Origins of some Math terms

**Abscissa** is the formal term for the x-axis in a graph. The word is a conjunction of *ab*(remove) + *scindere* (tear). Literally then, to tear or cut apart, as the x-axis does to the coordinate plane. The main root is closely related to the Latin root from which we get the word scissor. The mathematical use of the term was apparently coined by Leibniz around 1855.

**Absolute Value ** The word absolute is from a variant of absolve and has a meaning related to free from restriction or condition. It seems that the mathematical phrase was first used by Karl Weierstrass in reference to complex numbers.

**Acute ** is from the Latin word *acus * for needle, with derivatives generalizing to anything pointed or sharp. The root persists in the words acupuncture (to treat with needles) and acumen (mentally sharp). An acute angle then, is one which is sharp or pointed.

**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.

**Algebra **comes from an Arabic book 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-Khowarazmi. The phrase Al-jebr at the start of the title became the word Algebra in western languages. The phrase means "the reunion of broken parts".

**Algorithm, **as it is used in mathematics means a systematic procedure to solve a problem. The word is derived from the name of the Hindu mathematician, al-Khowarazmi (See algebra). The first use of the word I am aware of was by G W Liebniz in the late 1600. It remained a little known and little used term in western mathematics until the Russian mathematician Andrei Markov (1856-1922) 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 which 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 which 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.

**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

**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).

**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.** **

**Asymptote **The asymptote of a function as it is now used is much narrower than original Greek meaning. The word is believed to have been known to Apollonius of Perga before 200 BC. Originally used for any two curves which did not intersect. Now it is used primarily for straight lines which serve as a limiting barrier for some curve as one of its parameters approaches infinity (+/-).

**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**.

**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

**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 is sort of heart shaped and thus the term is from the Greek root for heart, *kardia*. The term appears to have first been used by Giovanni Salvemini de Castillon (1708-1791)

**Catenary** the name for the curve formed by a hanging rope is actually from the Latin root, *catenareus*, for chain. The word was developed in correspondence between Leibniz and Huygens around 1690, but there seems to be some disagreement about which used the term first.

**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 centurian.

**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.

**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 dates only from an article in American Mathematical Monthly in 1975, "Period Three Implies Chaos".

**Chi Square **The statistical test, and the name for it are both credited to Karl Pearson around the year 1900.

**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 Greek word congruent meant "coming together" or "working together". Whether applied to a geometric shape, or a military unit, it meant that all the parts fit together.

**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 fromt the Greek root *diakonos* for a servant. According to John Conway, the literal meaning is "one who raises the dust".

**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.

**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 sepreration. In widow the meaning is obvious, one who is seperated 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 meaning apart or away, as in the process of division in which equal parts are seperated from each other.

**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 which is about twelve inches long.

**Exponent** is the union of the Latin roots *exo*(out of) + *ponere* (place). The literal interpretation is to make something visible or obvious. The English word expound from the same source means to make clear. An exponent is also used in English to describe a person who explains or interprets. Exponent, as a math term, was introduced by Michael Stifel (1487-1567) in his book, Arithmetica Integra, in 1544.

**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, representing 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.

**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

** Geometry** is derived from the conjuction of the Greek word for the Earth, *Geos*, and the term for "to measure", *metros*. Literally then, Geometry means "to measure the Earth".

**Googol** A number invented by the nine 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.

**Hectare** A unit of land equal to 100 ares or about 2.47 acres. 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 from the Greek root *histo*, for tissue, and gram, for write or draw. E.S. Pearson, the first known user, apparently thought of each vertical bar as a cell.

**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 these ancient measures still persist in the *hundredweight* measures 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.

**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 meterology (lines connecting points of equal barometric pressure).

**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) in his work on logarithms, originally in the Latin form, and subsequently in English in correspondence with Briggs.

** 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. 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. 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.

**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.

**Noon** is NOT related to the number twelve, but to the number nine. It is derived from the Latin *nona* (ninth) and originally refered to the ninth hour after sunrise, which was closer to the present 3PM.

**Obtuse ** is from the Latin formation *ob* (against) + *tundere* (to beat) and literally means to beat against. An object thus beaten becomes blunt, dull, or rounded, and the application to an obtuse angle is in this sense.

**Plus** (as in two plus four) comes from the early latin word meaning "more". Extensions of the root were used for related ideas like fill, full, and abundant.
Common words of today related to the same root are plenty, complement (meaning complete or fill, as in complementary angles, the amount needed to complete a right angle), plural (more than one), and surplus (abundant, more than enough). The word is closely related to the Greek root *poly* for many. [see polygon]

**Polygon** is from the Greek roots *poli* (many) and *gonus* (knees) and, interprets literally as many angled. The relation between knee and angle relates to the flexed position of the knee. *Poly* appears in many words, and *gonus* remains mainly in its Latin derivative, genus, from which we get genuflect (to bend the knee). According to John Conway, terms like gnaw are from the same root, perhaps because the line of the jaw forms the same shape as the bent knee.

**Polyhedron** is the name for a solid with "many faces", The joining of *poli* (many) with *hedros*( face or seat). The hedros originally referred to any flat surface. Later, in the latin, hedra was used for a chair, flat places are good to sit on, and the root is preserved in our words for cathedra (the Bishop's chair) and the Cathedral where it is kept.

**Positive **comes from the root word *posit* which means to place or set. This probably refers to the ancient method of counting with markers and counting tables. Positive numbers could be set out one by one.

**Prime ** is from the Latin word for first, *primus*. Prime numbers are thus the first, or most basic, of numbers in a multiplicative sense. Other words drawn from the same root include primary [first in rank or order], Primitive [first of its kind], prima donna [literally, first lady] and Principal [first in power, from the derivative form *princeps* for ruler]. The word Prince, again from *princeps*, originally referred to a king, but later the French applied the term to any male member of the royal family.

**Quadratic** is the Latin root for "to make square". The word square was itself derived from this root [see square ].

**Random** comes to us from the old French root *Randir*, to gallop. Perhaps the idea is that, in full gallop, the horse or rider has abandoned control.

**Secant** is from the Latin root *Secare*, to cut. It is a proper name for for a segment that cuts through the circle. The word was introduced by Thomas Fincke in 1583 in Latin.

**Second** When the *pars minutia* [see Minute of an arc needed to be divided into even smaller parts, the 1/60 part of 1/60 of a degree needed a name also. Since it was the second small part, what could be more appropriate than *pars minuta secundus*. Later the term was shortened down to seconds and generalized to units of time which preserved the base 60 system. Secundus is from the Latin root *sequi* [see Sequence] for "to follow", and thus second was the natural term for the ordinal following First

**Sequence** is from the Latin root *sequi*, to follow. In mathematics it refers to a series of terms in order. The root is the source of such modern words as consequence (the results that follow an event), suitor (one who follows a lover), and second (the one after the first).

**Slope** is derived from the latin root *sleubh* for slip. The relation seems to be to the level or ground slipping away as you go forward. The root is also the progenitor of sleeve (the arm slips into it) and, by dropping the s in front we get lubricate and lubricious (a word describing a person who is "slick", or even "slimy"). The mathematical meaning of slope was first used by V. F. Rickey around 1850. Both the word, and the use of the letter *m* as its symbol, seem to have originated in the US.

**Square** is derived from the Latin phrase *Exquadrare*, like a quadratic. Over time the term was contracted into its present form, and came to mean the regular quadrilateral.

**Subtract **joins two easy to understand roots, the *sub* which commonly means under or below, and the *tract* from words like tractor and traction meaning to pull or carry away. Subtraction then, literally means to carry away the bottom part.

**SURD** The original meaning of surd was mute, or voiceless. The word still retains that meaning today in phonetics for an unvoiced consonant (as opposed to a voiced consonant, a sonant). The reference is to a root that could not be expressed (spoken) as a rational number.

** Symmetry** is from the Greek roots *sum + metros*. The prefix refers to things which are alike, and *metros* is the Greek word for measure. *Metros* is the root of the word Geometry also.

**Tangent** is from the Latin *tangere*, to touch, aptly describing two curves which meet at a single point. Tangent is another creation of Thomas Fincke, and was first written by him in Latin around 1583.

**Tangram ** is a name of a chinese puzzle of seven pieces that became popular in England around the middle of the 19th century. It seems to have been brought back to England by Sailors returning from Hong Kong. The origin of the name is not definite. One theory is that it comes from the Cantonese word for chin. A second is that it is related to a mispronunciation of a Chinese term that the sailors used for the ladies of the evening from whom they learned the game. A third suggestion is that it is from the archaic Chinese root for the number seven, which still persists in the Tanabata festival on July seventh.

**Tessellation** The root of tessellation is, *tessera*, the old Ionic (Greek) root for four. *Tessera* is the name of the square chips of stone or glass that are used to form a mosaic. *Tessela* is the dimunitive form, and is used to describe smaller *tessera*. Tiles, bricks and larger similar items were called *testa*, which is preserved in the name for the hard outer shell of seeds. The completed project, then, became a tessellation.

**Thousand** Our number for one thousand comes from an extension of hundred. The roots are from the Germanic roots *teue* and *hundt*. *Teue* refers to a thickening or swelling, and *hundt* is the root of our present day hundred. A thousand, then, literally means a swollen or large hundred. The root *teue* is the basis of such common words today as thigh, thumb, tumor, and tuber.

**Topology** comes from the Greek root *topos* (place). Before it was used in mathematics, it was applied to the geographic study of a place in relation to its history. The word was introduced in English by Solomon Lefschetz in the late 1920s. It appears that the word was originated around 1847 by Johann Benedict Listing in place of the earlier usage *analysis situs*"

**Torus** is from the Latin word for bulge and was first used to describe the molding around the base of a column. Although it is usually used to describe the rotation of a circle about a line in its plane, the definiton applies to the rotation of any conic section.

**Trapezoid and Trapezium** Both words come originally from the Greek word for table. Today, in the USA, the term trapezoid refers to a quadrilateral with one pair of sides parallel and a trapezium to one with NO parallel sides. This is exactly the reverse of the original meanings (and the meanings in some countries, particularly England, today. Here is a short comment on how this came about from Jeff Miller, a teacher at Gulf High School in New Port Richey, Florida, who maintains an excellent page on the first use of some common mathematical terms: "TRAPEZIUM and TRAPEZOID. The early editions of Euclid 1482-1516 have the Arabic helmariphe; trapezium is in the Basle edition of 1546. Both trapezium and trapezoid were used by Proclus (c. 410-485). From the time of Proclus until the end of the 18th century, a trapezium was a quadrilateral with two sides parallel and a trapezoid was a quadrilateral with no sides parallel. However, in 1795 a Mathematical and Philosophical
Dictionary by Charles Hutton (1737-1823) appeared with the definitions of the two terms reversed:
Trapezium...a plane figure contained under four right lines, of which both the opposite pairs are not parallel. When this figure has two of its sides parallel to each other, it is sometimes called a trapezoid.
No previous use the words with Hutton's definitions is known. Nevertheless, the newer meanings of the two words now prevail in U. S. but not necessarily in Great Britain (OED2).
Some geometry textbooks define a trapezoid as a quadrilateral with at least one pair of parallel sides, so that a parallelogram is a type of trapezoid."

**Truncated** means to shorten by cutting off and is related to the Old English *truncheon*, which means a club or staff. Both words are derived from the Early French *truncus* which referred to a cutting from a tree used for grafting stock. A truncated solid is usually cut off by non-parallel lines as opposed to a Frustum.

**Vector** is derived from the Latin root *vehere*, "to carry". The root is also the source of everyday words like vehicle. The mathematical use of vector was introduced by W R Hamilton

**Zero** comes to us from the Arabs, the inventors of zero, and the Arabic word *sifre*, from which we also get the word cipher