Math Words, pg 16

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   Mathematicians are often concerned with accuracy, and the key to being accurate is to take care.  In fact the origin of the word is from the Latin roots ad (to) and cura (care) which together formed the Latin word accuratus for "done with care". Cure, curio, curator, secure and sure are all from the same root and relate to work done with care. The word curator was created, in English, by the Royal Society in England in 1660 for the members who would supervise the experiments at each meeting. One of the first was Robert Hooke. A similar word existed in Latin for a manager or overseer, and in the church where a curate was an assistant to the vicar.

Aeon or Eon

Although a proposal was made in 1957 to define an aeon to be a unit of time equal to one billion years (1 Ga), the idea was not approved as a unit of scientific measure and is seldom used for a specific period of time. Its more common usage is for any long, indefinite, period of time. The origin is from the Greek root aion for age or life force. A similar Latin word aevum for age is still present in words such as Longevity and medieval.

Apollonius Problem
      Given three circles in the plane, find another circle tangent to the first three.  This problem is often called Apollonius Problem because it is first recorded in Tangencies, written around 200 BC by Apollonius of Pergia.  Actually Apollonius showed how to find a circle tangent to any three objects in the plane where each of the three objects could be either a point, a line, or a circle.  A very good explanation of the problem and the solution can be found at the Cut-the-Knot web site.
    Clark Kimberling, Hidetosi Fukagawa, and Shiko Iwata found that for a particular configuration there exists a central point where three lines from the points of tangency intersect.  They have called the point the Apollonius Point.
  If you start with three mutually tangent circles and create a fourth nested between the original three you can repeate this with three more circles nested between sets of three of these.  You can continue to do this for an infinite progression of circles.  The resulting creation is called Apollonius Gasket, and is a type of fractal.  A good illustration of the development of the gasket is at this Mathworld site.  The configuration of four circles all mutually tangent is often called Soddy circles.


The most common use of commutative in mathematics is when the order in which the objects are operated upon does not change the result. For example, addition is commutative because 2+5 =5+2. The idea was well known, but seemed so general in early mathematics that it seems the name was not applied until mathematics and algebra reached the point of considering operations which were not commutative. The responses to a question I submitted to Historia Matematica make me now think that the first non-commutative operation to be recognized as such, was the Quaternion

The first known use of the word commutative in its modern mathematical sense was "in a French-language memoir by François Joseph Servois (1768-1847) in _Annales de Gergonne_ (volume V, no. IV, October 1, 1814), pages 98-99." This last is the result of the research of Professor Gonzalez Cabillon. An excerpt from the text can be found at Earliest Known Uses of Some of the Words of Mathematics.

The roots of commutative are from the union of the Latin roots com, for mutual, and mutare, to change. The literal translation is an exchange between two things. The same two roots form the basis of words like commuter (one who travels back and forth to work). James Landau supplied an example of the early legal roots of commutative on another Historia Matematica posting. He wrote:

" 'commutative' and 'distributive' also appear in legal terminology. The Oxford English Dictionary 2nd edition traces these terms back to Aristotle.
.... "commutative justice, a term used by Aquinas and others as equivalent to Aristotle's 'the justice which is corrective in transactions between man and man.'"

Frequency is a term used most often for the number of times an event occurs per unit of time. In statistics we often use it as count of a variable, e.g., the number of women in a sample of 100 citizens, etc. A graph or table that shows the number of occurances of each of two or more categorical variables is called a frequency distribution.

For a periodic function if the independent variable is units of time, then the frequency is the reciprocal of the period, f= 1/p, expressed in units of cycles per unit time. The term is some times applied when time is not the independent variable. For example, if the independent variable is a dimensionless number, such as radian measure, then the frequency of a standard sine wave would be 1/(2pi). Some folks object strongly to the use of frequency independent of time, but heck, .... some folks don't even like grits...

The word comes from an Indo-european root that means "pack together or enclose tightly". Related terms include farce, fence, and diaphram.

Fundamental Counting Theorem

The fundamental counting theorem is one of the earliest rules of discrete mathematics that students often learn. It is sometimes called the fundamental principal of counting, as it is in the textbook For All Practical Purposes where it is defined as

If there are a ways of counting one thing, b ways of choosing a second after the first is chosen...., and z ways of choosing the last item after the earlier choices, then the total number of choice patterns is a x b x c x .... x z

Gunter's scale

Gunter's scale is a mechanical device used to multiply logarithms using a pair of divider. It was an early precursor to the development of the slide rule . A picture of a wooden Gunter scale can be found at this link .

Edmund Gunter was a professor of astronomy at Gresham College in London and a friend of Briggs. He published tables of the logs of Sines and Tangents and is credited with the creation of the terms Cosine and Cotangent. Gunter also invented a surveying instrument called Gunter's Chain which was 22 yards long and divided into 100 links. An acre of land is equal to ten square chains and a mile was 80 chains long. The chain is now obsolete as a unit of measure but was once very common in laying out townships and mapping the US along the train routes in the 19th century. In America there was a federal law passed in 1785 that all official government surveys must be done with a Gunter Chain. It was also called the Surveyor's Chain. On a visit to Stratford on Avon while at Hall's croft, the home of Shakespeare's daughter Susanna and her husband, Dr John Hall, I came across an early map of the town and the only legend shown was in Gunter's Chains. Watching an English Cricket match the other day (Dec of 2006) I realized that the length of the bowling area (between the two wickets) is one chain also.

There were other types of chains used for surveying. The image at the right is from Milton B. Goff's First Book in Arithmetic from 1876. The lengths shown clearly indicate that Gunter's chain is used. Another popular type was Ramsden's chain which is also called the engineer's chain and has a length of 100 feet.

The rod represented in the picture was another common unit of measure in England and the US. I had written a couple of notes on this when I came across Russ Rowlett's definition on "How Many: A Dictionary of Units of Measure". I have copied the definition here, and this link will take you directly to his page where you can follow his links to some of the other terms.

a traditional unit of distance equal to 5.5 yards (16 feet 6 inches or exactly 5.0292 meters). The rod and the furlong were the basic distance units used by the Anglo-Saxon residents of England before the Norman conquest of 1066. The Saxons generally called this unit the gyrd, a word which comes down to us as the name of a different unit, the yard. "Rod" is another Saxon word which meant just what it means today: a straight stick. The Normans preferred to call the gyrd a pole or a perch (a word of French origin, meaning a pole; see perche). The length of the rod was well established at least as early as the eighth century. It may have originated as the length of an ox-goad, a pole used to control a team of 8 oxen (4 yokes). Scholars are not sure how the rod was related to shorter units. It may have been considered equal to 20 "natural" feet (actual foot lengths; see foot), or it may have been measured "by hand" as 30 shaftments. In any case, when the modern foot became established in the twelfth century, the royal government did not want to change the length of the rod, since that length was the basis of land measurement, land records, and taxes. Therefore the rod was redefined to equal 16.5 of the new feet. This length was called the "king's perch" at least as early as the time of King Richard the Lionheart (1198). Although rods and perches of other lengths were used locally in Britain, the king's perch eventually prevailed. The relationship between the rod and the other English distance units was confirmed again by the Parliamentary statute of 1592, which defined the statute mile to be either 320 rods or 1760 yards, thus forcing the rod to equal exactly 5.5 yards or 16.5 feet.

In Stephen Stigler's Statistics on the Table he shows an photo image of the picture dated 1535 by Jacob Kobel in which the determination of a rod (then called a rood) was established. According to Stigler, "The picture shows how a 'right and lawful rood,' or rod, of 16 feet should be determined by measuring the feet of sixteen men as they leave church." From this it seems that the rod retained a standard 16 foot length sometime later than the English switch to 16.5 feet.

Hex, the game.

The game of hex is most often credited to the Danish mathematician, philosopher, and poet, Piet Hein. Supposedly he invented the game while at the Niels Bohr Institute working on the Four-color theorem. The game also seems to have been independently invented by Nobel Prize winner John Nash, while he was a graduate student at Princton. Although the game was originally conceived on a square board it is more commonly played on a non-square rhombus shaped board with hexagonal shaped spaces. You can download an interactive game of Hex to play against, but be warned the program is very good. The program, and the web page, is by Vadim Anshelevich.

In 1949 John Nash proved that the first player should always win, but the proof is non-constructive, and a winning strategy for boards larger than 7x7 is still unknown.

Piet Hein is also associated with employing the "Super Ellipse", and the three dimensional extension called the "Super Egg". Some additional information about both topics is at the Mathsyear 2000 web site

Another of Hein's more popular creations was the Soma Cube, a collection of twenty-seven cubes combined into seven solid pieces that could be put together to form a wonderous array of figures. The original challenge was to put them together into the 3x3x3 cube as they came packaged. Here is a very good page where you can lose yourself for hours with the soma cube

Hein's incredible variety of work also includes design and poetry. He is remembered for wonderful short poems called "grooks". One of the most popular is printed here, and more grooks can be found at a web page devoted to the topic.

A grook, by Piet Hein

Problems worthy
of attack
prove their worth
by hitting back.

Piet Hein was born on Dec 16 in 1905 and Died on April 17 of 1996. I recently read that his father was a civil engineer, well known as the designer of a roller coaster at Tivolo Gardens in Copenhagen.


In mathematics the word horizontal is used for lines that run from left to right across the page and thus seeming to be parallel to the horizon, the apparent line that appears to act as a boundary between earth and sky. In fact, it is this sense of boundary that is the root of the word. The early source of the word is the Greek root horos for a boundary or limit. Both Greek and Latin had words very nearly the same as the present English word for horizon.

Inverse Square Law

The first statement of a physical law involving an inverse square relationship was the decrease in the intensity of light according to the square of the distance from the source which was presented by Johannes Kepler between 1615 and 1625

Nine Men's Morris

   The game of Nine Men's Morris is also known as Merels and in Holland as Molenspel, the game of mills. It may be one of the oldest games in history. Depictions of the game are found carved into stone in the eastern Alps and in Egyptian temples dating back to as early as 1400 BC. A good introduction to the game and its rules is at the web page of Ms. Hebert's Math class. You can also play against the computer at this web site.

In Shakespeare's life the game was often played on a game "board" cut out of turf with stones for the markers. If the weather turned ugly, as Titania tells us in A Midsummer Night's Dream, "The nine men's morris is fill'd up with mud."

David Singmaster has found a suggestion that the game was known in China,

"On p. 12 of Fawdry is a scene, apparently from the Hundred Sons scroll of the Ming period, in which some children appear to be playing on a Twelve Men's Morris board. In Culin's Korean Games, p. 102, section 80: Kon-tjil - merrells is a description of the usual Nine Men's Morris. He states the Chinese name is S m-k'i (Three Chess) and continues: "I am told by a Chinese merchant that this game was invented by Chao Kw'ang-yin (917-975), founder of the Sung dynasty." This is the only indication of an oriental source of these games that I have seen."


Optimum comes directly from the same word in Latin, meaning best. Although the Latin word is ancient, the use in English, according to the OED, seems to have come into use between 1800 and 1850. As it is generally used in mathematics, to find the best solution under a given set of circumstances, is first cited in 1885.[1885 S. H. VINES Lect. Physiol. Plants 276 The minimum or zero point is the point at which the performance is just possible; the optimum point, at which it is carried on with the greatest activity]


As a gas, oxygen is colorless, odorless, and tasteless. The liquid and solid forms are a pale blue color. Oxygen is the eighth element in the periodic table and has an atomic weight of 16. Oxygen constitutes about 21% of the Earth's atmosphere. You can learn more about the properties of oxygen at this web site.

In the west, air was treated as a single element until it was seperated from lead and mercury oxides by Joseph Priestly, who recognized it as a unique element but in terms of phlogiston theory which he defended to his death. Lavoisier disagreed with the phlogiston theory based on his experimental evidence, and chose the name oxygen because he thought oxygen was a fundamental constituent of all acids. Many non-metallic elements such as carbon and suphur when combined with oxygen and dissolved in water will produce an acid. Lavoisier falsly assumed that all acids were created in combination with oxygen. The name oxygen comes from the roots oxy for sharp or acidic, and gen for to create or begat. The gen root remains in such words as gender, genes, and generation. Oxy is a variation of the ac root in acute which also means sharp. I am told that in Germany oxygen is called sauerstoff which literally means "acid stuff".

The true credit for the discovery of oxygen is not clear, as Lavoisier, Priestly, and Scheele all three may have some claim to the credit. A recent play, "Oxygen", written by two of the world's foremost chemists, has a plot based on this question.

Phlogiston Theory was popular for most of the 18th century. The word was created by George E Stahl (1660-1734) from the Greek word "to burn". It suggested that there was an element in all combustible substances, called phlogiston, that allowed it to burn. When it ran out of phlogiston, it would cese to burn. The work of Lavoisier and emergence of Dalton's atomic theory finally overcame the dogmatic adherance to phlogiston. The word was built on the Latin derivative of the Greek word phlogistos for combustible. The root phlox is preserved in the plant of the same name, and in words like conflagration.

Lavoisier was one of the greatest scientist of his age, but because of his aristocratic upbringing (and perhaps some jealous retribution) he was guillotined shortly after the French Revolution. [You can find more on the story here]. Lagrange is often quoted as having said "It took them only an instant to cut off that head,but France may not produce another like it in a century."

Q. E. D

Students (and teachers) are often confused by the appearence of the notation "QED" at the end of proofs. The use dates back to a time when the official scientific language was Latin. When a proposition was presented, and then proven, the last statement in the proof was often the Latin expression "quod erat demonstrandum", which means something like "which is that which was to be demonstrated." It was also common to abbreviate common expressions, and so it became common to write "QED" at the end of a proof. As the language of choice changed from Latin to French or English, the QED just stayed there.

In a post to math news group, John Conway explained that this was only one of three abbrevations that used to appear at the end of proofs

This is perhaps the least known of the triad:
Q.E.D. (Quod Erat Demonstrandum) "Which was to be demonstrated (proved)"
Q.E.F. (Quod Erat Faciendum) "Which was to be done (constructed)"
Q.E.I. (Quod Erat Inveniendum) "Which was to be found (discovered)"
The first two were in regular use in my high-school days, but I've only met the third in older books.

John Conway

No Royal Road to Geometry

Once upon a time SOMEONE was teaching geometry to a royal student. The student asked for a shorter way and the teacher replied "There is no royal road to geometry." There seem to be three common versions of this famous math tale. The most common, and the most likely to be true, is that Euclid spoke the words to Ptolemy I. This version is repeated in a page on famous anecdotes. Other versions say the words were from Menaechmus to Alexander the Great. Another common version is that it was told by some unnamed 'wise philosopher' to an equally unidentified royal. The question is, "Are any of them true?". The only direct reference I can find to support any of them is from Proclus who writes that the words passed between Euclid and Ptolemy. Heath gives the following details

Heath, History p. 354:
Proclus (410-485, an Athenian philosopher, head of the Platonic school) on Eucl. I, p. 68-20:

Not much younger than these is Euclid, who put together the Elements, collecting many of Eudoxus's theorems, perfecting many of Theaetetus's, and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors. This man lived in the time of the first Ptolemy. For Archimedes, who came immediately after the first, makes mention of Euclid; and further they say that Ptolemy once asked him if there was in geometry any shorter way that that of the Elements, and he replied that there was no royal road to geometry. He is then younger than the pupils of Plato, but older than Eratosthenes and Archimedes, the latter having been contemporaries, as Eratosthenes somewhere says.


While trying to design a more effective time piece, Christiann Huygens was confronted with a problem related to the free swinging pendulum. He knew that a mass moving along a circular arc under the influence of gravity would not have the same period for different amplitudes. The same mass swinging through 40% of arc would have a slightly different period than when it swung through 10% of arc. Prompted, perhaps, by a public contest to submit solutions to problems related to the cycloid, Huygens wondered about the oscillation of a clock constrained to the path of a cycloid. To his suprise, the period was the same no matter what the amplitude of the swing. He had discovered that the cycloid was the path of a tautochrone; a curve with the property that a mass sliding from any point along the curve will reach the bottom in the same time. You have to read that carefully. It means if you placed the mass 4 inches up the curve from the lowest point it would take the same time to get to the bottom as if you placed it 2 inches up the path.. The results were published in 1673 under the title Horologium Osiallatorium

The roots of the word are the Greek roots tauto, for same, and the chron that stands for time. Even today a clock is sometimes called a chronometer. The taut remains in terms such as tautology, which loosly translates to "same meaning". The cycloid is also the path of the brachistochrone.

Twin Primes

One unsolved mystery of the Prime numbers concerns the fact that it seems that there are prime numbers occuring in pairs, one on each side of some even number, and these pairs continue into the largest numbers. Examples of twin primes are 3 & 5; 11 & 13, 17 & 19, 29 & 31, but also pairs like 1427 & 1429, and 1451 & 1453. Here is a link to an internet site where the twin primes have been computed up to some huge value (it was 1014 when I posted this note).

you may notice as you glance at any long list that, except for 3,5, all the twin pairs occur as numbers on each side of a multiple of six. This fact, and the fact that the product of any two twin primes will be one less than a perfect square can be proven easily with some hints from a page at Cambridge University.

If you add up all the reciprocals of the twin primes, (1/3 + 1/5) + (1/5 + 1/7) + ... it turns out that the sum is just a tiny bit greater than 1.9 (some folks don't include 1/5 twice, so they get a slightly lower total). This is significant since we know that the sum of the reciprocals of all primes diverge to infinity, so this means the twins are significantly less dense than the primes in general. The constant is often called Brun's constant after the mathematician who proved the value was computable. The term "twin prime" was coined by Paul Stäckel around 1900 near the end of a short life.


A student challenged me with this word one morning, and I had NO idea what it might mean. The Oxford English dictionary suggested that the word was so obsolete that it was not unusual that I did not know it. The word appeared in Robert Recorde's, The Whetstone of Wit, published in 1557. It is used to refer to the eighth power of a number.

The word seems to have come from the Latin root censo which is related to the word census, and the assessment of value. It took on a gereral meaning of property, and was used as a translation of the Arabic word mal in many translations of Arabic math books into Latin. This was all before the common use of letters for variables, and it was common for the Arabic mathematicians to use mal for a squared variable. Where we would write 3x2, they would write 3 mal. When this was translated into Latin, it would appear as 3 censo. It seems that a "Z" version of this made its way for a while in English and German, as zenzic. When Recorde needed to write a fourth power, he simply sqared the square, zenizenzic which is the English translation of censo di censo as used by Fibonaci in his Liber Abaci. What would be more expected for the eighth power, the square of a square of a square, than to use zenzizenzizenzic. There seems to be no current residue of the German or English form of the words, but many versions of the Latin censo still remain.