Math Words, pg 3

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Astroid The Astroid is the path of a point on a circle rolling inside another circle with a radius four times as large. The path is sometimes called a hypocycloid of four cusps because it is rolling under (hypo) the larger circle.
The word astroid, which seems to be a different spelling of asteroid, comes from the Greek aster for Star. It seems that the name was applied as late as the 1800's.
The parametric equation for an astroid with a circle of radius r rolling inside a circle of radius 4r is given by x(t)= 3r cos(t)+ r cos(3t), and y(t)= 3r sin(t)-r sin(3t). The figure can also be drawn with the equation x2/3 + y2/3 = R2/3 where R is the radius of the fixed circle, and R/4 is the radius of the rolling circle. This formula was known to Liebniz in 1715. The area of the astroid is 3/2 times the area of the rolling circle.

It seems that the name astroid was not used until around 1836. Before that, and even afterward, it has gone by many names including tetracuspid (four cusps), cubocycloid, and paracycle. The length of the astroid is 6R and its area is 3pR2/8. Additional information and images may be found at Xuh Lee's web page for this curve.

Noon is NOT related to the number twelve, but to the number nine. It is derived from the Latin nona (ninth) and originally refered to the ninth hour after sunrise, which was closer to the present 3PM.

Obtuse is from the Latin formation ob (against) + tundere (to beat) and literally means to beat against. An object thus beaten becomes blunt, dull, or rounded, as in the application to an obtuse angle, one having more than 90o but less than 180o. A trianlge with an obtuse angle is called an obtuse triangle.

You may (very rarely) encounter the name amblygon used for an obtuse triangle. It is also sometimes spelled ambligon. Amblygon is drawn from the Greek roots for blunt amblu preceeding the root gon for angle. The use in English probably first occured in Billingsley's translation of Euclid in 1570, although he wrote "amblygonum". Billingsley's translation was the first translation of Euclid's "Elements" in English. It was published at London in 1570 with the title The Elements of Geometric of the most auncient Philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, Citizen of London.

Parallelepiped This word for a solid made by intersecting pairs of parallel planes forming six faces that are each parallelograms is rapidly becoming obsolete, although no good word has emerged to replace it. A rectangular or orthogonal parallelepiped is the shape of a room or a shoe box. The word is condensed from the Greek word parallelepipedon for the same shape. The roots are para (beside) + allel (other) + epi (on) and pedon (ground). Parallelepipedon was the word used by Billingsley in his 1570 translation of Euclid, the first known use of the word in English. According to John Conway, this was the common term in use until around 1870 to 1900 when it gave way to parallelepiped; although the OED lists its use by John Playfair as early as 1812. A posting from John Albree of Auburn University cited an earlier use. [In Charles Hutton's *Dictionary* (volume 2, 1795, p.199), the terms "parallelopiped" and "parallelopipedon" are presented equally, and he remarks that such a polyhedron "is only a particular species" of a prism.]

This newer term now seems headed for demise due to changes in school curriculum and the reduced coverage of solid geometry, although one correspondent suggests that "parallelepipedo" would be known by most Spanish students. I was somewhat surprised to find that parallelepiped is present in my computer spell check, which I find often omits technical terms.

The word is pronounced with the accent on the epi syllable. The para root is common in math words and is related to other words like parlor, paragraph, and parable. Allel became our alter, for other, and gives us alternate and alternative. The epi root shows up in epidermis (on the skin), epitaph (over the grave), epicycle (on the circle), and epidemic (on the people). Pedon is from ped for foot, and was also generalized for plane.

Plus (as in two plus four) comes from the early Latin word meaning "more". Extensions of the root were used for related ideas like fill, full, and abundant. Common words of today related to the same root are plenty, complement (meaning complete or fill, as in complementary angles, the amount needed to complete a right angle), plural (more than one), and surplus (abundant, more than enough). The word is closely related to the Greek root poly for many. [see polygon]The verbal use of the words plus and minus date back to the Romans when the terms were used much as we use the English words more and less. Most printed works before the 15th century used the letters "p" and "m" for additon or subtraction.

The "+" and "-" symbols were first used to indicate excess and deficiency and only later became an operational symbol. The first known occurance of the symbols in a printed arithmetic was in the 1489 Rechnung uff allen Kauffmanschafften by Johann Widman, one of the earliest of the German Cossists. The "+" and "-" symbols became commonplace in English after the use by Robert Recorde in The Whetstone of Wit around 1550.

The + sign was actually used as a symbol of negation in the Arabic document called the Bakhshali manuscript, after the village in Pakistan where it was found. The document dates to the period 200-400 AD.

Polygon is from the Greek roots poli (many) and gonus (knees) and, interprets literally as many angled. The relation between knee and angle relates to the flexed position of the knee. Poly appears in many words, and gonus remains mainly in its Latin derivative, genus, from which we get genuflect (to bend the knee). According to John Conway, terms like gnaw are from the same root, perhaps because the line of the jaw forms the same shape as the bent knee. Another current word with the "gen" connection to knee is genuine. In the early days of Rome, a father legally claimed his newborn by placing the child on his knee.

Polyhedron is the name for a solid with "many faces", The joining of poli (many) with hedros( face or seat). The hedros originally referred to any flat surface. Later, in the Latin, hedra was used for a chair, flat places are good to sit on, and the root is preserved in our words for cathedra (the Bishop's chair) and the Cathedral where it is kept. An excellent source of information and graphics about polyhedra is the Virtual Polyhedra page of George W. Hart.

Euclid used the term, in the words of Ken Pledger, "Euclid smuggles it in without a proper definition,.... (EUCLID) XII.17 uses "polyhedron" as a descriptive expression for a solid with many faces, then more or less adopts it as a technical term." He adds that, "the first English use in Billingsley's translation of Euclid (1570). .... Early in the proof (folio 377) Billingsley amplifies it to '...a Polyhedron, or a solide of many sides,'...".

Polynomial, a general term for algebraic terms joined by plus or minus signs, also uses the "poly" root for many. The word is a hybrid of Greek and Latin roots. Polynomial means "many names" and is an extension of binomial, literally two names, and monomial which means "one name". Nomen is a Latin root related to "name" and is also found in nominate (to name a candidate) , misnomer (wrongly named), nomenclature (the names of things in a discipline or science) and of course, the French nom de Plume or pen name.

In Florian Cajori's Hisotry of Mathematics he says that Francois Vieta coined the term, and the OED credits the first use in English to the Arithmetic of Samuel Jeake in 1674.

Binomial is, according to the Miriam Webster 0n-line dictionary, from "New Latin binomium, from Medieval Latin, neuter of binomius having two names, alteration of Latin binominis, from bi- + nomin-, nomen name ". Jeff Miller's web site on the first use of mathematical terms says the first English use of the term was in The Whetstone of Witte in 1577 by Robert Recorde. He quotes from the OED "The nombers that be compound with + be called Bimedialles... If their partes be of 2 denominations, then thei named Binomialles properly. Howbeit many vse to call Binomialles all compounde nombers that have +". Miller credits the first English use of monomial to "a 1706 dictionary".

Positive comes from the root word posit which means to place or set. This probably refers to the ancient method of counting with markers and counting tables. Positive numbers could be set out one by one.

Prime is from the Latin word for first, primus and related to the Greek protos. Prime numbers are thus the first, or most basic, of numbers in a multiplicative sense. The mathematical meaning is essentially unchanged from the ancient times and was apparently in use at the time of Pythagoras.

In 1960 a bone believed to date back to 9000 BC was discovered near Lake Edward in the Mountains of Zaire. It contains lists of scratch marks indicating the prime numbers and is believed to be the oldest record of prime numbers in still existing. You can see a picture, and find more detail about the Ishango Bone at this link.

Today it is common to say that two is a prime number and that one is not. This is justified on grounds as simple as the prime factorization theorem and as complex as the abstract relations of rings. Mathematicians have at times treated one as a prime. A popular American textbook, Standard Arithmetic by Milne, printed in 1892 defined one as a prime number. Earlier, the math historian Heath points out, there was some controversy as to whether 2 was prime, or even a number at all [many of the ancients divided quantity into three types, units (1), duals (2), and numeros or number (>2); zero of course did not exist in most ancient cultures]. Aristotle seems to have been the first to regard 2 as a prime number. Euler seems to have clearly said that one was NOT a prime. This link to the Math History web site at St Andrews University in Scotland gives excellent information about the history of prime numbers.
A link devoted to prime numbers is The Prime Pages

Other words drawn from the same root include primary [first in rank or order], Primitive [first of its kind], prima donna [literally, first lady] and Principal [first in power, from the derivative form princeps for ruler]. The word Prince, again from princeps, originally referred to a king, but later the French applied the term to any male member of the royal family. The Greek proto is the root for such words as protocol (literally, first sheet) and protozoa (first living thing).

Quadratic is the Latin root for "to make square". The word square was itself derived from this root [see square ].

Random comes to us from the old French root Randir, to gallop. Perhaps the idea is that, in full gallop, the horse or rider has abandoned control.

John Von Neumann once quipped, "Anyone who considers arithmetical methods of produing random digits is, of course, in state of sin."

Some folks have no fear of living in Von Neumann's proclaimed state of sin, and spend their time finding methods to generate random, or at least pseudo-random numbers. Here is a good site to learn more.

Rotate / Rotation A rotation is a rigid transformation in which every point of a set, or object, is moved along a circular arc centered on a specific point (the center of rotation) or axis (the axis of rotation). The word comes from the Latin rota, for wheel. The more ancient root ret related to running or rolling. It is alive today in some unexpected places. Rodeo, a sport that emerged during "round up", when the cattle were gathered and shipped is derived from the Spanish word for surround. A rotunda is a building or room that is round, usually with a domed roof, and if someone calls you rotund, they mean your shape is round. The French word roulette, for a cycloid, also comes from the same root.

Secant is from the Latin root Secare, to cut. It is a proper name for a segment that cuts through the circle. The word was introduced by Thomas Fincke in 1583 in Latin.

Second When the pars minutia [see Minute of an arc needed to be divided into even smaller parts, the 1/60 part of 1/60 of a degree needed a name also. Since it was the second small part, what could be more appropriate than pars minuta secundus. Later the term was shortened down to seconds and generalized to units of time which preserved the base 60 system. Secundus is from the Latin root sequi [see Sequence] for "to follow", and thus second was the natural term for the ordinal following First

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 words sequence and series are often confused by students (and teachers), and for good reason. Both words have similar meanings in everyday usage. Series is from the Latin serere which meant "to join". The root is found in words like sermon (joining together words or phrases), and insert.

Today in mathematics we define a sequence as a group of terms in a row such as 2,4,6,8. A series is the sum of such a sequence, 2+4+6+8. In simple words, a series is the sum of a sequence.

If a sequence is infinite, then we can construct another sequence called the sequence of partial sums. For example, using the sequence of even terms above, the sequence of partial sums would be 2, 2+4, 2+4+6, etc. When the sequence of partial sums of an infinite sequence has a limit, then we say the sum of the infinite series is the limit of the sequence of partial sums.

Sine The name sine came 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 a Sanskrit word. The actual first use seems to be a source of some disagreement. Here are the details according to the "Earliest Known Uses of Some of the Words of Mathematics " web site of Jeff Miller

Aryabhata the Elder (476-550) used the word jya for sine in Aryabhatiya, which was finished in 499.

According to Cajori (1906), the Latin term sinus was introduced in a translation of the astronomy of Al Battani by Plato of Tivoli (or Plato Tiburtinus).

According to some sources, sinus first appears in Latin in a translation of the Algebra of al-Khowarizmi by Gherard of Cremona (1114-1187). For example, Eves (page 177) writes:
The origin of the word sine is curious. Aryabhata called in ardha-jya ("half-chord") and also jya-ardha ("chord-half"), and then abbreviated the term by simply using jya ("chord").

From jya the Arabs phonetically derived jiba, which, following Arabian practice of omitting vowels, was written as jb. Now jiba, aside from its technical significance, is a meaningless word in Arabic. Later writers, coming across jb as an abbreviation for the meaningless jiba, substituted jaib instead, which contains the same letters and is a good Arabic word meaning "cove" or "bay." Still later, Gherardo of Cremona (ca. 1150), when he made his translations from the Arabic, replaced the Arabian jaib by its Latin equivalent, sinus, whence came our present word sine. However, Boyer (page 278) places the first appearance of sinus in a translation of 1145. He writes:
It was Robert of Chester's translation from the Arabic that resulted in our word "sine." The Hindus had given the name jiva to the half chord in trigonometry, and the Arabs had taken this over as jiba. In the Arabic language there is also a word jaib meaning "bay" or "inlet." When Robert of Chester came to translate the technical word jiba, he seems to have confused this with the word jaib (perhaps because vowels were omitted); hence he used the word sinus, the Latin word for "bay" or "inlet." Sometimes the more specific phrase sinus rectus, or "vertical sine," was used; hence the phrase sinus versus, or our "versed sine," was applied to the "sagitta," or the "sine turned on its side."

Smith (vol. 1, page 202) writes that the Latin sinus "was probably first used in Robert of Chester's revision of the tables of al-Khowarizmi."

Fibonacci used the term sinus rectus arcus.

Regiomontanus (1436-1476) used sinus, sinus rectus, and sinus versus in De triangulis omnimodis (On triangles of all kinds; Nuremberg, 1533) [James A. Landau].

Copernicus and Rheticus did not use the term sine (DSB).

The earliest known use of sine in English is by Thomas Fale in 1593:

It was when Leonardo de Fibonacci used the term in his writing, it became permanent. According to Carl Boyar'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 Leonid Euler [pronounced Oiler] that it became common to use the sine as a ratio.

Slope is derived from the Latin root slupan for slip.[Ooops, Charles Wells, Emeritus Professor from Case Western Reserve Univ. and author of the Blog "Gyre and Gimble" wrote to tell me that "slupan" is Anglo-Saxon, not Latin... Apparently "lubricate" and "slope" both come from the IndoEuropean root sleubh.] The relation seems to be to the level or ground slipping away as you go forward. The root is also the progenitor of sleeve (the arm slips into it) and, by dropping the s in front we get lubricate and lubricious (a word describing a person who is "slick", or even "slimy").

Many variations of where the idea of M for slope originated seem to be mostly myth. One of the most common is that the letter was used by Descarte because it was the first letter of some French word or another that related. In a recent post to the AP Stats discussion list, Hector Hirigoyen shared the following story:
I was told by Mary Dolciani herself, that the SMSG group "decided "to use y=mx+b because of the French (Descartes, I presume)-"montant"; I found it strange because the "logical word" would be "pente"(which is slope (and the standard term in Spanish is pendiente, which matches this). However, several years ago, while visiting a French high school, I noticed the teacher used y=sx+b. I inquired, and she said because of the "American" word "slope." I believe they are using ax+b for the most part these days.
. Here are several other clips from postings about the topic on a discussion group about math history.
In his "Earliest Uses of Symbols from Geometry" web page, ... Jeff Miller gathered the following information: Slope. The earliest known use of m for slope is an 1844 British text by Matthew O'Brien entitled _A Treatise on Plane Co-Ordinate Geometry_ [V. Frederick Rickey]. George Salmon (1819-1904), an Irish mathematician, used y = mx + b in his _A Treatise on Conic Sections_, which was published in several editions beginning in 1848. Salmon referred in several places to O'Brien's Conic Sections and it may be that he adopted O'Brien's notation.

According to Erland Gadde, in Swedish textbooks the equation is usually written as y = kx + m. He writes that the technical Swedish word for "slope" is "riktningskoefficient", which literally means "direction coefficient," and he supposes k comes from "koefficient."
According to Dick Klingens, in the Netherlands the equation is usually written as y = ax + b or px + q or mx + n. He writes that the Dutch word for "slope" is "richtingscoefficient", which also means "direction coefficient." In Austria k is used for the slope, and d for the y-intercept. In Uruguay the equation is usually written as y = ax + b or y = mx + n, and the "slope" is called "pendiente", coeficiente angular", or "parametro de direccion".

It is not known why the letter m was chosen for slope; the choice may have been arbitrary. John Conway has suggested m could stand for "modulus of slope." One high school algebra textbook says the reason for m is unknown, but remarks that it is interesting that the French word for "to climb" is monter. However, there is no evidence to make any such connection. Descartes, who was French, did not use m. In _Mathematical Circles Revisited_ (1971) mathematics historian Howard W. Eves suggests "it just happened."

Jeff Miller's web site cited above now has updated the earliest use of m for slope.

The earliest known use of m for slope appears in Vincenzo Riccati?s memoir De methodo Hermanni ad locos geometricos resolvendos, which is chapter XII of the first part of his book Vincentii Riccati Opusculorum ad res Physica, & Mathematicas pertinentium (1757):
Propositio prima. Aequationes primi gradus construere. Ut Hermanni methodo utamor, danda est aequationi hujusmodi forma y = mx + n, quod semper fieri posse certum est. (p. 151)
The reference is to the Swiss mathematician Jacob Hermann (1678-1733). This use of m was found by Dr. Sandro Caparrini of the Department of Mathematics at the University of Torino.

Square is derived from the Latin phrase Exquadrare, like a quadratic. Over time the term was contracted into its present form, and came to mean the regular quadrilateral. The word is also the root of the military term squadron, which originally meant a fighting square from the early practice of fighting in square formations.

In an 1847 paper to the Philosophical Magazine, Augustus De Morgan says that, "the original English sense of the word square applies to an angle, not a figure... and to this day the carpenter's right angle is called a square." He also includes that it was common in the sixteenth century to use "square" when one considers the figure, but to write "squire" when one refers to the instrument. The etymology seems to have no relation to the use of "squire" for one who attends a knight, which is derived from Esquire.

Stanine & Hollerith Cards 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 form of "Standard of Nine".

Here is a little more information about stanines from a posting by Lee Creighton of the Statistical Instruments Division of SAS Institute Inc. to the AP statistics discussion group.

Basically, the transformation from raw scores to stanine scores is pretty simple: (1)rank the scores from lowest to highest (2)assign the lowest four percent a stanine score of 1, the next 7 percent the stanine of 2, etc, according to the following table:
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 would come in just at 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 in the era when Hollerith punch cards were _de riguer_ in the computer industry. The score of a person could be coded in a single column on one of these cards.

Hollerith Cards are named for Herman Hollerith (1860-1929), the founder of Tabulating Machine Company, which was the predecessor of IBM. Hollerith's idea was founded on the Jacquard loom, an automated weaving process using heavy cards for controlling weaving looms. You can find more about his life and invention at this link

Well before Herman Hollerith copied Jacquard's card method, an earlier computing pioneer, Charles Babbage, had use a similar approach in his Analytial Engine. There are some nice pictures about Jacquard's loom method at the University of Houston.

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, literally means to carry away the bottom part. The "-" symbol for subtraction was first used as markings on barrels to indicate those that were underfilled. 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 Whetstone 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 applied statistical uses it is also called the residue, or residual, that which remains. In statistical uses it may also be called a deviation.

The term subduction was often used in older English books up until about 1800. John Wallis uses the term in his "Treatise on Arithmetic", 1685 in describing subtraction... "Supposing a man to have advanced or moved forward, (from A to B,) 5 yards; and then to retreat (from B to C) 2 yards: If it be asked, how much he had advanced (upon the whole march) when at C? Or how many yards he is now forwarder than when he was at A? I find (by subducting 2 from 5,) that he is advanced 3 yards. Samuel Johnson's 1768 dictionary defines both terms, but includes "substraction" as part of the definition of subduction.

In the same dictionary, Johnson defines both subtract and substract, but for subtraction, the reader is referred to "see substraction" so I assume that was the more common term.


The method of subtraction commonly called Borrowing or decomposition seems to go back at least to the 1200s. In The Art of Nombrynge by John of Hollywood (Sacrobosco) subtraction was taught "entirley like the method of today, 'borrowing' and all." [E. R. Slight from "The Craft of Nombrynge, Mathematics Teacher, Oct. 1939]. The word, "borrow" may not have been used until around 1600 as the earliest listing in the OED associated with subtraction is "[1594 BLUNDEVIL Exerc. I. (ed. 7) 91] Take 6 out of nothing, which will not bee, wherefore you must borrow 60." The borrowing of 60 suggests the exercise may have been about time. Here is a link with an image of a page from the Arithmetic of John Ayres in 1695 in which the word borrow is used in subtraction, although the method is more like what Ross and Pratt-Cotter (below) call the "equal additions" method.

Susan Ross and Mary Pratt-Cotter [Subtraction in the United States: An Historical Perspective, from The Mathematics Educator] show that prior to about 1940 in the US there were three common approaches to subtraction in arithmetic texts. The Borrow method, a method they call "equal additions" which also seems to date back to the 15th century (and is probably a logic based alternative to the borrowing approach), and a method of subtracting by adding on from the subtrahend, which is sometimes called the Austrian algorithm. (images of all three types can be found in the link above to Ross and Pratt-Cotter's document)

The research by Ross and Pratt-Cotter indicated that before 1937 there were few illustrations in American textbooks that show any physical "marking through or numbers being rewritten". Their work states that almost overnight, after a study by William Brownell, "most textbooks used the decomposition (borrow) method for describing borrowing in subtraction, and the use of the crutch described by Brownell became very popular. Today this method of subtraction is used in most textbooks that teach subtraction." The study also states,

Only one example was found, from a text published in 1857(Ray's Practical Arithmetic), where markings were used to keep track of the renaming process. This was done in only one problem in the text, with all other problems worked without any markings. Brownell was not aware, however, of any textbook employing this technique.
This statement, which I assume to be true, and the existence of a clear example of the "borrow" with markings in an 1898 copy of Gill's Oxford and Cambridge Practical Arithmetic , shown at right, make me suspect that the borrowing crutch first appeared in England and then made it's way to America. {if you have knowledge of any earlier appearance in textbooks in ANY country, please write)

I posted a request for information about texts or other sources of the use of the "crutch" and received the following from Ralph Raimi of the University of Rochester:

" I entered kindergarten in 1929, ten years before the Brownell article, and while I don't remember distinctly just which grade introduced me to the "borrowing" scheme for subtractions, it was surely in my schoolwork by 1933, probably 1932, in the Ferry School, Detroit, Michigan; and the crutch pictured in Ballew's example was standard procedure for us. We did (and every grocer did, too) the corresponding thing when adding a column of figures, as a grocer would do on his brown paper bag, listing and summing the item prices in a vertical column on the side of the bag before filling it with the items themselves. If the sum of the right-hand column (in cents) was, say 126, he would enter the 6 below the line, as part of the ultimate sum, and enter the 2 above the top of the tens column and the 1 above the top of the hundreds column, etc. In practical commercial sums and differences the place of the decimal point was implicit (dollars and cents) and disregarded in grocery stores until the end. In school we were careful to make vertical things *beautifully* vertical, and to preserve the decimal points throughout. Of course this made no difference, except to our understanding. Later, probably by the fifth grade, we were encouraged to *imagine* the crutch in a subtraction problem; writing it down was a sign of weakness, akin to moving the lips while reading. I'm sorry I have no documents from that era in my education, but I do know that my work was always supervised by my older brother, who was five years older than me, and he never showed any surprise at anything I did in arithmetic, so I imagine that by 1925 or 1926 he had also been learning to write subtractions (also in Ferry School) in the same way. Some years ago I was studying a facet of the history of the Detroit Central High School in the period 1898 - 1950 (it was debating clubs that interested me, not math lessons, but no matter), and I found that the Detroit Public Library had several archives of random materials of no particular importance that they had filed under the names of certain (dead)teachers, but cross-referenced so as to make it easy to know what they were. How those particular teachers, or their heirs, got these memorabilia into the library I can't imagine, for the files contained only old school newspapers, club meeting minutes, letters and so on, and the teachers themselves had not been notable; but I believe that if you go to any big city public library and ask for archives of local school teachers of a certain era (1900-1940, say), you might find a sheet of homework or a set of exams or answers, written out in that teacher's hand, or a student's, in some one of them. That might tell you more about the arithmetic style of the time than even the popular textbooks would. "
I have not yet followed up on Mr Raimi's suggestion in my brief visits to the US, but if anyone else finds information on the use of this "crutch" before 1937, I would appreciate a note.

It may be that the use of the "crutch" markings were commonly taught, but not found in books because disagreement about whether they should be used. In The Teaching of Arithmetic by Paul Klapper (1934), he gives an example both with and without the markings, and calls the form without the markings the "recommended form --- no 'crutches' should be permitted." The very use of the word crutch seems to confirm Professor Raimi's assertion that the marks were viewed as a weakness to be avoided or overcome.

However in the article Klapper states that, "This method is the favorite of many teachers who hold that it is very simple because it can be demonstrated objectively with dimes and cents and that it can be habituated quickly. Others are opposed to it because it requires a second set of number facts -- the subtraction combinations." The evidence seems to suggest that the use of a the borrow markings were common in America well before the publication of Brownell, but it may not have been common in textbooks because, as stated by both Professor Raimi and the Klapper book, it was viewed as a weakness.

I have also found another early use of supplemental marking of a problem. This example, using the equal additions method, comes from a 1873 copy of Charles S Venable's A Practical Arithmetic. Here is a copy of the paragraph from page 25

SURD The original 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 reference is to a root that could not be expressed (spoken) as a rational number. It has been 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.

Randy K Schwartz of Schoolcraft College in Michigan (USA) sent a nice explanation to a discussion list

Al-Khowarizmi (Baghdad c. 825) referred to rational and irrational numbers as 'audible' and 'inaudible', respectively (see, for example, D. E. Smith, History of Mathematics Vol. 2, 1925, page 252). In translating the Greek term alogos into Arabic, al-Khowarizmi chose the expression jathr asamm, literally "deaf root." His choice emphasized the earlier Greek connotations of alogos "without words," "inexpressible," instead of the later ones "without reason," "without ratio." Then, when the works of al-Khowarizmi and his fellow Arabs were translated into Latin, the Latin word surdus ("deaf") was called into play. Smith states that the first known European to adopt this terminology was Gherardo of Cremona (c. 1150). He used the Latin translation surde (a variant of surdus: deaf, silent, stupid). Smith also notes that Fibonacci (1202) adopted the same term to refer to a number that has no root [presumably a negative number]. The Merriam-Webster Dictionary cites the first English use of "surd" for irrational root in 1557, without giving a specific reference.

Jeff Miller's site on the first use of some math words adds some details:

The Arabic translators in the ninth century translated the Greek rhetos (rational) by the Arabic muntaq (made to speak) and the Greek alogos (irrational) by the Arabic asamm (deaf, dumb). See e. g. W. Thomson, G. Junge, The Commentary of Pappus on Book X of Euclid's Elements, Cambridge: Harvard University Press, 1930 [Jan Hogendijk].

This was translated as surdus ("deaf" or "mute") in Latin.

As far as is known, the first known European to adopt this terminology was Gherardo of Cremona (c. 1150).

Fibonacci (1202) adopted the same term to refer to a number that has no root, according to Smith.

Surd is found in English in Robert Recorde's The Pathwaie to Knowledge (1551): "Quantitees partly rationall, and partly surde" (OED2).

According to Smith (vol. 2, page 252), there has never been a general agreement on what constitutes a surd. It is admitted that a number like sqrt 2 is a surd, but there have been prominent writers who have not included sqrt 6, since it is equal to sqrt 2 X sqrt 3. Smith also called the word surd "unnecessary and ill-defined" in his Teaching of Elementary Mathematics (1900).

G. Chrystal in Algebra, 2nd ed. (1889) says that "...a surd number is the incommensurable root of a commensurable number," and says that sqrt e is not a surd, nor is sqrt (1 + sqrt 2).

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. Long ago the word symmetric and commensurable meant the same thing, measureable by integral measures of the same unit. Dionysus of Alexandria used the term to refer to objects in which two sides off the object "measure the same" and probably was the first to use it with todays meaning. According to the Oxford English Dict., the earliest English example of "symmetry" with respect to an axis of symmetry is 1823.

There are two major types 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, and that means that they will have a reflective image which is also chiral. The capital letter A has a line of symmetry, lower case b and d do not. The reflected image of d is b. Two chiral objects which are reflected images of each other are called enantiomorphic, from the Greek words for opposite, anti'os, and body, morph. Chiral comes from the Latin root, chiro through the Greek word for hand, cheir, which also gives us chiropractor and chiromancy (a fancy word for palm reading).

Teachers and students may learn more about The Four Types of Symmetry in the Plane at a site created by Susan Addington and Suzanne Alejandre. The Math Forum also hosts a web page on Symmetry and Pattern, The Art of Oriental Carpets that includes additional Educational resources.

A recent Plus Maths article on the implications of symmetry, and handedness, in chemistry is here .

In a recent discussion on a news list John H Conway and Malcolm Browne shared the following notes about the development of symmetry and Chiral

Malcolm T Browne wrote:
> A colleague has asked me about the origins of the words "symmetry" and "chirality".
Around 1848, Pasteur discovered that molecules have a handedness while he was studying salts of acids that had been separated from the bottom of old wine casks. Some years later, Kelvin defined the notion of a "chiral object" and "chirality".

<<<< And JHC responded >>>>>
The word "symmetry" has a much older history. Etymologically, it means "measuring together", and is, oddly, a Greek-derived doublet of the Latinate "commensurable", which now has a totally different meaning. The history of these words is tangled up with that of "proportion", and can be found in older works on art history. "Chiral" was indeed invented by Lord Kelvin, but distinctly later than your words above might suggest. He used it in his celebrated Baltimore Lectures of (I think) 1896, having introduced it a few years earlier in some less-read papers. Etymologically, it means "handed", but is a hybrid, since it combines a Greek stem with the Latin suffix "-al". I have the impression that the opposite word, "achiral", meaning "not handed", has only come into use quite recently (later than 1950). I have never much liked it, because it is a negative word for a positive property, but am slowly getting used to it. An older equivalent is "amphichiral" ("both-handed"), which was used a few times before 1950 by Coxeter (though I think he spelled it "amphicheiral"). He translated it as "fitting either hand". John Conway

The complete discussion can be found at the Math Forum discussion page.

Tangent is from the Latin tangere, to touch, aptly describing two curves which meet at a single point. Tangent is another creation of the Danish Mathematician Thomas Fincke, of Flensborg, and was first written by him in Latin around 1583. At the time, Fincke was only 22 years of age.

In a letter by Augustus De Morgan in the Philosophical Magazine (May, 1846, pg 382) he states that "The words tangent and Secant,... have an origin which is not mentioned by historians, Nobody, in fact, knows where they came from; very few people care." He then proceeds to explain when Fincke created the terms, and why it may have escaped popular notice until the middle of the nineteenth century. The title, greatly shortened, was Geometrie Rotundi libri xiiii. Historian Jurgen Schonbeck, described the book this way, "In this influentual work, in which Fincke introduced the terms tangent and secant and probable first noticed the Law of Tangents and the so-called Newton-Oppel-Mauduit-Simpson-Mollweide-Gauss-formula, he showed himself to be ,abreast of the mathematics of his time".

De Morgan adds that, " is obvious that Finck gives the words for the first time, and defends them as suggestions of his own." Later he eplains that it was the conflict between Catholic and Protestant beliefs that caused his name to disappear from the record. De Morgan states that Clavius had used Fincke's tables in Theodosius and that the "celebrated, Clavius... excludes none but persons whom a reputable Jesuit could not name as protestants and Copernicans."

Prior to the creation of Tangent most writers still used the terms umbra recta, vertical shadow, and umbra versa, turned shadow, although De Morgan, in the paper mentioned above, says that Viete and Regiomontus used the term Facunda. These terms referred to a vertical shadow left by a gnomen (sundial) attached to the wall, and one placed parallel to the ground. The Latin root for shadow, umbra which is still used for the darkest part of a sunspot or a complete solar eclipse, remains in the more common word umbrella (small shade). Whew, now that I have worked the word umbrella into a page, I can show you a picture of my grandaughter, Zoe, with an umbrella,of course.

Tangram is a name of a Chinese puzzle of seven pieces that became popular in England around the middle of the 19th century. It seems to have been brought back to England by Sailors returning from Hong Kong. The origin of the name is not definite. One theory is that it comes from the Cantonese word for chin. A second is that it is related to a mispronunciation of a Chinese term that the sailors used for the ladies of the evening from whom they learned the game. [Concubines on the floating brothels of Canton, Hong Kong, and many other ports belonged to an ethnic group called the Tanka whose ancestors came from the interior of the country to become fishermen and pearl divers. They were considered as non-chinese by the govenments of China until 1731. They were unique among Chinese women in refusing to have their feet bound. ] A third suggestion is that it is from the archaic Chinese root for the number seven, which still persists in the Tanabata festival on July seventh in Japan which celebrates the reunion of the weaver (vega) and the herdsman (altair). Whatever the origin of the name, the use of the seven shapes as a game in China were supposed to date back to the origin of the Chou dynasty over one thousand years before the common era. The Chinese name is Ch'i ch'iao t'u which translates, so I am told, as "ingenious plan of seven".

It appears however, that the game and the name are both much more modern than believed. From the website, I found that " The Tangram was invented between 1796 and 1802 in China by Yang-cho-chu-shih. He published the book Ch'i ch'iao t'u (Pictures using seven clever pieces). The first European publication of Tangrams was in 1817. The word Tangram itself was coined by Dr. Thomas Hill in 1848 for his book Geometrical Puzzles for the Young. He became the president of Harvard in 1862, and also invented the game Halma.

Here is some history of the game by David Singmaster, one of the world's foremost authorities on recreational mathematics,

TANGRAMS. These are traditionally associated with China of several thousand years ago, but the earliest books are from the early 19C and appear in the west and in China at about the same time. Indeed the word 'tangram' appears to be a 19C American invention (probably by Sam Loyd). A slightly different form of the game appears in Japan in a booklet by Ganreiken in 1742. Takagi says the author's real name is unknown, but Slocum & Botermans say it was probably Fan Chu Sen. There is an Utamaro woodcut of 1780 showing some form of the game (not yet seen by me). I have seen a 1786 print - Interior of an Edo House, from The Edo Sparrows or Chattering Guide - that may show the game. Needham says there are some early Chinese books, and van der Waals' historical chapter in Elffers' book Tangram cites a number with the following titles.
Ch'i Ch'iao ch'u pien ho-pi. >1820.
Ch'i Ch'iao hsin p'u. 1815 and later.
Ch'i Ch'iao pan. c1820.
Ch'i Ch'iao t'u ho-pi. Introduction by Sang-Hsi Ko. 1813 and later. remarks inserted from a description of the book in Tangram by Joost Elffers, {Located in the Leiden Library #6891: This book, with an introduction by Sang-Hsi Ko, is, as far as is known, the oldest example of a Chinese game-book. ] I would like to see some of these or photocopies of them. I would also be interested in seeing antique versions of the game itself. The only historical antecedent is the 'Loculus of Archimedes', a 14 piece puzzle known from about -3C to 6C in the Greek world. Could it have travelled to China? I found a plastic version of the Loculus on sale in Xian, made in Liaoning province. I wrote to the manufacturer to get more, but have had no reply.
For the 10th International Puzzle Party, Naoki Takashima sent a reproduction of a 1881 Japanese edition of an 1803 Chinese book on Tangrams which he says is the earliest known Tangram book.
Jean-Claude Martzloff found some some drawings of tangram-like puzzles from a 1727 booklet Wakoku Chie-kurabe, reproduced in Akira Hirayama's T“zai S–gaku Monogatari Heibonsha of 1973. Takagi has kindly sent his reprint of this booklet, but I am unsure as to the author, etc.

The "Ganriken" mentioned in Dr. Singmaster's post was a pseudonym, and it seems unclear who the actual author was. The book was titled "Sei-Shonagon Chie-no-Ita", which translates to "the ingenious pieces of Sei Shonagon". From Wikipedia, "Sei Shonagon was a lady-in-waiting at the Japanese Imperial Court in the beginning of the 11th Century. She kept a personal diary of sorts in which she wrote down her experiences but mainly her feelings. Such diaries were common at the time and were called pillow books because these books were often kept next to people's pillows in which they would write their experiences and observations. The Pillow Book of Sei Shonagon gives an invaluable insight into the world of the Imperial Court of Kyoto a thousand years ago. Sei Shonagon's observations are witty, wry, poignant, and at times condescending." The image below shows the seven Sei Shonagon pieces as pictured in Shigeo Takagi's article in The Mathemagician and Pied Puzzler, a tribute to Martin Gardner.

Tangrams received a boost in popularity when Charles Dodgson, writing as Lewis Carroll, used them to create illustrations of the Characters in the "Alice" books. In the Penguin Books translation of Tangram by Joost Elffers he states that English Puzzle writer H. E. Dudney purchased a copy of a play-book called The Fashionable Chinese Puzzle from Dodgeson's estate. This book seems to be the most common source of the assertion that Napoleon was an avid Tangram player,

You can download "OOG, The Object Orientation Game" from MCM Software which allows the player to solve puzzles using tangrams, pentominoes and more. Here is another nice site for exploring tangrams.

It is also known that Archimedes created a similar game with the disection of a square into 14 pieces. See one at this page from the MAA which also includes a nice history about the puzzle and the discovery of an Archimedian Palimpset recently. The object of the puzzle is to put the pieces back together to form a square. A more difficult question, unknown for over 2000 years, is how many unique ways are there of putting the pieces together to form a square. Bill Cutler used a computer program to show that there are 536 unique ways to assemble the pieces not counting similar rotations and reflections. All 536 solutions are visible in the article mentioned above.

The puzzle goes under the names of Stomachion or Loculus. Stomachion is from the Greek word for stomach. Loculus seems to be a word related to the division of a tomb area into small chambers for different bodies and is related to the diminutive of locus for a point or place, thus "a little place". (I don't yet get the connection between the puzzle and stomach.)

Tessellation The root of tessellation is, tessera, the old Ionic (Greek) root for four. Tessera is the name of the square chips of stone or glass that are used to form a mosaic. Tessela is the diminutive form, and is used to describe smaller tessera. Tiles, bricks and larger similar items were called testa, which is preserved in the name for the hard outer shell of seeds. The completed project, then, became a tessellation.

Here is a link to a web site created by ,Suzanne, at the Math Forum that is an excellent web site on tesselations.

Thousand Our number for one thousand comes from an extension of hundred. The roots are from the Germanic roots teue and hundt. Teue refers to a thickening or swelling, and hundt is the root of our present day hundred. A thousand, then, literally means a swollen or large hundred. The root teue is the basis of such common words today as thigh, thumb, tumor, and tuber.

Topology comes from the Greek root topos (place). Before it was used in mathematics, it was applied to the geographic study of a place in relation to its history. The word was introduced into English by Solomon Lefschetz in the late 1920s. It appears that the word was originated around 1847 by Johann Benedict Listing in place of the earlier usage analysis situs ...From Jeff Miller's web site, "The word TOPOLOGY was introduced in German in 1847 by Johann Benedict Listing (1808-1882) in "Vorstudien zur Topologie," Vandenhoeck und Ruprecht, Göttingen, pp. 67, 1848. However, Listing had already used the word for ten years in correspondence." Listing was also the first to write about the one-sided strip now better known as the Mobius Strip.

Torus is from the Latin word for bulge and was first used to describe the molding around the base of a column. Although it is usually used to describe the rotation of a circle about a line in its plane, the definition applies to the rotation of any conic section. The image is produced with WINPLOT, a freeware program created by Richard Parris of Phillips Exeter Academy

Trapezoid and Trapezium Both words come originally from the Greek word for table. Today, in the USA, the term trapezoid refers to a quadrilateral with one pair of sides parallel and a trapezium to one with NO parallel sides. Actually, the term for the case with no parallel sides is almost never used, so trapezium is an archaic term at best in the US. This is exactly the reverse of the original meanings and the meanings in some countries, particularly England, today. Here is a short comment on how this came about from Jeff Miller, a teacher at Gulf High School in New Port Richey, Florida, who maintains an excellent page on the first use of some common mathematical terms:

"TRAPEZIUM and TRAPEZOID. The early editions of Euclid 1482-1516 have the Arabic helmariphe; trapezium is in the Basle edition of 1546. Both trapezium and trapezoid were used by Proclus (c. 410-485). From the time of Proclus until the end of the 18th century, a trapezium was a quadrilateral with two sides parallel and a trapezoid was a quadrilateral with no sides parallel. However, in 1795 a Mathematical and Philosophical Dictionary by Charles Hutton (1737-1823) appeared with the definitions of the two terms reversed: Trapezium...a plane figure contained under four right lines, of which both the opposite pairs are not parallel. When this figure has two of its sides parallel to each other, it is sometimes called a trapezoid. No previous use of the words with Hutton's definitions is known. Nevertheless, the newer meanings of the two words now prevail in U. S. but not necessarily in Great Britain (OED2).
John Conway recently pointed out in a post on the use of the terms that;
What is true now is that the thing with two parallel sides is called a "trapezium" in England and a trapezoid in America, and that neither term is used in either country for the thing with no parallel sides. (The latest date for which I'vbe seen either of them so used was in a geometry book of 1912, which however, was a reprint of a 19th-century one.) Instead, to avoid confusion, the term "quadrilateral" is now standard for the general case. (This has a few earlier uses, dating back to about 1500, but was decidedly uncommon - like "trilateral" - before 1900.)
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.

Some geometry textbooks define a trapezoid as a quadrilateral with at least one pair of parallel sides, so that a parallelogram is a type of trapezoid. Euclid did not define the shape we now call a trapezoid, and the "trapezia" is defined by default.... "let quadrilaterals other than these be called trapezia" [from the Heath translation]. Heath's translation states that the language used implies that Euclid may have been creating a new word, or using an existing one in a new way. Proclus seperated out trapeziums and trapezoids (backwards to what we now do as explained in the quote above from Jeff Miller's page) but it seems clear he meant that a trapezium had exactly one pair of sides parallel (an exclusive definition) rather than at least one pair of sides parallel(an inclusive defintion). Many mathematicians today prefer the inclusive definition so that a parallelogram is a special case of trapezoid. Apparently this has been a question in geometry for a while as I recently read a note from Neal Silverman which suggests that the inclusive definition has appeared in some books back at least to 1900.

" I recently acquired a copy of "New Plane & Solid Geometry" by Wooster Woodruff Beman and David Eugene Smith (Ginn 1900), a revision of their earlier 1895 work. ... But as to quadrilaterals, consider their definition of trapezoid: "A quadrilateral that has one pair of opposite sides parallel is called a trapezoid." They go on to state that "[b]y the definition of trapezoid here given it will be seen that the parallelogram may be considered a special form of the trapezoid. (section 97 at p. 59).

It seems that the inclusive definition was not well received as in the same post Neal adds, "D. E. Smith went on to write many more books on geometry, some of which were revisions of the old Wentworth books. I have never seen this statement in any of his later books."

John Conway recently posted a note about the etymology of trapezoid;

The etymology of "trapezoid" is quite interesting. It's a corruption of "tetra-pes-oid", whose three parts mean "four-leg-shaped", or perhaps more familiarly "table-shaped", since "tetra-pes" was a familiar Greek name for a small table.

Truncated means to shorten by cutting off and is related to the Old English truncheon, which means a club or staff. Both words are derived from the Early French truncus which referred to a cutting from a tree used for grafting stock. A truncated solid is usually cut off by non-parallel lines as opposed to a Frustum.

Vector is derived from the Latin root vehere, "to carry". The root is also the source of everyday words like vehicle. The vector idea of a parallelogram of forces dates back to the works of Archimedes and Heron of Alexandria. Vectors in the modern sense, however, probably came about as late as the beginning of the 19 th century. Mobius work with barycenters present the idea of directed distances, and Argand presentated complex numbers as points or ordered pairs of real numbers. Sir W R Hamilton in his work on Quaternions developed a system of addition and a non-commutative multiplication of ordered quadruplets. Hamilton introduced the independent term "vector" in mathematics. This link at St Andrews University in Scotland has a more extensive history of the development of vectors .

A few years ago I was playing around trying to learn Java Scripts and I wrote this Vector Calculator script. It will do several simple vector operations, but I do not guarantee correct home work answers. You may want to actually learn how to work with vectors on your own. Jenny Olive has a nice tutorial called Math Help, Working with Vectors

Zero comes to us from the Hindus, the inventors of zero, through the Arabs and the Arabic word sifre, from which we also get the word cipher. According to Edward MacNeal, the author of MathSemantics, Making Numbers Talk Sense, the Hindu's used the word shunya to refer to a blank or empty space. When the Arabic method was introduced to Europe, the Roman system was already in place. Perhaps because the hand written contracts would have been much easier to forge or alter using the Arabic numbers, there was a strong resistance to their use. During a brief forced underground, the use of the cipher came to represent something done in secret or code. Eventually the Latinized form of cipher, zepherium came to be the common term, which eventually was reduced to zero in English.

Recent discoveries have led some math historians to give greater attention to the Early Egyptians for their contributions to the development of zero. In a recent discussion on the Historia Matematica newsgroup, Bea Lumpkin explains two areas where the Egyptian use of zero is evident...

The two applications of the zero concept used by ancient Egyptian scribes were 1) as a zero reference point for a system of integers used on construction guidelines, and 2) as a value that resulted from subtracting a number from an equal number. These are the achievements I believe should be acknowledged by historians.

It may well be that the first use of a zero was by the Mayans of Central America. Although little is known of the Mayan mathematics apart from calendric systems, we do know that they used a base 20 system with place value that included the use of a zero for a place holder. Most of what is known comes from the pen of Father Diego de Landa, an early missionary to Mexico. Landa is also responsible for the destruction of many of the documents that existed of the civilization at that time, having destroyed them as heretical documents.