Pick Two and Get One of Each?

 Another from my archives: Nice for Alg II level or so, I think.

A colleague from Colorado sent me an interesting probability problem the other day. I like it because it illustrates one of those serendipitous qualities of mathematics. Here is the problem. A Jar has a mixture of Red and White balls so that if you withdraw two, the probability of getting two alike, or two of different color are both equal to one-half. You may want to stop and try it before you read on.
Ok, so we let r = the number of red ones and w be the number of white ones, and the total is r+w. So how could we draw one of each color? Well, red first, and then white, or white first and then red. If we find the probability of each of these conditional events and add them up, that will have to equal 1/2. Ok, the probability of red on the first draw is r/(r+w), and on the second ball the probability is w/(r+w-1) since one of the red balls will be missing. The opposite order is exactly the same with the w and r reversed, so the probability of getting one of each color is 2rw/[(r+w)(r+w-1)]. Setting that equal to 1/2 we get 

If we exapand (r+w)² and subtract the 4rw we get 0=r²-2rw +w²-r-w. NOTICE the symmetry, we could exchange r and w and get the same equation. We know right off that any solution (r,w) will have another solution (w,r).

One of the things that is often hard for students is to think of one variable as a constant and the other as a variable. I like to use the word "pronumeral", like a pronoun only instead of him or her we say "that number". It is like a variable that doesn't vary, we just don't know what it is in a particular case. So think of w as if it were fixed. We have that many white balls in the jar and we are wondering how many red can be put in to make the problem work... see it.. w is a "fixed" unknown, but r is going to "vary". That makes the equation a quadratic in r; Ar² +Br+c=0 where A=1, B= -2w-1, and C=w²-w.
We can solve this using the quadratic formula, but if this solution is going to be a rational number, and the number of balls in a jar must be rational, then the discriminant, the expression under the squre root radical in the quadratic formula, B²-4AC, must be a perfect square.
 B²= 4w²+4w+1 and 4AC= 4w²-4w; so B²-4AC= 8W+1. If there is a rational solution, it must be when 8W+1 is a perfect square.

 Wait, I know this one! That's a problem from number theory. The numbers that make 8W+1 a perfect square are called triangular numbers; 1, 3, 6, 10, 15. They are the sum of the first n counting numbers. But a neat thing happens if we plug 1 in for W, the solution for r is 3.... and if we use 3 for w, r=6. Each time we substitute one of the triangular numbers into the quadratic, the next comes out as a solution. So the probability of drawing two balls of the same color, (or of two that are not alike) will equal 1/2 whenever the number of balls of each color are consecutrive triangular numbers. A very geometric solution to a very algebraic question. 

Volume T-Z

T-Z

Tally/Tally Sticks Tally comes from the name of a stick on which counts were made to keep a count or a score.  THe Latin root is talea and is closely related to the origin of tailor, "one who cuts."  Many math words have origins that reflect back to the earliest and most primitive uses of number.  Compare the origins of compute, score, ect.  The first record existing of tally marks is on a leg bone of a baboon dating prior to 30,000 BC.  The bone has 29 clear notches in a row.  It was discovered in a cave in Southern Africa. 
The American historian Henry Schoolcraft (who was married to an Ojibwe woman for a time) reported that Ojibwe grave markers formerly used tally marks to indicate the number of certain kinds of important events that had occurred in the deceased individuals life.  In Northern Minnesota it is recorded that the Ojibwe village members carved wooden census records, using a photographic totem symbol to represent each family with tally marks next to each totem to show the number of members of that family.  
 For more on the history of Tally sticks, including how they destroyed the houses of Parliament in England, read this.


Tangent is from the Latin tangere, to touch, aptly describing two curves which meet at a single point.  Tangent is another creation of the Danish Mathematician Thomas Fincke, and was first written by him in Latin around 1583.  Prior to the creation of Tangent most writers still used the terms umbra recta, vertical shadow, and umbra versa, turned shadow. These terms refered to the vertical shadow and the ground shadow of a Nomen (sundial) on a wall, and one on the ground. The Latin root for shadow, umbra,which is still used for the darkest part of a sunspot or a complete solar eclipse, remains in the more common word umbrella (small shade).     

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 suggestions is that it is from the archaic Chinese root for the number seven, which still persists in the Tanabata festival on July seventh.  Whatever the origin of the name, the use of the seven shapes as a game in China dated back to teh origin of the Chou dynasty over one thousand year before the common era.  
You can download "OOG, The Object Orientation Game from MCm Software which allows the player to solve puzzles using tangrams, pentominos and more.      

Tessellation The root of tessellation is tessera, the old Ionic (Grkeek 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 diminutive form, and is used to describe smaller tessera.  Tiles, bricks and larger similar items were called testa, which is preserved in the name of the hard outer shell of seeds.  The completed project, then, became a tessellation, which covered the object plain. 

Here is a link to Totally Tessellated, This website has a nice discussion with lots of nice visuals to introduce tessellations.  

Tetrahedron A tetrahedron is the most simple of three space

shapes since it consists of only four faces and four vertices, the smallest number you can have to avoid all of them being on the same plane. The Greek tetra stands for four, and can still be found in some science words such as tetrachloride or tetravalent.  The hedra is from the Greek base, or seat, and is found in Cathedral, which is where the Bishop's seat is. See Polyhedra.   

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 of 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 Our modern mathematics of topology comes from teh 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 into English by Solomon Lefschetz in the late 1920's .  It appears that the word was originated around 1847 by Johann Benedict Listing in place of the earlier usage "analysis situs". The OED credits this first mathematical use to 1 Feb 1883, when Nature Magazine wrote, "The term topology was introduced by Listing to distinguish what may be called qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated. (I bristle a little at the use of "ordinary geometry.") 

Torus is from the Latin word for bulge and was first used to

*Wikipedia

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 definition applies to the rotation of any conic section. The circular torus reminds most students of the common donut, but if the circle is made to rotate about a line tangent to the circle, then there is no hole, and it is called a horn torus. If the line of rotation is a chord of the circle, then the form is called a spindle torus.     

Totient/Euler's Totient Function The totient of a positive integer, n, is the number of positive integers less than n which are relatively prime to n, that is, they share no common factors.  The symbol for the totient of n is usually the Greek letter \$\phi\$.  We would write \$ \phi (10) = 4\$ to indicate that there are four numbers less than 10 which have no common factor with it, 1, 3, 7, and 9. These four numbers are then called the "totitives" of ten, a term first used by J J Sylvester in 1879 (OED) apparently a few years before he came up with totient in 1883.   
There is an interesting an unproven conjecture about the totients.  It seems that any number that appears in the sequence of \$ \phi(n)\$ must appear at least twice.  It has been proven that if there is an exception, it must have more than 10,000 digits.    

Towers of Hanoi Back awhile, in a blog about Fibonacci, I mentioned that Edouard Lucas had created the "Tower of Hanoi" game and received comments and mail from people who thought I must be mistaken because the game was "really old". Turns out, it really isn't, but just the creation of a master mathematical story teller.
The game was created by Lucas, a French mathematician of the later half of the nineteenth century. He created a sequence similar to the Fibonacci sequence and used his sequence and the Fibonacci sequence to develop techniques for testing for prime numbers. 

Lucas was also the creator of a popular puzzle called The Tower of Hanoi in 1883. You can see the original box cover above. Note that the author on the box cover is Professor N. Claus de Siam, an anagram of Lucas d' Amiens (his home). The professors college, Li-Sou-Stian, is also an anagram for "Lycee Saint-Louis" where Lucas worked. France was building an Empire in Indochina (the peninsula stretching from Burma to Viet Nam and Malaysia) and the "mysterious East" was a very fashionable topic. Lucas created a legend (some say he embellished an existing one, but I can find no earlier record of one) of monks working to move 64 gold disks from one of three diamond points to another after which the world would end. The solution for a tower of n disks takes 2n -1 moves, so the game often had less than the 64 disks of the legend. Solving the 64 disks at one move a second would require 18,446,744,073,709,551,615 seconds, which at 31,536,000 seconds a year would take 584 Billion years. (and you thought Monopoly took a long time to finish).  The reference in his instructions to Buddhist monks in a temple in Bernares(Varanasi),  India seems, even now, to make people believe there was such an activity taking place.  Varanasi is considered the holiest of the seven sacred cities (Sapta Puri) in Hinduism, and Jainism, and is important to Buddhism because it was in nearby Sarnath that Buddha gave his first teaching after attaining enlightenment, in which he taught the four noble truths and the teachings associated with it. There is a Buddhist temple there with many relics of the Buddha, but so far as I can find, no monks moving golden disks on needles. Students/teachers interested in further explorations of the history and math of the famous game should visit the work of Paul K Stockmeyer who maintains the page with the cover illustration mentioned above, and his Papers and bibliography on the Tower of Hanoi problem. Lucas developed several other mathematical games of his on, including the well known children's pastime of dots and boxes (which he called  La Pipopipette), which on large boards is still essentially unsolved, I believe.  He also (probably) invented a Mancala type game called Tchuka Ruma. Lucas is also remembered for his unusual death, caused by a waiter dropping a plate which shattered sending a piece of plate into his neck. Lucas died several days later from a deadly inflammation of the skin and subcutaneous tissue caused by streptococcus. The disease, officially listed as erysipelas (from the Greek for "red skin") was more commonly known as "Saint Anthony's Fire".  

Trajectory The trajectory of a particle or a point on a plane is the set of point which determine its path.  The root is from the Latin trajectus which unites trans for across and ject for throw, with a literal meaning of "to throw across."  Other modern words drawing on the ject root include project (to throw forward) and adjective, (to throw to... share with your English teacher, I bet she doesn't know) among a host of others, inject, reject, object.... heck, I could go one for seconds.  

Transcendental  A transcendental number is a number that can not be described by algebraic equations with rational coefficients.  They were named because they "transcend" the bounds of algebra.  The first number to be proven transcendental was found, or invented, by Liouville in 1844.  The number is a made up number that has all zeros except for the digits a positions 1, 2, 6, 24, 120... n!, which are all ones. The trans is from the Latin for "over" or "across".  The second part is from the Latin root scandere, to climb.  Literally then, the transcendental numbers climb over the algebraic boundary that held the other irrationals .  The more ancient root skand gives rise to interesting words such as scandal.  The Greek skandalon meant a snare and then more generally anything you could trip on .  And if you trip over a moral snare, you may become involved in a scandal.  The Latin scalce from the same root meant ladder or steps to climb on, and worked its way into the modern words echelon, escalate, and scale.  Transcendental mathematical constants you may now know include Pi, Euler's famous e, and the Golden ratio, Phi.    

Trapezoid/Trapezium Both words come originally fromt he Greek word for table.  Today, in the USA, teh term trapezoid refers to a quadrilater with one pair of sies parallel and a trapezium to a general quadrilateral with no parallel sides. This is exactly the opposite of the original meaning, (and the meaning in some countries, particularly England, today)  Here is a short explanation of how this contradiction came into existence. The Early editions of Euclid have the Arabic helmariphe; trapezium is in the Basle edition of 1546.  Both trapezium and trapezoid were used by Porclus (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 none.  However, in 1795 a Mathematical and Philosophical dictionary by Charles Hutton 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 with each other, it is sometimes called a trapezoid.  No previous use of the words with Hutton's definitions is known.  Nevertheless, the newer meanings of the two words now prevail in the US, but not necessarily in Great Britain.  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, just like a rectangle is a type of parallelogram.  Some geometers are adamant about inclusive definitions, and some just as adamantly protest for exclusive definitions.  

Trigonometry The root of trigonometry comes from the union of the Greek trigonon for triangle, and metron for measure.  Although the roots are nestled in ancient Greek, the word seems to have been the creation of Bartholomaeus Pitiscus, who used in in the title of a book in 1595.  
The mathematical ideas we call trigonometry though, have been studied as far back as 140 BC when Hipparchus produced the first table of "chords.  Early work in trigonometry was often more concerned with the triangles on spheres, like the Earth, than they were with those on idealized flat planes.  The Babylonians however, seem to have used ratios and relations to flat planes as far back as the Plimpton Papyrus, 1800 BC when they studied the volume of pyramids and frustra of pyramids, and their slopes and the relationship between sides of Right Triangles. The emergence of ratios in western trigonometry was much later.   

Truncated means to shorten by cutting off and is related to the old English truncheon, which means a club or staff.  Both words derived from the French truncus which referred to a cutting from a tree used for grafting stock. A truncated section is usually cut of by a non-parallel plane.  The choice for a cone or pyramid cut off by a parallel plane is a frustum of that object.   

Two is descended from the Greek root dyo and the Lain root duo thorough the old English twa.  Early languages often had both feminine and masculine forms for two and so there are abundant and diverse roots related to "two-ness".  Many "two" words use the Greek root bi; biannual, binary, biscuit, and biceps are examples. Others come from the Old English twa, such as between, twilight, twist, and twin.  From duo we get dual, duet, dubious (of two minds) , duplex and double.  The Latin di gives us diploma (two papers) and dihedral.  The earlier Greek dyo produces dyad , composed of two parts.  Where am precedes bi, as in ambivalent, it means "either of two." 
The word didymous is Greek for twin and is used in scientific terms to represent things which occur in paris.  Students of the bible may remember that didymous is aloso the nickname of the disciple Thomas (John 11:16).  Thomas itself is from the old Aramaic word for twin, t'oma, which was also used by the Greeks and later made its way into Latin.    

U-V

Undecagon the now, mostly defunct, term for an eleven sided polygon.  It appears as early in the OED as 1728 in Chambers Cylopedia, and 1879 in Cassell's Technical Educator.  It is derived from the Latin undecim, and apparently may still be used in Portugase and Spanish as undecagono. Similarly the Greek term is hendecagon. 

  (The name hendecagon, from Greek hendeka "eleven" and –gon "corner", is often preferred to the hybrid undecagon, whose first part is formed from Latin undecim "eleven".)

Union  The word union goes way back in English but the mathematical meaning only dates back to about 1912.  The OED 1912 definition gives, "The aggregate formed of the points present in at least one of hte sets ... is called their Union.  The usual notation is a bold U, and the union of all the elements of set A and set B, is given by A U B.  Prior to this constructed usage, the word sum was often used. 

Variable/Variance The word vary, and its many variants (oops, there is one now) come from the speckled fur of animals used in early apparel. At first applied to changes of color, the word was eventually used for things that involve changes of any kind.  The originator meaning is preserved in the word minevar.  Today miniver is the term for the ermine trim on the ceremonial robes of British Peers, but the term originally referred to any decorative fur trim on a robe, and was common in the medieval times.   Sir Ronald Fisher created the statistical term variance around 1018 as a measure of the variability of a data set.
The mathematical use of variable is first cited in the OED in 1710.  
While the idea and use of variables goes back to Diaphontus (250 AD)it was Leibniz who created the term variable, as well as constant and function, although his use of function was a very limited approach and nothing like how we define the term today. He deserves credit for coordinate and parameter as well.    The OED gives the earliest use of "variable" as a noun in 1763 in Emerson's Method of Increments.   

Vector is derived from the Latin root vehere, "to carry".  The root is also the source of everyday words like vehicle.  W R Hamilton introduced the use of vector in mathematics.  This link at McGill University has a more extensive history of Vectors.

Velocity In mathematics, velocity is the rate of change of position with respect to time.  In more general terms it can be thought of as the speed with which an object moves along its path. The difference is the idea of a speed at a direction, whether the direction is in a straight line or a curve.  The ancient origin of the word is from an Indo-European root for health or strong, and the more modern usage comes from the Latin velox, for fast.  The French adopted this to velocite which, with minor changes produced the word we use in English today.  Related words in use today include vigil, vigor, and vegetable (cause they are good for you vigor), and more uniquely, awake.  

Venn Diagram Venn diagrams are named after their creator John Venn(1834-1923). Venn was a lecturer at Cambridge and worked mainly in logic and probability theory.  He used diagrams of circles to represent the unions and intersections of subset of a universal set in non-overlapping regions. It seems the first person to call these diagrams Venn diagrams was Clarence Irving in his work, A Survey of Symbolic Logic in 1918,  The figure below shows a Venn Diagram of two subsets A and B, the shaded portion is the intersection of A and B.  If A represented the set of all prime numbers, and B represented the set of all even numbers, then the intersection of A and B would be the number which is both even and prime, 2.  

Versine The versine of an angle, A, is an almost extinct expression for the quantity 1-Cos(A).  Up to the 1600's this was

*Wikipedia

probably the second most common trigonometric value used. The Latin word versed relates to turning, and the "versed sine" was, in essence the sine turned 90 degrees. The mathematical terms converse and inverse are both from the same root. Many other words come less directly from this root. A plow turns dirt up and over a creates a furrow, a straight line of dirt along the ground.  Things laid out along a straight line were sometimes said to resemble the furrow and called verses, and thus words in a line of poetry became a verse. To reverse is to turn back, and the obverse side is the side you see when you turn something over, and your vertebra are joints that allow you to turn.

Vertex/Vertices vertex is from the Latin vertere, to turn, and had meanings related to highest or foremost.  Vertical is from the same root as are verse, verterbra, and wreath (see also Versine)..  The vertex of an angle, and the vertices of a polygon are the points where the perimeter suddenly changes direction, turning toward the next turning point, or vertex.

Vincula/"Repeat bar"  As a "math Doctor", one of the most common questions students ask is "what is the name of the bar over repeating decimal fractions?" I always answer that "repeat bar" seems the best name to communicate what it does, but I know they want the classic Latin name for the bar, "vinculum".   The word is from the diminutive of vincere, to tie.  Vinculum referred to a small cord for binding the hands or feet.  The meaning in math is mostly unchanged from the original meaning, as the purpose of the repeat is to bind together the sequence of repeating digits.  The symbol was once used in much the same way we now use brackets and parentheses, to bind together a group of numbers and operators to form a combined operation.  The vincula was usually written under the items to be grouped.  Where today we would write (2x+3)*5 with the vincula they would write 2x+3 *5.  Some prefer to call the horizontal fraction bar a vinculum as well, since it binds the numerator and  the denominator into a single fractional quantity.  Similarly the long bar after the raidx in the square root symbol is a vinculum, containing the entire representation of the power to be reduced.

Virgule The slanted bar , "/" that is used fro fractions, and that also probably appears on the division key on your calculator and computer, is often referred to as a virgule. The Latin, and later French, word had the meaning of a small rod.  It shares the same Indo-European root as out common English word "verge" with a meaning of lean toward.  The symbol is also used outside of mathematics to indicate choices (male/female) and to serve as a line break when verses are printed in a continuous string.  (Roses are red/Violets are blue).  The word solidus is also often used for this symbol.

  WXYZ

Wilson's Theorem Wilson's Thm. states that any number p is prime if, and only if, the value of (p-1)! + 1 is a multiple of P.  This is often written in a congruence format as (p-1)! = -1 mod p.  The theorem is named for Sir John Wilson (17412-1793), who discovered the theorem while he was still a student at Peterhouse College, Cambridge, although it seems he did not have a formal proof.  Wilson went onto be a Judge and seems to have done little else in mathematics.  The theorem was first published by Edward Waring around 1770 and it was he who attached Wilson's name to teh theorem, although it is now known that the result was known to Leibniz, and perhaps was known to Ibn al-Haytham (965-1040) much earlier.  The first known proof was by Lagrange.  For all composite numbers n, except four, it is true that (n-1)! = 0 mod n.

Yard In ancient times before paved roads and public services, a staff or rod had many uses. It could help you keep your balance on uneven cobblestone paths. It could be used to push uncooperative livestock out of your way, or defen an attack by rogues and bandits. It could also be used as a rough measure of distance by laying them end to end and counting the number of lengths to cover a distance. In the old Germanic languages gazdaz was the name for a staff or stick carried for such purposes. The English changed this to gierd and eventually to YARD.  Variations and remnants of the early meaning still persist. One example is "goad", to push with the staff, and sailors know that a tapered Spar long straight piece of wood or metal) is used to support a square sail, and this is also called a yard. The area around your house where you may have grass and a garden is also called a yard, but came from a non-related root.

Xenon , atomic number 54, is a colorless, oderless, and very non-reactive gas.  It was first isolated in 1898 by M. W. TRavers and William Ramsey.  It is the only non-radioactive noble gas which forms stable chemical compounds at room temperature.  The name xenon was given by Ramsey and comes fromt eh Greek word xenos for strange.  The gas has been usedin TV tubes and in bactericidal lamps. (I wonder how long the term "TV tubes" will have meaning to young people!)

Zenzizenzizenzic Before we launch into all that, let's start with zensi, which was found in and before the 16th century in England.  But long before that, the roots of zensi, and multiple zensi was laid down by, among others, Luca Pacioli.  Pacioli used cos for the unknown quantity, our x, and censo for the square of the unknown.  cubo for its cube, so censo-censo was a fourth power, censo-cubo was a sixth power, and these were picked up by others including Tartaglia, Cardan, and the Portugese scholar Pedro Nunez.
Somehow, the c became a harder z as it passed into English, and so zenzi was the square, zenzizenzic was a fourth power,  zenzicube was a sixth power in Robert Record's Whetstone of Witte, and jsut to be confusing, it was sometimes used as a root as well.  And that monster, zenzizenzizenzic, is just the square of the square of the square, or the eighth power, and yes, Recorde tucked that in his epic arithmetic as well,  but shortened a little to zenzizenzike, but  shortly later S Jeake used zenzizenzicube for a 12th power.
These repetitive uses by the early modern algebraists were drawn from the methods of the premodern algebraists, including Diophantus himself who used "dynamus"(power) for a squared unknown and dynamodynamis for the fourth power> and was copied by the Arabic writers who used "mal" (a unit of money)  for a square, and "mal mal" for a fourth power.

Zero comes to us from the Hindus, the inventors of zero, through the Arabs and the Arabic word sifre, from which we also get the word cipher. According to Edward MacNeal, the author of MathSemantics, Makin Number Talk Sense, the Hindus used the word shunya to refer to a blank or empty space.  When the Arabic method was introduced to Europe, the Roman system was already in place.  Perhaps because the hand written contracts would have been much easier to forge or alter using the Arabic numbers, there was a strong resistance to their use.  During a brief forced underground the use of the cipher came to represent something done in secret or code.  Eventually the Latinized form of cipher, zepherium came to be the common term. which was eventually reduced to zero in English. Recent investigations have led some historians to give greater attention to the Egyptians for their contributions to the development of zero.   In a recent discussion the Historia Matematica newsgroup, Bea Lumpkin explains two areas where the Egyptians use of zero is evident.

The two applications of the zero concept used by ancient Egyptian scribes were 1) as a zero reference point for a system of integers used on construction guidelines, and 2) as a value that resulted from subtracting a number from an equal number.  These are the achievements I believe should be acknowledged by historians.

Zero is first cited in the OED in 1598, and the entry is "Zero, a sipher of naught, a nullo." suggesting how many different terms existed for zero then, and still do... "zero, zip, null, none."
Symbols for Zero Positional number systems seem to have been invented around 2000 BC, but the idea of using something, a symbol, to represent nothing, the absence of a quantity, did not arrive until the time of Alexander, around 300 BC. babylonians tried several different zero symbols. Usually these constituted counting slanted to the left or right. Sometime around 100 AD teh Greek astronomer Ptolemy left one of the first records of the use of an open circle as a zero marker. Robert Kaplan suggest in The Nothing That Is that the merchant class may have used the open circle as far back as Alexander for commerce, but this is not noted in the writing of that time because of the Greek intellectuals' scorn for hte commercial applications of math. He suggests that perhaps the symbol spread to India with teh movement of Alexander's army.
Whether they obtained it from the Greek's or invented it on their own, by the 9th century AD the mathematicians of India were using a dot, bindu, and the open circle, shunya-bindu, to represent the empty space left be a marker on a sand counting table. Our present day ellipses (...)which represents something missing, is from the same source
The Arabs conquest of Western Asia and North Africa spread the "nine Arabic numbers and the single cipher" and by the 1200's they had been translated into Latin to begin to revolutionize (but very slowly) European mathematics. Tor the next six-hundred or so years, the "0" symbol under several names would emerge from the status as representative of somethings missing to gain full and equal stature as a number.

Volume S

S

Sagitta  The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit Aryabhatia, Section I) trigonometric tables. The versine of an angle is 1 minus its cosine.

This figure also illustrates the reason why the versine was sometimes called the sagitta, Latin for arrow, from the Arabic usage sahem of the same meaning. This itself comes from the Indian word 'sara' (arrow).  The term Sagitta was used in Latin by Fibonacci to mean the versed sine.

In geometry, the sagitta (sometimes abbreviated as sag) of a circular arc is the distance from the center of the arc to the center of its chord. This usage first appears in the OED in 1726  "The .. Line .. from the middle Point of the Chord up to the Arch, leaving equal Angles on each Side, is call'd the Sagitta", Alberti’s Archit.

The 1909 Webster's New International Dictionary credits the use of the word to Kepler, and adds the term is also an obsolete term for "abscissa".

Satellite During the decline of the Roman Empire, the rich and important were under constant threat. To protect themselves from attack they hired bodyguards, called satelles. The practice, and the name spread, and by the 1600's every Prince had his satellites revolving around him wherever he went. When Johannes Kepler used the word to describe the moons of Jupiter, he may have meant int in jest, but the term stuck. Now we refer to any body orbiting another primary mass as a satellite.

Scalene Most students associate the word scalene with triangles, but there are also scalene cylinders and cones, more often called oblique cones or cylinders; and there is even a scalene muscle. In each case the word scalene has the same meaning, uneven. A scalene triangle is a triangle in which no two of the sides (or angles) are congruent. Scalene comes from an early Indo-European root that is related to chopping and almost surely came into English from a same word in French. Such chopped up edges were often uneven. The earliest OED citation is for 1734 in the Builder's Dictionary. "scalenum Triangle." It seems clear the term was in some common usage before it appeared in a dictionary.

Science is from the Latin root scire, to know. The earliest origin of the word is related to cutting or splitting apart. Knowing is, in a sense, the are of being able to separated ideas from each other. Related terms include conscious, omniscient (all knowing) and less closely related to schism and schedule.

Score Like compute and tally (see each) score is a reminder of the primitive counting and record keeping technique of cutting notches in a piece of wood or bone. The idea of making one mark for each member of a collection is an ancient ideas as evidenced by inscribed marks on ice age bones, and ocher dots on the walls of Magdalenian caves. Because a long string of identical marks were difficult to count quickly, groups were sometimes marked off with a heavier or different mark. The Old Norse word for this practice was skor and the mark was used to mage groups of twenty. Many European primitive cultures used a base of twenty and so the word found purpose, and thus persisted in language. The base twenty is apparent residue of the English (20 shillings to the pound), as well as the Irish, French and Danish. A base twenty system was also used by the Mayans of Central American and the Ainu, the indigenous people of the Japanese Islands. The Indo-European root of score is sker. The word is related to cutting or slicing and is the progenitor of dozens of words sharing this common theme. Some examples include shears, scissors, and skirmish. The Old English version sceort for "to cut" gave us not only the word short, but shirt and skirt as well. The Latin word dropped the s to produce words like carnage, carnival, and carnation, a flower named because it was the color of flesh. Today our most popular pastimes remind us of our mathematical beginnings as they report the sports scores, the number of marks for each team.

Secant is from the Latin root secare, to cut. It is a proper name for a segment that cuts through a circle or

curve, but was actually used for cutting pairs of straight lines, like the sides of a triangle, "They call the line secant, the hipothenuse", Blundeville, 1594. The word was introduced by Thomas Fincke in 1583. In the same book, Blundeville (1594) used secant as a trigonometric function, but the first citation in the OED that mentions a circle is in 1684, in Elementary Geometry: “From the Center D, draw the Secant DC.” It is said that Viete did not like the term secant for a trigonometric name, fearing it would be confused with the geometric object*Jeff Miller.

Second When the "pars minuta" (see minute) needed additional dividing, they created a "second little part", the pars secundus
dividing each 1/60th of a minute into 60 additional divisions each. Later the name was shortened down to seconds, and then transferred along with the base 60 angle system into time keeping.
Second is also the ordinal name for the one following the first, or initial object counted.
Chaucer's Treatise on the Astrolabe, 1400, described the arc system of 60 minutes divided into 60 seconds. The first appearance in the OED for a minute of time, was in a 1588 book on "Cathechism or Short Inst." The first citation mentioning the second-hand of a clock is in 1759 in the Philosophical Transactions. Second as an ordinal adjective was used much earlier in 1297.

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). The mathematical sequence was a late-comer to the language, first appearing in 1910 in the Algebra entry of the Encyclopedia Brittanica; (with the exception of a notation of "sequence of natural numbers in order", by Sylvester in 1882) although entry in an encyclopedia makes me think it had been used in classrooms and between professionals for some time. Sure enough I found the use of sequence as Sylvester had used it several more times, and in 1890 H. B. Fine describes the "sequence of rationals," for the decimal sequence defining an irrational number.

Sesquicentennial The prefix sesqui is from the Latin and means one and one-half of whatever follows. Sesquicentennial refers to a period of one and on-half centuries, or 150 years. The prefix is actually a contraction of two parts, semi, for half, and qui for "and". About the only other surviving word with this prefix is the usually derivisive sesquipedalian. The ped root is from foot, and refers to very long words; words that a literally one and one-half feet long. Usually it is used to poke fun at the people who use them.

Similar comes from the Latin word similis (like), and refers to things which share some common characteristic. Similar triangles, for example, share a common shape, but are not necessarily the same size. The word probably dates from the earliest Indo-European languages and the sanskrit root sem which refers to a quantity of one. Symmetry employs the same root. Related English words include simple (one fold), resemble, simulate, and single. The US Marine Corp slogan, "semper fidelis" (always faithful) uses the Latin compound form semper which literally means "once for all".

Simson (Wallace) line  In geometry, given a triangle ABC and a point P on its circumcircle, the three closest points to P on lines AB, AC, and BC are collinear. The line through these points is the Simson line of P, named for Robert Simson. The concept was first published, however, by William Wallace in 1799.

The Simson line of a vertex of the triangle is the altitude of the triangle dropped from that vertex, and the Simson line of the point diametrically opposite to the vertex is the side of the triangle opposite to that vertex.

If P and Q are points on the circumcircle, then the angle between the Simson lines of P and Q is half the angle of the arc PQ. In particular, if the points are diametrically opposite, their Simson lines are perpendicular and in this case the intersection of the lines lies on the nine-point circle.

Letting H denote the orthocenter of the triangle ABC, the Simson line of P bisects the segment PH in a point that lies on the nine-point circle.

Sine comes to us from the Latin sinus, a term related to a curve, fold, or hollow. It is often interpreted as the fold of a garment, which was used as we would use a pocket today. The use in mathematics probably comes about through the incorrect translation of the Sanskrit word jiva, for bowstring. When Leonardo de Fibonaacci (and you are reminded no one ever called him that when he was alive) used the term in his writing, it became permanent. According to Carl Boyer's "A History of Mathematics:, the idea of the sine of an angle came from an Indian book written around the year 400. The early use of sine referred to a length of the chord in a circle. It was not until the 1700's and Leonard Euler that it became common to used trigonometric ratios. [But I read recently that Plimpton 322 clay tablet from 1800 BC contains a column of triangle lengths, the short side, a long side, and their ratio squared (tan^2) in 15 columns from about 1/4 short to long up to nearly equal (45 degrees).]

Sin of David In the bible stories in Samuel and Chronicles, God sent a plaque on the people of Israel because of David's "numbering the poeple". It is not clear what is meant in this context, or why God ws displeased, but thousands of years later religious people fought to defeat the idea of a public numberation (census) based on the fewr that it would bring God's Wrath. Some fundamentalist religions in the US even today avoid completing the census over this issue.

Skew The word skew seems to be a shortening of the word eschew, which means to avoid or turn away from something, or askew, for not in order. Skew lines in geometry are lines that do not intersect, but are not in the same plane. If you imagine a cube, a line on one side of the base, and a second line on a non-parallel edge of the upper square would be skew lines. Likewise the same base and a vertical edge not in the same face would be skew lines. These examples are all skew perpendicular, but you can imagine two such lines in a pyramid, for example, which are skew also. The terms are sometimes called oblique, a French borrowing, that is used for something that is not vertical or diverges from an expected path. It is also in the drill commands for marching military formations (or was in my youth), for turning a column in a 45 degree direction. Oblique actually shows up in English a little earlier than skew.
In statistics, a distribution is skew (or skewed) if it is not symmetrical about its mean. The earliest citation in the OED for the statistical meaning of Skew is in the Philosophical Transactions in 1895.

Slope seems to have come from an old English term like aslupen, slipping away. It seems uniquely English and not anything like words in any other language, as far as I could find.
Why M for Slope? Interestingly, m for slope has led to more mis-history speculation in classrooms than any other topic, with the possible exception of the life of that Great American-Indian Mathematician, Chief Soh Cah Toa. Fortunately I have collected a lot of information and the best answer I can think of, it just happened. What I've learned is at Why M for Slope.

Smith Numbers A Smith Number is a number for which the sum of its digits is equal to the sum of the digits of its prime factors, using repeated factors as often as they appear. The numbers were named for Harold Smith, the brother-in-law of Albert Wilansky of Lehigh University, who created them because Smith's phone number was such a number, 493-7775. Go ahead, call him and check. The sequence of Smith Numbers begins 4, 22, 27, 58.....

Solidus The slanted bar, "/", that is used for fractions, and division is often called a solidus. If you think it looks too much like 'solid" to be a coincidence, you're right. The word comes from the same root. From the glory days of Rome to the Fall of the Byzantine Empire, the solidus was a gold coin. The origin of the modern word "soldier" is from the custom of paying them in solidus. According to Steven Schwartzman's The words of Mathematics, the coins reverse carried a picture of a spear bearer, with the spear going across from lower left to upper right. He suggest that this is the relation to the slanted bar. This symbol is also sometimes called a Virgule.

Soliton is a shortened word to represent the solution of a differential equation representing a stationary wave. The term seems to date to about 1965 and was invented by Martin Kruskal of Rutgers Univ and printed in the Physical Review of Letters.
Here is a quote from the site of Kanehisa Takasaki about the naature of solitons:

...the scientific research of solitons had started in the 19th century when John Scott Russell observed a large solitary wave in a canal near Edinburgh. It the days of Russell there was much debate concerning the very existence of this kind of solitary waves. Nowadays, many model equations of nonlinear phenomena are known to possess soliton solutions. Solitons are very stable solitary waves in a soluton of those equations. As the term 'soliton' suggests, these solitary waves behave like particles. When they are located mutually far apart, eaach of them is apprximately a traveling wave with constant shape and velocity. As two such solitary waves get closer, they gradually deform and finally merge into a single wave packet; this wave packet, however, soon splits into two solitary waves with the same shape and velocity as before 'collision'.

Solve / Solution The middle root of solve and solution finds its way to us from the Greek luiin which meant to loosen or release. The Latin prefix seu in front came to represent actions like pulling apart or untie, and gave us our word for solve. A solution was something then, that allowed us to pull apart the problem. The same roots appear in absolute, dissolve, and soluble.

Sphere The word sphere is from the Greek sphaira for a globe or ball shape. The word is is little changed in use or application from its earliest usage.

Square is derived from the Latin phrase exquadra, like a quadratic. Over time the term was contracted into its present form, and then came to mean the regular quadrilateral. The word is also the origin of the military unit called a squadron, from an early form of fighting in squares called a Phalanx by the Greeks. The French term esquire is also from the same source.
The geometric term was first used in English by Robert Recorde in his 1551 Pathway to Knowledge, but around the same time there were still uses of square for what we would not call a rectangle, and in the same book Recorde writes of a "... whose use commeth often in geometry, and is called a squire, is made of two long squares ioyned together...". Chaucer mentions the carpenters square in his Treatise on the Astrolabe in 1400.
The square of a number appears in Record's Whetstone of Witte in 1557, "Twoo multiplications doe make a cubike nomber. Likewaies .3. multiplications doe giue a square of squares."
The first mention of the statistical method of least squares is recorded in the Philosophical Magazine in 1812, "The principle of the small squares."

Statistics The word we now use for numerical data and their analysis was originally from the German statistik and was "political science". The first appearance of the word in an American dictionary was in 1803. At first statistics did not have to be numbers, but any information about the state of government or the governed. It was derived from the Latin root status for position which was from the Greek statos, to stand. In the early days of the American republic, staatist was used as a synonym for statesman. Status is preserved in its original meaning and also is the root of state, as in nations, statue, stay, instant, and literally hundreds of other words, including steed and stud.
Bea Lumpkin, in a posting to the Historia Mathematica Newsgroup, has indicated that some early Egyptian records may be the oldest statistical tables in existence.

What can be claimed for early Egyptian collection and presentation of data is still impressive. For example, the Palermo stone, named for the museum where it is displayed, is one of the earliest (or earliest) historical and statistical records extant. It lists the reigns of kings from c-3100 pre-dynastic to c-2300, 5th dynasty. The record is in tabular form, ruled into rows and columns. The columns are arranged in chronological order, itself a concept not to be taken for granted at the early date. The top row gives the king's name, the middle row events of that year and enumeration of wealth including a biennial count of cattle. The bottom row gives the height of the Nile flood. If the ancients were looking for a pattern in the annual floods, none has been found to date. On Palermo Stone, see Sir Alan Gardiner, Egypt of the Pharaohs, Oxford U. Press, 1963, 62-4

Standard Deviation The creation of this statistical measure is credited to Karl Pearson around 1893. The term deviation is from the Latin roots de and via. Via is the Latin word for road, and deviate literally means "away from the road." This fits the statistical meaning of a distance measure away from the mean of the distribution. The Standard part is to make a fit for all normal distributions, adjusting for the size and spread of the numbers through standardizing the distance to a single unit. Any normal distribution then, will mark of a unit on each side of the mean which is 68.27% of the total population.

Stanine Scale Stanine Ratings are a nine point statistical scale. The word appears to have been created during WWII by someone in the Air Force where the idea was developed. The word was created as a shortened from of "standard of nine". Colonel Lawrence F Shaffer wrote shortly after the war that, "The origin of the word is somewhat hazy. I have complete certainty only with regard to two facts: that the word was originated at PRU #1 at Maxwell Field, and that the date was in the month of February, 1942. According to PRU #1 tradition, the word first appeared in the form stand-nine as a shortening of the phrase standard nine-point scale that occurred in area directives. This was soon shortened to stanine "
I was ably informed about stanines by Lee Creighton of the Statistical Instruments Division of SAS Institute Inc. in a posting to the Ap Statistics Discussion Group.

Basically, the transformation from raw scored to stanine scores is pretty simple: (1) rank the sores from lowest to highest (2) assign the lowest four percent a stanine score of 1, the next 7 percent the staninine of 2, etc according to the following table: 4..7..12..17..20..17..12..7..4.. to Stanines 1...2...3..... etc. So, they are assigned on a scale from 1 to 9, but there are not necessarily the same percentage in each "bin". Someone in the 88th percentile wold come in just a the top of the range for a stanine score of 7.
The reason this scale was developed was primarily to convert scores to a single digit number--a considerable asset when Hollerith punch cards were _de riguer_ in the computer industry. The scores could be coded in a single column on one of these cards.

Student's t The story of the name for this statistical distribution and test is almost legend, and some version of hte tale is remembered by Intro Stats students long after they forgot the purpose of the t-test. A dialogue between Randy Schwartz and James Landau in the Historia Matematica discussion group gives both one of the folk versions, and a brief history. Randy Schwartz writes, "the distribution now known st 'Student's t distribution' was first discussed in print in a paper by William S Gossett in the journal Biometrika in 1908, published under the pseudonym 'Student.' The paper solved a problem from the Guinness Brewery concerning how large a sample of people should be used in its tastings of beer. Apparently Gossett was embarrassed to be working on a problem stemming from the liquor industry, thus the pseudonym."
James Landau responds "The story of 'Student' has been told so many times that it has become folklore, and like all folklore variant versions exist until it is difficult to determine which is the original. The variant you tell is one I had not encountered before. Gossett was an employee of Guinness Brewery (a brew-master, I believe) who went to study statistics under Karl Pearson. Gossett eventually discovered a result that he published in Biometrika under the pen name of 'Student'. Why did he choose to use a pseudonym? Here is where the folklore kicks in . The most common story is that Guinness wanted to keep it secret that they were using statistics in their business and ordered Gossett not to reveal his identity.
In any event, Gossett published all his statistical work as 'Student', even though his identity became well known. Why he continued to use the pseudonym is not part of the folklore, and I have never heard a plausible story. Perhaps it is because he became famous as 'Student' and did not want to have to re-established his professional reputation under his real name. Perhaps he liked the notoriety.
Ok, now we know why he used 'Student', but why t? "

Subfactorial the name subractorial was created by W. A Whitworth in The Messenger of Mathematics in May of 1877.  The symbol for the subfactorial is !n, a simple reversal of the use of the exclamation for n-factorial. This was not the symbol used by Whitworth, as at this time many people preferred what is called the Jarret symbol for the factorial. Whitworth added an extra line in the L to make the subfactorial. This symbol for the factorial persisted into the 1950's.

.The subractorial, or derangement is about counting the number of ways to take objects which have some order, and arranging them so that none is in its right ordered place.  The numbers 1, 2, 3 can be arranged for example, as 2,3,1, or 3,1,2.  The problem was first considered by Pierre Raymond de Montmort in 1708, and first solved by him in 1713. 

Cajori mentioned the use by George Chrystal (1851-1911) of a subfactorial symbol using N with an upside down exclamation point, but does not mention at all the !n that is the common present symbol, leading me to believe it was created after 1929.

Crystal's books into the fifties continued to use the inverted exclamation symbol and the National Academy of Sciences used the symbol in 1967.
The earliest used of the !n symbol I have ever found is from 1958, In the MAA questions section:

This was obviously not an instant hit, as I received several comments like the following after a post in 2009.
"  I have several books on my shelf, none of which use !n notation.
D(n)
- Matoušek and Nešetřil, 1998
- Niven, 1965. I teach from this book.

D_n
- Chen and Koh, 1992. Interestingly, they use the notation D(n,r,k) to denote the number of r-permutations of N_n with k fixed points, and (good for them) cite Hanson, Seyffarth and Weston 1982 as the originators of this notation.
- Martin, 2001

- d_m
Goulden and Jackson, 1982.-----------------------------------------------------------------------------------------
The Niven book is his well known Mathematics of Choice, and he uses the symbol D(n) . In 1997 Robert Dickau used\$D_n\$ for derangements, another common name for subfactorials.    John Baez used !n  in 2003 without indicating that it was an uncommon symbol.
The formula for subfactorial, also called derangements of a set, is given by \$!n = n!( 1- \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!}..... \frac{1}{n!} )\$,  The quick approximation is !n = n!/e.

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, means to carry away the bottom part. The "-" symbol for subtraction was first used as markings on barrels to indicate those that were under-filled. Around the 1500's it began to be used as an operational symbol and it became common in English after it was used by Robert Recorde in the Whetsotne of Witte in 1557.
In a subtraction relationship, a - b = c, all three numbers have a special name. The first number, a, is called the minuend, from the same root as minus and literally means that which is to be made smaller. The part to be removed, b,is called the subtrahend and means that which is to be pulled from below.  The answer, c, is most often called the difference or result, but in many statistical uses it is also called the residue, or residual, that which remains.  is statistical uses it may also be called a deviation.
If you wish to know more about the history of subtraction, check out Subtraction, Borrowing, Carrying, and other Naughty Words, A Brief History Subtractions earliest usage in English, according to the OED, is from Steele's Craft of Nombryng in 1425.

Sum The root of sum is the Latin summus for highest. The root itself seems to be a contraction of the word supremus and is closely related to supra from which we get super, supreme and other superlatives (pun intended). Sum is drawn from the practice of the Greeks and Romans of adding columns of numbers from bottom to top and writing the result at the top. The answer then became the summus, the number at the top. In a similar way we get the word summit for the top of a hill or mountain.
It is interesting to me, that the earliest reference to the word in Jeff Miller's web site is a reference to Nicholas Chuqet in 1488 in which the word is spelled "some". The root of this homonym of sum if from the root sem which related to the quantity one.

Supplement The supplement of an angle is the angle that must be added to "fill up" a semi-circle. The sup root is a variation of the common sub for below or under. The ple is the same root that gives us the math word plus for "to increase or add to" something. Together they suggest the addition of something to fill the "low" amount. Several other English words are formed from the same roots. Supply is an alternative of the same word. The word supplicate, meaning beg or implore, if from one who needs to be supplied. Supple, for limber, is perhaps and early variation of "beggars can't be choosers"; those who need should remain flexible.
Similar mathematical words to see are complement, and explement.

Surd Surd is a polished word for numbers that are irrational roots, like the square root of two. The original Latin 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 refernce is to a root that could not be expressed (spoken) as a rational number. It has be reported that al-Khowarizmi (see ALGEBRA)referred to rationals and irrationals as sounded and unsounded in his writings. When these were translated into Latin in the 12th century, the word surdus was used. The earliest citation in the OED is from Recorde's Pathway to Knowledge in 1551, "quantities partly rationall and partly surde."

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.
There are tow major typse of self symmetry, rotational (point), and reflective (line). Reflective symmetry is sometimes called mirror symmetry because one part of the object looks like the reflection of the other half. Objects which do not have reflective symmetry are called Chiral ((handed) and their reflections will be the opposite handed chiral shape. Two chiral objects which are reflections of each otehr are called enantiomorphic, from the Greeek words for opposite body. Think of the image of your right hand in a mirror, which looks like your left hand. Chiral comes from the Latin root, chiro, for hand, which also gives us chiropractor and chiromancy, a fancy name for palm reading. Although the word appeared in general usage in England by the 14th century, the mathematical usage of symmetry did not emerge unitl 1888 when the American Journal of Mathematics had an article on "Notes on Geometric Inferences from Algebraic Symmetry.

Volume Q-R

Q

"Q.E.A 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”)".  *http://jeff560.tripod.com/q.html    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."

*Wikipedia

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.

Quadratic Formula (The unusual forms)   So much has been written so well on the basic quadratic formula that I have no interest in attempting to repeat or improve them here. The student looking for such might well look to the term in St. Andrews Math History site. However recently I acquired two old books that had versions of the quadratic formula which are not often seen, and I thought I would post them here. The first is from the 1895, Elements of Analytic Geometry by Simon Newcomb which is shown here:

Another, which I had never seen before appears in the 1913 Analytic Geometry of Ziwet and Hopkins. The book also is unusual in the use of p and q for half the linear term and the constant term respectively. The more typical formula and the more traditional A, B, C lettering is introduced in an exercise for the reader.

For those who want a fuller history, "there must be 50 ways to leave your lover", according to the old Paul Simon song, but I only found 18 (maybe 20) ways to solve a quadratic equation, here is my document with some notes about the solutions and their history.

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."

R

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."

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 subtended 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]

Roger Cotes died in 1716. By 1722, his cousin Robert Smith had collected and published Cotes' mathematical writings in a book, Harmonia mensurarum … . In a chapter of editorial comments by Smith, he gives, for the first time, the value of one radian in degrees. After stating that 180° corresponds to a length of π (3.14159…) along a unit circle (i.e., π radians), Smith writes: "Unde Modulus Canonis Trigonometrici prodibit 57.2957795130 &c. " (Whence the unit of trigonometric measure, 57.2957795130… [degrees per radian], will appear.)

Radius The word radius is the Latin word for the spoke of  a wheel, or a staff. The earliest English 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 used for taking the altitude of a star, was called a "radius Astonmical" in 1592 by John Dee. 

In Euclid Book 1, postulate 3 the term radius was not used, instead  distance was used. Archimedes called the radius "ek tou kentrou" (the [line] from the center) [Samuel S. Kutler].  

The term “semidiameter” appears in Latin in the Ars Geometriae of Boethius (c. 510), according to Smith (vol. 2, page 278). The English word appears in 1551 in Pathway to Knowledge by Robert Recorde: “Defin., Diameters, whose halfe, I meane from the center to the circumference any waie, is called the semidiameter, or halfe diameter” (OED).*Jeff Miller

  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 meant 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 perpendicular to an edge, what we now call the apothem as well. The term radiys of an regular polygon now refers to the distance from the center to a vertex of the polygon.  

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

*MacTutor

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.  From such fun, came the name for a geometric object that made its way into our mathematical landscape.

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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. 

If you have ever been to a rodeo, you have probably been impressed with the agility and courage of the rodeo clowns as they distract the bull after the rider departs from the bull. You may even think it was a sport created by cowboys of the old American west. The truth however, is that playing tag with a bull may date back to the ancient Greek civilizations around 2000 BC. Archiologists working on Minoan ruins found pots with illustrations that seemed to show that taunting a bull was a popular pastime for young males of that culture. The picture below is one found when the castle of King Minos was excavated.

It seems that sometimes, however, the bull was bored by the whole routine. It is hard to be macho if the bull is doing his "Ferdinand" routine and smelling the daisies so, to prod the animal into more ferocious activity, the young men began twirling an object on a string around their heads that made a roaring noise.

You may have seen other objects used to make a sound in a similar manner. Hopi Indians use something like that in their dances and you may have seen your science teacher twirl a length of plastic tube to make various "roaring" sounds. Such objects today are called bull roarers. I always thought it was because they were presumed to sound like a bull. Now I am less sure. The ancient Minoan object that twisted as it twirled and made the roaring sound was called a rhomb. The root began to be used in words that suggested rotation or twisting motions, such as spinning tops, but none of the others seem to have made it into modern language. The use that did prevail was for shapes that looked like the four sided object that they swung on the end of the string. This is how we came to call the equilateral quadrilateral a rhombus... and that's no bull.

Euclid uses the word rombos and in his translation Heath says it is apparently drawn from the Greek word rembw, to turn round and round. He also points out that Archimedes used the term solid rhombus for two right circular cones sharing a common base. Euclid extended the idea in using rhomboid to name the shape we more commonly call a parallelogram. Since the definition of rhomboid (romboeides) comes before the definition of parallel lines, Euclid defines the rhomboid as (in Heath's words)," that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled."

The term rhomb is often used for the same shape, and many people, particularly young students, refer to a rhombus as a diamond. Some use the term only for non-squre rhombi (or rhombuses). It is especially interesting to work with early elementary students who will identify a shape as a square when the sides are horizontal and vertical, and then call it a diamond when the shape is rotated 45 degrees, even while they watch. The word diamond seems to be a mutation of the word adamant. A person is adamant if they are firm or unyielding in their attitude or position. The origin seems to be from the common root anti and the Greek word deme which meant to force or break (as in training an animal) which is the root of our present word domesticate. Together the two roots meant unbreakable, and the word was commonly used for hard metals and gems, and extremely difficult people.

The French word lozenge is also used for the non-square rhombus by some people, although I have never seen the term in a current math book. The word comes from the Gaulish word lausa for "flat stone". 

Lozenge was used by Robert Recorde in 1551 in Pathway to Knowledge: "Defin., The thyrd kind is called losenges or diamondes whose sides bee all equall, but it hath neuer a square corner" *J Miller

As with many geometric terms, there are two common definitions that are still in use for a rhombus. Some think of geometric names with an inclusive approach, and they usually define rhombus as it is defined in The Concise Oxford English Dictionary, "rhombus n. ( pl. rhombuses or rhombi / -b / ) Geometry a quadrilateral whose sides all have the same length." Notice that in this definition, a square would be a rhombus also. Others, who want definitions to describe how things are different from each other will define it the way it is defined in The Oxford American Dictionary of Current English, "rhombus n. a parallelogram with oblique angles and equal sides." Note that the Oblique angles rules out the case of a square.

The word Rhomboid which means rhom-like was commonly used in the 19th century for a parallelogram which was neither a rectangle nor a rhombus. Today it is more often used for a solid figure with six faces in which each face is a parallelogram and opposite faces in pairs lie in parallel planes. Some crystals are formed in 3D rhomboids. It is also sometimes called a rhombic prism. The term shows up frequently in science terminology referring to both its two and three dimensional meaning.  

My thanks to Mary O'Keeffe for the suggestions to explore the origins of this word.

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.