Sunday 10 May 2020

Volume A-B Etymological and Historical Dictionary of Mathematics

Abacus The early Greeks used to design small trays and cover the bottome with sand to help hold pebbles steady when they were calculating.  The Greeks word for these trays was abax, and this became the term abacus.  Because it's in Greek, the plural is abacii.

Abcissa  is the formal term for the x-coordinate axis and for the x-value of a point on a coordinate graph.  The abscissa of the point [3,5] is three.  The word is a conjunction of ab, remove, and scindere, tear. Literally then, to tear off or cut apart, as a line perpendicular to the x-axis would do to the coordinate plane. It appears that Leibniz coined the term around 1692. The word was first used in the 18th century in the general "distance" meaning.  The first use of the word in English was in 1706.

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, and the use of parallel lines symbol, |z|,  was first used by Karl Weierstrass in reference to complex numbers. Steven Schwartzman suggests that the use of the word for real values only became common in the middle of the 20th Century. Common or not, it appears in the 1919 edition of Practical Mathematics for Home Study by Claude Irwin Palmer, "The absolute or numerical value of a number is the value which it has without reference to its sign, Thus +5 and -5 have hte same absolute value, 5."

The symbol for absolute value is usually a pair of vertical lines containing the number.  |3| is read as "The absolute value of three".  The absolute value of a real number is its distance from zero on the number line, so |3| = |-3| = 3.  For complex numbers, the absolute value is often called the magnitude, or the length of the complex number.  Complex numbers are sometimes drawn as a vector using an Argand Diagram, and the length of the vector of z=a+bi, is written as |a+bi| but the value of |a+bi| is z2 = ( a^2+b^2)1/2.  The term was used in Latin by Euler, valuer absolue, but the first English appearance seems to have been in 1816 in a translation of Lacroix.

Abundant:  Abundant Numbers are numbers whose proper divisors total to more than the number.  Twelve, 12, is the smallest abundant number, because the sum of its proper divisors, 1 + 2 + 3 + 4 + 6 = 16.

The word is from the union of the Latin roots ab, away, and undare, to flow. Literally then, the word means overflowing.  The ancient Greeks applied mystical properties to numbers and used them to predict futures and personal fortunes.  They worked with perfect, abundant, and deficient numbers as early as 100 years before the Christian era (Theon of Smyrna about A. D. 130). The first known use of "abundant number" in English was by Robert Recorde in Whetstone of Whet(1557)  "
Imperfecte nombers be suche, whose partes added together, doe make either more or lesse, then the whole nomber it self... And if the partes make more then the whole nomber, then is that nomber called superfluouse, or abundaunt. "
Accumulate comes into English from the Latin roots ad, to, and cummulus, a heap.(nice word to remember in identifying clouds). The Accumulation Function in finance is defined by the ratio of the future value and present value.

Acre The source of our word for land measure is from the Greek agros, field.  The same root gives us many modern words related to agriculture.  Although the original root described any open area, by the time of the Anglo-Saxons it was used more specifically as the amount of land a yoke of oxen could plow in one day.  Sometime in the middle ages it was set equal to its current size, a unit of land equal to 4840 yeards or 43,560 square feet.  (This seems strange to me, as neither is a perfect square).

Acumulate See Cumulative

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 less than 90 <sup>o</sup>.

Acute Triangle Impress your teacher/students by informing then that acute angled triangles were often referred to as "oxygons". The term appears in translations of Euclid into English in 1570, 1685, and by William Rowan Hamilton in 1838.  An acute triangle is simply one that has all three angles less than 90 degrees.  The root "oxy", as in oxygen, is from the Greek for acidic, or sharp.

Add/Addition The arithmetic word addition is from the Latin root addere, to give or to do.The dere part of the root is the same root that gives us Data/Datum and the name for dice. Donation and condone also share the same root.  The first recorded use of the word in English is from "The Craft of Nombrynge" according to Jeff Miller's web site.  The book was one of the first English language documents dealing with mathematics.  The plus symbol, +, was used for surplus, with - for deficit long before it began to be used as an arithmetic operator.  It has been shown that the symbol ws used on barrels to mark them as over or under weight prior to the earliest known use in manuscripts.  Some even suggest it dates back to the early Greeks, but others suggest it was a short form of the script word et, Latin for "and".  The word seems to have been first used in the 1300's in The Craft of the Nombrynge, but the symbol only came into common usage after it was used by Robert Recorde in his 1557 book, The Whetstone of Whet.

Addend The word we now use for the numbers to be added together in arithmetic is shortened from the word addendum. Originally the word was applied only to the values after the first, for example, in 4+3, the three was considered an addendum or addend, but not the four, since only the three was being added.(See Augend) Acording to the Oxford English Dictionary, the word was first used in English in 1969 by Samuel Jeake, "Place the Addends in rank and file one directly under another"

Additive Identity like other formal names for group and field operations emerged in the 1950's with the move to rigorous notation.  The first use I can find is in 1953 in Johnson's Abstract Algebra. In real numbers, the additive identity is zero, since a + 0 = a, preserving the identity of the addend.

Additive Inverse  The additive inverse of a number is the amount that is added to it to reduce it to zero, in effect, the additive inverse of any n, is -n.  The term was not used until the ascent of abstract algebra and formalized axiomized mathematics.  It appears in Birkoff and Maclean's Survey of Modern Algebra in 1953.

Adjacent The word adjacent is used to represent things that are next to, or close to, each other.  This closely reflects the origin of the word.  The ad prefix implies near or toward.  The major root is from the Latin word for "being thrown", jacere.  Literally then, adjacent angles are "thrown beside" each other. This math word might impress your English teacher if you tell her it is also the same base roots as the word adjective.  Adjectives are often placed between an article and the noun it modifies, thus, thrown near each other.  The "throw" base of the root is more evident in words like project and eject.

Affine is from the Latin affinis, for related, and gives us the English word affinity.The first use in Mathematics was in Latin by Leohnard Euler (1748). Jeff Miller's web page reports that "Affine is found in English in 1895 in Annals of Mathematics: “Let us find the differential equations of the first order which are invariant under the G1 of the affine transformations in the plane.” [OED]. There are Affine Geometry, Affine Transformations, Affine Spaces.In the coordinate plane, all triangles can be formed with an affine transformation of an equilateral triangle.

Affine geometry is the study of measures that are preserved uner a transformaton that carries each point (x,y) to a new point (ax+cy+c, bx+dy+f) This is sometimes described as a parallel projection from one plane to another.  Euler ws one of the first sto study affine geometry.  In an affine geometry, the transformation does not necessarily preserve angle size, or distance, although it does preserve proportions of the division of a line. 

The principle root of affine comes from the Latin root finis for end, or border.  The prefix, ad, has the meaning of near, so the combination was used to mean "sharing a common border.  This was generalized over time to sharing a common interest of any kind.  Today, affinity refers to an attraction of almost any kind.  I have an affinity for math.

Aliquot parts The Latin word aliquot meant "how many".  The aliquot parts of an integer n are all the numbers less than n that divide into it evenly with no remainder.  The Romans used the term in a similar way when they would divide land into equal areas.  Aliquot parts play a role in many number theory ideas, including perfect Numbers, Amicable numbers, abundant and deficient numbers, and finding primes.  In Billingsly's English translation of Euclid's Elements, her writes, according to the OED, "This is called a measuring part... and of the barbarous it is called an aliquote part.  (most young students might agree, I prefer the more common terms, proper divisors).

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 Mnusa (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 prob ably the greatest living mathematician of his period.  The phrase Al0jebr at the start of the title became the word algebra in western languages.  The term means, "the reunion of broken parts:., Abu Ja'far Ben Musa is often mistakenly labeled as an Arabian, 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."

Algorithm As it is used in mathematics means a systematic procedure to solve a problem. The word is derived from the name of hte Persian mathematician al-Khowarazmi (see Algebra).  The first use of the word that I am aware of was by G W Liebniz in the late 1600's. It 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 computations.

Alphametric/Alphametic Puzzles.  see Some History about Alphametics (SEND+MORE=MONEY), and More

Alternate angle When two lines are cut by a third line, they create eight angles at the intersections.  The line cutting the other two is called a transversal.  If the two cut lines are not parallel, then any of the three lines can be considered a transversal.  Most geometry uses of the term involve two parallel lines, and any two of the angles that are on opposite sides of the transversal are called opposite angles.  The terms are almost never applied unless the two angles either both lie inside the parallel (or cut) lines, called alternate interior angles, or both outside the parallel lines, called exterior angles.  The OED gives the first use as the 1570 translation of Euclid's Elements by Billingsley.

Altitude is the geometric term for the perpendicular distance from one base of a geometric figure to an opposite vertex, or parallel edge or face.  The same word is often used for a line representing this distance in a figure. It is frequently interchangeable with the word height.   The word comes from the Latin, altus, for high, with relations to words about growth and nourishment.  English words like elder, exalt, and adult are derived from the same root.  In astronomy altitude is used for the of a celestial object above the horizon.

Amicable Numbers The amicable number pairs are numbers x and y so that the sum of the proper divisors of x is y, and the sum of the proper divisors of y is x.  The pair 220, 284 were known as early as the Pythagoreans, about 500 BC. No others were found for over 2000 years according to Oystein Ore.  This seems surprising since the next discoveries were by Fermat, after he read a translation of a work by Arabic mathematician Thabit ibn Qurra, (836-901).  Others claim there is Arabic work from the 14th C regarding a second pair, 17296 and 18416.  In 1636 Fermat found that one, and also a third pair, 9363584 and 9,437056.  That was where things stood, until Euler took up the task. In 1789 alone he found 30 pairs, and continued in his life to find 30 more.  It seems almost unthinkable that after all this discovery, a much smaller pair was found in 1866 by a 16 year old Italian lad named Nicolo Paganini when he published 1184 and 1210. They are also sometimes called "amiable" numbers.  Hutton's 1796 mathematical dictionary says the first use of amicable was by Frans van Schooten.

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

Analogy The word comes from the union of the early Greek roots, ana + logos.  Logos was the 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.

Analysis – Latin from Greek analusis, from analuein ‘unloose’, from ana- ‘up’ + luein ‘loosen’. The branch of mathematics involved with limits, including differentiation, integration, measure, infinite series, and analytic functions. Formally began in the 17th century with the Scientific Revolution but many Greek mathematicians have already looked into this branch, in particular starting with geometric series. Also, Archimedes and Eudoxus looked into limits and convergence when studied areas and volumes of shapes. The explicit use of infinitesimals began with Aristotle in his book “The Method of Mechanical Theorems”. Chinese mathematician Liu Hui used proof by exhaustion to find the area of a circle in the 3rd century. Zu Chongzhi developed what would be called “Cavalieri’s Principle” in 5th century. Indian mathematician Bhaskara II used derivatives and what would be called “Rolle’s Theorem” in the 12th century. In the 1300s, Madhava of Sangamagrama used developed infinite series for trigonometric functions, long before Brook Taylor and power series, and he also found errors on these series. Then, in the 17th century, Descartes and Fermat independent developed analytic geometry and Newton and Leibniz independently developed infinitesimal calculus. Calculus then grew even into the 18th century with calculus of variations, ordinary and partial differential equations, and Fourier analysis.
In the 18th century, Euler developed the notion of a mathematical function. Real analysis really emerged with Bernard Bolzano with the modern definition of a continuity in 1816 but his work became recognized in the 1870s. Cauchy in 1821 though rejected the “generality of algebra” widely accepted and used, in particular my Euler. Instead, he formulated calculus in terms of geometry and infinitesimals. He also introduced Cauchy sequences and formalized complex analysis. Poisson, Liouville, and Fourier studied partial differential equations and harmonic analysis. These mathematicians as well as Weierstrass’ epsilon-delta definition created a breakthrough for modern day analysis. Further, in the 19th century, Weierstrass, Riemann, and Cauchy studied the theory of analytic functions.
In the mid 1800s, Riemann began with the concept of an integral. Mathematicians began to worry about the continuum of numbers without any proof. Dedekind swooped in with his “Dedekind cuts” to formalize exactly this, although Simon Stevin already developed this in terms of decimal expansions. Riemann integration also led to the study of discontinuity sets and measure theory from Cantor and Baire. Finally in the early 20th century, calculus was formalized using axiomatic set theory. With this, Lebesgue cleared up confusion about measure and Hilbert discussed important spaces with an inner product and in the 1920s, Banach began functional analysis. *Derek Orr

Analytic Geometry is a branch of algebra that is used to model geometric objects - points, (straight) lines, and circles being the most basic of these. Analytic geometry is a great invention of Descartes and Fermat. *Cut the Knot.  The term, according to math historian Carl Boyer, was used (created?) by Michel Rolle in 1709 (Since the term appears in the Title of the work, it was probably in some use by other mathematicians) .   The OED gives the earliest written use in English as Oct of 1817 in the American Mathematical Monthly:  "The question resolved by Analytic Geometry..."  

 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.  The Latin g word, genuflect is related to the English knee.  and refers to the Roman act of adopting a person(even an adult) by setting them on your knee to proclaim the adoptions.

Annulus When two circles share a common center, the area between the smaller and the larger is called the annulus. Think of it as a two dimensional doughnut. It may remind you of the annual rings you see on a cut tree stump.  It is hard not to try to connect annual, annul, and annulus.  Any connection that exists must relate to the early root of the word null,from the Latin  nullus, for none, or nothing.  The symbol for the null quantity is the circular zero, clearly looking like an annulus, and the annual ring; and something that is annulled is treated as if it never happened.  However remote the connection of the these words, if there is one, they can help you remember all of them better if you link them together in your mind.  The term first appears in English in 1803 in a translation of Newton's Principia, and again in 1834 in the Penny Cyclopedia.

Ansatz German word meaning "approach" or more specifically, "initial placement of a tool at a workplace". It is used as an educated guess to help solve a physics/mathematics problem such as a differential equation or a boundary value problem. By solving the problem, the ansatz is then justificed as an appropriate assumption/eduated guess. They have been known for centruies back to Euler in 1743 when he guessed y= Ce^(rx) as a solution to a constant coefficient second order linear differential equation. The OED gives the earliest citation of the term in 1936 by the Transactions of the American Mathematical Society. It is known that Hilbert used the term in the 1920's in German. *Derek Orr

Anti-parallels Almost every geometry student learns that a line through two sides of a triangle parallel to the third side (DE in the figure) will cut off a triangle ADE, that is similar to the original, ABC.  Almost none of them are shown that the reflection of such a parallel line segment in the angle bisector will produce another segment cutting off a triangle AE'D' that is also similar to ABC.  Such a segment is called an anti-parallel.  The anti parallel also creates a quadrilateral of the remaining piece of the triangle which is always cyclic, and therefore can be circumscribed.  The opposite sides of any cyclic quadrilateral are also said to be anti-parallels since extending either pair will produce a pair of similar triangles.


Anti-prism ----  See Prism  

Apeirogon A term, seemingly created around 1950, and still not widely in use, which is a general term for an infinite sided regular polygon.  The earliest I can find the word is in the Proceeding of the Royal Society, where it is described as inscriptable (it's circumcircle has infinite radius).  It seems to be an extension of the term perigon which is from the Greek for angle, and, it seems, and English term, perhaps related to periscope.  The OED simply calls "peri" in each case, a prefix, most often used in relation to astronomy. Perigon is defined as an angle swept out by a line in turning one complete circle around one end fixed as center.  Perigon dates back to 1868, but surely was in use earlier as the entry describes both the hemiperion and the hemisemiperigon (90 degree turn) .

a posteriori is drawn from the Latin for "what comes after."  Basicly drawing knowledge from experience, rather than from foundation axioms and postulates.  First entry in OED is for 1624.

Apothem The distance from the center of a regular polygon perpendicular to the sides, the apotherm, 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 for Earth). Apothem seems to be of modern origin, and seems to have appeared in English in the mid 1800's.  The word first appears in English in a translation of Legendre's Trigonometry by David Brewster(1822)

a priori is from the Latin for "from before" and essentially describes reasoning from foundation elements, like axioms and postutulates.  compare to a posteriori.

Arbelos  In geometry, an arbelos is a plane region bounded by three semicircles with three apexes such that each corner of each semicircle is shared with one of the others (connected), all on the same side of a straight line (the baseline) that contains their diameters.The term is Greek for  a shoemaker's knife, and was first found in Archimedes Book of Lemmas.  The area of the shade arbelos in the image is equal to the area of a circle with a diameter equal to the  height of the vertical line from the intersection of the two smaller circles up to the larger cirlce.  

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

Are An are is a unit of measure for areas 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 many countries.

Area The general term for a measure of two dimensional spaces comes unchanged from the Latin.  The more common meaning of area applies, as it did 2000 years ago, to a flat expanse of open unoccupied land. The French shortened the word to are as a name for a measure of land equal to 100 square meters.

Argand Diagram the method of drawing complex numbers as vectors on a coordinate plane.  They are named for Jean R. Argand (1768-1822), an amateur mathematician who described them in a paper

in 1806.  A similar method had been suggested as early as 120 years before by John Wallis, and developed extensively by Casper Wessell (1745-1818), a Norwegian surveyor (Actually, at the time the area where he lived was part of Denmark. Norway became an independent government in 1905 after years of domination by Denmark. and Sweden. )  It may be that even then, the method was unknown to Gauss and he had to discover it for himself in 1831.

    Wessel's paper was published in Danish, and was not circulated in the languages more common to mathematics at that time.  It was not until 1895 that his paper came to the attention of the mathematical community, long after the name Argand diagram had stuck. The earliest appearance in English seems to be around 1890.

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

Arithmetic was the Greek word for number, and it closely related to the root of reckon, which is becoming an obsolete term for compute, except of course here in the South where they "reckon" almost anything, as in "I reckon I'll have a sip of moon shine while I read me some Math.."  The beginning of the word is from the Indo-European root ar which is related to "fitting together" and gives us words like army, and art, order, adorn and rate. The final root is the same as metres, to measure,   There was a period where the spelling of arithmetic picked up another r in reflection of this root, and occasionally the leading a was dropped as it was by Fibonacci who wrote, "rismeterca".

Arithmetic Mean   See Mean

Arithmetic Triangle/Pascal's Triangle The short life of Blaise Pascal was rich with mathematical invention.  Unfortunately, most young students know him only through this triangular array of the binomial coefficients which bear his name, but which was not his creation.  When he spoke of the array,, Blaise Pascal used the term  "arithmetic triangle" (triangle arithmetique).  In Italy it is called Tartaglia's triangle, after the nickname of Nocoli Fontana, who had been nicknamed Tartaglia, the stutterer, after his was struck in the face with the sword of a French soldier at the age of 12.The second volume of his General Treatise on Number and Measure, included extensive investigations  into the triangle, and using it to raise binomials  to powers using the binomial expansion.

In parts of Asia it is called Yang Hui's triangle after a minor Chinese official who wrote tow books, dated 1261 and 1275, which uses decimal fractions (long before Western use).  Omar Khayyam, the Persian poet, mathematician, and astronomer, also wrote on the triangle and may have preceded Yang Hui, but the exact years of his life are uncertain.

   I highly recommend students trying to expand their understanding the arithmetic triangle, its construction and applications, should look at the excellent article at Wikipedia.    

Associative The root of the word associative is the Greek word for social, soci.The first use of the word was probably by W. R. Hamilton around 1850.  Until his inventions of Quaterninons, there were

simply almost no operations which did not have both the associative and commutative properties.  Associative means you can group the values anyway you want to compute them so (3+4) + 5 is the same as 3 +(4 + 5).  Multiplication in the Octonions, created by Graves as an extension of Hamiltons, quartionions, is non-associative.

Astroid The astroid is the path of a point on a circle rolling inside another circle with a diameter four times as large.  Such a path is sometimes called a hypocycloid of four cusps, because it is rolling under (hypo) the larger circle.  The word astroid, which seems to be a variant of the spelling of asteroid, comes from the Greek aster, for star. It seems the name astroid does not appear before 1836 (or 1838).  Before that, and even afterward, it has gone by many names, including tetracuspid (four cusps), cubocycloid, and paracycle.

  The parametric equation for an astroid with a circle of radius r rolling inside a circle of radius 4r is given by  x(t) = 3r Cos(t) + r Cos(3t), and y(t)= 3rSin(t) - rSin(3t).  The figure can also be drawn with the equation \$ x^{2/3} + y^{2/3} = R^{2/3} \$, where R is the radius of the fixed circle.  The length of its path is 6R. This formula was known to Liebniz in 1715.  The area of the astroid is 3/2 the area of the rolling circle.   The length of the astroid is 6R..

The word astronaut (sailor on the stars) is related, but is even a newer creation.  It seems to have been created around 1929 and never popularly used until 1961.
 A general term for equations of the form |xn| + |xn| =|rn| is a Lame curve, after French Mathematician Gabriel Lame (1795-1870).

Asymptote The asymptote of a function as it is now used is much narrower than the original Greek meaning.  The word joins the roots a(not), with sum (together) + pipten (to fall) and literally means "not falling together.", or not meeting.  The word is believed to have known to Apollonius of Perga before 200 B C.  Originally used for any two curves that did not intersect.  Now it is used primarily for straight lines that serve as a limiting barrier for some curve as one of its parameters approaches infinity.  The pet base of the root piptein gives us words like petition, petal, petite and propitious. The  OED cites the first use in English to Hobbes, "Asymptotes... come still nearer and nearer, but never touch."

Atom The early Greek word for something that was so small it could not be subdivided was atmos.   They used the term in reference to objects of time as well as matter.  In the 1800's chemists adopted the term atom to refer to what they thought was the indivisible unit of every element.  The word actually is made up of two parts, the negative prefix, a, meaning not, and the root tom for cut, hence the meaning "cannot be cut".  The Greek root is preserved in the word anatomy, "cut into pieces".

Augend is the term for the original term in an addition operation. In something like 4+7 the 7 is the addend and although many people now use addend for both terms, an earlier term for the four would be the augend, which came form both Latin and German roots meaning the amount to be added to or increased.

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 deductions was made from each persons share, based on their declared value, for a portion of the loss, their Average.

Axiom In the language of mathematics and logic, an axiom is a statement that is considered to be true without need for proof.  The word is often used interchangeably with postulate.  The origin comes from the Greek root axios for worthy.  An axiom is something that is accepted as worthy of its own accord, without proof or risk of refutation.  The root shares a common origin with words relating to leadership or influence, such as agitate, which originally meant a great conflict or contest, and ambassador, one who is sent around.  Although we often speak of Euclidean axioms or Euclidean postulates, math historian D. E. Smith has written that Euclid seems to have uase a more general phrase meaning "common notion".  Certainly he would have known of the word axiom as it was used by many of the ancient Greeks, including Aristotle.

Axis seems to be one of the older math words.  The early Greek root aks, for a point of turning or rotation.  Words like axle and axon are derived from the same root. The Latin diminutive forms, axilla, axla, ala for the shoulder or a wing.  From this extension we get aisle, which separates the wingof a build, and axial, for the point where a bud grows from a branch.  The word in English was used by Thomas Digges in 1571 in relation to the axis of a cone.


Bar chart/bar Graph The earliest known example of a bar chart  was by William Playfair in a Commercial and Political Atlas in 1786. The name was not affixed until the 20th Century.  The term did not emerge untill 1914 according to the OED which gives Nov issue of Engineering Magazine with a reference to a "bar chart above."

Barycenter The word is another term for the center of gravity or centroid. The Greek root is baras which generally refers to weight or heavy. The more ancient Indo-European root seems to have come from a word like '"gwerus" and has relatives in our words for gravity, and grave.  Another word derived from the same root is baryon, the name for a family of particles that are heavier(more massive) than mesons.  The word barometer also comes from the same root and is so named because, in a sense, it measures how heavy the air is.  Another related word still in current use is baritone, which literally means heavy voiced.  The science names for hte chemical barium and the ore from which we obtain it, barite, also called "heavy spar" are both from the same root.  The History of Math web site at St. Andrews University in Scotland credits the creation of barycenters to August Mobius (1790-1868).

Baryon See Barycenter

Base (geometry)  The word is drawn from the classical Latin, basis, which is the lowest part of an architectural structure. The same word is more generally used today to mean the foundation of any idea or argument.  In Geometry it is used as a geometric object (a line or plane..)  upon which in which some other object is standing.  The altitude of a triangle, for example, reaches from a vertex, to the opposite side, which is called the base.  One of the earliest uses in English was by Thomas Digges in 1571 in his Geometry Text.

Base of an isosceles triangle is defined in the same text by Digges by defining the third side of a triangle with two sides equal, as the base, and then stating that the angles joined by the base are equal.

Base (logarithms) /Base (of Number system)   The word "base" in both of these uses was often interchanged with the term radix, which is from the Latin radic, for root or base of a plant. The symbol for radix, ℞ , which was commonly used for a base of  an exponential quantity, even serving as the symbol for a variable, as we would use x, or y, today.  This symbol was sometimes used as a script r, and Euler thought that was the foundation of the now more comnon √ also called a radix, or radical (another root related word).  In 1560 Leonard Digges used, "The Radix Quadrate of the Product, is the Hypotenusa."   In 1772 J. Finn uses, "We can find the value of x such that \$p^x=b; this value is called the logarithm of b and p is the base of the logarithm."

 In 1864 J. Wilson  wrote, "there is some evidence to show that base four has at times been used as the base of a number system" in Phrasis 67.  The use of Radix for base continues into the 20th century, as in 1946 when J W Mauchly said, "Since it is now customary to use the decimal system, there is little likelihood that a system on the radix 8, or 16 can be brought into common use.  (I find it interesting that at the time he wrote this, the British monetary system, which one might think was a common use, was based on a system of one pound was divided into twenty shillings, which were each worth 12 pence.)

Bell Curve  See Normal Distribution

Bernoulli Numbers Students often learn to find the sum of the counting numbers, 1 + 2 + 3 +...+n by using the formula \$\frac{n*n+1}{2} \$   Later they may find another for summing the squares of those integers,  \(\frac{1}{3}(n^3 + \frac{3}{2} n^2 + \frac{1}{2}n) \)

Around 1631, a mathematician named Johann Faulhaber (1580-1635) derived a formula for sums of any power.  His method used an expansion that looked very similar to a binomial expansion with a sequence of special exponents.  These constants, 1, 1/2, 1/6, 0, -1/30, 0, + 1/42.. could be continued indefinitely to find the sum of the nth power of  the first k integers.   82 years later in his Ars Conjectandi, a masterful document written by Jacob Bernoulli (1654-1705) and published posthumously described this method in great detail giving full credit to Faulhaber.  Bernoulli claimed that he ws able to add the tenth powers of the integers up to 1000 in 7.5 minutes.  And in the strange, but common method of mathematical crediting, these numbers are known today as Bernoulli numbers.  Ivo Shcneider, in a note to Historia Matematica newsgroup, stated that de Moivre and Euler began calling them Bernoulli Numbers.  Between Faulhaber and Bernoulli, it seems that a Japanese mathematician named Takakazu Seki (?1642-1708) also may have discovered the sequence independently.

Bifurcation – It comes from medieval Latin bifurcat- ‘divided into two forks’, from the verb bifurcare, from Latin bifurcus ‘two-forked’,
from bi- ‘having two’ + furca ‘a fork’. Literally means “the division into two forks or branches”. Mainly used with “bifurcation theory” which studies the topological/qualitative changes to a system of differential equations as a parameter varies. It was coined by Henri Poincaré in 1885 who also named/defined types of bifurcation. Today, there are many types such as saddle-node/fold bifurcation, Hopf bifurcation, transcritical bifurcation, homoclinic bifurcation, pitchfork bifurcation and period-doubling bifurcation.
The image shows the pattern of bifurcation leading to the eventual chaotic behavior of the Logistic Curve, x' = r x(1-x) where x is in the interval 0-1.   *Derek Orr

Billion seems to have been a French creation, and was originally bi-million. The term originally meant \( 10^{12}\) oe onw 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 shows 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 --------Billiard -------- Quadrillion

BiMedian of Quadrilateral If you find the midpoint of each of two opposite sides of a quadrilateral, the segment that connects them is called the bimedian. If you construct the bimedian betwen the other two sides, the two bimedians will bisect each other.  They are related in an interesting way to the diagonals of the quadrilateral.

A)  The bimedians are of equal length if, and only if, the diagonals are perpendicular

B)  The bimedians are perpendicular if, and only if, the diagonals are equal in length.

Some people also refer to the segment joinging the midpoints of the two diagonals of a quadrilateral as a bimedian as well. This third bimedian passes through the other two and the point bisects that bimedian as well.
Indeed, the point of intersection of the medians is nothing but the barycenter -- the center of gravity -- of a system of four equal weights (or material points) placed at the vertices of the quadrilateral.

Binary Numbers   The word binary refers to something that has two main parts.  The root is from the Latin binarius, which literally meant two by two.  Binary numbers are numbers built up with the use of only two numerals, zero and one.  The counting sequence 0, 1, 2, 3, 4, 5 in binary would be 0, 1, 10, 11, 100, 101.  According to the Oxford English Dictionary the word first appeared in English in 1796.  The system of counting out things in a binary style is first known in the writings of  Pingala, an Indian mathematician from the 300  BC, who used a binary system of short and long beats to describe the poetry of Sanskrit.

Binormal – The line/vector which is normal (perpendicular) to both the normal and tangent lines/vectors to a point on a curve. These three vectors form the axes for 3D Cartesian space. *Derek Orr

Binomial Coefficient
Binomial Distribution/ Trials/ Bernoulli Trials.  A binomial trial is an experiment which is the result of a repeated number of events, each independent of others, each having one of two possible outcomes, and with the probability of success (usually called p) constant from trial to trial.  Flipping a coin ten times and seeing how many heads you get is such a trial.  The term binomial is from the fact that each trial has two outcomes, bi from the root for 2, and nomen from the Greek root for name. Nomen is the root for denominator and the French nom de plume. The term binomial was first used in English by Robert Recorde in 1557. The trials are now often called Bernoulli trials, after Jacob Bernoulli who wrote of them in 1713.  The actual term "Bernouli trial" seems not to have been used before 1937.

The results of such a trial should happen according to the Binomial Distribution which gives the expected outcome for any number of successes. The outcome for k successes in n trials is given by \( P(k) =\binom{n}{k} p^k (1-p)^{n-k} \).  these values preceeding the variables using p and k, are called the binomial coefficients. Students familiar with Pascal's Arithmetic Triangle can  find the expected outcomes for coin flips and other events of equal probability of success and failure by looking at the correct row of the triangle.  For example, in 5 flips, the row 1,  5, 10, 10, 5, 1 The numbers give you the probability of getting 0,1,2,3,4,or 5 successes when each term is divided by the row sum, 32.  So if you flip a coin five times, the probability of getting only one head, is 5/32. As the number of trials approaches infinity, the binomial distribution approaches the Normal Distribution, with a mean of np, and a standard deviation of \( \sqrt{np(1-p)} \)

Bisect  comes from Latin bisectus literally meaning to cut into two parts.  The roots are bi, for two, and secare, to cut.  Bi appears in such terms of binary,  and biscuit, which if from the French for "twice cooked".  Secare is the root for words like section.  In mathematics and geometry, bisectors usually cut into two equal pieces, such as the bisector of a line segment, which divides it into two congruent lengths, or the angle bisector which divides the angle into two angle with equal measure.  The word seems to have been an English creation, and first appeared around 1560. Barrow's translation of Euclid's Elements uses "bisect a right line".

Bit/byte  Claude E Shannon first used the word bit in his seminal 1948 paper, "A Mathematical Theory of Communication.  He attributed the word to John W. Tukey , Who had written a Bell Labs memo on January 9, 1947 in which he reduced the term "binary information digit" to bit.*Wik

Byte was created by Werner Buchholz in 1956 while working at IBM.  It was an respelling of bite to avoid any confusion with bit.

Bode's Law is a sequence which predicts the distances of the planets from the sum in astronomical units (AU).  The distance from the Earth to the Sun is defined as one AU.  The law was published in 1772 by German astronomer Johann Elert Bode, and was based on the work of German mathematician Johann David Titius.

When the law was first published there were only six known planets Mercury, Venus, Earth Mars Jupiter and Saturn.  The formula states that the distances from the sum should correspond to the terms of the sequence given by \( dis(N) = \frac{3*2^N + 4}{10} \) except for the initial term for Mercury, which he set at .4. The first few terms of the sequence , .4,  .7, 1, 1.6, 2.8, 5.2, 10, 19.6 predicted the known planets very well, except there was no planet at the distance 2.8 between Mars and Jupiter.  In 1781 the planet Uranus was discovered and its distance of 19.2 AU seemed close to the predicted 19.6.  Bode encouraged astronomers search for a planet at the missing 2.8 AU distance.  In 1801 Ceres, the first of the many asteroids at the 2.8 AU distance was found and thought to be the missing planet.  Hundreds more asteroids were found at that distance and that they might be the remains which had broken up, or the un-assembled clutter of a planet that had not yet formed.

I recently read that the French philosopher Hagel had begun to circulate his thesis to "prove" there could only be seven planets, when the news of Ceres discovery forced a quick change of the document.  The discovery of Neptune in 1846 at 30.1 AU in stead of the predicted 38.8 and then Pluto in 1930 which was at 77.2 AU instead of 39.6 led to discrediting of Bode's law, although many astronomers believe it may have some physical foundation.

Boolean – Credited to mathematician, philosopher, logician George Boole (1815 – 1864). The word pertains mostly to data that have the characteristic of being “true” or “false” (similarly “1” or “0”) and this type of mathematics is called “Boolean algebra”. It was introduced in an 1847 pamphlet titled “Mathematical Analysis of Logic” by George Boole. It seems “Boolean algebra” was first suggested by Henri Sheffer in 1913 however Charles Peirce gave the title “A Boolian Algebra with One Constant” (incorrectly spelled) as Chapter 1 in this 1880 book “The Simplest Mathematics”. The OED has an earlier suggestion for the term in the Cambridge and Dublin Mathematical Journal in 1851, "The Hessian, or as it ought to be termed, the first Bolian Determinant." *Derek Orr

Bootstrap (statistics) 

Borrow (in subtraction) Jeff Miller in his web site on the Earliest known use of some of the words of Mathematics has, "BORROW is found in English in 1594 in Blundevil: "Take 6 out of nothing, which will not bee, wherefore you must borrow 60" (OED2)." The writer appears to have been working with base sixty. For more detail, see my Borrowing in Subtraction, a Brief History

Bowditch Curves 

The curves are also called Bowditch curves for the early American mathematician, Nathanial Bowditch, who worked with them aournd 1800. In general, a parametric curve with equations x= A sin(k t ); y= B sin(m t), the curves can describe things as simple as a circle or ellipse to more complex open and closed curves. If the ratio of k/m is rational, the curve will eventually close.


When I introduced these to my students I often call them the Chinese finger cuff curve (I am still waiting for the rest of the mathematical world to adopt this term, fall into line people) As I neared retirement it seemed that many of the students had never heard of finger cuffs, but there were always a few who knew of them, and often at least one student who would produce one from home over the next few days.

 They are now more often called Lissajous curves (See ) after Jules Antoine Lissajous (4 March 1822 in Versailles, France - 24 June 1880 in Plombières, France) was a French mathematician best known for the Lissajous figures produced from a pair of sine waves. Lissajous was interested in waves and developed an optical method for studying vibrations. He wanted to be able to see the waves that were created by vibrations, usually expressed in the form of sound. At first he studied waves produced by a tuning fork in contact with water, studying the ripples that were caused. Working on these ideas, he published Sur la position des noeuds dans les lames qui vibrent transversalement (1850). In 1855 he described a way of studying acoustic vibrations by reflecting a light beam from a mirror attached to a vibrating object onto a screen.*Saint Andrews University

Brachystochrone  In the last part of the 15th Century one of the science questions of great interest was the question of what path a particle should move from point A to point B under the influence of gravity in the shortest time.  The curve was called the brachystrochrone problem.  The word joins two Greek roots, brakhus for short, and the more common khronos for time. The first root only exists in a few scientific terms, such as brachypterous for short winged insect.  Chron appears far more commonly in words like chronic (all the time) and chronological (in time order)

In 1695 John Bernoulli challenged the world to find the path, and Newton returned the solution anonymously.  It is said that upon receiving the solution, Bernoulli replied, "We recognize the lion by his claw.". Newton's solution showed that the path ws along part of a cycloid.

*wik  The curve of fastest descent is not a straight or polygonal line (blue) but a cycloid (red).

Briggsian Logarithms Shortly after John Napier published his great work on Logarithms, he was visited by an admiring English mathematician, Hernry Briggs. Briggs had started to develop a table of logarithms based on the powers of 10, with log(1)= 0 and log(10)=1.  With Napier's blessing and encouragement, he would expand them.  These logarithms, as we would say today, are base ten logarithms, and often called "common Logarithms", but the word base seems not to have been developed until much later.  The first recorded citation in the OED is in 1772 by a J. Fenn, "We can find the value of x such that px=b; this value of x is called the logarithm of b, and p is the base of the Logarithm.

Bureau  is another word that relates, in part, back to the wide spread use of counting tables in business transactions.  Today the word may refer to a dresser table, a writing table, or a department of the government, and how this came to be is our story.  The root of the word comes from the Latin burra for a tuft of wool (as a side note, the Romans also used the word to describe clownish or slapstick behavior, and that is the origin of the word burlesque).  The French made a rough wool cloth which they called bural.  The cloth, often in a checkered design, was spread across a counting table, and eventually, the cloth, table and all became the bureau.  In time, the room in which the counting took place, the money and debts being counted, and the entire agency responsible for the accounting became the bureau.  Records of wills and estates from the middle ages show that these tables, their coverings, and the counting instruments which necessitated the drawers, were passed along as heirlooms.  Eventually they lost their value as counting tables, but were still serviceable as tables to hold writing materials in the study, or my-ladies fine toiletries in the boudoir.  Counting boards also made their way into the kitchen, but here they more often keep a name that was a contracted for of the original purpose, counters.

In England , the Bureau responsible for the collection of taxes, distribution of national money, and the general keeping track of the government revenue was called the Exchecquer, for the checkered cloths that been used for counting, and similar to the ones used for playing chess and checkers.

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