Math Words, pg 20


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Bertrand's Postulate or Tchebyshef's Thm

When I was a student in college (long ago) I came across a little rhyme that I still have posted above the board in my class. It reads,

Tchebyshef said it
So I'll say it again
There's always a prime
between N and 2N
The theorem is more often referred to as the Postulate of Bertrand after the French mathematician Joseph Louis François Bertrand. In 1845 he made the conjecture that for any number greater than three there is a prime between any n and 2n (actually 2n-2) after examining the current tables of Primes. According to Mathworld the postulate became a theorem when it was proven by Tchebyshef five years later (1850)[It was proved in 1850 by Chebyshev (Havil 2003, p. 25; Derbyshire 2004, p. 124) using elementary methods, and is therefore sometimes known as Chebyshev's theorem.].

The Russian name has been translated as both Chebyshev and Tschebyshev which can lead to confusion. It is said that Besicovitch used to declare we use the letter T for the class of T-polynomials becuase it is the first letter of Chebyshev, which he was also known to claim had no letter T.



Chilihedron

In the spring of 2005 DeAnna K McDonald sent the following interesting post to the AP Stats list with another math word I had never heard.

"I found this word in a book called "The Word Museum--The Most Remarkable English Words Ever Forgotten" by Jeffrey Kacirk. I thought it especially appropriate to those of use living in the southwest.
Chilihedron--a figure of one thousand equal sides."
I presume the use of hedron means it had 1000 faces, and I don't understand the need for them to be "all equal". The "chili" root is derived from the Greek chilioi and is the root of the more common prefix, kilo. The root shows up in a few other words such as chiliad - A period of one thousand years - and chiliasm - The doctrine that Christ will return and reign for a thousand years.

Napier's book on logarithms, Chilias Logarithmorum contains the logs of the numbers from 1 to 1000. (Briggs had published a book with almost the same title, Logarithmorum chilias prima some seven years earlier in 1617.



Class or Set

A series of posts recently (Sep 2004) on the Historia Matematica list called my attention to the background of these terms. I quote here from those posts:

Dean Buckner gives a nice sumamry:

I was surprised by Peirce's remark, that that word "class" came to English in the 17C, so looked at this some more. The dictionary agrees, it is a 17C import.{[my addition] the OED gives " 1656 BLOUNT Glossogr. s.v. Classical, He divided the Romans into six great Armies or Bands which he called Classes; The valuation of those in the first Classe was not under two hundred pounds."} I managed to locate 5 instances of the word in Book 3 of Locke's Essay on Human Understanding (1690). He makes it plain that a "class" is really nothing different from what the medieval notion of a genus or species or universal. These correspond to the English "kind" and "sort". He uses "class" to make it clear that being in a class is not essential to the objects that make up the class. A ("kind" of thing, by contrast, suggests some essential feature that belongs to each thing of that kind, and which makes it of that kind).

Hobbes does not use the word "class" at all.

100 years later, Reid says, in his essay on the Intellectual Powers of Man, that we can divide infinitely things into "a limited number of *classes*, which are called kinds and sorts; and, in the scholastic language, genera and species. Observing many individuals to agree in certain attributes, we refer them all to one class, and give a name to the class." (Essay V, "Of General Conceptions").

60 years later (1847) in *The Mathematical Analysis of Logic", Boole says "that all reasoning ultimately depends on an application of the dictum of Aristotle, do omni at nullo. 'Whatever is predicated universally of a *class* of things, may be predicated in like manner of any thing comprehended in that class'". Clear evidence that, whatever nineteenth century writers meant by "class", they meant something similar to what they thought the medieval writers meant by a "kind" or "sort" of thing. That, of course, doesn't imply they understood the medieval writers properly!

The move from "class" to "set" is clearly connected with the formalism of set theory. Russell, as mentioned earlier, uses "class" as synonymous with "set".

Samuel S Kutler adds a note about the Greek terms that are the intellectual progenitors of our mathematical terms for class and set, "In ancient Greek the words for kind and form: idea (iota, delta, epsilon, alpha) and eidos (epsilon, iota, delta, omicron, sigma) are very important--especially in Plato. The idea of the good is said by Plato's Socrates to be beyond being! Idea is not like our word which is something we have, rather it is what gives intelligibility to things. Cantor thought that his word for set was like those Greek words! "

The OED suggests that "set" as used in mathemtical logic, "A number of things grouped together according to a system of classification or conceived as forming a whole. " was first used in 1690 by Locke. The word has been drawn from variations of the verb form of set, as in a physical placing of the objects, although it suggests that the usage may have been influenced by the German word gesette, " set or suite (of pieces), whence app. G. gesetz set of knitting-needles, etc., Da. sæt set of china, suit of clothes."



Notes on Counting boards in History

The oldest existing counting table is the Salamis Tablet. The white marble tablet, discovered on the island of Salamis (The largest of a group of islands near Athens) in 1899 is believed to date back to the third century BC. The tablet is presently in the National Museum in Athens. The major calculating feature was the group of eleven parallel lines crossed by a perpendicular line through their middle. Also marked along the two longer sides of the table and one of the shorter sides are Greek letters used to represent the denominations of ancient coins ranging from the talent down to the chalkos or one-eighth of an obel. In Karl Menninger's Number Words and Number Symbols he indicates that there were two columns for each unit. A counter in the left of the two columns indicates a value of five, while a counter in the right column indicates a value of one. Values such as seven Drachmas would be indicated by a single counter in the left column, and two in the right (5 + 2 = 7). By the middle ages the counting boards seem to have evolved into a base ten model with a single column for each unit. (see the image on the woodcut below, or the photos linked below) Menniger adds that the Greek term psephizein ("to pebble") is still the Greek term for "to compute".

Two other ancient artifacts confirm the use of counting boards in antiquity. One is an image on the "Darius Vase" located in the Museo Nationale in Naples. Among the images of the members of Darius' court is the image at right showing the royal treasurer at his counting board. A very similar image appears in an Etruscan Cameo shown on page 304 of the Menninger book mentioned earlier. This time the reckoner holds a tablet with Etruscan numerals on it.

Pictures of counting boards from the fourteenth century are at Convergence , the on-line magazine from the MAA.

A different type of counting board, at least the design is different, is shown in the classic illustration of the competition between the abacist and the cossist in Margarita Philosophica (1508) of Gregor Reisch. The muse of arithmetic stands in the center holding books about the two methods. The Margarita Philosophica (Pearl of Wisdom) was, in the words of the New Advent Catholic Encyclopedia, "... an encyclopedia of knowledge intended as a text-book for youthful students, and contains in twelve books Latin grammar, dialectics, rhetoric, arithmetic, music, geometry, astronomy, physics, natural history, physiology, psychology, and ethics. The usefulness of the work was increased by numerous woodcuts and a full index. The form is catechetical: the scholar questions and the teacher answers. The book was very popular on account of its comparative brevity and popular form, and was for a long time a customary textbook of the higher schools.". The figure calculating with the board is Pythagoreus (according to the banner) and the one using the new arabic calculating method is Boethius.

Menniger has suggested that the practice may have been to display the results of a computation above the middle line as a "head number" as the line is called the kephale or head, in Greek. This also shows up in the Latin summus, for highest, which gives our present word for sum.



Four

The English number four comes to us from the Gothic fidwor which is probably from the same proto root that gave the Latin quattuor. The Greek root for four which still shows up in many mathematical names is tetra. The Pythagoreans had a prayer addressed to four, or at least to fourfoldness, "O holy, holy tetraktys.." which was thought to represent the four elelments: fire, water, air, and earth.

Bill Bryson, in Mother Tounge (pgs 113,114) explains the loss of the "u" in forty. Why is there no "u" in forty? Four, fourteen, and fourth all have it. So did fourty in the writings of Chaucer. In fact, until the 17th Century when, as Bryson says, "But then, as if by universal decree, it just quietly vanished. No one seems to have remarked on it at the time."

The way four was written has changed over time more than many numbers. Wikipedia's page on the number four shows the following progression for the development of our current 4.


Karl Menninger in Number Words and Number Symbols points out that several of the shapes were used at the same time, and gives a photo (Fig. 260) from the Bamberger Rechembuch of 1483 in which two of the forms are used in the same example of a multiplication problem. He also has this graphic showing three different variants of a four in succesive years by Durer.

The Wikipedia page also gives several interesting tidbits about the uniqueness of four. Among them are the following; "Four is the only English number that, when spelled out, has the number of letters that it names", "Four is the first positive non-Fibonacci number", and "Each natural number divisible by 4 is a difference of squares of two natural numbers, i.e. 4x = y2 - z2". Four is also the smallest composite number, and a solution to Brocard's Problem (For what values of n is n! + 1 a perfect square) since 4! + 1 = 52

The fact that the word four has as many letters as the number it names was extended by Martin Gardner into an observation about all numbers. Take any number, use the number of letters in the name to get a new number. Repeat with the new number and eventually, it seems (I have not seen a proof, nor have I ever found an exception) you come to four. For example if we start with thirty, it has 6 letters, so we take six, which has 3 letters, so we write three, which has 5 letters, and at last we come to five, which has four letters.

Barney Hughes posted a note about the use of four as a base for counting. "Ed's remark about base-four would have me note that several tribes of California Indians, noteably those living in and around Ventura, counted by fours. A complete list of their words for the numbers, 1 to 64, is in Old Mission Library-Archives, Santa Barbara, a holograph by an early nineteenth century missionary. (I have seen it.) While I am at it: there were some twenty linguistic families in California and they used 4,5, 8, 10, and 20 for their bases, all depending on where they lived rather than on the language or dialect they spoke."



If the part of the curve y=1/x for x > 1 is rotated around the x-axis, the figure created resembles a trumpet with the mouth piece at infinity on the positive x-axis. The curve has been called Gabriel's Horn, in relation to the archangel Gabriel, and Toricelli's trumpet, in honor of the mathematician Evangelista Torricelli ( 1608 - 1647 ) who first pointed out the unusual relationship between surface area and volume of the shape. A quote from the National Curve Bank page at the University of California expresses the paradox well, "Torricelli's own words fully describe his amazement at discovering an infinitely long solid with a surface that calculates to have an infinite area, but a finite volume. 'It may seem incredible that although this solid has an infinite length, nevertheless none of the cylindrical surfaces we considered has an infinite length but all of them are finite.'" All done prior to the creation of the calculus.



Gauging

Many early arithmetic books contained sections on the practical art of Gauging or determining the volume of a barrell, cask, or keg. Robinson's The Complete Arithmetic (1880) defines, "Gauging is the process of finding the capacity or volume of casks and other vessels." [although only casks were given in any of the examples]. "A cask is equivalent to a cylinder having the same length and a diameter equal to the mean diameter of the cask." The text then goes on to explain that, "To find the mean diameter of the cask (nearly) Add to the head diameter 2/5, or, if the staves are but little curved .6 of the difference between the head and bung diameters." The actual formula given to find the volume is Volume (in gallons) = .0034 x length x (mean diameter)2 where the length and diameter are in inches.

Historically, gauging was done by pushing a rod through the bung hole to the opposite edge of the barrell head. Kepler was drawn to study this problem when he noted a difference in the shape of Austrian barrells and the Rhineland barrells for which the gauging rods were designed. Here is a note about Kepler's study from the web page of Franz Pichler, of Linz:

"Kepler may also be counted, with Archimedes and Pappus, as one of the founding fathers of the calculus of integration which later found its final formulation with Leibniz and Newton and their invention of the infinitesimal calculus.

He was first drawn to the subject of integration by the problem of determining the capacity of a wine-barrel of the Austrian type, which was done using a gauging-rod. He observed that the same method (with the same rod) was used regardless of the shape of a particular barrel. The volume was read off from the calibrations on the rod, which was positioned so as to measure from a bung-hole half-way up the barrel to the opposite edge of the barrelhead.

Kepler tackled this problem in his important book, "Nova Stereometria Doliorum Vinariorum" (Linz, 1615), and developed a complete mathematical theory in relation to it. This was based on the stereometry of Archimedes, who had already succeeded in calculating the volume of the sphere, spheroid and conoid, and on that of Pappus, who had calculated the volume of rotational bodies of any shape. Kepler extended Archimedes' stereometry to include new rotational bodies generated by means of conic sections - the "apple", "lemon", "spindle" and others. For wine-casks in Austria, where it was the rule among coopers that the radius of the barrelhead should be one-third of the length of the staves, Kepler showed that the volume of the "truncated lemon" (hyperbolic spindle), which can be successfully calculated, gave the best approximation. The same theory can also be used to show that Austrian wine-casks (unlike for instance the Rhineland casks, where the radius of the barrelhead is equal to one-half of the length of the staves) are "maximal", meaning that the accuracy of the method of measurement is not affected by the different shapes of barrel which result from the extent to which the staves are bent.

Kepler's "Messekunst Archimedis" (Linz, 1616) sets out the results obtained in "Nova Stereometria Doliorum Vinariorum" in a popular form and in German. In this book Kepler also introduces new German mathematical terms (equivalents for the Latin ones) which are still in use today. But, as F. Hammer points out, the "Messekunst Archimedis" also contains important new material. Kepler presents a procedure for calculating the content of a partly filled cask; and he also contributes to the practical side of stereometry by a systematic treatment of different units of measurement. He was to do further work in this latter area in connection with the construction of the "kettle of Ulm" ("Ulmer Masskessel") in 1627.

Kepler's work on the mathematics of gauging as contained in his "Nova Stereometria Doliorum Vinariorum" and the "Messekunst Archimedis" was to be carried further by the German mathematician Lambert, in his "Beyträge zum Gebrauche der Mathematik und deren Anwendung, Abhandlung II, Die Visierkunst" (Berlin, 1765).

"



Halma and Chinese Checkers

The popular American game of Chinese Checkers seems to have come from an early English game with the simple name Hoppity. The American Doctor and Harvard Graduate George W. Monks seems to have been told about the game of hoppity from his brother, Robert, who was traveling in England. George Monks father in law was Dr. Thomas Hill, a former President of Harvard College, and it was Hill who came up with the name Halma for the new game, drawn from the Greek root for "jump". Hill also seems to have been the inventor of the name Tangram for the Chinese seven-piece puzzle. Within only a year or two the game had been published by the E.I. Horsman Company. A copy of a Horsman version box for the game can be seen here. The game was published in England by the Spears company also using the name Halma. The game, was then (and may still be) also sometimes called by the old name hoppity. I found a copy of a game (circa 1950, I think) at a boot sale (British version of an American flea market) which is shown at right.

The game seems to have been produced in different size boards and played with multiple set ups of the intial pieces. Star Halma, was created with a hexagon shape to allow six players. This emerged into the game we now call Chinese Checkers. According to one game history sight,

The first Chinese Checkers game to be published in the United States was 'Hop Ching Checkers' in 1928 by J. Pressman & Co. This was exact(ly) the same game as the 1892 Star-Halma. The brothers Bill and Jack Pressman made up the name 'Chinese Checkers' during or shortly after 1928. The game was given a Chinese name and theme in keeping with the current interest in all things oriental (among them the discovery of King Tut's tomb in 1922 and the 'mah jongg' game that was introduced in 1923). "
An image of a "chinese checker game can be seen here.



Homoscedastic indicates that two or more statistical distributions are (approximately) evenly spread. It is often an assumption in regression that the response variable is equally spread across all ranges of the prediction variable. The term is drawn from combining the Greek terms homo for similar or alike, with skedannynai for to spread or discperse. The term is sometimes spelled with "homeo" for the prefix, but less commonly. If the distributions are not equally spread, they are called Heteroscedastic.

I wasn't sure when the term made its way into statistics, so I posted a request to the AP Stats discussion list. Within hours David Bee had responded with "Thanks to H.A. David's paper in the May 1995 issue of TAS [The American Statistician] I can. Karl Pearson was the first to use it, using it in a 1905 paper titled 'On the General Theory of Skew Correlation and Nonlinear Regression' " . Here is a clip from the reference on page 22.


The entire book can be found at Google Books.

A few days later, Dennis Roberts posted a pun taken from the Worm Runners Digest (May, 1972) which is the OLD name for the Journal of Biological Psychology.. anyway, the pun..

"homoscedastism"
The phenomenon of one person giving the very same response to 5 totally different questions.
The Worm Runners Digest is interesting enough to comment on in its own right, so here are some notes from a Nov 1966 Time Magazine article about the "official" demise of the digest
Among the fast-proliferating journals that report on assorted scientific specialties, few are even remotely comprehensible to the average layman. And many a literate scientist admits to being all but stupefied by their jargon-filled contents. One notable exception among such somber publications is the sprightly Worm Runner's Digest, which serves up its well-edited and important scientific papers along with side dishes of humorous satires, poems and cartoons.
Now even the seven-year-old Digest has had to retreat before the tide of scientific conformity. Beginning next year, its editor announced last week, the name of the Digest will be the Journal of Biological Psychology.
Proud of It The original choice of title was not made lightly, says Psychologist-Editor James McConnell, who heads the University of Michigan planarian (flatworm) research group, which publishes the W.R.D, "In psychological jargon," he explains, "those who experiment with rats are called 'rat runners,' and those who work with insects are called 'bug runners.' So we are 'worm runners'—and we're proud of it." Not enough scientists dig McConnell's logic—or humor. Some will not publish their work in a journal with so frivolous a name. Editors of other psychological journals refuse to allow their contributors to make any reference, however valid, to the W.R.D. "We even had trouble with librarians," says McConnell. "Many of them will not order journals with odd names for their science sections." The new name, he hopes, will make the old W.R.D. more acceptable to the entire scientific community.


The Lune of Hippocrates In geometry, the lune of Hippocrates, named after Hippocrates of Chios, is a lune bounded by arcs of two circles, the smaller of which has as its diameter a chord spanning a right angle on the larger circle. Equivalently, it is a non-convex plane region bounded by one 180-degree circular arc and one 90-degree circular arc. It is the first curved figure to have its exact area calculated mathematically.

Hippocrates wanted to solve the classic problem of squaring the circle, i.e. constructing a square by means of straightedge and compass, having the same area as a given circle. He proved that the lune bounded by the arcs labeled E and F in the figure has the same area as does triangle ABO. This afforded some hope of solving the circle-squaring problem, since the lune is bounded only by arcs of circles. Heath concludes that, in proving his result, Hippocrates was also the first to prove that the area of a circle is proportional to the square of its diameter

This lune is related to the Lune of Alhazen. As Hippocrates showed using a similar proof to the one above, if two lunes are formed on the two sides of a right triangle, whose outer boundaries are semicircles and whose inner boundaries are formed by the circumcircle of the triangle, then the areas of these two lunes add to the area of the triangle. The quadrature of the lune of Hippocrates is the special case of this result for an isosceles right triangle. The lunes formed in this way from a right triangle are known as the lunes of Alhazen, named after the 10th and 11th century Arabic and Persian mathematician Alhazen.



Knots, and Nautical Mile

The Nautical Mile is a measure of length. Here is the description given by "How Many? A Dictionary of Units of Measurement" which is produced by Russ Rowlett and the University of North Carolina at Chapel Hill:

a unit of distance used primarily at sea and in aviation. The nautical mile is defined to be the average distance on the Earth's surface represented by one minute of latitude. This may seem odd to landlubbers, but it makes good sense at sea, where there are no mile markers but latitude can be measured. Because the Earth is not a perfect sphere, it is not easy to measure the length of the nautical mile in terms of the statute mile used on land. For many years the British set the nautical mile at 6080 feet (1853.18 meters), exactly 800 feet longer than a statute mile; this unit was called the Admiralty mile. Until 1954 the U.S. nautical mile was equal to 6080.20 feet (1853.24 meters). In 1929 an international conference in Monaco redefined the nautical mile to be exactly 1852 meters or 6076.115 49 feet, a distance known as the international nautical mile. The international nautical mile equals about 1.1508 statute miles. There are usually 3 nautical miles in a league. The unit is designed to equal 1/60 degree [2], although actual degrees of latitude vary from about 59.7 to 60.3 nautical miles.
The unit of speed associated with the nautical miles is called the Knot, usually abbreviated with Kn. In the Winter 2002-2003 Newsletter of the British Society for the History of Mathematics I found that Edmund Gunter, an inventor of the Slide rule, and Gunter's Chain had also "devised the system of measuring speed using knots".

Into the 20th Century, ships carried an actual line, called a log-line for measuring their speed. One is described, with the process of measuing the speed in knots in the 1875 textbook, Elements of Plane and Spherical Trigonometry, with their applications to Mensuration, Surveying, and Navigation by Elias Loomis.



Oval and Ovoid

The Latin word for egg, ovum is the root of the words oval and ovoid, both of which mean "egg sahped". Although it is becoming common to refer to ellipses as ovals, the true oval or egg-shape is usually a shape with a single line of symmetry.

The Mathworld site gives a nice illustration of an oval and says that the term is not formally defined in mathematics.

The graph of the equation 1= |x/a|n + |y/b|n is called a Lame Oval, after Gabriel Lame the French Mathematician. The example at right shows a=b=2 and n=1/2,

When n = 2, and a=b the equation describes a circle. The curves were also studied and popularized by Piet Hein, the Danish Architecht, scientist and poet, who called the graphs with n>2 a super-ellipse. Hein is also the creator of the game of Hex

Sometimes the term ovoid is reserved for three dimensional analogies of the oval or an ellipse, but there seems to be no standardization of the usage.



PreferrenceVoting

In some Elections with more than two candidates, voters are allowed to rank the candidates in order of preferrence. These methods are called preferrence voting. There are many ways to select the winners of these elections that are used, or proposed for use around the world.

In Borda voting each voter ranks all the candidates from most favored to least favored. The candidates then receive "points" based on their ranking. The lowest ranked candidate receives zero votes, the next lowest receives one, and so on incrementally until the most favored of n candidates gets n-1 points. The winner is the candidate who receives the most points. Several methods are used in the event of a tie.

. The Borda system is named for Jean-Charles de Borda although it seems that Nicholas of Cusa developed the idea first. Borda was an emminent French scientist and mathematicain, and also served as a captain on a French ship during the American Revolution in support of the American Colonies. At the Wikipedia web link on Borda counts I found that "This form of voting is extremely popular in determining awards for sports in the United States. It is used in determining the Most Valuable Player in Major League Baseball, the national championship of college football, as well as many others. Used for parliamentary elections in countries of Nauru and Slovenia."

One very early voting system that is receiving popular support today, especially from minority candidates, is called approval voting. In approval voting the voters may vote for as many of the candidates as they would approve of for office. The candidate who receives the most approval votes is the winner. The method was first used in the Venetian State in the 12th and 13th century.

Another suggestion for voting methods was created by Ramon Llull. Llull described a method for holding "fair" elections which was to be used in electing the abbess of a convent. A translation of some of Llul's writing on elections alongside the Latin Original (for purists) is here. Llul's method was to have a vote pairing each pair of candidates. A winner would be the choice which won against every other candidate in a head to head election. Today this method is named for a later advocate, Marie-Jean-Antoine-Nicolas de Caritat, who is better remembered as the Marquis de Condorcet. Today the head to head method of Llull is often called Condorcet voting. It is easy (easier?) to determine Condorcet head to head winners with preferrence voting. Sometimes there is not a pure Condorcet winner, that is, someone who defeats each of the others in a head to head ballot. It can even happen that in a three person race between A, B, and C, A will be preferred by a majority of the population over B, while B is preferred over C, and C is preferred over A. This non-transitive result seems counter intuitive to most people, and is often called the Condorcet Paradox. Several methods were developed to break the circular relations that might exist if there was no pure Condorcet winner. Charles L. Dodgeson, better know as Lewis Carroll, the author of the Alice books, developed one of these. Dodgeson's Method was to select the candidate of the group whose largest margin of defeat in any pairing was the smallest.

Many of these voting methods were developed much earlier than the people whose names are given to them. In an article by I. McLean { The Borda and Condorcet principles : three medieval applications, Soc. Choice Welf. 7 (2) (1990), 99-108} we find, " We report three medieval works, hitherto unknown to social choice, which discuss procedures for elections when there are more than two candidates." Two of the three propose Borda methods and the third a Condorcet method of successive pairwise comparison. All three discuss problems of manipulation. One of them displays a matrix for pairwise comparisons; this is a work written in 1299, nearly 600 years before the matrix notation was believed to have been invented by C L Dodgson. "



Quadratic Formulas

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.

Sign was once used as a measure of 1/12th of a circle or 30o. The idea must have come from the idea of the 12 "signs' of the zodiac in the sky, and thus 1/12 of a full circle was called a Sign. The image below shows the Sign appearing in a table of measures from the 1850 edition of Frederick Emerson's North American Arithmetic (pg 29)



Zipf's Law and Lotka's Law

Zipf's Law relates to an observation made by Harvard Linquist George Kingsley Zipf that says that the frequency with which an English language word is used is (approximately) inversly proportional to its ranking in a list of words. For example, if it were true that "and" is the third most popular word in the English language (I have no idea if that is even close), then it would occur about 1/3 as often as the first. Zipf's law is a specific version of what is often called "Pareto's Principal". Zipf originally stated his law as a product. If the word's rank is given by r, the frequency by f, then the product of f(r) = K for some constant. His original example used Joyce's Ulysses and found the constant to be about 26,000. Examples of distributions that (more or less) follow Zipf's law include distribution of incomes, size of earthquakes, and the notes used in a musical performance.

A similar type of function, called Lotka's Law. The difference here is that the frequency depends inversly with the square of the rank. I found the following really clear explanation of the Law at a page at the Univ of Texas called "Bibliometrics.

Lotka's Law describes the frequency of publication by authors in a given field. It states that " . . . the number (of authors) making n contributions is about 1/n² of those making one; and the proportion of all contributors, that make a single contribution, is about 60 percent" (Lotka 1926, cited in Potter 1988). This means that out of all the authors in a given field, 60 percent will have just one publication, and 15 percent will have two publications (1/2² times .60). 7 percent of authors will have three publications (1/3² times .60), and so on. According to Lotka's Law of scientific productivity, only six percent of the authors in a field will produce more than 10 articles. Lotka's Law, when applied to large bodies of literature over a fairly long period of time, can be accurate in general, but not statistically exact. It is often used to estimate the frequency with which authors will appear in an online catalog (Potter 1988).