purposes".

Hectare A unit of land equal to 100 ares or about 2.47 acres. An are is the area of a square with sides of a dekameter (ten meters). A hectare is equal to a square made up of a ten by ten array of ares or, in more modern words, a square with sides of 100 meters. The prefix hecto is from the Greek word for one hundred, hekaton. The prefix is common in units of measure, such as hectogram or hectometer.

Helix is preserved from the Greek and has maintained its meaning since antiquity. The Greek word seems to have been used generally to apply to ideas about wrapping or twisting, but only its mathematical meaning seems to have survived.

Histogram The root of histogram is probably from the Greek root histo, for tissue, and gram, for write or draw. The suffixes gram and graph are almost interchangeable, and both have to do with the act of writing or drawing. Karl Pearson, the first known user, apparently thought of each vertical bar as a cell. Some have suggested that the root is from the word "history" since a histogram provides a record, and certainly Pearson knew of this meaning also.

In a recent post to a math history news group, John Aldrich of the Department of Economics of the University of Southampton wrote:

I do not know the history of the technique but the _term_ histogram was coined by Karl Pearson to refer to a "common form of graphical representation". The _OED_ quotes from _Philos. Trans. R. Soc._ A. CLXXXVI, (1895) 399 [Note. The word "histogram" was] introduced by the writer in his lectures on statistics as a term for a common form of graphical representation, i.e., by columns marking as areas the frequency corresponding to the range of their base. Stigler (_History of Statistics_) identifies the lectures as the 1892 lectures on the geometry of statistics. The Greek root of history is from histor, a learned man. The implication is that a learned man is aware of history, but it is more direct than just good advice. The Indo-European root of the word is the same root that gives us wise.

Hour and year are both derived from the Greek root horo, which was applied to ideas about time and the seasons. In the Old Germanic horo became yero and year was thus derived from the same root which gave us hour. Today horoscope refers to fortune telling, but the practice is rooted in the original meaning, measuring the aspect of the stars and planets to measure the seasons. Horology is still the name for a maker of timepieces.

Hundred is from the German root hundt. The quantity that it represents has not been consistent over the years and has ranged from its present value, 100, to 112, 120, 124, and 132 at different times in different areas. The remnants of these old measures still persist in the hundredweight of some countries representing 112 or 120 pounds, depending on the country. A hundred has also been used to represent an area of land equal to 100 hides (of cattle?). The measure of area was frequently used in colonial US, and parts of England in place of "Shire" or "Ward". A curious custom related to one hundred as a unit of land occurs in England when a member of the House of Commons wishes to resign his seat, which is illegal. An MP accepts stewardship of the

"Chiltern Hundreds", an area of chalk hills near Oxford and Buckingham, and effects his release from Parliament.

Hypotenuse comes from the common Greek root hypo(for under, as in hypodermic -under the skin) and the less common tein or ten, for stretch. This last is the source of our modern word tension. The hypotenuse was the line segment "stretched under" the right angle.

The other two sides of a right triangle are generally called legs, but the term is also applied to any side of a triangle with the idea that they are standing upon a "base". The term, leg, is also applied to either of the two parts on each side of the vertex of a curve such as a parabola or hyperbola. More formally, I have recently seen the term cathetus used to describe the two non-hypotenuse sides of a right triangle. Cathetus actually means a straight line falling perpendicularly on another straight line or surface, and was used by Euclid in this fashion in his tenth definition in the first book of The Elements

When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.

The literal meaning of the Greek root is "to let down". The medical term cathetor, and the electrical term cathode both come from the same root.

At Jeff Miller's website I found :

Cathetus occurs in English in 1571 in A Geometricall Practise named Pantometria by Thomas Digges (1546?-1595) (although it is spelled Kathetus). Cathetus is found in English in the Appendix to the 1618 edition of Edward Wright's translation of Napier's Descriptio. The writer of the Appendix is anonymous, but may have been Oughtred.

[[My thanks to Steve Earth, math teacher at the Kehillah

Jewish High School, for suggesting this term]]

Isosceles is the union of the Greek iso (same or equal) and skelos (legs) and refers to two sides of a object as being the same length, as in isosceles triangles and isosceles trapezoids. The root iso shows up in many scientific and mathematical words such as isometry (same measure), and isomorphic (same shape). Isobar is used both in chemistry (two atoms with equal atomic weight) and in meteorology

(lines connecting points of equal barometric pressure). The two equal length sides are called legs (see above), and the other side is called the base.

I recently (2009) became aware that the term "arms" is sometimes used instead of legs. Here is a link to a

January The first month of the year was originally a period of festival between the end of one year and the beginning of the next in honor of the Roman god Janus. Janus was the god of

beginnings and endings and is portrayed with two faces, one looking forward and one looking back.

Logarithm is the combination of two Greek roots, Logos(reason or ratio) + artihmus(number). The ratio refers to the original method of constructing logarithms by geometric sequences. The name was introduced by John Napier (1550-1617), the inventor of logarithms, in his 1614 work on logarithms, Mirifici logarithmorum canonis descriptio, [Description of the wonderful canon of logarithms .... but it is usually called "The Descripto"]. It was originally

written in Latin and subsequently translated into English. Here is a site where you can find a digital copy of the English text

It seems that Pietro Mengoli (1625-1686) was the first to use the term "natural logarithm". Boyer writes, "Mercator took over from Mengoli the name 'natural logarithm' for values that are derived by means of this series." The term Mengoli and Mercator actually used was "logarithmus naturalis". In a discussion group, Jeff Miller suggested that it might be this use of noun before adjective that prompted the use of the symbol "ln" for natural log rather than "nl". According to Cajori, the symbol "ln" was first used for the natural logarithm (log base e) in 1893 by Irving Stringham (1847-1909). Stringham introduces the notation without comment in a list of symbols following the table of contents, then uses it for the first time on page 41, shown below.

Thanks to Dave Renfro for help in getting this digital pic.

I also recently heard in a correspondence from George Zeliger that when he was a student in Russia (around 1989) it was common to use "lg" for the common logarithm (log base ten).

When Napier constructed his tables he used a base that was slightly smaller than one (1-10-7) and so as the number, n, got bigger, the logarithm, l, got smaller. It was common at the time in trigonometry tables to divide the radius of a circle into 10,000,000 parts. Because the main intention of his creation was focused on addressing the difficulty in performing trigonometric computations, Napier also divided his basic unit into 107 parts. Then to avoid having to use fractions, he multiplied each value by 107. In notation of today's mathematics, the form of Napier's logs would look like : 107 (1-10-7)L=N. Then L is the Naperian logarithm of N.

According to e: The Story of a Number by Eli Maor,

In the second edition of Edward Wright's translation of Napier's Descripto (London, 1618), in an appendix probably written by William Oughtred, there appears the equivalent of the statement that loge10 = 2.302585. Since the actual tables contains no decimals it was probably given as 2302585 without the decimal point.

In a famous meeting between Napier and Henry Briggs, Briggs suggested the use of a base of 10 instead of 1- 10-7 and to have the logarithm of one equal to zero. This Napier agreed to but the task of constructing tables of "common" logarithms fell to Briggs, and they were often called Brigg's Logarithms in his honor.

Robin Wilson, in his Gresham College lecture on the number e, that "Early ideas of logarithms are given in works of Chuquet and Stifel around the year 1500. They listed the first few powers of 2 and noticed that to multiply any two of them it is enough to add their exponents." Maor notes that Joost Burgi of Switzerland probably created a table of logarithms before Napier by several years, but did not publish until later, and he is almost forgotten today. Burgi may also have independently discovered the method of Prosthaphaeresis and gave it to Tycho Brahe. Burgi is also remembered as the person who taught Kepler Algebra.

The impact of logartihms on the working scientist of the period is hard to appreciate, but one may get an idea from this quote by Pierre Laplace, "Logarithms, by shortening the labors, doubled the life of the astronomer." While it is Napier's work on logartihms that he is remembered for today, in his own time he was famous for the calculating method called Napiers rods and a method of calculating spherical right triangle trigonometry. He thought his most important work had been published 21 years earlier in 1593. In that year he published a mathematical analysis of the book of Revelations in the Bible, A Plaine Discovery of the Whole Revelation of Saint John. In the book he revealed that the Pope was the antichrist, and that the world would end in the year 1786. Fortunately for us, he was wrong on at least that one point. To his credit, he more accurately predicted the development of the machine gun, the submarine, and the tank.

Gordon Fisher recently posted a time line of the development of the use of the abbreviation "log" for lograrithms. Here is his post with a few notes thrown in Log. (with a period, capital "L") was used by Johannes Kepler (1571-1630) in 1624 in Chilias logarithmorum (Cajori vol. 2, page 105) log. (with a period, lower case "l") was used by Bonaventura Cavalieri (1598-1647) in Directorium generale Vranometricum in 1632 (Cajori vol. 2, page 106). log (without a period, lower case "l") appears in the 1647 edition of Clavis mathematicae by William Oughtred (1574-1660) (Cajori vol. 1, page 193). Kline (page 378) says Leibniz introduced the notation log x (showing no period), but he does not give a source. loga was introduced by Edmund Gunter (1581-1626) according to an Internet source. [I do not see a reference for this in Cajori.] Many students (and teachers) have heard colorful legends

about the reasoning behind the use of "ln" for the natural

logarithm (from the French for something, or something about

the name Napier). Most of them seem to me to be more myth

than fact. The facts, as best I know them, is that the first

use of the terms "natural" and "logarithm" together was by

Nicholas Mercator (not the cartographer) in 1668 in his

logarithmo technica in which he used the Latinized "log

naturalis". [[[In early 2005 a post from Jeff Miller pointed

out that, according to Carl Boyer, Pietro Mengoli used the

term before Mercator. Both were working with values derived

from a series, Mercator with the expansion of log(1+x)]]]

The first use of "ln" as a symbol was, as Gordon points

out(below), by Stringham (I have not seen this book and do

not know if he gives an explanation). As to the correct

pronunciation of "ln(x)", whatever your teacher says is

correct, but high school students should be aware that many

college mathematicians find the symbol disturbing. In his

1984 biography, Paul Halmos described the symbol as

"childish". It is, however, very commonly used in computer

science. ln (for natural logarithm) was used in 1893 by Irving Stringham (1847-1909) in Uniplanar Algebra (Cajori vol. 2, page 107). The same note from Jeff Miller mentioned above pointed out

that Anton Steinhauser used the abbreviation "log.nat." in

1875 William Oughtred (1574-1660) used a minus sign over the characteristic of a logarithm in the Clavis Mathematicae (Key to Mathematics), "except in the 1631 edition which does

not consider logarithms" (Cajori vol. 2, page 110). The Clavis Mathematicae was composed around 1628 and published in 1631 (Smith 1958, page 393). Cajori shows a use from the 1652 edition.

I also recently saw a post that suggested that in computer classes it is sometimes common to use "lg" for the base two log.

In 1647 the French mathematician Saint-Vincent showed that the area under the hyperbola y = 1/x were like the logarithm function, that is, the area from 1 to 2 plus the area from 1 to 3 was equal to the area from 1 to 6, 2x3.

Minute When the early sailors from the Eastern Mediterranean chose to cut an arc into parts, they chose fractions in the sexagesimal (base 60) system that was common to their period in history. A nice article and illustrations of the Babylonian system of numerals is found at the St. Andrews Math-History page. Later when Latin writers described these small parts of an arc, they used the Latin phrase pars minuta, Latin for small parts. Our unit of time for 1/60 of an hour adopted and contracted this phrase into minute. The Conjugate word with the same spelling but different accent and pronunciation (mi nyoot') continues to refer to something very small. The word MINUS for subtraction is drawn from the same root and refers to making something smaller. The verbal use of the words plus and minus date back to the Romans when the terms were used much as we use the English words more and less. Other related words are minor (smaller of two), minced (cut into small pieces), miniature (on a small scale) and menu (a small list).

Multiply comes from the combined roots of multi, many, and pli, for folds, as in a number folded on itself many times. The first use I have found of the word as a verb, as in "multiply two by three" is credited to Chaucer in his 1391 work, A Treatise on the Astrolabe.

The two numbers that are multiplied together are most often called factors and the result is called the Product. Although they are not used much anymore, you may still find the two parts that are multiplied together called the multiplicand [that which is multiplied, or how many in each group] and the multiplier[that which does the multipling, or how many groups in all].

One of the earliest notations to indicate multiplication was by juxtaposition, placing the numbers adjacent to each other as we do for algebraic characters today. Cajori cites this as the method used to indicate multiplication on some ancient Indian manuscripts from the 10th century or earlier. Jeff Miller has a note that "In 1553, Michael Stifel brought out a revised edition of Rudolff's Coss, in which he showed multiplication by juxtaposition and repeating a letter to designate powers (Cajori vol. 1, pages 145-147)."

The use of an "x" to indicate the operation of multiplication seems to have been originated by William Oughtred in his Clavis Mathematicae (Key to Mathematics, 1631). The use of a dot, as in 6 .4 = 24, is sometimes credited to Leibniz with the first use attributed to a letter from Leibniz to John Bernoulli :

The dot was introduced as a symbol for multiplication by G. W. Leibniz. On July 29, 1698, he wrote in a letter to John Bernoulli: "I do not like X as a symbol for multiplication, as it is easily confounded with x; ... often I simply relate two quantities by an interposed dot and indicate multiplication by ZC LM. Hence, in designating ratio I use not one point but two points, which I use at the same time for division." [A History of Mathematical Notation, Vol 1, art. 233; F. Cajori] From Jeff Millers web page on "Earliest Uses of Symbols of Operation" I found the following correction to Cajori; "Cajori shows the symbol as a raised dot. However, according to Margherita Barile, consulting Gerhardt's edition of Leibniz's Mathematische Schriften (G. Olms, 1971), the dot is never raised, but is located at the bottom of the line. She writes that the non-raised dot as a symbol for multiplication appears in all the letters of 1698, and earlier, and, according to the same edition, it already appears in a letter by Johann Bernoulli to Leibniz dated September, 2nd 1694 (see vol. III, part 1, page 148). Some people credit the first use of a dot for multiplication to Thomas Harriot. He used a dot in Analytica Praxis ad

Aequationes Algebraicas Resolvendas, which was published posthumously in 1631. Cajori suggests these were not acutally intended as symbols for the operation of multiplication but "Scott (page 128) writes that Harriot was 'in the habit of using the dot to denote multiplication.' And Eves (page 231) writes, 'Although Harriot on occasion used the dot for multiplication, this symbol was not prominently used until Leibniz adopted it." [from Jeff Miller's page]. The use of a * instead of a dot appeared in Teutsche Algebra (1659) by Johann Rahnn.

Some notes on notation for multiplicaton, The ancient Greeks and Egyptians seemed to have no special symbol for multiplication. Sometimes a word or phrase was used as we might say "times" to indicate multiply. In the 16th Century Stifel used the capital M and D for multiply and divide in his Deutsche Arithmetica (1548). Other German writers did not follow his lead, and it seems that Stifel quickly dropped the symbols himself. Simon Stevin adopted the M and D in L' arithmetique(1634). Cajori credits the use by Christian Wolf and Euler in the 18th Century with making the dot popular in Europe, and the strong influence of Oughtred led to the more common use of the "x" in England, and in America. In America today it seems that "x" is more common through the teaching of arithmetic, and the dot is introduced for awhile in the early algebra teaching; but eventually the use of juxtaposition of variables, and parentheses for numbers becomes the most common indiation of multiplication. 3 x 4 = 3; 3 . 4; 3 (4)

Negative

Negative numbers, and the equivalent word for negative were introduce by Brahmagupta, a Hindu mathematician around 600 AD. The Latin root of today's word is negare, to deny. The negative numbers, in this sense, denying or invalidating an equivalent positive quantity.

The negative numbers were themselves denied for a long part of mathematical history, and only slowly came to be accepted. The first record of the operational rules for what we today call positive and negative numbers came from the pen of Diophantus (around 250 AD) who referred to them as "forthcomings" and "wantings". His work may have been drawn from proposition five in Euclid's Book II of the Elements in

which Euclid demonstrates with geometric figures what we would write in modern algebra as (a+b)(a-b)+b2 = a2. This, of course, is easily recast as the more common identity (a+b)(a-b)= a2 - b2. Diophantus would accept negatives only as a way of diminishing a greater quantity, but did not accept them as independent quantities and would not accept a solution that was negative. Al-Khwarizmi (850 AD), whose writings brought Arabic numerals to the west, used a similar approach with negatives allowed in-process but not as a final result.

Descartes, around 1636, used the French fausse, false, for negative solutions. Thomas Harriot had described negative roots as the solution to an alternate form of the equation with the signs of the odd powers changed. Today his idea would be expressed by saying that the appearance of -c as a root of f(x) was only to be understood to mean that c is a root of f(-x).

In Mathematics: The Loss of Certainty, by Morris Kline includes the following argument against negative numbers by Antoine Arnauld (1612-1694), mathematician, theologian, and friend of Blaise Pascal; "Arnauld questioned that -1:1 = 1:-1 because, he said, -1 is less than +1; hence, how could a smaller be to a greater as a greater is to a smaller?"

Franz Lemmermeyer wrote in a posting to the Historia-Matematica newsgroup that Gleanings from the History of the Negative Number by PGJ Vrendenduin suggests that a number line with both positive and negative numbers could be found in the work of Wallis (1657)[This is certainly true as seen here]. Another posting to the same list quoted Kline's "Mathematical Thought from Ancient to Modern Times":

"Though Wallis was advanced for his times and accepted negative numbers, he thought they were larger than infinity but not less than zero. In his 'Arithmetica Infinitorum' (1665), he argued that since the ratio a/0, when a is positive, is infinite, then, when the denominator is changed to a negative number, as in a/b with b negative, the ratio must be greater than infinity." Even as late as 1831, De Morgan would still write that one "must recollect that the signs + and - are not quantities, but directions to add and subtract." [ Albrecht Heeffer refutes this position, held by Kline and many others, in a

post to the math-history list. ] In a recent book by Gert Schubring see clips here he also supports a view that

Wallis' understanding of negatives was much broader than generally credited.

According to a post from Laura Laurencich, the Incas had a method of indicating both positive and negative numbers on their quipus as documented by the Jesuit Priest Blas Valeria in 1618