Saturday 16 May 2020

Volume S


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

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

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

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

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

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

Second When the "pars minuta" (see minute) needed additional dividing, they created a "second little part", the pars secundus

dividing each 1/60th of a minute into 60 additional divisions each. Later the name was shortened down to seconds, and then transferred along with the base 60 angle system into time keeping.

Second is also the ordinal name for the one following the first, or initial object counted.

Chaucer's Treatise on the Astrolabe, 1400, described the arc system of 60 minutes divided into 60 seconds. The first appearance in the OED for a minute of time, was in a 1588 book on "Cathechism or Short Inst." The first citation mentioning the second-hand of a clock is in 1759 in the Philosophical Transactions. Second as an ordinal adjective was used much earlier in 1297.

Series - See Summation

Sequence is from the Latin root sequi, to follow. In mathematics it refers to a series of terms in order.
The root is the source of such modern words as consequence (the results that follow an event), suitor (one who follows a lover), and second (the one after the first). The mathematical sequence was a late-comer to the language, first appearing in 1910 in the Algebra entry of the Encyclopedia Brittanica; (with the exception of a notation of "sequence of natural numbers in order", by Sylvester in 1882) although entry in an encyclopedia makes me think it had been used in classrooms and between professionals for some time. Sure enough I found the use of sequence as Sylvester had used it several more times, and in 1890 H. B. Fine describes the "sequence of rationals," for the decimal sequence defining an irrational number.

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

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

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

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

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

In statistics, a distribution is skew (or skewed) if it is not symmetrical about its mean. The earliest citation in the OED for the statistical meaning of Skew is in the Philosophical Transactions in 1895.

Slope seems to have come from an old English term like aslupen, slipping away. It seems uniquely English and not anything like words in any other language, as far as I could find.

Why M for Slope? Interestingly, m for slope has led to more mis-history speculation in classrooms than any other topic, with the possible exception of the life of that Great American-Indian Mathematician, Chief Soh Cah Toa. Fortunately I have collected a lot of information and the best answer I can think of, it just happened. What I've learned is at Why M for Slope.

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

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

Soliton is a shortened word to represent the solution of a differential equation representing a stationary wave. The term seems to date to about 1965 and was invented by Martin Kruskal of Rutgers Univ and printed in the Physical Review of Letters.

Here is a quote from the site of Kanehisa Takasaki about the naature of solitons:

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

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

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

Square is derived from the Latin phrase exquadra, like a quadratic. Over time the term was contracted into its present form, and then came to mean the regular quadrilateral. The word is also the origin of the military unit called a squadron, from an early form of fighting in squares called a Phalanx by the Greeks. The French term esquire is also from the same source.

The geometric term was first used in English by Robert Recorde in his 1551 Pathway to Knowledge, but around the same time there were still uses of square for what we would not call a rectangle, and in the same book Recorde writes of a "... whose use commeth often in geometry, and is called a squire, is made of two long squares ioyned together...". Chaucer mentions the carpenters square in his Treatise on the Astrolabe in 1400.

The square of a number appears in Record's Whetstone of Witte in 1557, "Twoo multiplications doe make a cubike nomber. Likewaies .3. multiplications doe giue a square of squares."

The first mention of the statistical method of least squares is recorded in the Philosophical Magazine in 1812, "The principle of the small squares."

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

Bea Lumpkin, in a posting to the Historia Mathematica Newsgroup, has indicated that some early Egyptian records may be the oldest statistical tables in existence.

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

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

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

I was ably informed about stanines by Lee Creighton of the Statistical Instruments Division of SAS Institute Inc. in a posting to the Ap Statistics Discussion Group.

Basically, the transformation from raw scored to stanine scores is pretty simple: (1) rank the sores from lowest to highest (2) assign the lowest four percent a stanine score of 1, the next 7 percent the staninine of 2, etc according to the following table: 4..7..12..17..20..17..12..7..4.. to Stanines 1...2...3..... etc. So, they are assigned on a scale from 1 to 9, but there are not necessarily the same percentage in each "bin". Someone in the 88th percentile wold come in just a the top of the range for a stanine score of 7.

The reason this scale was developed was primarily to convert scores to a single digit number--a considerable asset when Hollerith punch cards were _de riguer_ in the computer industry. The scores could be coded in a single column on one of these cards.

Student's t The story of the name for this statistical distribution and test is almost legend, and some version of hte tale is remembered by Intro Stats students long after they forgot the purpose of the t-test. A dialogue between Randy Schwartz and James Landau in the Historia Matematica discussion group gives both one of the folk versions, and a brief history. Randy Schwartz writes, "the distribution now known st 'Student's t distribution' was first discussed in print in a paper by William S Gossett in the journal Biometrika in 1908, published under the pseudonym 'Student.' The paper solved a problem from the Guinness Brewery concerning how large a sample of people should be used in its tastings of beer. Apparently Gossett was embarrassed to be working on a problem stemming from the liquor industry, thus the pseudonym."

James Landau responds "The story of 'Student' has been told so many times that it has become folklore, and like all folklore variant versions exist until it is difficult to determine which is the original. The variant you tell is one I had not encountered before. Gossett was an employee of Guinness Brewery (a brew-master, I believe) who went to study statistics under Karl Pearson. Gossett eventually discovered a result that he published in Biometrika under the pen name of 'Student'. Why did he choose to use a pseudonym? Here is where the folklore kicks in . The most common story is that Guinness wanted to keep it secret that they were using statistics in their business and ordered Gossett not to reveal his identity.

In any event, Gossett published all his statistical work as 'Student', even though his identity became well known. Why he continued to use the pseudonym is not part of the folklore, and I have never heard a plausible story. Perhaps it is because he became famous as 'Student' and did not want to have to re-established his professional reputation under his real name. Perhaps he liked the notoriety.

Ok, now we know why he used 'Student', but why t? "

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

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

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

Crystal's books into the fifties continued to use the inverted exclamation symbol and the National Academy of Sciences used the symbol in 1967.

The earliest used of the !n symbol I have ever found is from 1958, In the MAA questions section:

This was obviously not an instant hit, as I received several comments like the following after a post in 2009.

"  I have several books on my shelf, none of which use !n notation.

- Matoušek and Nešetřil, 1998
- Niven, 1965. I teach from this book.

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

- d_m
Goulden and Jackson, 1982.

The Niven book is his well known Mathematics of Choice, and he uses the symbol D(n) . In 1997 Robert Dickau used\$D_n\$ for derangements, another common name for subfactorials.    John Baez used !n  in 2003 without indicating that it was an uncommon symbol.

The formula for subfactorial, also called derangements of a set, is given by \$!n = n!( 1- \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!}..... \frac{1}{n!} )\$,  The quick approximation is !n = n!/e.

Subtract joins two easy to understand roots, the sub which commonly means under or below, and the tract from words like tractor and traction meaning to pull or carry away. Subtraction then, means to carry away the bottom part. The "-" symbol for subtraction was first used as markings on barrels to indicate those that were under-filled. Around the 1500's it began to be used as an operational symbol and it became common in English after it was used by Robert Recorde in the Whetsotne of Witte in 1557.

In a subtraction relationship, a - b = c, all three numbers have a special name. The first number, a, is called the minuend, from the same root as minus and literally means that which is to be made smaller. The part to be removed, b,is called the subtrahend and means that which is to be pulled from below.  The answer, c, is most often called the difference or result, but in many statistical uses it is also called the residue, or residual, that which remains.  is statistical uses it may also be called a deviation.

If you wish to know more about the history of subtraction, check out Subtraction, Borrowing, Carrying, and other Naughty Words, A Brief History Subtractions earliest usage in English, according to the OED, is from Steele's Craft of Nombryng in 1425.

Sum The root of sum is the Latin summus for highest. The root itself seems to be a contraction of the word supremus and is closely related to supra from which we get super, supreme and other superlatives (pun intended). Sum is drawn from the practice of the Greeks and Romans of adding columns of numbers from bottom to top and writing the result at the top. The answer then became the summus, the number at the top. In a similar way we get the word summit for the top of a hill or mountain.

It is interesting to me, that the earliest reference to the word in Jeff Miller's web site is a reference to Nicholas Chuqet in 1488 in which the word is spelled "some". The root of this homonym of sum if from the root sem which related to the quantity one.

Summation\Series – The act of adding finitely many (or infinitely) many terms together. The summation symbol (Σ) was first introduced in 1755 by Euler. In general, the index is “i” and travels from “m” to “n”. A well known (and possible false) story of Gauss (1777 – 1855) was when he was in primary school. One day Gauss’ teacher asked the students to sum up the integers from 1 to 100, with the idea that it would take them a long time to finish. However, after a few seconds of thought, 8-year-old Gauss wrote down 5050. The teacher was flabbergasted but Gauss mentioned it was quite simple. 1+100 = 101, 2+99 = 101, 3+98 = 101, and so on until 50+51 = 101. There were 50 of these additions and so 50*101 = 5050.

Infinite sums were trickier to understand and for a long time, adding infinitely many things appeared to be paradoxical. It wasn’t until the notion of a “limit” came in the 19th century and then it was better understood. Zeno’s paradox of “Achilles and the tortoise” describes the counterintuitive behavior of infinite sums. Achilles runs after the tortoise but once Achilles reaches the tortoise at time t1, the tortoise is at position 2. One Achilles reaches position 2 in time t2, the tortoise is now at position 3, and so on. Zeno concluded that Achilles will never reach the tortoise in finite time. However, this is false because the sum of these infinite time intervals can be a finite number, meaning Achilles can catch up to the tortoise.

Greek mathematician Archimedes was the first to produce the summation of an infinite series that uses the method of exhaustion. He was calculating the area under the arc of a parabola and in turn, it gave a very good approximation of π. Later, mathematicians from Kerala, India studied infinite series in 1350 CE. In the 17th Century, James Gregory worked with infinite series and published several Maclaurin series. In 1750, Brook Taylor extended the notion of Maclaurin series and hence formed Taylor series. Leonhard Euler in the 18th century developed the theory of q-series and hypergeometric series.

The validity of infinite series didn’t begin until Gauss in the early 19th century. More on this in “Convergence”.

Fourier series was also studied around this time in 1807, where a function could be written as an infinite series of sines and cosines. His work was later published in 1822 but he did not discuss convergence. *Derek Orr

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

Similar mathematical words to see are complement, and explement.

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

Symmetry is from the Greek roots sum + metros. The prefix refers to things which are alike, and metros is the Greek word for measure. Metros is the root of the word Geometry also.

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

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