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Math Words, pg 5


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

In 1827 Möbius published Der barycentrische Calcul, a geometrical book which studies transformations of lines and conics. The novel feature of this work is the introduction of barycentric coordinates. Given any triangle ABC then if weights a, b and c are placed at A, B and C respectively then a point P, the center of gravity, is determined. Möbius showed that every point P in the plane is determined by the homogeneous coordinates [a,b,c], the weights required to be placed at A, B and C to give the center of gravity at P. The importance here is that Möbius was considering directed quantities, an early appearance of vectors.


Cumulative / Accumulate The words accumulate and cumulative both refer to the grouping together of other objects or sets and both come from the ancient Indo-European root kou from which we also get words like cave, cage, cavity and excavate. By the time the Romans adopted cumulare for a heap or collection of objects, the meaning had apparently extended to the pile made in creating a cave or cavity. Today cumulative often refers to successive additions, as in the statistical use, while accumulate often suggests collecting all at once. A cumulus cloud is so called because it looks like a heaped mass, a pile of cloud. Which prompts me to provide a note on cloud names from The Weather DoctorCirrus, stratus, cumulus. We take these terms to describe the clouds above us pretty much for granted these days, often assuming they are names of ancient origin. While great age can be ascribed to their Latin roots, the specific terms for various basic cloud forms are not yet two centuries old and were only accepted as the international standards during the Twentieth Century. The man whom we must credit with developing the basis for our cloud classification system was Luke Howard (1772-1864), an English manufacturing chemist and pharmacist. Like many who observed and studied the workings of the atmosphere at that time, Howard was an amateur meteorologist. Despite that fact he produced several landmark works including On the Modification of Clouds, The Climate of London, and Seven Lectures on Meteorology, the first textbook on weather.



Discount

In the great markets of Italy, the competition could be harsh, and so the enterprising businessman would offer a little incentive to the reluctant buyer. If a Merchant buying spices for shipment surveyed the counting board and seemed hesitant, the seller would sweep a few stones to the side, away from the count. These would be included free if the buyer bought the ones counted. The Latin roots of discount reflect the early counting table origins, the dis indicating a reversal or movement away from, and the count from computare. When the French picked up the practice they called it d'escompte. By the middle of the 17th century, the English had begun to apply the practice in the pepper trade with Holland using the current word discount.

Three special types of discounts took on their own names, and problems regarding them were common in arithmetic books in the 18th and 19th centuries. Tare represented a discount, or adjustment against the weight of the container. The Latin root tara for waste. The same word is used for several types of weeds that grow among grain crops but are worhtless and thus "thrown away" at harvest. Tret is so old that even the OED states, "The reason or ground of the allowance was apparently forgotten already in the 17th c., and has been variously given since." Tret was a discount or reduction of the weight to allow for possible loss in shipping, or unusable product such as dust, defective or poor quality product, etc. It seems to have been established in England as about 1/26 of the net weight after the gross had been reduced by tare. Cloff is also of obscure origin. The OED defines cloff as "An allowance (now of 2 lbs. in 3 cwt., or 1/168), given with certain commodities, in order that the weight may hold good when they are sold by retail." All three terms appear in many popular arithmetics in England and the US. Here is a copy of a page addressing all three topics from Hill's Arithmetic from 1772 at the Pitt Digital Library.



Ether Long before the first use of the anesthetic we now call Ether, it was a name for the heavens, and later for the medium through which electromagnetic waves could propagate (after all, the reasoning went, everything must have something in which to move). We can trace the use of ether as a name for the heavens back to Aristotle's explanation of the nature of matter (about 350 BC). Earthly things, such as a stone, fell to the Earth because that was their natural place, the philosopher proclaimed. Fiery things would rise to the sky; witness the smoke. But the stars in the heavens did not move either up or down. They seemed to move in circles around the sky, so they must be made up of something very different than the objects of earth and sky. The sun, moon, stars and comets all seemed to be ablaze, and so Aristotle called the heavenly material the aether, that which is ablaze. Eventually, when scientists needed something to explain how the light got here from the stars, they used Aristotle's word for a mass-less medium through which the light waves moved.



Formula The origin of formula is from the Latin forma from which we get the word form for shape or style. The Romans were very formal in many of their religious and social practices, and the correct forma or procedure was taught to the young. Short or small rituals used the diminutive of form, formula. Over time short lists of directions and instructions began to be called formulas also. Today we apply the word to any collection of words or symbols used in a ceremony or procedure, from logging on to Netscape to the chemical description for water.



Idiot In spite of the way some students feel at times in math class, idiot is not directly a math word, but its origin was interesting enough to include. In Greece, public service was a high honor. It had to be, for it was seldom a paid position. The person who refused or was unworthy of a public position was an idiota, a private person. Eventually the term was applied more often to people who lacked the capacity to participate in the public process. I guess any student who turns down a chance at a free public education in the beauties of mathematics really is an idiot.

The OED explains the development of variations of the word this way, "In the 16th c., instances of the word are found with initial n, transferred from an (a nidiot:an idiot); nidiot was further popularly corrupted to NIDGET (q.v.). With the latter cf. the modern vulgar pronunciation, sometimes graphically represented as idget."



Kite


The geometric description of a quadrilateral with two pairs of congruent adjacent sides (some definitions require the two pairs to be distinct, so that a rhombus is NOT a kite) is drawn from the name of the flying toy which it resembles (although in many cultures the flying kites are not at all shaped like the ones in the U.S.). The toy itself probably drew its name from the bird commonly called a kite, or kyte, in England. The old English form of the word, cyte, is probably from an early German name for an Owl. The OED dates the word kete for the bird as early as 725 and the first reference for its use in reference to a toy was in 1664. John Conway has stated that the use of "kite" for the mathematical object is only a few decades old. He writes, "a few decades ago most authors would say something like ...a "kite-shaped" quadrilateral .... leaving those quotation marks in to show that they felt a bit uneasy about using this informal term in a technical sense. The toy "kite" was in term named after the bird, and about a century ago, people would have put explicit quote marks around this word when using it for the toy!" Given the early dates for its first use for a toy in the OED, I have some reservations about this last part.

It is not clear whether it was the shape of the bird, or its flight behavior that was responsible for its name being given to the toy. A non-convex kite is often called a dart, which I credit to Roger Penrose who used the name in a proof on a non-periodic tilling of the plane. Proclus calls this shape the four-sided triangle and speaks of it as a geometric paradox. In Ancient writings it is also called barb-like and hollow-angled.

You can tryout a JAVA SKETCHPAD exploration here.

In a recent discussion group, John Conway responded to the question "Is there a name (other than "kite") for a quadrilateral that looks like a kite --with no parallel sides, but with two pairs of equal sides?" His response was to describe a suggestion for a new word, Strombus. Here is the message in Mr Conway's own words.

I was trying to coin an acceptable word for this for a long time, without success until after being prompted by some considerable discussion on the net about a year ago, I eventually came up with "Strombus", which is derived from the Greek word for a spinning top.
I think it's the best of the terms that were suggested. It's interesting that the word "rhombus" is ultimately derived from the same source, a fact that lends the new term some respectability.
John Conway

I also have an English textbook that uses trapezion (note the n ending) for the shape we more commonly call a kite. In A Junior Geometry by Noel S. Lydon published in 1903 the definition on page 55 states A trapezion is a four-sided figure having two pairs of adjacent equal sides. It goes on to show the method of construction.

During the summer of 2002 while visiting a Chinese exhibit, Land of the Dragon, at the Dennos museum in Traverse City, Michigan, I happened upon the following historical tidbit about kites. I have paraphrased from notes scribbled at the scene.

In 206 BC, General Han Hsin sought to take the castle of an enemy. He flew a kite over the center of the castle, and measured the amount of string that was used. Then he had his scholars calculate the distance to the center of the castle on ground level, and dug a tunnel under the walls of the castle to the described distance and emerged to take the fortress by complete surprise. This little military anecdote is the first historical mention of a kite.

The name for "kite" in Chinese is fengzheng . Feng relates to the wind, and I had originally translated the Kanji for zheng to mean 'to oppose, to resist'. From this I had interpreted a literal meaning something like "wind fighter". Actually, zheng in fengzheng is either 1) a kind of harpsichord, or 2) a kite. A fengzheng then literally means a kite, from the aeolian (wind) harps sometimes attached to the string. [Thanks to Jeffrey Hayden for his corrections in July 2005, and for providing the following graphic.]

The museum display suggested that the kite made its way out of China during the Tang dynasty (around 700 AD) and spread through Asia and Europe. From the site of "http://www.kiteman.co.uk" I found:

"Its thought that kites were first introduced into Japan by Buddhist missionaries who travelled from China in the Nara period (649-794 AD) and were mainly used in religious and thanks giving ceremonies.

A Japanese dictionary dated 981 AD was the first to record the Japanese word for kite and used the characters for "Kami Tobi" meaning paper hawk - which suggests that the first kites were bird shaped."



Lemniscate The word lemniscate comes from the Greek word lemniskus for ribbon. The mathematical curve, a sort of figure eight, does look somewhat like the bow for a package made from a twisted ribbon [see figure]. The word is beginning to disappear from textbooks, and is completely missing in my high school edition of the American Heritage Dictionary. The only closely related term I could find was lemniscus, a term for a nerve bundle in the brain. No picture was available, but it may be this also looks like a ribbon.
The Mathematical curve [formulas below]is related to the rectangular hyperbola through the following relationship. If a tangent is drawn to the hyperbola and the perpendicular to the tangent is drawn through the origin, the point where the perpendicular meets the tangent is on the lemniscate.



I recently saw a picture of a chemical periodic table in the shape of a lemniscate created by William Crookes in 1888. The picture is on page 107 of The Ingredients: A Guided Tour of the Elements by Philip Ball.



Loxodrome

The shortest path between two points on the surface of the Earth is along a great circle arc, but this path is often not possible for ships. One reason is that a great circle arc takes constant changes of compass heading. Because it is not much longer in the middle latitudes, ships often sail a path of constant compass heading, called a loxodrome (and sometimes a rhumb line). Loxodrome comes from the Greek roots loxos for slanted, and drome which means path or course. The word rhumb was derived from the old Spanish term rumb for room or space, and the "h" seems to have crept in over confusion with the word rhombus. The first person to study the rhumb lines, and realize that they were not the shortest path between two points on the globe, was Pedro Nunez, the Portugese mathematician. In 1537, Nunez, the Royal Cosmographer of Portugal published his studies in Tratado da Sphera. In 1541 Gerard Mercator published the first globe with rhumb lines marked on it. The Globe was also the largest globe made up to that time. Today one of the globes may be found at the National Maritime Museum in Greenwich in Great Britain. Another is at the Harvard University Map Collection. You can view selected images of the globe at this web link.

The term "loxodrome" first appears in Tiphys batavus in 1624 by the Dutch Scientist Willebrord van Roijen Snell who is also known for his optical law of refraction.

Until the middle of the 18th century, finding a ships longitude at sea was nearly impossible. This forced seamen to try to navigate along a constant direction and use their estimated speed to "dead reckon" their position. It was 1569, almost 28 years after Mercator's use of rhumb lines on a globe, that he hit upon his most marvelous creation, a flat map in a new projection that would make navigation much easier, the type of map we now call a Mercator or cylindrical projection (at right). A straight line drawn on a Mercator projection map is a loxodrome.

The word loxodrome is also sometimes used for a logarithmic spiral because it always cuts a line through the origin with the same angle. A true complete loxodromic spiral on a sphere will endlessly circle the poles without reaching them. For that reason, the loxodrome is also called a spherical spiral.One of the first to show this was true was the English Mathematician Thomas Harriot. "He exhibited the logarithmic spiral as the stereographic projection of a loxodrome on a sphere, a projection he proved to be conformal." [St Andrews Univ math history web site]



Lune

The word lune is from Luna, the Roman Goddess of the Moon. The more ancient Indo-European root is leuk, which relates to light. Lunatic is from the same root, perhaps because the ancients believed it was the effects of the moon that accounted for the crazy behavior.
Lune is used in mathematics to describe two different ideas. [see figure below] The most common is the area on a sphere between two semicircles with endpoints at the same poles. Imagine two lines of longitude running from the North Pole to the South pole. The surface of the Earth between the two lines is called a lune. There is also a two dimensional shape called a lune. The crescent area formed between two excentric circles that share a common chord is also called a lune. This shape plays an important part in the mathematics of the ancient Greeks. The definition I have used is from "The Mechanic's Assistant: A Thorough Practical Treatise on Mensuration" by D. M. Knappen, published in 1849. The Wolfram Mathworld web site, which is usually one of the top sources for math topics defines the planer lune this way, "A lune is a plane figure bounded by two circular arcs of unequal radii, i.e., a crescent." These two definitions do NOT seem to be the same to me. Imagine the crescent formed by two excentric circles of the same radius. I wonder why this must be excluded from the Wolfram definition. Under either construction, there are two crescent regions formed. Students will not see many planer lunes in their math experience, but the ones they do seem almost all to be constructed so that the smaller radius circle has its center on the common chord of the two circles. I do not think this is a property of lunes in genral. To be sure, I asked the memebers of the Philomathes Yahoo discussion group, and Colin McLarty responded with, "Neither arc needs to have its center on the common chord, and Hippocrates did a case where neither one does. Since he was interested in squaring lunes, and he was so far as we know pretty much limited to Euclidean means, he necessarily dealt with only very special cases. It is unlikely he ever worried about a precise general definition of "lune" -- esp. whether or not the case of equal radii would qualify to be called a lune." [see a suggested definition from David W. Cantrell farther below]

A spherical lune is sometimes also called a digon, since it is a two-sided polygon (and less appropriately called a bigon since that mixes both Latin and Greek roots... and leads too often to the quip, "let bigons be bigons".)

The plane lune was studied in depth by the Greek mathematician Hippocrates of Chios (around 400 BC) in an attempt to find a way to find a square with the same area as a given circle using only the classic tools of compass and straightedge. He discovered that a lune formed by the chord which is the hypotenuse of a central right angle has an area equal to the area of the right triangle. This lune is still called the Lune of Hippocrates. Later he found two other plane lunes that could also be "squared". A nice site on the lunes which can be squared is here. The problem of "Squaring the Circle" was one of the three great unsolved problems of Greek mathematics. Much later it was proven using algebraic tools that the task was impossible, but the Greeks discovered lots of important mathematical ideas in their search to prove the impossible.

Hippocrates of Chios, the mathematician, is often confused with Hippocrates of Cos, who is considered to be the father of medicine, and for whom the Hippocratic Oath is named. This should not be two unexpected since they both lived at the same time, and came from relatively close islands off the coast of modern day turkey. This link shows a map of the area. There is an arrow pointing out the island of Cos, just northwest of Rhodes. North of Cos just south of Lesbos is the island of Chios. Between them is the Island of Samos, which was the birthplace of Pythagorus.

Two similar looking objects that may be included here are "lens" and "gibbous". The area of intersection between two circles is often called a lens, because it resembles a biconvex lens. The word lens actually is drawn from the word lentil, for its symmilarity to the lentil seed. Gibbous is derived from the Latin root for "hunchback" and can be applied to any object that is convex or rounded, but is most commonly used to describe the phases of the moon between a half-moon and a full-moon.

There is, or once was, a special name for the intersection between two identical circles and is called the Vesci Pescies. Here is a description from "Geometry in Art" by Hilton Andrade De Mello. "The 'Vesica Piscis' is an important symbol based on the circle. It is generated by the intersection of two identical circles, as shown in Figure 9.2 with one circle centered at point A and the other centered at point B. The area common to both circles, shown in blue, is called 'Vesica Piscis' because it resembles a vesicle, i.e., a receptacle. As seen later in this chapter, the “Piscis”, which is Latin for 'fish', is associated with the symbol adopted by early Christianity." Images below are from the e-text of the book.

David W. Cantrell suggested what I think are very clear definitions of lune and lens in a discussion on another list, and graciously sent me a copy: ”Given two circular disks, A and B, having a nonempty intersection and such that neither is entirely contained in the other, three regions are formed. (Think of a Venn diagram.) One of the regions, A intersect B, is convex; I suggest that "lens" be used to name that kind of shape. Neither of the other two regions, A-B and B-A, is convex; I suggest that "lune" be used to name that kind of shape. “



Mile The name for the distance we now call a mile comes from a shortening of the Latin phrase mille passes which means one thousand paces. Since the paces (one step with each foot) of the Roman Army were supposed to be 2 steps of 2.5 feet long, one thousand paces is very close to the length we now give to a statute mile, 5,280 feet. A nautical mile was developed to be a distance equal to one minute of arc (1/60 of a degree) distance along a great circle. The length of a nautical mile usually given as 6076 feet.



Monte Carlo Method During WWII the scientists and mathematicians working on the Manhatten project to develop the atomic bomb were forging into untested waters in both science and math. To answer some of the scientific questions, they would repeatedly sample from their best estimates of the partial results, then apply the math they knew to the interactions and study the range of results. This process, which they named after the famous Monaco gaming casino town of Monte Carlo, was created by Jon von Neumann and Stan Ulam. The term and a description of the method seems not to have been published until some time after the war.



Nebula Nebula comes to us directly from the same word in Latin representing a cloud. The Greek root seems to be nephos and is related to both clouds and rain. By the Roman period the word nimbus was used for rain or specific types of clouds, which we now call nimbostratus. Nimbus is still used as a term to describe the aura or cloudy luminescence that appears in pictures of saints and of the Deity.
Somewhere in the late 1600's or early 1700's astronomers began to describe the faint patches of diffused light in the night sky as nebulae, probably because they resembled clouds. Nebula also was used as a medical term from the 1400's on to describe cloudy spots on the cornea. Around the 1700's the term nebulizer was used to describe what we now call an atomizer in English because the spray came out in a fine mist or cloud. If neither term is familiar to you, this is the rubber bulb spray that you may recall on your grandmothers dressing table for spraying perfume. Lister used such an instrument to spray ether antiseptic throughout his operating rooms in the 1860's as part of his pioneering work in antiseptic surgery.



Protractor

To the thousands of geometry students who could never figure out which end to measure from, or what point to put where, a protractor will forever be remembered as an instrument of torture, but in reality it seems to have originated as a tool for surveying. The Latin roots of the word are from protractus, to draw out. The tract root is the common root for subtract, contract, traction, and tractor.

One current use of protract means to make something take longer, or to prolong it. The less current meaning of protract comes from the surveying activity of drawing a scale picture of a plot of land, and the tool they used to reproduce the angles when they protract the land, was called a protractor.

Teachers may find it useful to use this Java Applet that will illustrate how a protractor works.



Quadrivium From the Greeks to the Middle Ages the education process focused on the seven Liberal Arts. The lower three, grammar, logic, and rhetoric, were called the trivium. The tri is the common root for three, and the rest is from via, the Latin for road. The trivium was literally the meeting of three roads. Our everyday words trivia and trivial comes from this root and reflect their "lower" position. The upper four studies, called the quadrivium, were music, arithmetic, astronomy, and Geometry. From the explanation of trivium, it is easy to see 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 Reneissance.



Rhind Papyrus The best source we have about Egyptian arithmetic comes from the Rhind Papyrus. The document is approximately 33 cm tall and 565 cm long. The document is named for Englishman A H Rhind who bought it in 1858. Papyrus refers to the type of paper made from the stems of the papyrus plant, a tall aquatic sedge common to North Africa and the Nile region. The Rhind Papyrus, shown at right, 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. In honor of this scribe, the document is sometimes called the Ahmes Papyrus. Struik points out that Ahmes is "the earliest personal name known to us in the history of mathematics." An enlarged version of part of the scroll can be found here

One of the most interesting images on the Rhind Papyrus is Problem 48 (shown lower right). The picture shows a circle inscribed in a square. Compared to other circles drawn by the scribe, this one seems poorly drawn. Perhaps he is trying to suggest the reason for the use of a formula for the area of a circle that is essentially the area of the regular octogon that would fit inside the square. The Egyptian formula for the circle, as illustrated in problem 48, was to find the area of the square circumscribing the circle, reduce it by 1/9, and then square the result. In terms of the radius, this gives an area of 256/81 r2, or a surprisingly accurate approximation of 3.1605 for Pi.

There are four other lesser documents preserving Egyptian arithmetic. The Moscow Papyrus and the Berlin 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 the Egyptian methods.

The Moscow papyrus is sometimes called the Golenischev papyrus after the Russian, V. S. Golenischev, who purchased it in 1893 from two Egyptian brothers who found the tomb at Deir el-Bahri. The scroll is only 8 cm high, and 540 cm wide. The papyrus consists of 25 problems and their solutions. The most unusual are the tenth and the fourteenth. The fourteenth gives the formula for the volume of the frustum of a pyramid and the tenth seems to be the area of the surface of a hemishpere, or perhaps a cylinder. (The scroll is not completely clear and in the view of Egyptologists it was written by a very poor scribe.) Both of these are well above the level of other Egyptian mathematics in the various scrolls found to date.

The formula for the frustum of a pyramid found in the Moscow Papyrus fourteenth problem is often referred to as the Prismoidal formula. The formula, as stated in the papyrus, is V = h/3 (a2 + ab + b2), where h is the vertical distance between the two parallel faces, and a and b are the side lengths of the two bases. It is easy to show that this is exactly the same as the calculus rule of integration called Simpson's Rule. See Prismoidal Formula

In a more general form the formula is expressed as V = h/6 (Bottom area + 4 Mid-area + Top area) Where the mid-area is the area of a cross section half way between the top and bottom. This formula will work not just for the frustum of a prism, but any prism, cone, pyramid or frustum of any of these. It also works for spheroids and any other figure whose area at a height h is, at most, a cubic polynoial function of the height. All the examples above are mearly quadratic functions of the height. According to a posting from James A. Landau to the Historia Matematica discussion group, Charles Davies translation of Legendre's Trigonometry and Mensuration suggests that the rule was "commonly used in computing volumes of earth-work in railroad cutting and embankments."



Robot The origin of robot comes directly to us from the Czechoslovakian word for compulsory labor. The Indo-European root 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 in 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 eliminating menial labor. Ultimately, however, the machines triumphed in a robot rebellion, destroyed humanity and created a new world of their own."



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 it in jest, but the term stuck. Now we refer to any body orbiting another primary mass as a satellite.



Student's t The story of the name for this statistical distribution and test is almost legend, and some version of the tale is remembered by Intro Stats students long after they forgot the purpose of the t-test. A recent dialogue between Randy Schwartz and James A Landau on the Historia Matematica discussion group gives both one of the folk versions, and a brief history.
Randy Schwartz writes "the distribution now known as 'Student's t distribution' was first discussed in print in a paper by Willam S Gosset 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 Gosset 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 have not encountered before.
Gosset was an employee of Guinness Brewery (a brewmaster, I believe) who went to study statistics under Karl Pearson. Gosset 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 Gosset not to reveal his identity.


In any event, Gosset published all his statistical works 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-establish his professional reputation under his real name. Perhaps he liked the notoriety."
OK, now we know why he used "Student", but why t?

About a year after I wrote the last question in the paragraph above, I picked up a copy of a wonderful book,The Lady Tasting Tea by David Salsburg, and found the answer in a footnote on page 30. He writes "Gosset used the letter "z" to indicate this ratio. Some years later, writers of textbooks began referring to normally distributed variables with the letter "z", and they began using the letter "t" for "Student's ratio".

While looking up some data on the Data and Story Library site, I came across a little historical note that should be included here also:

W.S. Gosset (1876-1937) was employed by the Guniess Brewing Company of Dublin. Sample sizes available for experimentation in brewing were necessarily small, and Gosset knew that a correct way of dealing with small samples was needed. He consulted Karl Pearson (1857-1936) of Universiy College in London about the problem. Pearson told him the current state of knowledge was unsatisfactory. The following year Gosset undertook a course of study under Pearson. An outcome of his study was the publication in 1908 of Gosset's paper on "The Probable Error of a Mean," which introduced a form of what later became known as Student's t-distribution. Gosset's paper was published under the pseudonym "Student." The modern form of Student's t-distribution was derived by R.A. Fisher and first published in 1925.

The t-distribution is a family of curves that discribe the pattern of possible errors when the average of a sample, , is used to estimate the mean of the population, m, from which the sample is taken if the underlying population is normally distributed. The distribution for every different sample size is slightly different, becoming more normal as the size of the sample gets larger. The distribution is given by the function . Think of repeatedly drawning samples of size n from a normal distribution with a mean of m and a standard deviation of s. After each sample is drawn you find the mean of the sample, and the sample standard deviation, s. With all of this in hand you use the formula above to construct a single value of t. If you repeated this operation many, many times, the ditribution of the different t values you get from each trial would form the curve called the t-distribution with n-1 degrees of freedom.

The image at right shows a comparison of the t-distribution with one degree of freedom (dark) and the standard normal distribution (dotted). You can experiment with the impact of the size of the sample, n, on the shape of the distribution with this java applet . For small values of n the curve is leptokurtic or flattened in comparison to the Standard Normal curve. It is lower at its peak than the Normal curve, but has higher density at the extreme regions of the tails.

The t-distribution with n=2 (or 1 Degree of freedom) was actually used at least as early as 1824 when Poisson wrote of it (not by that name). The distribution was also briefly explored by F Y Edgeworth (1883) where he calls it the subexponential distribution.

The t-distribution with only one degree of freedom is also called a Cauchy distribution. It can be written as . The curve was frequently studied in the early stages of statistics as an alternative to the normal curve for the distribution of errors. Although the distributions are named for Cauchy, they were earlier studied by Fermat, and numerous others including Huygens, Poisson, and Maria Agnesi.

In the form it was studied by Agnesi, it has often been called the "Witch of Agnesi.". A nice visual for how the "witch" is derived geometrically is found at Mathworld.com The origin of the name seems to have two explanations. One is that the Agnesi used the term "averisera" for the versed (turned) sine curve, and was subsequently mis-translated by John Colson who was Lucasian Professor of Mathematics at Cambridge. [See the text box below for the story according to Shirley Gray of California State Univ.] According to Stephen Stigler, however, Agnesi had used the term "la versiera" and stated that it meant witch or she-devil. Stigler also adds that she was not the first to use that term for the curve. Guido Grandi had used the same term in 1718, "explaining that the curve had arisen from a consideration of the 'semi versi' versed sine, and that he would call it 'versiera' after the latin word 'versoria'." In fact, says Stigler, "Although some have interpreted this to mean that versoria meant versed sine, the only relevant definition of versoria given in dictionaries of that time is 'a turning or twisting around'... Versiera is not the Italian word for Versoria but, rather, is the feminine form of 'avversario', sometimes used to mean devil..." Stigler goes on to suggest that Guido may have been indulging in a little humorous play on words and may have been suggesting that the curve resembled a woman's breast.



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. 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 on the earliest known use of some words is to a reference by Nicholas Chuquet in 1484 in which the word is spelled "some". The origin of this homonym of sum is from the root sem which relates to the quantity one.



Venn Diagrams Venn Diagrams are named for 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 subsets of a Universal set in non-overlapping regions. You can find more about his life at this page from the Electronic Journal of Combinatorics.
It appears that the first person to call these types of diagrams "Venn diagrams" was Clarence Irving in his work, A Survey of Symbolic Logic in 1918. The figure below left shows a Venn Diagram for two subsets A and B. The shaded portion is the intersection of A and B.

At the Euler Project web site, maintained by Prof. Ed Sandifer, I found a note suggesting that the diagrams are actually the creation of Euler.

Letters to a German Princess is likely to be the source of much of what people attribute to Euler. For example, I know that what we call Venn diagrams first appear in there. (Venn himself first called them "Eulerian Circles", but then managed to get them called Venn Diagrams later on.)
In his book Cogwheels of the Mind, The Story of Venn Diagrams, Professor Anthony Edwards of Cambridge explains that the Venn Diagram were much broader in scope than Euler's, and in a comparing Venn's work to previous, and sometimes similar, work he states, "Venn's own contribution, which fully justifies our attaching his name to the general diagram was the first to see that the diagram could and should be generalized to any number of sets..." Professor Edwards is a Fellow of Gonville and Caius college as was Venn, and played a part in the design of the commemorative glass shown above, which is at the college. The Glass is part of a set of six that are all commemorative of math and science people.

Professor Edwards came up with a method of extending set diagrams to any indefinite size by drawing them on a sphere, and sterographically projecting them back onto a plane to create the cogwheels of the title of his book. Here is an example of a simple four set diagram by this method.

The Oxford mathematician, Charles Dodgson, writing under the pseudonym of Lewis Carroll, approached set diagrams with rectangles. The right image below shows an example of how Carroll's diagram might look for three sets(above the middle, right or left of middle, and inside or outside the inner rectangle).


Carroll probably was most influential in his use of a finite set for the Universal set, as Venn often simply used the infinite plane outside the boundaries drawn to indicate the set of things not beloging to any group. Carroll's book, The Game of Logic can be found free on the web at the Guttenburg Project. Perhaps one of the great logic statements of all time occurs in the beginning of Carroll's book when he writes, "Besides the nine Counters, it also requires one Player, AT LEAST. I am not aware of any Game that can be played with LESS than this number: while there are several that require MORE: take Cricket, for instance, which requires twenty-two. How much easier it is, when you want to play a Game, to find ONE Player than twenty-two."

I recently came across another piece of Carroll's cross between wit and logic in a 2008 Gresham lecture by Robin Wilson:

Honoured Sir,

Understanding you to be a distinguished algebraist (that is, distinguished from other algebraists by different face, different height, etc.), I beg to submit to you a difficulty which distresses me much.
If x and y are each equal to 1, it is plain that

2 × (x2 - y2) = 0, and also that 5 × (x - y) = 0.

Hence 2 × (x2 - y2) = 5 × (x - y).

Now divide each side of this equation by (x - y).

Then 2 × (x + y) = 5.

But (x + y) = (1 + 1), i.e. = 2. So that 2 × 2 = 5.

Ever since this painful fact has been forced upon me, I have not slept more than 8 hours a night, and have not been able to eat more than 3 meals a day.

I trust you will pity me and will kindly explain the difficulty to Your obliged, Lewis Carroll.

Venn's grave and memorials can be found at the "Find a Grave" website

For more information about Venn Diagrams check this Survey of Venn Diagrams from the Dept of Computer Science at the University of Victoria.



virgule The slanted bar, "/", that is used for 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 our 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.