Friday 15 May 2020

Volume Q-R


The acronym for quod est absurdum (“which is absurd“) — a Latin phrase used in the old days to conclude a proof by contradiction. In the modern days, its usage is increasingly being replaced by the symbol ⨳, though other symbols, such as ※ or \$ \Rightarrow \Leftarrow  \$, are equally well-adopted as well. "  *The Definitive Glossary of Higher Mathematical Jargon

Q. E. D. "In the Elements Euclid concluded his proofs with ὅπερ ἔδει δεῖξαι “that which was to be shown”: see e.g. the end of the proof of Proposition 4 on p. 11 of Fitzpatrick’s Greek-English Euclid. Medieval geometers translated the expression as quod erat demonstrandum (“that which was to be proven”)".  *    Paul Halmos in his 1950 book on Measure Theory introduced a new version for showing the end of a proof.  It is called the tombstone, and sometimes, the Halmos. Halmos said he arrived at the idea when he saw it used at the end of articles to show the end of a story.  Halmos is also credited for the iff to replace "if and only if."  

The earliest citation in the OED is for 1614 by William Bedwell in De Numeris Geometricis, "...12 by 2 and of 6 by 4, are equal, q. e. d."  The term was often used when no actual proof was involved, as in an 1892 quote from Haddon Hall by S. Grundy, "Tho' the world be bad, It's the best to be had; and therefore Q.E.D."

Quadratrix of Hippias The three classic problems of Greek Geometry were to produce a square equal to the area of a given circle, to create a cube with twice the volume of a given cube, and to trisect a general angle.  They could not, and still can not, be done under the challenges of the conventional use of Compass and straightedge.  But at least two of them were done by the use of a kinematic curve called the quadratrix of Hippiss of Elis, who used the construction to trisect any angle in 420 BC.  The same curve was later used by Dinostratus around 350 BC to square the circle.

The line is constructed by drawing a circle in a unit square, the point E moves the same proportion of the arc BD as the point F moves along the segment AD.  The intersection of the horizontal line at F to the ray AE forms the point S.

Once an angle was set to E, the horizontal F could be drawn, and trisecting the segment AF allows the construction of the lines F' and F'' at equal distances, and the angle BAF" will be 1/3 of the measure of the angle BAE.

Quadrilateral  is the name for a closed polygon with four sides and four angles.  It may be convex or

concave, and includes the group of all closed four sided polygons that have no crossing sides.  It is occasionally called a quadrangle in analogy to the triangle, and very rarely, tetragon.  Quadrilaterals with none of the conditions to make one of the specific named quadrilaterals, are sometimes called irregular quadrilaterals, to indicate this lack of other special features.  In both the US and UK, the term was seldom used before 1900, and the term trapezoid in the UK, and trapezium in the US, was used for these generic four sided polygons.  The two terms were also reversed in naming the quadrilateral with one pair of parallel sides.

Trapezoid and trapezium are from the Greek word for a table.  The division between which word to use for the generic shape, and which for the one with a pair of parallel sides seems to date back to the Greeks.  Euclid used trapezium for the generic case, and Proclus in the other.

Quadrilateral is from the Latin, quadrilaterus,  for Four-sided.


Quadrant/Quadratic/Quadrature/Quadratic Equation Quadrant is from the Latin word  quadratis, for square, As, or from a new coin created to be 1/4 of the As.  It was called a quandras.  Later the word was generalized to quadrant and its meaning broadened to apply to many things that were shaped like a quarter-circle, or arc of a circle.  Quadrants with special markings (like a protractor for the sky) were used by astronomers and navigators long into history.  The layout of many cities, including Washington D.C. is into four quadrants, and of course, when the time came to name the four corners of the coordinate plane created by the axes, they called them the Quadrants.  The English (1450)  picked up the 1/4 part of money in their own, for a quarter of a penny but it was overshadowed by farthing (from the Norse, 1335) and a quarter of the day appeared in 1582..

 Quadratic is the Latin root for "to make square."  It is used for ideas like squares, or making things square, and also for algebraic equations and expressions that represent square areas, with a highest exponent of 2.  .  The original geometric problems students now see as a quadratic were often presented as a square and a rectangle that formed a given area.  By completing the square, they were finding a side length which would be the correct length of a square equal to the total area.  The term shows up as a part of the name of many mathematical ideas from simple algebra, statistics, calculus, number theory, and computer science.

The earliest citation in English in the OED was a Nov 4, 1647 was by John Pell in a letter to C. Cavendish, "Not so high as a quadratic equation.".  In 30 March, 1668 John Wallis in a letter to Oldenburg wrote, "He tells us that these two forms of Quadratick Equations.... are both affirmitave."

The earliest citation of the term used to describe shape in English was by T Stanley in 1656, "A pyramid of quadratick base." An algebraic application in 1668 by Bishop J Watkins, "Those algebraic notions of the absolute, lineary, quadratiks, & ..`."  And in 1674 John Wallis, in a 14 Feb letter to Oldenburg, included, "spirals, quadratics, and the like..."  speaking of curves, not equations.

Quadrature became the Latin name for the Greek geometrical studies in which they sought to construct rectangular areas for those enclosed by circular arcs.  Squaring the circle, but using only the conventional Greek tools of compass and straightedge, was pursued by geometers around the globe, until advanced algebra allowed us to show it could not be done.  The earliest OED citation for this usage was in 1569 by J. Sanford, "Yet no geometrician hath founde out the true Quadrature [L. quadraturum]  of the circle. " 

Many students may only recognize the word quadratic as part of  quadratic Equation, so a brief note on the early history of quadratic equations.   The earliest known recording of a quadratic equation is in the Berlin Papyrus (named for the city of the museum where it is housed) from the 19th dynasty about 1300-1200 BC.  The problem presented two squares with an area of 100, and the smaller had sides 3/4 as long as the larger.  How they were solved is unknown, perhaps by inspecting tables of squares to use trial and error.  Later, by the time of Euclid in 300 BC, the Greeks had discovered how to solve such equations with straight edge and compass, in the method that comes down to you in modern times as "completing the square", something they did by geometric construction rather than analytics.

Quadrivium  From the Greeks to the Middle Ages the education process focused on the seven liberal arts.  The lower three, grammer, logic, and rhetoric; were called the trivium, Latin for "the meeting of three roads".  Trivia and trivial come from the same word and reflect there lower position.  The upper four studies, called the quadrivium, were music, arithmetic, astronomy, and geometry.   From which we would realize that quadrivium means "the meeting of four roads."  The Pythagoreans (550 BC) thought of the parts of the quadrivium as the four branches of mathematics, and they persisted as a course of study into the Renaissance. The OED cites the term in 1657, "That so famous Quadrivium of the Mathematicks..."


Quindecagon  "QUINDECAGON is found in English in 1570 in Henry Billingsley’s translation of Euclid: "In a circle geuen to describe a quindecagon or figure of fiftene angles" (OED2). *Jeff Miller's Web Page

Quotient   "How many times?", is the question in a division problem. How many times can you make a group of that many from this?  How many times can you subtract this quantity, repeatedly, from that?  And the name for that answer came from the Latin word quot and the Classical Latin quotiens,  for that very question, how many.  R. Steele used the term around 1450 when he wrote in his Art Nombryng, "The number that showeth be quocient."


Radians The word radians is believed to be a made up word.  Some suggest it may have been intended as an abbreviation for RADIusANgle.  Here is a quote from Cajori's History of Mathematical Notations.  "An isolated matter of interest is the origin of the term 'radian', used with trigonometric functions.  It first appeared in print on June 5 1873, in examination questions set by James Thomson at Queen's College, Belfast. James was the brother of Lord Kelvin.  He used the term as early as 1871, while in 1869 Thomas Muir, then of St. Andrew's University, hesitated between 'rad', 'radial' and 'radian'.  In 1874 Muir adopted 'radian' after a consultation with James Thompson."

One radian is a measure of approximately 57 degrees, and there are 2 Pi of them in a full circle.

Radian is first cited in the OED in 1877 in the Philosophical Transactions.

The concept of the radian was created and used much earlier by Roger Coates in everything except the name in 1714.  The use of the length of the circular arc was used even much earlier by the Persian al-Kashi around 1400.  Before Thompson invented 'radians', the common term was 'circular measure'.  [I have a bright shiny dime, maybe two, for anyone who would send me a digital image of that exam page......I can dream, can't I]

Radius The word radius is the Latin word for the spoke of  a wheel. The earliest citation I can find was for anatomy in 1565, in which the bones in the arm were given both the Greek name, and the equivalent Latin name. The Latin seems to have won out as the radius in now the most common name for this bone.  An astronomical instrument use for taking the altitude of a star, was called a "radius Astonmical" in 1592 by John Dee..  In 1590 Hood's Elementary Geometry gave, "A radius is the right line drawn from the centre to the perimeter.".  The definition may not have intended the  center and Apothem. 

perimeter to be the circumference of a circle, since the term radius was also used in classical geometry to refer to the line from the center of any circumscribable polygon as well.  by 1590 circumference was regularly used in England for 200 years.  The distance from the center perpendicular to the perimeter is called the

The term is also used for the length variable in polar coordinates and also a measure of minimal distance in graph theory. There are also radius of curvature, radius of conversion,

Ratio The ancient root of ratio comes from the same early Indo-European root, ar, or ree,  that gave us arithmetic, through the same word, ratio, in Latin. The word appears in English as early as 1536 in relation to "logical reasoning", but seems to have awaited Isaac Barrow's Euclid in 1660 for the mathematical relationship between the magnitudes of two quantities.  Rate is a synonym, usually used with dimensions on the quantities.  A rational number, is a number that may be expressed as a ratio of two integers.  If not, it is not rational, and called irrational.   

Reciprocal is from the very similar Classical Latin reciprocus, and the al ending is an English mantissa. It means returning, or alternating.  The OED cites Billingsley's translation of the Elements in 1570 as the earliest source.  The mathematical meaning of the reciprocal of a number is the quotient when 1 is divided by the number, or 1/n. Another way of defining it is that the reciprocal is the multiplicative inverse of the given.  More than just numbers, the term is sometimes applied to Matrices, and to equations.  The word is sometimes used a as a prefix for standard units of measure, and some of them develop names of their own.  The measure of electrical resistance, the Ohm, was such an example when the use of the 'reciprocal ohm' for conductance became common enough to earn it's own title, the mho.

Recursive/Recursion The mathematical meaning of recursive is to describe a process or function for which the output at each level depends on the output at the previous step.  Think of a machine in which the output is fed around to the entry to be input of the next machine next process[Such machines are common in electronic controls and are called feedback loops.] The re is the common Latin for back, and the root currere, to run.  The literal meaning is "to turn back" or "to run back", aptly describing the reuse of the output as an input.  Things as simple as counting sequences (2,5,8,...)  or the Fibonacci sequence (1, 1, 2, 3, 5, 8, .....) are examples.

Although the roots are ancient, the word made it's way into English only in the early 20th century.  The earliest citation of the OED is 1916 in Science magazine.

Other related terms from modern English from the currere root are currency and curriculum, and of course, the happy little line dancing ahead of my screen writings, the cursor.

Regression is from the Latin roots, re, back, + gradus, to go, with the literal meaning "to go back".   The general meaning, to return to an earlier or more general pattern, fits well with the mathematics and statistics.  The OED indicates that Karl Pearson used the term coefficient of regression in a paper dated 1897.  Two years later he used the term regression line for the line of least fit.


Repeat Bar    See Vinculum

Repunit is a term for a number made up of all unit digits, 1, 11, 111.... It was created by Albert Beilier in Recreations in the Theory of Numbers in 1964 as a contraction of repeated digit. Repunits may be extended to any base, so in base two, the repunits would be numbers of the form 2^n - 1,

In decimal numbers, students may pursue the prime repunits beyond 11.  The next is R19, a string of nineteen ones.  No number can be a prime repunit unless its number of digits is prime.

Until recently, repunit was not found in most dictionaries.  The newer companion, repdigit, still suffers that lack of recognition.  A repdigit is a repetition of any of the digits with no variation, 333, 77777, 6666 are examples.  Beilier had called them monodigits as early as 1966, but around 1974 the name repdigit surpassed it.  There is even a special variation of them called Brazilian numbers which are the base ten that can be written as a repdigits in any other base, except as a repunit.  Repunits can trivially be created from any number n, in the base n-1, so they are outlawed.  (If anyone knows the origin of this unusual variation, I would love a note.)

Residual Sit back, stay right there, and I will tell  you the meaning of residual.  Wait! I just did.  The common re prefix means back, and the sid is from the Latin sedere which means to sit, so the literal meaning or residual is one who sits back or, more appropriately, stays seated. In statisitcs we used it in the same sense as residue, that which remains(stays seated) when something else is taken away; what remains from teh observed amount when the prdicted amount is removed.  Another closely related word is residence.  Other words drawn from the sedere root include sedentary [one who sits around a lot], sediment[stuff that settles], and sedative [something that keeps you from moving around]. 

Rhind Papyrus The Rhind papyrus is a famous document from around 1650 BC. It is named for
Henry Rhind who purchased it, and his estate donated it to the British Museum in 1864.  This document is one of the best sources we have about Egyptian arithmetic and geometry in the Middle Kingdom.  The document is about 33cm tall, 565 cm long.  The papyrus is believed to have been copied by the scribe Ahmes around 1600 BC from an older document that may have dated back to 2000 BC. 

There are four other lesser documents preserving Egyptian mathematis.  The Moscow Papyrus and the Berline Papyrus are named for the places they are kept.  The Kahun Papyrus, named for the place it was found, and the Leather Roll, named for its composition.  Although there are other scraps of Egyptian mathematics preserved, these are the bulk of what we know about Egyptian mathematics. 

Rhombus The rhombus is a quadrilateral with four sides of congruent length(and whether you include squares, or don't, is a matter of taste.) It is sometimes called a rhomb, and sometimes a diamond and sometimes, if you are French especially, a lozenge. In classical Latin, a rhombus was a diamond shaped instrument that was whirled on a string to make a whirring sound, aha, but why.  For that we have to go way back to the Minoean period, and the male mating ritual of putting your

life on the line to attract a pretty maid.  The image at the right shows a Minoean lad doing a handstand over the back of a bull from a pottery from the island of Crete.  But what if the bull was passive, and not interested in supporting your romantic endeavors. That's when the rhombus came in.  Apparently whirling the spinning rhombus made a noise that aggravated the bulls, and promoted them to attack.  For such fun, came the name for a geometric object that made its way into our mathematical landscape.

Robot The word robot comes directly to us from the Czechoslovakian word for compulsory labor.  The Indo-European seems to be orbh which also gives us orphan and the old Slavic words orbu and the later form rabu for slave. Perhaps the common origin of orphan and slave gives insight into the plight of orphans in earlier (but perhaps not much earlier) times. 
The first use of the word as we now know it was the Creation of a Czech playwright named Karel Capek.  Here is a quote from Chrysti the Wordsmith from the Montana Public Radio webpage, "Early in the 20th century, Czechoslovakian playwright and author Karel Capek envisioned a world ruled by automated machines.  Capeck crystallized this dark vision in 1920 into a play called R. U. R.  The initials stood for Rossum's Universal Robots.  Rossum's Universal Robots was a British firm that mass-produced robots designed to work as mechanical slaves.  The grand scheme behind this design was to create a better world for humankind by elimination menial labor.  Ultimately, however, the machines triumphed in a robot rebellion [Don't they always?], destroyed humanity and created a new world of their own."  

Robust  A statistical test is robust if it holds true even when the underlying assumptions are not narrowly met.  The word comes from the Indo-Eruopean root reudh for things that are red or reddish, but its meaning comes from the Latin robur, for the Red Oak.  Whether it was because the Oak is such a hardy, vigorous tree, or the idea that red cheeks are a sign of health, the word began to take on meanings related to strong or healthy.  The word corroborate, to support, is drawn fromt he relation to strength.  Many other words from the original root and related to red are still common in our language.  Rust and ruby are examples of objects named for their reddish color.  

The word came into the English language around the end of the 16th century with meanings of strong or healthy.  By the end of the 19th century it was being applied to economies.  The first OED citation in relation to statistics is for 1955 in an article by George E P Box and S L Anderson for the Journal of the Royal Society of Statistics, "to fill the needs of the experimenter, statistical criteria should (1) be sensitive to change in the specific factors tested, (2) be insensitive to changes, of a magnitude likely to occur in practice, in extraneous factors.  A test which satisfies the first requirement is said to powerful and we shall typify a test which satisfies the second by calling it 'robust'.

Root/Radical/Radix The Indo-European root werad was used for the branches or roots of plants.  Later it was generalized to meant the origins or beginnings of something whether it was physical or mental.  In arithmetic the root of a number is the number of factor that is used to build up another number by repeated multiplication.  Since 8 = 2x2x2, we may say that 2 is the third root of 8.  The word is also used in the study of functions to indicated the value that will produce a zero (a ground level number) for the function.  If f(x) = x^2 - 9 then x=3 is one of the roots of the function.  The word was  used by Al-khowarizmi in his writings and was translated as radix in the Latin translation of the Algebra.  The word also gave us the word radical, which is used for the symbol indicating a root, √ , (Euler suggested that this symbol was created from exaggeration of an r, the first letter of radix). Students often call this the square root sign, but should be made to know that the long bar across the top of the number is a separate symbol of its own, known as a vincula, and serves the same purpose as the "repeat bar" over a repeating decimal fraction.  Jeff Miller writes that, "The radical symbol first appeared in 1525 in Die Coss by Christoff Rudolff (1499-1545)."

Other word related to the Indo-European root werad are rutabaga, radish, race, and eradicate.  From the Greek equivalent we get rhizome and licorice (honest).

Root was used in Crafte Nombrynge by R. Steele in 1425, "The seven is called

The radical sign first appeared in the Philosophical Transactions in 1781.  For a while in the 1700's a name for irrational numbers was "radical numbers."  It was  used for the solutions of an equation in 1671 by James Gregory in a letter to Newton.  The root is sometimes used in reference to a critical node on a graph.  The Digges, father and son, writing in 1579 used, "To find the Radix, or Roote of any number." Radical is used in geometry for some geometric object sharing a common relation with two (generally intersecting) circles.  G Salmon in 1848 wrote, "The line S-S'=0 has been called the radical axis of the two circles."

Rotate/Rotation  A rotations is a rigid transformation in which every point of a set, or object, is moved along a circular arc centered on a specific point (the center of rotation) or about an axes, the axis of rotation.  The word comes from the Latin rota, for wheel.  The more ancient root ret relates to running or rolling. It is alive today in some unexpected places.  Rodeo, a sport that emerged during "round up", when cattle were gathered and shipped is derived from the Spanish word surround.  A rotunda is a building or room that is round, usually with a domed roof, and if someone says I am rotund, they mean my body shape is round.  The French word roulette, for a cycloid, also comes from the root, as does the game with the same spelling that spins around at a casino.

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