## 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.

Theorem – via late Latin from Greek theōrēma ‘speculation, proposition’, from theōrein ‘look at’, from theōros ‘spectator’. It is a non-intuitive statement that has been proven to be true using previously established theorems or statements such as axioms. Popular quote: “A mathematician is a device for turning coffee into theorems” most likely from Alfred Renyi, probably talking about Paul Erdos and his love for coffee. *Derek Orr

Thousand Our number for one thousand comes from an extension of hundred.  The roots are from the Germanic roots teue and hundtTeue 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.

Three Hares The symbol of the three hares range from China to England, and their are numerous guesses about their meaning and origin.  The images show three running rabbits in a circle so that the three all seem to have two ears, but only three ears are shown.  Like the triquetra or the triskelion, they illustrate rotational symmetry, but unlike the others, they seem to invoke visual illusion.  Whatever they were meant to be, it seems they started in China around the sixth century.  How that became the "Tinner's Rabitts of England is anybody's Guess. My first view of one was while I worked in England, and visited Long Melford in Suffolk, where the cathedral has a stained glass with the illustration.

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 RumaLucas 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".

Trace - See Matirx

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.

Triquetra Anther example of a motif with rotational symmetry, although like unlike the Three Hares or the Triskelion, this one
also has lines of bilateral symmetry.  The shape is made with three intersecting lenses, or vesicae Piscis, (bladder of a fish) formed by two intersecting circles of equal raidii.
The triquetra is often circumscribed, and other examples have the circle passing through the three lenses. The three arcs forming the three lenses complete a trefoil knot.
The figure originated around the 4th century BC and probably originated in Persia, but early examples spread to northern Europe, and Japan, where it is known as Musubi Mitsugashiwa.

 *Wikipedia
Triskelion -  The triskelion is more mystical than mathematical, and the designs range from the very simple to the ornate.  The earliest example, dating back to the bronze age before 400 BC in Malta, is the triple spiral.  The simplest would be three 120 degree angles joined at one end of each, which is a simplification of the three-legged triskelion version that was used on a Sicilian coin. Imagine aversion of a swastika but with only three angled branches instead of four.
The OED gives the first English citation as 1857 in an article on the history of ancient pottery. The mathematical attraction of the figures is that they have only rotational symmetry.  Many examples of rotational symmetry also have some other form of symmetry.
The three legged Triskelion is on the logo of the Tau Gamma Phi fraternity.

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.

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 radixx 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

Wallis' product  for Pi/2-  In his Arithmetica Infiintorium, British Mathematician John Wallis gave the infinite product for Pi/2,  $\frac{\pi}{2} = \frac{2}{1}* \frac{2}{3}* \frac{4}{3}* \frac{4}{5}* \frac{6}{5}* \frac{6}{7}^{...}$

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 used in TV tubes and in bactericidal lamps. (I wonder how long the term "TV tubes" will have meaning to young people!)

Zeno's Paradoxes Zeno of Elia (490-430 BC) supposedly wrote his paradoxes to counter other paradoxes written to discount the work of Parimedes.  Unfortunately, none of his original writing survives, and we are left to resolve his work by the words of others.  All of his paradoxical presentations seem to suggest that motion is an illusion, and could not exist.
The first, the reason that Achilles could never catch up to a turtle with a head start, says that in the time that Achilles can run from is initial position to the position the turtle origianlly assumed, the turtle will have moved some distance X ahead, and the same in each initial time frame.
The others argue similar constraints on motion, the Wikiepedia page on  the paradoxes of Zeno will provide the level of confusion to satisfy the most curious.

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 just 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 mone)  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.